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Copyright SFA InterNoise 2000 2, Figure 1 Example of friction coefficient for rolling contact V 0 20 m s. with s0 G Young modulus of steel C 22 Kalkar constant C22 2 39 1 36 a b. For squealing wheels the vertical contact force F0 fx t the lateral contact force F0 s0 fy t and. the lateral slip s0 vsy t V0 consist of a constant and a time variant part Their relation is. Fy F0 Fx F0 Fx vsy 3, If the situation of starting squeal is considered in which the vibration amplitude is still very small the. non linear friction coefficient s can be linearized and equation 3 becomes. Fy F0 Fx F0 Fx vsy 4, Transformation of the time variant part from the time domain to the frequency domain gives. Fy Fx s0 Vsy 5, This equation gives the lateral force as a result of vertical force and creepage In turn the vertical force. and the creep velocity are a function of the lateral force as described in the next section. 3 WHEEL AND RAIL DYNAMICS, The TWINS software package 2 is used the to calculate the wheel rail and contact spring mobilities.

YW xx YW yy YW yx vertical lateral and cross mobility for the wheel in the contact point. YRxx YRyy YRyx vertical lateral and cross mobility for the rail in the contact point. YCx YCy vertical and lateral mobility of the contact spring. Copyright SFA InterNoise 2000 3, The squeal noise appears at wheel resonance frequencies At these frequencies the following relations. hold YW yx YRyx YW yy YRyy and YW yy YCy As a consequence the rail vibrations can be. neglected compared to the wheel vibrations Using these simplified relations to describe velocity over. the friction element see figure 2 yields,Fx YW xx YRxx YCx Fy YW xy YRxy 0 6. Fx YW yx Fy YW yy Fy YCy 0 7, Figure 2 Modelling of wheel and rail dynamics contact springs and friction elements. The mobilities YW xx and YW yx depend on the lateral offset of contact position on the wheel xyW. YW yx YW yx0 xyW YW yz0 8,YW xx YW xx0 2xyW YW xz0 x2yW YW zz0 9. Y Wyx 0 Y Wyz 0 Y Wxx 0 Y Wxz 0 Y Wzz 0 cross mobilities between vertical and lateral forces and moment. about the z axis in the nominal lateral contact position on the wheel thread. Substitution of equations 6 9 gives the transfer functions for Fx Fy and Vxy Fy Figures. 3 and 4 show the influence of the contact position on the transfer functions Vxy Fy and Fx Fy These. transfer functions determine the loop gain H and the dominant squeal noise frequency see sections. 4 NYQUIST CRITERIUM FOR STABILITY, Combination of the equations for contact mechanics and wheel dynamics gives a loop gain for the lateral.

contact force,Fy F0 Fx F0 Fx vsy 10, The instability of the system can be determined by the Nyquist criterion which states that the system is. unstable the wheel squeals for frequencies where the Nyquist contour H passes the real axis at the. right side of 1 corresponding to a loop gain H larger than 1 and a phase shift of 0 The Nyquist. contour shows that squeal noise can occur at the frequencies of the axial wheel modes. 5 DOMINANT FREQUENCY AMPLITUDE AND HIGHER HARMONICS. Heckl 3 shows that squeal noise occurs only at the resonance frequency of the mode with the largest. growth factor The growth factor is proportional to the ratio of the loop gain over the modal mass of. Copyright SFA InterNoise 2000 4, Figure 3 Transfer function ReFx Fy for axial modes dependent of contact position xyW. the corresponding mode The modal masses of the axial modes are approximately equal Hence the. frequency of the mode with the largest loop gain becomes the dominant squeal noise frequency 1. The non linear relationship between friction coefficient s and lateral creepage s causes higher har. monics in the force and vibration spectra The lateral vibration of the wheel will therefore consist of the. dominant frequency and higher harmonics k 1 with k 1 2 3. The amplitude of the squeal oscillations is calculated in the time domain The relationship vsy Fy is. sampled only for the frequencies k 1 and transformed to the time domain The relation Fx 1 Fy 1. is only used for the dominant frequency 1 This sampling reduced calculation time and is only allowed. after determination of the dominant frequency As a consequence the spectrum of the squeal noise. consists of the dominant frequency and higher harmonics of the dominant frequency Several parameters. as lateral contact position on the wheel thread determine which of the wheel resonant frequencies becomes. predominant,6 CONCLUSIONS, Based on existing models for contact mechanics contact dynamics wheel dynamics and rail dynamics. a frequency domain model is developed An essential step in the transformation from time to frequency. domain is the linearization of the friction coefficient which what is allowed for small vibration amplitude. The frequency domain model is used to determine whether or not squeal noise occurs and at which. frequency The model provides a short calculation time and an insight in related frequency dependent. phenomena The calculation of the dynamic force and vibration amplitudes and higher harmonics are. performed in the time domain The model predicts squeal noise for given wheel rail wheel load lateral. contact position on wheel thread rolling velocity and rolling angle The model is validated and used as. a design tool for measures against squeal noise, Wheel dynamics play a significant role in squeal noise Squeal occurs at one of the resonant frequencies. of axial wheel modes The lateral contact position on the wheel tyre influences which resonant frequency. becomes dominant,Copyright SFA InterNoise 2000 5, Figure 4 Transfer function ReVsy Fy for axial modes dependent of contact position xyW.

ACKNOWLEDGEMENTS, The research projects were funded by the Dutch Railways Rail Infrastructure Management NS RIB. Information and Technology Centre for Transport and Infrastructure CROW Dutch Ministry of Housing. Spatial Planning and the Environment VROM and the tram companies of Amsterdam GVBA Rotterdam. RET and Den Haag HTM Institute of Sound and Vibration Research NS Technisch Onderzoek and. NedTrain Consulting were research partners,REFERENCES. 1 U Fingberg A model of wheel rail squealing noise Journal of Sound and Vibration Vol 143. pp 365 377 1990, 2 D J Thompson Wheel rail noise generation part II wheel vibration Journal of Sound and. Vibration Vol 161 3 pp 401 419 1993, 3 M A Heckl Curve squeal of train wheels part 2 which wheel modes are prone to squeal. Journal of Sound and Vibration Vol 229 3 pp 695 707 2000. 4 P P Kooijman and al Curve squeal of railbound vehicles part 2 set up for measurement of. creepage dependent friction coefficient Internoise 2000 2000. 5 M H A Janssens and al Curve squeal of railbound vehicles part 3 measurement techniques. for wheel rail contact velocities and forces at squeal noise frequencies In Internoise 2000 2000. inter noise 2000 The 29th International Congress and Exhibition on Noise Control Engineering 27 30 August 2000 Nice FRANCE I INCE Classi cation 1 3 CURVE SQUEAL OF RAILBOUND VEHICLES PART 1 FREQUENCY DOMAIN CALCULATION MODEL F De Beer M Janssens P P Kooijman W J Van Vliet TNO Institute of Applied Physics P O Box 155 2600 AD Delft Netherlands Tel 31 15 2692398 Fax 31

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