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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
J. Vib. Acoust. 2018;140(5):051011-051011-12. doi:10.1115/1.4039571.

This article presents a novel moving isosurface threshold (MIST) method for designing flexible structures using graded materials with multivolume fractions and constraints and viscous or hysteretic damping under harmonic loadings. By employing a unit dynamic load with the same frequency of an applied load, the displacement amplitude at chosen degrees-of-freedom (DOFs) can be expressed in an integral form in terms of mutual modal strain and kinetic energy densities over the entire design domain. Such integrals enable the introduction of novel physical response functions for solving a range of topology optimization problems, including single and multiple objectives with single and multiple volume fractions and/or constraints, e.g., single-input and single-output (SISO) and multi-input and multi-output (MIMO). Numerical examples are presented to validate the efficiency and capability of the present extended MIST method. Experiments are also conducted on rectangular plates with and without damping layer, fully and optimally covered, to demonstrate the benefits of the optimal damping layer design.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;140(5):051012-051012-15. doi:10.1115/1.4039569.

Evaluating driving safety of moving vehicles on slender coastal bridges as well as bridge safety is important to provide supporting data to make decisions on continuing or closing the operations of bridges under extreme weather conditions. However, such evaluations could be complicated due to the complex dynamic interactions of vehicle-bridge-wind-wave (VBWW) system. The present study proposes a comprehensive evaluation methodology on vehicle ride comfort and driving safety on the slender coastal bridges subject to vehicle, wind, and wave loads. After a brief introduction of the VBWW coupling dynamic system and obtaining the dynamic responses of the vehicles, the vehicle ride comfort is evaluated using the advanced procedures as recommended in the ISO 2631-1 standard based on the overall vibration total value (OVTV). The vehicle driving safety is analyzed based on two evaluation criteria, i.e., the roll safety criteria (RSC) and the sideslip safety criteria (SSC), through the vehicle contact force responses at the wheels. Finally, the proposed methodology is applied to a long-span cable-stayed bridge for the vehicle ride comfort and driving safety evaluation.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;140(5):051013-051013-10. doi:10.1115/1.4039568.

A large array of elastically coupled micro cantilevers of variable length is studied experimentally and numerically. Full-scale finite element (FE) modal analysis is implemented to determine the spectral behavior of the array and to extract a global coupling matrix. A compact reduced-order (RO) model is used for numerical investigation of the array's dynamic response. Our model results show that at a given excitation frequency within a propagation band, only a finite number of beams respond. Spectral characteristics of individual cantilevers, inertially excited by an external piezoelectric actuator, were measured in vacuum using laser interferometry. The theoretical and experimental results collectively show that the resonant peaks corresponding to individual beams are clearly separated when operating in vacuum at the third harmonic. Distinct resonant peak separation, coupled with the spatially confined modal response, make higher harmonic operation of tailored, variable-length cantilever arrays well suited for a variety of resonant-based sensing applications.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;140(5):051014-051014-11. doi:10.1115/1.4039537.

This paper develops an adjustable high-static-low-dynamic (AHSLD) vibration isolator with a widely variable stiffness. By adjusting deformations of its horizontal springs, the natural frequency of the isolator can be substantially changed starting from a quasi-zero value. In this paper, the nonlinear static and dynamic analyses of the AHSLD isolator are presented. Effects of horizontal adjustments on the variation range of the stiffness and nonlinear dynamic characteristics are investigated. Good performance of the stiffness variation is validated by free-vibration tests. The wide-range variable stiffness from 0.33 N/mm to 23.2 N/mm is achieved in tests, which changes the natural frequency of the isolator from an ultra-low value of 0.72 Hz to 5.99 Hz. Besides, its nonlinear dynamic characteristics are also experimentally identified by applying the Hilbert transform. Both analytical and experimental results demonstrate the weakly hardening nonlinearity in the tested AHSLD isolator, which will not degrade its performance in practical applications.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;140(5):051015-051015-12. doi:10.1115/1.4039725.

The American Petroleum Institute (API) level II vibration stability analysis for impellers requires higher fidelity models to predict the dynamic forces of the whirling impeller. These forces are in turn required to predict the vibration stability, critical speeds, and steady-state vibration response of the shaft-bearing-seal-impeller system. A transient computational fluid dynamics (CFD)-based approach is proposed which is applicable to nonaxisymmetric turbomachinery components, such as the volute and/or diffuser vanes, unlike its predecessor models like the bulk-flow or the quasi-steady model. The key element of this approach is the recent advancements in mesh deformation techniques which permit less restrictive motion boundary conditions to be imposed on the whirling impeller. The results quantify the contributions of the volute and/or the diffuser to the total forces which guides the analyst on whether to include these components in the model. The numerical results obtained by this approach are shown to agree well with experimental measurements and to be superior to the earlier quasi-steady alternative in terms of accuracy. Furthermore, several volute shapes were designed and analyzed for the sensitivity of the solution to the geometrical properties of the volute. The design flow rotordynamic forces show a significant dependence on the presence of the volutes in the model, with the specific shape of the volute having a lesser influence. The dimensionless forces are shown to be almost independent of the spin speed.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;140(5):051016-051016-9. doi:10.1115/1.4039800.

Expansion of real-time operating data from limited measurements to obtain full-field displacement data has been performed for structures in air. This approach has shown great success, and its main advantage is that the applied forces do not need to be identified. However, there are applications where structures may be immersed in water and the full-field real-time response may be needed for design and structural health assessments. This paper presents the results of a structure submersed in water to identify full-field response using only a handful of measured data. The same approach is used to extract the full-field displacements, and the results are compared to the actual full-field measured response. The advantage of this approach is that the force does not need to be identified and, most importantly, the damping and fluid–structure interaction does not need to be identified in order to perform the expansion. The results show excellent agreement with the measured data.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;140(5):051017-051017-11. doi:10.1115/1.4039726.

The vibration control of a sandwich beam with supported mass subjected to random support motion excitations can be performed using magnetorheological visco-elastomer core with adjustable dynamic properties. The periodic distributions of geometrical and physical parameters of the sandwich beam can improve its vibration response characteristics. To further improve characteristics or reduce responses, the quasi-periodic sandwich beam with supported mass under random excitations is studied. The facial layer thickness and core layer modulus of the sandwich beam are considered as quasi-periodic distributions. The partial differential equations for the horizontal and vertical coupling motions of the sandwich beam are derived and converted into ordinary differential equations for multi-degrees-of-freedom (DOFs) vibration. The expressions of frequency response and response spectral densities of the sandwich beam are obtained. Numerical results are given to illustrate the greatly improvable vibration response characteristics of the sandwich beam and the outstanding relative reduction localization of antiresonant responses. The proposed quasi-periodic distribution and analysis method can be used for the vibration control design of sandwich beams subjected to random excitations.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;140(5):051018-051018-9. doi:10.1115/1.4039801.

Unbalance is one of the most common malfunctions found in rotating machines generating high vibration amplitudes that can lead to fatigue and wear of rotor elements. There are several well-known balancing techniques wherein one of the most widespread approaches is the so-called influence coefficients method (IC method). Aiming to increase the robustness of the standard IC method, in this paper, a revised IC balancing methodology for rotating machines is proposed. In this sense, a preprocessing stage is applied to access the uncertainties affecting the rotating machine. In this sense, measurement data sets evaluated under the fuzzy logic approach are used. Thus, the rotor vibration responses measured over a long period are considered by means of a fuzzy transformation (defining unbalance fuzzy sets). The unbalance condition of the rotating machine is determined through a defuzzification process. This unbalance condition is then introduced in the IC method algorithm aiming at obtaining correction weights and associated angular positions that increase the balancing robustness as compared with the classical approach. Numerical and experimental studies are used to evaluate the effectiveness of the proposed methodology. The obtained results illustrate the capacity to increase the balancing overall robustness.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;140(5):051019-051019-11. doi:10.1115/1.4039868.

It is of high importance to be able to decouple a system to obtain the dynamic characteristics of its substructures; however, the necessary frequency response functions (FRFs) of the coupling interface are usually challenging to measure due to the limited accessible space and complex geometries. In this paper, a measurement technique in the decoupling process of a coupled system is proposed in order to obtain the FRFs at coupling interface. Specifically, a variable cross section rod is adopted to transmit the dynamic behavior of coupling interface. The proposed technique has three advantages: (a) the thick end with large cross section can provide enough area for applying excitation force like using impact hammer and/or setting up sensors; (b) the slender end with small cross section can break through the spatial limitation more easily; and (c) the convenience that no additional experimental setup is required but just using an available variable cross section rod. Vibrational equation of the variable cross section probe method is derived and then combined with the existing decoupling theories. Finally, the proposed probe method and the new decoupling theory combining probe theory are validated through numerical simulations (FEM) and laboratory experiments, respectively. The results show its great practicability in decoupling process especially in low frequency range.

Commentary by Dr. Valentin Fuster