Quantum Mechanics and a Completely Integrable Dynamical System

1987 ◽  
Vol 42 (8) ◽  
pp. 819-824 ◽  
Author(s):  
W.-H. Steeb ◽  
A. J. van Tonder
Keyword(s):  
Differential Equations ◽  
Symmetric Matrix ◽  
Energy Levels ◽  
Equations Of Motion ◽  
Integrable Dynamical System ◽  
Finite Dimensional ◽  
Open Questions ◽  
The Matrix ◽  
Constants Of Motion ◽  
Energy Dependent

From the eigenvalue equation (H0 + λV) \ ψn(λ)) = En (λ) | ψn( λ ) ) one can derive an autonomous system of first order differential equations for the eigenvalues En (λ) and the matrix elements Vmn(X) = ( ψm(λ) \ V \ ψn(λ), where λ is the independent variable. If the initial values En (λ = 0) and ψn (λ = 0) are known the differential equations can be solved. Thus one finds the “motion” of the energy levels En(λ). Here we give two applications of this technique. Furthermore we describe the connection with the stationary state perturbation theory. We also derive the equations of motion for the extended case H = H0 + λ\ Vt + λ2 V2. Finally we investigate the case where the Hamiltonian is given by a finite dimensional symmetric matrix and derive the energy dependent constants of motion. Several open questions are also discussed.

Dynamic Behavior of Spatial Linkages: Part 1—Exact Equations of Motion, Part 2—Small Oscillations About Equilibrium

1969 ◽  
Vol 91 (1) ◽  
pp. 251-265 ◽  
Author(s):  
J. J. Uicker
Keyword(s):  
Differential Equations ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Dynamic Force ◽  
Lagrange Equations ◽  
Oscillation Analysis ◽  
Restricted Problems ◽  
The Matrix ◽  
Laplace Transformations ◽  
External Forces

Part 1: Over the past several years, the matrix method of linkage analysis has been developed to give the kinematic, static and dynamic force, error, and equilibrium analyses of three-dimensional mechanical linkages. This two-part paper is an extension of these methods to include some aspects of dynamic analysis. In Part 1, expressions are developed for the kinetic and potential energies of a system consisting of a multiloop, multi-degree-of-freedom spatial linkage having springs and damping devices in any or all of its joints, and under the influence of gravity as well as time varying external forces. Using the Lagrange equations, the exact differential equations governing the motion of such a system are derived. Although these equations cannot be solved directly, they form the basis for the solution of more restricted problems, such as a linearized small oscillation analysis which forms Part 2 of the paper. Part 2: This paper is a direct extension of Part 1 and it is assumed that the reader has a thorough knowledge of the previous material. Assuming that the spatial linkage has a stable position of static equilibrium and oscillates with small displacements and small velocities about this position, the general differential equations of motion are linearized to describe these oscillations. The equations lead to an eigenvalue problem which yields the resonant frequencies and associated damping constants of the system for the equilibrium position. Laplace transformations are then used to solve the linearized equations. Digital computer programs have been written to lest these methods and an example solution dealing with a vehicle suspension is presented.

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.

Dynamics of Rolling-Element Bearings—Part I: Cylindrical Roller Bearing Analysis

1979 ◽  
Vol 101 (3) ◽  
pp. 293-302 ◽  
Author(s):  
P. K. Gupta
Keyword(s):  
Differential Equations ◽  
Normal Force ◽  
Equations Of Motion ◽  
Roller Bearing ◽  
Rolling Element Bearings ◽  
Analytical Formulation ◽  
Rolling Element ◽  
Cylindrical Roller Bearing ◽  
Cylindrical Roller ◽  
Bearing Analysis

An analytical formulation for the roller motion in a cylindrical roller bearing is presented in terms of the classical differential equations of motion. Roller-race interaction is analyzed in detail and the resulting normal force and moment vectors are determined. Elastohydrodynamic traction models are considered in determining the roller-race tractive forces and moments. Formulation for the roller end and race flange interaction during skewing of the roller is also considered. Roller-cage interactions are assumed to be either hydrodynamic or fully metallic. Simple relationships are used to determine the churning and drag losses.

Mulit-Pulse Orbits and Chaotic Motion of Composite Laminated Plates

2008 ◽  
Author(s):  
Xiangying Guo ◽  
Wei Zhang ◽  
Ming-Hui Yao
Keyword(s):  
Numerical Simulation ◽  
Normal Form ◽  
Differential Equations ◽  
Thin Plate ◽  
Equations Of Motion ◽  
Laminated Plates ◽  
Phase Method ◽  
Rectangular Thin Plate ◽  
Partial Differential ◽  
Chaotic Motions

This paper presents an analysis on the nonlinear dynamics and multi-pulse chaotic motions of a simply-supported symmetric cross-ply composite laminated rectangular thin plate with the parametric and forcing excitations. Firstly, based on the Reddy’s three-order shear deformation plate theory and the model of the von Karman type geometric nonlinearity, the nonlinear governing partial differential equations of motion for the composite laminated rectangular thin plate are derived by using the Hamilton’s principle. Then, using the second-order Galerkin discretization approach, the partial differential governing equations of motion are transformed to nonlinear ordinary differential equations. The case of the primary parametric resonance and 1:1 internal resonance is considered. Four-dimensional averaged equation is obtained by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is used to give the explicit expressions of normal form. Based on normal form, the energy phase method is utilized to analyze the global bifurcations and multi-pulse chaotic dynamics of the composite laminated rectangular thin plate. The results obtained above illustrate the existence of the chaos for the Smale horseshoe sense in a parametrical and forcing excited composite laminated thin plate. The chaotic motions of the composite laminated rectangular thin plate are also found by using numerical simulation. The results of numerical simulation also indicate that there exist different shapes of the multi-pulse chaotic motions for the composite laminated rectangular thin plate.

PATH INTEGRAL QUANTIZATION OF SUPERSTRING

2010 ◽  
Vol 25 (02) ◽  
pp. 135-141
Author(s):  
H. A. ELEGLA ◽  
N. I. FARAHAT
Keyword(s):  
Phase Space ◽  
Differential Equations ◽  
Path Integral ◽  
Equations Of Motion ◽  
Singular System ◽  
Constrained Systems ◽  
Classical Structure ◽  
Path Integral Quantization ◽  
Total Differential

Motivated by the Hamilton–Jacobi approach of constrained systems, we analyze the classical structure of a four-dimensional superstring. The equations of motion for a singular system are obtained as total differential equations in many variables. The path integral quantization based on Hamilton–Jacobi approach is applied to quantize the system, and the integration is taken over the canonical phase space coordinates.

Concise Equations for Rotor Dynamics Analysis

1995 ◽  
Author(s):  
W. J. Chen
Keyword(s):  
Equations Of Motion ◽  
Rotor Dynamics ◽  
Memory Storage ◽  
The Other ◽  
Dynamics Analysis ◽  
Real Domain ◽  
The Matrix ◽  
System Equations ◽  
Ordering Algorithms ◽  
Matrix Bandwidth

Abstract Concise equations for rotor dynamics analysis are presented. Two coordinate ordering methods are introduced in the element equations of motion. One is in the real domain and the other is in the complex domain. The two proposed ordering algorithms lead to more compact element matrices. A station numbering technique is also proposed for the system equations during the assembly process. This numbering technique can minimize the matrix bandwidth, the memory storage and can increase the computational efficiency.

Flexible Multibody Dynamics Formulation by Hamilton’s Equations

2010 ◽  
Author(s):  
Martin M. Tong
Keyword(s):  
Multibody Dynamics ◽  
Multibody System ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Flexible Multibody Dynamics ◽  
Flexible Body ◽  
Flexible Multibody System ◽  
Generalized Coordinates ◽  
Flexible Multibody ◽  
The Matrix

Numerical solution of the dynamics equations of a flexible multibody system as represented by Hamilton’s canonical equations requires that its generalized velocities q˙ be solved from the generalized momenta p. The relation between them is p = J(q)q˙, where J is the system mass matrix and q is the generalized coordinates. This paper presents the dynamics equations for a generic flexible multibody system as represented by p˙ and gives emphasis to a systematic way of constructing the matrix J for solving q˙. The mass matrix is shown to be separable into four submatrices Jrr, Jrf, Jfr and Jff relating the joint momenta and flexible body mementa to the joint coordinate rates and the flexible body deformation coordinate rates. Explicit formulas are given for these submatrices. The equations of motion presented here lend insight to the structure of the flexible multibody dynamics equations. They are also a versatile alternative to the acceleration-based dynamics equations for modeling mechanical systems.

Sign in / Sign up

Export Citation Format

Share Document