Implementation of a non-Hertzian contact model for railway dynamic application

Shrouds with a frictional interface are used to reduce the dynamic stresses in turbine blades. Due to dry friction energy is dissipated, which can be used to decrease vibration amplitudes and, hence, to increase the life time of turbine blades. The spatial motion of the blades results in a spatial motion of the contact planes. Due to the non-linearity of the problem, the contact planes are discretized. For each contact area, the developed contact model is used to calculate the corresponding tangential and normal contact forces. This contact model includes the roughness of the contact surfaces, the normal pressure distribution due to roughness, the stiffnesses in normal and tangential direction and dry friction. Due to the roughness of the contact planes the normal contact forces and the contact stiffnesses in normal and tangential direction are nonlinear dependent on the relative displacements in the normal direction. This effect is verified by experiments. An experiment with one shrouded blade and two non-Hertzian contacts is used to verify the developed contact model and the calculation method. The comparison between measured and calculated frequency response functions for bending and torsional vibrations of the blade show a very good agreement. A bladed disk assembly with shrouds is investigated and optimized with respect to the vibration amplitudes and alternating stresses. Varying the normal contact force best damping effects are obtained. Separation of the contacts leads to an increase of the alternating stresses and, thus, has to be avoided.

In this paper we consider the contact between two rectangular rough surfaces that provide normal and tangential contact force as well as contact moment to counteract the net moment imposed by the applied forces. The surfaces are permitted to develop slight angular misalignment and thereby contact moment is derived. Through this scheme it is possible to also define elastic contribution to friction since the half-plane tangential contact force on one side of an asperity is no longer balanced by the half-plane tangential force component on the opposite side. The elastic friction force however is shown to be of a much smaller order than the contact normal force. Approximate closed form equations are found for contact force and moment as functions of separation, asperity radius of curvature sum, mean plane slope and nominal contact dimension. The approximate equations are shown to give error within seven percent.

Irregularities in the geometry and flexibility of railway crossings cause large impact forces, leading to rapid degradation of crossings. Precise stress and strain analysis is essential for understanding the behavior of dynamic frictional contact and the related failures at crossings. In this research, the wear and plastic deformation because of wheel–rail impact at railway crossings was investigated using the finite-element (FE) method. The simulated dynamic response was verified through comparisons with in situ axle box acceleration (ABA) measurements. Our focus was on the contact solution, taking account not only of the dynamic contact force but also the adhesion–slip regions, shear traction, and microslip. The contact solution was then used to calculate the plastic deformation and frictional work. The results suggest that the normal and tangential contact forces on the wing rail and crossing nose are out-of-sync during the impact, and that the maximum values of both the plastic deformation and frictional work at the crossing nose occur during two-point contact stage rather than, as widely believed, at the moment of maximum normal contact force. These findings could contribute to the analysis of nonproportional loading in the materials and lead to a deeper understanding of the damage mechanisms. The model provides a tool for both damage analysis and structure optimization of crossings.

The occurrence of adhesion-induced failures at microelectromechanical systems (MEMSs), also referred to as stiction, can be triggered by a variety of environmental, geometrical, and operational parameters, such as compressive contact forces acting in normal direction between two micromechanical surfaces. The knowledge of these parameters is essential for conducting and evaluating stiction-related reliability tests at MEMS. This paper investigates the operational parameter of tangential contact forces, which can occur in case of multi-directional overloads like shocks or vibrations during manufacturing, handling, or final application of MEMS inertial sensors. Tangential contact forces act in parallel to a contact area and are shown to affect the stiction-induced failure behavior of MEMS. For this purpose, three different types of test devices have been exposed to electrical as well as mechanical stimuli with and without cross-axes excitation contents, stimulating the occurrence of tangential forces during contact. The transient system responses of the test devices were calculated based on reduced-order impact models, allowing a detailed comparison of load conditions between different stimuli and test devices. The obtained results show that tangential contact forces lead to higher stiction rates, which indicates increased adhesion forces compared with excitations without tangential forces. [2015-0320]

Accurately modeling contact interactions in mechanical systems with complex geometries is a critical step towards predicting the dynamic response of the system. In particular, toroidal surfaces are especially challenging in contact mechanics due to their nontrivial geometry involving both convex and concave surfaces, which often results in complex contact interactions when dealing with planar surfaces or other curved bodies. Such shapes frequently appear in several engineering applications, including tires, seals, and various ring components. Hence, the present study focuses on developing a robust and computationally efficient methodology to simulate the contact dynamics between a toroidal surface and a plane by employing a penalty-based compliant force model. This work not only provides a framework for the standalone modeling of toroidal-plane interactions, but also demonstrates broader applicability to systems with more complex surfaces in which some parts can be approximated by toroidal geometry. The toroidal surface is represented parametrically, involving a set of two angular parameters: one defining the angle along the torus cross-section and another denoting the rotation around the axis of the torus. The proposed procedure facilitates the precise location of points on the torus geometry and enables the use of efficient contact detection methods. Unlike simpler geometries, such as spheres or ellipsoids, the torus requires specialized techniques to accurately identify the contact points and assess pseudo-penetration during contact scenarios. In order to address these challenges, the torus surface is discretized into multiple slices, each treated as a local cylindrical approximation. This slicing approach simplifies the contact region analysis while preserving the geometric fidelity. The interaction between the torus and the plane is evaluated for each slice, and suitable normal and tangential contact force models are adopted. The normal forces are computed using a Hunt and Crossley-based model, which takes into account the properties of the contacting materials, such as stiffness and damping, while the tangential forces are evaluated with a regularized static friction model. The methodology developed in the context of this work is applied to the dynamic simulation of a generic toroidal body contacting a fixed plane. The obtained results highlight the system's sensitivity to key parameters, including the stiffness and damping of the contact interface and the number of slices utilized to discretize the torus geometry. While increasing the number of slices improves the system’s accuracy, it also leads to an increase in the computational cost, imposing a trade-off between precision and efficiency. Additionally, the proposed approach has been able to capture the complex variations in contact regions as the torus orientation changes, showcasing its versatility in handling dynamic scenarios. Future work will focus on refining the model, exploring different contact models, and extending its application to more complex multibody systems. In addition, the extension of this work to supertoroidal shapes is intended.

An investigation on the dynamic modeling and analysis of spatial mechanisms with spherical clearance joints including friction is presented. For this purpose, the ball and the socket, which compose a spherical joint, are modeled as two individual colliding components. The normal contact-impact forces that develop at the spherical clearance joint are determined by using a continuous force model. A continuous analysis approach is used here with a Hertzian-based contact force model, which includes a dissipative term representing the energy dissipation during the contact process. The pseudopenetration that occurs between the potential contact points of the ball and the socket surface, as well as the indentation rate play a crucial role in the evaluation of the normal contact forces. In addition, several different friction force models based on the Coulomb's law are revisited in this work. The friction models utilized here can accommodate the various friction regimens and phenomena that take place at the contact interface between the ball and the socket. Both the normal and tangential contact forces are evaluated and included into the systems' dynamics equation of motion, developed under the framework of multibody systems formulations. A spatial four-bar mechanism, which includes a spherical joint with clearance, is used as an application example to examine and quantify the effects of various friction force models, clearance sizes, and the friction coefficients.

A time domain model is presented for the dynamic interaction between a railway wheel and rail, which takes into account the non-linear processes in the contact zone and aims at predicting both normal and tangential contact forces. The model follows an approach that has been used successfully, for instance for the modelling of the interaction between road and tyre. Track and wheel are described as linear systems by the means of impulse response functions. The contact zone is modelled by non-linear contact springs with stiffnesses depending on the roughness of rail and wheel. Here, the method of the area of real contact is applied in order to obtain the required spring characteristics. For the tangential contact, a characteristic function for the friction coefficient is applied. In a first stage, the approach is demonstrated for the calculation of normal contact forces. For validation, the results from the model are compared with an existing time domain model that itself has been validated by field testing. Very good agreement is found for different types of roughness excitation.

Experimental study and discrete element analysis on dynamic mechanical behaviour of railway ballast bed in windblown sand areas

Particle morphology and size are fundamental characteristics that significantly influence the mechanical behavior of granular materials. This study introduces key parameters—aspect ratio (Ω), sphericity (S), and equivalent diameter (Dₑ)—into a modified Hertz-based contact model to conduct a multiscale study using contact mechanics theory and the discrete element method (DEM). A series of two-particle tests and triaxial compression simulations were performed. The results show strong agreement between numerical simulations and theoretical predictions at the particle scale, validating the modified contact model. At the sample scale, the peak deviatoric stress increased by approximately 15–40% as aspect ratio decreased from 1.00 to 0.33 and sphericity decreased from 1.00 to 0.11. Similarly, increasing the equivalent diameter from 3.78 mm to 8.82 mm led to a 20–35% rise in peak stress. At the particle scale, both normal and tangential contact forces increased with larger equivalent diameters but exhibited complex dependencies on morphology due to varied contact patterns. These findings enhance the understanding of how particle-scale characteristics influence macroscopic mechanical properties.

Finite element based analysis of full-stick contact between a functionally graded hemispherical asperity and a rigid flat is carried out under tangential loading. In the adopted gradation model, Young’s modulus and tangent modulus (considering bilinear isotropic hardening model) are varied according to an exponential function in the radial direction of the hemisphere. Under post-flattening tangential loading, the FGM hemispherical asperity contact is investigated for different values of gradation parameters. Different aspects of contact behavior, i.e., tangential and normal contact forces, contact area, contact pressure, etc. are investigated under the influence of varying gradation parameters. The evolution of stresses and deformation in the asperity are also studied for different gradation parameters.

Dynamics investigation of spatial parallel mechanism considering rod flexibility and spherical joint clearance

Simulated granular packings with different particle friction coefficient mu are examined. The distribution of the particle-particle and particle-wall normal and tangential contact forces P(f) are computed and compared with existing experimental data. Here f identical with F/(-)F is the contact force F normalized by the average value (-)F. P(f) exhibits exponential-like decay at large forces, a plateau/peak near f=1, with additional features at forces smaller than the average that depend on mu. Additional information beyond the one-point force distribution functions is provided in the form of the force-force spatial distribution function and the contact point radial distribution function. These quantities indicate that correlations between forces are only weakly dependent on friction and decay rapidly beyond approximately three particle diameters. Distributions of particle-particle contact angles show that the contact network is not isotropic and only weakly dependent on friction. High force-bearing structures, or force chains, do not play a dominant role in these three-dimensional, unloaded packings.

The purpose of this study was to examine how contact forces normal to the skin surface and shear forces tangential to the skin surface are deployed during tactile exploration of a smooth surface in search of a tactile target. Six naive subjects participated in two experiments. In the first experiment, the subjects were asked to explore a series of unseen smooth plastic surfaces by using the index finger to search for either a raised or recessed target. The raised targets were squares with a height of 280 micro m above the background surface and that varied in side lengths from 0.2 mm to 8.0 mm. A second series of smooth plastic surfaces consisted of small recessed squares (side lengths: 2.0, 3.0, 4.0 and 8.0 mm) that were etched to a depth of 620 micro m. Although made of an identical material, the plastic substrate had a lower coefficient of friction against the skin because only the recessed square had been subjected to the electrolytic etching process. The surfaces were mounted on a six-axes force and torque sensor connected to a laboratory computer. From the three axes of linear force, the computer was able to calculate the instantaneous position of the index finger and the instantaneous tangential force throughout the exploratory period. When exploring for the raised squares, the subjects maintained a relatively constant, average normal force of about 0.49 N with an average exploration speed of 8.6 cm/s. In contrast, all subjects used a significantly higher average normal force (0.64 N) and slightly slower mean exploration speed (7.67 cm/s) when searching for the small recessed squares. This appeared to be an attempt to maximize the amount of skin penetrating the recessed squares to improve the probability of target detection. In a second experiment, subjects were requested to search for an identical set of raised squares but with the fingertip having been coated with sucrose to impede the scanning movement by increasing the friction. Overall, the subjects maintained the same constant normal force that they used on the uncoated surface. However, they increased the tangential force significantly. The similarity of the search strategy employed by all subjects supports the hypothesis that shear forces on the skin provide a significant stimulus to mechanoreceptors in the skin during tactile exploration. Taken together, these data suggest that, in active tactile exploration with the fingertip, the tangential finger speed, the normal contact force, and the tangential shear force are adjusted optimally depending on the surface friction and whether the target is a raised asperity or a recessed indentation.

This paper describes a practical wheel/rail tangential contact force model for railway dynamics analysis under running in rainy conditions. In our previous studies, it was confirmed that experimental results of tangential contact force measurement are agree well with Kalker’s rolling contact theory under dry condition with low humidity around contact surfaces. Here, it can be considered that if the ambient humidity around the contact surface become high infinity, the contact surface is under wet lubrication conditions. On the basis of this assumption, in this study, a practical estimation model for characteristics of tangential contact force under running in rainy conditions is proposed by combining the relationship between the adhesive coefficient and velocity measured on running experiments using actual railway vehicles and Kalker’s linear rolling contact theory. Therefore, to verify the proposed model, the tangential contact force measurement experiment using a twin-disk rolling sliding machine was carried out. As a result, it is confirmed that estimated formulae based on the proposed model are well agreed with experimental results even under different velocity conditions, comparing the relationship between tangential contact force coefficient and slip ratio. From these verified results, the proposed model is considered to be a practical wheel/rail tangential contact force model for vehicle dynamics analysis under rain conditions, because the proposed model has the advantage of low calculation cost and can be easily applied to vehicle dynamics analysis based on multi-body dynamics theories.

A 3D three-equation linear spring-dashpot model considering the normal contact force, the tangential contact force, and the rolling friction torque, and the corresponding programs for granular dynamics were developed. The flow and the segregation of the S-type binary granular mixtures in a thin rotating horizontal drum were simulated by using the new model. The influences of the rotation speed and the load of the drum on the pattern formation were discussed. The simulation results show that the pattern formation of the S-type binary granular mixtures in the rotating drum can be predicted accurately. As the rotation speed is high, the moon pattern appears,in which the small particles concentrate in a central core and the large particles distribute in the periphery. As the rotation speed is lowered, the petal pattern is found in thin drums, and the number of petals increases as the rotation speed decreases. The process of pattern formations was obtained by the simulation. As the petal pattern forms, the wavy flow of granules in the drum appears. Finally, the necessary condition and mechanisms of the pattern formation in the thin drums are discussed in the paper.