Global Analysis and Experimental Dynamics of the 2:1 Internal Resonance in the Higher-Order Modes of a MEMS Microbeam

Abstract

In this work, we consider a MEMS microbeam, and we investigate the experimental response of the device at the third-mode dynamics. By forward and backward sweeping, the data acquired via the laser Doppler vibrometer show the occurrence of a 2:1 internal resonance, where the coupling mode is the fifth. The experimental response is simulated via shooting technique and attractor basins. We focus on the metamorphoses of the basins of attraction scenario induced in the phase space by the activation of the internal resonance.

Appendix: Mechanical Model

We model the device as a parallel plate capacitor. The upper electrode is represented by a clamped-clamped microbeam with a rectangular cross section and straight configuration. To model the residual stresses, we assume a constant axial load P. To represent the half-electrode configuration, only half of the microbeam length is supposed to provide the electric contribution. The governing equation of motion is [29]

$$ \ddot{v}+\xi \dot{v}+{v^{{\prime\prime\prime}}}^{\prime }+\alpha {v}^{{\prime\prime} }=-{\gamma F}_{\mathrm{e}} $$

(A.1)

with

$$ \alpha =n- ka{\int}_0^1\frac{1}{2}{\left({v}^{\hbox{'}}\right)}^2 dz\kern1em {F}_{\mathrm{e}}=\frac{{\left({V}_{\mathrm{DC}}+{V}_{\mathrm{AC}}\cos \left(\Omega t\right)\right)}^2}{{\left(1-v\right)}^2}\left[H\left({z}_1\right)-H\left({z}_2\right)\right] $$

(A.2)

with H(z ₁) and H(z ₂) unit step functions, which are used to define the length and the position of the lower electrode. The considered nondimensional parameters are

$$ {\displaystyle \begin{array}{ccc} ka=(EA){d}^2/(EJ)& n=(EA){Lw}_{\mathrm{B}}/(EJ)& \xi ={cL}^4/\left( EJ T\right)\\ {}\gamma =\frac{1}{2}{\varepsilon}_0{\varepsilon}_{\mathrm{r}}{A}_{\mathrm{c}}{L}^3/\left({d}^3 EJ\right)& T=\sqrt{\left(\rho\ {AL}^4\right)/(EJ)}& \tilde{\Omega}=\Omega T\end{array}} $$

(A.3)

where L is length, A and J are respectively area and moment of inertia, ρ is the material density, E is the effective Young’s modulus, c is damping coefficient, d is the capacitor gap, ε ₀ is dielectric constant in the free space, ε _r is the relative permittivity of the gap space medium (air) with respect to the free space, and A _c is the area overlapping between the electrodes.

The dynamics of the microbeam are approximated as $v\left(z,t\right)\cong {\sum}_{i=1}^n{\phi}_i(z){u}_i(t)$, where ϕ _i(z) are the mode shapes under consideration (third and fifth). We derive the reduced-order model by applying the Galerkin procedure, which yields the 2 d.o.f. system [25]

$$ {\displaystyle \begin{array}{c}{\ddot{u}}_n+c{\dot{u}}_n+{\omega}_n^2{u}_n- ka\left({a}_{n1}{u}_1^3+{a}_{n2}{u}_2^3+{a}_{n3}{u}_1{u}_2^2+{a}_{n4}{u}_2{u}_1^2\right)\\ {}=-{\gamma V}^2{\int}_0^{0.5}\frac{\phi_n}{{\left(1-{\phi}_1{u}_1-{\phi}_2{u}_2\right)}^2} dz\kern2em \mathrm{for}\kern0.5em n=1,2\end{array}} $$

(A.4)

where numerical integration is applied to evaluate the electric force term. Table A.1 reports the coefficients used in the model.

Table 1 Coefficients of the model in Eq. (A.4)