scholarly journals Numerical Solution of High-Dimensional Nonlinear Rotor-Bearing Dynamic System with Precise Integration

2014 ◽  
Vol 8 (1) ◽  
pp. 264-269
Author(s):  
Guangtian Shi
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Solution ◽  
High Precision ◽  
Integration Method ◽  
Rotor System ◽  
Degree Of Freedom ◽  
High Dimensional ◽  
Precise Integration ◽  
Precise Integration Method ◽  
Rotor Bearing

Through an example of rotor system which has multi-degree of freedom mounted on the nonlinear fluid film bearings, this paper analyzes the precise integration algorithm, a new numerical solution for high–dimensional nonlinear dynamics system. The precise integration method has advantages of long step, high precision and absolute stability for solving nonlinear dynamics equations. To make good use of the method, firstly, the precise integral iterative formula is given and then the mechanism of controlling high precision and efficiency is discussed. The evolution of precise integration method is an algorithm with explicit, simple form, self-start, and fast to solve nonlinear dynamics equations. High power of athwart of Hamiltonian matrix is not needed, so it is convenient in this case. The stability of period response of nonlinear rotor-bearing system is analyzed by employing the precise integration method. The bifurcation rules of the period response of the elastic rotor system with multi-degree of freedom are obtained and the chaos of the system is determined according to the fractal dimension of Poincare mapping of phase space at a certain speed.

An Innovative Precise Integration Method in Solving Structural Dynamic Problems

2013 ◽  
Author(s):  
Chiun-lin Wu ◽  
Ching-Chiang Chuang
Keyword(s):  
Exact Solution ◽  
High Precision ◽  
Numerical Stability ◽  
Integration Method ◽  
Time Integration ◽  
Computer Hardware ◽  
Integration Algorithm ◽  
Precise Integration ◽  
Precise Integration Method ◽  
Stability And Accuracy

An innovative time integration method that incorporates spurious high-frequency dissipation capability into the so called “high precision direct integration algorithm” is presented, and its numerical stability and accuracy is discussed. The integration algorithm is named “high precision” to emphasize its numerical capability in reaching computer hardware precision. The proposed procedure employs the well-known state space approach to solve the simultaneous ordinary differential equations, the exact solution of which contains an exponential matrix to be efficiently computed using the truncated Taylor series expansion together with the power-of-two algorithm. The proposed method, belonging to the category of explicit methods, is found to provide better accuracy than many other existing time integration methods, and the integration scheme remains numerically stable over a wide range of frequencies of engineering interest. This paper is also devoted to study numerical accuracy of the Precise Integration Method in solving forced vibration problems, particularly near resonance conditions. The numerically obtained transfer functions are then compared with the analytical exact solution to detect spurious resonance. Finally, numerical examples are used to illustrate its high performance in numerical stability and accuracy. The proposed method carries the merit that can be directly applied to solve momentum equations of motion with exactly the same procedure.

scholarly journals Precise Integration Method for Solving Noncooperative LQ Differential Game

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Hai-Jun Peng ◽  
Sheng Zhang ◽  
Zhi-Gang Wu ◽  
Biao-Song Chen
Keyword(s):  
Differential Equation ◽  
Differential Game ◽  
Integration Method ◽  
Transformation Method ◽  
Linear Quadratic ◽  
Riccati Differential Equation ◽  
Precise Integration ◽  
Precise Integration Method ◽  
Direct Integration Method ◽  
Two Parameters

The key of solving the noncooperative linear quadratic (LQ) differential game is to solve the coupled matrix Riccati differential equation. The precise integration method based on the adaptive choosing of the two parameters is expanded from the traditional symmetric Riccati differential equation to the coupled asymmetric Riccati differential equation in this paper. The proposed expanded precise integration method can overcome the difficulty of the singularity point and the ill-conditioned matrix in the solving of coupled asymmetric Riccati differential equation. The numerical examples show that the expanded precise integration method gives more stable and accurate numerical results than the “direct integration method” and the “linear transformation method”.

Dynamic Behavior Analysis of Three-Span Rotor-Bearing System with Rub-Impact Fault

2012 ◽  
Vol 460 ◽  
pp. 160-164 ◽  
Author(s):  
Song He Zhang ◽  
Yue Gang Luo ◽  
Bin Wu ◽  
Bang Chun Wen
Keyword(s):  
Nonlinear Dynamics ◽  
Behavior Analysis ◽  
Dynamic Model ◽  
Dynamic Behavior ◽  
Rotor System ◽  
System Response ◽  
Rotor Bearing ◽  
Set Up ◽  
Bearing System ◽  
Main Influence

The dynamic model of the three-span rotor-bearing system with rub-impact fault was set up. The influence to nonlinear dynamics behaviors of the rotor-bearing system that induced by rub-impact of one disc, two discs and three discs were numerically studied. The main influence of the rotor system response by the rub-impact faults are in the supercritical rotate speed. There are mutations of amplitudes in the responses of second and third spans in supercritical rotate speed when rub-impact with one disc, and there are chaotic windows in the response of first span, and jumping changes in second and third spans when rub-impact with two or three discs.

Extension of the Wittrick-Williams Algorithm to Mixed Variable Systems

1997 ◽  
Vol 119 (3) ◽  
pp. 334-340 ◽  
Author(s):  
Zhong Wanxie ◽  
F. W. Williams ◽  
P. N. Bennett
Keyword(s):  
Timoshenko Beam ◽  
Stiffness Matrix ◽  
Integration Method ◽  
Dynamic Stiffness ◽  
Integration Algorithm ◽  
Precise Integration ◽  
Matrix Computations ◽  
Precise Integration Method ◽  
Count Method ◽  
Mixed Variable

A precise integration algorithm has recently been proposed by Zhong (1994) for dynamic stiffness matrix computations, but he did not give a corresponding eigenvalue count method. The Wittrick-Williams algorithm gives an eigenvalue count method for pure displacement formulations, but the precise integration method uses a mixed variable formulation. Therefore the Wittrick-Williams method is extended in this paper to give the eigenvalue count needed by the precise integration method and by other methods involving mixed variable formulations. A simple Timoshenko beam example is included.

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