1 Introduction

Targeted energy transfer (TET) is a widespread phenomenon in nature as concluded in Ref. [1]. In mechanical systems, TET is also noted as energy pumping, and is described as a large amount of one-way energy flow from a linear primary oscillator (LPO) to a designed nonlinear energy sink (NES). A properly designed NES could absorb almost most energy (94%) of the LPO and dissipate the energy under weak damping conditions, making it a very ideal vibration absorber. Gendelman et al.^[2] studied the basic dynamics of a cubic NES, and showed that 1:1 resonance capture was mainly responsible for the TET in the underlying Hamiltonian system. Vakakis and Gendelman^[3] proved that energy pumping was due to the 1:1 resonance manifold of the weakly damping system. These two series^[2-3] laid the foundation of analyses on the dynamics of NES. Lee et al.^[4] concluded that the structure of the undamped system greatly affected the dynamics of the underlying Hamiltonian system.

The configuration of cubic NES can be divided into two categories according to the structure, i.e., grounded NES and ungrounded NES. The system is excited by an impulse on an LPO, and almost all energy can be transferred to the NES if the strength of the impulse is above a certain level. That is to say, TET occurs on a certain range of input energy. Therefore, the efficiency of TET has been widely studied. Wei et al.^[5] studied the optimal TET, and improved the efficiency by a variable stiffness from the perspective of active control. The optimal stiffness changes with the input energy according to the analysis of the optimal TET. For single frequency excitation, cubic NES is unsatisfactory, while linear vibration absorber works better on the natural frequency of the LPO.

Vibro-impact (VI), as an effective measure of vibration suppression, has been studied systematically in the literature^[6]. The mechanism of VI NES is to induce the effects of LPO and NES and then dissipate the energy by incomplete elastic collision. For normal linear VI, the impact could be periodic, and the impact modes have been studied extensively to give an analytical treatment^[7-9]. However, for many kinds of broadband excitations except impulse, the response is not periodic, and thus numerical simulation is usually used to analyze the dynamics of VI NES. Lee et al.^[10] conducted a detailed study on VI NES, and found that VI NES could absorb and dissipate the energy of the LPO in a faster time scale than normal continuous NES^[2]. Karayannis et al.^[11] analyzed VI NES, as a vibration absorber, and obtained that, at most 73.9% of the energy of the LPO could be dissipated by VI NES under the given condition. That is to say, for broadband excitations, optimal designed cubic NES works better than optimal designed VI NES. However, VI NES could dissipate the system energy in a fast time scale. Therefore, this paper aims to propose a parallel NES with a VI NES and a cubic NES to combine both the advantages of the two kinds of nonlinearity.

2 TET

The system considered in this study is composed of a linear oscillator, a cubic absorber, and a VI absorber with three degrees of freedom. The configuration of this system is depicted in Fig. 1.

The dynamic equation without impact is described as follows:

(1)

where x₁, m₁, c₁, and k₁ denote the displacement, mass, damping, and stiffness of the LPO, respectively. x₂, m₂, c₂, and k₂ denote the displacement, mass, damping, and stiffness of the cubic NES, respectively. x₃ and m₃ denote the displacement and mass of the VI NES, respectively. We regard the impact as rigid body collision, and introduce the law of conservation of momentum as follows:

(2)

where |x₁-x₃|=Δ, r is the restitution coefficient of the rigid body collision, and 0 < r≤ 1.

We use the superscript ⁺ to represent a variable just after impact and the superscript ^- to represent a variable just before impact, i.e.,

(3)

For the convenience of calculations, we introduce the parameters

(4)

and the new variables

(5)

and use dots to denote variables with respect to τ. Then, we have

(6)

with which Eq. (3) can be rewritten as follows:

(7)

where |w|=Δ. To simplify the analysis, we assume that the masses of the two absorbers are equal, i.e., ε=ε₁=ε₂. Then, we carry out multiple-scale expansion as follows:

(8)

According to Refs. [8] and [12], we only use the first two orders. On the zeroth order, we have

(9)

Then, we have

(10)

Here, we introduce the zig-zag function to represent the relative variable w₀ as in Ref. [8], and then carry out a simulation to show the representation. The parameter settings are given as follows:

As a comparison, 2α arcsin (cos(τ₀-θ))/π is depicted to show the zig-zag function.

The response of the system described by Eq. (1) is calculated by MATLAB. The results are shown in Fig. 2 together with the zig-zag function. It appears that the zig-zag function sketches the rough outline of the displacement response of the relative variable w₀ in the early time of simulation. For a further analysis, one should expand the second equation of Eq. (10) with harmonics. Here, we just focus on the underlying Hamilton dynamics (without damping and the restitution coefficient r=1) of the system.

We rewrite Eq. (6) as follows:

(11)

As what has been done in Ref. [1], the impact does not change the system energy. Therefore, we can conclude the optimal stiffness through the calculation of Hamilton conservatism as follows:

(12)

When we introduce VI in the parallel NES, there is also an optimal clearance for the impact as what has been concluded for the optimal cubic stiffness. However, the state-of-the-art research on VI is limited to the periodic impact, and the optimal analysis is generally carried out with numerical simulation. Therefore, in this section, we will analyze the periodic part of the impact.

Let us reconsider Eq. (10). According to the assumption made in the zig-zag function and the saw-tooth transformation, we have that the impact occurs at

(13)

Replace Eq. (10) with Eq. (13). Then, we have

(14)

where Δ is the clearance of VI as defined before. Combining the derivation of the third equation of Eq. (10) at the impact point of time with Eq. (7) yields

(15)

Then, we put Eqs. (14) and (15) together, and obtain the basic relationship of impact at τ₀ as follows:

(16)

indicating that Δ plays a critical role in the dissipation of the amplitude of the LPO.

3 Numerical evaluation and analysis of TET 3.1 Evaluation metrics

The energy dissipation measure (E_DM) is used to evaluate the ability of NES to absorb and dissipate the energy of the LPO. It is defined as follows:

(17)

That is to say, E_DM is the ability of NES to dissipate the energy percentage of the LPO. For the parallel NES we discuss here, we need to include the energy dissipation of incomplete elastic collision into the E_DM. Then, Eq. (17) becomes

(18)

where i indicates the ith collision of the VI NES with the LPO, and the series sum gives the energy dissipated by the VI NES. The integration and series sum are infinity in time as it implies. Actually, we calculate E_DM only in the first 20-80 periods. The denominator of Eq. (18) means the strength of the input impulse on the LPO. For the arbitrary excitation F, we change the expression to

(19)

where t_i and t_d means the start time and the stop time of the external excitation. Next, we compare the E_DM results of the parallel NES and with those of the cubic NES, where the parameter settings are the same as those in Section 2, with different impulse strength X. It should be noted that the parameter settings are optimal for the normal cubic NES. We depict E_DM in Fig. 3. From the figure, we can see that there is a larger improvement for the parallel NES compared with the cubic NES. This is due to the well cooperation of the VI NES and the cubic NES. It is known that TET works well on a wide range of frequency, but it is sensitive to the input energy. TET emerges only when the input energy exceeds a certain level and the corresponding periodic orbit is excited. When the input energy increases, the efficiency of TET declines. However, the parallel NES keeps high performance in a large range of impulse X.

Here, we focus on the energy of the LPO, which is the goal of vibration suppression. If the vibration suppression mechanism is more efficient, the energy of the LPO would be more quickly absorbed and dissipated. Therefore, we propose the following new concept, named the integration of energy, to evaluate the performance for passive vibration control:

(20)

This metric is a direct way to evaluate the efficiency of NES. The integration of energy for both the parallel NES and the cubic NES is depicted in Fig. 4. The difference is too big, and logarithmic coordinates are used to describe the energy curves. If the absorber works well, the vibration would be dissipated quickly, and the integration would be small.

3.2 Analysis of TET

TET describes an intense energy exchange between LPO and NES. Therefore, we depict the energy percentage of NES along with time in Fig. 5.

The energy percentage is defined as follows:

(21)

From Fig. 5, we can see that the energy percentage curve is very special since the impact exists all the simulation time and the energy exchange for certain impact is very large. We depict the displacement response in Fig. 6. The nearly vertical lines are disordered, which means that the impact is not periodic.

In this section, we show the difference of the TET between the parallel NES and the cubic NES. This makes the energy exchange larger and earlier. Then, the energy dissipates more quickly, and the vibration suppression becomes more efficient.

3.3 Efficiency of the TET

Previously, we use the impulse to explain the TET phenomenon, and show that the VI and the cubic stiffness work well together to suppress the vibration. In this section, we will show that the good cooperation mechanism makes the parallel NES work well on many kinds of excitations. The parameter settings are the same with those in Section 2 if there is no special explanation.

Harmonic excitation is considered on the parallel NES, and the displacement response of the LPO is depicted in Fig. 7. Strongly modulated response (SMR) is found in the parallel NES. The SMR is regarded as a symbol of the TET phenomenon in the system under narrow band excitation. In Fig. 8, we depict the displacement response of the LPO in the steady time for comparison. It should be noted that the parameter settings in Figs. 7 and 8 are the same. It can be seen that the displacement of the LPO is suppressed a lot, compared with the displacement of the system without NES. Moreover, the performance of the parallel NES works better than the normal cubic NES. The pictures together show that VI and cubic stiffness work well in a good cooperation mechanism.

Compared with harmonic signals, excitations with noises are more common in industrial production processes. Here, we assume that the noises are zero mean white Gaussian noises. Then, the excitation force becomes

(22)

where ξ(t) is the Gaussian noise, and γ is the strength. We set γ=1, and f=0.01. The frequency ω₀=1 is just the same with the natural frequency of the LPO. In this case, the noise is a little larger, and the excitation can be regarded as colored noise. We show the response in a long period of time to express the evolution of all the three systems. In Fig. 9, we intercept a small portion. From the figure, we can see that the response is similar to that under pure harmonic excitation.

We depict the response of the parallel NES, the normal cubic NES, and the system without NES together in Fig. 10. The results show that the amplitudes of both the cubic NES and the parallel NES are strongly modulated. It implies that TET occurs in both the parallel NES and the cubic NES under colored noise excitations. Moreover, SMR exists all the time in the parallel NES, and is quicker than that in the cubic NES because of impact. As a result, the parallel NES works more efficiently, and absorbs most of the energy of the LPO. The amplitude is suppressed to a very low stage as concluded in Fig. 9.

We find SMR phenomena in both of the two kinds of narrow band excitations. It should be pointed out that the cubic stiffness is set the same as that in Section 2. According to the energy dependence relationship with nonlinear stiffness, there is also an optimal stiffness region for these narrow band excitations. In this section, the main purpose is to analyze the efficiency of the TET under different excitations. Therefore, the optimization problems of stiffness and clearance are not considered.

A random excitation f(t) is defined by a second-order filter as follows:

(23)

where ξ(t) is the pure white Gaussian noise, η=0.002, and ω₀²=1. The excitation f(t) can be regarded as the response of a linear oscillation system under random excitation. The response is depicted in Fig. 11. SMR phenomena are found in both the parallel NES and the normal cubic NES. The amplitude is modulated greatly in the cubic NES, and there is no sign of stability through all the simulation time. It is indicated that the TET is realized all the time in the cubic NES. Moreover, the parallel NES works better through the energy exchange of impact.

4 Conclusions

A parallel NES is proposed to combine both the advantages of VI NES and cubic stiffness NES. Both nonlinear stiffness and discontinuity nonlinearity are considered in the parallel NES. The evaluation metrics of both E_DM and energy integration are used to evaluate the performance of the parallel NES. Numerical simulation shows that the parallel NES combines both the quicker response of VI NES and the high efficiency of cubic NES. The values of E_DM are larger than those of the optimal TET in cubic NES. Besides, the high efficiency region is enlarged with respect to the input impulse strength.

There are also several problems left on the study of parallel NES, e.g., the numerical analyses of the periodic orbits and bifurcations with respect to the clearance of the impact and the optimization problems. The findings here are expected to enrich the study on energy absorption and vibration suppression.