An official website of the United States government
Abstract The inerter has been integrated into various vibration mitigation devices, whose mass amplification effect could enhance the suppression capabilities of these devices. In the current study, the inerter is integrated with a pendulum vibration absorber, referred to as inerter pendulum vibration absorber (IPVA). To demonstrate its efficacy, the IPVA is integrated with a linear, harmonically forced oscillator seeking vibration mitigation. A theoretical investigation is conducted to understand the nonlinear response of the IPVA. It is shown that the IPVA operates based on a nonlinear energy transfer phenomenon wherein the energy of the linear oscillator transfers to the pendulum vibration absorber as a result of parametric resonance of the pendulum. The parametric instability is predicted by the harmonic balance method along with the Floquet theory. A perturbation analysis shows that a pitchfork bifurcation and period doubling bifurcation are necessary and sufficient conditions for the parametric resonance to occur. An arc-length continuation scheme is used to predict the boundary of parametric instability in the parameter space and verify the perturbation analysis. The effects of various system parameters on the parametric instability are examined. Finally, the IPVA is compared with a linear benchmark and an autoparametric vibration absorber and shows more efficacious vibration suppression.
more »
« less
Instability in large bounded domains—branched versus unbranched resonances
Abstract We study transitions from convective to absolute instability near a trivial state in large bounded domains for prototypical model problems in the presence of transport and negative nonlinear feedback. We identify two generic scenarios, depending on the nature of the linear mechanism for instability, which both lead to different, universal bifurcation diagrams. In the first, classical case of a linear branched resonance the transition is hard, that is, small changes in a control parameter lead to a finite-size state. In the second, novel case of an unbranched resonance, the transition is gradual. In both cases, the bifurcation diagram is determined by interaction of the leading edge of an invasion front with upstream boundary conditions. Technically, we analyze this interaction in a heteroclinic gluing bifurcation analysis that uses geometric desingularization of the trivial state.
more »
« less
- Award ID(s):
- 1907391
- PAR ID:
- 10349922
- Date Published:
- Journal Name:
- Nonlinearity
- Volume:
- 34
- Issue:
- 11
- ISSN:
- 0951-7715
- Page Range / eLocation ID:
- 7916 to 7937
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
n/a (Ed.)Two types of resonance-derived interfacial instability are reviewed with a focus on recent work detailing the effect of side walls on interfacial mode discretization. The first type of resonance is the mechanical Faraday instability, and the second is electrostatic Faraday instability. Both types of resonance are discussed for the case of single-frequency forcing. In the case of mechanical Faraday instability, inviscid theory can forecast the modal forms that one might expect when viscosity is taken into account. Experiments show very favourable validation with theory for both modal forms and onset conditions. Lowering of gravity is predicted to shift smaller wavelengths or choppier modes to lower frequencies. This is also validated by experiments. Electrostatic resonant instability is shown to lead to a pillaring mode that occurs at low wavenumbers, which is akin to Rayleigh Taylor instability. As in the case of mechanical resonance, experiments show favourable validation with theoretical predictions of patterns. A stark difference between the two forms of resonance is the observation of a gradual rise in the negative detuning instability in the case of mechanical Faraday and a very sharp one in the case of electrostatic resonance. This article is part of the theme issue ‘New trends in pattern formation and nonlinear dynamics of extended systems’.more » « less
-
A power take-off based on the inerter pendulum vibration absorber (called IPVA-PTO) is integrated with a spar-floater system to study its hydrodynamic response suppression and wave energy conversion capabilities in regular waves. The hydrodynamics of the spar-floater system is computed using the boundary element method with linear wave theory. With the wave height and wave frequency as the bifurcation parameters, it is found that the system can undergo two bifurcations: period-doubling bifurcation around the first resonance frequency (spar mode) and secondary Hopf bifurcation around the second resonance frequency (floater mode). The period-doubling bifurcation results in an energy transfer between the spar-floater system and the IPVA-PTO for small electrical damping values. As a result, the IPVA-PTO system simultaneously reduces the maximum response amplitude operator (RAO) of the spar and increases the normalized capture width in comparison with the optimal linear benchmark. Experiments performed on a ‘‘dry’’ single-degree-of-freedom system integrated with the IPVAPTO where the base excitation is substituted for the wave excitation verify the simultaneous performance enhancement due to the period-doubling bifurcation. The system performance beyond the period-doubling bifurcation is also experimentally investigated. On the other hand, as the wave height approaches and passes the secondary Hopf bifurcation, the pendulum responses transition from primary harmonic responses to quasi-periodic responses to rotations. When the rotations occur, the IPVA-PTO system increases the maximum normalized capture width threefold to fivefold compared with the optimal linear benchmark, yet slightly increases the RAO around the second resonance frequency. Nevertheless, the RAO remains smaller than the global maximum RAO of the optimal linear benchmark. Finally, parametric studies are performed to study the effects of parameters on the bifurcations. It is observed that by varying the electrical damping, the wave height required for achieving the period-doubling bifurcation can be changed significantly, which can be exploited to stabilize the spar.more » « less
-
Abstract Systems of activator–inhibitor reaction–diffusion equations posed on an infinite line are studied using a variety of analytical and numerical methods. A canonical form is considered, which contains all known models with simple cubic autocatalytic nonlinearity and arbitrary constant and linear kinetics. Restricting attention to models that have a unique homogeneous equilibrium, this class includes the classical Schnakenberg and Brusselator models, as well as other systems proposed in the literature to model morphogenesis. Such models are known to feature Turing instability, when activator diffuses more slowly than inhibitor, leading to stable spatially periodic patterns. Conversely in the limit of small feed rates, semi-strong interaction asymptotic analysis shows existence of isolated spike-like patterns. This paper describes the broad bifurcation structures that connect these two regimes. A certain universal two-parameter state diagram is revealed in which the Turing bifurcation becomes sub-critical, leading to the onset of homoclinic snaking. This regime then morphs into the spike regime, with the outer-fold being predicted by the semi-strong asymptotics. A rescaling of parameters and field concentrations shows how this state diagram can be studied independently of the diffusion rates. Temporal dynamics is found to strongly depend on the diffusion ratio though. A Hopf bifurcation occurs along the branch of stable spikes, which is subcritical for small diffusion ratio, leading to collapse to the homogeneous state. As the diffusion ratio increases, this bifurcation typically becomes supercritical and interacts with the homoclinic snaking and also with a supercritical homogeneous Hopf bifurcation, leading to complex spatio-temporal dynamics. The details are worked out for a number of different models that fit the theory using a mixture of weakly nonlinear analysis, semi-strong asymptotics and different numerical continuation algorithms.more » « less
-
Wave-particle interaction plays a crucial role in the dynamics of the Earth’s radiation belts. Cyclotron resonance between coherent whistler mode electromagnetic waves and energetic electrons of the radiation belts is often called a coherent instability. Coherent instability leads to wave amplification/generation and particle acceleration/scattering. The effect of wave on particle’s distribution function is a key component of the instability. In general, whistler wave amplitude can grow over threshold of quasi-linear (linear) diffusion theory which analytically tracks the time-evolution of a particle distribution. Thus, a numerical approach is required to model the nonlinear wave induced perturbations on particle distribution function. A backward test particle model is used to determine the energetic electrons phase space dynamics as a result of coherent whistler wave instability. The results show the formation of a phase space features with much higher resolution than is available with forward scattering models. In the nonlinear regime the formation of electron phase space holes upstream of a monochromatic wave is observed. The results validate the nonlinear phase trapping mechanism that drives nonlinear whistler mode growth. The key differences in phase-space perturbations between the linear and nonlinear scenarios are also illustrated. For the linearized equations or for low (below threshold) wave amplitudes in the nonlinear case, there is no formation of a phase-space hole and both models show features that can be characterized as linear striations or ripples in phase-space.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1088/1361-6544/ac2a15
