scholarly journals A form of equations of motion of a mechanical system in quasi-coordinates

2000 ◽  
Vol 21 (1) ◽  
pp. 45-56
Author(s):  
Do Sanh
Keyword(s):  
Differential Equations ◽  
Mechanical System ◽  
Equations Of Motion ◽  
Simple Algorithm ◽  
Nonholonomic Systems ◽  
Closed Set ◽  
Algebraic Differential Equations ◽  
Reaction Forces ◽  
Holonomic And Nonholonomic Systems

In [3, 4, 5] the form of equations of motion in holonomic coordinates has constructed. The equations obtained give an effective tool for investigating complicated systems. In the present paper the form of equations of motion is written in quasi-coordinates. These equations are solved with respect to quasi-accelerations, which allow to define the motion of a holonomic and nonholonomic systems by a closed set of algebraic – differential equations. The reaction forces of constraints imposed on the system under consideration are calculated by means of a simple algorithm. For illustrating the effectiveness of this form of equations an example is considered.

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. 

Unified Approach for Holonomic and Nonholonomic Systems Based on the Modified Hamilton’s Principle

2009 ◽  
Author(s):  
Keisuke Kamiya ◽  
Junya Morita ◽  
Yutaka Mizoguchi ◽  
Tatsuya Matsunaga
Keyword(s):  
Equations Of Motion ◽  
Time Derivative ◽  
Nonholonomic Systems ◽  
Hamilton’S Principle ◽  
Virtual Power ◽  
Unified Approach ◽  
Hamilton's Principle ◽  
Basic Principles ◽  
Principle Of Virtual Power ◽  
Holonomic And Nonholonomic Systems

As basic principles for deriving the equations of motion for dynamical systems, there are d’Alembert’s principle and the principle of virtual power. From the former Hamilton’s principle and Langage’s equations are derived, which are powerful tool for deriving the equation of motion of mechanical systems since they can give the equations of motion from the scalar energy quantities. When Hamilton’s principle is applied to nonholonomic systems, however, care has to be taken. In this paper, a unified approach for holonomic and nonholonomic systems is discussed based on the modified Hamilton’s principle. In the present approach, constraints for both of the holonomic and nonholonomic systems are expressed in terms of time derivative of the position, and their variations are treated similarly to the principle of virtual power, i.e. time and position are fixed in operation with respect to the variations. The approach is applied to a holonomic and a simple nonholonomic systems.

A Variationally Consistent Approach to Constrained Motion

2016 ◽  
Vol 83 (5) ◽  
Author(s):  
John T. Foster
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
A Priori ◽  
Nonholonomic Systems ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Constrained Motion ◽  
Consistent Approach ◽  
Holonomic And Nonholonomic Systems ◽  
D'alembert's Principle

A variationally consistent approach to constrained rigid-body motion is presented that extends D'Alembert's principle in a way that has a form similar to Kane's equations. The method results in minimal equations of motion for both holonomic and nonholonomic systems without a priori consideration of preferential coordinates.

Methods for Automated Dynamic Analysis of Discrete Mechanical Systems

1973 ◽  
Vol 40 (3) ◽  
pp. 809-811 ◽  
Author(s):  
Y. O. Bayazitoglu ◽  
M. A. Chace
Keyword(s):  
Differential Equations ◽  
Dynamic Analysis ◽  
Ordinary Differential Equations ◽  
Mechanical System ◽  
Dynamic Systems ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Constrained Systems ◽  
Lower Pair ◽  
Linear And Nonlinear

The equations of motion for any discrete, lower pair mechanical system can be obtained by analyzing a branched, three-dimensional compound pendulum of indefinite length. In this paper, a set of expressions which provides the equations of motion of arbitrary mechanical dynamic systems directly as ordinary differential equations are presented. These expressions and the associated technique is applicable to linear and nonlinear unconstrained dynamic systems, kinematic systems and multidegree-of-freedom constrained systems.

scholarly journals Analysis of the brachistochronic motion of a variable mass nonholonomic mechanical system

2016 ◽  
Vol 43 (1) ◽  
pp. 19-32 ◽  
Author(s):  
Bojan Jeremic ◽  
Radoslav Radulovic ◽  
Aleksandar Obradovic
Keyword(s):  
Differential Equations ◽  
Mechanical System ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Variable Mass ◽  
Nonholonomic Constraints ◽  
Initial Position ◽  
Mass Particle ◽  
Control Forces ◽  
And Control

The paper considers the brachistochronic motion of a variable mass nonholonomic mechanical system [3] in a horizontal plane, between two specified positions. Variable mass particles are interconnected by a lightweight mechanism of the ?pitchfork? type. The law of the time-rate of mass variation of the particles, as well as relative velocities of the expelled particles, as a function of time, are known. Differential equations of motion, where the reactions of nonholonomic constraints and control forces figure, are created based on the general theorems of dynamics of a variable mass mechanical system [5]. The formulated brachistochrone problem, with adequately chosen quantities of state, is solved, in this case, as the simplest task of optimal control by applying Pontryagin?s maximum principle [1]. A corresponding two-point boundary value problem (TPBVP) of the system of ordinary nonlinear differential equations is obtained, which, in a general case, has to be numerically solved [2]. On the basis of thus obtained brachistochronic motion, the active control forces, along with the reactions of nonholonomic constraints, are determined. The analysis of the brachistochronic motion for different values of the initial position of a variable mass particle B is presented. Also, the interval of values of the initial position of a variable mass particle B, for which there are the TPBVP solutions, is determined.

scholarly journals Noether Theorem for Nonholonomic Systems with Time Delay

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Shi-Xin Jin ◽  
Yi Zhang
Keyword(s):  
Time Delay ◽  
Differential Equations ◽  
Lagrange Multiplier ◽  
Equations Of Motion ◽  
Nonholonomic Systems ◽  
Noether Theorem ◽  
Nonconservative Systems ◽  
Hamilton Principle ◽  
Multiplier Rules ◽  
Holonomic Systems

The paper focuses on studying the Noether theorem for nonholonomic systems with time delay. Firstly, the differential equations of motion for nonholonomic systems with time delay are established, which is based on the Hamilton principle with time delay and the Lagrange multiplier rules. Secondly, based upon the generalized quasi-symmetric transformations for nonconservative systems with time delay, the Noether theorem for corresponding holonomic systems is given. Finally, we obtain the Noether theorem for the nonholonomic nonconservative systems with time delay. At the end of the paper, an example is given to illustrate the application of the results.

Multibody Dynamics: A Formulation Using Kane’s Method and Dual Vectors

1993 ◽  
Vol 115 (4) ◽  
pp. 833-838 ◽  
Author(s):  
S. K. Agrawal
Keyword(s):  
Multibody Dynamics ◽  
Multibody Systems ◽  
Nonholonomic System ◽  
Equations Of Motion ◽  
Nonholonomic Systems ◽  
Dynamic Equations ◽  
Kane’S Method ◽  
Kane's Method ◽  
Holonomic Systems ◽  
Holonomic And Nonholonomic Systems

This paper proposes a formulation based on Kane’s method to form the dynamic equations of motion of multibody systems using dual vectors. Both holonomic and nonholonomic systems are considered in this formulation. An example of a holonomic and a nonholonomic system is worked out in detail using this formulation. This algorithm is shown to be advantageous for a class of holonomic systems which has cylindrical joints.

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