Euler Bernoulli Microbeams Research Articles

Vibration behavior of a rotating non-uniform FG microbeam based on the modified couple stress theory and GDQEM

In this paper, for the first time, the size dependent vibration behavior of a rotating non-uniform functionally graded (FG) Timoshenko and Euler–Bernoulli microbeam based on the modified couple stress theory is presented. Also, the impact of the shear deformation on natural frequencies of the microbeam considering different values of the material length scale parameter, the angular velocity and the rate of cross section change is investigated. The mechanical and physical properties of the FG microbeam are varying along the thickness according to a power law equation. To determine governing equations, Hamilton’s principle and a generalized differential quadrature element method (GDQEM) are applied. The natural frequencies of the microbeam for cantilever and propped cantilever boundary conditions are calculated. The accuracy and validity of the results are shown by several numerical examples. The influences of some parameters such as the small scale, the rate of cross section change, the angular velocity and the gradient index of the FG material on the first two natural frequencies, are plotted in several diagrams and are shown in tables. The results of this study can be used in designing and optimizing elastic and rotary type micro-electro-mechanical systems (MEMS) like micro-motors and micro-robots included rotating parts.

On size-dependent vibration of rotary axially functionally graded microbeam

Study of the mechanical properties of axially functionally graded (AFG) microbeams is a challenging work due to the varying mechanical properties of these microbeams along the axis. In the present study, the transverse vibration of a rotary tapered AFG Euler–Bernoulli microbeam is studied based on the modified couple stress theory by considering the axial forces which are due to the rotation, in the form of true spatial variation. The governing equations and boundary conditions are derived according to the Hamilton's principle and the governing equations are solved with the aid of the generalized differential quadrature element method (GDQEM). The effects of the small-scale parameter, length and width of the beam, rate of cross-section change and nondimensional angular velocity on the vibration behavior of the non-uniform microbeam are studied for cantilever and propped cantilever boundary conditions. The vibrational behavior of AFG microbeam is also compared with pure metal and pure ceramic. The results are useful in designation of micromachines such as micromotors and micro-rotors.

Adaptive boundary control of the size-dependent behavior of Euler–Bernoulli micro-beams with unknown parameters and varying disturbance

In this paper, modeling and vibration control of a strain-gradient clamped-free Euler–Bernoulli micro-beam exposed to varying disturbance is studied. A strain-gradient model of the Euler–Bernoulli micro-beam is represented in this paper and consisted of one partial differential equation and six ordinary equations as governing motion equation and boundary conditions, respectively. A boundary controller is proposed to suppress the system’s vibration. The controller is derived based on the direct Lyapunov method. An adaptation law is devised to assure system’s stability under parametric uncertainties. With the proposed adaptive robust boundary control, uniform boundedness under environmental and input disturbances could be achieved. The state of the system is proven to converge to a small neighborhood of zero by appropriately choosing design parameters. Simulations are provided to illustrate the applicability and effectiveness of the proposed controller.

Size-dependent electro-elasto-mechanics of MEMS with initially curved deformable electrodes

The aim of this paper is to examine the nonlinear size-dependent electro-elasto-statics/dynamics of a MEMS with an initially curved deformable electrode, based on the modified couple stress theory; in particular, the nonlinear motion characteristics of an initially curved deformable electrode actuated by a combination of DC and AC voltages are examined, taking into account small-size effects through use of the modified couple stress theory. The deformable electrode is modelled by means of an initially curved Euler–Bernoulli microbeam theory; the coupling between the electrical field and restoring microbeam force is demonstrated by electrical/displacement nonlinearities in the continuous model of the system, which is obtained by means of Hamilton׳s principle. The Galerkin method is employed to obtain a high-dimensional reduced-order model of the continuous system. The reduced-order model is solved by means of the pseudo-arclength continuation technique in order to analyse pull-in instabilities, snap-through motions, and nonlinear dynamical behaviour. Particularly, the electro-elasto-static deformation of the deformable electrode and pull-in instabilities are analysed when the system is subject to a DC voltage. The electro-elasto-dynamic motions of the deformable electrode are also analysed when the system is subject to both DC and AC voltages; the AC frequency-motion and AC amplitude-motion characteristics of the system are examined in the presence of coupled mechanical and electrical nonlinearities. A stability analysis is conducted via the Floquet theory. The importance of employing the modified couple stress theory, rather than the classical continuum theory, is highlighted by comparing the system response based on each of these theories.

Size dependent nonlinear free vibration of an axially functionally graded (AFG) microbeam using He’s variational method

Nonlinear free vibration of axially functionally graded (AFG) Euler–Bernoulli microbeams with immovable ends is studied by using the modified couple stress theory. The nonlinearity of the problem stems from the von-Kármán’s nonlinear strain–displacement relationships. Elasticity modulus and mass density of the microbeam vary continuously in the axial direction according to a simple power-law form. The nonlinear governing partial differential equation and the associated boundary conditions are derived by Hamilton’s principle. By using Galerkin’s approach, the nonlinear governing partial differential equation is reduced to a nonlinear ordinary differential equation. He’s variational method is utilized to obtain the approximate closed form solution of the nonlinear ordinary governing equation. Pinned–pinned and clamped–clamped boundary conditions are considered. The influences of the length scale parameters, material variation, vibration amplitude, and boundary conditions on vibration responses are examined in detail.

Vibration analysis of electrostatically actuated nonlinear microbridges based on the modified couple stress theory

In this paper natural frequency of electrostatically actuated microbridges is investigated based on the modified couple stress theory. Nonlinear formulation of Euler–Bernoulli microbeam is derived using Hamilton’s principle. By considering the von-Karman strain, the nonlinearities caused by the mid-plane stretching are included in the formulation. To confirm the model, results of static deflection and natural frequency of microbeams are calculated using modified couple stress theory and compared to those evaluated based on the classical theory and experimental observations. At first, from experimental results of static deflection of a microcantilever, estimation for length scale parameter of polysilicon is presented. Using this value of length scale, natural frequency of microbridges under electrostatic actuation is calculated on the basis of the modified couple stress theory and compared with experimental results. The results of the modified couple stress theory are in very good agreement with experimental findings while the difference between the results predicted based on the classical theory and experimental findings is considerable.

Transient analysis of nonlinear Euler–Bernoulli micro-beam with thermoelastic damping, via nonlinear normal modes

In this paper an Euler–Bernoulli model has been used for vibration analysis of micro-beams with large transverse deflection. Thermoelastic damping is considered to be the dominant damping mechanism and introduced as imaginary stiffness into the equation of motion by evaluating temperature profile as a function of lateral displacement. The obtained equation of motion is analyzed in the case of pure single mode motion by two methods; nonlinear normal mode theory and the Galerkin procedure. In contrast with the Galerkin procedure, nonlinear normal mode analysis introduces a nonconventional nonlinear damping term in modal oscillator which results in strong damping in case of large amplitude vibrations. Evaluated modal oscillators are solved using harmonic balance method and tackling damping terms introduced as an imaginary stiffness is discussed. It has been shown also that nonlinear modal analysis of micro-beam with thermoelastic damping predicts parameters such as inverse quality factor, and frequency shift, to have an extrema point at certain amplitude during transient response due to the mentioned nonlinear damping term; and the effect of system׳s characteristics on this critical amplitude has also been discussed.

Nonlinear bending and post-buckling of extensible microscale beams based on modified couple stress theory

This study proposes a computationally efficient approach to nonlinear bending and thermal post-buckling problems in Euler–Bernoulli microbeams based on modified couple stress theory under geometrically accurate relationships. The governing equations, which consider the size effect and the axis extensibility, are formulated via the equilibrium of an infinitesimal element. The proposed model, which encompasses the size-independent and Von Kármán nonlinear theory, is solved using the shooting technique after transformation into a two-point boundary value problem. The proposed method was validated based on comparisons with several case studies using existing simulations. The influences of the length scale parameter and the Poisson ratio on the bending and thermal post-buckling behaviors of microbeams are discussed in detail. The numerical results show that the intrinsic size dependency of the material and the Poisson ratio make the microbeam behave in a relatively stiff manner, thereby leading to smaller deformations and greater increases in the buckling temperature.

Exact boundary controllability of vibrating non-classical Euler–Bernoulli micro-scale beams

This study investigates the exact controllability problem for a vibrating non-classical Euler–Bernoulli micro-beam whose governing partial differential equation (PDE) of motion is derived based on the non-classical continuum mechanics. In this paper, it is proved that via boundary controls, it is possible to obtain exact controllability which consists of driving the vibrating system to rest in finite time. This control objective is achieved based on the PDE model of the system which causes that spillover instabilities do not occur.

Estimation of Static Pull-In Instability Voltage of Geometrically Nonlinear Euler-Bernoulli Microbeam Based on Modified Couple Stress Theory by Artificial Neural Network Model

In this study, the static pull-in instability of beam-type micro-electromechanical system (MEMS) is theoretically investigated. Considering the mid-plane stretching as the source of the nonlinearity in the beam behavior, a nonlinear size dependent Euler-Bernoulli beam model is used based on a modified couple stress theory, capable of capturing the size effect. Two supervised neural networks, namely, back propagation (BP) and radial basis function (RBF), have been used for modeling the static pull-in instability of microcantilever beam. These networks have four inputs of length, width, gap, and the ratio of height to scale parameter of beam as the independent process variables, and the output is static pull-in voltage of microbeam. Numerical data employed for training the networks and capabilities of the models in predicting the pull-in instability behavior has been verified. Based on verification errors, it is shown that the radial basis function of neural network is superior in this particular case and has the average errors of 4.55% in predicting pull-in voltage of cantilever microbeam. Further analysis of pull-in instability of beam under different input conditions has been investigated and comparison results of modeling with numerical considerations show a good agreement, which also proves the feasibility and effectiveness of the adopted approach.

Nonlinear forced vibration of strain gradient microbeams

In this paper, the strain gradient theory, a non-classical continuum theory able to capture the size effect happening in micro-scale structures, is employed in order to investigate the size-dependent nonlinear forced vibration of Euler–Bernoulli microbeams. The nonlinearities are caused by mid-plane stretching and nonlinear external forces such as van-der-Waals force. The nonlinear governing equations of the microbeams are solved analytically utilizing the perturbation techniques. The primary, super-harmonic and sub-harmonic resonances of a microbeam are studied and the size-dependency of the frequency responses is assessed. The results indicate that the nonlinear forced vibration behavior of microbeams is size-dependent and the ratio of the microbeam thickness to the material length scale parameter, an additional material property appearing in the strain gradient theory, plays an important role.

On the tunability of a MEMS based variable capacitor with a novel structure

In this work a novel MEMS based variable capacitor has been presented. To increase the tunability and decrease the applied voltage, the conventional fixed-fixed beam used in CPW lines has been changed to a fixed-simple supported beam. The proposed structure is a simple cantilever micro-beam in the first step of deflection and is changed to a fixed-simple supported micro-beam in the second step of motion. In the capacitive micro-structures increasing the applied voltage decreases the equivalent stiffness of the structure and leads the system to an unstable condition by undergoing to a saddle node bifurcation. In the proposed structure to avoid pull-in instability and increase the capacitance tuning range, mechanical stiffness of the structure is increased by changing boundary conditions by locating a pedestal in the end of the cantilever beam. The governing nonlinear equation for static deflection of the micro-beam, based on Euler–Bernoulli micro-beam theory has been presented. The results show that the proposed structure increases the capacitance tuning range and decreases the applied voltage. The results also show that the position of the pedestal affects the tunability and the threshold voltage of the structure.

Corotational Nonlinear Finite Element Formulation and Modeling of Electrostatically Actuated Microbeams

A complete nonlinear finite element model for coupled-domain microelectromechanical system devices with electrostatic actuation and squeeze film effect was developed. for this purpose, a corotational finite element formulation for the dynamic analysis of planer Euler-Bernoulli microbeams considering geometrical nonlinearities due to both large structural deformation and electrostatic actuation is developed. In this method, the internal forces due to deformation and residual stresses, the elemental inertias, and the damping effect of the squeeze-film are systematically derived by consistent linearization of the fully geometrically nonlinear beam theory using d'Alembert and the virtual work principles. An incremental-iterative method based on the Newmark direct integration procedure and the Newton-Raphson algorithm is used to solve the nonlinear dynamic equilibrium equations. Several numerical examples are presented and their results are compared with those found experimentally, which indicate a very close agreement.