On Theoretical and Numerical Aspects of Bifurcations and Hysteresis Effects in Kinetic Energy Harvesters

The piezoelectric energy-harvesting system with double-well characteristics and hysteresis in the restoring force is studied. The proposed system consists of a bistable oscillator based on a cantilever beam structure. The elastic force potential is modified by magnets. The hysteresis is an additional effect of the composite beam considered in this system, and it effects the modal solution with specific mass distribution. Consequently, the modal response is a compromise between two overlapping, competing shapes. The simulation results show evolution in the single potential well solution, and bifurcations into double-well solutions with the hysteretic effect. The maximal Lyapunov exponent indicated the appearance of chaotic solutions. Inclusion of the shape branch overlap parameter reduces the distance between the external potential barriers and leads to a large-amplitude solution and simultaneously higher voltage output with smaller excitation force. The overlap parameter works in the other direction: the larger the overlap value, the smaller the voltage output. Presumably, the successful jump though the potential barrier is accompanied by an additional switch between the corresponding shapes.

Accelerated Multifidelity Emulator Modeling for Probabilistic Rotor Response Study

This paper presents an efficient methodology to build a modal solution emulator for the probabilistic study of geometrically mistuned bladed rotors by using the newly developed localized-Galerkin multifidelity (LGMF) modeling and eigensolution reanalysis (ER) with the symmetric successive matrix inversion (SSMI) methods. The key idea of the mistuned blade emulator is to establish a reduced functional relationship between the stochastic geometric variations and the disturbed modal responses. The prediction accuracy of an emulator generally depends on how many training samples of modal solutions are available and how well the potential modal switching due to stochastic mistuning is captured. To reduce the computational costs of generating training samples without sacrificing accuracy, this paper introduces the collaborative framework of the new approaches of multifidelity (MF) modeling and ER. The proposed framework is demonstrated for its computational benefits with several numerical examples including the point-cloud scanned mistuned blade problem.

Accelerated Multi-Fidelity Emulator Modeling for Probabilistic Rotor Response Study

This paper presents an efficient methodology to build a modal solution emulator for the probabilistic study of geometrically mistuned bladed rotors by using the newly developed localized-Galerkin multi-fidelity modeling and eigensolution reanalysis with the symmetric successive matrix inversion methods. The key idea of the mistuned blade emulator is to establish a reduced functional relationship between the stochastic geometric variations and the disturbed modal responses. The prediction accuracy of an emulator depends on how many training samples of modal solutions are available and how well the potential modal switching due to stochastic mistuning is captured. To reduce the computational costs of generating training samples without sacrificing accuracy, this paper introduces the collaborative framework of the new approaches of multi-fidelity modeling and eigensolution reanalysis. The proposed framework is demonstrated for its computational benefits with several numerical examples including the point-cloud scanned mistuned blade problem.

A Multi-Tone Sound Absorber Based on an Array of Shunted Loudspeakers

It has been demonstrated that a single shunted loudspeaker can be used as an effective low frequency sound absorber in a duct, but many shunted loudspeakers have to be used in practice for noise reduction or reverberation control in rooms, thus it is necessary to understand the performance of an array of shunted loudspeakers. In this paper, a model for the parallel shunted loudspeaker array for multi-tone sound absorption is proposed based on a modal solution, and then the acoustic properties of a shunted loudspeaker array under normal incidence are investigated using both the modal solution and the finite element method. It was found that each shunted loudspeaker can work almost independently where each unit resonates. Based on the interaction analysis, multi-tone absorbers in low frequency can be achieved by designing multiple shunted loudspeakers with different shunt circuits respectively. The simulation and experimental results show that the normal incidence sound absorption coefficient of the designed absorber has four absorption peaks with values of 0.42, 0.58, 0.80, and 0.84 around 100 Hz, 200 Hz, 300 Hz, and 400 Hz respectively.

Natural frequency of bending vibration for stepped beam of different geometrical characters and materials

Based on the bending vibration theory of the beam with equal cross-sectional area, the free bending vibration equation of each subsection of stepped beam is given. The modal function of the stepped beam is assumed by the method of separating variable. First, substitute the modal function into free vibration equation so as to obtain the corresponding vibration shape function and then the modal solution with undetermined coefficients and natural frequency is developed. Next, substitute the modal solution into the boundary condition and the continuity condition to get the corresponding equations. Solving the equations by elimination method, the frequency equation of the step beam is obtained by the condition that the system of linear equations has non-zero solution. Taking the cantilever step beam and simplified supported step beam as examples, the first-order natural frequency of transverse bending vibration of stepped beams with different cross-sectional shapes, different materials, different areas, and different lengths for the two portions was obtained by solving the frequency equation with Mathcad software, and the corresponding curves of natural frequency for the circular and rectangular cross-section beam have been given.

Stability transitions of an axially moving string subjected to a distributed follower force

The transverse vibrations of an axially moving string that is subjected to a distributed follower force are examined here. This model provides an insight into the complex dynamics of seemingly simpler systems such as silicon wafer cutting using wire saws, and aerial or marine towing, where a relatively long flexible structure is dragged through fluid. The equation of motion is derived and it includes the axial variation in the tension that arises due to acceleration and the follower force. As the exact analytical solution of this equation is difficult to determine, the approximate closed-form modal solution of a non-travelling counterpart of the system is obtained using the asymptotic technique, which is then used as a basis to obtain the numerical solution for the axially moving string. The effect of the follower force and viscous dissipation on the eigenstructure of the system is investigated. Mathematical operations such as the Hermite form and the Routh–Hurwitz criterion are applied to the characteristic polynomial to investigate the dynamic behaviour of these modes. The semi-analytical approach presented explains the ‘mathematical’ instability (in the absence of damping) that arises when both axial transport and follower force are simultaneously present. An unusual transition of the dynamic behaviour from the stable to the overdamped and then directly to the unstable regime is observed.