3.1. Static Behavior vs. dc Voltage
A convergence analysis was conducted in order to ascertain the minimal quantity of modes needed in the Galerkin expansion. This analysis was accomplished by comparing the static deflection achieved from reduced-order models (ROMs) using two to five symmetrical mode shapes; Equation (
11). We found that for a shallow arch beam with 3.5 µm mid-point rise, the locations of the bifurcation points named snap-through, snap-back, and pull-in instability vary with the number of modes retained in the Galerkin expansion.
Table 1 presents a comparison of the discrepancies in accuracy observed among the three ROMs at the critical bifurcation points of snap-through, snap-back, and pull-in stability. It is important to highlight that employing an odd number of symmetrical mode shapes results in a more rapid and precise convergence compared to utilizing an even number. Therefore, for the remainder of the analysis, we will adopt the approximation of the five-mode ROM.
Here, the maximum static equilibrium position diagrams, which were generated by tracing the static dc voltage parameter, can be examined. The study is centered on examining how the initial curvature’s (shape and maximum elevation rise) affects the configuration of the static equilibrium diagram. It’s important to highlight that the static results depict the deflection value at a quarter of the beam length for the second mode shape profile, while for the first and third mode shapes, it shows the deflection at the shallow arch mid-point in addition to the quarter of the beam length.
For this purpose, we consider the shallow arch beam illustrated in
Figure 1 with dimensions mentioned in
Section 2 and initial mid-point rise varying from [2.5–4] µm and increasing with a step size of 0.5 µm. Then, the variation of the microbeam static equilibria as functions of dc voltage were obtained considering two initial curvature profiles (
curved-up and
curved-down) and three initial profiles,
,
, and
.
Figure 3.
Mid-point and quarter of the beam length deflections as a function of dc voltage obtained for an initial rise of µm and a curved-up initial profile of (a) , (b) , and (c) . The corresponding static profiles at several dc voltages for each case are shown in (d–f).
Figure 3.
Mid-point and quarter of the beam length deflections as a function of dc voltage obtained for an initial rise of µm and a curved-up initial profile of (a) , (b) , and (c) . The corresponding static profiles at several dc voltages for each case are shown in (d–f).
Figure 3 shows the static deflection at mid-point and quarter of the beam length and the static profiles as a function of dc voltage considering a curved-up beam configuration with an initial rise of
µm. Three initial profiles were utilized in this study. The results indicate that only one stable branch of equilibria exists considering the first, second, and third profiles, as shown in
Figure 3a–c, respectively. Additionally, it demonstrates that the deflection of the beam increases along a stable solution branch as the dc voltage rises, reaching an unstable solution branch through a saddle-node bifurcation known as ‘pull-in’.
We note that the deflections at the off-center points are equal in magnitude for the first profile,
Figure 3a, and third profile,
Figure 3c, respectively. Nonetheless, this does not hold true for the second profile, as depicted in
Figure 3b. The results also show that the maximum deflection occurs at the mid-point for the first and third cases while at the off-center points for the second case. This, in fact, is expected because the second mode has a node located at its span mid-point.
On the other hand, the static bifurcation diagram of the concave-down profile of the shallow arch exhibits behavior analogous to that of the concave-up configuration. Nevertheless, it displays a restricted stable travel range across all scenarios, as illustrated in
Figure A1. This limitation arises from the beam’s proximity to the stationary electrode (SWE), making it susceptible to voltage-induced destabilization, as detailed in
Appendix A.
Alternatively, increasing the initial mid-point rise to
µm shows that the beam exhibits two stable equilibria considering the first mode shape (
) profile and a curved-up curvature, as illustrated in
Figure 4a. The generated numerical data suggest that the bistability region is relatively restrained and narrow, implying that the shallow arch beam has the capability to transition between states, “jumping down” (snap-through) and reverting to its initial configuration (snap-back) at almost the same identical excitation dc loads. However, the existence of two stable equilibria disappears as the beam’s curvature mimics the second (
) and the third (
) mode shapes as shown in
Figure 4b,c, respectively.
To further illustrate this phenomenon, we evaluate the corresponding static profile for the first case and subjected to different dc voltages; see
Figure 4d. We note that for lower dc load, (>100 V), and for static deflection lower than the initial gap, the beam is deflecting in a standard manner of an elastic beam and around a single equilibrium. Once its mid-point deflection reaches the central line, the beam profile jumps to the counter configuration, characterized by a snap-through (ST), with mid-point deflection increasing as dc voltage increases. The profiles corresponding to the counter-curvature are marked as dashed lines in
Figure 4d. However, as depicted in the same figure, this phenomenon does not hold when the beam emulates the second and third shapes.
It is important to mention that the nodes of the second and third modes are disappearing as the voltage reaches the pull-in value, illustrated in
Figure 4e,f. Alternatively, the static bifurcation diagrams when the beam is curved-down are shown in
Figure 5. We note that only a single stable equilibrium exists for all three cases because the beam is set closer to the stationary electrode.
Figure 6a–c and
Figure 7a–c, respectively, depict the bifurcation diagrams of the beam static deflection including both stable and unstable branches as a function of dc voltages. The beam initial mid-point is set to
µm for the first case and
µm for the second case. These diagrams take into account three distinct initial curvature profiles: the first, second, and third natural mode shapes of a doubly-clamped beam similar to those obtained above. In this analysis, the deflections are simulated at the mid-point and quarter node positions of each respective initial curvature shape. This is conducted to investigate the effect of a higher initial rise on the static response of the shallow arch beam such that its curvature mimics the first three mode shapes of the clamped–clamped beam.
Furthermore, when the shallow arch is concave-up,
Figure 6 and
Figure 7a–c, as compared to that where the arch is concave-down,
Figure 6 and
Figure 7g–i, the beam exhibits two stable equilibria when considering the shallow arch mimicking the concave-up first mode-shape-like arrangement. This bistability refers to the existence of two stable equilibrium positions at the same presumed dc voltage resulting in a hysteric band delimited by the snap-through (ST) and snap-back (SB) nodes, both shown in
Figure 6a and
Figure 7a, respectively.
In this particular case, the arch experiences snap-through followed by pull-in as the applied dc voltage increases. However, for all other cases, the microarch undergoes immediate pull-in. Indeed, when the beam presumes an initial curvature corresponding to the second and third mode shapes of a clamped–clamped beam, only a single bistability behavior is recorded for various dc voltage; see
Figure 6b,c for the first case and
Figure 7b,c for the second case.
Figure 6 and
Figure 7d–f display the static profiles of the shallow arch with curved-up curvature excited by various dc voltages. They are describing consistent symmetry without notable options of symmetry-breaking behavior. This means that the beam maintains its symmetric (first and third) and asymmetric (second) shapes.
It is worth noting that in the third-mode-like shape case, the simulated figures indicate a trend in which the stroke decreases, while the pull-in voltage increases, when considering both concave-up and concave-down profiles. This trend suggests that the effective stiffness of the shallow arch increases with the clamped–clamped shallow arch stretching while considering it concave-down.
With an increase in the assumed maximum rise amplitude to 4 µm, there is a noticeable enhancement of the bistable behavior accompanied by a broader hysteresis band in the concave-up first mode shape arrangement, as illustrated in
Figure 7a. Furthermore, the third mode shape, as depicted in
Figure 7c, also begins to demonstrate bistable behavior under these conditions, while the second mode remains unaffected by the initial rise value, as shown in
Figure 7b. Nothing can be highlighted when considering the concave-down arrangement of
Figure 7d–f as compared to the previous case.
3.2. Eigenvalues vs. dc Voltage
In this section, we evaluate the variation of the natural frequencies of the shallow arch with various initial rise levels, three initial profiles, and under several dc voltages. Toward this, we substitute the static deflections obtained above for each case in Equation (
13) and then solve for the corresponding eigenvalues.
Figure 8 shows the first five natural frequencies versus the initial rise considering the first mode (
), second mode (
), and third mode (
) profiles obtained using five-modes ROM and FEM, respectively.
The FEM software COMSOL Multiphysics (5.3a) [
37] was also employed to solve for the variation of the eigenfrequencies as a function of the static voltage for all cases. A three-dimensional model was generated following the sensor’s dimensions. The beam is fixed at its two ends. Tetrahedral elements were employed to mesh the 3D model with number of elements approximately reached (35 K). The elements size ranges from 10 to 70 µm. The Solid Mechanics Interface module was utilized to solve for the variation of the first five eigenfrequencies with the beam profile and mid-point rise change.
The results obtained using the developed ROM are shown in solid lines while those obtained using the FEM model are marked with symbols. As seen in
Figure 8a, there is a crossing between the frequency corresponding to the first symmetric
, marked as a dark yellow line (
—), and the frequency corresponding to the first antisymmetric
, marked as a dark magenta line (
—) that occurs at
µm. We note that other frequencies do not cross nor veer.
On the other hand, for shallow arch mimicking the concave-up second mode-shape-like arrangement, several crossings between the odd and even eigenfrequencies were triggered, as shown in
Figure 8b. The first crossing occurs between the first antisymmetric
and the frequency corresponding to the second symmetric
, marked as a light blue line (
—), at
µm.
Another mode transition is observed between the second antisymmetric mode (indicated by the green line —) and the frequency associated with the third symmetric mode (indicated by the black line —). However, this transition occurs at a higher initial rise. Here, it should be emphasized that the eigenvalues corresponding to the symmetric modes remain unaltered when the beam curvature imitates the second mode shape of a clamped–clamped beam.
Furthermore, when we initially configured the shallow arch with a third-mode-like shape, the eigenvalues began to demonstrate distinct mode-veering and -crossing behaviors. The mode-veering and mode-crossover occurrences are controlled by intricate underlying mechanisms originating from the interaction of material characteristics, boundary conditions, and structural dynamics. When distinct vibration modes’ resonance frequencies move too close to one another, a nonlinear interaction between them happens. On the contrary, mode-crossing happens when two or more modes cross paths in frequency space, which results in an abrupt alteration of the resonator’s dominant vibrational behavior. These phenomena are clearly a consequence of nonlinearity present in specific parameters within the equations of motion, which inevitably lead to these mode interactions. It is important to mention that the third frequency
increases drastically as the initial rise increases until it crosses the second antisymmetric, as illustrated in
Figure 8c. A good agreement between both models was achieved, expect for higher initial rise and a shallow arch with a third-mode-like shape. Following that, the eigenvalue consistently increases until it approaches the fifth frequency, at which both modes undergo a mode-veering process.
Indeed, if the initial arrangement of a shallow arch beam closely resembles a mode shape that is symmetrical, the eigenvalues that are antisymmetric remain constant because the position of the central node remains unchanged. Similarly, when the initial shape of the arch beam resembles a mode that is antisymmetric, the eigenvalues that are symmetrical do not change since the positions of the interior nodes remain unaltered.
Subsequently, we examined the effect of a static voltage on the fundamental frequencies of the beam, considering various mid-point elevations and initial profiles. To achieve this, we replaced the previously obtained static equilibrium with a reduced-order model (ROM) featuring three symmetric and two antisymmetric modes. We then solved the coupled linear eigenvalue problem to determine the corresponding eigenvalues under the influence of a static dc load.
The first four fundamental frequencies were computed using the ROM and they correspond to the first and second in-plane symmetric and antisymmetric mode, respectively.
Figure 9a,c,e show the variation of the resonance frequencies as a function of dc voltage considering a curved-up beam configuration with an initial rise of
µm and three initial profiles. Considering the first profile (
)-like arrangement, we found that the beam initial rise was not sufficient enough to activate the snap-through, although it showed a peak at 90 V. In addition, the eigenvalues did not show any sign of snap-through behavior for a beam curvature mimicking the second profile, as illustrated in
Figure 9c.
On the other hand, we note that for a beam-curvature-like third-mode profile arrangement, the first and second resonance frequencies (
and
) decrease as the voltage increases. However, this is not the case for the third and fourth frequencies. The results show that the third frequency
, marked as a sky blue line (
—), increases as the dc voltage increases until it crosses the fourth frequency
, marked as a green line (
—), at 84 V, as shown in
Figure 9e.
Figure 10.
The variation of the fundamental frequencies of the beam as a function of the dc voltage for a mid-point initial rise of µm and a curved-up initial profile of (a) , (c) , (e) , and µm, and a curved-up initial profile of (b) , (d) , and (f) .
Figure 10.
The variation of the fundamental frequencies of the beam as a function of the dc voltage for a mid-point initial rise of µm and a curved-up initial profile of (a) , (c) , (e) , and µm, and a curved-up initial profile of (b) , (d) , and (f) .
Alternatively, increasing the mid-point rise to
µm shows a possibility for the snap-through phenomenon as long as the beam curvature mimics the first profile, as shown in
Figure 9b. The figure indicates a relatively restrained and narrow snap-through band. This, in fact, is consistent with the static bifurcation diagram in
Figure 4a.
We note that the third frequency is lower than the second frequency as the beam curvature follows the second mode profile
. These two values are moving away from each other as the dc voltage increases, as shown in
Figure 9d. This is a classical behavior of veering phenomenon. On the other hand, a mode-crossing phenomenon is observed between the third and fourth frequencies when the shallow arch initial profile transitions to the third mode like shape, as depicted in
Figure 9d.
Similar nonlinear phenomena were also observed for the beam with mid-point rise of
µm and
µm, respectively. Nevertheless, it’s worth noting that the extent of the snap-through instability grows as the mid-point elevation increases, and the curvature aligns with a shape resembling the first mode, as illustrated in
Figure 10a for the first initial rise and
Figure 10b for the second initial rise. Furthermore, an instantaneous snap-through event, characterized by a narrow dc voltages band, occurs when the mid-point rise is 4 µm, and mode-veering becomes evident as the curvature of the beam closely resembles the third mode shape, as depicted in
Figure 10f.
Appendix A provides a comprehensive presentation of the fundamental frequency fluctuations of the beam with the dc voltage, and for initial profiles characterized by a downward curvature, ensuring a comprehensive analysis.
It is important to acknowledge that the shallow arched MEMS resonators are highly sensitive to various parameters, such as residual stresses, geometric imperfections, and material characteristics. Material qualities like Young’s modulus, density, and Poisson’s ratio can significantly influence the mechanical behavior of the resonator under different loading conditions. Alterations in these properties can lead to changes in the overall performance, damping behavior, and resonant frequency of the device. Moreover, the residual stresses present in the fabricated structure can modify its mechanical response and affect metrics like stiffness, mode shapes, and stability [
38,
39]. These stresses, induced by factors like thermal fluctuations or the manufacturing process, need to be carefully considered in the design and optimization phase to mitigate any adverse effects on the resonator’s functionality. Additionally, the performance of shallow arched MEMS resonators is strongly impacted by geometric irregularities.
Moreover, the behavior of shallow arched MEMS resonators is significantly influenced by geometric imperfections. Anomalies in performance may arise due to deviations such as surface roughness, dimensional variations, and errors in fabrication, introducing asymmetries and irregularities into the system. These imperfections can not only affect the overall functioning and reliability of the device but also influence mode shapes, energy dissipation mechanisms, and resonance frequencies. It is essential to optimize the design and operation of shallow arched MEMS resonators for a variety of applications in the fields of sensing, signal processing, and communication systems. This optimization process requires a thorough understanding and assessment of the effects of material characteristics, residual stresses, and geometric irregularities.