Variational formulation of the Stevens-Tiersten equation and application in the analysis of rectangular trapped-energy quartz resonators

Shi, Junjie; Fan, CuiYing; Zhao, Minghao; Yang, Jiashi

doi:10.1121/1.4829535

Cited by 18 publications

(10 citation statements)

References 22 publications

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“…The displacement distribution for the thickness modes of a plate resonator with curved surfaces can be calculated from the Stevens-Tiersten theory [46][47][48]. The theory establishes a partial differential equation for the dominant component of the displacement u d , for the piezoelectric contoured BAW cavity with slowly varying thickness in the x-y plane due to the large radius of curvature (Fig.…”

Section: Acoustic Wave Distribution In a Curved Cavitymentioning

confidence: 99%

Gravitational wave detection with high frequency phonon trapping acoustic cavities

2014

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There are a number of theoretical predictions for astrophysical and cosmological objects, which emit high frequency (10 6 −10 9 Hz) Gravitation Waves (GW) or contribute somehow to the stochastic high frequency GW background. Here we propose a new sensitive detector in this frequency band, which is based on existing cryogenic ultra-high quality factor quartz Bulk Acoustic Wave cavity technology, coupled to near-quantum-limited SQUID amplifiers at 20 mK. We show that spectral strain sensitivities reaching 10 −22 per √ Hz per mode is possible, which in principle can cover the frequency range with multiple (> 100) modes with quality factors varying between 10 6 − 10 10 allowing wide bandwidth detection. Due to its compactness and well established manufacturing process, the system is easily scalable into arrays and distributed networks that can also impact the overall sensitivity and introduce coincidence analysis to ensure no false detections.

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Section: Acoustic Wave Distribution In a Curved Cavitymentioning

confidence: 99%

Gravitational wave detection with high frequency phonon trapping acoustic cavities

2014

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“…Also, a substantial amount of work is dedicated to the analysis of contoured resonators with the ability to trap acoustic energy inside the plate [36][37][38]. One of the most influential achievements of this field is known as Stevens-Tiersten theory [11,39,40]. This theory gives a partial differential equation for the dominant component of the displacement u d , for the piezoelectric spherically contoured BAW cavity, which has slowly varying thickness in the x-y plane due to the large radius of curvature (figure 1) [11].…”

Section: General Description Of Curved Baw Resonatorsmentioning

confidence: 99%

Effects of geometry on quantum fluctuations of phonon-trapping acoustic cavities

Goryachev

Tobar

2014

New J. Phys.

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This work presents some peculiarities of the near quantum ground state behaviour of curved (phonon trapping) bulk acoustic wave (BAW) cavities when compared to a conventional mechanical resonator. The curved cavity system resolves the quandary of the conventional mechanical system where the Bose-Einstein distribution requires higher frequencies for lower quantum occupation factors contrary to the constraint of an inverse frequency dependence of the quantum fluctuations of displacement. We demonstrate how the nontrivial cavity geometry can lead to better phonon trapping, enhancing the variance of zero-point-fluctuations of displacement. This variance becomes independent of overtone (OT) number (or BAW resonance frequency) overcoming the constraint and allowing better observation of quantum effects in a mechanical system. The piezoelectric electro-mechanical coupling approach is qualitatively compared to the parametric optomechanical technique for the curved BAW cavities. In both cases the detectible quantity grows proportional to the square root of the OT number, and thus the resonance frequency. Also, the phonon trapping improves with higher OT numbers, which allows the electrode size to be reduced such that in the optimal case the parasitic capacitive impedance becomes independent of the OT number, allowing effective coupling to very high frequency OTs.

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“…Shi, et al analyzed trapped-energy quartz resonators with the Ritz method based on the variational formulation of Tiersten’s equation. Results in their paper show the existence of energy trapping for a rectangular plate, which is significant for the understanding of quartz resonator [ 17 ].…”

Section: Introductionmentioning

confidence: 99%

Coupling Vibration Analysis of Trapped-Energy Rectangular Quartz Resonators by Variational Formulation of Mindlin’s Theory

Wang

Qian

2018

Sensors

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Mindlin’s two-dimensional theory has been derived and applied to research on quartz resonators for a long time. However, most works have focused on vibrations varying only in two directions, including thickness direction, while the effect of other directions like the length or width direction is normally neglected. Besides, researchers often model quartz resonators as fully electroded plates because of the resulting simplicity. Since a real device is finite in all directions and is only centrally electroded, results obtained in such works cannot offer quantitative information on vibrations with enough accuracy. In this paper, a theoretical analysis of a rectangular trapped-energy resonator of AT-cut quartz is studied using the Ritz method, associated with the variational formulation of Mindlin’s first-order equations. Frequency spectra and mode shapes of a real-scaled trapped-energy resonator, which is finite in all directions, are obtained with the consideration of mode couplings among thickness-shear mode, thickness-twist mode, and flexural mode. Results show the existence of an energy-trapping and coupling phenomenon and are helpful for thorough and accurate understanding of quartz resonator vibrations. Detailed discussions on the effects of structural parameters on mode couplings and energy trapping are provided, and the results can helpfully guide the selection of aspect ratio, length/thickness ratio, and electrode inertia in device design.

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Variational formulation of the Stevens-Tiersten equation and application in the analysis of rectangular trapped-energy quartz resonators

Cited by 18 publications

References 22 publications

Gravitational wave detection with high frequency phonon trapping acoustic cavities

Gravitational wave detection with high frequency phonon trapping acoustic cavities

Effects of geometry on quantum fluctuations of phonon-trapping acoustic cavities

Coupling Vibration Analysis of Trapped-Energy Rectangular Quartz Resonators by Variational Formulation of Mindlin’s Theory

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