Flexural Vibration of an Atomic Force Microscope Cantilever Based on Modified Couple Stress Theory

2015 ◽  
Vol 15 (07) ◽  
pp. 1540025 ◽  
Author(s):  
Li-Na Liang ◽  
Liao-Liang Ke ◽  
Yue-Sheng Wang ◽  
Jie Yang ◽  
Sritawat Kitipornchai
Keyword(s):  
Size Effect ◽  
Couple Stress ◽  
Scale Parameter ◽  
Couple Stress Theory ◽  
Flexural Vibration ◽  
Modified Couple Stress Theory ◽  
Length Scale ◽  
Stress Theory ◽  
Atomic Force ◽  
Modified Couple Stress

This paper is concerned with the flexural vibration of an atomic force microscope (AFM) cantilever. The cantilever problem is formulated on the basis of the modified couple stress theory and the Timoshenko beam theory. The modified couple stress theory is a nonclassical continuum theory that includes one additional material parameter to describe the size effect. By using the Hamilton's principle, the governing equation of motion and the boundary conditions are derived for the AFM cantilevers. The equation is solved using the differential quadrature method for the natural frequencies and mode shapes. The effects of the sample surface contact stiffness, length scale parameter and location of the sensor tip on the flexural vibration characteristics of AFM cantilevers are discussed. Results show that the size effect on the frequency is significant when the thickness of the microcantilever has a similar value to the material length scale parameter.

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.

Thermal Effect on Static Bending, Vibration and Buckling of Reddy Beam Based on Modified Couple Stress Theory

2013 ◽  
Vol 332 ◽  
pp. 331-338 ◽  
Author(s):  
Ali Reza Daneshmehr ◽  
Mostafa Mohammad Abadi ◽  
Amir Rajabpoor
Keyword(s):  
Thermal Effect ◽  
Couple Stress ◽  
Scale Parameter ◽  
Couple Stress Theory ◽  
Modified Couple Stress Theory ◽  
Length Scale ◽  
Bending Vibration ◽  
Stress Theory ◽  
Micro Scale ◽  
Modified Couple Stress

A microstructure-dependent Reddy beam theory (RBT) which contain only one material length scale parameter and can capture the size effect in micro-scale material unlike the classical theory is developed .using the variational principle energy the governing equation of motion is derived based on modified couple stress theory for the simply supported beam. the equations obtained are solved by Fourier series and the influence of the length scale parameter and thermal effect on static bending, vibration and buckling analysis of micro-scale Reddy beam is investigated.

scholarly journals Nonlinear Vibration Analysis of Axially Functionally Graded Microbeams Based on Nonlinear Elastic Foundation Using Modified Couple Stress Theory

2020 ◽  
Vol 64 (2) ◽  
pp. 97-108
Author(s):  
Mehdi Alimoradzadeh ◽  
Mehdi Salehi ◽  
Sattar Mohammadi Esfarjani
Keyword(s):  
Differential Equation ◽  
Forced Vibration ◽  
Couple Stress ◽  
Scale Parameter ◽  
Couple Stress Theory ◽  
Modified Couple Stress Theory ◽  
Functionally Graded ◽  
Length Scale ◽  
Stress Theory ◽  
Modified Couple Stress

In this study, a non-classical approach was developed to analyze nonlinear free and forced vibration of an Axially Functionally Graded (AFG) microbeam by means of modified couple stress theory. The beam is considered as Euler-Bernoulli type supported on a three-layered elastic foundation with Von-Karman geometric nonlinearity. Small size effects included in the analysis by considering the length scale parameter. It is assumed that the mass density and elasticity modulus varies continuously in the axial direction according to the power law form. Hamilton's principle was implemented to derive the nonlinear governing partial differential equation concerning associated boundary conditions. The nonlinear partial differential equation was reduced to some nonlinear ordinary differential equations via Galerkin's discretization technique. He's Variational iteration methods were implemented to obtain approximate analytical expressions for the frequency response as well as the forced vibration response of the microbeam with doubly-clamped end conditions. In this study, some factors influencing the forced vibration response were investigated. Specifically, the influence of the length scale parameter, the length of the microbeam, the power index, the Winkler parameter, the Pasternak parameter, and the nonlinear parameter on the nonlinear natural frequency as well as the amplitude of forced response have been investigated.

Determination of vibration of single-layered graphene sheets using nonlocal modified couple stress theory

2021 ◽  
pp. 2250011
Author(s):  
F. Attar ◽  
R. Khordad ◽  
A. Zarifi
Keyword(s):  
Couple Stress ◽  
Plate Theory ◽  
Couple Stress Theory ◽  
Nonlocal Parameter ◽  
Modified Couple Stress Theory ◽  
Length Scale ◽  
Vibration Modes ◽  
Classical Plate Theory ◽  
Stress Theory ◽  
Modified Couple Stress

The free vibration of single-layered graphene sheet (SLGS) has been studied by nonlocal modified couple stress theory (NMCS), analytically. Governing equation of motion for SLGS is obtained via thin plate theory in conjunction with Hamilton’s principle for two cases: (1) using nonlocal parameter only for stress tensor, (2) using nonlocal parameter for both stress and couple stress tensors. Navier’s approach has been used to solve the governing equations for simply supported boundary conditions. It is found that the frequency ratios decrease with increasing nonlocal parameter and also with enhancing vibration modes, but increase with raising length scale parameter. The nonlocal and length scale parameters are more prominent in higher vibration modes. The obtained results have been compared with the previous studies obtained by using classical plate theory, the modified couple stress theory and nonlocal elasticity theory, separately.

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