Chaotification in an internal-resonance-based electromagnetic vibration energy harvester

This paper explores a 2:1 internal resonance of a bistable vibration energy harvester (BVEH) to enhance the harvesting performance. Two linear natural frequencies are tuned to meet the nearly commensurate ratio, i.e., 1:2. Thus, the 2:1 internal resonance could be activated when the first mode is directly excited under the same vibration frequency as the first linear natural frequency. The amplitude-frequency responses under small vibration amplitudes are obtained through the multi-scale method. The analytic results reveal the double-jumping characteristics, which could expand the bandwidth of the BVEH. Among them, an intriguing “ flower pattern” amplitude-frequency curve is observed. Besides, the frequency spectrums are introduced to demonstrate the mode coupling and energy exchange between the first two modes. With small vibration amplitude, only intra-well responses could be obtained, and the energy exchange between modes obeys strict commensurate relation, i.e., 2:1. However, it does not obey the 2:1 ratio when the inter-well responses are activated, and the energy transfers to several low-frequency modes which results in a chaotic inter-well response. Afterward, the bifurcation diagrams and basin-of-attraction maps are explored by the numerical method to develop insights into the nonlinear responses of the BVEH. The results quantitatively prove the enhancement of the inter-well responses by internal resonance, and the chaotic inter-well response dominates the nonlinear behaviors. This paper explores the application of 2:1 internal resonance in bistable vibration energy harvester (BVEH). The BVEH combines with an inclined-spring-based bistable oscillator, an auxiliary oscillator, a linear track winded by coils, and the coils are connected with an electrical resistor. Two linear natural frequencies are tuned to meet the nearly commensurate ratio, i.e., 1:2. Thus, the 2:1 internal resonance could be activated when the first mode is directly excited. Amplitude-frequency responses under small vibration amplitudes are derived by the multi-scale method. Besides, the numerical method is employed to investigate the time-domain responses, frequency spectrums, bifurcation behaviors, and the basin-of-attraction maps of the BVEH. Intriguing nonlinear phenomena are revealed, and internal resonance has been proved to increase the occurring probability for the BVEH of exhibiting large-amplitude inter-well responses. The specific conclusions are:


Introduction
Nonlinearity is becoming more attractive and ubiquitous in terms of the dynamic characteristics compared to linearity. Unlike the linear systems, which are strongly affected by the resonance, some nonlinear systems are known as the non-resonant structures, and their dynamic behaviors could be desirable even the excitation frequency is far away from the linear natural frequencies [1]. Thus, nonlinearity has been employed in the design of mechanical equipment that requires great kinetic energy, such as actuators [2], locomotion robots [3], energy harvesters [4], etc.
The vibration energy harvester (VEH) is an electromechanical device based on a linear or nonlinear oscillator, which could convert ambient wasted vibration energy into electrical power, and the harvesting efficiency is highly dependent on the oscillator's kinetic energy. In the past two decades, energy harvesting has attracted much attention due to its convenience and cleanness, and it is an effective solution to solve the power supply problems of wireless self-sustained electrical electronics [5]. In recent researches, scholars have found that the VEHs based on nonlinearity outperform those based on linearity in terms of output displacement and harvesting efficiency. Fully utilizing the nonlinear characteristics of a structure could greatly improve the harvesting performance. Among them, the bistable nonlinearity draws much attention, and the snap-through motion could significantly improve the oscillator's kinetic energy, which contributes to high efficiency in transducing vibration energy into electrical power. Yang et al. [6] proposed a doublebeam piezo-magneto-elastic wind energy harvester which consisted of two piezoelectric beams, and both beams were supported by a prism-like bluff embedded with a permanent magnet. The bistability was induced by the magnetic repulsion force, and the harvesting power was enhanced by the bistable nonlinearity. Furthermore, employing an additional oscillator or dynamic magnifier could transfer a 1-DOF VEH to multi-DOF VEH to obtain multi-modal characteristic, which has been proven to be an effective solution to broaden the bandwidth [7][8][9]. Harne et al. [10] proposed an inclined-spring-based bistable energy harvester attached to an additional linear oscillator for harvesting enhancement. They found the additional DOF of the linear oscillator was an effective approach to enhance the VEH harvesting performance and robustness.
Internal resonance, which is one of the unique phenomena in the multi-DOF nonlinear dynamics, occurs when the linear natural frequencies of the first two vibration modes are in commensurate or nearly commensurate ratio, i.e., 1:2 or 1:3, etc. There are several significant investigations on the internal resonance：fluid-conveying pipe [11], marine riser [12], arch beams [13], composite plates [14], nanoscale rods [15], rotor systems [16], magnetic resonance force microscopy [17], and the unique phenomenon such as mode interaction was revealed by Galerkin truncation, multi-scale method, harmonic balance method, etc. Nonlinear behaviors such as bifurcation and chaos were detected as well. Wang et al. [18] investigated the 1:3 internal resonance of two coupled micro-resonators, and the frequency locking phenomenon was discovered for the first time once the coupling strength exceeded a critical value. Christopher and Jae [19] reported the multi-stable energy exchange between modes of a crystal palate, and the new insight into the complex interactions was presented by a mixed analytical-numerical method.
The internal resonance characteristics provide mode coupling and energy exchange [20]. The amplitude-frequency response of an internal-resonance-based VEH bends to both directions which could broaden the bandwidth [21,22]. As a result, internal resonance has been explored as a possible control mechanism to enhance the harvesting efficiency for the VEHs [23][24][25][26][27][28].
Recently, Chen and Jiang [29] explored the internal resonance as a possible approach to enhance the vibration energy harvesting efficiency. The amplitude-frequency responses were derived through the multiscale method, and two peaks bending to both the left and right directions were obtained. The averaged power was calculated, which outperformed the linear energy harvester under the same conditions. Xiong et al. [30] induced the unique internal resonance in a nonlinear VEH to pursue wider bandwidth. The nonlinearity was introduced by the magnetic interaction, and an additional oscillator was attached to a conventional 1-DOF VEH. Thus, the internal resonance could be induced when the first two linear frequencies were tuned to be commensurable ratio. Yang and Towfighian [31] presented a broadband internal-resonance-based energy harvester, and the nonlinearity was induced by the magnetic force. The results showed that the bandwidth of the proposed energy harvester was enlarged by 110% compared to the linear system when the internal resonance happened. Combining the nonlinear energy sink and the internal resonance, Hou et al. [32] proposed a 3:1 internal resonance system as an enhancement approach for energy harvesting. The results demonstrated the feasibility of the internal resonance characteristics. In Ref [33], Lan et al. found the dynamic instability a determinant for the internal resonance and the mode coupling. And the internal resonance could transfer the excitation energy between the first and the second modes, which could bring a large output power. Nie et al. [34] exploited the 1:2 internal resonance in an L-shaped piezoelectric energy harvester, several dynamic responses such as periodic responses, quasi-periodic responses, and chaotic responses were observed. They found the internal resonance enabled the energy harvester to harvest ambient energy more efficiently over a broader bandwidth. Inspired by the above researches, this paper investigates the 2:1 internal resonance of a 2-DOF bistable VEH (BVEH) to pursue broadband harvesting and intriguing nonlinear phenomena. The bistable nonlinearity is introduced by an one DOF inclined-spring-based bistable oscillator, and the second DOF is introduced by an auxiliary oscillator which could operate as an elastic boundary of the inclined-springbased oscillator. The two DOFs intersect vertically, which could bring strong mode interaction and energy exchange. Under small vibration excitation amplitude, the double-jumping phenomenon manifests, and the bandwidth is enlarged. The energy exchange obeys strict commensurate relation. With the increase of the vibration amplitude, large-amplitude inter-well responses occur and the harvesting efficiency could be improved. The basin-of-attraction map is presented to demonstrate that the internal resonance does enhance the occurring probability of exhibiting inter-well responses, and the chaos is dominant. Figure. 1 presents the schematic diagram of the BVEH. The BVEH combines with an inclined-springbased bistable oscillator, an auxiliary oscillator, a linear track winded by coils, and the coils are connected with an electrical resistor . The bistable oscillator has an embedded magnet mass ( ! ) and an inclined spring whose stiffness and undeformed length are ! and ! , respectively. The auxiliary oscillator comprises a linear spring with stiffness " and undeformed length " , a linear damper " , and an auxiliary mass block " . One end of the inclined spring is boned on ! , while the other end is connected to the auxiliary oscillator. There are two positions for the " needed to be emphasized. The first one is the gray translucent position where " is undeformed, and is the vertical displacement from the top of ! to the bottom of the gray translucent " position. The second one is the initial equilibrium position of " which is the yellow block in the vertical direction in figure 1. ! slides on the linear track with a linear damping constant ! , and the current could be obtained by cutting the magnetic field. When the BVEH is exposed to an acceleration vibration excitation ̈= − (Ω ), the governing equations of the BVEH could be expressed by

Governing equation
,̇+ , −̇= 0 (1c) where represents the relatively horizontal displacement from ! to the centerline, is the variation of " (relative displacement of " to the position where " is undeformed), is the gravitational acceleration, and are the acceleration vibration excitation amplitude and frequency, respectively. is the current flow through resistor , , , is the electromagnetic harvester inductance, is the electromagnetic coupling constant, and the operator ( ) is the derivative with respect to time .

Linear natural frequencies and modes
The static equilibria of the BVEH under the nonlinear restoring force could be obtained by letting the time derivatives to zero, and the satisfy the algebraic equations three static equilibrium positions could be obtained, i.e., (0,0, are the two stable static equilibrium displacements of ! and " , which are symmetrical about the centerline, i.e., bistability [2,35]. Under small vibration amplitude, the BVEH will exhibit the smallamplitude intra-well oscillations. Introducing the relative displacements from ( -, -) ,̇+ , −̇= 0 (4c) where D = − -is defined. By multivariate Taylor series expansion at ( = 0, = 0) and keeping the quadratic terms, the Eq. (4) is rewritten as In addition, as the quadratic terms dominate the nonlinearity of Eq. (5), thus, the 2:1 internal resonance could be activated when the BVEH is appropriately tuned. Considering the undamped free vibration, the first two linear natural frequencies and the corresponding two modes of the BVEH are 2.3 Model validation using the MSC Adams software The experimental prototype of the BVEH, which does not be tuned to have internal resonance characteristics, is presented in the appendix. The accuracy of the governing equations, i.e., Eq. (1), is experimentally validated. Besides, the MSC Adams software is also employed to validate the mathematical model of the BVEH which is tuned to have internal resonance characteristics. The 3D virtual prototype is established shown in Figure 2 and the simulation parameters are listed in Table 1.  Table 1 The simulation parameters parameters values Two time trajectories are presented in figure 3 to validate the accuracy of the governing equations. Two different vibration amplitudes are employed, the first one is a small amplitude with = 0.01m/s " , the second one uses a larger amplitude with = 10m/s " . ! = 0.001Ns/m, " = 0.002Ns/m, and Ω = 3Hz are set as well. It is seen that the two results obtained by different methods match well. In figure 3(a), only the low-efficiency intra-well responses are obtained when vibration amplitudes are small. With a larger amplitude, the large-amplitude inter-well responses could be activated, as shown in figure 3(b). In general, the accuracy of the mathematical model is validated, and the theoretical analyses of this paper are based on the governing equations.

Multi-scale method
Introduce two new modal coordinates ! and " , the and could be expressed as Substituting Eq. (8) into Eq. (5), and the linear parts of Eq. (5) are decoupled in the modal coordinates using the orthogonality of modes Small bookkeeping parameter is introduced to characterize the small vibration amplitude, the small displacement amplitude, the small current amplitude, the small damping, and the small electromagnetic coupling constant. Besides, the secular terms generated by the quadratic nonlinearities appear in the order " , thus, rescale the coefficients as The rescaled governing equations become ,̇+ , − ( ! + " ) = 0 (12c) The right hand of the Eq. (12) defines a nonlinear system dominated by the quadratic terms, thus, the 2:1 internal resonance could be activated when ω " = 2ω ! . Define the fast and slow time scales F = , and ! = . The multi-scale method assumes the solutions to Eq. (12) to be The derivatives with respect to the old timescale are changed to where ? ( = 1,2 …) represents the partial differential operator / 9 ( = 0,2 …). Substituting Eq. (13) and (14) into Eq. (12), and equating coefficients of F and ! lead to where stands for the complex conjugate of the preceding terms.

First primary resonance under small vibration amplitude
The detuning parameters σ F and σ ! are introduced to describe the nearness of the vibration frequency to ! and " to 2 ! , respectively. Thus, the frequency relations for the 2:1 internal resonance when the first linear frequency is directly excited are described by The complex functions ? ( ! ) ( = 1,2) could be expressed in the polar form where Λ ? and ? are real functions with respect to ! . Substituting Eq. (22) into Eq. (20) and separating the real and the imaginary parts yield For the steady-state response, Λ ? and ? ( = 1,2) should be independent of ! . Therefore " + (Ξ !2 ) " ) = 0 (25d) Eliminating ! , " , and Λ ! from Eq. (25), the amplitude-frequency response relationship in the first mode is obtained (27d) The amplitude-frequency response relationship in the second mode is According to Eq. (8) and Eq. (13), the displacement ( ) and the current ( ) amplitude could be expressed as where 9 = arctan ( /( 9 , )).

Stability analysis
The stability of the above steady-state responses could be determined via the Lyapunov theory [35]. The assumed polar form solution is not suitable for the stability analysis due to the presence of Λ ! ! Λ ! and Λ " ! Λ " . Thus, the complex amplitude is introduced by the Cartesian coordinate where ? = Λ ? ? , ? = Λ ? ? , and ? are chosen such that the solutions are not explicitly dependent on the time.
For the first primary resonance (i.e., the first linear frequency is directly excited), ! = 2σ F ! − σ ! ! and " = σ F ! . As a result, the Cartesian form of Eq. (23) can be expressed as The disturbance equation of Eq. (32) could be expressed as: where the superscript denotes the transpose of each matrix and is the Jacobian matrix of the fix points determined by Eq. (25). If all the eigenvalues of have a negative real part, then the corresponding trivial solution is asymptotically stable [35].

Parametric study
A series of parametric studies are conducted to explore the dynamic behaviors of the internalresonance-based BVEH in this section. Parameters of the BVEH are set as: ! = 0.1kg, " = 0.1kg, ! = 56N/m, " = 50N/m, = 0.05m, = 0.0495m, ! = 0.002Ns/m, " = 0.001Ns/m, , = 0.005H, , = 10Ω, = 0.5Tm, and = 0.15m/s " . The linear natural frequencies are ω ! = 2.3181Hz, ω " = 4.6343Hz, thus, the 2:1 internal resonance could be activated when the vibration frequency is appropriately selected. The two modes are P ! = 0.5499, P " = −1.8186, and the corresponding detuning parameter σ ! is -0.0128rad/s. In the amplitude-frequency response figures, the steady-state responses are presented in solid lines, while the unstable steady-state responses are illustrated in the dotted lines. Figure 4 shows the amplitude-frequency response curves of and for different vibration amplitudes . The typical double-jumping phenomenon is obtained for both and . Two separated multiple-valued ranges and two peaks are bending in two opposite directions, which could expand the harvesting bandwidth. Besides, it is seen that the unstable responses spring up with the increase of , and it may induce more chaotic responses. In a word, both and amplitude could be enlarged with a greater , as well as the bandwidth.  Figure 5 illustrates the amplitude-frequency response curves of and for different damping constant ! . It is seen that large ! leads to low-amplitude responses. Stable response is obtained in the entire vibration frequency range when ! = 0.1 Ns/m. With the decrease of ! , the double-jumping phenomenon manifests, and the peak deviates further. Furthermore, the unstable ranges appear when ! is sufficiently small, and the bandwidth could also be expanded. ! has the opposite effects on the dynamic behaviors of the BVEH as . The influence of " on the amplitude-frequency response is investigated as well. A similar trend is obtained as the ! , thus it is not presented in this paper.   figure 6(b). With a small , the amplitude-frequency response curve is on the outermost side of the "flower pattern", like the petals. With a greater , the amplitude-frequency response curve is in the center of the "flower pattern", like the stamens. The influence of , on the amplitude-frequency response of and is investigated as well, as shown in figure 7. Only two distinct curves are observed, though five different , are employed in this analysis. With a small , , i.e., , < 0.05H in figure 7, no significant difference in amplitude-frequency response of and could be obtained. When , is up to 0.5H, the response changes a lot. The amplitude is greater, and the peak bends further in the opposite direction. It could be concluded that , is negligible when it is sufficiently small.  Figure 8 shows the amplitude-frequency response curves of and for different resistor , . It is seen that , has a significant effect on the amplitude-frequency responses. Though the double-jumping phenomenon is ubiquitous, there exists a linear response when , = 1. In figure 8(b), it is apparent that the peak appears with the completely tuned vibration frequency when , = 1. The amplitude drops rapidly under other vibration frequencies. With a greater , , two peaks appear immediately, and the bandwidth is enlarged due to the internal resonance characteristic. Furthermore, numerical simulations under swept vibration excitation are investigated as well to verify the accuracy of the steady-state responses obtained by the multi-scale method. Both the forward sweep and backward sweep simulations are presented in figure 9. It is seen that the double-jumping phenomenon appears in the BVEH, which could expand the harvesting bandwidth. Besides, the swept simulation results show great agreement with the analytical results, especially the forward sweep simulation, which demonstrates the accuracy of the analytical approach.

Numerical analysis
Numerical analyses are conducted in this section to further uncover the internal resonance characteristics of the BVEH. Both the small amplitude and large amplitude of the vibration are employed in the numerical analyses. Furthermore, the bistable nonlinearity of the BVEH is activated by increasing the vibration amplitude and the large-amplitude inter-well responses are obtained by the numerical analysis under large vibration amplitude.

Frequency spectrum and time-domain response
The dynamic responses of the BVEH at specific vibration frequencies are investigated to show the harmonic components in the response spectrums. A small vibration amplitude = 0.1m/s " is employed at first, and the responses are presented in figure 10. It is seen that when the first mode is excited harmonically far away from the first linear natural frequency (i.e., Ω =1.818Hz, or 2.818Hz), only one distinct peak appears in the response spectrum, and the x amplitude is sufficiently small. However, when the first mode is excited harmonically at the first linear natural frequency (i.e., Ω =2.318Hz), an additional peak appears at nearly 2 ! (i.e., 4.682Hz), and the x amplitude is much greater. These results demonstrate that there exists strong mode coupling and energy exchanges between the first two modes when the 2:1 internal resonance is activated. The energy exchange between the first two modes obeys strict commensurate relation (i.e., in this case, 1:2), and the displacement amplitude could be increased witch could enhance the harvesting performance.
Increase the vibration amplitude to = 0.5m/s " , some intriguing phenomena could be observed, as shown in figure 11. The response spectrums and time-domain responses for the BVEH when Ω =1.818Hz, or 2.818Hz are similar to those depicted in figure 10. Only one dominant peak appears at the same frequency as the vibration frequency, and the x amplitude is small. When Ω = ! , the nonlinear response changes a lot for the BVEH. Large-amplitude inter-well response is obtained, and the x amplitude increases several times. Furthermore, there is no distinct peak in the response spectrums. Instead, a multi-frequency response is obtained and the energy transfers to different modes, which leads to chaotic inter-well response. Figure  12 shows the phase portraits and the Poincare maps when Ω = 1.818, 2.318, and 2.818Hz, respectively.
The period-1 intra-well response is obtained for both Ω = 1.818Hz and Ω = 2.818Hz, which contributes to low-efficiency. Differently, figure 12(b) presents a large-amplitude chaotic inter-well response. Two strange attractors at the two equilibria are observed, and plenty of irregular dispersed points are depicted in the Poincare map. ! oscillates around the two equilibria in a disordered way but with a clear boundary. The displacement amplitude increases multiple times compared with the intra-well responses, which has a positive impact on the energy harvesting performance of the BVEH.

Bifurcation and basin-of-attraction
To develop the insights into the nonlinear dynamic responses, the bifurcation diagrams of the BVEH under specific vibration frequencies versus are investigated, shown in figure 13. ! = 0.0002Ns/m and " = 0.0001Ns/m are employed. It is seen that greater might lead to inter-well responses when Ω = 1.818 and 2.318 Hz. And the vibration amplitude needed when Ω = 2.318 Hz is smaller, which demonstrates the internal resonance could increase the occurring probability for the large-amplitude interwell responses. Besides, the inter-well responses could not be activated for the BVEH under a higher offresonance vibration frequency, i.e., Ω = 2.818Hz. The basin-of-attraction maps of the BVEH are explored to quantitatively investigate the inter-well response occurring probability increment by the internal resonance. = 0.5m/s " , ! = 0.0002Ns/m and " = 0.0001Ns/m are employed. Figure 14 shows the basin-of-attraction maps under specific vibration frequencies, i.e., Ω = 1.818, 2.318, and 2.818Hz. The red, yellow, and blue squares in this figure represent different initial conditions of ! , from where ! could realize period-1 inter-well, chaotic inter-well, and intra-well oscillation, respectively. Extremely different phenomena are depicted in figure 14. It is seen that the inter-well responses could not be activated when Ω = 2.818Hz no matter what the initial conditions the ! are chosen, and a complete blue-banded region is observed in figure 14(c). When Ω = 1.818Hz, the period-1 and chaotic inter-well responses emerge with the occurring probabilities 8.76% and 0.61%. The intra-well response dominates with the occurring probability 90.62%. Thus, the BVEH might be inefficient when excited under an off-resonant vibration frequency. Furthermore, the chaotic inter-well response is ubiquitous when the BVEH is excited by a vibration frequency the same as the first linear natural frequency. The occurring probability of the chaotic inter-well response is 56.74%, which might improve the harvesting performance significantly.

Conclusion
This paper explores the application of 2:1 internal resonance in bistable vibration energy harvester (BVEH). The BVEH combines with an inclined-spring-based bistable oscillator, an auxiliary oscillator, a linear track winded by coils, and the coils are connected with an electrical resistor. Two linear natural frequencies are tuned to meet the nearly commensurate ratio, i.e., 1:2. Thus, the 2:1 internal resonance could be activated when the first mode is directly excited. Amplitude-frequency responses under small vibration amplitudes are derived by the multi-scale method. Besides, the numerical method is employed to investigate the time-domain responses, frequency spectrums, bifurcation behaviors, and the basin-ofattraction maps of the BVEH. Intriguing nonlinear phenomena are revealed, and internal resonance has been proved to increase the occurring probability for the BVEH of exhibiting large-amplitude inter-well responses. The specific conclusions are: (1) The double-jumping phenomenon could be obtained in all the amplitude-frequency curves, two peaks bend in opposite directions, which expands the bandwidth of the BVEH. Among them, an intriguing "flower pattern" amplitude-frequency curve is observed. With a small resistor, the BVEH exhibits linear behavior, and the peak appears at the completely tuned vibration frequency. (2) The frequency spectrums are presented to show the mode coupling and energy exchanges between the first two modes. The x and I amplitude of the BVEH is much greater when excited at the first linear natural frequency than other off-resonance excitation frequencies. This result demonstrates that the harvesting performance of the BVEH could be enhanced by the internal resonance. (3) With small vibration amplitude, the BVEH exhibits intra-well responses, and the energy exchange between modes obeys strict commensurate relation, i.e., 2:1. Increase the vibration amplitude, and the inter-well responses are activated. The energy transfers to several low-frequency modes, and it does not obey the 2:1 ratio. (4) The bifurcation diagrams and basin-of-attraction maps demonstrate the BVEH is easy to realize inter-well responses when the internal resonance is activated under large vibration amplitude. Furthermore, the chaotic inter-well response dominates the nonlinear behaviors, and the occurring probability is 56.74%.

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Research involving Human Participants and/or Animals
This work is about the mechanical engineering. This research doesn't involve any human participants or animals.

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Appendix: Experimental validation
An experimental prototype shown in figure 15 is employed to validate the governing equations of the BVEH. The experimental parameters are: ! ≈ 0.5kg, " ≈ 0.066kg, ! ≈ 2000N/m, " ≈ 980N/m, ≈ 0.109m, ≈ 0.1025m. Slowly swept base acceleration excitation from 1Hz to 8Hz at a rate of 0.025Hz/s is generated by the shaker, and two vibration excitation amplitudes are employed which are = 1m/s " and = 3.53m/s " , respectively. The acceleration of ! is measured by the accelerometer, and the RMS accelerations are presented in figure 16. It is seen that the BVEH could only realize small-amplitude intra-well oscillations when = 1m/s " . However, the large-amplitude inter-well oscillations are obtained when a greater vibration excitation amplitude = 3.53m/s " is employed. The RMS acceleration amplitude increases multi-times when the BVEH realizes inter-well oscillations. Besides, it could be concluded that the experimental results are similar to the numerical results. The bandwidth and RMS acceleration amplitude of the experimental and numerical results are in great agreement. In conclusion, the accuracy of the governing equations are validated, the results presented in this paper are reliable. Fig. 15 Experimental prototype. 1-linear track; 2-embedded magnet " ; 3-inclined spring " ; 4-frame; 5accelerators; 6-linear spring ! ; 7-auxiliary mass block ! ; 8-shaker.