Equations of motion of a system with nonlinear nonholonomic constraints

1962 ◽  
Vol 11 (1) ◽  
pp. 114-116 ◽  
Author(s):  
Q. K. Ghori
Keyword(s):  
Equations Of Motion ◽  
Nonholonomic Constraints

scholarly journals Numerical Integration of Multibody Dynamic Systems Involving Nonholonomic Equality Constraints

2021 ◽  
Author(s):  
Sotirios Natsiavas ◽  
Panagiotis Passas ◽  
Elias Paraskevopoulos
Keyword(s):  
Dynamic Systems ◽  
Equations Of Motion ◽  
Time Integration ◽  
Nonholonomic Constraints ◽  
Coupled System ◽  
Weak Form ◽  
Integration Scheme ◽  
Motion Constraints ◽  
Multibody Dynamic ◽  
Time Integration Scheme

Abstract This work considers a class of multibody dynamic systems involving bilateral nonholonomic constraints. An appropriate set of equations of motion is employed first. This set is derived by application of Newton’s second law and appears as a coupled system of strongly nonlinear second order ordinary differential equations in both the generalized coordinates and the Lagrange multipliers associated to the motion constraints. Next, these equations are manipulated properly and converted to a weak form. Furthermore, the position, velocity and momentum type quantities are subsequently treated as independent. This yields a three-field set of equations of motion, which is then used as a basis for performing a suitable temporal discretization, leading to a complete time integration scheme. In order to test and validate its accuracy and numerical efficiency, this scheme is applied next to challenging mechanical examples, exhibiting rich dynamics. In all cases, the emphasis is put on highlighting the advantages of the new method by direct comparison with existing analytical solutions as well as with results of current state of the art numerical methods. Finally, a comparison is also performed with results available for a benchmark problem.

Concurrent Multiple-Time-Scale Simulations With Improved Numerical Dissipation for Structural Dynamics

2013 ◽  
Author(s):  
Tejas Ruparel ◽  
Azim Eskandarian ◽  
James Lee
Keyword(s):  
Time Scale ◽  
Equations Of Motion ◽  
Time Integration ◽  
Nonholonomic Constraints ◽  
Numerical Dissipation ◽  
Multiple Time ◽  
Multiple Time Scale ◽  
Constraint Forces ◽  
Conservation Of Energy ◽  
Coupled Equations

Work presented in this paper describes the formulation for implementation of a concurrent multiple-time-scale integration method with improved numerical dissipation capabilities. This approach generalizes the previous Multiple Grid and Multiple Time-Scale (MGMT) Method [1] implemented for the Newmark family of algorithms. The framework is largely based upon the fundamental principles of Lagrange multipliers used to enforce workless nonholonomic constraints and Domain Decomposition Methods (DDM) to obtain coupled equations of motion for distinct regions of a continuous domain. These methods when combined together systematically yield constraint forces that not only ensure conservation of energy but also enforce continuity of velocities across the interfaces. Multiple grid connections between (non-conforming) sub-domains are handled using Mortar elements whereas coupled multiple-time-scale equations are derived for the Generalized-α Method [2]. We show that MGMT Method can be easily extended to incorporate the Generalized-α family of time integration algorithms, hence allowing selective discretization in space and time along with controlled numerical dissipation for distinct grids. We also show that interface energy across connecting sub-domains is identically zero, further assuring global energy balance and continuity of velocities across connecting sub-domains.

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.

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.

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.

Linear Stability Analysis of a Waveboard Multibody Model With a Minimal Set of Equations

2021 ◽  
Author(s):  
A. G. Agúndez ◽  
D. García-Vallejo ◽  
E. Freire ◽  
A. M. Mikkola
Keyword(s):  
Stability Analysis ◽  
Multibody System ◽  
Equations Of Motion ◽  
Nonholonomic Constraints ◽  
Design Parameters ◽  
Multibody Model ◽  
Aspect Ratios ◽  
Holonomic And Nonholonomic Constraints ◽  
The Stability ◽  
Nonlinear Equations Of Motion

Abstract In this paper, the stability of a waveboard, the skateboard consisting in two articulated platforms, coupled by a torsion bar and supported of two caster wheels, is analysed. The waveboard presents an interesting propelling mechanism, since the rider can achieve a forward motion by means of an oscillatory lateral motion of the platforms. The system is described using a multibody model with holonomic and nonholonomic constraints. To perform the stability analysis, the nonlinear equations of motion are linearized with respect to the forward upright motion with constant speed. The linearization is carried out resorting to a novel numerical linearization procedure, recently validated with a well-acknowledged bicycle benchmark, which allows the maximum possible reduction of the linearized equations of motion of multibody systems with holonomic and nonholonomic constraints. The approach allows the expression of the Jacobian matrix in terms of the main design parameters of the multibody system under study. This paper illustrates the use of this linearization approach with a complex multibody system as the waveboard. Furthermore, a sensitivity analysis of the eigenvalues considering different scenarios is performed, and the influence of the forward speed, the casters’ inclination angle and the tori aspect ratios of the toroidal wheels on the stability of the system is analysed.

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