# Power Flow in a Two-Stage Nonlinear Vibration Isolation System with High-Static-Low-Dynamic Stiffness

Authors: Ze-Qi Lu; Dong Shao; Hu Ding; Li-Qun Chen
Journal: Shock and Vibration
 (2018)

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## Abstract

The manuscript concerns the power flow characterization in a two-stage nonlinear vibration isolator comprising three springs, which are configured so that each stage of the system has a high-static-low-dynamic stiffness. To demonstrate the distinction of evaluation for vibration isolation using power flow, force transmissibility is used for comparison. The dynamic behavior of the isolator subject to harmonic excitation, however, is of interest here. The harmonic balance method (HBM) could be used to analyze the frequency response curve (FRC) of the strong nonlinear vibration system. A suggested stability analysis to distinguish the stable and the unstable HBM solutions is described. Increasing both upper and lower nonlinear stiffness could bend the first resonant peak to the left. The isolation range in the power and the force transmissibility plot could be extended to the lower frequencies when the nonlinear stiffness is increased, but the rate of roll-off for the power transmissibility is twice the rate for the force transmissibility at each horizontal stiffness setting. An explanation for this phenomenon is given in the paper.


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## Body

## 1. Introduction

Power flow could be used to quantify the isolation performance and reflect more information of the dynamical system [1–6]. In many cases, the mounting base was flexible. There exists a strong coupling effect between the isolator and the foundation structure [2, 4]. Sciulli and Inman [5] designed a linear vibration isolator for a vibration system supported on the flexible base. The strong interaction between the modes was concerned. More than one mode could be decayed by this damped mount. Later, many literatures concerned the usage of power flow in quantifying the nonlinear vibration isolation with flexible base [7–9].Vibration isolation with high-static-low-dynamic stiffness (HSLDS) was widely concerned [10–21]. For that, the linear isolation can only occur when the excitation frequency is above 2× natural frequency of the system. To increase isolation region, the natural frequency of the system needs to be set as low as possible. This can be achieved by a low dynamic stiffness. But the low dynamic stiffness could cause a large static deflection, which is undesirable. This problem can be overcome by a high-static-low-dynamic stiffness (HSLDS). However, single-stage vibration isolation remains a limitation on single-stage vibration isolators’ performance, regardless of having the linear or the nonlinear vibration isolation. That is, the transmissibility at which vibration isolation occurs reduces at a maximum rate of 40 dB/decade. This problem can be overcome by the two-stage nonlinear vibration isolation [22]. In this case, the maximum roll-off rate of the transmissibility doubled. Lu et al. [22, 23] investigated a two-stage nonlinear isolator to promote the HSLDS mechanism. HSLDS in each stage has a profound positive effect on isolation performance. In presence of the nonlinearity, the response at the lower natural frequency bends to right but at the second natural frequency it does not. The overall effect of the nonlinearity is to improve the vibration isolation performance compared to a linear two-stage isolator. Wang et al. [24] investigated a two-stage vibration isolation system with quasi-zero stiffness. The isolator with heavy damping in the upper stage, high intermediate mass, and soft spring in the lower stage has more desirable advantages. Heavy damping in the upper stage could eliminate the second resonance; thus a broader effective frequency range of isolation can be achieved. Lu et al. [25] experimentally investigated a novel two-stage HSLDS nonlinear vibration isolation system. The positive stiffness in each stage was realized by a metallic plate and the corresponding negative stiffness was realized by a bistable carbon fiber (CF) metal composite plate; a reduction in the displacement transmissibility of about 13 dB at 100 Hz was achieved, compared with the two-stage isolator with the bistable composite plates removed.To enrich the knowledge of the analysis on the HSLDS vibration isolation and explore the feasibility of the power flow analysis, it is worth studying the behavior of HSLDS vibration isolation systems by power flow analysis. To address the lack of research in these aspects, the present paper focuses on power flow analysis of the two-stage vibration isolation system using HSLDS mechanisms. The harmonic balance method combined with arc-length continuation is used to analyze the power flow of the strong nonlinear vibration. Here, only theoretical model is considered, which explores the advantages of horizontal springs. Alternatively, the horizontal stiffness could be adjusted in practical application, which could overcome the system imperfection caused by mistuning of the mass load. A similar analysis to that conducted in this paper could be carried out for a two-stage HSLDS isolator incorporating such a vertical spring deflection adjustment device [26], but this is outside the scope of the work presented here.


## 2. Formalizations

Figure1 shows a two-stage HSLDS vibration system mounted on a linear flexible base. It consists of a primary mass m1 connected to the secondary mass m2 that is connected to the flexible base. The oscillators have geometrical nonlinear stiffness; each of the oscillators is modeled by vertical springs kv1 and kv2, two horizontal springs kh1 and kh2, and vertical dampers cv1 and cv2. The linear flexible base consists of a mass ms, a linear spring with stiffness ks, and a linear damper with damping cs.Figure 1
Schematic of a two-stage HSLDS vibration isolation system mounted on a linear flexible base. The mass,m1, is the suspended mass, m2 is the mass of the intermediate stage, and ms is the base mass.The stiffness geometrical nonlinearity of the system is generated by the horizontal spring’s action in the direction of vertical motion. The total stiffness force of the isolator in each stage can be expressed by(1a)fupper=kv1x1-x2+2kh11-lox1-x22+l2x1-x2(1b)flower=kv2x2-xs+2kh21-lox2-xs2+l2x2-xsand the force transfer to the flexible base ms can be given by(2)ft=kv2x2-xs+2kh21-lox2-xs2+l2x2-xs+cv2x˙2-x˙s,where lo are the initial lengths of the lateral springs and l are their lengths when they are in the horizontal position. The actual and approximate stiffness terms are very similar for the relatively small motion of the mass where x1≤0.2l and x2≤0.2l. In the following analysis, it is therefore assumed that the small displacement (less than about 0.2) approximations are valid. Note that this is not a severe restriction as the displacements of a practical system are unlikely to be 20% of the length of the horizontal spring. It is also assumed that the maximum value of the excitation force is such that analytical results are valid [12, 23]. Using Taylor expansion, (1a) and (1b) and (2) can be approximated by(3a)fupper=k11x1-x2+k13x1-x23(3b)flower=k21x2-xs+k23x2-xs3(3c)ft=k21x2-xs+k23x2-xs3+cv2x˙2-x˙s,where k11=kv1-2(lo/l-1)kh1, k13=(lo/l3)kh1, k21=kv2-2(lo/l-1)kh2, and k23=(lo/l3)kh2.The equation of motion for the system in Figure1 is given by(4)Mx¨+Cx˙+Kx+K3x=f,where(5)M=m1000m2000ms,C=cv1-cv10-cv1cv1+cv2-cv20-cv2cv2+cs,K=k11-k110-k11k11+k21-k210-k21ks+k21,x=x1x2xs,K3x=k13x1-x23k23x2-xs3-k13x1-x23-k23x2-xs3,f=Fcos⁡ωt00.Equation (4) can be written in nondimensional form as(6)M^x^′′+C^x^′+K^x^+K^3x^=f^,where (7)M^=1000μ1000μ2,C^=2ζv1-ζv10-ζv1ζv1+μ1λ1ζv2-μ1λ1ζv20-μ1λ1ζv2μ1λ1ζv2+μ2λ2ζs,x^=x^1x^2x^s,K^=k^11-k^110-k^11k^11+k^21-k^210-k^21k^s+k^21,F^=k^13x^1-x^23k^23x^2-x^s3-k^13x^1-x^23-k^23x^2-x^s3,f^=F^cos⁡Ωτ00,x0=l012-l121/2,x^1=x1x0,x^2=x2x0,x^s=xsx0,μ1=m2m1,μ2=msm1,ω1=kv1m1,ω2=kv2m2,ωs=ksms,λ1=ω2ω1,λ2=ωsω1,k^11=1-21l^-1k^h1,k^21=k^v2-21l^-1k^h2,k^13=1-l^2l^3k^h1,k^23=1-l^2l^3k^h2,k^s=kskv,k^h1=kh1kv1,k^v2=kv2kv1,k^h2=kh2kv1,ζv1=cv12m1ω1,ζv2=cv22m2ω2,ζs=cs2msωs,l^=llo,F^=Fkv1x0,Ω=ωω1.and τ=ω1t, which is the time scale used for dimensionless treatment [13]. By the dimensionless method, ωt is replaced by Ωτ, and ·′=d·/dτ.


## 3. Power Flow Characteristics

### 3.1. Harmonic Balance Method

Using the HBM, the vector of nondimensional displacement can be assumed as a solution of the form(8)x^=∑k=1mA1kcos⁡kΩτ+A2ksin⁡kΩτ∑k=1mB1kcos⁡kΩτ+B2ksin⁡kΩτ∑k=1mD1kcos⁡kΩτ+D2ksin⁡kΩτ.The high-order harmonics (m>3) are neglected. The result of the amplitude-frequency response can be written as a group of the nonlinear algebraic equations, which could be given in the implicit form as(9)giΩ,A11,A21,A13,A23,B11,B21,B13,B23,D11,D21,D13,D23,μ1,μ2,λ1,λ2,ζv1,ζv2,ζn1,ζn2,k^11,k^21,k^13,k^23=0i=1⋯12.The transmitted force is given by(10)f^tτ=k^21x^2-x^s+k^23x^2-x^s3+2μ1ζv2λ1x^2′-x^s′.The force transmissibility defined as the ratio of the root mean square of the transmitted force to the excitation force is given by(11)TF=20log10RMSf^tτRMSf^eτ.The nondimensional instantaneous excited power into the system is the product of the excitation force with the corresponding velocity; it yields(12)pinτ=x^1′τf^eτ.The excited power has higher-order harmonics rather than only foundational harmonic; thus the magnitude of the excited power can be determined by root mean square (RMS) of the excited power; this yields(13)Pin=RMSx^1′τf^eτ.For vibration isolation, the interest is the amount of power transmission from the vibration source to the flexible base; the nondimensional instantaneous transmitted power is the product of the transmitted force with the corresponding velocity; this yields(14)ptτ=x^s′τf^tτ.The magnitude of the transmitted power is given as the same way in (14):(15)Pt=RMSx^s′τf^tτ.To quantify the effectiveness of the isolation system, power transmissibility may be introduced and defined as the ratio of the transmitted power to the input power; this yields(16)TP=20log10PtPin.


### 3.2. Stability Analysis

The fundamental harmonic solution can be written as(17)x^∗=x^1∗τx^2∗τx^s∗τ=A11τcos⁡Ωτ+A21τsin⁡ΩτB11τcos⁡Ωτ+B21τsin⁡ΩτD11τcos⁡Ωτ+D21τsin⁡Ωτ.The first and second derivatives of (17) versus τ yield(18)x^′∗=x^1′∗τx^2′∗τx^s′∗τ=A11′+ΩA21cos⁡Ωτ+A21′-ΩA11sin⁡ΩτB11′+ΩB21cos⁡Ωτ+B21′-ΩB11sin⁡ΩτD11′+ΩD21cos⁡Ωτ+D21′-ΩD11sin⁡Ωτ,(19)x^′′∗=x^1′′∗τx^2′′∗τx^s′′∗τ=A11′′+2ΩA21′-Ω2A11cos⁡Ωτ+A21′′-2ΩA11′-Ω2A21sin⁡ΩτB11′′+2ΩB21′-Ω2B11cos⁡Ωτ+B21′′-2ΩB11′-Ω2B21sin⁡ΩτD11′′+2ΩD21′-Ω2D11cos⁡Ωτ+D21′′+2ΩD11′-Ω2D21sin⁡Ωτ.Substituting (17), (18), and (19) into (8) and equating the coefficient of cos⁡Ωτ and sin⁡Ωτ to zero lead to six differential-algebraic equations with HBM solution as fixed points. (20a)A11′′+2ΩA21′-Ω2A11+FA11,A21,B11,B21,A11′,A21′,B11′,B21′,ζv1,k^11,k^13,Ω=0(20b)A21′′-2ΩA11′-Ω2A21+FA11,A21,B11,B21,A11′,A21′,B11′,B21′,ζv1,k^11,k^13,Ω=0(20c)μ1B11′′+2μ1ΩB21′-μ1Ω2B11+Fμ1,A11,A21,B11,B21,D11,D21,A11′,A21′,B11′,B21′,D11′,D21′,ζv1,ζv2,k^11,k^21,k^13,k^23,λ1,Ω=0(20d)μ1B21′′-2μ1ΩB11′-μ1Ω2B21+Fμ1,A11,A21,B11,B21,D11,D21,A11′,A21′,B11′,B21′,D11′,D21′,ζv1,ζv2,k^11,k^21,k^13,k^23,λ1,Ω=0(20e)μ2D11′′+2μ2ΩD21′-μ2Ω2D11+Fμ1,μ2,B11,B21,D11,D21,B11′,B21′,D11′,D21′,ζs,ζv2,k^s,k^13,k^23,λ1,λ2,Ω=0(20f)μ2D21′′-2μ2ΩD11′-μ2Ω2D21+Fμ1,μ2,B11,B21,D11,D21,B11′,B21′,D11′,D21′,ζs,ζv2,k^s,k^13,k^23,λ1,λ2,Ω=0.x^∗ is HBM solution; Δx^ is the deviation value of the HBM solution, so(21)x^=x^∗+Δx^,where(22)Δx=ΔA11cos⁡Ωτ+ΔA21sin⁡ΩτΔB11cos⁡Ωτ+ΔB21sin⁡ΩτΔD11cos⁡Ωτ+ΔD21sin⁡Ωτ.Linearizing the six differential-algebraic equations at the fixed points gives perturbation equation: (23)Iu′′+Cu′+Ku=0,where (24)u′′=ΔA11′′ΔA21′′ΔB11′′ΔB21′′ΔD11′′ΔD21′′,I=10000001000000μ1000000μ1000000μ2000000μ2,u′=ΔA11′ΔA21′ΔB11′ΔB21′ΔD11′ΔD21′,u=ΔA11ΔA21ΔB11ΔB21ΔD11ΔD21,C=2ζv12Ω-2ζv1000-2Ω2ζv10-2ζv100-2ζv102ζv2λ1μ1+2ζv12μ1Ω-2ζv2λ1μ100-2ζv1-2μ1Ω2ζv2λ1μ1+2ζv10-2ζv2λ1μ100-2ζv2λ1μ102ζsλ2μ2+2ζv2λ1μ12μ2Ω000-2ζv2λ1μ1-2μ2Ω2ζsλ2μ2+2ζv2λ1μ1K=K11K12K13K14K15K16K21K22K23K24K25K26K31K32K33K34K35K36K41K42K43K44K45K46K51K52K53K54K55K56K61K62K63K64K65K66,K11=k11-Ω2+k1334B212+94B112+34A212+94A112-92A11B11-32A21B21K12=32k13B11B21+A11A21-A11B21-A21B11K13=-k11+k13-34B212-94B112-34A212-94A112+92A11B11+32A21B21K14=32k13-B11B21-A11A21+A11B21+A21B11K15=0K16=0K21=32k13A11A21-A11B21+B11B21-A21B11K22=k11-Ω2+k1334A112+94A212+34B112+94B212-92A21B21-32A11B11K23=32k13A11B21+A21B11-A11A21-B11B21K24=-k11+k13-34A112-94A212-34B112-94B212+92A21B21+32A11B11K25=0K26=0K31=-k11-k1334B212+94B112+34A212+94A112-92A11B11-32A21B21K32=-32k13B11B21+A11A21-A11B21-A21B11K33=k21-μ1Ω2+k11-k13-34B212-94B112-34A212-94A112+92A11B11+32A21B21+k2334D212+94D112+34B212+94B112-92B11D11-32B21D21K34=32k23D11D21+B11B21-B11D21-B21D11-32k13-B11B21-A11A21+A11B21+A21B11K35=-k21+k23-34D212-94D112-34B212-94B112+92B11D11+32B21D21K36=32k23-D11D21-B11B21+B11D21+B21D11K41=-32k13A11A21-A11B21+B11B21-A21B11K42=-k11-k1334A112+94A212+34B112+94B212-92A21B21-32A11B11K43=32k23B11B21-B11D21+D11D21-B21D11-32k13A11B21+A21B11-A11A21-B11B21K44=k21+k11-μ1Ω2+k2334B112+94B212+34D112+94D212-92B21D21-32B11D11-k13-34A112-94A212-34B112-94B212+92A21B21+32A11B11K45=32k23B11D21+B21D11-B11B21-D11D21K46=-k21+k23-34B112-94B212-34D112-94D212+92B21D21+32B11D11K51=0K52=0K53=-k21+k23-34D212-94D112-34B212-94B112+92B11D11+32B21D21K54=32k23-D11D21-B11B21+B11D21+B21D11K55=ks+k21-μ2Ω2+k2334D212+94D112+34B212+94B112-92B11D11-32B21D21K56=32k23D11D21+B11B21-B11D21-B21D11K61=0K62=0K63=32k23B11B21-B11D21+D11D21-B21D11K64=-k21+k23-34B112-94B212-34D112-94D212+92B21D21+32B11D11K65=32k23-D11D21-B11B21+B11D21+B21D11K66=ks+k21-μ2Ω2+k2334B112+94B212+34D112+94D212-92B21D21-32B11D11.Eigenvalues can be determined by (25)θ2I+θC+K=0,where θ is the eigenvalue, the coefficients in the linearized perturbation equation A11, A21, B11, B21, D11, and D21 can be determined by (9). Equation (9) is a group of the nonlinear algebraic equations. These unknown parameters A11, A21, B11, B21, D11, and D21 can be calculated by Newton iteration method. Calculate the eigenvalues to determine the stability. If the real part of the eigenvalue is negative, the HBM solution is stable. If the real part of the eigenvalue is positive, the HBM solution is unstable.


### 3.3. Results and Discussion

To check whether the method of harmonic balance correctly calculates the frequency response curve for the parameters chosen, the amplitudes of the displacement of the primary, the secondary, and the base mass are plotted in Figure2 together with numerical method using Runge-Kutta scheme. The mass ratio 1 : 0.2 : 1 is chosen in this case. The numerical solutions in the figures are shown as black “o.” The analytical and the numerical results perfectly agree in the stable portions and so the method of harmonic balance can be used for further investigation of the dynamic behavior. The unstable solution shown as a red dotted line cannot be validated as this solution cannot be reached by numerical integration of the equations of motion.Figure 2
The frequency response curve (FRC) of the two-stage HSLDS isolation system. The parameters areμ1=0.2, μ2=1, λ1=5, λ2=1, k^s=1, k^h1=k^h2=0.9, l^=0.7, F^=0.01, and ζv1=ζv2=ζs=0.01. (a) FRC of m1, (b) FRC of m2, and (c) FRC of ms. HBM solution: stable solution (blue line) and unstable solution (red dashed line). Numerical solution: black “o.”
(a) (b) (c)This article addressed the positive effects of using horizontal stiffness in the two stages. The horizontal stiffnesses in two stages are the same,k^h1=k^h2. Such assumption is convenient to analyze the problem but does not affect the results of the analysis of this article. Indeed, the comparison of the effects of the horizontal stiffness between upper and lower stages may be desirable. An analytical method to that conducted in this paper could be applied when the horizontal stiffnesses between upper and lower stages are different, but this is outside the scope of the work presented here.To demonstrate the reduction of the power through the two-stage nonlinear vibration isolator, input and transmitted power are plotted together in Figure3(a), with k^h1=k^h2=0.9. It can be observed that the power is reduced dramatically in almost all ranges of the frequencies except the frequencies around the resonance. Figure 3(b) illustrates the comparison of power transmissibility between the two-stage HSLDS isolator, the two-stage linear HSLDS isolator, the single-stage HSLDS isolator, and the single-stage linear isolator. The linear isolators with horizontal stiffness removed are used for comparison. Horizontal stiffness could enlarge the vibration isolation bandwidth. Power transmissibility of the two-stage isolator is reduced at higher roll-off rate compared with the single-stage isolator. Of particular interest, the two-stage linear isolator outperforms the single-stage HSLDS isolator.Figure 3
Power flow in the two-stage isolation system. The parameters areμ1=0.2, μ2=1, λ1=5, λ2=1, k^s=1, k^h1=k^h2=0.9, l^=0.7, F^=0.01, and ζv1=ζv2=ζs=0.01. (a) Comparison of power P between upper mass m1 and base mass ms. (b) Comparison of the power transmissibility TP between the single-stage linear, the single-stage HSLDS, two-stage linear, and two-stage HSLDS isolators.
(a) (b)A parameter study was carried out to investigate how changes in the nonlinear stiffness, mass ratio, and length ratio can affect the power flow. Figure4(a) shows the effects on the power transmissibility when horizontal stiffness is changed. It can be seen that the power transmissibility peak bends to the right around the first natural frequency but the peaks around the second and the third natural frequency do not. Increasing upper and lower stage horizontal stiffness can extend the isolation range to the lower frequencies and improve the vibration isolation performance. Figure 4(b) shows the effects on force transmissibility when horizontal stiffness is changed. Force transmissibility peak shifts to the lower frequencies when upper and lower stage horizontal stiffness is increased. For high frequency, the rate of roll-off for the power transmissibility is twice the rate for the force transmissibility at each horizontal stiffness setting. Analysis of the definitions of the power and force transmissibility can give the reason for this. From (25), derive (26)TP=20log10RMSx^s′τf^tτRMSx^1′τf^eτ=20log10RMSx^s′τRMSf^tτRMSx^1′τRMSf^eτ=20log10RMSΩx^sτ+π/2RMSΩx^1τ+π/2·RMSf^tτRMSf^eτ=20log10RMSx^sτ+π/2RMSx^1τ+π/2+20log10RMSf^tτRMSf^eτ=TD+TF,where TD is the displacement transmissibility. For high frequency excitation, the displacement transmissibility equals the force transmissibility as shown in [23]. So, (26) becomes(27)TP=2TF.For low frequency, there is a new peak before the first resonance peak in power transmissibility plot. The reason for this is that the relative displacement at the high-order natural frequency between the upper and the lower masses and between the lower mass and the base is small, as is the displacement of the base, so the nonlinear effects on the power flow are minimal at this frequency. However, the relative displacement is relatively large at the first resonance frequency and so the nonlinear effect is more significant at this frequency. This is why the first resonant frequency for the power flow is affected much more by the horizontal springs, as seen in Figure 4. The effects of the mass ratio and length are investigated, respectively, when the two-stage isolator has only horizontal stiffness fixed at k^h1=k^h2=0.9. Figure 5 shows the effects on the power and force transmissibility when mass ratio is changed in the two-stage isolator. The second peak shifts to the lower frequencies but the first peak remains almost the same, when μ1 is increased. Magnitudes of the force transmissibility are reduced significantly at high frequencies. Figure 6 shows the variations of the power and force transmissibility against the length ratio. It can be seen that peaks of both power and force transmissibility shift to lower frequencies, the isolation region is extended to lower frequencies, but the frequency for the antiresonance is not affected.Figure 4
Power transmissibility (a) and force transmissibility (b) of the two-stage HSLDS system with different horizontal stiffness (k^h1,k^h2). The parameters are μ1=0.2, μ2=1, λ1=5, λ2=1, k^s=1, l^=0.7, F^=0.01, and ζv1=ζv2=ζs=0.01. Red solid line is shown as the linear system; blue dashed line, k^h1=k^h2=0.5; green dashed-dotted line, k^h1=k^h2=0.9.
(a) (b)Figure 5
Power transmissibility (a) and force transmissibility (b) of the two-stage HSLDS system with different mass ratioμ1. The parameters are μ2=1, λ1=5, λ2=1, k^s=1, k^h1=k^h2=0.9, l^=0.7, F^=0.01, and ζv1=ζv2=ζs=0.01. Red solid line is shown as the mass ratio μ1=0.2; blue dashed line, μ1=0.6; black-dotted line, μ1=1.
(a) (b)Figure 6
Power transmissibility (a) and force transmissibility (b) of the two-stage HSLDS system with different ratio ofl^. The parameters are μ1=0.2, μ2=1, λ1=5, λ2=1, k^s=1, k^h1=k^h2=0.9, F^=0.01, and ζv1=ζv2=ζs=0.01. Red solid line is shown as the ratio l^=0.7; blue dashed line, l^=0.8; black-dotted line, l^=0.9.
(a) (b)As shown in Figures3–6 for the examples’ simulations, some benefits on practical applications could be explored. Increasing nonlinear stiffness can extend the isolation range to the lower frequencies and improve the vibration isolation performance. Mass ratio and length ratio can enhance the benefits of the nonlinearity. The parameter setting is derived from the model of the artificial satellite vibration isolation. These results could give general rules for the designing of the vibration isolation of artificial satellite.


## 3.1. Harmonic Balance Method

Using the HBM, the vector of nondimensional displacement can be assumed as a solution of the form(8)x^=∑k=1mA1kcos⁡kΩτ+A2ksin⁡kΩτ∑k=1mB1kcos⁡kΩτ+B2ksin⁡kΩτ∑k=1mD1kcos⁡kΩτ+D2ksin⁡kΩτ.The high-order harmonics (m>3) are neglected. The result of the amplitude-frequency response can be written as a group of the nonlinear algebraic equations, which could be given in the implicit form as(9)giΩ,A11,A21,A13,A23,B11,B21,B13,B23,D11,D21,D13,D23,μ1,μ2,λ1,λ2,ζv1,ζv2,ζn1,ζn2,k^11,k^21,k^13,k^23=0i=1⋯12.The transmitted force is given by(10)f^tτ=k^21x^2-x^s+k^23x^2-x^s3+2μ1ζv2λ1x^2′-x^s′.The force transmissibility defined as the ratio of the root mean square of the transmitted force to the excitation force is given by(11)TF=20log10RMSf^tτRMSf^eτ.The nondimensional instantaneous excited power into the system is the product of the excitation force with the corresponding velocity; it yields(12)pinτ=x^1′τf^eτ.The excited power has higher-order harmonics rather than only foundational harmonic; thus the magnitude of the excited power can be determined by root mean square (RMS) of the excited power; this yields(13)Pin=RMSx^1′τf^eτ.For vibration isolation, the interest is the amount of power transmission from the vibration source to the flexible base; the nondimensional instantaneous transmitted power is the product of the transmitted force with the corresponding velocity; this yields(14)ptτ=x^s′τf^tτ.The magnitude of the transmitted power is given as the same way in (14):(15)Pt=RMSx^s′τf^tτ.To quantify the effectiveness of the isolation system, power transmissibility may be introduced and defined as the ratio of the transmitted power to the input power; this yields(16)TP=20log10PtPin.


## 3.2. Stability Analysis

The fundamental harmonic solution can be written as(17)x^∗=x^1∗τx^2∗τx^s∗τ=A11τcos⁡Ωτ+A21τsin⁡ΩτB11τcos⁡Ωτ+B21τsin⁡ΩτD11τcos⁡Ωτ+D21τsin⁡Ωτ.The first and second derivatives of (17) versus τ yield(18)x^′∗=x^1′∗τx^2′∗τx^s′∗τ=A11′+ΩA21cos⁡Ωτ+A21′-ΩA11sin⁡ΩτB11′+ΩB21cos⁡Ωτ+B21′-ΩB11sin⁡ΩτD11′+ΩD21cos⁡Ωτ+D21′-ΩD11sin⁡Ωτ,(19)x^′′∗=x^1′′∗τx^2′′∗τx^s′′∗τ=A11′′+2ΩA21′-Ω2A11cos⁡Ωτ+A21′′-2ΩA11′-Ω2A21sin⁡ΩτB11′′+2ΩB21′-Ω2B11cos⁡Ωτ+B21′′-2ΩB11′-Ω2B21sin⁡ΩτD11′′+2ΩD21′-Ω2D11cos⁡Ωτ+D21′′+2ΩD11′-Ω2D21sin⁡Ωτ.Substituting (17), (18), and (19) into (8) and equating the coefficient of cos⁡Ωτ and sin⁡Ωτ to zero lead to six differential-algebraic equations with HBM solution as fixed points. (20a)A11′′+2ΩA21′-Ω2A11+FA11,A21,B11,B21,A11′,A21′,B11′,B21′,ζv1,k^11,k^13,Ω=0(20b)A21′′-2ΩA11′-Ω2A21+FA11,A21,B11,B21,A11′,A21′,B11′,B21′,ζv1,k^11,k^13,Ω=0(20c)μ1B11′′+2μ1ΩB21′-μ1Ω2B11+Fμ1,A11,A21,B11,B21,D11,D21,A11′,A21′,B11′,B21′,D11′,D21′,ζv1,ζv2,k^11,k^21,k^13,k^23,λ1,Ω=0(20d)μ1B21′′-2μ1ΩB11′-μ1Ω2B21+Fμ1,A11,A21,B11,B21,D11,D21,A11′,A21′,B11′,B21′,D11′,D21′,ζv1,ζv2,k^11,k^21,k^13,k^23,λ1,Ω=0(20e)μ2D11′′+2μ2ΩD21′-μ2Ω2D11+Fμ1,μ2,B11,B21,D11,D21,B11′,B21′,D11′,D21′,ζs,ζv2,k^s,k^13,k^23,λ1,λ2,Ω=0(20f)μ2D21′′-2μ2ΩD11′-μ2Ω2D21+Fμ1,μ2,B11,B21,D11,D21,B11′,B21′,D11′,D21′,ζs,ζv2,k^s,k^13,k^23,λ1,λ2,Ω=0.x^∗ is HBM solution; Δx^ is the deviation value of the HBM solution, so(21)x^=x^∗+Δx^,where(22)Δx=ΔA11cos⁡Ωτ+ΔA21sin⁡ΩτΔB11cos⁡Ωτ+ΔB21sin⁡ΩτΔD11cos⁡Ωτ+ΔD21sin⁡Ωτ.Linearizing the six differential-algebraic equations at the fixed points gives perturbation equation: (23)Iu′′+Cu′+Ku=0,where (24)u′′=ΔA11′′ΔA21′′ΔB11′′ΔB21′′ΔD11′′ΔD21′′,I=10000001000000μ1000000μ1000000μ2000000μ2,u′=ΔA11′ΔA21′ΔB11′ΔB21′ΔD11′ΔD21′,u=ΔA11ΔA21ΔB11ΔB21ΔD11ΔD21,C=2ζv12Ω-2ζv1000-2Ω2ζv10-2ζv100-2ζv102ζv2λ1μ1+2ζv12μ1Ω-2ζv2λ1μ100-2ζv1-2μ1Ω2ζv2λ1μ1+2ζv10-2ζv2λ1μ100-2ζv2λ1μ102ζsλ2μ2+2ζv2λ1μ12μ2Ω000-2ζv2λ1μ1-2μ2Ω2ζsλ2μ2+2ζv2λ1μ1K=K11K12K13K14K15K16K21K22K23K24K25K26K31K32K33K34K35K36K41K42K43K44K45K46K51K52K53K54K55K56K61K62K63K64K65K66,K11=k11-Ω2+k1334B212+94B112+34A212+94A112-92A11B11-32A21B21K12=32k13B11B21+A11A21-A11B21-A21B11K13=-k11+k13-34B212-94B112-34A212-94A112+92A11B11+32A21B21K14=32k13-B11B21-A11A21+A11B21+A21B11K15=0K16=0K21=32k13A11A21-A11B21+B11B21-A21B11K22=k11-Ω2+k1334A112+94A212+34B112+94B212-92A21B21-32A11B11K23=32k13A11B21+A21B11-A11A21-B11B21K24=-k11+k13-34A112-94A212-34B112-94B212+92A21B21+32A11B11K25=0K26=0K31=-k11-k1334B212+94B112+34A212+94A112-92A11B11-32A21B21K32=-32k13B11B21+A11A21-A11B21-A21B11K33=k21-μ1Ω2+k11-k13-34B212-94B112-34A212-94A112+92A11B11+32A21B21+k2334D212+94D112+34B212+94B112-92B11D11-32B21D21K34=32k23D11D21+B11B21-B11D21-B21D11-32k13-B11B21-A11A21+A11B21+A21B11K35=-k21+k23-34D212-94D112-34B212-94B112+92B11D11+32B21D21K36=32k23-D11D21-B11B21+B11D21+B21D11K41=-32k13A11A21-A11B21+B11B21-A21B11K42=-k11-k1334A112+94A212+34B112+94B212-92A21B21-32A11B11K43=32k23B11B21-B11D21+D11D21-B21D11-32k13A11B21+A21B11-A11A21-B11B21K44=k21+k11-μ1Ω2+k2334B112+94B212+34D112+94D212-92B21D21-32B11D11-k13-34A112-94A212-34B112-94B212+92A21B21+32A11B11K45=32k23B11D21+B21D11-B11B21-D11D21K46=-k21+k23-34B112-94B212-34D112-94D212+92B21D21+32B11D11K51=0K52=0K53=-k21+k23-34D212-94D112-34B212-94B112+92B11D11+32B21D21K54=32k23-D11D21-B11B21+B11D21+B21D11K55=ks+k21-μ2Ω2+k2334D212+94D112+34B212+94B112-92B11D11-32B21D21K56=32k23D11D21+B11B21-B11D21-B21D11K61=0K62=0K63=32k23B11B21-B11D21+D11D21-B21D11K64=-k21+k23-34B112-94B212-34D112-94D212+92B21D21+32B11D11K65=32k23-D11D21-B11B21+B11D21+B21D11K66=ks+k21-μ2Ω2+k2334B112+94B212+34D112+94D212-92B21D21-32B11D11.Eigenvalues can be determined by (25)θ2I+θC+K=0,where θ is the eigenvalue, the coefficients in the linearized perturbation equation A11, A21, B11, B21, D11, and D21 can be determined by (9). Equation (9) is a group of the nonlinear algebraic equations. These unknown parameters A11, A21, B11, B21, D11, and D21 can be calculated by Newton iteration method. Calculate the eigenvalues to determine the stability. If the real part of the eigenvalue is negative, the HBM solution is stable. If the real part of the eigenvalue is positive, the HBM solution is unstable.


## 3.3. Results and Discussion

To check whether the method of harmonic balance correctly calculates the frequency response curve for the parameters chosen, the amplitudes of the displacement of the primary, the secondary, and the base mass are plotted in Figure2 together with numerical method using Runge-Kutta scheme. The mass ratio 1 : 0.2 : 1 is chosen in this case. The numerical solutions in the figures are shown as black “o.” The analytical and the numerical results perfectly agree in the stable portions and so the method of harmonic balance can be used for further investigation of the dynamic behavior. The unstable solution shown as a red dotted line cannot be validated as this solution cannot be reached by numerical integration of the equations of motion.Figure 2
The frequency response curve (FRC) of the two-stage HSLDS isolation system. The parameters areμ1=0.2, μ2=1, λ1=5, λ2=1, k^s=1, k^h1=k^h2=0.9, l^=0.7, F^=0.01, and ζv1=ζv2=ζs=0.01. (a) FRC of m1, (b) FRC of m2, and (c) FRC of ms. HBM solution: stable solution (blue line) and unstable solution (red dashed line). Numerical solution: black “o.”
(a) (b) (c)This article addressed the positive effects of using horizontal stiffness in the two stages. The horizontal stiffnesses in two stages are the same,k^h1=k^h2. Such assumption is convenient to analyze the problem but does not affect the results of the analysis of this article. Indeed, the comparison of the effects of the horizontal stiffness between upper and lower stages may be desirable. An analytical method to that conducted in this paper could be applied when the horizontal stiffnesses between upper and lower stages are different, but this is outside the scope of the work presented here.To demonstrate the reduction of the power through the two-stage nonlinear vibration isolator, input and transmitted power are plotted together in Figure3(a), with k^h1=k^h2=0.9. It can be observed that the power is reduced dramatically in almost all ranges of the frequencies except the frequencies around the resonance. Figure 3(b) illustrates the comparison of power transmissibility between the two-stage HSLDS isolator, the two-stage linear HSLDS isolator, the single-stage HSLDS isolator, and the single-stage linear isolator. The linear isolators with horizontal stiffness removed are used for comparison. Horizontal stiffness could enlarge the vibration isolation bandwidth. Power transmissibility of the two-stage isolator is reduced at higher roll-off rate compared with the single-stage isolator. Of particular interest, the two-stage linear isolator outperforms the single-stage HSLDS isolator.Figure 3
Power flow in the two-stage isolation system. The parameters areμ1=0.2, μ2=1, λ1=5, λ2=1, k^s=1, k^h1=k^h2=0.9, l^=0.7, F^=0.01, and ζv1=ζv2=ζs=0.01. (a) Comparison of power P between upper mass m1 and base mass ms. (b) Comparison of the power transmissibility TP between the single-stage linear, the single-stage HSLDS, two-stage linear, and two-stage HSLDS isolators.
(a) (b)A parameter study was carried out to investigate how changes in the nonlinear stiffness, mass ratio, and length ratio can affect the power flow. Figure4(a) shows the effects on the power transmissibility when horizontal stiffness is changed. It can be seen that the power transmissibility peak bends to the right around the first natural frequency but the peaks around the second and the third natural frequency do not. Increasing upper and lower stage horizontal stiffness can extend the isolation range to the lower frequencies and improve the vibration isolation performance. Figure 4(b) shows the effects on force transmissibility when horizontal stiffness is changed. Force transmissibility peak shifts to the lower frequencies when upper and lower stage horizontal stiffness is increased. For high frequency, the rate of roll-off for the power transmissibility is twice the rate for the force transmissibility at each horizontal stiffness setting. Analysis of the definitions of the power and force transmissibility can give the reason for this. From (25), derive (26)TP=20log10RMSx^s′τf^tτRMSx^1′τf^eτ=20log10RMSx^s′τRMSf^tτRMSx^1′τRMSf^eτ=20log10RMSΩx^sτ+π/2RMSΩx^1τ+π/2·RMSf^tτRMSf^eτ=20log10RMSx^sτ+π/2RMSx^1τ+π/2+20log10RMSf^tτRMSf^eτ=TD+TF,where TD is the displacement transmissibility. For high frequency excitation, the displacement transmissibility equals the force transmissibility as shown in [23]. So, (26) becomes(27)TP=2TF.For low frequency, there is a new peak before the first resonance peak in power transmissibility plot. The reason for this is that the relative displacement at the high-order natural frequency between the upper and the lower masses and between the lower mass and the base is small, as is the displacement of the base, so the nonlinear effects on the power flow are minimal at this frequency. However, the relative displacement is relatively large at the first resonance frequency and so the nonlinear effect is more significant at this frequency. This is why the first resonant frequency for the power flow is affected much more by the horizontal springs, as seen in Figure 4. The effects of the mass ratio and length are investigated, respectively, when the two-stage isolator has only horizontal stiffness fixed at k^h1=k^h2=0.9. Figure 5 shows the effects on the power and force transmissibility when mass ratio is changed in the two-stage isolator. The second peak shifts to the lower frequencies but the first peak remains almost the same, when μ1 is increased. Magnitudes of the force transmissibility are reduced significantly at high frequencies. Figure 6 shows the variations of the power and force transmissibility against the length ratio. It can be seen that peaks of both power and force transmissibility shift to lower frequencies, the isolation region is extended to lower frequencies, but the frequency for the antiresonance is not affected.Figure 4
Power transmissibility (a) and force transmissibility (b) of the two-stage HSLDS system with different horizontal stiffness (k^h1,k^h2). The parameters are μ1=0.2, μ2=1, λ1=5, λ2=1, k^s=1, l^=0.7, F^=0.01, and ζv1=ζv2=ζs=0.01. Red solid line is shown as the linear system; blue dashed line, k^h1=k^h2=0.5; green dashed-dotted line, k^h1=k^h2=0.9.
(a) (b)Figure 5
Power transmissibility (a) and force transmissibility (b) of the two-stage HSLDS system with different mass ratioμ1. The parameters are μ2=1, λ1=5, λ2=1, k^s=1, k^h1=k^h2=0.9, l^=0.7, F^=0.01, and ζv1=ζv2=ζs=0.01. Red solid line is shown as the mass ratio μ1=0.2; blue dashed line, μ1=0.6; black-dotted line, μ1=1.
(a) (b)Figure 6
Power transmissibility (a) and force transmissibility (b) of the two-stage HSLDS system with different ratio ofl^. The parameters are μ1=0.2, μ2=1, λ1=5, λ2=1, k^s=1, k^h1=k^h2=0.9, F^=0.01, and ζv1=ζv2=ζs=0.01. Red solid line is shown as the ratio l^=0.7; blue dashed line, l^=0.8; black-dotted line, l^=0.9.
(a) (b)As shown in Figures3–6 for the examples’ simulations, some benefits on practical applications could be explored. Increasing nonlinear stiffness can extend the isolation range to the lower frequencies and improve the vibration isolation performance. Mass ratio and length ratio can enhance the benefits of the nonlinearity. The parameter setting is derived from the model of the artificial satellite vibration isolation. These results could give general rules for the designing of the vibration isolation of artificial satellite.


## 4. Conclusions

In this manuscript, power flow of a two-stage nonlinear isolator mounted on a linear flexible base has been investigated. Both the input and transmitted power frequency responses are considered. The vertical stiffness in the both stages is superposition of the horizontal stiffness that acts as negative stiffness. Hence, the natural frequency is reduced. The effects of the resulting nonlinear isolator have then been investigated. It has been found that the first resonant peak in the input and transmitted power frequency response curve bends to the left when increasing the horizontal stiffness in both stages. The overall effects of the nonlinearity on power flow are to improve the vibration isolation designing compared with linear system.



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*Source: 1697639-2018-05-15.xml*