Literature review of models on tire-pavement interaction noise

Journal of Sound and Vibration xxx (2018) 1e89

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Literature review of models on tire-pavement interaction noise Tan Li*, Ricardo Burdisso, Corina Sandu Department of Mechanical Engineering, Virginia Tech, Blacksburg, VA 24061, USA

a r t i c l e i n f o

a b s t r a c t

Article history: Received 29 April 2017 Received in revised form 18 December 2017 Accepted 10 January 2018 Available online xxx

Tire-pavement interaction noise (TPIN) becomes dominant at speeds above 40 km/h for passenger vehicles and 70 km/h for trucks. Several models have been developed to describe and predict the TPIN. However, these models do not fully reveal the physical mechanisms or predict TPIN accurately. It is well known that all the models have both strengths and weaknesses, and different models fit different investigation purposes or conditions. The numerous papers that present these models are widely scattered among thousands of journals, and it is difficult to get the complete picture of the status of research in this area. This review article aims at presenting the history and current state of TPIN models systematically, making it easier to identify and distribute the key knowledge and opinions, and providing insight into the future research trend in this field. In this work, over 2000 references related to TPIN were collected, and 74 models were reviewed from nearly 200 selected references; these were categorized into deterministic models (37), statistical models (18), and hybrid models (19). The sections explaining the models are self-contained with key principles, equations, and illustrations included. The deterministic models were divided into three sub-categories: conventional physics models, finite element and boundary element models, and computational fluid dynamics models; the statistical models were divided into three sub-categories: traditional regression models, principal component analysis models, and fuzzy curve-fitting models; the hybrid models were divided into three sub-categories: tire-pavement interface models, mechanism separation models, and noise propagation models. At the end of each category of models, a summary table is presented to compare these models with the key information extracted. Readers may refer to these tables to find models of their interest. The strengths and weaknesses of the models in different categories were then analyzed. Finally, the modeling trend and future direction in this area are given. © 2018 Elsevier Ltd. All rights reserved.

Keywords: Tire noise Deterministic model Statistical model Hybrid model

1. Introduction The term tire-pavement interaction noise was first found in the literature in 1976 (Thrasher et al., 1976 [1] and Miller and Thrasher, 1976 [2]), and later in 1986 (Pottinger et al., 1986 [3]) and 1998 (Donavan et al., 1998 [4]). The acronym TPIN (for tirepavement interaction noise) was first formed in 2008 (Khazanovich and Izevbekhai, 2008 [5]), and later found in 2013 (Syamkumar et al., 2013 [6]). Tire-pavement interaction noise is also known as tire-road interaction noise, tire/pavement

* Corresponding author. E-mail addresses: [email protected] (T. Li), [email protected] (R. Burdisso), [email protected] (C. Sandu). https://doi.org/10.1016/j.jsv.2018.01.026 0022-460X/© 2018 Elsevier Ltd. All rights reserved.

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noise, tire/road noise (TRN), or tire noise. Tire-pavement interaction noise is defined as the noise emitted from a rolling tire as a result of the interaction between the tire and the road surface (Sandberg and Ejsmont, 2002 [7]). Modern research on TPIN started around mid-1970's as engines became less noisy and the aerodynamic design of the vehicle body was optimized. Sandberg and Ejsmont (1995) [8] have compiled a computerized bibliography termed “TRN Bibliography” totaling 1280 documents as of August 1994. However, most of these references were from special conference proceedings and industrial/government reports, and not from archival journals. To date, a larger number of references can be found (totaling 2654 published during the period 1970e2015, as displayed in Fig. 1), which have greatly advanced the modeling of tirepavement interaction noise. However, TPIN is still not fully understood, and it is difficult to describe or predict TPIN accurately; there are large differences in various findings, probably because of the many complex generation mechanisms involved. Although a tire seems to be a simple structure at the first glance, the combination of its geometrical properties and multi-layer composite structure leads to a highly complex vibrational behavior (Sabiniarz and Kropp, 2010 [9]). Tire-pavement interaction is the dominant source of noise at speeds above 40 km/h for passenger vehicles and 70 km/h for trucks. There are several reasons for reducing tire-pavement interaction noise in different countries: to pass the legal approval for exterior noise level; to reduce the annoyance level in the interior space of the car and increase passenger comfort (Harrison, 2004 [10]); to avoid the import of foreign tires (Chen, 2014 [11]). It is certain that accurate and reliable models on TPIN are desired by vehicle manufacturers and tire companies, as these models will be able to reduce the development time (Mohamed et al., 2013 [12]). Several models have been developed to describe and predict tire vibrations and TPIN. However, all these models have restricted applications in some sense (Dare, 2012 [13]). For example, some models describe only one or two generation mechanisms; some assume two-dimensional tire and pavement; some include only a limited number of tire/pavement combinations, such as a slick tire on a rough road; some calculate the noise level at the source without considering sound propagation; some are concerned only with the structural dynamics of the tire, and neglect the effect of the nonlinear aerodynamics surrounding the tire. It is difficult to evaluate the accuracy of a specific model because of the different objectives and lack of validation (Kuijpers and van Blokland, 2001 [14]). In addition, most of the current models focus only on passenger car tires. Very few studies have reported noise models for truck tires that might have more influence on traffic noise. Because of the increased popularity of porous asphalt pavement, a large number of models have included pavement absorption and mechanical impedance as the parameters of interest. In general, there are three acoustic modeling approaches: wave, energy, and ray acoustics (Crocker, 2007 [15]). The wave acoustics approach, commonly adopted by finite element software, is used for a small enclosed space, especially if its shape is simple and has well-defined boundary conditions. The energy acoustics approach is often used as a statistical approach for measuring sound intensity. The ray acoustics approach, together with the multiple-image-source concept for reflection calculations, can be used to investigate sound propagation in the atmosphere. Most of the current TPIN models have used one or two approaches mentioned above. In this literature review work, the present authors collected over 2000 references related to tire noise, published during the period from 1960's to 2010's (Li, 2017 [16]). The present authors cannot guarantee that all the references have been covered, but a vast majority has been included. In the end, 74 models were reviewed from nearly 200 selected references. The models were divided into three categories based on the characteristics of the approach used: deterministic model, statistical model, and hybrid model. A brief introduction of the three categories is presented in Table 1. The deterministic models normally focused on low frequencies (below 500 Hz). However, the main spectral content of the actual tire noise is located in the range of 600e1200 Hz. Therefore, these models tend to have considerable errors. The statistical and hybrid models normally cover the entire dominant frequency of tire noise, resulting in better accuracy. However, most of the models investigate only the pavement parameters. Both advantages and disadvantages exist for all the three categories of TPIN models. In fact, one specific model generally focuses only on a few points of interest (physics or

Fig. 1. Count of records in literature (generated via Engineering Village on August 12, 2016, using search query for “tire” and “noise” in subject/title/abstract from the databases of Compendex, Inspec, and NTIS).

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Table 1 Introduction of three categories of TPIN models. Deterministic

Statistical

Hybrid

Method

Physics without experimental data

Example

Brinkmeier et al., 2008 [17]: structurally and aerodynamically radiated noise components Low frequency only, computationally intensive, 17 dB error

Correlation of measured noise data and tire-pavement parameters Khazanovich et al., 2008 [5]: different pavements and same tire with curve fitting

Incorporating physical principles into empirical model Domenichini et al., 1999 [18]: effect of pavement texture on TPIN Mostly pavement parameters only, without knowledge of tire inputs; intensive data for training Optimization of the combination of tire and pavement

Drawback

Major application [14]

Tire design (qualitative)

No knowledge of noise generation mechanism

Pavement design (quantitative)

parameters) owing to the complexity of the TPIN mechanism. For different research purposes and/or under different conditions, one model might have certain advantages over the others in the same category of models. The purpose of this paper is to present the existing models from literature, compare these models, state their strengths and weaknesses, and provide insight into future research direction in this area. The organization of this paper is as follows. In section 2, the deterministic models are reviewed; in section 3, the statistical models are reviewed; in section 4, the hybrid models are reviewed; in section 5, instructions for readers to use the models are given; in section 6, conclusions of this literature review are given. 2. Deterministic models 2.1. Introduction Deterministic models are also called analytical or numerical models. Recent methods, such as finite element method (FEM) (Lou, 2007 [19]) and boundary element method (BEM) (Wang et al., 2011 [20]; Nakajima et al., 1993 [21]), require large computational effort. A large number of parameters are also required, such as tire geometry, deformation of the tire, material parameters, and boundary conditions (Yang et al., 2013 [22]). On the other hand, it is possible to investigate the influence of a single parameter on TPIN using deterministic models (Biermann et al., 2007 [23]). In general, deterministic models need several material or construction parameters as inputs for tire modeling. For example, Larsson and Kropp model (1992) [24e26] requires 24 input parameters (6 geometrical parameters and 18 material parameters), some of which are presented below as an example in Table 2. Anderson et al., (2001) [27] also presented a measurement method for collecting these tire material data. In this section, 37 deterministic models were reviewed; they consist of 16 conventional physics models, 16 finite element and boundary element models, and 5 computational fluid dynamics (CFD) models. The model name, method, output noise parameter with the noise measurement technique used, noise generation mechanisms investigated, and input parameters of each model are listed and compared in tables as a summary in section 2.5. The summary tables can be used as a reference to direct the readers to the models of their interest. 2.2. Conventional physics models Conventional physics models typically use traditional analytical methods and classic mathematics, such as modal analysis, to present and investigate the dynamic behavior of the tire that is modeled as a ring, shell, or plate. Traditional acoustic equations are sometimes applied to analyze the structural wave propagation mode in the tire tread and wall, as well as the acoustical mode in the tire cavity. Sixteen models are reviewed in this subsection. In this paper, the heading for the corresponding model is written in this form: model name (authors, year); the models in each subsection are ordered chronologically. 2.2.1. Monopole (Hayden, 1971) Hayden (1971) [29] developed the first air pumping model, focusing on the noise generation at the leading edge of the tire contact patch where the tread is compressed. The model is based on the monopole theory, whose governing equation is given below.

pðr; uÞ ¼

ruQ 4pr

(1)

where p is the sound pressure (root mean square value), r is the distance between the receiver and the monopole, u is the angular frequency, r is the air density, and Q is the air volume change rate (root mean square value). The initial groove volume V0 is given by Please cite this article in press as: T. Li et al., Literature review of models on tire-pavement interaction noise, Journal of Sound and Vibration (2018), https://doi.org/10.1016/j.jsv.2018.01.026

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Table 2 Input tire parameters (slick 205/60R15) for Larsson and Kropp model (the data in brackets refer to loss factors for the corresponding parameters) (Modified from Andersson and Larsson, 2005 [28], Table 1; reprinted with permission from Ms. Brigitte Badelt of S. Hirzel Verlag on behalf of the European Acoustics Association). Category

Parameter

Value

Geometry

Length of tread plate Width of tread plate Thickness of tread layer Thickness of belt layer Density Young's modulus (real part) Poisson's ratio Density Young's modulus (real part) Poisson's ratio Radial bedding Circumferential bedding Lateral bedding Circumferential tension Lateral tension

2p  0.314 m 0.355 m 0.010 m 0.001 m 1175 kg/m3 25 MPa [0.05e0.70] 0.499 3720 kg/m3 580 GPa [0.02] 0.4 1.02  106 N/m3 [0.09] 1.02  106 N/m3 [0.09] 1.02  106 N/m3 [0.09] 42000 N/m2 [0.055] 42000 N/m2 [0.055]

Tread layer

Belt layer

V0 ¼ LDW

(2)

where L is the groove length in the circumferential direction, D is the groove depth, and W is the groove width. The air volume change rate Q is calculated using the equation

Q ðvÞ ¼

DV dV0 dLDW ¼ dDWv ¼ ¼ L=v Dt Dt

(3)

where v is the tire forward speed, DV is the volume change, Dt is the time duration, and d is the fractional change in the initial volume. The frequency u is determined by the repetition between consecutive groove cavities.




uðvÞ ¼ 2pv lp

(4)

where lp is the pitch distance between the groove cavities. Combining the above equations gives

pðr; vÞ ¼

rv2 dDW 2lp r

n

(5)

where n is the number of groove cavities along the tire width. 2.2.2. Unified (Plotkin and Stusnick, 1981) Plotkin and Stusnick (1981) [30] presented a unified set of analytical models to investigate the major noise generation mechanisms (impact and air pumping) systematically, considering the basic tire structural properties. Bohm's thin-shell equations (Bohm, 1967 [31]) were used to model the motion of a tire. The air pumping mechanism was analyzed based on the physical behavior (Plotkin et al., 1980 [32]). The influence of pavement texture was also included using Nilsson's linear excitation model (Nilsson, 1976 [33]). Then, sound radiation analysis was employed using Eberhardt's model (Eberhardt, 1979 [34]). The nomenclature for the model is listed in Table 3. 2.2.2.1. Thin shell equations for the motion of the tire. The radial tire belt was modeled as a uniform shell on a uniform elastic foundation. The two-dimensional shell equations for the tangential and radial displacements of the belt are as follows (Bohm, 1967 [31]).

  EA 8 2 00 0 0 _ € _ _ ¼ pt ðf; tÞ v þ 2 m U ð v þ wÞ þ U ðv þ 2w  vÞ  2 ðv00 þ w0 Þ þ kt v  dt ðv0 U þ vÞ > > R0 > > <   00  EA  T EI  0 2 0 € _ _ > w w þ 2 m U ð w  vÞ þ U  2v  ðw þ R Þ þ 2 ðv0 þ wÞ þ 4 wiv þ 2w00 þ w  02 ðw þ w00 Þ 0 > > R0 R0 R0 > : 0 _ ¼ pr ðf; tÞ þkr w þ dr ðw U þ wÞ

(6)

For a stationary tire, after setting all time derivatives to zero, the solutions of the above linear equations are Please cite this article in press as: T. Li et al., Literature review of models on tire-pavement interaction noise, Journal of Sound and Vibration (2018), https://doi.org/10.1016/j.jsv.2018.01.026

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Table 3 Nomenclature for unified model. Denotation

Meaning

a A Ab Ac A(f0) An, Bn, Cn, Dn Cp d d r, d t jdetj E, F, G, H ER Es EA EI f(f1-f) F(f) G G(rjr0 ) GR h H(f1-f) kr, kt kA, kT l lb ls lv n N Nv p pi p0(t) p r, p t prs, prc, pts, ptc P Pavg Q r R R0 s s0 S t tb ts tt T0 U un V v, w wb wt wv W x, y, z yg ygi zs Dz

Speed of sound in air Tread area function Cross-sectional area of belt Cross-sectional area of cords in belt Function describing tread area Coefficients of Fourier series describing tire shape Phase velocity in belt Tread depth Radial and tangential damping, per unit length Magnitude of determinant Coefficients of inverted matrix Elastic modulus of rubber Elastic modulus of steel Extensional stiffness of tread and belts Bending stiffness of tread and belts; (EI)b is the value for the belt plies alone, and (EI)t is the value for the tread alone Function defining radial displacement at f due to unit force at f1 Force associated with tread-road interface Geometric amplification factor Green's function Shear modulus of rubber Distance from neutral surface of bending to center of tread Unit step function Radial and tangential bedding coefficients Wave number in air and tire Tread pitch length Tread block length An invariant to calculate bedding stiffness Tread void length Circumferential mode number Number of belt plies Number of voids Acoustic pressure Inflation pressure Sound pressure for a tread void with pitch l0, depth d0, and width w0 Radial and tangential external pressures Sine and cosine integrals of pr and pt; e.g., prs ¼ 1/p$!20pprsinnfdf Point force Average contact pressure Resonant Q factor; inverse of loss factor Vector from source to receiver; jrj ¼ r Local radius of curvature Unperturbed radius of tire Circumferential distance coordinate, measured from contact patch edge Coordinate on pavement Surface area Time Thickness of belt ply Sidewall thickness Tread thickness Tension in belt due to inflation pressure Speed of tire Normal velocity Tread void volume Tangential and radial displacement of belt Width of belt Effective width of tread Width of tread void Amplitude of harmonic component of radial displacement w, determined from the forced motion of the contact patch edge Cartesian coordinates with respect to contact patch center Shape of tread groove Shape of i-th groove in full tread pattern Pavement texture profile Average vertical compression of rubber Cord angle Matrix coefficients Dirac delta function Flow blockage factor Load distribution due to pavement texture Damping coefficient

a a, b, g, d, ε d(f-f1) εb

z h

(continued on next page)

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Table 3 (continued ) Denotation

Meaning

q0 k k0 m r f fa, fb f0 Df Dfcp u U

Angle between sidewall and belt Local belt curvature ¼ l/R Curvature of unperturbed belt ¼ l/R0 Mass density (per unit length) of tread and belts Density of air Circumferential coordinate Coordinates of the leading edge and trailing edge of contact patch Circumferential coordinate fixed to tire; corresponds to f at t ¼ 0 Angle with respect to contact patch center Angle from center to edge of contact patch Angular frequency Angular velocity of tire Time derivative Spatial derivative; either v/vf or v/vs

(_) ()0

∞ 8 X > ðAn sin nf þ Bn cos nfÞ >
(7)

∞ X > > :w ¼ ðCn sin nf þ Dn cos nfÞ n¼0

where the coefficients are calculated in the following matrix format

2 3 E An 6 6 Bn 7 6 7 ¼ 1 6 F 4 Cn 5 jdetj 4 G H Dn 2

F E H G

G H I J

32 3 pts H 6 7 G7 76 ptc 7 J 54 prs 5 I prc

(8)

where the parameters in capital letters are given by

  8 2 E ¼ a d þ ε2  dg2 > > >   > > 2 > > F ¼ b d þ ε2 þ εg2 > > > > > G ¼ ðbd þ aεÞg >   > < H ¼ g2  ad þ bε g   > > > I ¼ d a2 þ b2  ag2 > > >   > > 2 2 2 > > > J ¼ ε a þ b þ bg > >    > : 2 d2 þ ε2 þ 2g2 ðbε  adÞ þ g4 jdetj ¼ a2 þ b

(9)

where the parameters in Greek letters are expressed in terms of tire properties as

! 8   > EA 2 > > a ¼  mU n2 þ kt  mU2 > 2 > > R0 > > > > > > b ¼ dt Un > > > ! > > < EA 2 g ¼ 2  2mU n > R0 > > > > ! > > > > EI 4 2EI T0 2 2 > > d ¼ 4n  þ mU  2 n þ > > > R0 R0 R40 > > > : ε ¼ dr Un

(10) EA R20

þ

EI R40



T0 R20

! 2

 mU þ kr

The study also presented equations to calculate extensional stiffness, bending stiffness, and bedding stiffness of the tread belt.

  EA ¼ 4ðA  Ac ÞGR 1  cot2 a  cot4 a

(11)

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8 EA ðNtb Þ3 > > > < ðEIÞb ¼ t w wb 12 b b > 3 > > : ðEIÞ ¼ E w tt R t t 3 8 cos q0 þ q0 sin q0 > > > < kr ¼ sin q  q cos q pi 0 0 0 > G t > s R > : kt ¼ þ p0 cot q0 ls

7

(12)

(13)

The damping coefficients for truck tires are approximately calculated using the following equations.

8 pffiffiffiffiffiffiffiffi > kr m > > < dr ¼ 10 pffiffiffiffiffiffiffi > > > : d t ¼ kt m 10

(14)

2.2.2.2. Air pumping. It was found that tread void compression occurs mainly through the lateral motion of the tread elements entering and leaving the contact patch, and not through the vertical compression of the tread rubber (Plotkin et al., 1980 [35]; Samuels, 1979 [36]). The reflection from the ground and tire sidewall plays an important role in noise generation by increasing the pressure by a factor of four, which is the geometric radiation factor G. Air pumping also depends on the flow blockage factor εb; e.g., for rib tires with continuous grooves, some of the displaced air moves within the grooves instead of being pumped out, in which case, εb z 0.35; however, for cross-bar tires, which generate more air pumping noise, εb z 1. To analyze the air pumping noise, the volume of a single void is calculated; this is given by

V ¼ wv d½lv  lb hk0 þ ðlv þ lb Þhk

(15)

The equation for sound pressure for a monopole source is

p¼r

V€ 4pr

(16)

Assuming that k is a function of f, and that all the other parameters in V are constant, 2

d k V€ ¼ V 00 ðfÞU2 ¼ wv dðlv þ lb Þh 2 U2 df

(17)

The final equation for the sound pressure is

pðfÞ ¼ 

  ∞   Gεb r U 2 l X wv dhðlv þ lb Þ 2 n2 n2  1 ðCn sin nf þ Dn cos nfÞ 4pr R0 R0 n¼1

(18)

The sound pressure for a tread void with pitch ll, depth dl, and width wl is given by

pl ðtÞ ¼

ll dl wl p ðtÞ l0 d0 w0 0

(19)

where p0(t) is the sound pressure for a tread void with dimensions l0, d0, and w0. The sound pressure of air pumping for a full tread pattern is given by wt

  Nv Z 2 ygi ðxÞ 1 X dx pðfÞ ¼ p0 f  wt i¼1 R0

(20)

wt 2

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2.2.2.3. Carcass vibration. The models are applicable only for steady uniform carcass on a smooth pavement. For the vibration of tire carcass on a textured surface, the circumferential coordinate was changed from f to s ¼ R0f, and ()0 denotes v/vs. For the contact patch edge excitation, the radial displacement of the belt is

w ¼ Weiðkt sutÞ ehs

(21)

8 u > > > kT ¼ > C > p > > > ! > > < 1 kr 2 2 Cp ¼ EIkT þ T0 þ 2 m kT > > > > > > > Cp kr > > > :h ¼ 4EIk2T þ 2T0

(22)

where

The motions at the leading edge fa and trailing edge fb due to the excitation of the tread pattern are

8 < € wðfa Þ ¼ P U2 f ð0ÞA0 ðfa  UtÞ : wðf € b Þ ¼ P U2 f ð0ÞA0 ðfb  UtÞ

(23)

For the pavement texture excitation, the load at point (x, y) is

  zs ðx; yÞ ¼ Pð1 þ zðx; yÞÞ pr ðx; yÞ ¼ P 1 þ Dz

(24)

Then, the equation for the motion at the trailing edge fb due to the excitation of the pavement texture is

 € b Þ ¼ P U2 f ð0Þ A0 ðfb  UtÞ þ Az0 ðs0 þ Rðfb  UtÞÞR wðf

(25)

2.2.2.4. Sound radiation. The sound pressure from a moving surface S (Morse, 1970 [37]) is given by

Z pðrÞ ¼

!  b 0 dS Gðrjr0 Þ V p$ n

(26)

S

Using Eberhardt's model (Eberhardt, 1979 [34]) and employing Green's function for a vibrating plate, considering the flat tread area of the tire as an infinite baffle, the final equation is obtained as wb

rW 2 iut u e pðr; uÞ ¼ 2p

Z 2 Z∞ wb 2

eika r ikt s hs e e ds r

(27)

0

2.2.3. Truck tire (Keltie, 1982) Keltie (1982) [38] modeled the truck tire as an infinitely long incomplete circular cylindrical shell, as shown in Fig. 2, to predict the sound power generated. The assumption of infinite length is based on the fact that the waves propagating circumferentially are effectively damped out before they can meet near the top of the tire. The tire is both a nonisotropic and nonhomogeneous material owing to the composite structure of the rubber and the laminated plies. However, in the model, it is considered as a homogeneous and isotropic material, considering only the ply region, as the ply contributes to most of the load-carrying ability. The boundary conditions of the tire model are pinned connections with the rim along the lines q ¼ 0 and q ¼ q0. Based on Flügge's thin shell theory, considering both bending and extensional effects, the equation of motion for the shell surface is given by Please cite this article in press as: T. Li et al., Literature review of models on tire-pavement interaction noise, Journal of Sound and Vibration (2018), https://doi.org/10.1016/j.jsv.2018.01.026

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Fig. 2. Shell model geometry and coordinate system (source from Keltie, 1982 [38], Fig. 3; reprinted with permission from AIP Publishing LLC).

2

L11 4 L12 L13

L12 L22 L23

32 3

 2 q ðs; q; tÞ 3 L13 uðs; q; tÞ 2 1  n2 x R 4 q ðs; q; tÞ 5 L23 54 vðs; q; tÞ 5 ¼ q Eh L33 wðs; q; tÞ qz ðs; q; tÞ

(28)

where L11, L12, …L33 are differential operators; u, v, and w are the deformations in the x, q, and z directions, respectively; qx, qq, and qz are the externally applied pressures in the x, q, and z directions, respectively; s is the normalized coordinate given by  is the complex Young's modulus. To compute the acoustic radiation, the radial displacement s ¼ x/R; n is Poisson's ratio, and E needs to be obtained based on the radial load; the relevant equations are given below.

8 > q ðs; q; tÞ ¼ ejut qðsÞHðsÞ > > < z > > wðs; q; tÞ ¼ ejut wðs; qÞ ¼ ejut > :

∞ X

  1  2 2 npq wn ðsÞ sin

n¼1

q0

(29)

q0

where q(s) and H(q) are polynomials that satisfy the differentiability and continuity conditions. The sound pressure p(x, r, q) is governed by the equation

! 2 v2 v2 1 v 2 v 2 þ r þ r þ þ K a pðx; r; qÞ ¼ 0 vr vr2 vq2 vx2

(30)

where r is the radial coordinate, and Ka is the acoustic wavenumber. The normal velocity of the shell surface vs(x, q) is related to the sound pressure p(x, r, q) through the boundary condition and Euler's equation, as given below.

jur0 vs ðx; qÞ ¼

vpðx; r; qÞ vr r¼R

(31)

where u is the angular frequency, and r0 is the air density. The surface acoustic intensity J(x, q) is calculated using the equation

Jðx; qÞ ¼

o 1 n * Re ½pðx; R; q; tÞ½vs ðx; q; tÞ 2

(32)

where Re indicates the real part of the complex value, * indicates the complex conjugate. The total radiated sound power P is given by

Z∞ Z2p

P¼

Jðx; qÞRdqdx

(33)

∞ 0

The tire parameters such as bending stiffness, inflation pressure, and structural damping were analyzed for their effects on the sound power level in 1/3 octave band and on the overall sound power level. 2.2.4. 2D ring (Kung et al., 1986) Kung et al. (1986) [39], and Huang and Soedel (1987) [40,41] considered the tire belt as a two-dimensional rolling ring, and the sidewalls as an elastic foundation with radial and tangential stiffness. The equations of motion are given below (Huang and Soedel, 1987 [40]). Please cite this article in press as: T. Li et al., Literature review of models on tire-pavement interaction noise, Journal of Sound and Vibration (2018), https://doi.org/10.1016/j.jsv.2018.01.026

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  8  p   K 0   00 D 000 p0  2 > 0 00 00 0 > €q þ 2Uu_ r ¼ qq ðq; tÞ þ u  2u þ k  u þ u r h U þ  u þ r h u u  u þ > q q r r r q q q < a4 a a a2       >



 

D K p0 p0 > 2 000 0 0 00 > € _ : 4 u000 u þ k  u þ u r h U þ  u þ r h u  2 U u þ u þ þ 2u r r r r q ¼ qr ðq; tÞ r r q q q a a a a2

(34)

where kr is the radial stiffness, kq is the tangential stiffness, ur is the radial displacement, uq is the tangential displacement, qr and qq are the external forces per unit area in the radial and tangential directions, respectively, D [¼ Eh3/12(1-n2)] is the bending stiffness, K [¼ Eh/12(1-n2)] is the membrane stiffness, E is Young's modulus, r is the ring density, n is Poisson's ratio, a is the mean radius of the tire, h is the thickness of the tread, p0 is the internal pressure, and U is the rotational speed. The dots and primes denote differentiation with respect to time t and circumferential coordinate q, respectively. The modal analysis showed good agreement between theory and measurement in the frequency range of 0e400 Hz. At higher frequencies, when the wavelength is close to the tread width of the tire, the model fails, because the tire tread band should be modeled as a 3D plate instead of a 2D ring. The naming convention for the mode shapes can be found in the work by Wheeler et al. (2005) [42]. €hm (1966) [43], where the tire was modeled as a ring under The fundamental 2D circular ring model was developed by Bo the tension caused by inflation pressure. 2.2.5. Various mechanisms (Heckl, 1986) Heckl (1986) [44] investigated multiple tire noise generation mechanisms, including tire vibrations, air pumping, pavement texture impact, stick/slip, and sound radiation from the tire. The circular ring model for tire vibrations, excluding tread block motion, is based on the equation proposed by Boehm (1966) [43].

8 ES B 000 00 0 0 > > < 2 ðu þ v Þ  4 ðv þ v Þ ¼ rSu€ þ kT u a a > > : T0 ðv00 þ vÞ  ES ðu0 þ vÞ  B ðv0000 þ v00 Þ ¼ rSu€ þ ka v  q a2 a2 a4

(35)

where a is the radius, E is Young's modulus of the tire material, B is the bending stiffness, r is the tire density, kT is the stiffness of tangential bedding, ka is the stiffness of radial bedding, T0 is the tension, S is the cross-sectional area, q is the external driving pressure in the radial direction, v is the radial displacement, and u is the tangential displacement. The time derivatives are indicated by dots and derivatives with respect to the angle f by primes. The circular ring model appears to be more useful than the commonly used modal analysis, especially when the frequencies and/or the damping levels are high. However, the wavelength on the tire surface must be longer than approximately four times the tread thickness h or the tread block distance d (whichever is larger). For typical tires (a z 0.28 m, d z 25 mm, cI z 80 m/s), this gives an upper frequency limit of 800 Hz. Above this frequency, the tread blocks have to be considered individually. The air pumping model is based on the relationship proposed by Hayden (1971) [29].



P¼

2

r0 vq 4pc0 vt

(36)

where P is the sound power, r0 is the air density, c0 is the speed of sound in air, q is the volume flow from the grooves, and t is the time. As q f V, where V is the vehicle speed, and v/vt f V, the relationship between the air pumping sound power and vehicle speed is obtained as P f V4. The pavement texture impact model represents the tread blocks by small springs of stiffness s. Each of them, indexed as n, generates a force given by

Fn ¼ sðvT n  vRn Þ

(37)

at the point fv which is determined by the difference between the tire displacement vT and the road surface amplitude vR at the vth point. Then, the resulting deflection at point fi is

vTi ¼

N X

Fn gðfi  fn Þ

(38)

n¼1

where N is the number of springs, and g is the Green's function for a unit force. For the direct air-borne sound excitation of a blank tire on a rough road, another theory was proposed (Ronneberger, 1984 [45]).

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Stick/slip at the trailing edge was discussed, but no detailed quantitative theory was provided. The author also indicated that spatial changes in the bending stiffness of the tire from one tread block to another are transformed into temporal changes, and therefore they can also be found in the sound signal, but for a slick tire (homogeneous), no such effect is observed. 2.2.6. Tire vibration (Bremner et al., 1997) Bremner et al. (1997) [46] developed a tire vibration model to correlate the vibration-radiated noise with tire construction and material properties. The tire was modeled as an orthotropically stiffened and pressurized cylindrical shell with boundary condition as simply supported. This model was based on the model developed by Mikulas and McElman (1965) [47] and extended by Rennison et al. (1979) [48] (geometric tension stiffness included). This model included membrane waves in the lower frequency range [49] and bending waves in the higher frequency range. Statistical energy analysis (SEA) with modal joint acceptance formulations was used to calculate the wave speed and radiation efficiency. All the main tire design parameters were analyzed for sensitivity study. Wavenumber decomposition was used to determine the dispersion characteristics of a stationary tire under radial point force excitation for experimental validation. The mode shapes for the tread and sidewall, considered as a uniform construction, are as follows.

8 Jmn ðx; qÞ ¼ sinðkm xÞsinðkn RqÞ > > > > < mp km ¼ L > > > n > :k ¼ n R

(39)

where L is the circumference of the wheel rim, R is the smaller radius of the tire cross section, and m and n are mode indexes. It was found that below 300 Hz, tension stiffness has a dominating effect, where the natural frequency is proportional to the circumferential wavenumber. Above 300 Hz, bending stiffness begins to show a dominating trend, where the natural frequency is proportional to the square of the circumferential wavenumber. SEA gives the following relationship. in

PDu ¼ uðht þ hrad ÞEt;Du

(40)

where Pin Du, with a bar above, is the time-averaged power input to the tire in each frequency band Du, u is the temporal frequency of vibration, ht is the structural and material damping coefficient, and hrad is the radiation coupling loss factor (much smaller than ht). Without considering the direct vibration field (valid up to 700 Hz), the energy level given by SEA is

Et;Du ¼ Mt 〈v2t 〉sp;Du

(41)

where Mt is the tire mass; sp,Du, with a bar above, is the space-averaged and time-averaged reverberant vibration level (particle velocity in the sound field). The power input is applied based on the study by Remington et al. (1983) [50], in which the contact point velocity is determined by the roughness profile of the two contact surfaces and their relative contact point impedances. Assuming that the impedance of the pavement is much larger than that of the tire, the power input is given by



in

PDu ¼ v2i;Du Re Zii∞



(42)

where v2i,Du, with a bar above, is the mean squared vibration velocity at the tire contact point, and Z∞ ii is the point “infinite impedance” of the tire. The term v2i,Du, with a bar above, is given by

v2i;Du ¼

u2 〈F2road ðkÞ þ F2tread ðkÞ〉 V

$Du

(43)

where F2road(k) and F2tread(k) are the power spectral density (PSD) values of the road roughness and tread roughness (the two spectra are uncorrelated), respectively; k is the spatial roughness wavenumber defined by

k¼

u V

(44)

where V is the tire rolling speed. The model showed good agreement with the experimental results for far-field noise measurements (sound pressure level or SPL at 7.5 m), including the effects of rolling speed, rubber damping, inflation pressure, tread thickness, and tread materials.

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2.2.7. Porous pavement (Berengier et al., 1997) rengier et al. (1997) [51] studied the acoustical performance of a porous pavement, and developed analytical models to Be describe the surface impedance and sound propagation. Two different models were developed for the impedance characterization, i.e., a phenomenological model and a microstructural model; both models included viscous and thermal effects in the porous structure. The two models have a general formulation as given below.

8 1

> rg ðuÞ 2 > > u k ¼ > > < Kg ðuÞ h i1 > 2 > > rg ðuÞKg ðuÞ > > :Z ¼ c

(45)

U

where k is the complex wavenumber, Zc is the characteristic impedance, u is the angular frequency, U is the porosity, rg is the complex dynamic density, and Kg is the bulk modulus. In the phenomenological model, three physical input parameters are included: airflow resistivity of the porous structure Rs, porosity of the air-filled connected pores U, and tortuosity q2. The complex dynamic density and bulk modulus are given by

  8 fm 2 > > 1 þ i r ð u Þ ¼ r q > g 0 > f > < 31 2 > > ð g  1Þ > 4 >  5 > : Kg ðuÞ ¼ gP0 1 þ 1  i ff

(46)

q

where r0 is the air density, g is the specific heat ratio, P0 is the atmospheric pressure, i is the imaginary unit, f is the frequency; fm and fq describe the viscous and thermal dependencies, and are given by

8 > > > < fm ¼ > > > : fq ¼

URs 2pr0 q2

(47)

Rs 2pr0 Npr

where Npr is Prandtl number. In the microstructural model, five physical input parameters are included: besides Rs, U, and q2, the other two pore shape parameters are sr, associated with viscous dependency, and sk, associated with thermal dependency. The complex dynamic density and bulk modulus are given by

 8 Rs U F l p > 2 > > < rg ðuÞ ¼ r0 q þ i

u

 > ðg  1ÞTðLk Þ 1 > > : Kg ðuÞ ¼ gP0 1 þ 2 Lk

(48)

where F(lp) is a function of sr, and T(Lk) is a function of sk. It was found that the phenomenological model with only three input parameters is accurate enough for acoustic characterization. In addition, it was found that Rs and U are more important than q2. The models can also be used for the detection of pavement clogging. 2.2.8. Wavenumber decomposition (Bolton et al., 1998) Bolton et al. (1998) [52] represented the tire vibration in the form of propagating waves instead of the traditional modal form. The authors claimed that the wavenumber domain form might be more closely related to the physics of wave propagation in tires. The radial vibration of the tire carcass is given by

ur ¼

N X

Am Jm ðsc ; uÞe±ikqm s

(49)

m¼1

where ur is the radial velocity, m is the wave type number, N is the number of contributing wave types, Am is the wave type participation coefficient, Jm(sc, u) is the cross-sectional mode shape, s is the path variable in the circumferential direction, sc is the path variable in the cross-sectional direction, and kqm is the circumferential wave number (real part indicating the propagation speed, and imaginary part indicating the spatial damping). The wave type number m ¼ 1 corresponds to the fast

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extensional mode (180 m/s), while m ¼ 3, 5, and 7 correspond to slow flexural modes (60e80 m/s), as shown in Fig. 3. In this figure, each curving trajectory indicates a single wave type (slope being the propagating speed). The model on the radial tire velocity can then be applied as the boundary condition to the acoustic radiation model. The study investigated a stationary passenger car tire suspended horizontally and driven radially by a shaker at a point on the tread band. It was reported that at low frequencies, a small number of slowly propagating flexural modes dominate, while at higher frequencies the fast modes of the tread band extension are dominant. The type of gas (helium, carbon dioxide, and air) inflated into the tire cavity was seen to have no effect on the carcass vibration, while the inflation pressure (20 psi and 40 psi) had major effects. The cut-on frequencies increased with inflation pressure. It was also found that only the modes m ¼ 3 and n ¼ 1 (cut-on circumferential mode of the first propagational flexural mode) can deliver a net transverse force to the wheel hub and contribute to the transmission of structure-borne vibration. 2.2.9. Orthotropic plate (Kropp, 1999) The orthotropic plate model was developed by Dr. Kropp (1999, Department of Applied Acoustics, Charlmers University of Technology, Sweden) [53]. This model analyzes a smooth tire rolling on a rough pavement. It also consists of three submodels very similar to the one in the study by Kropp (3D two-plate that will be discussed later) (1992) [25,26], and addresses only the structure-borne noise. The differences are the following: (1) In the first submodel (tire model), the tire was simplified as a 2D orthotropic plate for vibration calculations. (2) In the second submodel (contact model), a bedding model was used for the tread, together with a time iterative solution method, to solve the nonlinear problem. (3) In the third submodel (sound radiation model), not only was the vibration of the tire belt considered, but the geometrical condition of the horn effect and local deformations of the tread were also included by using the multipole synthesis method (Cremer, 1984 [54]; Heckl, 1989 [55]). The advantage of this model is that it maintains sufficient accuracy as in the case of the Kropp (3D two-plate) model, and reduces computational effort. As a result, the measurement and calculation showed better agreement over a broader frequency range (125e2500 Hz). A brief introduction to the three submodels is given below. The tire model considers the belt and the two sidewalls as flat plates with different bending stiffnesses in the longitudinal and lateral directions, as illustrated in Fig. 4. The differential equation for the vertical motion of the plate is given by

! 8 00 00 d2 x d2 x d4 x d4 x d4 x > > þ 2 þ Bx 4 þ 2Bxy 2 2 þ By 4  m u2 x þ sx ¼ F0 < T0 2 dx dy dx dx dy dy > qffiffiffiffiffiffiffiffiffiffi > : Bxy ¼ Bx By

(50)

where T0 is the tension caused by the inflation pressure; Bx, By, and Bxy are the circumferential, axial, and cross stiffnesses, respectively; x is the vertical displacement, s is the spring stiffness of the elastic foundation, m00 is the mass per unit area, and F000 is the force acting per unit area. The submodel was validated based on measurements such as the complex point mobility (or admittance, i.e., velocity divided by force) on a smooth, unloaded, stationary tire. A drawback of this model is that it neglects the tire curvature and thickness of the plate, and hence there is a shift in the resonance frequencies below the ring frequency of 400 Hz (circular structure), and large errors at frequencies above 3000 Hz (wavelength is comparable to plate thickness) (Übler, 1994 [56]).

Fig. 3. Frequency-wavenumber characteristics of tire carcass vibration (source from Bolton et al., 1998 [52], Fig. 8; reprinted with permission from Dr. J. Stuart Bolton of Purdue University, USA).

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Fig. 4. Schematic of an orthotropic plate tire model (source from Kropp, 1999 [53], Fig. 2; reprinted with permission from Dr. Jeanette Brooks on behalf of Inderscience Enterprises Limited).

The contact model uses Green's function to calculate the displacement response at a given location as the result of a force applied at another arbitrary location. The discrete time domain equation is

xe ðNDtÞ ¼

X

Fm ðN DtÞgm;e ð0Þ þ

m

X N1 X m

Fm ðNDtÞgm;e ½ðN  nÞDt

(51)

n¼0

where xe(t) is the displacement at the discrete point e at time t, Fm(t) is the driving force at the discrete point m at time t, gm,e(t) is the Green's function representing the relationship between the input at the discrete point m and the output at the discrete point; Dt is the time increment, and N is the time step index. The local deformation of the tread is not considered in the Green's function, but will be compensated for later. The sound radiation model assumes two multipoles located symmetrically on either side of the road surface, as illustrated in Fig. 5. The sound pressure at point E is

pðxE ; yE ; tÞ ¼

X n

ð2Þ

an Hn ðkr1 Þejnf1 þ

X

ð2Þ

bn Hn ðkr2 Þejnf2

(52)

n

where r1, f1, and r2, f2 are the coordinates of point E(xE, yE) with respect to the two coordinate systems. H(2) n is the Hankel function of the second kind and nth order; an and bn are unknown coefficients; k is the wavenumber, j is the imaginary unit, and n is the circumferential mode index. The particle velocity at point E is given by

1 X ð2Þ0 an kHn ðkr1 Þejnf1 jur n 8  9 > sin f1 r1 b cos f1 > > > = <  r sin f 1 1 r2 1 X r  b cos f1 ð2Þ0 r23 ð2Þ0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ rn a n k 1 Hn ðkr2 Þ þ jnHn ðkr2 Þ ejnf2 2  > > jur n r2 > > r1 sin f1 ; : 1

vE ðr1 ; f1 Þ ¼ 

(53)

r2

0 where H(2) n (kr) is the derivative of the Hankel function with respect to its argument, b is the distance between the multipoles, rn is the reflection factor of the road surface for normal incidence, r is the air density, and u is the angular frequency. The

Fig. 5. Schematic of the two multipoles for the sound radiation model (source from Kropp, 1999 [53], Fig. 6; reprinted with permission from Dr. Jeanette Brooks on behalf of Inderscience Enterprises Limited).

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boundary conditions, i.e., the velocities normal to the tire surface, vzn, are determined by the velocity distribution calculated using the tire model and contact model. To calculate the sound power P(u), the tire was modeled as a finite cylinder of width b, as shown in Fig. 6. The equation for P(u) is

PðuÞ ¼

Z∞ ∞ rca X jvrn ðu; kz Þj2 sn ðu; kz Þdkz 8p2 n¼∞

(54)

∞

where a is the tire radius, c is the speed of sound in air, kz is the wavenumber in the axial direction, and vrn(u, kz) is the Fourier coefficient of the radial velocity. The equation for vrn(u, kz) is given by vrn(u, kz) ¼ vrn(u)$sin(kzb)/b. sn(u, kz) is the sound radiation efficiency, given by

sn ðu; kz Þ ¼

2u 1 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 apc u2 2 ð2Þ0 2  kz H c2 a uc2  k2z n

(55)

The full model was validated using the laboratory drum method (DR), as illustrated in Fig. 7, by measuring three parameters: (1) The radial accelerations were measured using accelerometers mounted in the tread grooves (tire vibration measurement, TVM) in the Lagrange reference; (2) The local velocities were measured using a laser Doppler vibrometer (LDV) in the Euler reference; (3) The radiated sound pressure level was measured using a microphone. The first

Fig. 6. Finite cylinder model (source from Kropp, 1999 [53], Fig. 8; reprinted with permission from Dr. Jeanette Brooks on behalf of Inderscience Enterprises Limited).

Fig. 7. Schematic of measurement facilities (source from Kropp, 1999 [53], Fig. 11; reprinted with permission from Dr. Jeanette Brooks on behalf of Inderscience Enterprises Limited).

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measurement showed higher values than those obtained by calculation, but the other two measurements showed good agreement. However, deviations were found to occur above 1500 Hz, because the local deformation of the tread, which acts like a small volume source, becomes more dominant than the belt vibrations. After adding the radiated sound, which is calculated as the integral over the displaced volume, the total third octave band showed very good agreement in the range of 125e2500 Hz. In the present authors' opinion, the Kropp model (orthotropic plate) might be currently the most advanced model, with a suitable approach for tire vibration modeling in tire noise studies. 2.2.10. Shell (Kim and Bolton, 2002) Kim and Bolton (2002) [57,58] analyzed the potential of a rotating tire to radiate sound from static tire dispersion using kinematic compensation. The tread band of the tire was modeled as a simply supported, rotating circular cylindrical shell (Matsubara et al., 2015 [59]) having a finite width to identify the effects of rotation. The governing equations for the tread band model are

D v v ¼ þU Dt vt vf

(56)

8

 Dux D2 ux > > > þ rh ¼ qx ðx; f; rÞ Lx ux ; uf ; ur þ l > > Dt > Dt 2 > > >   2 <

 Duf D uf Dur þ rh  U2 uf ¼ qf ðx; f; rÞ Lf ux ; uf ; ur þ l þ 2U 2 > Dt Dt Dt > > > >   2 >

>  Du D ur Duf > 2 > : Lr ux ; uf ; ur þ l r þ rh   2 U U u r ¼ qr ðx; f; rÞ Dt Dt Dt 2

(57)

8

 vNxx 1 vNfx > > > > Lx ux ; uf ; ur ¼  vx  a vf > > > > > <

 vNxf vNff Qfr   Lf ux ; uf ; ur ¼  vx vf a > > > > > r 2 2 > > > L u ; u ; u  ¼ vQxr  1 vQfr þ Nff  N r v ur  Nff v ur > : r x f r xx 2 2 a vf vx a vx a vf2

(58)

where the subscripts x, f, and r denote axial, circumferential, and radial directions, respectively, of the tire, and the superscript r denotes the residual force. U is the angular rotational speed, f is the circumferential angle in the reference frame, and u is the displacement in the corresponding direction; N and Q are the resultant normal and shear forces, respectively; q is the external force applied in the corresponding direction, r is the density of the tread band, h is its thickness, l is the damping constant, and a is the tire radius. Nrff is the resultant circumferential normal force given by r Nff ¼ ap þ rha2 U2

(59)

where p is the inflation pressure. The linear operators Li(i ¼ x, f, r) are related to the system's stiffness. The rotation of the tire was found to have two effects: stiffening of the tread band and “tilting” of the dispersion curves. However, at normal speeds, inflation pressure has greater influence on the stiffness than rotation. In contrast, the second effect (kinematic tilting effect) was found to be significant. The kinematic relationship is given by

~ mn ¼ umn  nU u

(60)

where ῶmn is the natural frequency in local (Lagrangian, tire-fixed) coordinates, umn is the natural frequency in global (Eulerian, vehicle-fixed) coordinates, and n is the wave mode number in the circumferential direction. 2.2.11. Coupled cavity (Molisani et al., 2003) Molisani et al. (2003) [60] modeled the tire as two shells of revolution and two annular plates, where only the outside shell is flexible, i.e., the tire sidewalls and rim are assumed to be rigid. The boundary conditions for the outer shell were assumed to be simply supported. This model provided an efficient tool to investigate the physical coupling mechanisms between the acoustic cavity and the tire structure without complicated computations such as finite element analysis (Richards, 1991 [61]), where these mechanisms are often hidden in the large numerical models. For the structural equation, the displacement vector of the outer shell was obtained by solving the eigenvalue problem of the self-adjoint Donnel-Mushtari operator. For the acoustic equation, a finite cylindrical cavity with rigid boundary conditions

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was developed. This model investigated the interaction between the acoustic mode and the structural radial mode. The equation of motion for the system, considering fluid loading, is

8 9 8 9 8 9 0 = nh io< u = < 0 = < 2 0  Lc  U ½I v ¼ 0 

: ; : ; : 0 ; p a; x0 q w fw

(61)

where Lc is the shell differential operator, U is the normalized frequency, I is the identity matrix, {u, v, w}T is the displacement vector, where the components u, v, and w are the displacements in the axial, tangential, and radial directions, respectively; fw is the vertical load, p(a, x0 , q0 ) is the cavity acoustic pressure acting on the tire. The equation of the coupled system is

n o ½K  U2 ½M fAmn g ¼ ffmn g  U2 ½afAmn g

(62)

where [K] and [M] are diagonal matrices, and their elements represent the modal stiffness and the mass of the radial modes, respectively; {Amn} is the vector of modal amplitudes, {fmn} is the vector of modal forces, and [a] is the fluid coupling matrix. [a] indicates the extent to which the acoustic mode will affect the structural mode. It was shown that the tire cavity resonance could largely increase the forces at the spindle, and thus increase the vehicle interior noise. It was also found that relatively high damping of the tire structure does not effectively damp out the energy in the acoustic resonance. However, the tire was assumed to be stationary without rotation, and the external excitation was assumed to be a point force acting normal to the shell, so it might not be able to represent the actual cavity resonances when the tire is in motion on a road. 2.2.12. Waveguide (Kim et al., 2007) Kim et al. (2007) [62] developed an analytical model for the tire vibration and structure-borne noise. Three models were developed to represent different vibrations or wave types (waveguide characteristics) at different frequencies: (1) springmass model for low frequencies, below 80 Hz; (2) elastically supported beam model for frequencies in the range of 80e300 Hz; (3) cylindrical shell for frequencies in the range of 300e2000 Hz. Below 80 Hz, the spring-mass model considers the tire sidewall as a spring, and the tread band as a mass. Four modes are found to be mostly related to sidewall stiffness. The eigen-frequencies of the first radial, lateral, and tractive modes, ur, ux, and ut, respectively, are given by

8 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > pR0 ðKr þ Kt Þ > > u ¼ > r > > M > > > rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < 2pR0 Kx ux ¼ > M > > > rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > > u ¼ 2pR0 Kt : t I

(63)

where R0 is the tire radius, Kr, Kx, and Kt are the sidewall spring constants in the radial, lateral, and tractive directions, respectively; M is the tire tread band mass, and I is the polar moment of inertia of the tire tread band. For the frequency range of 80e300 Hz, the tire tread band is modeled as a tensioned beam (R0 ¼ ∞) supported elastically by the sidewall springs with flexural waves propagating in the circumferential direction. The dispersion relationship of the free flexural waves is given by

8 2 > U ¼ b2 ðR0 kÞ4 þ g2 ðR0 kÞ2 þ U2r > > > > >
u

0 > > > > h > > : b ¼ pffiffiffiffiffiffi 12R0

(64)

where U is the non-dimensional frequency, b is the non-dimensional thickness, h is the tread band thickness, and u0 is the ring frequency. For frequencies over 300 Hz, the second and third radial modes occur, and the one-dimensional waveguide turns into two-dimensional waveguide, i.e., cylindrical shell/plate. Waveguide behavior and shell curvature are the two important properties that determine the characteristics of the 2D flexural waves. The approximate dispersion relationship is given by

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 8  2 n2 ð4  nÞ  2  n  m4 2 2 > 2 2 2 > m U ¼ 1  n þ b þ n  >  2 > 2ð1  nÞ > < m2 þ n2 > > > m ¼ kz R 0 > > : n ¼ ks R0

(65)

where n is Poisson's ratio; and kz and ks are circumferential and axial wavenumbers, respectively. The cross-sectional resonant frequencies (cut-off frequencies) are obtained by applying kz ¼ 0 in the above equation. The dispersion relationship accounting for the shell curvature that results in the coupling between radial, axial, and circumferential motions is given by the following equations.

8 > U2 ¼ b2 ðR0 kcs Þ4 þ cos4 f > > > > < 2 kcs ¼ k2z þ k2s   > > > > 1 ks > : f ¼ tan kz

(66)

It was also concluded that the tire waves discussed above are inefficient sound radiators, since their wavenumbers are larger than the acoustic wave numbers. Acoustically excited wave motion of the tire tread band is an important factor for sound radiation. 2.2.13. Cavity (Feng et al., 2009) Feng et al. (2009) [63] investigated the effects of tire rotating speed and tire load on the cavity resonance mechanism. The lowest two cavity modes (splitting into two modes because of the tire load) for a non-rotating tire are defined as horizontal (fore/aft) acoustic mode and vertical acoustic mode. The equations of motion for the two modes are

  8 < X€ þ 2zu0 X_ þ u2 þ af0 X ¼ 0 0   : Z€ þ 2zu Z_ þ u2 þ bf Z ¼ 0 0 0 0

(67)

where X and Z are the coordinates of the center of mass of the air molecules, u0 is the natural frequency of the acoustic cavity of the unloaded tire, z is the damping ratio, f0 is the static load on the tire outer surface in the Z direction; a (negative) and b (positive) represent the effects of tire load on the natural frequencies of the two acoustic modes. The equations for the natural frequencies of the two modes are

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 < u ¼ u2 þ af H 0 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 uV ¼ u0 þ bf0

(68)

where uH and uV (uH < uV) are the resonance frequencies for the horizontal and vertical directions, respectively. For a rotating tire with a rotation angle f, the transformation between the tire-fixed coordinate (X, Z) and the vehicle-fixed coordinate (x, z) is given by



x ¼ X cos f þ Z sin f z ¼ X sin f þ Z cos f

(69)

It may be noted that the rotation rate is slow compared to the lowest acoustic mode frequencies. The equations for the natural frequencies for the rotating tire are

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 2 h  i > > aþb ab 2 2 2 2 > > f f u ¼ u þ U þ  þ 2ða þ bÞf þ 4 u U2 > 0 0 0 1 0 0 < 2 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi >  2 h > i > a þ b ab > > u2 ¼ u2 þ U2 þ : f0 þ f0 þ 2ða þ bÞf0 þ 4u20 U2 2 0 2 2

(70)

where U is the rotation speed of the tire. Even though the natural frequencies of the two modes vary with U, they have the constant average frequency ῶ0 as given below, where the peak SPL occurs.

~0 ¼ u

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 1 2 u20 þ ða þ bÞf0 ¼ u þ u2V 2 2 H

(71)

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Further, Lagrange's equations were implemented into the model (Feng and Gu, 2011 [64]). 2.2.14. 3D multilayer (O'Boy and Dowling, 2009) O'Boy and Dowling (2009) [65] provided a multilayer viscoelastic cylindrical representation of the tire belt, consisting of thin orthotropic layers of different materials of varying thicknesses laid on each other, as illustrated in Fig. 8. It overcame the disadvantage of inaccuracy at high frequencies when the shortest wavelength in the tire becomes comparable to the thickness. Each layer is considered to be isotropic in the circumferential and axial directions, but orthotropic in the radial direction. The model assumed the tire side walls to be pinned, and tire belt to be periodic in the axial direction. The model made it possible to understand the vibration characteristics of a stationary tire, including resonant frequencies, mode shapes and wave speeds, using the parameters of the belt layer structure (density, Young's modulus, Poisson's ratio, and thickness), defined solely by design data without experimental testing. The stress s and displacement u on the lower surface of layer j are related to the coefficients within that layer (O'Boy and Dowling, 2009 [65]).

    s~ G ¼ W S C ~ u H r rj rj rj r j

(72)

j

where ~ denotes Fourier transform in the frequency domain, G and H contain all the information on how the material properties in the viscoelastic layer affect the stresses and displacements for all wavenumbers and frequencies, W has only diagonal terms depending on the Wronskian expressions for the Bessel functions, and S is a summation matrix. Furthermore, the components of stress and displacement on one face of the elastic layer can be related to the components on the other face, as given below.

  s~ ~ u

¼

rNþ1

N Y

Xj

  s~ ~ u

(73)

r1

j¼1

where Xj is the overall material matrix of layer j, which is a function of frequency, wavenumber, material properties of the layer, and its inner and outer radii. Xj is expressed as

 Xj ¼

G H

 rjþ1

Wrjþ1 SMrjþ1;j S1 W1 rj



G H

1 (74) rj

where Mr is the scaling factor that depends on the thickness of the layer. The most useful vibration data for noise prediction, ~ as given below. the displacement u, can be obtained from the inverse Fourier transform of u

uðr; t; q; yÞ ¼

Z∞ Z∞

1 3

ð2pÞ

∞ ∞

∞ X



~ r; n; ky ; u einq eiky y eiut dky du u

(75)

n¼∞

where r is the radius of the belt layer; t is the time, q is the circumferential angle, and y is the axial position; u, n, and ky are the corresponding variables in the frequency domain, i.e., angular frequency, angular order, and axial wavenumber, respectively. The model can provide the displacement or velocity response of the tire belt when excited by a radial or tangential force. The response can be displayed in both the frequency-wavenumber and time-spatial domains. It was shown that the dominant flexural wave under radial excitation travelled at 50e65 m/s, while the compression wave under tangential excitation travelled at 150 m/s.

Fig. 8. Complete viscoelastic cylindrical model of the tire belt (source from O'Boy and Dowling, 2009 [66], Fig. 1; reprinted with permission from Elsevier).

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O'Boy and Dowling (2009) [66] also introduced an approximation using a flat bending plate for the tire belt. The vertical displacement w(x, y, t) at position (x, y) at time t is governed by (Graff, 1991 [67]) the following equation.

pffiffiffiffiffiffiffiffi v v2 w TV2 w  DV4 w þ p þ hp DM V2 w ¼ M 2 vt vt

(76)

where T is the in-plane tension with the unit N/m, p(x, y, t) is the load per unit area, M is mass per unit area (area density), hp is a constant representing additional damping, and V is the gradient operator resulting in a vector like v/vx1 and v/vx2; V2 is the Laplace operator resulting in a scalar like v2/vx21 and v/vx22; D is the bending stiffness given by

D¼

Eh3

 12 1  n2

(77)

where E is Young's modulus, and n is Poisson's ratio. This corresponds to the case for a tire belt without the tread rubber layer. For the case with tread layer included, the equation becomes

r v2 TV w þ V  2 k m vt 2 2

2

!

rh3 v2 12 vt 2

!  DV

2

pffiffiffiffiffiffiffiffi v w þ ks w  hp DM V2 w þ vt

! DV2 r h 2 v2 v2 w 1 2 þ p¼M 2 vt k mh 12k2 m vt 2

(78)

 constant that accounts for where k is the Timoshenko shear coefficient that accounts for shear deformation, m is the Lame rotary inertia, ks is the stiffness of the equivalent distributed springs representing the tire sidewalls, and r is the rubber density. Then, the transfer function in the frequency domain between displacement w (with tilde above) and pressure p (with tilde above) is expressed as

I0 k2y þ I1 ~ w ¼ ~ ðu;n;k Þ J1 k4y þ ðJ2 þ I2 Þk2y þ ðJ3 þ I3 Þ p y

(79)

where J is the material term associated with the bending plate, I is the correction term that accounts for the shear deformation and rotary inertia. These are expressed by the following equations.

8 > > > > J1 ¼ D > > > > > pffiffiffiffiffiffiffiffi > 2Dn2 > > J2 ¼ 2 þ T þ iuhp DM > > > R > > > pffiffiffiffiffiffiffiffi > 2 > > Dn4 Tn2 iuhp DMn > > J ¼ þ þ þ ks  Mu2 > 3 > 4 2 2 > R R R > > > > > D > > < I0 ¼ 2 k mh > > > > Dn2 u2 rh2 > > I1 ¼ 1 þ 2  > > > k mhR2 12k2 m > > > >  3  > > > Dr > 2 rh > þ I ¼  u > 2 > 12 k2 m > > > > >   > 3 > Dr n2 u4 r2 h3 > 2 rh > > þ ¼  u þ I > 3 > 12 k2 m R2 > 12k2 m :

(80)

A numerical method was also presented to determine the parameters for the equivalent plate model which closely matches the multilayer viscoelastic cylindrical model. 2.2.15. Green's function (Arteaga, 2011) Arteaga (2011) [68] derived a closed-form expression for the Green's function for a loaded rotating tire expressed in the Eulerian reference coordinate, based on the modal-arbitrary Lagrangian-Eulerian (M-ALE) approach (Lopez et al., 2007 [69]). It addresses the contact problem directly in the Eulerian reference frame, and includes local tire softening due to nonlinear deformation without increasing computational cost. The expression for the Green's function gjl between the excitation at a given location and direction l, and the response at another location and direction j in the Eulerian reference frame is given below. Please cite this article in press as: T. Li et al., Literature review of models on tire-pavement interaction noise, Journal of Sound and Vibration (2018), https://doi.org/10.1016/j.jsv.2018.01.026

T. Li et al. / Journal of Sound and Vibration xxx (2018) 1e89

gjl ðU; tÞ ¼

N X

"

Fjk

2N X

PN vkr

p¼1 wpr Flp sr t

r¼1

k¼1

ar

21

#

e

(81)

where U is the rotating velocity of the tire, t is the time, Fjk and Flp are the corresponding elements of the eigenvector matrix, vkr and wpr are the corresponding elements of the relative eigenvectors of transformed uncoupled first order differential equations, sr is the corresponding eigenvalue (same for vkr and wpr), and ar is an intermediate variable. It may be noted that the only details needed to compute these Green's functions are the eigenvalues and eigenvectors of the statically loaded tire in the tire reference frame and the mass matrix, which can be easily obtained from standard finite element modeling. The full excitation response of the tire is then calculated from the convolutions between the Green's functions and the contact forces. It was shown that the modal density increases as the rotation speed increases for both the unloaded and loaded tires. 2.2.16. Coupled Mode (Bolton and Cao, 2013) Bolton and Cao (2013) [70] investigated the coupled tire structural-acoustic mode. The tire tread band is unwrapped and modeled as a tensioned 1D string, while the rim is modeled as a hard wall, as illustrated in Fig. 9. In the figure, R is the tire radius, H is the tire section height, and L is the circumferential length. The tire cavity lies between the string and the hard wall. The air that rotates with the tire in the cavity is assumed to be uniform with inviscid flow, governed by the following linearized acoustic wave equation.

 v2 p v2 p  2 v2 p v2 p ¼ c  v2 þ c2 2  2v vxvt vt 2 vx2 vy

(82)

where p is the sound pressure, c is the adiabatic speed of sound, v is the flow speed, and t is the time; x and y are the circumferential and radial coordinates shown in Fig. 9. The wave equation for a tensioned moving string is given below.

rL

v2 x v2 x þ 2v 2 vtvx vt

!

  v2 x þ rL v2  T ¼0 vx2

(83)

where rL is the linear density of the string, x is the radial displacement of the string in the y-direction, and T is the string tension force; v is the speed of the moving string, assumed to be the same as the air flow speed mentioned above. The solutions assumed are given below.

(

  p ¼ Aejky y þ Bejky y ejkx x ejut

(84)

x ¼ Cejks x ejut

where A, B, and C are the amplitudes, j is the imaginary unit, kx and ky are the acoustic wavenumbers of the air flow in the xand y-directions, respectively; ks is the wavenumber of the string, and u is the angular frequency. Assuming kx ¼ ks, and with boundary conditions applied to the fluid-rigid body coupled system, the characteristic equation is



u2  2vukx þ v2 

T



rL

k2x ¼

r0 u2

rL ky tan ky H



(85)

The dispersion equation is

Fig. 9. Tire-rim assembly modeled as tensioned string-hard wall (source from Bolton and Cao, 2013 [70]; reprinted with permission from Dr. J. Stuart Bolton of Purdue University, USA).

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u2 c2

 ¼

1

 v2 2 vu k þ k2y  2 2 kx c2 x c

(86)

In this study, frequency-split phenomenon was observed during tire rotation in the coupled mode. The phase velocity of each mode approximately equals the associated static phase velocity plus/minus flow velocity. It was found that the plane wave mode is significantly influenced by the coupling effect, while the higher modes are less influenced. In the further study, Cao and Bolton (2015) [71] improved this model significantly by considering the tread band as a 2D ring, and including flexural and longitudinal waves. The displacement of the ring occurs in both circumferential and radial directions. The solution of acoustic pressure in the air cavity is assumed to be harmonic in the circumferential direction, and follows the Bessel function in the radial direction. Bolton and Cao (2015) [72] verified this model by applying harmonic point input excitation both radially and circumferentially. 2.3. Finite element and boundary element models The finite element method (FEM), also known as finite element analysis (FEA), is a numerical technique for solving problems in partial differential form. FEM subdivides the entire body, including the surrounding boundary, into smaller/ simpler parts called finite elements. A system of linear equations that model these finite elements are then integrated to generate the solution equations for the entire body. Unlike the FEM, boundary element method (BEM) subdivides only the boundary of the body into finite elements, and is computationally more efficient. With the advancement of computers, BEM and FEM have become the two dominant numerical techniques in computeraided engineering (CAE). BEM and FEM are often coupled with other methods, and have several derivations, such as waveguide finite element method (WFEM), infinite/finite element method (IFEM), wave finite element (WFE), energy boundary element analysis (EBEA), and energy finite element analysis (EFEA); these methods will be discussed in the following sections. The above methods are often used to calculate the vibrations of the tire, and are used together with fluidstructure interaction (FSI) method to analyze the sound generation and propagation into the air. Sixteen models were reviewed in this subsection. 2.3.1. Toroidal membrane (Saigal et al., 1986) Saigal et al. (1986) [49] modeled the tire as a toroidal membrane under low internal pressure, with the tire mounted on the rim, to investigate its free vibrations. A 12 degree-of-freedom (DOF) membrane finite element was used. To verify the validity of the finite element formulations, the study presented the analytical solutions for three types of models: a flat rectangular membrane, a circular cylindrical membrane, and a toroidal membrane. The final solution of the mode shapes and natural frequencies for the toroidal membrane model is given by

8 h n i XX > ð6f  pÞ eiumn t Amn cosðmqÞsin wðq; f; tÞ ¼ > > > 10 > m n < vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u  pffiffiffi 2 2 > u > > > umn ¼ 1 t m þ 6 2n þ G > : 10 x h

(87)

where w is the displacement normal to the tire surface, q is the coordinate along the toroid circle, f is the coordinate along the meridian circle, and t is the time; m and n are mode numbers, and u is natural frequency; x, h, and G are coefficients given by

8 a2 rh > x¼ > > > Tq > > > > < b h¼ a > > > > > Eh > >  > :G ¼  Tq 1  n2

(88)

where a is the radius of the meridian circle, b is the radius of the toroid circle, r is the mass density, h is the thickness of the element; E is Young's modulus, and n is Poisson's ratio; Tq is the internal tension force per unit length in the direction of q; this force equals pa/2, where p is the internal pressure. The above equations are derived under the assumption that b is much greater than a, such that the toroid can be considered as a cylindrical shell of radius a and length 2pb, with the two ends attached to each other. Good agreement with measurements was found for the symmetrical and twisting modes of vibration. It may be noted that this model is restricted only to lower frequencies, as the membrane model cannot consider bending waves at higher frequencies. Hence, it will probably not apply to a steel belted radial tire, but might apply to certain bias ply tires and aircraft Please cite this article in press as: T. Li et al., Literature review of models on tire-pavement interaction noise, Journal of Sound and Vibration (2018), https://doi.org/10.1016/j.jsv.2018.01.026

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tires. In addition, the model considers the tire as nonrotating and unloaded. Diaz et al. (2016) [73] developed an analytical tire model that includes all the material behavior for modal analysis under different operating conditions such as speed, load, and inflation pressure. In this study, the effect of Coriolis accelerations was investigated; it was found that real and standing waves are dominant for a nonrotating tire, while complex and travelling waves occur because of speed bifurcation effect. Tire loading generally shifts down the respective modal frequencies. Increasing the inflation pressure increases the structural modal frequencies, but does not influence acoustic modes. 2.3.2. Acoustic radiation (Wright and Koopmann, 1986) The acoustic radiation model (Wright and Koopmann, 1986 [74]) predicted the sound pressure and intensity field around a stationary point-excited tire. First, the surface velocity field of the tire was obtained through conventional modal analysis methods. Then, the sound field was numerically calculated using Helmholtz integral formulation with the information on the tire geometry. The modal analysis was conducted using FEM. The number of elements was 336, which was determined by the wavelength of the structural vibrations being investigated. The wave speeds of the modes are shown in Fig. 10. It can be seen that the wave speed decreases exponentially with increasing mode number. In addition, all the structural wave speeds are well below the speed of sound in air, indicating that in these modes, the tire is an inefficient acoustic radiator, as the tire surface vibrations will not couple very well with the surrounding air. The Helmholtz integral formulation considers the vibrating tire body as a continuous series of monopole and dipole sources. It is shown that the sound intensity field has both positive and negative values, meaning that the tire is absorbing as well as generating sound. This again proves that the non-propagating responses of the tire are very inefficient at radiating the sound. It is suggested that it is the decaying, propagating flexural wave field on the tire that generates the majority of the structure-borne sound. Therefore, the most effective method of reducing structure-borne tire noise is to reduce the forcing inputs at the tire-road interface. 2.3.3. FEA coupling (Richards, 1991) Richards (1991) [61] numerically analyzed the structural-acoustic coupling cavity using FEA. The structural formulation was based on an existing FEA model for various aspects of tire analysis (Richards, 1987 [75]), consisting of isotropic rubber, cord layers, inflation loading, and forces at the fixed nodes. The cylindrical space co-ordinates of each of the nodes are expressed as r, q, and z, while the material coordinate which is attached to the structure is expressed as f. The governing equation is

  ½K0  þ in½K1  þ n2 ½K2   u2 ½M fag ¼ 0

(89)

where [K0], [K1], and [K2] are the stiffness matrices, [M] is the mass matrix, i is the imaginary unit, n is the integer representing the circumferential wavenumber, u is the natural frequency, {a} is the mode shape containing nodal displacement, element stresses, and the reactions at the fixed nodes. The acoustic formulation is governed by the following wave equation.

V2 j 

1 v2 j ¼0 c2 vt 2

(90)

where c is the speed of sound; j is the velocity potential (particle velocity v ¼ Vj), which is related to the acoustic pressure p, by the equation

p ¼ r

vj vt

(91)

Fig. 10. Variation in structural wave speed with frequency for unloaded tire (source from Wright and Koopmann, 1986 [74], Fig. 6; reprinted with permission from Mr. Michael Kaliske of TU Dresden, Germany on behalf of Tire Science and Technology).

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where r is the gas density. Employing the weight residual method to obtain an approximate solution leads to the following equation.

Z S

vu v2 j E v2 j $dS þ E2 2  20 2 ¼ 0 vt c vt vq

(92)

where S is part of a thin slice [q, q þ dq] on the cavity surface, and u is the displacement of the enclosing structural surface; E0 and E2 are calculated as follows.

8 E0 ¼ > > > < > > > : E2 ¼

Z dV V

Z V

1 r

(93)

dV 2

The resonance frequencies are

sffiffiffiffiffi E un ¼ nc 2 E0

(94)

where √(E2/E0) is considered as the effective radius of the cavity. The modified equation for the structural-acoustic coupling is

  ½K0  þ in½K1  þ n2 ½K2   u2 ½M fag  fRgP ¼ 0

(95)

where {R} includes the complex transformation from f to q, and depends on the circumferential wavenumber n; P is the scaling factor for acoustic pressure. Two additional equations related to these parameters are

8 E > > fRgþ fag  02 P  inE0 c0 ¼ 0 > < rc 2 > > > : inE0 P  u2 r E0 c0 ¼ 0 E2

(96)

where c0 is the average angular displacement of the gas particles in the cavity, and has the following relationship.

vc v E vj ¼ c einq cos ut ¼ 2 vt vt 0 E 0 vq

(97)

It was found that fluids with high density and pressures may have a significant effect on structural resonances. 2.3.4. 3D two-plate (Larsson and Kropp, 1992) Larsson and Kropp model (1992) [25,26] combines three different modules: (1) a tire model where the dynamic properties of the tire are considered; (2) a contact model where the nonlinear contact forces (both radial and tangential) and the corresponding vibration velocities on the tire surface are calculated; (3) a radiation model where the sound radiation from the tire due to the velocity field on the tire surface is calculated. This is one of the most commonly used TPIN models (Dare, 2012 [13]). It is known that at higher frequencies (above 500 Hz), the curvature of the tire can be neglected, while the importance of the internal structure (i.e., multi-layers of steel and rubber) increases (Larsson and Kropp, 1999 [26]). As a result, Kropp suggested a model of two coupled elastic orthotropic layers/plates (the top layer is the tread, and the bottom layer is the belt) under tension, supported by a linear bedding in the x, y, and z directions, as shown in Fig. 11. The tension in the layer/plate is caused by the internal inflation pressure, and the bedding represents both the side walls and the enclosed air in the tire cavity. The model shows good agreement with measurements up to 2 kHz for a slick 205/60R15 tire (Larsson and Kropp, 1999 [26]). Based on Hamilton's principle (conservation of kinetic and potential energies of a system), the equations of motion for each of the layers are given by Please cite this article in press as: T. Li et al., Literature review of models on tire-pavement interaction noise, Journal of Sound and Vibration (2018), https://doi.org/10.1016/j.jsv.2018.01.026

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25

Fig. 11. Tire model consisting of two coupled elastic layers under tension on an elastic bedding (source from Larsson and Kropp, 1999 [26], Fig. 1; reprinted under fair use provision).

8 "  # > > v2 x v2 x v2 x 1 v vx vh vz v2 x v2 x > > > þ þ þ s0 2 ¼ r 2 þ 2þ 2þ G > 2 > 1  2 vx n vx vy vz > vx vy vz vx vt > > > > # > " 2   < v h v2 h v2 h 1 v vx vh vz v2 h v2 h þ þ þ s0 2 ¼ r 2 G þ 2þ 2þ 2 > 1  2 vy n vx vy vz vx vy vz vx vt > > > > >  # > " 2 2 2 2 > > v vx vh vz v z v2 z >G v z þ v z þ v z þ 1 > þ þ þ s ¼ r > 0 > : vx2 vy2 vz2 1  2n vz vx vy vz vx2 vt 2

(98)

where G is Young's modulus, and n is Poisson's ratio; x, h, and z are the displacements inside the material in the x, y, and z directions, respectively; s0 is the constant external tension (here, only in the x-direction), r0 is the density of each layer, and t is the time. Modal superposition method is used to calculate the response of a finite structure. However, for high frequencies, a large number of modes is needed, which results in time-consuming calculations and difficulty in modeling the damping. The model was used to compare TPINs on stone mastic asphalt (SMA) and ISO surfaces. TPIN on the SMA surface was predicted to be 2e3 dB louder than that on the ISO surface, while it was 7e8 dB louder in reality [76]. Another study on pass-by TPIN on four different pavements showed accurate results within 5 dB, only in the range of 500e1250 Hz. The model was validated against measured radial and tangential point and transfer mobilities, and good agreement was observed in both magnitude and phase (Andersson and Larsson, 2005 [28]). 2.3.5. TRIAS (de Roo and Gerretsen, 2000) TRIAS (Tyre-Road Interaction Acoustic Simulation, 2000 [77,78]) includes generation mechanism modeling for both structure-borne (tread/texture impact) noise and airborne (air pumping) noise, and sound radiation modeling. The generation mechanism model uses FEM and BEM to predict one-third octave band pressure spectra near the tire. The model includes multiple pavement parameters, such as texture depth, porosity, structure factor, flow resistance, and/or sound absorption, which can be measured experimentally or acquired from the supporting simulation model RODAS (ROad Design Acoustic Simulation), which generates the physical pavement characteristics from the material composition of the mixture (chipping size, binder content, and layer thickness) and analyzes the texture spectrum. The schematic of the full model is shown in Fig. 12. The input pavement parameters include the following: (1) dimensional texture profiles measured in the direction of the road, (2) dimensional texture field, wavelength spectrum, mean profile depth, sound absorption coefficient, distribution of the dimensions of the aggregate particles, binder type and quantity (for porous pavement), thickness of the surface layer, porosity of the surface layer, and specific air flow resistance (de Roo et al., 1999 [78]). Another supporting submodel, TYDAS (TYre Design Acoustic Simulation) generates model inputs from known tire parameters, if all the necessary inputs are not known. The input tire parameters include cross-sectional drawing with dimensions (unloaded and loaded), tread pattern, tread depth, total mass of the tire, density per layer or the average density of the tire, inflation pressure, load, rolling speed, overall stiffness of the tire, and bending stiffness of the combination of tread and carcass in the x (circumferential) and y (axial) directions (de Roo et al., 1999 [78]). The TRIAS model was validated on dense and porous HMA pavements and ISO test tracks. The overall TPIN on dense asphalt pavements was predicted to be within 2 dB, and the TPIN on porous pavements was predicted to be within 5 dB. The model mostly over-predicted the noise. The measured and predicted TPIN spectra have not been presented. Please cite this article in press as: T. Li et al., Literature review of models on tire-pavement interaction noise, Journal of Sound and Vibration (2018), https://doi.org/10.1016/j.jsv.2018.01.026

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Fig. 12. Schematic diagram of the mathematical simulation model TRIAS (Tyre-Road Interaction Acoustic Simulation); the supporting model RODAS (ROad Design Acoustic Simulation) is indicated in the upper part of the scheme (source from de Roo and Gerrestsen, 2000 [77], Fig. 1; reprinted under fair use provision).

2.3.6. WFEM (Nilsson, 2004) The application of WFEM to tire vibration analysis was presented first by Nilsson (2004) [79], and later by Fraggstedt (2006) [80], both at KTH Royal Institute of Technology, Sweden. The analysis of the wave field on a stationary tire (no contact with ground) was presented by Sabiniarz and Kropp (2010) [9] at Chalmers University of Technology, Sweden. A waveguide is a structure that guides the waves in a single direction, such as the circumferential direction of the tire, with constant physical properties in that direction. WFEM uses a 2D finite element model [81] over the cross section of the waveguide, and describes the response of the tire in terms of a set of waves in this direction (Sabiniarz and Kropp, 2010 [9]). The response of the nodes in direction X at time t is governed by the following equation.

"

# v2 v v2 K2 2 þ K1 þ K0 þ M 2 VðX; tÞ ¼ FðX; tÞ vX vX vt

(99)

where Ki is the stiffness matrices, M is the mass matrix, F(X, t) is a vector originating from the external load, and V(X, t) contains all the nodal degrees of freedom (DOF). The results of dispersion relations for the waves in the tire are shown in Fig. 13 (Sabiniarz and Kropp, 2010 [9]). There are basically three groups of modes: (1) cross-section modes involving strong out-of-plane motion (Group 1 corresponds to symbols ‘⊥’, ‘’, ‘:’, ‘;’, ‘⋄’, ‘▫’); (2) waves involving significant extensional deformation of the whole cross section (Group 2 corresponds to symbols ‘=’); (3) waves with strong in-plane motion of the sidewalls and weak out-of-plane motion of the

Fig. 13. Dispersion relations with various symbols indicating different groups of modes (Continental 205/55R16, source from Sabiniarz and Kropp, 2010 [9], Fig. 13; reprinted with permission from Elsevier).

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belt (Group 3 corresponds to symbols ‘B’, ‘þ’, ‘<’). It was also concluded that Group I has a low wave speed; Group II would be much better radiators; Group III has a higher phase speed, but very low response in the direction normal to the tire surface. 2.3.7. IFEM (Biermann et al., 2007) Biermann et al. (2007) [23] developed a computational tool to predict the sound radiation from rolling tires. The computation process could be divided into three steps: (1) computation of nonlinear tire stationary rolling process; (2) analysis of tire dynamics caused by the road texture; (3) computation of sound radiation using IFEM. The third step employs the structural response calculated by using the first two steps as the boundary conditions. For the sound radiation analysis, the inner region close to and enveloping the tire was discretized with finite elements, while the outer region was discretized with infinite elements. The governing equation for the elements (Astley-Leis elements) is

h i KA þ ðiuÞCA  u2 MA p ¼ f A

(100)

where KA, CA, and MA are frequency-independent acoustic system matrices, referred to as inverse mass, admittance, and compressibility operator, respectively; p is the unknown pressure at the nodes of the acoustical mesh, and fA is the load vector representing the excitation of the acoustic fluid due to tire vibration. The model was validated with two tests, a standing tire excited by a shaker and a tire rolling on a drum; a good agreement was found up to 600 Hz. 2.3.8. FEM & BEM (Brinkmeier et al., 2008) Brinkmeier et al. (2008) [17] model is composed of a finite element model of the nonlinear stationary rolling tire and a boundary element model to predict the radiated sound using a modal superposition approach. The overall model uses a combined finite/infinite element approach, where the AstleyeLeis elements method is employed; this ensures a stable and extremely computationally efficient solution process. The finite element model uses the arbitrary Lagrange-Eulerian formulation (ALE). The final equation is

b¼ ðt K  WÞD f

tþDt

bf þ t bf  t bf e s i

(101)

where tK is the stiffness matrix, and W is the ALE-inertia matrix; Df, with a hat above, is the incremental displacement; tþDtf, with a hat above, is the force resulting from external loads; tfi, with a hat above, is the ALE-inertia; tfs, with a hat above, is the internal stress. The equation for the boundary element model is

i h AðuÞp :¼ KA þ ðiuÞCA  u2 MA p ¼ f A

(102)

where KA is the inverse mass operator or acoustic stiffness, CA is the admittance operator or acoustic damping, MA is the compressibility operator or acoustic mass matrix, p is the sound pressure, and fA is the load vector representing the tire dynamics on the surface. The symbol “:¼” indicates that it is a definition. The model successfully predicted the acoustic ranking of tires, but the error between the predicted and experimental results was in the range of 6e17 dB, probably because the model considers only the structural-borne noise at low frequencies (100e850 Hz), and ignores pavement absorption. In addition, only vertical contact forces are considered, and friction forces are not included. 2.3.9. WFE (Waki et al., 2011) Waki et al. (2011) [82] applied WFE method (Waki et al., 2009 [83]) for primitive estimation of tire vibration and noise radiation. The study provided insight into the locations of the radiating parts on the tire. In contrast to FEA, which could investigate vibrations only at low frequencies, WFE could predict vibrations up to high frequencies around 1 kHz. A short section of a tire, having a uniform structure, was modeled using FEM (Díaz et al., 2012 [84]). The equation of motion is



Dq ¼ f D ¼ K þ juC  u2 M

(103)

where D is the dynamic stiffness matrix, q is the nodal displacement vector, and f is the nodal force vector; K, C, and M are the stiffness, damping matrix, and mass matrix, respectively. Then, the periodicity condition, as shown below, was applied to the equations of motion of the short section to formulate the eigenproblem.



qR ¼ lqL f R ¼ lf L

(104)

where the superscripts L and R indicate the left- and right-hand sides, respectively, of the section. l is related to the wavenumber k as given below.

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li ¼ expðjki DÞ

(105)

where D is the length of the short section. It was found that the sound emission field could be very different from the vibration field. 2.3.10. Road input (Kido et al., 2011) Kido et al. (2011) [85] correlated the actual road profile with excitation at the tire contact patch. The spatial road profile was transformed into time domain profile, considering the driving speed, and was then transformed into frequency domain data. The profile data were two-dimensional, which means that the entire two-dimensional tread area could be considered. Further, an FE model was developed to correlate the tire patch excitation and the axle forces under steady-state conditions. Both Coriolis force and centrifugal force due to tire rotation were considered. The response of the wheel assembly x, excited by the external force f with tire rotational angular velocity of ur (rad/s), is represented by the following equation.

  Mx€ þ ðC þ ur M1 Þx_ þ K  u2r M2 x ¼ f

(106)

where M, K, and C are the mass matrix, stiffness matrix, and viscous damping matrix, respectively, based on the tire vibrational properties; M1 is the skew-symmetric matrix, and M2 is the symmetric matrix that depends on the inertia property and rotational axis of the tire. The Coriolis force is in proportion to the velocity urM1ẋ, and the centrifugal force is in proportion to the displacement u2r M2x. The model also considers the tire stiffness drop caused by tire rotation. The calculations of the longitudinal/lateral forces on the axle, and the accelerations at low frequencies (0e120 Hz) were found to have good agreement with experimental measurements. It was also found that the influences of Coriolis force and centrifugal force are especially important in the frequency range of 80e100 Hz. 2.3.11. FSI (Yang et al., 2013) The FSI model developed by Yang et al. (2013) [22] consists of two submodels: the rolling tire-pavement interaction submodel and the near-field sound propagation submodel. The two submodels are coupled using the FSI method, where the equilibrium conditions of displacement and stress are fulfilled. The tire-pavement interaction is assumed to be a two-body contact problem, where the pavement surface is assumed to be rigid, and the body forces are ignored. The relevant equation is given below.

Z t

V

rT u€T $duT dt V þ

Z t

V

cT u_ T $duT dt V þ

Z t t

V

Z

tT deT dt V ¼ t

Sf

t S f T $duS dt V

Z  t

t

f C $duT dt S

(107)

Sc

where T represents the tire, rT is the tire density, cT is the tire damping ratio, ttT is the stress tensor, deT is the strain corresponding to the virtual displacement duT, tfTS is the surface force per unit area, and tfC is the surface force per unit area on contact surface. The near-field sound propagation submodel is described using the large eddy simulation (LES) model (Wagner et al., 2007 [86]). The governing equations in the LES model for an arbitrary moving cell with a volume V(t) and a cell-face area S(t) are given by

d ∭ rdV þ ∬ SðtÞ rðU  WÞ$dS ¼ 0 dt VðtÞ

(108)

d ∭ rUdV þ ∬ SðtÞ ½rUðU  WÞ  s$dS ¼ ∭ VðtÞ f B dV dt VðtÞ

(109)

where r is the air density, U is the flow velocity vector, W is the moving mesh velocity vector, fB is the air body force, and s is the total stress tensor. This model focuses only on the PIARC passenger car tire (PIARC, Permanent International Association of Road Congresses) (Sandberg, 1990 [87]; PIARC, 2004 [88]). The A-weighted one-third octave band sound pressure level showed good agreement between simulation and experimental results in the range of 500e2500 Hz, with vehicle speed in the range of 50e90 km/h. 2.3.12. Wear (Tong et al., 2013) The wear model developed by Tong et al. (2013) [89] is composed of a steady rolling tire model, a tire wear model, and a tire acoustic model. The steady rolling tire model used the ALE approach, by decomposing the motion into a pure rigid body motion and superimposing the deformation. The expressions for the velocity and acceleration of the particle in both Lagrangian and Eulerian reference frame were derived.

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The tire wear model included an FE model of the 3D tread pattern prepared using Pro/E and Abaqus FE software. The tire acoustic model perceived the sound field as an ideal homogeneous fluid medium with small-amplitude waves. The acoustic excitation was obtained from steady rolling tire model. A modal test with random signal excitation was performed to compare the measured modal frequency with the simulation results. The results showed good agreement with a maximum deviation of 4.48% in the frequency range of 0e500 Hz. The model investigated four factors that potentially influenced TPIN: speed, tire load, inflation pressure, and tire wear. 2.3.13. EFEA/EBEA (Vlahopoulos et al., 2013) Vlahopoulos et al. (2013) [90] investigated the airborne noise inside a vehicle, originating from exterior components such as tires, engine, and exhaust. These exterior components were assumed to be the acoustic source, and then EBEA was applied to compute the exterior acoustic loading on the vehicle. Further, EFEA was employed to calculate the interior airborne noise with the exterior acoustic loading as the excitation, i.e., EFEA dealt with acoustic transmission through the vehicle mass panel, constraint layer damping (CLD), and acoustic absorption treatment (or noise reduction) from the exterior to the interior. The power transfer mechanism is accounted for by the mass barrier effect of the panel. The governing equation is given below.

2

tnormal ¼

3

ri ci 6 4 rj cj 

4

7

2  2 5 B 1 þ rri ccij þ u rs trs þm c j j j

(110)

where tnormal is the normal mass barrier transmission coefficient from acoustic enclosure i to acoustic enclosure j through the panel; ri, rj, and rs are the densities of the acoustic enclosure i, acoustic enclosure j, and panel s, respectively; ci and cj represent the speed of sound in acoustic enclosures i and j; ts is the thickness of the panel separating the two acoustic spaces; u is the angular frequency, and mB is the surface mass density of the mass barrier. It is known that FEA typically models only structure-borne noise at low frequencies, and a detailed FE model is needed in the path between the excitation and the receiver. As compared to FEA, EFEA captures the physics associated with the path between the exterior acoustic excitation and the interior acoustic cavity, and covers the entire frequency range of 200e8000 Hz. In this vehicle model, it was found in the contribution analysis, expressed in terms of the power flow into the interior acoustic space of the vehicle, that the front floor contributed the most, up to 70.8%. 2.3.14. Cavity (Mohamed and Wang, 2015) The cavity model developed by Mohamed and Wang (2015) [91] focuses on the noise generated from the tire torus cavity resonance. The tread plate is assumed to be flexible, while the other walls are considered as rigid. A modal series solution of the sound pressure frequency response inside the tire cavity is derived from the wave equation using modal superposition. This work also provides a noise reduction method, which involves placing a trim layer on to the inner surface of the tire tread plate by adding a damping loss term in the sound pressure frequency response function (sound pressure peak reduction is 10e15 dB). The cavity natural frequencies can be approximated by the following equation.

fmnl

c ¼ 2p

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2  2 lp 2np 4m þ þ Do þ Di W Do  Di

(111)

where fmnl is the cavity natural frequency at the mode corresponding to the indexes m, n, and l in the axial, radial, and azimuthal direction, respectively; c is the speed of sound, W is the tire width, Do is the outer diameter of the tire torus, Di is the inner diameter of the tire torus. The calculated results are also compared with FEM results. It may be noted that only a minimum of six elements per wavelength is required in the acoustic space, and hence FEM involves relatively less computation. The Donnel shell theory [92] is used for the calculation of tire tread vibration. The tread surface particle velocity is defined by

upr ¼ iu

F0 eiut r ht

∞ X B2ml sinðlpz0 =WÞcosðmq0 Þsinðlpz=WÞcosðmqÞ   u2lm 1 þ ihp  u2 m¼0;l¼0

(112)

where upr is the solid particle velocity of the plate surface in the r direction, r is the mass density of the top tire tread pattern plate, ht is the plate thickness, m and l are the plate wave numbers in the q and z directions, respectively; q0 and z0 are the location coordinates of the harmonic point excitation force, F0 is the amplitude of the harmonic point excitation force, ulm is the natural circular frequency of the annular outer plate, hp is the plate damping loss factor, and Bml is the normalized plate mode shape coefficient.

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2.3.15. MLE (Wei et al., 2016) Wei et al. (2016) [93] developed a model using mixed Lagrangian-Eulerian (MLE) method to predict the tire noise in the time domain. It was pointed out that the tire vibrations must be analyzed in the Lagrangian reference, while the acoustic analysis is normally performed in the Eulerian reference. The MLE model transformed the tire vibration data (velocity/acceleration) in the Lagrangian mesh to the Eulerian mesh (finite element model). The radiated noise was then calculated from the information on the vibration of the tire outer surface (tread pattern and groove surface) using acoustic BEM. It was found that the accelerations are largely distributed near the footprint (approximately double the contact length). The accelerations at the trailing edge are lower than those at the leading edge, but with a much wider distribution, as shown in Fig. 14; this explains the reason for the noise near the trailing edge to be greater than at the leading edge. The model considered only the tire noise due to vibrations, such as impact between the tread pattern and the road texture, and did not include air pumping noise. 2.3.16. Time domain BEM (Banz et al., 2016) Time model (Banz et al., 2016) [94] used a time domain Galerkin boundary element method based on a time-dependent integral equation of the second kind to formulate the acoustic wave equation for tire noise radiation with the tire dynamics given. The governing equation to be solved is given below.

ðI þ K 0 Þ4ðt; xÞ ¼ 2

vp v2 un vpI ðt; xÞ ¼ 2rc2 2 ðt; xÞ  2 ðt; xÞ vn vn vt

(113)

where  I þ K0 is an operator that accounts for the partially absorbing, acoustic boundary conditions on the road, 4 is the unknown density, t is the scaled time given by t ¼ ct, in which c is the speed of sound and t is the real time; x is the 3D spatial coordinate vector; p is the sound pressure, n is the direction normal to the tire surface; r is the air density, un is the displacement of the tire in the outer normal direction, and pI denotes the incident wave. The model was validated in three applications: (1) the sound emission of steady-state vibrating tires (205/55R16 passenger car tire and 315/80R22.5 rib truck tire); (2) noise amplification in the horn-like geometry between the road and the tire (205/55R16 slick tire); (3) Doppler effect (2% shift) of a moving tire (artificial signal). 2.4. Computational fluid dynamics models CFD uses numerical algorithms to solve and analyze fluid flow problems in computer-aided engineering (CAE). It can be applied to the modeling of airborne TPIN. Five models are reviewed in this subsection. 2.4.1. Shock wave (Gagen, 1999) Gagen (1999 [95], 2000 [96]) modeled the acoustic sources of air pumping from the squeezed tread groove cavities using the shock wave theory (Kinsler et al., 1999 [97]) instead of the commonly used monopole theory (Hayden, 1971 [29]). It was pointed out that at high speeds, as the tire/pavement impact causes the groove walls to move with velocities of the order of 1 m/s and accelerations of 103 m/s2 for a period more than 1 ms, the groove volume decreases (and then increases at the trailing edge) rapidly by around 10%, which is much faster than the rate at which air can evacuate from or suck into the squeezed/expanded groove. This leads to significant pressure and density fluctuations, which makes the linear approximations (small amplitude monopole theory) invalid. As per the nonlinear acoustic phenomenon, both thin-width grooves and large-width grooves generate low noise, while medium-width grooves generate large noise (Ejsmont et al., 1984 [98]).

Fig. 14. Left - Footprint and contact pressure distribution at 70% load (source from Wei et al., 2016 [93], Fig. 6; reprinted with permission from Mr. Michael Kaliske of TU Dresden, Germany on behalf of Tire Science and Technology); right - Acceleration at the footprint at 0.0024 s (source from Wei et al., 2016 [93], Fig. 7; reprinted with permission from Mr. Michael Kaliske of TU Dresden, Germany on behalf of Tire Science and Technology).

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The monopole equation for a simple harmonic source is expressed as

grV€ gru2 V ¼ 4pr 4pr

p1 ðr; tÞz

(114)

where p1 is the sound pressure, r is the distance between the sound source and receiver, t is the time, g is the ratio of specific heats (1.4 for air), r is the air density, V is the air volume, V with double dots is the second order derivative of the air volume with respect to time, u is the frequency of sound. All the parameters are normalized to be dimensionless, so that the speed of sound is unity. For non-simple harmonic source, Fourier decomposition into frequency components should be performed before applying the monopole theory. The shock wave equation for propagation of density disturbance is given by

0 ¼ vtt r 

vxx rg

g

 





 _c   € v r _ f c t

 f_vc rvy v r v v v r v v vcc rg f 2f vt r cx x y cc y c 2   v   þ   v r v r v v r þ  xx x c x c x f f f f f gf 2 f2 (115)

where v is the derivative symbol, vx and vy are the fluid velocities in the x and y coordinates, respectively; c is the computational coordinate where the groove walls are stationary, and is given by y ¼ f(t)c and vc ¼ (vy  ḟc)/f. In this study, the kinetic energy of the emitted jet was also investigated to obtain the sound intensity in the far field. 2.4.2. Hybrid CFD (Kim et al., 2006) The hybrid CFD model developed by Kim et al. (2006) [99] consisted of three submodels: (1) small-scale noise generation submodel; (2) noise propagation submodel; (3) far-field acoustic pressure prediction submodel. In the first submodel, the air-pumping noise source was simulated as piston/sliding-door/cavity geometry, as shown in Fig. 15 (Kim et al., 2006 [99]), using Navier-Stokes finite volume solver in CFD, which considers the motion of the road surface relative to the tire. The second submodel transferred the flow properties in the tire grooves, obtained in the first submodel, as nonlinear airpumping sources to the flow simulation of the full tire/road domain considering mid-scale acoustic scattering process. Finally, the third submodel predicted the far-field acoustic pressure by the linear Kirchhoff integral method with the flow properties around the rolling tire and road surface geometry, acquired from CFD calculation. The linear Kirchhoff integral equation is given below.

p0 ðx; tÞ ¼

1 4p

Z Z

 cos q 0 1 vp0 cos q vp0 þ  dSðy; tÞ p  r vn cr vt r2

(116)

S

where p0 is the perturbed pressure, x and t are the observer's location vector and time respectively; y and t are the source location vector and retarded time variables, respectively; q is the angle between the normal vector (n) on the surface and the radiation vector (r), and r is the distance between an observer and a source at the retarded time. The pressure and its derivatives are calculated at the retarded time. It was found that TPIN at frequencies in the range of 3e8 kHz was suppressed by the ventilation of the tire groove.

Fig. 15. Schematic diagrams of the piston/sliding-door/cavity source model (source from Kim et al., 2006 [99], Fig. 3; reprinted with permission from AIP Publishing LLC).

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2.4.3. CFD (Chen et al., 2014) Chen et al. (2014) [11] used CFD to investigate airborne TPIN, such as air pumping, air resonant radiation, and pipe resonance. Different tread patterns (hand carved) were simulated and compared with measured near-field noise using the laboratory drum method, and the deviation was found to be a maximum of 4.1% in the frequency range of 0e2000 Hz. 2.4.4. Air pumping CFD (Gautam and Chandy, 2016) Gautam and Chandy (2016) [100] investigated the air pumping noise generation due to the compression and expansion of air in the tread grooves using an approach based on LES. The compression and expansion of air was assumed to result from the change in the groove volume. The study suggested two types of tire groove deformation that resulted in the change in the groove volume: (1) a commonly used piston-type motion of the bottom wall and (2) a more realistic bulging in/out of the side walls of a tire groove, as shown in Fig. 16. In this figure, the white box refers to the groove, and the black box refers to the road. Numerical CFD analysis was performed using ANSYS FLUENT™ for both types of deformation. It was found that the piston deformations result in a distinct frequency peak around 2e4 kHz, while the side wall deformations result in a more broadband noise spectrum. However, the study assumed that the expansion period begins immediately after the compression period, which is not the case in reality. A further study by Gautam and Chandy (2016) [101] investigated both the near-field and far-field acoustic characteristics using LES and Ffowcs-Williams and Hawkings (FW-H) approach, respectively, considering several airborne mechanisms, such as air pumping, pipe resonance, Helmholtz resonance, and rotational turbulence. The simulation environment was a slick tire (215/ 60R16), with two transverse slots closely spaced in the tread, rotating on a smooth drum at 40 km/h in a semi-anechoic chamber. The change in the slot volume was assumed to be 22% (Takami and Furukawa, 2015 [102,103]) at the tire-pavement interface. It was concluded that the air pumping noise is focused at higher frequencies around 3e5 kHz, especially at the trailing edge of the contact patch. The horn amplification effect can be up to 20 dBA starting from 1.6 kHz, which is proved by the acoustic directivity analysis. However, validation experiments corresponding to the same conditions as in the case of simulation were not reported. 2.4.5. Pipe resonance (Fabrizi, 2016) In this study (Fabrizi, 2016 [104]), the pipe resonance mechanism was modeled using CFD and computational aeroacoustics. Only the aeroacoustic noise-generation mechanism of the pipe resonance phenomenon was covered; i.e., this is different from the amplification mechanism of pipe resonance, where vibroacoustic waves are also considered. The analysis of aeroacoustics in the longitudinal tread grooved is based on Lighthill acoustic analogy (Lighthill, 1952 [105]; Lighthill, 1954 [106]).

1 v2 p0 c20

v2 p0

 vfi v2

v2  ¼ r u u  t þ  i j ij vxi vxj vxi vt 2 vt 2 vx2i

p0 ¼ p  p0

p0 c20

! 0

r

(117)

r0 ¼ r  r0 where c0 is the speed of sound, t is the time, p and p0 are the instantaneous and equilibrium air pressures, r and r0 are the instantaneous and equilibrium air densities, ui and uj are air velocities in the i direction (flow direction) and j direction (perpendicular to flow direction), respectively; tij is the shear stress tensor, fi is the fluctuating external force per unit volume.

Fig. 16. Schematic diagram showing different deformation models with (a) piston-like motion of bottom wall (b) side walls bulging inward (source from Gautam and Chandy, 2016 [100], Fig. 3; reprinted with permission from Ms. Beth Darchi on behalf of ASME).

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Assuming that the flow can produce sound only at high velocities, the following equation is obtained.

vTij ðy; tÞ 1 xi xj ∭ dVy 4pc40 kxk3 vt2   Tij ¼ rui uj þ p  rc20 dij  tij

rðx; tÞz

(118)

where t is the time, x is the observer point (xi, xj), y is the source point, Tij is the Lighthill stress tensor or quadrupole strength density, and dij is the compressive stress tensor coefficient. The tire model used in the aerodynamic analysis is of size 205/55R16 with two longitudinal tread grooves, with a rotating speed of 80 km/h. Through LES, three phenomena related to the tire aerodynamic noise have been identified: (1) At the pipe inlet, the air flow is squeezed laterally into the groove from both sides, each with a high swirl component (counter-rotating vortices); (2) at the pipe outlet, the flow creates jets that develop inside the recirculation bubble of the wake; (3) the flow accelerates laterally to a speed of around 300 km/h and detaches, generating a high-energy shear layer. The sound pressures at four positions were analyzed; the results are shown in Fig. 17. Only for the positions of the pipe and wake, a clear resonance peak was observed at 1700 Hz, which matches the theory, if the correction length is set to 10 mm (two times the pipe diameter). This suggests that the resonant acoustic wave could propagate (acoustic energy) only in the direction of the flow (fluid energy). This directivity is similar to that of a dipole oriented in the longitudinal direction. The peak at 2100 Hz for the positions on the side of the leading edge is possibly the effect of the jet flow generated at the shoulder rib by the lateral squeezing of air out of the rolling tire. It may be noted that a peak around 600e800 Hz occurs for all positions, which is likely to be the result of the vibrations of the tire structure. 2.5. Summary The conventional physics models are summarized in Table 4. It can be seen that these models have been continuously improved since 1971. The tire was modeled as a ring, shell, or plate for vibration investigation; the tire/rim assembly was modeled as a structure coupled with an air cavity; the sound source was modeled as a monopole. However, most models focused only on structure-borne noise, especially the noise due to the impact between the tire and pavement. Therefore, the tire vibration was the output parameter instead of sound pressure level. The frequency range was typical below 500 Hz; some models claimed to be able to predict higher frequencies, but still they were limited to structure-borne noise. In addition, many of the models considered only a stationary tire, which is quite different from the normal driving conditions. Only one specific tire or one specific pavement was investigated in some of the models. The finite element and boundary element models are summarized in Table 5. FEM has been applied to TPIN modeling since 1986. Similar to the conventional physics model, the finite element and boundary element models also generally focused on structure-borne noise, and considered the tire carcass as a membrane or plate. However, the near-field or far-field sound pressure level could be easily predicted with the structure-fluid coupling technique. The frequency range was almost exclusively below 500 Hz or even lower, such as 150 Hz, because of the inherent drawback of the FEM. Vehicle speed was typically included in the modeling as an input. Tire parameters were investigated in detail, but not much attention has been paid to the pavement parameters. The CFD models are summarized in Table 6. The use of CFD models started in 1999, i.e., later than the previous two categories. With the help of CFD, the TPIN modeling was extended from structure-borne noise to include airborne noise (air

Fig. 17. On the left, A-weighted sound pressure levels of the pressure signals recorded with four probes placed around and inside the contact patch of the tire with the road surface; on the right, the position of the four probes with the reference of the footprint shape (source from Fabrizi, 2016 [104], Fig. 12; reprinted with permission from Mr. Michael Kaliske of TU Dresden, Germany on behalf of Tire Science and Technology).

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34

No. Model name

Author

Year

1

Monopole

Hayden

1971 [29] Monopole

2

Unified

Plotkin and Stusnick 1981 [30] Thin shell

3

Truck tire

Keltie

1982 [38] Thin shell

4 5

2D ring Various mechanisms Tire vibration

Kung et al. Heckl

1986 [39] Modal expansion 1986 [44] Various

Near-field, normalized radiation 1/3 octave band sound power level Near-field, SPL LDV, vibration, SPL

Bremner et al.

1997 [46] SEA, wavenumber decomposition 1997 [51] Image source

Far-field (7.5 m), 1/3 octave band SPL Impedance, absorption

1998 [52] Wavenumber decomposition

6

rengier et al. Be

Ref. Method

Output

Frequency Mechanism range [Hz]

Sound pressure, pure tone N/A

Air pumping

0e8000

Driver parameter Tire parameter

Pavement Environment parameter parameter

Speed

N/A

N/A

Texture

N/A

Impact, air pumping 100e2000 Structure-borne

Speed, tire load

0e400 0e2000

Tire load Rotating, loaded

Structure-borne Various

Inflation

Groove geometry Various

Damping, N/A N/A bending stiffness Various Texture N/A Tread pattern Roughness N/A

100e4000 Impact

Speed, 40 psi

Tread pattern, materials N/A

Texture

N/A

100e5000 Pavement absorption

N/A

Point mobility, wavenumber

0e1000

215/70R14

N/A

N/A Rough, 1  30 mm step N/A N/A

100/50R12

N/A

N/A

Geometry, structure

N/A

N/A

N/A Airflow resistivity, porosity, tortuosity N/A N/A

7

Porous pavement

8

Wavenumber Bolton et al. decomposition

9

Orthotropic plate

Kropp

1999 [53] Multipole synthesis

Vibration, 1/3 octave band SPL

125e2500 Structure-borne

Stationary, inflation, gas type Unloaded

10

Shell

Kim and Bolton

Vibration

0e4000

Point impact

Speed

11

Molisani et al. Kim et al.

Modal amplitude, force at spindle Vibration, flexural wave, eigen-frequency

0e1250

12

Coupled cavity Waveguide

0e2000

Cavity resonance Stationary without rotating Structure-borne Stationary

13

Cavity

Feng et al.

Cavity resonance Speed, tire load

N/A

N/A

Temperature

3D multilayer

O'Boy and Dowling

Cavity SPL, narrowband Vibration

100e300

14

2002 [57] Wave number decomposition 2003 [60] Donnell-Mushtari theory, structural acoustic 2007 [62] Spring-mass, elastically supported beam, cylindrical shell 2009 [63] Coordinate transformation 2009 [65] Bessel equation

0e2000

Structure-borne

225/45ZR17

N/A

20  C

15

Green's function Coupled mode

Arteaga

2011 [68] Modal-ALE

Vibration

0e300

Impact

185/70SR14

N/A

N/A

Bolton and Cao

2013 [70] Structural-acoustic coupling

LDV, vibration

0e6000

Structure-borne

225/55R17

N/A

N/A

16

Structure-borne

Tire load, break/acc. Tire load, speed Stationary & 50 m/s, 3 bar

Slick

T. Li et al. / Journal of Sound and Vibration xxx (2018) 1e89

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Table 4 Summary of conventional physics models on TPIN.

No.

Model name

Author

Year

Ref.

Method

Output

Frequency range [Hz]

Mechanism

Driver parameter

Tire parameter

Pavement parameter

Environment parameter

1

Toroidal membrane

Saigal et al.

1986

[49]

Membrane FE

Vibration

Low

Structure-borne

Geometry, structure

N/A

N/A

2

Wright and Koopmann Richards

1986

[74]

N/A

N/A

N/A

[61]

Near-field, sound pressure, intensity Vibration

Vibration, radiation

1991

0e400

Cavity resonance

185/70R14

N/A

N/A

4

3D two-plate

1992

[25]

Pass-by, SPL

0e2000

Structure-borne

Tire load

N/A

TRIAS

2000

[77]

FEA, BEM

N/A

Impact, air pumping

N/A

WFEM

2004

[79]

WFEM

0e1500

Structure-borne

Stationary

Geometry, structure

Various, RODAS N/A

N/A

6

N/A

7

IFEM

Biermann et al.

2007

[23]

0e1400

Impact

Speed

Slick

Texture

N/A

8

FEM & BEM

2008

[17]

Structure-borne

40-120 km/h

N/A

2011

[82]

0e1500

Impact

10 11

Road input FSI

Kido et al. Yang et al.

2011 2013

[85] [22]

FE, modal decomposition FE FEA

0e150 500e2500

Texture impact TPIN

225/45R17 265/70R17 Patch area PIARC

Texture N/A

N/A N/A

12

Wear

Tong et al.

2013

[89]

Acoustic FE

Axle vibration Near-field, 1/3 octave band SPL Near field, SPL

Inflation: 220/180 kPa Speed, tire load 70-90 km/h

Texture spectrum N/A

N/A

WFE

Pass-by, 1/3 octave band SPL Vibration

100e850

9

Brinkmeier et al. Waki et al.

IFEM, Astley-Leis elements FEA, BEM, ALE

Near field, 1/3 octave band SPL; pass-by, SPL Point mobility, wavenumber Far-field (1 m), 1/3 octave band SPL

Various, slick 205/60R15 Various, TYDAS

Texture

5

Larsson and Kropp De Roo and Gerretsen Nilsson

FE, Helmholtz integral Weighted residual approximation FE

80e200

3

Acoustic radiation FEA coupling

Inflation, nonrotating, unloaded Stationary, point-excited N/A

0e500

Impact

Wear

N/A

N/A

13

EFEA/EBEA

Vlahopoulos et al.

2013

[90]

Interior noise

200e8000

Airborne

N/A

N/A

N/A

14

Cavity

2015

[91]

Tire cavity SPL

150e500

Torus cavity resonance

N/A

Tire size

N/A

N/A

15

MLE

Mohamed and Wang Wei et al.

2016

[93]

Near field, 1/3 octave band SPL

Time domain, 0e1250

Vibration

70 km/h

N/A

Time domain BEM

Banz et al.

2016

[94]

1/3 octave band SPL

200e2000

Radiation

Steady-state moving

Truck/bus, 315/60R22.5 385/65R22.5 Passenger/truck, slick

N/A

16

EFEA, EBEA, panel contribution analysis FEM, Bessel root method Mixed Lagrangian-Eulerian, FE, BEM BEM

Tire load, speed, inflation N/A

N/A

N/A

N/A

T. Li et al. / Journal of Sound and Vibration xxx (2018) 1e89 35

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Table 5 Summary of finite element and boundary element models on TPIN.

36

T. Li et al. / Journal of Sound and Vibration xxx (2018) 1e89

Table 6 Summary of computational fluid dynamics models on TPIN. No. Model name

Author

Year Ref.

Method

1

Gagen

1999 [95]

CFD

2

3 4

5

Shock wave Hybrid CFD

Output

Frequency Mechanism range [Hz]

Near-field, sound N/A, time Air pumping pressure/energy domain Far-field, 1/3 250e8000 Air pumping, Kim et al. 2006 [99] CFD, octave band SPL pipe Kirchhoff resonance integral CFD Chen 2014 [11] CFD Near-field, 1/3 0e2000 Airborne et al. octave band SPL 1000 Airborne, air Near-/far-field, Gautam 2016 [100] CFD Air e10000 pumping drum, pumping and narrowband SPL Chandy CFD 1500 Pipe Pipe Fabrizi 2016 [104] CFD, LES, Pass-by/ine2000 resonance resonance aeroacoustics groove, narrowband SPL

Driver Tire parameter Pavement Environment parameter parameter parameter Speed

Groove angle, geometry 100 km/h Ventilation groove, 165/ 70SR13 N/A Tread pattern 40 km/h

80 km/h

215/60R16, transverse slots 205/55R16, longitudinal grooves

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

Drum with N/A Safety Walk

pumping mechanism). Near-field sound pressure level was widely accepted as the output parameter of the model. The tire tread pattern was the most important input parameter investigated. However, none of the current models included pavement parameters; the authors believe that these are also important, because air voids are formed within the contact patch between the tread pattern and the pavement texture.

3. Statistical models 3.1. Introduction As the physics of TPIN generation is uncovered to a great extent, statistical models are more common, especially in the industry. Statistical or semi-empirical models are based on correlation of measured noise data for various tire-pavement parameters. For example, the measured TPIN can be correlated with pavement texture, including texture levels at different wavelengths in the longitudinal direction (Rasmussen et al., 2006 [107]) and overall texture mean profile depth (MPD) (Fujikawa et al., 2009 [108]). The early models, shown in Tables 7 and 8, were used to predict pass-by traffic noise. They used L50 as the indicator during the period between the 1950's and 1960's, and then Leq prevailed as the indicator (Quartieri et al., 2009 [109]). All these models essentially have the following format (Murphy and King, 2014 [118]).

Lp ¼ E  Atot þ C

(119)

where Lp is the noise level to be predicted, E is the emission of the source (sound power of a single point source, or sound power per unit length of a line source), Atot is the total sound attenuation (geometric divergence, atmospheric absorption, ground absorption, the barrier, and miscellaneous effects), and C is the correction factor (façade reflection, different pavement/vehicle types). Lp can be in the form of L10,18h, LAeq or Lden. L10,18h is the noise level just exceeded for 10% of the 18-h measurement period from 06:00 to 24:00 h. LAeq is the A-weighted equivalent sound level. Lden is the day-evening-night equivalent level (A-weighted equivalent sound level), measured over the 24-h period, with a 10-dB penalty added to the levels between 23:00 and 07:00 h and a 5-dB penalty added to the levels between 19:00 and 23:00 h to reflect people's extra sensitivity to noise during the night and the evening. These statistical models have better accuracy than deterministic models, as the model parameter values are obtained from the actual measured data. However, empirical measurements can never account for all potential conditions, and hence

Table 7 Nomenclature of early models used for traffic noise prediction (Quartieri et al., 2009 [109]). Symbol

Meaning

Unit

LXX Leq Q Qeq d v P n

Noise level exceeded for XX% of the measurement period Equivalent (energy) sound level Traffic volume (QM for motorcycles, QBUS for buses) Equivalent vehicular flows Distance from the traffic lane center to the receiver Vehicle speed Percentage of heavy vehicles in the traffic Acoustical equivalent of heavy vehicles, indicating the number of light vehicles that generate the same acoustic energy of a heavy vehicle

[dB] or [dBA] [dB] or [dBA] [Vehicles per hour] [Vehicles per hour] [ft] [mph] [%] []

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37

Table 8 Early models used for traffic noise prediction (Quartieri et al., 2009 [109]). No.

Reference

Author

Year

Equation

1

[110]

Anonymous

1952

2

[111]

Nickson

1965

3 4

[112] [113]

Johnson and Saunders Griffiths and Langdon

1968 1968

5 6 7

[114] [115] [116]

1969 1977 1991

8

[117]

Galloway Burgess Centre Scientifique et Technique du Batiment (France) Fagotti and Poggi

L50 ¼ 68 þ 8.5log10(Q)  20log10(d), for speed of 55e75 km/h and d > 20 L50 ¼ C þ 10log10(Q/d), where C is a constant that is determined from experimental data under given traffic conditions (vehicle and pavement types) L50 ¼ 3.5 þ 10log10(Qv3/d) Leq ¼ L50 þ 0.018(L10  L90)2 L10 ¼ 61 þ 8.4log10(Q) þ 0.15P  11.5log10(d) L50 ¼ 44.8 þ 10.8log10(Q) þ 0.12P  9.6log10(d) L10 ¼ 39.1 þ 10.5log10(Q) þ 0.06P  9.3log10(d) L50 ¼ 20 þ 10log10(Qv2/d) þ 0.4P Leq ¼ 55.5 þ 10.2log10(Q) þ 0.3P  19.3log10(d) Qeq ¼ Q[1 þ P(n  1)/100] L50 ¼ 11.9log10(Qeq) þ 31.4 Leq ¼ 0.65L50 þ 28.8 Leq ¼ 10log10(QL þ QM þ 8QP þ 88QBUS) þ33.5

1995

statistical confidence level and confidence interval are normally analyzed. In addition, some of the parameters in these models are not real physical parameters, but derived parameters such as a texture spectrum [14], which means that they are not adjustable directly. To change these parameters, they have to be translated back to the original physical parameters, which are not always explicit or even impossible for nonlinear relationships. Various statistical analysis algorithms can be used. In this section, 18 statistical models are reviewed; they consist of 10 traditional regression models, 3 principal component analysis models, and 5 fuzzy curve-fitting models. The model name, method, output noise parameter with noise measurement technique, noise generation mechanisms investigated, and the input parameters of each model are listed and compared in the tables as a summary in section 3.5. The summary tables can be used as a reference to direct the readers to the models of their interest. 3.2. Traditional regression models This subsection covers the traditional regression models, which include linear regression models and nonlinear regression models. The linear regression is easier to model, while the nonlinear regression is more flexible. 3.2.1. Germany (RLS, 1990) €rmschutz an Strassen, or Guideline for noise protection on streets) was The first version of RLS (Richtlinien für den La released by Der Bundesminister für Verkehr (now BMVI, Federal Ministry of Transport and digital Infrastructure) in 1981 [119]. The current version is RLS 90 [120], released in 1990. It is used to predict the sound level at a distance of 25 m from the center of the traffic lane. The nomenclature for this model is presented in Table 9 (RLS, 1990 [120]). First, the average level (reference value) is calculated as ð25Þ

Lm;E ¼ 37:7 þ 10 log10 ðQ ð1 þ 0:082PÞÞ

(120)

Accounting for the vehicle and pavement conditions, the real average sound level at 25 m distance is given by ð25Þ

Lm ¼ Lm;E þ RSL þ RRS þ RRF þ RE þ RDA þ RGA þ RTB

(121)

where RRS ¼ 0.6jgj  3 (set to 0 if negative). RSL is given by

RSL ¼ LPkw  37:3 þ 10 log10

 ! 100 þ 100:1D  1 P 100 þ 8:23P

(122)

where

8 < LLkw ¼ 23:1 þ 12:5log10 h ðVLkw Þ i LPkw ¼ 27:7 þ 10log10 1 þ ð0:02vPkw Þ3 : D ¼ LPkw  LLkw

(123)

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T. Li et al. / Journal of Sound and Vibration xxx (2018) 1e89

Table 9 Nomenclature for RLS 90 model (RLS, 1990 [120]). Symbol

Meaning

Unit

L(25) m,E Lm Q P RSL RRS RRF RE RDA RGA RTB g vLkw vPkw

Reference average sound level at 25 m distance under prescribed/idealized conditions Real average sound level at 25 m distance under given conditions Traffic volume Percentage of heavy vehicles in the traffic (weight > 2.8 t) Correction for the speed limit Correction for the pavement Correction for the rises and falls along the road Correction for the absorption characteristics of building surfaces Correction for the propagation absorption due to the receiver distance Correction for the attenuation due to the ground and atmospheric conditions Correction for the attenuation due to the topography and building dimensions Road gradient Speed limit for light vehicles (normally 30e130 km/h) Speed limit for heavy vehicles (normally 30e80 km/h)

[dB] [dB] [vehicles per hour] [%] [dB] [dB] [dB] [dB] [dB] [dB] [dB] [%] [km/h] [km/h]

3.2.2. Italia (CNR, 1991) The Italian CNR model (Consiglio Nazionale delle Ricerche, or National Research Council in English) was published in 1983 (Canelli et al., 1983 [121]) and then improved by Cocchi et al., in 1991 [122]. The model is a modification of the RLS (Germany) model where the traffic flow is considered as a line source. The nomenclature for this model is presented in Table 10 (Cocchi et al., 1991 [122]). The A-weighted equivalent sound level for the traffic is given by

 LAeq ¼ a þ 10 log10 ðQL þ bQP Þ  10 log10

d d0

 þ DLV þ DLF þ DLB þ DLS þ DLG þ DLVB

3.2.3. US (TNM, 1998) The Traffic Noise Model (TNM) is a software to predict pass-by noise levels (at a distance of 15 m) near highways. It is the only validated model approved by Federal Highway Administration (FHWA, USA) to test compliance with Code of Federal Regulations “Traffic Noise Prediction” (23 CFR 772.9(a)) (FHWA, 2012 [123]). The first version of TNM was released in 1998 (FHWA, 1998 [124]) based on extensive measurement data (over 6000 pass-by events) taken from 1993 to 1995 (Fleming et al., 1995 [125]). After several revisions, the current and mature version is TNM 2.5 (software) released in 2004. TNM Version 3.0 is under development, with 2D graphics and GIS functionality implemented (FHWA, 2013 [126]). The SPL at the receiver is adjusted from a reference sound level termed as reference energy mean emission level (REMEL) that describes the maximum sound level from a vehicle under given conditions. The possible conditions are listed in Table 11 (Rochat et al., 2012 [127]). The energy mean level (overall A-weighted) associated with the maximum level during pass-by (REMEL, for one vehicle) is given by

LA;max ðSÞ ¼ 10 log10

h   i SA=10 10ðBþDEB Þ=10 þ 10ðCþDEC Þ=10

(124)

Table 10 Nomenclature for CNR model (Cocchi et al., 1991 [122]). Symbol

Meaning

Unit

LAeq QL QP d d0 DLV

A-weighted equivalent sound level for the given traffic Light vehicles in 1 h Heavy vehicles in 1 h Distance between the lane center and the observer Reference distance (25 m) Correction for the mean flux velocity (0 for 30e50 km/h; þ1 for 60 km/h; þ2 for 70 km/h; þ3 for 80 km/h; þ4 for 100 km/h) Correction for the presence of reflective facade near the observer (þ2.5) Correction for the presence of reflective facade in the opposite direction (þ1.5) Correction for the pavement type (0.5 for smooth asphalt; 0 for rough asphalt; þ1.5 for cement; þ4 for rough pavement) Correction for the road gradient Correction for the presence of traffic lights (þ1) or slow traffic (1.5) Coefficient related to noise emission from a single vehicle (default ¼ 35.1 in Italy) Coefficient related to the weightage factor for higher noise emitted by heavy vehicles (default ¼ 6 in Italy)

[dBA] [vehicles per hour] [vehicles per hour] [m] [m] [dBA]

DLF DLB DLS DLG DLVB a b

[dBA] [dBA] [dBA] [dBA] [dBA] [dBA] []

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39

Table 11 REMEL conditions in TNM (Rochat et al., 2012 [127]). Category

Condition

Vehicle type Pavement type

Automobile; medium truck; heavy truck; bus; motorcycle Dense-graded asphalt concrete (DGAC); open-graded asphalt concrete (OGAC); Portland cement concrete (PCC); average (a combination of DGAC and PCC) Cruise; full-throttle

Operating condition

where S is the vehicle speed in mph; A, B, and C are coefficients related to the vehicle type, pavement type, and operating conditions; DEB and DEC are corrections to convert the level mean to the energy mean, and are given by

0

1 N X Li Lavg 1 C DE ¼ 10 log10 B 10 10 A @ N i¼1

(125)

where N is the number of samples, Li is the overall A-weighted level for the ith sample, and Lavg is the linear average over all samples. The REMEL for one-third octave band spectra is given by

LA;max ðS; f Þ ¼ 10 log10

h   i SA=10 10ðBþDEB Þ=10 þ 10ðCþDEC Þ=10 þ ðD1 þ D2 SÞ þ ðE1 þ E2 SÞlog10 ðf Þ þ ðF1 þ F2 SÞlog10 ðf Þ2

þ ðG1 þ G2 SÞlog10 ðf Þ3 þ ðH1 þ H2 SÞlog10 ðf Þ4 þ ðI1 þ I2 SÞlog10 ðf Þ5 þ ðJ1 þ J2 SÞlog10 ðf Þ6 (126) where f is the nominal center frequency of the one-third octave band; D1, D2, …J2 are the REMEL model coefficients. The adjusted sound level at the receiver is calculated using the following equation that considers traffic flow Atraffic,i, propagation distance Ad, and shielding As.

LAeq;1h ¼ ELi þ Atraffic;i þ Ad þ As

(127)

where ELi is the vehicle noise emission for each vehicle type i depending on LA,max; Atraffic,i for each vehicle type i is given by

Atraffi;i ¼ 10 log10

  Vi  13:2 Si

(128)

3.2.4. Dense-graded (Hanson and James, 2004) Hanson and James (2004) [128,129] model was developed by National Center for Asphalt Technology (NCAT) in Auburn University, sponsored by Colorado Department of Transportation (CDOT) and Federal Highway Administration (FHWA). Fortysix different pavements (each 61 m in length, 2.7 km in total) in the NCAT test track were tested, most of which were densegraded hot mix asphalt (HMA). The sound levels were measured using two approaches, close-proximity trailer (CPX) and close-proximity sound intensity (CPI). The study focused on the mixture properties of the pavements. For each pavement, seven tires were tested at a speed of 72 km/h, and the average noise level was calculated as the model output. The final regression equation for the tire noise level is

LA ¼ 93:4  2:56  AirVoids þ 0:53  FinenessModulus  AirVoids

(129)

where LA is the overall A-weighted sound pressure level (OASPL), and R2 ¼ 0.64. The fineness modulus (FM) is an empirical factor obtained by adding the total percentages of a sample of the aggregate retained on each of a specified series of sieves, and dividing the sum by 100 (TXDOT, 1999 [130]; ASTM, 1980 [131]). The sieve sizes used are No. 100 (150 mm), No. 50 (300 mm), No. 30 (600 mm), No. 16 (1.18 mm), No. 8 (2.36 mm), No. 4 (4.75 mm), 3/8 in. (9.5 mm), 3/4 in. (19.0 mm), 1-1/2 in. (38.1 mm), and larger sizes, increasing in multiples of 2. The FM values of the fine aggregates are in the range of 2.00e4.00, while the course aggregates smaller than 38.1 mm have values in the range of 6.50e8.00. The pavements tested in the NCAT study have combinations of fine and coarse aggregates, and hence they have intermediate values (4.07e5.06). In addition, the separate influences of the following parameters were analyzed: fineness modulus (R2 ¼ 0.51), air voids (%) (R2 ¼ 0.11), and pavement age (R2 ¼ 0.78). The study also ranked the noise levels for four types of pavements as open-graded mixes (coarse gradations) > stone matrix asphalt matrix > dense-graded HMA > open-graded mixes (fine gradations). 3.2.5. Concrete texture (Rasmussen, 2009) Rasmussen (2009) [132] correlated TPIN with typical concrete pavement textures such as burlap drag, rake tining, and diamond grinding. 3D texture profiles were measured over 1000 sections representing approximately 400 nominal textures Please cite this article in press as: T. Li et al., Literature review of models on tire-pavement interaction noise, Journal of Sound and Vibration (2018), https://doi.org/10.1016/j.jsv.2018.01.026

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and totaling 70 km, with RoboTex laser measurement system. Enormous amount of data (1 GB for 1.6 km) were collected owing to the high sampling frequency (1000 Hz) and high resolution (spatial resolution of approximately 0.4 mm2 and height resolution of 0.01 mm). 3D texture bridging (envelopment) filter was employed to achieve data reduction. The noise data were collected using the OBSI (On-board Sound Intensity) method (96 kHz), except that there were two additional channels for the data acquisition system (DAQ). One of the channels was used to record the event markers, i.e., signal from an optical transceiver (reflective delineator) indicating the start and end of the test section. Thus, the texture data and noise data could be synchronized after compensating for the travel time between the optical transceiver (at the road side) and the OBSI microphone (at the tire). The study conducted measurements in the frequency range of 500e5000 Hz, but focused more on 630, 800, and 1000 Hz third-octave bands. Equations were developed for the various types of concrete pavements, i.e., they have different input texture parameters and coefficients, as shown below. It was concluded that noise cannot be predicted by texture depth (Rasmussen, 2010 [133]).

 8 OBSITT ¼ f Ltx;160 ; Ltx;25 ; Rk;tr > >

 < OBSILT ¼ f Ltx;40  > OBSIDG ¼ f Ltx;80þ63þ50þ40 ; Skew tr > : OBSID ¼ f Ltx;50þ40 ; Skewtr ; Rk;tr

(130)

where OBSI is the overall A-weighted sound intensity level (OASIL), the subscripts TT, LT, DG, and D indicate transverse tining, longitudinal tining, diamond grinding, and drag, respectively; Ltx,XX is the texture level for third-octave band centered at a texture wavelength of XX (mm) according to ISO 13473-4 (2008) [134]; Skewtr is the texture skew in the transverse direction (unitless), according to ASME B46.1 (2009) [135]; Rk,tr is the core roughness depth in the transverse direction (mm), according to ISO 13565-2 (1996) [136]. The complete equation for the drag concrete pavement is

 OBSID ¼ 91:0 þ 0:315 Ltx;40 þ Ltx;50  1:37Skewtr  3:11Rk;tr

(131)

where R2 is 0.57, and the standard error is 0.6 dBA. It may be noted that the tire used is not the current SRTT, but a Goodyear Aquatred III test tire. 3.2.6. Asperity (Fujikawa et al., 2009) Fujikawa et al. (2009) [108] developed a statistical model correlating the near-field TPIN levels with pavement roughness parameters. The roughness parameters in the final model are asperity spacing xA and asperity height unevenness hd, as illustrated in Fig. 18. Thirteen pavement profiles were measured with a single-head laser profile meter to cover 2e4 m, covering no less than 80 asperities for each sample. The specifications of the thirteen pavements are listed in Table 12. The TPIN levels were measured with a microphone placed under the vehicle beside the tire. The individual correlation between TPIN and asperity spacing was reported as R2 ¼ 0.62, with a standard error of 1.4 dB, while the individual correlation between TPIN and asperity height unevenness was reported as R2 ¼ 0.53, with a standard error of 1.5 dB. Then, multiple regression analysis was employed to correlate the TPIN levels with both the parameters. The final equation is

LR ¼ 0:25〈xA 〉 þ 1:55〈hd 〉 þ 91:8

(132)

where LR is the regressed A-weighted sound pressure level in dBA, is the average asperity spacing in mm, and is the average asperity height unevenness. The coefficient of determination was R2 ¼ 0.83 with a standard error of 0.9 dB. It was considered that the reduced spacing and height unevenness of the pavement asperities will lead to reduced TPIN. In an earlier study by the same authors, Fujikawa et al. (2004) [137], another pavement parameter, mean profile depth (MPD, in mm) was also included. The regression equation is given below.

LR ¼ 0:34〈xA 〉 þ 1:80〈hd 〉  1:15MPD þ 93:2 for block pattern tire; LR ¼ 0:33〈xA 〉 þ 1:07〈hd 〉  0:22MPD þ 91:1 for rib pattern tire:

(133)

Fig. 18. Illustration of pavement roughness parameters in Fujikawa et al. model (source from Fujikawa et al., 2009 [108], Fig. 1; reprinted under fair use provision).

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41

Table 12 Specifications of pavements measured in Fujikawa et al. model (source from Fujikawa et al., 2009 [108], Table 2; reprinted under fair use provision). No.

Pavement

Max. Age Asperity spacinsg aggregate [year] size [mm] Average 95% Confidence xA [mm] range [mm]

Asperity height unevenness

Remarks

Average 95% hd [mm] Confidence range [mm]

B0

Resin crushed stone pavement 13

3

16.7

1.2

2.73

0.28

B1

Shot blasted dense asphalt

20

3

22.2

1.8

0.74

0.09

B2 B3 B4 B5 B6 B7 B8 B9 B10

Semi-flexible pavement Semi-flexible pavement Semi-flexible pavement Dense asphalt max. 20 mm Dense asphalt max. 13 mm Dense asphalt max. 13 mm Dense asphalt max. 13 mm Dense asphalt max. 13 mm Dense asphalt max. 13 mm

13 13 13 20 13 13 13 13 13

6 6 6 3 1 5 10 1 3

17.4 20.6 20.6 17.1 10.3 13.4 11.8 11.4 11.6

1.5 2.1 2.1 1.2 1.0 1.1 0.9 1.2 1.1

0.51 0.52 0.58 0.61 0.20 0.40 0.44 0.16 0.19

0.07 0.09 0.10 0.06 0.03 0.05 0.06 0.02 0.02

B11 Dense asphalt max. 8 mm B12 Dense asphalt max. 13 mm

8 13

2 1

8.7 8.1

0.4 0.5

0.17 0.14

0.01 0.01

Sharp-edged aggregates are bound with resin. Asphalt matrix is roughened by shot blasting. Open-graded aggregates Open-graded aggregates Open-graded aggregates Aggregates with large size. Ordinary type of Japanese public road. Ordinary type of Japanese public road. Test track with reinforced asphalt. Ordinary type of Japanese public road. Rate of small aggregates is relatively higher than ordinary type. Aggregates with small size. Rate of small aggregates is relatively higher than ordinary type.

The model also analyzed tire tread vibration. However, it was measured only on three types of pavements, and no equations for the relationships were given in the study. 3.2.7. Pavement core (Reyes and Harvey, 2011) Reyes and Harvey (2011) [138] developed a regression model to predict OBSI noise levels using laboratory pavement cores. The advantage of the method is that it is easier to test a pavement core in the laboratory than to test a full-scale section in the field. The sixteen field pavement cores included those commonly used in California: five open-graded asphalt concrete (OGAC), six open-graded rubberized asphalt concrete (RAC-O), three gap-graded rubberized asphalt concrete (RAC-G), and two open-graded rubberized asphalt concrete with large aggregates (RAC-O F-mixes). The age of the pavements ranged from a few months to six years, and the air void content ranged from 9% to 22%. There were two typical diameters of the cores: 101 mm and 152 mm. Airflow resistance Rf, air void content, mean profile depth (MPD), root mean square (RMS) of the texture profile, and coefficient of uniformity Cu were investigated for their relationships with sound intensity levels. Air void content was found to have strong correlation with airflow resistance; root mean square of the texture profile was found to have strong correlation with mean profile depth; coefficient of uniformity was found to have poor correlation with sound intensity levels. The specific airflow resistance Rf (in Pa$s/m) and MPD (in m) were selected in the final model, as given below (R2 ¼ 0.52e0.86).

  SIf ¼ af þ bf $MPD þ cf $log10 Rf

(134)

where SIf is the sound intensity level for the frequency band f in dBA referenced to 1012 W/m2, af, bf, and cf are coefficients, whose values are listed in Table 13 (Reyes and Harvey, 2011 [138]). The specific airflow resistance is defined the pressure difference across a core specimen divided by the airflow velocity (Rf ¼ Dp/u, in Pa$s/m). It was measured only in the region of laminar flow (0.2e10 L/min), where Rf is constant. The range of Rf Table 13 Coefficients for Reyes and Harvey model (source from Reyes and Harvey, 2011 [138], Table 1; reprinted under fair use provision). 1/3 Octave band center frequency [Hz]

af

bf

cf

500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 Overall A-weighted sound intensity level

91.83 90.46 93.67 90.69 83.02 79.83 82.36 79.75 76.16 71.09 64.86 96.95

5.45 6.36 3.38 2.59 3.04 1.90 0.50 0.88 2.29 2.75 3.88 3.51

1.69 1.32 0.03 0.56 1.52 2.15 1.38 0.97 1.04 1.41 1.56 0.28

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T. Li et al. / Journal of Sound and Vibration xxx (2018) 1e89

tested was 51e76075 Pa s/m. The MPD was obtained using a laser texture scanner (Model 9200) developed by AMES Engineering. The author has noted that only one core for each section was scanned, owing to limited funding and time available, and recommended that at least three cores should be measured according to ISO 13473-1 (1997) [139]. It is also noted that the laboratory MPD and the measured values in the field have a good correlation (R2 ¼ 0.88). It was shown that airflow resistance has a positive correlation (R2 ¼ 0.64e0.81) with sound intensity in the range of 1250e2500 Hz (high frequency). MPD has a positive correlation (R2 ¼ 0.3) with sound intensity in the range of 500e630 Hz (low frequency). However, neither of the two parameters was able to explain the noise at the 800 Hz and 1000 Hz frequency bands (R2 z 0), where most spectral content exists, and tire contributions might dominate. In addition, neither of the two parameters showed satisfying relationships with overall A-weighted sound intensity levels (R2 ¼ 0.1). 3.2.8. EU (CNOSSOS, 2012) The EU model is the proposed common European method to predict noise emission for road traffic, and was first released in 2012 (Kephalopoulos et al., 2012 [140]). The noise emission is expressed in terms of sound power level, assuming that each point source of a vehicle is 0.05 m above the surface. The model presents separate calculations for TPIN and propulsion noise, for five types of vehicles (m ¼ 1 for light vehicle, 2 for medium heavy vehicle, 3 for heavy vehicle, 4 for powered two-wheeler, and 5 for other customized vehicles). For the TPIN, the sound power level LWR,i,m for the vehicle type m and octave frequency band i (125e4000 Hz) is given by

LWR;i;m ¼ AR;i;m þ BR;i;m log10

!

v vref

þ DLWR;i;m ðvÞ

(135)

where v is the vehicle speed, vref is the reference speed of 70 km/h; AR and BR are coefficients for the respective vehicle type m and octave frequency band i; DLWR,i,m is the total correction factor considering the pavement type, studded tires, speed variation, and air temperature. For propulsion noise, the sound power level LWR,i,m for the vehicle type m and octave frequency band i is given by

LWP;i;m ¼ AP;i;m þ BP;i;m log10

v  vref vref

! þ DLWP;i;m

(136)

It was shown that speed has much smaller influence on the propulsion (engine) noise than TPIN for the first three vehicle types. The overall sound power level for the vehicle is the energy sum of TPIN and propulsion noise.

0 B LW;i;m ¼ 10 log10 @10

1 LWR;i;m 10

þ 10

LWP;i;m 10

C A

(137)

The equivalent sound power level for a steady traffic flow Q (vehicle per hour) with average vehicle speed vm is given by

 LW;eq;i;m ¼ LW;i;m þ 10 log10

Q 1000vm

 (138)

3.2.9. Mode-wise (Syamkumar et al., 2013) Syamkumar et al. (2013) [6] developed statistical equations to predict pass-by TPIN for each combination (mode) of vehicle type and pavement. Six vehicle types were covered: heavy commercial vehicle (HCV), bus, light commercial vehicle (LCV), passenger car, auto-rickshaw (AR), and two-wheeler (2W). The pavements included eight asphalt concrete pavements (AC) and four cement concrete pavements (CC). The investigated parameters are listed in Table 14. The general form of the statistical equations is

TPINðmodeÞ ¼ a0 þ a1 p1 þ a2 p2 þ a3 p3 þ a4 p4 þ a5 m2

(139)

where TPIN is in the unit of dBA, and as an example, the coefficients for passenger cars on asphalt concrete pavement and cement concrete pavement are presented in Table 15. It was found that the TPIN levels on cement concrete pavements are 2 dB higher than those on asphalt pavements. Wet pavements increase TPIN by 3 dBA in the case of HCVs compared to dry pavements. The wind direction has influence on TPIN to the extent of 1e2 dBA in the case of HCVs. Please cite this article in press as: T. Li et al., Literature review of models on tire-pavement interaction noise, Journal of Sound and Vibration (2018), https://doi.org/10.1016/j.jsv.2018.01.026

T. Li et al. / Journal of Sound and Vibration xxx (2018) 1e89

43

Table 14 Investigated parameters in Syamkumar's model (Syamkumar et al., 2013 [6]). No.

Parameters

Symbol

Unit

Value

1 2 3 4 5 6 7

Vehicle type Pavement type (texture depth) Loading condition Vehicle speed Wind direction Pavement temperature Pavement condition

m1 m2 p1 p2 p3 p4 p5

[] [mm] [kg] [km/h] [ ] [ C] []

HCV, bus, LCV, car, AR, 2W AC, CC Loaded (L), unloaded (UL) 10-100 km/h Towards, away (0e180 ) 20-40  C Wet, dry

Table 15 Statistical equation coefficients and R2 for passenger cars in mode-wise model (Syamkumar et al., 2013 [6]).

[TPIN(Car)]AC [TPIN(Car)]CC

a0 (Intercept)

a1 (Load)

a2 (Speed)

a3 (Wind)

a4 (Temperature)

a5 (MPD)

R2

79.10 75.43

0.0013 0.0028

0.2817 0.4343

0.0059 0.02

0.016 0.0015

0.1260 0.3137

0.918 0.911

3.2.10. FDOT (Wayson et al., 2014) The 10 FDOT model (Wayson et al., 2014 [141]) was developed by researchers at the John A. Volpe National Transportation Systems Center (Cambridge, MA), sponsored by Florida Department of Transportation (FDOT). Various types of pavements at over 45 locations in Florida (especially rigid pavements, dense-graded and open-graded pavements) were measured and ranked by the OBSI trailer (near-field) and a roadside device (far-field). The sound levels measured by the two devices have the following relationship (i.e., propagation model).

SPLWayside ¼ SILOBSI  D

(140)

where SPLWayside is the road side sound pressure level in dBA, SILOBSI is the sound intensity level measured by the OBSI trailer in dBA; D is the correction term, and is given by

D ¼ 32:57 þ 0:0349$FN þ 18:094$MPD  0:0493$AG4

(141)

where FN is the average coefficient of friction for a ribbed tire, MPD is the mean profile depth, AG4 is the percent aggregate at screen No. 4 (4.74 mm). A residual standard error of 1.0 dBA was reported with R2 ¼ 0.73. For the FDOT OBSI system, the repeatability check showed an error of 0.26 dB at 95% confidence level. The pavement aging effect on TPIN was quantified as 0.2 dB/year. The frequency components of TPIN were also investigated in the study. It should be pointed out that the OBSI system was affixed to the wheel casing instead of affixing to the wheel hub, and hence the measurement might not be accurate owing to suspension compliance. Further, the direct relationship between the pavement parameters and the OBSI or pass-by levels were not reported. 3.3. Principal component analysis models Principal component analysis (PCA) uses an orthogonal transformation to identify a smaller number of uncorrelated variables, called principal components, from a larger number of possibly correlated variables. Principal component regression (PCR) is a regression analysis technique based on PCA. The advantage of PCA models is that the least number of variables (principal components) can be used to explain the maximum amount of variance. 3.3.1. Asphalt pavement (Ongel et al., 2008) Ongel et al. (2008) [142] combined the predictions for TPINs of different asphalt pavements into one universal statistical equation. OBSI tests as per the AASHTO standard (2008) [143] were conducted on 72 pavements, including open-graded, dense-graded, rubberized open-graded and rubberized gap-graded asphalt pavements. The initial parameters investigated in the study are listed in Table 16. Ten input parameters were selected based on the order of importance to develop the model. The statistics of the ten input parameters and the output parameter (OBSI) for all the measurements are listed in Table 17. The correlation matrix was generated using ordinary least-squares (OLS) multiple regression; the details are presented in Table 18. However, correlations for an input of 6 MPD were not reported. The final linear regression equation for OBSI prediction is

OBSI ¼ 107:6 þ 0:172Age  1:74Type  0:10AV  0:48IRI  1:25FM  0:005Cu þ 0:004MPD þ 1:84PR  0:09RTC þ 0:003SLT (142)

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T. Li et al. / Journal of Sound and Vibration xxx (2018) 1e89

Table 16 Initial parameters investigated in Ongel model (Ongel et al., 2008 [142]). Category

Parameter

Symbol

Unit

Description

Pavement mixture

Mix type

Type

[]

Categorical variable: 1 for open-graded mixes and 0 for gapand dense-graded mixes

Age Nominal maximum aggregate size Fineness modulus

Age NMAS FM

[year] [mm] []

Coefficient of uniformity

Cu

[]

Coefficient of curvature

Cc

[]

Rubber inclusion

RI

[]

Air void content Mean profile depth Root mean square of profile deviation International roughness index British pendulum number Surface layer thickness Well gradation

AV MPD RMS IRI BPN SLT WG

[%] [mm] [mm] [m/km] [] [mm] []

Presence of transverse cracking

PTC

[]

Presence of fatigue cracking

PFC

[]

Pavement distress condition

Environmental

Presence of raveling

PR

[]

Pavement temperature

T

[ C]

On-board sound intensity level

OBSI

A measure of aggregate gradation (larger indicates coarser grade) Cu ¼ D60/D10, where D60 is the sieve size associated with 60% passing, and D10 is the sieve size associated with 10% passing Cc ¼ D30/D10, where D30 is the sieve size associated with 30% passing Categorical variable: 1 for rubberized mixes, and 0 for nonrubberized mixes A measure of macrotexture A measure of macrotexture A measure of macrotexture Categorical variable: 1 for well graded mixes, and 0 for poorly graded mixes; If Cu > 4 and 1 < Cc < 3, the mix is well graded; otherwise poorly graded Categorical variable: 1 if the total length of transverse cracks  5 m in 150 m section, and 0 otherwise Categorical variable: 1 if the total fatigue cracking  5% of the wheel path area in 150 m section, and 0 otherwise Categorical variable: 1 if the total raveled area ¼ 5% of the total area in 150 m section, and 0 otherwise Pavement temperature measured during the OBSI measurements Output parameter, corrected and reported with their equivalent values at 96 km/h

[dBA]

Table 17 Statistics of parameters in Ongel model (Ongel et al., 2008 [142]).

Input 1 Input 2 Input 3 Input 4 Input 5 Input 6 Input 7 Input 8 Input 9 Input 10 Output

Parameter

Symbol

Unit

Mean

Standard deviation

Minimum

Maximum

Mix type Age Fineness modulus Coefficient of uniformity Air void content Mean profile depth International roughness index Surface layer thickness Presence of transverse cracking Presence of raveling On-board sound intensity level

Type Age FM Cu AV MPD IRI SLT PTC PR OBSI

[] [year] [] [] [%] [mm] [m/km] [mm] [] [] [dBA]

1.5 12.4 0.04 974.6 651.4 57 5 21.1 3.8 43.8 37.01

0.7 5 0.06 328.4 196.1 7.8 0.5 14.9 2.4 22.1 15.1

0.7 4.1 0.000007 405.6 282.3 41 3.8 2.9 <0.01 10.4 8.35

3.8 21.6 0.31 1807.00 1121.60 75 5.9 58 8.7 112.5 56.4

where R2 ¼ 0.73, and the standard error is 0.93 dBA. In Table 18, it can be seen that many parameters have strong correlations with each other. Cu has significant correlations with Type, AV, and FM (absolute value > 0.7). Thus, PCR was then employed to divide these input parameters into three orthogonal (uncorrelated) groups in the order of importance. The first group is “flow properties”; it includes AV, Cu, Type, FM, and SLT (in the order of importance); the second group is “surface roughness”; it includes PR, IRI, and Age; the third group is “crack resistance”; it includes PTC and Age. The statistical equation is

OBSI ¼ 100:87  1:28  FlowProperties þ 0:79  SurfaceRoughness  0:43  CrackResistance

(143)

where R2 ¼ 0.61, and the standard error is 1.04 dBA.

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45

Table 18 Correlation matrix for nine input parameters and PBSI output in Ongel model (source from Ongel et al., 2008 [142], Table 3; reprinted with permission from American Society of Civil Engineers).

OBSI Age Type AV IRI FM Cu PR PTC SLT a b

OBSI

Age

Type

AV

IRI

FM

Cu

PR

PTC

SLT

1.000 0.284a 0.477b 0.416b 0.384b 0.347b 0.446b 0.300a 0.273a 0.281a

0.287a 1.000 0.069 0.061 0.477b 0.010 0.101 0.206 0.235 0.129

0.479b 0.069 1.000 0.767b 0.139 0.607b 0.730b 0.063 0.321b 0.555b

0.417b 0.061 0.767b 1.000 0.071 0.640b 0.763b 0.062 0.238 0.687b

0.364b 0.477b 0.139 0.071 1.000 0.018 0.113 0.410b 0.146 0.076

0.349b 0.010 0.607b 0.640b 0.018 1000 0.720b 0.049 0.306a 0.465b

0.439b 0.101 0.730b 0.763b 0.113 0.720b 1.000 0.041 0.180 0.508b

0.297a 0.206 0.063 0.062 0.410b 0.049 0.041 1.000 0.095 0.199

0.274a 0.235 0.321b 0.238 0.146 0.306a 0.180 0.095 1.000 0.165

0.283a 0.129 0.555b 0.687b 0.076 0.465b 0.508b 0.199 0.165 1.000

Correlation is significant at 0.05 level (2-tailed). Correlation is significant at 0.01 level (2-tailed).

In a further study, Ongel and Harvey (2010) [144] investigated the influence of pavement characteristics on the frequency content of tire pavement noise. The effects of pavement parameters on different 1/3 octave band frequencies are presented in Table 19. 3.3.2. MTD (Zhang et al., 2014) Zhang et al. (2014) [145,146] developed a model to predict the mean texture depth (MTD) of the pavement macrotexture based on TPIN and vehicle speed. The TPIN cannot be explicitly obtained using this model, but it provides sufficient insight into the relationship between TPIN and MTD. The noise was measured using a microphone placed underneath the vehicle behind the driver side rear tire (behind the tire, BTT method); the noise measurement included TPIN and the noise caused by wind and vehicle vibrations. The acoustic energy was obtained by integration over the frequency spectrum (DC to 2 kHz), and then PCA was applied for reduction of the effects of noise and speed (Zhang et al., 2014 [145]; Zhang et al., 2012 [147]); thus, the combined method is termed as PCA energy method. PCA is a successful method to extract features from measurements (Wang et al., 2002 [148]; Hotelling, 1933 [149]; Jolliffe, 2002 [150]). It is a statistical procedure that uses an orthogonal transformation to convert possibly correlated variables into a set of values of linearly uncorrelated variables called principal components. All the tests (using the same tire) were conducted at the National Center for Asphalt Technology (NCAT) in Lee County, Alabama (Zhang et al., 2014 [145]), as in the case of Hanson model. Employing Taylor expansion and least-squares analysis (LS), the final equation was given by

MTD ¼ 0:2 þ 1:55En þ 0:44Vn þ 0:76En Vn

(144)

where MTD is the predicted mean texture depth in mm; En and Vn are normalized PCA energy and driving speed (dimensionless), respectively, and are given by

Table 19 Effects of pavement parameters on different 1/3 octave band frequencies (Modified from Ongel and Harvey, 2010 [144], Table 3). Frequency [Hz]

500

630

800

1000

1250

1600

2000

2500

3150

4000

5000

Type Age FM Cu AB MPD a RMS IRI SLT PTC PR b Rubber c NMAS d PFC

[

[

Y

Y

Y

Y

Y [ Y

Y [ Y

Y [ Y Y Y

Y [ Y Y Y

Y [ Y Y Y

Y [ Y [ Y Y Y

Y

[ Y [ [ [ [ Y

Y [ Y [ Y Y Y

Y

[ Y [ [ [ [ Y

Y [ Y [ Y

Y [ Y Y Y

[

[ [

[ [ [

[ [

[ [

[ [

[ [

[ [

[ [

[

[ [

[ [

[ [

[

a b c d

Y

RMS is the root mean square value of profile deviation in mm. Rubber is the pavement material with rubber inclusion (coded as 1 for rubberized mixes and 0 for non-rubberized mixes). NMAS is the nominal maximum aggregate size in mm. PFC shows the presence of fatigue cracking (coded as 1 if the total fatigue cracking is 5% of the wheel path area in a 150-m section, and 0 otherwise).

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T. Li et al. / Journal of Sound and Vibration xxx (2018) 1e89

En ¼

E  Emin Emax  Emin

(145)

Vn ¼

V  Vmin Vmax  Vmin

(146)

where E is the PCA energy in Hz$Pa, V is the driving speed in m/s; the subscripts min and max indicate the minimum and the maximum values, respectively, through all the tests. It was reported that the accuracy of this model is 83.61% with good repeatability when MTD is in the range of 0.2e3 mm, which is 10% higher than the results obtained with energy method (40e400 Hz) without PCA. On the other hand, a database is always needed for PCA to perform calibration and acquire the limits of the values. 3.3.3. Thin surfacing (Li et al., 2015) Li et al. (2015) [151] developed a statistical tire/road noise model considering the texture and absorption for thin-layer surfacings. Based on different combinations of input pavement parameters, three models were developed using PCR. The three combinations are presented in Table 20. The 12 output parameters are overall noise levels and 1/3 octave band noise levels in the frequency range of 315e3150 Hz. The coefficient of determination (R2) for the three models are presented in Table 21. In general, the more the number of parameters, the more accurate the statistical model is. 3.4. Fuzzy curve fitting models A fuzzy curve fitting model is used to solve a nonlinear complex system by a few simple “If … Then” fuzzy rules (Roychowdhury, 1998 [152]). The increase in such rules gives better modeling capability, especially with the advancement of computer technology. The advantage is that no specific form of equations is needed to fit the parameters. The fuzzy logic imitates the human learning process, and has a couple of derivative applications, such as artificial neural network (ANN), neuro-fuzzy system, and fuzzy genetic arithmetic (F-GA), which will be discussed in the following sections. In this sense, the fuzzy curve fitting model can also be called neural computation model. 3.4.1. CPB ANN (Fry et al., 1999) Fry et al. developed an ANN model to correlate vehicle design parameters with vehicle-controlled pass-by (CPB) noise (Fry et al., 1999 [153]; Fry and Jennings, 2003 [154]). As per ISO 362 [155,156], the vehicle should enter the test track with an initial speed of 50 km/h, and is to be driven through with 100% acceleration. The measurements are conducted in both directions, in 2nd and 3rd gears, while the maximum sound pressure levels (Max2nd and Max3rd) are recorded for each test run. The final average drive-by noise level DBN is given by

DBN ¼

Max2nd þ Max3rd  1dBA 2

(147)

The input parameters of the ANN model cover seven functional categories: powertrain, intake, exhaust, tire, test conditions, engine bay, and vehicle performance. The binary and categorical inputs were encoded to numeric values. Seventy-five

Table 20 Input pavement parameter combinations of surface texture and sound absorption (modified from Li et al., 2015 [151], Table 3). Model

Number of input parameters

Surface texture

Sound absorption

1 2 3

15 7 10

TL250, TL125, TL63, TL32, TL16, TL8, TL4, TL2, TL1 Mean profile depth (MPD) TL250, TL125, TL63, TL32, TL16, TL8, TL4, TL2, TL1

AL1000, AL1250, AL1600, AL2000, AL2500, AL3150 AL1000, AL1250, AL1600, AL2000, AL2500, AL3150 Maximum absorption coefficient

TLx is the texture level (dB) at the wavelength of x mm (ref. ¼ 106 m). ALf is the sound absorption level at the frequency of f Hz.

Table 21 Coefficient of determination for the three models (Li et al., 2015 [151]). Model

LA,eq

L315

L400

L500

L630

L800

L1000

L1250

L1600

L2000

L2500

L3150

1 2 3

0.87 0.66 0.80

0.68 0.49 0.70

0.76 0.53 0.77

0.82 0.66 0.81

0.87 0.85 0.87

0.90 0.82 0.85

0.92 0.72 0.87

0.89 0.65 0.80

0.82 0.59 0.74

0.94 0.71 0.89

0.84 0.78 0.93

0.90 0.81 0.89

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47

parameters were on the short list, refined from over 200 parameters by five vehicle noise experts. However, the details of these parameters were not published. The only output of the ANN model is the pass-by noise level. The suggested number of hidden neurons for an I-J-K ANN with I inputs, J hidden neurons, and K output neurons is given by (Tarassenko, 1998 [157])

J¼

pffiffiffiffiffi IK

(148)

The number of examples N should not be too small to capture the trend of the data, and also not too large to avoid overtraining or over-fitting when the ANN tries to fit the random noise. The rule of thumb is given below.

W ¼ ðI þ 1ÞJ þ ðJ þ 1ÞK N ¼ ð3  30ÞW

(149)

where W is the number of free variables, including weights and biases. The lower limit of the number of examples is given by

Nlower

pffiffiffiffiffi ¼ 3 IK I þ K þ

! rffiffiffiffi K þ1 I

(150)

Normally, K is much smaller than I; then, the equation becomes

Nlower z3I 1:5 K 0:5

(151)

It can be seen that I is more important than K. Based on the equations above, at least 2000 test runs were needed (Fry et al., 1999 [153]). To reduce this number, the ANN was divided into two submodels, as shown in Fig. 19. The vehicle acceleration was first used as the output of the vehicle performance network (20-5-1) with a mean percentage error (MPE) of ±7% (R2 ¼ 0.9) (Fry et al., 1999 [153]). Then, the output was changed to engine speed at the exit of the test track, because pass-by noise has a strong correlation with the engine speed (9 dBA per 1000 rpm). The error was found to decrease to ±4.8%. The type of neural network used is multi-layer perceptron (MLP), which is the most widely used network. The training algorithm is error back-propagation, or back-propagation. 85% of the published neural network applications has been created using some variant of the back-propagation algorithm (Wasserman, 1993 [158]). The corresponding algorithm in MATLAB is RPROP (resilient back-propagation), where the inputs were transformed to zero mean and unit variance to avoid “stuck units” (Tarassenko, 1998 [157]). Both the logsig (logistic sigmoid) and tansig (hyperbolic tangent sigmoid) transfer functions in MATLAB were tried for the hidden neurons. The best network found for the vehicle performance network and pass-by noise network is presented in Table 22. This model provides an insight into the vehicle acoustic performance in the early design stage before a prototype is built; this will save cost and engineering effort to a great extent. The prediction error of the model is within ±1.4% (i.e., ±1.1 dBA for the noise limit of 74 dBA), which is better than value that would be guessed by an experienced engineer (±3 dBA). The model also works for diesel and turbo-charged vehicles, but it is not valid for an automatic transmission vehicle, as its gear ratio will not be constant during the test. The authors have suggested a technique that combines neural networks with Bayesian statistics, called automatic relevance determination (ARD) to refine the input selection [159], and thereby potentially obtain a better prediction accuracy.

Fig. 19. Two-stage ANN models for pass-by noise (source from Fry and Jennings, 2003 [154], Fig. 3; reprinted with permission from Springer).

Table 22 Metrics for two-stage ANN models for pass-by noise (Fry and Jennings, 2003 [154]). Sub-ANN

Structure

Set for training and validation

Set for test

Accuracy

Vehicle performance network Pass-by noise network

25-15-1 58-19-1

2503 cases from 24 vehicles 4282 cases from 21 vehicles

997 cases from 13 vehicles 1424 cases from 7 vehicles

±4.8% ±1.4%

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3.4.2. Pavement aging (Khazanovich and Izevbekhai, 2008) Khazanovich and Izevbekhai (2008) [5] presented statistical equations to predict OBSI levels for different types of pavements using the data from 51 test sections. All the tests were conducted using a standard reference test tire (SRTT) (ASTM, 2014 [160]) at 97 km/h. The input parameters were estimated single axle load (ESAL), pavement age (Age), international roughness index (IRI), and surface rating (SR). The general equations have the following form.

I ¼ ESALa Ageb IRI m SRy

(152)

where I is the predicted OBSI sound intensity level; a, b, m, and n are the model exponents. For different types of pavements, such as asphalt pavements, astroturf drag, transversely tined white topping, and turf dragged white topping, these exponents have different values, which are calculated using the universal Levenberg-Marquardt algorithm (LMA), i.e., the hybrid of steepest-descent and least-squares nonlinear model fitting techniques. Some examples are presented in Table 23. However, no global values for all pavements were presented. It was found that pavement aging increases the TPIN on asphalt pavements, but decreases the TPIN on rigid concrete pavements. It should be noted that pavement age does not have a direct influence on TPIN, but it affects IRI and SR because of the texture degradation caused by traffic loads and environmental weathering. In a later study (Izevbekhai, 2012 [161]), Izevbekhai dropped pavement age, ESAL, and SR as the input parameters, but added temperature T, asperity interval which measures the texture wavelength ASP, texture direction relative to the traffic direction DIR (categorical variable, with values 0 or 1), texture depth TD, and texture spikiness which measures the texture amplitude SP (categorical variable, with values 0 or 1). The final equation is

  

       293  T IRI ASPT TD þC þD ðD  DIR þ EÞ þ ðF  DIR þ GÞ þ ½ðSPÞðJ  DIR þ KÞ I ¼AþB T IRIT ASP MPD

(153)

where the capital letters A, B, C, D, E, F, G, J, K are the model constants, MPD is the mean profile depth; and IRIT and ASPT are the corresponding values at the reference temperature. The errors for the model were found to be within 1.5 dBA for 90% of the field measurements. It was also found that friction number (FN) and mean profile depth (MPD) are not significant parameters for TPIN levels. 3.4.3. F-GA (Li et al., 2009) Similar to Che et al. model (ANN), Li et al. (2009) [162] used fuzzy genetic arithmetic (F-GA) to statistically correlate the tread pattern parameters with noise levels. The input tread pattern parameters include the width and length of the tread block, the width and length of the grooves, the total number of base pitches, pitch sequencing, and row offset. The output noise level is given below.

d¼

k

X

Lp;i  Lp;i;M



(154)

i¼1

where i is the index for each frequency band, Lp,i is the sound pressure level corresponding to specific tread pattern input, Lp,i,M is the sound pressure level in the noise standard line (curve M) provided by the American Test Center (ATC), and d is the model output. It was claimed that the model successfully optimized the parameters of the tread pattern for low noise tires, using fitness function iteration. However, further details on the optimized parameters and the accuracy of the model were not reported. It may be noted that both Che et al. model (ANN) and Li et al. model (F-GA) assumed that the tread pattern noise is independent of the tread block shape (only size matters). Kumar et al. (2013) [163] used multi-objective genetic arithmetic (MOGA) to optimize the tire tread pitch sequence, considering several performance parameters (noise, handling, and wear). 3.4.4. ANN (Che et al., 2012) Che et al. (2012) [164] used ANN to correlate the tire parameters with tire noise. ANN is a powerful tool to solve complex and highly nonlinear problems such as TPIN. Fig. 20 shows a schematic representation of a typical neural network. The ANN consists of multiple layers of neurons. A neuron can be mathematically interpreted as a nonlinear transfer function controlled by weights. The ANN receives the input data (tire and pavement parameters) in the “input layer”. In the

Table 23 Exponents for Khazanovich model (Khazanovich and Izevbekhai, 2008 [5]). Pavement Type

a (ESAL)

b (Age)

m (IRI)

n (SR)

Cracked white topping Bituminous (asphalt) Transversely tined

0.0075 0.188 0.00166

2.208 1.245 1.7678

0.0696 0.041 -.001858

0 0.025 0

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49

Fig. 20. Schematic of a typical artificial neural network (ANN) (source from Che et al., 2012 [164], Fig. 2; reprinted with permission from Trans Tech Publications Ltd).

same way, the layer that outputs the results (noise spectrum in narrowband, 1/1 octave or 1/3 octave) is called the “output layer”. The input and output parameters used in this study are listed in Table 24. It may be noted that the tests were conducted using the laboratory drum (DR) method (Yong et al., 2009 [165]); the surface was considered to have no variations or runout, and hence there were no inputs of pavement parameters. The ANN statistical equation is given below.

 Lp ðf Þ ¼ g Tw ; Tp ; TR ; PA ; PC ; PN ; PT ; PR ; PL ; PS ; PD ; V; L; F

(155)

where f is the center frequency of the corresponding octave band. The layers between the input layer and the output layer are normally called hidden layers. The simplest neural network is composed of an input layer and an output layer. The hidden layers improve the capacity of the network to solve complex problems. This model had one hidden layer. The number of neurons in the hidden layers does not follow any special rule, and it is chosen by the user. If the number is very small, it will be difficult for the ANN model to capture all the relationships; if the number is very large, the time for training will increase and “over-match” problem might occur. The model has the following empirical equation.

m¼

pffiffiffiffiffiffiffiffiffiffi xþlþa

(156)

where m is the number of hidden neurons, x is the number of neurons in the input layer, l is the number in output neurons, and a is a constant with value ranging from 1 to 10. The study employed back-propagation (BP) network training based on genetic algorithms (GA) that use crossover and mutation operations to determine the number of hidden neurons (6), number of training epochs (6000), and the initial weights (global optimization).

Table 24 Input and output parameters of ANN (Che et al., 2012 [164]).

Input

Parameter

Symbol

Description

Tire structure (size)

Tw Tp TR PA PC PN PT PR PL PS PD V L F Lp,125 Lp,250 Lp,500 Lp,1000 Lp,2000 Lp,4000

Tire width Aspect ratio Tire radius Area of single-pitch tread pattern block Area of single-pitch tread pattern groove Number of tread blocks per pitch Total number of pattern pitches Arrangement rule of the pitches Displacement (offset) between the pattern strips Symmetrical characteristics of tread patterns Groove depth Vehicle speed Tire load Inflation pressure Sound pressure level at the octave band of 125 Hz Sound pressure level at the octave band of 250 Hz Sound pressure level at the octave band of 500 Hz Sound pressure level at the octave band of 1000 Hz Sound pressure level at the octave band of 2000 Hz Sound pressure level at the octave band of 4000 Hz

Tread pattern

Operation

Output

Sound pressure level

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T. Li et al. / Journal of Sound and Vibration xxx (2018) 1e89

While the network is trained, the bias and the connections (weights) between the elements are adjusted. For a specific input, the network produces a specific output. Then, the output is compared with an expected result or target. This process is repeated by adjusting the weights, until the difference between the output and the target is less than the maximum permissible error. In this study, the training data obtained from the tests were divided into two parts: training samples (50 groups) and test samples (10 groups). It was reported that the average error was 1.4%. The main advantages of the ANN over nonlinear regression analysis used in the current hybrid methods are twofold. Firstly, no knowledge of the functional relationship between the input parameters and the noise spectrum output is needed. Secondly, the amount of data needed for training is less in the ANN than in the nonlinear regression. The ANN approach can be applied to various research areas. Another study (Kostial et al., 2013 [166]) correlated the angular frequency (output) of the tires with the tire structural parameters (input), such as nominal width, profile number, rim diameter, load index, and speed index. Additional input parameters were also included, such as radial static stiffness, circumferential static stiffness, lateral static stiffness, and static torsional stiffness. It was reported that the ANN results were qualitatively comparable with FEM simulation results with a root mean square error of 4.314 Hz. Yet another study (Kongrattanaprasert et al., 2009 [167]; Kongrattanaprasert et al., 2010 [168]) correlated the road surface states (output), such as snowy, slushy, wet, and dry states, with the measured tire noise of passing vehicles (input). An accuracy of approximately 90% was reported. A similar study (Kandpal et al., 2013 [169]) correlated the vehicle categories (output) with the measured noise (input). ANN was also used for tire contact optimization to improve the handling and wearing performance (Nakajima, 1999 [170]). The input is the sidewall and tread band contour; the output is the uniformness of contact pressure distribution. El-Gindy et al (1999 [171], 2001 [172]) developed a tire/pavement contact-stress model based on ANN for truck tires. The input parameters considered were vertical load and inflation pressure, and no tire parameters were included. However, separate ANN models were developed for one radial-ply tire and one bias-ply tire. 3.4.5. Bayesian (Yu and Lu, 2013) Yu and Lu (2013) [173] developed empirical models for four asphalt pavements: dense-graded asphalt concrete (DGAC), open-graded asphalt concrete (OGAC), rubberized open-graded asphalt concrete (RAC-O), and rubberized gap-graded asphalt concrete (RAC-G). The study presented two types of models: (1) overall model for all the asphalt pavements; (2) individual model for each type of asphalt pavement. The noise data were collected over a span of four years (296 observations covering 74 sections); certain observations and circumstances were missing or not considered for unavoidable reasons. To address this issue, a multiple imputation (MI) algorithm was used to capture the unobserved heterogeneity due to the missing parameters or the lack of knowledge on the relevant parameters. A Monte Carlo Markov chain (MCMC) sampling was used to deal with the panel structured data (Lu et al., 2009 [174]; Lu et al., 2010 [175]). Bayesian simulation was adopted to allow potential parameters to be random but include all the existing knowledge on the parameters. The noise level was measured using the OBSI method (A-weighted, 500e5000 Hz). The final prediction equation for each pavement type is given by

OBSI ¼ a0 þ a1  Age þ a2  AirVoid þ a3  MPD þ a4  AADTC þ a5  Pre þ a6  Thickness þ ε

(157)

where Age is the number of years since the pavement was open to traffic, AirVoid is the air void content (%), MPD is the mean profile depth (mm, ASTM E1845 [176]), AADTC is the average daily traffic on the main lane, Pre is the annual total precipitation (mm), Thickness is the thickness of the surface mix (mm), ε is an error term; a0 to a6 are coefficients to be estimated; the results are presented in Table 25. For the overall model, the variable Thickness is replaced by NewThickness that equals the product of the thickness of the given surface mix and the corresponding probability for this mix to be chosen. It was concluded that TPIN increases with pavement age and MPD, and decreases with air void content. The TPIN levels vary in the ascending order as RAC-O < OGAC < RAC-G < DGAC. From Table 25, it can be seen that the ability to resist noise level (small a1) increase with age rank varies as RAC-O > OGAC > RAC-G. The efficiency of air-void content increase for noise absorption (large absolute value of a2) varies as DGAC > RAC-G > RAC-O > OGAC. The efficiency of mix thickness increase for noise reduction (large absolute value of a6) varies as RAC-O > OGAC > DGAC > RAC-G.

Table 25 Coefficients for Yu and Lu (Bayesian) model (Yu and Lu, 2013 [173]). Coefficient

DGAC

OGAC

RAC-O

RAC-G

Overall

a0 (Intercept) a1 (Age) a2 (AirVoid) a3 (MPD) a4 (AADTC) a5 (Pre) a6 (Thickness)

103.6 0.0137 0.28 2.29  103 3  105 4.1  105 0.029

101.2 0.177 0.12 7.11  104 6.43  104 2.17  104 0.034

102.5 0.07 0.25 1.58  103 1.55  104 1.52  104 0.02

101.1 0.187 0.25 2.55  103 4.88  104 4.2  105 0.0276

101.4 0.083 0.199 1.41  103 1.33  104 1.69  104 0.021

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No.

Model name

Author

Year

Ref.

Method

Output

Frequency range [Hz]

Mechanism

Driver parameter

Tire parameter

Pavement parameter

Environment parameter

1

RLS (Germany)

BMVI

1990

[120]

Pass-by, Leq

N/A

Traffic

Slope

CNR (Italia)

CNR

1991

[122]

Pass-by, Leq

N/A

Traffic

N/A

Slope

3

TNM (US)

FHWA

1998

[124]

500e5000

N/A

Type

Dense-graded

2004

[128,129]

N/A

N/A

FM, voids, age

N/A

5

Concrete texture

Hanson and James Rasmussen

Vehicle and traffic TPIN

Propagation attenuation Propagation attenuation N/A

4

2009

[132]

Pass-by, 1/3 octave CPX, CPI, OASPL OBSI, 1/3 octave

Speed, vehicle type Speed, vehicle type Speed, vehicle type, acceleration 72 km/h

N/A

2

500e5000 630e1000

Texture impact

Buick Century, 97 km/h

SRTT

Texture, skewness

N/A

6

Asperity

Fujikawa et al.

2009

[108]

Nonlinear regression Nonlinear regression Nonlinear regression Nonlinear regression Linear regression, bridging filter Linear regression

BTT, OASPL, octave

500e4000

TPIN

80 km/h, 3.2 kN, 170 kPa

185/65R15, rib pattern

N/A

7

Pavement core

Reyes and Harvey

2011

[138]

Linear regression

500e5000

TPIN

N/A

SRTT

8

CNOSSOS (EU)

N/A

2012

[140]

Nonlinear regression

OBSI, 1/3 octave, OASIL Pass-by, LW, octave

Asperity spacing, asperity height unevenness, MPD Airflow resistance, MPD

125e4000

Speed, vehicle type, acceleration

Studded

Type, age

Air temperature

9

Mode-wise

Syamkumar et al.

2013

[6]

Linear regression, sensitivity analysis

CPB, OASPL

N/A

TPIN, vehicle and traffic TPIN

Speed, tire load, vehicle type

N/A

Type, wetness

10

FDOT

Wayson et al.

2014

[141]

Linear regression

OBSI, OASIL, pass-by, OASPL

500e4000

TPIN

88 km/h 97 km/h

SRTT

FN, MPD, AG4

Pavement temperature, wind direction Wind < 4 km/h

N/A

T. Li et al. / Journal of Sound and Vibration xxx (2018) 1e89 51

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Table 26 Summary of traditional regression models on TPIN.

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3.5. Summary The traditional regression models are summarized in Table 26. The regression analysis was the earliest one applied to statistical TPIN modeling. It has been widely used for traffic noise prediction by government transportation agencies. Nonlinear regression was dominant before the early 2000's, while linear regression has prevailed thereafter, probably because linear regression has easy form of equations, and needs less data to improve the accuracy. Besides pass-by noise measurement, more advanced and professional noise measurement techniques such as OBSI have been frequently used, where the 1/3 octave band noise spectrum was typically the output with a large frequency range of 500e5000 Hz. The vehicle speed was normally included in the modeling. However, unlike most deterministic models, the tire load or other tire parameters were not of much interest. Various types of pavement parameters were included, such as pavement type, slope, texture, friction, age, and wetness. Environmental parameters such as temperature and wind were also considered. However, the traditional regression model was unlikely to provide insight into the noise generation mechanisms or separate the different mechanisms. The principal component analysis models are summarized in Table 27. The input and output parameters of the model are similar to those of the traditional regression model. The principal component regression is more advanced than linear or nonlinear regression, and may include more input parameters in one model. The fuzzy curve fitting models are summarized in Table 28. Fuzzy logic technique, such as ANN, has been widely applied to various engineering problems, including TPIN modeling, since the late 1990's. The input and output parameters are similar to those of the regression models. However, the tire parameters such as tread pattern are more commonly included as input parameters. 4. Hybrid models 4.1. Introduction Hybrid models can be divided into deterministic submodels and statistical submodels. In the deterministic submodels, physical preprocessing and conditioning of input quantities are conducted to acquire intermediate parameters as inputs to the following statistical submodels. In most cases, the relationships in the statistical submodels can be significantly improved by including the preprocessing part of the data. The purpose of hybrid models is to find a compromise between the level of physical details and statistical accuracy. During recent years, the hybrid models appear to have become more popular, because they retain the accuracy of the statistical models and also provide insight into the mechanisms like the physical models.

Table 27 Summary of principal component analysis models on TPIN. No. Model name 1 2 3

Author Year

Ref.

Method

Asphalt Ongel 2008 [142] OLS, PCR pavement et al. MTD Zhang 2014 [145,146] PCA energy et al. Thin Li et al. 2015 [151] Principal surfacing component regression

Output

Frequency Mechanism Driver Tire Pavement Environment range [Hz] parameter parameter parameter parameter

OBSI, OASIL

N/A

TPIN

97 km/h

SRTT

Various

BTT, energy

0e2000

TPIN

Speed

Same tire

MTD

80 km/h

N/A

N/A Texture, MPD, absorption

CPX, 1/3 octave 315e3150 TPIN band SPL, OASPL

Pavement temperature N/A

Table 28 Summary of fuzzy curve fitting models on TPIN. No. Model name

Author

Year Ref.

1

CPB ANN

Fry et al.

1999 [153] Resilient BP

2

Pavement Khazanovich 2008 [5] Levenberg- OBSI, Aging and Izevbekhai Marquardt OASIL F-GA Li et al. 2009 [162] F-GA Octave band SPL ANN Che et al. 2012 [164] GA-BP DR, octave band SPL Bayesian Yu and Lu 2013 [173] Bayesian, OBSI, 1/3 MCMC, MI octave band, OASIL

3 4 5

Method

Output

Frequency Mechanism Driver range [Hz] parameter

CPB, OASPL

N/A

Vehicle

N/A

TPIN

125e4000 TPIN 125e4000 TPIN 500e5000 TPIN

Tire Pavement parameter parameter

Material, ISO 10844 geometry, slick included 97 km/h SRTT Type, ESAL, Age, IRI, SR N/A Tread N/A pattern Speed, tire Size, tread Standard load, inflation pattern surface Traffic N/A Age, voids, volume MPD, thickness

50 km/h (accel.), gear

Environment parameter N/A

30  C N/A N/A Precipitation

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53

In this section, 19 hybrid models are reviewed; they consist of 9 tire-pavement interface models, 6 mechanism separation models, and 4 noise propagation models. The model name, method, output noise parameter with the noise measurement technique, noise generation mechanisms investigated, and input parameters of each model are listed and compared in tables as a summary in section 4.5. The summary tables can be used as a reference to direct the readers to the models of their interest. 4.2. Tire-pavement interface models These models calculate the excitation characteristics of the interface between the tire and pavement within the contact patch. For example, the pavement texture is converted to a roughness parameter with reference to the tire using the envelope technique. The contact force/pressure spectrum is then obtained for further analysis of the tire vibrations. The TPIN is finally correlated with the contact force/pressure spectrum (intermediate parameters) using experimental data. 4.2.1. Truck contact (Fong, 1998) Fong (1998) [177] developed a model to predict the near-field sound pressure level from the computed value of contact pressure induced by the road surface texture. On-road measurements were conducted for 175/70R13 passenger tires (smooth and patterned), as well as 7.00R15 truck tires (radial and cross-ply) at a constant speed of 45 km/h using an on-board sound pressure technique (OBSP). This model is one of the few that included truck tires. The pavements were chip seal surfaces ranging from coarse-textured Grade 2 to finetextured Grade 6 (MTD in the range of 0.85e2.63 mm). The pavement texture was measured using laser profilometer (sampling distance of 3 mm, vertical resolution of 8 mm). The sound pressure was measured around the circumference of the rear right tire at four positions, one of which is close to the center of the contact patch. The sound pressure was measured using microphones at 10 kHz with a time duration of 104 s. The contact pressure was calculated by assuming that the contact load due to the interaction with the texture asperities was equal to the tire inflation pressure. It was found that the contact pressure increased with texture levels, as was expected. Transfer functions were calculated between the output (sound levels) and input (contact pressure). It was found that the transfer functions were independent of the pavement textures, which means that the key factor that influences tire noise was actually contact pressure. A good agreement was obtained between the predicted and measured noise levels, generally within 2 dBA across all the frequencies. 4.2.2. DGAC texture (Domenichini et al., 1999) The DGAC texture model (Domenichini et al., 1999 [18]) characterized the dense-graded asphalt concrete (DGAC) pavement texture, and related it to the near-field and far-field TPINs. The porosity of the wearing course and thickness of the surface layer were kept constant or ignored in the study. The study was part of a TINO research program, the other task of which was to identify the relationship between the mix composition and the pavement texture (D'Andrea, 1999 [178]). Combining the two studies, it will be possible to control the TPIN at the pavement design phase by suitable selection of aggregate type/shape/dimensions, aggregate grading curve, binder type, etc. Eleven types of pavements were tested, including one porous pavement as a reference and two TINO prototype pavements. The texture measurement was performed according to ISO 13473-2 (2002) [179] and the Italian standard CNR BU 125/88. (1) The microtexture was measured using a British portable pendulum tester (low speed friction test); (2) the macrotexture was measured using a TINO 3D profilometer; (3) the mega-texture was measured using a TINO 3D profilometer (2e2000 cycle/m or 0.5e500 mm) and a “Dipstick” walking profilometer (0.014e3.33 cycle/m, or 0.3e70 m). Both far-field and near-field noise tests were conducted with four microphone positions. Two tires were tested (a slick tire and a normal tire) at two speeds (50 km/h and 80 km/h). The correlation between the texture at different spatial frequencies (cycle/m) and noise levels at different frequencies (Hz) is represented in the form of correlation chromatic plots as shown in Fig. 21. Absolute values between 0.75 and 1 indicate good correlations. The preliminary model used was to correlate the near-field noise with the Huschek coefficient or Huschek texture index (Huschek, 1996 [180]) (R2 ¼ 0.79). However, the Huschek coefficient did not consider pavement unevenness. As a result, a new texture index (TI) was proposed; TI is given by

TI ¼ 2:5$PSP þ LUNE

(158)

where PSP is the “peak-slope parameter”, given by

8 LPEAK $i > > < PSP ¼ 100 > lfPEAK  þ lfPEAK þ lfPEAK þ > :L PEAK ¼ 3

(159)

LPEAK is the first peak of the texture level above the wavelength of 2 mm (or below the spatial frequency of 500 cycle/m), smoothed over the nearest three spectral lines, as illustrated in Fig. 22. Subscript fPEAK is the spatial frequency corresponding Please cite this article in press as: T. Li et al., Literature review of models on tire-pavement interaction noise, Journal of Sound and Vibration (2018), https://doi.org/10.1016/j.jsv.2018.01.026

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T. Li et al. / Journal of Sound and Vibration xxx (2018) 1e89

Fig. 21. Example of pavement texture-rolling noise correlation chromatic plot (source from Domenichini et al., 1999 [18], Fig. 10; reprinted under fair use provision).

Fig. 22. Texture index (TI) illustration for Domenichini et al. model (source from Domenichini et al., 1999 [18], Fig. 14; reprinted under fair use provision).

to wavelength lPEAK where the peak occurs. Subscripts fPEAK and fPEAKþ are the previous and following spatial frequencies, respectively. lxx indicates the texture level at the spatial frequency of xx cycle/m. The letter i refers to the slope of the descending portion above fPEAK in the macrotexture region, and is given by

i¼

LPEAK  L500 log10 ðlPEAKÞ  log10 ð2Þ

(160)

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where L500 is the texture level at the spatial frequency of 500 cycle/m (wavelength of 2 mm) smoothed over the nearest three spectral lines. LUNE is the unevenness level, and is given by

LUNE ¼

l1:26 þ l1:58 þ l2:00 3

(161)

LUNE represents the average texture level centered on 630 mm (500e800 mm). The final regression equations for the far-field sound pressure level at 50 km/h and 80 km/h are



SPL50km=h ¼ 0:47$TI þ 42:94 SPL80km=h ¼ 0:47$TI þ 49:39

(162)

where R2 ¼ 0.91. 4.2.3. SPERoN (Beckenbauer and Kuijpers, 2001) Beckenbauer and Kuijpers (2001) [181] developed a statistical and physical model to predict coast-by (CB) noise levels based on pavement parameters. The model is called SPERoN, which is the short form for statistical physical explanation of rolling noise. The noise data were based on the “Sperenberg project” (Influence of the road surface texture on the tire/road noise) for Federal Highway Research Institute of Germany (BASt), which included coast-by spectra of 840 tire-pavement combinations (21 dense surfaces; 12 normal, 2 slick, and 2 grooved passenger car tires) at speeds in the range of 50e120 km/h. The coast-by noise was measured 7.5 m away from the center of the vehicle, 1.2 m above the ground. The model was developed on a stepwise basis; hence, one additional input variable was included to expand the model for each step, and multivariate regression analysis was employed. The following requirements have to be met for each step: (1) the variables are independent; (2) the addition of a new variable must increase the coefficient of determination R2; (3) the influence of the variables on the noise spectra must be explainable physically. The final equation of the model includes four input variables (R2 ¼ 0.8): speed, frequency-transformed contact pressure spectrum, tire width, and shape factor; the equation is given below.

LpAF;i ¼ cvAF;i $10 log10

  v þ ccAF;i $LcF;i þ cwAF;i $w þ cgAF;i $Lg v0

(163)

where the subscript i is the index for the third octave band, LpAF,i is the A-weighted sound pressure level in dB, v is the vehicle speed in km/h, v0 is the reference vehicle speed (80 km/h), LcF,i is level of the frequency-transformed contact pressure in dB, w is the tire width in mm, Lg is the shape factor level in dB, cvAF,i is the regression coefficient for the speed term, ccAF,i is the regression coefficient for the contact pressure term, cwAF,i is the regression coefficient for the tire width term, and cgAF,i is the regression coefficient for the shape level term. The model is hybrid, because input parameters LcF,i and Lg are not available at the beginning, but deterministically calculated based on the other tire-pavement parameters, as shown in Fig. 23.

Fig. 23. Scheme of SPERoN model (source from Beckenbauer and Kuijpers, 2001 [181], Fig. 1; reprinted under fair use provision).

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T. Li et al. / Journal of Sound and Vibration xxx (2018) 1e89

The shape factor g is derived from the Abbot curve, and is used to identify convex and concave (or positive and negative) textures. As illustrated in Fig. 24(a), a shape factor of 25% in convex shape means that if an imaginary horizontal plane is lowered from the top of the surface by a distance R/2, it will be in contact with 25% (PCL, percentage contact length) of the texture. The linear scale shape factor g has the following relationship with the log scale shape level Lg (dB).

Lg ¼ 20 log10

g 1%

(164)

The contact pressure distribution was calculated based on the algorithm developed by Clapp et al. (1988) [182]. A contact patch length of 100 mm was assumed for a stationary tire. The contact pressure pc(x) as a function of the longitudinal position x was derived from the pavement texture profile and the average tire tread radial stiffness s. Assuming the tire rolling speed is v, the spatial domain of the contact pressure was transformed to time domain.

pc ðtÞ ¼ pc

x v

(165)

Then, the frequency domain of the contact pressure pc(f) was obtained using Fourier transformation of pc(t). The linear scale pc(f) has the following relationship with the log scale level of contact pressure LcF(f) in dB.

LcF ðf Þ ¼ 20 log10

  pc ðf Þ pc0

(166)

where pc0 ¼ 106 N/m2. It was concluded that the spectral energy decreases in the range of 800e1250 Hz as the speed increases, especially for rough pavement, as shown in Fig. 25(b). The coefficient of determination R2, the regression coefficient for the shape level term cgAF, and the regression coefficient for the speed term cvAF are shown in Fig. 26. It can be seen in Fig. 26(a) that slick tires have the highest coefficient of determination (R2 z 0.9) for most of the frequencies, while for the patterned tires, the values are relatively low, especially in the range of 400e800 Hz (R2 z 0.7), where the tread impact is assumed to have more effect. It is also concluded that the introduction of shape level Lg improves the reliability (R2 increases by 0.04) especially at low frequencies (<500 Hz), as shown in Fig. 26(a). Beckenbauer and Kuijpers (2001) [181] also concluded that the shape level Lg is independent of other input variables such as the level of the frequency transformed contact pressure LcF, because different values of Lg do not change the contact pressure regression coefficient ccAF. In Fig. 26(c), cvAF,i can be considered as the speed exponent that indicates the type of noise and corresponding generation mechanism. For slick tires, the speed exponent is greater than 4, indicating air pumping as the dominant mechanism. For patterned tires, the speed exponent is much lower, especially in the range of 400e800 Hz (cvAF z 3), again indicating the tread impact mechanism. In summary, the first version of SPERoN has the best predictability for slick tires, relatively good for groove tires, but only fair for normally profiled tires. Thus, the model is valid only on dense pavements with isotropic textures for slick tires. 4.2.4. Enhanced SPERoN (Kuijpers and Blokland, 2003) Kuijpers and Blokland (2003) [183] improved the SPERoN model by including the effects of tire tread profiles. The original goal of the SPERoN model was to evaluate acoustic performance of pavements, and to develop low-noise pavements, and not low-noise tires. After the incorporation of tread profile parameters, the model became more universal and unified.

Fig. 24. Illustration of shape factor (a. asphalt with surface dressing; b. hot rolled asphalt) (source from Beckenbauer and Kuijpers, 2001 [181], Fig. 2; reprinted under fair use provision).

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Fig. 25. Contact pressure level spectra for different speeds (a. stone mastic asphalt 0/8; b. surface dressing 3/5) (source from Beckenbauer and Kuijpers, 2001 [181], Fig. 3; reprinted under fair use provision).

Fig. 26. (a) Coefficient of determination R2, (b) Regression coefficient for the shape level cgAF, and (c) Regression coefficient for the speed term cvAF depending on third-octave canter frequency fT. Parameter: type of tire. The dashed line in (a) is valid for the statistical model without the variable Lg (shape level) for patterned tires (Source from Beckenbauer and Kuijpers, 2001 [181], Fig. 4; reprinted under fair use provision).

The contact model in the enhanced SPERoN model is shown in Fig. 27. The tread profile was divided into parallel slices across the tire width. The static contact pressure distribution pc(x, y) for each slice at the lateral position of y was calculated using the corresponding tread profile and pavement texture profile with the same algorithm used by Clapp et al. (1988) [182]. The texture profile (both 2D texture profile and 3D tread profile) was assumed to be identical for all slices. Then, the contact pressure in the frequency domain for each slice pc(f, y) was calculated using Fourier transformation. The input parameter pc(f) for the later statistical regression model was the average of pc(f, y) for all slices. It was concluded that the introduction of tread profile parameters reduced the prediction error by approximately 0.25 dBA in the frequency range of 500e1000 Hz. The tests were conducted on a very smooth grinded concrete surface with a specific profiled tire, where the tire profile was assumed to have more importance. However, the model prediction ability decreased at other frequencies (125e400 Hz and 1000e1500 Hz). In the case of tires with different tread profiles (slick, grooved, and normal), the overall prediction ability improved (the prediction error decreased by approximately 0.5 dBA) over the entire frequency range above 400 Hz. It can be stated that as a unified model for all types of passenger car tires, the enhanced SPERoN is promising.

Fig. 27. 3D contact model in the enhanced SPERoN model (source from Kuijpers and Blokland, 2003 [183], Fig. 1; reprinted under fair use provision).

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The enhanced SPERoN also includes mechanical impedance of the pavement and tire rubber hardness as input parameters for the statistical regression model. In the same study, the static contact model was compared with the dynamical contact model by Kropp et al. (2001) [184], where the variation in contact stiffness was substituted for the tire profile effect. To be more specific, if the contact area reduces because of changes in the tread profile, the deformation amplitude increases, i.e., the contact stiffness decreases. It was found that the static contact model and dynamical contact model gave comparable results, and a speed exponent of 3 in the dominant frequency range was predicted for both models. However, it may be noted that Kuijpers and Blokland (2003) [183] also concluded that tread profiles do not have influence on the rating of acoustic qualities of pavements. Caution should be exercised before using this enhanced SPERoN model, because tread profile analysis is very complicated and time consuming. As another enhancement for SPERoN, Blokland and The (2007) [185] included more types of pavements, other than dense surfaces. Several further development attempts were performed by Chalmers and Müller-BBM within the German project “Quiet Traffic”, the EC sixth framework project ITARI and the DEUFRAKO project P2RN [186]. The information on the new advanced model (version 3.0) can be found in Table 29. This advanced model includes four TPIN mechanisms, i.e., vibration due to impact (p2vibr), airflow due to air pumping (p2airflow), cavity due to cavity resonance (p2cavity), and aerodynamic noise due to air turbulence (p2aerodyn). The mean square value of coast-by sound pressure p2coast-by is given by

8 > p2coastby ¼ p2vibr þ p2airflow þ p2cavity þ p2aerodyn > > > > > p2 ¼ aFc2 Ga1 Ba2 Sa3 > > < vibr   p2airflow ¼ b Fc2 G1:5 S2 > > > g1 2 > > > pcavity ¼ cGpattern > > d1 : p2 aerodyn ¼ dV

b1

Ba2 V 4

(167)

where a, b, c, and d are the regression coefficients for the corresponding 3rd octave band, a1, a2, a3, b1, g1, and d1 are exponents; Fc is the contact force in N, G is the contact air flow resistance in Pa$s/m, B is the tire width in m, S is the tire tread stiffness in N/m, Gpattern is the spectral power for the tread pattern variation, and V is the vehicle speed in m/s. The most recent version of SPERoN has also included parameters of acoustic absorption and mechanical impedance. 4.2.5. HyRoNE (Klein and Hamet, 2007) Very similar to SPERoN, the HyRoNE (hybrid rolling noise estimation) model developed by Klein and Hamet (2007) [187] is also a hybrid model to predict pass-by noise levels as operational tools with less computational effort. The work was performed by INRETS (Hamet and Klein, 2004 [188]) and LCPC in the French PREDIT project “Texture&Bruit” and in the DEUFRAKO project P2RN (Beckenbauer et al., 2008 [186]). Both HyRoNE and SPERoN models physically analyze the raw texture of the road surface and convert it to the relevant roughness parameter with reference to the tire. The difference is that the former uses shape factor, while the latter uses the Table 29 Comparison of SPERoN and HyRoNE models (Beckenbauer et al., 2008 [186]; Klein et al., 2008 [190]).

Vehicle and tire Pavement parameter

SPERoN

HyRoNE

Passenger car tires (34 types), truck tires (Kuijpers et al., 2007 [191]) Airflow resistance, shape factor

One vehicle/tire combination (Renault Scenic 2.0 with Michelin Energy XH1/E3A tires) Acoustic absorption, porosity, shape factor and airflow resistance, porous layer thickness Twelve independent profiles (evenly distributed on both sides of the noise measurement point and characterize a 12-m long section, 6 profiles on each wheel track spaced by 1 m) Envelopment One specific tire (Michelin Energy XH1/E3A)

Texture profile

Quasi 3D texture (more than six 2D texture profiles of the pavement in parallel, 2 m each, covering 100 mm width in total, 1 mm resolution)

Texture analysis Tire parameter

Shape factor Tire type, load, tension, bending stiffness in circumferential and cross directions, tread stiffness (shore hardness) and tread pattern, radial uncoupled contact forces, inflation pressure Impulse response functions Average 3rd octave band spectrum of the contact forces As input parameter, arbitrary between 30 and 120 km/h Dense pavementsa Coast-by, 1/3 octave Maximum 2 dBA error

Tire model Contact model Speed Limitation Output Accuracy Computing time Software application Application a

210 s (69e120 km/h); 330 s (50e68 km/h) Windows (version 3.0), results in xml file Design of existing or artificial surfaces, at laboratory or test track scale

One specific tire (Michelin Energy XH1/E3A) Envelopment before spectral analysis Three nominal speeds: 70, 90, 110 km/h Isotropic or quasi-isotropic surface Controlled pass-by, 1/3 octave ±2.4 dBA, ±2.0 dBA, and ±2.1 dBA at 70, 90, and 110 km/ h, respectively 25 s (110 km/h); 45 s (90 km/h); 65 s (70 km/h) Windows (version 1.0), results in xls file Procurement process and in-situ monitoring at road scale

The most recent version of SPERoN has included acoustic impedance, and hence it is also applicable to porous pavements.

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method of envelopment before spectral analysis. The envelopment procedure is based on a static contact model between the pavement texture and an elastic half-space representing the tire rubber [189], as shown in Fig. 28. The tire rubber is characterized by its Young's modulus E. In the texture profile processing, wavelengths longer than 1 m and the upward peaks are removed. Another important difference between the two models is that HyRoNE also addresses attenuation mechanism (pavement absorption). HyRoNE mainly includes two TPIN mechanisms, tire vibration (100e1250 Hz) and air pumping (1250e5000 Hz), which are clearly separated in the physical and statistical submodels. In general, SPERoN is more complicated and universal than HyRoNE. A comparison of SPERoN and HyRoNE models is presented in Table 29 (Beckenbauer et al., 2008 [186]; Klein et al., 2008 [190]). 4.2.6. Road excitation (Rustighi et al., 2008) Rustighi et al. (2008) [192] developed a three-dimensional element model to predict the stochastic tire vibration due to tire/road excitation and the interior and exterior noises. In this model, it is assumed that a smooth soft tire is travelling on a road that is not very rough. The interaction between tire and the ground is modeled as a linear Winkler bedding (Johnson, 1985 [193]). The force f on the finite element points within the contact patch, as illustrated in yellow circles in Fig. 29, is given by

f ¼ Kðd  wÞ

(168)

where d is the vertical displacement of the road, w is the vertical displacement of the tire, and K is the linear contact stiffness matrix. The nodal displacement vector v is governed by the following equation.

Fig. 28. Illustration of envelopment procedure in HyRoNE model (Beckenbauer et al., 2008 [186], Fig. 5; reprinted with permission from Mr. Philippe Herzog on behalf of SFA, France).

Fig. 29. Mesh used for the tire model. The circles indicate the chosen contact points (Source from Rustighi et al., 2008 [192], Fig. 2; reprinted with permission from Elsevier).

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" K2

# v2 v 2 þ K þ K  j u G  u M vðf; uÞ ¼ fðf; uÞ 1 0 vf vf2

(169)

where f is the tire circumferential angular coordinate, u is the angular frequency, K2, K1, and K0 are the elastic force matrices, M is the inertial force matrix, and G is the fluid-structure coupling matrix. In the frequency domain, the vertical displacement d of the road is expressed as an excitation spectral density Sd(f), given by

Sd ðf Þ ¼

  c f 2:5 v v

(170)

where f is the excitation frequency in Hz, c is a constant, and v is the vehicle speed. A two-dimensional Gaussian model is used to represent the road surface, of which the roughness is homogeneous and isotropic (Robson, 1979 [194]). This approximation is experimentally determined, which is the reason why this model falls under the category of hybrid models. The calculated rms velocities of the tire wall and tread surface are shown in Fig. 30. The calculated sound pressure level (SPL) is shown in Fig. 31. The calculated interior sound pressure levels at eight hearing points inside the car cavity are shown in Fig. 32. The peaks around 60 Hz and 100 Hz may correspond to the first two internal air cavity resonances of the car, while the significant peak at approximately 230 Hz could be related to the first tire cavity resonance mode. 4.2.7. Viscoelastic contact (Dubois et al., 2011) The viscoelastic contact model [195] investigated the frictionless viscoelastic contact between a viscoelastic half-space (tire) and a rigid rough multi-indenter surface (pavement) using a macroscale approach (Cesbron et al., 2011 [196]). The viscoelastic behavior of the tire has to be considered owing to the time-dependent behavior of rubber, and the large contact area that includes many road asperities in different positions during rolling. The viscoelastic contact law for the kth asperity was described in terms of the convolution integral according to Radok's technique, as given below.

Zt Pk ðtÞ ¼

jðt  tÞ

d ½f ðd ðtÞÞdt dt k k

(171)

0

where Pk is the contact force on the summit of the kth asperity, j is the relaxation function, fk the elastic force-displacement law of the kth asperity, and dk is the local approaching displacement. Then, the TPIN levels were statistically correlated with the contact forces calculated above. The linear regression equation is

Fig. 30. Root mean squared velocities of the tire elements at 80 km/h. Note that the distances ±0.1 m in the radial direction correspond to the tread section of the tire, and the distances from ±0.1 to ±0.2 m correspond to the sidewalls (Source from Rustighi et al., 2008 [192], Fig. 7; reprinted with permission from Elsevier).

Fig. 31. Calculated value of sound pressure level (SPL) at the surface of the tire elements at 80 km/h (source from Rustighi et al., 2008 [192], Fig. 9; reprinted with permission from Elsevier).

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Fig. 32. Sound pressure level at eight passengers' ears at 20 (continuous line), 40 (dotted line), and 80 (dashed line) km/h (source from Rustighi et al., 2008 [192], Fig. 13; reprinted with permission from Elsevier).

LN ðfto Þ ¼ aj LF ðfto Þ þ bj

(172)

where LN is the TPIN level, LF is the contact force in the frequency domain, and fto is the center frequency of each third-octave band; aj and bj are the regression parameters. The TPIN levels and contact forces are positively correlated in the frequency range of 315e1000 Hz (R2 ¼ 0.93, better than the correlation between TPIN levels and pavement texture). It was found that the viscoelastic contact forces are greater than those obtained using the elastic contact method, and the contact duration is shorter, which fits better with the experimental results and gives better prediction for tire noise. A previous study (Cesbron et al., 2009 [197]) correlated the pavement texture spectrum, contact force spectrum, and the noise level spectrum in 1/3 octave band for a slick tire on six different pavements, as shown in Fig. 33. It was found that at 30 km/h, the levels of the resultant contact forces around 800 Hz are positively correlated with the texture wavelengths in the range of 4e100 mm (macrotexture) and with the noise levels in the range of 500e1000 Hz. It was also noted that the contact length under rolling condition is 20% smaller than that under the static condition, which is because of the viscoelasticity effect of the tire rubber. 4.2.8. Contact distribution (Dubois et al., 2013) Dubois et al. (2013) [198] improved the viscoelastic contact model by using two-scale iterative method (TIM) to obtain the contact pressure distribution. The tire/road noise measurements were conducted with close proximity (CPX) method on ten surfaces of the reference test track of IFSTTAR (French institute of science and technology for transport, spatial planning, development and networks) in Nantes, France. Two tires were tested: (1) patterned tire (Michelin Energy E3A 195/60R15); (2) slick tire (Michelin Racer Slick 186/57R15). The speed effect was expressed using the following regression equation.

Lm* N

  fj ¼ aj log10

V Vref

! þ bj

(173)

where Lm* N is the reprocessed measured noise level at any speed for frequency fj, V is the speed, and Vref is the reference speed m (90 km/h). An example of the data set for coefficients aj, bj, standard deviation sm j , and correlation coefficient rj is presented in Table 30. The 3D pavement texture profiles were used to calculate the contact pressure distribution. This process consists of two steps: (1) calculate the contact forces at the tips of the asperities (macroscale) (Dubois et al., 2012 [199]); (2) calculate the contact pressure within the tire-road interface (microscale). To obtain the spectral contact force, the pressure distribution p(x, y, t) in the time domain is integrated to get the total contact force F(y, t) in the time domain. dy yþ Z 2

dx

Z2

Fðy; tÞ ¼

pðx; h; tÞdxdh ydy 2

(174)

dx 2

The narrow band contact force spectrum SF(f) is given by Please cite this article in press as: T. Li et al., Literature review of models on tire-pavement interaction noise, Journal of Sound and Vibration (2018), https://doi.org/10.1016/j.jsv.2018.01.026

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Fig. 33. Iso-correlation curves between (a) contact force and noise level; (b) pavement texture and contact force; (c) pavement texture and noise level (source from Cesbron et al., 2009 [197], Fig. 17e19; reprinted with permission from Elsevier). Dy

1 SF ðf Þ ¼ Dy

Z2

SF ðy; f Þdy

(175)

D2y

where Dy is the scaling coefficient. Then, the third octave contact force level in dB is given by

LeF ðfi Þ ¼ 20 log10

SF ðfi Þ Fref

! (176)

where fi is the center frequency of the third octave band, and Fref is the reference force (1 107 N). The relationship between the noise level and contact force is given by

LeN ðfi Þ ¼ ai LeF ðfi Þ þ bi

(177)

It was found that the relationship is valid (R2 > 0.64) only at low frequencies (315e1250 for a slick tire, and 315e800 Hz for a patterned tire). The coefficients ai, bi, standard deviation sei , and correlation coefficient rei are presented in Table 31 for a slick tire and in Table 32 for a patterned tire. Please cite this article in press as: T. Li et al., Literature review of models on tire-pavement interaction noise, Journal of Sound and Vibration (2018), https://doi.org/10.1016/j.jsv.2018.01.026

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Table 30 Parameters of linear regression for speed effect on surface DAC 0/10 (source from Dubois et al., 2013 [198], Table 1; reprinted with permission from Elsevier). fj [Hz]

315

400

500

630

800

1000

aj bj [dB] sm j [dB] rm j

43.3 84.6 ±1.57 0.88

39.0 83.8 ±1.40 0.88

30.2 84.9 ±1.26 0.84

19.5 87.0 ±1.24 0.72

18.3 90.8 ±1.1 0.74

35.0 93.7 ±1.1 0.9

Table 31 Parameters for the correlation between noise level and contact force for slick tire (error ε ¼ 1.50%) (source from Dubois et al., 2013 [198], Table 2; reprinted with permission from Elsevier). fi [Hz]

315

400

500

630

800

1000

1250

ai bi [dB] sei [dB]

1.59 39.6 ±1.8 0.94

1.38 25.2 ±1.5 0.94

1.68 50.1 ±1.8 0.92

1.73 52.8 ±1.3 0.97

1.88 62.8 ±1.3 0.97

2.10 80.1 ±2.3 0.95

1.86 61.3 ±3.5 0.83

rei

Table 32 Parameters for the correlation between noise level and contact force for patterned tire (error ε ¼ 1:34%) (source from Dubois et al., 2013 [198], Table 3; reprinted with permission from Elsevier). fi [Hz]

315

400

500

630

800

ai bi [dB] sei [dB]

1.47 33.3 ±2.0 0.83

1.51 38.1 ±1.6 0.90

1.86 65.9 ±1.7 0.90

1.84 61.8 ±1.3 0.94

1.38 21.5 ±0.9 0.96

rei

It was also concluded that the estimation of noise levels is better by using contact force as input parameter than by using pavement texture. 4.2.9. 3D envelopment (Klein and Cesbron, 2016) The 3D envelopment model (Klein and Cesbron, 2016 [200]) is an improvement over HyRoNe model (Klein and Hamet, 2007 [187]), where the 2D enveloped texture profiles were used to calculate the tire-pavement contact forces without considering the lateral/transverse texture inhomogeneities. It was found that the 2D model did not work well for smooth pavements such as the ISO surface. To address this issue, in Ref. [200], both the 3D pavement texture and tread pattern profile were considered, and a new 3D envelopment procedure was developed for the prediction of the tire belt radiation noise. The input for the 3D envelopment calculation is the 3D combined tire-pavement roughness, obtained by subtracting the tire tread height from the road texture height, as shown in Fig. 34. The tire-pavement contact is modeled as an elastic half-space (tire) between a rigid body (pavement). The deflection of an elastic body w is given by the Boussinesq's formula.

wðx; yÞ ¼ ∬ pðu; vÞgðx; y; u; vÞdudv C

gðx; y; u; vÞ ¼

1  n2 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pE 2 ðx  uÞ þ ðy  vÞ2

(178)

Fig. 34. Top left: tire tread pattern; top right: pavement roughness; bottom: combined tire-pavement roughness (longitudinal/lateral spatial resolution ¼ 1 mm, source from Klein and Cesbron, 2016 [200], Fig. 10; reprinted under fair use provision).

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where p(x,y) is the pressure distribution within the contact patch C, E is the Young's modulus of the tread rubber, and n is Poisson's ratio. The global equilibrium condition is given by

∬ pðx; yÞdxdy ¼ F

(179)

C

where F is the total tire load. The iterative algorithm can be used to solve the contact patch C, the contact pressure p, and the half-space surface displacement w. An example of the calculated contact pressure is shown in Fig. 35. The surface displacement w can be considered as the 3D enveloped combined tire-pavement roughness, which can then be spectrally correlated with the tire noise. Good correlations were observed in the frequency range of 250e1250 Hz. However, the accuracy of the model at higher frequencies (over 1250 Hz), where the air pumping mechanism is dominant, needs to be improved. The error in the 1/3 octave noise level is within 3 dB, while the error in the overall noise level is within 1 dB. The results for one patterned tire (Michelin Energy E3A 195/60R15) tested on 11 non-absorptive pavements were presented, suggesting that more tires of different tread patterns need to be validated. Some other studies on the modeling of pavement envelopment can be found in Pinnington, 2012 [201]; Pinnington, 2013 [202]; and Pinnington, 2016 [203]. 4.3. Mechanism separation models In these models, an equation form for each noise generation mechanism is assumed based on its physics. Then, the total TPIN is obtained by summing up the values from all the generation mechanisms. Finally, the parameters in the assumed equations are statistically determined via experimental data. 4.3.1. Tire/road coupling (Cao et al., 2008) Cao et al. (2008) [204] studied the coupling noise during tire-road interaction in both time domain and frequency domain. The prediction model for tire tread pattern noise assumed a patterned tire running on a smooth road. Five generation mechanisms were included, which are discussed below. (1) Air pumping at the leading and trailing edges of the contact patch: The sound pressure due to this mechanism is given by

P1 ðtÞ ¼ gb $Ab $sinðub t þ qÞ$jB$sin ux tj

(180)

where

( gb ¼

1 sinðub t þ qÞ  0

x

sinðub t þ qÞ < 0

x2ð0; 1Þ

  1 1 B2 Ab ; Ab 5 9

ux 2ð8ub ; 10ub Þ

(2) Air pumping in the tread grooves:

Fig. 35. Contact pressure distribution (red curve: patterned E3A tire; black curve: slick tire) (source from Klein and Cesbron, 2016 [200], Fig. 13; reprinted under fair use provision). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

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P2 ðtÞ ¼ ðPmax $sin aÞ$gs $sinðus t þ qÞ

65

(181)

where a is the groove angle, Pmax is the maximum sound pressure, and



1 sinðus t þ qÞ  0 h sinðus t þ qÞ < 0 h2ð0; 1Þ gs ¼

(3) Pipe resonance:

P3 ðtÞ ¼

pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi sin ut d$l$A$ZðdÞ $sinð10uÞ $ d$l$A$ZðdÞ$sinf½0:8u±0:2RðtÞ$tg þ jsin utj 10

(182)

where d is the groove width, l is the groove length, Z(d) is a function of d, and R(t) is a random number between 1 and 1.

8 < 1; A¼ 1 : ; 3

sin ut  0 sin ut < 0

(183)

(4) The sound pressure due to cavities and pores in the contact patch is P4(t). (5) The sound pressure due to air turbulence is P5(t). The prediction model for road noise assumed a smooth tire running on a normal road. Four mechanisms were included. (6) Smooth tire on a smooth road (base noise):

P6 ðtÞ ¼

   

  R ðtÞ R ðtÞ $A1 $sin u1 0:5 þ 2 $t 0:5 þ 1 2 2

(184)

(7) Stick/slip (friction):

P7 ðtÞ ¼

   

  R ðtÞ A2 R ðtÞ $ $t $sin u2 0:5 þ 4 0:5 þ 3 2 2 x,k

(185)

(8) Road unevenness:

P8 ðtÞ ¼

    

 R ðtÞ R ðtÞ $ðA3 $h$jÞ $sin u3 0:5 þ 6 $t 0:5 þ 5 2 2

(186)

(9) Texture impact:

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P9 ðtÞ ¼

    

 R ðtÞ R ðtÞ $ðA4 $h$gÞ $sin u4 0:5 þ 8 $t 0:5 þ 7 2 2

(187)

Where the terms R1(t) to R8(t) are random numbers between 1 and 1, the terms A1 to A4 are the sound pressure amplitudes, and the terms u1 to u4 are the sound pressure frequencies. Then, total sound pressure in time domain is given by

PTR ðtÞ ¼ z1 P1 ðtÞ þ z2 P2 ðtÞ þ z3 P3 ðtÞ þ z4 P4 ðtÞ þ z5 P5 ðtÞ þ z6 P6 ðtÞ þ z7 P7 ðtÞ þ z8 P8 ðtÞ þ z9 P9 ðtÞ ¼ ½ z1 / z9 ½ P1 / P9 T

(188)

where z1 to z9 are the best-fit coefficients determined by experiments. Then, the total sound pressure in frequency domain HAll(f) after accounting for horn effect and pavement absorption is given by

HAll ðf Þ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  aðf Þ$Kðf Þ$HTR ðf Þ

(189)

where a(f) is the pavement absorption coefficient, K(f) is the horn effect coefficient, and HTR(f) is the Fourier transform of PTR(t). However, the experimental data and model accuracy have not been reported. 4.3.2. Mechanism decomposition (Dare, 2012) Dare (2012) [13] decomposed the OBSI spectrum into three constituent spectra, representing the contributions from sidewall vibration, tread band vibration, and tangential vibration, as illustrated in Fig. 36. It is assumed that the shapes of the constituent spectra will not change significantly, though the magnitudes may change. The general constituent spectra were derived using SRTT on six different pavements at five speeds (16e42 km/h). For the sidewall vibration constituent, the gain function GIV,sidewall for the relationship between the measured sound intensity and vibration level is given by

GIV;sidewall ðf Þ ¼

Isidewall ðf Þ jVsidewall ðf Þj2

(190)

where f is the frequency, Isidewall is the sound intensity for sidewall vibration constituent measured with tread band covered in sound-absorbing insulation, and Vsidewall is the velocity spectrum measured with an accelerometer mounted on the sidewall. Similarly, the gain function for the tread band vibration is given by

Fig. 36. Constituent spectra and predicted total spectrum compared to measured spectrum for cell 1, leading probe. Blue: tread band spectrum. Red: sidewall spectrum. Green: tangential spectrum. Dotted black: predicted total spectrum. Solid black: measured spectrum (Source from Dare, 2012 [13], Figure 5.10; reprinted with permission from Dr. Tyler Dare of Pennsylvania State University, USA). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

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T. Li et al. / Journal of Sound and Vibration xxx (2018) 1e89

GIV;treadband ðf Þ ¼

67

Itreadband ðf Þ

(191)

jVtreadband ðf Þj2

where Itreadband is measured with the sidewall covered with sound-absorbing insulation, and Vtreadband is measured with an accelerometer mounted on the tread groove. The tangential vibration is the result of stick-slip and tangential impact. The tread block was modeled as a dipole source, with two out-of-phase monopole sources at the front and rear edges of the tread block. The sound pressure P generated by a dipole is given by Ref. [97].

1 Q P ¼ r0 c 2 l

! 1 ejud þ r rþd

(192)

where r0 is the air density, c is the speed of sound, l is the wavelength of sound, r is the distance between one monopole source and receiver (2 cm), and d is the distance between the two monopole sources (length of the tread block). Q is the source strength, given by Q ¼ whV, where w is the tread block width (2.54 cm), h is the tread block height (0.64 cm), and V is the tangential velocity of the tread block. The gain function is

Gdipole

! 1 wh 1 ejud þ ¼ r0 c 2 l r rþd

(193)

The sound intensity equations for the three constituent spectra at the leading edge are given by

8 2 > > < Isidewall;lead ¼ GIV;sidewall Vsidewall;lead 2 Isidewall;trail ¼ GIV;sidewall Vsidewall;trail     > > :I tangential;lead ¼ Gdipole Vtangential;lead Gtrain Ghorn;close þ Gdipole Vtangential;trail Gtrain Ghorn;far

(194)

where Gtrain is the calculated impulse train spectrum due to the frequency of tread block impact, Ghorn,close and Ghorn,far are the measured horn effect gain functions for close and far OBSI probe locations, respectively. The sound intensity levels for the three constituent spectra at the leading edge are given by

! 8 Isidewall;lead > > > ¼ 10log þ Dsidewall;lead L > sidewall;lead 10 > Iref > > > > > ! > < Itreadband;lead Ltreadband;lead ¼ 10log10 þ Dtreadband;lead > Iref > > > > ! > > > Itangential;lead > > > þ Dtangential;lead : Ltangential;lead ¼ 10log10 Iref

(195)

where Iref is the reference sound intensity equal to 1 pW/m2, and D is the prediction offset/correction for a specific pavement. The total sound intensity level at the leading edge is

0 Ltreadband;lead B Lsidewall;lead 10 10 Ltotal;lead ¼ 10 log10 B þ 10 þ2 @10

1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Lsidewall;lead

10

10

Ltreadband;lead

$10

10

cos f þ 10

Ltangential;lead 10

C C A

(196)

where f is the measured phase difference between the sidewall and tread band sources. The average error for one-third octave band between the predicted and measured values is 2.3 dB for the leading probe, and 2.8 dB for the trailing probe. The sound levels at the trailing probe are calculated in a similar manner. To predict the total sound intensity for different pavements, a statistical model was developed.

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8 > > > Ltreadband þb1 xþb2 > > > 10 Itreadband ¼ 10 > > > < Lsidewall þb3 xþb4 10 Isidewall ¼ 10 > > > Ltangential þb5 xþb6 > > > 10 > Itangential ¼ 10 > > :

(197)

where b1 to b6 are best-fit coefficients, and x is a pavement parameter such as MPD, texture level at a given wavelength, or absorption. It was found that MPD, texture level at 12.5 mm wavelength, texture level at 3.15 mm wavelength, and skewness are the best predictors for this one-parameter OBSI model. The statistical model can also include two parameters simultaneously, as given below.

8 > > > Ltreadband þb1 x1 þb2 x2 þb3 > > > 10 Itreadband ¼ 10 > > > < Lsidewall þb4 x1 þb5 x2 þb6 10 Isidewall ¼ 10 > > > Ltangential þb7 x1 þb8 x2 þb9 > > > 10 > I ¼ 10 > > tangential :

(198)

It was found that texture levels at 12.5 mm and 3.15 mm wavelengths are the best two predictors, with an error within 0.9 dB at the leading probe and 1.1 dB at the trailing probe for the overall sound intensity levels. It may be noted that the experimental data did not cover tests at high speeds (the speeds were lower than 42 km/h). Extrapolation with speed exponent was used to account for the effect at high speeds. 4.3.3. Air pumping separation (Winroth et al., 2013) Winroth et al. (2013) [205] separated the noise component due to tire vibrations and the component due to air pumping. It was pointed out that the sound generation due to tire vibrations is proportional to the square of vehicle speed, while the sound generation due to air pumping is proportional to the fourth power of vehicle speed. Therefore, the sound pressure can be expressed as

p2rms ðf ; VÞ ¼ A2 ðf Þ$V 2 þ A4 ðf Þ$V 4

(199)

where prms is the rms value of sound pressure, f is the frequency, and V is the vehicle speed; A2 and A4 are coefficients as a function of frequency; A2 and A4 are statically determined by curve fitting from the spectra of controlled pass-by (CPB) levels in the speed range of 50e120 km/h. Several pavements were investigated: mastic asphalt, stone mastic asphalt sealed with synthetic resin, and a sand paper surface. However, only slick tire was considered in this model. It was found that air pumping noise is dominant at frequencies over 1000 Hz and at some low frequencies around 400 Hz. 4.3.4. Deterministic SEA (Mohamed and Wang, 2016) Mohamed and Wang (2016) [206] investigated tire cavity resonance using a combination of deterministic analysis (impedance compact mobility matrix method, ICMM) and statistical energy analysis (SEA). The deterministic part focuses on the cavity pressure response and the compliant wall vibration velocity response at low frequencies, while the statistical part focuses on the responses at high frequencies. In the mid-frequency range, the two methods are combined to identify the characteristics of tire-cavity coupling. The authors claimed that the same analysis method could be applied to other similar structural-acoustic systems with toroidal shape. The tire-cavity coupling system is modeled as an annular cylinder, where the side and inner walls are assumed to be rigid, while the outer shell (referred to as tire structure hereafter) is assumed to be flexible. The vibration energy E1 of the tire structure (large modal density) is statically calculated, while the acoustic energy E2 of the tire cavity (small modal density) is deterministically predicted.

E1 ¼

P1 þ uE2 hh12 ððuuÞÞh12 uðh1 þ h12 Þ

(200)

where P1 is the power input to the tire structure, u is the angular frequency, h1 and h2 are the damping loss factors of the tire structure and tire cavity, respectively; h12 is the coupling loss factor from the tire structure to the tire cavity. Please cite this article in press as: T. Li et al., Literature review of models on tire-pavement interaction noise, Journal of Sound and Vibration (2018), https://doi.org/10.1016/j.jsv.2018.01.026

T. Li et al. / Journal of Sound and Vibration xxx (2018) 1e89

E2 ¼

1 1 ∭ V V2

Zu2 2 pWðR  R Þ2 2 o i dudV pu ðr; q; zÞ r0 c2

69

(201)

u1

where V is the volume of the cavity, u1 and u2 are the lower and upper limits of the frequency band, respectively; pu is the time- and space-averaged rms pressure of the cavity obtained from experiment; r, q, and z are the cylindrical coordinates; W is the tire width, Ro and Ri are the outer and inner radii of the tire cavity, respectively; r0 is the air density, and c is the speed of sound in air. It can be seen that the coupling in this model is a strong coupling, where the structure and the fluid influence each other (the equation for E1 contains E2). 4.3.5. Statistical vibration (Le Bot et al., 2017) The statistical vibration model (Le Bot et al., 2017 [207]) was based on the deterministic model of a prestressed orthotropic plate on visco-elastic foundation (Kropp, 1999 [53]), as illustrated in Fig. 37. Both modal analysis and statistical analysis were performed to investigate the tire vibration characteristics at high frequencies. The plate is of length a and width b, unwrapped in the tire circumferential and transverse directions, respectively. The governing equation for out-of-plane vibrations is

2 4

qffiffiffiffiffi v2 pffiffiffiffiffi v2 Bx þ By vx2 vy2

!2 

v2 v2 Tx 2 þ Ty 2 vx vy

!

3 v v2 5 v ¼ f ðx; y; tÞ þSþc þm vt vt

(202)

where v(x, y, t) is the displacement field (output response) in the direction normal to the plate, and f(x, y, t) is the contact force field (input excitation). In Eq. (202), m is the mass per unit area in kg/m2; Tx and Ty are the x-component and y-component, respectively, of the tensile force (in N/m) imposed by the air pressure (always positive); Bx and By are the bending stiffnesses (in N$m) in the x-direction and y-direction, respectively; S is the stiffness per unit area (in N/m3) of the elastic foundation, and c the viscous damping coefficient (in N$s/m3). The periodic boundary conditions are applied for the x-direction, while simply supported boundary conditions are applied for the y-direction. The mode shapes are obtained as

x   y sin jp ; a b    x y jij2 ðx; yÞ ¼ bij sin 2ip sin jp ; a b qffiffiffiffiffiffiffiffiffiffiffi 8 < aij ¼ 2=ab; i¼0 . p ffiffiffiffiffi ffi : aij ¼ bij ¼ 2 ab; is0 

jij1 ðx; yÞ ¼ aij cos 2ip

i ¼ 0; 1; / j ¼ 1; 2; / i ¼ 1; 2; / j ¼ 1; 2; /

(203)

The eigenvalues are given by

mu2ij

" #2 "    2 # pffiffiffiffiffi 2ip2 qffiffiffiffiffi jp2 2ip 2 jp ¼ Bx þ By þ Tx þ Ty þS a b a b

(204)

Fig. 37. Tyre-road model. Left: the tyre is virtually unwrapped. Right: equivalent prestressed orthotropic plate under visco-elastic foundation (Source from Le Bot et al., 2017 [207], Fig. 1; reprinted with permission from Elsevier).

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The transfer mobility, i.e., frequency response function between a force applied at x0(x0, y0) and the vibrational velocity at x(x, y), is given by

   x  x0 y  y  sin jp sin jp 0 cos 2i p X 4εi b a b   Yðx; x0 ; uÞ ¼ iu ab m u2ij þ ihij uuij  u2 i0;j1 ( 1=2; i ¼ 0 εi ¼ 1; i>0

(205)

The identification of the physical parameters can be performed using a simplex search method from the point mobility measurement results. The dispersion relationship is given by

kðuÞ ¼

ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ffi  u 2 2 uT þ T  4B S  mu t

2B qffiffiffiffiffi 2 pffiffiffiffiffi BðqÞ ¼ Bx cos2 q þ By sin2 q

(206)

TðqÞ ¼ Tx cos2 q þ Ty sin2 q where k is the wavenumber, and q is the wave propagation angle with respect to x-direction. The cut-on frequency is given by

ucuton ¼

rffiffiffiffiffi S m

(207)

The vibration characteristics are defined by three statistical parameters: modal density n(u) that indicates the size of the population of resonant modes, modal overlap M(u) that measures the overlapping of successive modes, and normalized attenuation factor m(u) that indicates the number of reflections undergone by a ray during its time life. These parameters are given by

nðuÞ ¼

2m

Z Z

Re½Yðx; x; uÞdudAx

pDu A Du

MðuÞ ¼ hðuÞunðuÞ 3 2   Z2p 1 hu exp  l dq5 mðuÞ ¼ ln4 2p cg

(208)

0

where A is the plate area ab, Du is the bandwidth of interest (e.g., octave band centered at 1 kHz, i.e., [707, 1414] Hz), Y(x, x; u) is the point mobility at x, and Re denotes the real part; h(u) is the mean damping loss factor at frequency u; cg is the group speed, defined as du/dk. Another statistical parameter that characterizes the uniformity of spatial repartition of energy is s, which is defined as the ratio of spatial standard deviation of local energy and mean energy; the equation for s is given below.

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u1 Z u s¼ Eðx; u0 Þ2 dAx  Eðu0 Þ2 t Eðu0 Þ A 1

1 Eðu0 Þ ¼ A

Z

A

Eðx; u0 ÞdAx

A

Eðx; u0 Þ ¼ S0

Z

(209)

mjYðx; x0 ; uÞj2 du

Du

where S0 is the power spectral density of the force f at x0. Based on these statistical parameters, the vibrations in tires can be classified into three regimes: modal, diffuse field, and free field regimes, as shown in Figs. 38 and 39. For real tires, the damping is relatively high, and increases with frequency from 0.05 to 0.3. Therefore, as the frequency increases, the modal regime directly transitions into the free field regime without reaching the diffuse field regime. It may be noted that at low frequencies below 400 Hz, the tire dynamics is dominated by a Please cite this article in press as: T. Li et al., Literature review of models on tire-pavement interaction noise, Journal of Sound and Vibration (2018), https://doi.org/10.1016/j.jsv.2018.01.026

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Fig. 38. Relative standard deviation of spatial energy repartition in the frequency, damping loss factor plane under point-excitation. Left side: modal regime with a few modes contributing to the dynamics. Top right-hand corner (black): free field regime where rays are highly absorbed. Bottom right-hand corner (bright): diffuse field regime. The broken lines are regime boundaries determined by N(u) ¼ n(u)$Du and m(u) (Source from Le Bot et al., 2017 [207], Fig. 7; reprinted with permission from Elsevier).

Fig. 39. Examples of energy fields corresponding to crosses in Fig. 38. (a) Regime of modal field (point A); (b) Regime of diffuse field (point B); (c) Regime of free field (point C) (Source from Le Bot et al., 2017 [207], Fig. 8; reprinted with permission from Elsevier).

few modes (typically six nodes), whose modal overlap is relatively low. At high frequencies, e.g., 1000 Hz, the dynamics is mainly driven by bending effects, where the modal density and modal overlap are so large (e.g., mode count N ¼ 192 for the octave at 1 kHz) that the behavior of individual modes is not distinguishable on point mobility. To investigate the non-modal domain, the geometrical acoustics approach is used instead of conventional modal analysis. The energy field may be described in terms of rays that propagate in straight lines with a speed depending on the direction. 4.3.6. Network resonator (Wang and Duhamel, 2017) In this work (Wang and Duhamel, 2017 [208]), the grooves between the tire tread and pavement texture were modeled as acoustic network resonators, as shown in Fig. 40. A numerical method for the calculation of resonant frequencies of network resonators was proposed. First, the end corrections for the open-ended pipes were calculated. The end correction dsq for the transverse pipe, assuming rectangular flat flanges as shown in Fig. 40, is given by Dalmont's method (Dalmont et al., 2001 [209]).

dsq ¼ dsq∞ þ

"  5 #  asq

a a dsq0  dsq∞ þ 0:057 sq 1  sq aeff bsq bsq bsq

dsq∞ ¼ 0:811aeff dsq0 ¼ 0:597aeff

(210)

2asq aeff ¼ pffiffiffi

p

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Fig. 40. A network (in red) between a round surface and a rigid plane surface (ABCD indicates the contact patch; 1 is a longitudinal pipe while 2 and 3 are transverse pipes; bsq is half the flange width for pipe 2) (Source from Wang and Duhamel, 2017 [208], Fig. 2; reprinted with permission from Elsevier). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

where 2asq (¼ 0.009 m) is the pipe width, and 2bsq is the flange width. The end correction d for the longitudinal pipe, assuming a cylindrical flange, is obtained by BEM, and is statistically given by

d ¼ 0:00808 þ 0:22128w  3:72112w2 þ 19:80897w3

(211)

where 2w is the flange width. It may be noted that this d is slightly dependent on frequency, and Eq. (211) gives the mean values in the frequency range of 0e2000 Hz. Next, the resonant frequencies of the network resonators were solved using finite element software Abaqus, where the end corrections were accounted for during physical modeling. Finally, the genetic algorithm (GA) was used to optimize the acoustic network (e.g., pipe positions and junction types) to obtain the targeted resonant frequencies, and the result was validated by experiments on wooden network resonators. It may be noted that the acoustic resonator was used in this work to attenuate the sound pressure level, which is different from the common understanding that pipe resonance amplifies the tire noise. 4.4. Noise propagation models In these models, the sound power of TPIN is acquired statistically. Sound propagation theory is then applied to obtain the sound pressure level (SPL) at a given location. 4.4.1. UK (CRTN, 1988) The UK model (CRTN, Calculation of road traffic noise) was issued by the UK Department of Transport (DOT) in 1988 [210], and is widely used for traffic noise calculation in United Kingdom, Ireland, Australia, New Zealand, and Hong Kong (Leung and Mak, 2008 [211]). A webpage calculator is also available (NPL, 2005 [212]). Like NMPB 2008, CRTN also includes emission and propagation models. The road is modeled in CRTN as a line source instead of a set of point sources (in NMPB); additional methods to deal with dual source lines, median barriers, and corrections for thin surfaces were presented in 2008 (UK Department for Transport, 2008 [213]). The sound level is expressed in terms of L10 (the noise level exceeded for 10% of the measurement period) for the 18 h period between 06:00 and 24:00 (LA,10,18h) without noise assessment during night between 24:00 and 6:00. The source emission level is expressed as follows.

L10;18h ¼ 29:1 þ 10lgðQ Þ þ CorrectionV&p þ CorrectionG þ CorrectionTD

(212)

where Q is the number of vehicles for the 18 h duration, including all vehicles (light and heavy); CorrectionV&p is correction for mean traffic speed V (km/h) and the percentage of heavy vehicles p (%); CorrectionG is the correction for the road gradient G (%), CorrectionTD is the correction for the texture depth (TD) in mm. The equations for the three corrections are

    500 5p þ 10lg 1 þ  68:8 CorrectionV&p ¼ 33lg V þ 40 þ V V

(213)

CorrectionG ¼ 0:3G

(214)



CorrectionTD;concrete ¼ 10lgð90TD þ 30Þ  20 CorrectionTD;asphalt ¼ 10lgð20TD þ 60Þ  20

(215)

CorrectionTD is 1 dBA for speeds less than 75 km/h. LA,10,18h must be converted to Lden and Lnight indicators for noise mapping assessment (Abbott and Nelson, 2002 [214]). However, the noise data at night were not collected in the CRTN method. Even though the noise prevails at night, the correlation between L10,night and Leq,night is not good enough owing to insufficient traffic volume. Another disadvantage of CRTN Please cite this article in press as: T. Li et al., Literature review of models on tire-pavement interaction noise, Journal of Sound and Vibration (2018), https://doi.org/10.1016/j.jsv.2018.01.026

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compared to NMPB 2008 is that it does not consider different traffic flow types, and the fact that speeds for light vehicles and heavy vehicles are normally different. In different locations, CRTN needs to be recalibrated, which is not an easy task. For example, Table 33 shows the several recalibrated CRTN models proposed by local researchers in Hong Kong (Leung and Mak, 2008 [211]). They all have certain errors. Based on the CRTN results, Abbott and Nelson (2002) [218] correlated Leq and L10 using regression method. O'Malley et al. (2009) [219] then adapted it to an Irish scenario [36], and expressed Lden as

Lden ¼ 0:86  LA;10;18h þ 9:86dB

(216)

Murphy and King (2011) [220] noted that on an average, it underpredicts noise levels by 0.01 dB with an rms. error of 1.6 dB for the case in Dublin, Ireland. 4.4.2. France (NMPB, 1996) NMPB (1996) [221] model was described in French standard XPS 31-133, and was recommended by Environmental Noise Directive (END) for traffic noise calculation. The first version of the model is “NMPB-Routes-96 (SETRA-CERTU-LCPCCSTB)” (1996) [222], which described only the sound propagation model, while the sound emission model was presented in “Guide du Bruit” (1980) [223]. The revised model NMPB 2008 [224] separates the TPIN and engine noise (Dutilleux, 2013 [225]), and tra, 2009 [226]). is currently used in France; both noise emission and noise propagation models are used (Se In NMPB 2008, each road is modeled as a line source, or several lines in the case of multiple lanes. Each line is then divided into uninform sections; each section is divided into a set of incoherent point sources. For the noise emission model, the sound emission level E in dB for a single vehicle is given by

E ¼ LW  10 log V  50

(217)

where LW is the A-weighted sound power per meter in a given octave band in dB, and V is the vehicle speed. LW depends on the traffic conditions, as presented in Table 34. The alternative equation for sound emission level is

 E ¼ E0 þ a log10

V V0

 (218)

where the coefficients E0 and a are determined by the combination of traffic conditions; V0 is the reference speed (20 km/h). The sound power level per meter of lane for each elementary period k is given by



  LAW=m;k ¼ Elv;k þ 10lg Qlv;k þ Ehv;k þ 10lg Qhv;k

(219)

where Q is the number of vehicles (traffic load); the subscripts lv and hv refer to light vehicle and heavy vehicle, respectively. After considering the pavement types, the average sound power level per meter of lane is given by

LAW=m;J ¼ 10lg

X

10

LAW=m;k þJ

k

10

$

hk hp

(220)

Table 33 Recalibrated CRTN models proposed by local researchers in Hong Kong (Leung and Mak, 2008 [211]). Reference

Author

Year

Equation

[210] [215] [216] [217]

CRTN Lam and Tam To et al. Tang et al.

1988 1998 2002 2004

LA,10,18h ¼ 10log10(Q) þ 33log10(V þ 40 þ 500/V) þ 10log10(1 þ 5p/V) þ 0.3G e 26.6 LA,10,18h ¼ 1.0534  LA,10,18h,CRTN  5.3347 LA,10,18h ¼ 10.6log10(Q) þ 7.1log10(pV) e 17log10(V) þ53 LA,10,18h ¼ 10log10(Q) þ 42.6log10(V þ 40 þ 500/V) þ 10log10(1 þ 5p/V) þ 0.46G e 49.2

Table 34 Traffic conditions in NMPB 2008 [224]. Conditions

Category

Vehicle type Speed Traffic flow type Road slope

Light < 3.5 t; heavy > 3.5 t <20 km/h set to 20 km/h; 20e120 km/h Fluid continuous; pulsed continuous; pulsed accelerating; pulsed decelerating Rising > 2% upward; falling > 2% downward; level  2%

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where J is the correction for pavement type, as presented in Table 35; hk is the length of the elementary period in hours and hp is the length of the reference period in hours. The sound power level of an elementary source i in a given octave band j (125e4000 Hz) is given by

LAWi ¼ LAW=m;J þ 20 þ 10 log10 ðli Þ þ Rj

(221)

where li is the length of the elementary source line (uniform section), and Rj is the spectral correction for each octave band [227], as listed in Table 36. For the noise propagation model, geometric divergence, atmospheric absorption (ISO 9613-1, 1993 [228]), ground absorption, diffraction, and reflection are considered, assuming that each elementary source is 0.5 m above the ground. The noise propagation is divided into two distinct types: meteorological conditions favorable to sound propagation, with subscript F; and neutral (homogeneous) meteorological conditions, with subscript H. The sound levels for the two conditions are given below.

LF ¼ LW  Adiv  Aatm  Agrd;F  Adif ;F LH ¼ LW  Adiv  Aatm  Agrd;H  Adif ;H

(222)

The sound level is often expressed as Lden, which is appropriate for noise mapping for environment assessment (Murphy and King, 2014 [118] [229]). 4.4.3. Highway (Dai and Lee, 2010) Dai and Lee model (2010) [230] is called highway prediction model (HPM), and is used to predict highway traffic levels on pavements of asphalt concrete (AC) and asphalt rubber concrete (ARC). This model was developed based on ray acoustics approach that is also applied in ISO 9613 (attenuation of sound during propagation outdoors, with accuracy up to ±3 dBA within a range of 1000 m) (ISO 9613-1, 1993 [228]; ISO 9613-2, 1996 [231]). The advantage of this model is that it is presented in the form of Microsoft® Office Excel® 2007, and the users are required to provide only the parameter inputs that are very easy to acquire, such as weather conditions, pavement types, pavement porosity and permeability, hourly traffic volume, vehicle nominal speed, and receiver's distance from the road. Three assumptions were applied in the model: (1) The road is assumed to be an infinite line source; (2) Free field condition is assumed for the source and the receiver; (3) There is no barrier between the source and the receiver. The sound pressure level Lft(DW) at a given distance is given by

Lft ðDWÞ ¼ LW þ DC  A

(223)

where the nomenclature is as presented in Table 37; the total attenuation is given by

A ¼ Adiv þ Aatm þ Agr þ Abar þ Amisc

(224)

4.4.3.1. Attenuation due to geometric divergence. The attenuation due to geometric divergence is calculated using the following equation.

 Adiv ¼ 10 log10

Q 4pd2

 (225)

Table 35 Correction for pavement type in NMPB 2008 [224]. Pavement

Noise level correction J [dB]

Porous surface Smooth asphalt (concrete or mastic) Cement concrete and corrugated asphalt Paving stones

1 for 0e60 km/h; 2 for 61e80 km/h; 3 for 81e130 km/h 0 þ2 þ3

Table 36 Values for spectral correction of each octave band [227]. j

1

2

3

4

5

6

Octave band center frequency [Hz] Rj

125 14.5

250 10.2

500 7.2

1000 3.9

2000 6.4

4000 11.4

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75

Table 37 Nomenclature for highway prediction model (HPM) (Dai and Lee, 2010 [230]). Symbol

Unit

Description

Lft(DW) LW DC A Adiv Aatm Agr Abar Amisc Q d

[dB] [dB] [dB] [dB] [dB] [dB] [dB] [dB] [dB] [m2] [m] [dB/m] [Hz] [K] [K] [Hz] [Hz] [kPa] [kPa] [] [] [kPa] [K] [dB] [dB] [dB] [] [] [] [] [dB] [vehicles/h] [%]

Sound pressure level at a receiver location for a given octave band center frequency f and temperature T Sound power level Directivity correction (set to 0 for free field condition) Octave-band attenuation due to multiple effects Attenuation due to geometric divergence Attenuation due to atmospheric absorption Attenuation due to ground absorption Attenuation due to barriers (set to 0 because of the assumption that there are no barriers) Attenuation due to miscellaneous effects Noise source directivity index (set to 2 for hemispherical noise source) Distance from source to receiver Atmospheric attenuation coefficient Central frequency for a specific octave band Ambient temperature Reference air temperature (293.15 K) Oxygen relaxation frequency Nitrogen relaxation frequency Atmospheric pressure Reference atmospheric pressure (101.325 kPa) Molar concentration of water vapor Ambient relative humidity Saturation vapor pressure Triple-point isotherm temperature (T0 þ 0.01 K) Ground attenuation in the source region Ground attenuation in the receiver region Ground attenuation in the middle region Ground absorption coefficient Pavement porosity Pavement permeability Material coefficients determined by measurements Attenuation due to traffic volume Traffic volume Percentage of trucks in the traffic volume

a f T T0 frO frN pa pr h hr psat T01 As Ar Am

ap fp kp A1, A2, A3 Atra Vtra Ptruck

4.4.3.2. Attenuation due to atmospheric absorption. The attenuation due to atmospheric absorption is calculated using the following equation.

ad 1000

Aatm ¼

(226)

where the atmospheric attenuation coefficient a depends on weather parameters (temperature and relative humidity), and is given by

"

a ¼ 8:686f

2

11

1:84  10



pa pr

1  0:5  2:5 #h  1  1 i 2239:1 33352 T T f2 f2 0:01275e T frO þ þ þ 0:1068e T frN þ T0 T0 frO frN (227)

where the oxygen relaxation frequency fro and nitrogen relaxation frequency frN are given below.

 frO ¼

pa pr

 frN ¼

  0:5  T 0:02 þ h 24 þ 4:04  104 h T0 0:391 þ h " 1 # 1 0

 0:5 4:17 B pa T B9 þ 280he @ T0 pr

(228)

3

T T0

1

C C A

(229)

The molar concentration of water vapor in the above equations is given by

 h ¼ hr

psat pr



pa pr

 (230)

where the saturation vapor pressure is calculated using the following equation. Please cite this article in press as: T. Li et al., Literature review of models on tire-pavement interaction noise, Journal of Sound and Vibration (2018), https://doi.org/10.1016/j.jsv.2018.01.026

76

T. Li et al. / Journal of Sound and Vibration xxx (2018) 1e89

 1:261 psat ¼ 10 pr

6:8346

T T01

þ4:6151

(231)

4.4.3.3. Attenuation due to ground absorption. The attenuation due to ground absorption is given by

Agr ¼ As þ Am þ Ar

(232)

The specific calculations vary for different central frequencies of the octave bands f. However, except at 31.5 Hz and 63 Hz, the ground absorption coefficient is important. It is related to pavement parameters and given by







ap ¼ A1 ln fp þ A2 ln kp  A3

(233)

4.4.3.4. Attenuation due to the barrier. Abar is set to 0, because it is assumed that there are no barriers between the source and the receiver. 4.4.3.5. Attenuation due to miscellaneous effects. The attenuation due to traffic volume is considered for two pavement types: asphalt concrete (AC) and asphalt rubber concrete (ARC). The equations are 2 2 Atra;AC ¼ 0:0001Vtra  0:0446Vtra  0:0026Ptruck þ 0:2284Ptruck þ 25:934

(234)

2 2 Atra;ARC ¼ 0:00009Vtra  0:0359Vtra  0:0011Ptruck þ 0:1132Ptruck þ 10:1336

(235)

4.4.3.6. Results. The model (HPM) was compared with the measured noise using statistical pass-by method (SPB), computeraided noise abatement (CadnaA) simulation software (CadnaA), and traffic noise model (TNM®) “LookUp” program developed by the Federal Highway Administration (FHWA), United States. The results are shown in Fig. 41 (Dai and Lee, 2010 [230]). Comparing HPM and SPB, larger errors (up to 4 dBA) occur at 7:00 a.m. and 8:00 p.m. It is probably due to drastic changes in ambient temperature, leading to atmospheric inversions (Sheadel et al., 2007 [232]). On an average, HPM is the most accurate model (having error within 1.65 dB to 1 dB) among the three prediction models. The TNM accounts for only densegraded asphalt concrete (DGAC, a type of AC), and is applicable only when the distance between the source and receiver is in the range of 10e300 m. The CadnaA always underpredicts the noise levels, which is possibly because it is based on “Road traffic noise pollution guidelines” (RLS-90, published by German Federal Ministry of Transportation), and not applicable in local areas (Canada).

Fig. 41. Comparisons between HPM and SPB, CadnaA, and TNM (source from Dai and Lee, 2010 [230], Fig. 1; reprinted with permission from Ms. Beth Darchi on behalf of ASME).

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No. Model name

Author

Year

Output

Frequency Mechanism range [Hz]

Driver parameter Tire parameter

Pavement parameter

Environment parameter

1

Truck contact

Fong

1998 [177] Transfer function

OBSP, 1/3 octave band

10e4000

TPIN

45 km/h

Passenger and truck

N/A

OBSP, pass-by, OASPL CB, LpA,max, 1/3 octave band SPL

22e5500

TPIN

Speed

Slick & normal

Beckenbauer and Kuijpers

Cross-correlation, regression 2001 [181] Multivariate regression analysis

Texture, contact pressure spectrum Texture index

2

DGAC texture

Domenichini et al. 1999 [18]

3

SPERoN

125e5000 TPIN

50-120 km/h

Slick, tire width

4

Enhanced SPERoN

Kuijpers and Blokland

2003 [183] Multivariate regression analysis

CB, LpA,max, 1/3 octave band SPL

125e5000 TPIN

50-120 km/h

5

HyRoNE

Klein and Hamet

2007 [187] Envelopment, regression

CPB, LpA,max, 1/3 octave band SPL

100e5000 TPIN

70, 90, 110 km/h

6

Road excitation

Rustighi et al.

2008 [192] FEM, BEM, 2D Gaussian

Vibration, SPL

10e400

7

Viscoelastic contact Dubois et al.

8 9

Ref.

Method

2011 [195] Macroscale, multiasperity Contact distribution Dubois et al. 2013 [198] Weighted linear regression 3D envelopment Klein and Cesbron 2016 [200] Envelopment, linear regression

Near field, 1/3 octave band SPL CPX, 1/3 octave band CPX, 1/3 octave band; CB, Lpmax

20-100 km/h Texture impact, radiation, interior noise 315e1000 Texture impact 90 km/h 315e1250 Impact

50-110 km/h

250e1250 Belt vibration

65-110 km/h

Shape factor, contact pressure spectrum Tread pattern, tire Shape factor, contact pressure width, rubber spectrum, hardness mechanical impedance Michelin Energy Enveloped XH1/E3A texture, absorption, airflow resistance Smooth Texture excitation Slick 186/57R15 Slick and Patterned Tread pattern

Texture, contact force spectrum Texture, contact force spectrum Texture

N/A N/A

20  C

20  C

N/A

N/A

T. Li et al. / Journal of Sound and Vibration xxx (2018) 1e89

N/A N/A

77

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Table 38 Summary of tire-pavement interface models on TPIN.

78

No.

Model Name

Author

Year

Ref.

Method

Output

Frequency range [Hz]

Mechanism

Driver parameter

Tire parameter

Pavement parameter

Environment parameter

1

Tire/road coupling Mechanism decomposition

Cao et al.

2008

[204]

N/A

SPL, narrowband

0e5000

Various

N/A

Tread pattern

Texture, absorption

N/A

Dare

2012

[13]

Nonlinear least squares

OBSI, 1/3 octave band

500e5000

Speed

SRTT

Texture levels, skewness, absorption, friction

N/A

Winroth et al. Mohamed and Wang Le Bot et al.

2013

[205]

Curve fitting

125e4000

50-120 km/h

Slick

Texture

N/A

2016

[206]

ICMM, SEA

N/A

N/A

[207]

0e4000

Slick 155/70R13

Point source

N/A

2017

[208]

Modal analysis BEM, FEM

Stationary, shaker 29 psi

205/65R15

2017

CPB, 1/3 octave band SPL SPL, energy, narrowband Mobility

Sidewall, tread band, tangential vibration Tire vibration, air pumping Cavity resonance Vibration

Lab, SPL

200e2000

Pipe resonance

Stationary

Tread pattern

Smooth

N/A

2

3 4 5 6

Air pumping separation Deterministic SEA Statistical vibration Network resonator

Wang and Duhamel

0e500

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Table 39 Summary of mechanism separation models on TPIN.

T. Li et al. / Journal of Sound and Vibration xxx (2018) 1e89

79

4.4.4. Air pumping (Wei et al., 2016) Wei et al. (2016) [233] developed a hybrid numerical-experimental model to predict the air pumping noise at any field point. This model is based on a hand carved tread pattern with two longitudinal grooves and one lateral groove. The groove is meshed into regular triangular or quadrilateral elements, which can be considered as small point acoustic sources. The volume velocity or source strength Q for each element i is given by

Qi ðf Þ ¼ Si $qðf Þ$hðfÞ

(236)

where f indicates frequency domain, S is the surface area of each element, and q is the sound source of unit area that is related to the vehicle speed; h(f) is a coefficient that is a function of f, which is the distance between the element and the leading edge/trailing edge of the contact patch. The velocity of the groove volume in the frequency domain is obtained by the summation of all the elements, as given below.

Qn ¼

X

Qi

(237)

i

where n is the index of sound sources in the groove. In this case, there are four sources: two (near the longitudinal grooves) at the leading edge and two at the trailing edge. The sound pressure spectrum P in the frequency domain is obtained as

Pm ¼

X

Zmn Qn

(238)

n

where m is the index of microphone position in the near field, Zmn is the transfer function between sound pressure Pm and source strength Qn. The reverse manipulation can yield Qn if Pm is experimentally measured and Zmn is numerically determined; this is reason for calling this a hybrid model. 4.5. Summary The summary of the tire-pavement interface models is presented in Table 38. Since the late 1990's, more efforts have been taken to determine the contact force/pressure distribution for TPIN modeling. The pavement texture was also converted to intermediate parameters such as enveloped profile that have high correlation with TPIN. For either near-field or far-field noise, all the models were able to predict the noise spectrum of a broad frequency range, and not just the overall noise level. Vehicle speed was typically included as an input parameter. However, the contact force/pressure spectrum was calculated based on pavement texture while assuming a slick tire tread, which might not be accurate for normal patterned tires. The summary of the mechanism separation models is presented in Table 39. With this type of modeling, it is possible to analyze each noise generation mechanism individually and identify each noise source; this has become popular since the late 2000's. Both structure-borne and airborne noises have been investigated. The output of the models was the noise spectrum in 1/3 octave band or in the narrowband. It is seen that pavement texture was more frequently included than the tire parameters, such as tread pattern. The summary of noise propagation models is presented in Table 40. These models have been used to predict the far-field noise level at a specific distance from the traffic. The input parameters typically included vehicle speed, vehicle type, and some pavement parameters. The environmental parameters influencing sound propagation were more important than in the other type of models and were normally included. 5. Other technical categories The readers can follow the paper structure based on the modeling approach categories (i.e., deterministic, statistical, hybrid, and corresponding sub-categories) to identify the models of interest. More importantly, the tables in each summary section for the three categories follow the same format, which can be considered as an index that the readers can refer to and generate other technical categories. For example, if the readers are interested in TPIN models that reveal the tread pattern effect, “tread pattern” or related key words can be searched for in the column of “Tire Parameter” in the summary tables. Then, the TPIN models on tread pattern effect can be extracted, as presented in Table 41. It can be seen that there are seventeen models related to tread pattern effect. Ten models fall under the category of deterministic models, two under statistical models, and five under hybrid models, which indicates that it is not easy to predict the tread pattern effect on tire noise using statistical methods, mainly because it is not easy to quantify the tread pattern, and a large amount of data for different tread patterns is normally not available for statistical analysis. However, the two statistical methods are both neural computation models, suggesting that neural network might be a feasible approach to correlating the tread pattern with the tire noise. The most popular modeling approach for airborne noise, e.g., air pumping, is CFD (five models) since 21st century. To investigate the tread pattern effect, most of the models reach a very high frequency up to 2000 Hz, because the tread pattern noise normally occurs at higher frequencies. The air pumping and tread impact are the most extensively investigated mechanisms related to the tread pattern. Please cite this article in press as: T. Li et al., Literature review of models on tire-pavement interaction noise, Journal of Sound and Vibration (2018), https://doi.org/10.1016/j.jsv.2018.01.026

80

No.

Model name

Author

Year

Ref.

Method

Output

Frequency range [Hz]

Mechanism

Driver parameter

Tire parameter

Pavement parameter

Environment parameter

1

CRTN (UK) NMPB (France) Highway

DOT

1988

[210]

Line source

Far field, L10

N/A

Traffic

N/A

1996

[221]

Traffic

2010

[230]

Far field, octave band, power, Lden Pass-by, octave band SPL

125e4000

Dai and Lee

Segmentation of line source Ray acoustics

Type, MPD, slope Slope

N/A

N/A

N/A

Traffic

Wei et al.

2016

[233]

Transfer function

Drum, SPL, narrowband

0e3500

Air pumping, radiation

Speed, vehicle type Speed, vehicle type, acceleration Speed, vehicle type, traffic volume 461 kg, 180 kPa, 80 km/h

2 3

4

Air pumping

N/A N/A

205/55R16, hand cut pattern

Type, porosity, permeability ISO 10844

Atmospheric absorption Temperature, pressure

N/A

T. Li et al. / Journal of Sound and Vibration xxx (2018) 1e89

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Table 40 Summary of noise propagation models on TPIN.

No. Model name Author

Year Ref.

Method

Output

1

Monopole

Hayden

1971 [29]

Monopole

2

Unified

1981 [30]

Thin shell

3

Plotkin and Stusnick Heckl

Various mechanisms Tire vibration Bremner et al.

1986 [44]

Various

Sound pressure, N/A pure tone 0e8000 Near field, normalized radiation LDV, vibration, SPL 0e2000

Air pumping CFD 10 Pipe resonance

Far field (7.5 m), 1/3 SEA, wavenumber octave band SPL decomposition De Roo and 2000 [77] FEA, BEM Near field, 1/3 Gerretsen octave band SPL; pass-by, SPL Gagen 1999 [95] CFD Near field, sound pressure/energy Far field, 1/3 octave Kim et al. 2006 [99] CFD, band SPL Kirchhoff integral Chen et al. 2014 [11] CFD Near field, 1/3 octave band SPL 2016 [100] CFD Near/far field, Gautam drum, narrowband and SPL Chandy Fabrizi 2016 [104] CFD, LES, Pass-by/in-groove, aeroacoustics narrowband SPL

11 F-GA

Li et al.

2009 [162] F-GA

Octave band SPL

12 ANN

Che et al.

13 Enhanced SPERoN

Kuijpers and Blokland

4

5

TRIAS

6

Shock wave Hybrid CFD

7

8

CFD

9

14 3D Klein envelopment and Cesbron 15 Tire/road Cao et al. coupling 16 Network Wang Resonator and Duhamel 17 Air pumping Wei et al.

1997 [46]

Frequency range [Hz]

Mechanism

Driver parameter Tire parameter

Pavement Environment Category parameter parameter

Air pumping

Speed

Groove geometry

N/A

N/A

Impact, air pumping

Speed, tire load

Various

Texture

N/A

Various

Rotating, loaded Tread pattern

Roughness N/A

100e4000

Impact

Speed, 40 psi

Tread pattern, materials

Texture

N/A

N/A

Impact, air pumping

N/A

Various, TYDAS

Various, RODAS

N/A

Deterministic FEM & BEM

N/A, time domain 250e8000

Air pumping

Speed

N/A

Deterministic CFD

Air pumping, 100 km/h pipe resonance

Groove angle, N/A geometry Ventilation groove, N/A 165/70SR13

N/A

Deterministic CFD

Airborne

N/A

Tread pattern

N/A

N/A

Deterministic CFD

40 km/h

215/60R16, transverse slots

N/A

N/A

Deterministic CFD

Drum with Safety Walk N/A

N/A

Deterministic CFD

N/A

Statistical

0e2000

1000e10000 Airborne, air pumping 1500e2000

Pipe resonance 80 km/h

205/55R16, longitudinal grooves

125e4000

TPIN

N/A

Tread pattern

2012 [164] GA-BP

DR, octave band SPL 125e4000

TPIN

Size, tread pattern

2003 [183] Multivariate regression analysis

CB, LpA,max, 1/3 octave band SPL

125e5000

TPIN

Speed, tire load, inflation 50-120 km/h

Tread pattern, tire width, rubber hardness

2016 [200] Envelopment, CPX, 1/3 octave band; CB, Lpmax linear regression 2008 [204] N/A SPL, narrowband

250e1250

Belt vibration

65-110 km/h

Tread pattern

0e5000

Various

N/A

Tread pattern

2017 [208] BEM, FEM

Lab, SPL

200e2000

Pipe resonance Stationary

2016 [233] Transfer function

Drum, SPL, narrowband

0e3500

Air pumping, radiation

Tread pattern

461 kg, 205/55R16, hand 180 kPa, 80 km/h cut pattern

Standard N/A surface 20  C Shape factor, contact pressure spectrum, mechanical impedance Texture N/A

Sub-category

Deterministic Conventional physics Deterministic Conventional physics Deterministic Conventional physics Deterministic Conventional physics

Statistical Hybrid

Fuzzy curve fitting Fuzzy curve fitting Tire-pavement interface

Hybrid

Tire-pavement interface

Hybrid

Mechanism separation Mechanism separation

Texture, N/A absorption Smooth N/A

Hybrid

ISO 10844 N/A

Hybrid

T. Li et al. / Journal of Sound and Vibration xxx (2018) 1e89

Noise propagation 81

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Table 41 Summary of TPIN models on tread pattern effect.

No. Model name

Author

Year Ref.

Method

Output

Frequency Mechanism range [Hz]

Driver parameter

Tire parameter

Pavement parameter

Environment Category parameter

1

Unified

1981 [30]

Thin shell

Speed, tire load

Various

Texture

N/A

Deterministic Conventional physics

2D ring

1986 [39]

Tire load

Various

Texture

N/A

3

Various mechanisms Tire vibration

Heckl

1986 [44]

Modal expansion Various

Near field, normalized radiation Near field, SPL

0e8000

2

Plotkin and Stusnick Kung et al.

Tread pattern Tread pattern, materials

N/A

1997 [46]

Rotating, loaded Speed, 40 psi

Roughness

Bremner et al.

LDV, vibration, 0e2000 SPL 100e4000 Impact Far field SEA, (7.5 m), 1/3 wavenumber decomposition octave band SPL Image source Impedance, 100e5000 Pavement absorption absorption

Texture

N/A

Deterministic Conventional physics Deterministic Conventional physics Deterministic Conventional physics

N/A

N/A

N/A

Deterministic Conventional physics

Various, slick 205/60R15 Various, TYDAS

Airflow resistivity, porosity, tortuosity Texture

N/A

Deterministic FEM & BEM

Various, RODAS

N/A

Deterministic FEM & BEM

4

0e400

Impact, air pumping Structureborne Various

Sub-category

5

Porous pavement

rengier Be et al.

1997 [51]

6

3D two-plate

Larsson and Kropp

1992 [25]

FE

Pass-by, SPL

0e2000

Structureborne

Tire load

7

TRIAS

De Roo and Gerretsen

2000 [77]

FEA, BEM

N/A

Impact, air pumping

N/A

8

IFEM

Biermann et al.

2007 [23]

IFEM, AstleyLeis elements

0e1400

Impact

Speed

Slick

Texture

N/A

Deterministic FEM & BEM

9

FEM & BEM

Brinkmeier et al.

2008 [17]

FEA, BEM, ALE

100e850

Structureborne

40-120 km/h

N/A

Texture spectrum

N/A

Deterministic FEM & BEM

10 Road input

Kido et al.

2011 [85]

FE

Near field, 1/3 octave band SPL; pass-by, SPL Far field (1 m), 1/3 octave band SPL Pass-by, 1/3 octave band SPL Axle vibration

Texture

N/A

Deterministic FEM & BEM

Rasmussen

2009 [132]

SRTT

Texture, skewness

N/A

Statistical

Traditional regression

12 Asperity

Fujikawa et al.

2009 [108]

Linear regression, bridging filter Linear regression

Speed, tire load Buick Century, 97 km/h

Patch area

11 Concrete texture

Texture impact OBSI, 1/3 octave 500e5000 Texture 630e1000 impact BTT, OASPL, octave

500e4000 TPIN

80 km/h, 3.2 kN, 185/65R15, 170 kPa rib pattern

N/A

Statistical

Traditional regression

500e5000 TPIN

N/A

SRTT

N/A

Statistical

Traditional regression

SRTT

Wind < Statistical 4 km/h Pavement Statistical temperature N/A Statistical

Traditional regression PCA PCA

N/A

PCA

0e150

13 Pavement core

2011 [138]

Linear regression

OBSI, 1/3 octave, OASIL

14

2014 [141]

Linear regression OLS, PCR

OBSI, OASIL, 500e4000 TPIN pass-by, OASPL OBSI, OASIL N/A TPIN

88 km/h 97 km/h 97 km/h

SRTT

Asperity spacing, asperity height unevenness, MPD Airflow resistance, MPD FN, MPD, AG4 Various

2014 [145,146] PCA Energy

BTT, energy

Speed

Same tire

MTD

2015 [151]

CPX, 1/3 octave 315e3150 TPIN band SPL, OASPL

80 km/h

N/A

Texture, MPD, absorption

15 16 17

Reyes and Harvey FDOT Wayson et al. Ongel Asphalt pavement et al. MTD Zhang et al. Thin surfacing Li et al.

2008 [142]

PCR

0e2000

TPIN

Statistical

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82

Table 42 Summary of TPIN models on pavement texture effect.

SRTT

Type, ESAL, Age, IRI, SR

30  C

OBSI, 1/3 octave 500e5000 TPIN band, OASIL

Traffic volume

N/A

Age, voids, MPD, thickness

Precipitation Statistical

Transfer function

OBSP, 1/3 octave band

10e4000

TPIN

45 km/h

Passenger and truck

N/A

Hybrid

Crosscorrelation, regression Multivariate regression analysis

OBSP, pass-by, OASPL

22e5500

TPIN

Speed

Slick & normal

Texture, contact pressure spectrum Texture index

N/A

Hybrid

CB, LpA,max, 1/3 125e5000 TPIN octave band SPL

50-120 km/h

Slick, tire width

N/A

Hybrid

2003 [183]

Multivariate regression analysis

CB, LpA,max, 1/3 125e5000 TPIN octave band SPL

50-120 km/h

20  C

Hybrid

Tirepavement interface

Klein and Hamet

2007 [187]

Envelopment, regression

CPB, LpA,max, 1/3 100e5000 TPIN octave band SPL

70, 90, 110 km/h

20  C

Hybrid

Tirepavement interface

25 Road excitation

Rustighi et al.

2008 [192]

FEM, BEM, 2D Gaussian

Vibration, SPL

20-100 km/h

N/A

Hybrid

Tirepavement interface

26 Viscoelastic contact

Dubois et al.

2011 [195]

Macroscale, multi-asperity

Near field, 1/3 octave band SPL

N/A

Hybrid

Tirepavement interface

27 Contact distribution

Dubois et al.

2013 [198]

N/A

Hybrid

28 3D envelopment

Klein and Cesbron

2016 [200]

N/A

Hybrid

29 Tire/road Cao et al. coupling 30 Mechanism Dare decomposition

2008 [204]

Weighted linear regression Envelopment, linear regression N/A

N/A

Hybrid

N/A

Hybrid

Tirepavement interface Tirepavement interface Mechanism separation Mechanism separation

N/A

Hybrid

N/A

Hybrid

Khazanovich 2008 [5] and Izevbekhai

LevenbergMarquardt

OBSI, OASIL

19 Bayesian

Yu and Lu

2013 [173]

Bayesian, MCMC, MI

20 Truck contact

Fong

1998 [177]

21 DGAC texture

Domenichini et al.

1999 [18]

22 SPERoN

Beckenbauer and Kuijpers

2001 [181]

23 Enhanced SPERoN

Kuijpers and Blokland

24 HyRoNE

31 Air pumping separation 32 CRTN (UK)

Winroth et al. DOT

2012 [13]

2013 [205] 1988 [210]

N/A

TPIN

10e400

Texture impact, radiation, interior 315e1000 Texture impact

Shape factor, contact pressure spectrum Shape factor, Tread pattern, tire contact pressure width, rubber spectrum, mechanical hardness impedance Enveloped Michelin texture, Energy absorption, XH1/E3A airflow resistance Smooth Texture excitation

90 km/h

Slick 186/57R15

CPX, 1/3 octave 315e1250 Impact band

50-110 km/h

Slick and Patterned

CPX, 1/3 octave 250e1250 Belt band; CB, Lpmax vibration

65-110 km/h

Tread pattern

N/A Speed

Tread pattern SRTT

50-120 km/h

Slick

Speed, vehicle type

N/A

SPL, 0e5000 Various narrowband Nonlinear least OBSI, 1/3 octave 500e5000 Sidewall, squares band tread band, tangential vibration Curve fitting CPB, 1/3 octave 125e4000 Tire vibration, band SPL air pumping Line source Far field, L10 N/A Traffic

Texture, contact force spectrum Texture, contact force spectrum Texture

Texture, absorption Texture levels, skewness, absorption, friction Texture Type, MPD, slope

Statistical

Fuzzy curve fitting Fuzzy curve fitting Tirepavement interface Tirepavement interface Tirepavement interface T. Li et al. / Journal of Sound and Vibration xxx (2018) 1e89

Mechanism separation Noise propagation 83

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97 km/h

18 Pavement aging

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If the readers are interested in a specific model in Table 41, the corresponding section that covers the details of the model can be easily found in the main body of the paper. As another example, the TPIN models on pavement texture effect can be extracted based on the keywords in the column of “Pavement parameter” in the summary tables, as presented in Table 42. It can be seen that there are thirty-two models covering the effect of pavement texture on tire noise: ten under deterministic models, nine under statistical models, and thirteen under hybrid models. It may be noted that the most popular modeling approach to investigate the pavement texture effect is tire-pavement interface (nine models), while there is no model using CFD, which may be because it is not easy for CFD to handle the randomness in the pavement texture. The output of these models is the overall noise level or 1/3 octave band SPL, and only one model (Tire/road coupling, Cao et al., 2008 [204]) is able to predict the noise in narrowband. The structureborne noise, e.g., noise due to pavement texture impact, is the main mechanism investigated. However, because the pavement texture is also related to many other mechanisms (such as air pumping, pavement absorption, and tire vibration), many models considered the tire noise due to pavement texture as a whole, not just the impact noise excited by the pavement texture. The same procedure to cross-reference the models from the summary tables discussed above can be applied to other topics of interest, such as speed effect, temperature effect, noise generation mechanism investigated, modeling technique, and output of the model. 6. Conclusions In this work, the models on the tire-pavement interaction noise (TPIN) were divided into three categories: deterministic models, statistical models, and hybrid models. The deterministic TPIN models include conventional physics models, finite element and boundary element models, and computational fluid dynamics models. They generally give insight into the noise generation and propagation mechanisms. The influence of each specific parameter on TPIN can be numerically analyzed. The conventional physics models typically use dynamic laws to calculate the tire vibration modes; the finite element and boundary element models are capable of obtaining the structure-borne tire noise under arbitrary excitations; the computational fluid dynamics models are intended for the airborne noise analysis. However, a large number of tire parameters are unavoidably needed for deterministic modeling, which is very difficult in practice. In addition, most of the models were limited to the frequency range of below 500 Hz, and few models went through experimental validation. Therefore, the accuracy of these models is in question. The statistical TPIN models include traditional regression models, principal component analysis models, and fuzzy curve fitting models. They generally correlate the noise levels with the influencing parameters using only experimental data without the knowledge of underlying mechanisms. They typically have better accuracy than the deterministic models. The traditional regression analysis, including linear and nonlinear regression, is widely used for models with easy form of equations and having not many variables; the principal component analysis can be used to reduce the number of variables; fuzzy logic has the advantage that no specific form of equations is needed for modeling. However, large amounts of data are needed for statistical modeling, especially when the number of variables is large; this results in high labor cost. In addition, most of the current models included only variables related to the pavement; few models investigated the influence of tire parameters except for some fuzzy curve fitting models. It was also unlikely or impossible to gain insight into the noise generation mechanisms or separate the different mechanisms. The hybrid TPIN models include tire-pavement interface models, mechanism separation models, and noise propagation models. They combine the advantage of the deterministic model and statistical model. The hybrid model sheds light on the noise generation mechanism as a deterministic model, and also has good accuracy as a statistical model. The tire-pavement interface model gives insight into the interaction and excitation between the tire and pavement; the mechanism separation model investigates the noise source individually; the noise propagation model is capable of predicting the TPIN at arbitrary locations. However, the hybrid model tends to be more complex than the deterministic or statistical model. The careful selection of intermediate parameters is often the key to the success of the subsequent statistical correlation procedure. The deterministic models were dominant before the 2000s, especially for conventional physics models. With the development of computing technology, FEM and CFD models are being used frequently thereafter. The use of these models has improved the understanding of TPIN generation mechanisms. Further, partly owing to the computing technology and data techniques, the statistical models have started playing an important role after the 2000s. The application of statistical algorithms to TPIN modeling been has found to be successful, and has led to fairly good prediction accuracy, which is very encouraging. In the 2010s, the hybrid models have become increasingly acceptable among researchers. The authors expect that the hybrid modeling approach will be the future direction in this subject. In addition, the on-board noise measurement techniques are more readily accepted as replacements for pass-by noise measurement techniques, because they are able to obtain noise data for a longer duration and with little environmental influence, which is favorable for the data analysis. As the technology of transducer, data acquisition, and computation advances, narrowband spectrum will be used more frequently for digital signal processing instead of octave, 1/3 octave, or 1/12 octave band; as a result, more precise spectral contents can be analyzed and mechanisms can be better identified. Acknowledgments Please cite this article in press as: T. Li et al., Literature review of models on tire-pavement interaction noise, Journal of Sound and Vibration (2018), https://doi.org/10.1016/j.jsv.2018.01.026

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This study (Project Code: MODL-2015-B3-8) has been partially supported by the Center for Tire Research (CenTiRe), an NSF-I/UCRC (Industry/University Cooperative Research Centers) program led by Virginia Tech. The authors hereby wish to thank the project mentors and the members of the industrial advisory board (IAB) of CenTiRe for their kind support and guidance.

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