Vibration Of Microbeam Research Articles

Pull-in instability and nonlinear vibration of microbeam with piezoelectric patch driven by electrostatic force using modified couple stress theory

The static and dynamic pull-in characteristics of electrostatically and piezo-electrically actuated microbeam are investigated. The static pull-in voltage is evaluated using analytical methods based on MCST, CCMT and validated through the use of FE based COMSOL multiphysics tools. The dynamic pull-in voltage obtained from numerical technique and verified with existing literature, lower than the static pull-in voltage. The piezoelectric patch has a significant influence on the pull-in voltage and an effect on the aspect ratio. Moreover, the pull-in band and instability zone of the microbeam with weak and hard forcing function have been investigated using multiple scale method.

Nonlinear Random Vibration of Micro-Beams Resting on Visco-Elastic Foundation via the Modified Couple Stress Theory

Nonlinear Random Vibration of Micro-Beams Resting on Visco-Elastic Foundation via the Modified Couple Stress Theory

Nonlinear vibration of microbeams subjected to a uniform magnetic field and rested on nonlinear elastic foundation

Abstract This study investigates the nonlinear vibration motions of the Euler–Bernoulli microbeam on a nonlinear elastic foundation in a uniform magnetic field based on Modified Couple Stress Theory (MCST). The effect of size, foundation, and magnetic field on the nonlinear vibration motion of microbeam has been examined. The governing equations related to the nonlinear vibration motions of the microbeam are obtained by using Hamilton’s Principle, and the Multiple Time Scale Method was used to obtain the solutions for the governing equations. The linear natural frequencies of microbeam are presented in the table according to nonlinear parameters and boundary conditions. The linear and nonlinear natural frequency ratio graphs are shown. The present study results are also compared with previous work for validation. It is observed that length scale parameters and magnetic force have a more significant effect on the natural frequency of microbeams. It is seen that when the linear elastic foundation coefficient, the Pasternak foundation and the magnetic force effects increase, the ratio of nonlinear and linear natural frequency decreases.

Size-dependent nonlinear forced vibration of microbeam using two-phase local/nonlocal viscoelastic integral model with thermal effects

Combination strain-driven (εD) and stress-driven (σD) two-phase local/nonlocal integral models (TPNIM) with classic Kelvin-Voigt viscoelastic model, εD- and σD-two-phase local/nonlocal viscoelastic integral models (TPNVIM) are proposed to study the viscoelastic nonlinear forced vibration behaviors of microbeams in thermal environment. Based on von-Karman large deformation theory and the Hamilton’s principle, the differential governing equations of motion and boundary conditions are derived for nonlinear forced vibration of microbeam. Several non-dimensional variables are introduced to simplify the mathematical formulation. Nonlocal viscoelastic equations in integral form are transferred unitedly into equivalent differential form with constitutive constraints. The axial force of nonlocal microbeam with immovable ends is derived explicitly. Vibration frequency and corresponding linear vibration mode shape are derived through Laplace transformation for linear thermo-elastic vibration of microbeam using εD- and σD-TPNIMs. The Ritz-Galerkin approach is utilized to derive the reduced differential governing equation for nonlinear vibration of microbeam, which has exact same form as single-degree-of-freedom oscillator with cubic nonlinearity when material viscosity is neglected. The method of multiple scales (MMS) based on trigonometric functions is utilized to solve the reduced nonlinear differential equation approximately. The frequency-amplitude responses are addressed for primary and super-harmonic resonances of microbeam. The effects of viscous coefficient, amplitude of excitation force and environmental temperature variation as well as nonlocal length parameter on the nonlinear vibration behavior of microbeam are investigated numerically for different boundary conditions.

Nonlinear Random Vibrations of Micro-Beams Resting on Visco-Elastic Foundation Via the Couple Stress Theory

Nonlinear Random Vibrations of Micro-Beams Resting on Visco-Elastic Foundation Via the Couple Stress Theory

Nonlinear random vibrations of micro-beams with fractional viscoelastic core

This work investigates the nonlinear vibration of a fractional viscoelastic micro-beam exposed to random excitation. For the micro-scale effect, the micro-beam model is established based on the Modified Couple Stress Theory (MCST). The fractional nonlinear partial differential equation is obtained using Von-Karman’s nonlinear strains and the Kelvin–Voigt fractional viscoelastic model. The micro-beam is exposed to a random force which its amplitude is considered as the Gaussian white noise. The Finite Difference Method (FDM) and the Galerkin method are applied to discretize the stochastic governing equation of motion. To solve the stochastic partial differential equation, a numerical discretization for the white noise excitation is applied. Moreover, the Statistical Linearization (SL) method and the Spectral Representation (SR) method are also employed and a comparison is done between these methods. The simulations show good agreement between the numerical method and other methods. In addition, it is shown that the proposed method can be used for different boundary conditions of the micro-beam. The impacts of the fractional derivative order, white noise intensity, viscoelastic model, and length-scale parameter on the statistical properties of the micro-beam are shown. The numerical simulations indicate that the high values of fractional-order can significantly affect the variance and the Root Mean Square (RMS) of the transverse amplitude, However, this influence for small values of the fractional-order is negligible; the impact of the fractional-order derivative on the statistical characteristics of the micro-beam is very different from 0<α<0.6, and 0.6<α<1.

Viscoelastic initially stressed microbeam heated by an intense pulse laser via photo-thermoelasticity with two-phase lag

This work aims to assess the response of viscoelastic Kelvin–Voigt microscale beams under initial stress. The microbeam is photostimulated by the light emitted by an intense picosecond pulsed laser. The photothermal elasticity model with dual-phase lags, the plasma wave equation and Euler–Bernoulli beam theory are utilized to construct the system equations governing the thermoelastic vibrations of microbeams. Using the Laplace transform technique, the problem is solved analytically and expressions are provided for the distributions of photothermal fields. Taking aluminum as a numerical example, the effect of the pulsed laser duration coefficient, viscoelasticity constants and initial stress on photothermal vibrations has been studied. In addition, a comparison has been made between different models of photo-thermoelasticity to validate the results of the current model. Photo-microdynamic systems might be monolithically integrated on aluminum microbeams using microsurface processing technology as a result of this research.

Size-Dependent Free Vibration of Microbeams Submerged in Fluid

The size-dependent free vibration of microbeams submerged in fluid is presented in this paper based on the modified couple stress theory. Two different cross-section shapes of microbeams are considered, i.e. the circular cross-section and rectangular cross-section. This nonclassical microbeam model is introduced for capturing the size effect of microstructures. In this fluid and structure coupled system, the effect of hydrodynamic loading on microbeams can be expressed by the added mass method. By using Hamilton’s principle and differential quadrature (DQ) method, we can derive governing equations of microbeams in fluid, and then rewrite them in the discretized form. The frequencies and mode shapes for microbeams are determined by proposing an iterative method. Numerical examples are given to show the effect of fluid depth, fluid density, length scale parameter, slenderness ratio, boundary condition and cross-section shape on the vibration characteristics.

Influence of Temperature Pulse on a Nickel Microbeams under Couple Stress Theory

In this paper, the vibration of microbeams due to a temperature pulse has been investigated. The thermoelastic coupled equations for microbeam resonator have been derived via the modified theory of couple stress in connection with the generalized thermoelasticity with relaxation time. The analytical expressions for studied fields due to modified couple stress for the microbeam have been obtained by applying the Laplace transform method. In addition, some comparisons have been displayed in graphs to estimate the effects of different parameters such as the couple stress parameter and pulse of temperature on the considered fields. Numerical conclusions demonstrate that the estimation of deflection expected by the new theory is lower than that of the classical one. Comparisons are made with the results of different models in the absence and presence of couple stress theory. Particular cases of interest are also derived.

Theoretical Analysis of Free Vibration of Microbeams under Different Boundary Conditions Using Stress-Driven Nonlocal Integral Model

Previous studies have shown that Eringen’s differential nonlocal model leads to some inconsistencies for both the Euler–Bernoulli and Timoshenko beams subjected to different boundary conditions. In this paper, the free vibration analysis of the Euler–Bernoulli and Timoshenko beams is performed theoretically using the stress-driven nonlocal integral model. The Fredholm-type integral constitutive equations of the first kind are transformed to Volterra integral equations of the first kind by simply adjusting the limit of integrals. Also, the general solutions to the deflection, bending moment and so on are derived by solving the integro-differential governing equations by the Laplace transformation, of which the unknown constants are determined by the boundary conditions and extra constraint equations related to the constitutive relationship. Then the characteristic equations for free vibration of the Euler–Bernoulli and Timoshenko beams are derived, from which the vibration frequency for different boundary conditions can be determined. The effects of nonlocal parameter and vibration order on the natural frequency of the Euler–Bernoulli and Timoshenko beams are investigated numerically. The results from the present model are validated against those existing in the literature, and demonstrated to be theoretically consistent.

Nonlinear vibration of microbeams based on the nonlinear elastic foundation using the equivalent linearization method with a weighted averaging

We investigate the nonlinear vibration of microbeams based on the nonlinear elastic foundation through the modified couple stress theory. The equivalent linearization method with a weighted averaging is used to solve approximately the ordinary differential equation that describes the equation of motion of the microbeam. The effects of length scale parameter, the flexural rigidity ratio, the slenderness ratio, the Winkler parameter, the Pasternak parameter and the nonlinear foundation parameter on the nonlinear vibration of the microbeam are studied and discussed.

Nonlinear Vibration of Microbeams under Magnetic Field Using the Modified Couple Stress Theory

Nonlinear Vibration of Microbeams under Magnetic Field Using the Modified Couple Stress Theory

Postbuckling and Free Nonlinear Vibration of Microbeams Based on Nonlinear Elastic Foundation

In this paper, post-buckling and free nonlinear vibration of microbeams resting on nonlinear elastic foundation subjected to axial force are investigated. The equations of motion of microbeams are derived by using the modified couple stress theory. Using Galerkin’s method, the equation of motion of microbeams is reduced to the nonlinear ordinary differential equation. By using the equivalent linearization in which the averaging value is calculated in a new way called the weighted averaging value, approximate analytical expressions for the nonlinear frequency of microbeams with pinned–pinned and clamped–clamped end conditions are obtained in closed-forms. Comparisons with previous solutions are showed accuracy of the present solutions. Effects of the material length scale parameter and the axial compressive force on the frequency ratios of microbeams; and effect of the material length scale parameter on the buckling load ratios of microbeams are investigated in this paper.

A Porous Microbeam Model for Bending and Vibration Analysis Based on the Sinusoidal Beam Theory and Modified Strain Gradient Theory

In this research, we analyze size-dependent bending and vibration of microbeams made of porous metal foams. The porous microbeam model is developed based on the sinusoidal beam theory and the modified strain gradient theory. Hamilton’s principle is employed to obtain the governing equations and boundary conditions of the porous microbeam. Analytical solutions are presented for deflections and natural frequencies of the porous microbeam by using Navier’s method. The influences of the porosity distribution, the porosity coefficient, the slenderness ratio, and the microbeam thickness are clarified on the static bending and free vibration of porous microbeams. These findings can be applied to the design of metal foam microstructures in engineering.

Nonlinear Vibration of Microbeams Based on The Elastics Foundation Using High-Order Energy Balance Method and Global Error Minimization Method

In this paper, nonlinear vibration of microbeams based on the nonlinear elastic foundation is investigated. The equation of motion of microbeams based on three-layered nonlinear elastic medium (shear, linear and nonlinear layers) is described by the partial differential equation by using the modified couple stress theory. The equation of motion of microbeams is transformed into the ordinary differential equation by using Galerkin method. The high-order Energy Balance method and the high-order Global Error Minimization method are used to get the frequency – amplitude relationships for the nonlinear vibration of microbeams with pinned-pinned and clamped-clamped end conditions. Comparisons between the present solutions and the privious solutions show the accuracy of the obtained results.

Bending and free vibration of functionally graded piezoelectric microbeams based on the modified couple stress theory

A size-dependent model for bending and free vibration of functionally graded piezoelectric (FGP) microbeam is developed by using modified couple stress theory and a unified higher order beam theory. This model can be specialized to various beam models, such as Euler–Bernoulli, Timoshenko as well as Reddy beam ones and vice versa. The governing equations of motion and associated boundary conditions are derived from Hamilton’s principle. Only one material length scale parameter is introduced to capture the size effect. The analytical solutions of simply supported FGP microbeam are presented by using Navier approach to bring out the effect of the material length scale parameter on the bending and free vibration of microbeam. Numerical simulations are presented to account for the effect of various parameters, such as material length scale parameters, volume fraction indexes, and slenderness ratios on the responses of static bending and free vibration of FGP microbeam.

Dynamical characteristics of fluid-conveying microbeams actuated by electrostatic force

In this paper, the dynamic characteristics and pull-in instability of electrostatically actuated microbeams which convey internal fluids are investigated. A theoretical model is developed by considering the elastic structure, laminar flow and electrostatic field to characterize the dynamic behavior. In addition, the energy dissipation induced by the fluid viscosity is studied through analyzing the fluid–structure interactions between the laminar fluid flow and oscillating microbeam by comprehensively considering the effects of velocity profile and fluid viscosity. The results indicate that the system is subjected to both the pull-in instability and the fluid-induced instability. It is demonstrated that as the flow velocity increases, both the static pull-in voltage and the dynamic one decrease for clamped–clamped microbeams while increase for clamped-free microbeams. It is also shown that the applied voltage and the steady flow can adjust the resonant frequency. The perturbation viscous flow caused by the vibration of microbeam is manifested to result in energy dissipation. The quality factor decreases with the increment of both the mode order and flow velocity. However, when the oscillating flow dominates, the flow velocity has no obvious effect.

A practical method for calculating eigenfrequencies of a cantilever microbeam with the attached tip mass

This paper is concerned with the free vibration of cantilever microbeams with attached tip mass in a systematical manner. Small size effects on the vibrations of the microbeam are taken into consideration by introducing a scale parameter. A Fourier sine series is used to represent the lateral displacement function. Stokes’ transformation is applied in the present formulation and corresponding derivatives are presented explicitly. The present formulations can be readily reduced to those for others classical elasticity models by setting corresponding small scale parameter to zero. Several parametric studies are performed to validate the present solutions and the effects of various important physical parameters (scale parameter, tip mass) are investigated.

Natural vibrations of micro beams with nonrigid supports

The present study aims to provide some new information for the design of micro systems. It deals with free vibrations of Bernoulli–Euler micro beams with nonrigid supports. The study is based on the formulation of the modified couple stress theory. This theory is a nonclassical continuum theory that allows one to capture the small-scale size effects in the vibrational behavior of micro structures. More realistic boundary conditions are represented with elastic edge conditions. The effect of Poisson’s ratio on the micro beam characteristics is also analyzed. The present results revealed that the characterization of real boundary conditions is much more important for micro beams than for macro beams, and this is an assessment that cannot be ignored.

A higher-order micro-beam model with application to free vibration

ABSTRACTThe free vibration of micro-beams is analyzed employing three different beam models. The previously obtained equations of motion for Bernoulli–Euler and Timoshenko models are solved analytically. A higher-order model is devised, which satisfies the lateral boundary conditions of micro-beams. The equations of motion with associated boundary conditions are derived by means of Hamilton's principle. The generalized differential quadrature method is used for the solution of the equations. The first five natural frequencies are obtained for micro-beams with three length-scale parameter/height ratios and five different boundary conditions.