Generalized-<italic>α</italic> projection method for stiff dynamic equations of multibody systems

2013 ◽  
Vol 43 (4) ◽  
pp. 572-578
Author(s):  
JieYu DING ◽  
ZhenKuan PAN
Keyword(s):  
Projection Method ◽  
Multibody Systems ◽  
Dynamic Equations

Dynamics of Serial Multibody Systems Using the Decoupled Natural Orthogonal Complement Matrices

1999 ◽  
Vol 66 (4) ◽  
pp. 986-996 ◽  
Author(s):  
S. K. Saha
Keyword(s):  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Inverse Dynamics ◽  
Equations Of Motion ◽  
Kinematic Chain ◽  
Rigid Bodies ◽  
Dynamic Equations ◽  
Orthogonal Complement ◽  
Forward Dynamics ◽  
Velocity Constraints

Constrained dynamic equations of motion of serial multibody systems consisting of rigid bodies in a serial kinematic chain are derived in this paper. First, the Newton-Euler equations of motion of the decoupled rigid bodies of the system at hand are written. Then, with the aid of the decoupled natural orthogonal complement (DeNOC) matrices associated with the velocity constraints of the connecting bodies, the Euler-Lagrange independent equations of motion are derived. The De NOC is essentially the decoupled form of the natural orthogonal complement (NOC) matrix, introduced elsewhere. Whereas the use of the latter provides recursive order n—n being the degrees-of-freedom of the system at hand—inverse dynamics and order n3 forward dynamics algorithms, respectively, the former leads to recursive order n algorithms for both the cases. The order n algorithms are desirable not only for their computational efficiency but also for their numerical stability, particularly, in forward dynamics and simulation, where the system’s accelerations are solved from the dynamic equations of motion and subsequently integrated numerically. The algorithms are illustrated with a three-link three-degrees-of-freedom planar manipulator and a six-degrees-of-freedom Stanford arm.

Spatial Dynamics of Deformable Multibody Systems With Variable Kinematic Structure: Part 2—Velocity Transformation

1990 ◽  
Vol 112 (2) ◽  
pp. 160-167 ◽  
Author(s):  
C. W. Chang ◽  
A. A. Shabana
Keyword(s):  
Multibody Systems ◽  
Robot Manipulator ◽  
Spatial Dynamics ◽  
Dynamic Equations ◽  
Velocity Transformation ◽  
Joint Forces ◽  
Joint Coordinates ◽  
Deformable Bodies ◽  
Spatial System ◽  
Velocity Transformation Technique

In Part 1 of these two companion papers, the spatial system kinematic and dynamic equations are developed using the Cartesian and elastic coordinates in order to maintain the generality of the formulation. This allows introducing general forcing functions and adding and/or deleting kinematic constraints. In control applications, however, it is desirable to determine the joint forces associated with the joint variables. On the other hand the use of the joint coordinates to formulate the dynamic equations leads to a complex recursive formulation based on loop closure equations. In this paper a velocity transformation technique applicable to spatial multibody systems that consist of interconnected rigid and deformable bodies is developed. The Cartesian variables are expressed in terms of the joint and elastic variables. The resulting kinematic relationships are then employed to determine the joint forces associated with the joint variables. A spatial robot manipulator that manipulates an object is presented as a numerical example to exemplify the development presented in this paper.

Impact and Bifurcation of Flexible Multibody Systems

2003 ◽  
Author(s):  
Evtim V. Zahariev
Keyword(s):  
Finite Elements ◽  
Eigenvalue Problem ◽  
Multibody Systems ◽  
Dynamic Equations ◽  
Loss Of Stability ◽  
Motion Simulation ◽  
Novel Approach ◽  
Spatial Systems ◽  
Relative Coordinates ◽  
Initial Forms

In the paper, the process of loss of stability of multibody systems and structures is analyzed. A novel approach is presented and applied as for the statically loaded spatial systems so for analysis of dynamic response of systems imposed on impact. The analysis is based on solution of the dynamic equations and eigenvalue problem of systems, and of resultant motion simulation. The flexible systems are discretized using the method of finite elements. The dynamic equations are derived with respect to the relative coordinates of the finite elements. Large flexible deflections due to loss of stability are simulated. The initial forms of the possible deformations are defined by the eigenvectors computed solving the eigenvalue problem for the system stiffness matrix. The critical forces and system deflections obtained due to percussive forces and impact are then analyzed. Examples of bifurcation of beam and beam structure imposed on compulsive motion, percussive forces and impact are presented.

A Comparative Study on Some Different Formulations of the Dynamic Equations of Constrained Mechanical Systems

1987 ◽  
Vol 109 (4) ◽  
pp. 466-474 ◽  
Author(s):  
J. Unda ◽  
J. Garci´a de Jalo´n ◽  
F. Losantos ◽  
R. Enparantza
Keyword(s):  
Comparative Study ◽  
Relative Efficiency ◽  
Multibody Systems ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Dynamic Equations ◽  
Natural Coordinates ◽  
Constrained Mechanical Systems ◽  
Computer Codes

This paper presents a comparative theoretical and numerical study on the efficiency of several numerical methods for the dynamic analysis of constrained mechanical systems, also called in the literature multibody systems. This comparative study has been performed between methods based on the use of “reference point” coordinates and those based on the use of “natural” coordinates. This study embraces different possibilities to formulate the differential equations of motion. The relative efficiency of the resulting algorithms has been analyzed theoretically in terms of the number of multiplications needed to evaluate the mechanism accelerations. This efficiency has also been studied implementing the methods into computer codes and testing them with different examples. Conclusions on the relative efficiency of the methods are finally presented.

Symbolic Closed-Form Modeling and Linearization of Multibody Systems Subject to Control

1991 ◽  
Vol 113 (2) ◽  
pp. 124-132 ◽  
Author(s):  
Junghsen Lieh ◽  
Imtiaz-ul Haque
Keyword(s):  
Closed Form ◽  
Multibody Systems ◽  
Space Form ◽  
Virtual Work ◽  
Dynamic Equations ◽  
Principle Of Virtual Work ◽  
Constraint Forces ◽  
Active Suspensions ◽  
Linear Dynamic ◽  
Curved Track

Symbolic closed-form equation formulation and linearization for constrained multibody systems subject to control are presented. The formulation is based on the principle of virtual work. The algorithm is recursive, automatically eliminates the constraint forces and redundant coordinates, and generates the nonlinear or linear dynamic equations in closed-form. It is derived with respect to principal body coordinates and a moving reference frame that allows one to generate the dynamic equations for multibody systems moving along curved track or road. The output equations may be either in syntactically correct FORTRAN form or in the form as derived by hand. A procedure that simplifies the trigonometric expressions, linearizes the geometric nonlinearities, and converts the linearized equations in state-space form is included. Several examples have been used to validate the procedure. Included is a simulation using a seven-DOF automobile ride model with active suspensions.

Nonlinear Dynamics of Rigid and Flexible Multibody Systems

1999 ◽  
Author(s):  
Evtim V. Zakhariev
Keyword(s):  
Finite Elements ◽  
Multibody Systems ◽  
Virtual Work ◽  
Space Station ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Numerical Approach ◽  
Dynamic Equations ◽  
Dynamics Modeling ◽  
Lagrange Equations

Abstract In the present paper a unified numerical approach for dynamics modeling of multibody systems with rigid and flexible bodies is suggested. The dynamic equations are second order ordinary differential equations (without constraints) with respect to a minimal set of generalized coordinates that describe the parameters of gross relative motion of the adjacent bodies and their small elastic deformations. The numerical procedure consists of the following stages: structural decomposition of elastic links into fictitious rigid points and/or bodies connected by joints in which small force dependent relative displacements are achieved; kinematic analysis; deriving explicit form dynamic equations. The algorithm is developed in case of elastic slender beams and finite elements achieving spatial motion with three translations and three rotations of nodes. The beam elements are basic design units in many mechanical devices as space station antennae and manipulators, cranes and etc. doing three dimensional motion which large elastic deflections could not be neglected or linearised. The stiffness coefficients and inertia mass parameters of the fictitious joints and links are calculated using the numerical procedures of the finite element theory. The method is called finite elements in relative coordinates. Its equivalence with the procedures of recently developed finite segment approaches is shown, while in the treatment different results are obtained. The approach is used for solution of some nonlinear static problems and for deriving the explicit configuration space dynamic equations of spatial flexible system using the principle of virtual work and Euler-Lagrange equations.

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