On the Lagrangian Formalism of Nonholonomic Mechanical Systems

2005 ◽  
Author(s):  
Hiroaki Yoshimura
Keyword(s):  
Mechanical System ◽  
Tangent Bundle ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Lagrangian System ◽  
Lagrangian Formalism ◽  
Nonholonomic Mechanical Systems ◽  
Points Of View ◽  
Configuration Manifold

The paper illustrates the Lagrangian formalism of mechanical systems with nonholonomic constraints using the ideas of geometric mechanics. We first review a Lagrangian system for a conservative mechanical system in the context of variational principle of Hamilton, and we investigate the case that a given Lagrangian is hyperregular, which can be illustrated in the context of the symplectic structure on the tangent bundle of a configuration space by using the Legendre transformation. The Lagrangian system is denoted by the second order vector field and the Lagrangian one- and two-forms associated with a given hyperregular Lagrangian. Then, we demonstrate that a mechanical system with nonholonomic constraints can be formulated on the tangent bundle of a configuration manifold by using Lagrange multipliers. To do this, we investigate the Lagrange-d’Alembert principle from geometric points of view and we also show the intrinsic expression of the Lagrange-d’Alembert equations of motion for nonholonomic mechanical systems with nonconservative force fields.

A geometric approach to the transpositional relations in dynamics of nonholonomic mechanical systems

2018 ◽  
Vol 15 (07) ◽  
pp. 1850112 ◽  
Author(s):  
Mahdi Khajeh Salehani
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Geometric Approach ◽  
Nonholonomic Mechanical Systems ◽  
Nonholonomic Dynamics ◽  
Geometric Objects

Exploring the geometry of mechanical systems subject to nonholonomic constraints and using various bundle and variational structures intrinsically present in the nonholonomic setting, we study the structure of the equations of motion in a way that aids the analysis and helps to isolate the important geometric objects that govern the motion of such systems. Furthermore, we show that considering different sets of transpositional relations corresponding to different transitivity choices provides different variational structures associated with nonholonomic dynamics, but the derived equations (being referred to as the generalized Hamel–Voronets equations) are equivalent to the Lagrange–d’Alembert equations. To illustrate results of this work and as some applications of the generalized Hamel–Voronets formalisms discussed in this paper, we conclude with considering the balanced Tennessee racer, as well as its modification being referred to as the generalized nonholonomic cart, and an [Formula: see text]-snake with three wheeled planar platforms whose snake-like motion is induced by shape variations of the system.

Hamiltonian formulation for mechanical systems with nonholonomic constraints

2014 ◽  
Vol 11 (03) ◽  
pp. 1450017
Author(s):  
G. F. Torres del Castillo ◽  
O. Sosa-Rodríguez
Keyword(s):  
Finite Number ◽  
Mechanical System ◽  
Infinite Number ◽  
Degrees Of Freedom ◽  
Hamiltonian Structure ◽  
Equations Of Motion ◽  
Hamiltonian Formulation ◽  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Hamilton Equations

It is shown that for a mechanical system with a finite number of degrees of freedom, subject to nonholonomic constraints, there exists an infinite number of Hamiltonians and symplectic structures such that the equations of motion can be written as the Hamilton equations, with the original constraints incorporated in the Hamiltonian structure.

Explicit Poincaré equations of motion for general constrained systems. Part I. Analytical results

2007 ◽  
Vol 463 (2082) ◽  
pp. 1421-1434 ◽  
Author(s):  
Firdaus E Udwadia ◽  
Phailaung Phohomsiri
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Constrained Systems ◽  
Analytical Results ◽  
Multi Body ◽  
D'alembert's Principle

This paper gives the general constrained Poincaré equations of motion for mechanical systems subjected to holonomic and/or nonholonomic constraints that may or may not satisfy d'Alembert's principle at each instant of time. It also extends Gauss's principle of least constraint to include quasi-accelerations when the constraints are ideal, thereby expanding the compass of this principle considerably. The new equations provide deeper insights into the dynamics of multi-body systems and point to new ways for controlling them.

On the Boltzmann-Hamel Equations of Motion: A Vectorial Treatment

1994 ◽  
Vol 61 (2) ◽  
pp. 453-459 ◽  
Author(s):  
J. G. Papastavridis
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Discrete Mechanical Systems ◽  
The Common ◽  
Boltzmann Hamel Equations ◽  
Nonlinear Velocity

This paper presents a direct vectorial derivation of the famous Boltzmann-Hamel equations of motion of discrete mechanical systems, in general nonlinear nonholonomic coordinates and under general nonlinear (velocity) nonholonomic constraints. The connection between particle and system vectors is stressed throughout, in all relevant kinematic and kinetic quantities/principles/theorems. The specialization of these results to the common case of linear nonholonomic coordinates and linear nonholonomic (i.e., Pfaffian) constraints is carried out in the paper’s Appendix.

Dynamics of a Multibody Mechanical System With Lubricated Long Journal Bearings

2004 ◽  
Vol 127 (3) ◽  
pp. 493-498 ◽  
Author(s):  
B. J. Alshaer ◽  
H. Nagarajan ◽  
H. K. Beheshti ◽  
H. M. Lankarani ◽  
S. Shivaswamy
Keyword(s):  
Dynamic Response ◽  
Mechanical System ◽  
Journal Bearing ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Journal Bearings ◽  
Connecting Rod ◽  
Revolute Joint ◽  
Order Of Magnitude

Clearances exist in different kinds of joints in multibody mechanical systems, which could drastically affect the dynamic behavior of the system. If the joint is dry with no lubricant, impact occurs, resulting in wear and tear of the joint. In practical engineering design of machines, joints are usually designed to operate with some lubricant. Lubricated journal bearings are designed so that even when the maximum load is applied, the joint surfaces do not come into contact with each other. In this paper, a general methodology for modeling lubricated long journal bearings in multibody mechanical systems is presented. This modeling utilizes a method of solving for the forces produced by the lubricant in a dynamically loaded long journal bearing. A perfect revolute joint in a multibody mechanical system imposes kinematic constraints, while a lubricated journal bearing joint imposes force constraints. As an application, the dynamic response of a slider-crank mechanism including a lubricated journal bearing joint between the connecting rod and the slider is considered and analyzed. The dynamic response is obtained by numerically solving the constraint equations and the forces produced by the lubricant simultaneously with the differential equations of motion and a set of initial conditions numerically. The results are compared with the previous studies performed on the same mechanism as well a dry clearance joint. It is shown that in a multibody mechanical system, the journal bearing lubricant introduces damping and stiffness to the system. The earlier studies predict that the order of magnitude of the reaction moment is twice that of a perfect revolute joint. The proposed model predicts that the reaction moment is within the same order of magnitude of the perfect joint simulation case.

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.

scholarly journals Two problems of movement of multi-link wheeled vehicles

2021 ◽  
Vol 2094 (2) ◽  
pp. 022063
Author(s):  
E A Mikishanina
Keyword(s):  
Mechanical System ◽  
Software Package ◽  
Equations Of Motion ◽  
Mathematical Formulation ◽  
Nonholonomic Constraints ◽  
Mechanical Parameters ◽  
Controlled Motion ◽  
The Law ◽  
Wheeled Vehicles ◽  
Derivatives Of

Abstract The paper presents the results of a study of the dynamics of multi-link wheeled vehicles with nonholonomic connections superimposed on this mechanical system. The article is of an overview nature. The mathematical formulation is a system of ordinary differential equations of motion in the form of Lagrange with undefined multipliers, solved together with derivatives of nonholonomic constraints. The results are presented in the case of controlled motion, when the law of motion of the first link is known, as well as in the case of uncontrolled motion, when the law of motion of the leading link is also the desired function of time. General equations are obtained for a mechanical system consisting of an arbitrary number of links. Numerical results are presented for the case of three coupled trolleys. The software package Maple was used to perform numerical calculations and plotting of the desired mechanical parameters.

Dynamics of a Multibody Mechanical System With Lubricated Long Journal Bearings

2001 ◽  
Author(s):  
B. J. Alshaer ◽  
H. M. Lankarani ◽  
S. Shivaswamy
Keyword(s):  
Dynamic Response ◽  
Mechanical System ◽  
Journal Bearing ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Journal Bearings ◽  
Connecting Rod ◽  
Revolute Joint ◽  
Order Of Magnitude

Abstract Clearances exist in different kinds of joints in multibody mechanical systems, which could drastically affect the dynamic behavior of the system. If the joint is dry with no lubricant, impact occurs, resulting in wear and tear of the joint. In practical engineering design of machines, joints are usually designed to operate with some lubricant. Lubricated journal bearings are designed so that even when the maximum load is applied, the joint surfaces do not come into contact with each other. In this paper, a general methodology for modeling lubricated long journal bearings in multibody mechanical systems is presented. This modeling utilizes a new method of solving for the forces produced by the lubricant in a dynamically loaded long journal bearing. A perfect revolute joint in a multibody mechanical system imposes kinematic constraints, while a lubricated journal bearing joint imposes force constraints. As an application, the dynamic response of a crank-slider mechanism including a lubricated journal bearing joint between the connecting rod and the slider is considered and analyzed. The dynamic response is obtained by numerically solving the constraint equations and the forces produced by the lubricant simultaneously with the differential equations of motion and a set of initial conditions numerically. The results are compared with the previous studies performed on the same mechanism as well a dry clearance joint. It is shown that in a multibody mechanical system, the journal bearing lubricant introduces damping and stiffness to the system. The earlier studies previous predict that the order of magnitude of the reaction moment is twice that of a perfect revolute joint. The proposed model predicts that the reaction moment is within the same order of magnitude of the perfect joint simulation case.

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