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IN THIS ISSUE

### Research Papers

J. Vib. Acoust. 2018;141(2):021001-021001-10. doi:10.1115/1.4041462.

The combined effect of a crack with unbalanced force vector orientation in cracked rotor-bearing-disk systems on the values and locations of critical whirl amplitudes is numerically and experimentally investigated here for starting up operations. The time-periodic equations of motion of the cracked system are formulated according to the finite element (FE) time-varying stiffness matrix. The whirl response during the passage through the critical whirl speed zone is obtained via numerical simulation for different angles of the unbalance force vector. It is found that the variations in the angle of unbalance force vector with respect to the crack opening direction significantly alters the peak values of the critical whirl amplitudes and their corresponding critical whirl speeds. Consequently, the critical speeds of the cracked rotor are found to be either shifted to higher or lower values depending on the unbalance force vector orientation. In addition, the peak whirl amplitudes are found to exhibit significant elevation in some zones of unbalance force angles whereas significant reduction is observed in the remaining zones compared with the crack-free case. One of the important findings is that there exists a specific value of the unbalance force angle at which the critical whirl vibration is nearly eliminated in the cracked system compared with the crack-free case. These all significant numerical and experimental observations can be employed for crack damage detection in rotor systems.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;141(2):021002-021002-15. doi:10.1115/1.4041078.

In this paper, transfer function of rotating shaft system for detecting transverse open crack is developed. Rotating shaft system is modeled using one-dimensional finite element method (1D-FEM), and quantitative analysis is performed. Open crack is modeled as weak asymmetry rotating with shaft's rotation. It is known that, when both open crack and support stiffness anisotropy coexist, various frequency components of shaft's vibration are generated through their successive interaction. This paper evaluates the order of these components, and concludes that first five main components are enough to investigate interaction of open crack and support stiffness anisotropy. Then, five sets of transfer functions for these components are derived. The validity of this set of transfer functions is confirmed by numerical simulation. Moreover, excitation experiment utilizing active magnetic bearing (AMB) is performed, and the validity of derived transfer function was verified experimentally.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;141(2):021003-021003-8. doi:10.1115/1.4041367.

The objective of this paper is to demonstrate the motion reconstruction and the parameter estimation of a vibro-impact (VI) system from limited experimental information. Based on the measured displacement and acceleration of its linear main system, the rest motion information such as the displacement and velocity of the attached VI energy sink can be calculated rather than difficult direct measurement, and therefore, different response regimes from the strongly modulated response to the classic regime with two impacts per cycle are reconstructed. Consequently, it provides comprehensive experimental data for the validation of analytical and numerical results and for any experimental bifurcation analysis. Moreover, a procedure to estimate the restitution coefficient from periodic impacts is demonstrated. This new experimental approach to estimate the value of the restitution coefficient is simple and this accurate value could play an important role in analytical and numerical study.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;141(2):021004-021004-8. doi:10.1115/1.4041368.

We describe herein a method for extending the load range of a vibration isolator using a foldable cylinder consisting of a torsional buckling pattern and evaluate the vibration isolating performance through excitation experiments. A previous study determined that the foldable cylinder is bistable and acts as a vibration isolator with nonlinear characteristics in a displacement region, where the spring stiffness is zero. Its spring characteristics and vibration isolating performance were clarified by numerical analysis and excitation experiments. The findings indicated that the vibration in a certain frequency range is reduced where the spring stiffness is zero. However, this vibration isolator has a disadvantage in that it can only support an initial load that transfers to the zero-spring-stiffness region. Therefore, in this research, we improve the position of the linear spring attached to the isolator. As a result, the initial load range is extended by two to four times that of the conventional vibration isolator. Furthermore, the isolating performance is maintained even when the initial load is changed within a given load range.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;141(2):021005-021005-13. doi:10.1115/1.4041369.

In this paper, time-delayed feedback (TD-FB) control is introduced for a nonlinear vibration isolator (NL-VI), and the isolation effectiveness features are investigated theoretically and experimentally. In the feedback control loop, compound control with constant and variable time delays is considered. First, a stability analysis is conducted to determine the range of control parameters for stable zero equilibrium without excitation. Next, the nonlinear resonance frequency and the nonlinear vibration attenuation are studied by the method of multiple scales (MMS) to demonstrate the mechanism of TD-FB control. The results of the nonlinear vibration performances show that large variable time delays can improve the vibration suppression. Additionally, the mechanism for the time delay is not only to tune the equivalent stiffness and damping but also to induce effective isolation bandgap at high frequency. Therefore, the variable time delay is assumed as the function of frequency to meet different requirements at different frequency bands. The relevant experiment verifies the improvement of designed variable time delay on isolation performances in different frequency bands. Due to the improvement of isolation performances by compound time delay feedback control on nonlinear systems, it can be applied in the fields of ships, flexible structure in aerospace and aviation.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;141(2):021006-021006-9. doi:10.1115/1.4041513.

A novel type of dynamic vibration absorber (DVA) is proposed, which consists of a tuned mass damper (TMD) and tuned sloshing damper (TSD) connected in series to the structure. The system enables the expensive viscous damping devices (VDDs) associated with traditional TMDs to be omitted from the design. A linearized equivalent mechanical model and a nonlinear multimodal model are developed to investigate the proposed system. A TMD–TSD is nonlinear due to the quadratic damping associated with liquid drag, which ensures the system performance is amplitude-dependent. Simple expressions for the optimal TSD–TMD mass ratio, tuning, and damping ratios are employed to design a TMD–TSD coupled to a single degree-of-freedom (SDOF) structure. Frequency response curves for the structure, TMD, and TSD degrees-of-freedom are created for several excitation amplitudes, and the nonlinear behavior of the system response is evident. The performance of the TMD–TSD is evaluated against traditional TMD and TSD systems—with the same total mass—by computing the effective damping produced by each system. The proposed system is found to provide a superior acceleration reduction performance and superior robustness against changes to the frequency of the primary structure. The proposed system is, therefore, an effective and affordable means to reduce the resonant response of tall buildings.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;141(2):021007-021007-9. doi:10.1115/1.4041305.

Synchronous modal oscillations, characterized by unisonous motions for all physical coordinates, are well known. In turn, asynchronous oscillations lack a general definition to address all the associated features and implications. It might be thought, at first, that asynchronicity could be related to nonsimilar modes, which might be associated with phase differences between displacement and velocity fields. Due to such differences, the modes, although still periodic, might not be characterized by stationary waves so that physical coordinates might not attain their extreme values at the same instants of time, as in the case of synchronous modes. Yet, it seems that asynchronicity is more related to frequency rather than phase differences. A more promising line of thought associates asynchronous oscillations to different frequency contents over distinct parts of a system. That is the case when, in a vibration mode, part of the structure remains at rest, that is, with zero frequency, whereas other parts vibrate with non-null modal frequency. In such a scenario, localized oscillations would explain modal asynchronicity. When the system parameters are properly tuned, localization may appear even in very simple models, like Ziegler's columns, shear buildings, and slender structures. Now, the latter ones are recast, but finite rotations are assumed, in order to verify how nonlinearity affects existing linear asynchronous modes. For this purpose, the authors follow Shaw–Pierre's invariant manifold formulation. It is believed that full understanding of asynchronicity may apply to design of vibration controllers, microsensors, and energy-harvesting systems.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;141(2):021008-021008-9. doi:10.1115/1.4041617.

This work presents an efficient way to calculate the added mass matrix, which allows solving for natural frequencies and modes of solids vibrating in an inviscid and infinite fluid. The finite element method (FEM) is used to compute the vibration spectrum of a dry structure, then the boundary element method (BEM) is applied to compute the pressure modes needed to determine the added mass matrix that represents the fluid. The BEM requires numerical integration which results in a large computational cost. In this work, a reduction of the computational cost was achieved by computing the values of the pressure modes with the required numerical integration using a coarse BEM mesh, and then, interpolation was used to compute the pressure modes at the nodes of a fine FEM mesh. The added mass matrix was then computed and added to the original mass matrix of the generalized eigenvalue problem to determine the wetted natural frequencies. Computational cost was minimized using a reduced eigenvalue problem of size equal to the requested number of natural frequencies. The results show that the error of the natural frequencies using the procedure in this work is between 2% and 5% with 87% reduction of the computational time. The motivation of this work is to study the vibration of marine mammals' ear bones.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;141(2):021009-021009-7. doi:10.1115/1.4041512.

The effect of imperfect interface on the coupled extensional and flexural motions in a two-layer elastic plate is investigated from views of theoretical analysis and numerical simulations. A set of full two-dimensional equations is obtained based on Mindlin plate theory and shear-slip model, which concerns the interface elasticity and tangential discontinuous displacements across the bonding imperfect interface. Some numerical examples are processed, including the propagation of straight-crested waves in an unbounded plate, the buckling of a finite plate, as well as the deflection of a finite plate under uniform load. It is revealed that the bending-evanescent wave in the composites with a perfect interface eventually cuts-on to a propagating shear-like wave with cutoff frequency when the two sublayers imperfectly bonded. The similar phenomenon has been verified once again for coupled face-shear and thickness-shear waves. It also has been pointed out that the interfacial parameter has a great influence on the performance of static buckling, in which the outcome can be reduced to classical buckling load of a simply supported plate when the interface is perfect.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;141(2):021010-021010-10. doi:10.1115/1.4041593.

Resonant ultrasound spectroscopy (RUS) is an experimental method to measure elastic and anelastic properties of materials. The RUS experiment is conducted by exciting a specimen with a simple geometry and measuring resonant frequencies. From the resonant behaviors, both elastic and anelastic properties of the sample material can be extracted. This paper investigates the sensitivities of measured resonant frequencies to changes in elastic constants for an isotropic material and anisotropic material with cubic symmetry. Also under investigation is whether different specimen geometries increase the sensitivity of RUS; in other words, a path for optimizing the reliability of RUS data is explored.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;141(2):021011-021011-9. doi:10.1115/1.4041592.

This paper offers two interlinked contributions in the field of vibration absorption. The first involves an active tuning of an absorber for spectral and spatial variations. The second contribution is a set of generalized design guidelines for such absorber operations. “Spectral” tuning handles time-varying excitation frequencies, while “spatial” tuning treats the real-time variations in the desired location of suppression. Both objectives, however, must be achieved using active control and without physically altering the system components to ensure practicality. Spatial tuning is inspired by the concept of “noncollocated vibration absorption,” for which the absorber location is different from the point of suppression. This concept is relatively under-developed in the literature, mainly because it requires the use of part of the primary structure (PS) as the extended absorber—a delicate operation. Within this investigation, we employ the delayed resonator (DR)-based absorber, a hybrid concept with passive and active elements, to satisfy both tuning objectives. The presence of active control in the absorber necessitates an intriguing stability investigation of a time-delayed dynamics. For this subtask, we follow the well-established methods of frequency sweeping and D-subdivision. Example cases are also presented to corroborate our findings.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;141(2):021012-021012-10. doi:10.1115/1.4041671.

A two degrees-of-freedom (2DOFs) single mass-on-belt model is employed to study friction-induced instability due to mode-coupling. Three springs, one representing contact stiffness, the second providing lateral stiffness, and the third providing coupling between tangential and vertical directions, are employed. In the model, mass contact and separation are permitted. Therefore, nonlinearity stems from discontinuity due to dependence of friction force on relative mass-belt velocity and separation of mass-belt contact during oscillation. Eigenvalue analysis is carried out to determine the onset of instability. Within the unstable region, four possible phases that include slip, stick, separation, and overshoot are found as possible modes of oscillation. Piecewise analytical solution is found for each phase of mass motion. Then, numerical analyses are used to investigate the effect of three parameters related to belt velocity, friction coefficient, and normal load on the mass response. It is found that the mass will always experience stick-slip, separation, or both. When separation occurs, mass can overtake the belt causing additional nonlinearity due to friction force reversal. For a given coefficient of friction, the minimum normal load to prevent separation is found proportional to the belt velocity.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;141(2):021013-021013-17. doi:10.1115/1.4041734.

In this paper, a finite strip for vibration analysis of rotating toroidal shells subjected to internal pressure is developed. The expressions for strain and kinetic energies are formulated in a previous paper in which vibrations of a toroidal shell with a closed cross section are analyzed using the Rayleigh–Ritz method (RRM) and Fourier series. In this paper, however, the variation of displacements u, v, and w with the meridional coordinate is modeled through a discretization with a number of finite strips. The variation of the displacements with the circumferential coordinate is taken into account exactly by using simple sine and cosine functions of the circumferential coordinate. A unique argument $nφ+ω t$ is used in order to be able to capture traveling modes due to the shell rotation. The finite strip properties, i.e., the stiffness matrix, the geometric stiffness matrix, and the mass matrices, are defined by employing bar and beam shape functions, and by minimizing the strain and kinetic energies. In order to improve the convergence of the results, also a strip of a higher-order is developed. The application of the finite strip method is illustrated in cases of toroidal shells with closed and open cross sections. The obtained results are compared with those determined by the RRM and the finite element method (FEM).

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;141(2):021014-021014-7. doi:10.1115/1.4041897.

Acoustic cloaking is an intriguing phenomenon that has attracted lots of attention. The required inhomogeneous and anisotropic properties of acoustic cloaks derived with transformation acoustics make them difficult to realize. In this paper, a new mapping relation is presented. An acoustic cloak can be divided into any number of arbitrary triangular patterns, which are mapped from similar patterns in virtual space. Transformation from one triangular domain to another leads to homogeneous properties using transformation acoustics. The resulting cloak is composed of homogeneous triangular parts, each having just two alternating layers of material. The manner of division of the cloak affects the properties of each triangular part dramatically, which can be leveraged to vary the properties of each triangular part for more realistic material properties. Simulations of models based on this method show good cloaking performance at reducing the reflected and scattered waves due to the cloaked obstacle.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2018;141(2):021015-021015-16. doi:10.1115/1.4041896.

This paper presents a mathematical model used to obtain the vertical vibration of a ballasted railway track when a wheel is passing at a certain speed over a fixed location of the rail. The aim of this simulation is to compare calculated root-mean-square (RMS) values of the vertical vibration velocity with measured RMS values. This comparison is the basis for a proposed time domain methodology for detecting potential wheel flats or any other singular defect on the wheel rolling bands of metropolitan trains. In order to reach this goal, a wheel–rail contact model is proposed; this model is described by the track vertical impulse response and the vertical impulse response of the wheel with the primary suspension, both linked through a Hertz nonlinear stiffness. To solve the model for obtaining the wheel–rail contact force, a double convolution method is applied. Several kinds of wheel flats are analyzed, from theoretical round edged wheel flats to different real wheel profile irregularities. Afterward, the vertical vibration velocity at a fixed point on the rail is obtained using a variable kernel convolution method. Running different simulations for different wheel flats, a study of the vertical vibration attenuation along the rail is carried out. Finally, it is proceeded to obtain the temporary evolution of the RMS value for the rail vertical vibration velocity in order to be used as a reference for detecting wheel flats or any other defect. This last aspect will be presented in more detail in a second paper.

Topics: Vibration , Rails , Wheels
Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2019;141(2):021016-021016-10. doi:10.1115/1.4042041.

We developed a turbo compressor that has water-lubricated bearings driven at 30,000 rpm in a saturation condition, where the ambient pressure is at the saturation point of the discharged lubricant water. The bearings are supported with nonlinear elastomeric O-rings. At rotational speed exceeding 15,000 rpm, the rotor showed many subharmonic vibrations that are nonlinear phenomena unpredictable from a linear equation of motion. Instead, a stability analysis with a bifurcation diagram is an effective method to tackle these problems. In this paper, we investigated these rotor vibrations by bifurcation diagrams of the vibrations measured in experiments of saturated water journal bearings. The angular velocity was used as a bifurcation parameter. The bifurcations among synchronous, subharmonic, and chaotic vibrations were shown. Next, the nonlinear dynamics of the rotating rigid shaft were analyzed numerically with the nonlinear stiffness obtained by a commercial code that utilizes the two-dimensional (2D) Reynolds equation. The dynamic properties of the supporting structure were modeled with a complex stiffness coefficient. As a result, a Hopf bifurcation was found and a subharmonic limit cycle appeared spontaneously as observed in the experiments. The parametric studies revealed the influences of the dynamic properties of the structural components, especially the sensitive effect of the damping of the bearing support on the onset frequency and the amplitude of these vibrations. Furthermore, linear eigenvalue analysis of the motion equations clarified the mechanism of the sensitive effects.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2019;141(2):021017-021017-11. doi:10.1115/1.4042124.

A miniature microphone array based on interaural time difference (ITD) is designed. This array contains four microphones with certain arrangement and aims for two-dimensional (azimuth and elevation) direction-of-arrival (DOA) estimation in the whole three-dimensional space. The array can be small because it uses a coupling algorithm that magnifies the time delay between the signals received by every two microphones. The coupling algorithm is built according to a pairwise-coupled multidimensional mechanical model inspired by the ears of the tiny parasitoid fly Ormia ochracea. It was verified that the time-delay magnification can be independent of the incident angle when the parameters in the model satisfy specific relationships. This paper further investigates the multidimensional coupled system and advocates to realize the magnification mechanism in algorithm, where the physical parameters can change according to sound frequency to ensure the time-delay magnification. Moreover, the arrangement of microphones is specially designed to help the array to achieve similar measuring accuracy for all directions in the three-dimensional space. Corresponding signal process procedures are also provided. Simulations that use such an array to estimate the azimuth and elevation angles of sound source are performed via general cross-correlation (GCC) method. Results verify the feasibility of the microphone array and show that the accuracy of the estimation increases after the signals are processed by the coupled system.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2019;141(2):021018-021018-13. doi:10.1115/1.4042100.

Accurately measured mesh stiffness (MS) of gear pair is used in the mathematical model of the gearbox for doing the vibration analysis. Mesh stiffness measurement by experimental techniques is an import aspect for measuring the MS accurately of the physical structure of gear pair. Analytical methods (AMs) of MS require large mathematical equations and modifications in these equations for different gear geometries and cracks. Finite element method (FEM) requires huge computational time. In this paper, a new experimental technique based on the experimental modal analysis (EMA) is proposed to measure the MS of spur gear pairs. EMA technique is also used in the paper for measuring the MS of cracked gear tooth pairs and found a significant difference in the MS of healthy and cracked tooth pair. In the last, the experimental MS results are compared with the MS results obtained by the AM and digital image correlation (DIC) experimental technique and found a good agreement. This shows the usefulness of the EMA for measuring the MS. So, it can be used as an alternative method of MS measurement or to validate the analytical and FEM MS results.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2019;141(2):021019-021019-11. doi:10.1115/1.4042103.

Acoustic–structure coupling can substantially alter the frequency response of air-filled structures. Coupling effects typically manifest as two resonance peaks at frequencies above and below the resonant frequency of the uncoupled structural system. Here, a dynamic substructuring approach is applied to a simple acoustic–structure system to expose how the system response depends on the damping in the acoustic subsystem. Parametric studies show that as acoustic damping is increased, the frequencies and amplitudes of the coupled resonances in the structural response undergo a sequence of changes. For low levels of acoustic damping, the two coupled resonances have amplitudes approximating the corresponding in vacuo resonance. As acoustic damping is increased, resonant amplitudes decrease dramatically while the frequency separation between the resonances tends to increase slightly. When acoustic damping is increased even further, the separation of the resonant frequencies decreases below their initial separation. Finally, at some critical value of acoustic damping, one of the resonances abruptly disappears, leaving just a single resonance. Counterintuitively, increasing acoustic damping beyond this point tends to increase the amplitude of the remaining resonance peak. These results have implications for analysts and experimentalists attempting to understand, mitigate, or otherwise compensate for the confounding effects of acoustic–structure coupling in fluid-filled test structures.

Commentary by Dr. Valentin Fuster

### Technical Brief

J. Vib. Acoust. 2018;141(2):024501-024501-7. doi:10.1115/1.4041398.

The coupled axial–torsional modes of vibration are examined for three common types of braided wires: the well-known (1 + 6) configuration, the trapezoidal configuration, and the so-called twisted-pair. Representative volume elements of these systems with angles of twist ranging from 0 deg to 30 deg are described using three-dimensional elasticity theory and subjected to pure axial deformation and then pure twist to determine the stiffness coefficients that are used to describe the force–displacement relationship. These are compared with the results of existing braided wire models for the (1 + 6) geometry. Both analytical and finite element models of all three wires are then introduced to determine the level of coupling between the axial and torsional modes of vibration for representative homogeneous and composite cables.

Commentary by Dr. Valentin Fuster