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IN THIS ISSUE

### Research Papers

J. Vib. Acoust. 2018;140(5):051001-051001-9. doi:10.1115/1.4039406.

In most parametrically excited systems, stability boundaries cross each other at several points to form closed unstable subregions commonly known as “instability pockets.” The first aspect of this study explores some general characteristics of these instability pockets and their structural modifications in the parametric space as damping is induced in the system. Second, the possible destabilization of undamped systems due to addition of damping in parametrically excited systems has been investigated. The study is restricted to single degree-of-freedom systems that can be modeled by Hill and quasi-periodic (QP) Hill equations. Three typical cases of Hill equation, e.g., Mathieu, Meissner, and three-frequency Hill equations, are analyzed. State transition matrices of these equations are computed symbolically/analytically over a wide range of system parameters and instability pockets are observed in the stability diagrams of Meissner, three-frequency Hill, and QP Hill equations. Locations of the intersections of stability boundaries (commonly known as coexistence points) are determined using the property that two linearly independent solutions coexist at these intersections. For Meissner equation, with a square wave coefficient, analytical expressions are constructed to compute the number and locations of the instability pockets. In the second part of the study, the symbolic/analytic forms of state transition matrices are used to compute the minimum values of damping coefficients required for instability pockets to vanish from the parametric space. The phenomenon of destabilization due to damping, previously observed in systems with two degrees-of-freedom or higher, is also demonstrated in systems with one degree-of-freedom.

Topics: Stability , Damping , Waves
Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;140(5):051002-051002-8. doi:10.1115/1.4039539.

Mistuning refers to variations in modal properties of blades due to manufacturing tolerances and material defects. This can result in the amplification of a blade vibratory amplitude. This paper deals with the design of vibration absorbers for a mistuned bladed disk. First, the basic theory is established for undamped vibration absorbers using a single-mode model for each blade. Then, it is extended to include a multiple mode model of each blade and disk dynamics. The impact of mistuning on the bladed disk vibration is examined in the presence of undamped absorbers via Monte Carlo simulations. It is found that vibration absorbers can be an effective method to counter the detrimental effects of mistuning.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;140(5):051003-051003-15. doi:10.1115/1.4039531.

In this Part 1 of a two-part series, the theoretical modeling and optimization are presented. More specifically, the effect of attachment location on the dynamics of a flexible beam system is studied using a theoretical model. Typically, passive/active resonators for vibration suppression of flexible systems are uniaxial and can only affect structure response in the direction of the applied force. The application of piezoelectric bender actuators as active resonators may prove to be advantageous over typical, uniaxial actuators as they can dynamically apply both a localized moment and translational force to the base structure attachment point. Assuming unit impulse force disturbance, potential actuator/sensor performance for the secondary beam can be quantified by looking at fractional root-mean-square (RMS) strain energy in the actuator relative to the total system, and normalized RMS strain energy in the actuator over a frequency band of interest with respect to both disturbance force and actuator beam mount locations. Similarly, by energizing the actuator beam piezoelectric surface with a unit impulse, one can observe RMS base beam tip velocity as a function of actuator beam position. Through such analyses, one can balance both sensor/actuator performance and make conclusions about optimally mounting the actuator beam sensor/actuator. Accounting for both sensing and actuation requirements, the actuator beam should be mounted in the following nondimensionalized region: $0.4≤e≤0.5$.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;140(5):051004-051004-14. doi:10.1115/1.4039532.

In this Part 2 of a two-part series, the experimental verification and comparison of this work are presented. In this paper, the effect of beam-type resonator position on flexible dynamics is determined experimentally. The system is excited using band-limited white noise via electrodynamic shaker, and the data are collected with several transducers and a high-speed camera for each actuator beam mounting location; the first four mode shapes and natural frequencies are determined, and a finite element model (FEM) is developed and updated using these data. An additional set of data is collected using a linear sine chirp forcing function and the updated/experimental frequency response functions (FRFs) and time responses for the base and actuator beam tips are found to correlate. Plots of experimentally determined percent modal strain energy versus attachment position for the first four modes is presented, and a final study is also performed showing the fractional root-mean-square (RMS) strain energy in the actuator with respect to the total system. A final set of data is collected in which the actuator beam is moved up the base beam, the piezoelectric patch of the actuator beam is energized with white noise, and the tip response of the base beam is measured; an RMS base beam velocity versus mount position plot was developed. From this work, it is determined that the most practical/optimal position for the secondary beam to serve as both a sensor and actuator to control base beam tip response over a wide frequency band is in the nondimensionalized range: $0.4≤e<0.6$.

Topics: Actuators
Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;140(5):051005-051005-13. doi:10.1115/1.4039535.

A piezoelectric thin-film microactuator in the form of an asymmetrically laminated diaphragm is developed as an intracochlear hearing aid. Experimentally, natural frequencies of the microactuator bifurcate with respect to an applied bias voltage. To qualitatively explain the findings, we model the lead-zirconate-titanate (PZT) diaphragm as a doubly curved, asymmetrically laminated, piezoelectric shallow shell defined on a rectangular domain with simply supported boundary conditions. The von Karman type nonlinear strain–displacement relationship and the Donnell–Mushtari–Vlasov theory are used to calculate the electric enthalpy and elastic strain energy. Balance of virtual work between two top electrodes is also considered to incorporate an electric-induced displacement field that has discontinuity of in-plane strain components. A set of discretized equations of motion are obtained through a variational approach.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;140(5):051006-051006-10. doi:10.1115/1.4039536.

In Part 2 of the two-paper series, the asymmetrically laminated piezoelectric shell subjected to distributed bias voltage as modeled in Part 1 is analytically and numerically investigated. Three out-of-plane degrees-of-freedom (DOFs) and a number of in-plane DOFs are retained to study the shell's snap-through phenomenon. A convergence study first confirms that the number of the in-plane DOFs retained affects not only the number of predicted equilibrium states when the bias voltage is absent but also the prediction of the critical bias voltage for snap-through to occur and the types of snap-through mechanisms. Equilibrium states can be symmetric or asymmetric, involving only a symmetric out-of-plane DOF, and additional asymmetric out-of-plane DOFs, respectively. For symmetric equilibrium states, the snap-through mechanism can evolve from the classical bidirectional snap-through and latching to a new type of snap-through that only allows snap-through in one direction (i.e., unidirectional snap-through), depending on the distribution of the bias voltage. For asymmetric equilibrium states, degeneration can occur to the asymmetric bifurcation points when the radii of curvature are equal. Finally, the unidirectional snap-through renders an explanation to the experimental findings in Part 1.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;140(5):051007-051007-8. doi:10.1115/1.4039540.

This paper extends the resonance frequency detuning (RFD) vibration reduction approach to cases of turbomachinery blade mistuning. Using a lumped parameter mistuned blade model with included piezoelectric elements, this paper presents an analytical solution of the blade vibration in response to frequency sweep excitation; direct numerical integration confirms the accuracy of this solution. A Monte Carlo statistical analysis provides insight regarding vibration reduction performance over a range of parameters of interest such as the degree of blade mistuning, linear excitation sweep rate, inherent damping ratio, and the difference between the open-circuit (OC) and short-circuit (SC) stiffness states. RFD reduces vibration across all degrees of blade mistuning as well as the entire range of sweep rates tested. Detuning also maximizes vibration reduction performance when applied to systems with low inherent damping and large electromechanical coupling.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;140(5):051008-051008-11. doi:10.1115/1.4039534.

This paper presents a broadband vibration energy harvester (VEH) which consists of a monostable Duffing oscillator connected to an electromagnetic generator via a mechanical motion rectifier. The mechanical motion rectifier converts the bidirectional vibratory motion of the oscillator induced by ambient environment vibrations into unidirectional rotation of the generator and causes the harvester to periodically switch between a large- and small-inertia system, resulting in nonlinearity in inertia. By means of analytical and numerical methods, this inertia nonlinearity is shown to have two advantages. First, it allows for more stiffness nonlinearity without inducing nonuniqueness of energy branches and enhances bandwidths of energy harvesting. The effect of mitigating nonuniqueness of energy branches occurs to steady-state and transient responses of the harvester and is experimentally verified by a prototype. The experimental results show a nearly 50% increase in the half power bandwidth via mechanical motion rectification (MMR). Second, it enlarges the basin of attraction of the high-energy branch when multiple energy branches are present. A numerical example shows that a more than 50% increase in the basin area can be achieved via MMR.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;140(5):051009-051009-8. doi:10.1115/1.4039533.

The ability to predict multistable structural dynamics challenges the development of future high-performance air vehicles that will be subjected to extreme multiphysics loads. To aid in the establishment of methodologies that characterize the response states of harmonically excited multistable structures, a catalog of empirical and practical evidence is necessary. Recent research has suggested that evolving aspects of mechanical impedance metrics may be correlated with measurable quantities, although their relation to bifurcations of the dynamic response remains incompletely understood. Motivated to begin establishing such knowledge base, this research seeks to construct a library of experimental evidence from which to draw generalized insights on the impedance- and spectral-changing trends of multistable structures undergoing severe nonlinear response due to harmonic loading. A connection between vanishing real and imaginary components of impedance and dynamic bifurcations is uncovered. In the process, a technique to forecast dynamic bifurcations is articulated, which utilizes mechanical impedance measurements in real-time to monitor the susceptibility of postbuckled structural components to undergo dynamic bifurcations. An examination of higher-order harmonics of the dynamic responses further illuminates the nearness to bifurcations and may help classify the precise response regime. Thus, by correlating the real-time impedance and spectral response with analytical predictions, a future tool may be established for condition monitoring and diagnosis.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;140(5):051010-051010-7. doi:10.1115/1.4039538.

We perform an investigation on the vibration response of a simply supported plate buried in glass particles, focusing on the nonlinear dynamic behaviors of the plate. Various excitation strategies, including constant-amplitude variable-frequency sweep and constant-frequency variable-amplitude sweep are used during the testing process. We employ the analysis methods of power spectroscopy, phase diagramming, and Poincare mapping, which reveal many complicated nonlinear behaviors in the dynamic strain responses of an elastic plate in granular media, such as the jump phenomena, period-doubling bifurcation, and chaos. The results indicate that the added mass, damping, and stiffness effects of the granular medium on the plate are the source of the nonlinear dynamic behaviors in the oscillating plate. These nonlinear behaviors are related to the burial depth of the plate (the thickness of the granular layer above plate), force amplitude, and particle size. Smaller particles and a suitable burial depth cause more obvious jump and period-doubling bifurcation phenomena to occur. Jump phenomena show both soft and hard properties near various resonant frequencies. With an increase in the excitation frequency, the nonlinear added stiffness effect of the granular layer makes a transition from strong negative stiffness to weak positive stiffness.

Commentary by Dr. Valentin Fuster