52
V. Marinca and N. Herisanu
Numerous analytical, numerical or experimental investigations have been
conducted on the electrostatic and electrodynamic behaviors of the microbeams.
Lin and Zhao [1] studied the bifurcation behavior of nanoscale electrostatic actuators
taking into consideration the presence of the Casimir force. Stability analysis showed
that one equilibrium point is Hopf point, and other is unstable saddle point when
there are two equilibrium points. Hu [2] derived the total system energy expressions
based on Euler–Bernoulli beam and by neglecting the fringing field capacitances.
The closed form solution based on the full-order model is obtained by means of the
minimum energy and the assumed mode methods. Using the reduced-order models,
he showed that the fourth- and third-order models are not as accurate as the fullorder one. Rhoads et al. [3] regarded a microbeam device which couples the inherent
benefits of a resonator with purely parametric excitation with the simple geometry of
a microbeam. An approximate analytical solution to the pull-in voltage of a microbridge considering the elastic boundary effect, fringing field capacitance, residual
stresses and the distributed flexibility of the bridge is proposed by Hu et al. [4]. The
accuracy of the obtained results is verified by comparison with FEM packages, other
solutions and with experimental measured data. The static pull-in instability of electrostatically actuated microbridges and microcantilevers is investigated by Mojahedi
et al. [5] using the homotopy perturbation method. Soroush et al. [6] introduced
Adomian decomposition method in the study of the pull-in behavior and the interval
stress resultants of the nano-actuator using a distributed parameter model. Also, the
effects of the van der Waals and Casimir forces are taken into account.
Yin et al. [7] established a new non-classical Bernoulli–Euler beam invoking
size effect for electrostatically actuated microbeams by using the modified couple
stress theory. Nonlinear terms associated with the mid-plane stretching and the electrostatical force are considered. Askari and Tahani [8] used the homotopy analysis
method and Galerkin decomposition procedure to determine analytical approximate
solutions for oscillatory behavior of a nanobeam under the effect of the Casimir
force. Numerical integration is utilized to find critical value of the Casimir parameter to describe the pull-in instability. Kong [9] gets the pull-in instability model for
Bernoulli–Euler microbeams based on a modified couple stress theory and presented
the approximate analytical solutions to the pull-in voltage and pull-in displacement
on the electrostatically actuated microbeam.
Caruntu et al. [10] proposed the reduced-order model method (ROM) to investigate
the nonlinear parametric dynamics of electrostatically actuated MEMS cantilever
resonators. Fringe effect and damping forces are included, and the method of multiple
scales and ROM are compared in this study. Younis [11] presented an exact analytical
solution of the electrostatically actuated initially deformed cantilever beam problem.
Simple analytical expressions are derived for two configurations: the curled and tilted
configurations for beams of tip deflection of few microns and for largely deformed
beams. The pseudo-spectral method is adopted by Maida and Bianchi [12] to numerically solve the problem of pull-in instability in a cantilever microbeam. They showed
that poor approximation leads to very unphysical oscillatory attraction/repulsion
forces along the cantilever.
V. Marinca and N. Herisanu
Numerous analytical, numerical or experimental investigations have been
conducted on the electrostatic and electrodynamic behaviors of the microbeams.
Lin and Zhao [1] studied the bifurcation behavior of nanoscale electrostatic actuators
taking into consideration the presence of the Casimir force. Stability analysis showed
that one equilibrium point is Hopf point, and other is unstable saddle point when
there are two equilibrium points. Hu [2] derived the total system energy expressions
based on Euler–Bernoulli beam and by neglecting the fringing field capacitances.
The closed form solution based on the full-order model is obtained by means of the
minimum energy and the assumed mode methods. Using the reduced-order models,
he showed that the fourth- and third-order models are not as accurate as the fullorder one. Rhoads et al. [3] regarded a microbeam device which couples the inherent
benefits of a resonator with purely parametric excitation with the simple geometry of
a microbeam. An approximate analytical solution to the pull-in voltage of a microbridge considering the elastic boundary effect, fringing field capacitance, residual
stresses and the distributed flexibility of the bridge is proposed by Hu et al. [4]. The
accuracy of the obtained results is verified by comparison with FEM packages, other
solutions and with experimental measured data. The static pull-in instability of electrostatically actuated microbridges and microcantilevers is investigated by Mojahedi
et al. [5] using the homotopy perturbation method. Soroush et al. [6] introduced
Adomian decomposition method in the study of the pull-in behavior and the interval
stress resultants of the nano-actuator using a distributed parameter model. Also, the
effects of the van der Waals and Casimir forces are taken into account.
Yin et al. [7] established a new non-classical Bernoulli–Euler beam invoking
size effect for electrostatically actuated microbeams by using the modified couple
stress theory. Nonlinear terms associated with the mid-plane stretching and the electrostatical force are considered. Askari and Tahani [8] used the homotopy analysis
method and Galerkin decomposition procedure to determine analytical approximate
solutions for oscillatory behavior of a nanobeam under the effect of the Casimir
force. Numerical integration is utilized to find critical value of the Casimir parameter to describe the pull-in instability. Kong [9] gets the pull-in instability model for
Bernoulli–Euler microbeams based on a modified couple stress theory and presented
the approximate analytical solutions to the pull-in voltage and pull-in displacement
on the electrostatically actuated microbeam.
Caruntu et al. [10] proposed the reduced-order model method (ROM) to investigate
the nonlinear parametric dynamics of electrostatically actuated MEMS cantilever
resonators. Fringe effect and damping forces are included, and the method of multiple
scales and ROM are compared in this study. Younis [11] presented an exact analytical
solution of the electrostatically actuated initially deformed cantilever beam problem.
Simple analytical expressions are derived for two configurations: the curled and tilted
configurations for beams of tip deflection of few microns and for largely deformed
beams. The pseudo-spectral method is adopted by Maida and Bianchi [12] to numerically solve the problem of pull-in instability in a cantilever microbeam. They showed
that poor approximation leads to very unphysical oscillatory attraction/repulsion
forces along the cantilever.