Spatial Transient Analysis of Inertia-Variant Flexible Mechanical Systems

1984 ◽  
Vol 106 (2) ◽  
pp. 172-178 ◽  
Author(s):  
A. A. Shabana ◽  
R. A. Wehage
Keyword(s):  
Differential Equations ◽  
Transient Analysis ◽  
Large Scale ◽  
Equations Of Motion ◽  
Integration Algorithm ◽  
System Equations ◽  
Element Technique ◽  
Coordinate Partitioning ◽  
Mode Technique ◽  
Generalized Coordinate

An analytical method for transient dynamic simulation of large-scale inertia-variant spatial mechanical and structural systems is presented. Multibody systems consisting of interconnected rigid and flexible substructures which may undergo large angular rotations are analyzed. A finite element technique is used to characterize the elastic properties of deformable substructures. A component mode technique is then employed to eliminate insignificant substructure modes. Nonlinear holonomic constraint equations are used to define joints between different substructures. The system equations of motion are written in terms of a mixed set of modal and physical coordinates. A generalized coordinate partitioning technique is then employed to eliminate redundant differential equations. An implicit-explicit numerical integration algorithm solves the remaining set of differential equations and the approximate physical system state is recovered. The transient analysis of a spatial vehicle with flexible chassis is presented to demonstrate the method.

Design Sensitivity Analysis of Large-Scale Constrained Dynamic Mechanical Systems

1984 ◽  
Vol 106 (2) ◽  
pp. 156-162 ◽  
Author(s):  
E. J. Haug ◽  
R. A. Wehage ◽  
N. K. Mani
Keyword(s):  
Sensitivity Analysis ◽  
Large Scale ◽  
Equations Of Motion ◽  
Design Sensitivity ◽  
Differential Algebraic Equations ◽  
Adjoint Equations ◽  
Design Sensitivity Analysis ◽  
Integration Algorithm ◽  
Coordinate Partitioning ◽  
Generalized Coordinate

A method for computer-aided design sensitivity analysis of large-scale constrained dynamic systems is presented. A generalized coordinate partitioning method is used for assembling and solving sets of mixed differential-algebraic equations of motion and adjoint equations required for calculation of derivatives of dynamic response measures with respect to design variables. The reduction in dimension of the equations of motion and associated adjoint equations obtained through use of generalized coordinate partitioning significantly reduces the computational burden, as compared to methods previously employed. Use of predictor-corrector numerical integration algorithms, rather than an implicit integration algorithm used in the past is shown to greatly simplify the equations that must be formulated and solved. Two examples are presented to illustrate accuracy of the design sensitivity analysis method developed.

Dynamic Response of Spinning Blades Subjected to Gyroscopic Motion

1994 ◽  
Vol 116 (1) ◽  
pp. 6-15 ◽  
Author(s):  
T. H. Young ◽  
G. T. Liou
Keyword(s):  
Differential Equations ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Parametric Instability ◽  
Spin Axis ◽  
System Response ◽  
First Order ◽  
System Equations ◽  
Element Technique ◽  
Gyroscopic Motion

This paper presents an investigation into the vibration and stability of a blade spinning with respect to a nonfixed axis. Due to the motion of the spin axis, parametric instability of the blade may occur in certain situations. In this work, the discretized equations of motion are first formulated by the finite element technique. Then the system equations are transformed, by a special modal analysis procedure, into independent sets of first-order simultaneous differential equations. Each set of differential equations is solved analytically by the method of multiple scales if the precessional speed of the spin axis is assumed to be small compared to the spin rate of the blade, yielding the system response and the expressions for the boundaries of the unstable regions. Finally, the effects of system parameters on the changes in these boundaries are studied numerically.

Implicit Numerical Integration of Constrained Equations of Motion Via Generalized Coordinate Partitioning

1992 ◽  
Vol 114 (2) ◽  
pp. 296-304 ◽  
Author(s):  
E. J. Haug ◽  
Jeng Yen
Keyword(s):  
Numerical Integration ◽  
Equations Of Motion ◽  
Integration Algorithm ◽  
Differentiation Formula ◽  
Generalized Coordinates ◽  
Multibody Dynamic ◽  
Constrained Equations ◽  
Automated Simulation ◽  
Coordinate Partitioning ◽  
Generalized Coordinate

An implicit, stiffly stable numerical integration algorithm is developed and demonstrated for automated simulation of multibody dynamic systems. The concept of generalized coordinate partitioning is used to parameterize the constraint set with independent generalized coordinates. A stiffly stable, Backward Differentiation Formula (BDF) numerical integration algorithm is used to integrate independent generalized coordinates and velocities. Dependent generalized coordinates, velocities, and accelerations, as well as Lagrange multipliers that account for constraints, are explicitly retained in the formulation to satisfy all of the governing kinematic and dynamic equations. The algorithm is shown to be valid and accurate, both theoretically and through solution of an example.

Implicit Numerical Integration of Constrained Equations of Motion via Generalized Coordinate Partitioning

1989 ◽  
Author(s):  
E. J. Haug ◽  
J. Yen
Keyword(s):  
Numerical Integration ◽  
Equations Of Motion ◽  
Integration Algorithm ◽  
Generalized Coordinates ◽  
Multibody Dynamic ◽  
Constraint Set ◽  
Constrained Equations ◽  
Automated Simulation ◽  
Coordinate Partitioning ◽  
Generalized Coordinate

Abstract An implicit, stiffly stable numerical integration algorithm is developed and demonstrated for automated simulation of multibody dynamic systems. The concept of generalized coordinate partitioning is used to parameterize the constraint set with independent generalized coordinates. A stiffly stable, backward difference numerical integration algorithm is applied to determine independent generalized coordinates and velocities. Dependent generalized coordinates, velocities, and accelerations, as well as Lagrange multipliers that account for constraints, are explicitly retained in the formulation to satisfy all of the governing kinematic and dynamic equations. The algorithm is shown to be valid and accurate, both theoretically and through solution of a numerical example.

Efficient Dynamic Computer Simulation of Simple-Closed Spatial Linkages

1990 ◽  
Author(s):  
L. T. Wang
Keyword(s):  
Computer Simulation ◽  
Computational Algorithm ◽  
Equations Of Motion ◽  
Joint Space ◽  
Kinematic Chains ◽  
Spatial Linkages ◽  
Dynamic Computer Simulation ◽  
The Stability ◽  
Coordinate Partitioning ◽  
Generalized Coordinate

Abstract A new method of formulating the generalized equations of motion for simple-closed (single loop) spatial linkages is presented in this paper. This method is based on the generalized principle of D’Alembert and the use of the transformation Jacobian matrices. The number of the differential equations of motion is minimized by performing the method of generalized coordinate partitioning in the joint space. Based on this formulation, a computational algorithm for computer simulation the dynamic motions of the linkage is developed, this algorithm is not only numerically stable but also fully exploits the efficient recursive computational schemes developed earlier for open kinematic chains. Two numerical examples are presented to demonstrate the stability and efficiency of the algorithm.

Automated Dynamic Analysis of Chain-Driven Mechanical Systems

1983 ◽  
Vol 105 (3) ◽  
pp. 362-370 ◽  
Author(s):  
Ting W. Lee
Keyword(s):  
Dynamic Analysis ◽  
Large Scale ◽  
Equations Of Motion ◽  
Dynamic Effect ◽  
General Purpose ◽  
Displacement Function ◽  
Analysis And Design ◽  
Driven System ◽  
Chain Dynamic ◽  
System Equations

A general approach to analyze the dynamics of chain-driven systems subjected to transient loads is developed and applied. The method suitable for many automated dynamic analysis techniques involves the simulation of the dynamic effect of chain by a displacement function and the introduction of this function as a kinematic constraint to couple with the system equations of motion. A general purpose dynamic analysis algorithm, the DADS code (Dynamic Analysis and Design Systems), is then employed to generate the set of system equations and to provide a computer-aided dynamic analysis of the overall chain-driven system. Two ways of formulating the chain displacement functions are described. One provides the displacement of the chain based on the pitch circles of chain sprockets; the other includes a consideration of the polygonal effect of the chain which contributes essentially to the dynamics of the chain. The latter involves the use of the principle of kinematic equivalency, i.e., modeling the chain dynamic effect by a four-bar linkage. Using the proposed displacement function, the kinematic motion of the chain can be taken into account. This procedure, therefore, makes the system adaptable to conventional dynamic analysis code in which the chain is usually not included as one of the standard elements. Moreover, pulsation and dynamic load of the chain as well as the system dynamic response due to chain effect may be estimated. A typical large-scale chain-driven system which is an externally powered machine gun is investigated to illustrate the potential usefulness of the approach.

Transient Analysis of Forced Vibrations of Complex Structural-Mechanical Systems

1962 ◽  
Vol 66 (619) ◽  
pp. 457-460 ◽  
Author(s):  
S. P. Chan ◽  
H. L. Cox ◽  
W. A. Benfield
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Finite Differences ◽  
Transient Analysis ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Forced Vibrations ◽  
Degree Of Freedom ◽  
Dynamic Forces ◽  
Complex Structural

This paper presents a numerical method, derived directly from the basic differential equations of motion and expressed in the form of recurrence-matrix of finite differences, that can be generally applied to all multi-degree-of-freedom structures subjected to dynamic forces or forced displacements on any masses at any instants of time. The movements of the system may be described by any form of generalised co-ordinates.

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.

Dynamic Analysis of Constrained System of Rigid and Flexible Bodies With Intermittent Motion

1986 ◽  
Vol 108 (1) ◽  
pp. 38-45 ◽  
Author(s):  
Y. A. Khulief ◽  
A. A. Shabana
Keyword(s):  
Dynamic Analysis ◽  
Large Scale ◽  
Equations Of Motion ◽  
Constrained System ◽  
Integration Algorithm ◽  
Flexible Bodies ◽  
Intermittent Behavior ◽  
Finite Set ◽  
Implicit Integration Algorithm ◽  
Intermittent Motion

A method for dynamic analysis of large-scale constrained system of mixed rigid and flexible bodies with intermittent motion is presented. The system equations of motion are written in the Lagrangian formulation using a finite set of coupled reference position and local elastic generalized coordinates. Equations of motion are computer generated and integrated forward in time using an explicit-implicit integration algorithm. Points in time at which sudden events of the intermittent behavior occur are monitored by an event predictor which controls the integration algorithm and forces a solution for the system impulse-momentum relation at those points. Solutions of impulse-momentum relations define the jump discontinuities in the composite velocity vector as well as the generalized impulses of the reaction forces at different joints of the mechanical system.

Generalized Coordinate Partitioning for Dimension Reduction in Analysis of Constrained Dynamic Systems

1982 ◽  
Vol 104 (1) ◽  
pp. 247-255 ◽  
Author(s):  
R. A. Wehage ◽  
E. J. Haug
Keyword(s):  
Differential Equations ◽  
Gaussian Elimination ◽  
Equations Of Motion ◽  
Influence Coefficient ◽  
Variable Order ◽  
Total System ◽  
Step Size ◽  
Integration Algorithm ◽  
Independent Variables ◽  
Dependent Variables

This paper presents a computer-based method for formulation and efficient solution of nonlinear, constrained differential equations of motion for mechanical systems. Nonlinear holonomic constraint equations and differential equations of motion are written in terms of a maximal set of Cartesian generalized coordinates, to facilitate the general formulation of constraints and forcing functions. A Gaussian elimination algorithm with full pivoting decomposes the constraint Jacobian matrix, identifies dependent variables, and constructs an influence coefficient matix relating variations in dependent and independent variables. This information is employed to numerically construct a reduced system of differential equations of motion whose solution yields the total system dynamic response. A numerical integration algorithm with positive-error control, employing a predictor-corrector algorithm with variable order and step size, is developed that integrates for only the independent variables, yet effectively determines dependent variables. Numerical results are presented for planar motion of two tracked vehicular systems with 13 and 24 degrees of freedom. Computational efficiency of the algorithm is shown to be an order of magnitude better than previously employed algorithms.

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