Identifying Sets of Constraint Forces by Inspection

2013 ◽  
Vol 80 (2) ◽  
Author(s):  
Carlos M. Roithmayr ◽  
Dewey H. Hodges
Keyword(s):  
Mechanical System ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
The Body ◽  
Vector Form ◽  
Constraint Forces ◽  
Constraint Force ◽  
Symbolic Algebra ◽  
Constraint Equations ◽  
Angular Velocities

A mechanical system is often modeled as a set of particles and rigid bodies, some of which are constrained in one way or another. A concise method is proposed for identifying a set of constraint forces needed to ensure the restrictions are met. Identification consists of determining the direction of each constraint force and the point at which it must be applied, as well as the direction of the torque of each constraint force couple, together with the body on which the couple acts. This important information can be determined simply by inspecting constraint equations written in vector form. For the kinds of constraints commonly encountered, the constraint equations are expressed in terms of dot products involving velocities of the affected points or particles and angular velocities of the bodies concerned. The technique of expressing constraint equations in vector form and identifying constraint forces by inspection is useful when one is deriving explicit, analytical equations of motion by hand or with the aid of symbolic algebra software, as demonstrated with several examples.

Molecular/Mechanical Dynamics Simulations With Nonholonomic Constraints

2010 ◽  
Author(s):  
Po-Chih Chen ◽  
Chia-Ou Chang ◽  
Wen-Tien Chang Chien ◽  
Chan-Shin Chou
Keyword(s):  
Kinetic Energy ◽  
Generalized Inverse ◽  
Equations Of Motion ◽  
Nonholonomic Constraints ◽  
Constraint Forces ◽  
Constraint Force ◽  
Constraint Matrix ◽  
Constraint Equations ◽  
Dynamics Simulations ◽  
Mechanical Dynamics

In this paper a new method of incorporating linear/nonlinear nonholonomic constraints into the mechanical/molecular dynamical systems is presented. We first introduce the mass-weighted coordinates such that acceleration and forces are scaled to have the same units, and can be operated in the same space. Then we use the projector formalism and Gauss’s principle of least constraint to derive the constraint force in the explicit form so that the equations of motion are free of Lagrange multipliers. The use of mass-weighted coordinates enable the equation of the constraint forces to be expressed in terms of first generalized inverse of constraint matrix rather than the two-time generalized inverse of matrices used in the recent works. An algorithm of numerical integration for ensuring the satisfaction of constraint equations and avoiding the numerical drift is proposed. Two simple examples, constant kinetic energy (or temperature) and time-varying prescribed kinetic energy of three-particle dynamical system effectively verify our method.

A Computational Model of Scapulo-Humeral-Clavicle Complex via Multibody Dynamics

2009 ◽  
Author(s):  
Shanzhong Shawn Duan ◽  
Keith M. Baumgarten
Keyword(s):  
Rigid Body ◽  
Soft Tissues ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
The Body ◽  
Constraint Forces ◽  
Upper Arm ◽  
Body Model ◽  
Rigid Body Model ◽  
Concentric Ball

The shoulder-upper arm complex has the most mobile joint in the body and is composed of three main bones: the collarbone (clavicle), the shoulder blade (scapula), and the upper arm bone (humerus). The shoulder joint is a non-concentric ball and socket joint. It differs from the hip, a highly stabilized, concentric ball and socket joint, that is constrained mostly by its osseous anatomy. Thus, the shoulder has more flexibility and less inherent stability than the hip because it is mainly stabilized by muscles, tendons, and ligaments. The relative decrease in stability of the shoulder compared to other joints puts the shoulder at increase risk of damage by disease or injury. The constraints added by muscles, tendons, and ligaments make modeling of the shoulder a challenge task. This paper presents a multi rigid body model to describe dynamical properties of the scapulo-humeral-clavicle complex. The bones are represented by rigid bodies, and the soft tissues (tendons, ligaments and muscles) are represented by springs and actuators attached to the rigid bodies. The rigid bodies are connected by ideal kinematic joints and have fixed centers of gravity. Equations of motion of the multi rigid body model are derived via Kane’s methods. Combination of springs and actuators includes independent variables for both motion and constraint forces, the sum of which determine the activation level.

Formulation of the Equations of Motion of RRPR Robot Manipulator

2000 ◽  
Author(s):  
Hazem Ali Attia ◽  
Tarek M. A. El-Mistikawy ◽  
Adel A. Megahed
Keyword(s):  
Dynamic Analysis ◽  
Efficient Solution ◽  
Robot Manipulator ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Second Law ◽  
Equivalent System ◽  
Constraint Forces ◽  
Dynamic Formulation ◽  
System Of Particles

Abstract In this paper the dynamic analysis of RRPR robot manipulator is presented. The equations of motion are formulated using a two-step transformation. Initially, a dynamically equivalent system of particles that replaces the rigid bodies is constructed and then Newton’s second law is applied to derive their equations of motion. The equations of motion are then transformed to the relative joint variables. Use of both Cartesian and joint variables produces an efficient set of equations without loss of generality. For open chains, this process automatically eliminates all of the non-working constraint forces and leads to an efficient solution and integration of the equations of motion. The results of the simulation indicate the simplicity and generality of the dynamic formulation.

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.

ENVELOPING SURFACES AND ADMISSIBLE VELOCITIES OF HEAVY RIGID BODIES

2004 ◽  
Vol 14 (08) ◽  
pp. 2525-2553 ◽  
Author(s):  
IGOR N. GASHENENKO ◽  
PETER H. RICHTER
Keyword(s):  
Three Dimensional ◽  
Rigid Bodies ◽  
First Integrals ◽  
The Body ◽  
Heegaard Splitting ◽  
Body Motion ◽  
Invariant Sets ◽  
Poisson Problem ◽  
Integral Manifolds ◽  
Angular Velocities

The general Euler-Poisson problem of rigid body motion is investigated. We study the three-dimensional algebraic level surfaces of the first integrals, and their topological bifurcations. The main result of this article is an analytical and qualitatively complete description of the projections of these integral manifolds to the body-fixed space of angular velocities. We classify the possible types of these invariant sets and analyze the dependence of their topology on the parameters of the body and the constants of the first integrals. Particular emphasis is given to the enveloping surfaces of the sets of admissible angular velocities. Their pre-images in the reduced phase space induce a Heegaard splitting which lends itself for a general choice of complete Poincaré surfaces of section, irrespective of whether or not the system is integrable.

scholarly journals MATHEMATICAL MODEL OF SPATIAL OSCILLATIONS OF A RAILWAY FOUR-AXLE AUTONOMOUS TRACTION MODULE

2021 ◽  
pp. 55-68
Author(s):  
František Bures
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Mechanical System ◽  
Traffic Safety ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Railway Track ◽  
Theoretical Studies ◽  
System Of Differential Equations ◽  
Spatial Oscillations

A description of the original mathematical model of spatial oscillations of a four-axle autonomous traction module during its movement along straight and curved sections of the railway track is proposed. In this case, the design of a four-axle autonomous traction module is presented as a complex mechanical system, and the track is considered as an elastic-viscous inertial system. The equations of motion were compiled using the Lagrange method of the ІІ kind. For each of the solids, the kinetic energy is determined by the Koenig theorem. The potential energy component is obtained by the Clapeyron theorem, as the sum of the energies accumulated in the elastic elements of the system during their deformations. The dissipative energy was also taken into account when compiling the equations of motion. Generalized forces that have no potential, in this case, include the forces of interaction between wheels and rails, which are determined using the creep hypothesis. It is important to note that the model takes into account the forces in the bonds between the bodies of the system and the spatial displacements of the rigid bodies of the mechanical system, taking into account possible restrictions. Moreover, the mathematical model developed by the author is a system of differential equations of the 100th order. This mathematical model is the base for further theoretical studies of the dynamics of railway four-axle autonomous traction modules in single motion or when moving as part of a train. To solve the resulting system of differential equations, the author develops special software that allows for complex theoretical studies of spatial oscillations of a four-axle autonomous tractionmodule to determine the indicators of its dynamic loading and traffic safety. 

THE MODELLING OF HOLONOMIC MECHANICAL SYSTEMS USING A NATURAL ORTHOGONAL COMPLEMENT

1989 ◽  
Vol 13 (4) ◽  
pp. 81-89 ◽  
Author(s):  
J. ANGELES ◽  
SANGKOO LEE
Keyword(s):  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Kinematic Chain ◽  
Chain Structure ◽  
Rigid Bodies ◽  
Orthogonal Complement ◽  
Constraint Forces ◽  
Constrained Mechanical Systems ◽  
Constraint Equations ◽  
The Matrix

A computationally efficient and systematic algorithm for the modelling of constrained mechanical systems is developed and implemented in this paper. With this algorithm, the governing equations of mechanical systems comprised of rigid bodies coupled by holonomic constraints are derived by means of an orthogonal complement of the matrix of the velocity-constraint equations. The procedure is applicable to all types of holonomic mechanical systems, and it can be extended to cases including simple nonholonomic constraints. Holonomic mechanical systems having a simple Kinematic-chain structure, such as single-loop linkages and serial-type robotic manipulators, are analysed regarding the derivation of the matrix of the constraint equations and its orthogonal complement, and the computation of the constraint forces.

Roll Maneuverability of Flapping Flight Winged Systems

2020 ◽  
Author(s):  
Hamid Vejdani
Keyword(s):  
Equations Of Motion ◽  
Flapping Flight ◽  
The Body ◽  
Aerodynamic Parameter ◽  
Angular Velocities

Abstract The goal of this paper is to study the effect of wing flapping kinematics on roll maneuverability of flapping flight systems. Inspired from birds maneuvering action, we study the effect of asymmetric flapping angular velocities of the wings on generating roll motions on the body. To expand the generality of the results, the equations of motion are written dimensionless. The effect of aerodynamic parameter, forward velocity and wing inertia are presented. The results show that applying asymmetric velocities during flight is useful for relatively larger wings.

scholarly journals On the Interpretation of the Lagrange Multipliers in the Constraint Formulation of Contact Problems; or Why Are Some Multipliers Always Zero?

2014 ◽  
Author(s):  
A. L. Schwab
Keyword(s):  
Mechanical System ◽  
Lagrange Multipliers ◽  
Mechanical Systems ◽  
Contact Problems ◽  
Constraint Forces ◽  
Basic Notion ◽  
Constraint Equations ◽  
Linear Dependency ◽  
Constraint Approach ◽  
Two Bodies

One method for modeling idealized contact between two bodies in mechanical system is based on the constraint approach, where Lagrange multipiers are introduced, which serve as constraint forces. In the usage of this formulation, there exists a linear dependancy between the Lagrange multipliers. Moreover, it has been observed that some Lagrange multipliers are always identical to zero. This sort of contradicts the basic notion that Lagrange multipliers in mechanical systems act as constraint forces which, when constraints are violated, push the system back in the desired configuration. In this paper it will be shown, by theory and example, that the above-mentioned linear dependency of the Lagrange multipliers, together with specific entries in the Jacobian of the constraint equations, results in some Lagrange multipliers being identical to zero.

Vibration Modes and Frequencies of Timoshenko Beams With Attached Rigid Bodies

1995 ◽  
Vol 62 (1) ◽  
pp. 193-199 ◽  
Author(s):  
M. W. D. White ◽  
G. R. Heppler
Keyword(s):  
Boundary Conditions ◽  
Rigid Body ◽  
Timoshenko Beam ◽  
Equations Of Motion ◽  
Frequency Equation ◽  
Rigid Bodies ◽  
The Body ◽  
Mode Shapes ◽  
Timoshenko Beams ◽  
First Moment

The equations of motion and boundary conditions for a free-free Timoshenko beam with rigid bodies attached at the endpoints are derived. The natural boundary conditions, for an end that has an attached rigid body, that include the effects of the body mass, first moment of mass, and moment of inertia are included. The frequency equation for a free-free Timoshenko beam with rigid bodies attached at its ends which includes all the effects mentioned above is presented and given in terms of the fundamental frequency equations for Timoshenko beams that have no attached rigid bodies. It is shown how any support / rigid-body condition may be easily obtained by inspection from the reported frequency equation. The mode shapes and the orthogonality condition, which include the contribution of the rigid-body masses, first moments, and moments of inertia, are also developed. Finally, the effect of the first moment of the attached rigid bodies is considered in an illustrative example.

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