EffectoftheTime-VaryingDampingontheVibrationIsolationofa ...downloads.hindawi.com/journals/sv/2020/4373828.pdfnonlinear damping on the vibration transmissibility. e group concluded

Research ArticleEffectof theTime-VaryingDampingontheVibrationIsolationofaQuasi-Zero-Stiffness Vibration Isolator

Xin Li ,1,2 Jinqiu Zhang,1 and Jun Yao3

1Army Academy of Armored Forces, Beijing, China2Naval Research Academy, Beijing, China3Beijing Institute of Tracking and Telecommunications Technology, China

Correspondence should be addressed to Xin Li; [email protected]

Received 13 January 2020; Accepted 16 April 2020; Published 8 May 2020

Academic Editor: Jean-Jacques Sinou

Copyright © 2020 Xin Li et al. *is is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*is study focuses on the effect of damping changes on the vibration isolation of a quasi-zero-stiffness vibration isolator. A nonlinear-vibration equation for the quasi-zero-stiffness vibration isolator is found and solved using the multiscale method.*en, the vibrationcharacteristics before, in the process of and after the damping change, are also examined.*e results show that time-varying dampingcan be equivalent to the addition of a stiffness term to the vibration system, which leads to a change of the vibration amplitudefrequency response, leakage of power spectrum, and corresponding linear spectrum features being weakened. When the dampingchanges rapidly, the vibration system tends to be divergent rather than stable. After the change, the number of stable focuses of theproposed quasi-zero-stiffness vibration isolator increases from one to two, and the system will see decline in its vibration stability.

1. Introduction

In a floating raft isolation system, the raft frame itself caneffectively isolate high-frequency vibration, but plays a limitedrole in isolating low-frequency vibration. Since the low-fre-quency vibration travels long distance and is easy to detect,there has been extensive research on the measures to effec-tively reduce it. Among them, a quasi-zero-stiffness vibrationisolator is needed for low-frequency vibration isolationthrough reducing the inherent frequency of the system. [1, 2].

Quasi-zero-stiffness vibration isolators have been ex-tensively studied as well. From the perspective of applica-tion, Valeev et al. designed a quasi-zero-stiffness vibrationisolator for oil/gas transporters and analyzed the low-fre-quency vibration isolation performance [3]. From the per-spective of vibration, Lan et al. designed and tested a kind ofvibration isolator with a compact structure that can beardifferent masses [4]. Cheng et al. analyzed the vibrationcharacteristics of a quasi-zero-stiffness vibration isolator atthe primary resonance point and the 1/3 resonance pointunder a constant external force. *ey found graduallysoftening properties near the primary resonance point anddecreasing 1/3 resonance band under the action of a

constant external force. [5]. Xu et al. designed an electro-magnetically adjustable quasi-zero-stiffness vibration iso-lator and studied its vibration isolation using boththeoretical and experimental results. According to theirstudy, the designed quasi-zero-stiffness isolator showed ahigher vibration isolation efficiency than linear isolators [6].Huang et al. examined the vibration isolation properties of aquasi-zero-stiffness isolator during vibration control [7]. Liet al. designed a device similar to a quasi-zero-stiffnessisolator and examined its characteristics [8]. Kovacica et al.investigated the vibration isolation of a quasi-zero-stiffnessisolator and analyzed the nonlinear-vibration features thatmay appear during the vibration process, such as bifurcationand chaos [9]. *ere are more examples, than the above-mentioned, of those studies on the vibration isolationcharacteristics of quasi-zero-stiffness isolators, includingsuch study in the context of underload/overload [10], subjectto sinusoidal and stochastic excitations [11], and with aquasi-zero-stiffness isolator consisting of compound-shapememory alloys [12]. All of these studies focused on the effectof stiffness changes on the vibration isolation performance,which is in line with the main features of quasi-zero-stiffnessisolators. However, damping is a parameter of equal

HindawiShock and VibrationVolume 2020, Article ID 4373828, 10 pageshttps://doi.org/10.1155/2020/4373828

mailto:[email protected]://orcid.org/0000-0002-2266-142Xhttps://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2020/4373828

importance to stiffness in a vibration system, which alsorequires detailed studies.

*ere has been very little research so far on the effect ofdamping changes on the vibration isolation performance ofthe quasi-zero-stiffness vibration isolator. Liu et al. exam-ined the effect of nonlinear stiffness and nonlinear dampingon the vibration isolation performance of a quasi-zero-stiffness isolator [13]. Cheng et al. discussed the effect ofnonlinear damping on the vibration transmissibility. *egroup concluded that nonlinear damping reduces the vi-bration transmissibility significantly [14]. Amabili derivedaccurately, for the first time, nonlinear damping from afractional viscoelastic solid standard model by consideringgeometric nonlinearity [15]. Mofidian and Bardaweel fo-cused on investigating the effect of nonlinear cubic viscousdamping in a vibration isolation system, which consists of amagnetic spring with a positive nonlinear stiffness and amechanical oblique spring with geometric nonlinear nega-tive stiffness [16]. Even though the effect of damping wasconsidered, damping was still only at a constant value in theabovementioned studies. In order to enhance the vibrationisolation performance of the isolator against low-frequencyvibration as much as possible, both damping and stiffnessshould be adjustable [17]. To this end, magnetorheological(MR) damper is the currently preferred option [18].

Magnetorheological fluid is “smart” material. Despite ad-justable, the damping with this material is significantly influ-enced by temperature [19, 20]. With the magnetorheologicalfluid as the absorber, when the magnetic field and the tem-perature field are in parallel, the heat transfer rate increases by105% [21]. As the temperature exceeds 100°C, the dampingdecreases rapidly [22]. When the fluid is used as a brake, thetemperature should be strictly controlled to maintain thedamping performance [23–25]. Likewise, magnetorheologicalelastomers have similar properties [26–29]. While previousresearch revealed a feature that damping is a variable as thetemperature changes, rather than a constant, it did not considerthis feature. To enhance the isolation performance of vibrationisolators against low-frequency vibration, this study combines aquasi-zero-stiffness vibration isolator with a magneto-rheological damper and analyzes the isolation performance ofthe quasi-zero-stiffness isolator against low-frequency vibra-tion considering damping as a time-varying parameter.

2. Vibration Model for a Quasi-Zero-StiffnessVibration Isolator

In this study, the designed quasi-zero-stiffness vibrationisolator consists of two parts: a quasi-zero-stiffness isolatorbody and a magnetorheological damper (the dashed linepart), whose structure is shown in Figure 1. Here, m is thebearing mass, k is the stiffness, δ is the nonlinear stiffness,c(τ) is the magnetorheological time-varying damping (whereτ is the slow-scale time), ξ is the nonlinear damping coef-ficient, F is the excitation force amplitude,Ω is the excitationfrequency, and z is the vertical displacement of the rotor.

*emechanism of the vibration isolator can be describedas follows. *e vibrator moves vertically upon excitation.*e magnetic teeth of the stator and the rotor produce a

relative displacement, the magnitude of the electromagneticforce changes, and the electromagnetic stiffness changes.*eelectromagnetic stiffness and spring stiffness together rep-resent a nonlinear stiffness via a cubic term.*eMR damperof the isolator provides damping. Due to current heating,damping changes over time.

Using Newton’s Second Law of Motion, the vibrationequation can be written as follows:

md2z

dt2+

d[c(τ)z]dt

+ ξdz

dt

3

+ kz + δz3 � F cos(Ωt). (1)

Since the slow-varying time can be formulated as τ � εt,where ε is the perturbation parameter, the following ex-pression can be derived: (d/dt) � (d/dτ)(dτ/dt) � ε(d/dτ).Accordingly, equation (1) is simplified to

m€z + c(τ)z.

+ εc(τ)z.

+ ξ(z.)3

+ kz + δz3 � F cos(Ωt),(2)

where €z � (d2z/dt2) and z.

� (dz/dt).By comparing the vibration equation with a constant

damping, the vibration equation of the quasi-zero-stiffnessvibration isolator using time-varying damping is added witha stiffness item εc(τ)z, and the stiffness coefficient is afunction of time-varying damping εc(τ). As the dampingchanges, the inherent vibrational frequency changes.

*rough further processing, equation (2) can be re-written as follows:

z..

+ 2ςwz.

+ 2μw(z.)3

+(2ες + 1).

wz + βz3 � f cos(Ωt).(3)

Here, ς � (c(τ)/2wm), ς.

� (c(τ).

/2wm), μ � (ξ/2wm),w2 � (k/m), β � (δ/m), and f � (F/m).

2.1. Solution to the Equation with a Constant Damping.When the damping is constant, the abovementionedequation can be solved using the multiscale method. Sinceequation (3) cannot be solved accurately adopting a

9

10

1 2 3 4 5 6

7

8

Figure 1: Structure of the proposed quasi-zero-stiffness vibrationisolator:① base;② stator;③ intermediate shaft;④ valve body;⑤shell; ⑥ rotor ⑦ casing; ⑧ supporting spring; ⑨ support; ⑩supporting plate.

2 Shock and Vibration

numerical method, this study uses the multiscale method,introduces a small perturbation parameter ε, and performsthe following scale transformation:

ς⟶ ες,μ⟶ εμ,β⟶ εβ,f⟶ εf.

(4)

By substituting equation (4) into equation (3) andretaining ε0 and ε1, the following expression can be derived:

z..

+ 2εςwz.

+ 2εμw(z.)3

+(2ες + 1).

wz + εβz3 � εf cos(Ωt).(5)

Using the multiscale method, it was assumed that thesolution to the equation could be expressed as follows:

z(t, ε) � z0 T0, T1( + εz1 T0, T1( + Ο ε2

. (6)

Here, T0 is the fast-time scale (T0 � t) and T1 is the slow-time scale (T1 � εt).

Near the quasi-zero-stiffness resonance point, the ex-citation frequency is

Ω � w + εσ. (7)

Here, σ is the adjusting parameter that causes Ω toapproach w.

*e differential operator can be written as follows:ddt

(·) � D0 + εD1( (·),

d2

dt2(·) � D

20 + 2εD0D1 (·),

(8a)

D0 �z

zT0,

D1 �z

zT1.

(8b)

By substituting equations (6) to (8a) and (8b) intoequation (5) and comparing the same-order coefficients of εfor both two sides, the following expression can be derived:

ε0: D20z0 + w2z0 � 0, (9)

ε1: D20z1 + w2z1 � − 2D0D1z0 − 2ςwD0z0 − 2μw D0z0(

3

− 2ς.wz0 − β z0(

3+ f cos ΩT0( .

(10)

*e general solution to equation (9) can be written asfollows:

z0 � AeiwT0 + Ae

− iwT0 , (11)

where A is a function of slow time T1 and A is the conjugatecomplex of A.

By substituting equation (11) into equation (10) andeliminating the secular term, the following expression can beacquired:

− 2A′iw − 2ςwAiw − 6μwiA2Aw3 − 2ς.wA

− 3βA2A +12

feiσT1 � 0.

(12)

A can be rewritten in the following polar form:

A �12

aeiθ

. (13)

Here, the real numbers a and θ are functions of slow timeT1.

By substituting equation (13) into equation (12) andseparating the real part from the imaginary part, the fol-lowing expressions can be obtained:

a′ � h1a + h2a3

+ h5 sin c, (14a)

ac′ � h3a + h4a3

+ h5 cos c. (14b)

Here, h1 � − wς, h2 � − (3/4)μw3, h3 � − σ + ς.,

h4 � (3/8)(β/w), h5 � − (1/2)(f/w), and c � θ − σT1.

2.2. Solution to the Equation with Varying Damping. In theprocess of damping changes, multiscaling is no longer ap-plicable. *e Runge–Kutta method is used to solve theamplitude frequency characteristic equation of quasi-zero-stiffness isolator.

3. Stability Analysis of the Solution

3.1.Amplitude FrequencyResponse and the Solution’s StabilityRegion. Let a′ � 0 and c′ � 0 in equations (14a) and (14b),and the solution corresponding to the stable state of thequasi-zero-stiffness vibration isolator can be obtained.Equations (14a) and (14b) can then be simplified as follows:

h52

− h1a + h2a3

2

− h3a + h4a3

2

� 0. (15)

*ere may be one or three solutions to equation (15). Inthe case of three solutions, a saddle-node bifurcationappeared and the frequency response curves jump duringthe vibration. In the case of only one solution, the criticalamplitude can be solved as follows:

h5 stable �

���������������������������������

827 h22 + h24(

2 h1h4h2 −

�3

√h4

h4 +�3

√h2

− h1h2

3

.

(16)

According to equation (16), the critical excitation am-plitude is related to damping coefficient and stiffness.

*e stability of the stationary vibration solution of thequasi-zero-stiffness vibration isolator can be treated as thestability of the autonomous system at the singular point(a, r).*erefore, the system can be treated as a linear system.

By linearizing equations (14a) and (14b) at the singularpoint (a, r), the autonomous differential equations of thedisturbing quantities Δa and Δr can be written as follows:

Shock and Vibration 3

dΔadT1

� h1Δa + 3h2a2Δa + h5 cos cΔc, (17a)

adΔcdT1

� h3Δa + 3h4a2Δa − h5 sin cΔc. (17b)

Considering that a′ � c′ � 0, c in equations (17a) and(17b) can be eliminated and the characteristic equation canbe written as follows:

deth1 + 3h2a2 − λ

M

N

h1 + h2a2 − λ

⎡⎣ ⎤⎦ � 0, (18)

where M � (1/a)[h3 + 3h4a2] and N � − a[h3 + h4a2].Expanding equation (18) yields

λ2 − 2h1 + 4h2a2

λ + h12

+ h32

+ 4 h1h2 + h3h4( a2

+ 3 h22

+ h42

a4

� 0.(19)

In the case of 2h1 + 4h2a2 < 0, the instability condition ofthe stationary solution can be written as follows:

h12

+ h32

+ 4 h1h2 + h3h4( a2

+ 3 h22

+ h42

a4 < 0. (20)

*e stable region of the solution can now be evaluatedaccording to equation (20).

3.2. Stability of the Bifurcation Solution. Excitation onlychanges the position of the dynamic bifurcation equilibriumpoint for a slow-time scale τ, during which the bifurcationproperties remain unchanged. *erefore, the effect of theexcitation is not considered when investigating the dynamicbifurcation properties of the quasi-zero-stiffness vibrationisolators. Let h5 � 0, and equations (14a) and (14b) can berewritten as follows:

a′ � h1a + h2a3, (21a)

c′ � h3 + h4a2. (21b)

Next, this paper focuses on equation (21a). Let

z(a) �def

h1a + h2a3

� 0. (22)

Two solution curves intersecting at (0, 0) can thus beacquired:

a � 0, (23a)

a �

���

−h1

h2

. (23b)

*e abovementioned equations provide dots and circlesin polar coordinates, which correspond, respectively, to theequilibrium point and the limit cycle of the two-dimensionalsystem of equations (21a) and (21b). Next, the stability isconsidered. Let

z′(a) �dz

da� h1 + 3h2a

2. (24)

For the trivial solution a � 0: z′(0) � h1, when h1 > 0, (0,0) is unstable; when h1 < 0, a � 0 is asymptotically stable.

For the nontrivial solution a �������− h1/h2

:z′(

������− h1/h2

) �

− 2h1, when h1 > 0 the system has an asymptotically stablesolution. However, when h1 < 0, the system has an unstablesolution.*e former case corresponds to supercritical pitchforkbifurcation, and the latter subcritical pitchfork bifurcation.

Magnetorheological damping drops slowly with in-creasing temperature. When damping of the quasi-zero-stiffness isolator is positive, the system has a trivial solution:a � 0. Assuming that damping can be reduced to a negativevalue, a nontrivial solution appears for the systema �

������− h1/h2

to form a limit cycle. In the nontrivial solution

zone, on account of the constant expansion of the limit cycle,the vibration state that originally approached the stable focustends to be divergent. Accordingly, the stability of the vi-bration system changes.

4. Numerical Analysis

*e related parameters of the designed quasi-zero-stiffnessvibration isolator are listed as follows:m� 75kg, k� 200,000N/m, δ � 90,000N/m, and ξ � 0.1Ns/m. During the vibration, thetemperature of the magnetorheological damper rises, whiledamping decreases with time and finally stabilizes.*e variationof damping can be expressed via the following function:

c(τ) � 50 − rτ2. (25)

Here, r is a parameter to describe the change of dampingwith temperature. Specifically, c(τ)> 0.

4.1.Analysis ofVibrationCharacteristics beforeAnyChange inDamping. According to equation (16), the maximum ex-citation acceleration corresponding to no bifurcation undera stable state is fstable � 2wh5_stable � 10.88m/s2. When theexcitation acceleration exceeds this maximum value, theamplitude frequency curve appears.

As shown in Figure 2, when the excitation acceleration isbelow fstable, each frequency in the frequency response curvecorresponds to an amplitude, and the amplitude frequencycurve, as the frequency sweeps downward, is identical with thecurve as the frequency sweeps upward. When the excitationacceleration exceeds fstable, a jump occurs in the frequencyresponse curve. As the frequency sweeps upward, the am-plitude increases and jumps vertically to a corresponding lowpoint upon arriving at the maximum and then drops grad-ually with increasing frequency. As the frequency sweepsdownward, the amplitude increases gradually and jumpsvertically to a corresponding high point when arriving at theinflection point and then decreases with decreasing frequency.In other words, a bifurcation-induced jump of the excitationfrequency can be found in the frequency response curve. *eamplitude frequency curves are different as the frequencysweeps downward or upward. As the excitation frequencyincreases, the vibration amplitudes in all frequency bandsincrease, and the unstable frequency band expands.

Figures 3 and 4 show the frequency response curves forf� 25m/s2. In Figure 3, δ � 90,000N/m, while in Figure 4

4 Shock and Vibration

ξ � 0.1Ns/m. According to Figure 3, the unstable frequencyband occupies a larger area at a smaller damping coefficient.Unlike Figure 2, as the damping coefficient drops, only theamplitude around the resonance frequency band increases,while the amplitudes of the other frequency bands remainalmost unchanged. *is behavior suggests that the change ofthe damping coefficient increases the amplitude around theresonant frequency point and expands the unstable fre-quency band and that it imposes slight effect on the vibrationof the frequency band far away from the resonance point.*e increase of nonlinear stiffness (Figure 4) does not lead tothe vibration amplitude increase, but shifts the resonancepoint towards the right, namely, the unstable frequency bandappears.

Figures 5 and 6 illustrates the results, assuming non-linear stiffness δs � 90,000N/m and nonlinear dampingξs � 0.1Ns/m, respectively. Based on Figure 3 analysis, thevibration amplitude decreases as the damping coefficientincreases, accompanied by fewer solutions in the unstable

region and enhanced system stability. As shown in Figure 5,when the nonlinear damping coefficient increases, the un-stable region decreases along the direction of the saddle

Increase of nonlinear damping coefficient

ξ = 0.06, 0.08, 0.10Ns/m

0.3

0.35

0.4

0.45

0.5

Am

plitu

de o

f sta

ble-

state

resp

onse

of a

(mm

)

51.75 51.8 51.8551.7Force frequency Ω (rad/s)

Figure 3: Effect of nonlinear damping on the stable vibrationsolution.

Increase of nonlinear stiffness

δ = 7, 8, 9 ×104N/m

0.25

0.3

0.35

0.4

0.45

Am

plitu

de o

f sta

ble-

state

resp

onse

of a

(mm

)

51.75 51.851.7Force frequency Ω (rad/s)

Figure 4: Effect of nonlinear stiffness on the stable vibration solution.

Increase of excitation

f = 10, 25, 40m/s2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Am

plitu

de o

f sta

ble-

state

resp

onse

of a

(mm

)

51.6 51.851.4 52Force frequency Ω (rad/s)

Figure 2: Effect of excitation acceleration on the stable vibrationsolution.

Increase of nonlinear damping coefficient

ξ = 0.06, 0.08, 0.10Ns/m

Stable Unstable

0.25

0.3

0.35

0.4

0.45

0.5

0.55A

mpl

itude

of s

tabl

e-sta

te re

spon

se o

f a (m

m)

51.72 51.74 51.76 51.78 51.8 51.82 51.8451.7Force frequency Ω (rad/s)

Figure 5: Effect of nonlinear damping on the solution’s stabilityregion.

Increase of nonlinear stiffness

δ = 7, 8, 9 ×104N/m

StableUnstable

0.2

0.3

0.4

0.5

0.6

Am

plitu

de o

f sta

ble-

state

resp

onse

of a

(mm

)

51.7551.7 51.8551.8Force frequency Ω (rad/s)

Figure 6: Effect of nonlinear stiffness on the solution’s stability region.

Shock and Vibration 5

node. As described above, based on Figure 4, the peak vi-bration amplitude in the amplitude frequency curve movesrightward as the nonlinear stiffness increases. In addition,the number of solutions that fall in the unstable regionincreases, and the system stability weakens. As shown inFigure 6, with the increase of nonlinear stiffness, the unstableregion as a whole moves downward and also expandsgradually.

*is study assumed that wdown and wupper are the bi-furcation critical upper-limit and lower-limit frequencies,respectively. As the excitation frequency Ω approacheswdown, the vibrations under all initial conditions are attractedto the stable focus P1; once Ω>wdown, another stable focusP3 appears (see Figure 7(a) and Figure 7(b)). According toFigure 7(c), when wdown

leakage becomes more obvious as the damping changesmore rapidly and vice versa.

Assuming that the damping coefficient can be negative,the vibration time domain and phase diagrams at differentchanging rates are plotted (see Figure 8). When the dampingchanges slowly (r� 7) and still higher than 0, the systemvibration tends to be stable and no limit cycle can be ob-served, see Figures 8(a) and 8(b). When damping changessubstantially and becomes negative after certain time, a limitcycle appears in the system. As shown in Figures 8(c) and

8(e), damping on the left side of the dashed line is positive,while damping on the right side is negative. Limit cyclesappear in Figures 8(d) and 8(f). Moreover, from Figure 8(c)and 8(e), we know the minimum amplitudes at differentdamping changing rates are different and the minimumamplitude is larger at a larger rate. After the damping be-comes negative, the limit cycle forms and the vibrationapproaches the limit cycle. *e gradual expansion of thelimit cycle results in the divergence of vibration, but thevibration never exceeds the limit cycle. By comparing

–0.5

0

0.5A

mpl

itude

of a

(mm

)

86 100 2 4Time t (s)

(a)

–30

–20

–10

0

10

20

30

Vel

ocity

of v

a (m

m/s

)

0.5–0.5 0Amplitude of a (mm)

(b)

–0.5

0

0.5

Am

plitu

de o

f a (m

m)

86 100 2 4Time t (s)

(c)

–30

–20

–10

0

10

20

30

Vel

ocity

of v

a (m

m/s

)

0.5–0.5 0Amplitude of a (mm)

(d)

–1.5

–1

–0.5

0

0.5

1

1.5

Am

plitu

de o

f a (m

m)

86 100 2 4Time t (s)

(e)

–100

–50

0

50

100

Vel

ocity

of v

a (m

m/s

)

0.5–0.5 1.5–1.5 0–1 1Amplitude of a (mm)

(f )

Figure 8: Time domain and phase diagrams: (a) r� 7, (b) r� 7, (c) r� 400, (d) r� 400, (e) r� 700, and (f) r� 700.

Shock and Vibration 7

Figures 8(a), 8(c), and 8(e), due to different damping changerates with a same period of time, the initially identical vi-bration finally undergoes different changes: one tends to bestable and the other becomes divergent.

4.3. Analysis of the Vibration Characteristics after theDamping Change Stabilizes. During the vibration process,heat production and heat dissipation of the damper even-tually reach a dynamic equilibrium. Damping stabilizeswhen temperature reaches a certain value. Although externalbearingmass, excitation amplitude, and excitation frequencyremain unchanged, damping changes. *erefore, the finalstable state of some initial vibrations changes. As thedamping changed from 50 to 20Ns/m, the vibration state inthe intermediate frequency band, within the blue dashedlines in Figure 10, also varies.

Figure 11 shows the vibration states, before and after thedamping changes, which is what is within the dashed lines inFigure 10. At the damping of 50Ns/m, the quasi-zero-stiffness vibration isolator has only one solution and a stable

vibration state. As the damping drops to 20Ns/m, the quasi-zero-stiffness vibration isolator has three solutions, and thesolution in/on the intermediate branch is unstable, i.e., thereare two possibilities for the vibration’s stable state. *isindicates that time-varying damping induces the change ofthe system’s vibration state.

5. Conclusions

*is study analyzed the effect of time-varying damping onthe vibration characteristics of a quasi-zero-stiffness vibra-tion isolator, by establishing a nonlinear vibration equationand solving this equation with a multiscale method. *efollowing main conclusions can be drawn:

(a) In a system with quasi-zero-stiffness vibration iso-lators, the smaller nonlinear damping or the highernonlinear stiffness or the higher excitation ampli-tude, the wider unstable frequency band and thelarger unstable region. *is results in weakenedsystem vibration stability.

(b) Time-varying damping is equivalent to the stiffnessterm of the vibration system in terms of effect.Because of the stiffness change, the amplitude of themain resonance peak drops, accompanied by the

r = 4 r = 7

51.4 51.6 51.8 52

0.5

0.4

0.3

0.2

0.1

0Am

plitu

de o

f sta

ble-

stat

e-re

spon

se o

f a (m

m)

Force frequency Ω (rad/s)

Figure 9: Amplitude frequency responses at different rates ofdamping change.

0.6

0.5

0.4

0.3

0.2

0.1

0Am

plitu

de o

f sta

ble-

stat

e res

pons

e of a

(mm

)

51.4 51.6 51.8Force frequency Ω (rad/s)

52

Figure 10: Amplitude frequency responses before and afterdamping changes.

c = 50Ns/mP3

–4

–2

0

2

4

6

8

Phas

e Υ (r

ad/s

)

0.2 0.4 0.6 0.80Amplitude of stable - state response of a (mm)

(a)

c = 20Ns/mP3

P1

–10

–5

0

5

Phas

e Υ (r

ad/s

)

0.2 0.4 0.6 0.80Amplitude of stable - state response of a (mm)

(b)

Figure 11: Stable focus before and after damping changes:(a) c� 50Ns/m and (b) c� 20Ns/m.

8 Shock and Vibration

leakage of energy and the less clear linear spectralcharacteristics. After the temperature stabilizes at acertain value, the vibration characteristics resume.

(c) Assuming that damping drops and turns negative, itschange rate affects the final stable state. At a greaterdamping-change rate, the minimum vibration am-plitude tends to be bigger. Furthermore, the finalvibration state, which originally would approachstabilization, tends to be divergent.

(d) Time-varying damping reduces the vibration sta-bility of the quasi-zero-stiffness vibration isolator.Under the condition of changing damping, thenumber of the final stable focus points, which cor-respond to the same initial vibration state, increasesfrom 1 to 2, and the system stability declines.

Data Availability

No data were used to support this study.

Disclosure

*e authors would like to declare, on behalf of co-authors,that the work described was original research which has notbeen published previously and is not under consideration forpublication elsewhere, in whole or in part.

Conflicts of Interest

*e authors declare that they have no conflicts of interest.

Authors’ Contributions

All listed authors have approved the manuscript.

References

[1] J. Zhou, X. Wang, D. Xu, and S. Bishop, “Nonlinear dynamiccharacteristics of a quasi-zero stiffness vibration isolator withcam-roller-spring mechanisms,” Journal of Sound and Vi-bration, vol. 346, pp. 53–69, 2015.

[2] Y. Han, Q. Cao, and J. Ji, “Nonlinear dynamics of a smoothand discontinuous oscillator with multiple stability,” Inter-national Journal of Bifurcation and Chaos, vol. 25, no. 13,Article ID 1530038, 2015.

[3] A. R. Valeev, “Vibration isolators for oil- and gas-transferequipment with a low vibration frequency,” Chemical andPetroleum Engineering, vol. 47, no. 5-6, pp. 374–377, 2011.

[4] C.-C. Lan, S.-A. Yang, and Y.-S. Wu, “Design and experimentof a compact quasi-zero-stiffness isolator capable of a widerange of loads,” Journal of Sound and Vibration, vol. 333,no. 20, pp. 4843–4858, 2014.

[5] C. Cheng, S. Li, Y. Wang, and X. Jiang, “Resonance responseof a quasi-zero stiffness vibration isolator considering aconstant force,” Journal of Vibration Engineering & Tech-nologies, vol. 6, no. 6, pp. 471–481, 2018.

[6] D. Xu, Q. Yu, J. Zhou, and S. R. Bishop, “*eoretical andexperimental analyses of a nonlinear magnetic vibrationisolator with quasi-zero-stiffness characteristic,” Journal ofSound and Vibration, vol. 332, no. 14, pp. 3377–3389, 2013.

[7] D. Huang, W. Xu, W. Xie et al., “Dynamical properties of aforced vibration isolation system with real-power nonline-arities in restoring and damping forces,”Nonlinear Dynamics,vol. 81, no. 1-2, pp. 641–658, 2015.

[8] H. Li and J. Zhang, “Design and analysis of a magnetic QZSvibration isolator,” Applied Mechanics andMaterials, vol. 470,pp. 484–488, 2014.

[9] I. Kovacic, M. J. Brennan, and T. P. Waters, “A study of anonlinear vibration isolator with a quasi-zero stiffnesscharacteristic,” Journal of Sound and Vibration, vol. 315, no. 3,pp. 700–711, 2008.

[10] L. Meng, J. Sun, and W. Wu, “*eoretical design and char-acteristics analysis of a quasi-zero stiffness isolator using adisk spring as negative stiffness element,” Shock and Vibra-tion, vol. 2015, Article ID 813763, , 2015.

[11] T. Yang and Q. Cao, “Nonlinear transition dynamics in atime-delayed vibration isolator under combined harmonicand stochastic excitations,” Journal of Statistical Mechanics:@eory and Experiment, vol. 2017, Article ID 043202, 2017.

[12] Y. Araki, K. Kimura, T. Asai, T. Masui, T. Omori, andR. Kainuma, “Integrated mechanical and material design ofquasi-zero-stiffness vibration isolator with superelastic Cu-Al-Mn shape memory alloy bars,” Journal of Sound andVibration, vol. 358, pp. 74–83, 2015.

[13] Y. Liu, L. Xu, C. Song, H. Gu, and W. Ji, “Dynamic char-acteristics of a quasi-zero stiffness vibration isolator withnonlinear stiffness and damping,” Archive of Applied Me-chanics, vol. 89, no. 9, pp. 1743–1759, 2019.

[14] C. Cheng, S. Li, Y. Wang, and X. Jiang, “Force and dis-placement transmissibility of a quasi-zero stiffness vibrationisolator with geometric nonlinear damping,” Nonlinear Dy-namics, vol. 87, no. 4, pp. 2267–2279, 2016.

[15] M. Amabili, “Nonlinear damping in large-amplitude vibra-tions: modelling and experiments,” Nonlinear Dynamics,vol. 93, no. 1, pp. 5–18, 2017.

[16] S. M. M. Mofidian and H. Bardaweel, “Displacement trans-missibility evaluation of vibration isolation system employingnonlinear-damping and nonlinear-stiffness elements,” Jour-nal of Vibration and Control, vol. 24, no. 18, pp. 4247–4259,2017.

[17] S. Sun, X. Tang, J. Yang et al., “A new generation of mag-netorheological vehicle suspension system with tunablestiffness and damping characteristics,” IEEE Transactions onIndustrial Informatics, vol. 15, no. 8, pp. 4696–4708, 2019.

[18] Y. H. Guo, E. W. Chen, Q. Wu, Y. M. Lu, and Z. Q. Xia, “*ecalculation of the equivalent linear damping coefficient of themagnetorheological damper,” Applied Mechanics and Mate-rials, vol. 336-338, pp. 475–479, 2013.

[19] M. McKee, X. Wang, and F. Gordaninejad, “Effects of tem-perature on performance of a compressible magneto-rheological fluid damper-liquid spring suspension system,”Active and Passive Smart Structures and Integrated Systems,vol. 2011, no. 7977, Article ID 797712, 2011.

[20] P. Mitrouchev, A. Klevinskis, V. Bucinskas et al., “Analyticalresearch of damping efficiency and heat generation of mag-netorheological damper,” Smart Materials and Structures,vol. 26, Article ID 065026, 2017.

[21] J. Maroofi and S. H. Hashemabadi, “Experimental and nu-merical investigation of parameters influencing anisotropicthermal conductivity of magnetorheological fluids,” Heat andMass Transfer, vol. 55, no. 10, pp. 2751–2767, 2019.

[22] S. Chen, J. Huang, K. Jian et al., “Analysis of influence oftemperature on magnetorheological fluid and transmission

Shock and Vibration 9

performance,” Advances in Materials Science and Engineering,vol. 2015, Article ID 583076, 2015.

[23] D. M. Wang, Y. F. Hou, and Z. Z. Tian, “A novel high-torquemagnetorheological brake with a water cooling method forheat dissipation,” Smart Materials and Structures, vol. 22,Article ID 025019, 2013.

[24] S. R. Patil, K. P. Powar, and S. M. Sawant, “*ermal analysis ofmagnetorheological brake for automotive application,” Ap-plied @ermal Engineering, vol. 98, pp. 238–245, 2016.

[25] R. Russo and M. Terzo, “Design of an adaptive control for amagnetorheological fluid brake with model parametersdepending on temperature and speed,” Smart Materials andStructures, vol. 20, Article ID 115003, 2011.

[26] J. Cazenove, D. A. Rade, A. M. G. Lima et al., “A numericaland experimental investigation on self-heating effects inviscoelastic dampers,” Mechanical Systems and Signal Pro-cessing, vol. 27, pp. 433–445, 2012.

[27] G. T. Ngatu, C. S. Kothera, W. Hu et al., “Effects of tem-perature onmagnetorheological fluid elastomeric lag dampersfor helicopter rotors,” 50th AIAA/ASME/ASCE/AHS/ASCstructures,” Structural Dynamics, and Materials Conference,vol. 17, pp. 1–16, 2009.

[28] V. P. Mikhailov, A. M. Bazinenkov, P. A. Dolinin, andG. V. Stepanov, “Research on the dynamic characteristics of acontrolled magnetorheological elastometer damper,” Instru-ments and Experimental Techniques, vol. 61, no. 3, pp. 427–432, 2018.

[29] Y. Zhong, J. Tu, Y. Yu, J. Xu, and D. Tan, “Temperaturecompensation in viscoelastic damper using magneto-rheological effect,” Journal of Sound and Vibration, vol. 398,pp. 39–51, 2017.

10 Shock and Vibration