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.

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”.

Stable and Accurate Computation of Dispersion Relations for Layered Waveguides, Semi-Infinite Spaces and Infinite Spaces

2019 ◽  
Vol 141 (3) ◽  
Author(s):  
Q. Gao ◽  
Y. H. Zhang
Keyword(s):  
Guided Waves ◽  
Integration Method ◽  
Dynamic Stiffness ◽  
Dispersion Relations ◽  
Energy Matrix ◽  
Counting Method ◽  
Precise Integration ◽  
Transcendental Equations ◽  
Different Levels ◽  
Methods Numerical

This paper studies the dispersion characteristics of guided waves in layered finite media, surface waves in layered semi-infinite spaces, and Stoneley waves in layered infinite spaces. Using the precise integration method (PIM) and the Wittrick–Williams (W-W) algorithm, three methods that are based on the dynamic stiffness matrix, symplectic transfer matrix, and mixed energy matrix are developed to compute the dispersion relations. The dispersion relations in layered media can be reduced to a standard eigenvalue problem of ordinary differential equations (ODEs) in the frequency-wavenumber domain. The PIM is used to accurately solve the ODEs with two-point boundary conditions, and all of the eigenvalues are determined by using the eigenvalue counting method. The proposed methods overcome the difficulty of seeking roots from nonlinear transcendental equations. In theory, the three proposed methods are interconnected and can be transformed into each other, but a numerical example indicates that the three methods have different levels of numerical stability and that the method based on the mixed energy matrix is more stable than the other two methods. Numerical examples show that the method based on the mixed energy matrix is accurate and effective for cases of waves in layered finite media, layered semi-infinite spaces, and layered infinite spaces.

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