# The Elliptic Harmonic Balance Method for the Performance Analysis of a Two-Stage Vibration Isolation System with Geometric Nonlinearity

Authors: Weilei Wu; Bin Tang
Journal: Shock and Vibration
 (2021)

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## Abstract

This study develops a modified elliptic harmonic balance method (EHBM) and uses it to solve the force and displacement transmissibility of a two-stage geometrically nonlinear vibration isolation system. Geometric damping and stiffness nonlinearities are incorporated in both the upper and lower stages of the isolator. After using the relative displacement of the nonlinear isolator, we can numerically obtain the steady-state response using the first-order harmonic balance method (HBM1). The steady-state harmonic components of the stiffness and damping force are modified using the Jacobi elliptic functions. The developed EHBM can reduce the truncation error in the HBM1. Compared with the HBM1, the EHBM can improve the accuracy of the resonance regimes of the amplitude-frequency curve and transmissibility. The EHBM is simple and straightforward. It can maintain the same form as the balancing equations of the HBM1 but performs better than it.


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## Body

## 1. Introduction

Nonlinearity has become the focus of recent research studies on improving vibration isolation performance [1, 2]. Many of them have the configuration of geometrically nonlinear springs [3–5] and dampers [6–8] whose arrangement can obtain low dynamic stiffness, thereby reducing the natural frequency without inducing large static deflection [9,10]. Furthermore, two-degree-of-freedom (2-DOF) vibration isolation systems with quasi-zero-stiffness (QZS) and geometrically nonlinear damping have been studied [11–14]. Lu et al. showed that both the force and displacement transmissibilities are mitigated in the isolation range as the horizontal stiffnesses in both stages are increased [11]. Comparing three kinds of nonlinear 2-DOF vibration isolation models, which are grounded-grounded, bottom-springs grounded, and top-springs grounded isolators, Wang et al. found that the bottom-springs grounded isolator has the best isolation performance when the excitation force amplitude is small [12]. The vibrational power flow method was applied to investigate the performance of a 2-DOF nonlinear isolation system [15]. Recently, Yang et al. illustrated that the addition of a nonlinear inertance mechanism to a QZS isolator could enhance vibration isolation performance [16]. Deng et al. illustrated that a multilayer QZS combined system might give distinguished isolation performance in low-frequency regime without losing supporting stability [17].The harmonic balance method (HBM) is a technique for solving the equations of motion of strongly nonlinear oscillators in the papers mentioned above. However, when these nonlinear equations of motion are solved using the HBM, the truncation error is induced when the high-order harmonic components are neglected. To improve the results’ accuracy, complex and numerous balance equations should be included [18–20]. Some modified methods, such as an incremental HBM [21], two-timescale HBM [22], and the HBM with prior linearization [23], have been presented. In this respect, Zhou et al. illustrated that an accurate approximation solution for the resonance response of harmonically forced strongly nonlinear oscillator could be obtained by linearizing the governing equation before harmonic balancing [24].Jacobi elliptic functions are widely used to solve the nonlinear dynamical systems [19, 25], especially for strongly nonlinear problems [26–28]. The exact solutions for certain conservative nonlinear solutions can be directly solved using the Jacobi elliptic functions, which can be equivalent to the Fourier series expansion with high orders [26]. A nonlinear vibration isolation system with a negative stiffness mechanism was investigated using the averaging method [29]. However, when the elliptic harmonic balance method is applied to solve two or multiple DOFs systems, the number of equations obtained by harmonic balancing is not equal to that of unknowns [30]. For this issue, Chen and Liu analysed a 2-DOF self-excited oscillator with strongly cubic nonlinearity by an additional equation prior to harmonic balancing using Jacobi elliptic functions [30]. Using an elliptic balance method, Elias-Zuniga and Beatty obtained the forced response of a 2-DOF undamped system with cubic nonlinearity [31]. Cveticanin proposed an approximate method for solving coupled strongly nonlinear differential equations with small nonlinearities based on the Krylov-Bogolyubov procedure with Jacobi elliptic functions [32]. Recently, Wu and Tang modified the harmonic components of the first-order terms of the damping and stiffness force of a single DOF nonlinear isolation system using Jacobi elliptic functions [33].Herein, we developed an elliptic harmonic balance method (EHBM) and used it to obtain the transmissibility of a two-stage geometrically nonlinear vibration isolation system. The geometrically nonlinear damping and stiffness are included in both the upper and lower stages to improve the isolation characteristics. The steady-state response of the two-stage vibration isolator is derived using Jacobi elliptic functions and trigonometric functions. Using Jacobi elliptic functions, we modified the first-order harmonic balance method (HBM1) according to the orthogonal relationship between force components. After calculating the amplitude-frequency characteristics, we can obtain the force and displacement transmissibility. The accuracy of the results is analysed.


## 2. A Two-Stage Geometrically Nonlinear Vibration Isolator

A two-stage nonlinear vibration isolator is shown in Figure1. In addition to the suspended mass and the vertical springs and dampers, horizontal springs and dampers are inserted in each stage to produce geometric nonlinearity [3–8]. The exciting force is applied in the vertical direction. The relative displacement between each adjacent stage are given by xr=x1−x2 and xm=x2−xe, respectively, where x1 is the displacement of the mass of the upper stage ms1, x2 is the displacement of the mass of the lower stage ms2, and xe is the displacement of the base. The force transmitted from ms1 to ms2 and the force transmitted from ms2 to the basement are given by(1a)frxr,x˙r=cv1x˙r+2ch1xr2xr2+l12x˙r+kv1xr+2kh11−l01xr2+l12xr,(1b)fmxm,x˙m=cv2x˙m+2ch2xm2xm2+l22x˙m+kv2xm+2kh21−l02xm2+l22xm,where cv1 and ch1 are the damping coefficients of the vertical and horizontal dampers of the upper stage, cv2 and ch2 are the damping coefficients of the vertical and horizontal dampers of the lower stage, kv1 and kh1 are the stiffnesses of the vertical and horizontal springs of the upper stage, kv2 and kh2 are the stiffnesses of vertical and horizontal springs of the lower stages, l1 and l2 are the lengths of the upper and lower horizontal springs between the suspended mass and host structure in the state of rest, and l01 and l02 are the free lengths of the horizontal springs of the upper and lower stages, respectively.Figure 1
Model of a two-stage nonlinear vibration isolator.If the relative displacement between each adjacent stage is less than 20 percent of the length of the horizontal spring, the series expansion of the forcesfr and fm shown in equations (1a) and (1b) can be approximated as [8, 33](2a)frxr,x˙r=fdr+fsr≈cv1x˙r+2ch1xrl12x˙r+kr1xr+kr3xr3,(2b)fmxm,x˙m=fdm+fsm≈cv2x˙m+2ch2xml22x˙m+km1xm+km3xm3,where fdr and fdm are the approximated damping forces when xr2/xr2+l12≈xr/l12 and xm2/xm2+l22≈xm/l22, fsr and fsm are the approximated spring forces when kv1xr+2kh11−l01/xr2+l12xr≈kr1xr+kr3xr3 and kv2xm+2kh21−l02/xm2+l22xm≈km1xm+km3xm3, where kr1=kv1−2kh1l01/l1−1, kr3=kh1l01/l13, km1=kv2−2kh2l02/l2−1, and km3=kh2l02/l23, and the percentage error is less than four percent [8,34]. It should also be noticed that if the amplitude of the excitation is large, so that the force-deflection characteristic of the two-stage system cannot be approximately described by a series expansion to the third order as in equations (2a) and (2b), the nonlinearity induces undesirable effects [35].For the force excitation case, where the base displacementxe=0, x1=xr+xm, and x2=xm, the equations of motion are written as(3a)ms1x¨r+x¨m+cv1x˙r+2ch1xrl12x˙r+kr1xr+kr3xr3=Fecosωet,(3b)ms2x¨m+cv2x˙m+2ch2xml22x˙m+km1xm+km3xm3−cv1x˙r−2ch1xrl12x˙r−kr1xr−kr3xr3=0,where Fe is the excitation force amplitude and ωe is its frequency.Settingl1=l2, the nondimensional forms of equations (3a) and (3b) become(4a)x^r″+x^m″+f^dr+f^sr=F^ecosΩτ,(4b)μx^m″+f^dm+f^sm−f^dr−f^sr=0,where xs=l012−l12=l022−l22, x^r=xr/xs, x^m=xm/xs, l^=l1/l01=l2/l02, τ=ωnt, ωn=kv1/ms1, Ω=ωe/ωn, F^e=Fe/kv1xs, μ=ms2/ms1, ζv1=cv1/2ms1ωn, ζv2=cv2/2ms1ωn, ζh1=ch1/2ms1ωn, ζh2=ch2/2ms1ωn, ζn1=ζh11−l^2/l^2, ζn2=ζh21−l^2/l^2, Ω12=kr1/kv1, Ω22=km1/kv1, γ1=kr3xs2/kv1, γ2=km3xs2/kv1, f^dr=2ζv1x^r′+4ζn1x^r2x^r′, f^sr=Ω12x^r+γ1x^r3, f^dm=2ζv2x^m′+4ζn2x^m2x^m′, and f^sm=Ω22x^m+γ2x^m3.When the excitation forcefe=0 and the base displacement xe=Xecosωet, where Xe and ωe are the base excitation amplitude and frequency, the equations of motion of the base excited system are given by(5a)x^r″+x^m″+f^dr+f^sr=Ω2X^ecosΩτ,(5b)μx^m″+f^dm+f^sm−f^dr−f^sr=μΩ2X^ecosΩτ,where X^e=Xe/xs.


## 3. Elliptic Harmonic Balance Method

### 3.1. HBM1

The solutions of equations (4a) and (4b) are assumed as(6a)x^rτ=X^rcosΩτ+ϕr,(6b)x^mτ=X^mcosΩτ+ϕm,where X^r is the nondimensional amplitude of the relative displacement between ms1 and ms2, X^m is the nondimensional amplitude of the relative displacement between ms2 and the base, ϕr is the phase difference between the relative displacement x^r and the excitation force, and ϕm is the phase difference between the relative displacement x^m and the excitation force.The nondimensional damping and stiffness force can be approximately written asf^dr≈α˜rX^rsinΩτ+ϕr, f^sr≈β˜rX^rcosΩτ+ϕr, f^dm≈α˜mX^msinΩτ+ϕm, and f^sm≈β˜mX^mcosΩτ+ϕm, where α˜r, β˜r, α˜m, and β˜m are the amplitudes of the first order of series expansion of f^dr, f^sr, f^dm, and f^sm at the steady-state response, respectively, which can be expressed as follows [18]:(7a)α˜rX^r≈Ωπ∫02π/Ωf^drX^rcosΩτ+ϕr,−X^rΩsinΩτ+ϕrsinΩτ+ϕrdτ,(7b)β˜rX^r≈Ωπ∫02π/Ωf^srX^rcosΩτ+ϕr,−X^rΩsinΩτ+ϕrcosΩτ+ϕrdτ,(7c)α˜mX^m≈Ωπ∫02π/Ωf^dmX^mcosΩτ+ϕm,−X^mΩsinΩτ+ϕmsinΩτ+ϕmdτ,(7d)β˜mX^m≈Ωπ∫02π/Ωf^smX^mcosΩτ+ϕm,−X^mΩsinΩτ+ϕmcosΩτ+ϕmdτ.Substituting equations (6a) and (6b) into equations (4a) and (4b) and equating coefficients of the harmonic components sinΩτ+ϕr, cosΩτ+ϕr, sinΩτ+ϕm, and cosΩτ+ϕm, we obtain(8a)−Ω2X^msinθΩX^rx^r′τ+f^dr=−F^esinϕrΩX^rx^r′τ,(8b)X^r+X^mcosθX^rx^r″τ+f^sr=F^ecosϕrX^rx^rτ,(8c)Ω2X^rsinθΩX^mx^m′τ+f^dm=−F^esinϕmΩX^mx^m′τ,(8d)1+μX^m+X^rcosθX^mx^m″τ+f^sm=F^ecosϕmX^mx^mτ,where x^r′τ=−ΩX^rsinΩτ+ϕr, x^r″τ=−Ω2X^rcosΩτ+ϕr, x^m′τ=−ΩX^msinΩτ+ϕm, x^m″τ=−Ω2X^mcosΩτ+ϕm, and θ=ϕm−ϕr.Combining equations (7a)–(7d) and (8a)–(8d), the amplitude-frequency equations of equations (4a) and (4b) are given by(9a)α˜r≈F^esinϕr+Ω2X^msinϕr−ϕm,(9b)β˜r−Ω2X^r≈F^ecosϕr+Ω2X^mcosϕr−ϕm,(9c)α˜m≈F^esinϕm+Ω2X^rsinϕm−ϕr,(9d)β˜m−1+μΩ2X^m≈F^ecosϕm+Ω2X^rcosϕm−ϕr.


### 3.2. Jacobi Elliptic Function Solutions

Using Jacobi elliptic functions, we assume that the steady-state solutions of equations (4a) and (4b) are in the following form:(10a)x^rτ=X^rcnψr|mr,(10b)x^r′τ=−X^rωrsnψr|mrdnψr|mr,(10c)x^mτ=X^mcnψm|mm,(10d)x^m′τ=−X^mωmsnψm|mmdnψm|mm,where ψr=ωrτ+φr and ψm=ωmτ+φm are the complete phases, mr and mm are the squares of the elliptic modulus, ωr and ωm are the Jacobi elliptic frequencies, and φr and φm are the Jacobi phases of the corresponding displacement responses, respectively. It should also be noticed that ∂cnψ|m/∂ψ=−snψ|mdnψ|m. Following a similar procedure in [33], the relationship between the second derivative and the linear and cubic terms of equations (10a) and (10c) are given by(11a)x^r″τ=−ωr21−2mrx^rτ−2mrωr2x^rτ3X^r2,(11b)x^m″τ=−ωm21−2mmx^mτ2mmωm2x^mτ3X^m2.Substituting equations (10a) and (10c) and equations (11a) and (11b) into equations (8b) and (8d), respectively, the equations of the cosine harmonic component become(12a)X^r+X^mcosθX^rωr21−2mrx^rτ+2mrωr2x^rτ3X^r2=Ω12−F^ecosϕrX^rx^rτ+γ1x^rτ3,(12b)1+μX^m+X^rcosθX^mωm21−2mmx^mτ+2mmωm2x^mτ3X^m2=Ω22−F^ecosϕmX^mx^mτ+γ2x^mτ3.Equating the coefficients ofx^rτ and x^rτ3 of equation (12a), the coefficients of x^mτ and x^mτ3 of equation (12b) give(13a)ωr2=Ω12X^r−F^ecosϕr+γ1X^r3X^r+X^mcosθ,(13b)mr=γ1X^r32Ω12X^r−F^ecosϕr+γ1X^r3,(13c)ωm2=Ω22X^m−F^ecosϕm+γ2X^m31+μX^m+X^rcosθ,(13d)mm=γ2X^m32Ω22X^m−F^ecosϕm+γ2X^m3.WhenΩ12>0, Ω22>0, γ1>0, and γ2>0, the relationship between the excited frequency Ω and the Jacobi elliptic frequencies ωr and ωm can be expressed as(14)Ω=πωr2Kmr=πωm2Kmm,where Kmr and Kmm are the complete elliptic integrals of the first kind when 0<mr,mm<1/2.


### 3.3. Modifying the Stiffness Force Components

Because equations (7a) and (7d) are the first-order harmonic components, both the damping force components α˜r and α˜m and the stiffness force components β˜r and β˜m are approximate terms. The stiffness force components β˜r and β˜m are first modified using Jacobi elliptic functions.Using equation (14), ϕr=πφr/2Kmr, and ϕm=πφm/2Kmm, equations (7b) and (7d) can be rewritten as(15a)β˜r≈ωr2Kmr∫04Kmr/ωrf^srx^r,x^r′cosπψr2Kmrdτ,(15b)β˜m≈ωm2Kmm∫04Kmm/ωmf^smx^m,x^m′cosπψm2Kmmdτ.Substituting the first Fourier series expansion ofcnψr|mr≈crcosπψr/2Kmr and cnψm|mm≈cmcosπψm/2Kmm, where πψr/2Kmr and πψm/2Kmm are the approximate phases of the displacement responses x^r and x^m respectively, into equations (15a) and (15b), the modified stiffness components become(16a)β˜re≈12crKmr∫04Kmrf^srx^r,x^r′cnψr|mrdψr,(16b)β˜me≈12cmKmm∫04Kmmf^smx^m,x^m′cnψm|mmdψm,where cr=2πqr/mrKmr1+qr, cm=2πqm/mmKmm1+qm, qr=exp−πK1−mr/Kmr, and qm=exp−πK1−mm/Kmm.


### 3.4. Modifying the Damping Force Components

Similar to the stiffness force components modification, equations (7a) and (7c) can be expressed as(17a)α˜r≈ωr2Kmr∫04Kmr/ωrf^drx^r,x^r′sinπψr2Kmrdτ,(17b)α˜m≈ωm2Kmm∫04Kmm/ωmf^dmx^m,x^m′sinπψm2Kmmdτ.Substituting the first Fourier series expansion ofsnψr|mrdnψr|mr≈arsinπψr/2Kmr and snψm|mmdnψm|mm≈amsinπψm/2Kmm into equations (17a) and (17b), the modified damping components become(18a)α˜re≈12arKmr∫04Kmrf^drx^r,x^r′snψr|mrdnψr|mrdψr,(18b)α˜me≈12amKmm∫04Kmmf^dmx^m,x^m′snψm|mmdnψm|mmdψm,where ar=π2qr/mrK2mr1+qr and am=π2qm/mmK2mm1+qm.


### 3.5. EHBM

When the modified stiffness and damping force components shown in equations (16a), (16b), (18a), and (18b) are obtained, we rewrite equations (9a)–(9d) as(19a)α˜re≈F^esinϕr+Ω2X^msinϕr−ϕm,(19b)β˜re−Ω2X^r≈F^ecosϕr+Ω2X^mcosϕr−ϕm,(19c)α˜me≈F^esinϕm+Ω2X^rsinϕm−ϕr,(19d)β˜me−1+μΩ2X^m≈F^ecosϕm+Ω2X^rcosϕm−ϕr.Using equations (9a)–(9d) and (19a)–(19d), we plot the orthogonal relationship between force components on ms1 and ms2 in Figures 2(a) and 2(b), respectively. It shows how the EHBM improves the accuracy of the results.Figure 2
The orthogonal relationship between force components.


## 3.1. HBM1

The solutions of equations (4a) and (4b) are assumed as(6a)x^rτ=X^rcosΩτ+ϕr,(6b)x^mτ=X^mcosΩτ+ϕm,where X^r is the nondimensional amplitude of the relative displacement between ms1 and ms2, X^m is the nondimensional amplitude of the relative displacement between ms2 and the base, ϕr is the phase difference between the relative displacement x^r and the excitation force, and ϕm is the phase difference between the relative displacement x^m and the excitation force.The nondimensional damping and stiffness force can be approximately written asf^dr≈α˜rX^rsinΩτ+ϕr, f^sr≈β˜rX^rcosΩτ+ϕr, f^dm≈α˜mX^msinΩτ+ϕm, and f^sm≈β˜mX^mcosΩτ+ϕm, where α˜r, β˜r, α˜m, and β˜m are the amplitudes of the first order of series expansion of f^dr, f^sr, f^dm, and f^sm at the steady-state response, respectively, which can be expressed as follows [18]:(7a)α˜rX^r≈Ωπ∫02π/Ωf^drX^rcosΩτ+ϕr,−X^rΩsinΩτ+ϕrsinΩτ+ϕrdτ,(7b)β˜rX^r≈Ωπ∫02π/Ωf^srX^rcosΩτ+ϕr,−X^rΩsinΩτ+ϕrcosΩτ+ϕrdτ,(7c)α˜mX^m≈Ωπ∫02π/Ωf^dmX^mcosΩτ+ϕm,−X^mΩsinΩτ+ϕmsinΩτ+ϕmdτ,(7d)β˜mX^m≈Ωπ∫02π/Ωf^smX^mcosΩτ+ϕm,−X^mΩsinΩτ+ϕmcosΩτ+ϕmdτ.Substituting equations (6a) and (6b) into equations (4a) and (4b) and equating coefficients of the harmonic components sinΩτ+ϕr, cosΩτ+ϕr, sinΩτ+ϕm, and cosΩτ+ϕm, we obtain(8a)−Ω2X^msinθΩX^rx^r′τ+f^dr=−F^esinϕrΩX^rx^r′τ,(8b)X^r+X^mcosθX^rx^r″τ+f^sr=F^ecosϕrX^rx^rτ,(8c)Ω2X^rsinθΩX^mx^m′τ+f^dm=−F^esinϕmΩX^mx^m′τ,(8d)1+μX^m+X^rcosθX^mx^m″τ+f^sm=F^ecosϕmX^mx^mτ,where x^r′τ=−ΩX^rsinΩτ+ϕr, x^r″τ=−Ω2X^rcosΩτ+ϕr, x^m′τ=−ΩX^msinΩτ+ϕm, x^m″τ=−Ω2X^mcosΩτ+ϕm, and θ=ϕm−ϕr.Combining equations (7a)–(7d) and (8a)–(8d), the amplitude-frequency equations of equations (4a) and (4b) are given by(9a)α˜r≈F^esinϕr+Ω2X^msinϕr−ϕm,(9b)β˜r−Ω2X^r≈F^ecosϕr+Ω2X^mcosϕr−ϕm,(9c)α˜m≈F^esinϕm+Ω2X^rsinϕm−ϕr,(9d)β˜m−1+μΩ2X^m≈F^ecosϕm+Ω2X^rcosϕm−ϕr.


## 3.2. Jacobi Elliptic Function Solutions

Using Jacobi elliptic functions, we assume that the steady-state solutions of equations (4a) and (4b) are in the following form:(10a)x^rτ=X^rcnψr|mr,(10b)x^r′τ=−X^rωrsnψr|mrdnψr|mr,(10c)x^mτ=X^mcnψm|mm,(10d)x^m′τ=−X^mωmsnψm|mmdnψm|mm,where ψr=ωrτ+φr and ψm=ωmτ+φm are the complete phases, mr and mm are the squares of the elliptic modulus, ωr and ωm are the Jacobi elliptic frequencies, and φr and φm are the Jacobi phases of the corresponding displacement responses, respectively. It should also be noticed that ∂cnψ|m/∂ψ=−snψ|mdnψ|m. Following a similar procedure in [33], the relationship between the second derivative and the linear and cubic terms of equations (10a) and (10c) are given by(11a)x^r″τ=−ωr21−2mrx^rτ−2mrωr2x^rτ3X^r2,(11b)x^m″τ=−ωm21−2mmx^mτ2mmωm2x^mτ3X^m2.Substituting equations (10a) and (10c) and equations (11a) and (11b) into equations (8b) and (8d), respectively, the equations of the cosine harmonic component become(12a)X^r+X^mcosθX^rωr21−2mrx^rτ+2mrωr2x^rτ3X^r2=Ω12−F^ecosϕrX^rx^rτ+γ1x^rτ3,(12b)1+μX^m+X^rcosθX^mωm21−2mmx^mτ+2mmωm2x^mτ3X^m2=Ω22−F^ecosϕmX^mx^mτ+γ2x^mτ3.Equating the coefficients ofx^rτ and x^rτ3 of equation (12a), the coefficients of x^mτ and x^mτ3 of equation (12b) give(13a)ωr2=Ω12X^r−F^ecosϕr+γ1X^r3X^r+X^mcosθ,(13b)mr=γ1X^r32Ω12X^r−F^ecosϕr+γ1X^r3,(13c)ωm2=Ω22X^m−F^ecosϕm+γ2X^m31+μX^m+X^rcosθ,(13d)mm=γ2X^m32Ω22X^m−F^ecosϕm+γ2X^m3.WhenΩ12>0, Ω22>0, γ1>0, and γ2>0, the relationship between the excited frequency Ω and the Jacobi elliptic frequencies ωr and ωm can be expressed as(14)Ω=πωr2Kmr=πωm2Kmm,where Kmr and Kmm are the complete elliptic integrals of the first kind when 0<mr,mm<1/2.


## 3.3. Modifying the Stiffness Force Components

Because equations (7a) and (7d) are the first-order harmonic components, both the damping force components α˜r and α˜m and the stiffness force components β˜r and β˜m are approximate terms. The stiffness force components β˜r and β˜m are first modified using Jacobi elliptic functions.Using equation (14), ϕr=πφr/2Kmr, and ϕm=πφm/2Kmm, equations (7b) and (7d) can be rewritten as(15a)β˜r≈ωr2Kmr∫04Kmr/ωrf^srx^r,x^r′cosπψr2Kmrdτ,(15b)β˜m≈ωm2Kmm∫04Kmm/ωmf^smx^m,x^m′cosπψm2Kmmdτ.Substituting the first Fourier series expansion ofcnψr|mr≈crcosπψr/2Kmr and cnψm|mm≈cmcosπψm/2Kmm, where πψr/2Kmr and πψm/2Kmm are the approximate phases of the displacement responses x^r and x^m respectively, into equations (15a) and (15b), the modified stiffness components become(16a)β˜re≈12crKmr∫04Kmrf^srx^r,x^r′cnψr|mrdψr,(16b)β˜me≈12cmKmm∫04Kmmf^smx^m,x^m′cnψm|mmdψm,where cr=2πqr/mrKmr1+qr, cm=2πqm/mmKmm1+qm, qr=exp−πK1−mr/Kmr, and qm=exp−πK1−mm/Kmm.


## 3.4. Modifying the Damping Force Components

Similar to the stiffness force components modification, equations (7a) and (7c) can be expressed as(17a)α˜r≈ωr2Kmr∫04Kmr/ωrf^drx^r,x^r′sinπψr2Kmrdτ,(17b)α˜m≈ωm2Kmm∫04Kmm/ωmf^dmx^m,x^m′sinπψm2Kmmdτ.Substituting the first Fourier series expansion ofsnψr|mrdnψr|mr≈arsinπψr/2Kmr and snψm|mmdnψm|mm≈amsinπψm/2Kmm into equations (17a) and (17b), the modified damping components become(18a)α˜re≈12arKmr∫04Kmrf^drx^r,x^r′snψr|mrdnψr|mrdψr,(18b)α˜me≈12amKmm∫04Kmmf^dmx^m,x^m′snψm|mmdnψm|mmdψm,where ar=π2qr/mrK2mr1+qr and am=π2qm/mmK2mm1+qm.


## 3.5. EHBM

When the modified stiffness and damping force components shown in equations (16a), (16b), (18a), and (18b) are obtained, we rewrite equations (9a)–(9d) as(19a)α˜re≈F^esinϕr+Ω2X^msinϕr−ϕm,(19b)β˜re−Ω2X^r≈F^ecosϕr+Ω2X^mcosϕr−ϕm,(19c)α˜me≈F^esinϕm+Ω2X^rsinϕm−ϕr,(19d)β˜me−1+μΩ2X^m≈F^ecosϕm+Ω2X^rcosϕm−ϕr.Using equations (9a)–(9d) and (19a)–(19d), we plot the orthogonal relationship between force components on ms1 and ms2 in Figures 2(a) and 2(b), respectively. It shows how the EHBM improves the accuracy of the results.Figure 2
The orthogonal relationship between force components.


## 4. Force and Displacement Transmissibility

This section solves the transmissibility of the two-stage nonlinear vibration isolator using the HBM1 and EHBM and discusses the results.


### 4.1. Amplitude-Frequency Characteristics

Integrating equations (7a)–(7d) and substituting the results into equations (9a)–(9d), we obtain(20a)−X^rΩ2ζv1+ζn1X^r2≈F^esinϕr+Ω2X^msinϕr−ϕm,(20b)Ω12X^r+3γ1X^r34−Ω2X^r≈F^ecosϕr+Ω2X^mcosϕr−ϕm,(20c)−X^mΩ2ζv2+ζn2X^m2≈F^esinϕm+Ω2X^rsinϕm−ϕr,(20d)Ω22X^m+3γ2X^m34−1+μΩ2X^m≈F^ecosϕm+Ω2X^rcosϕm−ϕr.Integrating equations (16a), (16b), (18a), and (18b), in which the nondimensional nonlinear stiffness forces f^sr=Ω12X^rcnψr|mr+γ1X^rcnψr|mr3 and f^sm=Ω22X^mcnψm|mm+γ2X^mcnψm|mm3 and the nondimensional nonlinear damping forces f^dr=2ζv1+4ζn1X^rcnψr|mr2−X^rωrsnψr|mrdnψr|mr and f^dm=2ζv2+4ζn2X^mcnψm|mm2−X^mωmsnψm|mmdnψm|mm, yields(21a)α˜re≈−4X^rωrarζv1R1+2ζn1X^r2R2,(21b)α˜me≈−4X^mωmamζv2R3+2ζn2X^m2R4,(21c)β˜re≈2X^rcrΩ12R5+γ1X^r2R6,(21d)β˜me≈2X^mcmΩ22R7+γ2X^m2R8,where R1 to R8 are given in Appendix A.Substituting equations (21a) and (21b) into equations (19a) and (19c), respectively, and combining equation (14), we can numerically solve the amplitudes (X^r and X^m) and phases (ϕr and ϕm) of the steady-state response of the two-stage nonlinear isolator.The amplitude-frequency solutions obtained using the HBM1 and EHBM are plotted in Figure3. The numerical solutions calculated using the fourth-order Runge-Kutta method with a step-size control algorithm are also plotted. The parameters are μ=0.2, ζv1=ζv2=0.01, ζh1=ζh2=0.1, kv2/kv1=1, kh1/kv1=kh2/kv1=0.5, l^=0.7, and F^e=1. The EHBM improves the accuracy of the amplitude-frequency curve around the peak region. In the low-amplitude regime, the solutions of both the EHBM and HBM1 are compared well with the numerical results.Figure 3
(a) Amplitude-frequency curves of the relative displacement betweenms1 and ms2. (b) Amplitude-frequency curves of ms2, when μ=0.2, ζv1=ζv2=0.01, ζh1=ζh2=0.1, kv2/kv1=1, kh1/kv1=kh2/kv1=0.5, l^=0.7, and F^e=1. Dashed-dotted line , stable solution by the HBM1; thin-dashed line , unstable solution by the HBM1; solid line , stable solution by the EHBM; thick-dashed line , unstable solution by the EHBM; dot , numerical results.
(a)(b)Whenl^=0.7 and the nondimensional linear stiffness terms kv2/kv1=1 and kh1/kv1=kh2/kv1=7/6, the system becomes a two-stage QZS system. The amplitude-frequency curves of the relative displacement between ms1 and ms2 of the two-stage geometrically nonlinear isolation system, when μ=0.2, ζv1=ζv2=0.01, ζh1=ζh2=0.1, kv2/kv1=1, kh1/kv1=kh2/kv1=7/6, l^=0.7, and F^e=0.01, 0.1, 1, and 10, respectively, are plotted in Figure 4. When the system becomes a QZS system, the solutions of the EHBM can still be compared well with the numerical solutions. To improve the accuracy of the harmonic balance method, we should include enough terms and the calculation complexity is simultaneously increased [25]. Because the base displacement xe is zero, the displacement of the upper mass ms2x2 = xm. The amplitude-frequency curves of ms2 of the two-stage geometrically nonlinear isolation system are plotted in Figure 5, and the accuracy of the EHBM can be seen.Figure 4
Amplitude-frequency curves of the relative displacement betweenms1 and ms2 when μ=0.2, ζv1=ζv2=0.01, ζh1=ζh2=0.1, kv2/kv1=1, kh1/kv1=kh2/kv1=7/6, l^=0.7, and (a) F^e=0.01, (b) F^e=0.1, (c) F^e=1, and (d) F^e=10, respectively. Dashed-dotted line , stable solution by the HBM1; thin-dashed line , unstable solution by the HBM1; solid line , stable solution by the EHBM; thick-dashed line , unstable solution by the EHBM; dot •, numerical results.
(a)(b)(c)(d)Figure 5
Amplitude-frequency curves ofms2 when μ=0.2, ζv1=ζv2=0.01, ζh1=ζh2=0.1, kv2/kv1=1, kh1/kv1=kh2/kv1=7/6, l^=0.7, and (a) F^e=0.01, (b) F^e=0.1, (c) F^e=1, and (d) F^e=10, respectively. Dashed-dotted line , stable solution by the HBM1; thin-dashed line , unstable solution by the HBM1; solid line , stable solution by the EHBM; thick-dashed line , unstable solution by the EHBM; dot , numerical results.
(a)(b)(c)(d)To investigate the accuracy of the EHBM at different damping cases, we chooseζv1=ζv2=0.02 and ζh1=ζh2=0.2 as Case 1 and ζv1=ζv2=0.03 and ζh1=ζh2=0.3 as Case 2. The system parameters are the same as those in Figure 5 and F^e=0.1. The results are shown in Figure 6. The EHBM works well for these two cases.Figure 6
(a) and (c) are amplitude-frequency curves of the relative displacement betweenms1 and ms2, and (b) and (d) are amplitude-frequency curves of ms2, when μ=0.2, (a-b) ζv1=ζv2=0.02, and ζh1=ζh2=0.2, (c-d) ζv1=ζv2=0.03 and ζh1=ζh2=0.3, and kv2/kv1=1, kh1/kv1=kh2/kv1=7/6, l^=0.7, and F^e=0.1. Dashed-dotted line , stable solution by the HBM1; thin-dashed line , unstable solution by the HBM1; solid line , stable solution by the EHBM; thick-dashed line , unstable solution by the EHBM; dot , numerical results.
(a)(b)(c)(d)This two-stage nonlinear isolation system extends the frequency range of isolation to low frequencies [13]. Furthermore, as shown in Figures 3–6, the values of the relative amplitude between upper and lower stages in the resonance region are at a similar level, but in the high frequency region, the isolation performance of the lower stage is better than that of the upper stage.


### 4.2. Force Transmissibility

The force transmissibility of the two-stage nonlinear isolator is expressed by [11](22)TF=F^mF^e,where the nondimensional force transmitted from ms2 to the ground can be written as(23)f^mx^m,x^m′=2ζv2x^m′τ+4ζn2x^mτ2x^m′τ+Ω22x^mτ+γ2x^mτ3.Substituting the amplitude of the first-order component ofF^m obtained from equation (23) into equation (22) yields(24)TF=1F^e−4ωmX^mamζv2R3+2ζn2X^m2R42+2X^mcmΩ22R7+γ2X^m2R82.When the numerical solution of equations (4a) and (4b) is obtained, the force transmissibility can be solved using equation (24) and is shown in Figure 7. It can be seen that the solution of the EHBM can be compared well with the numerical results. More detailed analyses of the force transmissibility are given in [11–15].Figure 7
Force transmissibility whenμ=0.2, ζv1=ζv2=0.01, ζh1=ζh2=0.1, kv2/kv1=1, kh1/kv1=kh2/kv1=7/6, l^=0.7, and F^e=10. Solid line , stable solution by the EHBM; dashed line , unstable solution by the EHBM; dot , numerical results.


### 4.3. Displacement Transmissibility

The solutions of equations (5a) and (5b) can be assumed as equations (6a) and (6b) or equations (10a)–(10d). The amplitude-frequency equations of the two-stage geometrically nonlinear isolation system with base excitation obtained by using the HBM1 are given by(25a)α˜r≈Ω2X^esinϕr−X^msinθ,(25b)β˜r−Ω2X^r≈Ω2X^mcosθ+X^ecosϕr,(25c)α˜m≈Ω21+μX^esinϕm+X^rsinθ,(25d)β˜m−1+μΩ2X^m≈Ω21+μX^ecosϕm+X^rcosθ,where the expressions of α˜r, β˜r, α˜m, and β˜m are the same as in equations (20a)–(20d).Substitutingα˜re, β˜re, α˜me, and β˜me, which are given in equations (21a)–(21d), into equations (25a)–(25d), respectively, we can obtain the results of the EHBM. The square of the elliptic modulus and the Jacobi frequency functions are given by mr=γ1X^r2/2Ω12+γ1X^r2, mm=γ2X^m2/2Ω22+γ2X^m2, ωr2=Ω12+γ1X^r2X^r/X^r+X^mcosθ+X^ecosϕr, and ωm2=Ω22+γ2X^m2/1+μ+1+μX^ecosϕm+X^rcosθ/X^m. Using Ω=πωr/2Kmr=πωm/2Kmm and equations (25a) and (25c), the amplitudes (X^r and X^m) and phases (ϕr and ϕm) of the steady-state response can be solved using the EHBM. The relative and absolute displacement transmissibilities can be determined by Tr=x^1−x^e/X^e=X^rsinϕr+X^msinϕm2+X^rcosϕr+X^mcosϕm2/X^e and Td=x^1/X^e=X^rcosϕr+X^mcosϕm+X^e2+X^rsinϕr+X^msinϕm2/X^e and are shown in Figure 8. The solutions of the EHBM are compared well with the numerical results. More detailed analyses of the isolation characteristics are given in [11–15].Figure 8
(a) Relative displacement transmissibility and (b) absolute displacement transmissibility whenμ=0.2, ζv1=ζv2=0.01, ζh1=ζh2=0.1, kv2/kv1=1, kh1/kv1=kh2/kv1=7/6, l^=0.7, and X^e=0.1. Solid line , stable solution by the EHBM; dashed line , unstable solution by the EHBM; dot , numerical results.
(a)(b)


## 4.1. Amplitude-Frequency Characteristics

Integrating equations (7a)–(7d) and substituting the results into equations (9a)–(9d), we obtain(20a)−X^rΩ2ζv1+ζn1X^r2≈F^esinϕr+Ω2X^msinϕr−ϕm,(20b)Ω12X^r+3γ1X^r34−Ω2X^r≈F^ecosϕr+Ω2X^mcosϕr−ϕm,(20c)−X^mΩ2ζv2+ζn2X^m2≈F^esinϕm+Ω2X^rsinϕm−ϕr,(20d)Ω22X^m+3γ2X^m34−1+μΩ2X^m≈F^ecosϕm+Ω2X^rcosϕm−ϕr.Integrating equations (16a), (16b), (18a), and (18b), in which the nondimensional nonlinear stiffness forces f^sr=Ω12X^rcnψr|mr+γ1X^rcnψr|mr3 and f^sm=Ω22X^mcnψm|mm+γ2X^mcnψm|mm3 and the nondimensional nonlinear damping forces f^dr=2ζv1+4ζn1X^rcnψr|mr2−X^rωrsnψr|mrdnψr|mr and f^dm=2ζv2+4ζn2X^mcnψm|mm2−X^mωmsnψm|mmdnψm|mm, yields(21a)α˜re≈−4X^rωrarζv1R1+2ζn1X^r2R2,(21b)α˜me≈−4X^mωmamζv2R3+2ζn2X^m2R4,(21c)β˜re≈2X^rcrΩ12R5+γ1X^r2R6,(21d)β˜me≈2X^mcmΩ22R7+γ2X^m2R8,where R1 to R8 are given in Appendix A.Substituting equations (21a) and (21b) into equations (19a) and (19c), respectively, and combining equation (14), we can numerically solve the amplitudes (X^r and X^m) and phases (ϕr and ϕm) of the steady-state response of the two-stage nonlinear isolator.The amplitude-frequency solutions obtained using the HBM1 and EHBM are plotted in Figure3. The numerical solutions calculated using the fourth-order Runge-Kutta method with a step-size control algorithm are also plotted. The parameters are μ=0.2, ζv1=ζv2=0.01, ζh1=ζh2=0.1, kv2/kv1=1, kh1/kv1=kh2/kv1=0.5, l^=0.7, and F^e=1. The EHBM improves the accuracy of the amplitude-frequency curve around the peak region. In the low-amplitude regime, the solutions of both the EHBM and HBM1 are compared well with the numerical results.Figure 3
(a) Amplitude-frequency curves of the relative displacement betweenms1 and ms2. (b) Amplitude-frequency curves of ms2, when μ=0.2, ζv1=ζv2=0.01, ζh1=ζh2=0.1, kv2/kv1=1, kh1/kv1=kh2/kv1=0.5, l^=0.7, and F^e=1. Dashed-dotted line , stable solution by the HBM1; thin-dashed line , unstable solution by the HBM1; solid line , stable solution by the EHBM; thick-dashed line , unstable solution by the EHBM; dot , numerical results.
(a)(b)Whenl^=0.7 and the nondimensional linear stiffness terms kv2/kv1=1 and kh1/kv1=kh2/kv1=7/6, the system becomes a two-stage QZS system. The amplitude-frequency curves of the relative displacement between ms1 and ms2 of the two-stage geometrically nonlinear isolation system, when μ=0.2, ζv1=ζv2=0.01, ζh1=ζh2=0.1, kv2/kv1=1, kh1/kv1=kh2/kv1=7/6, l^=0.7, and F^e=0.01, 0.1, 1, and 10, respectively, are plotted in Figure 4. When the system becomes a QZS system, the solutions of the EHBM can still be compared well with the numerical solutions. To improve the accuracy of the harmonic balance method, we should include enough terms and the calculation complexity is simultaneously increased [25]. Because the base displacement xe is zero, the displacement of the upper mass ms2x2 = xm. The amplitude-frequency curves of ms2 of the two-stage geometrically nonlinear isolation system are plotted in Figure 5, and the accuracy of the EHBM can be seen.Figure 4
Amplitude-frequency curves of the relative displacement betweenms1 and ms2 when μ=0.2, ζv1=ζv2=0.01, ζh1=ζh2=0.1, kv2/kv1=1, kh1/kv1=kh2/kv1=7/6, l^=0.7, and (a) F^e=0.01, (b) F^e=0.1, (c) F^e=1, and (d) F^e=10, respectively. Dashed-dotted line , stable solution by the HBM1; thin-dashed line , unstable solution by the HBM1; solid line , stable solution by the EHBM; thick-dashed line , unstable solution by the EHBM; dot •, numerical results.
(a)(b)(c)(d)Figure 5
Amplitude-frequency curves ofms2 when μ=0.2, ζv1=ζv2=0.01, ζh1=ζh2=0.1, kv2/kv1=1, kh1/kv1=kh2/kv1=7/6, l^=0.7, and (a) F^e=0.01, (b) F^e=0.1, (c) F^e=1, and (d) F^e=10, respectively. Dashed-dotted line , stable solution by the HBM1; thin-dashed line , unstable solution by the HBM1; solid line , stable solution by the EHBM; thick-dashed line , unstable solution by the EHBM; dot , numerical results.
(a)(b)(c)(d)To investigate the accuracy of the EHBM at different damping cases, we chooseζv1=ζv2=0.02 and ζh1=ζh2=0.2 as Case 1 and ζv1=ζv2=0.03 and ζh1=ζh2=0.3 as Case 2. The system parameters are the same as those in Figure 5 and F^e=0.1. The results are shown in Figure 6. The EHBM works well for these two cases.Figure 6
(a) and (c) are amplitude-frequency curves of the relative displacement betweenms1 and ms2, and (b) and (d) are amplitude-frequency curves of ms2, when μ=0.2, (a-b) ζv1=ζv2=0.02, and ζh1=ζh2=0.2, (c-d) ζv1=ζv2=0.03 and ζh1=ζh2=0.3, and kv2/kv1=1, kh1/kv1=kh2/kv1=7/6, l^=0.7, and F^e=0.1. Dashed-dotted line , stable solution by the HBM1; thin-dashed line , unstable solution by the HBM1; solid line , stable solution by the EHBM; thick-dashed line , unstable solution by the EHBM; dot , numerical results.
(a)(b)(c)(d)This two-stage nonlinear isolation system extends the frequency range of isolation to low frequencies [13]. Furthermore, as shown in Figures 3–6, the values of the relative amplitude between upper and lower stages in the resonance region are at a similar level, but in the high frequency region, the isolation performance of the lower stage is better than that of the upper stage.


## 4.2. Force Transmissibility

The force transmissibility of the two-stage nonlinear isolator is expressed by [11](22)TF=F^mF^e,where the nondimensional force transmitted from ms2 to the ground can be written as(23)f^mx^m,x^m′=2ζv2x^m′τ+4ζn2x^mτ2x^m′τ+Ω22x^mτ+γ2x^mτ3.Substituting the amplitude of the first-order component ofF^m obtained from equation (23) into equation (22) yields(24)TF=1F^e−4ωmX^mamζv2R3+2ζn2X^m2R42+2X^mcmΩ22R7+γ2X^m2R82.When the numerical solution of equations (4a) and (4b) is obtained, the force transmissibility can be solved using equation (24) and is shown in Figure 7. It can be seen that the solution of the EHBM can be compared well with the numerical results. More detailed analyses of the force transmissibility are given in [11–15].Figure 7
Force transmissibility whenμ=0.2, ζv1=ζv2=0.01, ζh1=ζh2=0.1, kv2/kv1=1, kh1/kv1=kh2/kv1=7/6, l^=0.7, and F^e=10. Solid line , stable solution by the EHBM; dashed line , unstable solution by the EHBM; dot , numerical results.


## 4.3. Displacement Transmissibility

The solutions of equations (5a) and (5b) can be assumed as equations (6a) and (6b) or equations (10a)–(10d). The amplitude-frequency equations of the two-stage geometrically nonlinear isolation system with base excitation obtained by using the HBM1 are given by(25a)α˜r≈Ω2X^esinϕr−X^msinθ,(25b)β˜r−Ω2X^r≈Ω2X^mcosθ+X^ecosϕr,(25c)α˜m≈Ω21+μX^esinϕm+X^rsinθ,(25d)β˜m−1+μΩ2X^m≈Ω21+μX^ecosϕm+X^rcosθ,where the expressions of α˜r, β˜r, α˜m, and β˜m are the same as in equations (20a)–(20d).Substitutingα˜re, β˜re, α˜me, and β˜me, which are given in equations (21a)–(21d), into equations (25a)–(25d), respectively, we can obtain the results of the EHBM. The square of the elliptic modulus and the Jacobi frequency functions are given by mr=γ1X^r2/2Ω12+γ1X^r2, mm=γ2X^m2/2Ω22+γ2X^m2, ωr2=Ω12+γ1X^r2X^r/X^r+X^mcosθ+X^ecosϕr, and ωm2=Ω22+γ2X^m2/1+μ+1+μX^ecosϕm+X^rcosθ/X^m. Using Ω=πωr/2Kmr=πωm/2Kmm and equations (25a) and (25c), the amplitudes (X^r and X^m) and phases (ϕr and ϕm) of the steady-state response can be solved using the EHBM. The relative and absolute displacement transmissibilities can be determined by Tr=x^1−x^e/X^e=X^rsinϕr+X^msinϕm2+X^rcosϕr+X^mcosϕm2/X^e and Td=x^1/X^e=X^rcosϕr+X^mcosϕm+X^e2+X^rsinϕr+X^msinϕm2/X^e and are shown in Figure 8. The solutions of the EHBM are compared well with the numerical results. More detailed analyses of the isolation characteristics are given in [11–15].Figure 8
(a) Relative displacement transmissibility and (b) absolute displacement transmissibility whenμ=0.2, ζv1=ζv2=0.01, ζh1=ζh2=0.1, kv2/kv1=1, kh1/kv1=kh2/kv1=7/6, l^=0.7, and X^e=0.1. Solid line , stable solution by the EHBM; dashed line , unstable solution by the EHBM; dot , numerical results.
(a)(b)


## 5. Conclusions

This study concerns an elliptic harmonic balance method (EHBM) used to calculate the force and displacement transmissibility of a two-stage nonlinear vibration isolator with geometric stiffness and damping. Through the first-order harmonic balance method (HBM1), we can obtain the amplitude-frequency equations of the two-stage nonlinear isolator. The orthogonal relationships between the nondimensional stiffness, damping, inertia, and excitation force components in the two-stage nonlinear isolator have been presented. The relationship between the Jacobi elliptic frequency and modulus and the excitation frequency is obtained during the steady-state response of the system. With the same number of balancing equations as HBM1, the EHBM can improve the accuracy of the harmonic components and perform better than the HBM1 for the solutions of amplitude-frequency response and transmissibility. Furthermore, multistage nonlinear vibration isolation systems can be analysed using the proposed procedure.



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