Variable Degree-of-Freedom Component Mode Analysis of Inertia Variant Flexible Mechanical Systems

1983 ◽  
Vol 105 (3) ◽  
pp. 371-378 ◽  
Author(s):  
A. Shabana ◽  
R. A. Wehage
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Mode Analysis ◽  
Variable Degree ◽  
Transient Dynamic Analysis ◽  
Flexible Bodies ◽  
Generalized Coordinates ◽  
Elastic Bodies ◽  
Body Reference ◽  
Angular Displacements

A method is presented for transient dynamic analysis of mechanical systems composed of interconnected rigid and flexible bodies that undergo large angular displacements. Gross displacement of elastic bodies is represented by superposition of local linear elastic deformation on nonlinear displacement of body reference coordinate systems. Flexible bodies are thus represented by combined sets of reference and local elastic generalized coordinates. Modal analysis and sub-structuring of the local elastic system identifies all modes and allows for elimination of insignificant modes. Equations of motion and constraints are formulated in terms of minimum mixed sets of modal and reference generalized coordinates, and dynamically adjusted to a time and interia-variant eigenspectrum. A flexible mechanical linkage and a tracked vehicle with a flexible gun tube are simulated to illustrate the method.

Dynamics of Flexible Mechanical Systems Using Vibration and Static Correction Modes

1986 ◽  
Vol 108 (3) ◽  
pp. 315-322 ◽  
Author(s):  
W. S. Yoo ◽  
E. J. Haug
Keyword(s):  
Finite Element ◽  
Dynamic Analysis ◽  
Elastic Deformation ◽  
Large Scale ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Nonlinear Dynamic Analysis ◽  
Mode Analysis ◽  
Lumped Mass ◽  
Static Correction

A finite-element-based method is developed and applied for geometrically nonlinear dynamic analysis of spatial mechanical systems. Vibration and static correction modes are used to account for linear elastic deformation of components. Boundary conditions for vibration and static correction mode analysis are defined by kinematic constraints between components of a system. Constraint equations between flexible bodies are derived and a Lagrange multiplier formulation is used to generate the coupled large displacement-small deformation equations of motion. A standard, lumped mass finite-element structural analysis code is used to generate deformation modes and deformable body mass and stiffness information. An intermediate-processor is used to calculate time-independent terms in the equations of motion and to generate input data for a large-scale dynamic analysis code that includes coupled effects of geometric nonlinearity and elastic deformation. Two examples are presented and the effects of deformation mode selection on dynamic prediction are analyzed.

An Adaptive Constraint Violation Stabilization Method for Dynamic Analysis of Mechanical Systems

1985 ◽  
Vol 107 (4) ◽  
pp. 488-492 ◽  
Author(s):  
C. O. Chang ◽  
P. E. Nikravesh
Keyword(s):  
Dynamic Analysis ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
General Purpose ◽  
Adaptive Mechanism ◽  
Stabilization Method ◽  
Constraint Violation ◽  
Transient Dynamic Analysis ◽  
Constrained Mechanical Systems ◽  
Feedback Control Theory

The transient dynamic analysis of equations of motion for constrained mechanical systems requires the solution of a mixed set of algebraic and differential equations. A constraint violation stabilization method, based on feedback control theory of linear systems, has been suggested by some researchers for solving these equations. However, since the value of damping parameters for this method are uncertain, the method is to some extent unattractive for general-purpose use. This paper presents an adaptive mechanism for determining the damping parameters. The results of the simulation for two examples illustrate the improvement in reducing the constraint violations when using this method.

Automated Vehicle Dynamic Analysis With Flexible Components

1984 ◽  
Vol 106 (1) ◽  
pp. 126-132 ◽  
Author(s):  
S. S. Kim ◽  
A. A. Shabana ◽  
E. J. Haug
Keyword(s):  
Dynamic Analysis ◽  
Equations Of Motion ◽  
Coupled System ◽  
Ride Quality ◽  
Vertical Acceleration ◽  
Transient Dynamic Analysis ◽  
Transient Dynamic ◽  
Generalized Coordinates ◽  
Rigid Body Model ◽  
Nonlinear Transient

A method is presented for nonlinear, transient dynamic analysis of vehicle systems that are composed of interconnected rigid and flexible bodies. The finite element method is used to characterize deformation of each elastic body and a component mode technique is employed to reduce the number of elastic generalized coordinates. Equations of motion and constraints of the coupled system are formulated in terms of a minimal set of modal and reference generalized coordinates. A Lagrange multiplier technique is used to account for kinematic constraints between bodies and a generalized coordinate partitioning technique is employed to eliminate dependent coordinates. The method is applied to a planar truck model with a flexible chassis and nonlinear suspension components. Simulation results for transient dynamic response as the vehicle traverses a bump, including the effect of bump-stops, and random terrain show that flexibility of the chassis can be routinely accounted for and predicts significant effects on vibratory motion of the vehicle. Compared with a rigid body model, flexibility of the chassis increases peak acceleration of the chassis and induces high-frequency vertical acceleration in the range of human resonance, measured in this paper as driver absorbed power, which deteriorates ride quality of off-road vehicles.

A Discrete Quasi-Coordinate Formulation for the Dynamics of Elastic Bodies

2006 ◽  
Vol 74 (2) ◽  
pp. 231-239 ◽  
Author(s):  
G. M. T. D’Eleuterio ◽  
T. D. Barfoot
Keyword(s):  
Equations Of Motion ◽  
Computational Cost ◽  
Dynamical Equations ◽  
Generalized Coordinates ◽  
Order Form ◽  
Elastic Bodies ◽  
Alternative Approach ◽  
Elastic Systems ◽  
Computationally Intensive ◽  
Derivatives Of

The discretized equations of motion for elastic systems are typically displayed in second-order form. That is, the elastic displacements are represented by a set of discretized (generalized) coordinates, such as those used in a finite-element method, and the elastic rates are simply taken to be the time-derivatives of these displacements. Unfortunately, this approach leads to unpleasant and computationally intensive inertial terms when rigid rotations of a body must be taken into account, as is so often the case in multibody dynamics. An alternative approach, presented here, assumes the elastic rates to be discretized independently of the elastic displacements. The resulting dynamical equations of motion are simplified in form, and the computational cost is correspondingly lessened. However, a slightly more complex kinematical relation between the rate coordinates and the displacement coordinates is required. This tack leads to what may be described as a discrete quasi-coordinate formulation.

Application of Euler Parameters to the Dynamic Analysis of Three-Dimensional Constrained Mechanical Systems

1982 ◽  
Vol 104 (4) ◽  
pp. 785-791 ◽  
Author(s):  
P. E. Nikravesh ◽  
I. S. Chung
Keyword(s):  
Differential Equations ◽  
Dynamic Analysis ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Influence Coefficient ◽  
Variable Order ◽  
Total System ◽  
Euler Parameters ◽  
Generalized Coordinates ◽  
Dependent Variables

This paper presents a computer-based method for formulation and efficient solution of nonlinear, constrained differential equations of motion for spatial dynamic analysis of mechanical systems. Nonlinear holonomic constraint equations and differential equations of motion are written in terms of a maximal set of Cartesian generalized coordinates, three translational and four rotational coordinates for each rigid body in the system, where the rotational coordinates are the Euler parameters. Euler parameters, in contrast to Euler angles or any other set of three rotational generalized coordinates, have no critical singular cases. The maximal set of generalized coordinates facilitates 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 matrix relating variations in dependent and indpendent 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, integrates for only the independent variables, yet effectively determines dependent variables.

Aspects of Symbolic Formulations in Flexible Multibody Systems

2014 ◽  
Vol 9 (4) ◽  
Author(s):  
Markus Burkhardt ◽  
Robert Seifried ◽  
Peter Eberhard
Keyword(s):  
Multibody Systems ◽  
Multibody System ◽  
Dynamic Properties ◽  
Equations Of Motion ◽  
Flexible Multibody Systems ◽  
Fem Model ◽  
Local Shape ◽  
Flexible Bodies ◽  
Elastic Bodies ◽  
Flexible Multibody

The symbolic modeling of flexible multibody systems is a challenging task. This is especially the case for complex-shaped elastic bodies, which are described by a numerical model, e.g., an FEM model. The kinematic and dynamic properties of the flexible body are in this case numerical and the elastic deformations are described with a certain number of local shape functions, which results in a large amount of data that have to be handled. Both attributes do not suggest the usage of symbolic tools to model a flexible multibody system. Nevertheless, there are several symbolic multibody codes that can treat flexible multibody systems in a very efficient way. In this paper, we present some of the modifications of the symbolic research code Neweul-M2 which are needed to support flexible bodies. On the basis of these modifications, the mentioned restrictions due to the numerical flexible bodies can be eliminated. Furthermore, it is possible to re-establish the symbolic character of the created equations of motion even in the presence of these solely numerical flexible bodies.

scholarly journals A form of equations of motion of constrained mechanical systems

2002 ◽  
Vol 24 (2) ◽  
pp. 123-132
Author(s):  
Do Sanh
Keyword(s):  
Mechanical System ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
The Other ◽  
Constrained Mechanical Systems ◽  
Generalized Coordinates ◽  
Minimum Number ◽  
Dependent Coordinates

In the present paper a form of equations of motion of a constrained mechanical system is constructed. These equations only contain a minimum number of accelerations. In the other words, such equations are written in independent accelerations while the configuration of the system is described by dependent coordinates. It is important that the equations obtained are applied conveniently for. the mechanisms in which the use of independent generalized coordinates is not suitable.

A unified tensor approach to the analysis of mechanical systems

1997 ◽  
Vol 211 (4) ◽  
pp. 313-322 ◽  
Author(s):  
A A Fogarasy ◽  
M R Smith
Keyword(s):  
Dynamic Analysis ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Equation Of Motion ◽  
Newtonian Mechanics ◽  
Kinematic Constraint ◽  
Lagrangian Equation ◽  
Generalized Coordinates ◽  
Lagrangian Equations

It is shown in this paper that all methods of dynamic analysis of mechanisms used in practice can be derived from an invariant formed from the Lagrangian equation of motion. For the dynamic analysis of mechanisms subjected to kinematic constraint conditions, the Lagrangian equations of motion are far more suitable than the Newtonian approach. Since the Lagrangian equations are tensor equations, they are valid irrespective of what kind of generalized coordinates are used. This is not so, however, when the Newtonian approach is used. It is demonstrated by a simple example that a careless use of Newtonian mechanics can lead to erroneous results.

Generalized Coordinate Partitioning for Analysis of Mechanical Systems with Nonholonomic Constraints

1983 ◽  
Vol 105 (3) ◽  
pp. 379-384 ◽  
Author(s):  
P. E. Nikravesh ◽  
E. J. Haug
Keyword(s):  
Differential Equations ◽  
Gaussian Elimination ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Variable Order ◽  
Step Size ◽  
Euler Parameters ◽  
Integration Algorithm ◽  
Generalized Coordinates

This paper presents a computer-based method for formulation and efficient solution of nonlinear, constrained differential equations of motion for spatial dynamic analysis of mechanical systems with holonomic and nonholonomic constraints. Holonomic and nonholonomic constraint equations and differential equations of motion are written in terms of a maximal set of Cartesian generalized coordinates, three translational and four rotational coordinates for each rigid body in the system, where the rotational coordinates are Euler parameters. The maximal set of generalized coordinates facilitates the general formulation of constraints and forcing functions. A Gaussian elimination algorithm with full pivoting decomposes the constraint Jacobian matrix and identifies independent coordinates and velocities. This information is employed to numerically construct a reduced system of differential equations of motion whose solution yields the system dynamic response. A numerical integration algorithm with positive-error control, employing a predictor-corrector algorithm with variable order and step size, integrates for only the independent variables, yet effectively determines dependent variables.

Flexible Multibody Dynamics Formulation by Hamilton’s Equations

2010 ◽  
Author(s):  
Martin M. Tong
Keyword(s):  
Multibody Dynamics ◽  
Multibody System ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Flexible Multibody Dynamics ◽  
Flexible Body ◽  
Flexible Multibody System ◽  
Generalized Coordinates ◽  
Flexible Multibody ◽  
The Matrix

Numerical solution of the dynamics equations of a flexible multibody system as represented by Hamilton’s canonical equations requires that its generalized velocities q˙ be solved from the generalized momenta p. The relation between them is p = J(q)q˙, where J is the system mass matrix and q is the generalized coordinates. This paper presents the dynamics equations for a generic flexible multibody system as represented by p˙ and gives emphasis to a systematic way of constructing the matrix J for solving q˙. The mass matrix is shown to be separable into four submatrices Jrr, Jrf, Jfr and Jff relating the joint momenta and flexible body mementa to the joint coordinate rates and the flexible body deformation coordinate rates. Explicit formulas are given for these submatrices. The equations of motion presented here lend insight to the structure of the flexible multibody dynamics equations. They are also a versatile alternative to the acceleration-based dynamics equations for modeling mechanical systems.

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