High-efficient internal resonance energy harvesting: Modelling and experimental study

Non-linear damped vibrations of a cylindrical shell subjected to the different conditions of the combinational internal resonance are investigated. Its viscous properties are described by Riemann-Liouville fractional derivative. The displacement functions are determined in terms of eigenfunctions of linear vibrations. The procedure resulting in decoupling linear parts of equations is proposed with the further utilization of the method of multiple time scales for solving nonlinear governing equations of motion, in so doing the amplitude functions are expanded into power series in terms of the small parameter and depend on different time scales. It is shown that the phenomenon of internal resonance can be very critical, since in a circular cylindrical shell the internal additive and difference combinational resonances are always present. The influence of viscosity on the energy exchange mechanism is analyzed. It is shown that each mode is characterized by its damping coefficient connected with the natural frequency by the exponential relationship with a negative fractional exponent.

Nonlinear internal resonance mechanism is exploited in piezoelectric vibration energy harvesting (PVEH) for the purpose of broadening the resonance band. Conventional linear energy harvester has narrow operating bandwidth. In this research, a buckled piezoelectric beam structure with preload under transverse excitation is investigated to demonstrate the superiority of internal resonance. The condition for 2:1 internal resonance could be established by truncating the continuum beam with geometrical nonlinearity. Integro–partial–differential equations are derived for governing transverse motion measured from a stable equilibrium position. At specific initial axial compressive force, two modes are coupled through the internal resonance interaction. For weak nonlinear perturbations, multiple scales method is used to explore the amplitude-frequency responses of the buckled beam system under primary resonance with 2:1 internal resonance. Numerical examples demonstrate that the resonance bandwidth is broadened thanks to the coexistence of softening and hardening nonlinear characteristics. Moreover, validity of the approximate analytical method is demonstrated by comparing with simulation. Furthermore, the optimal resistance is discussed with a pure resistive load. This research on the internal resonance of buckled beam provides a basis for structure design and optimization in broadband PVEH.

In vibrating structures, nonlinear effects can play a decisive role. For example, stiffening responses can occur in integral components with low damping, but softening effects with a shift of the resonance to lower values by an increase of the vibration energy can be observed. Often these systems are considered only in a characteristic nonlinear vibration mode, allowing well‐known phenomena to be investigated. However, it may be the case that more than one nonlinear mode has a significant effect on the structural response of multi‐degree‐of‐freedom systems. If these modes take on integer multiples of frequency values, we speak of an internal resonance. In this work, the phenomenon of internal resonance is explained on the basis of a two‐mass oscillator. For this purpose, nonlinear modal analysis (NMA) is used to determine the amplitude‐dependent behavior of the resonant frequencies, referred to as nonlinear modes. These nonlinear modes already display the internal resonance in the form of characteristic trajectories, which are examined and explained in more detail here. A computational framework consisting of harmonic balance and continuation is used to calculate the nonlinear mode curves. In addition, nonlinear frequency responses are determined and their trajectories and properties in the internal resonance region are shown. To further understand the oscillatory behavior time plots are considered. Finally, a possible use for the observed properties is discussed as well and current approaches in research for the use of internal resonances are pointed out.

Interplay between internal resonance and nonlinear magnetic interaction for multi-directional energy harvesting

The transformation of waste vibration energy into low-power electricity has been heavily researched to enable self-sustained wireless electronic components. Monostable and bistable nonlinear oscillators have been explored by several researchers in an effort to enhance the frequency bandwidth of operation. Linear two degree of freedom (2-DOF) configurations as well as combination of a nonlinear single-DOF harvester with a linear oscillator to constitute a nonlinear 2-DOF harvester have also been explored to develop broadband energy harvesters. In the present work, the concept of nonlinear internal resonance in a continuous frame structure is explored for broadband energy harvesting. The L-shaped beam-mass structure with quadratic nonlinearity was formerly studied in the nonlinear dynamics literature to demonstrate modal energy exchange and the saturation phenomenon when carefully tuned for two-to-one internal resonance. In the current effort, piezoelectric coupling is introduced, and electromechanical equations of the L-shaped energy harvester are employed to explore the primary resonance behaviors around the first and the second linear natural frequencies for bandwidth enhancement. Simulations using approximate analytical frequency response equations as well as time-domain numerical solutions reveal that 2-DOF configuration with quadratic and two-to-one internal resonance could extend the bandwidth enhancement capability. Both electrical power and shunted vibration frequency response curves of steady-state solutions are explored in detail. Effects of various electromechanical system parameters, such as piezoelectric coupling and load resistance, on the overall dynamics of the internal resonance energy harvesting system are reported.

Harvesting energy from nonlinear systems has been at the centre of research in the energy harvesting community. Many such proposed systems are single nonlinear harvester. While these systems show an increase in bandwidth of harvesting frequency, overall, they are not effective enough in power generation. This article studies power harvesting and frequency bandwidth characteristics of an array of harvesters. Multiple harvesters are considered with linear and nonlinear coupling between the harvesters. The phenomena of internal resonance (IR) and stochastic resonance (SR) are reported. The IR in multiple coupled nonlinear harvesters is explored using multiple-scale analysis. A parametric study is conducted to demonstrate the effect of coupling strength, frequency mistuning, innate nonlinearity and other parameters. The parametric study helped establish effective ways to increase bandwidth. Moreover, a stochastically loaded linearly coupled bistable harvester array is numerically analysed to find the effect of coupling strength and array size on the phenomenon of SR and on the system's harvesting performance. Through these studies, the potential of multiple coupled nonlinear harvesters in enhanced energy harvesting is demonstrated under both harmonic and stochastic excitation.This article is part of the theme issue 'Celebrating the 15th anniversary of the Royal Society Newton International Fellowship'.

A body hair-inspired multi-directional piezoelectric energy harvester with spatial internal resonance effect

The transformation of waste vibration energy into low-power electricity has been heavily researched over the last decade to enable self-sustained wireless electronic components. Monostable and bistable nonlinear oscillators have been explored by several research groups in an effort to enhance the frequency bandwidth of operation. Linear two-degree-of-freedom (2-DOF) configurations as well as the combination of a nonlinear single-DOF harvester with a linear oscillator to constitute a nonlinear 2-DOF harvester have also been explored to develop broadband energy harvesters. In the present work, the concept of nonlinear internal resonance in a continuous frame structure is explored for broadband energy harvesting. The L-shaped beam-mass structure with quadratic nonlinearity was formerly studied in the nonlinear dynamics literature to demonstrate modal energy exchange and the saturation phenomenon when carefully tuned for two-to-one internal resonance. In the current effort, piezoelectric coupling and an electrical load are introduced, and electromechanical equations of the L-shaped energy harvester are employed to explore primary resonance behaviors around the first and the second linear natural frequencies for bandwidth enhancement. Simulations using approximate analytical frequency response equations as well as numerical solutions reveal significant bandwidth enhancement as compared to a typical linear 2-DOF counterpart. Vibration and voltage responses are explored, and the effects of various system parameters on the overall dynamics of the internal resonance-based energy harvesting system are reported.

A frequency up-converting, wide band vibration energy harvesting (VEH) mechanism via 1:2 internal resonance is proposed for the first time in this work. By introducing a constant magnetic field to break the symmetry of the structure and introducing the secondary nonlinear coupling stiffness, the internal coupling of the first two bending modes is realized. Using the frequency up-converting effect of 1: 2 internal resonance, the vibration energy collection efficiency is improved exponentially; using the quadratic stiffness nonlinearity to realize the bilateral frequency broadening, the effective bandwidth of the vibration energy collection is broadened significantly. Numerical and experimental results indicate that the peak-splitting effect of the 1: 2 internal resonance can bidirectionally widen the relative bandwidth of vibrational energy harvesting, with a 200% increase in energy harvesting efficiency owing to frequency up-conversion.

Owing to their locally resonant mechanism, internal resonators are usually used to provide band gaps in low-frequency region for many types of periodic structures. In this study, internal resonators are used to improve the vibration attenuation ability of finite periodic tetra-chiral coating, enabling high reduction of the radiated sound power by a vibrating stiffened plate. Based on the Bloch theorem and finite element method, the band gap characteristics of tetra-chiral unit cells filled with and without internal resonators are analysed and compared to reveal the relationship between band gaps and vibration modes of such tetra-chiral unit cells. The rotational vibration of internal resonators can effectively strengthen the vibration attenuation ability of tetra-chiral lattice and extend the effective frequency range of vibration attenuation. Two tetra-chiral lattices with and without internal resonators are respectively designed and their vibration transmissibilities are measured using the hammering method. The experimental results confirm the vibration isolation effect of the internal resonators on the finite periodic tetra-chiral lattice. The tetra-chiral lattice as an acoustic coating is applied to a stiffened plate, and analysis results indicate that the internal resonators can obviously enhance the vibration attenuation ability of tetra-chiral lattice coating in the frequency range of the band gap corresponding to the rotating vibration mode of internal resonators. When the soft rubber with the internal resonators in tetra-chiral layers has gradient elastic modulus, the vibration attenuation ability and noise reduction of the tetra-chiral lattice coating are basically enhanced in the frequency range of the corresponding band gaps of tetra-chiral unit cells.

Design and optimization of quasi-zero-stiffness dual harvester-absorber system

Vortex-induced vibrations have been recently employed to capture scalable flow energy harvesters, which can attain the maximum power when the wind speed is in the lock-in region where the vortex-shedding frequency is close to the structural frequency. Nevertheless, the dynamical mechanism of the synchronization phenomenon has not been reported. To solve this critical problem, in this paper we explore a novel internal resonance to scavenge flow energy from vortex-induced vibrations, the mechanism of synchronous oscillations is introduced by the amplitude-frequency relationship and confirmed by the energy function. To show the capturing performance, an electromagnetic energy harvester with one-to-one internal resonance is proposed. Based on the harmonic balance method, the electromechanical coupling equations are decoupled, and the first order approximate harmonic responses of displacement and current are established. The modulation equations are derived, the amplitude-frequency curves of displacement and current are plotted with different detuning parameters. The advantage of the proposed one-to-one internal resonance is compared to the noninternal resonance case, the results express that the internal resonance scheme can enhance flow energy capture. The effects of physical parameters on the scavenged power are discussed. The accuracy and efficiency of the approximate analytical results are checked by numerical simulations.

A high-performance mini-generator with average power of 2 W for human motion energy harvesting and wearable electronics applications

Harmonic resonances in a continuous rotating shaft with distributed mass are discussed. The restoring force of the shaft has geometric stiffening nonlinearity due to the extension of the shaft centerline. The effect of a distributed lateral force, such as the gravity, is assumed. The possibility of the occurrences of harmonic resonances, the shapes of resonance curves, and internal resonance phenomena are investigated.

The Jeffcott Rotor used widely in the analysis of resonancies is a two-degree-of-freedom model with a disk at the midspan of a massless-shaft. Natural frequency of a forward whirling mode pf and that of a backward whirling mode pb have the relation of ps:pb=1:-1. Consequently, subharmonic resonancies of a forward and a backward whirling modes and a combination resonance occur at the same rotational speed of twice the major critical speed and internal resonance phenomena appear. In this paper, nonlinear phenomena in the neighborhood of this rotational speed are investigated. The following are clarified theoretically and experimentally. (a) The shape of resonance curves becomes far more complex than that of a single resonance. (b) A quasi-periodic motion occurs. (c) These phenomena are influenced greatly by the misalignment in assembly and misattachment of rotor position.