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.

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.

On the Use of the Canonical Equations of Motion for the Dynamic Analysis of Constrained Multibody Systems

1992 ◽  
Author(s):  
E. Bayo ◽  
J. M. Jimenez
Keyword(s):  
Dynamic Analysis ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Numerical Algorithms ◽  
Constrained Multibody Systems ◽  
Constrained Mechanical Systems ◽  
First Order ◽  
Canonical Variables ◽  
Lagrange’S Multipliers

Abstract We investigate in this paper the different approaches that can be derived from the use of the Hamiltonian or canonical equations of motion for constrained mechanical systems with the intention of responding to the question of whether the use of these equations leads to more efficient and stable numerical algorithms than those coming from acceleration based formalisms. In this process, we propose a new penalty based canonical description of the equations of motion of constrained mechanical systems. This technique leads to a reduced set of first order ordinary differential equations in terms of the canonical variables with no Lagrange’s multipliers involved in the equations. This method shows a clear advantage over the previously proposed acceleration based formulation, in terms of numerical efficiency. In addition, we examine the use of the canonical equations based on independent coordinates, and conclude that in this second case the use of the acceleration based formulation is more advantageous than the canonical counterpart.

Simulation of Planar Dynamic Mechanical Systems With Changing Topologies: Part I — Characterization and Prediction of the Kinematic Constraint Changes

1987 ◽  
Author(s):  
B. J. Gilmore ◽  
R. J. Cipra
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Rigid Bodies ◽  
State Variables ◽  
Kinematic Constraint ◽  
Kinematic Constraints ◽  
Force Closure ◽  
Simulation Strategy ◽  
Reaction Forces ◽  
Discontinuous Equations

Abstract Due to changes in the kinematic constraints, many mechanical systems are described by discontinuous equations of motion. This paper addresses those changes in the kinematic constraints which are caused by planar bodies contacting and separating. A strategy to automatically predict and detect the kinematic constraint changes, which are functions of the system dynamics, is presented in Part I. The strategy employs the concepts of point to line contact kinematic constraints, force closure, and ray firing together with the information provided by the rigid bodies’ boundary descriptions, state variables, and reaction forces to characterize the kinematic constraint changes. Since the strategy automatically predicts and detects constraint changes, it is capable of simulating mechanical systems with unpredictable or unforeseen changes in topology. Part II presents the implementation of the characterizations into a simulation strategy and presents examples.

Analysis and Optimal Design of Spatial Mechanical Systems

1987 ◽  
Author(s):  
H. Ashrafeiuon ◽  
N. K. Mani
Keyword(s):  
Sensitivity Analysis ◽  
Optimal Design ◽  
Dynamic Analysis ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Design Sensitivity ◽  
General Purpose ◽  
Design Sensitivity Analysis ◽  
Analysis And Design ◽  
Spatial Systems

Abstract This paper presents a new approach to optimal design of large multibody spatial mechanical systems. This approach uses symbolic computing to generate the necessary equations for dynamic analysis and design sensitivity analysis. Identification of system topology is carried out using graph theory. The equations of motion are formulated in terms of relative joint coordinates through the use of velocity transformation matrix. Design sensitivity analysis is carried out using the Direct Differentiation method applied to the relative joint coordinate formulation for spatial systems. Symbolic manipulation programs are used to develop subroutines which provide information for dynamic and design sensitivity analysis. These subroutines are linked to a general purpose computer program which performs dynamic analysis, design sensitivity analysis, and optimization. An example is presented to demonstrate the efficiency of the approach.

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.

A Sparsity-Oriented Approach to the Dynamic Analysis and Design of Mechanical Systems—Part 1

1977 ◽  
Vol 99 (3) ◽  
pp. 773-779 ◽  
Author(s):  
N. Orlandea ◽  
M. A. Chace ◽  
D. A. Calahan
Keyword(s):  
Computer Simulation ◽  
Dynamic Analysis ◽  
Numerical Solutions ◽  
Sparse Matrix ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Mechanical Systems ◽  
Numerical Algorithms ◽  
Front Suspension ◽  
Analysis And Design

The work described herein is an extension of sparse matrix and stiff integrated numerical algorithms used for the simulation of electrical circuits and three-dimensional mechanical dynamic systems. By applying these algorithms big sets of sparse linear equations can be solved efficiently, and the numerical instability associated with widely split eigenvalues can be avoided. The new numerical methods affect even the initial formulation for these problems. In this paper, the equations of motion and constraints (Part 1) and the force function of springs and dampers (Part 2) are set up, and the numerical solutions for static, transient, and linearized types of analysis as well as the modal optimization algorithms are implemented in the ADAMS (automatic dynamic analysis of mechanical systems) computer program for simulation of three-dimensional mechanical systems (Part 2). The paper concludes with two examples: computer simulation of the front suspension of a 1973 Chevrolet Malibu and computer simulation of the landing gear of a Boeing 747 airplane. The efficiency of simulation and comparison with experimental results are given in tabular form.

The Incorporation of Arc Boundaries and Stick/Slip Friction in a Rule-Based Simulation Algorithm for Dynamic Mechanical Systems with Changing Topologies

1993 ◽  
Vol 115 (3) ◽  
pp. 423-434 ◽  
Author(s):  
Inhwan Han ◽  
B. J. Gilmore ◽  
M. M. Ogot
Keyword(s):  
High Speed ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Coefficient Matrix ◽  
Design Tool ◽  
Simulation Algorithm ◽  
Kinematic Constraint ◽  
Stick Slip ◽  
Rule Based ◽  
Dynamic Mechanical

Many dynamic mechanical systems, such as parts-feeders and percussive power tools, are described by equations of motion which are discontinuous. The discontinuities result from kinematic constraint changes which are difficult to foresee, especially in presence of impact and friction. A simulation algorithm for these types of systems must be able to algorithmically predict and detect the kinematic constraint changes without any prior knowledge of the system’s motion. This paper presents a rule-based approach to the prediction and detection of kinematic constraint changes between bodies with arc and line boundaries. A new type of constraint change, constraint exchange, is characterized. When arc contact exists, stick/slip friction is the difference between pure rolling and rolling with slip. Therefore, stick/slip friction is included in the algorithm. A force constraint is applied to the equations of motion when additional kinematic constraints due to friction would render the coefficient matrix singular. The efficacy of the rule-based simulation algorithm as a design tool is demonstrated through the design and experimental validation of a parts-feeder. The parts-feeder design is validated through two means: (1) a frame-by frame comparison of simulation results with the part motion recorded by high speed video and (2) actual testing.

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.

Inverse dynamic analysis of mechanical systems in joint coordinate space

2003 ◽  
Vol 217 (1) ◽  
pp. 29-37 ◽  
Author(s):  
B H Lee
Keyword(s):  
Dynamic Analysis ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Coordinate Space ◽  
Inverse Dynamic ◽  
Velocity Transformation ◽  
Inverse Dynamic Analysis ◽  
Joint Reaction Forces ◽  
System Equations ◽  
Cartesian Coordinate

An inverse dynamic analysis algorithm for spatial flexible mechanical systems with closed loops is developed in the relative joint coordinate space. System equations of motion and constraint acceleration equations are derived using the velocity transformation technique. An inverse velocity transformation operator, which transforms the Cartesian velocities to the relative velocities, is derived systematically, corresponding to the types of kinematic joint connecting the bodies. Using the resulting matrix, the joint reaction forces and moments are analysed in the Cartesian coordinate space. The joint coordinates and the deformation modal coordinates are used as the generalized coordinates of a flexible mechanical system. The algorithm is verified by means of two numerical examples.

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