Periodic Motion and Frequency Energy Plots of Dynamical Systems Coupled with Piecewise Nonlinear Energy Sink

The frequency-energy plots (FEPs) of two-degree-of-freedom linear structures attached to a piecewise nonlinear energy sink (PNES) are generated here and thoroughly investigated. This study provides the FEP analysis of such systems for further understanding to nonlinear targeted energy transfer (TET) by the PNES. The attached PNES to the considered linear dynamical systems incorporates a symmetrical clearance zone of zero stiffness content where the boundaries of the zone are coupled with the linear structure by linear stiffness elements. In addition, linear viscous damping is selected to be continuous during the PNES mass oscillation. The underlying nonlinear dynamical behaviour of the considered structure-PNES systems is investigated by generating the fundamental backbone curves of the FEP and the bifurcated subharmonic resonance branches using numerical continuation methods. Accordingly, interesting dynamical behaviour of the nonlinear normal modes (NNMs) of the structure-PNES system on different backbones and subharmonic resonance branches has been observed. In addition, the imposed wavelet transform frequency spectrums on the FEPs have revealed that the TET takes place in multiple resonance captures where it is dominated by the nonlinear action of the PNES.

Non-planar Dynamics of a CFRP Cable with an External 1/3-order Subharmonic Resonance

Based on the non-planar vibration equations of a cable made of carbon fiber reinforced polymer (CFRP), the nonlinear behaviors of the cable are studied. The one-to-one internal resonance of the lowest in-plane and out-of-plane modes of the cable is investigated. Three different cases, namely, 1/3-order subharmonic resonance of the in-plane mode, 1/3-order subharmonic resonance of the out-of-plane mode and 1/3-order subharmonic resonance of both of the in-plane and out-of-plane modes are examined. The vibration equations of the cable are discretized by using Galerkin’s method. In this way, the ordinary differential equations (ODEs) are obtained. The multiple time scale method is employed to solve the ODEs and the corresponding modulation equations are derived. Then, the response curves of the cable are obtained by using Newton-Raphson method and pseudo arclength algorithm. Meanwhile, the response curves of the CFRP cable are compared with those of the steel cable to explore the differences in nonlinear behaviors of the cables made of different materials. The results show that the CFRP cable has potential advantages over the steel cable from a nonlinear point of view.

Theoretical Study on Widening Bandwidth of Piezoelectric Vibration Energy Harvester with Nonlinear Characteristics

In order to make a piezoelectric vibration energy harvester collect more energy on a broader frequency range, nonlinearity is introduced into the system, allowing the harvester to produce multiple steady states and deflecting the frequency response curve. However, the harvester can easily maintain intra-well motion rather than inter-well motion, which seriously affects its efficiency. The aim of this paper is to analyze how to take full advantage of the nonlinear characteristics to widen the bandwidth of the piezoelectric vibration energy harvester and obtain more energy. The influence of the inter-permanent magnet torque on the bending of the piezoelectric cantilever beam is considered in the theoretical modeling. The approximate analytical solutions of the primary and 1/3 subharmonic resonance of the harvester are obtained by using the complex dynamic frequency (CDF) method so as to compare the energy acquisition effect of the primary resonance and subharmonic resonance, determine the generation conditions of subharmonic resonance, and analyze the effect of primary resonance and subharmonic resonance on broadening the bandwidth of the harvester under different external excitations. The results show that the torque significantly affects the equilibrium point and piezoelectric output of the harvester. The effective frequency band of the bistable nonlinear energy harvester is 270% wider than that of the linear harvester, and the 1/3 subharmonic resonance broadens the band another 92% so that the energy harvester can obtain more than 0.1 mW in the frequency range of 18 Hz. Therefore, it is necessary to consider the influence of torque when modeling. The introduction of nonlinearity can broaden the frequency band of the harvester when it is in primary resonance, and the subharmonic resonance can make the harvester obtain more energy in the global frequency range.

Effects of phase difference between instability modes on boundary-layer transition

Phase effect on the modal interaction of flow instabilities is investigated for laminar-to-turbulent transition in a flat-plate boundary-layer flow. Primary and secondary three-dimensional (3-D) oblique waves at various initial phase differences between these two instability modes. Three numerical methods are used for a systematic approach for the entire transition process, i.e. before the onset of transition well into fully turbulent flow. Floquet analysis predicts the subharmonic resonance where a subharmonic mode locally resonates for a given basic flow composed of the steady laminar flow and the fundamental mode. Because Floquet analysis is limited to the resonating subharmonic mode, nonlinear parabolised stability equation analysis (PSE) is conducted with various phase shifts of the subharmonic mode with respect to the given fundamental mode. The application of PSE offers insights on the modal interaction affected by the phase difference up to the weakly nonlinear stage of transition. Large-eddy simulation (LES) is conducted for a complete transition to turbulent boundary layer because PSE becomes prohibitively expensive in the late nonlinear stage of transition. The modulation of the subharmonic resonance with the initial phase difference leads to a significant delay in the transition location up to $\Delta Re_{x, tr} \simeq 4\times 10^5$ as predicted by the current LES. Effects of the initial phase difference on the spatial evolution of the modal shape of the subharmonic mode are further investigated. The mechanism of the phase evolution is discussed, based on current numerical results and relevant literature data.

Subharmonic Resonance of Two Thirds Order of Electrostatically Actuated Bio-MEMS Circular Plates: Amplitude-Frequency Response

This paper deals with subharmonic resonance of two thirds order or electrostatically activated BioMEMS circular plates. Specifically, this paper investigates the frequency amplitude response of this resonance. The system consists of a clamped flexible circular plate above a parallel electrode situated at a distance, and under an AC voltage of frequency near three fourths of the natural frequency of the plate. The method of multiple scales is used to model the hard excitations in the system, hard excitations that are necessary to produce this secondary resonance. This work predicts the response, as well as the effects of parameters such as voltage and damping on the response. This paper predicts that the subharmonic resonance of two thirds order consists of a near zero steady state amplitude, and higher values steady state amplitudes consisting of a stable branch and an unstable branch, and a saddle-node bifurcation point. The predictions regarding the effects of voltage and damping on the response of the BioMEMS plate shows that as the voltage increases, the bifurcation point is shifted to lower frequencies and lower amplitudes, while as the damping increases the bifurcation point is shifted to lower frequencies and high amplitudes. Therefore, the increase of damping leads to a case in which it is harder to reach higher amplitude steady-state solutions since it requires large initial amplitudes of the plate.

Subharmonic Resonance of One Fourth Order of Electrostatically Actuated MEMS Circular Plates: Amplitude-Frequency Response

This work deals with the amplitude-frequency response subharmonic resonance of 1/4 order of electrostatically actuated circular plates. The method of multiple scales is used to model the hard excitations and to predict the response. This work predicts that the steady state solutions are zero amplitude solutions, and non-zero amplitude solutions which consist of stable and unstable branches. The effects of parameters such as voltage and damping on the response are predicted. As the voltage increases, the non-zero amplitude solutions are shifted to lower frequencies. As the damping increases, the non-zero steady-state amplitudes are shifted to higher amplitudes, so larger initial amplitudes for the MEMS plate to reach non-zero steady-state amplitudes.

Vibration Characteristics of Hot Rolling Mill Rolls Based on Elastoplastic Hysteretic Deformation

Based on the analysis of the influence of roll vibration on the elastoplastic deformation state of a workpiece in a rolling process, a dynamic rolling force model with the hysteresis effect is established. Taking the rolling parameters of a 1780 mm hot rolling mill as an example, we analyzed the hysteresis between the dynamic rolling force and the roll vibration displacement by varying the rolling speed, roll radius, entry thickness, front tension, back tension, and strip width. Under the effect of the dynamic rolling force and considering the nonlinear effect between the backup and work rolls as well as the structural constraints on the rolling mill, a hysteretic nonlinear vertical vibration model of a four-high hot rolling mill was established. The amplitude-frequency equations corresponding to 1/2 subharmonic resonance and 1:1 internal resonance of the rolling mill rolls were obtained using a multi-scale approximation method. The amplitude-frequency characteristics of the rolling mill vibration system with different parameters were studied through a numerical simulation. The parametric stiffness and nonlinear stiffness corresponding to the dynamic rolling force were found to have a significant influence on the amplitude of the subharmonic resonance system, the bending degree of the vibration curve, and the size of the resonance region. Moreover, with the change in the parametric stiffness, the internal resonance exhibited an evident jump phenomenon. Finally, the chaotic characteristics of the rolling mill vibration system were studied, and the dynamic behavior of the vibration system was analyzed and verified using a bifurcation diagram, maximum Lyapunov exponent, phase trajectory, and Poincare section. Our research provides a theoretical reference for eliminating and suppressing the chatter in rolling mills subjected to an elastoplastic hysteresis deformation.

Frequency Energy Plots of Dynamical Systems Attached to Piecewise Nonlinear Energy Sink

The frequency-energy plots (FEPs) of two-degree-of-freedom linear structures attached to piecewise nonlinear energy sink (PNES) are generated here and thoroughly investigated. This study provides the FEP analysis of such systems for further understanding of nonlinear targeted energy transfer (TET) by the PNES. The attached PNES incorporates a symmetrical clearance zone of zero stiffness content about its equilibrium position where the boundaries of the zone are coupled with linear structure by linear stiffness elements. In addition, linear viscous damping is selected to be continuous during PNES mass oscillation. The underlying nonlinear dynamical behaviour of the considered structure-PNES systems is investigated by generating the fundamental backbone curves of the FEP and the bifurcated subharmonic resonance branches using numerical continuation methods. Accordingly, interesting dynamical behaviour of the nonlinear normal modes (NNMs) of the structure-PNES system on different backbones and subharmonic resonance branches has been observed. In addition, the imposed wavelet transform frequency spectrums on the FEPs have revealed that the TET takes place where it is dominated by the nonlinear action of the PNES.