system dynamics of radial and lateralelastic railway wheels

20
Concluding Colloquium of DFG-Priority-Program, Stuttgart, 13-15 March 2002, H.Claus and W. Schiehlen->1/20 Concluding Colloquium of DFG–Priority–Program, Stuttgart, 13–15 March 2002 System Dynamics of Radial– and Lateralelastic Railway Wheels H. Claus , W . Schiehlen Contents: ` Motivation ` Vertical Motion ` Lateral Motion and Stability Analysis ` Parameter Studies ` Conclusion Institute B of Mechanics Prof. Dr.-Ing. Schiehlen University of Stuttgart

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Page 1: System Dynamics of Radial and Lateralelastic Railway Wheels

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Concluding Colloquium of DFG–Priority–Program, Stuttgart, 13–15 March 2002

System Dynamics of Radial– andLateralelastic Railway Wheels

H. Claus, W. Schiehlen

Contents: � Motivation

� Vertical Motion

� Lateral Motion and Stability Analysis

� Parameter Studies

� Conclusion

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Page 2: System Dynamics of Radial and Lateralelastic Railway Wheels

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� Rigid wheelsets result at high speeds in droning noiseand polygonalization by inhomogeneous wear.

� Radialelastic wheelsets reduce the generation andtransmission of noise as well as the dynamic loads towheelrim and rail.

� First design for German ICE train was not reliable.

� However, the fundametal principle is still valid.

Radialelastic versus Rigid Wheelsets for High–Speed Trains

Page 3: System Dynamics of Radial and Lateralelastic Railway Wheels

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Design Variation for Radialelastic Wheels

Rubber suspendedsteel ring on wheel disc

Elastically suspended reinforcedsteel ring on double wheel disc

Page 4: System Dynamics of Radial and Lateralelastic Railway Wheels

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estimated mass and inertia

�++ � �,� �-.mass of wheel rim:

�// � �0 �*-.��inertia of wheel rim:��� � �� ��-.��

mass and inertia of the wheelset inreference configuration correspond

with the rigid wheelset

How to choose stiffness and damping???

Proposed Design Features of Radial and Lateral Elasticity

�++��

�++��

�++��

�++��

�++��

�+1��

�+1��

�+1�2

�+1�2

radial– andlateralelastic

wheel

Page 5: System Dynamics of Radial and Lateralelastic Railway Wheels

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carbody

bogie frame

trackexcitations

vertical eigenmodes starting with

3�-�2� 4�-�2� 4*-�2�- ���

flexible bogie frame

FE Modeling

Modal Reduction

Multibody Formalism

radialelastic wheelset

Model of a quarter passenger coach witha half bogie frame and one wheelset

Modeling

joint ofbogie pin

Page 6: System Dynamics of Radial and Lateralelastic Railway Wheels

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Soundpowerlevel

Evaluation of Droning Noise and Wheel/Rail Forces

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Noisetransmission

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����2

,��2

�� 5�6-��

mean square value,restricted in frequency range

mean square valueof wheel/rail forces

carbody

bogie frame

track ���+�

��

�-��5�6 ��-��-��

Page 7: System Dynamics of Radial and Lateralelastic Railway Wheels

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Spectral Density of Vertical Acceleration

10–12

10–8

10–4

10 0

Spe

ctra

l den

sity

2/s

3]

�[m

10 10020 30 40 60 80 200

Model shows the positive effect of radialelastic wheels for droning noise reduction

Acceleration amplitudes decrease considerably between 70 and 100 Hz

1/��������7 8� ����. ���#�������#! ����

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�-#��!��� � � ���*-��!�

Reference: Claus, On Dynamics of Radialelastic Railway Wheelsets, VSDIA’2000

Page 8: System Dynamics of Radial and Lateralelastic Railway Wheels

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Design variables

Applied algorithms

Design vektor

Limits for design variables

— Evolutionary strategy

— MCDM (NEWOPT/AIMS, PVM)

— NEWSIM (Simulation)

� ���+1�2-�+1�2-��+�2-��+�2-����2

=

������ #��!������ +1�<������ !�!< �+�!#������� !�!< ��

Static deflection 5mm :

other parameters may vary between0.1 and 10 of the initial value of rubbersuspended wheel

carbody

bogie frame

track

��+�2

��+�2

�+1�2

�+1�2

����2����2

�=>�2

�=>�2

�+1�2 � � ? ��,@��

Page 9: System Dynamics of Radial and Lateralelastic Railway Wheels

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Optimization of carbody accelerationand wheel/rail force

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�+1-:����;

��+-:����;

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4 ,#�

� �#*

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* ,#�

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4 ,#�

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� �#*

� �#*

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�� 0� ,� 3� 4������ ���

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�� �4

�� �,

����

�<����2#��#!��

�� :��

�!�;

���8��� ���#�#������

� � � � * � � � � � *� � � � * ��

��

���

�+1 :.@;

9�##�%���� ����#

��������#!��

���� �<����2#��������

Reduction of droning noise and wheel/rail force

by optimization of the radial elasticity of the wheels

�+1-:���; � �#,� �#3

��#�#��� :�2;

���# :!;

Page 10: System Dynamics of Radial and Lateralelastic Railway Wheels

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Wheelset suspension of ICE

��+�/ � � � ? ��,-@����+�� � , � ? ��0-@��

��+�2 � � � ? ��0-@��

��+�2 � � � ? ��*-@!��

3D–Modeling

carbody

bogie frame

��+�2��+�2 ��+�2��+�2

��+�/ ��+�/

Track parameterTrack stiffness (for contact force)

��� � � ? ��4-@��

������

��

Primary stiffness coefficients:

Primary damping coefficients:

Page 11: System Dynamics of Radial and Lateralelastic Railway Wheels

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Modeling of Suspension and Damping

��

5�6 � �5�6-�5�6--9���--�5�6 � �5�� 6��

5�6 � ���5�6--9���--� � �5�6

Springs/Dampers fixed atcoordinate systems:

–> not rotating–> general damping

Springs/Dampers fixedat material points:

–> rotating–> specific damping

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9�##� ��!.

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9�##���!.

Page 12: System Dynamics of Radial and Lateralelastic Railway Wheels

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Bogie frame with 2 rolling rigid wheelsets,i=1,2

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��

��

10 root locus plots over velocity

Re(�i)

��

–400 –300 –200 –100 0

���

���

Degrees of freedom

Bogie frame: 2

Wheelset: 2x4

Im(�i)

0200400600800

10001200carbody

bogie frame

Im(�i)

Re(�i)

vgLin � 360 km�h

–30 –20 –10 00

20

40

60

Eigenvaluesof the

hunting motion

Page 13: System Dynamics of Radial and Lateralelastic Railway Wheels

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Re(�i)–400 –300 –200 –100 0

Im(�i)

Re(�i)

vgLin � 360 km�h

–30 –20 –10 00

20

40

60

Im(�i)

0200400600800

10001200

Root locus plot over velocity

Bogie frame with 2 rolling elastic wheelsets i=1,2

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���

Degrees of freedomBogie frame: 2 Wheelset: 2x4

carbody

bogie frame

Wheel rim: 4x5

Stiffness and damping coefficents

�+1�� � �+1�2 � � � ? ����-@��

�+1�� � �+1�2 � � � ? ����-@!��

Verification of model

Results correspond to the rigid model–> rigid and elastic model are consistent

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Page 14: System Dynamics of Radial and Lateralelastic Railway Wheels

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�+1�2 � � � ? ��,-@��

�+1�2 � � � ? ��*-@!��

Parameter of radial suspension:

Large stiffness required to guarantee stability.

105

106

107

108

109

103

104

105

106

1070

100

200

300

400

]m/N[ y,ERc tiekgifietS

Kritische Geschwindigkeit

]m/sN[ y,ERd gnuf

v krit [

km/s

]

pmaD

linear critical speed

damping coeff. dRE,y [Ns�m] stiffness coeff. cRE,y [N�m]

vIgLin � 285, 5 km�h

Springs/Dampers fixedat coordinate systems:

Variation of Lateral Stiffness and Damping (cRE,y, dRE,y)

/h]

[km

vIIgLin � 284, 8 km�h

according to optimizationresults of vertical dynamics

9�##� ���

9�##� ��!.

��

5�6 � ���5�6--9���--� � �5�6

��+1�2��+1�2

Page 15: System Dynamics of Radial and Lateralelastic Railway Wheels

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�+1�� � � � ? ��3-@��

�+1�� � � � ? ���-@!��

Parameter of lateral, yaw androll suspension:

Optimal parameters of radial stiffness result in robust behavior

105

106

107

108

109

103

104

105

106

1070

100

200

300

400

]m/N[ z,ERc tiekgifietS

Kritische Geschwindigkeit

]m/sN[ z,ERd gnuf

v krit [

km/s

]

pmaD

linear critical speed

damping coeff. dRE,z [Ns�m] stiffness coeff. cRE,z [N�m]

�&&A��

� �3*� 3-.���

/h]

[km

Variation of Radial Stiffness and Damping (cRE,z, dRE,z)

�+1���+1��

Springs/Dampers fixedat coordinate systems:

��

5�6 � ���5�6--9���--� � �5�6

Page 16: System Dynamics of Radial and Lateralelastic Railway Wheels

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��������� �������� ���� �����������

Linear systems with periodic coefficients

��

5�6 � �5�6�5�6-9���-�5�6 � �5�� 6

Floquet’s Theory:

�5�� ��6 � �5�6��:� ��;-9���-�5�6 � �5�� 6

���-� � �5�6-�

�5��� ��6 � � � �5��6

���-� � 5�� � 6-����!

�56 � �5�� � � ��6 � �-��

B#�5�566B � �

Transition matrix �(T) through integration of

��

5�� ��6 � �5�6-�5�� ��6

Computation time for �(T)

Hill’s eqn.: � � ��

� :�� � �� ��!5��6;� � �

Floquet (Multiplication,Friedman 1977)

Floquet (Integration)

Time integration of eqn.of motion (5 Maxima)

0.06 s

0.14 s

2.00 s

Wheelset (8 dof, no primary suspension):

Floquet (Multiplication)

Time integration (withsuitable initial conditions)

3 days

15 min

Stability tests by time integration–> better computation efficiency

System is asymptotically stabil for

Page 17: System Dynamics of Radial and Lateralelastic Railway Wheels

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Comparison with general damping:At lower stiffness values increased critical speed

Variation of Lateral Stiffness and Damping (cRE,y, dRE,y)Springs/Dampers fixedat material points:

�+1�2 � � � ? ��,-@��

�+1�2 � � � ? ��*-@!��

Parameter of radial suspension:

according to optimizationresults of vertical dynamics

9�##� ���

9�##���!.

�+1�2

�+1�2

106

107

108

103

104

105

106

1070

100

200

300

400

linear critical speed

damping coeff. dLE,y [Ns/m] stiffness coeff. cLE,y [N/m]

vcr

it [

km/h

]

��

5�6 � �5�6�5�6--9���--�5�6 � �5�� 6

Page 18: System Dynamics of Radial and Lateralelastic Railway Wheels

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High critical speed even for low damping coefficients,instability for low stiffness coefficients

Variation of Radial Stiffness and Damping (cRE,z, dRE,z)Springs/Dampers fixedat material points:

–> rotating

�+1���+1��

�+1�� � � � ? ��3-@��

�+1�� � � � ? ���-@!��

Parameter of lateral, yaw androll suspension:

106

107

108

103

104

105

106

1070

100

200

300

400

linear critical speed

damping coeff. dRE,z [Ns/m] stiffness coeff. cRE,z [N/m]

vcr

it [

km/h

]

Page 19: System Dynamics of Radial and Lateralelastic Railway Wheels

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Comments on Design Parameter for Elastic Wheels

Damping coefficients turn out to be less critical

Stability behavior very robust for�+1�� � � ? ��,-@��

For comparison, stiffness in vertical direction

�+1�2 � � � ? ��,-@��

The lateral stiffness has to be considerably higherthan the radial stiffness

Page 20: System Dynamics of Radial and Lateralelastic Railway Wheels

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Variation of lateral and radialsuspension coefficients ofthe wheels (general andspecific damping)

Outlook: Nonlinear criticalspeed for radial– und laterale-lastic wheels

Reduction of droning noiseand wheel/rail force

Damping coefficients less critical

Verification of the results ofthe linear model

Optimization of the verticalmodel (with radialelasticwheels)

Korr–Opt.MCDS

Optimal parameters of radialstiffness –> robust behavior

Small stiffness coefficients notsuitable