scholarly journals Transverse Free Vibration of Axially Moving Stepped Beam with Different Length and Tip Mass

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Guoliang Ma ◽  
Minglong Xu ◽  
Liqun Chen ◽  
Zengyong An
Keyword(s):  
Free Vibration ◽  
Beam Theory ◽  
Bending Moment ◽  
Frequency Equation ◽  
Eigenvalue Equation ◽  
Stepped Beam ◽  
Complex Modal Analysis ◽  
Axially Moving ◽  
Semianalytical Method ◽  
Tip Mass

Axially moving stepped beam (AMSB) with different length and tip mass is represented by adopting Euler-Bernoulli beam theory, and its characteristics and displacements of transverse free vibration are calculated by using semianalytical method. Firstly, the governing equation of the transverse free vibration is established based on Hamilton’s principle. The equation is cast into eigenvalue equation through the complex modal analysis. Then, a scheme is proposed to derive the continuous condition accordingly as the displacement, rotation, bending moment, and shear force are all equal at the connections of any two segments. Another scheme is to derive frequency equation from the given boundary conditions which contain a tip mass in the last segment. Finally, the natural frequency and modal function are calculated by using numerical method according to the eigenvalue equation and frequency equation. Due to the introduction of modal truncation, displacement and, the free vibration solution can be obtained by adopting modal superposition after Hilbert transform. The numerical examples illustrate that length, velocity, mass, and geometry affect characteristics and displacements significantly; the series of methods are effective and accurate to investigate the vibration of the AMSB with different length and tip mass after comparing several results.

scholarly journals Application of Laplace Transform to the Free Vibration of Continuous Beams

2017 ◽  
Vol 63 (1) ◽  
pp. 163-180 ◽  
Author(s):  
H.B. Wen ◽  
T. Zeng ◽  
G.Z. Hu
Keyword(s):  
Free Vibration ◽  
Laplace Transform ◽  
Reaction Force ◽  
Bending Moment ◽  
Frequency Equation ◽  
Simplified Model ◽  
Continuous Beam ◽  
Transform Method ◽  
Continuous Beams ◽  
Vibration Problems

AbstractLaplace Transform is often used in solving the free vibration problems of structural beams. In existing research, there are two types of simplified models of continuous beam placement. The first is to regard the continuous beam as a single-span beam, the middle bearing of which is replaced by the bearing reaction force; the second is to divide the continuous beam into several simply supported beams, with the bending moment of the continuous beam at the middle bearing considered as the external force. Research shows that the second simplified model is incorrect, and the frequency equation derived from the first simplified model contains multiple expressions which might not be equivalent to each other. This paper specifies the application method of Laplace Transform in solving the free vibration problems of continuous beams, having great significance in the proper use of the transform method.

FREE VIBRATION ANALYSIS OF CROSS-PLY LAMINATED COMPOSITE BEAMS BY MIXED FINITE ELEMENT FORMULATION

2013 ◽  
Vol 13 (02) ◽  
pp. 1250056 ◽  
Author(s):  
ATİLLA ÖZÜTOK ◽  
EMRAH MADENCİ
Keyword(s):  
Finite Element ◽  
Free Vibration ◽  
Vibration Analysis ◽  
Mixed Finite Element ◽  
Laminated Composite ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Bending Moment ◽  
Vertical Displacement ◽  
Laminated Composite Beams

In this study, a mixed-finite element method for free vibration analysis of cross-ply laminated composite beams is presented based on the "Euler–Bernoulli Beam Theory" and "Timoshenko Beam Theory". The Gâteaux differential approach is employed to construct the functionals of laminated beams using the variational method. By using these functionals in the mixed-type finite element method, two beam elements CLBT4 and FSDT8 are derived for the Euler–Bernoulli and Timoshenko beam theories, respectively. The CLBT4 element has four degrees of freedom (DOFs), containing the vertical displacement and bending moment as unknowns at the nodes, whereas the FSDT8 element has eight DOFs, containing the vertical displacement, bending moment, shear force and rotation as unknowns. A computer program is developed to execute the analyses for the present study. The numerical results of free vibration analyses obtained for different boundary conditions are presented and compared with results available in the literature, which indicates the reliability of the present approach.

Free Vibration Analysis of Shafts on Resilient Bearings Using Timoshenko Beam Theory

1999 ◽  
Vol 121 (2) ◽  
pp. 256-258 ◽  
Author(s):  
S. Karunendiran ◽  
J. W. Zu
Keyword(s):  
Free Vibration ◽  
Timoshenko Beam ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Frequency Equation ◽  
Timoshenko Beam Theory ◽  
Mode Shapes ◽  
Complex Eigenvalues ◽  
First Time

This paper presents an analytical method adopted for the free vibration analysis of a shaft, both ends of which are supported by resilient bearings. The shaft is modeled by Timoshenko beam theory. Based on this model exact frequency equation to calculate complex eigenvalues is derived and presented in complex compact form for the first time. Explicit expressions to compute the corresponding mode shapes are also presented.

Free vibration analysis of angle-ply laminate composite beams by mixed finite element formulation using the Gâteaux differential

2014 ◽  
Vol 21 (2) ◽  
pp. 257-266 ◽  
Author(s):  
Atilla Ozutok ◽  
Emrah Madenci ◽  
Fethi Kadioglu
Keyword(s):  
Finite Element ◽  
Free Vibration ◽  
Timoshenko Beam ◽  
Mixed Finite Element ◽  
Beam Theory ◽  
Bending Moment ◽  
Vertical Displacement ◽  
Composite Beams ◽  
Differential Method ◽  
Element Formulation

AbstractFree vibration analyses of angle-ply laminated composite beams were investigated by the Gâteaux differential method in the present paper. With the use of the Gâteaux differential method, the functionals were obtained and the natural frequencies of the composite beams were computed using the mixed finite element formulation on the basis of the Euler-Bernoulli beam theory and Timoshenko beam theory. By using these functionals in the mixed-type finite element method, two beam elements, CLBT4 and FSDT8, were derived for the Euler-Bernoulli and Timoshenko beam theories, respectively. The CLBT4 element has 4 degrees of freedom (DOFs) containing the vertical displacement and bending moment as the unknowns at the nodes, whereas the FSDT8 element has 8 DOFs containing the vertical displacement, bending moment, shear force and rotation as unknowns. A computer program was developed to execute the analyses for the present study. The numerical results of free vibration analyses obtained for different boundary conditions were presented and compared with the results available in the literature, which indicates the reliability of the present approach.

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.

Free vibration analysis of composite foils with different ply angles based on beam theory

2021 ◽  
Vol 226 ◽  
pp. 108854
Author(s):  
Hanzhe Zhang ◽  
Qin Wu ◽  
Yunqing Liu ◽  
Biao Huang ◽  
Guoyu Wang
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Beam Theory

Centrifuge Modeling on Seismic Behavior of Pile in Liquefiable Soil Ground

2013 ◽  
Vol 479-480 ◽  
pp. 1139-1143
Author(s):  
Wen Yi Hung ◽  
Chung Jung Lee ◽  
Wen Ya Chung ◽  
Chen Hui Tsai ◽  
Ting Chen ◽  
...  
Keyword(s):  
Seismic Behavior ◽  
Soil Structure ◽  
Lateral Displacement ◽  
Bending Moment ◽  
Soil Liquefaction ◽  
Centrifuge Modeling ◽  
Shaking Table ◽  
Pile Head ◽  
Liquefiable Soil ◽  
Tip Mass

Dramatic failure of pile foundations caused by the soil liquefaction was founded leading to many studies for investigating the seismic behavior of pile. The failures were often accompanied with settlement, lateral displacement and tilting of superstructures. Therefore soil-structure interaction effects must be properly considered in the pile design. Two tests by using the centrifuge shaking table were conducted at an acceleration field of 80 g to investigate the seismic response of piles attached with different tip mass and embedded in liquefied or non-liquefied deposits during shaking. It was found that the maximum bending moment of pile occurs at the depth of 4 m and 5 m for dry sand and saturated sand models, respectively. The more tip mass leads to the more lateral displacement of pile head and the more residual bending moment.

Effect of Dynamic Soft Elasticity on Vibration of Embedded Nematic Elastomer Timoshenko Beams

2018 ◽  
Vol 10 (05) ◽  
pp. 1850058 ◽  
Author(s):  
Dong Zhao ◽  
Ying Liu
Keyword(s):  
Low Frequency ◽  
Beam Theory ◽  
Timoshenko Beams ◽  
Analysis Method ◽  
Anisotropic Parameter ◽  
Nematic Elastomer ◽  
Dissipation Mechanism ◽  
Viscous Media ◽  
Complex Modal Analysis ◽  
Soft Elasticity

This paper addresses the transverse vibration of a nematic elastomer (NE) beam embedded in soft viscoelastic surroundings with the aim to clarify a new dissipation mechanism caused by dynamic soft elasticity of this soft material. Based on the viscoelasticity theory of NEs in low-frequency limit and the Timoshenko beam theory, the governing equation of motion is derived by using the Hamilton principle and energy method, and is solved by the complex modal analysis method. The dependence of vibration property on the intrinsic parameters of NEs (director rotation time, rubber relaxation time, anisotropic parameter) and foundation (spring, shear and damping constants) are discussed in detail. The results show that dynamic soft elasticity leads to anomalous anisotropy of energy transfer and attenuation. The relative stiffer foundation would restraint the rubber dissipation of viscoelastic beams, but has less influence on the director rotation dissipation, which is particular for NE beams. This study would provide a useful guidance in the dynamic design of NE apparatus embedded in soft viscous media.

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