To gain an understanding of the electro-elastic properties of tactile piezoelectric sensors used in the characterization of soft tissue, the frequency-dependent electric impedance response of thick piezoelectric disks has been calculated using finite element modeling. To fit the calculated to the measured response, a new method was developed using harmonic overtones for tuning of the calculated effective elastic, piezoelectric, and dielectric parameters. To validate the results, the impedance responses of 10 piezoelectric disks with diameter to thickness ratios of 20, 6, and 2 have been measured from 10 kHz to 5 MHz. A two-dimensional, general purpose finite element partial differential equation solver with adaptive meshing capability run in the frequency-stepped mode, was used. The equations and boundary conditions used by the solver are presented. Calculated and measured impedance responses are presented, and resonance frequencies have been compared in detail. The comparison shows excellent agreement, with average relative differences in frequency of 0.27%, 0.19%, and 0.54% for the samples with diameter to thickness ratios of 20, 6, and 2, respectively. The method of tuning the effective elastic, piezoelectric and dielectric parameters is an important step towards a finite element model that describes the properties of tactile sensors in detail.
To gain an understanding of the high-frequency elastic properties of silicone rubber, a finite element model of a cylindrical piezoelectric element, in contact with a silicone rubber disk, was constructed. The frequency-dependent elastic modulus of the silicone rubber was modeled by a fourparameter fractional derivative viscoelastic model in the 100 to 250 kHz frequency range. The calculations were carried out in the range of the first radial resonance frequency of the sensor. At the resonance, the hyperelastic effect of the silicone rubber was modeled by a hyperelastic compensating function. The calculated response was matched to the measured response by using the transitional peaks in the impedance spectrum that originates from the switching of standing Lamb wave modes in the silicone rubber. To validate the results, the impedance responses of three 5-mm-thick silicone rubber disks, with different radial lengths, were measured. The calculated and measured transitional frequencies have been compared in detail. The comparison showed very good agreement, with average relative differences of 0.7%, 0.6%, and 0.7% for the silicone rubber samples with radial lengths of 38.0, 21.4, and 11.0 mm, respectively. The average complex elastic moduli of the samples were (0.97 + 0.009i) GPa at 100 kHz and (0.97 + 0.005i) GPa at 250 kHz.
The 3ω-method is used to study time dependent processes through measurements of dynamic heat capacity. The 3ω-sensor is a thin (∼0.1 µm) metal strip which is evaporated onto a substrate. The sample is probed by periodic diffusive thermal waves of frequency 2ω emitted from the strip. The heater temperature T 0 measured at frequency 3ω yields the dynamic heat capacity. The validity of a one-dimensional heat flow model, assuming an infinitely thin heater, is here studied using a finite element modelling (FEM) technique as well as experiments. To obtain results within 1% of the theory, FEM shows that the ratio between the heater width and the heat wave penetration depth (= D/ω, where D is the thermal diffusivity) must be greater than 30, which sets a low-frequency limit for the model. At high frequencies, the finite thickness of the heater causes a deviation from the model. For a thickness of 0.1 µm, the deviation is <1% at 200 Hz, reaching 5% at 5 kHz. A small 3ω-component intrinsic to the electronics together with a thermal resistance between heater and sample can explain deviations from T 0 ∝ ω −υ , at high frequencies, where υ = 0.5 is predicted by the model but experiments generally show smaller values.
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