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.

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.

Design of the Ball Screw Mechanism for Optimal Efficiency

1989 ◽  
Author(s):  
M.-C. Lin ◽  
S. A. Velinsky ◽  
B. Ravani
Keyword(s):  
Closed Form ◽  
Static Analysis ◽  
Load Capacity ◽  
Equations Of Motion ◽  
Closed Form Solution ◽  
Form Solution ◽  
Ball Screw ◽  
Exact Theory ◽  
Screw Mechanism ◽  
Extensive Computation

Abstract This paper develops theories for evaluating the efficiency of the ball screw mechanism and additionally, for designing this mechanism. Initially, a quasi-static analysis, which is similar to that of the early work in this area, is employed to evaluate efficiency. Dynamic forces, which are neglected by the quasi-static analysis, will have an effect on efficiency. Thus, an exact theory based on the simultaneous solution of both the Newton-Euler equations of motion and the relevant kinematic equations is employed to determine mechanism efficiency, as well as the steady-state motion of all components within the ball screw. However, the development of design methods based on this exact theory is difficult due to the extensive computation necessary and thus, an approximate closed-form representation, that still accounts for the ball screw dynamics, is derived. The validity of this closed-form solution is proven and it is then used in developing an optimum design methodology for the ball screw mechanism based on efficiency. Additionally, the self-braking condition is examined, as are load capacity considerations.

Closed-Form Transient Response of Distributed Damped Systems, Part II: Energy Formulation for Constrained and Combined Systems

1996 ◽  
Vol 63 (4) ◽  
pp. 1004-1010 ◽  
Author(s):  
Bingen Yang
Keyword(s):  
Closed Form ◽  
Transient Response ◽  
Differential Operators ◽  
Equations Of Motion ◽  
Closed Form Solution ◽  
Form Solution ◽  
Response Analysis ◽  
Lumped Parameter ◽  
Damped Systems ◽  
Energy Formulation

The transient response analysis presented in Part I is generalized for distributed damped systems which are viscoelastically constrained or combined with lumped parameter systems. An energy formulation is introduced to regain symmetry for the spatial differential operators, which is destroyed in the original equations of motion by the constraints, and the coupling of distributed and lumped elements. As a result, closed-form solution is systematically obtained in eigenfunction series.

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.

Determination of Eigenstresses from Curvature Data

1992 ◽  
Vol 276 ◽  
Author(s):  
Mauro Ferrari ◽  
Marie Weber
Keyword(s):  
Structural Analysis ◽  
Closed Form ◽  
Mechanical Systems ◽  
Closed Form Solution ◽  
Form Solution ◽  
Curvature Measurements

ABSTRACTCurvature measurements are generally employed in conjunction with elementary structural analysis to estimate deposition stresses in miniaturized electro-mechanical systems. In this paper the validity of this procedure is discussed by presenting a closed form solution for a bilayer subject to nonuniform intrinsic straining, and comparing the exact stress-curvature relations with the oft-used formulae of Stoney and Brenner-Senderoff.

Motion of a Bored Sphere in a Spinning Spherical Cavity

1981 ◽  
Vol 103 (4) ◽  
pp. 389-394 ◽  
Author(s):  
R. H. Nunn ◽  
J. W. Bloomer
Keyword(s):  
Numerical Methods ◽  
Closed Form ◽  
Predictive Model ◽  
Spherical Cavity ◽  
Equations Of Motion ◽  
Closed Form Solution ◽  
Form Solution ◽  
Analytical Expressions ◽  
Theory And Experiment

Theory and experiment are combined to develop a predictive model for the motion of a bored sphere within a spinning spherical cavity. The motion is gyroscopic in nature with the sphere eventually aligning its hole with the axis of spin of the cavity. Analytical expressions are derived for the applied moments on the sphere due to its motion relative to that of the cavity, and the resulting equations of motion are solved by numerical methods. An approximate closed-form solution is also obtained. Experiments are described in which the measured nutation of the sphere substantiates the analytical predictions.

scholarly journals Closed-Form Solution of Adiabatic Particle Trajectories in Axis-Symmetric Magnetic Fields

2021 ◽  
Author(s):  
F. Sattin ◽  
D.F. Escande
Keyword(s):  
Magnetic Field ◽  
Charged Particle ◽  
Closed Form ◽  
Equations Of Motion ◽  
Functional Relation ◽  
Closed Form Solution ◽  
Form Solution ◽  
Low Energy ◽  
Particle Orbit ◽  
Pass Through

The dynamics of a low-energy charged particle in an axis-symmetric magnetic field is known to be a regular superposition of periodic–although possibly incommensurate–motions. The projection of the particle orbit along the two non-ignorable coordinates (x,y) may be expressed in terms of each other: y=y(x), yet–to our knowledge–such a functional relation has never been directly produced in literature, but only by way of a detour: first, equations of motion are solved, yielding x=x(t),y=y(t), and then one of the two relations is inverted, x(t)→t(x). In this paper we present a closed-form functional relation which allows to express coordinates of the particle’ orbit without the need to pass through the hourly law of motion.

scholarly journals Closed-Form Solution of Adiabatic Particle Trajectories in Axis-Symmetric Magnetic Fields

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1784
Author(s):  
Fabio Sattin ◽  
Dominique Franck Escande
Keyword(s):  
Magnetic Field ◽  
Charged Particle ◽  
Closed Form ◽  
Equations Of Motion ◽  
Functional Relation ◽  
Closed Form Solution ◽  
Form Solution ◽  
Low Energy ◽  
Particle Orbit ◽  
Pass Through

The dynamics of a low-energy charged particle in an axis-symmetric magnetic field is known to be a regular superposition of periodic—although possibly incommensurate—motions. The projection of the particle orbit along the two non-ignorable coordinates (x,y) may be expressed in terms of each other: y=y(x), yet—to our knowledge—such a functional relation has never been directly produced in literature, but only by way of a detour: first, equations of motion are solved, yielding x=x(t),y=y(t), and then one of the two relations is inverted, x(t)→t(x). In this paper, we present a closed-form functional relation which allows us to express coordinates of the particle’s orbit without the need to pass through the hourly law of motion.

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