Research paperNonlinear energy harvester with coupled Duffing oscillators
Introduction
In vibration theory, the concept of dynamic absorbers has a long history and it was initiated in [1] as a device based on additional mass and spring attached to the main structure. The main property of such devices is the transfer of vibration energy from the main structure to the tuned vibration absorber. It was demonstrated that nonlinear vibration absorbers are more effective in vibration absorption than the linear ones since they can cover a wider frequency range. The term nonlinear energy sink (NES) was introduced in [2] for a single degree of freedom lumped-mass system with strong nonlinear stiffness and weak linear damping attached to the primary structure with liner or nonlinear properties. Due to their exceptional properties and robust design, nonlinear energy sinks have been widely used in engineering practice over the last two decades with applications ranging from vibration control and suppression to energy harvesting purposes.
Subsequently, in [3] the authors proposed many different configurations of nonlinear absorbers based on the continuous (rod, beam, plate) and discrete structures (lumped mass models) with nonlinear single and multiple degrees of freedom attachments. By using analytical, numerical, and experimental methods, they have studied the targeted energy transfer for different models of a nonlinear energy sink in coupled discrete and continuous structural vibration systems. The key concept of a nonlinear energy sink device is to transfer energy from a primary structure to the nonlinear attachment i.e. nonlinear energy sink, for the entire frequency band, especially in the resonance regime where the energy and vibration amplitudes are at the highest level [4]. This phenomenon is also known as energy pumping. Moreover, in [4] the authors proposed a wide spectrum of applications of this physical phenomenon as a concept for passive energy transfer in complex fluid flow, nonlinear acoustics, or simple mechanical models. On the other hand, the authors in [5] performed a systematic review of different approaches used in the design and analysis of nonlinear dissipative devices, known as ”nonlinear dampers”, that appear in structural vibration control practices. Based on the aforementioned, NES belongs to a subgroup of ”nonlinear dampers”, which is a more general term for nonlinear models used in vibration absorption of complex mechanical systems. The authors in [5] divided ”nonlinear dampers” into three main groups, (i) Nonlinear Energy Sinks (NESs), (ii) Particle Impact Dampers (PIDs), and Nonlinear Viscous Dampers (NVDs), and provided a detailed explanation on how damping phenomenon occurs in all three subgroups. In the following, the focus will be kept on the NES model and its application in vibration suppression and energy harvesting (EH).
Genedelman and colleagues [6], [7], [8] proposed systematic analytical and numerical approaches for the analysis of a harmonically excited NES system. By introducing the nonlinear normal mode concept, Genedelman [6] investigated the energy distribution in a highly asymmetric two-degree of freedom system, which consists of coupled linear and highly nonlinear damped oscillators employed as NES. Special attention is devoted to the mechanisms of subharmonic resonance and energy transfer to the nonlinear normal modes. Subsequently, in [7], [8] the authors performed bifurcation analyses of NES models by using the combination of the invariant manifold approach and multiple scales methods, as well as the averaging method. On the other hand, the numerical continuation technique in combination with the harmonic balance and shooting method was introduced in [9] to investigate the bifurcation and stability of periodic orbits of NES with weak nonlinearity and linear damping. They traced frequency - amplitude response curves and detected different bifurcation points such as fold, Hopf, symmetry braking, and period of doubling bifurcation, which leads to the conclusion that energy transfer is bounded i.e. it is not possible when damping and nonlinearity in weakly coupled systems have small values. In [10], the authors suggested the NES model for passive flutter control of a rigid wing with structural nonlinearity, which belongs to a special class of NESs used in aeroelasticity problems. They focused their investigation on the multi-degree of freedom NES by explaining the details of an energy transfer to NES through the phenomenon of nonlinear modal interaction. In another study [11], a normal form analytical approach is used to analyse the flutter control and suppression of limit cycle oscillations in an aeroelastic system with NES. Furthermore, in [12] the authors performed advanced numerical simulations to determine the NES limits and provide the effective control of transonic flutter. Moreover, vortex-induced vibrations are one of the most interesting problems where the concept of NES was introduced to absorb vibration energy from a primary structure, as presented in [13], [14].
It should be noted that there are two main categories of nonlinear energy harvesting devices, the first one belongs to mono-stable and the second one to bi-stable harvesters. The general difference between these two concepts is in the potential energy function of the system, where the mono-stable system has a single stable equilibrium while the bi-stable system has two stable equilibrium points separated by a potential barrier. An outstanding monograph [15] was published as a comprehensive study of energy harvesting problems based on the piezoelectric effect. They investigated a wide spectrum of EH models ranging from linear, to stochastic and nonlinear regimes of vibration, which represents a remarkable source of fundamental research and application of EH systems. Because of the unique mechanical concept, NES models can be very useful in EH [16], [17], [18], where it is possible to have simultaneous vibration suppression and energy harvesting. There are two main directions in the application of NES in combination with EH. The first one includes NES with piezoelectric effects [19] while the second one is based on the electromagnetic effect [20]. In [21], the authors suggested a novel piezoelectric EH device based on the NES concept to achieve simultaneous broadband energy harvesting from a nonlinear energy sink and vibration suppression of the primary structure. By using the numerical method and nonlinear normal form, they investigated the transient behaviour and energy transfer at 1: 1 resonance and demonstrated effective performance of NES with quasi-essentially nonlinear stiffness. By introducing the mixed multi-scale and harmonic balance methods as well as the Newton - Raphson harmonic balance method, nonlinear dynamic behaviour of a piezoelectric vibration energy harvester working in conjunction with the NES mechanism was investigated in [22]. The authors demonstrated remarkable properties of the proposed nonlinear energy sink - energy harvesting (NES-EH) model, where the energy localized branch in the multi-valued region looks very promising in terms of the energy absorbed from a primary structure and the improvement of energy conversion efficiency. Recently, some authors investigated a special class of NES-EH model based on the oscillating magnet between fixed coils, where both functions of such a device, vibration suppression and energy harvesting, are fulfilled [23], [24]. For the investigation of complex dynamic behaviour, the MATCONT software is often used to obtain frequency responses of EH models or to validate numerical results experimentally. Introducing the analytical approximation method such as the complexification-averaging method, the steady-state regime of a giant magnetostrictive harvester integrated with a nonlinear energy sink was investigated in [25] based on the previous study [20]. Moreover, by investigating the stability of periodic orbits, they detected the saddle-node bifurcation and derived the instability boundaries. On the other hand, the application of the Lyapunov function in the stability analysis with additional control can be used to extend the analysis of nonlinear discrete NES-EH systems, as given in [26], [27].
The main objective of this paper is to provide a reliable methodology to analyse the complex dynamic behaviour of a strongly nonlinear NES-EH with varying the amplitude of external base excitation. The presented NES-EH model is based on a physically realized system given in [23] exhibiting both energy harvesting and vibration suppression through a nonlinear energy sink device. Herein, numerical simulations are carried out by the incremental harmonic balance (IHB) method in combination with the continuation technique to generate the response amplitude - base amplitude and frequency-amplitude response curves (i.e. bifurcation diagrams) that reveal a unique dynamic behaviour of the NES-EH system. Moreover, to detect stable and unstable periodic solution branches, the Floquet theory is introduced in combination with IHB. The main attention is pointed to the detection of amplitude response functions for different values of excitation frequency and system parameters, where multiple periodic solutions and unstable branches of periodic responses are detected. The application of the IHB method in combination with continuation technique for determination of the response amplitude - base amplitude is the key difference compared to the previous research [23]. The focus is directed to the determination of voltage responses of load resistors, which are shown to be significantly influenced by the excitation frequency, the nonlinear stiffness parameter, and the resistance load. Further analysis demonstrates how the system parameters affect the localization of the system’s energy to the NES mass. The results obtained by the IHB method are verified against the Runge-Kutta method and also against the relevant results from the literature.
Section snippets
Nonlinear energy sink - energy harvester model
Let us consider the NES-EH device model formed as a two-mass system coupled by a nonlinear spring and a linear damper, which is subjected to external harmonic base excitation (see Fig. 1). In general, the device is based on two connected oscillators where mp represents a primary mass of the main structure (linear oscillator) while ma is the attached mass of the nonlinear absorber i.e. the nonlinear energy sink (nonlinear attachment). The terms kp and cp represent, respectively, the stiffness
The incremental harmonic balance method
The main advantages of the IHB method are the possibility for obtaining the semi-analytical solution of a system of nonlinear differential equations and an easy implementation of the continuation algorithm for tracing the responses curves. Moreover, in comparison to standard perturbation methods, there is no need to introduce a small scale parameter for systems with strong nonlinearity. In [28], the authors developed a methodology based on IHB method, Floquet theory and continuation method to
Stability of the periodic solution - Floquet theory
After finding the periodic solution in the form of Fourier series given in Eq. (9), the stability of thus obtained solution can be investigated by introducing a methodology based on the Floquet theory [31], [32]. By using the Floquet theory in for the system of nonlinear differential equations, the general form can be written aswhere is the N - dimensional displacement vector and . By inserting small perturbations Δy(τ) in a neighbourhood of
Continuation method in combination with IHB
To trace the frequency-amplitude or response amplitude - base amplitude curves, the solution procedure starts from an arbitrary initial vibration state (it can be started from some linear solution far away from the resonance state or with a small value of excitation force amplitude). Then, point-to-point calculation is performed by using the ΔY incremental equation Eq. (15) or ΔΩ incremental equation Eq. (18) to obtain the corresponding response curves, as shown in [40]. However, this solution
Parametric study
This section is divided into two main parts. In the first part, a comparative study of the results obtained in the present work against the results given by Kremer and Liu [23] is presented. Based on the IHB and continuation methodology described in previous sections, the nonlinear dynamic behaviour of the proposed NES-EH model is studied. Special attention is devoted to investigation of the periodic responses in the form of frequency-amplitude curves. The periodic orbits obtained by the IHB
Summary and conclusion
In this communication, the nonlinear dynamic behaviour of a two-degree of freedom NES-EH system subject to the periodic base excitation is investigated. The corresponding mathematical model of the system is presented in the form of two Duffing type nonlinear differential equations, where nonlinearity is introduced through a nonlinear spring. The electromagnetic phenomenon is employed for the energy harvester mechanism based on a magnet oscillating between two coils placed on the primary
Credit author statement
DK and MC developed the computational program and generated the results. SP wrote the discussions on the results. SA secured the funding, supervised the work and contributed towards the writing of the paper. All authors developed the scientific idea underpinning the paper together.
Declaration of Competing Interest
The authors declare that they have no conflict of interest.
Acknowledgements
Dr. D. Karličić and Prof. S. Adhikari were supported by the Marie Skdowska-Curie actions | Horizon 2020 - European Commission: 799201-METACTIVE. Dr. M.Cajić and S. Paunović were funded by the Serbian Ministry of Education, Science and Technological Development through the Mathematical Institute Serbian Academy of Science and Art.
References (43)
- et al.
Nonlinear dissipative devices in structural vibration control: a review
J Sound Vib
(2018) - et al.
Response regimes of linear oscillator coupled to nonlinear energy sink with harmonic forcing and frequency detuning
J Sound Vib
(2008) - et al.
Integration of a nonlinear energy sink and a giant magnetostrictive energy harvester
J Sound Vib
(2017) - et al.
A nonlinear energy sink with an energy harvester: harmonically forced responses
J Sound Vib
(2017) - et al.
A nonlinear energy sink with an energy harvester: transient responses
J Sound Vib
(2014) - et al.
Non-linear vibration of coupled duffing oscillators by an improved incremental harmonic balance method
J Sound Vib
(1995) - et al.
Periodic response predictions of beams on nonlinear and viscoelastic unilateral foundations using incremental harmonic balance method
Int J Solids Struct
(2016) - et al.
Nonlinear analysis of a parametrically excited beam with intermediate support by using multi-dimensional incremental harmonic balance method
Chaos Soliton Fract
(2016) - et al.
Nonlinear dynamics of heave motion of the sandglass-type floating body with piecewise-nonlinear, time-varying stiffness
Mar Struct
(2018) - et al.
Applications of incremental harmonic balance method combined with equivalent piecewise linearization on vibrations of nonlinear stiffness systems
J Sound Vib
(2019)
Nonlinear dynamics of a spur gear pair with time-varying stiffness and backlash based on incremental harmonic balance method
Int J Mech Sci
Inducing passive nonlinear energy sinks in vibrating systems
J Vib Acoust
Nonlinear targeted energy transfer in mechanical and structural systems, volume 156
Passive nonlinear targeted energy transfer
Philos Trans R Soc A
Transition of energy to a nonlinear localized mode in a highly asymmetric system of two oscillators
Nonlinear Dyn
Bifurcations of nonlinear normal modes of linear oscillator with strongly nonlinear damped attachment
Nonlinear Dyn
Steady-state dynamics of a linear structure weakly coupled to an essentially nonlinear oscillator
Nonlinear Dyn
Enhancing the robustness of aeroelastic instability suppression using multi-degree-of-freedom nonlinear energy sinks
AIAA J
Effectiveness of a nonlinear energy sink in the control of an aeroelastic system
Nonlinear Dyn
Passive control of transonic flutter with a nonlinear energy sink
Nonlinear Dyn
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2023, Communications in Nonlinear Science and Numerical SimulationCitation Excerpt :Compared with TMD, NES has a wider vibration suppression bandwidth [7]. The NES is connected to the main structure to absorb and dissipate its vibrational energy [8–10]. A properly designed NES has a higher energy transfer efficiency than TMD.