Energy Harvesting and Nonlinear Dynamics of a Two-Degree of Freedom Master-Slave System Integrated with a Piezoelectric Actuator

—This study explores the dynamics of a piezoelectric energy harvester coupled with a master-slave system. The entire system is modeled as a two-degree-of-freedom system (2-DOF), along with a first-order differential equation governing the dynamics of the harvested voltage. The perturbation method is applied to derive the slow-flow modulating equations that govern the master-slave oscillation amplitudes and phases. By analyzing various response curves, the effects of different system parameters on both vibration amplitudes and harvesting voltage are investigated. The findings indicate that the system can be controlled as a vibration control or energy harvester. Optimal parameters for energy harvesting as well as for vibration control purposes are reported based on analytical investigations. The results are validated through numerical simulations with MATLAB algorithms (ODE45) using time response, phase plane, Poincaré map, and bifurcation diagram, where an excellent agreement between the numerical solutions and analytical investigations has been demonstrated.


Energy Harvesting and Nonlinear Dynamics of a Two-Degree of Freedom Master-Slave System Integrated with a Piezoelectric Actuator
Mohamed Nagah, Wedad A. Elganini, Nasser. A. Saeed Abstract-This study explores the dynamics of a piezoelectric energy harvester coupled with a master-slave system.The entire system is modeled as a two-degree-offreedom system (2-DOF), along with a first-order differential equation governing the dynamics of the harvested voltage.The perturbation method is applied to derive the slow-flow modulating equations that govern the master-slave oscillation amplitudes and phases.By analyzing various response curves, the effects of different system parameters on both vibration amplitudes and harvesting voltage are investigated.The findings indicate that the system can be controlled as a vibration control or energy harvester.
Optimal parameters for energy harvesting as well as for vibration control purposes are reported based on analytical investigations.The results are validated through numerical simulations with MATLAB algorithms (ODE45) using time response, phase plane, Poincaré map, and bifurcation diagram, where an excellent agreement between the numerical solutions and analytical investigations has been demonstrated.

I. I . INTRODUCTION
ecent advancements in remote monitoring and integrated circuit technology have led to a growing interest in selfpowered wireless low-power electronic components.These devices can be powered by harnessing energy from the surrounding environment using vibration energy harvesting technology.Piezoelectric vibration energy harvesting technology offers several excitation types for collectors and different working modes, making it a promising approach to power such devices.
Despite the significant progress made in this area and extensive reviews, there are still several challenging issues to address, such as improving the efficiency of energy harvesting and addressing power management issues to ensure continuous operation, we added a piezoelectric , RC circuit and nonlinear stiffness between master and slave system In order to enhance the versatility of the system, efforts were made to expand its functionality beyond being solely a vibration controller, allowing it to also function as an energy harvester.
. Future research directions in this field include developing novel materials with improved piezoelectric properties, designing more efficient energy harvesting systems, and developing advanced power management strategies to ensure reliable and continuous operation of such devices [1][2][3].The addition of a nonlinear coupling spring element to the vibration absorber can improve its performance by offering more design flexibility, accommodating larger amplitude vibrations, and reducing resonant peaks in the system.It is worth noting, however, that nonlinearities in the system can cause dynamic instabilities, leading to an increase in vibration amplitudes instead of their reduction, which can facilitate more energy harvesting [4,5].Combinational resonance can occur in a system due to its nonlinear behavior, where multiple frequencies interact nonlinearly, resulting in nearly periodic vibrations with significant amplitudes.Similarly, one-toone internal resonance can arise when different vibration modes exhibit a specific mathematical relationship, causing nonlinear interaction and energy exchange.Both of these situations can cause dynamic instabilities and amplify vibration amplitudes, requiring careful system design and analysis to mitigate these challenges [6].There has been a significant interest in using piezoelectric vibration-based energy harvesters as potential power sources for electronic devices, as indicated by several literature reviews on the subject [7].New and improved techniques for harvesting energy using piezoelectric materials have been explored by scientists.However, the efficient vibration energy harvesting using these methods remains an ongoing area of research.In the past, the focus was mainly on increasing power density by targeting a single linear resonance frequency [8].Although the performance of energy harvesters is generally limited to a narrow frequency range, it can be improved by utilizing a linear anti-resonance regime and optimizing the resistive load to achieve higher power output [9].The primary aim of designing energy harvesting devices is to achieve resonance at multiple frequencies, resulting in increased efficiency and power output.To improve device efficiency, R researchers have proposed an L-shaped piezoelectric structure that can resonate at two frequencies.In addition to linear resonance, nonlinearity has also been explored by researchers to enhance the performance of energy harvesting systems.By selecting suitable parameters, researchers can achieve broadband energy harvesting by enhancing the electrical power harvested at various frequencies of ambient vibrations [10].
To broaden the operational frequency bandwidth for a given drive acceleration amplitude and increase power density, an alternative approach for vibration energy harvesting is to utilize parametric resonance.However, the presence of nonlinear attachments in the harvester can lead to hysteresis behavior in the frequency response near resonance [11].For extracting energy across a wide frequency range, it is advisable to operate in the vicinity of quasi-periodic vibrations away from resonance.To achieve optimal energy efficiency in both singledegree-of-freedom (SDOF) and multi-degree-of-freedom (MDOF) mechanical systems, various techniques have been utilized [12].Multi-degree-of-freedom (MDOF) systems have a wide range of applications in various fields, including industrial machinery [13][14][15], Vehicles used for transportation [16,17], Vibration induced by wind and human movement can be harnessed to generate energy.Dual-mass or multi-mass vibration energy harvesters have been shown to have superior energy harvesting capabilities compared to their singlemass counterparts [18].Engineers have recently been concentrating on introducing nonlinearities in vibration absorbers or tuned mass damper systems to improve energy harvesting performance in engineering structures such as vehicle suspensions, tall buildings, and large flexible bridges.This technique is especially beneficial in dealing with random forces or motion excitations [19,20].
Numerous innovative designs for energy harvesting and vibration suppression (NES) have been suggested, and both experimental and analytical investigations have been conducted to explore these methods.One study investigated a new piezoelectric energy harvesting approach that utilized NES and demonstrated its ability to achieve a broad range of power output [21]. in a different study, integrated vibration suppression and piezoelectric energy harvesting were examined for a primary oscillator exposed to harmonic excitation.The study noted quasi-periodic responses near the resonance frequency, leading to decreased resonant amplitude, wider energy harvesting bandwidth, and targeted energy transfer [22,23].Two additional studies were centered on the utilization of piezoelectric materials for the suppression of mechanical vibration and energy harvesting [24,25].
The effectiveness of a vibration absorber consisting of a simple mass-spring-damper system in suppressing nonlinear vibrations in a weakly nonlinear oscillator subjected to periodic external excitation was investigated in the first study, we compare our work with [24] Based on the findings, it was determined that adding a piezoelectric and nonlinear stiffness enhance the flexibility of the system to act not only as a vibration controller but also as an energy harvester.
. In the second study, a nonlinear energy sink (NES) absorber coupled to an electric circuit through a piezoelectric mechanism was explored for periodic and quasi-periodic vibration-based energy harvesting.Both studies utilized mathematical techniques to analyze the systems and showed the efficacy of the suggested solutions.
This paper investigates a two-degree-of-freedom system coupled with a first-order differential equation to harvest electrical voltage.The perturbation method is utilized to derive the slow-flow modulating equations governing the oscillation amplitudes.We examine how different system parameters affect the vibration amplitudes and harvesting voltage through response curves, showing that the system can display either periodic or quasi-periodic motion depending on these parameters.We conduct analytical investigations to determine the optimal system parameters for energy harvesting and vibration control purposes.Additionally, we conduct numerical simulations using various techniques and find that the results obtained numerically agree very well with the analytical findings.

A. Mathematical model
The equations of motion that describe the nonlinear oscillations of the master-slave system coupled with a piezoelectric circuit as shown in Figure (1), can be expressed as follows [31,32]:  (1.c) where and are the displacement, velocity, and acceleration of the master system, and denote the displacement, velocity, and acceleration of the slave system, represents the harvesting voltage of piezoelectric circuit, and denote the master and slave masses, respectively, and are the linear damping coefficients of the master and slave systems, respectively.and are the linear and nonlinear damping coefficients of the master system, and are the linear and nonlinear damping coefficients of the slave system, and denote the excitation force amplitude and its frequency, respectively.
, and are the electro-mechanical coupling coefficients.and represent the electrical capacitance and resistance of the electrical circuit.
By introducing the dimensionless parameters and (where is a representative length) into Equations (1a)-(1c), one can obtain the following dimensionless equations of motion:

B. Perturbation Analysis
To investigate the dynamical behaviors of the nonlinear system given by Equations (2a) -(2c), an approximate solution has been sought applying multiple time scales method as follows [26,27]: ) where is the perturbation parameter, and are the fast and slow time scales of the system motion.In terms of and , the ordinary derivatives and can be expressed as follows: To make damping, nonlinearities, and primary resonance force appear in the same perturbation equations, we scale the equation parameters as follows: The solution of the homogeneous differential equation given by Equation ( 6) can be expressed as follows: where denotes the complex conjugate of the preceding terms.The coefficients ) and are unknown coefficients in at this stage of the analysis.They will be determined by eliminating the secular and small-divisor terms at the next approximation order.Inserting Equations ( 8) into Equations (7c), we get Where Inserting Equations ( 9) into Equations (7a) and (7b), we have ) 3 (2 , ˆ (11) Substituting Equation (11) into Equations (10a) and (10b), we can obtain the following solvability conditions of Equations (10): To investigate Equations (12a) and (12b), let us express and in the polar forms as follows [26,27]: where and are the steady-state amplitudes of the instantaneous displacements of the coupling piezo system, while and are the phases of the two motions.Inserting Equation ( 13) into Equations ( 12), with separating the real and imaginary parts, yields:          8), ( 9), (11), and (13) into Equation (3), we have.
The coupled master-slave-harvester system can be analyzed by solving Equation (15).Accordingly, at the steady-state oscillation of the master-slave system, we have .Setting this condition into Equations ( 15) yields the flowing nonlinear algebraic system:     17) stability, let the solution of Equation ( 17) be ( ) and suppose ( ) denote small perturbations, so one can express the following:  18) and ( 19) into Equation ( 15), one can obtain the following linearized model: The coefficients of the above Jacobian matrix are given in the appendix.Accordingly, to investigate the stability of the nonlinear system given by Equation ( 15) can be explored by checking the eigenvalues of the linear system given by Equation ( 20) [28].

A. Results
Given the obtained solution of Equation of motion (2) as given in Equation ( 16), designing the slave characteristics to function as a vibration absorber or energy harvester requires deep investigations for the system dynamics.The introduced analyses are obtained following the dimensionless system parameters: and , unless otherwise stated.Figure (2) illustrates the response curve of the master-slave system at three different values of the excitation force amplitudes .The figure shows that the oscillation of the master system is directly proportional to the excitation amplitudes.It is clear from the figure also that the harvester response curve is a difference between the master response curves and the slave response curves because piezoelectric material lies between the master and the slave.The effect of the parameter on the system response curve is investigated through three different values of (i.e., and ) as illustrated in Figure (3).By examining Figure (3), one can deduce that increasing from to shifts the response curve to the right making the master and harvester minimum vibration amplitude occur at , while decreasing from to shifts the response curve to the left to a minimum oscillation of both the master and harvester system occurring at .Accordingly, by altering the mass ratio of the master and slave , depending on the desired purpose of the presented system to function as an absorber or energy harvester, one can obtain the lowest oscillation amplitude of the master system or the maximum harvesting voltage.). it is clear from the figure that the master system oscillates with a very small amplitude at , while the slave system exhibits the maximum vibration amplitude at .In this case (i.e., ), To prevent or lessen the nonlinear oscillation of the master system, the slave system acts as an absorber, absorbing all surplus energy brought on by the excitation force .However, the same figure demonstrates that the increase of the mass ratio from to maximizes the master oscillation amplitude by using the slave system as an exciter, which also raises the harvesting voltages.therefore, the presented system can be employed to operate as an absorber or energy-harvesting device by controlling the mass ratio.The effect of the master-slave linear stiffness ratio on the system response curve has been illustrated in The effect of the nonlinear stiffness coefficients of the master system on the whole system dynamics is explored through Figure ( 6), where the system response curves at and is depicted.It is clear from the figure that the positive values of bend the response curve to the right to act as the hard spring Duffing oscillator, while the negative values of bend the response curves to respond as a soft spring Duffing oscillator having multiple solutions.To validate the accuracy of the plotted response curves in Figure ( 6), the whole system response curve is plotted again as in Figure (7) when against the numerical solution of Equation (2), where the numerical solution is distinguished as small circles.It is obvious that the analytical solution and the result have excellent concordance.given by Equations ( 16) and the numerical solution obtained by applying the ODE45 algorithm.It is clear from Figure (7) also that the masterslave-harvester system may oscillate with a nonperiodic motion at .The behavior of the proposed system for various absorber nonlinear stiffness coefficient values are investigated through Figures ( 8)- (10), where Figure (8) shows the system response curves at and .The figure shows how the magnitudes of the nonlinear stiffness coefficient of the absorber have a significant impact on the dynamical behaviors of the entire system , where the system may perform aperiodic motion on both sides of according to the sign of either positive or negative.In order to verify the response curves shown in Figure (8), The analytical solution is plotted against the numerical solution, represented by little circles given by Equation ( 16) when as shown in Figures (9a), (9b), and (9c), where it is reported that the numerical and analytical studies are in excellent agreement.However, Figures (9a), (9b), and (9c) depicted that the master-slave system has three solutions when , in which two of these solutions are stable and one is unstable.The figures clearly show that of the two stable solutions, one has a high-oscillation amplitude and the other a small-oscillation amplitude, where the initial conditions determine whether the system under consideration will respond with a high or low oscillation.Finally, the instantaneous oscillations of the considered system according to Figure (9) when at two initial sets (i.e., and ) are simulated in Figure (10) from the graphic, it is obvious that the system exhibits low oscillations at , while at , the system produces strong vibrations and hence harvests high voltage.Hence, it is possible to increase the generated power by controlling the initial conditions for a high harvesting energy.

B. Comparison study
We compared the performance of our proposed system with existing methods based on several parameters: 1. Function: this metric describes how the system act.2. System parameters: parameters that control the dynamical system behavior.
3. System components.And the results are shown in table 1.

Ref [24]
Proposed paper System components Main system and absorber ( only amass damper system with linear stiffness ) Main system and absorber ( with linear and nonlinear stiffness, piezoelectric and RC circuit)

IV. C ONCLUSIONS
The objective of this study is to examine the nonlinear dynamics of a master-slave system that is linked to a piezoelectric energy harvester.Perturbation methods are used to analyze the mathematical model of the whole system.The investigation concentrates on how different system parameters affect the vibration amplitudes and harvesting voltage.Through analytical results, the study identifies the ideal system parameters for both energy harvesting and vibration control objectives.In summary, this work presents optimal system parameters for energy harvesting and vibration control purposes, where the following concluded points can be summarized: 1. Depending on the desired goal, the slave system can be utilized as either a vibration absorber or an energy harvester by modifying the master-slave mass ratio in accordance with the excitation frequency.2. Adjusting the master-slave linear stiffness ratio allows the slave system to operate as either a vibration absorber or an energy harvester depending on the excitation frequency, in line with its intended purpose.

Changing the nonlinear stiffness coefficient of either
the master or slave system can cause instability throughout the system, leading to significant excitation amplitudes for energy harvesting, even with a small excitation force.4. By taking advantage of the inherent nonlinearity of the system, it is possible to induce strong oscillation for energy harvesting purposes using only the initial conditions.

5.
The conclusions drawn from this study emphasize the importance of modifying system parameters and initial conditions to achieve efficient energy harvesting and vibration control in the system under investigation.

Figure 1 :
Figure 1: Two-degree-of-freedom mechanical system coupled to a piezoelectric circuit.

( 5 )
Substituting Equations (3)-(5) into Equations (2a)-(2c) with equating the coefficients of the same power of , we have investigate the system dynamics at the simultaneous primary-internal resonance conditions (), let us introduce the parameters and to describe the considered resonance conditions as follows: d) By solving the above nonlinear equations in terms of the system parameter, one can obtain the bifurcation diagrams given in the following section.In addition, to investigate the stability of Equation (

Figure 2 .
Oscillation amplitudes of the master-slave-harvester system at three different values of excitation and : (a) master, (b) slave, and (c) harvester.Based on Figure (3) when , the system's instantaneous oscillation and harvesting voltage are illustrated as shown in Figure (4) at three distinct master-slave mass ratio values (i.e., when and

Figure ( 5 )Figure 3 .Figure 4 .
Figure(5).It is clear from the figure that acts like the mass ratio coefficient , but increasing from to shifts the response curve to the right, while decreasing from to shifts the response curve to the left.

Figure 5 .Figure 6 .Figure 7 .
Oscillation amplitudes of the master-slave-harvester system at three different values of and : (a) master, (b) slave, and (c) harvester.Oscillation amplitudes of the master-slave-harvester system at three different values of and : (a) master, (b) slave, and (c) harvester.Oscillation amplitudes of the master-slave-harvester versus the numerical solution (small circle) according to Figure (6) when : (a) master, (b) slave, and (c) harvester.

FunctionFigure 8 .Figure 9 .Figure 10 .
Acting only as a vibration control Not only vibration control but also energy harvester Oscillation amplitudes of the master-slave-harvester system at three different values of and : (a) master, (b) slave, and (c) harvester.Oscillation amplitudes of the master-slave-harvester versus the numerical solution (small circle) according to Figure (9) when : (a) master, (b) slave, (c) harvester.High and low harvesting voltage depending on the initial conditions according to Figure (10)) at : (a) master, (b) slave, (c) harvester, (d)master orbit plot and Poincare-map, (e) slave orbit plot and Poincare-map.