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.     

DYNAMIC PULL-IN INSTABILITY OF DOUBLE-SIDED ACTUATED NANO-TORSIONAL SWITCHES

A nonlinear frequency-amplitude relation is developed to investigate the vibrational amplitude effect on the dynamic pull-in instability of double-sided-actuated nano-torsional switches. The governing equation of a nano-electro-mechanical system pre-deformed by an electric field contains the quintic nonlinear term. The influences of basic parameters on the pull-in instability and natural frequency are investigated using a powerful analytical approach called the homotopy perturbation method. It is demonstrated that two terms in series expansion are sufficient to produce an acceptable solution. The numerical results obtained have verified the soundness of the asymptotic procedure. The phase portraits of the double-sided nano-torsionalactuator exhibit periodic, homoclinic and heteroclinic orbits.     

Primary resonance excitation of electrically actuated clamped circular plates

We investigate the response of an electrically actuated clamped circular plate to a primary resonance excitation of its first axisymmetric mode using an analytical reduced-order model (macromodel). We discuss the inf luence of the number of modes retained in the discretization on the predicted solutions. The reduced-order model, which is a system of coupled nonlinear ordinary-dif ferential equations, accounts for general residual stress and strain hardening and allows for general material and geometric design variables. Our reduced-order model is robust up to the pull-in instability and is general enough to be an ef fective design tool for capacitive micromachined ultrasonic transducers.     

Reduced-order models for microelectromechanical rectangular and circular plates incorporating the Casimir force

We consider the von Kármán nonlinearity and the Casimir force to develop reduced-order models for prestressed clamped rectangular and circular electrostatically actuated microplates. Reduced-order models are derived by taking flexural vibration mode shapes as basis functions for the transverse displacement. The in-plane displacement vector is decomposed as the sum of displacements for irrotational and isochoric waves in a two-dimensional medium. Each of these two displacement vector fields satisfies an eigenvalue problem analogous to that of transverse vibrations of a linear elastic membrane. Basis functions for the transverse and the in-plane displacements are related by using the nonlinear equation governing the plate in-plane motion. The reduced-order model is derived from the equation yielding the transverse deflection of a point. For static deformations of a plate, the pull-in parameters are found by using the displacement iteration pull-in extraction method. Reduced-order models are also used to study linear vibrations about a predeformed configuration. It is found that 9 basis functions for a rectangular plate give a converged solution, while 3 basis functions give pull-in parameters with an error of at most 4%. For a circular plate, 3 basis functions give a converged solution while the pull-in parameters computed with 2 basis functions have an error of at most 3%. The value of the Casimir force at the onset of pull-in instability is used to compute device size that can be safely fabricated.     

Size-dependent dynamic pull-in instability of hydrostatically and electrostatically actuated circular microplates

In the present study, the dynamic pull-in instability and free vibration of circular microplates subjected to combined hydrostatic and electrostatic forces are investigated. To take size effects into account, the strain gradient elasticity theory is incorporated into the Kirchhoff plate theory to develop a nonclassical plate model including three internal material length scale parameters. By using Hamilton’s principle, the higher-order governing equation and the corresponding boundary conditions are obtained. Afterward, a generalized differential quadrature (GDQ) method is employed to discritize the governing differential equations along with simply supported and clamped edge supports. To evaluate the pull-in voltage and vibration frequencies of actuated microplates, the hydrostatic-electrostatic actuation is assumed to be calculated by neglecting the fringing field effects and utilizing the parallel plate approximation. Also, a comparison between the pull-in voltages predicted by the strain gradient theory and the degenerated ones is presented. It is revealed that increasing the dimensionless internal length scale parameter or decreasing the applied hydrostatic pressures leads to higher values of the pull-in voltage. Moreover, it is found that the value of pull-in hydrostatic pressure decreases corresponding to higher dimensionless internal length scale parameters and applied voltages.     

Pull-in instability of a typical electrostatic MEMS resonator and its control by delayed feedback

Pull-in instability of the electrostatic microstructures is a common undesirable phenomenon which implies the loss of reliability of micro-electromechanical systems. Therefore, it is necessary to understand its mechanism and then reduce the phenomenon. In this work, pull-in instability of a typical electrostatic MEMS resonator is discussed in detail. Delayed position feedback and delayed velocity feedback are introduced to suppress pull-in instability, respectively. The thresholds of AC voltage for pull-in instability in the initial system and the controlled systems are obtained analytically by the Melnikov method. The theoretical predictions are in good agreement with the numerical results. It follows that pull-in instability of the MEMS resonator can be ascribed to the homoclinic bifurcation inducing by the AC and DC load. Furthermore, it is found that the controllers are both good strategies to reduce pull-in instability when their gains are positive. The delayed position feedback controller can work well only when the delay is very short and AC voltage is low, while the delayed velocity feedback will be effective under a much higher AC voltage and a wider delay range.     

Lyapunov exponents as a criterion for the dynamic pull-in instability of electrostatically actuated microstructures

The dynamic pull-in instability of double clamped microscale beams actuated by a suddenly applied distributed electrostatic force and subjected to non-linear squeeze film damping is investigated. A reduced order model is built using the Galerkin decomposition with undamped linear modes as base functions and verified through comparison with numerical finite differences solution. The stability analysis of a beam actuated by one and two electrodes symmetrically located at two sides of the beam and operated by a step-input voltage is performed by evaluating the largest Lyapunov exponent, the sign of which defines the character of the response. It is shown that this approach provides an efficient quantitative criterion for the evaluation of dynamic pull-in instability, especially when combined with compact reduced order models. Based on the Lyapunov exponent criterion, the influence of various parameters on the beam dynamic stability is investigated.     

Experiments and modeling of carbon nanotube-based NEMS devices

In this paper, carbon nanotube-based nanoelectromechanical systems (NEMS) are nanofabricated and tested. In-situ scanning electron microscopy measurements of the deflection of the cantilever under electrostatic actuation are reported. In particular, a cantilever nanotube suspended over an electrode (nanoswitch), or two symmetric cantilever nanotubes (nanotweezers), from which a differential in electrical potential is imposed, are studied. The finite deformation regime investigated here is the first of its kind. An analytical model based on the energy method in both small deformation and finite kinematics (large deformation) regimes is used to interpret the measurements. The theory overcomes limitations of prior analysis reported in the literature towards the prediction of the structural behavior of NEMS. Some of the simplifying hypotheses have been removed. Furthermore, the theory takes into account the cylindrical shape of the deflected nanotube in the evaluation of its electrical capacitance, the influence of the van der Waals forces as well as finite kinematics. In addition, tip charge concentration and a quantum correction of the electrical capacitance are also considered. The energy-based method is used to predict the structural behavior and instability of the nanotube, corresponding to the on/off states of the nanoswitch, or to the open/close states of the nanotweezers—at the so-called pull-in voltage. Accuracy of the derived formulas is assessed by comparison of the theoretical prediction and experimental data in both small deformation and finite kinematics regimes. The results reported in this work are particularly useful in the characterization of the electromechanical properties of nanotubes as well as in the optimal design of nanotube-based NEMS devices.     

An analytical molecular structural mechanics model for the mechanical properties of carbon nanotubes

An analytical molecular structural mechanics model for the prediction of mechanical properties of defect-free carbon nanotubes is developed by incorporating the modified Morse potential with an analytical molecular structural model. The developed model is capable of predicting Young’s moduli, Poisson’s ratios and stress–strain relationships of carbon nanotubes under tension and torsion loading conditions. Results on the mechanical properties of single-walled carbon nanotubes show that Young’s moduli of carbon nanotubes are sensitive to the tube diameter and the helicity. Young’s moduli of both armchair and zigzag carbon nanotubes increase monotonically and approach Young’s modulus of graphite when the tube diameter is increased. The nonlinear stress–strain relationships for defect-free nanotubes have been predicted, which gives a good approximation on the ultimate strength and strain to failure of nanotubes. Armchair nanotubes exhibit higher tensile strength than zigzag nanotubes but their torsion strengths are identical based on the present study. The present theoretical investigation provides a very simple approach to predict the mechanical properties of carbon nanotubes.     

Suppression of pull-in in a microstructure actuated by mechanical shocks and electrostatic forces

This work investigates the effect of a high-frequency voltage (HFV) on the pull-in instability in a microstructure actuated by mechanical shocks and electrostatic forces. The microstructure is modelled as a single-degree-of-freedom mass-spring-damper system. The method of direct partition of motion is used to split the fast and slow dynamics. Analysis of steady-state solutions of the slow dynamic allows the investigation of the influence of the HFV on the pull-in. The results show that adding HFV rigidifies the system, creates new stable equilibria and suppresses the pull-in instability for adequate high-frequency voltages. To illustrate the applicability of the result, a specific capacitive microelectromechanical system consisting of a clamped-clamped microbeam is considered.     

Dynamic analysis and design of electrically actuated viscoelastic microbeams considering the scale effect

Viscoelastic phenomena widely exist in MEMS materials, which may have certain effects on quasi-static behaviors and transition mechanism of nonlinear jumping phenomena. The static and dynamic behaviors of a doubly clamped viscoelastic microbeam actuated by one sided electrode are investigated in detail, based on a modified couple stress theory. The governing equation of motion is introduced here, which is essentially nonlinear due to its midplane stretching effect and electrostatic force. Through quasi-static analysis, the equilibrium position, pull-in voltage and pull-in location of the system are obtained with differential quadrature method and finite element method. The equivalent geometric nonlinear parameter is presented to explain the influence of the scale effect on the pull-in location. Different from elastic material, there are two kinds of pull-in voltages called as instantaneous pull-in voltage and the durable pull-in voltage in viscoelastic system. Then, Galerkin discretization and the method of multiple scales are applied to determine the response and stability of the system for small vibration amplitude. A new perturbation method to deal with viscoelastic term is presented. Theoretical expressions about the parameter spaces of linear-like vibration, hardening-type vibration and softening-type vibration are then deduced. The influence of viscoelasticity and scale effect on nonlinear dynamic behavior is studied. Results show that the viscoelasticity can reduce the effective elastic modulus and make the system tend to softening-type vibration; the scale effect can increase effective elastic modulus and make the system tend to hardening-type vibration. And most of all, simulation results of case studies are used to realize parameter optimization. Then parameter conditions of linear-like vibration, which is desired for many applications, are obtained. In this paper, the results of multi-physical field coupling simulation are used to verify the theoretical analysis.     

Thermal effects on interfacial stress transfer characteristics of carbon nanotubes/polymer composites

The paper presents an analytical method to investigate thermal effects on interfacial stress transfer characteristics of single/multi-walled carbon nanotubes/polymer composites system under thermal loading by means of thermoelastic theory and conventional fiber pullout models. In example calculations, the mechanical properties and the thermal expansion coefficients of carbon nanotubes and polymer matrix are, respectively, treated as the functions of temperature change. Numerical examples show that the interfacial shear stress transfer behavior can be described and affected by several parameters such as the temperature field, volume fraction of CNT, and numbers of wall layer and the outermost radius of carbon nanotubes. From the results carried out it is found that mismatch of thermal expansion coefficients between the carbon nanotubes and polymer matrix may be more important in governing interfacial stress transfer characteristics of carbon nanotubes/polymer composite system.     

Critical velocities in fluid-conveying single-walled carbon nanotubes embedded in an elastic foundation

The problem of stability of fluid-conveying carbon nanotubes embedded in an elastic medium is investigated in this paper. A nonlocal continuum mechanics formulation, which takes the small length scale effects into consideration, is utilized to derive the governing fourth-order partial differential equations. The Fourier series method is used for the case of the pinned–pinned boundary condition of the tube. The Galerkin technique is utilized to find a solution of the governing equation for the case of the clamped–clamped boundary. Closed-form expressions for the critical flow velocity are obtained for different values of the Winkler and Pasternak foundation stiffness parameters. Moreover, new and interesting results are also reported for varying values of the nonlocal length parameter. It is observed that the nonlocal length parameter along with the Winkler and Pasternak foundation stiffness parameters exert considerable effects on the critical velocities of the fluid flow in nanotubes.     

Upper and lower bounds for the pull-in parameters of a micro- or nanocantilever on a flexible support

An analytical method is proposed to accurately estimate the pull-in parameters of a micro- or nanocantilever beam elastically constrained by a rotational spring at one end. The system is actuated by electrostatic force and subject to Casimir or van der Waals forces according to the beam size. The deflection of the beam is described by a fourth-order nonlinear boundary value problem, or equivalently in terms of a nonlinear integral equation. New a priori analytical estimates on the solution from both sides are first derived and then lower and upper bounds for the pull-in parameters are obtained, with no need of solving the nonlinear boundary value problem. The lower and upper bounds turn out to be very close each other and in excellent agreement with the numerical results provided by the shooting method. The approach also provides accurate predictions for the pull-in parameters of a freestanding nanoactuator.     

Pull-in instability analyses for NEMS actuators with quartic shape approximation

The pull-in instability of a cantilever nano-actuator model incorporating the effects of the surface, the fringing field, and the Casimir attraction force is investigated. A new quartic polynomial is proposed as the shape function of the beam during the deflection, satisfying all of the four boundary values. The Gaussian quadrature rule is used to treat the involved integrations, and the design parameters are preserved in the evaluated formulas. The analytic expressions are derived for the tip deflection and pull-in parameters of the cantilever beam. The micro-electromechanical system (MEMS) cantilever actuators and freestanding nanoactuators are considered as two special cases. It is proved that the proposed method is convenient for the analyses of the effects of the surface, the Casimir force, and the fringing field on the pull-in parameters.     

An investigation on primary resonance phenomena of elastic medium based single walled carbon nanotubes

In this paper, the geometrically nonlinear free and forced oscillations of simply supported single walled carbon nanotubes (SWCNTs) are analytically investigated on the basis of the Euler–Bernoulli beam theory. The nonlinear frequencies of SWCNTs with initial lateral displacement are discussed. Equations have been solved using an exact method for free vibration and multiple times scales (MTS) method for forced vibration and some analytical relations have been obtained for natural frequency of oscillations. The numerical results reveal that the nonlinear free and forced vibration of nanotubes is effected significantly by both surrounding elastic medium and CNT aspect ratio.     

Application of homotopy analysis method in studying dynamic pull-in instability of microsystems

In this study, homotopy analysis method is used to derive analytic solutions to predict dynamic pull-in instability of electrostatically-actuated microsystems. The model considers midplane stretching, initial stress, distributed electrostatic force and fringing fields effect. Influences of different parameters on dynamic pull-in instability are investigated. Results are in good agreement with numerical and experimental findings.     

In-plane and out-of-plane free vibration and stability of a curved rod in flow

This paper investigates the free vibration and stability of a curved rod in flow. The equations of the three-dimensional motions of the rod are derived by the Newtonian approach. The differential quadrature method (DQM) is introduced to formulate the discrete forms of the governing equations of the inextensible rod with clamped–clamped supports. Based on numerical calculations, the effects of several system parameters, especially the flow velocity, on the natural frequencies and stability of the system are discussed. Buckling and flutter instability are detected as the flow velocity is varied in a certain range. Moreover, a derivation of the generalized slender-body theory for such a deformable curved rod is given in Appendix A.     

Influence of size effect and elastic boundary condition on the pull-in instability of nano-scale cantilever beams immersed in liquid electrolytes

In this study, the static pull-in instability of nanocantilever beams immersed in a liquid electrolyte is theoretically investigated. In modeling the nanocantilever beam, the effects of van der Waals forces, elastic boundary condition and size dependency are considered. The modified couple stress theory, containing material length scale parameter, is used to interpret the size effect which appears in micro/nanoscale structures. The modified Adomian decomposition (MAD) method is used to gain an approximate analytical expression for the critical pull-in parameters which are essential for the design of micro/nanoactuators. The results show that the beam can deflect upward or downward, based on the values of the non-dimensional parameters. It is found that the size effect greatly influences the beam deflection and is more noticeable for small thicknesses. Neglecting size effect overestimates the deflection of the nanobeam. The findings reveal that the increase of ion concentration increases the pull-in voltage but decreases the pull-in deflection. Furthermore, an increase in ion concentration increases the influence of size-dependent effect on pull-in voltage.     

Dynamics of solid propellant motor composite casing under impact pressure

A dynamic behavior of solid propellant motor composite casing under the action of an internal impact pressure is treated. This pressure describes the engine operation. The thin-walled casing consists of the cylindrical shell and two bottoms. These bottoms are truncated hemisphere. The casing is clamped along two edges of the bottoms. A shear, a rotary inertia and stress–strain relations for an orthotropic material are accounted. Semi analytical method is suggested to analyze the structure stress–strain state. The thin-walled casing dynamic is described by large dimension system of the ordinary differential equations.