Exact closed form solution for the analysis of the transverse vibration mode of a Nano-Timoshenko beam with multiple concentrated masses

2017 ◽  
Vol 131-132 ◽  
pp. 728-743 ◽  
Author(s):  
A. Jafarzadeh Jazi ◽  
B. Shahriari ◽  
K. Torabi
Keyword(s):  
Closed Form ◽  
Timoshenko Beam ◽  
Transverse Vibration ◽  
Vibration Mode ◽  
Closed Form Solution ◽  
Form Solution ◽  
Concentrated Masses

Wave-Induced Vibrations of an Axial-Loaded Immersed Timoshenko Beam Carrying an Eccentric Tip Mass with Rotary Inertia

2010 ◽  
Vol 54 (01) ◽  
pp. 15-33
Author(s):  
Jong-Shyong Wu ◽  
Chin-Tzu Chen
Keyword(s):  
Closed Form ◽  
Timoshenko Beam ◽  
Forced Vibration ◽  
Natural Frequencies ◽  
Closed Form Solution ◽  
Form Solution ◽  
Rotary Inertia ◽  
Mode Shapes ◽  
Mode Superposition Method ◽  
Tip Mass

Under the specified assumptions for the equation of motion, the closed-form solution for the natural frequencies and associated mode shapes of an immersed "Euler-Bernoulli" beam carrying an eccentric tip mass possessing rotary inertia has been reported in the existing literature. However, this is not true for the immersed "Timoshenko" beam, particularly for the case with effect of axial load considered. Furthermore, the information concerning the forced vibration analysis of the foregoing Timoshenko beam caused by wave excitations is also rare. Therefore, the first purpose of this paper is to present a technique to obtain the closed-form solution for the natural frequencies and associated mode shapes of an axial-loaded immersed "Timoshenko" beam carrying eccentric tip mass with rotary inertia by using the continuous-mass model. The second purpose is to determine the forced vibration responses of the latter resulting from excitations of regular waves by using the mode superposition method incorporated with the last closed-form solution for the natural frequencies and associated mode shapes of the beam. Because the determination of normal mode shapes of the axial-loaded immersed "Timoshenko" beam is one of the main tasks for achieving the second purpose and the existing literature concerned is scarce, the details about the derivation of orthogonality conditions are also presented. Good agreements between the results obtained from the presented technique and those obtained from the existing literature or conventional finite element method (FEM) confirm the reliability of the presented theories and the developed computer programs for this paper.

scholarly journals A closed-form solution for free vibration of multiple cracked Timoshenko beam and application

2017 ◽  
Vol 39 (4) ◽  
pp. 315-328
Author(s):  
Nguyen Tien Khiem ◽  
Duong The Hung
Keyword(s):  
Free Vibration ◽  
Closed Form ◽  
Timoshenko Beam ◽  
Crack Depth ◽  
Closed Form Solution ◽  
Beam Theory ◽  
Frequency Equation ◽  
Form Solution ◽  
Mode Shapes ◽  
Timoshenko Beams

A closed-form solution for free vibration is constructed and used for obtaining explicit frequency equation and mode shapes of  Timoshenko beams with arbitrary number of cracks. The cracks are represented by the rotational springs of stiffness calculated from the crack depth.  Using the obtained frequency equation, the sensitivity of natural frequencies to crack of the beams is examined in comparison with the  Euler-Bernoulli beams. Numerical results demonstrate that the Timoshenko beam theory is efficiently applicable not only for short or fat beams but also for the long or slender ones. Nevertheless, both the theories are equivalent in sensitivity analysis of fundamental frequency to cracks and they get to be different for higher frequencies.

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