# Dynamics of Nonlinear Primary Oscillator with Nonlinear Energy Sink under Harmonic Excitation: Effects of Nonlinear Stiffness

Authors: Min Sun; Jianen Chen
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

The dynamics of a system consisting of a nonlinear primary oscillator, subjected to a harmonic external force, and a nonlinear energy sink (NES) are investigated. The analytical solutions for the steady-state responses are obtained by the complexification-averaging method and the analytical model is confirmed by numerical simulations. The results indicate that the introduction of the NES can effectively suppress the vibrations of the primary oscillator. However, as the excitation amplitude increased, the NES may lose its efficiency within certain frequency range due to the appearance of the high response branches. Following the results analysis, it is concluded that this failure can be eliminated by reducing the nonlinear stiffness of the NES properly. The effects of nonlinear stiffness of the primary oscillator on the corresponding responses are also studied. The increase in this nonlinear stiffness can reduce the response amplitude and alter the frequency band where the high branches exist.


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## Body

## 1. Introduction

The nonlinear energy sink, consisting of a small mass and a pure nonlinear spring, constitutes an effective solution for vibration suppression over a broad frequency range [1–3]. It can result in a one-way irreversible energy transfer from the primary system to the NES. Compared to traditional linear and weakly nonlinear absorbers, the NES has the higher suppression efficiency under many conditions [4, 5].The targeted energy transfer (TET) in mechanical systems is one of the attractive issues in the past decades. The introduction of the strong nonlinearity to the primary system can lead to complicated dynamic responses. The mechanisms and phenomena of the nonlinear TET, such as the transient and sustained resonance captures, energy localization, and weakly and strongly modulation responses, were of high concern to researchers [6–8]. Applications of the NES for the vibration suppression in various structures were reported in several papers. Theoretical and experimental studies indicated that the vibrations of beams [9, 10], plates [11, 12], circular cylinders [13], and stay-cables [14] can be effectively suppressed by the NES attachment. With the deepening of research, NES has been applied increasingly to suppress vibration of complicated engineering systems such as rotor-blisk-journal bearing system [15], helicopter [16], and whole-spacecraft [17].The investigation on energy transfer efficiency of the NES when the primary system is excited by harmonic load is an indispensable part of applications for this absorber [18, 19]. Following a series of optimization, the essentially nonlinear absorber exhibited apparent advantages over linear or weakly nonlinear counterparts. However, a main drawback in NES exists: several unwanted additional branches of response may become apparent under certain conditions. The strategy which can eliminate these branches is a key issue for robust increase of this absorber [20, 21]. In addition, in the vast majority of cases, the controlled primary systems were linear models. These researches contributed significantly to the characteristics of the nonlinear TET. However, many engineering structures present nonlinear characters, and their nonlinear phenomena must be considered [22–25]. The dynamics of the TET system where the nonlinear factors of the primary system were taken into account had been paid more attentions [26–29].In the current study, the steady-state responses of a nonlinear primary oscillator coupled to a NES are investigated by the complexification-averaging technique. The nonlinear frequency responses of the system under various excitation amplitudes are considered. The nonlinear stiffness of the NES is adjusted to attempt to eliminate these unwanted branches. Also, the effects of the nonlinear stiffness of both the NES and primary oscillator on the vibration suppression are analyzed and compared.


## 2. Dynamic System

A nonlinear primary oscillator with a single degree of freedom essentially nonlinear attachment is presented in Figure1. A harmonic external force is imposed on the primary oscillator. The dynamic equation for the integrated system is derived as follows:(1a)w¨+γ1w˙+Kw+k1w3+k2w-v3+γ2w˙-v˙=fcosΩt,(1b)εv¨+k2v-w3+γ2v˙-w˙=0,where K, k1, and γ1 are the linear stiffness, nonlinear stiffness, and linear damping of the primary oscillator, respectively. k2 and γ2 are the nonlinear stiffness and linear damping of the NES. f and Ω represent the excitation amplitude and frequency, respectively. ε is the mass ratio between the NES and the primary oscillator and 0<ε≪1. For convenience, we let M=1.Figure 1
Nonlinear primary oscillator with a nonlinear energy sink.The complexification-averaging technique is utilized to obtain the slow-flow model of the integrated system, whereas the following transformations are introduced:(2)ut=wt-vt,w˙+iΩw=α1eiΩt,w˙-iΩw=α-1e-iΩt,u˙+iΩu=α2eiΩt,u˙-iΩu=α-2e-iΩt,where α-n is the conjugate of the αn,n=1,2,i=-1. Based on expression (2), we have(3)w=α1eiΩt-α-1e-iΩt2iΩ,w˙=α1eiΩt+α-1e-iΩt2,w¨=12α˙1eiΩt+iΩα1eiΩt+α-˙1e-iΩt-iΩα-1e-iΩt,u=α2eiΩt-α-2e-iΩt2iΩ,u˙=α2eiΩt+α-2e-iΩt2,u¨=12α˙2eiΩt+iΩα2eiΩt+α-˙2e-iΩt-iΩα-2e-iΩt.Substituting (3) into (1a) and (1b) and retaining the slow-flow parts, we obtain(4a)α˙1+iΩα1+γ1α1+Kα1iΩ+3k1α12α-14iΩ3+3k2α22α-24iΩ3+γ2α2=f,(4b)εα˙1+iΩα1-α˙2-iΩα2-3k2α22α-24iΩ3-γ2α2=0.Letting,(5)α1=a1+ib1,α2=a2+ib2.We substitute (5) into (4a) and (4b) and separate real and imaginary parts, yielding(6a)a˙1=-a1γ1+b1Ω-Kb1Ω-3k2b2a22+b224Ω3-γ2a2-3k1b1a12+b124Ω3+f,(6b)b˙1=-a1Ω-b1γ1+Ka1Ω+3k2a2a22+b224Ω3-γ2b2+3k1a1a12+b124Ω3,(6c)εa˙2-εa˙1=εΩb2-εΩb1-3k2b2a22+b224Ω3-γ2a2,(6d)εb˙2-εb˙1=εΩa1-εΩa2+3k2a2a22+b224Ω3-γ2b2.The steady-state responses of the system are investigated through lettinga˙1=0, b˙1=0, a˙2=0, b˙2=0. Solving the resulting nonlinear algebraic equations, the response amplitude of the primary oscillator and the relative motion between two oscillators are obtained and the expressions are, respectively,(7)w=a12+b12Ω,w-v=a22+b22Ω.In order for the stability of the steady-state solutions to be analyzed, the small perturbationsδnn=1,2,3,4 are introduced:(8)a1=a10+δ1,b1=b10+δ2,a2=a20+δ3,b2=b20+δ4,where a10, b10, a20, b20 are the steady-state solutions of the system. Substituting (8) into (6a), (6b), (6c), and (6d) yields(9a)δ˙1=-γ1δ1+Ωδ2-KΩδ2-3k24Ω3a202+3b202δ4+2a20b20δ3-γ2δ3-3k14Ω3a102+3b102δ2+2a10b10δ1,(9b)δ˙2=-γ1δ2-Ωδ1+KΩδ1+3k24Ω33a202+b202δ3+2a20b20δ4-γ2δ4+3k14Ω33a102+b102δ1+2a10b10δ2,(9c)δ˙3=Ωδ4-Ωδ2-3k24εΩ3a202+3b202δ4+2a20b20δ3-γ2δ3ε+δ˙1,(9d)δ˙4=Ωδ1-Ωδ3+3k24εΩ33a202+b202δ3+2a20b20δ4-γ2δ4ε+δ˙2.The coefficient matrix of (9a), (9b), (9c), and (9d) is presented as follows; the eigenvalues of the matrix can be used to determine the stability of the solutions:(10)J=J11J12J13J14J21J22J23J24J31J32J33J34J41J42J43J44,where(11)J11=-γ1-3k1a10b102Ω3,J12=Ω-KΩ-3k1a102+3b1024Ω3,J13=-γ2-3k2a20b202Ω3,J14=-3k2a202+3b2024Ω3,J21=-Ω+KΩ+3k13a102+b1024Ω3,J22=-γ1+3k1a10b102Ω3,J23=3k23a202+b2024Ω3,J24=-γ2+3k2a20b202Ω3,J31=-γ1-3k1a10b102Ω3,J32=-KΩ-3k1a102+3b1024Ω3,J33=-γ2ε-3k2a20b202εΩ3-γ2-3k2a20b202Ω3,J34=Ω-3k2a202+3b2024εΩ3-3k2a202+3b2024Ω3,J41=KΩ+3k13a102+b1024Ω3,J42=-γ1+3k1a10b102Ω3,J43=-Ω+3k23a202+b2024εΩ3+3k23a202+b2024Ω3,J44=-γ2ε+3k2a20b202εΩ3-γ2+3k2a20b202Ω3.


## 3. Validation

The numerical results are utilized for the predictions confirmation of the analytical model. The Runge-Kutta method is directly attacking (1a) and (1b) to obtain the numerical results. It is demonstrated in Figure 2(a) that the results obtained from these two methods agree well. In Figure 2(a), squares and circles represent stable and unstable solutions obtained from complexification-averaging technique, respectively. Pluses represent numerical solutions obtained from Runge-Kutta method. Moreover, the stability of the analytical results is also presented in this figure. In order for the general applicability of the results to be ensured, all parameters in (1a) and (1b) are dimensionless values. The stable responses when Ω=14.8 and Ω=15.7 as well as the unstable responses when Ω=15.1 and Ω=15.4 are shown in Figures 2(b)–2(e), respectively. The unstable responses demonstrate a kind of relaxation oscillation, usually referred to as strong modulation response. These parameters are selected, respectively, as K=225, k1=50, k2=700, γ1=0.1, γ2=0.2, ε=0.1, f=1.Figure 2
Analytical model validation.
(a) (b) (c) (d) (e)


## 4. Results and Discussions

### 4.1. Effects of NES Nonlinear Stiffness

The vibration suppression efficiency is of most concern. In this section, the efficiency is investigated under various harmonic excitation amplitudes by the analytical method, whereas all parameters except the nonlinear stiffnessk2 are selected as the parameters in Section 3. Figure 3 illustrates the nonlinear frequency responses of the primary oscillator without the NES and with various NESs under f=1. When the nonlinear stiffness k2 is selected as 100, the good vibration suppression effect is achieved, the value of the resonance peak is reduced exceeding three times with respect to that of the primary oscillator without the NES attached. In this case, the responses in the entire frequency band are stable. When the nonlinear stiffness k2 is increased to 400, the unstable responses appear approximately at Ω=15 and the suppression efficiency of this NES is enhanced compared to that of the NES with k2=100. Generally, the weakly/strongly modulation responses are preferable for vibration suppression. When k2=700, the efficiency is further enhanced and the frequency band where the unstable response occurs is broadened. When k2=1000, two high branches of the frequency response curve appear in the frequency band 13.4,14, one is stable and the other is unstable. The appearance of the high stable branch is interpreted that the NES intensifies the vibration of the primary oscillator in this frequency band. When k2=1300, compared to those demonstrated in Figure 3(e), the frequency band where the high stable branch exists is broadened and the maximum response amplitude is larger.Figure 3
Frequency responses of the primary oscillator with various NESs asf=1 and k1=50.
(a) (b) (c) (d) (e) (f)The frequency responses of the primary oscillator under the harmonic excitation amplitudef=3 are presented in Figure 4. When k2=50, the unstable response appears approximately at Ω=15 and a large amount of vibration energy is dissipated. When k2=150 and k2=250, the high branches appear, whereas the frequency band where the high stable branch appears to be broader as the nonlinear stiffness increased. When the excitation amplitude f=5, the variations in the frequency response along with the nonlinear stiffness k2 are the same as those demonstrated in Figures 3 and 4. It is noted that the high branches appear as k2=50 in this case. Comparing Figures 3–5, it is discovered that, prior to the appearance of the high response branches, the unstable response occurs approximately at the linear natural frequency of the primary oscillator firstly. In addition, as the excitation amplitude increased, the lower stiffness should be selected to prevent the appearance of the high branches of the frequency response.Figure 4
Frequency responses of the primary oscillator with various NESs asf=3 and k1=50.
(a) (b) (c) (d)Figure 5
Frequency responses of the primary oscillator with various NESs asf=5 and k1=50.
(a) (b) (c) (d)Then, the bifurcations for the primary oscillator via the nonlinear stiffness of the NES under a fixed excitation frequency and various excitation amplitudes are studied. WhenΩ=15, the response amplitude always remains at the relatively low value as f=2 in the studied value range of the nonlinear stiffness. As the excitation amplitude increased, the high branches will appear at certain stiffness. For example, when f=3, the response amplitude decreases with increasing the nonlinear stiffness k2 at first, and the response of the primary oscillator becomes unstable as k2=93. Next, the high branches of the response amplitude appear until the nonlinear stiffness k2=147. Comparing the four subfigures in Figure 6, it is worth being noted that the larger the excitation amplitude, the lower the stiffness which corresponds to the bifurcation point. Figure 7 shows that, when Ω=15.5, the response amplitude always remains the relatively lower value as f=2; however, a bifurcation occurs as the nonlinear stiffness k2=93 and the response becomes unstable after the nonlinear stiffness k2=132. Similarly, the stiffness which corresponds to the bifurcation point is reduced as the excitation amplitude increased.Figure 6
Bifurcations of primary oscillator asΩ=15.
(a) (b) (c) (d)Figure 7
Bifurcations of primary oscillator asΩ=15.5.
(a) (b) (c) (d)


### 4.2. Effects of Primary Oscillator Nonlinear Stiffness

The effects of the primary oscillator nonlinear stiffness on the frequency response are investigated. All parameters exceptk1 and k2 are the same as those in Figure 4. Figure 8 is depicted as k1=0 (i.e., the primary oscillator is linear). The change rates of the response amplitude, within the high stable branches in Figures 8(c) and 8(d), are quite different from the counterparts when k1≠0. Comparing the responses of the primary oscillator without the NES, which are demonstrated in Figures 4(a), 8(a), 9(a), and 10(a), it is indicated that the value of the resonance peak decreases and the frequency which corresponds to the resonance peak increases with the raise of the nonlinear stiffness k1. It can also be concluded by comparing the rest figures in Figures 4, 9, and 10 that the integrated systems with the various values of k1 and the same value of k2 have the similar response. Similar to the effects of the nonlinear stiffness k1 on the dynamics of the primary oscillator without the NES, an increase in the nonlinear stiffness k1 brings about the decrease of the unwanted branches when the NES is coupled to the primary oscillator. In addition, the nonlinear stiffness k1 has a small effect on the frequency range of the high branches of the frequency response.Figure 8
Frequency responses of the linear primary oscillator with various NESs asf=3.
(a) (b) (c) (d)Figure 9
Frequency responses of the primary oscillator with various NESs ask1=200 and f=3.
(a) (b) (c) (d)Figure 10
Frequency responses of the primary oscillator with various NESs ask1=350 and f=3.
(a) (b) (c) (d)


## 4.1. Effects of NES Nonlinear Stiffness

The vibration suppression efficiency is of most concern. In this section, the efficiency is investigated under various harmonic excitation amplitudes by the analytical method, whereas all parameters except the nonlinear stiffnessk2 are selected as the parameters in Section 3. Figure 3 illustrates the nonlinear frequency responses of the primary oscillator without the NES and with various NESs under f=1. When the nonlinear stiffness k2 is selected as 100, the good vibration suppression effect is achieved, the value of the resonance peak is reduced exceeding three times with respect to that of the primary oscillator without the NES attached. In this case, the responses in the entire frequency band are stable. When the nonlinear stiffness k2 is increased to 400, the unstable responses appear approximately at Ω=15 and the suppression efficiency of this NES is enhanced compared to that of the NES with k2=100. Generally, the weakly/strongly modulation responses are preferable for vibration suppression. When k2=700, the efficiency is further enhanced and the frequency band where the unstable response occurs is broadened. When k2=1000, two high branches of the frequency response curve appear in the frequency band 13.4,14, one is stable and the other is unstable. The appearance of the high stable branch is interpreted that the NES intensifies the vibration of the primary oscillator in this frequency band. When k2=1300, compared to those demonstrated in Figure 3(e), the frequency band where the high stable branch exists is broadened and the maximum response amplitude is larger.Figure 3
Frequency responses of the primary oscillator with various NESs asf=1 and k1=50.
(a) (b) (c) (d) (e) (f)The frequency responses of the primary oscillator under the harmonic excitation amplitudef=3 are presented in Figure 4. When k2=50, the unstable response appears approximately at Ω=15 and a large amount of vibration energy is dissipated. When k2=150 and k2=250, the high branches appear, whereas the frequency band where the high stable branch appears to be broader as the nonlinear stiffness increased. When the excitation amplitude f=5, the variations in the frequency response along with the nonlinear stiffness k2 are the same as those demonstrated in Figures 3 and 4. It is noted that the high branches appear as k2=50 in this case. Comparing Figures 3–5, it is discovered that, prior to the appearance of the high response branches, the unstable response occurs approximately at the linear natural frequency of the primary oscillator firstly. In addition, as the excitation amplitude increased, the lower stiffness should be selected to prevent the appearance of the high branches of the frequency response.Figure 4
Frequency responses of the primary oscillator with various NESs asf=3 and k1=50.
(a) (b) (c) (d)Figure 5
Frequency responses of the primary oscillator with various NESs asf=5 and k1=50.
(a) (b) (c) (d)Then, the bifurcations for the primary oscillator via the nonlinear stiffness of the NES under a fixed excitation frequency and various excitation amplitudes are studied. WhenΩ=15, the response amplitude always remains at the relatively low value as f=2 in the studied value range of the nonlinear stiffness. As the excitation amplitude increased, the high branches will appear at certain stiffness. For example, when f=3, the response amplitude decreases with increasing the nonlinear stiffness k2 at first, and the response of the primary oscillator becomes unstable as k2=93. Next, the high branches of the response amplitude appear until the nonlinear stiffness k2=147. Comparing the four subfigures in Figure 6, it is worth being noted that the larger the excitation amplitude, the lower the stiffness which corresponds to the bifurcation point. Figure 7 shows that, when Ω=15.5, the response amplitude always remains the relatively lower value as f=2; however, a bifurcation occurs as the nonlinear stiffness k2=93 and the response becomes unstable after the nonlinear stiffness k2=132. Similarly, the stiffness which corresponds to the bifurcation point is reduced as the excitation amplitude increased.Figure 6
Bifurcations of primary oscillator asΩ=15.
(a) (b) (c) (d)Figure 7
Bifurcations of primary oscillator asΩ=15.5.
(a) (b) (c) (d)


## 4.2. Effects of Primary Oscillator Nonlinear Stiffness

The effects of the primary oscillator nonlinear stiffness on the frequency response are investigated. All parameters exceptk1 and k2 are the same as those in Figure 4. Figure 8 is depicted as k1=0 (i.e., the primary oscillator is linear). The change rates of the response amplitude, within the high stable branches in Figures 8(c) and 8(d), are quite different from the counterparts when k1≠0. Comparing the responses of the primary oscillator without the NES, which are demonstrated in Figures 4(a), 8(a), 9(a), and 10(a), it is indicated that the value of the resonance peak decreases and the frequency which corresponds to the resonance peak increases with the raise of the nonlinear stiffness k1. It can also be concluded by comparing the rest figures in Figures 4, 9, and 10 that the integrated systems with the various values of k1 and the same value of k2 have the similar response. Similar to the effects of the nonlinear stiffness k1 on the dynamics of the primary oscillator without the NES, an increase in the nonlinear stiffness k1 brings about the decrease of the unwanted branches when the NES is coupled to the primary oscillator. In addition, the nonlinear stiffness k1 has a small effect on the frequency range of the high branches of the frequency response.Figure 8
Frequency responses of the linear primary oscillator with various NESs asf=3.
(a) (b) (c) (d)Figure 9
Frequency responses of the primary oscillator with various NESs ask1=200 and f=3.
(a) (b) (c) (d)Figure 10
Frequency responses of the primary oscillator with various NESs ask1=350 and f=3.
(a) (b) (c) (d)


## 5. Concluding Remarks

The effects of the nonlinear stiffness of both the NES and primary oscillator on the dynamics of the primary oscillator under various harmonic excitations are investigated. The complexification-averaging technique is utilized to obtain the analytical model of the integrated system, and the coefficient matrix of the integrated system with small perturbation is obtained to analyze the stability of the dynamic responses.The predictions of the analytical model are confirmed by numerical simulations. The analytical results demonstrate that the vibration suppression can be enhanced by increasing the NES nonlinear stiffness. However, the high branches appear following the stiffness excessive increase. As the stiffness remains fixed, the high branches also can appear with the excitation amplitude increases. Actually, for preventing the appearance of the high branches of the frequency response under the large amplitude external force, the nonlinear stiffness should be properly adjusted to a relatively low value. Therefore, the utilization of a semiactive control is an efficient method for the robustness improvement of the nonlinear absorber.The NES brings an extra nonlinear factor to the nonlinear primary oscillator. A distinction should be made between the contributions of the two nonlinear factors. The results indicate that the introduction of the strongly nonlinear factor can not only suppress the vibrations of the primary oscillator but also can result in the bifurcations and instability of the responses under certain conditions. In the system studied in this paper, by contrast, the effects of the nonlinear stiffness of the primary oscillator are relatively simple. The increase in this nonlinear stiffness can reduce the response amplitude and alter the frequency band where the high branches exist. Also, the stiffness of the primary oscillator can affect the value of the NES optimal stiffness. In this work, the cubic nonlinearity of the primary oscillator is considered only. The effects of the NES on the dynamics of the primary oscillator which has more complex nonlinear factors are required to be further investigated.



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