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IN THIS ISSUE

### Research Papers

J. Vib. Acoust. 2017;139(5):051001-051001-13. doi:10.1115/1.4036210.

A new locking-free formulation of a three-dimensional shear-deformable beam with large deformations and large rotations is developed. The position of the centroid line of the beam is integrated from its slope that is related to the rotation of a corresponding cross section and stretch and shear strains. The rotation is parameterized by a rotation vector, which has a clear and intuitive physical meaning. Taylor polynomials are used for certain terms that have zero denominators to avoid singularity in numerical implementation. Since the rotation vector can have singular points when its norm equals $2mπ$, where $m$ is a nonzero integer, a rescaling strategy is adopted to resolve the singularity problem when there is only one singular point at a time instant, which is the case for most engineering applications. Governing equations of the beam are obtained using Lagrange's equations for systems with constraints, and several benchmark problems are simulated to show the performance of the current formulation. Results show that the current formulation does not suffer from shear and Poisson locking problems that the absolute nodal coordinate formulation (ANCF) can have. Results from the current formulation for a planar static case are compared with its exact solutions, and they are in excellent agreement with each other, which verifies accuracy of the current formulation. Results from the current formulation are compared with those from commercial software abaqus and recurdyn, and they are in good agreement with each other; the current formulation uses much fewer numbers of elements to yield converged results.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2017;139(5):051002-051002-10. doi:10.1115/1.4036212.

The root-mean-square (RMS) response of various points in a system comprised of two parallel plates coupled at a point undergoing high frequency, broadband transverse point excitation of one component is considered. Through this prototypical example, asymptotic modal analysis (AMA) is extended to two coupled continuous dynamical systems. It is shown that different points on the plates respond with different RMS magnitudes depending on their spatial relationship to the excitation or coupling points in the system. The ability of AMA to accurately compute the RMS response of these points (namely, the excitation point, the coupling points, and the hot lines through the excitation or coupling points) in the system is shown. The behavior of three representative prototypical configurations of the parallel plate system considered is: two similar plates (in both geometry and modal density), two plates with similar modal density but different geometry, and two plates with similar geometry but different modal density. After examining the error between reduced modal methods (such as AMA) to classical modal analysis (CMA), it is determined that these several methods are valid for each of these scenarios. The data from the various methods will also be useful in evaluating the accuracy of other methods including statistical energy analysis (SEA).

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2017;139(5):051003-051003-13. doi:10.1115/1.4036501.

Recent studies have presented first-order multiple time scale approaches for exploring amplitude-dependent plane-wave dispersion in weakly nonlinear chains and lattices characterized by cubic stiffness. These analyses have yet to assess solution stability, which requires an analysis incorporating damping. Furthermore, due to their first-order dependence, they make an implicit assumption that the cubic stiffness influences dispersion shifts to a greater degree than the quadratic stiffness, and they thus ignore quadratic shifts. This paper addresses these limitations by carrying-out higher-order, multiple scales perturbation analyses of linearly damped nonlinear monoatomic and diatomic chains. The study derives higher-order dispersion corrections informed by both quadratic and cubic stiffness and quantifies plane wave stability using evolution equations resulting from the multiple scales analysis and numerical experiments. Additionally, by reconstructing plane waves using both homogeneous and particular solutions at multiple orders, the study introduces a new interpretation of multiple scales results in which predicted waveforms are seen to exist over all space and time, constituting an invariant, multiharmonic wave of infinite extent analogous to cnoidal waves in continuous systems. Using example chains characterized by dimensionless parameters, numerical studies confirm that the spectral content of the predicted waveforms exhibits less growth/decay over time as higher-order approximations are used in defining the simulations' initial conditions. Thus, the study results suggest that the higher-order multiple scales perturbation analysis captures long-term, nonlocalized invariant plane waves, which have the potential for propagating coherent information over long distances.

Topics: Stability , Waves , Chain
Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2017;139(5):051004-051004-5. doi:10.1115/1.4036465.

We present a prototype vibration isolator whose design is inspired by origami-based foldable cylinders with torsional buckling patterns. The vibration isolator works as a nonlinear spring that has quasi-zero spring stiffness in a given frequency region, where it does not transmit vibration in theory. We evaluate the performance of the prototype vibration isolator through excitation experiments via the use of harmonic oscillations and seismic-wave simulations of the Tohoku-Pacific Ocean and Kobe earthquakes. The results indicate that the isolator with the current specification is able to suppress the transmission of vibrations with frequencies of over 6 Hz. The functionality and constraints of the isolator are also clarified. It has been known that origami-based foldable cylinders with torsional buckling patterns provide bistable folding motions under given conditions. In a previous study, we proposed a vibration isolator utilizing the bistability characteristics and numerically confirmed the device's validity as a vibration isolator. Here, we attempt prototyping the isolator with the use of versatile metallic components and experimentally evaluate the isolation performance.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2017;139(5):051005-051005-10. doi:10.1115/1.4036502.

Many mechanical systems often show nonlinear behavior related to particular operating conditions or to the nonlinear characteristic of the elements (springs, dampers, etc.) making up the system. In these cases, common engineering practice is to linearize the equation of motion around a particular operating point and to design a linear controller. Although this approach is simple, its main disadvantage is that stability properties and validity of the controller are only local. For these reasons, over the last decades, nonlinear control techniques have been investigated more and more in order to improve control performance. In particular, in this paper, sliding mode control (SMC) technique, which is based on the model of the system (model-based), is considered because of its easy implementation, especially on simple mechanical systems, and the considerable robustness of the controller even under significant model uncertainties. This technique is analyzed numerically with respect to the pendulum system to better understand the influence of the control action on the system dynamics. A nonlinear sliding surface is also considered, recalling the terminal sliding mode (TSM) control already analyzed in the scientific literature. This sliding surface is characterized for the numerical system, and then it is applied experimentally in order to control a highly nonlinear system, consisting of a three-link flexible manipulator. For this system, a nonlinear modal model is developed, and a nonlinear observer is designed. Finally, results of experimental tests on the manipulator are reported, in order to compare the performances of the linear embedded control and the sliding mode controllers with the linear and nonlinear sliding surface.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2017;139(5):051006-051006-8. doi:10.1115/1.4036466.

Periodic structures have interesting acoustic and vibration properties making them suitable for a wide variety of applications. In a periodic structure, the number of frequencies for each wavevector depends on the degrees-of-freedom of the unit cell. In this paper, we study the number of wavevectors available at each frequency in a band diagram. This analysis defines the upper bound for the maximum number of wavevectors for each frequency in a general periodic structure which might include damping. Investigation presented in this paper can also provide an insight for designing materials in which the interaction between unit cells is not limited to the closest neighbor. As an example application of this work, we investigate phonon dispersion curves in hexagonal form of boron nitride to show that first neighbor interaction is not sufficient to model dispersion curves with force-constant model.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2017;139(5):051007-051007-13. doi:10.1115/1.4036499.

This work proposes a method for controlling vibration using compliant-based actuators. The compliant actuator combines a conventional actuator with elastic elements in a series configuration. The benefits of compliant actuators for vibration control applications, demonstrated in this work, are twofold: (i) vibration reduction over a wide frequency bandwidth by passive control means and (ii) improvement of vibration control performance when active control is applied using the compliant actuator. The vibration control performance is compared with the control performance achieved using the well-known vibration absorber and conventional rigid actuator systems. The performance comparison showed that the compliant actuator provided a better flexibility in achieving vibration control over a certain frequency bandwidth. The passive and active control characteristics of the compliant actuator are investigated, which shows that the control performance is highly dependent on the compliant stiffness parameter. The active control characteristics are analyzed by using the proportional-derivative (PD) control strategy which demonstrated the capability of effectively changing the respective effective stiffness and damping of the system. These attractive dual passive–active control characteristics are therefore advantageous for achieving an effective vibration control system, particularly for controlling the vibration over a specific wide frequency bandwidth.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2017;139(5):051008-051008-15. doi:10.1115/1.4036503.

Vibration energy harvesting can be an effective method for scavenging wasted mechanical energy for use by wireless sensors that have limited battery life. Two major goals in designing energy harvesters are enhancing the power scavenged at low frequency and improving efficiency by increasing the frequency bandwidth. To achieve these goals, we derived a magnetoelastic beam operated at the transition between mono- and bi-stable regions. By improving the mathematical model of the interaction of magnetic force and beam dynamics, we obtained a precise prediction of natural frequencies as the distance of magnets varies. Using the shooting technique for the improved model, we present a fundamental understanding of interesting combined softening and hardening responses that happen at the transition between the two regimes. The transition regime is proposed as the optimal region for energy conversion in terms of frequency bandwidth and output voltage. Using this technique, low-frequency vibration energy harvesting at around 17 Hz was possible. The theoretical results were in good agreement with the experimental results. The target application is to power wildlife biologging devices from bird flights that have consistent high power density around 16 Hz (Shafer et al., 2015, “The Case for Energy Harvesting on Wildlife in Flight,” Smart Mater. Struct., 24(2), p. 025031).

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2017;139(5):051009-051009-12. doi:10.1115/1.4036468.

The impulsive behavior of the piston in the cylinder liner plays a key role in the noise, vibration, and harshness (NVH) of internal combustion engines. There have been several studies on the identification and quantification of piston impact action under various operation conditions. In the current study, the dynamics of the piston secondary motion are initially explored in order to describe the aggressive oscillations, energy loss, and noise generation. The control of piston secondary motion (and thus, impacts) is investigated using a new passive approach based on energy transfer of the highly transient oscillations to a nonlinear absorber. The effectiveness of this new method for improving the piston impact behavior is discussed using a preliminary parametric study that leads to the conceptual design of a nonlinear energy absorber.

Topics: Pistons , Engines
Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2017;139(5):051010-051010-13. doi:10.1115/1.4036469.

The finite element (FE) method offers an efficient framework to investigate the evolution of phononic crystals which possess materials or geometric nonlinearity subject to external loading. Despite its superior efficiency, the FE method suffers from spectral distortions in the dispersion analysis of waves perpendicular to the layers in infinitely periodic multilayered composites. In this study, the analytical dispersion relation for sagittal elastic waves is reformulated in a substantially concise form, and it is employed to reproduce spatial aliasing-induced spectral distortions in FE dispersion relations. Furthermore, through an anti-aliasing condition and the effective elastic modulus theory, an FE modeling general guideline is provided to overcome the observed spectral distortions in FE dispersion relations of infinitely periodic multilayered composites, and its validity is also demonstrated.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2017;139(5):051011-051011-5. doi:10.1115/1.4036470.

The scattered acoustic waves by a transmissive half-plane, which is illuminated by a line source, are investigated. The high-frequency diffracted wave expressions are obtained by taking into account a resistive half-screen that is defined in electromagnetics. The uniform diffracted fields are expressed in terms of the Fresnel cylinder functions. The behavior of the waves is compared with the case when the uniform theory of diffraction is considered. The geometrical optics and diffracted fields are examined numerically.

Topics: Diffraction , Waves
Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2017;139(5):051013-051013-15. doi:10.1115/1.4036633.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2017;139(5):051014-051014-10. doi:10.1115/1.4036680.

This paper presents a study of multi-objective optimization of elastic beams with minimum weight and radiated sound power. The goal of this research is to discover the potentials to design multi-objective optimal elastic structures for better acoustic performance. We discuss various structural-acoustic properties of the Pareto solutions of the multi-objective optimization problem (MOP). We have found that geometrical and dynamic constraints can substantially reduce the volume fraction of feasible solutions in the design space, which can make it difficult to search for the optimal solutions. Several case studies with different boundary conditions are studied to demonstrate the multi-objective optimal designs of the structure.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2017;139(5):051015-051015-9. doi:10.1115/1.4036722.

This paper describes a theoretical and experimental investigation into an electrical Helmholtz resonator (EHR): that is, an active noise control (ANC) loudspeaker used in conjunction with a microphone and a feedback controller for suppressing resonant noise in an acoustic cavity. The microphone is collocated with the loudspeaker and a band pass filter of second-order is used as the control filter inside the controller. The EHR is configured as such in order to suppress an acoustic mode that is within the volume velocity drive frequency range of the loudspeaker used. The concepts of impedance and passivity are used to develop the mathematical model as well as to study its dynamics. From these, it is theoretically shown that the EHR for single-mode suppression is an extremely low-impedance acoustic damping device that electrically realizes the pressure neutralization mechanism of a conventional Helmholtz resonator (HR). Experimental work is also presented, in which an EHR is constructed to suppress the Helmholtz mode of an acoustic cavity at about 40 Hz by more than 40 dB, to justify the mathematical model and also to verify the dynamic control mechanism.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2017;139(5):051016-051016-13. doi:10.1115/1.4036500.

Nonlinear axisymmetric free vibration analysis of liquid-filled spherical shells with volume constraint condition using membrane theory is presented in this paper. The energy functional of the shell and contained liquid can be expressed based on the principle of virtual work using surface fundamental form and is written in the appropriate forms. Natural frequencies and the corresponding mode shapes for specified axisymmetric vibration amplitude of liquid-filled spherical shells can be calculated by finite element method (FEM). A nonlinear numerical solution can be obtained by the modified direct iteration technique. The results indicate that the Lagrange multiplier is a parameter for adapting the internal pressure in order to sustain the shell in equilibrium state for each mode of vibration with the volume constraint condition. The axisymmetric mode shapes of the liquid-filled spherical shells under volume constraint condition were found to be in close agreement with those in existing literature for an empty spherical shell. Finally, the effects of support condition, thickness, initial internal pressure, bulk modulus of internal liquid, and elastic modulus on the nonlinear axisymmetric free vibration and change of pressure of the liquid-filled spherical shells with volume constraint condition were demonstrated. The parametric studies showed that the change of pressure has a major impact on the fundamental vibration mode when compared with the higher vibration modes.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2017;139(5):051017-051017-8. doi:10.1115/1.4036498.

To identify sound sources situated in a fluid flow, an equivalent source method (ESM)-based nearfield acoustic holography (NAH) in a moving medium is proposed, and two types of acoustic inputs, pressure and particle velocity, are considered. In particular, an analytical relationship between the particle velocity perpendicular to the flow direction and the equivalent source strength is deduced, which makes it possible to realize the reconstruction with particle velocity input. Compared to the planar NAH in a moving medium, the proposed method is applicable to sound sources with more complicated geometries. Numerical simulations with sound sources distributed over two types of geometries including planar geometry and nonplanar one are conducted to test the performances of the proposed method. The results indicate that the proposed method provides satisfactory reconstructed results whatever with pressure input or with particle velocity input, and it is valid and robust over a wide range of flow velocities and frequencies and under different levels of background noise.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2017;139(5):051018-051018-7. doi:10.1115/1.4036467.

In this paper, new exact closed-form solutions for free longitudinal vibration of a one-parameter countable family of cantilever rods with one end tip mass are obtained. The analysis is based on the reduction of the equation governing the longitudinal vibration to the Sturm–Liouville canonical form and on the use of double Darboux transformations. The rods for which exact eigensolutions are provided are explicitly determined in terms of an initial rod with known closed-form eigensolutions. The method can be also extended to include longitudinally vibrating rods with tip mass at both ends.

Commentary by Dr. Valentin Fuster

### Technical Brief

J. Vib. Acoust. 2017;139(5):054501-054501-8. doi:10.1115/1.4036471.

Acoustic liner optimization calls for very efficient simulation methods. A highly efficient and straightforward algorithm is proposed here for the Wiener–Hopf solver, which also takes advantage of the parallel processing capability of the emerging graphics processing unit (GPU) technology. The proposed algorithm adopts a simple concept that re-arranges the formulations of the Wiener–Hopf solver to appropriate matrix forms. This concept was often overlooked but is surprisingly succinct, which leads to a stunningly efficient computational performance. By examining the computational performance of two representative setups (lined duct and duct radiations), the current study shows the superior performance of the proposed algorithm, particularly with GPU. The much improved computational efficiency further suggests the potential of the proposed algorithm and the use of GPU for practical low-noise aircraft engine design and optimization.

Commentary by Dr. Valentin Fuster