scholarly journals Nonlinear free vibration of size-dependent microbeams with nonlinear elasticity under various boundary conditions

2021 ◽  
Vol 37 ◽  
pp. 380-403
Author(s):  
F Lin ◽  
J S Peng ◽  
S F Xue ◽  
L Yang ◽  
J Yang
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Nonlinear Elasticity ◽  
Couple Stress ◽  
Couple Stress Theory ◽  
Differential Quadrature Method ◽  
Nonlinear Free Vibration ◽  
Euler Beam ◽  
Stress Strain ◽  
Constitutive Relationships

Abstract In this study, nonlinear couple stress–strain constitutive relationships in the modified couple stress theory (MCST) are derived on the basis of previous classical stress–strain constitutive relationships of nonlinear elasticity materials. Hamilton's principle is employed to obtain higher-order nonlinear governing equations within the framework of the updated MCST, von Kármán geometric nonlinearity and Bernoulli–Euler beam theory. These mathematical formulations are solved numerically by the differential quadrature method together with an iterative algorithm to determine the nonlinear dynamic features of microbeams with four groups of boundary conditions. A detailed parametric study is conducted to analyze the influences of nonlinear elasticity properties on the nonlinear free vibration characteristics of the microbeams. Results show that these microbeams exhibiting nonlinear couple constitutive relationships have lower frequencies than their approximately simplified linear couple constitutive relationships. In addition, the frequencies of microbeams with nonlinear elasticity properties decrease as the vibration amplitude increases.

scholarly journals Nonlinear Free Vibration Analysis of Micro-beams Resting on Viscoelastic Foundation Based on the Modified Couple Stress Theory

2017 ◽  
Vol 64 (2) ◽  
pp. 239-256 ◽  
Author(s):  
Jafar Eskandari Jam ◽  
Milad Noorabadi ◽  
Nader Namdaran
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Couple Stress ◽  
Couple Stress Theory ◽  
Free Vibration Analysis ◽  
Modified Couple Stress Theory ◽  
Viscoelastic Foundation ◽  
Nonlinear Free Vibration ◽  
Stress Theory ◽  
Modified Couple Stress

AbstractIn this paper, nonlinear free vibration analysis of micro-beams resting on the viscoelastic foundation is investigated by the use of the modified couple stress theory, which is able to capture the size effects for structures in micron and sub-micron scales. To this aim, the gov-erning equation of motion and the boundary conditions are derived using the Euler–Bernoulli beam and the Hamilton’s principle. The Galerkin method is employed to solve the governing nonlinear differential equation and obtain the frequency-amplitude algebraic equation. Final-ly, the effects of different parameters, such as the mode number, aspect ratio of length to height, the normalized length scale parameter and foundation parameters on the natural fre-quency-amplitude curves of doubly simply supported beams are studied.

Dynamics of a micro scale Timoshenko beam subjected to a moving micro particle based on the modified couple stress theory

2016 ◽  
Vol 24 (3) ◽  
pp. 527-548 ◽  
Author(s):  
RA Jafari-Talookolaei ◽  
M Abedi ◽  
M Şimşek ◽  
M Attar
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Timoshenko Beam ◽  
Couple Stress ◽  
Couple Stress Theory ◽  
Modified Couple Stress Theory ◽  
Stress Theory ◽  
Micro Scale ◽  
Pasternak Elastic Foundation ◽  
Modified Couple Stress

In this study, the free and forced vibration analysis of a micro scale Timoshenko beam resting on a Pasternak elastic foundation and subjected to a moving micro particle is presented. Based on the modified couple stress theory and employing Hamilton’s principle, the governing equations along with the boundary conditions are derived. A semi-analytical solution is obtained for the free vibration of the problem by expressing the dynamic lateral displacement and cross-section rotation in terms of the series of Legendre polynomials and extremizing the objective functional of the problem with respect to the unknown displacements and Lagrange multipliers. Correspondingly, the computed eigenvalue information of the system is utilized in the modal expansion technique to obtain the transient dynamic response. For comparison purposes, the free vibration frequencies of the micro beam and the dynamic deflections using the classical Timoshenko beam theory are compared with previously published studies and very good agreements have been observed. Furthermore, more numerical examples for natural frequencies and dynamic deflection of the beam are presented and the effects of some parameters, such as the material length scale parameter, the velocity of micro particle, the Pasternak elastic foundation parameters, shear deformation effects and boundary conditions are examined.

Large Amplitude Free Vibration Analysis of Nanobeams Based on Modified Couple Stress Theory

2021 ◽  
pp. 2150129
Author(s):  
Ehsan Raeisi Estabragh ◽  
Gholam Hossein Baradaran
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Large Amplitude ◽  
Vibration Analysis ◽  
Couple Stress ◽  
Couple Stress Theory ◽  
Free Vibration Analysis ◽  
Modified Couple Stress Theory ◽  
Stress Theory ◽  
Modified Couple Stress

In this study, the large amplitude free vibration of nanobeams based on the modified couple stress theory was developed by using Total Lagrangian finite element formulation. In this study, Timoshenko beam theory has been used in free vibration analysis of nanobeams. Minimal kinematic assumptions have been used to model nanobeams. With this model, free vibration of nanobeams with small to large amplitude and with arbitrary boundary conditions can be analyzed. The numerical results obtained for free vibration based on the modified couple stress theory with small amplitude and the results obtained for free vibration with large amplitude without considering the modified couple stress theory are in good agreement with the similar results reported in previous research. Effects of the dimensionless length scale parameter, slenderness ratio, vibration amplitude and different boundary conditions on the nonlinear frequency ratio of nanobeams have been investigated. The results show the importance of considering nonlinear and size effects in the free vibration analysis of nanobeams with large amplitude.

Dynamic stiffness formulation for a micro beam using Timoshenko–Ehrenfest and modified couple stress theories with applications

2021 ◽  
pp. 107754632110482
Author(s):  
J Ranjan Banerjee ◽  
Stanislav O Papkov ◽  
Thuc P Vo ◽  
Isaac Elishakoff
Keyword(s):  
Free Vibration ◽  
Couple Stress ◽  
Scale Parameter ◽  
Couple Stress Theory ◽  
Dynamic Stiffness ◽  
Modified Couple Stress Theory ◽  
Length Scale Parameter ◽  
Length Scale ◽  
Stress Theory ◽  
Modified Couple Stress

Several models within the framework of continuum mechanics have been proposed over the years to solve the free vibration problem of micro beams. Foremost amongst these are those based on non-local elasticity, classical couple stress, gradient elasticity and modified couple stress theories. Many of these models retain the basic features of the Bernoulli–Euler or Timoshenko–Ehrenfest theories, but they introduce one or more material scale length parameters to tackle the problem. The work described in this paper deals with the free vibration problems of micro beams based on the dynamic stiffness method, through the implementation of the modified couple stress theory in conjunction with the Timoshenko–Ehrenfest theory. The main advantage of the modified couple stress theory is that unlike other models, it uses only one material length scale parameter to account for the smallness of the structure. The current research is accomplished first by solving the governing differential equations of motion of a Timoshenko–Ehrenfest micro beam in free vibration in closed analytical form. The dynamic stiffness matrix of the beam is then formulated by relating the amplitudes of the forces to those of the corresponding displacements at the ends of the beam. The theory is applied using the Wittrick–Williams algorithm as solution technique to investigate the free vibration characteristics of Timoshenko–Ehrenfest micro beams. Natural frequencies and mode shapes of several examples are presented, and the effects of the length scale parameter on the free vibration characteristics of Timoshenko–Ehrenfest micro beams are demonstrated.

scholarly journals Vibration Characteristics of Piezoelectric Microbeams Based on the Modified Couple Stress Theory

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
R. Ansari ◽  
M. A. Ashrafi ◽  
S. Hosseinzadeh
Keyword(s):  
Boundary Conditions ◽  
Couple Stress ◽  
Natural Frequencies ◽  
Couple Stress Theory ◽  
Equations Of Motion ◽  
Modified Couple Stress Theory ◽  
Beam Model ◽  
Governing Equations ◽  
Stress Theory ◽  
Modified Couple Stress

The vibration behavior of piezoelectric microbeams is studied on the basis of the modified couple stress theory. The governing equations of motion and boundary conditions for the Euler-Bernoulli and Timoshenko beam models are derived using Hamilton’s principle. By the exact solution of the governing equations, an expression for natural frequencies of microbeams with simply supported boundary conditions is obtained. Numerical results for both beam models are presented and the effects of piezoelectricity and length scale parameter are illustrated. It is found that the influences of piezoelectricity and size effects are more prominent when the length of microbeams decreases. A comparison between two beam models also reveals that the Euler-Bernoulli beam model tends to overestimate the natural frequencies of microbeams as compared to its Timoshenko counterpart.

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