Research ArticleDynamics of Nonlinear Primary Oscillator withNonlinear Energy Sink under Harmonic Excitation:Effects of Nonlinear Stiffness

Min Sun 1 and Jianen Chen 2,3

1School of Science, Tianjin Chengjian University, Tianjin 300384, China2Tianjin Key Laboratory of the Design and Intelligent Control of the Advanced Mechatronical System,Tianjin University of Technology, Tianjin 300384, China3National Demonstration Center for Experimental Mechanical and Electrical Engineering Education,Tianjin University of Technology, Tianjin 300384, China

Correspondence should be addressed to Min Sun; sunmin0537@163.com

Received 31 March 2018; Revised 15 July 2018; Accepted 13 August 2018; Published 2 September 2018

Academic Editor: Viktor Avrutin

Copyright © 2018 Min Sun and Jianen Chen. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The dynamics of a system consisting of a nonlinear primary oscillator, subjected to a harmonic external force, and a nonlinearenergy 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 ofthe NES can effectively suppress the vibrations of the primary oscillator. However, as the excitation amplitude increased, the NESmay lose its efficiency within certain frequency range due to the appearance of the high response branches. Following the resultsanalysis, it is concluded that this failure can be eliminated by reducing the nonlinear stiffness of the NES properly. The effects ofnonlinear stiffness of the primary oscillator on the corresponding responses are also studied.The increase in this nonlinear stiffnesscan reduce the response amplitude and alter the frequency band where the high branches exist.

1. Introduction

The nonlinear energy sink, consisting of a small mass anda pure nonlinear spring, constitutes an effective solution forvibration suppression over a broad frequency range [1–3]. Itcan result in a one-way irreversible energy transfer from theprimary system to the NES. Compared to traditional linearand weakly nonlinear absorbers, the NES has the highersuppression efficiency under many conditions [4, 5].

The targeted energy transfer (TET) inmechanical systemsis one of the attractive issues in the past decades. Theintroduction of the strong nonlinearity to the primary systemcan lead to complicated dynamic responses.Themechanismsand phenomena of the nonlinear TET, such as the transientand sustained resonance captures, energy localization, andweakly and strongly modulation responses, were of highconcern to researchers [6–8]. Applications of the NES forthe vibration suppression in various structures were reported

in several papers. Theoretical and experimental studies indi-cated that the vibrations of beams [9, 10], plates [11, 12],circular cylinders [13], and stay-cables [14] can be effectivelysuppressed by the NES attachment. With the deepeningof research, NES has been applied increasingly to suppressvibration 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 NESwhen the primary system is excited by harmonic load is anindispensable part of applications for this absorber [18, 19].Following a series of optimization, the essentially nonlinearabsorber exhibited apparent advantages over linear or weaklynonlinear counterparts. However, a main drawback in NESexists: several unwanted additional branches of responsemay become apparent under certain conditions. The strategywhich can eliminate these branches is a key issue for robustincrease of this absorber [20, 21]. In addition, in the vast

HindawiMathematical Problems in EngineeringVolume 2018, Article ID 5693618, 13 pageshttps://doi.org/10.1155/2018/5693618

2 Mathematical Problems in Engineering

M m

K

w(t)

v(t)

k2

k1

12

Figure 1: Nonlinear primary oscillator with a nonlinear energy sink.

majority of cases, the controlled primary systems were linearmodels. These researches contributed significantly to thecharacteristics of the nonlinear TET. However, many engi-neering structures present nonlinear characters, and theirnonlinear phenomena must be considered [22–25]. The dy-namics of the TET system where the nonlinear factors of theprimary system were taken into account had been paid moreattentions [26–29].

In the current study, the steady-state responses of a non-linear primary oscillator coupled to a NES are investigatedby the complexification-averaging technique. The nonlinearfrequency responses of the system under various excitationamplitudes are considered.The nonlinear stiffness of the NESis adjusted to attempt to eliminate these unwanted branches.Also, the effects of the nonlinear stiffness of both the NES andprimary oscillator on the vibration suppression are analyzedand compared.

2. Dynamic System

A nonlinear primary oscillator with a single degree of free-dom essentially nonlinear attachment is presented in Figure 1.A harmonic external force is imposed on the primaryoscillator. The dynamic equation for the integrated system isderived as follows:�� + 𝛾

1�� + 𝐾𝑤 + 𝑘

1𝑤3 + 𝑘

2 (𝑤 − V)3 + 𝛾2 (�� − V)= 𝑓 cosΩ𝑡, (1a)

𝜀V + 𝑘2 (V − 𝑤)3 + 𝛾2 (V − ��) = 0, (1b)

where𝐾, 𝑘1, and 𝛾

1are the linear stiffness, nonlinear stiffness,

and linear damping of the primary oscillator, respectively.𝑘2and 𝛾

2are the nonlinear stiffness and linear damping of

the NES. 𝑓 and Ω represent the excitation amplitude andfrequency, respectively. 𝜀 is the mass ratio between the NESand the primary oscillator and 0 < 𝜀 ≪ 1. For convenience,we let𝑀 = 1.

The complexification-averaging technique is utilized toobtain the slow-flowmodel of the integrated system, whereasthe following transformations are introduced:

𝑢 (𝑡) = 𝑤 (𝑡) − V (𝑡) ,�� + 𝑖Ω𝑤 = 𝛼1𝑒𝑖Ω𝑡,�� − 𝑖Ω𝑤 = 𝛼1𝑒−𝑖Ω𝑡,�� + 𝑖Ω𝑢 = 𝛼2𝑒𝑖Ω𝑡,�� − 𝑖Ω𝑢 = 𝛼2𝑒−𝑖Ω𝑡,

(2)

where 𝛼𝑛is the conjugate of the 𝛼

𝑛, (𝑛 = 1, 2), 𝑖 = √−1.

Based on expression (2), we have𝑤 = 𝛼1𝑒𝑖Ω𝑡 − 𝛼1𝑒−𝑖Ω𝑡2𝑖Ω ,�� = 𝛼1𝑒𝑖Ω𝑡 + 𝛼1𝑒−𝑖Ω𝑡2 ,�� = 12 (��1𝑒𝑖Ω𝑡 + 𝑖Ω𝛼1𝑒𝑖Ω𝑡 + ��1𝑒−𝑖Ω𝑡 − 𝑖Ω𝛼1𝑒−𝑖Ω𝑡) ,𝑢 = 𝛼2𝑒𝑖Ω𝑡 − 𝛼2𝑒−𝑖Ω𝑡2𝑖Ω ,�� = 𝛼2𝑒𝑖Ω𝑡 + 𝛼2𝑒−𝑖Ω𝑡2 ,�� = 12 (��2𝑒𝑖Ω𝑡 + 𝑖Ω𝛼2𝑒𝑖Ω𝑡 + ��2𝑒−𝑖Ω𝑡 − 𝑖Ω𝛼2𝑒−𝑖Ω𝑡) .

(3)

Substituting (3) into (1a) and (1b) and retaining the slow-flow parts, we obtain��1+ 𝑖Ω𝛼

1+ 𝛾1𝛼1+ 𝐾𝛼1𝑖Ω + 3𝑘1𝛼21𝛼14𝑖Ω3 + 3𝑘2𝛼22𝛼24𝑖Ω3+ 𝛾

2𝛼2= 𝑓, (4a)

𝜀 (��1+ 𝑖Ω𝛼

1− ��2− 𝑖Ω𝛼

2) − 3𝑘2𝛼22𝛼24𝑖Ω3 − 𝛾2𝛼2 = 0. (4b)

Letting, 𝛼1= 𝑎1+ 𝑖𝑏1,𝛼2 = 𝑎2 + 𝑖𝑏2. (5)

Mathematical Problems in Engineering 3

We substitute (5) into (4a) and (4b) and separate real andimaginary parts, yielding

𝑎1= −𝑎1𝛾1+ 𝑏1Ω − 𝐾𝑏1Ω − 3𝑘2𝑏2 (𝑎22 + 𝑏22 )4Ω3

− 𝛾2𝑎2− 3𝑘1𝑏1 (𝑎21 + 𝑏21)4Ω3 + 𝑓, (6a)

��1= −𝑎1Ω − 𝑏1𝛾1+ 𝐾𝑎1Ω + 3𝑘2𝑎2 (𝑎22 + 𝑏22)4Ω3

− 𝛾2𝑏2+ 3𝑘1𝑎1 (𝑎21 + 𝑏21)4Ω3 , (6b)

𝜀 𝑎2− 𝜀 𝑎1= 𝜀Ω𝑏

2− 𝜀Ω𝑏

1− 3𝑘2𝑏2 (𝑎22 + 𝑏22)4Ω3 − 𝛾

2𝑎2, (6c)

𝜀��2− 𝜀��1= 𝜀Ω𝑎

1− 𝜀Ω𝑎

2+ 3𝑘2𝑎2 (𝑎22 + 𝑏22)4Ω3 − 𝛾

2𝑏2. (6d)

The steady-state responses of the system are investigatedthrough letting 𝑎

1= 0, ��

1= 0, 𝑎

2= 0, ��

2= 0. Solving

the resulting nonlinear algebraic equations, the responseamplitude of the primary oscillator and the relative motionbetween two oscillators are obtained and the expressions are,respectively,

𝑤 = √𝑎21 + 𝑏21Ω ,𝑤 − V = √𝑎22 + 𝑏22Ω . (7)

In order for the stability of the steady-state solutions tobe analyzed, the small perturbations 𝛿

𝑛(𝑛 = 1, 2, 3, 4) are

introduced: 𝑎1= 𝑎10+ 𝛿1,𝑏

1= 𝑏10+ 𝛿2,𝑎

2= 𝑎20+ 𝛿3,𝑏2 = 𝑏20 + 𝛿4,

(8)

where 𝑎10, 𝑏10, 𝑎20, 𝑏20

are the steady-state solutions of thesystem. Substituting (8) into (6a), (6b), (6c), and (6d) yields𝛿1= −𝛾1𝛿1+ Ω𝛿2− 𝐾Ω𝛿2− 3𝑘24Ω3 [(𝑎220 + 3𝑏220) 𝛿4 + 2𝑎20𝑏20𝛿3] − 𝛾2𝛿3− 3𝑘14Ω3 [(𝑎210 + 3𝑏210) 𝛿2 + 2𝑎10𝑏10𝛿1] ,

(9a)

𝛿2= −𝛾1𝛿2− Ω𝛿1+ 𝐾Ω𝛿1+ 3𝑘24Ω3 [(3𝑎220 + 𝑏220) 𝛿3 + 2𝑎20𝑏20𝛿4] − 𝛾2𝛿4+ 3𝑘14Ω3 [(3𝑎210 + 𝑏210) 𝛿1 + 2𝑎10𝑏10𝛿2] ,

(9b)

𝛿3= Ω𝛿4− Ω𝛿2− 3𝑘24𝜀Ω3 [(𝑎220 + 3𝑏220) 𝛿4 + 2𝑎20𝑏20𝛿3]− 𝛾2𝛿3𝜀 + 𝛿1,

(9c)

𝛿4= Ω𝛿1− Ω𝛿3+ 3𝑘24𝜀Ω3 [(3𝑎220 + 𝑏220) 𝛿3 + 2𝑎20𝑏20𝛿4]− 𝛾2𝛿4𝜀 + 𝛿2.

(9d)

The coefficient matrix of (9a), (9b), (9c), and (9d) ispresented as follows; the eigenvalues of thematrix can be usedto determine the stability of the solutions:

𝐽 = (𝐽11 𝐽12 𝐽13 𝐽14𝐽21𝐽22𝐽23𝐽24𝐽

31𝐽32𝐽33𝐽34𝐽

41𝐽42𝐽43𝐽44

), (10)

where 𝐽11= −𝛾1− 3𝑘1𝑎10𝑏102Ω3 ,𝐽

12= Ω − 𝐾Ω − 3𝑘1 (𝑎210 + 3𝑏210)4Ω3 ,

𝐽13= −𝛾2− 3𝑘2𝑎20𝑏202Ω3 ,𝐽

14= −3𝑘2 (𝑎220 + 3𝑏220)4Ω3 ,

𝐽21= −Ω + 𝐾Ω + 3𝑘1 (3𝑎210 + 𝑏210)4Ω3 ,

𝐽22= −𝛾1+ 3𝑘1𝑎10𝑏102Ω3 ,𝐽

23= 3𝑘2 (3𝑎220 + 𝑏220)4Ω3 ,

𝐽24= −𝛾2+ 3𝑘2𝑎20𝑏202Ω3 ,𝐽

31= −𝛾1− 3𝑘1𝑎10𝑏102Ω3 ,

4 Mathematical Problems in Engineering

𝐽32= −𝐾Ω − 3𝑘1 (𝑎210 + 3𝑏210)4Ω3 ,

𝐽33= −𝛾2𝜀 − 3𝑘2𝑎20𝑏202𝜀Ω3 − 𝛾2 − 3𝑘2𝑎20𝑏202Ω3 ,𝐽34= Ω − 3𝑘2 (𝑎220 + 3𝑏220)4𝜀Ω3 − 3𝑘2 (𝑎220 + 3𝑏220)4Ω3 ,

𝐽41= 𝐾Ω + 3𝑘1 (3𝑎210 + 𝑏210)4Ω3 ,

𝐽42= −𝛾1+ 3𝑘1𝑎10𝑏102Ω3 ,𝐽

43= −Ω + 3𝑘2 (3𝑎220 + 𝑏220)4𝜀Ω3 + 3𝑘2 (3𝑎220 + 𝑏220)4Ω3 ,

𝐽44= −𝛾2𝜀 + 3𝑘2𝑎20𝑏202𝜀Ω3 − 𝛾2 + 3𝑘2𝑎20𝑏202Ω3 .

(11)

3. Validation

The numerical results are utilized for the predictions con-firmation of the analytical model. The Runge-Kutta methodis directly attacking (1a) and (1b) to obtain the numericalresults. It is demonstrated in Figure 2(a) that the resultsobtained from these two methods agree well. In Figure 2(a),squares and circles represent stable and unstable solutionsobtained from complexification-averaging technique, respec-tively. Pluses represent numerical solutions obtained fromRunge-Kutta method. Moreover, the stability of the analyticalresults is also presented in this figure. In order for the generalapplicability of the results to be ensured, all parameters in(1a) and (1b) are dimensionless values. The stable responseswhen Ω = 14.8 and Ω = 15.7 as well as the unstable re-sponses when Ω = 15.1 and Ω = 15.4 are shown in Fig-ures 2(b)–2(e), respectively. The unstable responses demon-strate a kind of relaxation oscillation, usually referred to asstrong modulation response. These parameters are selected,respectively, as𝐾 = 225, 𝑘

1= 50, 𝑘

2= 700, 𝛾

1= 0.1, 𝛾

2= 0.2,𝜀 = 0.1, 𝑓 = 1.

4. Results and Discussions

4.1. Effects of NES Nonlinear Stiffness. The vibration sup-pression efficiency is of most concern. In this section, theefficiency is investigated under various harmonic excitationamplitudes by the analytical method, whereas all parametersexcept the nonlinear stiffness 𝑘

2are selected as the parameters

in Section 3. Figure 3 illustrates the nonlinear frequency re-sponses of the primary oscillator without the NES and withvarious NESs under 𝑓 = 1. When the nonlinear stiffness 𝑘2 isselected as 100, the good vibration suppression effect isachieved, the value of the resonance peak is reduced exceed-ing three times with respect to that of the primary oscillatorwithout the NES attached. In this case, the responses inthe entire frequency band are stable. When the nonlinear

stiffness 𝑘2is increased to 400, the unstable responses appear

approximately at Ω = 15 and the suppression efficiency ofthis NES is enhanced compared to that of the NES with 𝑘

2=100. Generally, theweakly/stronglymodulation responses are

preferable for vibration suppression. When 𝑘2= 700, the

efficiency is further enhanced and the frequency band wherethe unstable response occurs is broadened. When 𝑘

2= 1000,

two high branches of the frequency response curve appearin the frequency band [13.4, 14], one is stable and the otheris unstable. The appearance of the high stable branch is inter-preted that the NES intensifies the vibration of the primaryoscillator in this frequency band.When 𝑘

2= 1300, compared

to those demonstrated in Figure 3(e), the frequency bandwhere the high stable branch exists is broadened and themaximum response amplitude is larger.

The frequency responses of the primary oscillator underthe harmonic excitation amplitude 𝑓 = 3 are presented inFigure 4. When 𝑘

2= 50, the unstable response appears

approximately at Ω = 15 and a large amount of vibrationenergy is dissipated. When 𝑘2 = 150 and 𝑘2 = 250, the highbranches appear, whereas the frequency band where the highstable branch appears to be broader as the nonlinear stiffnessincreased. When the excitation amplitude 𝑓 = 5, the varia-tions in the frequency response along with the nonlinearstiffness 𝑘

2are 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, priorto the appearance of the high response branches, the unstableresponse occurs approximately at the linear natural frequencyof the primary oscillator firstly. In addition, as the excitationamplitude increased, the lower stiffness should be selected toprevent the appearance of the high branches of the frequencyresponse.

Then, the bifurcations for the primary oscillator viathe nonlinear stiffness of the NES under a fixed excitationfrequency and various excitation amplitudes are studied.When Ω = 15, the response amplitude always remains at therelatively low value as 𝑓 = 2 in the studied value range of thenonlinear stiffness. As the excitation amplitude increased, thehigh branches will appear at certain stiffness. For example,when 𝑓 = 3, the response amplitude decreases with increas-ing the nonlinear stiffness 𝑘

2at first, and the response of

the primary oscillator becomes unstable as 𝑘2 = 93. Next,the high branches of the response amplitude appear until thenonlinear stiffness 𝑘

2= 147. Comparing the four subfigures

in Figure 6, it is worth being noted that the larger the excita-tion amplitude, the lower the stiffness which corresponds tothe bifurcation point. Figure 7 shows that, when Ω = 15.5,the response amplitude always remains the relatively lowervalue as𝑓 = 2; however, a bifurcation occurs as the nonlinearstiffness 𝑘

2= 93 and the response becomes unstable after

the nonlinear stiffness 𝑘2= 132. Similarly, the stiffness

which corresponds to the bifurcation point is reduced as theexcitation amplitude increased.

4.2. Effects of Primary Oscillator Nonlinear Stiffness. Theeffects of the primary oscillator nonlinear stiffness on the fre-quency response are investigated. All parameters except 𝑘

1

Mathematical Problems in Engineering 5

12 14 16 18 2010

0.00

0.03

0.06

0.09

(a)

Ω=14.8−0.1

−0.05

0

0.05

0.1

35 40 45 5030t

(b)

Ω=15.1−0.1

−0.05

0

0.05

0.1

35 40 45 5030t

(c)

Ω=15.435 40 45 5030

t

−0.1

−0.05

0

0.05

0.1

(d)

Ω=15.735 40 45 5030

t

−0.1

−0.05

0

0.05

0.1

(e)

Figure 2: Analytical model validation.

6 Mathematical Problems in Engineering

without NES

0.0

0.2

0.4

0.6

12 14 16 18 2010

(a)

=100

12 14 16 18 2010

0.00

0.05

0.10

0.15

0.20

(b)

12 14 16 18 2010

0.00

0.03

0.06

0.09

0.12

=

(c)=700

12 14 16 18 2010

0.00

0.02

0.04

0.06

0.08

(d)

=1000

12 14 16 18 2010

0.0

0.1

0.2

0.3

(e)

=1300

12 14 16 18 2010

0.0

0.1

0.2

0.3

0.4

(f)

Figure 3: Frequency responses of the primary oscillator with various NESs as 𝑓 = 1 and 𝑘1= 50.

without NES

0.0

0.4

0.8

1.2

1.6

12 14 16 18 2010

(a)

=50

0.0

0.1

0.2

0.3

12 14 16 18 2010

(b)

=150

0.0

0.2

0.4

0.6

0.8

1.0

12 14 16 18 2010

(c)

=250

12 14 16 18 2010

0.0

0.3

0.6

0.9

1.2

(d)

Figure 4: Frequency responses of the primary oscillator with various NESs as 𝑓 = 3 and 𝑘1= 50.

Mathematical Problems in Engineering 7

without NES

14 16 18 20 2212

0.0

0.5

1.0

1.5

2.0

2.5

(a)

=30

12 14 16 18 2010

0.0

0.1

0.2

0.3

0.4

(b)

=50

0.0

0.3

0.6

0.9

1.2

12 14 16 18 2010

(c)

=70

0.0

0.4

0.8

1.2

1.6

12 14 16 18 2010

(d)

Figure 5: Frequency responses of the primary oscillator with various NESs as 𝑓 = 5 and 𝑘1= 50.

and 𝑘2are the same as those in Figure 4. Figure 8 is

depicted as 𝑘1 = 0 (i.e., the primary oscillator is linear). Thechange rates of the response amplitude, within the high stablebranches in Figures 8(c) and 8(d), are quite different from thecounterparts when 𝑘

1= 0. Comparing the responses of the

primary oscillator without the NES, which are demonstratedin Figures 4(a), 8(a), 9(a), and 10(a), it is indicated that the

value of the resonance peak decreases and the frequencywhich corresponds to the resonance peak increases with theraise of the nonlinear stiffness 𝑘

1. It can also be concluded

by comparing the rest figures in Figures 4, 9, and 10 thatthe integrated systems with the various values of 𝑘

1and

the same value of 𝑘2have the similar response. Similar to

the effects of the nonlinear stiffness 𝑘1on the dynamics

8 Mathematical Problems in Engineering

f=2

100 200 30002

0.0

0.4

0.8

1.2

(a)

f=3

100 200 30002

0.0

0.4

0.8

1.2

(b)

f=4

100 200 30002

0.0

0.4

0.8

1.2

(c)

f=5

0.0

0.4

0.8

1.2

100 200 30002

(d)

Figure 6: Bifurcations of primary oscillator asΩ = 15.of the primary oscillator without the NES, an increase inthe nonlinear stiffness 𝑘

1brings about the decrease of the

unwanted branches when the NES is coupled to the primaryoscillator. In addition, the nonlinear stiffness 𝑘

1has a small

effect on the frequency range of the high branches of thefrequency response.

5. Concluding Remarks

The effects of the nonlinear stiffness of both the NES andprimary oscillator on the dynamics of the primary oscillatorunder various harmonic excitations are investigated. Thecomplexification-averaging technique is utilized to obtain the

Mathematical Problems in Engineering 9

f=2

0.0

0.3

0.6

0.9

1.2

100 200 30002

(a)

f=3

100 200 30002

0.0

0.3

0.6

0.9

1.2

(b)

f=4

100 200 30002

0.0

0.3

0.6

0.9

1.2

(c)

f=5

0.0

0.3

0.6

0.9

1.2

100 200 30002

(d)

Figure 7: Bifurcations of primary oscillator asΩ = 15.5.analytical model of the integrated system, and the coefficientmatrix of the integrated system with small perturbation isobtained to analyze the stability of the dynamic responses.

The predictions of the analytical model are confirmedby numerical simulations. The analytical results demonstratethat the vibration suppression can be enhanced by increasingthe NES nonlinear stiffness. However, the high branches

appear following the stiffness excessive increase. As the stiff-ness remains fixed, the high branches also can appear with theexcitation amplitude increases. Actually, for preventing theappearance of the high branches of the frequency responseunder the large amplitude external force, the nonlinearstiffness should be properly adjusted to a relatively low value.Therefore, the utilization of a semiactive control is an efficient

10 Mathematical Problems in Engineering

without NES

0.0

0.5

1.0

1.5

2.0

12 14 16 18 2010

(a)

=50

0.0

0.1

0.2

0.3

12 14 16 18 2010

(b)

=150

12 14 16 18 2010

0.0

0.4

0.8

1.2

(c)

=250

0.0

0.3

0.6

0.9

1.2

1.5

12 14 16 18 2010

(d)

Figure 8: Frequency responses of the linear primary oscillator with various NESs as 𝑓 = 3.method for the robustness improvement of the nonlinearabsorber.

TheNES brings an extra nonlinear factor to the nonlinearprimary oscillator. A distinction should be made betweenthe contributions of the two nonlinear factors. The resultsindicate that the introduction of the strongly nonlinear factorcan not only suppress the vibrations of the primary oscillator

but also can result in the bifurcations and instability of theresponses under certain conditions. In the system studied inthis paper, by contrast, the effects of the nonlinear stiffnessof the primary oscillator are relatively simple. The increasein this nonlinear stiffness can reduce the response amplitudeand alter the frequency band where the high branches exist.Also, the stiffness of the primary oscillator can affect the

Mathematical Problems in Engineering 11

without NES

0.0

0.4

0.8

1.2

12 14 16 18 20 22 2410

(a)

=50

0.0

0.1

0.2

0.3

12 14 16 18 2010

(b)

=150

0.0

0.2

0.4

0.6

12 14 16 18 2010

(c)

=250

0.0

0.2

0.4

0.6

0.8

1.0

12 14 16 18 2010

(d)

Figure 9: Frequency responses of the primary oscillator with various NESs as 𝑘1= 200 and 𝑓 = 3.

12 Mathematical Problems in Engineering

without NES

0.0

0.3

0.6

0.9

1.2

12 14 16 18 20 22 2410

(a)

=50

0.0

0.1

0.2

0.3

12 14 16 18 2010

(b)

=150

0.0

0.1

0.2

0.3

0.4

12 14 16 18 2010

(c)

=250

12 14 16 18 2010

0.0

0.2

0.4

0.6

0.8

(d)

Figure 10: Frequency responses of the primary oscillator with various NESs as 𝑘1= 350 and 𝑓 = 3.

value of the NES optimal stiffness. In this work, the cubicnonlinearity of the primary oscillator is considered only. Theeffects of the NES on the dynamics of the primary oscillatorwhich has more complex nonlinear factors are required to befurther investigated.

Data Availability

Thedatasets used during the current study are available fromthe corresponding author on reasonable request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the National Natural Sci-ence Foundation of China (nos. 11402165 and 11402170) andTianjin Natural Science Foundation of China (no. 17JCY-BJC18800).

References

[1] O. V. Gendelman, “Transition of energy to a nonlinear localizedmode in a highly asymmetric system of two oscillators,”Nonlin-ear Dynamics, vol. 25, pp. 237–253, 2001.

[2] G. Kerschen, Y. S. Lee, A. F. Vakakis, D. M. McFarland, and L.A. Bergman, “Irreversible passive energy transfer in coupledoscillatorswith essential nonlinearity,” SIAM Journal on AppliedMathematics, vol. 66, no. 2, pp. 648–679, 2005.

[3] A. F. Vakakis, O. V. Gendelman, L. A. Bergman, D. M. McFar-land, G. Kerschen, and Y. S. Lee, Nonlinear Targeted EnergyTransfer in Mechanical and Structural Systems, Springer, Berlin,Germany, 2009.

[4] A. Tripathi, P. Grover, and T. Kalmar-Nagy, “On optimal per-formance of nonlinear energy sinks in multiple-degree-of-free-dom systems,” Journal of Sound andVibration, vol. 388, pp. 272–297, 2017.

[5] F. S. Samani and F. Pellicano, “Vibration reduction on beamssubjected to moving loads using linear and nonlinear dynamicabsorbers,” Journal of Sound and Vibration, vol. 325, no. 4-5, pp.742–754, 2009.

Mathematical Problems in Engineering 13

[6] G. Kerschen, J. J. Kowtko, D. McFarland, L. A. Bergman, and A.F. Vakakis, “Theoretical and experimental study of multimodaltargeted energy transfer in a systemof coupled oscillators,”Non-linear Dynamics, vol. 47, pp. 285–309, 2007.

[7] E. Gourdon, C. H. Lamarque, and S. Pernot, “Contribution toefficiency of irreversible passive energy pumping with a strongnonlinear attachment,” Nonlinear Dynamics, vol. 50, no. 4, pp.793–808, 2007.

[8] D. D. Quinn, O. Gendelman, G. Kerschen, T. P. Sapsis, L. A.Bergman, and A. F. Vakakis, “Efficiency of targeted energytransfers in coupled nonlinear oscillators associated with 1:1 re-sonance captures: Part I,” Journal of Sound and Vibration, vol.311, no. 3-5, pp. 1228–1248, 2008.

[9] Y.-W. Zhang, Z. Zhang, L.-Q. Chen, T.-Z. Yang, B. Fang, andJ. Zang, “Impulse-induced vibration suppression of an axiallymoving beam with parallel nonlinear energy sinks,” NonlinearDynamics, vol. 82, no. 1-2, pp. 61–71, 2015.

[10] M. Kani, S. E. Khadem, M. H. Pashaei, and M. Dardel, “Designand performance analysis of a nonlinear energy sink attachedto a beam with different support conditions,” Proceedings of theInstitution of Mechanical Engineers, Part C: Journal of Mechani-cal Engineering Science, vol. 230, no. 4, pp. 527–542, 2016.

[11] Y.-W. Zhang, H. Zhang, S. Hou, K.-F. Xu, and L.-Q. Chen,“Vibration suppression of composite laminated plate withnonlinear energy sink,” Acta Astronautica, vol. 123, pp. 109–115,2016.

[12] J. E. Chen, W. Zhang, M. H. Yao, J. Liu, and M. Sun, “Vibrationreduction in truss core sandwich plate with internal nonlinearenergy sink,” Composite Structures, vol. 193, pp. 180–188, 2018.

[13] H. L. Dai, A. Abdelkefi, and L. Wang, “Vortex-induced vibra-tions mitigation through a nonlinear energy sink,” Communi-cations in Nonlinear Science and Numerical Simulation, vol. 42,pp. 22–36, 2017.

[14] M. Izzi, L. Caracoglia, and S. Noe, “Investigating the use ofTargeted-Energy-Transfer devices for stay-cable vibrationmiti-gation,” Structural Control and HealthMonitoring, vol. 23, no. 2,pp. 315–332, 2016.

[15] S. Bab, S. E. Khadem, M. Shahgholi, and A. Abbasi, “Vibrationattenuation of a continuous rotor-blisk-journal bearing systememploying smooth nonlinear energy sinks,”Mechanical Systemsand Signal Processing, vol. 84, pp. 128–157, 2017.

[16] B. Bergeot, S. Bellizzi, and B. Cochelin, “Passive suppressionof helicopter ground resonance using nonlinear energy sinksattached on the helicopter blades,” Journal of Sound and Vibra-tion, vol. 392, pp. 41–55, 2017.

[17] K.Yang,Y.W.Zhang,H.Ding, andL.Q.Chen, “Nonlinear ener-gy sink for whole-spacecraft vibration reduction,” Journal ofVibration and Acoustics, vol. 139, 2017.

[18] J. Zang and L.-Q. Chen, “Complex dynamics of a harmonicallyexcited structure coupled with a nonlinear energy sink,” ActaMechanica Sinica, vol. 33, no. 4, pp. 801–822, 2017.

[19] J. E. Chen, W. He, W. Zhang, M. H. Yao, J. Liu, and M. Sun,“Vibration suppression and higher branch responses of beamwith parallel nonlinear energy sinks,” Nonlinear Dynamics, vol.91, no. 2, pp. 885–904, 2018.

[20] Y. Starosvetsky and O. V. Gendelman, “Attractors of harmoni-cally forced linear oscillatorwith attachednonlinear energy sinkII: Optimization of a nonlinear vibration absorber,” NonlinearDynamics, vol. 51, no. 1-2, pp. 47–57, 2008.

[21] Y. Starosvetsky and O. V. Gendelman, “Vibration absorption insystems with a nonlinear energy sink: Nonlinear damping,”

Journal of Sound and Vibration, vol. 324, no. 3-5, pp. 916–939,2009.

[22] A.H.Nayfeh andD.T.Mook,NonlinearOscillations, JohnWiley& Sons, New York, NY, USA, 1979.

[23] W. Zhang, “Global and chaotic dynamics for a parametricallyexcited thin plate,” Journal of Sound and Vibration, vol. 239, no.5, pp. 1013–1036, 2001.

[24] W. Zhang, Y. X. Hao, and J. Yang, “Nonlinear dynamics ofFGM circular cylindrical shell with clamped-clamped edges,”Composite Structures, vol. 94, no. 3, pp. 1075–1086, 2012.

[25] R. Q. Wu,W. Zhang, andM. H. Yao, “Nonlinear dynamics nearresonances of a rotor-active magnetic bearings system with 16-pole legs and time varying stiffness,” Mechanical Systems andSignal Processing, vol. 100, pp. 113–134, 2018.

[26] R. Viguie and G. Kerschen, “Nonlinear vibration absorbercoupled to a nonlinear primary system: a tuning methodology,”Journal of Sound and Vibration, vol. 326, no. 3–5, pp. 780–793,2009.

[27] R. Viguie, M. Peeters, G. Kerschen, and J.-C. Golinval, “Energytransfer and dissipation in a duffing oscillator coupled to anonlinear attachment,” Journal of Computational and NonlinearDynamics, vol. 4, 2009.

[28] M. Kani, S. E. Khadem, M. H. Pashaei, and M. Dardel, “Vi-bration control of a nonlinear beam with a nonlinear energysink,” Nonlinear Dynamics, vol. 83, no. 1, pp. 1–22, 2016.

[29] M. Parseh, M. Dardel, M. H. Ghasemi, and M. H. Pashaei,“Steady state dynamics of a non-linear beam coupled to a non-linear energy sink,” International Journal of Non-Linear Me-chanics, vol. 79, pp. 48–65, 2016.

Hindawiwww.hindawi.com Volume 2018

MathematicsJournal of

Hindawiwww.hindawi.com Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwww.hindawi.com Volume 2018

Probability and StatisticsHindawiwww.hindawi.com Volume 2018

Journal of

Hindawiwww.hindawi.com Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwww.hindawi.com Volume 2018

OptimizationJournal of

Hindawiwww.hindawi.com Volume 2018

Hindawiwww.hindawi.com Volume 2018

Engineering Mathematics

International Journal of

Hindawiwww.hindawi.com Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwww.hindawi.com Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwww.hindawi.com Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwww.hindawi.com Volume 2018

Hindawi Publishing Corporation http://www.hindawi.com Volume 2013Hindawiwww.hindawi.com

The Scientific World Journal

Volume 2018

Hindawiwww.hindawi.com Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwww.hindawi.com Volume 2018

Hindawiwww.hindawi.com

Di�erential EquationsInternational Journal of

Volume 2018

Hindawiwww.hindawi.com Volume 2018

Decision SciencesAdvances in

Hindawiwww.hindawi.com Volume 2018

AnalysisInternational Journal of

Hindawiwww.hindawi.com Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwww.hindawi.com