Dynamic Stiffness Formulation and Free Vibration Analysis of a Moving Timoshenko Beam

2008 ◽  
Author(s):  
J. Ranjan Banerjee ◽  
Huijuan Su ◽  
W Gunawardana
Keyword(s):  
Free Vibration ◽  
Timoshenko Beam ◽  
Vibration Analysis ◽  
Dynamic Stiffness ◽  
Free Vibration Analysis

An Exact Solution to the Free Vibration Analysis of a Uniform Timoshenko Beam Using an Analytical Approach

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Exact Solution ◽  
Free Vibration ◽  
Closed Form ◽  
Timoshenko Beam ◽  
Vibration Analysis ◽  
Analytical Approach ◽  
Dynamic Stiffness ◽  
Free Vibration Analysis ◽  
Winkler Foundation ◽  
Mass System

This study presents an exact solution to the free vibration analysis of a uniform Timoshenko beam using an analytical approach, a harmonic vibration being assumed. The Timoshenko beam theory covers cases associated with small deflections based on shear deformation and rotary inertia considerations. In this paper, a moment-shear force-circular frequency-curvature relationship was presented. The complete study was based on this relationship and closed-form expressions of efforts and deformations were derived. The free vibration response of single-span systems, as well as that of spring-mass systems, was analyzed; closed-form formulations of matrices expressing the boundary conditions were presented and the natural frequencies were determined by solving the eigenvalue problem. Systems with intermediate mass, spring, or spring-mass system were also analyzed. Furthermore, first-order dynamic stiffness matrices in local coordinates were derived. Finally, second-order analysis of beams resting on an elastic Winkler foundation was conducted. The results obtained in this paper were in good agreement with those of other studies.

scholarly journals An Exact Solution to the Free Vibration Analysis of a Uniform Timoshenko Beam Using an Analytical Approach

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Free Vibration ◽  
Closed Form ◽  
Timoshenko Beam ◽  
Vibration Analysis ◽  
Dynamic Stiffness ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Winkler Foundation ◽  
Closed Form Solutions ◽  
Stiffness Matrices

This study presents an analytical solution to the free vibration analysis of a uniform Timoshenko beam. The Timoshenko beam theory covers cases associated with small deflections based on shear deformation and rotary inertia considerations. A material law combining bending, shear, curvature, and natural frequency is presented. This complete study is based on this material law and closed-form solutions are found. The free vibration response of single-span systems, as well as that of spring-mass systems, is analyzed. Closed-form formulations of matrices expressing the boundary conditions are presented; the natural frequencies are determined by solving the eigenvalue problem. First-order dynamic stiffness matrices in local coordinates are determined. Finally, second-order analysis of beams resting on an elastic Winkler foundation is conducted.

FREE VIBRATION ANALYSIS OF NON-UNIFORM TIMOSHENKO BEAMS USING DIFFERENTIAL TRANSFORM

1998 ◽  
Vol 22 (3) ◽  
pp. 231-250 ◽  
Author(s):  
Cha’o Kuang Chen ◽  
Shing Huei Ho
Keyword(s):  
Free Vibration ◽  
Timoshenko Beam ◽  
Vibration Analysis ◽  
Present Method ◽  
Series Solution ◽  
Free Vibration Analysis ◽  
Timoshenko Beams ◽  
Differential Transform ◽  
Vibration Problems ◽  
Algebraic Operations

This study introduces using differential transform to solve the free vibration problems of a general elastically end restrained non-uniform Timoshenko beam. First, differential transform is briefly introduced. Second, taking differential transform of a non-uniform Timoshenko beam vibration problem, a set of difference equations is derived. Doing some simple algebraic operations on these equations, we can determine any i-th natural frequency, the closed form series solution of any i-th normalized mode shape. Finally, three examples are given to illustrate the accuracy and efficiency of the present method.

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