Sharma. 2013. Int. J. Vehicle Structures & Systems, 5(1), 9-14 ISSN: 0975-3060 (Print), 0975-3540 (Online) doi: 10.4273/ijvss.5.1.02 © 2013. MechAero Foundation for Technical Research & Education Excellence
Inter nati onal Jour nal of Vehicle Structures & Systems Available online at www.ijvss.maftree.org
Stability and Eigenvalue Analysis of an Indian Railway General Sleeper Coach using Lagrangian Dynamics Rakesh Chandmal Sharma M.M.University, Mullana Ambala India. Email:
[email protected]
ABSTRACT: This research paper determines the eigenvalues of main rigid bodies i.e. car body, bolsters, bogie frames and wheel axles of a 37 DoF coupled vertical-lateral model of a General Sleeper ICF coach of Indian Railway formulated using Lagrangian dynamics. The primary and secondary hunting speeds of the railway vehicle are determined to investigate the dynamic stability. Critical parameters which influence the railway vehicle dynamic stability are analysed. KEYWORDS: Railway vehicle; Eigenvalue; Dynamic stability; Hunting speed; Lagrangian dynamics CITATION: R.C. Sharma. 2011. Stability and Eigenvalue Analysis of an Indian Railway General Sleeper Coach using Lagrangian Dynamics, Int. J. Vehicle Structures & Systems, 5(1), 9-14. doi:10.4273/ijvss.5.1.02
tB
NOMENCLATURE: mC , B
Mass of car body and bolster respectively.
mBF , W
Mass of bogie frame and wheel axle respectively.
I Cx , y , z
Roll, pitch and yaw mass moment of inertia of car body respectively. Roll, pitch and yaw mass moment of inertia of bolster respectively. Roll, pitch and yaw mass moment of inertia of Bogie frame respectively. Roll, pitch and yaw mass moment of inertia of wheel axle respectively. Vertical (½ part) and lateral (½ part) stiffness between Car body and bolster respectively. Vertical (½ part) and lateral (½ part) damping coefficient between car body & bolster respectively. Vertical (¼ part) and lateral (½ part) stiffness between bolster and bogie frame respectively. Vertical and lateral damping coefficient between bolster and bogie frame respectively (½ part). Vertical (¼ part) and lateral (½ part) stiffness between bogie frame and corresponding wheel axle. Vertical (¼ part) and Lateral (½ part) damping coefficient between bogie frame and corresponding wheel axle. Vertical and lateral stiffness of rail respectively.
I Bx , y , z I
x, y , z BF
I Wx , y , z z, y kCB
z, y cCB
z, y k BBF
z, y c BBF
z, y k BFWA
z, y c BFWA
k Rz , y
cRz , y
tW
tC
z12 z24 z46 lA
a f 11 f 12 f 22 f 33
Lateral distance from bolster c.g. to vertical. suspension between bolster and bogie frame. Vertical distance between the Car body c.g. and the bolster c.g. Vertical distance between the bolster c.g. and the bogie frame c.g. Vertical distance between the bogie frame c.g. and the corresponding wheel axle c.g. longitudinal distance from wheel axle set c.g. to vertical suspension between corresponding bogie frame and wheel axle set. Half of wheel gauge. Lateral creep coefficient. Lateral/spin creep coefficient. Spin creep coefficient. Longitudinal creep coefficient.
1. Introduction The railway vehicle running along a track is one of the most complex dynamical systems in engineering. It has many degrees of freedom and the study of rail vehicle dynamics is a difficult task. On tangent track, at slower speeds of operation, rock and roll problems occur. At higher speeds, a vehicle may hunt and bounce severely. While negotiating on a curved track, wheel tends to climb the rail, excessive lateral forces may develop and rail vehicle may roll over. The response study is concerned with the prediction of dynamic behaviour of the system due to external inputs. On the other hand, the system behaviour is analyzed for performance parameters under different operating conditions. Dynamic stability and eigenvalue analysis of rail vehicle provide information about critical speed, system’s relative stability, damping ratios and natural frequencies.
Vertical & lateral damping coefficient of rail respectively. Lateral distance from bogie frame c.g. to corresponding vertical suspension between bogie frame and wheel axle. Lateral distance from car body c.g. to side bearings.
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Sharma. 2013. Int. J. Vehicle Structures & Systems, 5(1), 9-14
The railway vehicle is symmetric along longitudinal plane. Vehicle is travelling at a constant speed such that the longitudinal DoF is not a rigid body motion. All springs and dampers are assumed to be linear. Creep forces are assumed to be linear [6]. Track is considered to be flexible. Car body is assumed to be rigid. The contact between wheel and rail is intact. The railway vehicle is modelled using 37 DoF considering the following rigid bodies: Car body with 5 DoF representing vertical, lateral, roll, pitch and yaw (yi, zi, i, i and i where i = 1 for car body) motions. Front and rear bolster with 3 DoF representing vertical, lateral and roll (yi, zi, and i, where i = 2 for front and 3 for rear bolster) motions. Front and rear bogie frames are modelled using 5 DoF representing vertical, lateral, roll, pitch and yaw yi, zi, i, i and i where i = 4 for front and 5 for rear bogie frame) motions. Four wheel axles are modelled using 4 DoF representing vertical, lateral, roll and yaw (yi, zi, i and i, where i = 6, 7, 8 and 9 for four wheel axles) motions.
The eigenvalue analysis for railway vehicles is of particular interest because it provides some information about the factors that cause the hunting motion of such vehicles. Solutions of the linear equations of motion will be unstable if the real portion of any eigenvalue is positive. When this occurs, solutions of the linear equations diverge and nonlinear effects limit the motion of the actual vehicle during hunting. The stability boundary for hunting is usually expressed in terms of the linear critical speed, VCRITICAL, at which the linear equations diverge and nonlinear effects limit the motion of the actual vehicle during hunting. The approach of finding approximate values for the linear critical speed through a stability analysis has been extensively used by Wickens [4, 5, 7, 8, 10, 14 and 15]. Blader and Kurtz [11] and Hadden and Law [9] have analyzed the stability of North American freight train. Matsudiara [13], Hadden and Law [9] and Cooperrider [12] have carried out such studies for conventional passenger dual axle trucks.
2. Mathematical modelling 2.1. Vehicle model The mathematical model of a general sleeper coach of Indian Railway shown in Fig. 1 is formulated using Lagrangian dynamics with following assumptions.
Fig. 1: Rail-vehicle model
stored in the system due to springs. ED is the Rayleigh’s dissipation function of the system. Qi are the generalized forces corresponding to the generalized coordinates, yi. The final equations of motion of rail vehicle are obtained in the following form: (2) [M ]{yi } [C ]{ y i } [ K ]{ yi } [ Fr ()]
The equations of motion describing coupled vertical-lateral dynamics of the rail-vehicle are obtained using the Lagrangian equations as follows: E P E D d L L Qi dt q i qi qi q i
(1)
Lagrangian operator L is defined as (T - Vg), where T is the kinetic energy and Vg is the potential energy due the gravity effect of the vehicle system. EP is the energy
Where [M], [K] and [C] are the mass, stiffness and damping matrices respectively for rail vehicle. 10
Sharma. 2013. Int. J. Vehicle Structures & Systems, 5(1), 9-14 [ Fr ( )]
is a force matrix of size 37×1 for displacement excitations at the eight wheel contact points, r = 1, 2,..., 8 due to the vertical and lateral irregularities of the track.
1 1 1 1 1 z z z c Rz cWz cSL cS cSS
(4)
Similar equations are written for lateral direction.
2.2. Track model
3. Eigenvalue analysis
The track may be divided into a super structure and a sub structure. The super structure includes rails, rail fastenings, pads, sleepers and ballast (i.e. soil). The subsoil is the sub structure of a track. The track in the present analysis is assumed to be flexible in both vertical and lateral directions. Its flexibility is accounted by considering wheel to be in series with sleeper, soil and subsoil as shown in Fig. 2.
The final equations of motion of rail vehicle as given in Eqn. (2) may also be expressed in the following form: [M ]{yi } [C]{ yi } [ K ]{ yi } 0
(5)
The linear 37 second order differential equations of motion is reduced to a set of 74 first order equations in the state space form by the introduction of following variables: X 1 y1 X 2 , X 2 y1 , X 3 y 2 X 4 , X 4 y 2 , …
X 71 8 X 72 , X 72 8 , X 73 9 X 74 , X 74 9
(6)
The equations of motions in the state space form are represented as: (7) [ A]( X ) [ B]( X ) ( X ) [ A]1[ B]( X )
where [A] and [B] are square, non symmetric matrices of order 74×74. X is [ X 1 , X 2 , X 3 .........X 74 ] and X is [ X 1 , X 2 , X 3 ........X 74 ] . The parameters of a loaded general sleeper coach, as listed in Table 1, are obtained from Indian Railways Research Department, Research Designs and Standards Organisation. The values of creep coefficients for the wheel-track interaction have been taken from Yuping & McPhee [3].
Fig. 2: Track model
The flexibilities of the track n vertical directions are given by: 1 1 1 1 1 z z z k Rz kWz k SL k S k SS
(8)
(3)
Table 2: Values of rail vehicle and track parameters
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Sharma. 2013. Int. J. Vehicle Structures & Systems, 5(1), 9-14
The eigenvalues of the matrix [A]-1 [B] are obtained to have an insight into system’s relative stability, damping ratios and natural frequencies. The rail vehicle is considered to be moving at a constant speed of 80 km/hr over a straight track. The 37 DoF rail vehicle model will yield as many natural frequencies and modes of vibration. However, due to the effects of coupling in a multi DoF system where one vibration is likely to influence others, the shape of the response curves may be remarkably changed. It is possible that some peaks are attenuated, or shifted to a different frequency or may even sometimes disappear. The general motion would be a superposition of all these modes. Since damping is presented in the system, some modes are complex and occur in conjugate pairs. The real part of the eigenvalue,
which are negative, gives the decay rate. The imaginary part gives the damped natural frequency. Out of the total eigenvalues obtained for loaded general sleeper railway coach, it is found that 15 eigenvalues are complex and occurred in conjugate pairs. The eigenvalue for a particular mode was identified on the basis of varying the parameters, which are likely to influence it most i.e. mass, mass moment of inertia, stiffness and damping coefficient. From the eigenvalue analysis, it is observed that the car body damped natural frequencies in vertical, lateral, roll, pitch and yaw mode is 0.65 Hz, 3.64 Hz, 3.15 Hz, 0.69 Hz and 3.64 Hz respectively. The eigenvalues of rigid bodies considered in 37 DoF coupled vertical-lateral rail vehicle system is listed in Table 2.
Table 1: Eigen values of 37 DoF rail vehicle system
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Sharma. 2013. Int. J. Vehicle Structures & Systems, 5(1), 9-14
better choice from stability point of view. Wheel conicity has relatively large effect on secondary hunting. With the other existing parameters, secondary hunting is observed at wheel conicity of 0.15 rad. A 2% increment in the conicity results in 5% reduction in secondary hunting speed. The secondary hunting is insensitive to Car body mass and secondary suspension parameters. However, it is very much a function of primary suspension parameters. A 10% increment in lateral damping coefficient between the wheel set and bogie frame can increase the critical speed up to 210 km/hr. Wheel set mass has also little influence on secondary hunting. Therefore, secondary hunting is also termed as wheel set hunting.
4. Stability analysis The hunting phenomenon is a self-exited lateral oscillation that is produced by the forward speed of the railway vehicle and wheel-rail interactive forces, which result from the conicity of the wheel-rail contours contact geometry and friction-creep characteristics of the wheel-rail. It is analysed from the present study that loaded general sleeper coach of Indian Railway does not exhibit any primary hunting because of proper amount of lateral damping between the car body & bolster and between the bolster & bogie frame. However it is observed that the railway vehicle at the speed of 55 km/hr becomes unstable in Car body yaw mode if these lateral damping values are reduced 25% from the existing values. The railway vehicle is stable within the range of ±30% of present primary suspension values before it reaches the speed of 195 km/hr. At this speed, it is unstable in lateral mode of the rear bogie front wheel axle. This speed corresponds to the secondary hunting speed of the railway vehicle. The locus of the eigenvalue in secondary hunting mode is shown in Fig. 3. This secondary hunting phenomenon is inherent condition and can not be totally eliminated. One possible solution is that its value i.e. 195 km/hr may be further increased by altering the rail vehicle and track design.
5. Conclusions In this study the eigenvalues of main rigid bodies of a General Sleeper ICF coach of Indian Railway using 37 DoF model is presented. These eigenvalues are helpful to the railway designer as they give an insight into system’s relative stability, damping ratios and natural frequencies. With the existing rail-vehicle and track parameters, the secondary hunting mode is observed at the critical speed of 195 km/hr. It is also observed that secondary hunting is inherent and can not be eliminated. Wheel conicity, lateral track damping along with lateral damping coefficient between wheel set and bogie frame are the most critical parameters influencing the stability. However, it is a well known fact that changes in parameter value which improve the stability most likely deteriorate the curving ability. Therefore the parameter change with the objective of improving the stability would be advantageous for the straight track. REFERENCES: [1] R.C. Sharma. 2011. Ride analysis of Indian railway coach using Lagrangian dynamics, Int. J. Vehicle Structures & Systems, 3(4), 219-224. http://dx.doi.org/10.4273 /ijvss.3.4.02 [2] W. M. Zhai, K. Wang and C. Cai. 2009. Fundamentals of vehicle-track coupled dynamics, Vehicle System Dynamics, 47(11), 1349-1376. http://dx.doi.org/10.1080/ 00423110802621561 [3] H. Yuping and J. McPhee. 2002. Optimization of the lateral stability of rail vehicles, Vehicle System Dynamics, 38 (5), 361-390. http://dx.doi.org/10.1076/vesd.38.5. 361.8278 [4] A.H. Wickens. 1991. Steering and stability of the bogie: vehicle dynamics and suspension design, Proc. Institution of Mechanical Engineers, Part F: J. Rail and Rapid Transit, 205(2), 109-122. http://dx.doi.org/10.1243/ PIME_PROC_1991_205_224_02 [5] A.H. Wickens. 1988. Stability optimization of multi-axle railway vehicles possessing perfect steering, J. Dynamic Systems, Measurements and Control, 110(1), 1-7. http://dx.doi.org/10.1115/1.3152642 [6] J. J. Kalker. 1979. Survey of Wheel-Rail Contact Theory, Vehicle System Dynamics, 8, 317-358. http://dx.doi.org/ 10.1080/00423117908968610 [7] A.H. Wickens. 1978. Stability criteria for articulated railway vehicles possessing perfect steering, Vehicle System Dynamics, 7(4), 165-182. http://dx.doi.org/ 10.1080/00423117808968561
Fig. 3: Locus of eigenvalue in secondary hunting mode
It is investigated from the present analysis that the secondary critical speed is insensitive to the lateral and longitudinal creep coefficients. It is observed that the hunting speed increases linearly with lateral/spin coefficient. An increment in the existing value of lateral track stiffness and lateral track damping increases the value of secondary hunting speed. A stiffer sub structure increases the secondary hunting speed. This may be a 13
Sharma. 2013. Int. J. Vehicle Structures & Systems, 5(1), 9-14 [8] A.H. Wickens. 1977. Static and dynamic stability of a class of three-axle railway vehicles possessing perfect steering, Vehicle System Dynamics, 6(1), 1-19. http://dx.doi.org/10.1080/00423117708968499 [9] J.A. Hadden and E.H. Law, 1977, Effects of truck design on hunting stability of railway vehicles, ASME J. Engineering for Industry, 99(1), 162-171. http://dx.doi.org/10.1115/1.3439133 [10] A.H. Wickens. 1976. Steering and dynamic stability of railway vehicles, Vehicle System Dynamics, 5(1-2), 15-46. http://dx.doi.org/10.1080/00423117508968404 [11] F.B. Blader and E.F. Kurtz. 1974. Dynamic stability of cars in long freight trains, ASME J. Engineering for Industry, 96(4), 1159-1167. http://dx.doi.org/10.1115/ 1.3438490 [12] N.K. Cooperrider. 1972. The hunting behavior of conventional railway trucks, ASME J. Engineering for Industry, 94(2), 752-762. http://dx.doi.org/10.1115/ 1.3428240
[13] T. Matsudaira. 1966, Hunting problem on high speed railway vehicles with special reference to bogie design for the new Tokaido line - Interaction between vehicle and track, Proc. Institution of Mechanical Engineers, Part F: J. Rail and Rapid Transit, 180, 58-66. [14] A.H. Wickens. 1965. The dynamic stability of a simplified four-wheeled railway vehicle having profiled wheels, Int. J. Solids and Structures, 1(4), 385-405. http://dx.doi.org/ 10.1016/0020-7683(65)90004-1 [15] A.H. Wickens. 1965. The dynamic stability of railway vehicle wheelsets and bogies having profiled wheels, Int. J. Solids & Structures, 1, 319-341. http://dx.doi.org/ 10.1016/0020-7683(65)90037-5 [16] A.H. Wickens. 2003. Fundamentals of Rail Vehicle Dynamics, Swets & Zeitlinger Publishers, Netherlands. http://dx.doi.org/10.1201/9780203970997
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