Lead zirconate titanate (PbZr_xTi_1-xO₃, or PZT) is a piezoelectric material widely used as sensors and actuators. For microactuators, PZT often appears in the form of thin films to maintain proper aspect ratios. One major challenge encountered is accurate measurement of piezoelectric coefficients of PZT thin films. In this paper, we present a simple, low-cost, and effective method to measure piezoelectric coefficient d₃₃ of PZT thin films through use of basic principles in mechanics of vibration. A small impact hammer with a tiny tip acts perpendicularly to the PZT thin-film surface to generate an impulsive force. In the meantime, a load cell at the hammer tip measures the impulsive force and a charge amplifier measures the responding charge of the PZT thin film. Then the piezoelectric coefficient d₃₃ is obtained from the measured force and charge based on piezoelectricity and a finite element modeling. We also conduct a thorough parametric study to understand the sensitivity of this method on various parameters, such as substrate material, boundary conditions, specimen size, specimen thickness, thickness ratio, and PZT thin-film material. Two rounds of experiments are conducted to demonstrate the feasibility and accuracy of this new method. The first experiment is to measure d₃₃ of a PZT disk resonator whose d₃₃ is known. Experimental results show that d₃₃ measured via this method is as accurate as that from the manufacturer's specifications within its tolerance. The second experiment is to measure d₃₃ of PZT thin films deposited on silicon substrates. With the measured d₃₃, we predict the displacement of PZT thin-film membrane microactuators. In the meantime, the actuator displacement is measured via a laser Doppler vibrometer. The predicted and measured displacements agree very well validating the accuracy of this new method.

This paper is to study how the vibration modes of a cyclic symmetric rotor evolve when it is assembled to a flexible housing via multiple bearing supports. Prior to assembly, the vibration modes of the rotor are classified as “balanced modes” and “unbalanced modes.” Balanced modes are those modes whose natural frequencies and mode shapes remain unchanged after the rotor is assembled to the housing via bearings. Otherwise, the vibration modes are classified as unbalanced modes. By applying fundamental theorems of continuum mechanics, we conclude that balanced modes will present vanishing inertia forces and moments as they vibrate. Since each vibration mode of a cyclic symmetric rotor can be characterized in terms of a phase index (Chang and Wickert, “Response of Modulated Doublet Modes to Travelling Wave Excitation,” J. Sound Vib., 242, pp. 69–83; Chang and Wickert, 2002, “Measurement and Analysis of Modulated Doublet Mode Response in Mock Bladed Disks,” J. Sound Vib., 250, pp. 379–400; Kim and Shen, 2009, “Ground-Based Vibration Response of a Spinning Cyclic Symmetric Rotor With Gyroscopic and Centrifugal Softening Effects,” ASME J. Vibr. Acoust. (in press)), the criterion of vanishing inertia forces and moments implies that the phase index by itself can uniquely determine whether or not a vibration mode is a balanced mode as follows. Let $N$ be the order of cyclic symmetry of the rotor and $n$ be the phase index of a vibration mode. Vanishing inertia forces and moments indicate that a vibration mode will be a balanced mode if $n≠1,N−1,N$. When $n=N$, the vibration mode will be balanced if its leading Fourier coefficient vanishes. To validate the mathematical predictions, modal testing was conducted on a disk with four pairs of brackets mounted on an air-bearing spindle and a fluid-dynamic bearing spindle at various spin speeds. Measured Campbell diagrams agree well with the theoretical predictions.

In this paper, we develop a mathematical analysis to gain insights of mode localization often encountered in nearly cyclic symmetric rotors that contain slight mistune. First, we conduct a Fourier analysis in the spatial domain to show that mode localization can appear only when a group of tuned rotor modes form a complete set in the circumferential direction. In light of perturbation theories, these tuned rotor modes must also have very similar natural frequencies, so that they can be linearly combined to form localized modes when the mistune is present. Second, the natural frequency of these tuned rotor modes can further be represented in terms of a mean frequency and a deviatoric component. A Rayleigh–Ritz formulation then shows that mode localization occurs only when the deviatoric component and the rotor mistune are about the same order. As a result, we develop an effective visual method—through use of the deviatoric component and the rotor mistune—to precisely identify those modes needed to form localized modes. Finally, we show that curve veering is not a necessary condition for mode localization to occur in the context of free vibration. Not all curve veering leads to mode localization, and not all modes in curve veering contribute to mode localization. Numerical examples on a disk–blade system with mistune confirm all the findings above.

In this paper, we conduct experimental, theoretical, and numerical studies of a lead-zirconate-titanate (PZT) thin-film microactuator probe submerged in water. The major component of the actuator is a PZT diaphragm anchored on four silicon sidewalls. There is also silicon residue at the juncture of the diaphragm and the sidewalls due to imperfect etching processes. In the experimental study, frequency response functions of actuator displacement are measured via a laser Doppler vibrometer and a spectrum analyzer. The measurements show that the first natural frequency of the microactuator reduces from 80 kHz in air to 20 kHz when the microactuator is submerged in water. A viable explanation is that the surrounding water induces significant added mass to the microactuator. Estimation of the added mass based on theories in fluid mechanics successfully reconciles the predicted frequency to the vicinity of 20 kHz confirming the effects of added mass. Finite element models are also created to study how the silicon sidewalls and residue affect the added mass. Simulations show that presence of the sidewalls or residue would modify the fluid flow thus altering the added mass and natural frequency. In general, the finite element predictions agree well with the experimental measurements within 10% difference.