On the Finite Deflection Dynamics of Thin Elastic Beams

1971 ◽  
Vol 38 (4) ◽  
pp. 961-963 ◽  
Author(s):  
S. Y. Lee
Keyword(s):  
Large Deflection ◽  
Dynamic Theory ◽  
Axial Stress ◽  
Small Strain ◽  
Beam Equation ◽  
Finite Deflection ◽  
Rotatory Inertia ◽  
Geometric Nonlinearities ◽  
Thin Beams ◽  
Timoshenko Beam Equation

A dynamic theory of thin beams undergoing large deflection but small strain is derived. Geometric nonlinearities are preserved but the material is assumed to behave linearly. Contributions due to rotatory inertia, shear deformation, and axial stress resultants are included. The resulting equations are analyzed by characteristic techniques. Wave-propagation speeds, jump properties, and their physical significances are discussed. A simplifying assumption generates a modified Timoshenko beam equation which is valid for large deformation.

scholarly journals Controlling Resonator Nonlinearities and Modes through Geometry Optimization

Micromachines ◽  
2021 ◽  
Vol 12 (11) ◽  
pp. 1381
Author(s):  
Amal Z. Hajjaj ◽  
Nizar Jaber
Keyword(s):  
Axial Stress ◽  
Geometry Optimization ◽  
Beam Equation ◽  
Design Parameters ◽  
Dynamical Response ◽  
Successful Implementation ◽  
Shape Effect ◽  
Mems Resonators ◽  
Wide Range ◽  
The Galerkin Method

Controlling the nonlinearities of MEMS resonators is critical for their successful implementation in a wide range of sensing, signal conditioning, and filtering applications. Here, we utilize a passive technique based on geometry optimization to control the nonlinearities and the dynamical response of MEMS resonators. Also, we explored active technique i.e., tuning the axial stress of the resonator. To achieve this, we propose a new hybrid shape combining a straight and initially curved microbeam. The Galerkin method is employed to solve the beam equation and study the effect of the different design parameters on the ratios of the frequencies and the nonlinearities of the structure. We show by adequately selecting the parameters of the structure; we can realize systems with strong quadratic or cubic effective nonlinearities. Also, we investigate the resonator shape effect on symmetry breaking and study different linear coupling phenomena: crossing, veering, and mode hybridization. We demonstrate the possibility of tuning the frequencies of the different modes of vibrations to achieve commensurate ratios necessary for activating internal resonance. The proposed method is simple in principle, easy to fabricate, and offers a wide range of controllability on the sensor nonlinearities and response.

Response of a Simply Supported Timoshenko Beam to a Purely Random Gaussian Process

1958 ◽  
Vol 25 (4) ◽  
pp. 496-500
Author(s):  
J. C. Samuels ◽  
A. C. Eringen
Keyword(s):  
Timoshenko Beam ◽  
Beam Theory ◽  
Beam Equation ◽  
Mean Square ◽  
Bending Stress ◽  
Random Loading ◽  
Simply Supported ◽  
The Mean ◽  
Timoshenko Beam Equation ◽  
Classical Beam Theory

Abstract The generalized Fourier analysis is applied to the damped Timoshenko beam equation to calculate the mean-square values of displacements and bending stress, resulting from purely random loading. Compared with the calculations, based on the classical beam theory, it was found that the displacement correlations of both theories were in excellent agreement. Moreover, the mean square of the bending stress, contrary to the results of the classical beam theory, was found to be convergent. Computations carried out with a digital computer are plotted for both theories.

Method of Initial Functions in Two-Dimensional Elastodynamic Problems

1970 ◽  
Vol 37 (1) ◽  
pp. 137-140 ◽  
Author(s):  
Y. C. Das ◽  
A. V. Setlur
Keyword(s):  
Time Dependent ◽  
Beam Equation ◽  
Deep Beam ◽  
Two Dimensional ◽  
Plane State ◽  
Infinite Beam ◽  
Transverse Loads ◽  
Layered Beams ◽  
Particular Solution ◽  
Timoshenko Beam Equation

The method of initial functions is formulated for the two-dimensional elastodynamic problems both in plane stress and plane strain. Knowing the stresses and displacements on a given layer of an elastic body in a plane state, the stresses and displacements at any other layer or point can be obtained by the method developed in this paper. The method is illustrated in the case of a deep beam subjected to time-dependent transverse loads. A particular solution for a straight-crested wave traveling along an infinite beam is obtained. Exact value of the dynamic shear coefficient for the well-known Timoshenko beam equation is derived. This method is also applied to the dynamic analysis of layered beams.

Nonlinear Analysis of the In-Plane Young’s Moduli of Two-Dimensional Cellular Materials with Negative Poisson’s Ratios

2007 ◽  
Vol 334-335 ◽  
pp. 157-160
Author(s):  
Hui Wan ◽  
Zhen Yu Hu ◽  
Wu Jun Bao ◽  
Guo Ming Hu
Keyword(s):  
Large Deflection ◽  
Small Deformation ◽  
Small Strain ◽  
Cellular Materials ◽  
Deformation Model ◽  
Two Dimensional ◽  
Young’S Moduli ◽  
Incomplete Elliptic Integrals ◽  
Poisson's Ratios ◽  
Poisson’S Ratios

This study deals with the in-plane Young’s moduli of two-dimensional auxetic cellular materials with negative Poisson’s ratios. The in-plane Young’s moduli of these cellular materials are theoretically analyzed, and calculated from the cell member bending with large deflection. Expressions for the in-plane Young’s moduli of the above-mentioned cellular materials are given by incomplete elliptic integrals. It is found that the in-plane Young’s moduli of two-dimensional cellular materials with negative Poisson’s ratios depend both on the geometry of the cell, and on the induced strain of these cellular materials. The in-plane Young’s moduli are no longer constants at large deformation. But at the limit of small strain, they converge to the results predicted by the small deformation model of flexure.

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.

Large-Deformation Theory of Shells of Revolution

1967 ◽  
Vol 34 (1) ◽  
pp. 56-58 ◽  
Author(s):  
W. Flu¨gge ◽  
S-C. Chou
Keyword(s):  
Internal Pressure ◽  
Large Deformation ◽  
Deformation Theory ◽  
Large Deflection ◽  
Small Strain ◽  
Strain Theory ◽  
Large Strains ◽  
Shells Of Revolution ◽  
Elastic Strains ◽  
Theory Of Shells

In this paper, nonlinear membrane equations are derived for a shell of revolution under the assumption that not only are the displacements and rotations large, but that, also, large strains are admitted. The equations, therefore, are aimed at shells which are not only very thin, but which are also made of a material which permits large elastic strains. The special difficulties resulting from this extension of the theory are discussed. As an example for the application of the equations, a circular toroid subjected to internal pressure is studied. Numerical results are given for a level of loading which lies clearly outside the domain of a large-deflection, small-strain theory.

Sign in / Sign up

Export Citation Format

Share Document