The stability maps of the Mathieu and Meissner equations have been extensively studied. These maps show regions of unstable response, a.k.a. instability windows. The boundaries of these windows are where the response is periodic. In contrast to the stability map of the Mathieu equation, the stability map of the Meissner equation includes crossover points which are intersections of two boundaries of an instability window. All these crossover points are located below the critical line, where the modulated stiffness is necessarily positive. Recently, it was shown that the stability map of the Whittaker-Hill equation includes crossover points, some which are located well above the critical line (i.e., where the stiffness is negative during specific durations of the modulation cycle). The present work extends previous studies of the Whittaker-Hill equation and reveals that crossover points are each engulfed by a pair of small regions of stable response. It is shown that inclusion of linear damping merges each pair into an island of stability. All these islands of stability are located above the critical line, where the response is expected to be predominantly unstable. It is further shown that all crossover points are located along watermark lines with distinct values of stiffness modulation. The Whittaker-Hill equation is generalized by adding a tuning parameter to the second term of the stiffness modulation. It is demonstrated that for sufficiently large values of this tuning parameter, crossover points may be relocated to the left side of the stability map. The relevance of this work to microelectromechanical systems (MEMS) is that electrostatic double-sided comb-drive actuators with tapered fingers are seemingly the only practical way to modulate the linear component of stiffness, without inducing any other unwanted nonlinearity. The Whittaker-Hill equation requires the modulation of linear stiffness with two harmonic components, and it appears that MEMS devices would be the most feasible way to implement this.

We present a rigorous analysis of the in-plane flexural vibrations of a thin rotating circular ring. The ring is made from an anisotropic material with cubic symmetry, as in (100) single-crystalline silicon. The ring is assumed to be sufficiently thin, such that it can be considered as an inextensible Euler-Bernoulli beam. In this study, both the natural frequencies and their related mode shapes are analytically derived using an asymptotic method, for the modes numbered n=2, 3 and 4. We show that due to a rate of rotation, these modes exhibit a precession-like response, which was previously shown to occur in isotropic rings. However, for rotating rings that are made from an anisotropic material, we show that the even-ordered modes n=2 and n=4, exhibit a ‘breathing’ phenomenon in the precessing mode. In deriving the asymptotic approximation of the frequencies and mode shapes, we adopt a technique used in quantum mechanics, and modify it appropriately for the problem at hand. The theoretical predictions are verified by comparing them to finite element simulations, showing good agreement. This work is relevant to the emerging technology of vibrating ring gyroscopes that are made from (100) single-crystalline silicon.

This letter considers the influence of the Poisson ratio and beam width on the fundamental frequency of cantilever beams, i.e., their natural frequency at infinitesimal amplitudes of vibrations. We show that the fundamental frequency of a cantilever increases with increasing Poisson ratio, and also increases with increasing beam width. Within the context of linear theory, we show that this effect is caused by a decrease in inertia rather than an increase in stiffness. This insight may be relevant to studies of the nonlinear response of cantilevers, where an increase in the fundamental frequency may be partially attributed to a decrease of inertia.

This work relates to piezoelectric vibrating energy harvesters, that are constructed from a unimorph cantilever with a massive edge block. The dynamic response of the cantilever is considered when it is excited into vibrations at its natural frequency, where its deformation amplitude is maximal. The optimal response of such a harvester is achieved when the amplitude of the axial strain in the piezoelectric layer, is uniform . Practical technological considerations dictate the thickness of the unimorph, but its planform geometry (i.e. the vareation of the width along the cantilever) is a design choice. The optimal planform of such a unimorph cantilever has been the focus of many previous studies, which included extensive simulations and experimental investigations. In these previous studies it was concluded that the optimal planform is a trapeze , where the cantilever tapers from its clamped edge towards the edge block. However, to date, no model with explicit predictive capabilities was proposed. In the present study we derive an analytic expression of the planform, that ensures a uniform axial strain over the top surface of a cantilever unimorph with an edge block. Our analysis provides a rational explanation why a trapeze planform is optimal, and provides an explicit functional form of the optimal geometrical parameters of this planform. The predictive capabilities of our model are validated by comparison to finite element simulations.

Electromechanical resonators are an essential component in many sensors. These systems are essentially mechanical resonators that are driven by harmonic electrostatic forces, and their response is often measured using capacitive sensing. There is some ambiguity in the literature with respect to the resonance response of mechanical resonators. Should resonance be associated with the peak amplitude of velocity or should it be associated with the peak amplitude of displacement? Another ambiguity relates to the definition of phase: Should phase relate velocity or displacement to the driving force? These two issues are addressed here, and it is shown why resonance should be associated with peak amplitude of velocity and not with peak amplitude of displacement.

We present a rigorous analysis of the in-plane flexural vibrations of a thin circular ring, made from anisotropic material with cubic symmetry (e.g. single-crystalline silicon). We do not make any assumptions on the mode shapes of vibration, but rather derive them as part of the solution. Rings in which the cross-section has a finite radius of gyration (i.e. with a finite radial thickness) are considered. In our analysis we present an analytic proof of the degeneracy of odd-ordered modes. The results are compared with previous studies and numerical simulations.

We demonstrate that a very thin NiCr adhesion layer may have a strong effect on the deformation of released structures. We consider MEMS switch sensors that are constructed from suspended perforated disks. The switches are fabricated by depositing a NiCr adhesion layer over a sacrificial layer, followed by deposition of a seed gold layer. This thin seed construction is then thickened by electrodeposition of a second gold layer. The last process step before releasing the structure is wet etching of the adhesion layer. We measured the curvature of many disk structures, that are identical in all geometry and process parameters, except for the level of NiCr overetch. The level of NiCr overetch strongly affects the curvature of the released structures. Much of the residual deformation occurs during packaging, which requires a relatively long period of elevated temperature. We suspect that interdiffusion during packaging determines the residual curvature. This interdiffusion should be different in regions where the adhesion layer has been overetched, which may explain the dependency of disk curvature on the level of NiCr overetch. It is suggested that careful attention should be given to the adhesion layer, which is often neglected as it is very thin. Such attention may help to improve the reliability and functionality of surface-micromachined metallic devices.

This work considers nonlinear effects in the in-plane free-vibrations of a thin circular ring. The investigation focuses on the flexural elliptic mode of vibrations. The ring thickness is finite, but is sufficiently small to allow the ring to be modeled as a curved Euler-Bernoulli beam. The governing equations are obtained by using the Hamilton principle, postulating specific mode shapes of the vibrating ring, and applying the Galerkin method to obtain reduced-order governing equations. To investigate the effect of vibration amplitude and the effect of ring thickness on the free-vibration frequency, we postulate general functional forms of the time-dependent variables, and apply the harmonic balancing technique to obtain a system of nonlinear algebraic equations. An explicit asymptotic solution is presented. The results show that for an increasing amplitude of vibrations, the behavior is initially of a softening type, as was suggested in many previous investigations. However, we show that a further increase of the amplitude of vibrations may result in a hardening behavior. The results of the asymptotic solution are validated with numerical computations.

We demonstrate a novel technique for operating a differentially-driven and differentially-sensed double sided comb-drive resonator. The resonator is driven by application of two out-of-phase ac signals to the drive stators, while the rotor is subjected to a pure harmonic signal at the same frequency as of the ac driving signals. In this mode of operation, no dc bias is necessary. This strategy results in two frequency mixing operations, such that a clear response can be sensed at the 3rd harmonic of the drive signal. Besides the responses in the 1st and 3rd harmonics of the drive frequency, no other higher-harmonic components are predicted from the analysis or observed in the measurements. With harmonic biasing, the resonator is insensitive to both feed-through currents that result from direct capacitive cross-coupling of sense and drive pads, and to feed-through currents that result from imbalanced sense ports due to packaging and setup. We show that a clear response can be achieved for low-power drive signals, whereas if the system were driven as a classic resonator with a dc bias on the rotor, the response would have been overshadowed by feed-through. We also demonstrate that the phase of the sensed response can be modified by varying the phase of the harmonic bias on the rotor. (c) 2021 Elsevier B.V. All rights reserved.

Recently we presented a parametric resonator which is constructed from a double-sided comb-drive transducer with an electrostatically floating rotor. That device had a natural frequency of ~2.3 kHz. In the present study we present a parametric resonator of the same type, but with a natural frequency of ~30 kHz, and a more compact design. The higher frequency is relevant for several applications, and the increased stiffness may contribute to enhancing fabrication yield. However, due to the more compact design, the electrostatic modulation of stiffness is less effective. Because of the drastic reduction of the ratio between modulated stiffness and average stiffness, the new resonator cannot be driven in high-order instability windows, without reverting to excessively high driving voltages. Since it could only be driven in the first instability window, the differential sensing signal is at the same frequency as the driving signal. This makes it difficult to distinguish between motional and feed-through currents. We demonstrate that the 3 rd harmonic of the differential current is unaffected by feed-through and is therefore preferable for sensing the device response. We show that this higher harmonic component of current, is a unique characteristic of the resonator, and it is not due to a nonlinear mechanical effect, such as Duffing stiffening. [2021-0032]

In this work we measure the response of an electrostatic MEMS resonator that is made from single-crystalline silicon. Our setup allows for simultaneous measurement of the resonator motion, using both capacitive sensing and optical sensing with a laser vibrometer. We demonstrate that both vibrometer laser illumination and microscope illumination, can affect the resonance frequency of the resonator. Furthermore, we demonstrate that this thermal effect may be non-monotonic. We demonstrate that for low levels of illumination power, the resonance frequency first increases, but then resonance frequency decreases when the illumination power is further increased. This peculiar response is attributed to the competing effects of prestress, and of thermal stress that is induced by illumination. We demonstrate that the measurement of device resistance is a good indicator of thermal heating.

When a piezoelectric material is subjected to a quasi-static electric (or mechanical) load, part of the input energy is converted to the mechanical (or electric) domain. The piezoelectric coupling factork(2)is defined as the ratio between the converted energy and the supplied energy. This factor is often considered as a measure of the transduction efficiency of the material. Another definition of the coupling factor is a non-dimensional ratio of material coefficients. To examine the compatibility between these two different definitions, we consider several quasi-static loading cycles of a simple one-dimensional problem. We show that in some specific cases, the two definitions are equivalent, but that in other cases they are incompatible. In addition, we show that in specific quasi-static loading cycles, the converted energy may be increased by a slight modification of the unloading part of the cycle.

For the first time ever, we experimentally demonstrate auto-resonance driving of an electrostatic resonator. This simple driving scheme, instantaneously locks on to the resonance frequency from the very first cycle, and the amplitude of the harmonic oscillation rapidly converges to the stable, fully-developed response. We demonstrate that even if the resonator is nonlinear and its resonance frequency is affected by motion amplitude, the auto-resonance driving scheme naturally tracks the nonlinear backbone of the dynamic response. We demonstrate that auto-resonance driving eliminates the bifurcation instability, which appears when a nonlinear resonator is driven in frequency sweeps. This means that nonlinear resonators can be operated at large amplitudes without any concern of instability. Auto-resonance is therefore a simple and practical alternative to phase-lock-loop driving of resonators.

We derive an analytic solution for a guided-guided beam that undergoes large deformations. Both edges of the beam are rotationally constrained, and they can only be translated along two non-parallel straight lines. The analytic solution is validated by a finite-elements simulation and experimental measurements. All solutions are in good agreement. This analytic solution is relevant for efficient optimization of compliant mechanisms. (C) 2020 Elsevier Ltd. All rights reserved.

For the first time ever, we demonstrate an Epi-Seal encapsulated, self-excited Franklin oscillator. This electromechanical switch oscillator, is driven by a constant input of 3.8 [V], sustains self-oscillations at a frequency of 88.6 [kHz], and requires a power of only 1.5 [μWRMS]. We demonstrate over 200 million switching cycles, with two events of contact/charge transfer in each cycle. We demonstrate that a high mechanical restoring force is crucial for long-term reliability of switch oscillators. Furthermore, we implement a new driving scheme that further increases the long-term reliability. The relevance of this device is its potential as a new concept for a low cost, low power, wafer level MEMS oscillator.

We present a novel way of using a double-sided comb-drive resonator, by disconnecting the rotor from any dc bias, and leaving it electrostatically floating. To demonstrate this, we first characterize the response of the resonator when the rotor is subjected to a dc bias, and demonstrate that as is widely known, increasing the dc bias on the rotor increases the resonator response to a small ac input signal. Then we demonstrate that even when the rotor is disconnected from any dc bias voltage, and it is left electrostatically floating, the resonator still responds to the input signal, and this response is considerably large. The explanation for this surprising phenomenon is that by disconnecting the rotor and keeping it electrostatically floating, we turn the common resonator into a parametric resonator. When the system is thus configured, small ac signals that drive the system are sufficient to produce a large amplitude of harmonic response, without requiring any dc bias. To validate the parametric response, we demonstrate that the system can be driven to resonance, by driving it at unit-fractions of the fundamental frequency. The relevance of this work is that it demonstrates that electrostatic resonators can be operated without requiring any dc bias. This may be beneficial for low-power sensors necessary for IoT technology. [2020-0066]

Vibrational wineglass modes in disk and rings that are made from (100) silicon are considered. It is shown that all odd-ordered wineglass modes are necessarily frequency matched, each with its orthogonal conjugate. In contrast, all even-ordered wineglass modes are likely not to be frequency matched with their orthogonal conjugate. Some specific cases are presented that show that many wineglass modes do not have a rotational periodic symmetry. This is relevant to disks and rings that are driven and sensed by surrounding gap-closing electrostatic actuators.

We provide, a rational explanation for mode ordering observed in some specific disk resonating gyros (DRG). In DRGs that are made from (100) single crystalline silicon (SCS), it is well known that the frequencies of the two orthogonal conjugate 2nd order (elliptic) wineglass modes, are distinct. This is because the material is anisotropic, and the elastic moduli in the [100] direction are different from those in the [110] direction. For DRGs that are made from (100) SCS, it is also well known that the frequencies of the two orthogonal conjugate 3rd order (rounded triangular) wineglass modes, are identical. In essence, the 3rd order wineglass mode is insensitive to material anisotropy. In most structures, it is often the case that higher order modes occur at a higher frequency. However, in a recent study of DRGs that are made from (100) SCS and are constructed from spoke-connected rings, it was surprisingly discovered that the frequency of the 3rd order wineglass mode is lower than the frequency of the 2nd order wineglass mode. In this work we analyze the problem and provide new insight, and for the first time provide a rational explanation of this phenomenon.

We present for the first time ever, a piezoelectric beam actuator that can be directly driven in pure torsion. The beam actuator is designed with interdigitated electrodes (IDEs), on both its top and bottom surfaces. The IDEs are used for both poling and driving. When the IDEs are driven by a voltage with the same polarity as used for poling, they induce a combination of shear and expansion strains, close to the top and bottom surfaces. The expansion strains at the top and bottom surfaces are similar, and hence they produce no net bending. However, the shear strains at the top and bottom surfaces are in opposite directions, and they produce a pure twist. If the IDEs on the top and bottom surfaces are driven by the same voltage but in opposite polarity, then a pure bending response is produced. We experimentally demonstrate that the actuator can be driven in either pure torsion or in pure bending, with negligible cross-coupling.