Journal of Vibration and Acoustics Newest Issue
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en-usMon, 17 Sep 2018 00:00:00 GMTMon, 17 Sep 2018 07:43:55 GMTSilverchaireditor@vibrationacoustics.asmedigitalcollection.asme.orgwebmaster@vibrationacoustics.asmedigitalcollection.asme.orgImproving Model Parameters in Vibrating Systems Using Neumann Series
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Mon, 17 Sep 2018 00:00:00 GMTLiem AT, Gregory McDaniel JJ, Wixom AS. <span class="paragraphSection">A method is presented to improve the estimates of material properties, dimensions, and other model parameters for linear vibrating systems. The method improves the estimates of a single model parameter of interest by finding parameter values that bring model predictions into agreement with experimental measurements. A truncated Neumann series is used to approximate the inverse of the dynamic stiffness matrix. This approximation avoids the need to directly solve the equations of motion for each parameter variation. The Neumman series is shown to be equivalent to a Taylor series expansion about nominal parameter values. A recursive scheme is presented for computing the associated derivatives, which are interpreted as sensitivities of displacements to parameter variations. The convergence of the Neumman series is studied in the context of vibrating systems, and it is found that the spectral radius is strongly dependent on system resonances. A homogeneous viscoelastic bar in longitudinal vibration is chosen as a test specimen, and the complex-valued Young's modulus is chosen as an uncertain parameter. The method is demonstrated on simulated experimental measurements computed from the model. These demonstrations show that parameter values estimated by the method agree with those used to simulate the experiment when enough terms are included in the Neumann series. Similar results are obtained for the case of an elastic plate with clamped boundary conditions. The method is also demonstrated on experimental data, where it produces improved parameter estimates that bring the model predictions into agreement with the measured response to within 1% at a point on the bar across a frequency range that includes three resonance frequencies.</span>http:///vibrationacoustics.asmedigitalcollection.asme.org/article.aspx?articleid=vibrationacoustics.asmedigitalcollection.asme.orgFree and Forced Vibrations of Monolithic and Composite Rectangular Plates With Interior Constrained Points
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Mon, 17 Sep 2018 00:00:00 GMTChattopadhyay AP, Batra RC. <span class="paragraphSection">The restriction of deformations to a subregion of a system undergoing either free or forced vibration due to an irregularity or discontinuity in it is called mode localization. Here, we study mode localization in free and forced vibration of monolithic and unidirectional fiber-reinforced rectangular linearly elastic plates with edges either simply supported (SS) or clamped by using a third-order shear and normal deformable plate theory (TSNDT) with points on either one or two normals to the plate midsurface constrained from translating in all three directions. The plates studied are symmetric about their midsurfaces. The in-house developed software based on the finite element method (FEM) is first verified by comparing predictions from it with either the literature results or those computed by using the linear theory of elasticity and the commercial FE software <span style="text-transform:lowercase;font-variant:small-caps;">abaqus</span>. New results include: (i) the localization of both in-plane and out-of-plane modes of vibration, (ii) increase in the mode localization intensity with an increase in the length/width ratio of a rectangular plate, (iii) change in the mode localization characteristics with the fiber orientation angle in unidirectional fiber reinforced laminae, (iv) mode localization due to points on two normals constrained, and (iv) the exchange of energy during forced harmonic vibrations between two regions separated by the line of nearly stationary points that results in a beats-like phenomenon in a subregion of the plate. Constraining translational motion of internal points can be used to design a structure with vibrations limited to its small subregion and harvesting energy of vibrations of the subregion.</span>http:///vibrationacoustics.asmedigitalcollection.asme.org/article.aspx?articleid=vibrationacoustics.asmedigitalcollection.asme.orgThe Energy Flow Analysis as a Tool for Identification of Damping in Tall Buildings Subjected to Wind: Contributions of the Foundation and the Building Structure
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Mon, 10 Sep 2018 00:00:00 GMTGómez SS, Metrikine AV. <span class="paragraphSection">In this paper, the energy dissipated in a tall building is identified by means of the energy flow analysis. This approach allows assessing energy dissipation within a specific domain or element of the structure. In this work, the focus is placed on the superstructure, which is the part of the building above the ground, and on the foundation. Damping operators for the superstructure and the foundation are formulated based on the identified energy dissipation in these parts of the building. The obtained damping operators are used to compute the modal damping ratios in a simplified model of the building. The modal damping ratios of the three lowest modes of vibration are compared to those identified in full-scale measurements by means of the half-power bandwidth method.</span>http:///vibrationacoustics.asmedigitalcollection.asme.org/article.aspx?articleid=vibrationacoustics.asmedigitalcollection.asme.orgVersatile Absolute Nodal Coordinate Formulation Model for Dynamic Folding Wing Deployment and Flutter Analyses
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Mon, 10 Sep 2018 00:00:00 GMTOtsuka K, Wang Y, Makihara K. <span class="paragraphSection">Aircraft performance can be improved using morphing wing technologies, in which the wing can be deployed and folded under flight conditions, providing a wide flight envelope, good fuel efficiency, and reducing the space required to store the aircraft. Because the deployment of the wing is a nonlinear-coupled motion comprising large rigid body motion and large elastic deformation, a nonlinear folding-wing model is required to perform the necessary time-domain deployment simulation, while a linear model is required to perform the frequency-domain flutter analysis. The objective of this paper is to propose a versatile model that can be applied to both the time-domain and frequency-domain analyses of a folding wing, based on flexible multibody dynamics (MBD) using absolute nodal coordinate formulation (ANCF) and unsteady aerodynamics. This new versatile model expands the application range of the flexible MBD using ANCF in time-domain simulation, allowing it to express the coupled motion of extremely large elastic deformations and large rigid body motions that arise in next-generation aircraft. The time-domain deployment simulation conducted using the proposed model is useful for parametric deployment-system design because the model has improved calculation time. In the frequency-domain flutter analysis of a folding wing, the flutter speed obtained from the proposed model agrees with that obtained from an experiment, with an error of 4.0%, showing promise for application in next-generation aircraft design.</span>http:///vibrationacoustics.asmedigitalcollection.asme.org/article.aspx?articleid=vibrationacoustics.asmedigitalcollection.asme.orgA Mode-Resolved Continuum Mechanics Model of Acoustic Wave Scattering From Embedded Cylinders
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Mon, 10 Sep 2018 00:00:00 GMTUnni V, Feser JP. <span class="paragraphSection">In this paper, we use continuum mechanics to develop an analytic treatment of elastic wave scattering from an embedded cylinder and show that a classic treatise on the subject contains important errors for oblique angles of incidence, which we correct. We also develop missing equations for the scattering cross section at oblique angles and study the sensitivity of the scattering cross section as a function of elastodynamic contrast mechanisms. We find that in the Mie scattering regime for oblique angles of incidence, both elastic and density contrast are important mechanisms by which scattering can be controlled, but that their effects can offset one another, similar to the theory of reflection at flat interfaces. In comparison, we find that in the Rayleigh scattering regime, elastic and density contrast are always complimentary toward increasing scattering cross section, but for sufficiently high density contrast, the scattering cross section for incident compressional and <span style="font-style:italic;">y</span>-transverse modes is nearly independent of elastic contrast. The solution developed captures the scattering physics for all possible incident elastic wave orientations, polarizations, and wavelengths including the transition from Rayleigh to geometric scattering regimes, so long as the continuum approximation holds. The method could, for example, enable calculation of the thermal conductivity tensor from microscopic principles which requires knowledge of the scattering cross section spanning all possible incident elastic wave orientations and polarizations at thermally excited wavelengths.</span>http:///vibrationacoustics.asmedigitalcollection.asme.org/article.aspx?articleid=vibrationacoustics.asmedigitalcollection.asme.orgA Thévenin-Inspired Approach to Multiple Scattering in Acoustics
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Mon, 10 Sep 2018 00:00:00 GMTWilliams RP, Hall NA. <span class="paragraphSection">We have previously shown how Thévenin's theorem may be used to solve problems in linear acoustic scattering from a mobile body, by forming the solution as a superposition of the field scattered from the body when held immobile and the solution for radiation from the body in a quiescent field (Williams, R. P. and Hall, N. A., 2016, “Thévenin Acoustics” J. Acoust. Soc. Am., <strong>140</strong>(6), pp. 4449–4455). For problems involving scattering from multiple mobile bodies, the approach can be extended by using multiport network formalism. The use of network formalism allows for the effects of multiple scattering to be treated using analogous circuit models, facilitating the integration of scattering effects into circuit-based models of acoustic transducers. In this paper, we first review Thévenin's theorem for electrical and linear acoustic systems, and discuss the Thévenin-inspired approach to scattering from one rigid, mobile cylinder. Two-port formalism is introduced as a way to address problems involving two scatterers. The method is illustrated using the problem of scattering from a pair of rigid, mobile cylinders in an ideal plane progressive wave. The velocities of the cylinders and the resultant pressure field in response to the incoming wave are found. Unique features of the method compared to more conventional approaches are discussed.</span>http:///vibrationacoustics.asmedigitalcollection.asme.org/article.aspx?articleid=vibrationacoustics.asmedigitalcollection.asme.orgUsing Liquid Metal in an Electromechanical Motor With Breathing Mode Motion
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Mon, 10 Sep 2018 00:00:00 GMTFarzbod F, Naghdi M, Goggans PM. <span class="paragraphSection">Electromechanical actuators exploit the Lorentz force law to convert electrical energy into rotational or linear mechanical energy. In these electromagnetically induced motions, the electrical current flows through wires that are rigid, and consequently, the types of motion generated are limited. Recent advances in preparing liquid metal alloys permit wires that are flexible. Such wires have been used to fabricate various forms of flexible connections, but very little has been done to use liquid metal as an actuator. In this paper, we propose and have tested a new type of motor using liquid metal conductors in which radial (or breathing) modes are activated.</span>http:///vibrationacoustics.asmedigitalcollection.asme.org/article.aspx?articleid=vibrationacoustics.asmedigitalcollection.asme.orgA Fast Analytic Method to Calculate the Dynamic Response of Railways Sleepers
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Mon, 13 Aug 2018 00:00:00 GMTTran LH, Hoang TT, Duhamel DD, et al. <span class="paragraphSection">Existing analytical models for railway tracks consider only one rail supported by a continuous foundation or periodic concentrated supports (called the periodically supported beam). This paper presents an analytical model for a railway track which includes two rails connected by sleepers. By considering the sleepers as Euler–Bernoulli beams resting on a Kelvin–Voigt foundation, we can obtain a dynamic equation for a sleeper subjected to the reaction forces of the rails. Then, by using the relation between the rail forces and displacements from the periodically supported beam model, we can calculate the sleeper responses with the help of Green's function. The numerical applications show that the sleeper is in flexion where the displacement at the middle of the sleeper is greater than those at the rail seats. Moreover, the deformed shape of the sleeper is nonsymmetric when the loads on the two rails are different. The model result agrees well with measurements performed using instrumented sleeper in situ</span>http:///vibrationacoustics.asmedigitalcollection.asme.org/article.aspx?articleid=vibrationacoustics.asmedigitalcollection.asme.orgAn Efficient Hybrid Analytical-Computational Method for Nonlinear Vibration of Spur Gear Pairs
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Mon, 13 Aug 2018 00:00:00 GMTDai X, Cooley CG, Parker RG. <span class="paragraphSection">This work develops a hybrid analytical-computational (HAC) method for nonlinear dynamic response in spur gear pairs. The formulation adopts a contact model developed in (Eritenel, T., and Parker, R. G., 2013, “Nonlinear Vibration of Gears With Tooth Surface Modifications,” ASME J. Vib. Acoust., <strong>135</strong>(5), p. 051005) where the dynamic force at the mating gear teeth is determined from precalculated static results based on the instantaneous mesh deflection and position in the mesh cycle. The HAC method merges this calculation of the contact force based on an underlying finite element static analysis into a numerical integration of an analytical vibration model. The gear translational and rotational vibrations are calculated from a lumped-parameter analytical model where the crucial dynamic mesh force is calculated using a force-deflection function (FDF) that is generated from a series of static finite element analyses performed before the dynamic calculations. Incomplete tooth contact and partial contact loss are captured by the static finite element analyses and included in the FDF, as are tooth modifications. In contrast to typical lumped-parameter models elastic deformations of the gear teeth, including the tooth root strains and contact stresses, are calculated. Accelerating gears and transient situations can be analyzed. Comparisons with finite element calculations and available experiments validate the HAC model in predicting the dynamic response of spur gear pairs, including for resonant gear speeds when high amplitude vibrations are excited and contact loss occurs. The HAC model is five orders of magnitude faster than the underlying finite element code with almost no loss of accuracy.</span>http:///vibrationacoustics.asmedigitalcollection.asme.org/article.aspx?articleid=vibrationacoustics.asmedigitalcollection.asme.orgDesign Space Exploration of a Beam Flexible Hub Concept for an Inside-Out Ceramic Turbine Using a Simplified Rotordynamic Finite Element Model
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Mon, 13 Aug 2018 00:00:00 GMTLandry C, Dubois PK, Plante J, et al. <span class="paragraphSection">This paper presents a new flexible hub design for the inside-out ceramic turbine (ICT) rotor configuration. This configuration is used in microturbines to integrate ceramic blades in order to increase turbine inlet temperature (TIT), which leads to higher cycle efficiency values. The ICT uses an outer composite rim to load the ceramic blades in compression by converting the centrifugal loads of the blades into hoop stresses in the composite rim. High stresses in the composite rim lead to high radial displacement of the blades. This displacement is compensated by using flexible hub in order to maintain the contact with the blades. However, hub flexibility can lead to rotordynamic problems as heavy hub deformation will induce high stresses in it. Thus, stresses in the hub are induced by both rotordynamics and centrifugation, requiring a multi-objective design process, which has yielded geometries that limited, until now, the blade tip speed to 358 m/s. In this paper, a simplified rotordynamics finite element model of a flexible hub is developed to allow quick design iterations. Using the model, a design space exploration of this hub concept is done while considering centrifugation and rotordynamics. Experimental validation is conducted on a simplified ICT prototype up to 129 krpm, i.e., an equivalent blade tip speed of 390 m/s. Finally, predictions from the experimentally calibrated model show that the tested prototype hub could reach a blade tip speed of 680 m/s.</span>http:///vibrationacoustics.asmedigitalcollection.asme.org/article.aspx?articleid=vibrationacoustics.asmedigitalcollection.asme.orgAn Online Active Balancing Method Using Magnetorheological Effect of Magnetic Fluid
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Mon, 13 Aug 2018 00:00:00 GMTZhang X, Xia X, Xiang Z, et al. <span class="paragraphSection">The improvement of machining efficiency and precision puts forward new requirements for the balancing performance of machine tool spindle. Work piece quality can be effectively improved by implementing the active balance on the spindle. In this paper, a new active balancing method using magnetorheological (MR) effect of magnetic fluid is proposed. The mechanism of forming compensation mass by changing the distribution of magnetic fluid under local magnetic field is expounded. Experiments are carried out to verify the feasibility of the proposed method. Profile lines of magnetic fluid surface shape at different positions are measured with linear laser projection measurement method in experiments. Three-dimensional (3D) surface shape of the magnetic fluid is reconstructed by the synthesis of the measured profile lines. Experiments demonstrate that mass center of the magnetic fluid increases with the strength of magnetic field. Thus, the feasibility of the proposed method is verified experimentally. In order to weaken the vibration of machine tool spindle using this method, a balancing device is designed, which includes magnetic fluid chambers and three conjugated C-type electromagnets arranged at 120 deg intervals. For each electromagnet, the relationship among compensation mass (the corresponding balancing mass), excitation current, and rotation speed is established. Also, the performance of the balancing device is further proved in experiments conducted on the experimental platform. The imbalance vibration amplitude of the test spindle decreased by an average of 87.9% indicates that the proposed active balancing method in this paper is promising.</span>http:///vibrationacoustics.asmedigitalcollection.asme.org/article.aspx?articleid=vibrationacoustics.asmedigitalcollection.asme.orgDesign of Magnetic Bearing Control System Based on Active Disturbance Rejection Theory
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Mon, 13 Aug 2018 00:00:00 GMTJin C, Guo K, Xu Y, et al. <span class="paragraphSection">At present, most of the magnetic bearing system adopts the classical proportional–integral–derivative (PID) control strategy. However, the external disturbances, system parameter perturbations, and many other uncertain disturbances result in PID controller difficult to achieve high performance. To solve this problem, a linear active disturbance rejection controller (LADRC) based on active disturbance rejection controller (ADRC) theory was designed for magnetic bearing. According to the actual prototype parameters, the simulation model was built in <span style="text-transform:lowercase;font-variant:small-caps;">matlab</span>/<span style="text-transform:lowercase;font-variant:small-caps;">simulink</span>. The step and sinusoidal disturbances with PID and LADRC control strategies were simulated and compared. Then, the experiments of step and sinusoidal disturbances were performed. When control parameters are consistent, the experiment showed that the rotor displacement fluctuation decreased by 28.6% using the LADRC than PID control under step disturbances and decreased by around 25.8% under sinusoidal disturbances. When the rotor is running at 24,000 r/min and 27,000 r/min, the displacement of rotor is reduced by around 15% and 13.7%, respectively. Rotate the rotor with step disturbances and sinusoidal disturbances. It can also be seen that LADRC has the advantages of fast response time and good anti-interference. The experiments indicate that the LADRC has better anti-interference performance compared with PID controller.</span>http:///vibrationacoustics.asmedigitalcollection.asme.org/article.aspx?articleid=vibrationacoustics.asmedigitalcollection.asme.orgActive Control of Longitudinal Vibration of a Time-Varying Shafting System With a Dynamic Interpolating Adaptive Method
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Mon, 13 Aug 2018 00:00:00 GMTZheng H, Hu F, Qin H, et al. <span class="paragraphSection">An adaptive control method with dynamic interpolation is proposed for the active longitudinal vibration control of propulsion shafting systems. In such systems, the dynamics of longitudinal vibration change with the speed-dependent stiffness, which can result in a time-varying system as the shaft speed changes with time. A longitudinal vibration model is established for the investigation of the dynamic interpolating adaptive method (DIAM). In this model, the longitudinal vibration is induced by the disturbance exerted on the propeller (the left mass) and the control force is exerted on the thrust bearing (the right mass), which defines the disturbance channel and the control channel. The proposed DIAM is used to suppress longitudinal vibration transmission from the propeller to the thrust bearing by applying an active force on the right mass. The interpolation technique in DIAM updates the parameter-dependent compensator dynamically and eliminates the influence of parameter-dependent dynamics on the stability of control. Simulation results have demonstrated that the proposed DIAM is effective in suppressing longitudinal vibration of the thrust bearing in comparison to conventional adaptive methods.</span>http:///vibrationacoustics.asmedigitalcollection.asme.org/article.aspx?articleid=vibrationacoustics.asmedigitalcollection.asme.orgStructural Intensity Analysis on Irregular Shells
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Mon, 13 Aug 2018 00:00:00 GMTPires FF, Vanlanduit SS, Dirckx JJ. <span class="paragraphSection">A method is presented to assess the transmission path of vibration energy and to localize sources or sinks on shells with arbitrary shape, constant thickness, and isotropic material properties. The derived equations of the structural intensity (SI) are based on the Kirchhoff–Love postulates and are formulated in terms of displacements, Lamé parameters, principal curvatures, and their partial derivatives with respect to the principal curvilinear coordinates (PCC). To test the accuracy of the method, two numerical models of thin shells with nonuniform curvatures were developed. The coordinates, displacements, and principal curvature directions (PCDs) at the shell's outer surface were used to estimate the SI vector fields and the energy density at the shell's middle surface. The power estimated from the surface integral of the divergence of the SI over the source was compared to the actual power injected in the shell. The absolute error in both models did not exceed 1%, showing that, in theory, the method is able to handle the high-order spatial derivatives of the displacement and geometry data. The qualitative effect of varying the internal damping in the models on the energy transmission was also investigated.</span>http:///vibrationacoustics.asmedigitalcollection.asme.org/article.aspx?articleid=vibrationacoustics.asmedigitalcollection.asme.orgHighly Directional Acoustic Waves Generated by a Horned Parametric Acoustic Array Loudspeaker
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Mon, 13 Aug 2018 00:00:00 GMTTong LH, Lai SK, Yan JW, et al. <span class="paragraphSection">Acoustic horns can enhance the overall efficiency of loudspeakers to emanate highly directional acoustic waves. In this work, a theoretical model is developed to predict difference frequency acoustic fields generated by a parametric array loudspeaker (PAL) with a flared horn. Based on this model, analytical solutions are obtained for exponentially horned PALs. A numerical analysis on the performance of horned PALs subject to various horn parameters (i.e., horn length and flare constant) is implemented. To compare with nonhorned parametric acoustic array (PAA) devices, it is able to generate highly directional acoustic wave beams for a wide range of difference frequencies, in which the generated sound pressure levels at low frequencies can be significantly enhanced. In addition, the equivalent radius of a nonhorned emitter that matches the directivity achieved by a horned one is also quantitatively investigated. The present research will provide useful guidelines for the design and optimization of horned parametric array equipment.</span>http:///vibrationacoustics.asmedigitalcollection.asme.org/article.aspx?articleid=vibrationacoustics.asmedigitalcollection.asme.orgExact Optimization of a Three-Element Dynamic Vibration Absorber: Minimization of the Maximum Amplitude Magnification Factor
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Tue, 24 Jul 2018 00:00:00 GMTNishihara O. <span class="paragraphSection">In this study, the maximum amplitude magnification factor for a linear system equipped with a three-element dynamic vibration absorber (DVA) is exactly minimized for a given mass ratio using a numerical approach. The frequency response curve is assumed to have two resonance peaks, and the parameters for the two springs and one viscous damper in the DVA are optimized by minimizing the resonance amplitudes. The three-element model is known to represent the dynamic characteristics of air-damped DVAs. A generalized optimality criteria approach is developed and adopted for the derivation of the simultaneous equations for this design problem. The solution of the simultaneous equations precisely equalizes the heights of the two peaks in the resonance curve and achieves a minimum amplitude magnification factor. The simultaneous equations are solvable using the standard built-in functions of numerical computing software. The performance improvement of the three-element DVA compared to the standard Voigt type is evaluated based on the equivalent mass ratios. This performance evaluation is highly accurate and reliable because of the precise formulation of the optimization problem. Thus, the advantages of the three-element type DVA have been made clearer.</span>http:///vibrationacoustics.asmedigitalcollection.asme.org/article.aspx?articleid=vibrationacoustics.asmedigitalcollection.asme.orgMultifidelity Recursive Cokriging for Dynamic Systems' Response Modification
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Tue, 24 Jul 2018 00:00:00 GMTBregant L, Parussini L, Pediroda V. <span class="paragraphSection">In order to perform the accurate tuning of a machine and improve its performance to the requested tasks, the knowledge of the reciprocal influence among the system's parameters is of paramount importance to achieve the sought result with minimum effort and time. Numerical simulations are an invaluable tool to carry out the system optimization, but modeling limitations restrict the capabilities of this approach. On the other side, real tests and measurements are lengthy, expensive, and not always feasible. This is the reason why a mixed approach is presented in this work. The combination, through recursive cokriging, of low-fidelity, yet extensive, numerical model results, together with a limited number of highly accurate experimental measurements, allows to understand the dynamics of the machine in an extended and accurate way. The results of a controllable experiment are presented and the advantages and drawbacks of the proposed approach are also discussed.</span>http:///vibrationacoustics.asmedigitalcollection.asme.org/article.aspx?articleid=vibrationacoustics.asmedigitalcollection.asme.orgBeam-Based Vibration Energy Harvesters Tunable Through Folding
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Tue, 24 Jul 2018 00:00:00 GMTPydah A, Batra RC. <span class="paragraphSection">We present a novel beam-based vibration energy harvester, and use a structural tailoring concept to tune its natural frequencies. Using a solution of the Euler–Bernoulli beam theory equations, verified with finite element (FE) solutions of shell theory equations, we show that introducing folds or creases along the span of a slender beam, varying the fold angle at a crease, and changing the crease location helps tune the beam natural frequencies to match an external excitation frequency and maximize the energy harvested. For a beam clamped at both ends, the first frequency can be increased by 175% with a single fold. With two folds, selective frequencies can be tuned, leaving others unchanged. The number of folds, their locations, and the fold angles act as tuning parameters that provide high sensitivity and controllability of the frequency response of the harvester. The analytical model can be used to quickly optimize designs with multiple folds for anticipated external frequencies.</span>http:///vibrationacoustics.asmedigitalcollection.asme.org/article.aspx?articleid=vibrationacoustics.asmedigitalcollection.asme.orgApproximate Floquet Analysis of Parametrically Excited Multi-Degree-of-Freedom Systems With Application to Wind Turbines
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Tue, 24 Jul 2018 00:00:00 GMTAcar GD, Feeny BF. <span class="paragraphSection">General responses of multi-degrees-of-freedom (MDOF) systems with parametric stiffness are studied. A Floquet-type solution, which is a product between an exponential part and a periodic part, is assumed, and applying harmonic balance, an eigenvalue problem is found. Solving the eigenvalue problem, frequency content of the solution and response to arbitrary initial conditions are determined. Using the eigenvalues and the eigenvectors, the system response is written in terms of “Floquet modes,” which are nonsynchronous, contrary to linear modes. Studying the eigenvalues (i.e., characteristic exponents), stability of the solution is investigated. The approach is applied to MDOF systems, including an example of a three-blade wind turbine, where the equations of motion have parametric stiffness terms due to gravity. The analytical solutions are also compared to numerical simulations for verification.</span>http:///vibrationacoustics.asmedigitalcollection.asme.org/article.aspx?articleid=vibrationacoustics.asmedigitalcollection.asme.org