Fractional nonlinear energy sinks

The cubic or third-power (TP) nonlinear energy sink (NES) has been proven to be an effective method for vibration suppression, owing to the occurrence of targeted energy transfer (TET). However, TET is unable to be triggered by the low initial energy input, and thus the TP NES would get failed under low-amplitude vibration. To resolve this issue, a new type of NES with fractional nonlinearity, e.g., one-third-power (OTP) nonlinearity, is proposed. The dynamic behaviors of a linear oscillator (LO) with an OTP NES are investigated numerically, and then both the TET feature and the vibration attenuation performance are evaluated. Moreover, an analogy circuit is established, and the circuit simulations are carried out to verify the design concept of the OTP NES. It is found that the threshold for TET of the OTP NES is two orders of magnitude smaller than that of the TP NES. The parametric analysis shows that a heavier mass or a lower stiffness coefficient of the NES is beneficial to the occurrence of TET in the OTP NES system. Additionally, significant energy transfer is usually accompanied with efficient energy dissipation. Consequently, the OTP NES can realize TET under low initial input energy, which should be a promising approach for micro-vibration suppression.


Introduction
Targeted energy transfer (TET), the irreversible flow of energy from a donor to an acceptor, is an interesting effect in passive energy transfer [1][2] . Gendelman [3] first investigated the TET phenomenon of a coupled non-conservative oscillator system. Then, Vakakis and Gendelman [4] and Gendelman et al. [5] conducted in-depth theoretical studies to better understand the TET phenomenon, and Mcfarland et al. [6] verified the theoretical results using a subsequent experimental study. These studies showed that TET occurs only when the excitation exceeds a certain value (called threshold), and resonance capture (mainly 1 : 1 internal resonances) is proven to be the main causation of energy transfer. In addition, TET or a similar phenomenon also prevalently exists in other systems, such as quantum systems [7][8] , multiple oscillators arrangement [9] , and acoustic meta-materials [10][11][12] .
Generally, a two-degree-of-freedom (2-DOF) system consisting of a linear oscillator (LO) and a lightweight nonlinear energy sink (NES) is always used to show the behavior and reveal the underlying mechanism of TET [13] . It indicates that the vigorous energy exchange exists between the NES and the LO as transient internal resonance. Moreover, the NES does not have a preferential resonant frequency, which enables it to interact resonantly with numerous modes of the primary system at arbitrary frequencies [3] . Furthermore, the stable and unstable periodic orbits, bifurcations, strong motion localization, nonlinear beat phenomena and so on can be observed in the case of resonance [14] . However, the initial input energy should reach or exceed the threshold to trigger the behavior of TET. Because the energy is localized in the NES when TET happens, the threshold was first evaluated from an energetic standpoint [4] . To better understand the principle of TET in the NES system, the topological characteristics of the underlying Hamiltonian system were revealed [15] . Additionally, Vakakis et al. [16] investigated the dynamical properties of a one-dimensional chain with NESs, which could remarkably arrest the propagating waves. The dynamic characteristics of bi-stable NES systems were also studied and some fundamental insights for the NES optimal design were proposed [17][18] . Besides, the parallel NES was shown to achieve more efficient TET [19] .
The irreversible flow of energy from the host oscillator to the NES enables considerable vibration attenuation of the host oscillator. Hence, the NES has been used to resolve the issue on vibration attenuation in the fields of mechanical engineering [20][21][22][23][24][25] , aerospace engineering [26][27][28][29][30] , and civil engineering [31][32] . Furthermore, new structures or materials based on the principle of NES were also constructed to obtain desired properties of vibration suppression [33][34][35] . The distinct characteristics of vibration reduction by the NES could be referred to the comprehensive and insightful review by Ding and Chen [36] . The parameters of the NES were shown to have a significant effect on vibration attenuation and design optimization was carried out to obtain desired vibration attenuation [37][38][39][40][41][42] . The NES combined with traditional vibration control methods could also achieve better vibration attenuation [23,43] . In addition, the concept of the NES cell was proposed to improve the versatility of the NES [44] . As mentioned early, the energy irreversibly flowing from the LO to the NES is induced by resonance capture when the vibration frequency of the NES is close to the natural frequency of the LO. For the cubic or third-power (TP) NES, the vibration frequency rises as the displacement amplitude increases. A considerable amount of initial input energy is needed to achieve a sufficiently high stiffness of the TP NES system and thus to trigger the internal resonance between the LO and the NES. Therefore, the NES with cubic (or TP) nonlinearity is workable only when the initial input energy is sufficiently high.
Nevertheless, in practice, micro-vibration will have a negative impact on the performance of precision devices, such as precision machining equipment, precision measuring instruments, and spacecraft [45] . For example, in spacecraft, micro-vibration is mainly generated by reaction/momentum wheel assemblies (R/MWAs), cryo-coolers, and thrusters, which would downgrade the precision of space optical instruments [46][47] . Since the vibration amplitude is much lower in these cases, the NES with TP nonlinearity would get failed. Fortunately, Zhang et al. [48] reported that the fractional-power nonlinearity is much more sensitive to small-amplitude excitation, indicating that the vibration attenuation bandwidth of a meta-material beam with one-third-power (OTP) nonlinearity under small-amplitude excitation was three times larger than that under large-amplitude excitation. Note that in Ref. [48], the fractional stiffness was fulfilled by using the piezoelectric patches shunted with a fractional digital oscillator. Admittedly, it is hard to realize fractional stiffness by natural materials in real world, and thus the studies about fractional stiffness are rare.
Inspired by the unique inherent property of fractional-power nonlinearity, a novel NES with OTP nonlinearity (abbreviated as OTP NES) is proposed here to achieve a sufficiently large stiffness near-zero displacement, and then to trigger TET under low initial input energy. As a result, the OTP NES can be employed to pump energy from the host oscillator to the NES when the system is under micro-vibration, which enables high-efficient attenuation of micro-vibration.
The main contribution of this paper is to propose the concept of the OTP NES, and deeply explore the TET characteristics of the OTP NES system. The threshold for the appearance of TET is determined from the energy aspect through numerical computations. The structure of the paper is as follows. In Section 2, we begin with an introduction of the OTP and TP nonlinearity, and subsequently describe the dynamic model of the OTP NES system. Then, the TET characteristics of the OTP NES system under small-amplitude excitation are demonstrated in Section 3. The effects of system parameters on TET are discussed in Section 4, and the circuit simulation verification is carried out in Section 5. Finally, the main conclusions are summarized in Section 6.
2 Model of OTP NES system

OTP nonlinearity
Consider a spring with OTP nonlinearity. The restoring force F and the stiffness K with respect to the deformation x can be described as where k otp denotes the stiffness coefficient of the OTP spring with the unit N · m − 1 3 . As a counterpart, the restoring force and the stiffness of the spring with TP nonlinearity are also given as where k tp denotes the stiffness coefficient of the TP spring with the unit N · m −3 . Then, the restoring force and the stiffness of the OTP spring are compared with those of the TP spring, when both the stiffness coefficients of the OTP and TP springs are set as the same quantitative value, i.e., k otp = 10 4 N · m − 1 3 and k tp = 10 4 N · m −3 , as shown in Fig. 1. Obviously, the stiffness of the OTP spring is much more sensitive to small deformation than that of the TP spring. It can be observed that the restoring force of the OTP spring rapidly increases with the deformation while the stiffness sharply declines from infinity. Hence, the OTP spring can fulfill a sufficiently large stiffness at a quite small deformation. On the contrary, in the range of small deformation, both the restoring force and the stiffness of the TP spring increase very slowly with the deformation. Particularly, when the deformation is less than 0.1 m, the restoring force and the stiffness of the TP spring can be considered as zero compared with those of the OTP spring. Clearly, the TP spring cannot realize a sufficiently large stiffness to trigger internal resonance at a small deformation.
Moreover, for a system with the TP NES, many interesting phenomena of TET caused by the TP nonlinearity could occur under relatively large-amplitude oscillation [13] . Consequently, we posit that in the OTP NES system, TET would be realized under micro-amplitude oscillation, owing to the unique stiffness feature that enables resonance to capture under low initial input energy. In subsequent sections, we will evaluate this hypothesis and reveal the performance of TET in the OTP NES system.

Dynamic equation and energy indicator
An OTP NES system is illustrated in Fig. 2, which consists of an LO and an OTP NES, and the equations of motion can be written as where m 1 and m 2 represent the masses, x 1 and x 2 denote the displacements, c 1 and c 2 represent the damping coefficients of the LO and the NES, respectively, k 1 is the linear stiffness coefficient, and k 2 is the nonlinear stiffness coefficient. Introduce a new time scale τ and the following dimensionless parameters: The equations of motion (3) can be rewritten in a non-dimensional form as follows: where l represents the natural length of the spring k 1 . In order to directly demonstrate the effect of the damping of the NES on TET, we set ζ 1 = 0 for further analysis, and assume that the LO is excited by impact, i.e., z where v 0 is the initial velocity of the LO induced by the impact.
Generally, some energy indicators are always used to illustrate the energy transfer from the LO to the NES. First, the total instantaneous energy of the system E(τ ) is calculated, consisting of the instantaneous energy of the LO E 1 (τ ) and the NES E 2 (τ ), which can be given by Then, the ratios h i (τ ) (i = 1, 2) of the instantaneous energy of the LO and the NES to the initial total energy, and the ratio h(τ ) of the instantaneous total energy to the initial total energy are defined to represent the corresponding energy dissipation efficiency when the system shows different energy transfer behaviors, In addition, the instantaneous energy ratios h i (τ ) (i = 1, 2) of the instantaneous energy of the LO and the NES to the instantaneous total energy are also defined to clearly observe the process of energy transfer, Then, the maximum value of the instantaneous energy ratio for the NES h 2 (τ ) is defined as the maximum energy transfer ratio h 2max , which is used to assess the efficiency of energy transfer. Moreover, up to time τ , the energy dissipation rate E NES of the energy, dissipated by the damper in the NES, to the initial total energy is calculated to quantitatively evaluate the energy dissipation performance of the NES, 3 Energy transfer of OTP NES system In this section, the energy threshold of TET is determined, and then the characteristics of energy transfer of the OTP NES system are investigated using numerical computation.

Threshold of TET
The maximum energy transfer ratio h 2max and the energy dissipation rate E NES of the OTP NES system under different input energy are calculated by using the parameters λ = 0.05, ζ 1 = 0, ζ 2 = 0.004, β = 0.01, and τ = 100, as depicted in Fig. 3(a). These two types of energy indicators for the TP NES system are also presented in Fig. 3(b) as a counterpart. Obviously, both the maximum energy transfer ratio and the energy dissipation rate sharply jump up, as the initial velocity (or input energy) increases to a critical value. Such a critical initial velocity (or input energy) is defined as the threshold for TET [49] . In practice, the intensive internal resonance of the NES system is triggered once the input energy reaches the threshold, leading to considerable energy transfer, and then the system energy is efficiently dissipated by the NES. More importantly, the threshold of the OTP NES system (around v 0 = 0.042) is much lower than that of the TP NES system (around v 0 = 1.028). It implies that the OTP NES has much better performance of energy transfer than the TP NES when the system is impacted by low initial input energy.  To demonstrate the characteristics of energy transfer in the OTP NES system in detail, the initial velocity is divided into three regions according to the threshold value, i.e., the pre-TET region (I: v 0 = 0-0.042), the TET region (II: v 0 = 0.042-0.127), and the post-TET region (III: v 0 = 0.127-1), as shown in Fig. 3(a). In the pre-TET region, both the maximum energy transfer ratio and the energy dissipation rate are small, owing to weak responses induced by ultra-low initial input energy. In the TET region, significant energy transfer is accompanied by efficient energy dissipation when the initial input energy is near the threshold. However, as the initial input energy increases into the post-TET region, the energy transfer and energy dissipation are no longer consistent with each other, and the energy dissipation is remarkably higher than that of the energy transfer. This is because the internal resonance of the NES system would disappear once the initial input energy exceeds the threshold too much, leading to inefficient energy transfer. Nevertheless, the relative velocity between the LO and the NES is still considerable to achieve efficient energy dissipation.

Energy transfer and dissipation behaviors of OTP NES system
To further reveal the energy transfer and dissipation behaviors of the OTP NES system, three representative initial velocities (v 0 = 0.02, v 0 = 0.08, and v 0 = 0.2) are separately selected from the three energy regions to impact the system. The dynamic responses are calculated and illustrated in Fig. 4, where the temporal evolutions of the displacement response, the energy ratio of the LO and the NES, and the energy transfer between the LO and the NES are presented.
When the system is impacted by the initial velocity v 0 = 0.02, which is selected from the pre-TET region, the characteristics of TET are depicted in Figs. 4(a)-4(c). In such a case, the relative displacement between the LO and the NES is quite small, leading to a very large stiffness for the OTP spring (see Eq. (1)), and thus the OTP spring between the LO and the NES can be regarded as a rigid connection. Therefore, the motion of the LO and the NES quickly becomes nearly synchronous after the system is impacted, as shown in Fig. 4(a). Meanwhile, there is almost no energy dissipation due to a near-zero relative velocity (see Fig. 4(b)), and the energy transfer is also insignificant. Specifically, the NES gains only a small amount of energy in the form of kinetic energy, and the instantaneous energy ratio of the NES does not exceed 25% of the total energy, as depicted in Fig. 4(c). Consequently, in the pre-TET region, the OTP NES is unable to fulfill remarkable energy transfer and efficient energy dissipation. However, in the TET region (v 0 = 0.08), the behavior of TET is markedly distinct from the case in the pre-TET region, as illustrated in Figs. 4(d)-4(f). It can be observed that the relative motion between the NES and the LO becomes more apparent (see Fig. 4(d)), and the energy is dissipated rapidly (see Fig. 4(e)). Meanwhile, significant energy transfer and efficient energy dissipation happen simultaneously (see Fig. 4(e)). Moreover, the process of energy transfer is presented in Fig. 4(f). Obviously, significant energy transfer occurs immediately as the system is impacted by the input energy, and the transient energy transferred from the LO to the NES reaches nearly 100% of the total energy. This phenomenon gives a piece of evidence again that notable energy transfer can bring about considerable energy dissipation. The underlying prin-ciple is that significant energy transfer is associated with internal resonance, and most energy of the LO is pumped into the NES, and then the transferred energy is efficiently dissipated by the damper in the OTP NES.
When the initial input energy is further increased into the post-TET region, such as v 0 = 0.2, the characteristics of energy transfer are demonstrated in Figs. 4(g)-4(i). It can be observed that the energy of the NES slightly fluctuates at the beginning, while the energy is mostly localized in the LO and then gradually decays with time (see Fig. 4(h)). Therefore, weak energy transfer from the LO to the NES happens only at the beginning, but energy dissipation always occurs. Remind that the stiffness of the OTP spring is softened as the deformation is enhanced (see Fig. 1(b)). Hence, under high-level input energy, the relative displacement between the LO and the NES becomes quite large, and the stiffness would be very low in the initial stage after impact, leading to a lower natural frequency than that of the LO. As a result, significant energy transfer does not happen due to the absence of internal resonance under high-level input energy. Nevertheless, the relative velocity between the LO and the NES is sufficiently high so that the total energy could be dissipated by the damper, but the efficiency of energy dissipation is lower than that in the TET region.
To sum up, TET can be realized under quite low input energy by the OTP NES compared with the conventional TP NES. Especially, energy dissipation is considerable when significant energy transfer happens, and thus the vibration of the LO can be effectively attenuated under low-energy impact. In addition, another interesting phenomenon can be observed from Fig. 4. That is, after most energy of the system is dissipated, the energy of the NES starts to decrease in the case of the TET region (see Fig. 4(f)). In contrast, under high-level input energy in the post-TET region, the energy of the NES gradually increases as most energy is dissipated (see Fig. 4(i)). To further figure out such an interesting behavior of energy transfer, long-time observation is conducted in the following subsection. Figure 5 demonstrates the long-time temporal evolutions of the displacement response and the energy transfer between the LO and the NES under high-level input energy (v 0 = 0.2). As depicted in Fig. 5(a), the vibration frequencies of the LO and the NES are almost identical, but the amplitude of the NES is much higher than that of the LO. Most importantly, the energy of the system is mainly localized in the NES, as shown in Fig. 5(b). These behaviors are similar to those in the TET region. In other words, as the energy decays, the vibration amplitude declines but the stiffness of the OPT NES increases, which would trigger internal resonance and thus enable significant energy transfer. Therefore, when the OTP NES system is impacted by high-level input energy in the post-TET region, the behavior of energy transfer that happens in the TET region can be observed over time. It can also be predicted that after a sufficiently long time, the system would experience the behavior of energy transfer that appears in the pre-TET region.

Long-time observation of energy transfer under high-level input energy
To exhibit the complete picture of the behavior of energy transfer, the temporal evolutions of the responses with lighter damping (ζ 2 = 0.002) under high-level input energy are demonstrated over a sufficiently long time, as shown in Fig. 6. Obviously, the system exhibits the behavior of energy transfer of the post-TET region first (τ = 0-100), and considerable energy transfer cannot be observed. Then, significant energy transfer happens (τ = 200-300) as the energy of the system is dissipated. Furthermore, when most of the energy is dissipated, the system eventually exhibits the characteristics of energy transfer of the pre-TET region (τ = 900-1 000), and energy transfer nearly does not occur in such a case. It is attributed to the fact that the relative displacement between the LO and the NES varies and the stiffness of the OTP spring changes as the system energy decays. The internal resonance between the LO and the NES would appear or disappear as the vibration frequency of the NES is close to or far away from the natural frequency of the LO. Therefore, the OTP NES can also fulfill TET and thus suppress the vibration of the host oscillator under high-level input energy in the post-TET region. However, the time for occurrence of significant energy transfer is much longer than that under the input energy in the TET region.

Effects of system parameters on TET
As mentioned earlier, TET always happens when the internal resonance is triggered, and the behaviors of TET are thereby closely related to the system parameters, such as the mass ratio λ, the stiffness ratio β, and the damping ratio ζ 2 . The influence laws of these parameters on the energy transfer are demonstrated in Fig. 7. In these contour maps, the maximum energy transfer ratio h 2max and the energy dissipation rate E NES are plotted with respect to the system parameters and the initial velocity. The red color represents remarkable energy transfer or efficient energy dissipation, and the lower edge of the red region denotes the threshold of TET. The area marked by 'T' (dashed area) indicates the parameter region for significant energy transfer and efficient energy dissipation. Figures 7(a)-7(c) demonstrate the effects of the mass ratio λ on TET. In practice, the mass of the LO keeps unchanged, while the mass of the NES varies. In a large section of the parameter region (0.148 λ 0.5, v 0 0.019), the behavior of TET is not active due to the absence of internal resonance, as depicted in blue in Fig. 7(a), and energy dissipation is also not remarkably efficient, as demonstrated in green in Fig. 7(b). However, in a certain region (0.148 λ 0.5, v 0 0.019), significant energy transfer appears but energy dissipation is not obviously efficient. The reason is that TET happens owing to the internal resonance, but increasing the mass of the NES reduces the ability of energy dissipation. In fact, there exists a mass region that is favorable for energy transfer and energy dissipation [13] , as marked by character T in Figs. 7(a) and 7(b). In such a region, significant energy transfer accompanied by efficient energy dissipation happens.
Moreover, the threshold of TET can be determined by the lower boundary of the red region marked by 'T'. The relationship between the threshold and the mass ratio is depicted in Fig. 7(c). Apparently, the mass of the NES has a remarkable effect on the threshold. That is, TET happens only when the mass ratio is larger than 0.02, and the threshold can be substantially reduced by increasing the mass ratio of the NES. However, the red region corresponding to TET declines with the mass ratio (see Figs. 7(a)-7(b)). It implies that the increase in the mass ratio enables the OTP NES to realize TET more easily, but the region of the initial input energy for significant TET would be shrunk.
The TET behaviors of the OTP NES system under different levels of damping are depicted in Figs. 7(d)-7(f). Clearly, a red triangle-like region for remarkable energy transfer can be observed in Fig. 7(d), and efficient energy dissipation occurs in a large section of the selected parameter region, as shown in Fig. 7(e). Moreover, the threshold can be determined by the left edge of this red triangle-like region, as depicted in Fig. 7(f). It can be found that the parameter region of significant energy transfer can be broadened by increasing the damping ratio of the NES, while the damping has few effects on the threshold. The reason is that increasing the damping can enhance the energy dissipation, which allows the total energy to drop quickly and thus to trigger internal resonance and achieve significant energy transfer when the system is under somewhat large input energy. An interesting phenomenon can also be found in the region of v 0 from 0.035 to 0.47. As the damping ratio increases, the energy dissipation nearly keeps unchanged, but the energy transfer changes notably with the damping ratio. Especially, when ζ 2 is greater than 0.034, heavy damping dissipates the total energy of the system quickly, and the behavior of energy transfer thereby disappears nearly completely.
The effects of the stiffness ratio (the nonlinear stiffness coefficient of the OTP NES to the linear stiffness coefficient of the LO) on the energy transfer are depicted in Figs. 7(g)-7(i). To highlight the effect of the nonlinear stiffness coefficient of the OTP NES, the linear stiffness coefficient of the LO is fixed here. It is obvious that the behavior of TET is highly related to the nonlinear stiffness coefficient. To trigger internal resonance for TET of the OTP NES system, larger input energy is needed to drive the system to oscillate with a higher amplitude and thus to achieve the same vibration frequency, as the nonlinear stiffness coefficient increases. Consequently, the parameter region for TET can be effectively broadened by increasing the nonlinear stiffness coefficient (see Fig. 7(g)), and the threshold keeps increasing with the stiffness coefficient (see Fig. 7(i)). In addition, efficient energy dissipation can also be observed in the region marked by the character 'T', as depicted in Fig. 7(h), which happens with the occurrence of remarkable TET.
By comparing these three system parameters, the stiffness and the mass of the OTP NES have more notable effects on the threshold of TET. To realize TET under low input energy, the mass of the OPT NES should be chosen as large as possible within the allowed range, but the stiffness coefficient should be as small as possible. This can be attributed to the fact that both the heavy mass and the low stiffness coefficient can facilitate internal resonance under low initial input energy in the OTP NES system, which is contrary to that of the conventional TP NES system. Therefore, the OTP NES system can realize TET under low initial input energy and thus attenuate the micro-amplitude vibration of the host oscillator.

Circuit simulation of OTP NES system
In this section, the TET characteristics of the OTP NES system are validated by circuit simulation. First, the analog circuit of the OTP NES system is designed, and then the simulations are carried out under different initial conditions. Finally, the numerical computations are compared with the circuit simulation results to confirm the unique energy transfer behaviors of the OTP NES.
The equations of motion (see Eq. (5)) for the system with the OTP NES in Fig. 2 are rewritten as the form of state equation, Then, the analog circuit can be designed for the system with the OTP NES, as depicted in Fig. 8. The analog circuit consists of integral circuits, inverse circuits, and multipliers, where the operational amplifiers OP07H are employed, the output gain of the multipliers is set as 1 V/V, and the stabilized voltage supply is set as ±18 V to power the circuit components. Importantly, the OTP operation is realized via a serial connection of two multipliers. By reasonable selection of the parameters of electronic components, the analog circuit can simulate the system with parameters of λ = 0.05, ζ 1 = 0, ζ 2 = 0.004, and β = 0.01, which are identical to the system considered in Section 3. The parameter values of the electronic components are marked in the schematic of the analog circuit, as shown in Fig. 8. In addition, since the circuit must follow Kirchhoff's law, the initial condition of the analog circuit should be set as (z 1 , y 1 , z 2 , y 2 ) = (v 0 , v 0 , −v 0 , −v 0 ), instead of (z 1 , y 1 , z 2 , y 2 ) = (0, v 0 , 0, −v 0 ). To make fair comparison, the initial condition for numerical computation is also set as (z 1 , z ′ 1 , z 2 , z ′ 2 ) = (v 0 , v 0 , −v 0 , −v 0 ). The circuit is connected to a constant-voltage source or to the ground at the beginning of simulation and then disconnected to simulate the initial condition. Finally, the analog results are obtained by signal acquisition at points VF 1 and VF 2 .
Three representative initial conditions of v 0 = 0.02, v 0 = 0.05, and v 0 = 0.2 are selected to illustrate the energy transfer characteristics of the system under different levels of initial input energy. The comparison of energy transfer behavior evolution between the circuit simulation and numerical computation is depicted in Fig. 9. The analog and numerical results are demonstrated in the first and second rows, respectively. It can be clearly seen that energy transfer becomes more significant and then declines as the initial input energy increases. Especially, in the case of v 0 = 0.2, with massive energy dissipated, the energy transfer is getting more significant, and the energy is gradually localized in the NES. More importantly, the energy transfer behavior evolution of circuit simulation is in good agreement with the numerical computation. This validates the TET behaviors of the system with an OTP NES.

Conclusions
This paper proposes a new type of NES with OTP nonlinear stiffness to realize significant TET under low initial input energy. The model of the OTP NES system is established by   Fig. 9 Comparison of circuit simulation results (the first row) and numerical computation results (the second row). The superscripts CS and NC represent circuit simulation results and numerical results, respectively (color online) attaching an OTP NES onto an LO. The equations of motion of the OTP NES system are numerically resolved, and the dynamic responses of the displacement and energy are obtained to evaluate the behavior of TET. Some interesting phenomena induced by the unique OTP nonlinearity are observed. Furthermore, the numerical results are verified via circuit simulation. First, the stiffness feature of the OTP NES is converse to that of the conventional TP NES, and a sufficiently high stiffness can be achieved at a quite small deformation of the OTP spring in the NES. Therefore, the internal resonance and thus TET can be triggered by low initial input energy. Specifically, the threshold of TET of the OTP NES system is two orders of magnitude smaller than that of the TP NES system. Additionally, significant energy transfer is usually accompanied with efficient energy dissipation when TET happens, which should be a beneficial characteristic for micro-amplitude vibration attenuation by the OTP NES.
Moreover, the parameters of the OTP NES, such as the mass, the stiffness, and the damping, have considerable effects on the behavior of TET. The threshold of TET can be effectively lowered by increasing the mass or declining the nonlinear stiffness coefficient. Therefore, a heavier mass or a lower stiffness coefficient is beneficial for the OTP NES to realize TET when the system undergoes micro-amplitude oscillation.