Dynamic Stability of Elastic Mechanisms

1979 ◽  
Vol 101 (1) ◽  
pp. 149-153 ◽  
Author(s):  
M. Badlani ◽  
W. Kleinhenz
Keyword(s):  
Shear Deformation ◽  
Dynamic Stability ◽  
Timoshenko Beam ◽  
Rotary Inertia ◽  
Connecting Rod ◽  
Crank Mechanism ◽  
Beam Theories

A study of the dynamic stability of a slider-crank mechanism with an undamped elastic connecting rod is presented using the Euler-Bernoulli and Timoshenko beam theories. It is shown that new regions of instability exist when rotary inertia and shear deformation effects are included in the analysis.

Dynamic Stability of Timoshenko Beams by Finite Element Method

1976 ◽  
Vol 98 (4) ◽  
pp. 1145-1149 ◽  
Author(s):  
J. Thomas ◽  
B. A. H. Abbas
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stability Analysis ◽  
Shear Deformation ◽  
Dynamic Stability ◽  
Timoshenko Beam ◽  
Dynamic Instability ◽  
Rotary Inertia ◽  
Timoshenko Beams ◽  
Buckling Loads

A Finite Element model is developed for the stability analysis of Timoshenko beam subjected to periodic axial loads. The effect of the shear deformation on the static buckling loads is studied by finite element method. The results obtained show excellent agreement with those obtained by other analytical methods for the first three buckling loads. The effect of shear deformation and for the first time the effect of rotary inertia on the regions of dynamic instability are investigated. The elastic stiffness, geometric stiffness, and inertia matrices are developed and presented in this paper for a Timoshenko beam. The matrix equation for the dynamic stability analysis is derived and solved for hinged-hinged and cantilevered Timoshenko beams and the results are presented. Values of critical loads for beams with various shear parameters are presented in a graphical form. First four regions of dynamic instability for different values of rotary inertia parameters are presented. As the rotary inertia parameter increases the regions of instability get closer to each other and the width of the regions increases thus making the beam more sensitive to periodic forces.

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.

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.

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.

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.

Dynamic stability analysis of linkages with elastic members via analog simulation

SIMULATION ◽  
1972 ◽  
Vol 18 (2) ◽  
pp. 67-74
Author(s):  
J.A. Seevers ◽  
A.T. Yang
Keyword(s):  
Stability Analysis ◽  
Dynamic Stability ◽  
Flexural Vibration ◽  
Parabolic Partial Differential Equation ◽  
Speed Ratio ◽  
Connecting Rod ◽  
Analog Simulation ◽  
Modal Equation ◽  
Planar Linkages ◽  
Crank Mechanism

Small-amplitude flexural vibration of a slightly deformable binary link in a planar linkage is governed by a fourth-order parabolic partial differential equation. Specifically, for a slider- crank mechanism with an elastic connecting rod, stability charts are obtained via analog simulation on the basis of the principal modal equation of vibration. It is found that stability of the mechanism depends on three nondimensional parameters - length ratio, mass ratio, and rigidity-speed ratio. The procedure of dynamic stability analysis via analog simulation for the slider-crank may be applied to other planar linkages with elastic binary links.

Vibration Analysis of the Flexible Connecting Rod With the Breathing Crack in a Slider-Crank Mechanism

2013 ◽  
Vol 135 (6) ◽  
Author(s):  
Yan-Shin Shih ◽  
Chen-Yuan Chung
Keyword(s):  
Governing Equation ◽  
Moving Boundary ◽  
Beam Theory ◽  
Crack Model ◽  
Breathing Crack ◽  
Connecting Rod ◽  
Bernoulli Beam ◽  
The Galerkin Method ◽  
Euler Bernoulli Beam ◽  
Crank Mechanism

This paper investigates the dynamic response of the cracked and flexible connecting rod in a slider-crank mechanism. Using Euler–Bernoulli beam theory to model the connecting rod without a crack, the governing equation and boundary conditions of the rod's transverse vibration are derived through Hamilton's principle. The moving boundary constraint of the joint between the connecting rod and the slider is considered. After transforming variables and applying the Galerkin method, the governing equation without a crack is reduced to a time-dependent differential equation. After this, the stiffness without a crack is replaced by the stiffness with a crack in the equation. Then, the Runge–Kutta numerical method is applied to solve the transient amplitude of the cracked connecting rod. In addition, the breathing crack model is applied to discuss the behavior of vibration. The influence of cracks with different crack depths on natural frequencies and amplitudes is also discussed. The results of the proposed method agree with the experimental and numerical results available in the literature.

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