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Vibrational modes of higher order in micromachined resonators exhibit low damping in liquid environments, which facilitates accurate sensing even in highly viscous liquids. A steady increment in mode order, however, results in sound dissipation effects at a critical mode number n crit , which drastically increases damping in the system. Basic under...
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In this study, the effects of temperature (T) and relative humidity (RH) of moist air are discussed on the quality factors (Q-factor) of micro-electro-mechanical-system (MEMS) cantilever resonators in atmospheric pressure (p = 101,325 Pa). The dominant squeeze film damping (SFD) of MEMS cantilever resonators is studied by solving the modified molec...
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This paper discussed the effect of environmental conditions (moisture and temperature) on the quality factors (Q-factor) of micro-electro-mechanical systems (MEMS) cantilever beam resonators in wide range of gas rarefaction (pressure (p), and accommodation coefficients (ACs)), and flexural mode of resonator. The modified molecular gas lubrication (...
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... However, reasonable for the considered resonators, this approximation may not hold for other geometries of interest where the mechanical wavelength becomes commensurate with the length scale of the flow. 23,24 A compressible fluid being elastic, it requires the concomitant treatment of compressibility and viscous effects and hence falls under the category of visco-elastic problems, even if relaxation effects in the stress are absent in the constitutive stress-strain relaxations. In a seminal paper, 25 the problem of a sphere oscillating in a compressible viscous liquid was solved exactly using a Voigt body description for elastic and viscous media. ...
... In the approximation of a thin disk (h ( a), we neglect the normal components of $ Â W on the sidewall, top, and bottom surfaces, for the reasons already discussed in the viscous case. The boundary conditions on the normal velocity are then solely imposed by $/, while those on the tangential velocity involve both $/ and $ Â W [Eqs. (22) and (23)]. This fixes a unique solution, and we can superpose $ Â W and $/ in order to calculate the complex power [Eq. ...
Radial mechanical modes of miniature disk-shaped resonators are promising candidates for probing the ultra-high-frequency rheological properties of liquids. However, the lack of an analytical fluid–structure model hinders the inference of liquid properties from the experimental measurement of such radial vibrations. Here, we develop analytical models for the case of a disk vibrating in a compressible viscous liquid. Closed-form expressions for the mechanical quality factor and resonant frequency shift upon immersion are obtained and compared with the results of numerical modeling for a few significant cases. At frequencies above 1 GHz, our model points out the significance of compressibility effects.
... 5,[22][23][24] The above-mentioned locality of the hydrodynamic load for slender beams deteriorates as the length-to-width aspect ratio of the cantilevered plate decreases, with the dynamic response of wide cantilevered plates differing markedly from their slender counterparts. Moreover, non-conventional transverse modes can be excited by low aspect ratio plate-type resonators in liquid, 25 with a recent numerical study reporting a distinct displacement spectrum for these modes. 26 Despite the behavioral shift as the aspect ratio is reduced and the existence of moderate-to-low aspect ratio cantilevers in practice, analytical results for wide cantilevered plates are scant. ...
Comprehensive theoretical models for the dynamic response of slender cantilevered beams immersed in fluid have been widely reported, while the distinct behavior of wide cantilevered plates has received comparatively little attention. In this article, we develop an exact analytical theory for the resonant response of rectangular cantilevered plates of zero length-to-width aspect ratio that are immersed in unbounded viscous fluids. Unlike the opposite slender limit of large aspect ratio, the hydrodynamic load experienced by zero-aspect-ratio cantilevered plates is inherently non-local, which can strongly affect the individual mode shapes of the plate. In addition, finite-element-method simulations are reported for two- and three-dimensional cases of zero and finite aspect ratio, respectively. Accuracy of the present theory and that of Atkinson and Manrique de Lara [J. Sound Vib. 300, 352 (2007)] for small viscosity and zero aspect ratio is assessed using the former simulations. The latter simulations are used to clarify the regime of validity of the present theory as a function of aspect ratio, along with that of existing theory for slender (large aspect ratio) beams. The results of this study are expected to be of practical importance to micro- and nano-electromechanical system design and their applications.
... only intrinsic damping in the case of high vacuum, or only viscous damping at atmospheric pressure), so that the influence of the other effects on the total quality factor can be neglected. For special geometries and flow regimes, analytical models for the quality factor are available from the literature [3][4][5][6][7][8][9][10]. However, the transition regime between molecular free flow and viscous flow is particularly difficult to model, since neither molecular theory nor continuum mechanics can be applied directly. ...
This study explores the damping behavior of micro-oscillators within the transitional flow regime between molecular flow and viscous flow, including the presence of a neighboring geometric boundary. The damping is described by a model based on standing thermal waves (STW), which form a thermal resonance depending on the pressure and on the gap width to the boundary. This concept was first introduced in a previous paper for a nitrogen atmosphere and is now examined for seven further gases (He, Ne, Ar, Kr, CO2, N2O, SF6). The distance between the micro-oscillator and the confining geometry, given by an Al plate, is in the range of 150 µm to 3500 µm. Fitting the thermal wave model to the experimental data shows very good agreement and explains the convex region for the measured quality factor Q in the transition from the molecular to the viscous flow regime. The evaluation of the thermal resonance model shows a strong correlation between the ambient pressure at which thermal resonance occurs and the thermal properties of the gas (i.e. thermal diffusivity a and adiabatic index κ) including the gap width h which is in agreement with the proposed theory. This opens up the possibility of using micro-oscillators to measure the thermal properties of gases in a very precise way, comparable to the thermal wave resonator cavity principle, but with a much simpler and smaller setup.
... For instance, Q-factors up to 366 in water were observed. [32][33][34] Plate out-of-plane vibrational modes differ from beam modes because plate modes may be two-dimensional. The plate's transverse velocity w may vary not only along the plate's length (x-direction) but also along the plate's width (y-direction), w ¼ w(x, y). Figure 3(a) shows an example of an out-of-plane vibrational mode of a plate where the displacement w is not constant along the y-direction. ...
... In fluid flows around MEMS, compressibility effects occur when the acoustic wavelength becomes smaller than the flexural wavelength. For micro-plates in liquids, compressibility occurs at frequencies above 3 MHz, 34 and in air above hundreds of kilohertz. 22,38 With the incompressibility hypothesis, the mass continuity equation is given by ...
... One important characteristic of wide micro plates is that their vibrational modes are not limited to the beam-like modes. Furthermore, according to experiments, 34,35 non-beam-like vibrational modes may exhibit very high Q-factors in viscous fluids. Figure 22 shows the free corner's (x ¼ l, y ¼ b=2) absolute displacement spectrum f c of a plate with r ¼ 8=16 up to 500 kHz in water due to symmetric and anti-symmetric excitations. ...
We numerically investigate the fluid-structure interaction of thin elastic cantilever micro-structures in viscous fluids. The Kirchhoff plate equation describes the dynamics of the structure, and a boundary integral formulation represents the fluid flow. We show how the displacement spectrum of the structures changes as the geometry is altered from a narrow beam to a wide plate in a liquid. For narrow beams, the displacement spectrum exhibits only a few resonance frequencies, which correspond to the vibrational modes described by the Euler-Bernoulli equation (Euler-Bernoulli modes). The spectrum of wide plates exhibits several additional resonance frequencies associated with the plate's torsional and higher-order vibrational modes. Wide plates in Euler-Bernoulli modes exhibit higher damping coefficients, but due to an increased added-mass effect, also higher Q-factors than slender beams. An investigation into the fluid flow reveals that for the Euler-Bernoulli modes of wider plates, the fluid flow and energy dissipation near the plate's edges increase, resulting in increased damping coefficients. Concomitantly, a region of minimal viscous dissipation near the plate's center appears for wider plates, resulting in an increased added-mass effect. Higher-order modes of wider plates exhibit lower Q-factor than the Euler-Bernoulli modes due to a decreased fluid flow at the plate's edges caused by the appearance of circulation zones on both sides of the plate. This decreased flow at the edge reduces the damping and the added-mass effect, yielding lower Q-factors. We anticipate that the results presented here will play a vital role in conceiving novel MEMS resonators for operation in viscous fluids.
... Recently it has been discovered experimentally that wide thin plates can exhibit very high Q-factors in liquids. For example, a Q-factor of 197 was obtained in water at 336 kHz when the plate was excited in the so-called roof tile-shaped modes [39,56,47,48]. Roof tile-shaped modes are characterized by having two or more nodal lines parallel to the plate's length (n y P 2), and only one nodal line parallel to the plate's width (n x ¼ 1), as depicted in Fig. 1. ...
... For micro-plates in liquids, this condition is fulfilled commonly up to at least 3 MHz [48], and in air up to hundreds of kilohertz [64,65]. Hence, incompressible continuum flow is a standard flow regime around micro-plates. ...
... The roof tile-shaped vibrational modes exhibit an increasing Q, reaching values as high as 200 for the (1,9) mode. This high Q-factor pattern is consistent with experimentally obtained quality factors of micro-plates in liquids [39,56,48]. ...
In this paper, we present a semi-numerical method for determining the dynamics of micro-resonators with finite width immersed in incompressible viscous fluids. The micro-resonator is modeled using Kirchhoff plate theory, and the hydrodynamic force acting on the plate is determined from a boundary integral formulation of the Stokes equations. The resulting equation of motion is solved with a continuous/discontinuous finite element method in which an interior penalty term imposes C1-continuity to the plate’s deflection. Numerical investigations show the method to be convergent with an exponent of the convergence rate equals 2. Examples demonstrate that the proposed method overcomes the limitations of existing semi-analytic methods, only applicable to beam geometries, considering arbitrary plate modes in the structure’s dynamics and their effects on the fluid flow. Different resonator geometries are investigated for which displacement spectrum, mode shapes, and quality factors are not determinable with existent semi-analytic methods. Moreover, we find excellent agreement between simulated and experimental data, which has not been achieved even with purely numerical methods. The present method will allow the understanding of high quality factors of wide micro-resonators in viscous fluids and facilitate new applications in liquid atomic force microscopy and gas sensing in ambient and low-pressure conditions.
... In another study the nonlinearity of other eigenmodes was investigated leading to the result that the RTS modes exhibit the highest nonlinearity, due to two reasons. The RTS modes are related to a high quality factor [5] leading to a large vibration amplitude and therefore to a high nonlinearity. Furthermore, due to the higher frequency, the modes form higher vibration velocities which favor nonlinear effects [6]. ...
This paper investigates the utilization of MEMS oscillators for absolute pressure measurement applications. For this purpose, the resonance behavior of piezoelectrically (AlN) driven micro-oscillators is characterized. The focus of this work is on so-called roof tile-shape (RTS) modes which exhibit a strong nonlinearity. These eigenmodes are investigated experimentally for two cantilever structures and for three AlN-layer coverage areas (33, 50, and 100 %), depending on the ambient pressure ranging from 10-3 to 10 3 mbar. As a special feature related to a Duffing oscilla-tor, nonlinear spring softening and spring hardening effects occur, depending on the size of the coverage. For the evaluation , the frequency shift of the peak amplitude and the frequency hysteresis are presented as two quantities, which correlate well with the ambient pressure.
