This paper is to propose a viable hybrid damping design that integrates active and passive dampings through intelligent constrained layer (ICL) treatments. This design consists of a viscoelastic shear layer sandwiched between a piezoelectric constraining cover sheet and the structure to be damped. According to measured vibration response of the structure, a feedback controller regulates axial deformation of the piezoelectric layer to perform active vibration control. In the meantime, the viscoelastic shear layer provides additional passive damping. The active damping component of this design will produce adjustable and significant damping. The passive damping component of this design will increase gain and phase margins, eliminate spillover, reduce power consumption, improve robustness and reliability of the system, and reduce vibration response at high frequency ranges where active damping is difficult to implement. To model the dynamics of ICL, an eighth-order matrix differential equation governing bending and axial vibrations of an elastic beam with the ICL treatment is derived. The observability, controllability, and stability of ICL are discussed qualitatively for several beam structures. ICL may render the system uncontrollable or unobservable or both depending on the boundary conditions of the system. Finally, two examples are illustrated in this paper. The first example illustrates how an ICL damping treatment, which consists of an idealized, distributed sensor and a proportional-plus-derivative feedback controller, can reduce bending vibration of a semi-infinite elastic beam subjected to harmonic excitations. The second example is to apply an ICL damping treatment to a cantilever beam subjected to combined axial and bending vibrations. Numerical results show that ICL will produce significant damping.

Lead zirconate titanate (PbZrxTi1-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.

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This paper studies natural frequencies and mode shapes of a spinning disk/spindle assembly consisting of multiple elastic circular plates mounted on a rigid spindle that undergoes infinitesimal rigid-body translation and rotation. Through use of Lagrangian mechanics, linearized equations of motion are derived in terms of Euler angles, rigid-body translation, and elastic vibration modes of each disk. Compared with a single rotating disk whose spindle is fixed in space, the free vibration of multiple disks with rigid-body motion is significantly different in the following ways. First of all, lateral translation of the spindle, rigid-body rotation (or rocking) of the spindle, and one-nodal diameter modes of each disk are coupled together. When all the disks (say N disks) are identical, the coupled disk/spindle vibration splits into N − 1 groups of “balanced modes” and a group of “unbalanced modes.” For each group of the balanced modes, two adjacent disks vibrate entirely out of phase, while other disks undergo no deformation. Because the out-of-phase vibration does not change the angular momentum, the natural frequencies of the balanced modes are identical to those of the one-nodal-diameter modes of each disk. For the group of the unbalanced modes, all disks undergo the same out-of-plane vibration resulting in a change of angular momentum and a steady precession of the spindle. As a result, the frequencies of the unbalanced modes are significantly lower than those of one-nodal-diameter modes of each disk. Secondly, axial translation of the spindle and the axisymmetric modes of each disk are couple together. Similarly, the coupled motion split into N − 1 groups of “balanced modes” and one group of “unbalanced modes,” where the frequencies of the balanced and unbalanced modes are identical to and smaller than those of the axisymmetric modes of each disk, respectively. Thirdly, the rigid-body motion of the spindle does not affect disk vibration modes with two or more nodal diameters. Response of those modes can be determined through the classical vibration analysis of rotating disks. Moreover, vibration response of the disk/spindle assembly from a ground-based observer is derived. Finally, a calibrated experiment is conducted to validate the theoretical predictions.

The purposes of this paper are to formulate active constrained layer (ACL) damping treatments through a variational approach, to study the work-energy relation of ACL, and to identify damping mechanisms of ACL treatments. Application of the extended Hamilton principle to ACL results in the equations of motion of ACL and the charge equation of electrostatics for the piezoelectric constraining layer. The work-energy equation together with the charge equation shows that the power dissipated through the active damping is the product of the electric field and the axial velocity of the piezoelectric constraining layer at the boundaries. This unique feature suggests that a self-sensing and actuating piezoelectric constraining layer may be an appropriate design in dissipating vibration energy without causing instability. To identify the damping mechanisms, a sensitivity analysis shows that the effectiveness of ACL damping primarily depends on the active and passive damping forces transmitted to the vibrating structure through the viscoelastic layer. The active damping force transmitted depends on the controller transfer function as well as a system parameter, termed active damping sensitivity factor, which depends entirely on the configuration of the passive constrained layer and the sensor. Finally, numerical results on ACL beams are obtained to illustrate the theoretical predictions above.

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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.

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