Model for a Quartz-Crystal Tuning Fork Using Torsion Spring at the Joint of the Arm and the Base and Analysis of Its Frequency

Aoshima, Y.; Egawa, Tamotsu

doi:10.1143/jjap.41.3422

Cited by 6 publications

(5 citation statements)

References 15 publications

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“…In Table I, the estimated frequency of 1.556 GHz is close to 1.49 GHz by considering the fact that the calculated frequency of a free-clamped bar using Bernoulli-Euler beam theory is slightly larger than the measured frequency of a quartz crystal tuning fork. 16) The frequency of 1.592 GHz for the 10th mode in Table II corresponds to an estimated frequency of 1.556 GHz for the unknown 10th mode in Table I. Therefore, the displacement of the antenna structure resonator at 1.592 GHz is used to calculate the quantized displacement in §3.2 and §3.…”

Section: Vibrational Modementioning

confidence: 99%

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Analysis of Transverse Vibration of Antenna Structure Resonator Using Bernoulli–Euler Beam Theory and Quantum Mechanical Examination of Its Quantized Displacement

Ishikawa¹,

Kobayashi²

2008

jjap

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We analytically investigated the vibration of a macroscopic antenna structure resonator of 10.7 mm length showing a quantized displacement at 110 mK using Bernoulli-Euler beam theory. According to our calculations on the vibrational displacement and mode of the resonator, the vibration mode increased step-by-step from the 1st to 10th mode and then repeated again from the 1st to 10th mode. We found the 10th mode of the resonator at 1.592 GHz when using every paddle length of 490 nm, which is 10 nm shorter than that of a usual antenna structure resonator, existed near a frequency of 1.49 GHz at which the quantized displacement was observed. In the 10th mode of the antenna structure resonator at 1.592 GHz, the displacement of the central beam was so extensively damped that its mechanical energy may be considered to be zero. Therefore, the mechanical energy of the antenna structure resonator could be approximately calculated from the displacement of forty paddles regarded as cantilever beams. We could explain the quantized displacement of the antenna structure resonator using the equivalent mass of a harmonic oscillator, to which the antenna structure resonator was approximately equivalent in mechanical energy, and the quantization of a harmonic oscillator in a textbook on quantum mechanics if the wave function of quantum mechanics can be applied to this harmonic oscillator.

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Section: Vibrational Modementioning

confidence: 99%

“…Using eqs. (14), (16), (20), and (21), and the fact that the upper elements are symmetrical with respect to the lower elements, the total mechanical energy of the antenna structure resonator can be written as…”

Section: Harmonic Oscillator Approximationmentioning

confidence: 99%

Analysis of Transverse Vibration of Antenna Structure Resonator Using Bernoulli–Euler Beam Theory and Quantum Mechanical Examination of Its Quantized Displacement

Ishikawa¹,

Kobayashi²

2008

jjap

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“…Figure 6 shows the relationship between the calculated resonant frequency of the quartz-crystal tuning fork using both the analytical model shown in Fig. 4(a) and the torsion spring constant R 14) in Fig. 4(b).…”

Section: Resultsmentioning

confidence: 99%

Analysis of Change in Motional Capacitance of the Electrical Equivalent Circuit of Quartz-Crystal Tuning-Fork Tactile Sensor at Resonance by Combining Analytical Models for Elastic Foundation and L-Shaped Bar with Torsion Spring

Hatakeyama

2010

Jpn. J. Appl. Phys.

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We derived the analytical formula of the motional capacitance C a of a quartz-crystal tuning-fork tactile sensor by use of the conservation law of energy that the electrostatic energy in C a is equal to the sum of the strain energy of its arm and the elastic energy of the material in contact with its base. In this analysis, the strain energy of the arm and the elastic energy of the material in contact with its base were obtained by applying the torsion spring model to the joint of the arm and the base, which were depicted as an L-shaped bar, and the elastic foundation to the materials in contact with its base, and the electrostatic energy of the arm was derived by applying the bimorph flexural model to the arm treated as a piezoelectric material. As a result of calculation, we found that the calculated value of the change in reciprocal motional capacitance Áð1=C a Þ of the quartz-crystal tuning-fork tactile sensor by our model is closer to the measured one for plastics than that given by the theoretical formula of motional capacitance of only one arm and Áð1=C a Þ is affected by the Young's moduli of materials in contact with the sensor's base.

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“…1. This calculated result shows that it is possible to make Q of the quartz-crystal tuning fork as high as about 3:27 Â 10 4 if the vibration leakage from the base could be suppressed and the tuning fork was designed and miniaturized in submillimeter size so as to appear the torsion spring behavior in the base (the torsion spring behavior in the base is a fundamental behavior of the quartz-crystal tuning fork [9][10][11] ). Our prediction seems to be effective for realizing a sensor of gas by optoacoustic spectroscopy 12) as an application of such a miniaturized quartz-crystal tuning fork.…”

Section: Derivation Of Size Dependence Of Approximatementioning

confidence: 92%

Analysis of Size Dependence of Quality Factor of Quartz-Crystal Tuning Fork

Seki¹

2011

Jpn. J. Appl. Phys.

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We derived an approximate formula for the quality factor (Q value or Q) of a quartz-crystal tuning fork by applying the approximate calculation method developed by Landau et al. and Lifshitz et al. to the thermoelastic coupling equations previously obtained from the viewpoint of the thermoelasticity for a cantilever beam. We found that our calculated Q obtained using the approximate formula is half of the measured Q for an etching-processed quartz-crystal tuning fork. Furthermore, we derived the equation of the size dependence of Q by further approximating the approximate formula for Q at a constant frequency and various frequencies. The predicted size dependence of Q from these obtained equations is confirmed by measuring the Q of seven commercially available quartz-crystal tuning forks. The equations, which show the size dependence of Q, and the approximate formula for Q obtained here are effective for explaining the decreases in Q of the miniaturized quartz-crystal tuning fork.

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Model for a Quartz-Crystal Tuning Fork Using Torsion Spring at the Joint of the Arm and the Base and Analysis of Its Frequency

Cited by 6 publications

References 15 publications

Analysis of Transverse Vibration of Antenna Structure Resonator Using Bernoulli–Euler Beam Theory and Quantum Mechanical Examination of Its Quantized Displacement

Analysis of Transverse Vibration of Antenna Structure Resonator Using Bernoulli–Euler Beam Theory and Quantum Mechanical Examination of Its Quantized Displacement

Analysis of Change in Motional Capacitance of the Electrical Equivalent Circuit of Quartz-Crystal Tuning-Fork Tactile Sensor at Resonance by Combining Analytical Models for Elastic Foundation and L-Shaped Bar with Torsion Spring

Analysis of Size Dependence of Quality Factor of Quartz-Crystal Tuning Fork

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