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The Rail Technology Unit

Validation of a Matlab Railway Vehicle Simulation using a Scale Roller Rig Iwnicki S.D, Wickens A.H.

This article was download from the Rail Technology Unit Website at MMU Rail Technology Unit, Manchester Metropolitan University, Department of Engineering & Technology, John Dalton Building, Chester Street M1 5GD, Manchester, United Kingdom http://www.railtechnologyunit.com

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VALIDATION OF A MATLAB RAILWAY VEHICLE SIMULATION USING A SCALE ROLLER RIG. Iwnicki S.D, Manchester Metropolitan University U.K. Wickens A.H. Loughborough University U.K. SUMMARY A 1/5 scale roller rig has been constructed for use in analysing the dynamic behaviour of railway vehicles. The roller rig includes a servo hydraulic system to allow a realistic input of track irregularities at the rollers and instrumentation is fitted to allow measurement of the position and acceleration of the bodies. This paper reports on the First stage in validating the behaviour of the roller rig using a relatively simple, linear computer model constructed in MATLAB. Initial results show good correlation between the behaviour seen on the roller rig and that predicted by the model. NOTATION The following notation is used in this paper: (see also Fig. 4b) A

inertia matrix

B

creep damping matrix

C

creep stiffness matrix

c

semi-distance between bogie centres

D

suspension damping matrix

E

elastic stiffness matrix

f

creep coefficient, f,, * is the lateral creep coefficient for the first wheelset, etc

h

semi-distance between wheelsets

I

moment of inertia of wheelset

k

factor on suspension stiffness

k,

suspension element stiffness

2 I

semi-distance between wheelset contact points

m

wheelset mass

mb , md

bogie frame mass, leading and trailing

mc

car body mass

q

column of generalised co-ordinates

ro

wheel radius, wheelset in central position

t

time

V

vehicle speed

yi

lateral displacement of ith wheelset

ì.

real part of eigenvalue

λ

equivalent conicity

λi

eigenvalue

ωi

imaginary part of ith eigenvalue

ψi

yaw displacement of ith wheelset

d1, d2, d3

vertical suspension distances

Suffices b

frame, leading bogie

c

car body

d

frame, trailing bogie 1. INTRODUCTION The dynamic behaviour of railway vehicles relates to the motion or vibration of all the parts of the vehicle and is influenced both by the vehicle design, particularly the suspension, and the track on which the vehicle runs. It has implications for the

comfort of passengers or damage to freight, for wheel and rail wear and noise generation and for the safety of the railway system. The Rail Technology Unit at Manchester Metropolitan University (MMU) is currently investigating potential improvements to suspension design for railway vehicles. The Group employs computer simulation techniques to evaluate the effects of these design changes. Analysis of railway vehicle dynamics is a subject which has advanced rapidly since the advent of digital computing and, more

3 recently, has been used to solve the problems posed by very high speed trains. Models of typical coaches have been set up and their response to track of varying quality simulated. Using these models it is possible to analyse the effect of design changes in the suspension and the results of improvements to track alignment. In order to validate their predictions the Group has also set up a dynamic testing laboratory. The testing of full size vehicles is prohibitively expensive for this type of advanced research so a 1/5 scale roller rig has been designed and manufactured in the Department of Mechanical Engineering, Design and Manufacture at MMU. The roller rig is being used to examine a number of novel ideas for improved suspension components and layout. Together with Professor Wickens from Loughborough University we have used a simple linear computer simulation model written with MATLAB to predict the behaviour of the roller rig and to confirm the calculated effect of the effects of scaling and of using rollers in place of rails. One of the most important and most complicated factors influencing the dynamic behaviour of a railway vehicle is the interface between the wheel and the rail. There is often a highly non-linear relationship between the motion of the vehicle and the forces on the wheels. In order to solve the equations of motion and thus predict the dynamic behaviour, knowledge of the wheel and rail geometry and an understanding of the material properties in the contact region are required. Analytical techniques and software are available to allow reliable prediction of the wheel-rail forces but it is still necessary to validate the models against experimental data to ensure a high degree of confidence in the results. 2.1. Specification for Roller Rig The main purpose of the rig was to simulate the behaviour of a railway vehicle and to include as accurate as possible a recreation of the forces within the contact between wheels and rails/rollers. These forces strongly affect the lateral behaviour of the vehicle but less so the vertical behaviour and therefore, to keep complexity as low as possible, the rig was designed to consider only lateral motion

4 and yaw of the wheelsets and rollers. The bogie and main body are additionally free to move vertically and to roll and pitch. To provide a realistic excitation the requirement was for lateral and yaw excitation at the rollers. In order to allow suspensions of high speed vehicles to be examined the maximum design speed of the vehicle being simulated was set at 250 mph. It was felt to be of great importance that flexibility in the selection of as many parameters as possible be allowed and ease of changing the type of vehicle simulated be maintained. 2.2. Design of Roller Rig The roller rig incorporates two sets of rollers which can be driven at up to 250 mph scale speed and can be exited laterally and in yaw to represent track irregularities. A 1/5 scale bogie sits on top of the rollers and is restrained in a way that simulates the constraints for the full size vehicle. The wheelbase and gauge of the rollers can be easily varied by a system of sliding bearings. The drive is provided by an electric motor and controller driving through a belt and pulleys. The rollers at either side of a wheelset are connected with splined and Hooke-jointed shafts to ensure a positive coupling across the wheelset. Excitation of the rollers to represent track irregularities is controlled by a digital controller and a system of hydraulic actuators. The primary suspension consists of a set of 8 rubber bushes mounted between the axleboxes and the bogie. These bushes can be changed to obtain suitable stiffness and damping parameters. The mass and inertia properties of all the bodies has been chosen to represent a typical high speed passenger coach (the BR Mk4 passenger coach). Secondary suspension is provided by small air springs and yaw dampers with a Watt linkage between the bogie and the coach body. In practice the secondary suspension seems to provide a rather hard connection and modifications are being made. Wheel profiles are machined scale versions of BR P8 and the rollers have a scale BS110a rail profile with no rail cant. The roller rig is shown in Figures 1 and 2.

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Fig.1 Side elevation of the roller rig.

Fig. 2. The roller rig.

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Fig. 3. Plan view of the roller rig showing the actuators.

2.3. Excitation of Rollers Each roller is mounted in a yoke which is connected to two lateral actuator shafts. This arrangement allows the rollers to be excited laterally and in yaw as required for accurate simulation of lateral track irregularities. All four rollers are held in a sub frame which may be modified to allow cant deficiency to be varied and (with a little more effort) vertical excitation of the rollers to be provided if required at a later stage. The actuator arrangement is shown in Figure 3. A digital controller and computer program follows an algorithm that has been developed to control the shakers which are required to excite each roller pair. These motions then accurately simulate the effect of the rail misalignments in the lateral direction. The maximum force capacity of each actuator is 0.5 kN. Oil flow in the system is sufficient to allow signals of over 20 Hz to be followed reliably.

7 2.4. Instrumentation Instrumentation is used to monitor the dynamic behaviour of the scale bogie and the results are stored for later analysis and comparison with the computer predictions. For acquisition of the data required from the roller rig a micro-computer based system has been set up. This uses four channels of analogue input which are simultaneously sampled and digitised. The data is required to compare the behaviour of the vehicle with mathematical simulations that have been carried out and to assess vehicle performance against safety and passenger comfort criteria. The effects of changes to the suspension of the .vehicle can therefore be assessed at a range of speeds and track conditions. The Computer models that have been set up produce results in the form of a frequency spectrum of vibrations or as a time history of the selected parameter. The roller rig is required to validate this data and therefore accelerometers and displacement transducers are mounted in suitable locations on the body and the wheelsets of the roller rig. The signals are conditioned and digitised and stored on the computer. The computer package 'Spiders' is used to monitor the four channels simultaneously. The signals are combined in the required way and can then be stored for presentation and comparison with the simulation results. Fast Fourier Transform routines are used to provide and display the frequency spectrum of the various signals. 2.5. Errors in Roller Rig The errors inherent in using a roller rig with rollers which have a finite radius have been analysed carefully and the work of many previous research groups studied. The errors fall into two groups: •

Errors caused by the scaling factor used.

•

Errors caused by use of rollers with a finite radius rather than rails.

A study of the magnitude of each error under these headings has been carried

8 out for the roller rig. A re-evaluation of these errors and their effects on the behaviour of the roller rig will be published soon. Others have also looked at these effects, see e.g. (1), (2), (3). The conclusions are that the errors in scaling can be quite small if care is taken but that errors due to the roller diameter are more difficult to eliminate. The cumulative effect of these errors could result in a change of about 10% in the critical speed and in other parameters that have been examined. The error, however, seems to be very sensitive to certain parameters and speed and it is to establish a rough value of the error in this case that the current project was started. THE MATLAB COMPUTER MODEL 3.1. General Description of Full Model A lateral model of an idealised four-axle vehicle with a body and two bogies has been set up as shown in Figure 4. Each bogie has a frame and two wheelsets. The wheelsets, bogie frames and car body are all assumed to be rigid and connected by massless elastic structures and massless damper elements.

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Fig. 4. (a) The full vehicle model, (b) Detail of the full vehicle model. The vehicle is symmetric about a longitudinal plane of symmetry but the bogies are not assumed to be symmetric. Therefore, for small displacements, motions which are symmetric about this plane of symmetry are decoupled from those which are

10 anti-symmetric and will not be considered here. Accordingly, the bogie frames and the car body each have three degrees of freedom, (lateral translation, roll and yaw) and each wheelset has two degrees of freedom, (lateral translation and yaw). Thus the motion of the vehicle is defined by the following set of seventeen generalised co-ordinates

q = { y1ψ 1 y bϕ bψ b y 2ψ 2 y c ϕ cψ c y 3ψ 3 y d ϕ dψ d y 4ψ 4 }.

(1)

yB , yD and y c refer to lateral translation of the bogie frames and vehicle body ψ b ,ψ d and ψ c refer to yaw of the bogie frames and vehicle body. The other y1 ,ψ1

are standard wheelset co-ordinates. For motions within the flangeway clearance the principal nonlinearity arises from the wheel-rail geometry. This has been measured for the rollers and wheels on the roller rig using a micro-topographer and analysed using the RSGEO routine in MEDYNA. The values of conicity at various lateral displacements of the wheelset were thus evaluated and are shown in Figure 5. A describing function approach has been used to give values of equivalent conicity as a function of lateral displacement amplitude. As the creepages are small in the motions of interest here linear creep coefficients have been derived using Kalker's linear table. In addition the terms associated with spin creep and gravitational stiffness largely cancel out and have been neglected, as have the small gyroscopic terms. Consequently, the equations of motion will be of the form [ As 2 + ( B / V + D) s + C + E ]{q} ={ Q}

(2)

where A is the inertia matrix, B and C the creep damping and stiffness matrices, and D and E are the suspension damping and elastic stiffness matrices describing the properties of the connections between wheelsets and vehicle body. V is the forward speed of the vehicle.

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Fig. 5. Plot of conicity against relative lateral displacement of the wheelset.

A=diag

[m1 I z1m b Ixb I zbm 2 I z 2m c IxcI zcm 3 Iz 3md IxdI zdm 4I zv ]. B11 = 2 f 22

B66 = 2 f 22

(1)

(2)

B22 = 2 f 11 12

B77 = 2 f11 12

B11,11 = 2 f 22

B16,16 = 2 f 22

(1)

(3)

B12,12 = 2 f 11 12 (3)

C12 = −2 f22

C11,12 = − 2 f 22

(2)

(3)

(4)

(4)

2)

C16,17 = − 2 f22

(4)

C21 = 2 f11 λ11 / r1

C76 = 2 f11 λ21 / r2

C12,11 = 2 f11 λ3 1/ r3

C17,16 = 2 f11 λ4 1 / r4

(1)

(3)

all other Bij and Cij = 0.

(4)

B17,17 = 2 f11 12

C67 = − 2 f 22

(1)

(3)

(2)

(4)

(5)

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The vehicle is considered to be a system of rigid bodies possessing 17 degrees of freedom connected by 18 massless elastic elements. The strain in these elastic elements can be represented by the 18x1 matrix ε , and is related to the generalised displacements q by the equation of compatibility ε = aq

(6)

so that a is the 18X17 compatibility matrix, given in Appendix 1. Each row in a defines a structural element. The stresses (T in the elastic elements are related to the strains by σ = kε

(7)

It follows from the principle of virtual work that the elastic stiffness matrix E is given by E=aTka,

(8)

where k is an 18x18 diagonal matrix of stiffnesses (also given in Appendix 1) corresponding to the strains represented by the rows of a. An exactly analogous analysis applies to the suspension damping matrix D. Stability may be considered by making a trial solution in which q is proportional to exp ( λ t ) and solving the resulting characteristic equation of Equations (2)

Aλ 2 + ( B / V + D) λ + C + E = 0.

3.2. Application to Roller Rig The parameters of the roller rig were measured and input into the general model described above (see Appendix 2). There is only one bogie on the roller rig and the car body is constrained by a pinned joint at its centre. Thus in the model Yc =0, and this is carried through in

(9)

13 the simulation. In addition, at this stage of the development of the rig, the lateral stiffness of the secondary suspension k yb was found to be very large and unrepresentative. In the simulation this was accounted for by setting k yb = 0. The result is now a 14 degree of freedom model. 4. RESULTS 4.1. Results of Simulation The MATLAB simulation was carried out at gradually increasing speeds and the eigenvalue locus plotted. This is shown in Figures 6 and 7. The critical speed is clearly seen to be at 8.3 m/s where the real part of the eigenvalue becomes positive. The imaginary part of the eigenvalue at the same point is 7 giving a frequency of I.I Hz.

Fig. 6. Plot of the real part of the eigenvalue against simulation speed.

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Fig. 7. Plot of the real part against the imaginary part of the eigenvalue

Fig. 8. Plot of the measured lateral acceleration (bogie & body) at 7 m/s

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Fig. 9. Plot of the measured lateral acceleration (bogie & body) at 8 m/s 4.2. Results Measured at the Roller Rig The roller rig was run at the same speeds as the MATLAB simulation and results at 6 m/s and 8 m/s are shown in Figures 8 and 9. Instability started to occur at just below 8 m/s and can be clearly seen at 1.2 Hz. 5. CONCLUSIONS The occurrence of the hunting instability in a 1/5 scale roller rig has clearly been predicted by a linear MATLAB model of a bogie vehicle. Both the speed of onset of the instability and the frequency at which the resulting oscillations occur match very well between the simulation and the measured behaviour at the roller rig. Although the model may not accurately represent aspects of the behaviour of the roller rig which are more strongly related to the non-linearities which are present and although the roller rig itself includes errors due to scaling and the finite radius of the roller, this work provides the basis for a more thorough re-examination of the effects of these errors and the usefulness of a roller rig for analysis of railway vehicle behaviour.

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REFERENCES 1. A. Jaschinski, On the application of similarity laws to a scaled railway bogie model, Diss. Delft University of Technology (1990). 2. A.D. de Pater and Yang Gu-ang, The geometrical contact between a pair of rollers and a wheelset in a railway roller rig, Proc. IUTAM symposium on Dynamical Problems of Rigid-Elastic systems and Substructures, Moscow, Springer (1991) pl79-189. 3. J.A. Elkins and N.G. Wilson, Train resistance measurements using a roller rig, Proc 9th IAVSD, Linkoping, Sweden 1985, Swets & Zeitlinger. APPENDIX I: MATRICES K AND A FOR A CONVENTIONAL BOGIE

17 APPENDIX 2: PARAMETERS FOR MODEL BOGIE VEHICLE k y = 72000 N / m

kψ = 28700 Nm / rad

kφ = 17507 Nm / rad

for 100% critical damping in the primary d y = 3470 Ns / m

dψ = 667 Nms / rad

dφ = 241Nms / rad

k yb = 144 N / m

kψ b = 3.69 Nm / rads

kφb = 768Nm / rad

for 25% critical damping in the secondary d yb = 54 Ns / m

dψ b = 244 Nms / rad

h = 0.2m

c = 1.6m

ro = 0.1m

1 = 0.143m

d1 = 0m

dψ b = 61Nms / rad

d 2 = 0.08m

f = 70000N

m = 12.8 kg

I = 0.32 kgm 2

mb = 20.9kg

I xb = 0.414 kgm2

I zb = 1.76kgm2

mc = 40kg

I xc = 9.6kgm2

I zc = 320kgm2

λ = 0.01

d 3 = 0.2m

The Rail Technology Unit The Rail Technology Unit based at Manchester Metropolitan University carries out research and consultancy into the dynamic behaviour of railway vehicles and their interaction with the track. We use state of the art simulation tools to model the interaction of conventional and novel vehicles with the track and to predict track damage, passenger comfort and derailment. Our simulation models are backed up by validation tests on vehicles and supported by tests on individual components in our test laboratory. We are developing methods to investigate the detailed interaction between the wheel and rail. January, 2004 Simon Iwnicki (RTU Manager)

Core expertise:

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Wheel/Rail Interface Engineering

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•Railway vehicle suspension dynamics –Computer simulation –Lab and field testing •Wheel-rail interface modelling –Profile design –Wear –Rolling contact fatigue

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The Rail Technology Unit Contact: General enquiries: [email protected] RTU Manager email: [email protected] RTU Manager Tel: +44 (0)161 247 6247 RTU Fax: +44 (0)161 247 1633 RTU Address: Rail Technology Unit, Manchester Metropolitan University, Department of Engineering & Technology, John Dalton Building, Chester Street M1 5GD, Manchester, United Kingdom The Rail Technology Unit Website: http://www.sci-eng.mmu.ac.uk/rtu or http://www.railtechnologyunit.com