Multiparameter Perturbation Solution of Algebraic Equations

1966 ◽  
Vol 33 (1) ◽  
pp. 218-219 ◽  
Author(s):  
W. F. Ames ◽  
J. F. Sontowski
Keyword(s):  
Power Series ◽  
Perturbation Method ◽  
Algebraic Equation ◽  
Timoshenko Beam ◽  
Perturbation Solution ◽  
Beam Equation ◽  
Algebraic Equations ◽  
Timoshenko Beam Equation

The classical perturbation method—the expansion of a solution of an algebraic equation as a power series in a parameter—is extended to an expansion in several parameters. An example concerning the Timoshenko beam equation is used to illustrate the ideas. Advantages of the procedure are discussed in the light of this example.

scholarly journals Modal Perturbation Method for the Dynamic Characteristics of Timoshenko Beams

2005 ◽  
Vol 12 (6) ◽  
pp. 425-434 ◽  
Author(s):  
Menglin Lou ◽  
Qiuhua Duan ◽  
Genda Chen
Keyword(s):  
Perturbation Method ◽  
Dynamic Characteristics ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Solution Process ◽  
Equation Of Motion ◽  
Timoshenko Beams ◽  
Algebraic Equations ◽  
Nonlinear Algebraic Equations ◽  
Shear Distortion

Timoshenko beams have been widely used in structural and mechanical systems. Under dynamic loading, the analytical solution of a Timoshenko beam is often difficult to obtain due to the complexity involved in the equation of motion. In this paper, a modal perturbation method is introduced to approximately determine the dynamic characteristics of a Timoshenko beam. In this approach, the differential equation of motion describing the dynamic behavior of the Timoshenko beam can be transformed into a set of nonlinear algebraic equations. Therefore, the solution process can be simplified significantly for the Timoshenko beam with arbitrary boundaries. Several examples are given to illustrate the application of the proposed method. Numerical results have shown that the modal perturbation method is effective in determining the modal characteristics of Timoshenko beams with high accuracy. The effects of shear distortion and moment of inertia on the natural frequencies of Timoshenko beams are discussed in detail.

Response of a Simply Supported Timoshenko Beam to a Purely Random Gaussian Process

1958 ◽  
Vol 25 (4) ◽  
pp. 496-500
Author(s):  
J. C. Samuels ◽  
A. C. Eringen
Keyword(s):  
Timoshenko Beam ◽  
Beam Theory ◽  
Beam Equation ◽  
Mean Square ◽  
Bending Stress ◽  
Random Loading ◽  
Simply Supported ◽  
The Mean ◽  
Timoshenko Beam Equation ◽  
Classical Beam Theory

Abstract The generalized Fourier analysis is applied to the damped Timoshenko beam equation to calculate the mean-square values of displacements and bending stress, resulting from purely random loading. Compared with the calculations, based on the classical beam theory, it was found that the displacement correlations of both theories were in excellent agreement. Moreover, the mean square of the bending stress, contrary to the results of the classical beam theory, was found to be convergent. Computations carried out with a digital computer are plotted for both theories.

Some Solutions of the Timoshenko Beam Equation for Short Pulse-Type Loading

1958 ◽  
Vol 25 (3) ◽  
pp. 379-385
Author(s):  
H. J. Plass
Keyword(s):  
Timoshenko Beam ◽  
Short Pulse ◽  
Sine Wave ◽  
Beam Equation ◽  
Transform Method ◽  
The Laplace Transform ◽  
Support Conditions ◽  
Time Required ◽  
The Impact ◽  
Timoshenko Beam Equation

Abstract A collection of solutions to the Timoshenko beam equation is presented. Various types of support conditions and impact conditions are included. In every case the impact is assumed to be a pulse in the form of a half-sine wave. The results were found numerically, using the method of characteristics, except for one case, which was done in addition by the Laplace transform method, for check purposes. Agreement with experiment is good except for a pulse of duration comparable to the time required for the bending-type wave to travel a distance of one diameter. Discussion is included of the differences among the various cases studied.

scholarly journals Flexural vibration property of periodic pipe system conveying fluid based on Timoshenko beam equation

2009 ◽  
Vol 58 (12) ◽  
pp. 8357 ◽  
Author(s):  
Shen Hui-Jie ◽  
Wen Ji-Hong ◽  
Yu Dian-Long ◽  
Wen Xi-Sen
Keyword(s):  
Timoshenko Beam ◽  
Flexural Vibration ◽  
Beam Equation ◽  
Pipe System ◽  
Timoshenko Beam Equation ◽  
Conveying Fluid
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