4.2. Modal Analysis
In this section, we conduct a free vibrations analysis of arch microbeams by solving the corresponding linear damped eigenvalue problem, Equation (
3), using the differential quadrature method (DQM) [
24]. The formulation of the eigenvalue problem is shown in
Appendix B. The number of DQM points is set to
to guarantee convergence of the numerical solutions. Taking into account the damping and applied DC voltage, we calculate the natural frequencies and the corresponding mode shapes of arch microbeams. The resulting modes are dubbed in terms of their initial configuration in the absence of initial rise and voltage as mode 1, the first symmetric mode; mode 2, the first asymmetric mode; and mode 3, the second symmetric mode.
We plot in
Figure 9 variations in the natural frequencies of the first three modes as functions of the initial rise
under a DC voltage of
V. We note that Mode 1 and Mode 2 refer to the first symmetric and first asymmetric mode shapes rather than the numerical value of their natural frequencies. The crossing between the first two natural frequencies, observed in the full-electrode case, changes into veering in the half-electrode case. Veering is observed near
µm, where the natural frequency of mode 1 approaches that of mode 2. These modes are similar in shape but out of phase, as can be seen in
Figure 10c. This is an indicator of a strong interaction between the two modes. The natural frequencies for the selected values of the initial rise
around the veering point are listed in
Table 4. Since the veering point varies with static transverse loads, the DC voltage (or RMS of the voltage waveform) was held constant at 15 V throughout the present numerical study. Beyond the crossing or veering point, the first symmetric mode is hybridized, as illustrated in
Figure 9. The impact of the initial rise
on the natural frequency of asymmetric mode (Mode 2) is insignificant throughout the range of interest. On the other hand, the natural frequency of the first symmetric mode (Mode 1) increases with the initial rise.
For comparison, we also show in
Figure 9 the natural frequencies of the first three modes under full-electrode actuation with
V. While the natural frequencies and mode shapes are similar to those of the half-electrode case, the symmetry of the electrical and mechanical domains result in crossing of modes 1 and 2 rather than the veering seen under asymmetry of the electrical field in the former case [
41].
We used the model to investigate the interaction between the mode shapes around the veering point under half-electrode actuation.
Figure 10, shows the first three mode shapes for five selected values of the initial rise under a DC voltage of
V. We note that mode 1 and mode 2 refer to the first symmetric and first asymmetric modes, respectively. The shape of mode 2 remains unchanged except at the veering point, where the two modes are symmetric with respect to the mid span point. On the other hand, mode 1 hybridizes gradually with mode 2 as it approaches and crosses the veering point [
31,
41]. It is important to note here that the veering and the subsequent hybridization phenomena are occurring in a continuous manner with a smoothly increasing significance before the veering point, and then their effect decreases as the initial rise is further increased. In the present study, our main focus is on the half electrode design. We examine the strength of the interaction between the mode shapes, which is controlled by the initial rise of the arch beam. The shape of mode 3 remains unchanged throughout this process.
4.3. Dynamic Analysis of Arch Microbeams
In this section, we obtain the frequency–response curves of half-electrode actuated arch microbeams. The response orbits are evaluated following the numerical procedure described in [
38]. The mode shapes of a straight clamped–clamped microbeam are used in a Galerkin expansion to discretize the equation of motion and obtain a set of ordinary differential equations (ODEs). The resulting dynamic reduced-order model is given in
Appendix C. The ODEs are numerically solved using the finite difference method and an arc-length continuation technique [
24,
39,
40].
We first determine the number of modes required for the convergence of the dynamic solution. In
Figure 11, we compare the frequency–response curves obtained using 1–5 mode Galerkin expansions for an arch microbeam with an initial rise of
µm in the frequency range
kHz. The curves describe the maximum deflection over the orbit of the mid-point with respect to a line connecting the supports
. The microbeam is excited by a the voltage waveform with
V. The RMS of this waveform is equivalent to
V resulting in natural frequencies identical to those shown in
Figure 9 and
Table 4. Under these conditions, the natural frequencies of modes 1 and 2 appear in this range and are closely spaced. As a result, modal interactions between their primary resonances are expected. This behavior is recovered using three or more modes expansions as shown in
Figure 11. The one and two mode expansions are qualitatively incorrect. On the other hand, the addition of the fourth and fifth modes does not seem to significantly improve the model accuracy. Therefore, we adopt a three-mode approximation for the rest of this paper. In this respect, the developed model is consistent with the literature on the required number of modes to achieve convergence [
35].
We show in
Figure 12 the frequency response of the arch microbeam for five values of the initial rise.
, 4,
, 5 and 6 µm, under the voltage waveform
V. The figures show the maximum displacement realized by the microbeams’s quarter-point
, mid-point
, and three quarters-point
over the orbit. The stability of each orbit is determined by computing its Floquet multipliers. Stable orbits are marked by solid lines and unstable orbits are marked by dashed lines The location of and type of bifurcation are determined by observing changes in the number and/or stability of orbits available over the frequency spectrum. In all cases, the asymmetric actuation via the half-electrode results in excitation of both the symmetric and asymmetric modes (modes 1 and 2).
For the low initial rise case of
, distinct primary resonances and the superharmonic resonances of order 2 of modes 1 and 2 can be identified near their two natural frequencies and half of their natural frequencies in
Figure 12a. The superharmonic resonance of order 2 of mode 3 is also observed near 38 kHz. While these resonances can be seen along the beam span, the relative magnitudes vary depending on the observation point. In the primary resonance of mode 1, for example, the response of the mid-point is larger than that of the quarter-point which is larger than that of the three quarters-point
. On the other hand, the response of the three quarters-point is slightly larger than that of the quarter-point and mid point in primary resonance of mode 2
. Bifurcation analysis identified seven cyclic-fold (CF
i) bifurcations, where a Floquet multiplier exits the unit circle through +1; two Hopf bifurcations (HP
i), where two complex conjugate multipliers exit the unit circle; and a period-doubling (PD) bifurcation, where a multiplier exits the unit circle through −1. The bifurcation points are marked accordingly on the mid-point curve. While the response of mode 1 shows evidence of softening, the response of modes 2 and 3 show evidence of hardening.
The primary and superharmonic resonances of modes 1 and 2 start to merge as the initial rise increases to
µm,
Figure 12b and the natural frequency of mode 1 approaches that of mode 2. The stronger interaction between the modes results in more elaborate dynamics with three additional cyclic-fold bifurcations CF
i, a third Hopf bifurcation HP
3, and second (reverse) period-doubling bifurcation PD
2 appearing on the merged primary resonance curve. The responses of the quarter-point and mid-point are similar and larger than that of the three quarters-point
. Additionally, we observe the fading of the superharmonic of order 2 of mode 3.
The merger between modes 1 and 2 is complete at the veering point
µm, where their natural frequencies are almost identical,
Table 4, and their mode shapes are anti-symmetric,
Figure 10c. In this case, their primary and superharmonic resonances are almost indistinguishable;
Figure 12c. The response along the beam span undergoes a reversal as the response at the three quarters-point and mid-point dominate that at the quarter-point
indicating a shift in the phase relationship between the symmetric and asymmetric modes.
Beyond the veering point.
µm, the modes assume distinct shapes,
Figure 10d, and their natural frequencies start separating as that natural frequency of mode 1 continues to increase beyond that of mode 2,
Table 4. While their primary and superharmonic resonances show evidence of strong interaction, their peaks start to pull-off from each other,
Figure 12d. However, the order of the response along the beam span is maintained with the response at the three quarters-point and mid-point continuing to dominate that at the quarter-point
.
As the natural frequency of mode 1 continues to increase beyond that of mode 2 with the initial rise,
µm, the primary and superharmonic resonances of the two modes become completely distinct,
Figure 12e, and their dynamics less complex. The overarching result of the hybridization of mode 1 is the dominance of the response at the quarter-point and three quarters-point over the mid-point
. Finally, comparison among the five cases shows that the primary resonance strengthens as the interaction between the modes becomes more active and is strongest on the lower side of the veering point, where the stiffness of mode 1 is relatively low, allowing for larger motions under a fixed excitation level.