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.

Time Integration Incorporated Sensitivity Analysis With Generalized-α Method for Multibody Dynamics Systems

2011 ◽  
Author(s):  
Guang Dong ◽  
Zheng-Dong Ma ◽  
Gregory Hulbert ◽  
Noboru Kikuchi
Keyword(s):  
Sensitivity Analysis ◽  
Topology Optimization ◽  
Dynamic Loading ◽  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Time Integration ◽  
Optimization Method ◽  
Sensitivity Analysis Method ◽  
System Equations ◽  
Consecutive Time

The topology optimization method is extended for the optimization of geometrically nonlinear, time-dependent multibody dynamics systems undergoing nonlinear responses. In particular, this paper focuses on sensitivity analysis methods for topology optimization of general multibody dynamics systems, which include large displacements and rotations and dynamic loading. The generalized-α method is employed to solve the multibody dynamics system equations of motion. The developed time integration incorporated sensitivity analysis method is based on a linear approximation of two consecutive time steps, such that the generalized-α method is only applied once in the time integration of the equations of motion. This approach significantly reduces the computational costs associated with sensitivity analysis. To show the effectiveness of the developed procedures, topology optimization of a ground structure embedded in a planar multibody dynamics system under dynamic loading is presented.

Parallel Time-Integration of Flexible Multibody Dynamics Based on Newton-Waveform Method

2017 ◽  
Author(s):  
Shilei Han ◽  
Olivier A. Bauchau
Keyword(s):  
Multibody Dynamics ◽  
Time Integration ◽  
Flexible Multibody Dynamics ◽  
Small Scale ◽  
Time Step ◽  
Current Time ◽  
Parallel Time ◽  
Previous Time ◽  
Flexible Multibody ◽  
Multi Body

Traditionally, the time integration algorithms for multibody dynamics are in sequential. The predictions of previous time steps are necessary to get the solutions at current time step. This time-marching character impedes the application of parallel processor implementation. In this paper, the idea of computing a number of time steps concurrently is applied to flexible multi-body dynamics, which makes parallel time-integration possible. In the present method, the solution at the current time step is computed before accurate values at previous time step are available. This method is suitable for small-scale parallel analysis of flexible multibody systems.

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.

A COMPARISON OF DIFFERENT SOLUTION ALGORITHMS FOR THE NUMERICAL ANALYSIS OF VEHICLE–BRIDGE INTERACTION

2014 ◽  
Vol 14 (02) ◽  
pp. 1350065 ◽  
Author(s):  
K. LIU ◽  
N. ZHANG ◽  
H. XIA ◽  
G. DE ROECK
Keyword(s):  
Numerical Analysis ◽  
Time Domain ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Good Choice ◽  
Contact Forces ◽  
Time Dependent ◽  
Time Step ◽  
Solution Algorithms ◽  
Coupled Equation

The interaction between a bridge and a train moving on the bridge is a coupled dynamic problem. The equations of motion of the bridge and the vehicle are coupled by the time dependent contact forces. At each time step, the motion of the bridge influences the forces transferred to the vehicle and this, in turn, changes the forces acting on the bridge. In this paper, a comparison of three different time domain solution algorithms for the coupled equation of motion of the train–bridge system is presented. Guidelines are given for a good choice of the time step.

Modeling Impacts Between a Continuous System and a Rigid Obstacle Using Coefficient of Restitution

2009 ◽  
Vol 77 (2) ◽  
Author(s):  
Chandrika P. Vyasarayani ◽  
John McPhee ◽  
Stephen Birkett
Keyword(s):  
Initial Conditions ◽  
Equations Of Motion ◽  
Continuous System ◽  
Coefficient Of Restitution ◽  
Continuous Systems ◽  
Rigid Obstacle ◽  
Unit Impulse ◽  
Before And After ◽  
System Equations ◽  
The Impact

In this work, we discuss the limitations of the existing collocation-based coefficient of restitution method for simulating impacts in continuous systems. We propose a new method for modeling the impact dynamics of continuous systems based on the unit impulse response. The developed method allows one to relate modal velocity initial conditions before and after impact without requiring the integration of the system equations of motion during impact. The proposed method has been used to model the impact of a pinned-pinned beam with a rigid obstacle. Numerical simulations are presented to illustrate the inability of the collocation-based coefficient of restitution method to predict an accurate and energy-consistent response. We also compare the results obtained by unit impulse-based coefficient of restitution method with a penalty approach.

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 Computational Investigation of the Disk-Housing Impacts of Accelerating Rotors Supported by Hydrodynamic Bearings

2010 ◽  
Vol 78 (2) ◽  
Author(s):  
Jaroslav Zapoměl ◽  
Petr Ferfecki
Keyword(s):  
Virtual Work ◽  
Equations Of Motion ◽  
Angular Speed ◽  
Contact Forces ◽  
Computational Time ◽  
Radial Clearance ◽  
Lagrange Equations ◽  
Time Step ◽  
Hydrodynamic Bearings ◽  
Nonlinear Force

As the radial clearance between disks and the casing of rotating machines is usually very narrow, excessive lateral vibration of accelerating rotors passing critical speeds can produce impacts between the disks and the housing. The computer modeling method is an important tool for investigating such events. In the developed procedure, the shaft is flexible and the disks are absolutely rigid. The hydrodynamic bearings and the impacts are implemented in the mathematical model by means of nonlinear force couplings. Most of the publications and computer codes from the field of rotor dynamics are referred only in the case when the rotor turns at a constant angular speed and in simple cases of disk-housing impacts. Moreover, if the disks turning at variable speed are investigated, the resulting form of the equations of motion derived by different authors slightly differs and the differences depend on the method used for their derivation. Therefore, particular emphasis in this article is given to the derivation of the motion equations of a continuous rotor turning with variable revolutions to explain the mentioned differences, to develop a computer algorithm enabling the investigation of cases when impacts between an arbitrary number of disks and the stationary part take place, and to analyze the mutual interaction between the impacts and the fluid film bearings. The Hertz theory is applied to determine the contact forces. Calculation of the hydrodynamic forces acting on the bearings is based on solving the Reynolds equation and taking cavitation into account. Lagrange equations of the second kind and the principle of virtual work are used to derive equations of motion. The Runge–Kutta method with an adaptive time step is applied for their solution. The applicability of the developed procedure was tested by computer simulations. The results show that it can be used for the modeling of complex rotor systems and that the short computational time enables carrying out calculations for a number of design and operation parameters.

Torsional Vibrations of Synchronous Motor Driven Trains Using p-Method

1995 ◽  
Vol 117 (1) ◽  
pp. 152-160 ◽  
Author(s):  
W. J. Chen
Keyword(s):  
Dynamic Equilibrium ◽  
Dynamic Stiffness ◽  
Equations Of Motion ◽  
Synchronous Motor ◽  
Time Step ◽  
Integration Algorithm ◽  
Damping Matrix ◽  
Nonlinear Transient ◽  
Direct Numerical Integration ◽  
Stiffness And Damping

Synchronous motors produce damaging oscillating torques during startup. If the system is not properly analyzed and designed, the torsional excitation can be very destructive. This paper presents a systematic approach to the dynamic analysis of synchronous motor driven rotating machinery. The p-version of the finite element method is used in the formulation of the equations of motion which provides a great deal of simplicity in the modeling process. The convergence is achieved by increasing the polynomial order of the basis functions of the geometric elements. The system damping matrix can be constructed from the element level or can be calculated by specifying the critical damping factors for a given number of modes of interest. A modified Newmark integration method is employed in the nonlinear transient response calculation. The nonlinearity of the flexible resilient couplings can be easily implemented into this direct numerical integration algorithm. The dynamic stiffness and damping of the resilient couplings are updated at each time step to ensure the dynamic equilibrium. Two examples have been employed to illustrate the validity of the proposed algorithm. The effectiveness, accuracy, and simplicity of the use of p-method on the torsional vibration of synchronous motor driven trains are demonstrated in this paper.

scholarly journals Formulation of Equations of Motion for a Simply Supported Bridge under a Moving Railway Freight Vehicle

2007 ◽  
Vol 14 (6) ◽  
pp. 429-446 ◽  
Author(s):  
Ping Lou ◽  
Qing-yuan Zeng
Keyword(s):  
Equations Of Motion ◽  
Contact Forces ◽  
Dynamic Responses ◽  
Railway Track ◽  
Time Step ◽  
Bogie Frame ◽  
Simply Supported ◽  
Car Body ◽  
Railway Freight ◽  
Interaction System

Based on energy approach, the equations of motion in matrix form for the railway freight vehicle-bridge interaction system are derived, in which the dynamic contact forces between vehicle and bridge are considered as internal forces. The freight vehicle is modelled as a multi-rigid-body system, which comprises one car body, two bogie frames and four wheelsets. The bogie frame is linked with the car body through spring-dashpot suspension systems, and the bogie frame is rigidly linked with wheelsets. The bridge deck, together with railway track resting on bridge, is modelled as a simply supported Bernoulli-Euler beam and its deflection is described by superimposing modes. The direct time integration method is applied to obtain the dynamic response of the vehicle-bridge interaction system at each time step. A computer program has been developed for analyzing this system. The correctness of the proposed procedure is confirmed by one numerical example. The effect of different beam mode numbers and various surface irregularities of beam on the dynamic responses of the vehicle-bridge interaction system are investigated.

Modeling of Impact Dynamics of a Tennis Ball With a Flat Surface

2005 ◽  
Author(s):  
Syed Muhammad Mohsin Jafri ◽  
John M. Vance
Keyword(s):  
Flat Surface ◽  
Equations Of Motion ◽  
Contact Forces ◽  
Rolling Contact ◽  
Tennis Ball ◽  
Computationally Efficient ◽  
Mass Model ◽  
Experimental Findings ◽  
Tennis Balls ◽  
The Impact

A model of impact of a tennis ball with a flat surface is developed based on a planar, two-mass, linear, four degree of freedom vibration system idealization. The impact is assumed to be incident on a flat surface with friction. The incident parameters of the ball include the centre of mass translational velocity, angle of impact with the surface and the incident angular spin of the ball. The linear, piecemeal vibration model predicts the corresponding rebound parameters of the tennis ball. The model also predicts the duration of contact of the tennis ball with the flat surface, the transition of motion of the tennis ball during contact with the ground from sliding to rolling contact, and the resulting contact forces developed between the tennis ball and the flat surface. The model is computationally efficient because the governing differential equations of motion are linear and their standard solutions can be easily implemented on a personal computer. Predictions of the rebound parameters from the model are compared with experimental findings on tennis balls which are incident on a flat surface with various angles, velocities and angular spins (zero spin, topspin and backspin). For selected parameters of the two-mass model, the comparisons show excellent agreement between the model and the measurements.

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