Tangential Contact Forces Research Articles

Interaction between super-quadric particles and triangular elements andits application to hopper discharge

Super-quadric elements based on continuous function representation can be used to construct the geometric shape of irregular particles accurately. Although the contact detection of these elements has been well established, in real industrial applications, the geometric boundaries are complex and cannot be described by a specific function. A common and popular approach is to use a standard triangular finite-element surface mesh. In this work, the interaction between super-quadric particles and triangular elements is discussed in detail. Because the shapes and surface curvatures differ under the various contact patterns between super-quadricparticles, the linear contact force model cannot be used to calculate the contact force accurately; therefore, a corresponding non-linear force model is extended to super-quadric particles. In this model, the equivalent radius of curvature at a local contact point is introduced to calculate the normal contact force, and the tangential contact force is simplified based on the contact model of the spherical elements. To examine the validity of the algorithms and this model, three tests are performed. The first consists of a comparison against theoretical results for a flat wall impacted by a single cylinder-like particle with different blockiness parameters. In the second, the continuity of the contact force when a particle slides from one triangular element through the shared edge or the shared vertex to the neighboring element is tested. In addition, the three contact modes are counted in a boundary model represented by several triangular elements, and compared with the previous numerical experiments. The last tests consist of comparisons against experimental results of the arch structure of cube-like particles and the dynamic hopper discharge of ellipsoids. These studies demonstrate that the proposed method and force model are reliable and applicable for the dynamic hopper discharge of non-spherical particles. Furthermore, the effects of particle shapes, base angles, and friction coefficients on the discharge rate are discussed. The results show that the discharge rate always decreases with decreasing base angle or increasing blockiness, particle friction, or aspect ratio (from 0.5 to 1.5). Moreover, the influences of the base angle, the friction coefficient, and the blockiness on the flow rate are the primary factors, whereas the aspect ratio has a secondary effect on the flow rate. Finally, the packing fraction and the probability density functions of the normal contact force at the initial moment are analyzed to demonstrate the effect of blockiness on the macroscopic discharge rate.

Study of railway curve squeal in the time domain using a high-frequency vehicle/track interaction model

Railway curve squeal is an intense tonal and annoying type of noise commonly attributed to self-excited vibrations during curving. The mechanisms for its generation remain unclear and it is still a subject of discussion among researchers. Most of them have considered the falling behaviour of the friction coefficient with the slip velocity essential for reenergising the system. Recently, some authors have found that squeal can also appear even for constant friction coefficient through the wheel modal coupling between the normal and tangential directions caused by the wheel/rail contact. This paper particularly evaluates whether the latter mechanism is sufficient to find squeal in curving conditions.The introduction of flexibility in the railway subsystems is required to widen the domain to the high-frequency range in which squeal occurs. One single flexible and rotatory wheelset is considered and suitable forces are prescribed at the primary suspension seats in the current investigation. The rails are modelled through the Moving Element Method (MEM), permitting to extend the range of validity of beam models usually utilised in the literature. This work extends the formulation to rails supported by a viscoelastic Winkler bedding. Both wheelset and track models are coupled by means of a non-linear and unsteady wheel/rail contact model based on Kalker's Variational Theory. Simulation results for different track curvatures and friction coefficients are presented and discussed, showing tonal peaks in the tangential contact forces of the inner wheel. These results can be associated with squeal according to the characterisation of this phenomenon, indicating that squeal can be found in curving conditions using advanced dynamic interaction models even with constant friction coefficient.

Vehicle–bridge interaction analysis modeling derailment during earthquakes

A realistic simulation of train derailment is crucial when assessing the safety of a train running over a bridge during earthquake excitation. This paper presents a seismic vehicle–bridge interaction analysis that simulates directly different wheel–rail contact states including flange contact, detachment, uplifting, wheel–rail climbing-up, recontact, and ultimately, derailment. The proposed model determines the contact point and the direction of the contact forces over practical nonlinear profiles of wheels and rails. It then classifies the wheel–rail contact, as double contact, single contact or double detachment, and tackles accordingly the kinematics. The modeling of the wheel–rail contact along the normal direction hinges upon the principles of nonsmooth dynamics and accounts for continuous contacts of finite duration, impacts (instantaneous duration) and transitions from continuous contacts to detachments. The modeling of the tangential contact forces follows the nonlinear creep theory. The results verify that well-known force-based metrics such as derailment factor and offload factor yield conservative estimations of train operational safety. The analysis stresses the key role of flange contact under a large contact angle that could lead to the detachment of the other wheel of the same wheelset, and underlines the importance of a more realistic train–bridge interaction modeling during earthquakes. For the examples examined, which involve a complete three-dimensional vehicle running on simply supported bridge units, derailment occurs when a wheel rolls over the rail head (wheel–rail climbing-up). The results unveil that both the amplitude and the frequency of the earthquakes are important to the safety of trains running over bridges.

A novel discrete element method based on the distance potential for arbitrary 2D convex elements

SummaryA new 2‐dimensional discrete element method, which is able to simulate a system involving a large number of arbitrary convex elements, is proposed. In this approach, a novel distance potential function is defined using a normalized format of the penetrated distance between contact couples, while a holonomic and precise algorithm for contact interaction is established, accounting for the influence of the tangential contact force. Furthermore, the new contact detection algorithm is well suited for nonuniform blocks unlike the common no binary search method that requires uniform elements. The proposed method retains the merit of the combined finite‐discrete element method and avoids its deficiencies. Compared with the existing finite‐discrete element method, the distance potential function has a clear physical meaning, where the calculation of contact interaction avoids the influence of the element shape. Accordingly, the new method completely gets rid of the restraint of uniform element type and can be applied to arbitrary convex elements. The new method is validated with well‐known benchmark examples, and the results are in very good agreement with existing experimental measurement and analytical solutions. Finally, the proposed method is applied to simulate the Tangjiashan landslide.

Influence of spin creepage and contact angle on curve squeal: A numerical approach

Curve squeal is a loud tonal sound that may arise when a railway vehicle negotiates a tight curve. Due to the nonlinear nature of squeal, time-domain models provide a higher degree of accuracy in comparison to frequency-domain models and also enable the determination of squeal amplitudes. In the present paper, a previously developed engineering time-domain model for curve squeal is extended to include the effects of the contact angle and spin creepage. The extensions enable the evaluation of more realistic squeal cases with the computationally efficient model. The model validation against Kalker's variational contact model shows good agreement between the models. Results of studies on the influence of spin creepage and contact angle show that the contact angle has a significant influence on the vertical-lateral dynamics coupling and, therefore, influences both squeal amplitude and frequency. Spin creepage mainly influences processes in the contact, therefore influencing the tangential contact force amplitude. In the combined spin-contact angle study the spin creepage value is kinematically related to the contact angle value. Results indicate that the influence of the contact angle is dominant over the influence of spin creepage. In general, results indicate that the most crucial factors in squeal are those that influence the dynamics coupling: the contact angle, wheel/rail contact positions and friction.

Study on polyurethane media for mass finishing process: Dynamic characteristics and performance

Aiming at the difficult finishing problem of special parts such as deep groove and narrow slit of high performance parts, a new type of polyurethane media was prepared with high strength, low elastic modulus, and good deformation adaptability. Based on vertical centrifugal mass finishing process, the dynamic behavior of the media with different mass ratio and hardness was analyzed using discrete element method, the experiments were carried out on aluminum alloy wheel hub specimens, the better performance parameters of polyurethane media were determined, and the material removal model of polyurethane media was obtained. The results show that the numbers of polyurethane media with different velocities and energies show Maxwell distribution in the processing. The whole velocity, energy, normal contact force and tangential contact force of the media increase with increasing hardness of the media and mass ratio of polyurethane raw rubber and the abrasive. It also shows that the roughness value Ra can be decreased from 1.209 µm to 0.195 µm, decreased by 83.87%. The maximum material removal ratio can reach 0.147 mg/g·min within 10 min. The experimental results are consistent with the results of the discrete element analysis. The research results in this paper provide a new idea and way for expanding the application range of mass finishing processes.

Three-Dimensional Wear Prediction of Four-Degrees-of-Freedom Parallel Mechanism With Clearance Spherical Joint and Flexible Moving Platform

Three-dimensional (3D) wear of the clearance spherical joint in four-degrees-of-freedom (DOF) parallel mechanism is predicted based on Archard's wear model. The flexible moving platform is treated as thin plate element based on absolute nodal coordinate formulation (ANCF). The tangent frame is introduced to formulate the constraint equation of universal joint. One of the spherical joints is treated as clearance joint. The normal and tangential contact forces are calculated based on Flores contact force model and modified Coulomb friction model. In order to predict 3D wear, the normal contact force, tangential contact velocity, and eccentricity vector are decomposed in the global coordinate system. Simulation results show that 3D wear occurred in three directions are not uniform each other.

Discrete element analysis of buffering capacity of non-spherical granular materials based on super-quadric method

Granular system commonly encountered in industry or nature is comprised of non-spherical grains. Comparing with spherical particles, high discretization and interlocking among non-spherical particles can effectively dissipate the system energy and improve the buffer capacity. The superquadric element based on continuous function envelop can form the geometric shape of irregular particles accurately, and then contact collision action between particles can be calculated easily. In this paper, we provide a comprehensive introduction to particle-particle and particle-boundary contact collision. In addition, considering different shapes and surface curvatures under various contact patterns between super-quadric particles, the linear contact force model cannot be applied to the accurate calculation of the contact force, and a corresponding non-linear viscoelastic force model is developed. In this model, the equivalent radius of curvature at a local contact point is adopted to calculate the normal contact force, and the tangential contact force is simplified based on the contact model of spherical elements. To examine the validity of the algorithm and this model, we compare the discrete element analytical results with the analytical results for a single cylinder impacting a flat wall and the previous experimental results for spherical granular material under impact load, and this method is verified by good agreement between the simulated results and the previous experimental results. According to the aforementioned method, we study the buffer capacity of non-spherical particles under impact load by the discrete element method, and the influences of granular thickness and particle shapes on the buffer capacity are discussed. The results show that a critical thickness Hc is obtained for different particle shapes. The buffer capacity is improved with increasing the granular thickness when H Hc, but is independent of the granular thickness and particle shapes when H Hc. Moreover, the impact peak and initial packing fraction increase significantly with increasing the blockiness. Rectangular particles account for the highest packing fraction, and the packing fraction of cylindrical particles is higher than the packing fraction of spherical particles. Therefore, Rectangular particles are more likely to form dense face-face contacts and ordered packing structures with high packing fraction. These denser packings prevent the particles from their relatively moving, and thus reducing the buffering capacity of the particles. Furthermore, the impact peak and initial packing fraction decrease with increasing or reducing the aspect ratio of cylindrical particles and the aspect ratio of rectangular particles. The aspect ratio of particle can be used to adjust the dense packing structure and reduce the stability of the system. It means that the particles have more effective buffer capacity for the non-spherical particle system.

Chaos phenomenon and stability analysis of RU-RPR parallel mechanism with clearance and friction

The chaos phenomenon often exists in the dynamics system of the mechanism with clearance and friction, which has obvious effect on the stability of the mechanism, then it is worthy of attention for identifying the relationship between the friction coefficient and the stability of the mechanism. Two rotational degrees of freedom decoupled parallel mechanism RU-RPR is taken as the research object. Considering the clearance existing in the revolute pair, Lankarani–Nikravesh contact force model is used to calculate the normal contact force, and the Coulomb friction force model is used to calculate the tangential contact force. The dynamics model is established using Newton–Euler equations, and the Baumgarte stabilization method is used to keep the stability of the numerical analysis. Then, the equations are solved using the fourth adaptive Runge–Kutta method, and the effect of the revolute pair’s clearance on the dynamic behavior is analyzed. Poincare mapping is plotted, and the bifurcation diagrams are analyzed with varying the friction coefficient corresponding to different values of clearance size. The research contents possess a certain theoretical guidance significance and practical application value on the analysis of the chaotic motion and its stability in the dynamics of the parallel mechanism.

Dynamics Model of 4-SPS/CU Parallel Mechanism With Spherical Clearance Joint and Flexible Moving Platform

Effects of flexible body and clearance spherical joint on the dynamic performance of 4-SPS/CU parallel mechanism are analyzed. The flexible moving platform is treated as thin plate based on absolute nodal coordinate formulation (ANCF). In order to formulate the parallel mechanism's constraint equations between the flexible body and the rigid body, the tangent frame is introduced to define the joint coordinate. One of the spherical joints between moving platform and kinematic chains is introduced into clearance. The normal and tangential contact forces are calculated based on Flores contact force model and modified Coulomb friction model. The dynamics model of parallel mechanism with clearance spherical joint and flexible moving platform is formulated based on equation of motion. Simulations show that the dynamic performance of parallel mechanism is not sensitive to the flexible body because of the inherent property of moving platform; however, when the clearance spherical joint is considered into the parallel mechanism with flexible body, the flexible moving platform exhibits cushioning effect to absorb the energy caused by clearance joint.

Force Transmission Modes of Non-Cohesive and Cohesive Materials at the Critical State.

This paper investigates the force transmission modes, mainly described by probability density distributions, in non-cohesive dry and cohesive wet granular materials by discrete element modeling. The critical state force transmission patterns are focused on with the contact model effect being analyzed. By shearing relatively dense and loose dry specimens to the critical state in the conventional triaxial loading path, it is observed that there is a unique critical state force transmission mode. There is a universe critical state force distribution pattern for both the normal contact forces and tangential contact forces. Furthermore, it is found that using either the linear Hooke or the non-linear Hertz model does not affect the universe force transmission mode, and it is only related to the grain size distribution. Wet granular materials are also simulated by incorporating a water bridge model. Dense and loose wet granular materials are tested, and the critical state behavior for the wet material is also observed. The critical state strength and void ratio of wet granular materials are higher than those of a non-cohesive material. The critical state inter-particle distribution is altered from that of a non-cohesive material with higher probability in relatively weak forces. Grains in non-cohesive materials are under compressive stresses, and their principal directions are mainly in the axial loading direction. However, for cohesive wet granular materials, some particles are in tension, and the tensile stresses are in the horizontal direction on which the confinement is applied. The additional confinement by the tensile stress explains the macro strength and dilatancy increase in wet samples.

Planar Multi-body Dynamics of a Tracked Vehicle using Imaginary Wheel Model for Tracks

<p class="p1">Off-road vehicles achieve their mobility with the help of a track system. A track has large number of rigid bodies with pin joints leading to computational complexity in modelling the dynamic behaviour of the system. In this paper, a new idea is proposed, where the tracks are replaced by a set of imaginary wheels connected to the road wheels using mechanical links. A non-linear wheel terrain interaction model considering longitudinal slip is used to find out the normal and tangential contact forces. A linear trailing arm suspension, where a road arm connecting the road wheel and chassis with a rotational spring and damper system is considered. The differential algebraic equations (DAEs) from the multi-body model are derived in Cartesian coordinates and formulated using augmented formulation. The augmented equations are solved numerically using appropriate stabilisation techniques. The novel proposition is validated using experimental measurements done on a tracked vehicle.</p>

An enhanced formulation to model spatial revolute joints with radial and axial clearances

The primary objective of this work is to present a new formulation to model spatial revolute joints with radial and axial clearances. The methodology developed is based on the relative motion between the journal and bearing elements, which accounts for all the possible contact scenarios within a revolute clearance joint. In this process, normal and tangential contact force models are reexamined to evaluate their suitability to represent the contact-impact interactions between the mechanical parts connected by clearance joints. Also the equations of motion that govern the dynamic response of multibody systems and incorporate the intra-joint contact forces are presented. The Newton-Euler approach is applied here for these procedures. Finally, several numerical examples of the application are presented to discuss the main assumptions and procedures adopted in this work. The obtained results show that the clearance in mechanical joints can be quite significant in terms of the dynamic response of the system, and, consequently, plays a crucial role in the analysis, design and control of multibody systems.

A Study on the Dynamics of Spatial Mechanisms With Frictional Spherical Clearance Joints

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.

Effects of particle asphericity on the macro- and micro-mechanical behaviors of granular assemblies

Particle shape has been regarded as a key factor affecting mechanical behaviors of granular assemblies. The present work proposes a shape descriptor, i.e., asphericity, to isolate the effect of aspect ratio and angularity. A series of numerical triaxial compression tests are conducted on mono-disperse assemblies with different asphericities using a superball-based discrete element model. The corresponding mechanical behaviors are investigated from both macroscopic and microscopic points of view. Effects of particle asphericity are examined as well. It demonstrates that particle asphericity is able to enhance the particle–particle inter-locking with higher locked-in forces, showing an increasing proportion of sliding contacts, thereby making a granular assembly stiffer and stronger. In addition, weak contacts have a dominant proportion in the overall contact networks, positively correlated with asphericity. However, asphericity has an insignificant effect on the mean coordination number at the critical state. Furthermore, an established analytical stress–force–fabric relationship is verified using the present data. Anisotropy within an assembly is quantified in terms of several measures, i.e., contact normals, normal and tangential branch vectors, and normal and tangential contact forces. It is found that anisotropy of granular fabric is strongly determined by anisotropy sources in strong contact networks, where a larger asphericity results in a more anisotropic granular fabric.

Parameter and contact force estimation of planar rigid-bodies undergoing frictional contact

This paper addresses the identification of the inertial parameters and the contact forces associated with objects making and breaking frictional contact with the environment. Our goal is to explore under what conditions, and to what degree, the observation of physical interaction, in the form of motions and/or applied external forces, is indicative of the underlying dynamics that governs it. In this study we consider the cases of passive interaction, where an object free-falls under gravity, and active interaction, where known external perturbations act on an object at contact. We assume that both object and environment are planar and rigid, and exploit the well-known complementarity formulation for contact resolution to establish a constrained optimization-based problem to estimate inertial parameters and contact forces. We also show that when contact modes are known, or guessed, the formulation provides a closed-form relationship between inertial parameters, contact forces, and observed motions, that turns into a least squares problem. Consistent with intuition, the analysis indicates that without the application of known external forces, the identifiable set of parameters remains coupled, i.e. the ratio of mass moment of inertia to mass and the ratio of contact forces to the mass. Interestingly the analysis also shows that known external forces can lead to decoupling and identifiability of mass, mass moment of inertia, and normal and tangential contact forces. We evaluate the proposed algorithms both in simulation and with real experiments for the cases of a free falling square, ellipse, and rimless wheel interacting with the ground, as well as a disk interacting with a manipulator.

A collision model for grain-resolving simulations of flows over dense, mobile, polydisperse granular sediment beds

We present a collision model for phase-resolved Direct Numerical Simulations of sediment transport that couple the fluid and particles by the Immersed Boundary Method. Typically, a contact model for these types of simulations comprises a lubrication force for particles in close proximity to another solid object, a normal contact force to prevent particles from overlapping, and a tangential contact force to account for friction. Our model extends the work of previous authors to improve upon the time integration scheme to obtain consistent results for particle–wall collisions. Furthermore, we account for polydisperse spherical particles and introduce new criteria to account for enduring contact, which occurs in many sediment transport situations. This is done without using arbitrary values for physically-defined parameters and by maintaining the full momentum balance of a particle in enduring contact. We validate our model against several test cases for binary particle–wall collisions as well as the collective motion of a sediment bed sheared by a viscous flow, yielding satisfactory agreement with experimental data by various authors.

Decomposition of stiffness and friction tangential contact forces during periodic motion

The purpose of this paper is to introduce techniques used in experimental analysis and simulation of the internal resistance force in mechanical contact. We are presenting an idea and a method to separate the internal resistance into two force parts, the spring stiffness and the dissipative friction. This separation is based on the proven dependency for the projections of the phase angle between the displacement and acceleration of the periodic motion. By combining the Hilbert transform analysis with the phase relations of the simultaneously measured applied force and the vibration response we are able to estimate the nonlinear separated spring stiffness and dissipative friction force parts in the contact interference. Theoretical analysis and experimental measurements are presented as well as simulated verification results.

Wheel–Rail Impact at Crossings: Relating Dynamic Frictional Contact to Degradation

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.

Dynamic Analysis of 4-SPS/CU Parallel Mechanism Considering Three-Dimensional Wear of Spherical Joint With Clearance

The dynamic performance of 4-SPS/CU (spherical joint, prismatic joint, cylindrical joint, and universal joint) parallel mechanism considering spherical joint with clearance is developed, and the three-dimensional (3D) wear property of the socket is based on the Archard's wear model. First, the kinematics model of spherical joint with clearance is established, and the updated procedure pertaining to the contact mechanics and wear state is explained via a flowchart. An improved contact force model was proposed by Flores et al. contact force model through a revised contact stiffness coefficient. The normal and tangential contact forces between ball and socket are calculated using the improved contact force model and a modified Coulomb friction model. Second, the dynamic model of the parallel mechanism considering spherical joint with clearance is formulated based on the multibody equations of motion. In order to obtain the 3D wear property of spherical joint with clearance, the contact force is decomposed into three components in the global coordinate system. The three components of sliding distance are computed based on the 3D revolute property of the parallel mechanism. Finally, the contact pressures in three different directions are calculated by the corresponding contact force and approximate contact area components for the sake of predicting the 3D wear depth of socket based on the Archard's wear model. The simulation results show that the wear depth in different directions along the socket surface is nonuniform, which affects the performance of 4-SPS/CU parallel mechanism.