#### Abstract

This paper presents a novel design by integrating geometrical and material nonlinear energy sink (NES) with a piezoelectric-based vibration energy harvester under shock excitation, which can realize vibration control and energy harvesting. The nonlinear spring and hysteresis behavior of the NES could reflect geometrical and material nonlinearity, respectively. Two configurations of the piezoelectric device, including the piezoelectric element embedded between the NES mass and the single-degree-of-freedom system or ground, are utilised to examine the energy dissipated by damper and hysteresis behavior of NES and the energy harvested by the piezoelectric element. Similar numerical research methods of Runge-Kutta algorithm are used to investigate the two configurations. The energy transaction measure (ETM) is adopted to examine the instantaneous energy transaction between the primary and the NES-piezoelectricity system. And it demonstrates that the dissipated and harvested energy transaction is transferred from the primary system to the NES-piezoelectricity system and the instantaneous transaction of mechanical energy occupies a major part of the energy of transaction. Both figurations could realize vibration control efficiently.

#### 1. Introduction

A nonlinear energy sink (NES) is a simple single or multi-degrees of freedom (MDOF) system, and it can dissipate primary system energy and realize vibration control through the use of a damper and a nonlinear stiffness spring. Extensive studies have investigated some methods to reveal the capability of NES and passively control the system [1–4]. Zhang et al. analysed the effectiveness of NES connected to an axially moving string and proved that NES can effectively suppress the vibration of the axially moving string with transverse wind loadings [5]. Starosvetsky and Gendelman studied a two-DOF system with NES in the resonance and proved that simultaneous targeted energy transfer from linear oscillator to a NES is feasible [6]. Yang et al. conducted a numerical study on a pipe-NES system and found that the system can efficiently transfer and dissipate the vibration energy generated by fluid movement in the pipe [7]. Luongo and Zulli considered a general, nonlinear, multi-DOF structure attached to a nonlinear oscillator with small mass and damping, and a mixed multiple scale/harmonic balance method was proposed to obtain differential equations describing the slow- and fast-flow dynamics of the entire structure [8]. Zhang et al. analysed the influence of flexible solar arrays on the passive multistrut vibration isolation platform of control moment gyroscopes for a satellite and discussed the reasonable parameters of flexible solar arrays [9]. Fang et al. presented an approach by integrating NES with giant magnetostrictive material (GMM) to realize vibration mitigation and energy harvesting. Parametric optimization was made to check how the values of NES mass, stiffness, and damper affect the performance of the integration of a NES and a GMM harvester [10]. Ebrahimzade et al. studied the performance of linear passive vibration absorbers and nonlinear passive vibration absorbers or nonlinear energy sink (NES) on the stability properties and nonlinear behaviors of an aeroelastic model [11].

The energy of vibration dissipated by a NES damper could be very large in several cases that make energy harvesting economically efficient. Given that a piezoelectric material can convert mechanical vibration into electrical energy with a simple structure, piezoelectric energy harvesting is considered as a self-power source in wireless sensor network systems [12]. Jacquelin et al. designed a piezoelectric harvester to harvest impact energy and discovered the influence of several mechanical parameters on harvesting output energy in order to optimize its performance [13]. Renno et al. presented a study of the effects of damping and electromechanical coupling on the power optimality of vibration energy harvesting. It is shown that mechanical dampers have an effect on optimal frequency ratios and optimal harvested power [14]. NES can absorb and dissipate energy through targeted energy transfers (TET) so that NES coupled with piezoelectric element could realize vibration control and energy harvesting. Ahmadabadi and Khadem utilised two configurations composed of NES and a piezoelectric element for vibration mitigation of a free-free beam subjected to shock excitation [15], and the NES they mentioned did not possess material nonlinearity which could realize vibration mitigation. Zhang et al. presented a piezoelectric energy harvesting device based on NES to achieve simultaneous broadband energy harvesting. The study shows that the apparatus has similar characteristics as NES as follows: 1 : 1 resonance, targeted energy transfer, and so forth [16].

In most cases, the actual behaviors of structures are not linear; these structures can present nonlinear behaviors in general and nonsmooth or hysteresis ones in particular (e.g., hysteresis behavior of magnetorheological fluid dampers and shape-memory alloys) [17]. The nonlinear and hysteresis behaviors of structures need to be studied to thoroughly understand the real responses of structures under external excitations. The Bouc-Wen model was proposed to describe the complex nonlinear characteristics of hysteretic systems [18, 19]. Ikhouane et al. performed an analytical study on how the parameters of the Bouc-Wen model influence the shape of the hysteresis loop, and some specific features of the hysteresis loop are defined formally [20]. Charalampakis comprehensively examined the response and dissipated energy of the Bouc-Wen model. New analytical and numerical solutions were derived through a generic model formulation without any parameter constraints [21]. Lamarque and Savadkoohi investigated time multiscale energy exchange between a main system which is of Bouc-Wen family models of hysteresis and a cubic NES, and application of the coupled NES in passive control of a main system with hysteresis behavior was demonstrated [22]. Savadkoohi et al. presented a general methodology to deal with time evolutionary energy exchanges between two oscillators with dual nonlinearities: a NES with smooth nonlinear geometrical (cubic) and nonsmooth hysteresis (Bouc-Wen) behaviors [23]. Based on the geometrical NES with material nonlinearity and the configurations composed of NES and a piezoelectric element [15], the research is made as follows.

The present study investigates a system comprised of geometrical NES with hysteresis behavior reflecting material nonlinearity and a piezoelectricity-based vibration energy harvester, which is attached to the single-degree-of-freedom primary system under shock excitation. Two configurations of the piezoelectric device, including the piezoelectric element embedded between the NES mass and the structure or ground, are studied through similar methods. The energy transaction measure (ETM) is adopted to study the instantaneous energy transaction between the primary and the NES-piezoelectricity system.

#### 2. Configuration Models

Figures 1 and 2 show the primary hysteresis system containing NES and a piezoelectric element. Figure 1 shows that the piezoelectric element is in the inset between the NES mass and ground. Figure 2 shows that the piezoelectric element is embedded between the NES mass and the primary system.

Mass is subjected to an external shock load, [15], which can be expressed in the following form: where , means Heaviside function, . The dynamic equations of motion of the two configurations are given in the following forms.

*Configuration 1*

*Configuration 2*where and are the rigid mass and NES mass, respectively; , , and are the displacements of the NES mass, the rigid mass, and the internal variable of the hysteresis model, respectively, which are all relative to the ground; and are the linear damping coefficients; is the linear spring stiffness; is the cubic nonlinear spring stiffness; and are the linear equivalent stiffness and clamped capacitance of the piezoelectric element, respectively; is the external resistance; is the voltage across the electrodes of the piezoelectric element; and is the electromechanical coupling coefficient. The hysteresis behavior is assumed to be of Bouc-Wen type, , with the following characteristics: is the internal variable of the hysteresis model, is the initial linear stiffness, the postyield stiffness is , is the ratio of the postyield stiffness () to the initial stiffness, , , and and are dimensionless Bouc-Wen parameters controlling the model behavior.

The parameters of the system are kg, kg, N/m, and Ns/m [10]. The parameters of the piezoelectric device are as follows: N/m, Ω, F, and C/m^{2} [24]. The parameters of the hysteresis model are as follows: N/m, , , m^{−2}, *γ* = 0.7 m^{−2}, and [25]. In the following numerical simulations, the initial values are given for a better energy harvesting performance: , , , and the time derivative of and are zero, while the time derivative of equals 0.4 m/s.

#### 3. Methodologies and Simulations

Runge-Kutta algorithm (ode45 function of Matlab) [26, 27] is used to investigate the nonlinear energy dissipated by the NES damper and hysteresis behavior and harvested by the piezoelectric device. The final calculation results must satisfy the relation provided in (2) and (3) to ensure that the calculation of the model is sufficiently accurate.

Equation (7) derives from (5) and (6) as follows:where and are the input and output energy up to time , is the harvested energy by the piezoelectric device, is percentage of the transient excitation energy dissipated by the piezoelectric device, and is the piezoelectric element displacement and defined by

The influence of spring stiffness and damper of NES on the percentage of harvested energy by the piezoelectric device is described by Figure 3.

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In Figure 3, different values of the spring stiffness and the damper of NES are considered for both configurations, namely, and . In the region of and , is obviously higher than others. The spring stiffness has no effect on except for the distinct boundary when is about 10^{8} N/m, while the damper is positively correlated with in both configurations.

Input energy is dissipated by the NES damper and the hysteresis behavior of the Bouc-Wen model and harvested by the piezoelectric device. The percentage of the transient excitation energy dissipated by the NES damper () and hysteresis behavior () and harvested by the piezoelectric device () is expressed in (9), (11), and (7), respectively. In consideration of improving energy harvesting, the NES damper was selected with a relative small parameter, namely, Ns/m.

From (4), the equation below could be derived:

Figures 4 and 5(a)–5(c) show the percentage of the captured energy dissipated by the NES damper and hysteresis behavior and harvested by the piezoelectric device, respectively. The curves in Figures 4 and 5(b) are fitted so that the tendency of energy dissipated by hysteresis behavior could be analysed better. Figures 4 and 5 show that the system vibration ceased in 4th second approximately. The captured shock energy dissipated by the NES damper and hysteresis behavior and harvested by the piezoelectric element amounts to about 66.63%, 10.13%, and 1.57% for configuration 1, respectively, while it is 64.28%, 12.10%, and 3.17% for configuration 2. This phenomenon illustrates that most of the vibratory energy is dissipated by the NES damper and harvested by the piezoelectric material, and a small part of the energy is dissipated by the hysteresis behavior. The values of dissipated and harvested energy by NES-piezoelectric system of the two configurations are quite close. If both configurations are feasible, configuration 2 is given the priority to be adopted because of the higher energy harvesting.

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The expression of instantaneous energy is presented below to show the energy change in the NES-piezoelectricity systems under external excitation.where is the instantaneous kinetic energy, is the instantaneous elastic potential energy included in NES and the piezoelectric element, and is the instantaneous elastic potential energy of the Bouc-Wen model. For both configurations, Figure 6 shows variation of the percentage of the instantaneous input energy captured in the NES-piezoelectricity system with different spring stiffness of NES as time goes on. The value of spring stiffness has no obvious correlation with the proportion of instantaneous input energy. And the instantaneous mechanical energy approaches 0 as time passes.

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The energy transaction measure (ETM) tremendously contributes to investigating the energy exchange [28], so it is adopted to study the instantaneous energy transaction between the primary and NES-piezoelectricity system in this paper. ETM, , between the two subsystems is defined inwhere represents the corresponding energy difference value among the subsystems. For instant, with time , is the kinetic energy in the NES; is the elastic potential energy in NES; is the dissipated energy up to time by NES, while is the elastic potential energy of hysteresis behavior; is the elastic potential energy in the piezoelectric device. ETM is favourable to the identification of the transient energy inflow or outflow between the primary and NES-piezoelectricity system. Positive ETM values indicate that the transient energy is transferred from the primary system to the NES-piezoelectricity system. Negative ETM values that indicate transient energy are transferred from the NES-piezoelectricity system to the primary system. For configuration 1, Figure 7(a) presents the mechanical energy transaction in the NES-piezoelectricity system, equal . Figure 7(b) depicts the kinetic and potential energy transactions of NES, namely, and , respectively. Figure 7(c) illustrates the potential energy transaction of the hysteresis behavior and piezoelectric device, namely, and , respectively. Figure 7(d) depicts the dissipated and harvested energy transactions in the NES-piezoelectricity system, namely, , , and .

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For configuration 2, Figure 8(a) presents the mechanical energy transaction in the NES-piezoelectricity system. Figure 8(b) depicts the kinetic and potential energy transaction of NES. Figure 8(c) illustrates the potential energy transaction of the hysteresis behavior and piezoelectric device. Figure 8(d) depicts the dissipated and harvested energy transactions in the NES-piezoelectricity system. Figures 7 and 8 illustrate that the energy in the primary structure and NES-piezoelectricity system is flowing back and forth, highlighting that the dissipated and harvested energy transaction between the primary structure and NES-piezoelectricity system is positive and the instantaneous transaction of mechanical energy between them is more significant than that of dissipated and harvested energy in both configurations.

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Comparison of the transient response (the displacement of ) of the primary system with and without NES-piezoelectricity system in both configuration 1 and configuration 2 indicates that the system can contribute to vibration control. For the two configurations, the process and effect of vibration control are similar, the amplitude of with NES-piezoelectricity system is little bigger than that without it in the first 2 seconds approximately, and after that the amplitude of with NES-piezoelectricity system rapidly becomes much smaller than that without the system. The response of primary structure with NES-piezoelectricity system approaches 0 at 4 seconds (Figure 9).

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#### 4. Conclusions

This study investigated a novel design for a system integrating NES including hysteresis behavior and an essential, strongly cubic, nonlinear stiffness with a piezoelectric-based vibration energy harvester under shock excitation to realize vibration mitigation and energy harvesting. Two piezoelectric device configurations, including the piezoelectric element embedded between the NES mass and the structure or ground, were studied through a similar numerical research method. Initially, a big part of shock energy was captured by the NES-piezoelectricity system in the form of mechanical energy. Over time, the captured shock energy was dissipated by the NES damper and the hysteresis behavior of the Bouc-Wen model and harvested by the piezoelectric device. Finally, more than half of the input energy is dissipated by the damper of NES while the energy harvested by the piezoelectric element and dissipated by hysteresis behavior is relatively small. The dissipated and harvested energy by NES-piezoelectric system of the two configurations is slightly different. ETM was applied to examine the instantaneous energy transaction between the primary and NES-piezoelectricity system**.** The instantaneous transaction of mechanical energy in the NES-piezoelectricity system was more significant than that of dissipated and harvested energy. The NES-piezoelectricity system could realize vibration control of the primary structure and its displacement approaches to zero rapidly in both configurations.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work is supported by the National Natural Science Foundation of China (Project nos. 11402151, 11572182, 51305421, and 11232009). The authors acknowledge the funding support of the Natural Science Foundation of Liaoning Province (Project no. 2015020106) and State Key Laboratory of Structural Analysis for Industrial Equipment Foundation of China (GZ15209).