A Symbolic Formulation for Linearization of Multibody Equations of Motion

1995 ◽  
Vol 117 (3) ◽  
pp. 441-445 ◽  
Author(s):  
A. G. Lynch ◽  
M. J. Vanderploeg
Keyword(s):  
Differential Equations ◽  
Equilibrium Point ◽  
Dynamic Systems ◽  
Closed Loop ◽  
Multibody System ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Multibody Dynamic ◽  
Constrained Multibody System ◽  
Symbolic Equations Of Motion

This paper presents a method for obtaining linearized state space representations of open or closed loop multibody dynamic systems. The paper develops a symbolic formulation for multibody dynamic systems which result in an explicit set of symbolic equations of motion. The symbolic equations are then used to perform symbolic linearizations. The resulting symbolic, linear equations are in terms of the system parameters and the equilibrium point, and are valid for any equilibrium point. Finally, a method is developed for reducing a linearized, constrained multibody system consisting of a mixed set of algebraic-differential equations to a reduced set of differential equations in terms of an independent coordinate set. An example is used to demonstrate the technique.

Multirate Integration for Multibody Dynamic Analysis With Decomposed Subsystems

1999 ◽  
Author(s):  
Sung-Soo Kim ◽  
Jeffrey S. Freeman
Keyword(s):  
Dynamic Analysis ◽  
Multibody System ◽  
Equations Of Motion ◽  
Low Frequency ◽  
Inertia Force ◽  
Integration Scheme ◽  
Integration Algorithm ◽  
Multibody Dynamic ◽  
Higher Order Derivative ◽  
Derivative Information

Abstract This paper details a constant stepsize, multirate integration scheme which has been proposed for multibody dynamic analysis. An Adams-Bashforth Moulton integration algorithm has been implemented, using the Nordsieck form to store internal integrator information, for multirate integration. A multibody system has been decomposed into several subsystems, treating inertia coupling effects of subsystem equations of motion as the inertia forces. To each subsystem, different rate Nordsieck form of Adams integrator has been applied to solve subsystem equations of motion. Higher order derivative information from the integrator provides approximation of inertia force computation in the decomposed subsystem equations of motion. To show the effectiveness of the scheme, simulations of a vehicle multibody system that consists of high frequency suspension motion and low frequency chassis motion have been carried out with different tire excitation forces. Efficiency of the proposed scheme has been also investigated.

MULTIBODY DYNAMIC MODELING AND FLUTTER ANALYSIS OF A FLEXIBLE SLENDER VEHICLE

2012 ◽  
Vol 12 (06) ◽  
pp. 1250049 ◽  
Author(s):  
A. RASTI ◽  
S. A. FAZELZADEH
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Dynamic Modeling ◽  
Equations Of Motion ◽  
The Body ◽  
Elastic Deformations ◽  
Strip Theory ◽  
Multibody Dynamic ◽  
Flutter Analysis ◽  
Rigid Body Motions

In this paper, multibody dynamic modeling and flutter analysis of a flexible slender vehicle are investigated. The method is a comprehensive procedure based on the hybrid equations of motion in terms of quasi-coordinates. The equations consist of ordinary differential equations for the rigid body motions of the vehicle and partial differential equations for the elastic deformations of the flexible components of the vehicle. These equations are naturally nonlinear, but to avoid high nonlinearity of equations the elastic displacements are assumed to be small so that the equations of motion can be linearized. For the aeroelastic analysis a perturbation approach is used, by which the problem is divided into a nonlinear flight dynamics problem for quasi-rigid flight vehicle and a linear extended aeroelasticity problem for the elastic deformations and perturbations in the rigid body motions. In this manner, the trim values that are obtained from the first problem are used as an input to the second problem. The body of the vehicle is modeled with a uniform free–free beam and the aeroelastic forces are derived from the strip theory. The effect of some crucial geometric and physical parameters and the acting forces on the flutter speed and frequency of the vehicle are investigated.

scholarly journals Numerical Integration of Multibody Dynamic Systems Involving Nonholonomic Equality Constraints

2021 ◽  
Author(s):  
Sotirios Natsiavas ◽  
Panagiotis Passas ◽  
Elias Paraskevopoulos
Keyword(s):  
Dynamic Systems ◽  
Equations Of Motion ◽  
Time Integration ◽  
Nonholonomic Constraints ◽  
Coupled System ◽  
Weak Form ◽  
Integration Scheme ◽  
Motion Constraints ◽  
Multibody Dynamic ◽  
Time Integration Scheme

Abstract This work considers a class of multibody dynamic systems involving bilateral nonholonomic constraints. An appropriate set of equations of motion is employed first. This set is derived by application of Newton’s second law and appears as a coupled system of strongly nonlinear second order ordinary differential equations in both the generalized coordinates and the Lagrange multipliers associated to the motion constraints. Next, these equations are manipulated properly and converted to a weak form. Furthermore, the position, velocity and momentum type quantities are subsequently treated as independent. This yields a three-field set of equations of motion, which is then used as a basis for performing a suitable temporal discretization, leading to a complete time integration scheme. In order to test and validate its accuracy and numerical efficiency, this scheme is applied next to challenging mechanical examples, exhibiting rich dynamics. In all cases, the emphasis is put on highlighting the advantages of the new method by direct comparison with existing analytical solutions as well as with results of current state of the art numerical methods. Finally, a comparison is also performed with results available for a benchmark problem.

Symbolic Linearized Equations for Nonholonomic Multibody Systems With Closed-Loop Kinematics

2018 ◽  
Author(s):  
Márton Kuslits ◽  
Dieter Bestle
Keyword(s):  
Lagrange Multipliers ◽  
Multibody Systems ◽  
Closed Loop ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Algebraic Equations ◽  
Computationally Efficient ◽  
Linearized Equations ◽  
Nonlinear Equations Of Motion ◽  
Loop Constraints

Multibody systems and associated equations of motion may be distinguished in many ways: holonomic and nonholonomic, linear and nonlinear, tree-structured and closed-loop kinematics, symbolic and numeric equations of motion. The present paper deals with a symbolic derivation of nonlinear equations of motion for nonholonomic multibody systems with closed-loop kinematics, where any generalized coordinates and velocities may be used for describing their kinematics. Loop constraints are taken into account by algebraic equations and Lagrange multipliers. The paper then focuses on the derivation of the corresponding linear equations of motion by eliminating the Lagrange multipliers and applying a computationally efficient symbolic linearization procedure. As demonstration example, a vehicle model with differential steering is used where validity of the approach is shown by comparing the behavior of the linearized equations with their nonlinear counterpart via simulations.

Methods for Automated Dynamic Analysis of Discrete Mechanical Systems

1973 ◽  
Vol 40 (3) ◽  
pp. 809-811 ◽  
Author(s):  
Y. O. Bayazitoglu ◽  
M. A. Chace
Keyword(s):  
Differential Equations ◽  
Dynamic Analysis ◽  
Ordinary Differential Equations ◽  
Mechanical System ◽  
Dynamic Systems ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Constrained Systems ◽  
Lower Pair ◽  
Linear And Nonlinear

The equations of motion for any discrete, lower pair mechanical system can be obtained by analyzing a branched, three-dimensional compound pendulum of indefinite length. In this paper, a set of expressions which provides the equations of motion of arbitrary mechanical dynamic systems directly as ordinary differential equations are presented. These expressions and the associated technique is applicable to linear and nonlinear unconstrained dynamic systems, kinematic systems and multidegree-of-freedom constrained systems.

scholarly journals Nonlinear vibration absorbers for ropeway roller batteries control

2020 ◽  
pp. 095440622095345
Author(s):  
Biagio Carboni ◽  
Andrea Arena ◽  
Walter Lacarbonara
Keyword(s):  
Passive Control ◽  
Multibody System ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Elastic Vibration ◽  
Vibration Absorbers ◽  
Response Curves ◽  
Control Effectiveness ◽  
De Algorithm ◽  
Linear Equations Of Motion

This work investigates a nonlinear passive control strategy designed to reduce the peak accelerations in ropeway roller batteries systems by deploying an array of nonlinearly visco-elastic vibration absorbers. The control effectiveness is compared with that of an equivalent array made of linearly visco-elastic absorbers. A nonlinear parametric model describing the interactions between the different parts of this mechanical multibody system previously developed by the present authors is here extended to include the passive vibration control system aimed to mitigate the acceleration peaks induced by the vehicles transit at different operational speeds. To this aim, a set of linearly visco-elastic vibration absorbers is first optimized through the Differential Evolution (DE) algorithm seeking to minimize the area below the frequency-response curves of the linear equations of motion. Then, a new group of nonlinearly visco-elastic absorbers, that can be largely tuned (i.e., they can exhibit either softening or hardening behaviors), is proposed to mitigate the accelerations induced in the roller by the vehicle transit. These nonlinearly visco-elastic absorbers are optimized by means of the DE algorithm and comparisons with the control achieved by the linear absorbers are carried out to show the higher performance of the proposed nonlinear device. A possible design of the nonlinearly visco-elastic absorber, based on the hysteresis of a wire rope assembly undergoing flexural cycles, is also proposed and discussed.

EQUATIONS OF MOTION OF COMPLEX MULTIBODY SYSTEMS USING KINEMATICAL DIFFERENTIALS

1989 ◽  
Vol 13 (4) ◽  
pp. 113-121 ◽  
Author(s):  
M. HILLER ◽  
A. KECSKEMETHY
Keyword(s):  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Multibody System ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Nonholonomic Systems ◽  
Automatic Generation ◽  
Minimal Order ◽  
Closed Form Solutions ◽  
Constraint Equations

In complex multibody systems the motion of the bodies may depend on only a few degrees of freedom. For these systems, the equations of motion of minimal order, although more difficult to obtain, give a very efficient formulation. The present paper describes an approach for the automatic generation of these equations, which avoids the use of LAGRANGE-multipliers. By a particular concept, designated “kinematical differentials”, the problem of determining the partial derivatives required to state the equations of motion is reduced to a simple re-evaluation of the kinematics. These cover the solution of the global position, velocity and acceleration problems, i.e. the motion of all bodies is determined for given generalized (independent) coordinates. For their formulation and solution, the multibody system is mapped to a network of nonlinear transformation elements which are connected by linear equations. Each transformation element, designated “kinematical transformer”, corresponds to an independent multibody loop. This mapping of the constraint equations makes it possible to find closed-form solutions to the kinematics for a wide variety of technical applications, and (via kinematical differentials) leads also to an efficient formulation of the dynamics. The equations are derived for holonomic, scleronomic systems, but can also be extended to general nonholonomic systems.

Linearization and Jacobian Evaluation of the Dynamics of Closed-Chain Mechanical Systems

1992 ◽  
Author(s):  
Tsung-Chieh Lin ◽  
K. Harold Yae
Keyword(s):  
Dynamic Systems ◽  
Large Scale ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Numerical Differentiation ◽  
Large Scale Systems ◽  
Closed Chain ◽  
Multibody Dynamic ◽  
The Finite Difference Method ◽  
Recursive Equations

Abstract This paper presents an analytical/numerical method for linearizing the equations of motion and evaluating the system Jacobian matrices of mechanical systems with closed chains. The linearization algorithm developed here first identifies and linearizes basic recursive kinematic relationships and then applies the chain rule to the derivation of the equations of motion under the framework of recursive formulation. This method can be incorporated into formulating recursive equations of motion for general multibody dynamic systems, to handle large scale systems. Since no numerical differentiation is used in the proposed algorithm, its accuracy is comparable to symbolic, closed-form linearization. Moreover, without the need of repetitious computation to select proper perturbation quantities, this method is computationally more efficient than the finite difference method.

Formulation of Multibody Dynamics Equations in Absolute or Relative Coordinates Using the Vector-Network Method

1994 ◽  
Author(s):  
John J. McPhee
Keyword(s):  
Closed Loop ◽  
Multibody System ◽  
Equations Of Motion ◽  
Open Loop ◽  
Graph Representation ◽  
Linear Graph ◽  
Relative Coordinates ◽  
Network Method ◽  
Complete Set ◽  
Theoretical Procedure

Abstract The objective of this paper is to show how a single linear graph representation of a multibody system can be used to derive the complete set of equations of motion in either absolute or relative coordinates, depending upon the elements selected into the spanning tree of the linear graph. Criteria for selecting a tree that gives the desired set of equations is given and the systematic nature of this graph-theoretical procedure, known as the Vector-Network Method, is demonstrated by means of two planar examples. The first is an open-loop compound pendulum, and the second is a closed-loop four-bar mechanism driven by a time-varying torque.

Recursive Dynamics of Overconstrained Mechanisms

1994 ◽  
Author(s):  
Udo Rein
Keyword(s):  
Closed Loop ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Open Loop ◽  
Loop Closure ◽  
Kinematic Chains ◽  
Reaction Forces ◽  
Overconstrained Mechanisms ◽  
Recursive Dynamics ◽  
Closure Constraints

Abstract Overconstrained mechanisms contain loop-closure constraints which are redundant due to a special geometry of the links. Some reaction forces of an overconstrained mechanism cannot be calculated from the dynamics of the mechanism. This means that an overconstrained mechanism is statically indeterminate. The recursive formalism was originally developed to derive the equations of motion for open-loop kinematic chains, but it has been extended by various authors to closed-loop mechanisms. This paper discusses the recursive formalism when it is applied to an overconstrained closed-loop mechanism. It will be shown that the redundant loop-closure constraints lead to rather small singular, but consistent sets of linear equations for the reaction forces at the corresponding cut joints. This means that the reaction forces at those cut joints are not unique for an overconstrained mechanism, but the variety of possible solutions does not affect the dynamics of the overconstrained mechanism. This behaviour of the recursive formalism can be used to perform an on-line investigation of the static indeterminacy of a mechanism, including singular positions, where joint constraints are redundant only at one specific position.

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