DYNAMICS OF FLEXIBLE MULTIBODY MECHANICAL SYSTEMS

1991 ◽  
Vol 15 (3) ◽  
pp. 235-256 ◽  
Author(s):  
X. Cyril ◽  
J. Angeles ◽  
A. Misra
Keyword(s):  
Equations Of Motion ◽  
Matrix Multiplication ◽  
Mechanical Systems ◽  
Dynamical Behaviour ◽  
Constraint Forces ◽  
Dynamical Equations ◽  
Simple Matrix ◽  
Kinematic Velocity ◽  
Velocity Constraints ◽  
The Individual

In this paper the formulation and simulation of the dynamical equations of multibody mechanical systems comprising of both rigid and flexible-links are accomplished in two steps: in the first step, each link is considered as an unconstrained body and hence, its Euler-Lagrange (EL) equations are derived disregarding the kinematic couplings; in the second step, the individual-link equations, along with the associated constraint forces, are assembled to obtain the constrained dynamical equations of the multibody system. These constraint forces are then efficiently eliminated by simple matrix multiplication of the said equations by the transpose of the natural orthogonal complement of kinematic velocity constraints to obtain the independent dynamical equations. The equations of motion are solved for the generalized accelerations using the Cholesky decomposition method and integrated using Gear’s method for stiff differential equations. Finally, the dynamical behaviour of the Shuttle Remote Manipulator when performing a typical manoeuvre is determined using the above approach.

Reduction of Hamiltonian Mechanical Systems With Affine Constraints: A Geometric Unification

2016 ◽  
Vol 12 (2) ◽  
Author(s):  
Robin Chhabra ◽  
M. Reza Emami ◽  
Yael Karshon
Keyword(s):  
Symmetry Group ◽  
Mechanical Systems ◽  
Hamiltonian Formalism ◽  
Geometrical Approach ◽  
Constraint Forces ◽  
Dynamical Equations ◽  
Relative Configuration ◽  
D’Alembert Equation ◽  
Dynamical Reduction ◽  
Configuration Manifold

This paper presents a geometrical approach to the dynamical reduction of a class of constrained mechanical systems. The mechanical systems considered are with affine nonholonomic constraints plus a symmetry group. The dynamical equations are formulated in a Hamiltonian formalism using the Hamilton–d'Alembert equation, and constraint forces determine an affine distribution on the configuration manifold. The proposed reduction approach consists of three main steps: (1) restricting to the constrained submanifold of the phase space, (2) quotienting the constrained submanifold, and (3) identifying the quotient manifold with a cotangent bundle. Finally, as a case study, the dynamical reduction of a two-wheeled rover on a rotating disk is detailed. The symmetry group for this example is the relative configuration manifold of the rover with respect to the inertial space. The proposed approach in this paper unifies the existing reduction procedures for symmetric Hamiltonian systems with conserved momentum, and for Chaplygin systems, which are normally treated separately in the literature. Another characteristic of this approach is that although it tracks the structure of the equations in each reduction step, it does not insist on preserving the properties of the system. For example, the resulting dynamical equations may no longer correspond to a Hamiltonian system. As a result, the invariance condition of the Hamiltonian under a group action that lies at the heart of almost every reduction procedure is relaxed.

Dynamical Equations With Non-Ideal Constraints: New and Old Alternatives

2007 ◽  
Author(s):  
Arun K. Banerjee ◽  
Mark Lemak
Keyword(s):  
Equations Of Motion ◽  
Alternative Form ◽  
Constraint Forces ◽  
Dynamical Equations ◽  
Constraint Force ◽  
Kane’S Method ◽  
Kane's Method ◽  
Ellipsoidal Surface ◽  
Ideal Constraint ◽  
The Ideal

This paper deals with the motion of mechanical systems with non-ideal constraints, defined as constraints where the forces associated with the constraint do work. The first objective of the paper is to show that two newly published formulations of equations of motion of systems with such non-ideal constraints are unnecessarily complex for situations where the non-ideal constraint force does not depend on the ideal constraint force, because they introduce and then eliminate these non-working constraint forces. We point out that a method already exists for nonideal constraints, namely, Kane’s equations, which are simpler because, among other things, they are based on automatic elimination of non-working constraints. The examples considered in these recent publications are worked out with Kane’s method to show the applicability and simplicity of Kane’s method for non-ideal constraints. A second objective of the paper is to present an alternative form of equations for systems where the non-ideal constraint force depends on the ideal constraint force, as in the case of Coulomb friction. The formulation is shown to lend itself naturally to also analyzing impact dynamics. The method is applied to the dynamics of a slug moving against friction on a moving ellipsoidal surface. Such a crude model may simulate, in essence, propellant motion in a tank in zero-g, or during docking of a spacecraft.

Explicit Equations of Motion for Mechanical Systems With Nonideal Constraints

2000 ◽  
Vol 68 (3) ◽  
pp. 462-467 ◽  
Author(s):  
F. E. Udwadia ◽  
R. E. Kalaba
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Lagrangian Mechanics ◽  
Constrained Motion ◽  
Constraint Forces ◽  
Constrained Mechanical Systems ◽  
Lagrangian Formulations ◽  
Explicit Equations Of Motion ◽  
D'alembert's Principle

Since its inception about 200 years ago, Lagrangian mechanics has been based upon the Principle of D’Alembert. There are, however, many physical situations where this confining principle is not suitable, and the constraint forces do work. To date, such situations are excluded from general Lagrangian formulations. This paper releases Lagrangian mechanics from this confinement, by generalizing D’Alembert’s principle, and presents the explicit equations of motion for constrained mechanical systems in which the constraints are nonideal. These equations lead to a simple and new fundamental view of Lagrangian mechanics. They provide a geometrical understanding of constrained motion, and they highlight the simplicity with which Nature seems to operate.

Equations of Motion for Nonholonomic, Constrained Dynamical Systems via Gauss’s Principle

1993 ◽  
Vol 60 (3) ◽  
pp. 662-668 ◽  
Author(s):  
R. E. Kalaba ◽  
F. E. Udwadia
Keyword(s):  
Dynamical Systems ◽  
Closed Form ◽  
Programming Problem ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Closed Form Solution ◽  
Nonholonomic Constraints ◽  
Form Solution ◽  
Constrained Motion ◽  
Constraint Forces

In this paper we develop an analytical set of equations to describe the motion of discrete dynamical systems subjected to holonomic and/or nonholonomic Pfaffian equality constraints. These equations are obtained by using Gauss’s Principle to recast the problem of the constrained motion of dynamical systems in the form of a quadratic programming problem. The closed-form solution to this programming problem then explicitly yields the equations that describe the time evolution of constrained linear and nonlinear mechanical systems. The direct approach used here does not require the use of any Lagrange multipliers, and the resulting equations are expressed in terms of two different classes of generalized inverses—the first class pertinent to the constraints, the second to the dynamics of the motion. These equations can be numerically solved using any of the standard numerical techniques for solving differential equations. A closed-form analytical expression for the constraint forces required for a given mechanical system to satisfy a specific set of nonholonomic constraints is also provided. An example dealing with the position tracking control of a nonlinear system shows the power of the analytical results and provides new insights into application areas such as robotics, and the control of structural and mechanical systems.

Inverse dynamics of the 6-dof out-parallel manipulator by means of the principle of virtual work

Robotica ◽  
2009 ◽  
Vol 27 (2) ◽  
pp. 259-268 ◽  
Author(s):  
Yongjie Zhao ◽  
Feng Gao
Keyword(s):  
Parallel Manipulator ◽  
Inverse Dynamics ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Principle Of Virtual Work ◽  
Constraint Forces ◽  
Dynamical Equations ◽  
Robot System ◽  
Lead Screw ◽  
Moving Platform

SUMMARYIn this paper, the inverse dynamics of the 6-dof out-parallel manipulator is formulated by means of the principle of virtual work and the concept of link Jacobian matrices. The dynamical equations of motion include the rotation inertia of motor–coupler–screw and the term caused by the external force and moment exerted at the moving platform. The approach described here leads to efficient algorithms since the constraint forces and moments of the robot system have been eliminated from the equations of motion and there is no differential equation for the whole procedure. Numerical simulation for the inverse dynamics of a 6-dof out-parallel manipulator is illustrated. The whole actuating torques and the torques caused by gravity, velocity, acceleration, moving platform, strut, carriage, and the rotation inertia of the lead screw, motor rotor and coupler have been computed.

What is the General Form of the Explicit Equations of Motion for Constrained Mechanical Systems?

2002 ◽  
Vol 69 (3) ◽  
pp. 335-339 ◽  
Author(s):  
F. E. Udwadia ◽  
R. E. Kalaba
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Analytical Mechanics ◽  
Constraint Forces ◽  
Constrained Mechanical Systems ◽  
Explicit Equations Of Motion ◽  
D'alembert's Principle

This paper presents the general form of the explicit equations of motion for mechanical systems. The systems may have holonomic and/or nonholonomic constraints, and the constraint forces may or may not satisfy D’Alembert’s principle at each instant of time. The explicit equations lead to new fundamental principles of analytical mechanics.

On the Transformation Properties of the Nonlinear Hamel Equations

1995 ◽  
Vol 62 (4) ◽  
pp. 924-929 ◽  
Author(s):  
J. G. Papastavridis
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Inertia Force ◽  
Discrete Mechanical Systems ◽  
Inertial Terms ◽  
The Individual ◽  
Boltzmann Hamel Equations ◽  
System Inertia

This paper discusses the transformation properties of the famous Johnsen-Hamel equations of motion of discrete mechanical systems in general nonlinear nonholonomic coordinates and constraints (i.e., the nonlinear extension of the well-known Boltzmann-Hamel equations), under general nonlinear (local) quasi-velocity transformations. It is shown that the individual kinematico-inertial terms making up the system inertia force, or system acceleration, such as the nonlinear nonholonomic Euler-Lagrange operator and nonholonomic correction (or deviation) terms, in general, do not transform as nonholonomic covariant vectors; although taken as a whole they do, as expected. This work extends and completes the work of Papastavridis (1994), and it is strongly recommended that it be read after that paper.

scholarly journals An Argument Against Augmenting the Lagrangean for Nonholonomic Systems

2009 ◽  
Vol 76 (3) ◽  
Author(s):  
Carlos M. Roithmayr ◽  
Dewey H. Hodges
Keyword(s):  
Nonholonomic System ◽  
Nonholonomic Constraint ◽  
Equations Of Motion ◽  
Nonholonomic Systems ◽  
Euler Method ◽  
Constraint Forces ◽  
Dynamical Equations ◽  
New Approach ◽  
Constraint Equations ◽  
Kane's Method

Although it is known that correct dynamical equations of motion for a nonholonomic system cannot be obtained from a Lagrangean that has been augmented with a sum of the nonholonomic constraint equations weighted with multipliers, previous publications suggest otherwise. One published example that was proposed in support of augmentation purportedly demonstrates that an accepted method fails to produce correct equations of motion whereas augmentation leads to correct equations. This present paper shows that, in fact, the opposite is true. The correct equations, previously discounted on the basis of a flawed application of the Newton–Euler method, are verified by using Kane’s method together with a new approach for determining the directions of constraint forces.

Integrals of Linearized Differential Equations of Motion of Mechanical Systems; Part II: Linearized Equations of Motion

1987 ◽  
Vol 54 (3) ◽  
pp. 661-667 ◽  
Author(s):  
T. R. Kane ◽  
S. Djerassi
Keyword(s):  
Differential Equations ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Dynamical Equations ◽  
Linearized Equations

Theorems derived in Part I are here applied to differential equations of motion of mechanical systems. The theorems are reformulated in terms of variables appearing in dynamical equations of motion, and their use is illustrated by means of an example.

Parallel O(log2N) Algorithm for the Motion Simulation of General Multi-Rigid-Body Mechanical Systems

1997 ◽  
Author(s):  
Kurt S. Anderson
Keyword(s):  
Temporal Integration ◽  
Equations Of Motion ◽  
Processing System ◽  
Iterative Solution ◽  
Rigid Bodies ◽  
Constraint Forces ◽  
Dynamical Equations ◽  
Highly Efficient ◽  
Time Optimal

Abstract This paper presents a new highly efficient procedure for the determination of the dynamical equations of motion for complex multibody systems and their subsequent temporal integration using parallel computing. The method is applicable to general systems of rigid bodies which may contain arbitrary joint types, multiple branches, and/or close loops. The method is based on the explicit determination of constraint forces at key joint locations and the subsequent highly efficient determination of system state time derivatives. The algorithm uses a novel hybrid direct and iterative solution scheme which allows a substantially higher degree of parallelization than is generally obtainable using the more conventional recursive O(N) procedures. It is shown that at the coarsest level the parallelization obtainable easily exceeds that indicated by the topology of the system. The procedure can produce a theoretical and time optimal O(log2N) performance on computational throughput with a processor optimal O(N) processors on a MDMD distributed architecture processing system.

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