Nonlinear Oscillations: Exact Solutions and their Approximations

A dual-functional metamaterial for integrated vibration isolation and energy harvesting

Vibration present on various levels in many engineering fields and hence vibration mitigation has become a subject of intense study. The nonlinear vibration isolation devices are effective for broad frequency bandwidth and can provide better vibration isolation than linear devices. The need for nonlinearity in stiffness and damping characteristics has motivated researchers to apply the nonlinearity found in mechanisms or materials in the passive vibration control devices. This review discusses the applications of nonlinearity in the passive vibration control devices to provide an understanding of how the nonlinearity is applied and useful in the implemented system. Further, applications for nonlinearity can also be extended in the energy harvesting devices, Nonlinear energy sink, metamaterials for the purpose of vibration isolation and energy harvesting. The need for nonlinearity also encouraged research work through inspiration from the nature called bio-inspired devices. The bio-inspired devices mimic the nonlinearity of the biological system to suppress the vibrations. The nonlinear passive isolation is effective for wide frequency bandwidth than the linear isolation system. Further, the nonlinear systems also reduce transmissibility much efficiently than the linear system. The nonlinear energy harvesting system shows a great scope to harvest energy from wide ranges of excitations. The bio-inspired devices also are proven to be effective in vibration isolation. Additionally the design of the metamaterial with nonlinearity in the microstructure, proves to be promising in the vibration suppression applications. Based on the review, the nonlinearity introduced into the systems has greater benefits than the linear systems.

Applications of Nonlinearity in Passive Vibration Control: A Review

An analytical framework has been proposed to analyze the effect of random structural irregularity in honeycomb core for natural frequencies of sandwich panels. Closed-form formulas have been developed for the out-of-plane shear moduli of spatially irregular honeycombs following minimum potential energy theorem and minimum complementary energy theorem. Subsequently an analytical approach has been presented for free-vibration analysis of honeycomb core sandwich panels to quantify the effect of such irregularity following a probabilistic paradigm. Representative results have been furnished for natural frequencies corresponding to low vibration modes of a sandwich panel with high length-to-width ratio. The results suggest that spatially random irregularities in honeycomb core have considerable effect on the natural frequencies of sandwich panels.

Free-Vibration Analysis of Sandwich Panels with Randomly Irregular Honeycomb Core

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Multistable vibration energy harvesters: Principle, progress, and perspectives

Recent focus has been given to spectro-spatial analysis of nonlinear metamaterials since they can predict interesting nonlinear phenomena not accessible by spectral analysis (i.e., dispersion relations). However, current studies are limited to a nonlinear chain with single linear resonator or linear chain with nonlinear resonator. There is no work that examines the combination of nonlinear chain with nonlinear resonators. This paper investigates the spectro-spatial properties of wave propagation through a nonlinear metamaterials consisting of nonlinear chain with multiple nonlinear local resonators. Different combinations of softening and hardening nonlinearities are examined to reveal their impact on the traveling wave features and the band structure. The method of multiple scales is used to obtain closed-form expressions for the dispersion relations. Our analytical solution is validated via the numerical simulation and results from the literature. The numerical simulation is based on spectro-spatial analysis using signal processing techniques such as spatial spectrogram, wave filtering, and contour plots of 2D Fourier transform. The spectro-spatial analysis provides a detailed information about wave distortion due to nonlinearity and classify the distortion into different features. The observations suggest that nonlinear chain with multiple nonlinear resonators can affect the waveform at all wavelength limits. Such nonlinear metamaterials are suitable for broadband vibration control and energy harvesting, as well as other applications such as acoustic switches, diodes, and rectifiers, allowing wave propagation only in a pre-defined direction.

Spectro-spatial analyses of a nonlinear metamaterial with multiple nonlinear local resonators

The planar vibrational response of a single conductor with an attached Stockbridge damper is investigated. The mathematical model accounts for the two-way coupling between the conductor and the damper, the flexural rigidity of both the damper and the conductor, and the mass of the two counterweights of the damper. Hence, the dynamic behaviors of the damper and conductor are simultaneously assessed. Both free and forced vibration analyses are implemented via the use of a finite element code developed in MATLAB. The results of the force vibration analyses show that the effectiveness of Stockbridge dampers depends on their location, mass, and excitation frequency.

Aeolian vibration of a single conductor with a Stockbridge damper

The vibration of a single-conductor transmission line with a Stockbridge damper is examined by modeling the system as a double-beam concept. The equations of motion are derived using Hamilton’s pri...

Analytical and experimental investigation of overhead transmission line vibration

The present paper deals with the nonlinear dynamics of a Stockbridge damper. The nonlinearity is from damping and the geometric stretching of the messenger. The Stockbridge damper is modeled as two cantilevered beams with tip masses. The equations of motion and boundary conditions are derived using Hamilton’s principle. The model is valid for both symmetric and asymmetric Stockbridge dampers. Explicit expressions are presented for the frequency equation, mode shapes, nonlinear frequency, and modulation equations. Experiments are conducted to validate the proposed model. [DOI: 10.1115/1.4029526]

Oumar Barry

Papers

Spectro-spatial analyses of a nonlinear metamaterial with multiple nonlinear local resonators

Aeolian vibration of a single conductor with a Stockbridge damper

Analytical and experimental investigation of overhead transmission line vibration

Nonlinear Dynamics of Stockbridge Dampers

Simultaneous energy harvesting and vibration control in a nonlinear metastructure: A spectro-spatial analysis