Research Papers

J. Vib. Acoust. 2016;139(1):011001-011001-10. doi:10.1115/1.4034631.

A three-dimensional (3D) dynamic gear model is presented which combines classic shaft, lumped parameter, and specific two-node gear elements. The mesh excitation model is based on transmission errors (TEs), and its mathematical grounding is briefly described. The validity of the proposed methodology is assessed for both spur and helical gears by comparison with experimental evidence. The model is then employed to analyze the relationship between dynamic transmission errors (DTE) and dynamic tooth loads (DF) or root stresses. It is shown that a linear dependency can be found as long as the system behavior is dominated by shaft torsion but that this linear relationship tends to disappear when bending cannot be neglected.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2016;139(1):011002-011002-7. doi:10.1115/1.4034626.

The multi-objective optimal control design usually generates hundreds or thousands of Pareto optimal solutions. How to assist a user to select an appropriate controller to implement is a postprocessing issue. In this paper, we develop a method of cluster analysis of the Pareto optimal designs to discover the similarity of the optimal controllers. After we identify the clusters of optimal controllers, we develop a switching strategy to select controls from different clusters to improve the performance. Numerical and experimental results show that the switching control algorithm is quite promising.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2016;139(1):011003-011003-8. doi:10.1115/1.4034618.

The Helmholtz-type hydraulic silencer is one of the most practical silencers for attenuating pressure pulsations in hydraulic systems owing to its simple structure and reasonable cost. Maximum attenuation performance can be attained at the resonance frequency in accordance with the principle of Helmholtz resonance. Therefore, it is extremely important to precisely determine the resonance frequency at the design stage. It was clarified in our previous study that the shape of the volume vessel affects the resonance frequency of the silencer because of the wave propagation of pressure pulsation inside the volume vessel. In this study, the attenuation characteristics and wave propagation in a silencer with a hemispherical vessel are investigated. A mathematical model that takes into account the propagation of a one-dimensional wave in the radial direction of the hemispherical vessel is proposed and compared with the step section approximation model and the classic lumped parameter model. Furthermore, the effectiveness of the theoretical analysis is verified by experiments wherein the dimensional specifications of the vessel and neck are adjusted.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2016;139(1):011004-011004-10. doi:10.1115/1.4034768.

The effects of typical machining errors on the dynamic features of rod-fastened rotor bearing system (RBS) are studied in this paper. Three micron-sized machining errors are considered in a three-dimensional (3D) rod-fastened model. The static effects of machining errors are investigated by applying finite element method. Results demonstrate that machining errors not only bring about mass eccentricity but also cause obvious rotor bending due to large pretightening force. Then, nonlinear dynamic features such as stability and bifurcation are analyzed by using target-shooting technique, track-following method, and Floquet theory. Analysis data indicate that rotor bending originated from machining errors reduces the system stability evidently. It is also observed that the vibration value continues to go up after critical speed as rotating speed increases. It is a particular property compared with integral rotor. It explains the reason why the machining precision of rod-fastened rotor is much higher than that of the corresponding integral rotor to some extent. Moreover, differences between machining errors are compared and the results show that the machining precision of axial assembly interfaces should be paid more attention in the rod-fastened rotor design.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2016;139(1):011005-011005-12. doi:10.1115/1.4034628.

The stability of an axially moving string system subjected to parametric excitation resulting from speed fluctuations has been examined in this paper. The time-dependent velocity is assumed to be a harmonically varying function around a (low) constant mean speed. The method of characteristic coordinates in combination with the two timescales perturbation method is used to compute the first-order approximation of the solutions of the equations of motion that governs the transverse vibrations of an axially moving string. It turns out that the system can give rise to resonances when the velocity fluctuation frequency is equal (or close) to an odd multiple of the natural frequency of the system. The stability conditions are investigated analytically in terms of the displacement-response and the energy of the system near the resonances. The effects of the detuning parameter on the amplitudes of vibrations and on the energy of the system are also presented through numerical simulations.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2016;139(1):011006-011006-11. doi:10.1115/1.4034771.

This paper investigates the problem of optimally locating passive vibration isolators to minimize unwanted vibration caused by exogenous disturbance forces. The stiffness and damping parameters of the isolators are assumed to be known, leaving the isolator locations, which are nonlinearly related to system states, as unknown optimization variables. An approach for reformulating the nonlinear isolator placement problem as a linear time-invariant (LTI) feedback control problem, by linking fictitious control forces to fictitious measured outputs using a nonzero feedforward term, is proposed. Accordingly, the isolator locations show up within a static output feedback gain matrix which can be optimized, using methods from optimal control theory, to minimize the H2 and/or H norms of transfer functions representing unwanted vibration. The proposed framework also allows well-established LTI control theories to be applied to the analyses of the optimal isolator placement problem and its results. The merits of the proposed approach are demonstrated using single and multivariable case studies related to isolator placement in precision manufacturing machines. However, the framework is applicable to optimal placement of passive isolators, suspensions, or dampers in automotive, aerospace, civil, and other applications.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2016;139(1):011007-011007-10. doi:10.1115/1.4034769.

Bridge noise and rail noise induced by passing trains should be included while estimating low- and medium-frequency (20–1000 Hz) noise in railway viaducts. However, the prediction of bridge noise and rail noise using a three-dimensional (3D) acoustic model is not efficient, especially for far-field points. In this study, a combined 2.5-dimensional (2.5D) and two-dimensional (2D) method is proposed to predict bridge noise and rail noise in both the near- and far-field. First, the near-field noise is obtained by combining the 2.5D acoustic model and a 3D vehicle–track–bridge interaction analysis. Then, the 2D method is used to estimate the attenuation of bridge noise and rail noise in the far-field, and the accuracy is validated through comparison with the 2.5D method. Third, the near-field points are treated as reference sources, and the noise at far-field points is predicted by combining the 2.5D and 2D methods. Finally, the proposed method is used to predict the bridge noise and rail noise for a box girder and a U-shaped girder. The spatial distribution of the bridge noise and rail noise is investigated. Generally, the rail noise is dominant above the bridge, and the bridge noise has a larger contribution to the total noise beneath the bridge. The rail noise from the U-shaped girder is much smaller than that from the box girder due to the shielding effect of the webs.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2016;139(1):011008-011008-10. doi:10.1115/1.4034777.

Particle damping has the promising potential for attenuating unwanted vibrations in harsh environments especially under high temperatures where conventional damping materials would not be functional. Nevertheless, a limitation of simple particle damper (PD) configuration is that the damping effect is insignificant if the local displacement/acceleration is low. In this research, we investigate the performance of a tuned mass particle damper (TMPD) in which the particle damping mechanism is integrated into a tuned mass damper (TMD) configuration. The essential idea is to combine the respective advantages of these two damping concepts and in particular to utilize the tuned mass damper configuration as a motion magnifier to amplify the energy dissipation capability of particle damper when the local displacement/acceleration of the host structure is low. We formulate a first-principle-based dynamic model of the integrated system and analyze the particle motion by using the discrete element method (DEM). We perform systematic parametric studies to elucidate the damping effect and energy dissipation mechanism of a TMPD. We demonstrate that a TMPD can provide significant vibration suppression capability, essentially outperforming conventional particle damper.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2016;139(1):011009-011009-13. doi:10.1115/1.4034250.

A dynamic model of an automotive belt-drive system with a noncircular sprocket instead of a round sprocket is developed in this work to study the effect of reducing the angular variation of camshafts. There are two submodels in the belt-drive system, which are an engine model and a belt-drive model, and they are decoupled to simplify the analysis. When the belt-drive system operates at a steady-state, it is described by a nonlinear model with forced excitation, which can be approximated by a linear model with combined parametric and forced excitations. Steady-state responses of the engine and belt-drive models are calculated by a modified incremental harmonic balance method that incorporates fast Fourier transform and Broyden's method, which is efficient and accurate to obtain a periodic response of a multi-degree-of-freedom system. Steady-state responses of the angular variation of camshafts with different values of sprocket parameters are compared to investigate their optimal values to reduce the angular variation of camshafts. The optimal eccentricity and installation angle are larger and smaller than those from the kinematic model, respectively, which is consistent with published experimental results. This study first shows from a dynamic point of view why use of a noncircular sprocket can reduce the angular variation of camshafts in the operating speed of an engine. Simulation of a speed-up procedure for different sprocket parameters shows results that are consistent with steady-state responses. The belt-drive model developed in this work can be used to select sprocket parameters to minimize the angular variation of camshafts and numerically evaluate the dynamic performance of a belt-drive system with a given design of sprocket parameters.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2016;139(1):011010-011010-12. doi:10.1115/1.4035132.

In this investigation, the pantograph/catenary contact is examined using two different formulations. The first is an elastic contact formulation that allows for the catenary/panhead separation and for the analysis of the effect of the aerodynamic forces, while the second approach is based on a constraint formulation that does not allow for such a separation by eliminating the freedom of relative translation in two directions at the catenary/panhead contact point. In this study, the catenary system, including the contact and messenger wires, is modeled using the nonlinear finite element (FE) absolute nodal coordinate formulation (ANCF) and flexible multibody system (MBS) algorithms. The generalized aerodynamic forces associated with the ANCF position and gradient coordinates and the pantograph reference coordinates are formulated. The new elastic contact formulation used in this investigation is derived from the constraint-based sliding joint formulation previously proposed by the authors. By using a unilateral penalty force approach, separation of the catenary and panhead is permitted, thereby allowing for better evaluating the response of the pantograph/catenary system to wind loading. In this elastic contact approach, the panhead is assumed to have six degrees-of-freedom with respect to the catenary. The coordinate system at the pantograph/catenary contact point is chosen such that the contact model developed in this study can be used with both the fully parameterized and gradient deficient ANCF elements. In order to develop a more realistic model, the MBS pantograph model is mounted on a detailed three-dimensional MBS rail-vehicle model. The wheel/rail contact is modeled using a nonlinear three-dimensional elastic contact formulation that accounts for the creep forces and spin moment. In order to examine the effect of the external aerodynamic forces on the pantograph/catenary interaction, two scenarios are considered in this investigation. In the first scenario, the crosswind loading is applied on the pantograph components only, while in the second scenario, the aerodynamic forces are applied on the pantograph components and also on the flexible catenary. For the configuration considered in this investigation, it was found that the crosswind assists the uplift force exerted on the pantograph mechanism, increasing the mean contact force value. Numerical results are presented in order to compare between the cases with and without the wind forces.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2016;139(1):011011-011011-15. doi:10.1115/1.4034844.

Nonlinear vibrations and their control are critical in improving the magnetic bearings system performance and in the more widely spread use of magnetic bearings system. Multiple objective genetic algorithms (MOGAs) simultaneously optimize a vibration control law and geometrical features of a set of nonlinear magnetic bearings supporting a generic flexible, spinning shaft. The objectives include minimization of the actuator mass, minimization of the power loss, and maximization of the external static load capacity of the rotor. Levitation of the spinning rotor and the nonlinear vibration amplitude by rotor unbalance are constraint conditions according to International Organization for Standardization (ISO) specified standards for the control law search. The finite element method (FEM) was applied to determine the temperature distribution and identify the hot spot of the actuator during steady-state operation. Nonlinearities include magnetic flux saturation, and current and voltage limits of power amplifiers. Pareto frontiers were applied to identify and visualize the best-compromised solutions, which give a most compact design with minimum power loss whose vibration amplitudes satisfy ISO standards.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2016;139(1):011012-011012-11. doi:10.1115/1.4034770.

Inspired by the mechanism of acoustic–elastic metamaterial (AEMM) that exhibits a stop band gap for wave transmission, simultaneous vibration suppression and energy harvesting can be achieved by integrating AEMM with energy-harvesting component. This article presents an analytical study of a multifunctional system based on this concept. First, a mathematical model of a unit-cell AEMM embedded with a piezoelectric transducer is developed and analyzed. The most important finding is the double-valley phenomenon that can intensively widen the band gap under strong electromechanical coupling condition. Based on the mathematical model, a dimensionless parametric study is conducted to investigate how to tune the system to enhance its vibration suppression ability. Subsequently, a multicell system is conceptualized from the findings of the unit-cell system. In a similar way, dimensionless parametric studies are conducted to optimize the vibration suppression performance and the energy-harvesting performance severally. It turns out that different impedance matching schemes are required to achieve optimal vibration suppression and energy harvesting. To handle this problem, compromising solutions are proposed for weakly and strongly coupled systems, respectively. Finally, the characteristics of the AEMM-based piezoelectric energy harvester (PEH) from two functional aspects are summarized, providing several design guidelines in terms of system parameter tuning. It is concluded that certain tradeoff is required in the process of optimizing the performance toward dual functionalities.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2016;139(1):011013-011013-8. doi:10.1115/1.4034775.

In this paper, a novel smart vibration energy harvester (VEH) is presented. The harvester automatically adjusts its natural frequency to stay in resonance with ambient vibration. The proposed harvester consists of two piezoelectric cantilever beams, a tiny piezomotor with a movable mass attached to one of the beams, a control unit, and electronics. Thanks to its self-locking feature, the piezomotor does not require energy to fix its movable part, resulting in an improvement in overall energy demand. The operation of the system is optimized in order to maximize the energy efficiency. At each predefined interval, the control unit wakes up, calculates the phase difference between two beams, and if necessary, actuates the piezomotor to move its mass in the appropriate direction. It is shown that the proposed tuning algorithm successfully increases the fractional bandwidth of the harvester from 4% to 10%. The system is able to deliver 83.4% of the total harvested power into usable electrical power, while the piezomotor uses only 2.4% of the harvested power. The presented efficient, autotunable, and self-sufficient harvester is built using off-the-shelf components and it can be easily modified for wide range of applications.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2016;139(1):011014-011014-7. doi:10.1115/1.4034843.

This paper addresses the Lyapunov functions and sliding mode control design for two degrees-of-freedom (2DOF) and multidegrees-of-freedom (MDOF) fractional oscillators. First, differential equations of motion for 2DOF fractional oscillators are established by adopting the fractional Kelvin–Voigt constitute relation for viscoelastic materials. Second, a Lyapunov function candidate for 2DOF fractional oscillators is suggested, which includes the potential energy stored in fractional derivatives. Third, the differential equations of motion for 2DOF fractional oscillators are transformed into noncommensurate fractional state equations with six dimensions by introducing state variables with physical significance. Sliding mode control design and adaptive sliding mode control design are proposed based on the noncommensurate fractional state equations. Furthermore, the above results are generalized to MDOF fractional oscillators. Finally, numerical simulations are carried out to validate the above control designs.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2016;139(1):011015-011015-16. doi:10.1115/1.4034776.

This paper considers the optimal design of double-mass dynamic vibration absorbers (DVAs) attached to an undamped single degree-of-freedom system. Three different optimization criteria, the H optimization, H2 optimization, and stability maximization criteria, were considered for the design of the DVAs, and a performance index was defined for each of these criteria. First, the analytical models of vibratory systems with double-mass DVAs were considered, and seven dimensionless parameters were defined. Five of these parameters must be optimized to minimize or maximize the performance indices. Assuming that all dimensionless parameters are non-negative, the optimal value of one parameter for a double-mass DVA arranged in series (series-type DVA) was proven to be zero. The optimal adjustment conditions of the other four parameters were derived as simple algebraic formulae for the H2 and stability criteria and numerically determined for the H criterion. For a double-mass DVA arranged in parallel (parallel-type DVA), all five parameters were found to have nonzero optimal values, and these values were obtained numerically by solving simultaneous algebraic equations. Second, the performance of these DVAs was compared with a single-mass DVA. The result revealed that for all optimization criteria, the performance of the series-type DVA is the best among the three DVAs and that of the single-mass DVA is the worst. Finally, a procedure for deriving the algebraic or numerical solutions for H2 optimization is described. The derivation procedure of other optimal solutions will be introduced in the future paper.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2016;139(1):011016-011016-10. doi:10.1115/1.4035029.

In common rotary piezoelectric (PZT) frequency up-converting energy harvesters, impact or nonimpact frequency up-conversion technologies are used. For low separation distances between the magnets in nonimpact cases, when weak excitation is applied, depending on some parameters such as separation distance between the magnets, eccentric proof mass may be unable to overcome the magnetic potential between the magnets, and thus, the extracted power of the harvester lowers. To increase the harvester power output, the use of an additional pair of magnets, called the assisting part, is proposed in this paper. For different harmonic excitations, the generated powers of the harvester with and without assisting part have been compared to each other. It is found that by appropriately adjusting the separation distance, the use of such part can increase the generated power in most cases. Using a real-world multifrequency multi-amplitude excitation, the ability of the proposed idea to increase the extracted power is investigated. It is found that the maximum generated power of the device can effectively increase to more than two times. In order to check the accuracy of the applied mathematical modeling, some experiments have been conducted.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2016;139(1):011017-011017-14. doi:10.1115/1.4035107.

Cyclically periodic structures, such as bladed disk assemblies in turbomachinery, are widely used in engineering systems. It is well known that small uncertainties exist among their substructures, which in certain situations may cause drastic change in the dynamic responses, a phenomenon known as vibration localization. Previous studies have suggested that the introduction of small, prespecified design modification, i.e., intentional mistuning, may alleviate vibration localization and reduce response variation. However, there has been no systematic methodology to facilitate the optimal design of intentional mistuning. The most significant challenge is the computational cost involved. The finite-element model of a bladed disk usually requires a very large number of degrees-of-freedom (DOFs). When uncertainties occur in a cyclically periodic structure, the response may no longer be considered as simple perturbation to that of the nominal structure. In this research, a suite of interrelated algorithms is proposed to enable the efficient design optimization of cyclically periodic structures toward alleviating their forced response variation. We first integrate model order reduction with a perturbation scheme to reduce the scale of analysis of a single run. Then, as the core of the new methodology, we incorporate Gaussian process (GP) emulation to conduct the rapid sampling-based evaluation of the design objective, which is a metric of response variation under uncertainties, in the parametric space. The optimal design modification can thus be directly identified to minimize the response variation. The efficiency and effectiveness of the proposed methodology are demonstrated by systematic case studies.

Commentary by Dr. Valentin Fuster

Technical Brief

J. Vib. Acoust. 2016;139(1):014501-014501-10. doi:10.1115/1.4034778.

The vibration of pipes supported by a flexible tank wall is analyzed taking into account the hydroelastic vibration of the tank and the nonlinearity caused by the support clearances. Because the support clearances increase the pipe displacement, it is important to examine whether the support clearances augment the pipe stress. We illustrate that the support clearances can cause an increase in the pipe stress not only due to the increase in pipe displacement but also to the difference between elastic behaviors of the tank wall and pipes. The tank wall and pipes are dominated by membrane and bending deformations, respectively. Furthermore, we illustrate that the support clearances render a stress reduction method ineffective. In this study, a semi-analytical method is applied, rather than a full finite element analysis. The semianalytical method is helpful not only for computationally efficient analysis but also for gaining physical insight into the clearance-nonlinearity-induced stress behaviors noted above.

Topics: Pipes , Displacement , Stress
Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2016;139(1):014502-014502-6. doi:10.1115/1.4035112.

The coupling of vibration and fatigue crack growth in a simply supported uniform Euler–Bernoulli beam containing a single-edge crack is analyzed. The fatigue crack length is treated as a generalized coordinate in a model for the mechanical system. This coupled model accounts for the interaction between the beam oscillations and the crack propagation dynamics. Nonlinear characteristics of the beam motion are introduced as loading parameters to the fatigue model to match experimentally observed failure dynamics. The method of averaging is utilized both as an analytical and numerical tool to: (1) show that, for cyclic loading, our fatigue model reduces to the Paris' law and (2) compare the predicted fatigue damage accumulation with the experimental data for chaotic and random loadings. A utility of the fatigue model is demonstrated in estimating fatigue life under irregular loadings.

Commentary by Dr. Valentin Fuster

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