Integrated equations of motion for direct integration methods

2002 ◽  
Vol 13 (5) ◽  
pp. 569-589 ◽  
Author(s):  
Shuenn-Yih Chang
Keyword(s):  
Equations Of Motion ◽  
Direct Integration ◽  
Integration Methods

Comportement dynamique des lignes aériennes de transport d'électricité dû aux bris de câbles. II. Problèmes numériques associés à la modélisation mathématique

1989 ◽  
Vol 16 (3) ◽  
pp. 354-374 ◽  
Author(s):  
Ghyslaine McClure ◽  
René Tinawi
Keyword(s):  
Transmission Lines ◽  
Nonlinear Dynamic ◽  
Numerical Models ◽  
Equations Of Motion ◽  
Nonlinear Dynamic Analysis ◽  
Small Scale ◽  
Convergence Criterion ◽  
Direct Integration ◽  
Time Step ◽  
Integration Methods

The evaluation of the response of an aerial power line section subjected to cable breakage is a complex dynamic problem in which geometric nonlinearities are important. The solution of the equations of motion of the model calls for direct integration methods for which the stability and behavior, in nonlinear situations, are difficult to predict. Seven numerical models are analyzed with the Wilson-θ and the Newmark-β algorithms and results are compared with those of small-scale tests of the American Electric Power Research Institute. This comparison emphasizes the importance of the high frequency modes in the response and the artificial damping induced by the Wilson-θ method. This method allows up to three times the time increment required by the trapezoidal rule that does not filter the contribution of higher frequencies but is limited by a convergence criterion. Many observations made in this particular study are also applicable to multi-degree-of-freedom nonlinear dynamic problems using direct time-step integration. Key words: Nonlinear dynamic analysis, direct integration methods, aerial electric transmission lines, cable breakage simulation.

An Improved Transfer Matrix-Direct Integration Method for Rotor Dynamics

1986 ◽  
Vol 108 (2) ◽  
pp. 182-188 ◽  
Author(s):  
Jialiu Gu
Keyword(s):  
Transfer Matrix ◽  
Integration Method ◽  
Equations Of Motion ◽  
Rotor Dynamics ◽  
Direct Integration ◽  
Rotating System ◽  
Direct Integration Method ◽  
Mass Moment ◽  
Mass Moment Of Inertia ◽  
Matrix Iteration

A transfer matrix-direct integration combined method is proposed, which employs the transfer matrix method to derive the equations of motion of a “characteristic disk,” and uses the direct integration method to determine the critical speeds, modes and unbalance response of a rotor-bearing system, and to analyze its stability. Despite the complexity of the system, the number of governing equations is not greater than eight. For a single-spool rotating system, the number of equations is only four. A transfer matrix for a uniform shaft is derived to consider its distributed mass, moment of inertia and the effect of shearing force. An impedance matrix iteration method is proposed to consider the effect of a complicated bearing-supporting system on the rotor dynamics. Two examples are given, and the results agree satisfactorily with the experiments.

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