Kinematics for unilateral constraints in multibody dynamics

2016 ◽  
Vol 22 (8) ◽  
pp. 1654-1687
Author(s):  
P Lidström
Keyword(s):  
Multibody Dynamics ◽  
Distance Function ◽  
Contact Surface ◽  
Equations Of Motion ◽  
Time Derivative ◽  
Unilateral Constraint ◽  
Contact Forces ◽  
Unilateral Constraints ◽  
Constraint Forces ◽  
Friction Forces

This paper is concerned with the kinematics of unilateral constraints in multibody dynamics. These constraints are related to the contact between parts and the principle of impenetrability of matter and have the property that they may be active, in which case they give rise to constraint forces, or passive, in which case they do not give rise to constraint forces. In order to check whether the constraint is active or passive a distance function between parts of the multibody is required. The paper gives a rigorous definition of the distance function and derives certain of its properties. The unilateral constraint may then be expressed in terms of this distance function. The paper analyses the transitions from passive constraints to active and vice versa. Sufficient regularity of the transplacements of the parts and their boundary surfaces will lead to specific properties of the time derivative of the distance function. When the unilateral constraint is active then the parts are geometrically in contact and there is a certain contact surface that, in specific cases, may degenerate into a point. If the parts are in mechanical contact over the contact surface then there will be an interaction between the parts given by contact forces, such as normal and friction forces. Parts in contact may be at rest relative to one another, over the contact surface, or they may be in relative sliding motion. The transition from non-sliding contact to sliding and from sliding to non-sliding is discussed and necessary conditions on the relative velocity and the traction vector are derived. Appropriate complementary conditions are then formulated. These are instrumental when the technique of linear complementarity is used in order to find solutions to the equations of motion.

A General Purpose Formulation for Nonsmooth Dynamics With Finite Rotations: Application to the Woodpecker Toy

2020 ◽  
Vol 16 (3) ◽  
Author(s):  
Alejandro Cosimo ◽  
Federico J. Cavalieri ◽  
Javier Galvez ◽  
Alberto Cardona ◽  
Olivier Brüls
Keyword(s):  
Finite Element ◽  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Horizontal Displacement ◽  
General Purpose ◽  
Nonsmooth Dynamics ◽  
Unilateral Constraints ◽  
Simulation Code ◽  
Frictional Contacts ◽  
Large Rotations

Abstract The aim of this work is to extend the finite element multibody dynamics approach to problems involving frictional contacts and impacts. The nonsmooth generalized-α (NSGA) scheme is adopted, which imposes bilateral and unilateral constraints both at position and velocity levels avoiding drift phenomena. This scheme can be implemented in a general purpose simulation code with limited modifications of pre-existing elements. The study of the woodpecker toy dynamics sets up a good example to show the capabilities of the NSGA scheme within the context of a general finite element framework. This example has already been studied by many authors who generally adopted a model with a minimal set of coordinates and small rotations. It is shown that good results are obtained using a general purpose finite element code for multibody dynamics, in which the equations of motion are assembled automatically and large rotations are easily taken into account. In addition, comparing results between different models of the woodpecker toy, the importance of modeling large rotations and the horizontal displacement of the woodpecker's sleeve is emphasized.

An Efficient Method for a Category of Contact/Impact Problems in Multibody Systems: Tree Topologies

2005 ◽  
Author(s):  
Michael J. Sadowski ◽  
Kurt S. Anderson
Keyword(s):  
Rigid Body ◽  
Dynamic Systems ◽  
Equations Of Motion ◽  
Momentum Balance ◽  
Unilateral Constraints ◽  
Constraint Forces ◽  
Rigid Body Dynamic ◽  
Tree Topologies ◽  
Contact Impact ◽  
Impact Problems

This paper presents an algorithm for the efficient numerical analysis and simulation of a category of contact/impact problems in multi-rigid-body dynamic systems with tree topologies. The algorithm can accommodate the jumps in structure which occur in the equations of motion of general multi-rigid-body systems due to a contact/impact event between bodies, or due to the locking of joints as long as the resulting system is a tree topology. The presented method uses a generalized momentum balance approach to determine the velocity jumps which take place across impacts in such multibody dynamic systems where event constraint forces are of the “non-working” category. The presented method does not suffer from the performance (speed) penalty encountered by most other momentum balance methods given its O(n) overall cost, and exact direct embedded consideration of all the constraints. Due to these characteristics, the presented algorithm offers superior computing performance relative to other methods in situations involving both large n and potentially many unilateral constraints.

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.

Implicit Integration of the Equations of Multibody Dynamics in Descriptor Form

1997 ◽  
Author(s):  
Edward J. Haug ◽  
Mirela Iancu ◽  
Dan Negrut
Keyword(s):  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Time Derivative ◽  
Kinetic Equations ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Integration Formulas ◽  
Multibody Dynamic ◽  
Constrained Equations

Abstract An implicit numerical integration approach, based on generalized coordinate partitioning of the descriptor form of the differential-algebraic equations of motion of multibody dynamics, is presented. This approach is illustrated for simulation of stiff mechanical systems using the well known Newmark integration method from structural dynamics. Second order Newmark integration formulas are used to define independent generalized coordinates and their first time derivative as functions of independent accelerations. The latter are determined as the solution of discretized equations obtained using the descriptor form of the equations of motion. Dependent variables in the formulation, including Lagrange multipliers, are determined to satisfy all the kinematic and kinetic equations of multibody dynamics. The approach is illustrated by solving the constrained equations of motion for mechanical systems that exhibit stiff behavior. Results show that the approach is robust and has the capability to integrate differential-algebraic equations of motion for stiff multibody dynamic systems.

The Influence of the Suspension on the Phenomena of Wheel-Rail Contact

2015 ◽  
Vol 809-810 ◽  
pp. 1061-1066
Author(s):  
Ioan Sebeşan ◽  
Valeriu Ştefan
Keyword(s):  
Contact Surface ◽  
Contact Problems ◽  
Contact Forces ◽  
Dynamic Loads ◽  
Friction Forces ◽  
Normal Contact ◽  
The Moment ◽  
The Relationship ◽  
Contact Pressures ◽  
Pivoting Friction

Efficient use of adhesion between wheels and rails involves a good knowledge of this phenomenon, in order to equip the vehicle with adequate facilities and systems that protect the vehicle and the rail. The loading of the vehicle's axle with dynamic loads in vertical and horizontal planes, are to be developed in the area of contact, both normal stress and shear distributed stress, their sum giving the friction force and the moment of pivoting friction (spin). This makes the wheel-rail contact problems take the two aspects of the study, namely the problem of normal and tangential contact issue. The normal contact problem involves regular geometric shape bodies, determining the size of the resulting contact surface, the distribution of the normal contact pressures and the relationship between the proximity of the bodies and the normal contact force. Solving the problem of the tangential wheel-rail contact is about to establish the correlation between the creepage, normal contact forces and friction forces, and also the ratio between the adherent contact surface and the nominal contact surface where the creepage ocurs.

A Methodology to Detect the Precise Instant of Contact in Multibody Dynamics

2011 ◽  
Author(s):  
Paulo Flores
Keyword(s):  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Contact Forces ◽  
Time Step ◽  
Dynamic Simulations ◽  
Integration Algorithm ◽  
Current Time ◽  
Impact Time ◽  
System Equations ◽  
The Impact

The main purpose of this work is to present a general and comprehensive approach to automatically adjust the time step for the contact and non contact periods in multibody dynamics. The basic idea of the described methodology is to ensure that the first impact within a multibody system does not occur with a large value for relative bodies’ penetration in order to avoid the artificially large contact forces associated. The detection of the instant of contact takes place when the distance between two bodies change the sign between two discrete moments in time. In fact, in theory, the contact starts when this distance is zero, or a very small value to prevent the round-off errors. Thus, during the numerical solution of the system equations of motion if the first penetration is below this small value previously specified, then the current time is taken as the impact time. On the other hand, if the first penetration is larger than the specified tolerance, then the current time step is beyond the impact time. In this case, integration algorithm is forced to go back and take a smaller time step until a step can be taken within the acceptable tolerance. The main features of this approach are the easiness to implement and the good computational efficiency. In addition, it can easily deal with the transitions between non contact and contact cases in multibody dynamics. Finally, results obtained from dynamic simulations are presented and discussed to study the validity of the methodology proposed in this work.

scholarly journals Modelling and Numerical Solution of the Problem with Unilateral Constraints and Friction under Dynamic Action of the Load

2021 ◽  
Vol 20 (1) ◽  
pp. 16-25
Author(s):  
A. A. Lukashevich ◽  
N. K. Lukashevich ◽  
N. V. Ostrovskaya
Keyword(s):  
Coulomb Friction ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Dynamic Contact ◽  
Point Of View ◽  
Contact Friction ◽  
Unilateral Constraints ◽  
Contact Conditions ◽  
Friction Forces ◽  
Dynamic Action

Problems with unilateral constraints are not uncommon in the practice of calculating  building construction and structures. Certain difficulties in solving them arise during contact friction, as well as the dynamic action of the load. It is known that such problems from a mathematical point of view s are not correct enough, their solution becomes more complicated and depends on the history of loading and deformation of the structure. At the same time, the ability to take into account the complex working conditions of the structure makes its calculation more complete and accurate. The paper considers the solution of  a dynamic contact problem on the basis of the finite element method and the step-by-step loading method. Unilateral constraints with Coulomb friction are modeled using contact finite elements of a frame-rod type. The method of compensating loads is applied in order to comply with the limitations under ultimate friction-sliding conditions. Based on the considered discrete contact model and the step-by-step analysis method, a numerical algorithm has been developed, which allows in one step-by-step process to integrate simultaneously the equations of motion and implement contact conditions with Coulomb friction. With the help of the proposed approach, numerical solutions of the problem pertaining to a structure contact with the base have been obtained and analyzed at various parameters of dynamic load. Comparison of the results with the solution obtained by the well-known iteration method on the ultimate friction forces allows to conclude about the efficiency and reliability of the developed algorithm under complex contact conditions and dynamic loading.

Nonlinear Dynamics and Vibrations of Three Dimensional Multibody Tracked Vehicles

1996 ◽  
Author(s):  
A. A. Shabana ◽  
J. H. Choi ◽  
H. C. Lee
Keyword(s):  
Nonlinear Dynamic ◽  
Large Scale ◽  
Three Dimensional ◽  
Nonlinear Dynamic Analysis ◽  
Contact Forces ◽  
Algebraic Equations ◽  
Constraint Forces ◽  
Tracked Vehicle ◽  
Friction Forces ◽  
Tracked Vehicles

Abstract A three dimensional model that includes significant details is developed for the nonlinear dynamic analysis of large scale multi-body tracked vehicle systems. In this model, the joint articulations of the track chains are taken into consideration so as to allow the developement of a computational procedure for the analysis of the vibrations and the contact forces of the multibody tracked vehicles. The three dimensional vehicle system is assumed to consist of three kinematically decoupled subsystems which include the chassis subsystem, and two track subsystems. A recursive approach for formulating the nonlinear equations of the vehicle based on the velocity transformation is used in this investigation in order to reduce the number of equations, avoid the solution of a system of differential and algebraic equations, and avoid the use of nonholonomic constraints to describe the rotations of the sprockets. The singular configurations of the closed kinematic chains of the tracks are also avoided by using a penalty function approach to define the constraint forces at selected secondary joints of the tracks. Detailed three dimensional nonlinear contact force models that describe the interaction between the track links and the vehicle components such as the rollers, sprockets, and idlers as well as the interaction between the track links and the ground are developed and used to define the generalized contact forces associated with the vehicle generalized coordinates. In particular, body and surface coordinate systems are introduced in order to define the spatial contact conditions that describe the dynamic interaction between the teeth of the sprockets and the track link pins. These conditions provide the forces necessary for driving the tracked vehicle. The effect of the tangential friction forces on the stability of the motion of the vehicle is also discussed in this investigation. A computer simulation of a tracked vehicle that consists of one hundred and six bodies and has one hundred and twenty degrees of freedom is presented in order to demonstrate the use of the formulations presented in this study. A simple formula that can be used to predict the steady state velocity of the vehicle when the sprockets rotate with a constant angular velocity is presented and used to verify the numerical results obtained from the nonlinear dynamic simulation of the multibody vehicle.

Dynamics of Multibody Systems Containing Dependent Unilateral Constraints with Friction

1996 ◽  
Vol 2 (2) ◽  
pp. 161-192 ◽  
Author(s):  
M. Wösle ◽  
F. Pfeiffer
Keyword(s):  
Multibody Systems ◽  
Convex Sets ◽  
Equations Of Motion ◽  
Mathematical Formulation ◽  
Differential Algebraic Equations ◽  
Mathematical Form ◽  
Algebraic Equations ◽  
Unilateral Constraints ◽  
Constraint Forces ◽  
Stick Slip

In couplings of machines and mechanisms, backlash and friction phenomena are always occurring. Whether stick-slip phenomena take place depends on the structure of such couplings. These processes can be modeled as multibody systems with a time-varying topology. Making use of Lagrange multiplier methods with a mathematical formulation of the contact problem is very efficient for large systems with many constraints. The differential-algebraic equations of a system are transformed into a resolvable mathematical form by means of contact laws. In the following, rigid multibody systems with dependent constraints and planar friction will be considered. For the evaluation of such problems, an iterative algorithm is presented. This method is based on transformations of the kinematic secondary conditions in the form of inequalities to equations. In mathematical sense, these transformations are projections of the constraint forces on convex sets. Ultimately, we have a solvable nonlinear system of equations consisting of the differential equations of motion, the constraint equations and the projections of the constraint forces.

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.

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