On Solvability of Multibody Contact Problems

2007 ◽  
Author(s):  
Alexander P. Ivanov
Keyword(s):  
Contact Problem ◽  
Equations Of Motion ◽  
Sufficient Conditions ◽  
Contact Problems ◽  
Point Of View ◽  
Contact Forces ◽  
Empty Intersection ◽  
Contact Points ◽  
Whole Space ◽  
Multibody Contact

The paper is devoted to dynamic multi-rigid-body contact problems with dry friction. It is known that such problem may have multiple solutions or none solution (so-called Painleve´ paradoxes). A great deal of works concerning overcoming of the paradoxes was published last century, but general conditions of existence and uniqueness were not derived yet. We consider systems with a finite numbers of contact points with well-defined contact directions and Coulomb friction. The equations of motion contain unknowns of two kinds: the accelerations and the contact forces. According to the friction law, some of these variables vanish, and remaining ones can be treated as a coordinate system in the space of the generalized forces. Thus, this space splits to a finite number of regions with different coordinates. From a geometrical point of view, the solvability of the multi-contact problem means that the union of these regions equals to the whole space. Furthemore, the solution is unique ⇔ any pair of regions has empty intersection, and the coordinates within any region are defined uniquely. We present some algebraic conditions, which are equivalent to these geometric properties. Therefore, necessary and sufficient conditions of correct solution to multibody contact problem are obtained for the first time. A number of mechanical examples are considered to illustrate new results.

Stability Analysis of High-Speed Railway Vehicle Based on Multibody Dynamics Analysis

2013 ◽  
Vol 392 ◽  
pp. 156-160
Author(s):  
Ju Seok Kang
Keyword(s):  
Stability Analysis ◽  
Multibody Dynamics ◽  
High Speed ◽  
Equations Of Motion ◽  
Contact Forces ◽  
Design Parameters ◽  
Railway Vehicle ◽  
Dynamics Analysis ◽  
Contact Points ◽  
The Stability

Multibody dynamics analysis is advantageous in that it uses real dimensions and design parameters. In this study, the stability analysis of a railway vehicle based on multibody dynamics analysis is presented. The equations for the contact points and contact forces between the wheel and the rail are derived using a wheelset model. The dynamics equations of the wheelset are combined with the dynamics equations of the other parts of the railway vehicle, which are obtained by general multibody dynamics analysis. The equations of motion of the railway vehicle are linearized by using the perturbation method. The eigenvalues of these linear dynamics equations are calculated and the critical speed is found.

scholarly journals A Lookup-Table-Based Approach for Spatial Analysis of Contact Problems

2014 ◽  
Vol 9 (4) ◽  
Author(s):  
Margarida Machado ◽  
Paulo Flores ◽  
Jorge Ambrósio
Keyword(s):  
Computation Time ◽  
Contact Problems ◽  
Contact Forces ◽  
Lookup Table ◽  
Direct Access ◽  
Normal Vector ◽  
Parametric Surface ◽  
Dynamic Simulations ◽  
Contact Points ◽  
Definition Of

The aim of this work is to present an efficient methodology to deal with general 3D-contact problems. This approach embraces three steps: geometrical definition of 3D surfaces, detection of the candidate contact points, and evaluation of the contact forces. The 3D-contact surfaces are generated and represented by using parametric functions due to their simplicity and ease in handling freeform shapes. This task is carried during preprocessing, which is performed before starting the multibody analysis. The preprocessing procedure can be condensed into four steps: a regular and representative collection of surface points is extracted from the 3D-parametric surface; for each point the tangent vectors to the u and v directions of the parametric surface and the normal vector are computed; the geometrical information on each point is saved in a lookup table, including the parametric point coordinates, the corresponding Cartesian coordinates, and the components of the normal, tangent, and bitangent vectors; the lookup table is rearranged such that the u-v mapping is converted into a 2D matrix being this surface data saved as a direct access file. For the detection of the contact points, the relative distance between the candidate contact points is computed and used to check if the bodies are in contact. The actual contact points are selected as those that correspond to the maximum relative indentation. The contact forces are determined as functions of the indentation or pseudopenetration, impact velocity, and geometric and material properties of the contacting surfaces. In general, lookup tables are used to reduce the computation time in dynamic simulations. However, the application of these schemes involves an increase of memory needs. Within the proposed approach, the amount of memory used is significantly reduced as a result of a partial upload into memory of the lookup table. A slider-crank mechanism with a cup on the top of the slider and a marble ball are used as a demonstrative example. A contact pair is considered between a cup and a marble ball, the contact forces for which are computed using a dissipative contact model.

scholarly journals A Lookup Table-Based Approach for Spatial Analysis of Contact Problems

2013 ◽  
Author(s):  
Maria Margarida Machado ◽  
Paulo Flores ◽  
Jorge Ambrósio
Keyword(s):  
Computation Time ◽  
Contact Problems ◽  
Contact Forces ◽  
Lookup Table ◽  
Direct Access ◽  
Normal Vector ◽  
Parametric Surface ◽  
Dynamic Simulations ◽  
Contact Points ◽  
3D Matrix

The aim of this work is to present an efficient methodology to deal with general 3D-contact problems. This approach embraces three steps: geometrical definition of 3D-surfaces; detection of the candidate contact points; evaluation of the contact forces. The 3D-contact surfaces are generated and represented by using parametric functions due to their simplicity and easiness to handle freeform shapes. This task is carried in preprocessing, performed preliminarily to the implementation of the multibody code. The preprocessing procedure can be condensed into four steps: a regular and representative surface collection of points is extracted from the 3D-parametric surface; for each point the tangent vectors to the u and v directions of the parametric surface and the normal vector are computed; the geometrical information on each point is saved in a lookup table, including the parametric point coordinates, the corresponding Cartesian coordinates and the Cartesian components of the normal, tangent and binormal vectors; the lookup table is rearranged such that the u-v mapping is converted into a 3D-matrix form. In the last step, the surface data is saved as a direct access file. Regarding the detection of the contact points, the relative distance between the candidate contact points are computed and used to check if the bodies are in contact. The actual contact points are selected as those that correspond to the maximum relative indentation. The contact forces are determined as functions of the indentation, impact velocity and geometric and material properties of the contacting surfaces. In general, lookup tables are used to reduce the computation time in dynamic simulations. However, the application of these schemes involves an increase of memory needs. Within the proposed approach, the amount of memory used is significantly reduced, as a result of a partial upload into memory of the lookup table. A slider-crank mechanism with a cup on the top of the slider and a marble ball is used as demonstrative example. A contact pair is considered between a cup and a marble ball, being the contact forces computed using a dissipative contact model.

Classification of Solutions to the Plane Extremal Distance Problem for Bodies With Smooth Boundaries

2016 ◽  
Vol 11 (6) ◽  
Author(s):  
Jochen Damerau ◽  
Robert J. Low
Keyword(s):  
Equations Of Motion ◽  
Time Integration ◽  
Contact Problems ◽  
Extremal Point ◽  
Point Problem ◽  
Transversality Conditions ◽  
Contact Points ◽  
Extremal Distance ◽  
Two Bodies

The determination of the contact points between two bodies with analytically described boundaries can be viewed as the limiting case of the extremal point problem, where the distance between the bodies is vanishing. The advantage of this approach is that the solutions can be computed efficiently along with the generalized state during time integration of a multibody system by augmenting the equations of motion with the corresponding extremal point conditions. Unfortunately, these solutions can degenerate when one boundary is concave or both boundaries are nonconvex. We present a novel method to derive degeneracy and nondegeneracy conditions that enable the determination of the type and codimension of all the degenerate solutions that can occur in plane contact problems involving two bodies with smooth boundaries. It is shown that only divergence bifurcations are relevant, and thus, we can simplify the analysis of the degeneracy by restricting the system to its one-dimensional center manifold. The resulting expressions are then decomposed by applying the multinomial theorem resulting in a computationally efficient method to compute explicit expressions for the Lyapunov coefficients and transversality conditions. Furthermore, a procedure to analyze the bifurcation behavior qualitatively at such solution points based on the Tschirnhaus transformation is given and demonstrated by examples. The application of these results enables in principle the continuation of all the solutions simultaneously beyond the degeneracy as long as their number is finite.

An enhanced multi-point dynamics methodology for collision and contact problem

2012 ◽  
Vol 227 (6) ◽  
pp. 1203-1223 ◽  
Author(s):  
Hua-Nan Yu ◽  
Jing-Shan Zhao ◽  
Fu-Lei Chu
Keyword(s):  
Contact Problem ◽  
Numerical Simulations ◽  
Multibody System ◽  
Contact Problems ◽  
Contact Forces ◽  
Point System ◽  
Collision System ◽  
Elastic Deformations ◽  
Real Contact ◽  
Collision Problem

The general methodologies for collision and contact problems of a multibody system often artificially divide the movements of the objects into the rigid displacements and contact deformations. Both the parts of the movements are often analyzed independently, which should be simultaneously processed in nature. Therefore, this article proposes an enhanced methodology to unify the rigid displacements and the contact elastic deformations of a collision system. The rigid displacements accompanying flexible deformations in collision are expressed as the displacements of a number of equivalent points of the colliding bodies. This method converts the collision problem into the collisions between the points within a multi-point system, and can therefore precisely model the real contact forces and deformations of the system. Numerical simulations for the collision-contact problems are carried out to demonstrate the effectiveness of the method.

scholarly journals Development of a Model for the Prediction of Wheel and Rail Wear in the Railway Field

2012 ◽  
Vol 7 (4) ◽  
Author(s):  
M. Ignesti ◽  
L. Marini ◽  
E. Meli ◽  
A. Rindi
Keyword(s):  
Fundamental Problem ◽  
Point Of View ◽  
Contact Forces ◽  
Dynamical Analysis ◽  
Railway Vehicle ◽  
Technical Documentation ◽  
Wear Model ◽  
Contact Points ◽  
Rail Wear ◽  
Dynamical Simulations

The wear prediction at the wheel-rail interface is a fundamental problem in the railway field, mainly correlated to the planning of maintenance interventions, vehicle stability, and the possibility of researching strategies for the design of optimal wheel and rail profiles from the wear point of view. The authors in this work present a model specifically developed for the evaluation of the wheel and rail wear and of the wheel and rail profiles evolution. The model layout is made up of two mutually interactive parts: a vehicle model for the dynamical analysis and a model for the wear estimation. The first one is a 3D multibody model of a railway vehicle where the wheel-rail interaction is implemented in a C/C++ user routine. Particularly, the research of the contact points between wheel and rail is based on an innovative algorithm developed by authors in previous works, while normal and tangential forces in the contact patches are calculated according to the Hertz and Kalker’s global theory, respectively. The wear model is mainly based on experimental relationships found in literature between the removed material by wear and the energy dissipated by friction at the contact. It starts from the outputs of the dynamical simulations (position of contact points, contact forces, and global creepages) and calculates the pressures inside the contact patches through a local contact model; then, the material removed by wear is evaluated and the worn profiles of wheel and rail are obtained. In order to reproduce the wear evolution, the overall mileage traveled by the vehicle is divided into discrete steps, within which the wheel and rail profiles are constant; after carrying out the dynamical simulations relative to one step, the profiles are updated by means of the wear model. Therefore, the two models work alternately until completing the whole mileage. Moreover, the different time scales characterizing the wheel and rail wear evolutions require the development of a suitable strategy for the profile update; the strategy proposed by the authors is based both on the total distance traveled by the vehicle and on the total tonnage burden on the track. The entire model has been developed and validated in collaboration with Trenitalia S.p.A. and Rete Ferroviaria Italiana (RFI), which have provided the technical documentation and the experimental results relating to some tests performed with the vehicle DMU Aln 501 Minuetto on the Aosta-Pre Saint Didier line.

Modeling Two-Point Wheel/Rail Contacts Using Constraint and Elastic-Force Approaches

2002 ◽  
Author(s):  
Ahmed A. Shabana ◽  
Khaled E. Zaazaa ◽  
Jose´ L. Escalona ◽  
Jalil R. Sany
Keyword(s):  
Contact Problem ◽  
Degrees Of Freedom ◽  
Point Contact ◽  
Elastic Force ◽  
Contact Forces ◽  
Force Model ◽  
Arc Length ◽  
Normal Contact ◽  
Contact Points ◽  
Constraint Approach

Two approaches are commonly used for solving the problem of wheel/rail contact in railroad dynamics. The first is the elastic approach in which the wheel is assumed to have six degrees of freedom with respect to the rail. The normal contact forces are defined using Hertz’s contact theory or in terms of assumed stiffness and damping coefficients. The second approach is the constraint approach in which nonlinear kinematic contact constraint equations are introduced, leading to a model in which the wheel has five degrees of freedom with respect to the rail. It is the objective of this investigation to present a new formulation for the wheel/rail contact problem based on the elastic force approach. Crucial to the success of any elastic force formulation for wheel/rail contact problem is the accurate prediction of the location of the contact points. To this end, features of multibody formulations that allow introducing arbitrary differential equations are exploited in this investigation in order to obtain a good estimate of the rail arc length traveled by the wheel set. In the formulation presented in this paper, four surface parameters are used to describe the wheel and the rail surfaces each with arbitrary geometry. In order to determine the location of the points of contact between the wheel and the rail, a first order differential equation for the rail arc length is introduced and is integrated simultaneously with the multibody equations of motion of the wheel/rail system. The method presented in this paper allows for multiple points of contact between the wheel and the rail by using an optimized search for all possible contact points. The normal contact forces are calculated and used with non-linear expressions for the creepages to determine the creep forces. The paper also discusses two different procedures for the analysis of the two-point contact in the wheel/rail interaction. Numerical results obtained using the elastic force model are presented and compared with the results obtained using the constraint approach.

scholarly journals The Indentation Rolling Resistance in Belt Conveyors: A Model for the Viscoelastic Friction

Lubricants ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 58 ◽  
Author(s):  
Nicola Menga ◽  
Francesco Bottiglione ◽  
Giuseppe Carbone
Keyword(s):  
Contact Problem ◽  
Finite Thickness ◽  
Numerical Procedure ◽  
Contact Problems ◽  
Accurate Solution ◽  
Rolling Contact ◽  
Viscoelastic Layer ◽  
Force Coefficient ◽  
Belt Conveyors ◽  
Energy Dissipated

In this paper, we study the steady-state rolling contact of a linear viscoelastic layer of finite thickness and a rigid indenter made of a periodic array of equally spaced rigid cylinders. The viscoelastic contact model is derived by means of Green’s function approach, which allows solving the contact problem with the sliding velocity as a control parameter. The contact problem is solved by means of an accurate numerical procedure developed for general two-dimensional contact geometries. The effect of geometrical quantities (layer thickness, cylinders radii, and cylinders spacing), material properties (viscoelastic moduli, relaxation time) and operative conditions (load, velocity) are all investigated. Physical quantities typical of contact problems (contact areas, deformed profiles, etc.) are calculated and discussed. Special emphasis is dedicated to the viscoelastic friction force coefficient and to the energy dissipated per unit time. The discussion is focused on the role played by the deformation localized at the contact spots and the one in the bulk of the thin layer, due to layer bending. The model is proposed as an accurate solution for engineering applications such as belt conveyors, in which the energy dissipated on the rolling contact of idle rollers can, in some cases, be by far the most important contribution to their energy consumption.

scholarly journals Slipping–rolling transitions of a body with two contact points

2021 ◽  
Author(s):  
Mate Antali ◽  
Gabor Stepan
Keyword(s):  
Rigid Body ◽  
Contact Point ◽  
Static Friction ◽  
Rigid Body Dynamics ◽  
Contact Forces ◽  
Friction Forces ◽  
Contact Points ◽  
Body Dynamics ◽  
Coulomb Model ◽  
Rigid Surfaces

AbstractIn this paper, the general kinematics and dynamics of a rigid body is analysed, which is in contact with two rigid surfaces in the presence of dry friction. Due to the rolling or slipping state at each contact point, four kinematic scenarios occur. In the two-point rolling case, the contact forces are undetermined; consequently, the condition of the static friction forces cannot be checked from the Coulomb model to decide whether two-point rolling is possible. However, this issue can be resolved within the scope of rigid body dynamics by analysing the nonsmooth vector field of the system at the possible transitions between slipping and rolling. Based on the concept of limit directions of codimension-2 discontinuities, a method is presented to determine the conditions when the two-point rolling is realizable without slipping.

Periodic Response of a Link Coupling With Clearance

1989 ◽  
Vol 111 (2) ◽  
pp. 253-259 ◽  
Author(s):  
Y. S. Choi ◽  
S. T. Noah
Keyword(s):  
Steady State ◽  
Boundary Conditions ◽  
Solution Procedure ◽  
Contact Forces ◽  
Steady State Response ◽  
Algebraic Equations ◽  
State Response ◽  
Contact Points ◽  
Integration Schemes ◽  
Nonlinear Algebraic Equations

The nonlinear, steady-state response of a displacement-forced link coupling with clearance with finite stiffness is determined. The solution procedure is derived from satisfying the boundary conditions at the contact points and then solving the resulting nonlinear algebraic equations by setting the duration of contact as a parameter. This direct approach to determining periodic solutions for systems with clearances with finite stiffness is substantially more efficient than numerical integration schemes. Results in terms of contact forces and durations of contact are pertinent to fatigue and wear studies. Parametric relations are presented for effects of the variation of damping, stiffness, exciting displacement, and gap length on the dynamic behavior of the link pair.

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