The application of Chebyshev polynomials to the solution of the nonprismatic Timoshenko beam vibration problem

2006 ◽  
Vol 296 (1-2) ◽  
pp. 243-263 ◽  
Author(s):  
P. Ruta
Keyword(s):  
Timoshenko Beam ◽  
Chebyshev Polynomials ◽  
Beam Vibration ◽  
Vibration Problem

COUPLED TIMOSHENKO BEAM VIBRATION EQUATIONS FOR FREE SYMMETRIC BODIES

1996 ◽  
Vol 195 (2) ◽  
pp. 348-352 ◽  
Author(s):  
A.N. Kathnelson
Keyword(s):  
Timoshenko Beam ◽  
Beam Vibration

Analysis of the Natural Vibration Properties of a Timoshenko Beam Supported by Springs (by Finite Integral Transform Method)

1999 ◽  
Author(s):  
Zhixiang Xu ◽  
Hideyuki Tamura ◽  
Kunisato Seto
Keyword(s):  
Timoshenko Beam ◽  
Natural Vibration ◽  
Natural Frequencies ◽  
Integral Transform ◽  
Frequency Equation ◽  
Transform Method ◽  
Vibration Problem ◽  
Vibration Properties ◽  
Finite Integral ◽  
Integral Transform Technique

Abstract This paper presents analytical results of transverse vibration of a Timoshenko beam supported by spring-spring which stiffness is variable, that is a simplified model of magnetically levitated vehicle body’s vibration problem and magnetic bearings support shaft’s vibration problem. By applying the finite integral transform technique, the analytical solution of this dynamic model is successfully obtained. Especially, by investigating the frequency equation, the effect of the stiffness of two supporting-springs to the natural frequencies is clarified. From the results, it is cleared that the natural frequencies of the beam system can be effectively controlled by changing the supporting-spring’s stiffness.

Teaching Vibration Analysis Using Symbolic Computation Software

1998 ◽  
Vol 26 (2) ◽  
pp. 75-88
Author(s):  
Hemanshu R. Pota ◽  
Thomas E. Alberts
Keyword(s):  
Boundary Conditions ◽  
Symbolic Computation ◽  
Vibration Analysis ◽  
Complete Solution ◽  
Linear Equations ◽  
Flexible Systems ◽  
Beam Vibration ◽  
Different Boundary Conditions ◽  
Vibration Problem ◽  
Forcing Functions

This paper presents a general approach to modelling and learning vibration analysis for simple beams using symbolic computation software. The emphasis here is on the fact that a complete solution of the beam vibrations problem, for different boundary conditions and arbitrary forcing functions, is made very simple by using symbolic computation software. The heart of the procedure is to convert the beam vibration problem into a system of simultaneous algebraic linear equations and then use symbolic computation software to solve it. The analysis here also shows how to obtain models suitable to design controllers for flexible systems.

Three-Dimensional Dynamics Analysis of Rotating Functionally Gradient Beams Based on Timoshenko Beam Theory

2019 ◽  
Vol 11 (04) ◽  
pp. 1950040 ◽  
Author(s):  
Ding Zhou ◽  
Jianshi Fang ◽  
Hongwei Wang ◽  
Xiaopeng Zhang
Keyword(s):  
Timoshenko Beam ◽  
Chebyshev Polynomials ◽  
Slenderness Ratio ◽  
Ritz Method ◽  
Three Dimensional ◽  
Beam Theory ◽  
Functionally Graded ◽  
Timoshenko Beam Theory ◽  
Strengthening Effect ◽  
Graded Materials

Through the Timoshenko beam theory (TBT), the 3D dynamics of a rotary functional gradient (FG) cantilever beam are investigated. Material capabilities alter continuously throughout the thickness obeying the power law. It is assumed that the Poisson’s ratio does not change. Based on the von Kármán nonlinearity, the governing equation is determined through the Hamilton principle, which includes the Coriolis effects. The couplings among the axial, flapwise and chordwise deformations caused by the usage of the functionally graded materials (FGMs) are revealed. Chebyshev polynomials are utilized to construct trial functions of deformations in the Rayleigh–Ritz method. The centrifugal strengthening effect caused by the rotational motion is described through the nonlinear axial shortening deformations derived from transverse deformations. The influences of the dimensionless angular velocity, FG index and slenderness ratio on vibration characteristics are studied. It is proved that the FG index significantly affects the dynamic response of deformation. For high-frequency external excitation cases, selection of Chebyshev polynomials as trial functions is more stable and effective than other polynomials.

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