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.

scholarly journals Natural Frequency Characteristics of the Beam with Different Cross Sections Considering the Shear Deformation Induced Rotary Inertia

2020 ◽  
Vol 10 (15) ◽  
pp. 5245
Author(s):  
Chunfeng Wan ◽  
Huachen Jiang ◽  
Liyu Xie ◽  
Caiqian Yang ◽  
Youliang Ding ◽  
...  
Keyword(s):  
Shear Deformation ◽  
Cross Sections ◽  
Timoshenko Beam ◽  
High Frequency ◽  
Natural Frequencies ◽  
Beam Theory ◽  
Equation Of Motion ◽  
Timoshenko Beam Theory ◽  
Rotary Inertia ◽  
Cross Sectional

Based on the classical Timoshenko beam theory, the rotary inertia caused by shear deformation is further considered and then the equation of motion of the Timoshenko beam theory is modified. The dynamic characteristics of this new model, named the modified Timoshenko beam, have been discussed, and the distortion of natural frequencies of Timoshenko beam is improved, especially at high-frequency bands. The effects of different cross-sectional types on natural frequencies of the modified Timoshenko beam are studied, and corresponding simulations have been conducted. The results demonstrate that the modified Timoshenko beam can successfully be applied to all beams of three given cross sections, i.e., rectangular, rectangular hollow, and circular cross sections, subjected to different boundary conditions. The consequence verifies the validity and necessity of the modification.

scholarly journals Effect of Dead Load on Dynamic Characteristics of Rotating Timoshenko Beams

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Yan-Qi Yin ◽  
Bo Zhang ◽  
Yue-ming Li ◽  
Wei-Zhen Lu
Keyword(s):  
Finite Element ◽  
Cantilever Beam ◽  
Dynamic Characteristics ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Experimental Result ◽  
Dead Load ◽  
Rotating Speed ◽  
Experimental Apparatus ◽  
The Dead

The dynamic characteristics of a rotating cantilever Timoshenko beam under dead load are investigated in this paper. Considering the predeformation caused by dead load and centrifugal force, governing equation of rotating cantilever Timoshenko beam is derived based on Hamilton’s principle, and the influence of the load on natural vibration is revealed. A suit of modal experimental apparatus for cantilever beam is designed and used to test the natural frequencies under the dead load, and the natural frequencies under rotation condition are calculated with a commercial finite element code. Both the experimental result and numerical result are utilized to compare with the present theoretical result, and the results obtained by present modeling method show a good agreement with those obtained from the experiment and finite element method. It is found that the natural frequencies of cantilever beam increase with both the dead load and the rotating speed.

Free Vibration of a Spinning Stepped Timoshenko Beam

2000 ◽  
Vol 67 (4) ◽  
pp. 839-841 ◽  
Author(s):  
S. D. Yu ◽  
W. L. Cleghorn
Keyword(s):  
Free Vibration ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Equation Of Motion ◽  
Mode Shapes ◽  
The Finite Element Method ◽  
Linear Transformations ◽  
Stepped Beam ◽  
Uniform Cross Section ◽  
Vibration Problems

The finite element method is employed in this paper to investigate free-vibration problems of a spinning stepped Timoshenko beam consisting of a series of uniform segments. Each uniform segment is considered a substructure which may be modeled using beam finite elements of uniform cross section. Assembly of global equation of motion of the entire beam is achieved using Lagrange’s multiplier method. The natural frequencies and mode shapes are subsequently reduced with the help of linear transformations to a standard eigenvalue problem for which a set of natural frequencies and mode shapes may be easily obtained. Numerical results for an overhung stepped beam consisting of three uniform segments are obtained and presented as an illustrative example. [S00021-8936(01)00101-5]

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 A Modal Perturbation Method for Eigenvalue Problem of Non-Proportionally Damped System

2020 ◽  
Vol 10 (1) ◽  
pp. 341
Author(s):  
Danguang Pan ◽  
Xiangqiu Fu ◽  
Qingjun Chen ◽  
Pan Lu ◽  
Jinpeng Tan
Keyword(s):  
Perturbation Method ◽  
State Space ◽  
Mode Shapes ◽  
Vibration Modes ◽  
Algebraic Equations ◽  
Engineering Structures ◽  
Damped System ◽  
Real Mode ◽  
Nonlinear Algebraic Equations ◽  
Practical Engineering

The non-proportionally damped system is very common in practical engineering structures. The dynamic equations for these systems, in which the damping matrices are coupled, are very time consuming to solve. In this paper, a modal perturbation method is proposed, which only requires the first few lower real mode shapes of a corresponding undamped system to obtain the complex mode shapes of non-proportionally damped system. In this method, an equivalent proportionally damped system is constructed by taking the real mode shapes of a corresponding undamped system and then transforming the characteristic equation of state space into a set of nonlinear algebraic equations by using the vibration modes of an equivalent proportionally damped system. Two numerical examples are used to illustrate the validity and accuracy of the proposed modal perturbation method. The numerical results show that: (1) with the increase of vibration modes of the corresponding undamped system, the eigenvalues and eigenvectors monotonically converge to exact solutions; (2) the accuracy of the proposed method is significantly higher than the first-order perturbation method and proportional damping method. The calculation time of the proposed method is shorter than the state space method; (3) the method is particularly suitable for finding a few individual orders of frequency and mode of a system with highly non-proportional damping.

scholarly journals Forced Vibration of a Timoshenko Beam Subjected to Stationary and Moving Loads Using the Modal Analysis Method

2017 ◽  
Vol 2017 ◽  
pp. 1-26 ◽  
Author(s):  
Taehyun Kim ◽  
Ilwook Park ◽  
Usik Lee
Keyword(s):  
Modal Analysis ◽  
Timoshenko Beam ◽  
Forced Vibration ◽  
Natural Frequencies ◽  
Dynamic Responses ◽  
Mode Shapes ◽  
Timoshenko Beams ◽  
Moving Loads ◽  
Analysis Method ◽  
Simply Supported

The modal analysis method (MAM) is very useful for obtaining the dynamic responses of a structure in analytical closed forms. In order to use the MAM, accurate information is needed on the natural frequencies, mode shapes, and orthogonality of the mode shapes a priori. A thorough literature survey reveals that the necessary information reported in the existing literature is sometimes very limited or incomplete, even for simple beam models such as Timoshenko beams. Thus, we present complete information on the natural frequencies, three types of mode shapes, and the orthogonality of the mode shapes for simply supported Timoshenko beams. Based on this information, we use the MAM to derive the forced vibration responses of a simply supported Timoshenko beam subjected to arbitrary initial conditions and to stationary or moving loads (a point transverse force and a point bending moment) in analytical closed form. We then conduct numerical studies to investigate the effects of each type of mode shape on the long-term dynamic responses (vibrations), the short-term dynamic responses (waves), and the deformed shapes of an example Timoshenko beam subjected to stationary or moving point loads.

FREE VIBRATION OF TAPERED TIMOSHENKO BEAMS BY DEFORMATION DECOMPOSITION

2013 ◽  
Vol 13 (02) ◽  
pp. 1250057 ◽  
Author(s):  
BYOUNG KOO LEE ◽  
SANG JIN OH ◽  
TAE EUN LEE
Keyword(s):  
Differential Equations ◽  
Free Vibration ◽  
Natural Frequencies ◽  
Search Method ◽  
Mode Shapes ◽  
Timoshenko Beams ◽  
Runge Kutta Method ◽  
Shear Distortion ◽  
Transverse Deflection ◽  
Regula Falsi Method

This paper deals with the free vibration of tapered Timoshenko beams. The simultaneous differential equations governing the free vibration of tapered Timoshenko beams are derived by decomposing the deformations of the beam into components as transverse deflection, bending rotation and shear distortion. The governing differential equations are first integrated by the Runge–Kutta method and then solved by the determinant search method, combined with the Regula–Falsi method, to obtain the natural frequencies of the beam along with their corresponding mode shapes. In the numerical examples, the effects of various parameters on the frequencies and mode shapes of the beam are extensively discussed.

Simplified Dynamic Analysis Methods for Metallic Bellows Expansion Joints

1991 ◽  
Vol 113 (4) ◽  
pp. 504-510 ◽  
Author(s):  
M. Morishita ◽  
N. Ikahata ◽  
S. Kitamura
Keyword(s):  
Dynamic Analysis ◽  
Seismic Response ◽  
Dynamic Characteristics ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Vibration Modes ◽  
Lateral Vibration ◽  
Expansion Joints ◽  
Structure Interaction ◽  
Lateral Vibrations

An investigation into dynamic characteristics and seismic response of bellows expansion joints is described. For axial and lateral vibrations of the bellows, simplified methods are developed to evaluate their natural frequencies and seismic response, based on analogies with those of a uniform rod and a Timoshenko beam, respectively. For the lateral vibration modes, effect of fluid-structure interaction between the convolutions and flow-sleeve is taken into account. The validity and applicability of the simplified methods are shown, by comparing with the results of vibration experiments using a simple bellows model and a piping model with bellows, along with corresponding detailed FEM analyses’ results.

Algorithm for diagnosing fastenings of pipe with liquid

2007 ◽  
Vol 5 ◽  
pp. 96-100
Author(s):  
A.M. Akhtyamov ◽  
F.F. Safina
Keyword(s):  
Frequency Spectrum ◽  
Natural Frequencies ◽  
Algebraic Equations ◽  
Bending Vibrations ◽  
Narrow Tube

An algorithm is considered for diagnosing fastening of a narrow tube filled with a fluid by a spectrum of natural frequencies of its bending vibrations. The constructed algorithm, based on the solution of systems of algebraic equations, allows one to determine any pipe fastenings by 9 values from the frequency spectrum of its vibrations when the liquid is flowing through the pipe.

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