# Effect of the Time-Varying Damping on the Vibration Isolation of a Quasi-Zero-Stiffness Vibration Isolator

Authors: Xin Li; Jinqiu Zhang; Jun Yao
Journal: Shock and Vibration
 (2020)

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## Abstract

This 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. Then, the vibration characteristics before, in the process of and after the damping change, are also examined. The results show that time-varying damping can be equivalent to the addition of a stiffness term to the vibration system, which leads to a change of the vibration amplitude frequency response, leakage of power spectrum, and corresponding linear spectrum features being weakened. When the damping changes rapidly, the vibration system tends to be divergent rather than stable. After the change, the number of stable focuses of the proposed quasi-zero-stiffness vibration isolator increases from one to two, and the system will see decline in its vibration stability.


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## Body

## 1. Introduction

In a floating raft isolation system, the raft frame itself can effectively isolate high-frequency vibration, but plays a limited role in isolating low-frequency vibration. Since the low-frequency vibration travels long distance and is easy to detect, there has been extensive research on the measures to effectively reduce it. Among them, a quasi-zero-stiffness vibration isolator is needed for low-frequency vibration isolation through reducing the inherent frequency of the system. [1, 2].Quasi-zero-stiffness vibration isolators have been extensively studied as well. From the perspective of application, Valeev et al. designed a quasi-zero-stiffness vibration isolator for oil/gas transporters and analyzed the low-frequency vibration isolation performance [3]. From the perspective of vibration, Lan et al. designed and tested a kind of vibration isolator with a compact structure that can bear different masses [4]. Cheng et al. analyzed the vibration characteristics of a quasi-zero-stiffness vibration isolator at the primary resonance point and the 1/3 resonance point under a constant external force. They found gradually softening properties near the primary resonance point and decreasing 1/3 resonance band under the action of a constant external force. [5]. Xu et al. designed an electromagnetically adjustable quasi-zero-stiffness vibration isolator and studied its vibration isolation using both theoretical and experimental results. According to their study, the designed quasi-zero-stiffness isolator showed a higher vibration isolation efficiency than linear isolators [6]. Huang et al. examined the vibration isolation properties of a quasi-zero-stiffness isolator during vibration control [7]. Li et al. designed a device similar to a quasi-zero-stiffness isolator and examined its characteristics [8]. Kovacica et al. investigated the vibration isolation of a quasi-zero-stiffness isolator and analyzed the nonlinear-vibration features that may appear during the vibration process, such as bifurcation and chaos [9]. There are more examples, than the abovementioned, of those studies on the vibration isolation characteristics of quasi-zero-stiffness isolators, including such study in the context of underload/overload [10], subject to sinusoidal and stochastic excitations [11], and with a quasi-zero-stiffness isolator consisting of compound-shape memory alloys [12]. All of these studies focused on the effect of stiffness changes on the vibration isolation performance, which is in line with the main features of quasi-zero-stiffness isolators. However, damping is a parameter of equal importance to stiffness in a vibration system, which also requires detailed studies.There has been very little research so far on the effect of damping changes on the vibration isolation performance of the quasi-zero-stiffness vibration isolator. Liu et al. examined the effect of nonlinear stiffness and nonlinear damping on the vibration isolation performance of a quasi-zero-stiffness isolator [13]. Cheng et al. discussed the effect of nonlinear damping on the vibration transmissibility. The group concluded that nonlinear damping reduces the vibration transmissibility significantly [14]. Amabili derived accurately, for the first time, nonlinear damping from a fractional viscoelastic solid standard model by considering geometric nonlinearity [15]. Mofidian and Bardaweel focused on investigating the effect of nonlinear cubic viscous damping in a vibration isolation system, which consists of a magnetic spring with a positive nonlinear stiffness and a mechanical oblique spring with geometric nonlinear negative stiffness [16]. Even though the effect of damping was considered, damping was still only at a constant value in the abovementioned studies. In order to enhance the vibration isolation performance of the isolator against low-frequency vibration as much as possible, both damping and stiffness should be adjustable [17]. To this end, magnetorheological (MR) damper is the currently preferred option [18].Magnetorheological fluid is “smart” material. Despite adjustable, the damping with this material is significantly influenced by temperature [19, 20]. With the magnetorheological fluid as the absorber, when the magnetic field and the temperature field are in parallel, the heat transfer rate increases by 105% [21]. As the temperature exceeds 100°C, the damping decreases rapidly [22]. When the fluid is used as a brake, the temperature should be strictly controlled to maintain the damping performance [23–25]. Likewise, magnetorheological elastomers have similar properties [26–29]. While previous research revealed a feature that damping is a variable as the temperature changes, rather than a constant, it did not consider this feature. To enhance the isolation performance of vibration isolators against low-frequency vibration, this study combines a quasi-zero-stiffness vibration isolator with a magnetorheological damper and analyzes the isolation performance of the quasi-zero-stiffness isolator against low-frequency vibration considering damping as a time-varying parameter.


## 2. Vibration Model for a Quasi-Zero-Stiffness Vibration Isolator

In this study, the designed quasi-zero-stiffness vibration isolator consists of two parts: a quasi-zero-stiffness isolator body and a magnetorheological damper (the dashed line part), whose structure is shown in Figure1. Here, m is the bearing mass, k is the stiffness, δ is the nonlinear stiffness, c(τ) is the magnetorheological time-varying damping (where τ is the slow-scale time), ξ is the nonlinear damping coefficient, F is the excitation force amplitude, Ω is the excitation frequency, and z is the vertical displacement of the rotor.Figure 1
Structure of the proposed quasi-zero-stiffness vibration isolator: ① base; ② stator; ③ intermediate shaft; ④ valve body; ⑤ shell; ⑥ rotor ⑦ casing; ⑧ supporting spring; ⑨ support; ⑩ supporting plate.The mechanism of the vibration isolator can be described as follows. The vibrator moves vertically upon excitation. The magnetic teeth of the stator and the rotor produce a relative displacement, the magnitude of the electromagnetic force changes, and the electromagnetic stiffness changes. The electromagnetic stiffness and spring stiffness together represent a nonlinear stiffness via a cubic term. The MR damper of the isolator provides damping. Due to current heating, damping changes over time.Using Newton’s Second Law of Motion, the vibration equation can be written as follows:(1)mdz2dt2+dcτzdt+ξdzdt3+kz+δz3=FcosΩt.Since the slow-varying time can be formulated asτ = εt, where ε is the perturbation parameter, the following expression can be derived: d/dt=d/dτdτ/dt=εd/dτ. Accordingly, equation (1) is simplified to(2)mz¨+cτz.+εcτz.+ξz.3+kz+δz3=FcosΩt,where z¨=d2z/dt2 and z.=dz/dt.By comparing the vibration equation with a constant damping, the vibration equation of the quasi-zero-stiffness vibration isolator using time-varying damping is added with a stiffness itemεc(τ)z, and the stiffness coefficient is a function of time-varying damping εc(τ). As the damping changes, the inherent vibrational frequency changes.Through further processing, equation (2) can be rewritten as follows:(3)z..+2ςwz.+2μwz.3+2ες+1.wz+βz3=fcosΩt.Here,ς=cτ/2wm, ς.=cτ./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 abovementioned equation can be solved using the multiscale method. Since equation (3) cannot be solved accurately adopting a numerical method, this study uses the multiscale method, introduces a small perturbation parameter ε, and performs the following scale transformation:(4)ς⟶ες,μ⟶εμ,β⟶εβ,f⟶εf.By substituting equation (4) into equation (3) and retaining ε0 and ε1, the following expression can be derived:(5)z..+2εςwz.+2εμwz.3+2ες+1.wz+εβz3=εfcosΩt.Using the multiscale method, it was assumed that the solution to the equation could be expressed as follows:(6)zt,ε=z0T0,T1+εz1T0,T1+Οε2.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 excitation frequency is(7)Ω=w+εσ.Here,σ is the adjusting parameter that causes Ω to approach w.The differential operator can be written as follows:(8a)ddt⋅=D0+εD1⋅,d2dt2⋅=D02+2εD0D1⋅,(8b)D0=∂∂T0,D1=∂∂T1.By substituting equations (6) to (8a) and (8b) into equation (5) and comparing the same-order coefficients of ε for both two sides, the following expression can be derived:(9)ε0:D02z0+w2z0=0,(10)ε1:D02z1+w2z1=−2D0D1z0−2ςwD0z0−2μwD0z03−2ς.wz0−βz03+fcosΩT0.The general solution to equation (9) can be written as follows:(11)z0=AeiwT0+A¯e−iwT0,where A is a function of slow time T1 and A¯ is the conjugate complex of A.By substituting equation (11) into equation (10) and eliminating the secular term, the following expression can be acquired:(12)−2A′iw−2ςwAiw−6μwiA2A¯w3−2ς.wA−3βA2A¯+12feiσT1=0.A can be rewritten in the following polar form:(13)A=12aeiθ.Here, the real numbersa and θ are functions of slow time T1.By substituting equation (13) into equation (12) and separating the real part from the imaginary part, the following expressions can be obtained:(14a)a′=h1a+h2a3+h5sinγ,(14b)aγ′=h3a+h4a3+h5cosγ.Here,h1=−wς, h2=−3/4μw3, h3=−σ+ς., h4=3/8β/w, h5=−1/2f/w, and γ=θ−σT1.


### 2.2. Solution to the Equation with Varying Damping

In the process of damping changes, multiscaling is no longer applicable. The Runge–Kutta method is used to solve the amplitude frequency characteristic equation of quasi-zero-stiffness isolator.


## 2.1. Solution to the Equation with a Constant Damping

When the damping is constant, the abovementioned equation can be solved using the multiscale method. Since equation (3) cannot be solved accurately adopting a numerical method, this study uses the multiscale method, introduces a small perturbation parameter ε, and performs the following scale transformation:(4)ς⟶ες,μ⟶εμ,β⟶εβ,f⟶εf.By substituting equation (4) into equation (3) and retaining ε0 and ε1, the following expression can be derived:(5)z..+2εςwz.+2εμwz.3+2ες+1.wz+εβz3=εfcosΩt.Using the multiscale method, it was assumed that the solution to the equation could be expressed as follows:(6)zt,ε=z0T0,T1+εz1T0,T1+Οε2.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 excitation frequency is(7)Ω=w+εσ.Here,σ is the adjusting parameter that causes Ω to approach w.The differential operator can be written as follows:(8a)ddt⋅=D0+εD1⋅,d2dt2⋅=D02+2εD0D1⋅,(8b)D0=∂∂T0,D1=∂∂T1.By substituting equations (6) to (8a) and (8b) into equation (5) and comparing the same-order coefficients of ε for both two sides, the following expression can be derived:(9)ε0:D02z0+w2z0=0,(10)ε1:D02z1+w2z1=−2D0D1z0−2ςwD0z0−2μwD0z03−2ς.wz0−βz03+fcosΩT0.The general solution to equation (9) can be written as follows:(11)z0=AeiwT0+A¯e−iwT0,where A is a function of slow time T1 and A¯ is the conjugate complex of A.By substituting equation (11) into equation (10) and eliminating the secular term, the following expression can be acquired:(12)−2A′iw−2ςwAiw−6μwiA2A¯w3−2ς.wA−3βA2A¯+12feiσT1=0.A can be rewritten in the following polar form:(13)A=12aeiθ.Here, the real numbersa and θ are functions of slow time T1.By substituting equation (13) into equation (12) and separating the real part from the imaginary part, the following expressions can be obtained:(14a)a′=h1a+h2a3+h5sinγ,(14b)aγ′=h3a+h4a3+h5cosγ.Here,h1=−wς, h2=−3/4μw3, h3=−σ+ς., h4=3/8β/w, h5=−1/2f/w, and γ=θ−σT1.


## 2.2. Solution to the Equation with Varying Damping

In the process of damping changes, multiscaling is no longer applicable. The Runge–Kutta method is used to solve the amplitude frequency characteristic equation of quasi-zero-stiffness isolator.


## 3. Stability Analysis of the Solution

### 3.1. Amplitude Frequency Response and the Solution’s Stability Region

Leta′=0 and γ′=0 in equations (14a) and (14b), and the solution corresponding to the stable state of the quasi-zero-stiffness vibration isolator can be obtained. Equations (14a) and (14b) can then be simplified as follows:(15)h52−h1a+h2a32−h3a+h4a32=0.There may be one or three solutions to equation (15). In the case of three solutions, a saddle-node bifurcation appeared and the frequency response curves jump during the vibration. In the case of only one solution, the critical amplitude can be solved as follows:(16)h5_stable=827h22+h422h1h4h2−3h4h4+3h2−h1h23.According to equation (16), the critical excitation amplitude is related to damping coefficient and stiffness.The stability of the stationary vibration solution of the quasi-zero-stiffness vibration isolator can be treated as the stability of the autonomous system at the singular pointa¯,r¯. Therefore, the system can be treated as a linear system.By linearizing equations (14a) and (14b) at the singular point a¯,r¯, the autonomous differential equations of the disturbing quantities Δa¯ and Δr¯ can be written as follows:(17a)dΔa¯dT1=h1Δa¯+3h2a¯2Δa¯+h5cosγ¯Δγ¯,(17b)a¯dΔγ¯dT1=h3Δa¯+3h4a¯2Δa¯−h5sinγ¯Δγ¯.Considering thata′=γ′=0, γ¯ in equations (17a) and (17b) can be eliminated and the characteristic equation can be written as follows:(18)deth1+3h2a¯2−λMNh1+h2a¯2−λ=0,where M=1/a¯h3+3h4a¯2 and N=−a¯h3+h4a¯2.Expanding equation (18) yields(19)λ2−2h1+4h2a¯2λ+h12+h32+4h1h2+h3h4a¯2+3h22+h42a¯4=0.In the case of2h1+4h2a¯2<0, the instability condition of the stationary solution can be written as follows:(20)h12+h32+4h1h2+h3h4a¯2+3h22+h42a¯4<0.The stable region of the solution can now be evaluated according to equation (20).


### 3.2. Stability of the Bifurcation Solution

Excitation only changes the position of the dynamic bifurcation equilibrium point for a slow-time scaleτ, during which the bifurcation properties remain unchanged. Therefore, the effect of the excitation is not considered when investigating the dynamic bifurcation properties of the quasi-zero-stiffness vibration isolators. Let h5=0, and equations (14a) and (14b) can be rewritten as follows:(21a)a′=h1a+h2a3,(21b)γ′=h3+h4a2.Next, this paper focuses on equation (21a). Let(22)za=defh1a+h2a3=0.Two solution curves intersecting at (0, 0) can thus be acquired:(23a)a=0,(23b)a=−h1h2.The abovementioned equations provide dots and circles in polar coordinates, which correspond, respectively, to the equilibrium point and the limit cycle of the two-dimensional system of equations (21a) and (21b). Next, the stability is considered. Let(24)z′a=dzda=h1+3h2a2.For the trivial solutiona=0: z′0=h1, when h1>0, (0, 0) is unstable; when h1<0, a=0 is asymptotically stable.For the nontrivial solutiona=−h1/h2:z′−h1/h2=−2h1, when h1>0 the system has an asymptotically stable solution. However, when h1<0, the system has an unstable solution. The former case corresponds to supercritical pitchfork bifurcation, and the latter subcritical pitchfork bifurcation.Magnetorheological damping drops slowly with increasing 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 negative value, a nontrivial solution appears for the system a=−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 focus tends to be divergent. Accordingly, the stability of the vibration system changes.


## 3.1. Amplitude Frequency Response and the Solution’s Stability Region

Leta′=0 and γ′=0 in equations (14a) and (14b), and the solution corresponding to the stable state of the quasi-zero-stiffness vibration isolator can be obtained. Equations (14a) and (14b) can then be simplified as follows:(15)h52−h1a+h2a32−h3a+h4a32=0.There may be one or three solutions to equation (15). In the case of three solutions, a saddle-node bifurcation appeared and the frequency response curves jump during the vibration. In the case of only one solution, the critical amplitude can be solved as follows:(16)h5_stable=827h22+h422h1h4h2−3h4h4+3h2−h1h23.According to equation (16), the critical excitation amplitude is related to damping coefficient and stiffness.The stability of the stationary vibration solution of the quasi-zero-stiffness vibration isolator can be treated as the stability of the autonomous system at the singular pointa¯,r¯. Therefore, the system can be treated as a linear system.By linearizing equations (14a) and (14b) at the singular point a¯,r¯, the autonomous differential equations of the disturbing quantities Δa¯ and Δr¯ can be written as follows:(17a)dΔa¯dT1=h1Δa¯+3h2a¯2Δa¯+h5cosγ¯Δγ¯,(17b)a¯dΔγ¯dT1=h3Δa¯+3h4a¯2Δa¯−h5sinγ¯Δγ¯.Considering thata′=γ′=0, γ¯ in equations (17a) and (17b) can be eliminated and the characteristic equation can be written as follows:(18)deth1+3h2a¯2−λMNh1+h2a¯2−λ=0,where M=1/a¯h3+3h4a¯2 and N=−a¯h3+h4a¯2.Expanding equation (18) yields(19)λ2−2h1+4h2a¯2λ+h12+h32+4h1h2+h3h4a¯2+3h22+h42a¯4=0.In the case of2h1+4h2a¯2<0, the instability condition of the stationary solution can be written as follows:(20)h12+h32+4h1h2+h3h4a¯2+3h22+h42a¯4<0.The stable region of the solution can now be evaluated according to equation (20).


## 3.2. Stability of the Bifurcation Solution

Excitation only changes the position of the dynamic bifurcation equilibrium point for a slow-time scaleτ, during which the bifurcation properties remain unchanged. Therefore, the effect of the excitation is not considered when investigating the dynamic bifurcation properties of the quasi-zero-stiffness vibration isolators. Let h5=0, and equations (14a) and (14b) can be rewritten as follows:(21a)a′=h1a+h2a3,(21b)γ′=h3+h4a2.Next, this paper focuses on equation (21a). Let(22)za=defh1a+h2a3=0.Two solution curves intersecting at (0, 0) can thus be acquired:(23a)a=0,(23b)a=−h1h2.The abovementioned equations provide dots and circles in polar coordinates, which correspond, respectively, to the equilibrium point and the limit cycle of the two-dimensional system of equations (21a) and (21b). Next, the stability is considered. Let(24)z′a=dzda=h1+3h2a2.For the trivial solutiona=0: z′0=h1, when h1>0, (0, 0) is unstable; when h1<0, a=0 is asymptotically stable.For the nontrivial solutiona=−h1/h2:z′−h1/h2=−2h1, when h1>0 the system has an asymptotically stable solution. However, when h1<0, the system has an unstable solution. The former case corresponds to supercritical pitchfork bifurcation, and the latter subcritical pitchfork bifurcation.Magnetorheological damping drops slowly with increasing 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 negative value, a nontrivial solution appears for the system a=−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 focus tends to be divergent. Accordingly, the stability of the vibration system changes.


## 4. Numerical Analysis

The related parameters of the designed quasi-zero-stiffness vibration isolator are listed as follows:m = 75 kg, k = 200,000 N/m, δ = 90,000 N/m, and ξ = 0.1 Ns/m. During the vibration, the temperature of the magnetorheological damper rises, while damping decreases with time and finally stabilizes. The variation of damping can be expressed via the following function:(25)cτ=50−rτ2.Here,r is a parameter to describe the change of damping with temperature. Specifically, cτ>0.


### 4.1. Analysis of Vibration Characteristics before Any Change in Damping

According to equation (16), the maximum excitation acceleration corresponding to no bifurcation under a stable state is fstable = 2wh5_stable = 10.88 m/s2. When the excitation acceleration exceeds this maximum value, the amplitude frequency curve appears.As shown in Figure2, when the excitation acceleration is below fstable, each frequency in the frequency response curve corresponds to an amplitude, and the amplitude frequency curve, as the frequency sweeps downward, is identical with the curve as the frequency sweeps upward. When the excitation acceleration exceeds fstable, a jump occurs in the frequency response curve. As the frequency sweeps upward, the amplitude increases and jumps vertically to a corresponding low point upon arriving at the maximum and then drops gradually with increasing frequency. As the frequency sweeps downward, the amplitude increases gradually and jumps vertically to a corresponding high point when arriving at the inflection point and then decreases with decreasing frequency. In other words, a bifurcation-induced jump of the excitation frequency can be found in the frequency response curve. The amplitude frequency curves are different as the frequency sweeps downward or upward. As the excitation frequency increases, the vibration amplitudes in all frequency bands increase, and the unstable frequency band expands.Figure 2
Effect of excitation acceleration on the stable vibration solution.Figures3 and 4 show the frequency response curves for f = 25 m/s2. In Figure 3, δ = 90,000 N/m, while in Figure 4 ξ = 0.1 Ns/m. According to Figure 3, the unstable frequency band occupies a larger area at a smaller damping coefficient. Unlike Figure 2, as the damping coefficient drops, only the amplitude around the resonance frequency band increases, while the amplitudes of the other frequency bands remain almost unchanged. This behavior suggests that the change of the damping coefficient increases the amplitude around the resonant frequency point and expands the unstable frequency band and that it imposes slight effect on the vibration of the frequency band far away from the resonance point. The increase of nonlinear stiffness (Figure 4) does not lead to the vibration amplitude increase, but shifts the resonance point towards the right, namely, the unstable frequency band appears.Figure 3
Effect of nonlinear damping on the stable vibration solution.Figure 4
Effect of nonlinear stiffness on the stable vibration solution.Figures5 and 6 illustrates the results, assuming nonlinear stiffness δs = 90,000 N/m and nonlinear damping ξs = 0.1 Ns/m, respectively. Based on Figure 3 analysis, the vibration amplitude decreases as the damping coefficient increases, accompanied by fewer solutions in the unstable region and enhanced system stability. As shown in Figure 5, when the nonlinear damping coefficient increases, the unstable region decreases along the direction of the saddle node. As described above, based on Figure 4, the peak vibration amplitude in the amplitude frequency curve moves rightward as the nonlinear stiffness increases. In addition, the number of solutions that fall in the unstable region increases, and the system stability weakens. As shown in Figure 6, with the increase of nonlinear stiffness, the unstable region as a whole moves downward and also expands gradually.Figure 5
Effect of nonlinear damping on the solution’s stability region.Figure 6
Effect of nonlinear stiffness on the solution’s stability region.This study assumed thatwdown and wupper are the bifurcation critical upper-limit and lower-limit frequencies, respectively. As the excitation frequency Ω approaches wdown, the vibrations under all initial conditions are attracted to the stable focus P1; once Ω>wdown, another stable focus P3 appears (see Figure 7(a) and Figure 7(b)). According to Figure 7(c), when wdown<Ω<wupper, stable focus P1 and P3 as well as the saddle-node P2 can be seen. Only the initial value under certain conditions can form the trajectory that arrives at the saddle node P2. The system stability, once slightly disturbed, will be attracted to P1 or P3. The dashed line in Figure 8(c) represents the boundary between two regions, the region outside the dashed line representing the states attracted to the stable focus P3, while the other region representing the states attracted to the stable focus P1. As the excitation frequency increases further and when Ω>wupper, the focus point P1 disappears, and all states are attracted to the focus P3, and the region, enclosed by a red dashed line, shrinks and disappears finally. All these phenomena indicate that upper and lower solutions of the amplitude frequency curves are asymptotically stable, while the intermediate solution is unstable. Since only asymptotic stability can be achieved in practical applications, jumping occurs. The abovementioned analysis results fit well with equation (24).Figure 7
Variations of phase diagrams for different excitation frequencies: (a) 51.82 rad/s, (b) 51.83 rad/s, (c) 51.84 rad/s, and (d) 51.86 rad/s.
(a)
(b)
(c)
(d)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.
(a)
(b)
(c)
(d)
(e)
(f)


### 4.2. Analysis of the Vibration Characteristics during Damping Changing Process

During the vibration, rising temperature and decreasing damping characterize the magnetorheological damper.According to equation (24), no limit cycle appears in the system vibration for the positive damping coefficient. Figure 9 shows the amplitude frequency response curves, which suggests that time-varying damping can induce a changing amplitude frequency response. Since the power spectrum density (PSD) of the vibration is proportional to the amplitude frequency response, it also explains that time-varying damping triggers the leakage of vibration power spectra, the decline of the corresponding linear spectral peak, the expansion of the frequency band, and less clear linear characteristics. The small box in Figure 9 highlights the leakage in detail. It is also shown in the figure that the vibration power leakage becomes more obvious as the damping changes more rapidly and vice versa.Figure 9
Amplitude frequency responses at different rates of damping change.Assuming that the damping coefficient can be negative, the vibration time domain and phase diagrams at different changing rates are plotted (see Figure8). When the damping changes slowly (r = 7) and still higher than 0, the system vibration tends to be stable and no limit cycle can be observed, see Figures 8(a) and 8(b). When damping changes substantially and becomes negative after certain time, a limit cycle 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 cycles appear in Figures 8(d) and 8(f). Moreover, from Figure 8(c) and 8(e), we know the minimum amplitudes at different damping changing rates are different and the minimum amplitude is larger at a larger rate. After the damping becomes negative, the limit cycle forms and the vibration approaches the limit cycle. The gradual expansion of the limit cycle results in the divergence of vibration, but the vibration never exceeds the limit cycle. By comparing Figures 8(a), 8(c), and 8(e), due to different damping change rates with a same period of time, the initially identical vibration finally undergoes different changes: one tends to be stable and the other becomes divergent.


### 4.3. Analysis of the Vibration Characteristics after the Damping Change Stabilizes

During the vibration process, heat production and heat dissipation of the damper eventually reach a dynamic equilibrium. Damping stabilizes when temperature reaches a certain value. Although external bearing mass, excitation amplitude, and excitation frequency remain unchanged, damping changes. Therefore, the final stable state of some initial vibrations changes. As the damping changed from 50 to 20 Ns/m, the vibration state in the intermediate frequency band, within the blue dashed lines in Figure10, also varies.Figure 10
Amplitude frequency responses before and after damping changes.Figure11 shows the vibration states, before and after the damping changes, which is what is within the dashed lines in Figure 10. At the damping of 50 Ns/m, the quasi-zero-stiffness vibration isolator has only one solution and a stable vibration state. As the damping drops to 20 Ns/m, the quasi-zero-stiffness vibration isolator has three solutions, and the solution in/on the intermediate branch is unstable, i.e., there are two possibilities for the vibration’s stable state. This indicates that time-varying damping induces the change of the system’s vibration state.Figure 11
Stable focus before and after damping changes: (a)c = 50 Ns/m and (b) c = 20 Ns/m.
(a)
(b)


## 4.1. Analysis of Vibration Characteristics before Any Change in Damping

According to equation (16), the maximum excitation acceleration corresponding to no bifurcation under a stable state is fstable = 2wh5_stable = 10.88 m/s2. When the excitation acceleration exceeds this maximum value, the amplitude frequency curve appears.As shown in Figure2, when the excitation acceleration is below fstable, each frequency in the frequency response curve corresponds to an amplitude, and the amplitude frequency curve, as the frequency sweeps downward, is identical with the curve as the frequency sweeps upward. When the excitation acceleration exceeds fstable, a jump occurs in the frequency response curve. As the frequency sweeps upward, the amplitude increases and jumps vertically to a corresponding low point upon arriving at the maximum and then drops gradually with increasing frequency. As the frequency sweeps downward, the amplitude increases gradually and jumps vertically to a corresponding high point when arriving at the inflection point and then decreases with decreasing frequency. In other words, a bifurcation-induced jump of the excitation frequency can be found in the frequency response curve. The amplitude frequency curves are different as the frequency sweeps downward or upward. As the excitation frequency increases, the vibration amplitudes in all frequency bands increase, and the unstable frequency band expands.Figure 2
Effect of excitation acceleration on the stable vibration solution.Figures3 and 4 show the frequency response curves for f = 25 m/s2. In Figure 3, δ = 90,000 N/m, while in Figure 4 ξ = 0.1 Ns/m. According to Figure 3, the unstable frequency band occupies a larger area at a smaller damping coefficient. Unlike Figure 2, as the damping coefficient drops, only the amplitude around the resonance frequency band increases, while the amplitudes of the other frequency bands remain almost unchanged. This behavior suggests that the change of the damping coefficient increases the amplitude around the resonant frequency point and expands the unstable frequency band and that it imposes slight effect on the vibration of the frequency band far away from the resonance point. The increase of nonlinear stiffness (Figure 4) does not lead to the vibration amplitude increase, but shifts the resonance point towards the right, namely, the unstable frequency band appears.Figure 3
Effect of nonlinear damping on the stable vibration solution.Figure 4
Effect of nonlinear stiffness on the stable vibration solution.Figures5 and 6 illustrates the results, assuming nonlinear stiffness δs = 90,000 N/m and nonlinear damping ξs = 0.1 Ns/m, respectively. Based on Figure 3 analysis, the vibration amplitude decreases as the damping coefficient increases, accompanied by fewer solutions in the unstable region and enhanced system stability. As shown in Figure 5, when the nonlinear damping coefficient increases, the unstable region decreases along the direction of the saddle node. As described above, based on Figure 4, the peak vibration amplitude in the amplitude frequency curve moves rightward as the nonlinear stiffness increases. In addition, the number of solutions that fall in the unstable region increases, and the system stability weakens. As shown in Figure 6, with the increase of nonlinear stiffness, the unstable region as a whole moves downward and also expands gradually.Figure 5
Effect of nonlinear damping on the solution’s stability region.Figure 6
Effect of nonlinear stiffness on the solution’s stability region.This study assumed thatwdown and wupper are the bifurcation critical upper-limit and lower-limit frequencies, respectively. As the excitation frequency Ω approaches wdown, the vibrations under all initial conditions are attracted to the stable focus P1; once Ω>wdown, another stable focus P3 appears (see Figure 7(a) and Figure 7(b)). According to Figure 7(c), when wdown<Ω<wupper, stable focus P1 and P3 as well as the saddle-node P2 can be seen. Only the initial value under certain conditions can form the trajectory that arrives at the saddle node P2. The system stability, once slightly disturbed, will be attracted to P1 or P3. The dashed line in Figure 8(c) represents the boundary between two regions, the region outside the dashed line representing the states attracted to the stable focus P3, while the other region representing the states attracted to the stable focus P1. As the excitation frequency increases further and when Ω>wupper, the focus point P1 disappears, and all states are attracted to the focus P3, and the region, enclosed by a red dashed line, shrinks and disappears finally. All these phenomena indicate that upper and lower solutions of the amplitude frequency curves are asymptotically stable, while the intermediate solution is unstable. Since only asymptotic stability can be achieved in practical applications, jumping occurs. The abovementioned analysis results fit well with equation (24).Figure 7
Variations of phase diagrams for different excitation frequencies: (a) 51.82 rad/s, (b) 51.83 rad/s, (c) 51.84 rad/s, and (d) 51.86 rad/s.
(a)
(b)
(c)
(d)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.
(a)
(b)
(c)
(d)
(e)
(f)


## 4.2. Analysis of the Vibration Characteristics during Damping Changing Process

During the vibration, rising temperature and decreasing damping characterize the magnetorheological damper.According to equation (24), no limit cycle appears in the system vibration for the positive damping coefficient. Figure 9 shows the amplitude frequency response curves, which suggests that time-varying damping can induce a changing amplitude frequency response. Since the power spectrum density (PSD) of the vibration is proportional to the amplitude frequency response, it also explains that time-varying damping triggers the leakage of vibration power spectra, the decline of the corresponding linear spectral peak, the expansion of the frequency band, and less clear linear characteristics. The small box in Figure 9 highlights the leakage in detail. It is also shown in the figure that the vibration power leakage becomes more obvious as the damping changes more rapidly and vice versa.Figure 9
Amplitude frequency responses at different rates of damping change.Assuming that the damping coefficient can be negative, the vibration time domain and phase diagrams at different changing rates are plotted (see Figure8). When the damping changes slowly (r = 7) and still higher than 0, the system vibration tends to be stable and no limit cycle can be observed, see Figures 8(a) and 8(b). When damping changes substantially and becomes negative after certain time, a limit cycle 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 cycles appear in Figures 8(d) and 8(f). Moreover, from Figure 8(c) and 8(e), we know the minimum amplitudes at different damping changing rates are different and the minimum amplitude is larger at a larger rate. After the damping becomes negative, the limit cycle forms and the vibration approaches the limit cycle. The gradual expansion of the limit cycle results in the divergence of vibration, but the vibration never exceeds the limit cycle. By comparing Figures 8(a), 8(c), and 8(e), due to different damping change rates with a same period of time, the initially identical vibration finally undergoes different changes: one tends to be stable and the other becomes divergent.


## 4.3. Analysis of the Vibration Characteristics after the Damping Change Stabilizes

During the vibration process, heat production and heat dissipation of the damper eventually reach a dynamic equilibrium. Damping stabilizes when temperature reaches a certain value. Although external bearing mass, excitation amplitude, and excitation frequency remain unchanged, damping changes. Therefore, the final stable state of some initial vibrations changes. As the damping changed from 50 to 20 Ns/m, the vibration state in the intermediate frequency band, within the blue dashed lines in Figure10, also varies.Figure 10
Amplitude frequency responses before and after damping changes.Figure11 shows the vibration states, before and after the damping changes, which is what is within the dashed lines in Figure 10. At the damping of 50 Ns/m, the quasi-zero-stiffness vibration isolator has only one solution and a stable vibration state. As the damping drops to 20 Ns/m, the quasi-zero-stiffness vibration isolator has three solutions, and the solution in/on the intermediate branch is unstable, i.e., there are two possibilities for the vibration’s stable state. This indicates that time-varying damping induces the change of the system’s vibration state.Figure 11
Stable focus before and after damping changes: (a)c = 50 Ns/m and (b) c = 20 Ns/m.
(a)
(b)


## 5. Conclusions

This study analyzed the effect of time-varying damping on the vibration characteristics of a quasi-zero-stiffness vibration isolator, by establishing a nonlinear vibration equation and solving this equation with a multiscale method. The following main conclusions can be drawn:(a)
In a system with quasi-zero-stiffness vibration isolators, the smaller nonlinear damping or the higher nonlinear stiffness or the higher excitation amplitude, the wider unstable frequency band and the larger unstable region. This results in weakened system vibration stability.(b)
Time-varying damping is equivalent to the stiffness term of the vibration system in terms of effect. Because of the stiffness change, the amplitude of the main resonance peak drops, accompanied by the leakage of energy and the less clear linear spectral characteristics. After the temperature stabilizes at a certain value, the vibration characteristics resume.(c)
Assuming that damping drops and turns negative, its change rate affects the final stable state. At a greater damping-change rate, the minimum vibration amplitude tends to be bigger. Furthermore, the final vibration state, which originally would approach stabilization, tends to be divergent.(d)
Time-varying damping reduces the vibration stability of the quasi-zero-stiffness vibration isolator. Under the condition of changing damping, the number of the final stable focus points, which correspond to the same initial vibration state, increases from 1 to 2, and the system stability declines.



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*Source: 4373828-2020-05-08.xml*