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.

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.

Equations of motion with the identity mass matrix for holonomic systems

2004 ◽  
Vol 218 (7) ◽  
pp. 731-736
Author(s):  
P Herman
Keyword(s):  
Differential Equations ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Dynamical Equations ◽  
First Order ◽  
Interesting Insight ◽  
Holonomic Systems ◽  
Insight Into ◽  
First Order Differential Equations

Some consequences concerning holonomic systems described in terms of the inertial quasi-velocities (IQV) are discussed in this note. Introducing the IQV vector into Lagrange's formulation leads to first-order equations with the identity mass matrix of the system. The first-order differential equations give an interesting insight into dynamics and some important properties. The two examples that are provided use the dynamical equations in terms of IQV.

On First-Order Decoupling of Equations of Motion for Constrained Dynamical Systems

1995 ◽  
Vol 62 (1) ◽  
pp. 216-222 ◽  
Author(s):  
T. A. Loduha ◽  
B. Ravani
Keyword(s):  
Dynamical Systems ◽  
Equations Of Motion ◽  
Nonholonomic Systems ◽  
Numerical Implementation ◽  
Dynamical Equations ◽  
Generalized Coordinates ◽  
First Order ◽  
Holonomic Systems ◽  
Constrained Dynamical Systems ◽  
Generalized Coordinate

In this paper we present a method for obtaining first-order decoupled equations of motion for multirigid body systems. The inherent flexibility in choosing generalized velocity components as a function of generalized coordinates is used to influence the structure of the resulting dynamical equations. Initially, we describe how a congruency transformation can be formed that represents the transformation between generalized velocity components and generalized coordinate derivatives. It is shown that the proper choice for the congruency transformation will insure generation of first-order decoupled equations of motion for holonomic systems. In the case of nonholonomic systems, or holonomic systems with unreduced configuration coordinates, we incorporate an orthogonal complement in conjunction with the congruency transformation. A pair of examples illustrate the results. Finally, we discuss numerical implementation of congruency transformations to achieve first-order decoupled equations for simulation purposes.

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.

scholarly journals A Type of Time-Symmetric Stochastic System and Related Games

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 118
Author(s):  
Qingfeng Zhu ◽  
Yufeng Shi ◽  
Jiaqiang Wen ◽  
Hui Zhang
Keyword(s):  
Nash Equilibrium ◽  
Time Delay ◽  
Differential Equations ◽  
Equilibrium Point ◽  
Stochastic Differential Equations ◽  
Stochastic System ◽  
Stochastic Systems ◽  
Necessary Condition ◽  
Nash Equilibrium Point ◽  
Doubly Stochastic

This paper is concerned with a type of time-symmetric stochastic system, namely the so-called forward–backward doubly stochastic differential equations (FBDSDEs), in which the forward equations are delayed doubly stochastic differential equations (SDEs) and the backward equations are anticipated backward doubly SDEs. Under some monotonicity assumptions, the existence and uniqueness of measurable solutions to FBDSDEs are obtained. The future development of many processes depends on both their current state and historical state, and these processes can usually be represented by stochastic differential systems with time delay. Therefore, a class of nonzero sum differential game for doubly stochastic systems with time delay is studied in this paper. A necessary condition for the open-loop Nash equilibrium point of the Pontriagin-type maximum principle are established, and a sufficient condition for the Nash equilibrium point is obtained. Furthermore, the above results are applied to the study of nonzero sum differential games for linear quadratic backward doubly stochastic systems with delay. Based on the solution of FBDSDEs, an explicit expression of Nash equilibrium points for such game problems is established.

Disk Position Nonlinearity Effects on the Chaotic Behavior of Rotating Flexible Shaft-Disk Systems

2012 ◽  
Vol 28 (3) ◽  
pp. 513-522 ◽  
Author(s):  
H. M. Khanlo ◽  
M. Ghayour ◽  
S. Ziaei-Rad
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamic Behavior ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Disk System ◽  
Partial Differential ◽  
Rayleigh Beam ◽  
Flexible Shaft

AbstractThis study investigates the effects of disk position nonlinearities on the nonlinear dynamic behavior of a rotating flexible shaft-disk system. Displacement of the disk on the shaft causes certain nonlinear terms which appears in the equations of motion, which can in turn affect the dynamic behavior of the system. The system is modeled as a continuous shaft with a rigid disk in different locations. Also, the disk gyroscopic moment is considered. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed modes method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work are inclusive of time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The effect of disk nonlinearities is studied for some disk positions. The results confirm that when the disk is located at mid-span of the shaft, only the regular motion (period one) is observed. However, periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for situations in which the disk is located at places other than the middle of the shaft. The results show nonlinear effects are negligible in some cases.

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