Medium-Frequency Vibration Analysis of Timoshenko Beam Structures

2020 ◽  
Vol 20 (13) ◽  
pp. 2041009
Author(s):  
Yichi Zhang ◽  
Bingen Yang
Keyword(s):  
Shear Deformation ◽  
Timoshenko Beam ◽  
Vibration Analysis ◽  
Beam Theory ◽  
Frequency Region ◽  
Bernoulli Beam ◽  
Beam Structure ◽  
Medium Frequency ◽  
Beam Structures ◽  
Euler Bernoulli Beam

Medium-frequency (mid-frequency) vibration analysis of complex structures plays an important role in automotive, aerospace, mechanical, and civil engineering. Flexible beam structures modeled by the classical Euler–Bernoulli beam theory have been widely used in various engineering problems. A kinematic hypothesis made in the Euler–Bernoulli beam theory is that the plane sections of a beam normal to its neutral axis remain planes after the beam experiences bending deformation, which neglects shear deformation. However, previous investigations found out that the shear deformation of a beam (even with a large slenderness ratio) becomes noticeable in high-frequency vibrations. The Timoshenko beam theory, which describes both bending deformation and shear deformation, would naturally be more suitable for medium-frequency vibration analysis. Nevertheless, vibrations of Timoshenko beam structures in a medium frequency region have not been well studied in the literature. This paper presents a new method for mid-frequency vibration analysis of two-dimensional Timoshenko beam structures. The proposed method, which is called the augmented Distributed Transfer Function Method (DTFM), models a Timoshenko beam structure by a spatial state-space formulation in the [Formula: see text]-domain. The augmented DTFM determines the frequency response of a beam structure in an exact and analytical form, in any frequency region covering low, middle, or high frequencies. Meanwhile, the proposed method provides the local information of a beam structure, such as displacement, shear deformation, bending moment and shear force at any location, which otherwise would be very difficult with energy-based methods. The medium-frequency analysis by the augmented DTFM is validated in numerical examples, where the efficiency and accuracy of the proposed method is demonstrated. Also, the effects of shear deformation on the dynamic behaviors of a beam structure at medium frequencies are examined through comparison of the Timoshenko beam and Euler–Bernoulli beam theories.

Medium Frequency Vibration Analysis of Beam Structures Modeled by the Timoshenko Beam Theory

2020 ◽  
Author(s):  
Yichi Zhang ◽  
Bingen Yang
Keyword(s):  
Shear Deformation ◽  
Timoshenko Beam ◽  
Vibration Analysis ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Bernoulli Beam ◽  
Beam Structure ◽  
Medium Frequency ◽  
Beam Structures ◽  
Euler Bernoulli Beam

Abstract Vibration analysis of complex structures at medium frequencies plays an important role in automotive engineering. Flexible beam structures modeled by the classical Euler-Bernoulli beam theory have been widely used in many engineering problems. A kinematic hypothesis in the Euler-Bernoulli beam theory is that plane sections of a beam normal to its neutral axis remain normal when the beam experiences bending deformation, which neglects the shear deformation of the beam. However, as observed by researchers, the shear deformation of a beam component becomes noticeable in high-frequency vibrations. In this sense, the Timoshenko beam theory, which describes both bending deformation and shear deformation, may be more suitable for medium-frequency vibration analysis of beam structures. This paper presents an analytical method for medium-frequency vibration analysis of beam structures, with components modeled by the Timoshenko beam theory. The proposed method is developed based on the augmented Distributed Transfer Function Method (DTFM), which has been shown to be useful in various vibration problems. The proposed method models a Timoshenko beam structure by a spatial state-space formulation in the s-domain, without any discretization. With the state-space formulation, the frequency response of a beam structure, in any frequency region (from low to very high frequencies), can be obtained in an exact and analytical form. One advantage of the proposed method is that the local information of a beam structure, such as displacements, bending moment and shear force at any location, can be directly obtained from the space-state formulation, which otherwise would be very difficult with energy-based methods. The medium-frequency analysis by the augmented DTFM is validated with the FEA in numerical examples, where the efficiency and accuracy of the proposed method is present. Also, the effects of shear deformation on the dynamic behaviors of a beam structure at medium frequencies are illustrated through comparison of the Timoshenko beam theory and the Euler-Bernoulli beam theory.

ANALYSIS OF AUXETIC BEAMS AS RESONANT FREQUENCY BIOSENSORS

2012 ◽  
Vol 12 (05) ◽  
pp. 1240027 ◽  
Author(s):  
TEIK-CHENG LIM
Keyword(s):  
Shear Deformation ◽  
Cross Sections ◽  
Timoshenko Beam ◽  
Poisson’S Ratio ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Poisson's Ratio ◽  
Bernoulli Beam ◽  
Bernoulli Beam Theory ◽  
Euler Bernoulli Beam

The mechanics of beam vibration is of fundamental importance in understanding the shift of resonant frequency of microcantilever and nanocantilever sensors. Unlike the simpler Euler–Bernoulli beam theory, the Timoshenko beam theory takes into consideration rotational inertia and shear deformation. For the case of microcantilevers and nanocantilevers, the minute size, and hence low mass, means that the topmost deviation from the Euler–Bernoulli beam theory to be expected is shear deformation. This paper considers the extent of shear deformation for varying Poisson's ratio of the beam material, with special emphasis on solids with negative Poisson's ratio, which are also known as auxetic materials. Here, it is shown that the Timoshenko beam theory approaches the Euler–Bernoulli beam theory if the beams are of solid cross-sections and the beam material possess high auxeticity. However, the Timoshenko beam theory is significantly different from the Euler–Bernoulli beam theory for beams in the form of thin-walled tubes regardless of the beam material's Poisson's ratio. It is herein proposed that calculations on beam vibration can be greatly simplified for highly auxetic beams with solid cross-sections due to the small shear correction term in the Timoshenko beam deflection equation.

Vibration Analysis of Multi-Cracked Beam Traversed by Moving Masses Using Discrete Element Technique

2013 ◽  
Vol 376 ◽  
pp. 220-223
Author(s):  
Reza Alebrahim ◽  
Nik Abdullah Nik Mohamed ◽  
Sallehuddin Mohamed Haris ◽  
Salvinder Singh Karam Singh
Keyword(s):  
Vibration Analysis ◽  
Beam Theory ◽  
Discrete Element ◽  
Maximum Deflection ◽  
Bernoulli Beam ◽  
Cracked Beam ◽  
Time To Build ◽  
Element Technique ◽  
Euler Bernoulli Beam ◽  
Good Agreement

The vibration analysis of a multi-cracked beam using discrete element technique (DET) was investigated in this study. Undamped simply supported beam was traversed by moving mass with constant speed and Euler Bernoulli beam theory was considered. Cracks are located in different positions and maximum deflection of mid-span was derived and compared. The results showed that increasing numbers of cracks in the beam causes more deflection while maximum deflection of beam takes longer time to build up. The results were validated by solving the equations generated using finite element method (FEM) and their comparison with already established results from previous similar studies (literatures) showed good agreement.

scholarly journals Euler-Bernoulli Beam Theory: First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Finite Difference Method ◽  
Differential Equations ◽  
Vibration Analysis ◽  
Beam Theory ◽  
Second Order ◽  
Bernoulli Beam ◽  
Order Analysis ◽  
First Order ◽  
The Finite Difference Method ◽  
Euler Bernoulli Beam

This paper presents an approach to the Euler-Bernoulli beam theory (EBBT) using the finite difference method (FDM). The EBBT covers the case of small deflections, and shear deformations are not considered. The FDM is an approximate method for solving problems described with differential equations (or partial differential equations). The FDM does not involve solving differential equations; equations are formulated with values at selected points of the structure. The model developed in this paper consists of formulating partial differential equations with finite differences and introducing new points (additional points or imaginary points) at boundaries and positions of discontinuity (concentrated loads or moments, supports, hinges, springs, brutal change of stiffness, etc.). The introduction of additional points permits us to satisfy boundary conditions and continuity conditions. First-order analysis, second-order analysis, and vibration analysis of structures were conducted with this model. Efforts, displacements, stiffness matrices, buckling loads, and vibration frequencies were determined. Tapered beams were analyzed (e.g., element stiffness matrix, second-order analysis). Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, the damping being considered. The efforts and displacements could be determined at any time.

An Analytical Model for Beam Flexure Modules Based on the Timoshenko Beam Theory

2017 ◽  
Author(s):  
M. H. Kahrobaiyan ◽  
M. Zanaty ◽  
S. Henein
Keyword(s):  
Timoshenko Beam ◽  
Axial Load ◽  
Compliant Mechanisms ◽  
Slenderness Ratio ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Bernoulli Beam ◽  
Model Based ◽  
Bernoulli Beam Theory ◽  
Euler Bernoulli Beam

Short beams are the key building blocks in many compliant mechanisms. Hence, deriving a simple yet accurate model of their elastokinematics is an important issue. Since the Euler-Bernoulli beam theory fails to accurately model these beams, we use the Timoshenko beam theory to derive our new analytical framework in order to model the elastokinematics of short beams under axial loads. We provide exact closed-form solutions for the governing equations of a cantilever beam under axial load modeled by the Timoshenko beam theory. We apply the Taylor series expansions to our exact solutions in order to capture the first and second order effects of axial load on stiffness and axial shortening. We show that our model for beam flexures approaches the model based on the Euler-Bernoulli beam theory when the slenderness ratio of the beams increases. We employ our model to derive the stiffness matrix and axial shortening of a beam with an intermediate rigid part, a common element in the compliant mechanisms with localized compliance. We derive the lateral and axial stiffness of a parallelogram flexure mechanism with localized compliance and compare them to those derived by the Euler-Bernoulli beam theory. Our results show that the Euler-Bernoulli beam theory predicts higher stiffness. In addition, we show that decrease in slenderness ratio of beams leads to more deviation from the model based on the Euler-Bernoulli beam theory.

Prediction of Changes in Modal Properties of the Euler–Bernoulli Beam Structures Due to the Modification of Its Spatial Properties

2017 ◽  
Vol 17 (05) ◽  
pp. 1740014 ◽  
Author(s):  
Milan Naď ◽  
Ladislav Rolník ◽  
Lenka Čičmancová
Keyword(s):  
Dynamical Behavior ◽  
Mode Shapes ◽  
Structural Elements ◽  
Bernoulli Beam ◽  
Beam Structure ◽  
Beam Structures ◽  
Spatial Properties ◽  
Modal Properties ◽  
Euler Bernoulli Beam ◽  
Operating Regimes

The beams can be considered as fundamental structural elements applied in many technical structures and equipment. In the operating regimes, these beam structures are very often loaded by the time-dependent forces which cause their undesirable dynamical behavior and the whole system is getting into critical resonant state. It is clear that the reduction of the level of unwanted vibrations or prevention of their occurrence should be one of the important objectives in the design of machine equipment and structures. To achieve these aims, the knowledge of modal properties of beam structures in relation to their internal structure is necessary. The insertion of reinforcing core into a beam body is one of the possible techniques to achieve the required changes in modal properties (mode shapes, natural frequencies) of the beam structure. The dependency of the modal properties of structurally modified beam structure on the geometric parameters and material properties of the inserted reinforcing core is studied in this paper.

Estimation of damping in riveted short cantilever beams

2020 ◽  
Vol 26 (23-24) ◽  
pp. 2163-2173
Author(s):  
Yemineni Siva Sankara Rao ◽  
Kutchibotla Mallikarjuna Rao ◽  
V V Subba Rao
Keyword(s):  
Timoshenko Beam ◽  
Damping Ratio ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Analytical Models ◽  
Cantilever Beams ◽  
Bernoulli Beam ◽  
Bernoulli Beam Theory ◽  
Half Power ◽  
Euler Bernoulli Beam

In layered and riveted structures, vibration damping happens because of a micro slip that occurs because of a relative motion at the common interfaces of the respective jointed layers. Other parameters that influence the damping mechanism in layered and riveted beams are the amplitude of initial excitation, overall length of the beam, rivet diameter, overall beam thickness, and many layers. In this investigation, using the analytical models such as the Euler–Bernoulli beam theory and Timoshenko beam theory and half-power bandwidth method, the free transverse vibration analysis of layered and riveted short cantilever beams is carried out for observing the damping mechanism by estimating the damping ratio, and the obtained results from the Euler–Bernoulli beam theory and Timoshenko beam theory analytical models are validated by the half-power bandwidth method. Although the Euler–Bernoulli beam model overestimates the damping ratio value by a very less fraction, both the models can be used to evaluate damping for short riveted cantilever beams along with the half-power bandwidth method.

Nonlinear Beam-Column Element Under Consistent Deformation

2015 ◽  
Vol 15 (05) ◽  
pp. 1450068 ◽  
Author(s):  
Yi Qun Tang ◽  
Zhi Hua Zhou ◽  
Siu Lai Chan
Keyword(s):  
Shear Deformation ◽  
Beam Theory ◽  
Displacement Function ◽  
Shear Locking ◽  
Bernoulli Beam ◽  
Equilibrium Equations ◽  
Nonlinear Beam ◽  
Euler Bernoulli Beam ◽  
Column Element ◽  
Force Equilibrium

A new nonlinear beam-column element capable of considering the shear deformation is proposed under the concept of consistent deformation. For the traditional displacement interpolation function, the beam-column element produces membrane locking under large deformation and shear locking when the element becomes slender. To eliminate the membrane and shear locking, force equilibrium equations are employed to derive the displacement function. Numerical examples herein show that membrane locking in the traditional nonlinear beam-column element could cause a considerable error. Comparison of the present improved formulae based on the Timoshenko beam theory with that based on the Euler–Bernoulli beam theory indicates that the present approach requires several additional parameters to consider shear deformation and it is suitable for computer analysis readily for incorporation into the frames analysis software using the co-rotational approach for large translations and rotations. The examples confirm that the proposed element has higher accuracy and numerical efficiency.

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