Nonlinear multimodal model for an annular tuned sloshing damper equipped with damping screens

Annular tuned sloshing dampers equipped with damping screens are studied experimentally and analytically. A nonlinear multimodal model is presented to simulate the coupled response among the lowest order sloshing modes in a tank equipped with damping screens, which leads to velocity-squared damping. Shake table tests are conducted on annular tanks with various inner radii, water depths, screen orientations, and base excitation amplitudes. The proposed model is evaluated by comparing the predicted and measured sloshing forces, energy dissipation per cycle, and wave heights. The predicted sloshing forces and energy dissipation per cycle are in good agreement with the measured results. The wave heights show larger discrepancies, including phase shifts; however, the peak amplitudes are captured with reasonable accuracy for the tests conducted. Secondary resonances lead to multiple peaks in the frequency response plots when higher order sloshing modes become excited through modal coupling. Plots created to indicate which secondary resonances are likely to occur for a given liquid depth ratio indicate that it may not be possible to avoid all secondary resonances. Radial damping screens can be strategically positioned within the tank to provide the desired level of damping to the fundamental sloshing modes, as well as a reasonable amount of damping to higher order modes that are susceptible to secondary resonance excitation. Since existing linearized models for annular tuned sloshing dampers equipped with damping screens do not capture the important nonlinear response characteristics of these devices, the proposed model fills an important research gap necessary to facilitate their effective design.

Bifurcation Analysis and Nonlinear Dynamics of a Capacitive Energy Harvester in the Vicinity of the Primary and Secondary Resonances

The impetus of the present study is to examine the effect of nonlinearity on the efficiency enhancement of a capacitive energy harvester. The model consists of a cantilever microbeam underneath which there is an electret layer with a surface voltage, which is responsible for the driving energy. The packaged device is exposed to unwanted harmonic mechanical excitation. The microbeam undergoes mechanical vibration and accordingly the energy is harvested throughout the output circuit. The dynamic formulation accounts for nonlinear curvature, inertia, and nonlinear electrostatic force. The efficiency of the device in the vicinity of the primary and super-harmonic resonances is examined and accordingly the output power is evaluated. Bifurcation analysis is carried out on the dynamics of the system by detecting the bifurcations in the frequency domain and diagnosing their types. One of the challenging issues in the design and analysis of energy harvesting devices is to broaden the bandwidth so that more frequencies are accommodated within the amplification region. In this study the effect of the nonlinearity on the bandwidth broadening, as well as efficiency improvement of the device, is studied.

Analytic Exploration of Safe Basins in a Benchmark Problem of Forced Escape

The paper presents an analytic approach for predicting the safe basins (SB) in a plane of initial conditions (IC) for escape of classical particle from the potential well under harmonic forcing. The solution is based on the approximation of isolated resonance, which reduces the dynamics to conservative flow on a two-dimensional resonance manifold (RM). Such a reduction allows easy distinction between escaping and non-escaping ICs. As a benchmark potential, we choose a common parabolic-quartic well with truncation at varying energy levels. The method allows accurate predictions of the SB boundaries for relatively low forcing amplitudes. The derived SBs demonstrate an unexpected set of properties, including decomposition into two disjoint zones in the IC plane for a certain range of parameters. The latter peculiarity stems from two qualitatively different escape mechanisms on the RM. For higher forcing values, the accuracy of the analytic predictions decreases to some extent due to the inaccuracies of the basic isolated resonance approximation, but mainly due to the erosion of the SB boundaries caused by the secondary resonances. Nevertheless, even in this case the analytic approximation can serve as a viable initial guess for subsequent numeric estimation of the SB boundaries.

Secondary Resonance Energy Harvesting with Quadratic Nonlinearity

Piezoelectric energy harvesters can transform the mechanical strain into electrical energy. The microelectromechanical transformation device is often composed of piezoelectric cantilevers and has been largely experimented. Most resonances have been developed to harvest nonlinear vibratory energy except for combination resonances. This paper is to analyze several secondary resonances of a cantilever-type piezoelectric energy harvester with a tip magnet. The conventional Galerkin method is improved to truncate the continuous model, an integro-partial differential equation with time-dependent boundary conditions. Then, more resonances on higher-order vibration modes can be obtained. The stable steady-state response is formulated approximately but analytically for the first two subharmonic and combination resonances. The instability boundaries are discussed for these secondary resonances from quadratic nonlinearity. A small damping and a large excitation readily result in an unstable response, including the period-doubling and quasiperiodic motions that can be employed to enhance the voltage output around a wider band of working frequency. Runge–Kutta method is employed to numerically compute the time history for stable and unstable motions. The stable steady-state responses from two different methods agree well with each other. The outcome enriches structural dynamic theory on nonlinear vibration.

The onset of instability in resonant chains

ABSTRACT There is evidence that most chains of mean motion resonances of type k:k − 1 among exoplanets become unstable once the dissipative action from the gas is removed from the system, particularly for large N (the number of planets) and k (indicating how compact the chain is). We present a novel dynamical mechanism that can explain the origin of these instabilities and thus the dearth of resonant systems in the exoplanet sample. It relies on the emergence of secondary resonances between a fraction of the synodic frequency 2π(1/P1 − 1/P2) and the libration frequencies in the mean motion resonance. These secondary resonances excite the amplitudes of libration of the mean motion resonances, thus leading to an instability. We detail the emergence of these secondary resonances by carrying out an explicit perturbative scheme to second order in the planetary masses and isolating the harmonic terms that are associated with them. Focusing on the case of three planets in the 3:2–3:2 mean motion resonance as an example, a simple but general analytical model of one of these resonances is obtained, which describes the initial phase of the activation of one such secondary resonance. The dynamics of the excited system is also briefly described. Finally, a generalization of this dynamical mechanism is obtained for arbitrary N and k. This leads to an explanation of previous numerical experiments on the stability of resonant chains, showing why the critical planetary mass allowed for stability decreases with increasing N and k.