Step-by-step Linearization Method Research Articles

Nonlinear dynamic stability of piezoelectric thermoelastic electromechanical resonators

This research work deals with analyzing instability and nonlinear behaviors of piezoelectric thermal nano-bridges. An adjustable thermo-elastic model with the ability to control stability conditions is developed to examine the system behavior at different temperatures. To increase the performance range and improve system characteristics, a piezovoltage is applied and a spring is connected to the sliding end of the deformable beam as design parameters. The partial differential equations (PDEs) are derived using the extended Hamilton’s principle and Galerkin decomposition is implemented to discretize the nonlinear equations, which are solved via a computational method called the step-by-step linearization method (SSLM). To improve the accuracy of the solution, the number of mode shapes and the size of voltage increments are analyzed and sufficient values are employed in the solution. The validity of the formulation and solution method is verified with experimental, analytical, and numerical data for several cases. Finally, the vibration and eigenvalue problem of the actuated nano-manipulator subjected to electrostatic and Casimir attractions are investigated. It is concluded that the fringing-fields correction changes the system frequency, static equilibrium, and pull-in characteristics significantly. The results are expected to be instrumental in the analysis, design, and operation of numerous adjustable advanced nano-systems.

Nonlinear Instability Modeling of a Nonlocal Strain Gradient Functionally Graded Capacitive Nano-Bridge in Thermal Environment

This paper develops a theoretical model directed towards investigation of the static pull-in instability of a functionally graded (FG) electrostatically actuated nano-bridge via nonlocal strain gradient theory (NLSGT) of elasticity and Euler–Bernoulli beam theory in thermal environment. The nano-beam is under the influence of electrostatic and van der Waals (vdW) forces. In addition to the nonlinear nature of the electrostatic force, the other type of nonlinearity namely geometric nonlinearity resulting from the mid-plane stretching is considered. Material properties of FG nano-beam are assumed to vary gradually along the thickness direction according to simple power-law form. With the purpose of eliminating the coupling between the stretching and bending due to the asymmetrical material variation along the thickness, a new surface reference is introduced. The nonlinear integro-differential governing equation is derived utilizing minimum total potential energy principle, linearized by means of the step-by-step linearization method (SSLM) and solved by Galerkin-based weighted residual method. The numerical investigations are performed while the emphasis is placed on studying the effect of various parameters including: nonlocal parameter, material characteristic length scale, material gradient index, thermal effect and intermolecular force on the static pull-in instability of FG nano-beam. To establish the validity of the present formulation, a comparison is conducted with experimental and numerical results reported in previous studies.

FGM micro-gripper under electrostatic and intermolecular Van-der Waals forces using modified couple stress theory

In this paper mechanical behavior of the functional gradient materials (FGM) micro-gripper under thermal load and DC voltage is numerically investigated taking into account the effect of intermolecular forces. In contrary to the similar previous works, which have been conducted for homogenous material, here, the FGM material has been implemented. It is assumed that the FGM micro-gripper is made of metal and ceramic and that material properties are changed continuously along the beam thickness according to a given function. The nonlinear governing equations of the static and dynamic deflection of microbeams have been derived using the coupled stress theory. The equations have been solved using the Galerkin based step-by-step linearization method (SSLM). The solution procedure has been evaluated against available data of literature showing good agreement. A parametric study has been conducted, focusing on the combined effects of important parameters included DC voltage, temperature variation, geometrical dimensions and ceramic volume concentration on the dynamic response and stability of the FGM micro-gripper.

Bifurcation analysis of an electro-statically actuated micro-beam in the presence of centrifugal forces

In this paper, the influence of centrifugal forces on the stability of an electro-statically actuated clamped–clamped micro-beam has been investigated. The non-dimensional governing static and dynamic equations have been linearized using the step by step linearization method (SSLM), then, a Galerkin-based reduced order model has been used to solve the linearized equations. For constant value of a bias DC voltage and different values of angular velocity the equilibrium points of the corresponding autonomous system including stable center points, unstable saddle points and singular points have been obtained using the equivalent mass-spring model. Subsequently the bifurcation diagram has been depicted using the obtained fixed point. The static pull-in voltage value for different values of angular velocity and the static pull-in angular velocity for different values of bias voltage have been calculated. The obtained results are validated using results of previous studies and a good agreement has been observed. The effect of the centrifugal force on the fixed points has been studied using the phase portraits of the system for different initial conditions. Moreover, the effects of centrifugal forces on the dynamic pull-in behavior have been investigated using time histories and phase portraits for different angular velocities.

Effect of Length-scale Parameter on Pull-in Voltage and Natural Frequency of a Micro-plate

This paper deals with the effect of the intrinsic material length-scale parameter on the stability and natural frequency of a rectangular micro-plate for two different cases; fully clamped and fully simply supported. A variational formulation based on Hamilton ’s principle and the modified couple stress theory is used to obtain the nonlinear governing equation of a micro-plate incorporating the stretching effect. In the static case, the nonlinear governing equation is solved using the step-by-step linearization method (SSLM) and in the dynamic case, is integrated using fourth-ordered Runge-Kutta method. The static and dynamic pull-in parameters, limiting the stability regions of capacitive MEMS devices, are calculated and compared to those obtained by the classical theory. The numerical results reveal that the intrinsic size dependence of materials is more significant for smaller thicknesses and in this case, the stretching effect can be neglected.

Static and dynamic stability modeling of a capacitive FGM micro-beam in presence of temperature changes

Stability of a functionally graded (FG) micro-beam, based on modified couple stress theory (MCST), subjected to nonlinear electrostatic pressure and thermal changes regarding convection and radiation, is the main purpose of this paper. It is assumed that the functionally graded beam, made of metal and ceramic, follows the volume fraction definition and law of mixtures, and its properties change as an exponential function through its thickness. By changing the ceramic constituent percent of the bottom surface, five different types of the micro-beams are investigated. The static pull-in voltages in presence of temperature changes are obtained by using step-by-step linearization method (SSLM) and, by adapting Runge–Kutta approach, the dynamic pull-in voltages are obtained numerically. Though the temperature distribution through the thickness of FG micro-beam (due to its too small measurement) is considered uniform, owing to the different thermal expansions of layers, temperature changes cause deflection in the micro-beam, and consequently affect pull-in values. Hence the profound effects of different material constituent over the pull-in voltages are illustrated and it is graphically displayed that how in some cases neglecting components of the couple stress leads to inaccurate results.

A comprehensive study of stability in an electro-statically actuated micro-beam

This paper studies the mechanical and bifurcation behavior of a capacitive micro-beam suspended between two conductive stationary plates, which can be used as a micro-switch or as a RF resonator. The equation of dynamic motion of the micro-switch is obtained using Euler–Bernoulli beam theorem. The equilibrium positions or the fixed points of the micro-switch are obtained by solving the equation of the static deflection using the step-by-step linearization method (SSLM) and discretizing by Galerkin weighted residual method. In order to study the global stability of the obtained fixed points a modified non-linear mass–spring model is used. Non-linear motion trajectories in phase portraits are given and regions of bounded and unbounded solutions separated by a homoclinic or heteroclinic orbits and positions of the stationary conductive plates are illustrated. Critical values of the applied voltage leading to qualitative changes in the micro-beam behavior through a saddle node or pitch fork bifurcations for different values of the gap and voltage ratios are obtained. The effects of different gaps and voltage ratios also are investigated.

MECHANICAL BEHAVIOR OF A BI-LAYER CANTILEVER MICRO-BEAM SUBJECTED TO ELECTROSTATIC FORCE, MECHANICAL SHOCK AND THERMAL MOMENT

In the present work, we study the static and dynamic behavior of a bi-layer cantilever micro-beam and its pull-in instability subjected to simultaneous effects of thermal moment, electrostatic force and mechanical shock. The nonlinear governing equation in the static case has been linearized using a Step-by-Step Linearization Method (SSLM) and the resulted linear equation has been solved using a Finite Difference Method (FDM). The obtained results for some cases have been compared to the existing results and good agreement achieved. The dynamic response of the system owing to application of a mechanical shock and a step DC voltage for thermally actuated micro-beam have been accomplished by a partial differential equation. A Reduced-Order Model (ROM) has been used to simulate the dynamic behavior of the micro-beam. The effects of temperature changes and mechanical shock on the pull-in behavior have been studied and obtained results show an earlier pull-in phenomenon experience when these effects are considered.

STATIC AND DYNAMIC RESPONSE OF CARBON NANOTUBE-BASED NANO-TWEEZERS

In this paper static and dynamic response of nanotweezers composed of two carbon nanotube (CNT) arms are investigated. Taking into account a continuum model and considering the electrostatic actuation as well as the presence of the van der Waals forces, the static nonlinear equations are solved by a step by step linearization and Galerkin projection method. Simulating the closing dynamics of a specific nanotweezer, the effective diameters of its nanotube cantilevers are specified and compared with existing experimental data. Then by imposing a step DC voltage and taking into account the inertia effects, dynamic responses and pull-in conditions of nanotweezers are studied. In the analyses, the effects of various parameters such as initial gap, nanotube length and diameter on the pull-in conditions are investigated. Effects of damping and asymmetric stiffness's of the arms on the pull-in voltages of a nanotweezer are also reported. Comparing the results with the published experimental data shows that the use of continuum model and employing the Galerkin based step by step linearization method (SSLM) could effectively simulate the response of nanotweezers.

Effect of temperature on pull-in voltage and natural frequency of an electrostatically actuated microplate

In this paper a model based on distributed parameters has been presented to study temperature effects on the mechanical behavior of an electrostatically actuated microplate. This work consists of two major parts. First part deals with the effect of temperature, stretching and residual stresses on the static instability of an electrostatically actuated microplate. To do this, the governing nonlinear integro differential equation has been derived using Kirchhoff thin plate theory and linearized using step-by-step linearization method (SSLM). The obtained linearized differential equation has been discretized applying finite difference method (FDM). The results obtained have been compared to other existing experimental results and good agreement is observed. In the second part, in order to study the natural or eigenfrequencies of the system, small vibrations of the electrostatically deflected microplate about the equilibrium position have been studied. Here, the governing linear eigenvalue partial differential equation has been solved using a Galerkin based reduced-order model, and the natural frequencies of the microplate have been determined. The results show that temperature changes and residual stress have considerable effects on the system characteristics such as pull-in voltage and natural frequencies.

Dynamic characteristics and forced response of an electrostatically-actuated microbeam subjected to fluid loading

In this paper flexural vibrations of an electrostatically actuated cantilever microbeam in an incompressible inviscid stationary fluid have been studied. By applying “Three dimensional aerodynamic theory” pressure jump across the microbeam has been investigated and the inertial effects of fluid on microbeam dynamics have been modeled as a mass added to microbeam mass. Magnitude of the added mass has been calculated for various aspect ratios of cantilever microbeams and compared with those of clamped-clamped microbeams. To investigate the dynamic characteristics, it has been considered that the microbeam has been deflected by a DC voltage, V DC and then the dynamic characteristics and forced response of the system have been considered about these conditions. Galerkin-based step by step linearization method (SSLM) and Galerkin-based reduced order model have been applied to solve the nonlinear static and dynamic governing equations, respectively. Water by neglecting viscidity effects, as an instant has been considered as a surrounding fluid and the frequency response of the microbeam has been compared with that of vacuum conditions. It has been shown that because of the added mass effects in watery environment, the natural frequencies of the microbeam decrease. Because of the higher dielectric coefficient and increasing electrical stiffness and decreasing total stiffness consequently, maximum amplitude of the microbeam vibrations increases in watery environment, compared with vacuum. Moreover, it has been shown that increasing the DC voltage, increases the electrical stiffness and maximum amplitude of the microbeam vibrations, consequently, It has been shown that in higher voltages (near pull-in voltage), the rate of variation of resonance frequency and maximum amplitude is stronger than lower voltages.