Dynamic response of railroad vehicles: a frequency ...

Int. J. Heavy Vehicle Systems, Vol. 15, No. 1, 2008

Dynamic response of railroad vehicles: a frequency domain approach K.V. Gangadharan Mechanical Department, National Institute of Technology Karnataka, Surathkal, 575 025, India E-mail: [email protected] E-mail: [email protected]

C. Sujatha* Mechanical Engineering Department, Indian Institute of Technology Madras, Chennai, 600 036, India E-mail: [email protected] *Corresponding author

V. Ramamurti Mechanical Engineering Department, Anna University, Chennai 600 025, India E-mail: [email protected] Abstract: A very elaborate Finite Element (FE) model and a rigid body model of a typical electrical multiple unit trailer coach are described. These models were used to find the dynamic response to track irregularities in the frequency domain. The Power Spectral Density (PSD) of track irregularities was used as input to the system. The influence of different track irregularities on dynamic response and coupling between vertical and lateral dynamics was investigated. Extensive experiments were carried out, and analytical results were compared with the measured response. Keywords: railroad vehicle; dynamic analysis; rail power spectral density; PSD; random vibration; railroad finite element model. Reference to this paper should be made as follows: Gangadharan, K.V., Sujatha, C. and Ramamurti, V. (2008) ‘Dynamic response of railroad vehicles: a frequency domain approach’, Int. J. Heavy Vehicle Systems, Vol. 15, No. 1, pp.65–81. Biographical notes: K.V. Gangadharan is an Assistant Professor in the Department of Mechanical Engineering, National Institute of Technology Karnataka, Surathkal, India. His areas of research are vibration and control, dynamics, finite element analysis, condition monitoring and experimental methods in vibration. He has teaching experience of 13 years and industrial experience of one year. He is actively engaged in industrial consultancy and sponsored research projects and has more than ten research papers to his credit.

Copyright © 2008 Inderscience Enterprises Ltd.

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K.V. Gangadharan et al. C. Sujatha is with the Machine Design Section of Mechanical Engineering Department at the Indian Institute of Technology Madras, Chennai, India. She was earlier with the Department of Applied Mechanics since 1984. She has around 26 years of experience in teaching and research in the areas of machine dynamics, vehicular vibration, acoustics, machinery diagnostics, instrumentation and signal processing. She has more than 70 research papers to her credit and has guided three PhD candidates. She has been very active in industrial consultancy projects. V. Ramamurti was a Professor in the Department of Applied Mechanics at the Indian Institute of Technology Madras, Chennai, India during the period 1977–2000. He is at present an Emeritus Professor at Anna University, Chennai. Prior to joining the teaching profession, he was a Machine Tools Engineer in Heavy Engineering Industry for three years. He has over 150 international publications and has guided 24 PhD candidates. He is actively engaged in industrial consultancy. He has written three books and is the recipient of a number of awards.

1

Introduction

The continuously varying displacement imposed at wheel rail contact point (due to track irregularities) of a railroad vehicle traversing a tangent track may be considered to be a random input exciting a dynamic system of masses, springs and dampers. By removing occasional events such as rail joints, switches, fogs, etc., the response may be determined by the theory of random vibration. The geometrical track irregularities arise from initial installation tolerances and degradation through usage. An accurate representation of these irregularities is essential for predicting the dynamic behaviour of the railroad vehicle. The four geometrical track parameters, namely, vertical profile, alignment, cross level and gauge, are commonly used to define the tangent track irregularities. Random track irregularities are well described by their PSD, and these have been used in the dynamic analysis in the frequency domain. There have been several studies on quantifying track irregularities. The Office of Research and Experiments (ORE) reports C116/RP 1−9 /EC 1971−1978 (1978), describe PSD functions of various track profiles obtained from four different railways. Iyengar and Jaiswal (1995) presented random field models of track irregularity data obtained from Indian Railways. Wickens and Gilchrist (1977) gave a detailed account of the emerging trends of railway vehicle dynamics theory and a good review of the state of art of research during 1970s. Knothe and Grassie (1993) presented a detailed review of dynamic modelling of railway track and vehicle/track interaction, with emphasis on different frequency ranges for various types of analyses. A detailed review of modelling methods for railway vehicle suspension components was presented by Eickhoff et al. (1995). These review papers give an idea of the history and present state of art of research in the field of railroad vehicles. There were many models proposed and validated using experiments; different analysis techniques have also been developed during the last few decades. Chang et al. (1979) presented a comparison of linear and non-linear mathematical models for locomotive response analysis and concluded that a linear model is sufficient for preliminary design analysis. Tanifuji (1991) described an analytical study on body

Dynamic response of railroad vehicles: a frequency domain approach

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bending vibration of a bogie vehicle for an evaluation of the ride quality. He assumed that the car body behaved like a uniform beam supported at the two bogie pivots. Published research papers specific to Indian Railways are very much limited. Wadhwa (1992) highlighted the design modifications carried out on high-tech metre-gauge coaches to improve their ride quality and to increase the maximum permissible speed to 120 kmph. Vehicles obtain track-induced input through the wheels, which generally number more than two. Analysis for vehicle response in a variable velocity run on a non-homogeneously profiled flexible track, supported by compliant inertial foundation, was presented by Yadav and Upadhyay (1992). Zhai et al. (1996) presented a new dynamic model of vehicle/track interaction, which includes both vertical and lateral dynamics. They highlighted the need to investigate the coupling between vertical and lateral dynamics using a combined model. This paper presents the dynamic response studies of the electrical multiple unit/trailer (EMU/T) coach, giving random track irregularities in the form of PSD as input to FE and rigid body models. Extensive experiments have been conducted and a comparison of measured response with predicted response has been presented. This is a part of an investigation to arrive at a suitable mathematical model, which can be used as a tool for design modification of existing suburban EMU/T coaches running at an average speed of 45 kmph.

2

Mathematical models of vehicle/track system

A typical Indian railway vehicle of the Alternating Current/Electrical Multiple Unit/Trailer (AC/EMU/T) type running on broad-gauge (1676 mm) track has been modelled. The vehicle consists of a car body, two bogies per car, four wheels and two axles for each bogie (Figure 1). The car body is connected to the bogies through the secondary suspension, which has four sets of coil springs and two dampers in the vertical direction. Wheels and axles are connected to the bogies through the primary suspension system. Four coil springs, two on either side of the wheel, constitute the primary suspension. These springs are vertically guided using dashpots fixed at the centre of the coil springs. Two approaches used for modelling the track vehicle system are those using the FE model and the rigid body model. Figure 1

A railway vehicle: component description

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K.V. Gangadharan et al.

2.1 Finite Element (FE) models The FE model of a railroad vehicle consists of substrate, track, wheels and axles, bogies and car body. To build an FE model by including all these components is a complex task; hence, a few assumptions have been made to simplify the model without compromising much on the accuracy of the model. Three different FE models were generated with increasing levels of sophistication. The simplest is the one where the underframe alone was modelled by lumping the superstructure mass and inertia. The next level was achieved by adding the superstructure framework on to the underframe model, neglecting the sheet metal covering the framework. The last and most elaborate FE model is the one where the superstructure framework and the sheet metal panels were also taken into account. A detailed account of modelling and relevant parameters was presented by Gangadharan (2001). A brief explanation of the most elaborate FE model has been presented here. In the FE model, the rail has been treated as a Beam on Elastic Foundations (BEF), i.e., the track has been considered as an Euler–Bernoulli beam resting on Winkler’s foundation (Jaiswal and Iyengar, 1997), with vertical translation and rotation about lateral axis being present. The sleeper mass was lumped at appropriate nodes and a foundation stiffness of 4 × 107 N/m as reported by Newton and Clark (1979) was used. Wheels were modelled as elements with vertical and lateral stiffness. The primary and secondary suspensions were modelled as springs with vertical degrees of freedom (dof) in the case of the vertical model and as springs with vertical and lateral dof in the case of the combined vertical and lateral model. The axle, bogie frame and underframe with sole bar, cross bearer and superstructure framework were modelled as 3D-beam elements, with all the six dof present. Sheet metal panels of end wall, side wall, roof and floor were modelled using triangular plate elements. FE formulation used for the triangular plate is as discussed by Zienkiewicz (2000). The model has 576 nodes, 866 beam elements and 768 triangular plate elements. It has 3080 active dof and a bandwidth of 216. Mass and stiffness matrices were assembled in banded form to save core and labour. The problem size is 3080 × 216 for the vertical and lateral combined model. Different views of the model are shown in Figure 2. Henceforth, this model will be denoted as Under Frame with superstructure Beam and Plate (UFBP) model.

2.2 Rigid body model An all-inclusive dynamic model of a railroad vehicle would be very large and complex; hence, size and complexity are the factors that tend to reduce physical insight into the system behaviour. It has been reported by many researchers (Garg and Dukkipati, 1984) that a relatively weak coupling exists between the vertical and lateral motions of a railroad vehicle. They suggest that it may not be required to include lateral dof in the vertical model and vertical dof in the lateral model. This helps in reducing computational time and core and makes interpretation of results easier, i.e., it would be adequate to consider the vertical translation (bounce z), rotation about x-axis (roll φ) and rotation about y-axis (pitch θ) alone for vertical response model. Similarly lateral translation (y), roll (φ) and rotation about z-axis (yaw ψ) would be sufficient for the lateral model.

Dynamic response of railroad vehicles: a frequency domain approach Figure 2

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FE model of vehicle/track system: UFBP model

In the present model, the car body, two bogies, four wheels and two axles are considered along with stiffness of primary and secondary suspensions. Dof considered are the bounce (z), pitch (θ) and roll (φ) of the car body and bogie. Besides, the wheel and axle’s bounce and roll are also taken into account, adding up to 17 dof (Figure 3). The equations of motion were written for all the masses and moments of inertia and are rearranged in matrix form as [ M ]{ x} + [C ]{x} + [ K ]{x} = {F }. Figure 3

Rigid body model of vehicle/track system

(1)

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Different FE models and rigid body models developed have been made use of for the dynamic analysis in the frequency domain. The eigenvalues and eigenvectors of all the models have been found out before attempting the dynamic analysis and has been reported (Gangadharan et al., 1999, 2001). Suspension damping of primary and secondary stages are included in the model for dynamic analysis, and no structural damping has been assumed. Damping used for each secondary suspension is 600 N s/cm and for primary suspension 200 N s/cm (details from coach manufacturer, Integral Coach Factory, Perambur, India). It has been found that the most elaborate FE model, i.e., UFBP model, predicts dynamic response close to the measured response, and all other models underestimate the response. Hence, even though studies were carried out with all the models, results are presented only for UFBP and rigid body models.

3

Track inputs for dynamic studies

The continuous track geometry variations, namely, vertical profile (Zv), alignment (Ya), cross level (Zc) and gauge (Yg) are the primary dynamic inputs to a railroad vehicle. These irregularities are random in nature and are well described by their PSD. The PSD of the track irregularities can be directly used as inputs to the models in the frequency domain. Track PSDs of the Indian rail tracks were obtained from Iyengar and Jaiswal (1995), the origin of which can be traced to Research and Design Standards Organisation, Government of India, Ministry of Railways. The track PSDs used for the present work have been shown in Figure 4 as spatial PSDs. Figure 4

Geometrical track irregularity PSDs in spatial domain (mm2/(cycle/m) vs. cycle/m)


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For analysis at different running speeds of the vehicle, these PSDs have to be converted to temporal PSDs. The spatial to temporal domain conversion is done using the relations shown in equations (2) and (3). f(Hz) = Ω(c/m) × v(m/s)

(2)

Sd(f) = Sd(Ω)/v

(3)

where v is the vehicle speed, Sd(f ) and Sd(Ω) are the temporal and spatial PSDs. Based on the experimental results reported by Garivaltis et al. (1980) and Nigam and Narayanan (1994), it is assumed that all irregularities (Zv, Zc, Ya and Yg) are homogeneous, Gaussian random processes with zero mean and are mutually uncorrelated.

4

Solution techniques

A railroad vehicle can be treated as a system with eight random loadings, i.e., random input disturbances due to the track irregularities at each of the eight rail wheel contact points. If the input from the left rail is completely correlated with that of the right rail, then the system can be simplified to a case of four random loadings. For a single random loading to a system, the response is given by the following equation: S x ( f ) = |α (if )|2 S p ( f ).

(4)

Equation (4) gives the simple relationship between the spectral densities of excitation and response. The spectral density of the displacement at any frequency is equal to the spectral density of the exciting force at that frequency, multiplied by the square of the modulus of the receptance at that frequency. Here Sx(f ) is the output displacement PSD, Sp(f ) is the input force PSD and α(if ) is the ratio of displacement at any point to a unit sinusoidal force as the input. Let p(t), q(t), r(t) and s(t) be the random loads acting simultaneously on the railroad vehicle at wheel rail contact points (Figure 5). Let the corresponding receptances for harmonic excitation be αxp, αxq, αxr and αxs. From random vibration theory (Robson, 1964), it can be shown that the spectral density of the response x(t) can be expressed in terms of the spectral densities of the input random loads, p(t), q(t), r(t) and s(t) as shown in equation (5). S x ( f ) = α *xp α xp S p ( f ) + α *xp α xq S pq ( f ) + α *xp α xr S pr ( f ) + α *xp α xs S ps ( f ) + α *xq α xp Sqp ( f ) + α *xq α xq S q ( f ) + α *xq α xr S qr ( f ) + α *xq α xs S qs ( f ) + α *xr α xp S rp ( f ) + α *xr α xq S rq ( f ) + α *xr α xr S r ( f ) + α *xr α xs S rs ( f )

(5)

+ α *xs α xp S sp ( f ) + α *xs α xq S sq ( f ) + α *xs α xr S sr ( f ) + α *xs α xs S s ( f )

where * denotes complex conjugation and f is the forcing frequency. Spq, Spr, Sps, Sqp, Sqr, Sqs, Srp, Srq, Srs, Ssp, Ssq and Ssr are cross spectral densities of forces p(t), q(t), r(t) and s(t). It is assumed that the input is space correlated between the successive wheels; hence, q(t), r(t) and s(t) reproduce p(t) after time lags of τ1, τ2 and τ3.

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Figure 5

Points of application of random load

Therefore Sp(f ) = Sq(f ) = Sr(f ) = Ss(f ). Cross spectral densities of the forces can be written as equation (6) S pq = ei 2π f τ1 S p ( f ), S pr = ei 2π f τ 2 S p ( f ), S ps = ei 2π f τ 3 S p ( f ) S qr = ei 2π f τ 4 S p ( f ), S qs = ei 2π f τ 5 S p ( f ), S rs = ei 2π f τ 6 S p ( f ).

(6)

Phase angle φ can be written as

φ1 = 2π fb1 / v, φ2 = 2π fb2 / v, φ3 = 2π fb3 / v φ4 = 2π fb4 / v, φ5 = 2π fb5 / v, φ6 = 2π fb6 / v

(7)

b1–b6 are the distances between the points of application of loads, i.e., distance between wheel rail contact points as shown in Figure 5 and Table 1. v is the speed of the vehicle in m/s and f is the forcing frequency. Table 1

Distances between the points of application of loads

Length of the coach

20.726 m

b1

2.896 m

b2

14.63 m

b3

17.526 m

b4

11.734 m

b5

14.63 m

b6

2.896 m

The time lag τ is given by

τ1 = b1 / v, τ 2 = b2 / v, τ 3 = b3 / v, τ 4 = b4 / v, τ 5 = b5 / v, τ 6 = b6 / v.

(8)


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Rewriting equation (5) by substituting equation (6) gives equation (9)

{

2

2

2

2

S x ( f ) = α xp + α xq + α xr + α xs + 2 α xp α xq cos φ1 + 2 α xr α xp cos φ 2 + 2 α xs α xp cos φ3 + 2 α xq α xr cos φ 4 + 2 α xq α xs cos φ5 + 2 α xr α xs cos φ6

(9)

} S ( f ). p

The track inputs as shown in Figure 4 were sampled at 256 spatial frequency steps and converted to corresponding temporal frequency steps, depending on the running speed of the vehicle. The force due to track irregularities was given as a base excitation. In order to find out the receptance, an input force of kpX sin ωt was given to the system at each primary suspension. Here kp is the stiffness of individual primary suspension (there are four such suspensions per bogie and eight per coach) and X is the unit displacement at frequency ω. The output corresponding to this input is the receptance. The square of the modulus of the receptance at a frequency multiplied by the spectral density of the exciting force yields the response PSD at that specific frequency.

5

Analytical results

The analytical studies were carried out using FE and rigid body models. The undulation (vertical profile) and cross level are the vertical inputs, and the alignment and gauge are the lateral inputs. These inputs were given as individual vertical and lateral inputs and also as combined vertical and lateral inputs. Analysis was conducted to find the contribution of each input. In the figures that follow, the notations used to represent input PSDs are U − undulation or vertical profile PSD, X − cross-level PSD, A − alignment PSD, G − gauge PSD.

5.1 Vertical dynamics As a first step towards studying the dynamic behaviour of the railroad vehicle in the frequency domain, random vertical track irregularities were given as input. The vertical undulation PSD was given as input and the responses at car cg, bogie cg and axle cg were found out. Both FE and rigid body models were used for analysis. Similarly, the dynamic response to combined vertical profile (undulation) and cross-level PSDs was determined. Figure 6 depicts the acceleration response of car cg and bogie cg to undulation PSD and cross-level PSD. The contribution of the response from each of these inputs can be clearly seen from the figure.

5.2 Lateral dynamics The alignment and gauge PSDs are the two inputs for lateral dynamic studies. The lateral dynamic responses at various points were predicted using FE model, and the dynamic response contribution from each of these inputs was found out. Figure 7 shows the influence of alignment and gauge PSDs on lateral response amplitude at car cg and bogie cg.

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Figure 6

Acceleration response of car cg and bogie cg with different vertical inputs: U: undulation or vertical profile PSD as input and X: cross-level PSD as input

Figure 7

Acceleration response of car cg and bogie cg with different vertical inputs: A: alignment PSD as input and G: gauge PSD as input


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5.3 Combined vertical and lateral dynamics In most of the dynamic studies of railroad vehicles reported, it is assumed that the vertical and the lateral dynamics are uncoupled. In the present work, the coupling between the vertical and lateral dynamics has been investigated. This was carried out using a combined vertical and lateral FE model (UFBP model) and studying the influence of each input on the response. Figure 8 clearly indicates how the vertical inputs influence the lateral response and vice versa. The vertical input along with the lateral inputs predicts higher vertical response acceleration. In the case of lateral response, there is no change in the spectral components irrespective of whether the input given is lateral alone or combined lateral and vertical. From the eigenvalue analysis (Gangadharan et al., 1999), it has been seen that there are only two predominant lateral modes within the 0−15 Hz frequency range; these modes get excited and can be seen in the response. On the other hand, in the vertical response PSD, along with increase in the response amplitude, there are some additional spectral components that appear when the lateral inputs are given along with the vertical inputs. Figure 8

Acceleration response of car cg: influence of vertical and lateral inputs: A: alignment PSD as input, G: gauge PSD as input, U: undulation or vertical profile PSD as input and X: cross-level PSD as input

The dynamic analysis using the rigid body model can predict the response at the car cg only, whereas with the FE model it is possible to find the dynamic response of each node on the car body. This is specifically useful to predict the ride index at various locations of a coach. The rms acceleration response of different points along the length of the coach has been calculated using the FE model and plotted for various speeds in Figure 9. This figure shows that rms acceleration level along the solebar is not the same at all points, as predicted by the rigid body model.

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Figure 9

6

The vertical rms acceleration response along solebar length

Experimental studies

A detailed experimental study on the dynamic behaviour of an AC/EMU/T running on a broad-gauge suburban track has been carried out. Details have been presented by the author. Acceleration and displacement levels of different points on the coach and bogies were measured when the vehicle was running at a speed of 45 kmph. Measurements of acceleration at each point in three directions (longitudinal, lateral and vertical) were made. Displacements were measured only on the floor of the coach. The measurements were required to bring out the response of the railroad vehicle in the low- and medium-frequency range. Hence, inductive type of accelerometers and linear variable differential transformer-type displacement pickups, with carrier frequency amplifier for signal conditioning, were made use of. The conditioned signals were recorded in an instrumentation tape recorder for further analysis. The analogue signals from the instrumentation tape recorder were digitised using an analogue-to-digital converter interfaced to a computer. Using MATLAB, the digitised time domain data were analysed and frequency spectrum and PSD were obtained.

7

Comparison with experimental results

The measurements were carried out at 45 kmph under normal running conditions. Figure 10 shows a typical measured acceleration response of a point near the car cg on the coach floor and analytical response at the car cg. Table 2 shows the spectral components of analytical and measured response. From the figure and the table, it can be seen that there is a very close match between the measured and predicted acceleration responses. Most of the spectral components in the measured response are reflected in the


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predicted response also, but the amplitude of measured response is higher than that of the predicted response at many frequencies as shown in Table 2. The analytical and measured lateral acceleration response of the car cg has been plotted in Figure 11. Corresponding spectral components are shown in Table 3. The measured response shows a strong peak at 0.916 Hz, whereas the corresponding analytical peak is small. The reason for this is that, in the UFBP model, the wheel and rail are assumed to have the same displacement at the contact point. From eigenvalue analysis, it is seen that the rigid body model (incorporating creep effects) clearly showed a natural frequency around 0.8 Hz. This is not reflected in the FE model due to the above-mentioned assumption (Gangadharan et al., 1999). Figure 10 Measured and analytical vertical acceleration response at car cg

Table 2

Spectral components of vertical acceleration PSDs at car cg Measured response

Frequency (Hz)

Acceleration PSD (g2/Hz) × 10−4

Analytical response Frequency (Hz)


0.976

1.494

0.966

5.576

1.831

4.463

1.812

2.051

3.051

1.839

2.658

0.584

4.028

3.345

4.409

3.240

9.033

3.418

8.456

2.914

11.96

0.681

11.90

1.179

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Figure 11 Measured and analytical lateral acceleration response of car cg

Table 3

Spectral components of lateral acceleration PSDs at car cg Measured response

Frequency (Hz)


Analytical response Frequency (Hz)

0.916

60.00

0.785

1.984

40.00

1.752

2.899

4.314

9.436

9.874

9.422

–

13.356

4.261

13.290

Acceleration PSD (g2/Hz) × 10−4 2.471 60.00 – 50.00 6.237

The comparison of analytical and measured displacement response at the car cg in the lateral direction is shown in Figure 12. Table 4 gives the details of spectral components in the measured and predicted responses. The comparison of predicted and measured response (Figures 10–12) shows qualitative agreement, with most of the measured spectral components being present in the predicted response and with a larger amplitude of measured value. This can be attributed to incomplete inputs used for analytical prediction, system parameter variation and possible small speed variations during measurement. Along with the track irregularities, irregularities on wheels such as wheel flats might have been present in the normal running condition. The coaches used for measurement were the same as the one modelled, but a difference in system properties due to extensive use might have been


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present. The track PSD used for analysis was measured on a continuously welded main line track, whereas the response measurements were carried out on suburban tracks with short-welded rails. Figure 12 Measured and analytical lateral displacement response of car cg

Table 4

Spectral components of lateral displacement PSDs at car cg Measured response Displacement PSD (mm2/Hz)

Frequency (Hz)

Displacement PSD (mm2/Hz)

0.785

43.269

0.725

41.03

1.221

21.900

–

–

1.709

19.439

1.752

39.86

2.035

28.135

–

–

Frequency (Hz)

8

Analytical response

Summary and conclusions

The dynamic analysis in the frequency domain of the railroad vehicle, with the random track irregularity PSD as input, has been carried out using different mathematical models. Comparison of predicted response of different models has been presented.

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Extensive comparison of analytical and experimental results has been shown. The spectral components present in measured and analytical results have been tabulated to give a quantitative comparison. It has been found that the rigid body model underestimates the response and the FE model predicts a response closer to the measured response. This indicates that inclusion of structural elasticity improves the response prediction. The measured response shows a higher amplitude and almost the same frequency components. This higher amplitude of measured response can be attributed to incomplete inputs used for analytical prediction, system parameter variation and possible small speed variations during measurement. There could be a small variation in the location of point on which measurement was done and the point on which response was predicted. The measured response shows some additional spectral components, which are not present in the predicted response. This can be due to the additional inputs such as wheel flats and other occasional track discontinuities, which are not included in the track PSD used. Besides, it is seen from the FE analysis of combined vertical and lateral model that there is coupling between the vertical and lateral dynamics, contrary to the results published in the past by many researchers using rigid body models.

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Robson, J.D. (1964) An Introduction to Random Vibration, Elsevier Publishing Company, New York. Tanifuji, K. (1991) ‘An analysis of the body bending vibration of a bogie vehicle for an evaluation of the ride quality with deflated air springs’, Journal of Rail and Rapid Transport, Vol. 205, pp.35−42. Wadhwa, K.B.L. (1992) ‘Design of high−tech metre gauge coaches on Indian railways’, Journal of Rail and Rapid Transit, Vol. 206, pp.137−143. Wickens, A.H. and Gilchrist, A.O. (1977) Railway Vehicle Dynamics − The Emergence of a Practical Theory, Council of Engineering Institute, MacRobert Award Lecture, London, February, pp.1−29. Yadav, D. and Upadhyay, H.C. (1992) ‘Dynamics of vehicle in variable velocity runs over non-homogeneous flexible track and foundation with two point input models’, Journal of Sound and Vibration, Vol. 156, No. 2, pp.247−268. Zhai, W.M., Cai, C.B. and Guo, S.Z. (1996) ‘Coupling model of vertical and lateral vehicle/track interactions’, Vehicle System Dynamics, Vol. 26, pp.61−79. Zienkiewicz, O.C. (2000) Finite Element Method in Engineering Science, McGraw-Hill, New York.