Celebrating the Centenary of Timoshenko's Study of Effects of Shear Deformation and Rotary Inertia

2015 ◽  
Vol 67 (6) ◽  
Author(s):  
Isaac Elishakoff ◽  
Julius Kaplunov ◽  
Evgeniya Nolde
Keyword(s):  
Differential Equation ◽  
Shear Deformation ◽  
Timoshenko Beam ◽  
Initial Conditions ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Rotary Inertia ◽  
Governing Differential Equation ◽  
Second Spectrum

This study revisits Timoshenko beam theory (TBT). It discusses at depth a more consistent and simpler governing differential equation. The so-called second spectrum is also addressed. Then, we provide the asymptotic justification of the aforementioned differential equation along with detailed discussion of the boundary and initial conditions. The paper also presents remarks of historical character, in the context of other pertinent studies.

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.

Nonlocal vibration of a carbon nanotube embedded in an elastic medium due to moving nanoparticle analyzed by modified Timoshenko beam theory-parametric excitation and spectral response

2014 ◽  
Vol 23 (3-4) ◽  
pp. 109-128 ◽  
Author(s):  
Ivo Senjanović ◽  
Marko Tomić ◽  
Neven Hadžić
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Elastic Medium ◽  
Timoshenko Beam ◽  
Parametric Excitation ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Inertia Force ◽  
Governing Differential Equation ◽  
Moving Nanoparticle

AbstractThe Timoshenko beam theory, which deals with the deflection and rotation in two partial differential equations of motion, is transformed into a single partial differential equation with pure bending deflection as a potential function for the determination of the total deflection, rotation angle, and sectional forces. Inclusion of a nonlocal stress parameter results in the extension of the governing differential equation from the 4th to the 6th order. A simply supported nanotube is considered, and the governing differential equation is decomposed into a system of ordinary differential equations by employing the modal superposition method, separation of variables, and the Galerkin method. Both moving nanoparticle gravity and inertia force are consistently taken into account, resulting in ordinary and parametric excitation, respectively. As a novelty, the parameters are split into a constant and a time-dependent part. The former is added to the ordinary system of equations, which is solved analytically in the frequency domain by the harmonic balance method, while the system with variable coefficients is solved in the time domain by the perturbation method. The effects of slenderness ratio, nonlocal parameter, stiffness of elastic medium, nanoparticle gravity and inertia force, and velocity on the free and forced nanotube response are also investigated. Special attention is paid to the influence of damping on resonance. Performed parametric analysis is physically transparent due to the obtained semi-analytical solution. Some analytical results of illustrative examples are compared with numerical ones from the relevant literature, and notable differences are discussed.

scholarly journals Who developed the so-called Timoshenko beam theory?

2019 ◽  
Vol 25 (1) ◽  
pp. 97-116 ◽  
Author(s):  
Isaac Elishakoff
Keyword(s):  
Shear Deformation ◽  
Correction Factor ◽  
Timoshenko Beam ◽  
World Literature ◽  
Beam Theory ◽  
Scientific Work ◽  
Timoshenko Beam Theory ◽  
Rotary Inertia ◽  
The World ◽  
First Time

The use of the Google Scholar produces about 78,000 hits on the term “Timoshenko beam.” The question of priority is of great importance for this celebrated theory. For the first time in the world literature, this study is devoted to the question of priority. It is that Stephen Prokofievich Timoshenko had a co-author, Paul Ehrenfest. It so happened that the scientific work of Timoshenko dealing with the effect of rotary inertia and shear deformation does not carry the name of Ehrenfest as the co-author. In his 2002 book, Grigolyuk concluded that the theory belonged to both Timoshenko and Ehrenfest. This work confirms Grigolyuk’s discovery, in his little known biographic book about Timoshenko, and provides details, including the newly discovered letter of Timoshenko to Ehrenfest, which is published here for the first time over a century after it was sent. This paper establishes that the beam theory that incorporates both the rotary inertia and shear deformation as is known presently, with shear correction factor included, should be referred to as the Timoshenko-Ehrenfest beam theory.

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.

Application of Timoshenko Beam Theory to the Dynamics of Flexible Legged Locomotion

1988 ◽  
Vol 110 (1) ◽  
pp. 28-34
Author(s):  
E. M. Bakr ◽  
A. A. Shabana
Keyword(s):  
Shear Deformation ◽  
Timoshenko Beam ◽  
Shape Function ◽  
Reference Conditions ◽  
Beam Theory ◽  
Legged Locomotion ◽  
Rotary Inertia ◽  
Stiffness Matrices ◽  
Reference Motion ◽  
Invariant Matrices

A method for the dynamic analysis of flexible legged locomotion systems that accounts for the rotary inertia and shear deformation effects is presented. The motion of the flexible components in the legged vehicle is described using a set of inertia-variant Timoshenko beams that undergo large rotations. A shape function that accounts for the combined effect of rotary inertia and shear is employed to describe the deformation relative to a selected component reference and the rigid-body modes of the shape function are eliminated using a set of reference conditions. Kinetic and strain energies are derived for each Timoshenko beam, thus identifying the beam mass and stiffness matrices which account for the rotary inertia and shear deformation effects. A new set of time-invariant matrices that describe the nonlinear inertia coupling between the reference motion and elastic deformation and account for the rotary inertia and shear is developed and it is shown that the form of these matrices as well as the mass and stiffness matrices are significantly affected by the inclusion of rotary inertia and shear. Numerical experimentations indicate that shear and rotary inertia can have a significant effect on the dynamics of flexible legged locomotion.

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.

Exact Dynamic Analysis of Space Structures Using Timoshenko Beam Theory

AIAA Journal ◽  
2004 ◽  
Vol 42 (4) ◽  
pp. 833-839 ◽  
Author(s):  
Jen-Fang Yu ◽  
Hsin-Chung Lien ◽  
B. P. Wang
Keyword(s):  
Dynamic Analysis ◽  
Timoshenko Beam ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Space Structures
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