Modelling and mitigation of wheel squeal noise amplitude

Journal of Sound and Vibration 413 (2018) 144e158

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Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi

Modelling and mitigation of wheel squeal noise amplitude Paul A. Meehan a, *, Xiaogang Liu b a b

University of Queensland, Brisbane, Australia Wuhan University of Technology, Wuhan, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 30 January 2017 Received in revised form 26 August 2017 Accepted 19 October 2017

The prediction of vibration amplitude and sound pressure level of wheel squeal noise is investigated using a concise mathematical model that is verified with measurements from both a rolling contact two disk test rig and a field case study. The model is used to perform an energy-based analysis to determine a closed form solution to the steady state limit cycle amplitude of creep and vibration oscillations during squealing. The analytical solution compares well with a numerical solution using an experimentally tuned creep curve with full nonlinear shape. The predicted squeal sound level trend also compares well with that recorded at various crabbing velocities (proportional to angle of attack) for the test rig at different rolling speeds. In addition, further verification is performed against many field recordings of wheel squeal on a sharp curve of 300 m. A comparison with a simplified modified result from Rudd [1] is also provided and highlights the accuracy and advantages of the present efficient model. The analytical solution provides insight into why the sound pressure level of squeal noise increases with crabbing velocity (or angle of attack) as well as how the amplitude is affected by the critical squeal parameters including a detailed investigation of modal damping. Finally, the efficient model is used to perform a parametric investigation into means of achieving a 6 dB decrease in squeal noise. The results highlight the primary importance of crabbing velocity (and angle of attack) as well as the creep curve parameters that may be controlled using third body control (ie friction modifiers). The results concur with experimental and field observations and provide important theoretical insight into the useful mechanisms of mitigating wheel squeal and quantifying their relative merits. © 2017 Elsevier Ltd. All rights reserved.

Keywords: Wheel squeal Vibration amplitude Predictive modelling Limit cycle analysis

1. Introduction Wheel squeal is a high-pitched tonal noise that can occur as a train negotiates a curve (corner) of a railway line. It often occurs in the frequency range where our ears are the most sensitive, and is therefore very annoying for receivers near the track. This phenomenon has plagued the railway industry for many years and continues to rise in importance as railway usage increases and subjective human noise tolerance decreases. For instance, wheel squeal is a major impact from freight rail operations on tight curves in Australia, particularly in the metropolitan areas. Although much research insight has been obtained into the mechanisms of squeal over the past decade the occurrences and amplitude of wheel squeal still appear unpredictable in the field as it appears to be dependent on a wide range of vehicle and track parameters. Also the squeal

* Corresponding author. E-mail address: [email protected] (P.A. Meehan). https://doi.org/10.1016/j.jsv.2017.10.032 0022-460X/© 2017 Elsevier Ltd. All rights reserved.

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amplitude is determined by nonlinear limit cycle oscillations that have remained difficult to model except via complex simulation. Much modelling of wheel squeal has been performed particularly following the renowned works of Rudd [1] and review by Remington [2] and Thompson et al. [3] in which the fundamental mechanism due to lateral creepage was consolidated. Curve squeal is believed to originate from the unstable vibratory response of a railway wheel when subject to large creep forces whilst negotiating corners. The conventional mechanism from the literature is that the unstable excitation of the squealing wheel originates from a lateral ‘stick-slip’ mechanism in the contact region analogous to the bowing of a violin string. In particular, when a bogie negotiates a curve of a track, there is a misalignment between the rolling velocity and the wheel velocity, namely angle of attack, leading to a crabbing velocity, i.e., lateral sliding velocity, of the wheel across the top of rail as shown in Fig. 1. Referring to Fig. 1, the squeal mechanism is analogous to playing a violin and depends on the behaviour of lateral creep force/traction and lateral creepage (zangle of attack) conditions during the excitation of a railway wheel [4,5]. The friction coefficient and shape and slope of the traction/creepage curve is affected by the so-called third body of the contact; an interfacial layer consisting of any lubricants, contaminants and material generated as a result of the contact interaction [6]. If the crabbing velocity (or angle of attack) is large enough its oscillations will occur in the full sliding region c). The negative slope in this region can be shown to be associated with negative damping of creep oscillations and hence squeal instability. This leads to self-excited “stick-slip” oscillations, which in turn excite wheel (or violin string) vibrations and radiated sound. It is noted that, conversely, some recent research contends that a modal coupling phenomena between the normal and tangential dynamics may cause the instability eg. Ref. [7]. Pure tone components of squeal, are generally related to wheel natural frequencies that correspond to the out-of-plane wheel bending (or axial) modes. Much research on the modelling of squeal has been performed in the past with differences in modelling details of wheel/ rail mechanical impedances (analytical [8e12], FEM [4,13,14]), vertical dynamics [4,14], contact forces and wheel sound radiation [4,13,14]. Some have also included wheel/rail roughness or wheel rotation effects [11,12]. Recently, a transient analysis of the lateral creepage of the wheel was performed to account for nonlinearities of friction forces and resultant excited wheel modes appeared to match field observations better [15]. Notably, a time domain model was presented by Heckl and Abrahams [11], which focused on the squeal noise generated by a flat round disc excited at one point along the edge by a dry-friction force dependent on the disc velocity. This paper concluded that curve squeal is an unstable wheel oscillation that grows to a limit cycle oscillation, whose velocity amplitude is equal or very close to the crabbing velocity. Furthermore, the simulation results of Chiello et al. [16] also showed that the vibration velocity stabilises below the lateral sliding velocity. Rudd [1] developed an approximation for squeal noise amplitude assuming particular simplified (exponential) creep and cornering mechanics that was limited to lower lateral sliding velocities (or angles of attack). For higher angles of attack, Rudd also stated the vibration velocity approaches the lateral sliding velocity (ie. crabbing velocity). The present authors investigated this further in Refs. [17,23] using a numerical power balance analysis, however, an analytical prediction and explanation was not achieved. Much recent research has also been focused upon experimental verification of model predicted conditions under which squeal occurs and the effect of friction modifiers [18] on the phenomena. Recent predictive modelling includes that of [4,19] which include detailed representation of the dynamic behaviour of the wheel and rail and creepage in the saturated region. Twin disk and bogie testrigs have been utilised for verification under controlled environments [20]. Experimental results reported on the rolling contact force conditions during squeal include those by de Beer et al. [4], Monk-Steel et al. [19] and Koch et al. [21]. In Monk-Steel et al. [19] the inclusion of longitudinal creep was shown to reduce the lateral creep force and thereby change the slope of the friction curve. This leads to a lower incidence of squeal in the presence of longitudinal creep, and an increase in the threshold of lateral creepage necessary for squeal. In Koch et al. [21], measurements were carried out on a 1/4 scale test rig including a mono-block wheelset, and tests of anti-squealing solutions. A relation between noise level, rolling speed and angle of attack was confirmed experimentally and the average friction coefficient as a function of lateral creep was measured/inferred in dry conditions and with water. In Ref. [20] novel instrumentation directly on twin disc wheels close to the contact patch was used to obtain more direct measurements of lateral force to provide some verification of

a)

Fig. 1. Lateral creepage characteristic of railway wheel-rail contact. a) region of slip/adhesion, b) critical point at which full sliding first occurs and c) negative slope region of increased sliding causing negative damping of creep oscillations.

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existing predictive models although the presence of a third body in the contact appears to affect the reliability of the testrig results. In Ref. [18] friction modifiers were shown to cause substantial (~12 dB) noise reductions associated with top-of-rail squeal and flanging noise at a range of European mass transit sites. Despite these considerable efforts, there has been uncertainty in fully understanding, predicting and verifying trends of how squeal noise varies with important parameters such as crabbing velocity and angle of attack. In particular, models have in general involved too much complexity to be able to efficiently predict squeal amplitudes and perform detailed theoretical and field investigations of the effects of critical parameters on squeal sound levels. The present research investigates the efficient prediction of vibration amplitude and sound pressure level of wheel squeal noise using a concise mathematical model that is verified with results from a rolling contact two disk test rig as well as field site measurements. The major contributions include: 1. Theoretical prediction of the limit cycle amplitude of squeal vibration and noise as a function of critical squeal parameters. 2. Verification of squeal noise amplitude trends with experimental and field measurements as well as comparison with simplified modified predictions from Rudd [1]. 3. Theoretical insight into why the sound pressure level of squeal noise is strongly dependent upon and increases with crabbing velocity as well as how the amplitude is affected by other parameters. 4. Identification and quantification of the required changes in critical parameters to achieve a substantial reduction in wheel squeal noise including a detailed investigation of the effect of modal damping. It is noted that this paper is focused on wheel squeal amplitude prediction and reduction in contrast to many previous papers focused on wheel squeal occurrence (ie the critical conditions at onset) only. This paper will first describe the testrig, field measurement and mathematical methodology used for squeal investigation. Subsequently, the limit cycle analysis of squeal is provided to obtain a closed form solution to the squeal vibration and noise amplitudes. These are then compared to numerical, experimental and field measurement trends. Finally, the efficient theoretical model is used to perform a critical parameter sensitivity analysis to identify means by which the squeal noise amplitude can be reduced before conclusions are made. 2. Methodologies The experimental results presented in this paper have been obtained previously in Ref. [24] using a rolling contact two disc test rig developed for the investigation of squeal noise (described in the following for convenience in section 2.1). Details of a field investigation of squeal are then described in section 2.2. A theoretical model in the time domain (introduced in Ref. [24]) is redescribed here for convenience in section 2.3. Subsequently the analytical methodology for determining the amplitude of squeal oscillation is detailed in section 2.4. The parameters used for numerical and analytical simulation are also derived from the characteristics of the test rig described. 2.1. Experimental methods A rolling contact two disc test rig is used to investigate the effect of crabbing velocity on squeal noise as described in Ref. [24] as demonstrated in Fig. 2. The lateral force between the upper and lower wheel can be measured with strain gauge bridges as marked in Fig. 2(b) and this method is introduced in detail in Ref. [22]. Parameters of this test rig are listed in Table 1. The angle of attack between the upper and lower wheel were adjusted and measured using the method introduced in Ref. [17]. The sound pressure levels of the test rig squealing were recorded using a microphone placed adjacent to (5 cm away

Fig. 2. Rolling contact two disc test rig used for the investigation of squeal noise (a) front view of the test rig, (b) the FEM model of the test rig structure showing strain gauges for load measurement [22].

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Table 1 Parameters of the test rig and simulation. Description

Value

Radii of longitudinal and tangential curvature for the lower wheel (R1, R1t) Thickness of the lower wheel (rim, web) Density (r) Inner radius of lower wheel (R1’) Young's modulus of upper and lower wheel (E) Radii of longitudinal and tangential curvature for the upper wheel (R2, R2t) Thickness of the upper wheel Poisson's ratio (n) Angle of attack range Creep coefficient (C22) Normal loading (W) Creep curve parameters Static friction coefficient (ms) Critical creep (zc) Creep curve slip region slope (k1) e analytical model Creep curve full sliding region slope (k2) e analytical model Contact parameter (patch size/friction coef) (k3) e simulation model Squeal vibration parameters Modal mass (m) Modal damping (cd) Modal stiffness (k) Squeal nominal conditions Crabbing velocity (Vc) Rolling velocity (Vo)

0.213 m, 0.300 m 0.026 m, 0.015 m 7800 kg/m3 0.0325 m 175 GPa 0.085 m, 0.040 m 0.080 m 0.28 0e26 mrad 3.14 1000 N 0.35 0.007 0.27
0.02 2.733E5 N 3.1 kg 42 Ns/m 1.6E8 N/m 0.39 m/s 17.8 m/s

from) the lower disk and 80 cm above the ground as displayed in Fig. 2. A conditioning amplifier, an analogue digital converter (ADC) and Labview's Signal Express 3.0 by National Instruments and Matlab were used to process the signal with the reference pressure set at 20 mPa RMS. The sound was recorded each time for 2 s, twice for each crabbing velocity increment at a sample rate of 8000 Hz. The vibration characteristics of the test rig were investigated with modal tests conducted with a hard tip impact hammer and analysed with the finite element method. The vibration characteristics of the lower wheel acquired from finite element analysis and modal tests correlate well with the results of sound recordings. More details are provided in Ref. [17]. 2.2. Field measurements In order to verify the efficient squeal model, noise and angle of attack (the angle of a wheelset relative to the rail) data were obtained from a condition monitoring installation on an Australian network. The system is located near the mid-section of a curve of 300 m radius on a main line carrying both freight and passenger trains. The data from the wayside system includes noise level, AoA, lateral position, and speed for each passing wheel. Both positive and negative angle of attacks were recorded where negative values indicate the wheel tends to attack the high rail, however in this paper the absolute value of angle of attack was considered (although similar results were found for both positive and negative cases alone). Wheel squeal was observed to occur at a wide range of noise levels, from the level equivalent to the background rolling noise, to well over 115 dBA when it is measured 1.2 m from the rail. Other noise sources, such as flanging noise, locomotive noise, or rolling noise, were present but rarely exceeded this level and therefore were not specifically excluded. A simple algorithm was developed and tested manually to identify squeal and flanging noise from other noise sources based on the frequency spectrum as detailed in Ref. [29]. Basically squeal is identified as the pure tonal, high frequency and high level noise, while flanging noise is identified as a broad-band high-frequency noise. By using this algorithm, noise events of 120 dBA or higher, measured at 1.2 m, were found to be exclusively squeal noise. Passenger trains do not generate severe squeal at this site, and therefore were excluded in the analysis. In order to compare with the analytical prediction, the sound pressure level exceedance for 10% of the passing wheels was used for the field measurement (ie L10). This measure is commonly used for traffic noise as it provides an indication of the upper limit and has been found to correlate well with the disturbance people feel. It is noted that although the sound pressure level exceedance for 10% of the passing wheels (ie L10) would typically exclude non-squeal noise, some extreme occurrences could have affected the field data analysis at lower levels and are discussed subsequently. A dataset, including more than 30,000 wheel passes over a four month period from 03/2013 to 07/2013, was analysed including 1517 events over 100dBA. Both rails of the monitoring site were not lubricated during the monitoring period. 2.3. Theoretical modelling The model used for the present analytical investigation has already been described in Refs. [23,24] and is reintroduced here for convenience. Wheel squeal may be concisely modelled based on the conceptual diagram of Fig. 3. An important

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Fig. 3. (a) 3D model of a squealing wheel rolling on rail top, (b) SDOF friction self-excited oscillation in lateral direction.

parameter, the crabbing velocity arises due to the misalignment between the wheel velocity and rolling velocity as shown in Fig. 3(a). The lateral force Q arises at the contact point due to the crabbing velocity according to the rolling frictional behaviour. In the lateral direction, the dominant vibration of the wheel is analogous to the friction self-excited oscillation of mass connected to a spring and damping on a rolling belt as demonstrated in Fig. 3(b). The wheel squeal vibration mode can be described by an effective spring-mass-damper system that can be expressed as,

€ þ cd yðtÞ _ þ kyðtÞ ¼ Q ðzÞ; myðtÞ

(1)

where m is the modal mass, cd is the modal damping coefficient, k is the modal stiffness that also account intrinsically for the modal participation from the lateral force Q. The modal parameters of the dominant mode were curve fitted from a receptance spectrum of a modal test as detailed in Ref. [24] and are listed in Table 1. The modal parameters could also be determined via finite element analysis. The lateral force Q is dependent on the normal loading, W, according to,

Q ¼
mðzÞW

(2)

where m(z) is the creepage dependent lateral adhesion ratio in rolling contact. Many models exist for this, however, de Beer et al. [4] developed and tested a lateral creep model for squeal that accounts for both positive slip and negative sloped, full sliding behaviour that can be expressed as,

 o 8  1 02 1 03 n 0 > 1
0:5e
0:138=jzV0 j
0:5e
6:9=jzV0 j < ms z
z þ z 3 27 mðzÞ ¼ n o > : ms 1
0:5e
0:138=jzV0 j
0:5e
6:9=jzV0 j

0

for z  3 0

;

(3)

for z > 3

where ms is the stationary friction coefficient. The lateral creepage z is the ratio of the lateral relative velocity between the wheel and rail divided by rolling speed V0,

_ z ¼ ðVc þ yðtÞÞ=V 0

(4)

_ where yðtÞ is the vibration velocity of the wheel and Vc is the crabbing velocity. The lateral crabbing velocity between two wheels can be calculated with the angle of attack, q, and rolling speed, V0, i.e., Vc ¼ V0sinq. As the angle of attack is less than 1, the crabbing velocity can be approximated accurately as,

Vc ¼ qV0 :

(5)

The z0 in Equation (3) is a normalised creepage that can be described with,

z0 ¼

zGabC22 zk3 ¼ ; ms W W

(6)

where G ¼ E/2(1þy). The values used for the simulated creep contact parameter k3, elastic modulus E, Poisson's ratio y, stationary friction coefficient ms, the constant C22 and the normal loading W are listed in Table 1. The dimensions of the elliptical contact patch, a and b, can be determined by the contact theory of Hertz [26] or other means. When the effect of viscous damping is considered, the power input due to the lateral force and damping dissipation can be expressed as,

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2 _
cd ðyðtÞÞ _ PQd ¼ Q yðtÞ

149

(7)

where cd is the modal damping of the dominant mode listed in Table 1. More details of the experimental and simulation models are provided in Refs. [17] and [24] (including experimental modal analysis) although the present paper includes all detail required to rebuild the simulations. 2.4. Analysis of wheel squeal to predict the steady state amplitude The Equations of motion (1)e(7) may be solved for limit cycle analysis to analytically determine the amplitude of wheel squeal under simplified assumptions. It is noted that Rudd [1] also provided a prediction for wheel squeal amplitude but was limited to a particular simplified exponential creep curve, simplified cornering geometry and a lower range of lateral sliding velocity. The present analysis provides an almost instantaneous solution that may be used to investigate a wide range of conditions and how the vibration amplitude compares with the crabbing velocity analytically. In order to obtain such an efficient solution, a simplification of the creep behaviour described in (3) is used. In particular, it is assumed that the full creep behaviour can be characterised by linear slopes, k1 and k2, for the slip and full sliding regions, respectively, as described by:

mðzÞ ¼



0 k1 z0  for z0  1  0 k1 þ k2 z
1 for z > 1

(8)

The parameters for this simplified creep law may be obtained by tuning k2 to the full sliding region slope at the nominal lateral creep and k1 to match the critical creep of a measured or more complex simulated creep curve (ie Equation (3)). In addition, it is assumed that the wheel vibration displacement is small and hence adequately represented by a sinusoid of amplitude A,

yðtÞ ¼
A sinðutÞ

(9)

where the phase is arbitrary but is chosen here to simplify the subsequent analysis. Wheel squeal is characterised by an initial growth in vibration until a steady state amplitude is reached. This process is governed by an energy balance process based on the effective positive or negative damping characteristic of the slopes of the creep curve. In particular, solving Equations (1), (2), (4) and (8) for small oscillations about a nominal crabbing velocity Vc , yields,

  € þ cd þ k1;2 k3 =V0 yðtÞ _ þ kyðtÞ ¼ 0; myðtÞ

(10)

where k1,2 represents the local slope of the creep curve under nominal conditions. Equation (10) is in the form of the wellknown second order vibrating system that has its stability determined by the sign of the effective system damping term cd þ k1,2k3/V0. A negative system damping term indicates instability and system energy input to vibrations causing them to grow in amplitude. In particular, for small vibrations, in the fully sliding region of the creep curve, the negative slope of k2 causes a positive power input to the squeal vibration which causes them to grow if it is greater than the power output due to modal damping cd. The vibration amplitude grows to cause larger creep oscillations that start to impinge on the positive slip side of the creep curve. This positive slope k1 of the creep cycle causes a negative power input (or power output) to the vibration which acts like additional damping to cd and causes them to decay. The vibration will continue to grow until a balance is achieved between the power input and output over the different slopes of the creep curve so that a steady state (limit cycle) amplitude is reached. Hence the wheel squeal steady state vibration amplitude may be determined by solving Equations of motion (1)e(8) to determine when the net energy E from the positive and negative sloped sides of the creep curve is zero, or mathematically,

0

Z E¼

2p

u

0

B PQd dt ¼ 2@

Z 0

tc

Z PQd dt þ

1 p u

tc

C PQd dt A ¼ 0;

(11)

where tc is the time in the vibration cycle at which critical creep occurs (between the positive and negative sides of the creep curve). This time tc may be determined as a function of vibration velocity amplitude by solving Equations (4) and (9) for the critical creep to obtain,

Au ¼ ðVc
zc V0 Þ=cosðutc Þ

(12)

where zc is the critical creep (between the positive and negative sloped sides of the creep curve). Substituting Equations (2), (4), (6)e(9) into Equation (11) and solving, after some algebra and calculus yields,

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Z 0

tc

Z PQd dt þ

p u

tc

PQd dt ¼ ½ðk2
k1 Þ½4ðV0 W
k3 Vc Þsinðutc Þ þ Auð2utc þ sinð2utc ÞÞk3 
2pAuðk2 k3 þ cd V0 Þ =4V0 ¼ 0 (13)

Solving (12) and (13) leads to

sinð2utc Þ
2utc ¼ 2pðV0 cd þ k2 k3 Þ=ðk3 ðk1
k2 ÞÞ;

0  utc  p

(14)

Equation (14) is a transcendental equation, which can be easily solved for the critical value of utc using any root finding function. Subsequently the steady state squeal vibration velocity amplitude Au can be obtained by substitution of the solution for utc into (12). Note that the condition for squeal occurrence is intrinsic in the solution of Equation (12) that contains the effective system damping term cd þ k2k3/V0 where k2 and k3 are determined by the wheel and rail/geometrical/mechanical and contact patch conditions. This term not only determines whether squeal occurs but also the amplitude of the squeal ie the more negative the system damping term, the larger the squeal amplitude. Solution for the squeal amplitude via Equations (12) and (14) is almost instantaneous and allows a wide range of parameter investigations to be carried out. In order to verify the model using sound recordings the radiated sound power from the squealing wheel vibrations may be calculated as (see Refs. [1,27]),

Psound ¼ s rco S〈v2 〉

(15)

where s is the radiation efficiency (or ratio), rco is the characteristic impedance of air, S is the surface area of the wheel and 〈v2 〉is the squared velocity normal to the wheel surface at the squealing frequency which is averaged both over time (ˉ) and surface area (〈 〉). The radiation and transmission parameters associated with a microphone recording are assumed to be constant for each of the experimental and field case studies such that the squeal sound pressure level may be calculated using Equations (15) and (9) as,

SPL ¼ 10 logðPsound Þ þ SPLtrans ¼ 20 logðAuÞ þ SPLconst

(16)

where SPLconst accounts for all the approximately constant transmission (SPLtrans ) and radiation components (ie from s, rco , S) that are independent of the vibration velocity. The transmission components include the distance from the microphone to the squealing wheel, any reflections and/or sound absorption, viewing angle, weightings and other corrections. The first part of Equation (16) may be used to obtain a non-calibrated prediction of the measured squeal noise amplitude if all radiation and transmission parameters can be accurately determined and or modelled a priori. For example, Boundary Element Modelling may be used to predict the radiation ratio s for complex wheel geometries as detailed in Ref. [27]. In the present case, it is assumed that the radiation ratio and transmission parameters are approximately constant over the many field and experimental recordings such that SPLconst may be simply calibrated once to each of the experimental and field measurement conditions. For clarity, a conceptual block diagram of the model is shown in Fig. 4.

Fig. 4. Concept block diagram of analytical model (thicker blocks) for wheel squeal noise amplitude.

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The block diagram highlights that the modelling has distilled the critical interaction of a small number of squeal parameters from the complex vehicle curving mechanics, rolling contact mechanics and vibrational modal dynamics conditions. Hence analysis of the fundamental mechanism of (conventional) wheel squeal has enabled a very efficient calculation of the corresponding vibration amplitude and noise level. 3. Results The results for the efficient analytical prediction of wheel squeal are first compared with the numerical solution using an experimentally tuned creep curve with full nonlinear shape in section 3.1. The predicted squeal sound level trends are then compared with that recorded at various crabbing velocities for the test rig at different rolling speeds. In addition, further verification is performed against field recordings of wheel squeal on a sharp curve of 300 m and a comparison with a simplified modified result from Rudd [1]. Finally in section 3.2, the efficient analytical model is used to perform a parametric investigation into means of reducing squeal noise amplitude including a detailed investigation of the effects of modal damping. 3.1. Prediction and verification of wheel squeal amplitude It was of interest to first compare the analytical results described by solution to Equations (14) and (12) to the numerical solution of the exact equations of motion (1)e(8). Numerical integration of Equations (1)e(8) were performed in Matlab using the explicit Runge-Kutta method “ode45” and confirmed in MathCad using the Radua method as part of the “Odesolve” function with sampling rate set at least 20 times the squealing frequency, u ¼ √(k/m), and initial velocity of 0.01 m/s. The parameters were chosen to be consistent with previous investigations (ie [17] and [24]) ie nominal crabbing velocity 0.39 m/s, rolling speed 800 rpm (17.8 m/s) with all parameters described in Table 1. The simplified creep law parameters k2 and k1 were tuned to the full sliding region slope at the nominal lateral creep and the critical creep of the measured and simulated creep curve (ie Equation (3)), respectively. Fig. 5 shows the exact solution of these conditions using the exact creep curve described by Equation (3). The steady state squealing vibration velocity is shown to grow and settle at approximately 0.35 m/s, a value close to but under the crabbing velocity (0.39 m/s). Alternatively, the analytical solution with an approximate creep curve described in Fig. 6 a) with k1 ¼ 0.27 and k2 ¼
0.02 is found by solving Equations (14) and (12) to be 0.33 m/s. The analytical solution is found to be very close to the numerical solution. Further evidence of this is provided subsequently in Figs. 7 and 13. Fig. 6 a) shows the experimentally measured [17], simulated (Equation (3)) and simplified (Equation (8)) creep laws as well as the predicted creep oscillation range. It highlights that the steady state squeal amplitude is achieved via creep oscillations spanning both the positive and negative sloped regions of the curve to enable a balance of energy. This is highlighted further by inspecting the power input for both numerical and analytical conditions as shown in Fig. 6 b). Fig. 6 b) highlights that although the analytical solution under predicts the simulated power input of the numerical solution, the trend is similar. In particular, there is a reduction and splitting of the positive peaks of the power due to the creep passing below the critical creep to the positive sloped side of the creep curve. This reduces the power input to the squeal vibration cycle and hence balances the energy to achieve a steady state amplitude. The difference in amplitude between the analytical and numerical solution is most likely due to the simplified estimate of the creep curve in the analytical model. Fig. 6 and Equation (4) also highlights that the steady state vibration velocity amplitude will always be less than the crabbing velocity if the creep remains positive. It was of interest to investigate whether this holds for a range of crabbing

Fig. 5. The simulated vibration velocity at a crabbing velocity of 0.39 m/s (from Liu & Meehan [24]).

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Fig. 6. a) The simulated (- . -) and approximated creep curve (-), analytical solution for creep oscillations range (-) quasistatic lateral creepage () and experimental measurements (,) from Ref. [17], b) The power input around the time of 0.4 s for both simulation (-) and analytical prediction (- -) (adapted from Liu & Meehan [24]).

Fig. 7. The vibration velocity amplitude (-) versus crabbing velocity; analytical prediction (..) simulation (A).

velocities using the efficient model. Fig. 7 shows a plot of solutions for the steady state squealing vibration amplitude versus crabbing velocity using the analytical and numerical solutions. Fig. 7 highlights that the vibration velocity amplitude approaches and is proportional to the crabbing velocity for values above the critical creep. The simulation results verify the analytical predictions however it is interesting to note that the analytical model predicts the vibration velocity amplitude becomes greater than the crabbing velocity for higher values (greater than approximately 5 times the critical slip). This was not confirmed by the simulations since the slope of the creep curve becomes significantly less negative (more stable) compared to the analytical model in this region (see Fig. 6(a)). These conditions correspond to large angles of attack greater than approximately 30 mrad and hence are not common under typical railway conditions. It was of interest to investigate the effect of crabbing velocity on recorded sound pressure level, SPL. The analytical predictions based on solution to Equations (12), (14) and (16) were plotted with real experimental recordings on the two disk testrig under the same squeal conditions. The sound spectrum of a typical recording, at conditions described in Table 1, was presented in Ref. [17] and showed a dominant squealing frequency at approximately 1100 Hz at over 120 dB with multiple decaying secondary peaks up to 10 kHz. The recorded sound data was analysed and presented in Fig. 8 along with the analytical solutions for vibration velocity. In this experimental case the sound pressure level measurement was calibrated with constant offset SPLconst ¼ 130 dB according to (16). The results in Fig. 8 show that the experimental trend of sound pressure level of squeal noise increasing with crabbing velocity is well predicted by the analytical solution for crabbing velocities greater than 0.15 m/s. This is consistent for squeal conditions. By inspection of the analytical solution (12), the squeal vibration amplitude is directly proportional to the difference between the crabbing velocity and the critical creep velocity. Below 0.15 m/s the trend is not well matched because the experimental noise level is no longer dominated by squeal ie other sources such as rolling noise are largely contributing that are not modelled by the analytical solution. In particular, the analytical solution drops off rapidly when squeal does not occur when the crabbing velocity is less than the critical creep velocity. The dropoff occurs at different values for different rolling speeds because it is assumed in this model that the critical creep is independent of speed. There is evidence that this spread is less in reality because the critical creep reduces slightly with increasing speed. To verify the efficient squeal prediction further, it was important to compare these results to field measurements of squeal noise. In particular, the analytical solution to Equations (12), (14) and (16) was compared with field recordings of freight trains under conditions described in section 2.2. For the field recordings the sound pressure level was calibrated with constant offset

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Fig. 8. Sound pressure level of squeal noise versus crabbing velocity; experimental (,, ◊, B) and analytical (.., _,- -) prediction.

Fig. 9. Sound pressure level spectrum of squeal noise; field measurements (-) analytical prediction (- -).

SPLconst ¼ 120 dB according to (16). A typical case of a field measured squeal spectrum and that predicted using the efficient analytical model is shown in Fig. 9. The spectrum was calculated using the Fast Fourier transform algorithm applied over a 0.125 s time interval of squeal sampled at 51.2 kHz. The spectrum indicates a field measured squeal noise level of 114 dB at 2125 Hz. The analytical model has been tuned to this frequency, although by inspection of Equations (12) and (14) it is interesting to note that the solution for vibration velocity amplitude is independent of squealing modal frequency. There is evidence of other tonal responses in the field recordings at approximately 4100 Hz and above but these are not determining the noise amplitude that is adequately represented analytically by a single modal frequency. The analytical solution to Equations (12), (14) and (16) as well as Rudds [1] simplified prediction was also plotted with field recordings of freight trains for a range of angle of attack as shown in Fig. 10. In this case, the creep curve of Fig. 5, damping constant of Table 1 and average speed of 45.9 km/h was used for the analytical prediction, however the trend was similar for a range of other speeds. In addition, a simplified prediction from Rudd [1] was also plotted for comparison in which a modification to incorporate angle of attack directly was used (instead of the estimate based on simplified cornering geometry used in Ref. [1]). For larger angle of attacks, the crabbing velocity was also plotted. Fig. 10 indicates the trend of field measured squeal noise level with angle of attack is well predicted by the analytical results. As was found for the experimental results (ie Fig. 8), for lower angle of attacks below 10 mrad, the field results trend diverges from the analytical prediction due to the presence of other sources of noise including flanging and rolling noise. By inspection of the analytical solution (12), the trend in Fig. 10 is due to the squeal vibration amplitude being directly proportional to the difference between the angle of attack and the critical creep ie the squeal noise amplitude is directly proportional to how much the angle of attack exceeds its critical creep value. The simplified modified prediction from Rudd [1] provides a reasonable prediction for low angles of attack but greatly diverges and over approximates the squeal amplitude for angles of attack approaching 3 times the critical angle of attack. This is primarily due to the use of the simplified exponential creep curve and its slope averaging approximation in Ref. [1]. For higher angles of attack the squeal amplitude approaches and then exceeds the crabbing velocity as predicted in Fig. 7. The combined simplified modified predictions of Rudd [1] and the crabbing velocity are shown to be reasonable, however the analytical prediction of (12) and (14) is more general, accurate and convenient to investigate a range of important parameters. The effect of speed on squeal noise was investigated directly as shown in Fig. 11. In this case, the analytical prediction was based on an average angle of attack of 11 mrad for noise levels exceeding 100 dB in the field, however the trend was similar for a range of other angles of attack. It is noted that no speeds below 20 km/h were

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Fig. 10. Sound pressure level of squeal noise versus angle of attack; field measurements (þ) analytical prediction (-) modified Rudd [1] prediction (- -) and crabbing velocity (- . -).

measured at the field site. The results highlight that the squeal noise increases monotonically with speed as predicted well by the analytical solution. In particular, the creepage relation (12) predicts that the squeal vibration velocity is approximately proportional to the speed when the modal damping is small compared to the negative damping from the creep curve slope (see (14)). This proportionality is consistent with the trend in Fig. 11 and with Rudd [1]. Variations in the field results are most likely mainly due to the non-uniform number of samples over the speed range, field parameter variations and the possible occurrence of non-squeal events in the data at lower sound pressure levels in Fig. 11. It was noted that the analytical predictions of Figs. 10 and 11 were relatively insensitive to the slopes of the creep curve and squeal frequency. This may explain why the comparison is good even though the analytical model was not tuned to the field creep curve or wheel geometry, which were not able to be measured due to practical reasons. However it is important that the critical creep value of zc ¼ 0.007 appears to be close to that inferred from the field measurements in Fig. 10 (ie critical angle of attack). These issues were investigated further in the subsequent sensitivity analysis. The good match between the results indirectly indicates that there was not a strong correlation between speed and angle of attack as found directly from the field results due to complexity of train cornering dynamics. By comparison of the y-axes of Figs. 10 and 11, it may be inferred that the squeal noise level is more strongly affected by angle of attack than speed. Higher angle of attacks more consistently lead to higher noise levels than higher speeds. This is confirmed by the analytical model results. The strong correlations between the predictions and field results provide a sound basis for use of the analytical prediction for efficiently investigating means of mitigating squeal noise.

3.2. Mitigation of wheel squeal amplitude It was of interest to use the model to parametrically investigate means of substantially reducing squeal noise via a sensitivity analysis. In particular, the analytical prediction was used to determine the change in squeal noise SPL due to a factor change in all the critical parameters ranging up to 3 times as shown in Fig. 12. Creep parameters were shown separately in Fig. 12 b) for convenience. Fig. 12 a) indicates that squeal vibration amplitude is most strongly dependent upon crabbing velocity (or angle of attack) and then rolling speed and critical creep as indicated in the experimental and field results. In particular, reductions in these

Fig. 11. Sound pressure level of squeal noise vs angle of attack; field measurements (þ) and analytical prediction (-).

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parameters below a factor of 1 for crabbing velocity and rolling speed and above a factor of 1 for critical creep result in relatively large reductions in squeal sound pressure level (in the order of 10 dB). Again this behaviour is due to the squeal vibration amplitude being directly proportional to how much the angle of attack exceeds its critical creep value. To confirm this, simplified solutions assuming the vibration amplitude is purely proportional to this (ie Au f Vc e zcVo) are plotted at marked points þ,  and o, respectively for comparison. The very good comparisons confirm that this simple relationship dominates changes in squeal noise levels for these parameters. Fig. 12 b) shows that relatively large changes in squeal vibration amplitude can be achieved by changing the creep curve for lower factors below 1 (ie reductions in parameter values). Physically this is achieved by using different top of rail lubricants such as friction modifiers. In particular, Fig. 12 b) shows reductions in the (magnitude of the) slope of the full sliding region of the creep curve, k2, below a threshold of approximately 0.27 stabilises the squeal instability by eliminating the system negative damping term (cd þ k1,2k3/V0) and hence any squeal vibration. However it is interesting that the squeal noise level as relatively insensitive to changes in slope, k2, above this threshold. This indicates that the steady state limit cycle amplitude of squeal is relatively insensitive to this parameter, although the critical instability squeal onset and growth rate may not be. Similar behaviour is found for changes in the other creep parameters k1 and k3 for similar reasons although the trend is reversed for k1 since the slope in the slip region is positive in contrast to the negatively sloped full sliding region k2. Conversely, Fig. 12 a) shows changes in modal damping cause relatively small changes in squeal vibration amplitude. This is investigated in more detail by also plotting the changes in squeal growth time to steady state squeal amplitude in Fig. 13. Fig. 13 confirms both analytically and numerically (denoted with symbol ,) that the squeal noise level is relatively insensitive to changes in the modal damping, cd, up to a threshold of approximately 3.8 at which point it stabilises the squeal instability by eliminating the system negative damping term (cd þ k1,2k3/V0) and hence any squeal vibration. However, conversely the squeal growth time (denoted with symbol o) increases substantially (at least exponentially according to dB scale) with increases in the modal damping. It is noted that the highest growth time plotted corresponds to 2.12s which is still within the order of time it takes for the wheel to traverse a typical curve and therefore to reach the full squeal noise level in

Fig. 12. Sensitivity of Sound pressure level of squeal noise using analytical prediction to; a) crabbing velocity (-) rolling velocity (..) modal damping (- -) critical creep (- -) and creep parameters b) slip region slope (-), full sliding region slope (..), contact parameter (- -). (þ,  and o) denote simplified solution proportional to the difference between the crabbing velocity and critical creep.

Fig. 13. Sensitivity of sound pressure level of squeal noise to modal damping; using analytical prediction (-) numerical model (,) and time to grow to steady state limit cycle amplitude (o).

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practice. Hence this analysis indicates that increased damping measures to control squeal noise levels are primarily effective, only if they can overcome the system negative damping completely, although they could have a significant effect on squeal exposure time. To summarise the reductions in squeal noise amplitude that can be achieved, the percentage changes in key squeal model parameters required to produce a 50% decrease in squeal vibration amplitude corresponding to a 6 dB reduction in squeal noise were determined and are presented in Table 2. The sensitivity results in Table 2 confirm the previous theoretical and experimental observations that the squeal vibration amplitude is strongly dependent upon crabbing velocity (or angle of attack). In this case only a 34% reduction in crabbing velocity (or angle of attack) is required to achieve a 50% reduction in squeal vibration amplitude. This result is illustrated on the creep curve of Fig. 14 and is due to the crabbing velocity moving closer to the critical slip velocity and lowering the amplitude of creep oscillations required to obtain the positive damping effect of the positive sloped slip region. In particular, Fig. 14 highlights how the creep oscillation range (indicated by the arrows) is reduced substantially as the quasistatic lateral creep (crabbing velocity on the rolling velocity) moves closer to the critical creep. Much larger percentage changes are required for the other parameters to achieve the same 6 dB reduction in squeal noise. For instance, a 51% decrease in rolling velocity is required to achieve the same. However, this result does not take into account the dependence of angle of attack on rolling velocity due to train curving dynamics, although this is small in the field results. A very large increase in squeal modal damping constant is required that is predicted from the stability analysis result of Equation (10). In particular, according to Equation (10) a 280% increase in the modal damping is required to stabilise and overcome the negative damping from the negatively sloped creep, k2, and eradicate the squeal entirely (as shown in Fig. 13). It is interesting that lesser increases in modal damping are not sufficient to reduce the squeal noise by 6 db. This suggests that practical methods to moderately increase wheel damping on a vehicle may have very small effects on squeal noise in practice (although squeal exposure time reduction may be significant). Similar behaviour is found for the effect of changing the slope of the creep curve in the full sliding region k2 and the contact parameter k3 whereby squeal is eliminated for large reductions of 73% and 50%, respectively, according to the squeal stability Equation (10). However smaller reductions of these parameters cannot reduce the squeal amplitude by 50%. In practice, such changes in the creep slope k2 is achieved by the use of friction modifiers and other lubricants including water under certain conditions. The results of Table 2 are consistent with what has been found with the use of such lubricants in that when successful, squeal seems to be eliminated entirely rather than

Table 2 Parameter change required to decrease squeal noise by 6 dB*. Parameter Description

Change required*

Crabbing velocity (& angle of attack, Vc/Vo) (Vc) Rolling velocity (for angle of attack, Vc/Vo constant) (Vo) Squeal modal damping constant (cd) Critical creep ðzc Þ Creep curve slip region slope (k1) Creep curve full sliding region slope (k2) Contact parameter (patch size/friction coef) (k3) Squeal nominal conditions: Vc Vo cd (m/s) (m/s) (Ns/m) 0.39 17.8 42


34%
51% > þ280% þ100% Not Possible <
73% <
50% W (kN) 1

zc e 0.007

k1 e 0.27

k2 e
0.02

* Note: Parameter change required for Squeal Noise reduction of 6 dB or vibration amplitude reduction of 50%.

Fig. 14. The approximated creep curve (-), analytical solution for creep oscillations range for nominal conditions (-) for crabbing velocity (A) and quasistatic lateral creepage for nominal conditions () for crabbing velocity reduced by 34% ().

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reduced in amplitude. However there are complications in using a quasistatic creep curve for squeal prediction under lubricated conditions as discussed in Ref. [28]. The critical creep is another parameter that can be controlled by specialized lubricants or other 3rd body conditions (ie wet track) and shows that large increases of the order of 100% may substantially reduce the amplitude of squeal noise. This is consistent with the squeal creep curve behaviour discussed previously in that as the critical creep moves closer to the angle of attack (crabbing velocity on the rolling velocity) the amplitude of creep oscillations required to encounter the stabilizing effect of the positive sloped slip region is lower and hence the steady state squeal vibration velocity is also lowered. 4. Conclusions A mathematical model integrating the curving and contact mechanics with the modal vibration of the wheel is used to theoretically predict the vibration velocity amplitude of wheel squeal. The limit cycle analysis results show that the squeal vibration velocity amplitude reaches a steady state approaching the crabbing velocity under reasonable conditions in agreement with previous research. Theoretically this is shown to be due to the limit cycle analysis demanding that steady state squeal oscillations be large enough to encompass both positive and negative sloped regions of the creep curve in order to obtain an energy balance for nonlinear steady state squeal amplitudes. However, it is also shown that the vibration velocity may theoretically exceed the crabbing velocity under very high angle of attack conditions. The efficient theoretical model is then used to predict the effect of crabbing velocity (and angle of attack) on wheel squeal noise amplitude and the results compare very favourably with the trend found experimentally. The model is further verified using field measurements of squeal from freight vehicles at a 300 m radius curve providing a sound basis for further investigation of squeal mitigation measures. The squeal noise amplitude is shown to be directly proportional to how much the angle of attack exceeds its critical creep value. A comparison with a simplified modified result from Rudd [1] is also provided and highlights the accuracy and advantages of the present efficient model. Finally, the efficient analytical model is used to perform a parametric investigation into means of reducing squeal noise amplitude including a detailed investigation of the effects of modal damping. Increased damping measures are shown to increase squeal growth times but have minimal effect in reducing squeal noise levels unless they overcome the system negative damping and hence eliminate squeal completely. The efficient model is then used to quantify measures to achieve a 6 dB reduction in squeal noise. The results highlight the primary importance of crabbing velocity (and angle of attack) as well as the creep curve parameters that may be controlled using third body control (ie friction modifiers). The results concur with experimental and field observations but also provide important theoretical insight into the useful mechanisms of controlling wheel squeal and quantifies their relative merits. Acknowledgements The authors acknowledge the assistance and advice provided by Dr. Jiandong Jiang in undertaking the work presented in this paper. The second author would like to acknowledge the funding from National Natural Science Foundation of China (Grant no. 51505352). References [1] M.J. Rudd, Wheel/rail noise Part II: wheel squeal, J. Sound Vib. 46 (3) (1976) 381e394. [2] P.J. Remington, Wheel/Rail squeal and impact noise: what do we know? What don't we know? Where do we go from here? J. Sound Vib. 116 (2) (1985) 339e353. [3] D.J. Thompson, C.J.C. 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