Kalker's Theory Research Articles

Applicability of half-space-based methods to non-conforming elastic normal contact problems

The half-space assumption has been employed in many solution methods for non-conforming contact problems in elasticity such as the Hertz theory and the Kalker's variational theory. It is generally believed that to guarantee acceptable accuracy in these half-space-based methods, the characteristic size (twice length of one semi-axis) of the contact patch should be much smaller than the significant dimensions (i.e. the height, width, length and the principal radii of curvature) of each body in contact. In engineering practice, the 3x rule is often employed, which requires that the significant dimensions be at least three times as large as the characteristic size. However, this requirement has not been justified. In this paper, the applicability of half-space-based methods is investigated by comparing the solutions obtained using the Hertz theory and the Kalker's theory with those of the Finite Element (FE) method which is not limited to the half-space assumption. Different combinations of significant dimensions in terms of height, width and length are studied. Various contact patch eccentricities and contact body shapes are considered. It is found that the half-space-based methods yield high-accuracy calculation for non-conforming contact problems. Even when the significant dimensions are as small as 1.1x the characteristic size, the differences between the solutions of the half-space-based methods and the FE method are within 9%. The findings of this paper indicate that the typically assumed 3x restriction can be greatly relaxed. Since a clear estimation of the deviation of the results of half-space-based methods from those of the FE method is provided, the applicability of half-space-based methods in mechanical engineering can be much better understood.

Dynamics Analysis of Wheel Rail Contact Using FEA

A rail wheel contact mechanism has been a keen interest area for railway engineers. The present study focuses on the influence of interacting wheel and rail profile topology. A standard rail UIC 60 and standard wheel profile, as per the standards of Indian railways, were taken for different rail profile radii, wheel profile radii and wheel profile tapers. The mathematical and numerical studies were done by using Quasi–Hertz and FE analysis to analyze the impact of the interacting wheel and rail profiles on the distribution of contact zones and stresses. Moreover, the effect of contact forces and contact stresses on the deformation of rolling wheels and rails were also evaluated. The effect of lateral movement in the evaluation of frictional forces was found using Kalker theory.

Consideration of nonlinear wheel–rail contact forces for dynamic vehicle–bridge interaction in high-speed railways

In this work models with nonlinear wheel–rail contact forces are considered for analysing the dynamic interaction between high speed trains and bridges, in order to study dynamic effects both in the bridge and in the vehicles resulting from the coupling. Nonlinear contact models may be necessary for evaluating the stability and the safety of running traffic in situations such as vehicle overturn when the train is crossing a bridge under strong lateral winds or when an earthquake occurs. For studying the coupled dynamic response of trains and bridges, models of multibody dynamics are used for vehicles and the finite element method for structures. Special relevance is given here to the consideration of contact interaction forces between railway vehicles and the track. Four different interaction models are compared in this work: (1) a model where the vehicle wheelset is considered to be rigidly coupled to the track; (2) a staggered uncoupled method in which vehicle and structure are analysed separately; (3) a linear contact model in which lateral relative displacements between rails and train wheels are allowed, assuming biconic wheel and rail profiles and linear Kalker theory for tangential contact; (4) a nonlinear model in which realistic wheel and rail profiles, Hertz's nonlinear theory for normal contact and Kalker's nonlinear theory for tangential contact are used. The different models are applied and compared to experimental measurements for a test case of a high-speed train in China.

A numerical model of a HIL scaled roller rig for simulation of wheel–rail degraded adhesion condition

Scaled roller rigs used for railway applications play a fundamental role in the development of new technologies and new devices, combining the hardware in the loop (HIL) benefits with the reduction of the economic investments. The main problem of the scaled roller rig with respect to the full scale ones is the improved complexity due to the scaling factors. For this reason, before building the test rig, the development of a software model of the HIL system can be useful to analyse the system behaviour in different operative conditions. One has to consider the multi-body behaviour of the scaled roller rig, the controller and the model of the virtual vehicle, whose dynamics has to be reproduced on the rig. The main purpose of this work is the development of a complete model that satisfies the previous requirements and in particular the performance analysis of the controller and of the dynamical behaviour of the scaled roller rig when some disturbances are simulated with low adhesion conditions. Since the scaled roller rig will be used to simulate degraded adhesion conditions, accurate and realistic wheel–roller contact model also has to be included in the model. The contact model consists of two parts: the contact point detection and the adhesion model. The first part is based on a numerical method described in some previous studies for the wheel–rail case and modified to simulate the three-dimensional contact between revolute surfaces (wheel–roller). The second part consists in the evaluation of the contact forces by means of the Hertz theory for the normal problem and the Kalker theory for the tangential problem. Some numerical tests were performed, in particular low adhesion conditions were simulated, and bogie hunting and dynamical imbalance of the wheelsets were introduced. The tests were devoted to verify the robustness of control system with respect to some of the more frequent disturbances that may influence the roller rig dynamics. In particular we verified that the wheelset imbalance could significantly influence system performance, and to reduce the effect of this disturbance a multistate filter was designed.

Dynamic Response of Railway Bogie Due to Change in Vertical Primary Suspension Stiffness Parameters

A mathematical model of a railway carriage moving on tangent tracks is constructed by deriving the equations of motion concerning the model in which single-point and two-point wheel-rail contact is considered. The presented railway carriage model comprises of carbody and front and rear simple conventional bogie with two leading and trailing wheelets attached to each bogie. The railway carriage is modeled by 31 degrees of freedom which govern vertical displacement, lateral displacement, roll angle and yaw angle dynamic response of wheelset whereas vertical displacement, lateral displacement, roll angle, pitch angle and yaw angle dynamic response of carbody and each of the two bogies. Linear stiffness and damping parameters of longitudinal, lateral and vertical primary and secondary suspensions are provided to the railway carriage model. Combination of linear Kalker's theory and nonlinear Heuristic model is adopted to calculate the creep forces in which introduced at wheel and rail contact patch area. Computer aided-simulation is constructed to solve the governing differential equations of the mathematical model using Runge-Kutta fourth order method. Principle of limit cycle and phase plane approach is applied to realize the stability and to evaluate the concerning critical hunting velocity at which railway carriage starts to hunt. Numerical simulation model is used to study the dynamic responses of a railway carriage bogie subjected to specific parameters of wheel conicity and primary suspension characteristics. A comparison to study the sensitivity of railway carriage bogie to dynamic responses is also presented at different vertical primary suspension stiffness parameters.

Railway Carriage Model to Study the Influence of Vertical Secondary Stiffness on Ride Comfort of Railway Carbody Running on Curved Tracks

A mathematical model of a railway carriage running on curved tracks is constructed by deriving the equations of motion concerning the model in which single-point and two-point wheel-rail contact is considered. The presented railway carriage model comprises of front and rear simple conventional bogies with two leading and trailing wheelets attached to each bogie. The railway carriage is modeled by 31 degrees of freedom which govern vertical displacement, lateral displacement, roll angle and yaw angle dynamic response of wheelset whereas vertical displacement, lateral displacement, roll angle, pitch angle and yaw angle dynamic response of carbody and each of the two bogies. Linear stiffness and damping parameters of longitudinal, lateral and vertical primary and secondary suspensions are provided to the railway carriage model. Combination of linear Kalker's theory and nonlinear Heuristic model is adopted to calculate the creep forces in which introduced at wheel and rail contact patch area. Computer aided-simulation is constructed to solve the governing differential equations of the mathematical model using Runge-Kutta fourth order method. Principle of limit cycle and phase plane approach is applied to realize the stability and evaluate the concerning critical hunting velocity at which railway carriage starts to hunt. The numerical simulation model is used to study the influence of vertical secondary suspension spring stiffness on the ride passenger comfort of railway carbody running with speeds under and at critical hunting velocity. High magnitudes of vertical secondary spring stiffness suspension introduce undesirable roll and yaw dynamic response in which affect ride passenger comfort at critical hunting velocity. Low critical hunting velocity with railway carriage running on curved tracks can be represented.

Dynamic properties of an individual wheelset drive of a railway vehicle

AbstractSpatial vibrations of an individual wheelset drive supported by rubber silent blocks in a two–axle bogie's frame of a railway vehicle are studied. The method, used for a creation of a mathematical model, is based on a decomposition of the wheelset drive into subsystems — a drive assembled from an engine and a gear transmission, a flexible composite hollow shaft and a flexible wheelset. Subsystems are modeled in local configuration spaces and they are mutually connected by deformable couplings. An adhesive characteristic in a wheel–rail contact is described using the Kalker's theory. The nonlinear mathematical model respects spatial vibrations of the drive's components and the flexible wheelset, a deformable rail ballast and it is used for investigating transient dynamic responses on different types of excitations. Experimentally investigated spatial bogie's frame displacements or displacements derived from a total model of the bogie caused by spatial irregularities of the rail track are main sources of excitations. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Effect of sleeper pitch on rail corrugation at a tangent track in vehicle hunting

The effect of sleeper pitch on the initiation and development of rail corrugation at tangent track is investigated using a numerical method in vehicle hunting. The numerical analysis considers a combination of Kalker's rolling contact theory with non-Hertzian form, linear frictional work model and the dynamics model of a half railway vehicle coupled with a tangent track. Kalker's theory is modified. The half vehicle has a two-axle bogie and doubled suspension systems. It is treated as a full dynamic rigid multi-body model. In the tangent track model, an Euler beam is used to model each rail, and the backward moving discrete supports of the rails are considered to simulate the action of the discrete sleeper supports on the wheels/rails in rolling contact, when the train running on the track. The sleepers are treated as rigid bodies and the ballast bed is replaced with equivalent mass bodies. The effect of the discrete sleeper supports on the normal loads of the wheels/rails is analyzed at different speeds. The easily excited resonant frequencies of the wheelset and the track at different speeds are investigated in detail. In order to find the conditions necessary for rail corrugation occurring at the tangent track through numerical simulation, the vehicle hunting at higher speed is caused by setting the appropriate initial displacement condition of the vehicle system. The numerical results show that when the yaw angle of the wheelset reaches its maximum value in the vehicle hunting the lateral creepage of the wheel and rail oscillates at sleeper passing frequency and higher track frequencies. Therefore, rail corrugation due to wear initiates with several frequencies including sleeper passing frequency, and grouped near the peaks of the wheelset yaw angle. With increasing passage number of the vehicle in hunting, the corrugation became serious and also extended further along the rail.

Short wavelength rail corrugation and non-steady-state contact mechanics

J. J. Kalker has been the first to consider non-steady-state or transient contact mechanics. Based on Kalker the second author developed a linear contact model for the non-steady-state rolling contact of a wheel running over slightly corrugated rails. The theoretical investigations are concentrated on linear, non-steady-state contact mechanics superimposed to a nonlinear reference state. The reference state is given by the running behaviour of a wheelset due to traction, curving or hunting. For the linear, non-steady-state analysis Kalker's theory has to be modified to predict wear rates in dependency of the corrugation wavelengths. As a result corrugations are only amplified in the range between 2 and 10 cm. Therefore, non-steady-state contact mechanics and wear are responsible for a wavelength fixing mechanism. Structural mechanics of the rail indicate that wavelength in this range is predominantly amplified.

QUASI STATIC BEHAVIOUR AND CONTACT FORCES OF A BOGIE WITH TWO WHEELSETS RUNNING ON A CURVED TRACK

SUMMARY The aim of this study is the modelisation of the behaviour of a railway vehicle and especially of a truck with two wheel set running on a curved track. Moreover, the contact forces at the contact point are investigated and will be defined from Kalker's theory. The relative position parameter equations are established, solved and numerically exploited in order to calculate the contact forces. The comparison of these results with those obtained from experimentation turns out to be correct.

Fundamental research on railway vehicle dynamics (3rd report, Creepage and creep force law)

The former part of this paper describes an experimental study on the creepage and creep force law. The study has been carried out on a one-third scale model test-rig. By the test-rig, creep forces which correspond to three creepages-longitudinal, lateral and spin-are measured. The results of the experiments give good agreement with the values that have been obtained by the Kalker's creep theory. In the later part of this paper a simple method which calculates creep forces is proposed. The creep forces calculated by the algorithm give good agreement with the values calculated by the Kalker's theory. This method can also represent the decreasing of creep forces in the high-creepage area which is caused by the difference between friction coefficients of stick and slip.