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.

Dynamic Modelling of a Three Degree-of-Freedom Planar Platform-Type Manipulator

1997 ◽  
Author(s):  
Hazem A. Attia ◽  
Maher G. Mohamed
Keyword(s):  
Differential Equations ◽  
Efficient Solution ◽  
Equations Of Motion ◽  
Translational Motion ◽  
Dynamic Modelling ◽  
Degree Of Freedom ◽  
Constraint Forces ◽  
Dynamic Formulation ◽  
System Of Particles ◽  
Three Degree Of Freedom

Abstract In this paper, the dynamic modelling of a planar three degree-of-freedom platform-type manipulator is presented. A kinematic analysis is carried out initially to evaluate the initial coordinates and velocities. The dynamic model of the manipulator is formulated using a two-step transformation. Initially, the dynamic formulation is written in terms of the Cartesian coordinates of a dynamically equivalent system of particles. Since there is no rotational motion associated with a particle, then the differential equations of motion are derived by applying Newton’s second law to study the translational motion of the particles. The constraint forces between the particles are expressed in terms of Lagrange multipliers. Then, the differential equations of motion are written in terms of the relative joint variables. This leads to an efficient solution and integration of the equations of motion. A numerical example is presented and a computer program is developed.

Differential equations of motion of mechanical systems with variable parameters

1974 ◽  
Vol 10 (6) ◽  
pp. 671-674
Author(s):  
V. A. Lazaryan ◽  
L. A. Manashkin ◽  
A. V. Yurchenko
Keyword(s):  
Differential Equations ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Variable Parameters

The Maggi or Canonical Form of Lagrange’s Equations of Motion of Holonomic Mechanical Systems

1990 ◽  
Vol 57 (4) ◽  
pp. 1004-1010 ◽  
Author(s):  
John G. Papastavridis
Keyword(s):  
Linear Algebra ◽  
Nonlinear Equations ◽  
Canonical Form ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Degree Of Freedom ◽  
Holonomic System ◽  
Holonomic Constraints ◽  
Constrained Mechanical Systems ◽  
New Variables

This paper formulates the simplest possible, or canonical, form of the Lagrangean-type of equations of motion of holonomically constrained mechanical systems. This is achieved by introducing a new special set of n holonomic (system) coordinates in terms of which the m ( < n) holonomic constraints are expressed in their simplest, or uncoupled, form: the first m of these new coordinates vanish; the remaining (n-m) (nonvanishing) new coordinates of the (n-m) degree-of-freedom system are then independent. From the resulting equations of motion: (a) The last (n-m) are reactionless canonical equations (the holonomic counterpart of the linear or nonlinear equations, either of Maggi (in the old variables), or of Boltzmann/Hamel (in the new variables)) whose solution yields the motion, while (b) the first m supply the system reactions, in the old or new coordinates, once the motion is known. Special forms of these equations and a simple example are also given. The geometrical interpretation of the above, in modern vector/linear algebra language is summarized in the Appendix.

scholarly journals Nonlinear pre and post-buckled analysis of curved beams using differential quadrature element method

2019 ◽  
Vol 14 (1) ◽  
Author(s):  
M. Zare ◽  
A. Asnafi
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Equations Of Motion ◽  
Differential Quadrature ◽  
Mode Shapes ◽  
Curved Beams ◽  
Buckling Loads ◽  
Differential Quadrature Element Method ◽  
Quadrature Element Method ◽  
Post Buckling

AbstractThis paper studied the in-plane elastic stability including pre and post-buckling analysis of curved beams considering the effects of shear deformations, rotary inertia, and the geometric nonlinearity due to large deformations. Firstly, the governing nonlinear equations of motion were derived. The problem was solved performing both the static and dynamic analysis using the numerical method of differential quadrature element method (DQEM) which is a new and efficient numerical method for rapidly solving linear and nonlinear differential equations. Firstly, the method was applied to the equilibrium equations, leading to a nonlinear algebraic system of equations that would be solved utilizing an arc length strategy. Secondly, the results of the static part were employed to linearize the dynamic differential equations of motion and their corresponding boundary and continuity conditions. Without any loss of generality, a clamped-clamped curved beam under a concentrated load was considered to obtain the buckling loads, natural frequencies, and mode shapes of the beam throughout the method. To validate the proposed method, the beam was modeled using a finite element simulation. A great agreement between the results was seen that showed the accuracy of the proposed method in predicting the pre and post-buckling behavior of the beam. The investigation also included an examination of the curvature parameter influencing the dynamic behavior of the problem. It was shown that the values of buckling loads were completely influenced by the curvature of the beam; also, due to the sharp change of longitudinal stiffness after bucking, the symmetric mode shapes changed more than it was expected.

Tracking Controllers for Uncertain Systems: Application to a Manutec R3 Robot

1989 ◽  
Vol 111 (4) ◽  
pp. 609-618 ◽  
Author(s):  
Martin Corless
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
General Class ◽  
Uncertain Systems ◽  
Mechanical Systems ◽  
Degree Of Freedom ◽  
Tracking Controllers ◽  
Asymptotic Tracking ◽  
Three Degree Of Freedom

We consider a class of uncertain dynamical systems described by ordinary differential equations and characterized by certain structural conditions and known bounding functions. For a feasible class of desired state motions we present a class of controllers which assure asymptotic tracking to within any desired degree of accuracy. The results are applied to a general class of mechanical systems and are illustrated by a simple example and by application to a three degree-of-freedom model of a Manutec r3 robot.

Alternative Approach for the Exact Solution of the Forced Vibrations of Beams

1995 ◽  
Vol 48 (11S) ◽  
pp. S96-S101 ◽  
Author(s):  
Carlos P. Filipich ◽  
Marta B. Rosales
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Equations Of Motion ◽  
Generalized Solution ◽  
Forced Vibrations ◽  
Three Dimensions ◽  
Boundary Problems ◽  
Important Theorem ◽  
Alternative Approach

The present work is an extension of a tool vastly used by the authors to solve static boundary problems in one, two, and even three dimensions. It consists in a so-called generalized solution with special trigonometric Fourier functions to solve the equations of motion of beams. An important theorem that guarantees that the classic answer is attained through an alternative way is demonstrated. In other words, it is a variational methodology to solve differential equations in engineering. An example solved numerically completes the present proposal.

New Formulations on Motion Equations in Analytical Dynamics

2016 ◽  
Vol 823 ◽  
pp. 49-54 ◽  
Author(s):  
Iuliu Negrean ◽  
Kalman Kacso ◽  
Claudiu Schonstein ◽  
Adina Duca ◽  
Florina Rusu ◽  
...  
Keyword(s):  
Differential Equations ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Higher Order ◽  
Analytical Dynamics ◽  
Lagrange Principle ◽  
Motion Equations ◽  
Differential Motion

Using the main author's researches on the energies of acceleration and higher order equations of motion, this paper is devoted to new formulations in analytical dynamics of mechanical multibody systems (MBS). Integral parts of these systems are the mechanical robot structures, serial, parallel or mobile on which an application will be presented in order to highlight the importance of the differential motion equations in dynamics behavior. When the components of multibody mechanical systems or in its entirety presents rapid movements or is in transitory motion, are developed higher order variations in respect to time of linear and angular accelerations. According to research of the main author, they are integrated into higher order energies and these in differential equations of motion in higher order, which will lead to variations in time of generalized forces which dominate these types of mechanical systems. The establishing of these differential equations of motion, it is based on a generalization of a principle of analytical differential mechanics, known as the D`Alembert – Lagrange Principle.

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.

Sign in / Sign up

Export Citation Format

Share Document