A new method to improve the numerical stability of the hierarchical equations of motion for discrete harmonic oscillator modes

2020 ◽  
Vol 153 (20) ◽  
pp. 204109
Author(s):  
Yaming Yan ◽  
Tao Xing ◽  
Qiang Shi
Keyword(s):  
Harmonic Oscillator ◽  
Numerical Stability ◽  
Equations Of Motion ◽  
New Method

NEW METHOD FOR TREATING RELAXATION PROCESSES: DAMPED HARMONIC OSCILLATOR AND THE JAYNES-CUMMINGS MODEL WITH CAVITY LOSSES

1996 ◽  
Vol 10 (20) ◽  
pp. 981-987
Author(s):  
J. SEKE
Keyword(s):  
Harmonic Oscillator ◽  
Equations Of Motion ◽  
Operator Method ◽  
New Method ◽  
New Technique ◽  
Relaxation Processes ◽  
Total System ◽  
Damped Harmonic Oscillator ◽  
A New Technique ◽  
Reduced Density

In the present paper a new technique for treating relaxation processes, based on the author’s self-consistent projection-operator method, is developed. This new technique, which yields (instead of a master equation for a reduced density operator) equations of motion for probability amplitudes for a reduced set of states of the total system, opens a new way for treating relaxation processes. The applicability of the new method is demonstrated in the case of a damped harmonic oscillator and in that of the Jaynes-Cummings model with cavity losses.

On the constant of motion of dissipative systems

2002 ◽  
Vol 80 (1) ◽  
pp. 1-5 ◽  
Author(s):  
A Patiño ◽  
H Rago
Keyword(s):  
Harmonic Oscillator ◽  
Equations Of Motion ◽  
Viscous Medium ◽  
Dissipative Systems ◽  
Damped Harmonic Oscillator ◽  
Constant Of Motion

We apply results on symmetries of equations of motion and equivalent Lagrangians to obtain a constant of motion for a particle travelling through a viscous medium and for the damped harmonic oscillator. PACS No.: 45.20Jj

A Recursive Householder Transformation for Complex Dynamical Systems With Constraints

1988 ◽  
Vol 55 (3) ◽  
pp. 729-734 ◽  
Author(s):  
F. M. L. Amirouche ◽  
Tongyi Jia ◽  
Sitki K. Ider
Keyword(s):  
Dynamical Systems ◽  
General Solution ◽  
Equations Of Motion ◽  
New Method ◽  
Systems Dynamics ◽  
Complex Dynamical Systems ◽  
Governing Equations ◽  
Householder Transformation ◽  
Constraint Equations ◽  
Computer Automation

A new method is presented by which equations of motion of complex dynamical systems are reduced when subjected to some constraints. The method developed is used when the governing equations are derived using Kane’s equations with undetermined multipliers. The solution vectors of the constraint equations are determined utilizing the recursive Householder transformation to obtain a Pseudo-Uptriangular matrix. The most general solution in terms of new independent coordinates is then formulated. Methods previously used for handling such systems are discussed and the new method advantages are illustrated. The procedures developed are suitable for computer automation and especially in developing generic programs to study a large class of systems dynamics such as robotics, biosystems, and complex mechanisms.

scholarly journals The stability of collocation methods for higher-order Volterra integro-differential equations

2005 ◽  
Vol 2005 (19) ◽  
pp. 3075-3089 ◽  
Author(s):  
Edris Rawashdeh ◽  
Dave McDowell ◽  
Leela Rakesh
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Numerical Stability ◽  
Polynomial Spline ◽  
Higher Order ◽  
Collocation Methods ◽  
New Method ◽  
Spline Collocation ◽  
Spline Collocation Method ◽  
The Stability

The numerical stability of the polynomial spline collocation method for general Volterra integro-differential equation is being considered. The convergence and stability of the new method are given and the efficiency of the new method is illustrated by examples. We also proved the conjecture suggested by Danciu in 1997 on the stability of the polynomial spline collocation method for the higher-order integro-differential equations.

A New Method for the Solution of a Differential Equation with Two-Point Boundary Conditions Applied to the Compressible Boundary Layer on a Yawed Infinite Wing

1956 ◽  
Vol 60 (552) ◽  
pp. 808-809
Author(s):  
L. F. Crabtree ◽  
E.R. Woollett
Keyword(s):  
Differential Equation ◽  
Boundary Layer ◽  
Exact Solution ◽  
Stagnation Point ◽  
Close Agreement ◽  
Equations Of Motion ◽  
New Method ◽  
Compressible Boundary Layer ◽  
Point Boundary ◽  
Direct Solution

The compressible laminar boundary layer on a yawed infinite wing is considered in Ref. 1, where it is shown that the problem may be solved by a direct solution of the linearised equations of motion under certain assumptions. As an example of this procedure the boundary layer near a stagnation point was calculated. Tinkler has published solutions of the exact equations for the general Falkner-Skan case (Ref. 1) obtained on the M.I.T. differential analyser for several values of the parameter involved. It has been found that the numerical results of Ref. 1 were in error and the corrected results obtained by a new method are tabulated below. Tinkler's exact solution of the stagnation point flow for ω = 0·10 is also given for comparison, and it will be seen that there is close agreement

A New Method of Stabilization for Holonomic Constraints

1983 ◽  
Vol 50 (4a) ◽  
pp. 869-870 ◽  
Author(s):  
J. W. Baumgarte
Keyword(s):  
Essential Feature ◽  
Equations Of Motion ◽  
New Method ◽  
Lagrangian Multipliers ◽  
Holonomic Constraints ◽  
Asymptotic Stabilization

A new method for the asymptotic stabilization of holonomic constraints is presented. The essential feature of this approach is the introduction of stabilizing momenta of constraint. The advantage of the method is the fact that to obtain the nonclassical Lagrangian multipliers in the equations of motion, the holonomic constraints need to be differentiated only once with respect to time.

Squeezing caused by a change of mass in a harmonic oscillator

1989 ◽  
Vol 67 (2-3) ◽  
pp. 152-154 ◽  
Author(s):  
Fan Hong-Yi ◽  
H. R. Zaidi
Keyword(s):  
Harmonic Oscillator ◽  
Time Evolution ◽  
Equations Of Motion ◽  
Time Dependent ◽  
Mass Change ◽  
Transformation Time

It is shown that a mass change in a harmonic oscillator generates a squeezing transformation. Time-independent as well as time-dependent transformations are investigated. An expression for the interaction Hamiltonian responsible for squeezing and the equations of motion for the time evolution are derived.

A Generalization of Effective Mass for Selecting Free–Free Target Modes

2006 ◽  
Vol 129 (1) ◽  
pp. 121-127
Author(s):  
Daniel C. Kammer ◽  
Joseph Cessna ◽  
Andrew Kostuch
Keyword(s):  
Effective Mass ◽  
Equations Of Motion ◽  
Ground Vibration ◽  
Singular Values ◽  
Element Model ◽  
New Method ◽  
Mode Shapes ◽  
Vibration Test ◽  
Aerospace Applications ◽  
Elastic Modes

One of the most important tasks in pretest analysis and modal survey planning is the selection of target modes. The target modes are those mode shapes that are determined to be dynamically important using some definition. While there are many measures of dynamic importance, one of the measures that has been of greatest interest to structural dynamicists, is the contribution of each mode to the dynamic loads at an interface. Dynamically important modes contribute significantly to the interface loads and must be retained in any reduced analytical representation. These modes must be identified during a ground vibration test to validate the corresponding finite element model. Structural dynamicists have used interface load based effective mass measures to efficiently identify target modes for constrained structures. The advantage of these measures of dynamic importance is that they are absolute, in contrast to other measures that only indicate the importance of one mode shape relative to another. However, in many situations, especially in aerospace applications, structures must be tested in a free–free configuration. In the case of free–free elastic modes, the effective mass values are zero, making them useless measures of dynamic importance. This paper presents a new effective mass like measure of absolute dynamic importance that can be applied to free–free structures. The new method is derived based upon the free–free modal equations of motion. The approach is shown to be directly related to ranking mode shapes based on approximate balanced singular values. But, unlike the approximate balanced singular value approach, it is an absolute measure of importance. A numerical example of a general spacecraft system is presented to illustrate the application of the new technique. Dynamically important mode shapes were easily identified for modal acceleration, velocity, and displacement output. The new method provides an efficient technique for selecting target modes for a modal vibration test, or the reduction of a modal based analytical model to the dynamically important mode shapes.

Sign in / Sign up

Export Citation Format

Share Document