A General and Efficient Method for Dynamic Analysis of Mechanical Systems Using Velocity Transformations

1986 ◽  
Vol 108 (2) ◽  
pp. 176-182 ◽  
Author(s):  
S. S. Kim ◽  
M. J. Vanderploeg
Keyword(s):  
Equations Of Motion ◽  
General Purpose ◽  
Rigid Bodies ◽  
Transformation Matrix ◽  
Cartesian Coordinates ◽  
Velocity Transformation ◽  
Joint Coordinates ◽  
Relative Coordinates ◽  
General Purpose Computer ◽  
New Formulation

This paper presents a new formulation for the equations of motion of interconnected rigid bodies. This formulation initially uses Cartesian coordinates to define the position of the system, the kinematic joints between bodies, and forcing functions on and between bodies. This makes initial system definition straightforward. The equations of motion are then derived in terms of relative joint coordinates through the use of a velocity transformation matrix. The velocity transformation matrix relates relative coordinates to Cartesian coordinates. It is derived using kinematic relationships for each joint type and graph theory for identifying the system topology. By using relative coordinates, the equations of motion are efficiently integrated. Use of both Cartesian and relative coordinates produces an efficient set of equations without loss of generality. The algorithm just described is implemented in a general purpose computer program. Examples are used to demonstrate the generality and efficiency of the algorithms.

Systematic Construction of the Equations of Motion for Multibody Systems Containing Closed Kinematic Loops

1989 ◽  
Author(s):  
P. E. Nikravesh ◽  
G. Gim
Keyword(s):  
Equations Of Motion ◽  
Transformation Matrix ◽  
Velocity Transformation ◽  
Constraint Equations ◽  
Advantages And Disadvantages ◽  
Lagrange Multiplier Technique ◽  
Joint Coordinates ◽  
Minimum Number ◽  
Kinematic Loops ◽  
Equations Of Motions

Abstract This paper presents a systematic method for deriving the minimum number of equations of motion for multibody system containing closed kinematic loops. A set of joint or natural coordinates is used to describe the configuration of the system. The constraint equations associated with the closed kinematic loops are found systematically in terms of the joint coordinates. These constraints and their corresponding elements are constructed from known block matrices representing different kinematic joints. The Jacobian matrix associated with these constraints is further used to find a velocity transformation matrix. The equations of motions are initially written in terms of the dependent joint coordinates using the Lagrange multiplier technique. Then the velocity transformation matrix is used to derive a minimum number of equations of motion in terms of a set of independent joint coordinates. An illustrative example and numerical results are presented, and the advantages and disadvantages of the method are discussed.

A Computational Method for the Dynamic Analysis of Planar Linkages With Applications

2001 ◽  
Author(s):  
Hazem A. Attia
Keyword(s):  
Closed Loop ◽  
Equations Of Motion ◽  
Open Loop ◽  
Rigid Bodies ◽  
Transformation Matrix ◽  
Computational Method ◽  
Constraint Forces ◽  
Constrained System ◽  
Velocity Transformation ◽  
Planar Linkages

Abstract This paper presents a computational method for generating the equations of motion of planar linkages consisting of interconnected rigid bodies. The formulation uses initially the rectangular Cartesian coordinates of an equivalent constrained system of particles to define the configuration of the mechanism. This results in a simple and straightforward procedure for generating the equations of motion. The equations of motion are then derived in terms of relative joint variables through the use of a velocity transformation matrix. The velocity transformation matrix relates the relative joint velocities to the Cartesian velocities. For the open loop case, this process automatically eliminates all of the non-working constraint forces and leads to an efficient integration of the equations of motion. For the closed loop case, suitable joints should be cut and few cut-joints constraint equations should be included for each closed loop. Examples are used to demonstrate the generality and efficiency of the proposed method.

Systematic Construction of the Equations of Motion for Multibody Systems Containing Closed Kinematic Loops

1993 ◽  
Vol 115 (1) ◽  
pp. 143-149 ◽  
Author(s):  
P. E. Nikravesh ◽  
Gwanghun Gim
Keyword(s):  
Equations Of Motion ◽  
Transformation Matrix ◽  
Velocity Transformation ◽  
Constraint Equations ◽  
Advantages And Disadvantages ◽  
Lagrange Multiplier Technique ◽  
Joint Coordinates ◽  
Minimum Number ◽  
Kinematic Loops ◽  
Equations Of Motions

This papers presents a systematic method for deriving the minimum number of equations of motion for multibody system containing closed kinematic loops. A set of joint or natural coordinates is used to describe the configuration of the system. The constraint equations associated with the closed kinematic loops are found systematically in terms of the joint coordinates. These constraints and their corresponding elements are constructed from known block matrices representing different kinematic joints. The Jacobian matrix associated with these constraints is further used to find a velocity transformation matrix. The equations of motions are initially written in terms of the dependent joint coordinates using the Lagrange multiplier technique. Then the velocity transformation matrix is used to derive a minimum number of equations of motion in terms of a set of independent joint coordinates. An illustrative example and numerical results are presented, and the advantages and disadvantages of the method are discussed.

Dynamics of Multirigid-Body Systems

1978 ◽  
Vol 45 (4) ◽  
pp. 889-894 ◽  
Author(s):  
R. L. Huston ◽  
C. E. Passerello ◽  
M. W. Harlow
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Rigid Bodies ◽  
Computer Algorithms ◽  
Euler Parameters ◽  
Closed Loops ◽  
Explicit Formulation ◽  
Relative Coordinates ◽  
D'alembert's Principle

New and recently developed concepts and ideas useful in obtaining efficient computer algorithms for solving the equations of motion of multibody mechanical systems are presented and discussed. These ideas include the use of Euler parameters, Lagrange’s form of d’Alembert’s principle, quasi-coordinates, relative coordinates, and body connection arrays. The mechanical systems considered are linked rigid bodies with adjoining bodies sharing at least one point, and with no “closed loops” permitted. An explicit formulation of the equations of motion is presented.

Formulation and Implementation of Nonholonomic Constraints in the Analysis of Multibody Systems Using Joint Coordinates

1996 ◽  
Author(s):  
Claus Balling
Keyword(s):  
Multibody Systems ◽  
Relative Velocity ◽  
Nonholonomic Constraint ◽  
Nonholonomic Constraints ◽  
General Purpose ◽  
Mutual Interaction ◽  
Cartesian Coordinates ◽  
Joint Coordinates ◽  
Moving Points ◽  
Two Bodies

Abstract A general formulation for analysis of spatial multibody systems subjected to nonholonomic constraints is presented. Nonholonomic constraints are usually formulated in Cartesian coordinates constraining the relative velocity or acceleration between some (fixed or moving) points on two bodies in mutual interaction or a point on one body with respect to ground. The formulation involves a specific type of nonholonomic constraint (rolling disk on a surface) in terms of joint coordinates and perform the implementation in a general purpose program.

Euler Parameters in Computational Kinematics and Dynamics. Part 1

1985 ◽  
Vol 107 (3) ◽  
pp. 358-365 ◽  
Author(s):  
P. E. Nikravesh ◽  
R. A. Wehage ◽  
O. K. Kwon
Keyword(s):  
Computer Program ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
General Purpose ◽  
Euler Parameters ◽  
Memory Space ◽  
Constrained Mechanical Systems ◽  
Limited Memory ◽  
General Purpose Computer ◽  
Purpose Computer

This paper presents useful and interesting identities between Euler parameters and their time derivatives. Using these identities, kinematic constraints and equations of motion for constrained mechanical systems are derived. These equations can be developed into a computer program to systematically generate all of the necessary equations to model mechanical systems. The compact form of these equations makes it possible to develop a general-purpose computer program for dynamic analysis of mechanical systems suitable for operation on small computers with limited memory space.

scholarly journals A Symbolic Approach to the Multibody Modeling of Road Vehicles

2017 ◽  
Vol 09 (05) ◽  
pp. 1750068 ◽  
Author(s):  
Roberto Lot ◽  
Matteo Massaro
Keyword(s):  
Equations Of Motion ◽  
Automatic Generation ◽  
General Purpose ◽  
Rigid Bodies ◽  
Simulation Code ◽  
Symbolic Form ◽  
Multibody Modeling ◽  
Mixed Coordinates ◽  
Computer Algebra Software ◽  
Symbolic Approach

This paper introduces MBSymba, an object-oriented language for the modeling of multibody systems and the automatic generation of equations of motion in symbolic form. MBSymba has built upon the general-purpose computer algebra software Maple and it is freely available for teaching and research purposes. With MBSymba, objects such as points, vectors, rigid bodies, forces and torques, and the relationships among them may be defined and manipulated both at high and low levels. Absolute, relative or mixed coordinates may be used, as well as combination of infinitesimal and noninfinitesimal variables. Once the system has been modeled, Lagrange’s and/or Newton’s equations can be derived in a quasi-automatic way, either in an inertial or noninertial reference frame. Equations can be automatically converted into Matlab, C/C++ or Fortan code to produce stand alone, numerically optimized simulation code. MBSymba is particularly suited for the modeling of ground, water or air vehicles; therefore, the mathematical model of a passenger car with trailer is illustrated as a case study. Time domain simulations, steady state analysis and stability results are also presented.

Velocity Transformation Matrix of a Revolute-Spherical Massless Link

2001 ◽  
Author(s):  
Wan-Suk Yoo ◽  
Jeong-Hyun Sohn ◽  
O-kaung Lim ◽  
Keum-Shik Hong
Keyword(s):  
Transformation Matrix ◽  
Analysis Program ◽  
Vehicle Dynamic ◽  
Spherical Joint ◽  
Transformation Technique ◽  
Inertia Effects ◽  
Velocity Transformation ◽  
Composite Joint ◽  
Relative Coordinates ◽  
Velocity Transformation Technique

Abstract Since the contribution of lightweight components in the total energy of a system is small, the inertia effects are sometimes ignored via replacing them to massless links. A massless link, which is sometimes called as a composite joint, connects two adjacent bodies keeping the relative degrees of freedoms. For a revolute-spherical massless link, one edge is connected to an adjacent body with a revolute joint and the other edge is linked to another body with a spherical joint. Using velocity transformation technique, it is possible to combine the generality of Cartesian coordinates in modeling and the efficiency of relative coordinates in simulation. In this paper, velocity transformation matrix of a revolute-spherical massless link is formulated and implemented as a joint module in a vehicle dynamic analysis program. Numerical examples are presented to verify the formulation.

A General Purpose Formulation for Nonsmooth Dynamics Including Large Rotations: Application to the Woodpecker Toy

2019 ◽  
Author(s):  
Javier Galvez ◽  
Alejandro Cosimo ◽  
Federico J. Cavalieri ◽  
Alberto Cardona ◽  
Olivier Brüls
Keyword(s):  
Finite Element ◽  
Equations Of Motion ◽  
General Purpose ◽  
Rigid Bodies ◽  
Nonsmooth Dynamics ◽  
Minimal Set ◽  
Frictional Contacts ◽  
Bilateral Constraints ◽  
Large Rotations ◽  
Element Approach

Abstract The aim of this work is to extend the finite element multibody dynamics approach to problems involving frictional contacts and impacts. Since rigid bodies and joints involve bilateral constraints, it is important to avoid any drift phenomenon. Therefore, the nonsmooth generalized-α method is used, which imposes the constraints both at position and at velocity levels. Its low intrusiveness allows one to reuse an existing library of elements without major modifications. The study of the woodpecker toy dynamics sets up a good example to show the capabilities of the nonsmooth generalized-α within the context of a general finite element framework. This example has already been studied by many authors who generally adopt a model with a minimal set of coordinates and small rotations. We show that using a finite element approach, the equations of motion can be assembled automatically, and large rotations can be easily considered.

Generalized Vector Derivatives for Systems With Multiple Relative Motion

1968 ◽  
Vol 35 (1) ◽  
pp. 20-24 ◽  
Author(s):  
T. A. Sherby ◽  
J. F. Chmielewski
Keyword(s):  
Reference Frame ◽  
Relative Motion ◽  
Reference Frames ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Main Body ◽  
Angular Velocity Vector ◽  
Relative Coordinates ◽  
Moving Body ◽  
Classical Vector

For the analysis of relative motion, classical vector mathematics is limited to the use of one moving reference frame when taking vector derivatives. However, many dynamical systems consist of a number of rigid bodies in motion relative to one another. The classical procedure requires the specification of the position of each body relative to a single “main body.” The use of relative coordinates allows a natural specification of the position of one moving body relative to another moving body in network fashion. To use relative coordinates in dynamic and kinematic analyses, it is necessary to use relative vector derivatives involving more than one moving reference frame. This paper presents general expressions for the kth-order derivative of a relative position and angular velocity vector measured in any moving reference frame in a system of m reference frames with multiple relative motion. These expressions are used to develop a procedure which generates the differential equations of motion for the system by routine substitution of the relative coordinates and their scalar derivatives. This procedure offers promise as an algorithm for machine generation of the system equations and eliminates the possibility of neglecting subtle accelerations due to relative motion. The use of the procedure is demonstrated by generating the equations of motion of an offset unsymmetrical gyroscope.

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