Prediction of wheel squeal noise under mode coupling

Journal of Sound and Vibration 465 (2020) 115025

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Prediction of wheel squeal noise under mode coupling Paul A. Meehan The University of Queensland, Brisbane, Australia

a r t i c l e i n f o

a b s t r a c t

Article history: Received 9 April 2019 Received in revised form 15 October 2019 Accepted 16 October 2019 Available online 22 October 2019 Handling Editor: D.J. Thompson

Wheel squeal vibration and noise amplitude under the effects of mode coupling and rising and falling creep behavior is investigated and predicted using an efficient analytical model. Based on previous research, a simplified modal dynamics model for wheel squeal noise amplitude is enhanced to include two coupled modes. Closed form analytical solutions for the vibration and squeal noise amplitude are determined under combined squeal mechanisms of full sliding falling friction and mode coupling. The analytical predictions are then compared with numerical time domain simulations of squeal vibrations with nonlinear creep and sound pressure level amplitude over a range of angle of attacks and contact angles for the inner and outer wheels in a curve. As an example, the effect of rail dynamics on squeal amplitude is quantified by comparing results with and without the mode coupling and the conditions and occurrences of stiffness and viscous mode coupling (where the stability is determined by the coupled structural complex stiffness, due to the closeness of the uncoupled modes, and damping, respectively) are clearly identified and quantified. The viscous mode coupling from the rail is shown to significantly reduce and/or eliminate conventional wheel squeal, for the inner and outer wheels, at higher contact angles defined by the friction coefficient. It can otherwise amplify (up to approximately 5 dB) or cause squeal, under high rail damping for the leading outer wheel. Conversely, it is shown that under only a small range of contact angles and closely matched uncoupled modes, stiffness mode coupling causes very high amplitude squeal beyond the scope of the simplified creep model. Rail mass and damping only models are shown to represent the occurrences of stiffness and viscous mode coupling, respectively, where the optimum amplitudes are dependent upon the complex stiffness and rail damping, respectively. The scope of the efficient analytical model is quantified and shown to be limited by excessive non-proportional damping and very high squeal amplitudes associated with reverse full sliding. The analytical model is shown to provide insight into the effects of mode coupling dynamics on wheel squeal noise amplitude and possibly help explain the enigma that squeal occurs seemingly unreliably in the field. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Wheel squeal Mode coupling Large creep oscillations

1. Introduction Wheel squeal (or curve squeal) is a high-pitched tonal noise in railways that remains still somewhat of an enigma due to seemingly distinctly different conditions and uncertain causes that it occurs under, in practice [1]. However, it commonly occurs when a train negotiates a curve (corner) of a railway line. Multiple mechanisms have been investigated as the cause of

E-mail address: [email protected]. https://doi.org/10.1016/j.jsv.2019.115025 0022-460X/© 2019 Elsevier Ltd. All rights reserved.

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wheel squeal, similar to brake squeal in automobiles. The conventional ‘falling friction’ mechanism of wheel squeal involves the negative slope of the full sliding traction-creepage at large angles of attacks causing a locally unstable excitation of the wheel [2e4]. Alternatively, modal coupling [5e12] has been investigated as a mechanism that may occur independently of the slope of the creep curve. The instability arises when mode coupling causes system energy input from the dynamics of the non-conservative friction force [6]. However, a full experimental validation of this theory has not yet been presented in the literature [11]. The investigation of wheel squeal using modelling, particularly bloomed following Rudd's [2] focus on the ‘falling friction’ mechanism and is captured by subsequent reviews in Remington [3] and Thompson et al. [4,5]. A small sample of notable modelling contributions include Heckl and Abrahams [13] and Chiello et al. [14], who confirmed that wheel squeal grows to a stable limit cycle vibration from a local instability. Using a transient analysis, Brunel et al. [15] identified that wheel damping had a smaller effect on squeal than the creep curve slope. Heckl [16] provided an approximate method based on complex eigenmode analysis for determining which wheel modes are prone to squeal. De Beer [17] included and identified squeal in a model with rail and normal contact force dynamics, although mode coupling was not explored. More recently, the investigation of mode-coupling as an alternative cause of wheel squeal in railways was most likely spawned from the research on mode coupling in automotive brake squeal [18]. Hoffman [6,7], first provided a simplified mathematical numerical analysis to understand the mechanism from which railway application developed. Brunel et al. [15] also found squeal limit cycle solutions due to the coupling of the normal and lateral dynamics of the wheel. Glocker et al. [19] used a Finite Element wheel model on a rigid rail with constant friction coefficient and showed squeal at 4.1 kHz due to coupling between closely spaced axial and radial modes. Pieringer [8] also identified squeal in a time-domain model with constant friction using pre-calculated impulse response functions that were derived from detailed finite element models. Ding et al. [11] assessed mode coupling and falling friction together using a simplified approach. They showed that both mechanisms can occur together, with mode coupling influencing the level of damping required to overcome the negative damping effect of falling friction. This work was extended more recently by including the rail, and a new coupled rail mechanism involving rail damping only, was identified as causing squeal [12]. Other recent research accounts for the effects of wheel rotation and unsteady longitudinal creepage [20]. This previous research on mode coupling was primarily focused upon numerically identifying and simulating the conditions of squeal occurrence as opposed to the prediction of squeal limit cycle vibration and noise amplitude. Alternatively, an efficient analytical model for prediction of squeal noise amplitude was developed by Meehan and Liu [21] recently and used for squeal mitigation investigation. The model was verified using a two-disk testrig [22,23] and field measurements, however, the effect of mode coupling was not investigated. In particular, an analytical solution for wheel squeal occurrence and amplitude due to mode coupling has not been developed. To address this, the prediction of wheel squeal vibration noise under mode coupling and ‘falling friction’ conditions is investigated using efficient closed form mathematical modelling. In particular, an analytical solution is developed and then compared with numerical time domain simulations of squeal amplitudes with nonlinear creep over a range of angle of attacks and contact angles for the inner and outer wheels in a curve. The major contributions include: 1. Closed form analytical solutions for wheel squeal occurrence, growth and steady state amplitude prediction under mode coupling and full sliding/slip creep conditions. 2. Insight into the effects of rail dynamics on wheel squeal noise amplitude and identification and quantification of significant suppression and amplification of squeal due to mode coupling. 3. Identification and quantification of stiffness and viscous mode coupling mechanisms with rail mass and damping only and the determination and quantification of optimum conditions for squeal growth and suppression. This paper first describes the mathematical modelling for wheel squeal including mode coupling and falling friction. Efficient closed form analytical solutions for the squeal vibration including mode coupling are then developed and discussed and optimum conditions for squeal amplification and suppression identified. Squeal predictions are then compared with numerical simulations under a range of angle of attacks and contact angles including a rail mode. Finally, the limitations of the analytical predictive models are determined and quantified.

2. Methodology The modelling of squeal in the time domain, including coupled dynamics, is first described and developed in section 2.1. This is primarily based on [12,21,23], but all details are provided here for completeness. Subsequently the analytical methodology for determining the occurrence and amplitude of squeal oscillation under dynamic coupling is detailed in section 2.2. The conditions and amplitudes of stiffness and viscous mode coupling (where the stability is determined by the coupled structural complex stiffness and damping, respectively) are also identified and quantified in this section. An analytical investigation of the effect of single rail parameter models on squeal is also provided. Squeal vibration including full nonlinear creep and mode coupling may be modelled based on a conceptual block diagram of the fundamental mechanics involved as shown in Fig. 1.

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Fig. 1. Model for wheel squeal vibration and noise amplitude including couped wheel/rail dynamics. (Modified from [21]).

Fig. 1 shows that squeal can be modelled as a limit cycle oscillation due to a complex interaction between the vehicle curving mechanics controlling the speed, angle of attack and contact position angle, the rolling contact mechanics controlling the traction-creep parameters and the vibrational modal dynamics including coupled wheel/rail dynamics. The coupled modal dynamics determines the system modal damping contributed from the structural coupling, csi, which is dependent upon the vehicle curving dynamics (that effects the contact angle, qc) and the contact mechanics (that effects the contact stiffness and friction). The squeal noise comes predominantly from the wheel vibrations that propagate sound pressure fluctuations through the air. More details are described in the following.

2.1. Model for wheel squeal simulations According to Fig. 1, the contact mechanics and coupled modal dynamics may be represented by the model of Fig. 2. In particular, Fig. 2a highlights that an angle of attack, qA, of a curving vehicle causes a lateral or crabbing velocity Vc perpendicular to the rolling velocity. This causes a lateral contact force Ff due to rolling friction which along with the normal contact force FN excites a dominant wheel lateral vibration mode at an angle q to another mode as shown, as an example, in Fig. 2b as a rail mode. The modes are represented by modal mass, stiffness and damping constants. The contact angle, qc is the angle

Fig. 2. (a) Model of a squealing wheel rolling on rail top with a non-zero angle of attack, (b) 2DOF coupled flanged wheel rail model for sliding direction of leading inner wheel in curve (Modified from [12]Fig. A1). Note q ¼ qc if the wheel mode vibration is axial.

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between the axial/lateral/horizontal and the contact plane. A non-zero angle q, induces coupling between the two modes. The general model of Fig. 2b may be used to represent other forms of friction induced mode coupling. The reduced structural wheel rail model is similar to Ref. [12] but additionally includes a contact angle. In particular, the reduced case of qc ¼ 0 (vertical rail mode) and a small value of q ¼ 2.58 (slightly non-axial wheel mode) under constant friction was evaluated numerically in Ref. [12]. The present model accounts for a range of contact angles and angle of attacks and can be used to obtain an analytical solution for squeal under mode coupling and falling friction. This means the wheel and rail parameters may vary with the contact angle according to the rotated receptance/mobility. A case study of this will be investigated in the results. According to Fig. 2, the wheel squeal modal dynamics can be expressed as,

mw y€w ðtÞ þ cw y_w ðtÞ þ kw yw ðtÞ ¼ FN sinðqÞ  Ff cosðqÞ

(1)

mr y€r ðtÞ þ cr y_r ðtÞ þ kr yr ðtÞ ¼ FN

(2)

where m, c and k are the modal masses, damping coefficients and stiffnesses, y are the modal displacements and the subscripts w and r represent the wheel and rail, respectively. The dominant modal parameters can be determined from experimental modal analysis [24] or via finite element analysis [12]. The rolling contact mechanics normal and frictional forces, FN and Ff are dependent on the sum of the static and dynamic normal loading, N þ FN, according to,

FN ¼ kH ðyw sinðqÞ  yr Þ

(3)

Ff ¼ dw mðzÞðN þ FN Þ

(4)

where dw defines the direction of the sliding in a curve (shown positively in Fig. 2 b) where dw ¼ 1 for a leading inner wheel and dw ¼ -1 for a leading outer wheel. This direction is defined as it is important for mode coupling as the frictional power flow is dependent upon the phase relationship between the frictional force and the vibrational displacements, although the mechanisms may vary, eg Refs. [6,7,12,21]. The lateral adhesion ratio m(z) is represented here (although many models are available) as a nonlinear curve describing slip and full sliding behaviour, according to the squeal modelling of de Beer et al. [17],

 8  o 1 02 1 03 n 0 > > for z0  3 1  0:5e0:138=jzV0 j  0:5e6:9=jzV0 j z < ms z  z þ 3 27 mðzÞ ¼ n o > > : m 1  0:5e0:138=jzV0 j  0:5e6:9=jzV0 j for z0 > 3 s

(5)

The lateral creepage, z, is the ratio of the lateral relative velocity between the wheel and rail and the rolling speed V0, in the contact plane,

z ¼ ððVc = cosðqc ÞÞ þ ðdw y_w ðtÞ = cosðqÞÞÞ = V0

(6)

where y_ w ðtÞ is the wheel vibration velocity and Vc is the crabbing velocity. Note that the lateral crabbing velocity, angle of attack, qA, the rolling speed, V0, and the nominal lateral creepage zo are quasistatic and approximately related, if the angle of attack is less than 1, according to,

Vc ¼ qA V0 and zo ¼ qA =cosðqc Þ:

(7)

Note the lateral adhesion ratio m(z) is represented as a function of z only since the normalised creepage z’ is directly proportional to z and defined as,

z0 ¼

zGabC22 zkz3 ¼ ms N N

(8)

where G ¼ E/2(1 þ y) and N is the normal loading. The simulated creep contact parameter kz3, encapsulates the shear modulus G, the stationary friction coefficient ms (of non-rolling contact), the Kalker constant C22 and the dimensions of the elliptical contact patch, a and b, as determined by Hertz contact theory [25] or other means. More details of the contact mechanics simulation model and associated experiments are provided in Refs. [23,26]. The wheel vibrations radiate sound pressure fluctuations at a level according to Refs. [1,27],

  SPL ¼ 10 log s rco SCv2 D þ SPLtrans ¼ 20 log ðAuÞ þ SPLconst

(9)

where s is the radiation ratio, rco is the characteristic impedance of air, S is the surface area of the wheel and Cv2 D is the squared velocity normal to the wheel surface at the squealing frequency which is averaged both over time (ˉ) and surface area (C D). SPLtrans represents the transmission component effects including the distance from the microphone to the squealing wheel, any reflections and/or sound absorption, viewing angle, weightings and other corrections. SPLconst is a constant for a

P.A. Meehan / Journal of Sound and Vibration 465 (2020) 115025

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given case study (or microphone setup) when the transmission and radiation components are independent of the vibration velocity. Hence numerical solution to equations (1)-(9) can provide the squeal noise amplitude under mode coupling assuming the structural and creep curve parameters can be determined experimentally or numerically. 2.2. Analytical model for wheel squeal occurrence and amplitude Although mode coupling substantially complicates the nonlinear limit cycle analysis of squeal, closed form solutions to equations (1)e(8), for wheel squeal occurrence and amplitude, may be obtained under simplified assumptions using a similar approach as [21] combined with complex eigenvalue analysis. It is first assumed that the squeal limit cycle behaviour is characterised by unstable coupled modes or, in mathematical terms, the complex eigenvalues. Specifically, a modal decoupling of the stiffness and mass terms of the coupled equations (1) and (2) can be performed to obtain

m1 y€1 ðtÞ þ c1 y_1 ðtÞ þ c12 y_ 2 ðtÞ þ k1 y1 ðtÞ ¼ F1

(10)

m2 y€2 ðtÞ þ c2 y_2 ðtÞ þ c21 y_ 1 ðtÞ þ k2 y2 ðtÞ ¼ F2

(11)

where the modal damping and forcing terms can be defined more explicitly as,

m1 ¼ mw þ pL1 mr p1 ; m2 ¼ mr þ pL2 mw p2 ; c1 ¼ cw þ pL1 cr p1 ; c2 ¼ cr þ pL2 cw p2 ; ki ¼ pL Kpi;i F1 ¼ dw mðzÞN cosðqÞ; F2 ¼ pL2 dw mðzÞN cosðqÞ

(12)

and the p variables are the left and right undamped eigenvector components defined by,

 pL ¼

1 pL1

  pL2 1 2 where pTL K ¼ l pTL M; p ¼ 1 p1

 p2 2 where Kp ¼ l Mp: 1

(13)

Note the left and right eigenvalues are required since the stiffness matrix is nonsymmetric due to the mode coupling. In particular, the stiffness and mass matrices of equations (1) and (2) are:

 K¼

kw þ kH sinðqÞðsinðqÞ þ dw m cosðqÞÞ kH sinðqÞ

  mw kH ðsinðqÞ þ dw m cosðqÞÞ ; M¼ 0 kr þ kH

 0 ; mr

(14)

Equations (10) and (11) are in a simpler form, for insight into instability, than the coupled equations (1) and (2) as the energy dissipation or growth appears explicitly through the damping and possibly the complex stiffness terms rather than through the coupled stiffness matrix K. The nonlinear limit cycle behaviour of the creep of (5) may be approximated as a bilinear curve with slopes, kz1 and kz2, for the slip and full sliding regions, respectively, as described by,



mðzÞ ¼

kz1 z0 for z0  1 where the bilinear approximation is bounded by z   zc 0 kz1 þ kz2 ðz  1Þ for z0 > 1

(15)

The squeal vibration is also assumed to be sinusoidal with a steady state amplitude Ai,

yi ðtÞ ¼  dw Ai sinðui tÞ:

(16)

This approximation has been shown to be reasonable for typical squeal vibration amplitudes avoiding negative full sliding conditions [21,22]. Solving Equations (10)e(12), (15) for small oscillations of a single dominant mode i ¼ 1,2, about a nominal crabbing velocity, Vc, yields,



mi y€i ðtÞ þ csi þ kFi kz1;2 kz3 V0 y_i ðtÞ þ ki yi ðtÞ ¼ 0;

(17)

where the modal forcing factors are defined as,

kF1 ¼ 1; kF2 ¼ pL2 p2 ;

(18)

and kz1,2 is the local slope of the creep curve under nominal conditions. Equation (17) has its stability determined by the sign of the effective system damping term csi þ kFikz1,2kz3/V0 that also determines the growth or decay of vibration amplitude y_ ampi ðtÞaccording to,

y_ampi ðtÞ ¼ y_ampi ð0Þeðcsi þkFi kz1;2 kz3 =V0 Þt =ð2mi Þ

(19)

where csi is the system modal damping contributed from the structural coupling taking into account possible contributions from complex stiffness during mode coupling. By inspection of (19), it can be seen that csi/(2mi) is simply the real part of the

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damped system eigenvalues that may be directly obtained from any complex modal analysis of the structure (not including the falling creep behaviour). Using the undamped eigenvectors, the system modal damping contributed from the structural coupling may be accurately approximated if the structural damping is small or sufficiently proportional as,

csi ¼ ci if pi ; pLi ¼ Reðpi Þ; ReðpLi Þ

(20)

Equation (20) defines the squeal growth or decay due to viscous mode coupling. However there is also the possibility of complex undamped eigenvectors pi and pLi due to stiffness mode coupling where the overall structural damping behaviour is determined from the structural complex stiffness,

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

 . csi ¼ Re ci = ð2mi Þ þ c2i ð2mi Þ2  ki =mi 2mi if pi ; pLi sReðpi Þ; ReðpLi Þ:

(21)

The terms viscous mode coupling and stiffness mode coupling identify that the stability behavior is determined by the coupled structural modal damping and (complex) stiffness, respectively. Viscous instability identified in Ref. [7] is the result of viscous mode coupling giving a negative modal damping term. The instability mechanism found in Ref. [12] can also be considered to be due to a special type of viscous mode coupling where only the modal damping of the rail is important. Stiffness mode coupling arises due to the closeness of the uncoupled modes and is the more commonly recognised mechanism of mode coupled squeal identified in Ref. [6]. The exponent in (19) is equivalent to the real part of the roots in a complex eigenvalue analysis including the effect of non-constant friction/creep. If the exponent (or effective system damping), determined by the creep slope kz1,2 and the system modal damping contributed from the structural coupling csi, becomes negative, energy from the friction/creep will feed into the system causing squeal vibrations to grow initially exponentially in amplitude according to (19). Therefore the effects of ‘falling friction’ from the negatively sloped creep curve and mode coupling can be directly quantified and superposed according to (19). In such a way the effects of modal coupling can be considered to be transformed into negative damping. In addition, equations (12), (13), (18), (20) and (21) can provide an analytical solution for squeal stability under both falling friction and two mode structural coupling since the eigenvectors for a two degree of freedom undamped system can be obtained in closed form. The system damping term indicates that the vibration amplitude will grow fastest in the negatively sloped region of the creep curve. This growth rate can be increased, decreased (or changed to decay) depending on the structural coupling and damping. This squeal growth from local instability reaches a steady state nonlinear limit cycle amplitude when the oscillations impinge on the positive slip region causing an energy balance between the effective positive and negative damping characteristics of the creep curve slopes and the system modal damping contributed from the structural coupling. Note it is also possible that the system negative damping may be so large as to overcome the positive damping in the steeply positive sloped slip region of the creep curve in which case the oscillations theoretically do not reach this limit cycle and continue to grow to very high amplitudes. In this case, additional nonlinear creep mechanisms not adequately modelled by (15) will limit the squeal amplitude in reality. Based on this understanding, an energy balance over a creep limit cycle can be performed to obtain a closed form solution for the squeal vibration within the scope of (15). For this purpose, the energy balance for the dominant squeal mode limit cycle may be expressed as, 2p

Zu Ei ¼

p

Ztc PQdi dt ¼ 2

0

Zu PQdi dt þ

0

! PQdi dt ¼ 0; PQdi ¼ Fi y_i ðtÞ  csi ðy_i ðtÞÞ2

(22)

tc

This vibration amplitude of the wheel due to the dominant mode Awui may be obtained by solving equations (10)e(12), (16), (22) to obtain [21],

Aw ui ¼ ððVc = cosðqc ÞÞ  zc V0 ÞcosðqÞ = cosðui tc Þ where sinð2ui tc Þ  2ui tc ¼ 2pðV0 csi þ kFi kz2 kz3 Þ = ðkFi kz3 ðkz1  kz2 ÞÞ; (23) where zc and tc are the critical creep and time in the squeal cycle (between the positive and negative sloped sides of the creep curve). Equation (23) can be solved numerically using any root finding function or alternatively in closed form using a Taylor series approximation as,

Aw ui z V0 DqA where

.



1=3  cos 3pCsys ð2K12 Þ

DqA ¼ ððqA = cosðqc ÞÞ  zc ÞcosðqÞ;

for K12 > 6 / DAu < 10% or 1dB Csys ¼ ðcsi þ kFi kz2 kz3 = V0 Þ ðkFi kz2 kz3 = V0 Þ;

Kz12 ¼ 1  kz1 kz2 ;

(24)

where DqA is a measure of how far the angle of attack is from the critical creep referenced to the horizontal plane. Csys is a ratio of the effective system damping including mode coupling effects to the damping component from the full sliding region slope.

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Kz12 is a ratio of the combined slopes of the creep curve to the component from the full sliding region. The approximation is within 1 dB error for typical railway creep curves [22]. Equation (24) provides a concise closed form solution for wheel squeal amplitude including falling friction and mode coupling mechanisms assuming the system modal damping contributed from the structural coupling can be obtained. The effects of mode coupling on squeal maybe determined through the effects of Csys via equation (24). In particular, equation (24) directly predicts that the squeal amplitude increases from 0 as the system negative damping increases from 0 but at a smaller rate than linearly. Mode coupling will cause a change to the system negative damping via csi and more subtly through the modal forcing factors kFi. By inspection of (12), (20) and (21), csi is determined by the amount of wheel and rail damping, the product of the left and right eigenvectors pL p and possibly the complex modal stiffness. Interestingly the eigenvectors may cause the contributions to be negative and therefore the seemingly contradictory conclusion that in some cases high values of structural rail damping may actually cause squeal instability. This behaviour has been identified in Refs. [7,12] and will be investigated further in this section. Any complex eigenvalue analysis, such as via Finite Element modelling or experimental modal analysis via impact testing, can be performed to obtain the system modal damping contributed from the structural coupling, csi, from which the squeal amplitude can be calculated via (23) or (24), but an efficient closed form analytical solution is lacking in the previous literature. However, the 2 mode squeal system described in equations (10)e(13) is fourth order and therefore maybe solved in closed form analytically using the undamped eigenvectors. In the next section, we determine this solution and the critical conditions and amplitudes for squeal generated by mode coupling. 2.2.1. Closed form solutions quantifying the effects of mode coupling on wheel squeal The modal equations of motion (10)e(14) can be solved in closed form analytically to provide efficient criteria for the occurrence of wheel squeal generated, amplified and or mitigated by mode coupling. These closed form solutions assume the undamped eigenvectors (mode shapes) are similar to the damped eigenvectors. This assumption holds true if the structural damping is small and or is sufficiently proportional. This assumption will be systematically tested in section 3. In particular, by solving the eigenvalue problem defined by equation (13), the left and right eigenvalues of each mode may be obtained in closed form as,

h pffiffiffiffi i.

  p1 ¼ u2w  u2r þ D 2f ðm; qÞkH mj sinðqÞ ; p2 ¼ 1=p1 ; pL1 ¼ mw =p1 mr ; pL2 pffiffiffiffi i2  2 .h 2 ¼ mr =p2 mw therefore pL pi ¼ 4f ðm; qÞ kH mj uw  u2r  D ; where f ðm; qÞ . . ¼ sinðqÞðsinðqÞ þ dw m cosðqÞÞ; u2w ¼ ðkw þ kH f ðm; qÞÞ mw ; u2r ¼ ðkr þ kH Þ mr ; D 2 .  ¼ u2w  u2r þ 4f ðm; qÞk2H ðmw mr Þ

(25)

Here represents the conjugate, i represents the mode number from lowest to highest natural frequency and j ¼ r,w. Equation (25) provides a complete closed form solution to the system modal damping contributed from the structural coupling, csi, via equations (12), (20) and (21), from which the squeal amplitude can be determined using (24). The type of squeal due to mode coupling is determined by the complexity of the mode shapes (eigenvectors) which is determined directly by the sign of the discriminant D in (25). In particular, squeal by viscous or stiffness mode coupling is investigated, respectively, in the following. 2.2.1.1. Viscous mode coupling. Equation (25) provides a complete closed form solution to the modal damping csi ¼ ci (equation (12)) when the eigenvectors are real and therefore the squeal amplitude (23) or (24) taking into account mode coupling. This is valid when the discriminant D in (25) is positive. By inspection of these equations we can determine the contribution to squeal amplification or decay via mode coupling when the modal damping ratio ci/kFi becomes greater than or less than the wheel damping cw alone. In particular, the sign of pLpi is determined by that of f(m,q) as the denominator in (25) is always positive, therefore the following criteria can be developed:

mode coupling amplification ci =kFi < cw when dw ¼ 1 and 0 < tanðqÞ < m mode coupling decay ci =kFi > cw when dw ¼ 1 or tanðqÞ > m D  0

D0

(26)

These mathematical criteria translate to some surprisingly simple physical conditions for the effects of viscous mode coupling on squeal. In particular, equation (26), means that mode coupling will amplify wheel squeal only for the leading outer wheelset (ie when dw ¼ 1) when the angle q is small ie in between 0 and tan1(m). For all the other conditions (excluding no effect) of equation (26) the mode coupling will add system positive damping and therefore decay wheel squeal. Whether the amplification or decay caused by the rail mode damping causes the existence or non-existence of squeal is determined by the effective system modal damping (see (24) or (17)) which also takes into account the negative damping from the slope of the creep curve. This mechanism of mode coupling squeal via rail damping has been numerically identified in Ref. [7] and in simpler form in Ref. [12].

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It is also of interest to determine the maximum viscous mode coupling amplification and decay values. The maximum amplification occurs when pLpi is most negative and can be determined by differentiation of (25). This shows the maximum amplification occurs halfway in between the extremes of (26) and the maximum decay values can be determined using a similar method as,

Max mode coupling amplification Max mode coupling decay

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 1 1  1 þ m2 D0 2 2 1 dw ¼ 1 at qmax a ¼ tan1 ðmÞ when f ðm; qmax d Þ ¼ 1 D0 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi  q  1 1 dw ¼ 1 at qmax d ¼ p  tan1 ðmÞ when f ðm; qmax d Þ ¼ 1 þ 1 þ m2 2 2 (27)

dw ¼ 1 at qmax a ¼ tan1 ðmÞ when f ðm; qmax a Þ ¼

Hence the maximum mode coupling amplification occurs at a small angle that is entirely dependent upon the friction coefficient. Conversely the maximum decay occurs at high angles. The maximum decay factor f(m,qmax) is greater than or equal to 1 while the corresponding amplification factor is close to zero. This suggests that viscous mode coupling has a greater effect in retarding wheel squeal as compared to amplifying. The amount of the maximum amplification or decay may be quantified in closed form directly using (27), (25), (12) and (20). By inspection of (25), it may be deduced that the closeness of the uncoupled modes only affects the magnitude of pLpi and therefore could affect the occurrence of squeal via Csys but not whether mode coupling has an amplifying or decaying effect on squeal according to (26). It is noted that the maximum decay occurs at high angles close to perpendicular. In the case of an axial wheel squeal mode coupled to a rail mode this translates to a vertical contact angle (ie flanging). Therefore in this case, wheel squeal is less likely to occur under high levels of flanging which is consistent with field observations that wheel squeal is generally a top of rail phenomenon. 2.2.1.2. Stiffness mode coupling. When the discriminant, D in (25), is negative, the undamped eigenmodes become complex. Physically this occurs when the modes of the undamped system √(ki/mi) converge and one mode of vibration pumps energy into the other mode via a phase difference between the wheel and rail motions (hence complex values). This is the more commonly recognised mechanism of mode coupled squeal without damping described in Ref. [6]. Here instability occurs due to the negative system modal damping contributed from the structural coupling, csi, that arises due to the complex modal stiffness in (21). Accordingly, the amplitude of csi is determined in closed form by using (25) to evaluate all the complex modal mass, mi, stiffness, ki, and damping, ci, terms by use of the closed form solutions for mode shapes of (12). By inspection of these equations we can determine that squeal amplification via stiffness mode coupling occurs when:

mode coupling amplification when dw ¼ 1; 0 < tanðqÞ < m and D < 0

(28)

These mathematical criteria translate to simple physical conditions for the effects of stiffness mode coupling. In particular, equation (28), means that stiffness mode coupling will amplify wheel squeal only under more restrictive conditions than viscous mode coupling ie for the leading outer wheelset (ie when dw ¼ 1) when the angle q is small ie in between 0 and tan1(m) as well as the uncoupled modes need to be close according to the last condition D<0. As previous, whether the amplification or decay caused by the stiffness mode coupling causes the existence or non-existence of squeal, is determined by the effective total system modal damping (see (24) or (17)) which also takes into account the negative damping from the slope of the creep curve. The stiffness mode coupling amplification is determined by the real part of the complex stiffness that is physically due to the closeness of the uncoupled modes causing coupling and a phase difference between the 2 mode motions. Specifically this is determined by the mode shape solutions, pLi pi, (25) on the system modal damping contributed from the structural coupling csi (21) and may be expressed as,

ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . h i 2 ðuw þ ur Þ  D  ðuw þ ur Þ Re ki =mi 2mi ¼ ±mw D 2f ðm; qÞk2H ðmw mr Þ

if D < 0

(29)

The extremes of these coupled stiffness damping components may be shown to occur when the discriminant D is most negative and the uncoupled modes are identical. This shows the maximum amplification occurs halfway in between the extremes of angle q (28) as found for viscous mode coupling and the maximum decay values are the opposite sign to the amplification,

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qffiffiffiffiffiffiffiffiffiffiffiffiffiffi    1 Max mode coupling amplification dw ¼ 1 at qmaxa ¼ tan1 ðmÞ; D ¼ 2 1  1 þ m2 k2H ðmw mr Þ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi sr ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi  . qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 

 

2  2 2 1  2 1  1 þ m kH mw mr 2min uw;r when Re 1 ki =mi 2mi ¼ ±2mw 2min uw;r

(30)

ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi   . 2 2 2mr min uw;r 1þm 1 z±2kH mw where minðuw;r Þ is the minimum undamped natural frequency of the wheel or the rail (with Hertzian stiffness included). The result of equation (30) is rather remarkable as it enables the maximum squeal amplitude due to stiffness mode coupling to be predicted a priori using a relatively simple function of the masses, undamped natural frequencies and the friction coefficient, to obtain insight into avoiding the phenomenon. The solution predicts that the maximum mode coupled negative damping component increases with increases in the Hertzian contact stiffness and wheel mass and friction coefficient and decreases in the matched undamped frequencies and the rail mass. The maximum decay is the same as the maximum amplification as the undamped eigenvalues have converged in frequency but polarised in effective damping in accordance with [6]. Physically one mode drives the other due to the phasing between the motions and energy is gained from the frictional force. The damped solution defined by (21) includes the effects of the structural damping from the wheel and rail. This tends to dampen (add positive damping) to both coupled modes in accordance with [7] which will cause lower growth rates. The amount of the maximum amplification or decay may be quantified in closed form directly using (30), (25), (12) and (24). By inspection of (25), it may be deduced that instability due to stiffness mode coupling is caused by the closeness of the uncoupled modes on the leading outer wheelset in a small range of low angles q defined by D < 0. 2.2.2. Closed form solutions for single rail parameter models Following Ding et al. [12], the model may be reduced further by considering the rail as only a modal mass, mr, stiffness, kr, or damping, cr, respectively, to obtain the following closed form results. Mass only

 2 pL pi ¼ 4f ðm; qÞ kH mj



u2w  u2r 

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 . 

u2w  u2r 2 þ 4f ðm; qÞk2H ðmw mr Þ

where u2r ¼ kH

.

mr

If D  0, c1 =kF1 ¼ cw therefore rail coupling has no effect on wheel squeal amplification/decay. If D < 0;

mode amplification

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  Re ki =mi 2mi ¼ mw D

max mode amplification Re

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ki =mi 2mi z 

sr ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  h . i   2 2 2 2 2f ðm; qÞk2H ðmw mr Þ uw þ ur  D  uw þ ur

ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 2kH mw 1 þ m2  1 when uw ¼ ur ; qmaxa ¼ tan1 ðmÞ 2 (31)

Stiffness only (one mode only)

. pL p1 ¼ f ðm; qÞ ð1 þ kr =kH Þ2 c1 =kF1 ¼ cw

therefore rail coupling has no effect on wheel squeal amplification=decay;

Damping only (one mode only)

(32)

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pL p1 ¼ f ðm; qÞ mode amplification

c1 =kF1 ¼ cw þ f ðm; qÞcr < cw when dw ¼ 1

and 0 < tanðqÞ < m

mode decay

c1 =kF1 ¼ cw þ f ðm; qÞcr < cw when dw ¼ 1 or tanðqÞ > m

1 1 1 Max mode amplification dw ¼ 1 at qmaxa ¼ tan1 ðmÞ when f ðm; qmaxa Þ ¼ 2 2 Max mode decay

dw ¼ 1 at qmaxd ¼ dw ¼ 1 at qmaxd ¼

p 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 þ m2

(33)

when f ðm; qmaxd Þ ¼ 1

 1 p  tan1 ðmÞ when 2

f ðm; qmaxd Þ ¼

1 1þ 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 þ m2

The analytical predictions for the reduced rail model cases described by equations (31)e(33) are interesting as they distil the important mechanisms of mode coupling on wheel squeal and identify and allow immediate quantification of the effects. The stiffness only model does not affect the occurrence of wheel squeal as the rail stiffness simply adds to the contact stiffness in series as a single mode and would therefore not affect the system modal damping contributed from the structural coupling (as found in [12]). The mass only result is somewhat less intuitive. According to (31) and (12), the rail mass can induce viscous mode coupling if D  0, which will change the modal damping of the second mode, c2, by a factor, pLp2 ¼ kF2 as the rail damping is zero. However the negative damping due to the slope of the creep curve will also change by the same factor and hence the factor, c2/kF2. Therefore the ratio of the effective system modal damping to the damping component from the full sliding region slope, Csys, in (24) remains unchanged. Essentially, although the rail mass can change the system modal damping, contributed from the structural coupling, due to viscous mode coupling, it does not change the effective system modal damping relative to the falling friction damping which determines the wheel squeal occurrence. It is however interesting to note that the transient exponential growth or decay behaviour described by (20) is dependent on the modal mass such that higher rail mass will tend to decrease the magnitude of the exponential rate towards zero under viscous mode coupling. This rail mass only behaviour will not be predicted by a constant friction model and is confirmed via full nonlinear time domain simulations. In addition, stiffness mode coupling is also possible with the rail mass only model, if D < 0 which will have the same general conditions of (29) and (30), but with the rail undamped natural frequency reduced to √(kH/mr). The simplicity of the analytical prediction for maximum stiffness mode coupling is intriguing as it depends on the Hertzian contact stiffness, the wheel mass and friction coefficient only. The maximum occurs when the rail mass acts as a tuned mass damper maximising the energy flow between the two masses which comes from the optimum phased friction force. Essentially the negative damping contribution from the complex stiffness is maximised. The effect of the rail damping only model encapsulates the viscous modal coupling behaviour that the mass and stiffness only models lack, according to (33). Squeal mode amplification or decay occurs under the same conditions predicted by the full rail model in (26), however the amount of amplification or decay can be directly determined by the friction coefficient and angle q through the simple function f(m,qmax). In particular, by inspection of (33), the negative damping due to coupling can be substantially amplified by large rail damping, compared to wheel damping, and this effect is maximised for the leading outer wheelset at the particular angle q ¼ ½ tan1(m). Essentially the mode coupling causes the phase of the rail damping to be completely out of phase with the wheel damping and hence act as negative damping of a proportion less than one depending on the angle q. Squeal mode decay occurs for larger angles q greater than tan1(m) for the leading outer wheelset, when the rail damping acts in phase with the wheel damping and is maximised at an angle q of 90 when the rail damping adds directly to the wheel damping. For the outer wheelset the maximum decay occurs close to 90 when the rail damping contribution is a little more than this. Hence the rail damping in viscous mode coupling has a bigger effect on squeal decay as compared to amplification, but it is interesting to note that high rail damping can both retard and grow squeal on the leading outer wheelset, depending on the angle q. It is also interesting to note that this damping only mechanism is independent of any coupling of modes; ie it is a single mode mechanism that doesn't depend on the closeness of the natural frequencies of uncoupled modes. These results concur with and provide analytical insight to the numerical findings in Ref. [12] based on complex eigenvalue analysis. It is of note that the viscous mode coupling has a large decaying effect on squeal that could suppress its occurrence for the outer wheel and the inner wheel for large angles q if the rail damping is large enough. In contrast stiffness mode coupling provides only a very high amplification of squeal depending on the complex (Hertzian) stiffness, but under more constrained conditions of a small range of angles, q, and closeness of the uncoupled modes. 3. Results In this section, the comparisons between the full nonlinear time domain model presented in section 2.1 and the analytical solutions of section 2.2 are presented for a railway wheel squeal case. Firstly, results for squeal are presented for a range of nominal lateral creepages (zangles of attacks) and contact angles, to identify the effect of mode coupling on conventional squeal amplitude. The efficient analytical model is then used to identify and quantify conditions under which mode coupling

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has a substantial effect on the occurrence or non-occurrence of wheel squeal. The Radua method as part of MathCad 15.0 was used for numerical integrations of equations (1)e(9) at a sampling rate of approximately 20 times the wheel natural frequency, √(kw/mw), and initial velocity of 0.02 m/s. The creep parameters were chosen to be consistent with the previous experimental and field investigations of squeal under dry conditions [23] while the structural parameters were estimated generically from modelling and measurements from Refs. [12,27,28] as detailed in Appendix A. Note in contrast with [12], where the case of constant friction and no contact angle was investigated ie qc ¼ 0, the rail modal parameters are estimated from approximate rotated vertical and lateral mobility/receptances in the range of 1e2 kHz (see)Appendix A to explore the case where the contact angle qc varies and the wheel mode is axial ie qc ¼ q. To this end, the modal parameters at any contact angle qc may be estimated from the vertical and lateral estimated mobilities/receptances according to,

. 2 1 2 1 1 ðiUÞ; j ¼ rv; rl Yq1 ¼ Y cos q þ Y sin q ; Y ¼ c þ m i U þ k c c j j j rv j rl c

(34)

where Yj, and kj, mj, and cj represent the mobility and estimated modal stiffness, mass and damping, respectively, for the contact angle, qc, vertical, rv, and lateral, rl, directions. iU is the complex frequency of vibration to model phase. In the following case, the wheel mode is also assumed to be dominated by an axial/lateral mode neglecting the influence of nonlateral deflections under flange contact. This assumes that the wheel squeal mode of interest in practice is primarily axial/lateral as it is the most efficient at providing sound power. It is noted that this simplified modal analysis ignores other complexities such as cross coupling and other multimodal interactions that would require a more detailed mobility/receptance analysis to model more accurately. These rail parameters and all others are described in Table 1 with a nominal angle of attack and rolling speed of 16 mrad and 17.8 m/s, respectively. The sound radiation and transmission constant was set at SPLconst ¼ 130 dB to be consistent with the previous study [21] but could vary depending on actual sound measurement conditions (see (9)). The full nonlinear simulated (5), simplified analytical (15) and experimental [23] creep curves for dry conditions are shown in Fig. 3. Note the static friction coefficient constant is tuned to be a little smaller (~5%) than that in Ref. [23] to be be more consistent with field conditions. The simplified creep law parameters kz1 and kz2 were tuned to the critical creep and full sliding region slope at the nominal lateral creepage, respectively, in an automated fashion [21] described by,

kz1 z mðzc Þ and kz2 zðmð2zo  zc Þ  mðzc ÞÞzc =ð2zo  2zc Þ:

(35)

Fig. 3 highlights that the dry creep curve has a negative sloped region associated with negative damping. In addition the simplified analytical approximation is within the scatter variations of the experimental measurements in the negatively sloped region. Near and below the critical creep the simplified approximation has a larger error. 3.1. Effects of viscous mode coupling on squeal Simulations and analytical predictions were then performed using the exact and approximately tuned creep curves, respectively, with and without viscous mode coupling over a range of nominal lateral creepages (zangles of attack) for the leading outer wheel. In particular, the nominal parameters were chosen such that the discriminant D in (25) was positive and viscous mode coupling was maximised (qc ¼ 0.16 rad) or eliminated. The undamped wheel uw and rail mode frequencies ur (including contact stiffness) weren't too close, (>18% difference) to avoid stiffness mode coupling. Fig. 4a) shows the numerical simulations of the time histories at a nominal lateral creepage of 16 mrad under no mode coupling (qc ¼ 0) and maximum viscous mode coupling at (qc ¼ 0.16 rad). The analytical predictions of squeal vibration steady state amplitude and amplitude growth are also shown in Fig. 4a).

Table 1 Parameters of the squeal modelling and simulation including mode coupling and nonlinear creep. Description Creep curve parameters: Static friction coefficient constant (ms) Critical creep (zc - dry) Creep curve slip region slope (kz1 e dry) e analytical model Creep curve full sliding region slope (kz2 e dry) e analytical model Contact parameter (patch size/friction coef - dry) (kz3) e simulation model Normal loading (N) Contact stiffness (kH) Structural vibration parameters: Modal mass: wheel, rail vertical, rail lateral (mw, mrv, mrl) Modal damping: wheel, rail vertical, rail lateral (cw, crv, crl) Modal stiffness: wheel, rail vertical, rail lateral (kw, krv, krl) Squeal nominal conditions: Nominal lateral creep (qA) Rolling velocity (Vo)

Value 0.478 0.007 0.380 0.030 2.733E5 N 1000 N 1.12E9 N/m 74.5, 57.6, 14.4 kg 103, 2.21E4, 1.547E4 Ns/m 3.57E9, 2.754E9, 6.885E8 N/m 16 mrad 17.8 m/s

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P.A. Meehan / Journal of Sound and Vibration 465 (2020) 115025

Fig. 3. The simulated () analytical (..) and experimentally measured (,) [23] creep curve at a rolling speed of 17.8 m/s.

Fig. 4 a) shows that squeal grows and settles on a higher steady state amplitude under viscous mode coupling. Hence, the squeal that is occurring due to the negatively sloped instantaneous creep curves has been amplified by viscous mode coupling by about 5 dB. This is also confirmed by the much faster growth that is well predicted by the analytical solution based on the real part of the damped system eigenvalue (19). The squeal growth is higher because the viscous mode coupling has reduced the system modal damping contributed from the structural coupling, csi, from the no-mode coupling case (cs1 ¼103 N/m) to the coupled case (cs1 ¼ 490.3 N/m), due the coupled phasing of the motion (25), causing the rail damping to provide a negative contribution. The zoomup during the squeal growth at 0.3s confirms the initially sinusoidal wheel vibration velocity is in phase with the frictional force (hence there is a 90 phase difference between the frictional force and the tangential deflections) to provide the system power input for squeal throughout the vibration cycle. In this case, the wheel and rail vibrations are approximately in phase (not shown) ie not complex. As the limit cycle is reached, the frictional force, in particular, becomes more non-sinusoidal due to the nonlinear creep oscillations. It is noted that the frequency of squeal (1102 Hz in the zoomup of Fig. 6 a) is determined by solution to the dominant system modal equation (17) and therefore is a result of the coupling of the wheel and rail mode frequencies including the contact stiffness, (uw ¼ 1097 Hz and ur ¼ 1309 Hz) and system damping. Further analytical predictions and simulations were performed under a range of nominal lateral creepage (f angles of attack) as shown in Fig. 4 b). The increase in squeal amplitude from no coupling conditions is relatively minimal (<5 dB) because the additional creep oscillations must predominantly balance the energy dissipation from the highly positively sloped region of the creep curve. The differences between the analytical predictions of the growth amplitudes shown in Fig. 4b) and the numerically simulated growth are due to the linearised representation of the slopes of the creep curve based on the approximated slopes kz1 and kz2 (see equation (20)) and nonlinear changes in the friction coefficient. In particular, the numerical simulation growth takes into account the continuously changing slope and friction coefficient as the

Fig. 4. a) Simulated time histories of squeal vibration growth under no mode coupling, qc ¼ 0 () and maximum viscous mode coupling, qc ¼ 0.16 rad ( ) conditions with zoomup at 0.3s including frictional force ( ) and corresponding analytical predictions of squeal amplitudes under no mode coupling (.-.) and maximum viscous mode coupling ( ) conditions, and for initial amplitude growth under no mode coupling (..) and maximum viscous mode coupling ( ) conditions, b) Sound pressure level vs nominal lateral creepage at a rolling speed of 17.8 m/s; under no mode coupling, qc ¼ 0, analytical () and simulated () result and under maximum viscous mode coupling, qc ¼ 0.16 rad, analytical ( ), simulated ( ) result. Leading outer wheel.

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amplitude grows while the analytical model simplifies these to constant initial values. This accentuates errors for squeal oscillations near the critical creep. However the analytical model provides an extremely efficient and reasonably accurate prediction of the initial growth and amplitude of the squeal under both conditions. The effect of contact angle and rail damping on these results was investigated and shown in Fig. 5. In particular, Fig. 5a) and b) show how the squeal sound pressure level (SPL) and system modal damping changes with contact angle. Viscous mode coupling causes amplification or decay of conventional falling friction squeal depending on the contact angle and wheel. For the leading outer wheel, a clear maximum in SPL occurs at a contact angle of approximately atan(m)/2 according to the analytical prediction of (27). This maximum amplification of 4.2 dB for nominal damping, simulated in Fig. 4, is shown to decrease with decreasing rail damping. It is noted that the nominal rail damping is at approximately 10% of the critical uncoupled rail damping. Fig. 5 b) confirms the maximum amplifications in squeal SPL are associated with minimums in the system modal damping contributed from the structural coupling in accordance with (24). These minimums lie half way in between points of no mode coupling contribution to squeal at qc ¼ 0 and atan(m) in accordance with (27). The reason the SPL is lower at qc ¼ atan(m) than at 0 is the non zero contact angle lowers the wheel vibration amplitude which is no longer aligned with the creep oscillations. This is also why the maximum SPLs occur at contact angles slightly less than atan(m)/2. By inspection of Fig. 4 b) it is also noted that the substantial negative damping contribution from viscous mode coupling may be large enough to cause wheel squeal on its own by reducing the wheel damping below the critical level defined by (19). This is evident at low nominal creepages in Fig. 4 b). Fig. 5 also confirms that viscous mode coupling has a stronger effect on decay and suppression of wheel squeal than amplification. In particular Fig. 5 a) confirms that wheel squeal caused by falling friction alone is firstly reduced for small nonzero contact angles and then completely suppressed (indicated by the vertical descents) for the leading inner wheel. Similar behaviour occurs for the leading outer wheel for contact angles greater than atan(m). Essentially this means that for contact angles greater than atan(m), viscous mode coupling has a substantial decaying and suppression effect on wheel squeal for both inner and outer wheels. For lower contact angles, viscous mode coupling will amplify (and possibly be the major cause) of wheel squeal for the leading outer wheel while decaying and/or completely suppressing wheel squeal for the inner wheel. These effects are all amplified as the rail damping increases in accordance with the predictions of (12), (25) and (26). The results of Fig. 5 were found to be relatively insensitive to the effects of changing rail modal parameters with contact angle (when comparing with results run with constant rail parameters). This is most likely because the range of contact angles in which squeal was amplified due to mode coupling is small (ie smaller than 0 to atan(m)) and viscous mode coupling occurrence is determined primarily by the rail damping (as investigated further in 3.3). The effects on decaying squeal are expected to be larger as this occurs over a much larger range of contact angles up to 90 . To show the application of the model to different mechanisms of mode coupled squeal, the wheel squeal amplitude is also predicted under falling friction and mode coupling for the reduced case described in Ref. [12] where the wheel mode is not axial due to cross coupling and the rail mode is vertical. This is represented in the present modelling with qc ¼ 0 (vertical rail mode) and a small value of q ¼ 2.58 (slightly non-axial wheel mode). Note the rail mode parameters are different to Ref. [12]. For these nominal conditions, the discriminant D in (25) is positive. Fig. 6a) shows the numerical simulations of the time histories under this case compared to no mode coupling (q ¼ qc ¼ 0) as well as analytical predictions of the growth and steady amplitudes. The simulations show that wheel squeal grows faster and to a higher amplitude under the cross-coupled wheel mode angle of 2.58 compared to no mode coupling. The squeal amplitude could be amplified further if the wheel mode angle was increased to the optimal angle of q ¼ 9.15 in accordance with (27) or if the rail damping is increased in accordance with (33). This is most likely why the squeal amplitude is only a little higher (<4 dB) across a range of lateral creepage (or angle of attack)

Fig. 5. a) Change in squeal sound pressure level and b) the system modal damping contributed from the structural coupling due to contact angle for different wheels and rail damping levels, at nominal lateral creep of 16 mrads and rolling speed of 17.8 m/s. Leading outer () and inner (.-.)wheel at nominal cr ¼ 2.21E4 Ns/m. Leading outer ( ) and inner ( )wheel at 10%cr . No mode coupling occurs at qc ¼ 0 and atan(m).

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Fig. 6. a) Simulated time histories of squeal vibration growth under no mode coupling, q ¼ 0 () and wheel mode angle, q ¼ 2.58 ( ) with zoomup at 0.6s including frictional force ( ) and corresponding analytical predictions of squeal amplitudes under no mode coupling (.-.) and wheel mode angle, q ¼ 2.58 ( ) and for initial amplitude growth under no mode coupling (..) and wheel mode angle, q ¼ 2.58 ( ) conditions, b) Sound pressure level vs nominal lateral creepage at a rolling speed of 17.8 m/s; under no mode coupling, q ¼ 0, analytical () and simulated () result and under wheel mode angle, q ¼ 2.58 , analytical ( ), simulated ( ) result. The contact angle qc ¼ 0 for both cases. Leading outer wheel.

than the no mode coupling case, as shown in Fig. 6 b). The zoomup in Fig. 6 a) during the squeal growth at 0.6s highlights again that the frictional force and wheel mode velocity are in phase although the wheel and rail modes were found to be slightly out of phase (not shown). More details of this particular mechanism of wheel-rail coupled squeal including the effects of friction and wheel damping may be found in Ref. [12]. 3.2. Effects of stiffness mode coupling on squeal Simulations and analytical predictions were then performed using the exact and approximately tuned creep curves, respectively, with and without stiffness mode coupling over a range of nominal lateral creepages (zangles of attack) for the leading outer wheel. In particular, the nominal rail mass and damping parameters were increased to 76.3 kg and decreased to 1097 N/m (as detailed in Appendix A) such that the discriminant D in (25) was negative, giving complex eigenvalues. Physically this means for the leading outer wheel, the undamped wheel and rail mode frequencies (including contact mechanics), uw and ur, were close (<3% difference) and the contact angle was less than atan(m). Stiffness mode coupling cannot occur for the leading inner wheel. Fig. 7a) shows the numerical simulations of the time histories at a nominal lateral creepage of 16 mrad under no mode coupling (qc ¼ 0) and stiffness mode coupling at (qc ¼ 0.16 rad) as well as analytical predictions of the growth and steady amplitudes. The squeal amplitude and growth rate under mode coupling greatly exceeds the no coupling case which is reshown as the same result of Fig. 4 a) but cutoff on the smaller time scale for comparison of the initial

Fig. 7. a) Simulated time histories of squeal vibration growth under no mode coupling, qc ¼ 0 () (identical result as Fig. 4 a) but on a smaller time scale) and stiffness mode coupling, qc ¼ 0.16 rad ( ) conditions with zoomup at 0.016s including frictional force ( ). Corresponding analytical predictions of squeal amplitudes under stiffness mode coupling ( ) conditions, and for initial amplitude growth under stiffness mode coupling ( ) conditions, b) Sound pressure level vs nominal lateral creepage at a rolling speed of 17.8 m/s; under no mode coupling, qc ¼ 0, analytical () and simulated () result and under stiffness mode coupling, qc ¼ 0.16 rad, analytical ( ), and simulated ( ) result which exceeded 130 dB. Leading outer wheel. Arrows indicate the analytical prediction exceeded the maximum limit of simplified creep model scope of applicability (15).

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growth rates. In particular, the mode coupled case reaches an offset amplitude approaching 0.4 m/s in approximately 0.4s whereas the no mode coupling case takes over 2.5s to reach a smaller amplitude of 0.23 m/s. The zoomup during the squeal growth at 0.016s again confirms the squeal is powered by the frictional force being approximately in phase with the wheel vibration velocity (which is close to being aligned with the tangential). In this case, the wheel and rail vibrations are not in phase (not shown) due to the complex solution to the eigenvalues as predicted in 2.2.1. The predicted analytical amplitude growth from (19) is shown to be very close to the numerical simulations prior to large nonlinear oscillations occurring around 0.02s. After this, the analytical prediction of squeal vibration steady amplitude for the mode coupled case encroaches the creep limit boundary into full negative sliding (Vc þ zcVocos(qc) shown as the dashed dot lines in Fig. 7 a)) which is beyond the scope of the bilinear approximation to the creep curve (15). In particular there is a nonlinear drifting of the lower negative vibration amplitude due to the nonlinear shape of the creep curve at higher amplitudes near negative critical creep (the bounds of the bilinear approximation (15)). In this case, the nonlinear drift converges to a constant amplitude at about 0.5s. This is also confirmed by the non-symmetric upper and lower bounds of the mode coupled time history as the amplitude grows due to nonlinear non-sinusoidal oscillations. Hence the simulations and analytical predictions show that stiffness mode coupling amplifies the squeal amplitude by at least 1.5 times for this case. Further analytical predictions and simulations were performed under a range of nominal lateral creepage (or angles of attack) as shown in Fig. 6b). Fig. 7 b) highlights that the simulated squeal settles to a higher steady state amplitude (at least 5 dB higher) under stiffness mode coupling for a wide range of angle of attacks. This occurs even for lower angles of attack below the critical creep, indicating the stiffness mode coupled negative damping overrides even the effective positive damping from the high positively sloped slip region of the creep curve. This also explains why there is a nonlinear drifting in the time history of Fig. 7 a) to much higher nonlinear amplitudes when this occurs. As found with the viscous mode coupling, the squeal growth is higher because the stiffness mode coupling has reduced the system modal damping contributed from the structural coupling csi from the no mode coupling case of the wheel alone (cs1 ¼ cw ¼ 103 N/m) to the coupled case (cs1 ¼ 3.05  104 N/m). This is due to the coupled complex phasing of the motion causing the complex modal stiffness (29) to provide a very large negative damping contribution. The analytical predictions of the growth amplitudes exceed the bounds of the creep law bilinear approximation (15) shown in Fig. 7b) which provide a close prediction of the numerically simulated growth that encroaches this boundary for this case. The error increases for low creepage below critical due to the linear approximation of the bilinear creep law (15) to the full nonlinear curve (5). To obtain more insight into the magnitude of the amplification caused by stiffness mode coupling, the effect of contact angle and closeness of the undamped modes was investigated and shown in Fig. 8. Fig. 8 a) shows how the squeal system modal damping changes with contact angle under stiffness mode coupling conditions. It confirms that stiffness mode coupling causes amplification of conventional falling friction squeal for the leading outer wheel within a small range of contact angles defined by D < 0 in accordance with the analytical predictions (28). Fig. 8 a) shows this range is less than that defined by 0
Fig. 8. a) Change in the system modal damping contributed from the structural coupling due to a) contact angle and b) closeness of the uncoupled modes for different wheels and rail damping levels, at nominal lateral creep of 16 mrads and rolling speed of 17.8 m/s. Negatively damped mode ()( ) and positively damped mode (.-.)( ) at nominal cr ¼ 1097 N/m and 10cr respectively. No stiffness mode coupling occurs at D0.

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uncoupled modes on the optimum modal dampings is shown in Fig. 8 b). The optimum values occur when the uncoupled frequencies are matched (ur ¼ uw) in accordance with (30). This leads to a maximum negative system modal damping of 51.2kNs/m in agreement with the analytical prediction without damping (30) of 52.4kNs/m. This is over 500 times the uncoupled wheel structural damping and is the reason why the stiffness mode coupled squeal has grown to nonlinear amplitudes encroaching the bounds of the simplified creep model of (15). Equation (30), is shown to be a powerful predictive tool for understanding and perhaps avoiding the maximum effects of stiffness mode coupling. By inspection of this simplified solution with (15) it may be deduced that optimum stiffness mode coupling amplifies squeal to much greater levels than the other types of squeal investigated presently, because the negative damping is provided by a complex system stiffness (f kH/ uw) of magnitude much higher than the negative damping from the slope of the creep curve or the structural damping. However, this only occurs under a more constrained set of conditions ie uncoupled modes closely matched and a small range of lower contact angles for the leading outer wheel. Since the stable and unstable modes are always present together under stiffness mode coupling, unless initial conditions are perfectly matched to the stable mode, the unstable mode will always eventually dominate the response as was found in the simulations. Hence stiffness mode coupling will only increase the likelihood and/or amplitude of wheel squeal for the leading outer wheel under these conditions of its existence. Fig. 8b) also shows a similar but smaller stabilising effect of increased rail damping on stiffness mode coupling. The lesser effect is because the magnitude of the system modal damping is even larger than that of Fig. 8 a) compared to rail damping. 3.3. Solutions for single rail parameter models The effects of single rail parameter modelling on wheel squeal were investigated over a range of nominal lateral creepages (zangles of attack) and contact angles to correlate the effects identified so far to specific rail dynamic properties. In particular, the rail was modelled separately as a mass, stiffness and damper only and numerical and analytical simulations performed for comparison as shown in Fig. 9. Fig. 9 a) repeats the full rail model simulations and analytical predictions vs nominal lateral creepage (f angle of attack) shown in Fig. 4 b) and 7 b) for the single rail parameter models. The results are very close to the full rail model results which closely overlayed them except for the mass only model which predicts a higher amplitude squeal but predicted beyond the bounds of the bilinear creep curve. This suggests the mechanisms of stiffness mode coupling and viscous mode coupling are associated with the rail mass and damper only models, respectively (in accordance with [12]). The mass only model SPL greatly exceeds the scope of applicability of the simplified creep model (15) and the creep oscillations were highly nonlinear (chaotic-like in some circumstances) with positive and negative ranges far exceeding both positive and negative critical creep values. These results are confirmed by the analytical predictions of system modal damping vs contact angle as shown in Fig. 9b). In particular the rail mass only model, corresponding to stiffness mode coupling, shows a minimum modal damping constant approximately 100 times more negative than the rail damping only result (corresponding to viscous mode coupling) and 275 times the wheel damping. For the stiffness mode coupling, the full rail model results do not match the rail mass only model because the rail mass is different in order to match ur. A maximum modal damping for the rail mass only model of 9.43E4Ns/m (not shown) was found when the rail uncoupled mode, ur, was tuned to the wheel's, uw, and this value matched the analytical prediction of (31) to within 1% error. Note this was less (a higher magnitude) than the value shown in Fig. 8 b) because the rail mass was lowered to match the nominal ur (not uw) in accordance with the prediction of (30). The results of Figs. 8 and 9b) were found to be relatively insensitive to the effects of changing rail modal parameters with contact angle

Fig. 9. a) Sound pressure level vs nominal lateral creepage at a rolling speed of 17.8 m/s and contact angle qc ¼ 0.16; and b) the system modal damping contributed from the structural coupling due to contact angle at a nominal lateral creepage of 16 mrads under rail stiffness only, analytical () and simulated () results, rail damping only, analytical ( ), and simulated ( ) results and rail mass only, analytical ( ) and simulated ( ) results which exceeded 130 dB. The previous complete rail model overlayed all single parameter results except for rail mass only ( ) which has a different rail mass mr ¼ 22.356 kg to match ur. All results are for the leading outer wheel unless specified otherwise. Arrows indicate the analytical prediction exceeded the maximum limit of simplified creep model scope of applicability (15).

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Fig. 10. Sound pressure level error of analytical prediction compared with numerical simulation vs non-proportional damping factor at a nominal lateral creepage of 16 mrads and rolling speed of 17.8 m/s for squeal under no mode coupling (, -), viscous mode coupling ( ) and contact angle qc ¼ 0.16 rad; and stiffness mode coupling ( ) and contact angle qc ¼ 0.266 rad. All results are for the leading outer wheel.

(when the results were run with constant rail parameters). In particular, the contact angle range and amplitudes of squeal amplification and decay were found to change by less then 8%. This small effect is most likely because the range of contact angles in which squeal was amplified due to stiffness mode coupling is small (ie smaller than 0 to atan(m)). The results show that a reasonable approximation of the system modal damping and squeal amplitude can be predicted analytically using the single rail parameter models without the need for lengthy nonlinear time domain simulations unless the amplitude is nonlinear and over approximately 125 dB. Another important limitation of the analytical modelling is the assumption of proportional damping. This was investigated systematically in the following. 3.4. Effect of non-proportional damping on accuracy To investigate the effect of non-proportional damping on the error between the analytical predictions and numerical simulations, a non-proportionality factor was derived from the system matrixes defined by (14) and the damping. In particular, proportional damping occurs if CM1K is symmetric so that a factor describing the ratio of the off-diagonal terms may be defined as,

fnp ¼ cr k21 mw = ðcw k12 mr Þ

(36)

where k12 and k21 are the off-diagonal stiffnesses defined in (14) and the non-proportionality factor, fnp varies from 1 which gives exact proportionality. Hence numerical and analytical simulations were used to quantify the prediction error as a function of the non-proportionality factor that was changed by increasing the rail damping. In order to simulate a range of squeal amplitudes for stiffness mode coupling within the scope of the simplified creep model, a contact angle of 0.266 rad was chosen in accordance with the results shown in Fig. 8 a) near D ¼ 0. Fig. 10 highlights that the analytical prediction error grows with non-proportional damping for the mode-coupled cases as the analysis assumes the damped coupled eigenvectors are close to the undamped ones. Hence the error in conventional squeal from the falling slope of the creep curve is unaffected as it doesn't rely on coupling. Values not shown for the stiffness mode coupled case were where squeal amplitude was beyond the scope of the simplified creep model. The results indicate that the analytical model may be used for predictions within 3 dB error up to a nonproportional damping factor of 500. This corresponds to a rail damping within the order of 100 times the wheel damping. The results show that the analytical prediction may be a very valuable tool for squeal investigations within a reasonable range of conditions. However, for mode coupled squeal of very high amplitudes (approximately >125 dB) or nonproportional damping factors less than 500, more complex nonlinear and eigenvalue analysis is required. 4. Conclusions The analytical prediction of wheel squeal noise occurrence and growth under ‘falling friction’ and mode coupling mechanisms has been performed using an efficient mathematical model of the nonlinear limit cycle and complex eigenmode analysis. Eigenmode analysis is used to determine the mode coupled conditions expressed as predictive closed form solutions for system modal damping and squeal amplitude as a function of critical structural and creep parameters. Conditions under which stiffness and viscous mode coupling occur and their growth rates were determined in closed form in order to confirm

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and identify maximum effects on squeal occurrence and amplitude. A particular case study of an axial wheel mode coupled to a rail mode at a contact angle is investigated in the results. The viscous mode coupling from the rail is shown to significantly reduce and/or eliminate conventional wheel squeal for the inner and outer wheels at higher contact angles defined by the friction coefficient and otherwise amplify (up to approximately 5 dB) or cause squeal under high rail damping for the leading outer wheel. Hence viscous mode coupling is likely reducing the occurrences of wheel squeal over a wide range of conditions in practice. Conversely, only under a small range of contact angles and closely matched uncoupled modes, stiffness mode coupling causes very high amplitude squeal beyond the scope of the simplified creep model. The very high amplitudes greater than 125 dB were predicted and shown to be due to analytically quantified negative damping from complex (Hertzian) stiffness being much greater than (>750 times) positive damping from the wheel and slip region creep curve damping (rising friction). The identification and quantification of stiffness and viscous mode coupling mechanisms with rail mass and damping only modelling was also analytically determined and optimum conditions and amplitudes for wheel squeal growth and suppression provided. The effects of changing rail modal parameters with contact angle on the occurrence of mode coupled squeal was minimal most likely because the range of contact angles in which squeal was amplified due to mode coupling is small (ie smaller than 0 to atan(m)). The effects on decaying squeal are expected to be larger as this occurs over a much larger range of contact angles up to 90 . Finally a systematic investigation of non-proportional damping identified the model accuracy is within <3 dB in squeal amplitude for nonproportional damping factors within 500. The results provide insight into prediction and quantification of wheel squeal under mode coupling by the development and use of closed form analytical solutions. In particular, the results highlight that mode coupling can retard and eliminate squeal in addition to causing and enhancing squeal. This may help explain the enigma that squeal occurs seemingly unreliably in the field. The modelling is limited by very large nonlinear creep behavior, large non-proportional damping and complex multimodal behavior including cross coupling. Further experimental investigation is also required, however insight has been gained to identify optimal conditions for squeal. Due to the generality of the structural model used, it is hoped that the analytical results presented may provide further insight into other types of friction induced mode coupling squeal including other railway and automotive brake disk squeal. Acknowledgments The authors acknowledge the assistance and advice provided by David Anderson from RailCorp and John Powell from QR amongst many other industry and academic collaborators on wheel squeal in undertaking the work presented in this paper. Appendix A. Estimation of rail parameters The rail modal parameters were estimated as functions of contact angle qc to take into account that the direction of the rail mobility will change as the contact angle varies. The change may be approximated by rotating the impedances (inverse of the mobilities) in a similar manner to rotation of inclined stiffnesses [29] to obtain,

. 2 1 2 1 1 Yq1 c ¼ Yrv cos qc þ Yrl sin qc ; Yj ¼ cj þ mj iU þ kj ðiUÞ; j ¼ rv; rl

(A.1)

where Yj, and kj, mj, and cj represent the mobility and estimated modal stiffness, mass and damping, respectively, for the contact angle, qc, vertical, rv, and lateral, rl, directions. iU is the complex frequency of vibration to model phase. It is noted that this simplified modal analysis ignores other complexities such as cross coupling and other multimodal interactions that would require a more detailed mobility/receptance analysis to model more accurately. The modal parameters needed to use (A.1) were estimated from a generic/approximate representation of the vertical and lateral mobility/receptances in the range of 1e2 kHz from Refs. [12,27,28] as shown in Fig. A1.

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Fig. A1. a)The estimated rail vertical ( ) and lateral ( ) mobilities in the range of 1e2 kHz using single modes with modal parameters tuned as listed in Table 1 b) the resulting modal parameters as a function of contact angle as a ratio of the vertical values mrv ¼ 57.6 kg, krv ¼ 2.754E9N/m and crv ¼ 2.21E4Ns/m; mqc/mrv ( ),kqc/krv ( ) and cqc/crv ( )

In particular the mobilities were tuned to have the same undamped natural frequency (without the contact stiffness) due to lateral offset coupling [12] of 1101 Hz and a magnitude and decay similar to the mobilities/receptances modelled and measured in Refs. [12,27,28] across the frequency range of 1e2 kHz as shown in Fig. A1a). Note that a single mode model can only approximate the rail dynamic behavior across a limited frequency range and measured mobilities/receptances are required for more accuracy. The resulting modal parameters as functions of contact angle are shown in Fig. A1b) as ratios of the vertical values. This figure shows a substantial reduction in the vertical rail modal parameters as the contact angle increases until the lateral values are reached at 90 . This is somewhat consistent with the lower stiffness of the rail in the lateral direction. To investigate stiffness mode coupling the nominal rail mass and damping parameters were increased to 76.3 kg (135%) and decreased to 1097 N/m and 10970 N/m (5% and 50%), respectively. The corresponding estimated rail mobilities are shown in Fig. A2.

Fig. A2. The estimated rail vertical at 5% ( ) and 50% ( ) nominal rail damping cr and corresponding rail lateral ( )( single modes with 135% nominal rail mass tuned to investigate stiffness mode coupling.

) mobilities in the range of 1e2 kHz using

By comparing Fig. A1 a) to Fig. A2 it is seen the higher nominal rail mass drops the modal natural frequencies and the lower rail damping mainly increases the magnitude at the peaks. The peak magnitude for the lowest rail damping is unlikely to be found in practice but is used to investigate the characteristics and sensitivity of stiffness mode coupling as described in section 3.2. References [1] D. Anderson, N. Wheatley, Mitigation of wheel squeal and flanging noise on the Australian rail network, in: Proceedings of the 9th International Workshop on Railway Noise, Notes on Numerical Fluid Mechanics and Multidisciplinary Design Book Series, vol. 99, NNFM, 2008, pp. 399e405. [2] M.J. Rudd, Wheel/rail noise Part II: wheel squeal, J. Sound Vib. 46 (3) (1976) 381e394. [3] 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. [4] D.J. Thompson, C.J.C. Jones, A review of the modelling of wheel/rail noise generation, J. Sound Vib. 231 (3) (2000) 519e536. [5] D.J., G. Thompson, B. Squicciarini, L. Baeza Ding, A state-of-the-art review of curve squeal noise: phenomena, mechanisms, modelling and mitigation, Notes on Numerical Fluid Mechanics and Multidisciplinary Design 139 (2018) 3e41. [6] N. Hoffmann, M. Fischer, R. Allgaier, L. Gaul, A minimal model for studying properties of the mode-coupling type instability in friction induced oscillations, Mech. Res. Commun. 29 (2002) 197e205. [7] N. Hoffmann, L. Gaul, Effects of damping on mode-coupling instability in friction induced oscillations, ZAMM Z. Angew. Math. Mech. 83 (8) (2003) 524e534. [8] A. Pieringer, A numerical investigation of curve squeal in the case of constant wheel/rail friction, J. Sound Vib. 333 (18) (2014) 4295e4313. [9] I. Zenzerovic, W. Kropp, A. Pieringer, Influence of spin creepage and contact angle on curve squeal: a numerical approach, J. Sound Vib. 419 (2018) 268e280. [10] X.L. Cui, G.X. Chen, H.G. Yang, Q. Zhang, H. Ouyang, M.H. Zhu, Effect of the wheel/rail contact angle and the direction of the saturated creep force on rail corrugation, Wear 330e331 (2015) 554e562. [11] B. Ding, G. Squicciarini, D. Thompson, R. Corradi, An assessment of mode-coupling and falling-friction mechanisms in railway curve squeal through a simplified approach, J. Sound Vib. 423 (2018) 126e140. [12] B. Ding, G. Squicciarini, D. Thompson, Effect of rail dynamics on curve squeal under constant friction conditions, J. Sound Vib. 442 (2019) 183e199. [13] M.A. Heckl, I.D. Abrahams, Curve squeal of train wheels, Part 1: mathematical model for its generation, J. Sound Vib. 229 (3) (2000) 669e693. [14] O. Chiello, J.-B. Ayasse, N. Vincent, J.-R. Koch, Curve squeal of urban rolling stockdPart 3: theoretical model, J. Sound Vib. 293 (3e5) (2006) 710e727. noy, M. Naït, J.L. Mun ~ oz, F. Demilly, Transient models for curve squeal noise, J. Sound Vib. 293 (3) (2006) 758e765. [15] J.F. Brunel, P. Dufre [16] M.A. Heckl, Curve squeal of train wheels, Part 2: which wheel modes are prone to squeal? J. Sound Vib. 229 (3) (2000) 695e707. [17] F.G. de Beer, M.H.A. Janssens, P.P. Kooijman, Squeal noise of rail-bound vehicles influenced by lateral contact position, J. Sound Vib. 267 (3) (2003) 497e507. [18] N.M. Kinkaid, O.M. O'Reilly, P. Papadopoulos, Automotive disc brake squeal, J. Sound Vib. 267 (1) (2003) 105e166. [19] C. Glocker, E. Cataldi-Spinola, R.I. Leine, Curve squealing of trains: measurement,modelling and simulation, J. Sound Vib. 324 (1) (2009) 365e386.

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€be, P.S. Heyns, R.D. Fro €hling, Frequency domain model for railway wheel squeal resulting from unsteady longitudinal creepage, J. [20] D.J. Fourie, P.J. Gra Sound Vib. 445 (2019) 228e246. [21] P.A. Meehan, X. Liu, Modelling and mitigation of wheel squeal noise amplitude, J. Sound Vib. 413 (2018) 144e158. [22] P.A. Meehan, X. Liu, Modelling and mitigation of wheel squeal noise under friction modifiers, J. Sound Vib. 440 (2019) 147e160. [23] X. Liu, P.A. Meehan, Investigation of squeal noise under positive friction characteristics condition provided by friction modifiers, J. Sound Vib. 371 (2016) 393e405. [24] X. Liu, P.A. Meehan, Investigation of the effect of crabbing velocity on squeal noise based on a rolling contact two disc test rig, in: Proceedings of the 10th International Conference of Contact Mechanics and Wear of Rail/wheel Systems CM2015, Colorado (USA). [25] K.L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, 1985. [26] X. Liu, P.A. Meehan, Investigation of the effect of relative humidity on lateral force in rolling contact and curve squeal, Wear 310 (1e2) (2014) 12e19. [27] D.J. Thompson, Railway Noise and Vibration: Mechanisms, Modelling and Means of Control, Elsevier, Oxford, 2009. [28] M. Oregui1, A. Núenez, R. Dollevoet, Z. Li, Sensitivity analysis of railpad parameters onVertical railway track dynamics, J. Eng. Mech. 143 (5) (2017), 04017011. [29] C.M. Harris, A.G. Piersol, Harris' Shock and Vibration Handbook, fifth ed., McGraw Hill, 2002.