# Dynamic Analysis of a High-Static-Low-Dynamic-Stiffness Vibration Isolator with Time-Delayed Feedback Control

Authors: Yong Wang; Shunming Li; Chun Cheng; Xingxing Jiang
Journal: Shock and Vibration
 (2015)

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## Abstract

This paper proposes the time-delayed cubic velocity feedback control strategy to improve the isolation performance of High-Static-Low-Dynamic-Stiffness (HSLDS) vibration isolator. Firstly, the primary resonance of the controlled HSLDS vibration isolator is obtained by using multiple scales method. The equivalent damping ratio and equivalent resonance frequency are defined to study the effects of feedback gain and time delay on the primary resonance. The jump phenomenon analysis of the controlled system without and with time delay is investigated by using Sylvester resultant method and optimization method, respectively. The stability analysis of the controlled system is also considered. Then, the 1/3 subharmonic resonance of the controlled system is studied by using multiple scales method. The effects of feedback gain and time delay on the 1/3 subharmonic resonance are also presented. Finally, force transmissibility is proposed to evaluate the performance of the controlled system and compared with an equivalent linear passive vibration isolator. The results show that the vibration amplitude of the controlled system around the resonance frequency region decreases and the isolation frequency band is larger compared to the equivalent one. A better isolation performance in the high frequency band can be achieved compared to the passive HSLDS vibration isolator.


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## Body

## 1. Introduction

Recently passive nonlinear vibration isolators with High-Static-Low-Dynamic-Stiffness (HSLDS) characteristic [1, 2] have drawn much attention by many researchers, since they can have a high-static stiffness with a small static displacement without sacrificing the load bearing capacity and a small dynamic stiffness to achieve a low natural frequency which can perform better than the linear ones. The HSLDS vibration isolator is comprised of a load bearing elastic element providing positive stiffness and special mechanisms providing negative stiffness called stiffness correctors. The load bearing elastic element is usually the vertical spring and the stiffness correctors have various types. Carrella, Kovacic, and Wang et al. [3–6] used inclined springs as stiffness correctors to build a HSLDS vibration isolator and studied the static and dynamic behaviors theoretically. Le and Ahn [7] considered a HSLDS vibration isolator composed of two symmetric negative stiffness structures and a positive stiffness structure for improving vibration isolation performance of the vehicle seat. Robertson, Zhou, Xu, and Shin et al. [8–11] built the HSLDS vibration isolators using electromagnetic springs or magnetic springs as stiffness correctors and analyzed the static and dynamic characteristics theoretically and experimentally. Liu et al. [12, 13] built a HSLDS vibration isolator by using Euler buckled beams as stiffness correctors and investigated the dynamic characteristics systematically. Shaw et al. [14, 15] used bistable composite plate as stiffness correctors to build a HSLDS vibration isolator and analyzed the steady state response detailedly. After being loaded at the static equilibrium position, the negative stiffness provided by the stiffness correctors is exactly balanced by the positive stiffness of the vertical spring; then a small dynamic stiffness can be obtained to achieve a low natural frequency.The dynamic motion of the HSLDS vibration isolator under force or base excitation can be approximately described by the Duffing equation with linear and cubic stiffness terms. Because of the inherent nonlinearity in the HSLDS vibration isolator, with the increase of the excitation force or base amplitude, the resonance peak can become large and jump phenomenon can occur at the resonant frequency which are undesirable conditions in practical engineering. Therefore, the HSLDS vibration isolator should be improved to obtain better isolation performance. Increasing the linear damping can reduce the vibration amplitude around the resonance frequency band, avoid the jump phenomenon, and obtain a larger isolation frequency band but results in a poor performance in the high frequency band where the isolation is required. Passive vibration isolators often meet the dilemma between these characteristics; this can be eliminated by using active control strategy. The active control of the Duffing oscillator has been studied by many researchers [16–20]; because of the unavoidable time delay in controllers and actuators, the effects of time delay are considered in the active control strategy. The active control strategy is usually the time-delayed linear velocity or time-delayed linear displacement control strategy. Few studies have drawn attention to the active control of the HSLDS vibration isolator; Sun et al. [21] used the time-delayed linear displacement control strategy to obtain better isolation performance around the resonance frequency band under both force and base excitation. Although these mentioned strategies can be used to achieve better isolation performance of the HSLDS vibration isolator around the resonance frequency band and obtain a larger isolation frequency band under both force and base excitation, the dilemma as the passive vibration isolator case can exist in these active control strategies. So in order to overcome this dilemma, this paper proposes the time-delayed cubic velocity feedback control strategy. Compared to the above-mentioned strategies, the time-delayed cubic velocity feedback control strategy can not only obtain better isolation performance around the resonance frequency band and obtain a larger isolation frequency, but also achieve a better performance in the high frequency band where the isolation is required.The organization of this paper is as follows. A HSLDS vibration isolator with time-delayed cubic velocity feedback control is presented and brief description of static analysis is shown in Section2. In Section 3, the primary resonance of the controlled HSLDS vibration isolator is obtained by using the multiple scales method. The jump phenomenon analysis of the controlled system without and with time delay is investigated by using the Sylvester resultant method and optimization method, respectively. The stability analysis of the controlled system is then studied. The effects of the feedback gain and time delay on the primary resonance, jump phenomenon, and stability are also considered. In Section 4, the 1/3 subharmonic resonance is studied by using the multiple scales method. The effects of the feedback gain and time delay on the existence regions where 1/3 subharmonic resonance occurs are also presented. In Section 5, the force transmissibility of the controlled HSLDS vibration isolator is obtained and compared with an equivalent linear passive vibration isolator. Conclusions are drawn in Section 6.


## 2. Modeling of a HSLDS Vibration Isolator

A HSLDS vibration isolator with time-delayed cubic velocity feedback control is shown in Figure1. The HSLDS vibration isolator is comprised of vertical spring used as load bearing element and two symmetric stiffness correctors providing negative stiffness.Figure 1
Scheme of a HSLDS vibration isolator with time-delayed cubic velocity feedback control.Figure1 also shows when loading a mass m the system is balanced at the static equilibrium position. The stiffness of the vertical and horizontal springs is Kv and Kh; the initial length of the horizontal springs is L0; the compression deformation of the horizontal springs when the system is at the static equilibrium position is λ; the length of the connecting element is L; x is the displacement of the mass from the static equilibrium position; the damping coefficient of the damping is C; and the excitation is force excitation with a harmonic force fe=Fecos⁡wt. The velocity of the mass is measured by the sensor; the velocity signal x˙ can be exported to a certain cubic velocity feedback signal through the nonlinear controller and actuator. Because of the unavoidable time delay in the controller and actuator, the effects of time delay should be considered; then the time-delayed cubic velocity feedback control strategy can be implemented.The force-displacement and stiffness-displacement relationships of the system are given as(1a)F=Kvx-2Khλ-L-L2-x2xL2-x2,(1b)K=Kv-2Kh+2KhL-λL2L2-x23/2.Equations (1a) and (1b) can be written in nondimensional form as(2a)F^=1-2kx^+2k1-λ^x^1-x^2,(2b)K^=1-2k+2k1-λ^1-x^23/2,where x^=x/L, λ^=λ/L, k=Kh/Kv, F^=F/KvL, and  K^=K/Kv.The nondimensional stiffness at the static equilibrium position can be obtained by substitutingx^=0 into (2b):(3)K^s=1-2kλ^.If the stiffness of the HSLDS vibration isolator is zero at the static equilibrium position, the quasi-zero-stiffness (QZS) characteristic can be obtained; then the value ofλ^ is given as(4)λ^qzs=12k.The nondimensional force-displacement and stiffness-displacement curves of the HSLDS vibration isolator for various values ofλ^ when k=1 are shown in Figure 2. It can be clearly seen that if λ^=λ^qzs, the negative stiffness provided by the stiffness correctors is exactly balanced by the positive stiffness of the vertical spring in the vertical direction at the static equilibrium position; then the QZS characteristic can be achieved. If λ^>λ^qzs, the stiffness is negative in the neighborhood of the static equilibrium position; then the system can be unstable which is an undesirable condition in practical engineering. Because of some inevitable errors in practical conditions, the QZS characteristic can be hard to achieve and in order to ensure the system stability, the stiffness of the system should keep a small positive value at the static equilibrium position, so the system can have a common HSLDS characteristic. In this paper, the case λ^<λ^qzs is considered.Figure 2
Nondimensional force-displacement and stiffness-displacement curves.When the amplitude of the displacement is small, the nondimensional force and stiffness can be expanded as a Taylor series at the static equilibrium positionx^=0 for simplicity(5a)F^ax^=1-2kλ^x^+k1-λ^x^3=αx^+γx^3,(5b)K^ax^=α+3γx^2,(5c)α=1-2kλ^,(5d)γ=k1-λ^.


## 3. Primary Resonance

### 3.1. Steady State Resonance

The dynamic equation of the HSLDS vibration isolator with time-delayed cubic velocity feedback control under harmonic force excitation using third-order Taylor series expansion is given as(6)mx¨+Cx˙+C3x˙3t-δ+Kvαx+γKvx3L2=Fecos⁡wt,where C3 and δ are the feedback gain and time delay, respectively; the dots denote derivatives with respect to time t. Equation (6) can be written in nondimensional form as(7)x^′′+2ζ1x^′+2ζ3x^′3T-τ+x^+βx^3=f1cos⁡ΩT,where x^=x/L, ζ1=C/2mwn, ζ3=C3L2wn/2m, β=γ/α, wn=αKv/m, Ω=w/wn, T=wnt, f1=Fe/αKvL, and τ=wnδ. The primes denote derivatives with respect to T. To analyze the primary resonance of the controlled system under force excitation by using the multiple scales method, it is assumed that(8a)ζ1=εζ^1,(8b)ζ3=εζ^3,(8c)β=εβ^,(8d)f1=εf,(8e)Ω2=1+εσ,where ε is a formal small parameter and σ is the detuning frequency. Then it is convenient to rewrite (7) as(9)x^′′+2εζ^1x^′+2εζ^3x^′3T-τ+Ω2-εσx^+εβ^x^3=εfcos⁡ΩT.For simplicity, assuming a two-scale expansion of (9) as(10)x^τ=x0T0,T1+εx1T0,T1+Oε2,Tn=εnT,n=0,1and using the following differential operators [22](11)ddT=∂∂T0+ε∂∂T1+Oε2=D0+εD1+Oε2,d2dT2=D02+2εD0D1+Oε2.Substituting (10) and (11) into (9) and equating the coefficients of the same power of ε, the following equations can be obtained:(12a)D02x0+Ω2x0=0,(12b)D02x1+Ω2x1=-2D0D1x0-2ξ^1D0x0-2ξ^3D0x0T0-τ3-β^x03+σx0+fcos⁡ΩT0.The solution of (12a) can be written in the form(13a)x0T0,T1=AT1ejΩT0+cc,(13b)AT1=12aT1ejΦT1,where cc denotes the conjugate term and the amplitude a and phase Φ are functions of the slow time scale T1. Substituting (13a) into (12b) and removing secular terms from (12b) require(14)-j2ΩD1A-j2ξ^1ΩA-3β^A2A--j6ζ^3Ω3A2A-e-jΩτ+σA+f2=0.Substituting (13b) into (14) and separating the real and imaginary terms, the following equations can be obtained:(15a)a′=-1ΩΩζ^1a+34ζ^3Ω3a3cos⁡Ωτ+f2sinΦ,(15b)aΦ′=1Ω-12σa+38β^a3+34ζ^3Ω3a3sinΩτ-f2cos⁡Φ.The steady state solutions of (9) for the primary resonance corresponding to the fixed points are given as(16a)Ωζ^1a+34ζ^3Ω3a3cos⁡Ωτ=-f2sinΦ,(16b)-12σa+38β^a3+34ζ^3Ω3a3sinΩτ=f2cos⁡Φ.Then the amplitude-frequency and phase-frequency relationships using the original structural parameter can be given by(17a)2Ωζ1a+32ζ3Ω3a3cos⁡Ωτ2+Ω2-1a-34βa3-32ζ3Ω3a3sinΩτ2=f12,(17b)tanΦ=Ω2-1a-3/4βa3-3/2ζ3Ω3a3sinΩτ2Ωζ1a+3/2ζ3Ω3a3cos⁡Ωτ.From (17a), it can be seen that the primary resonance of the controlled system is greatly influenced by the nondimensional feedback gain ζ3 and time delay τ when the other structural parameters and excitation force amplitude are chosen. The effects of these two parameters on the primary resonance can be seen by defining the equivalent damping ratio and equivalent resonance frequency as follows:(18a)ζeq=ζ1+34ζ3Ω2a2cos⁡Ωτ,(18b)weq2=1+34βa2+32ζ3Ω3a2sinΩτ.Figure3 shows the amplitude-frequency and phase-frequency relationships of the primary resonance for the controlled system under different feedback gain ζ3 without time delay. When the time delay τ=0, the feedback gain ζ3 only appears in the equivalent damping ratio term and (18a) and (18b) simplify into(19a)ζeq=ζ1+34ζ3Ω2a2,(19b)weq2=1+34βa2.Figure 3
Amplitude-frequency and phase-frequency relationships of the primary resonance for the controlled system under different feedback gainζ3 without time delay (ζ1=0.025, f1=0.05, k=1, and λ^=0.4).The effect of the feedback gainζ3 on the primary resonance can be clearly seen in Figure 3. When the linear damping ratio ζ1 is small and the system is uncontrolled, the peak amplitude of the primary resonance takes a larger value and the jump phenomenon occurs. When the feedback gain ζ3 increases, the equivalent damping ratio becomes larger, the peak amplitude of the primary resonance decreases, the equivalent resonance frequency decreases, and the jump phenomenon can be avoided.When the time delayτ exists in the controlled system, the following inequalities of the equivalent damping ratio and equivalent resonance frequency can be obtained:(20a)ζ1-34ζ3Ω2a2≤ζeq=ζ1+34ζ3Ω2a2cos⁡Ωτ≤ζ1+34ζ3Ω2a2,(20b)1+34βa2-32ζ3Ω3a2≤weq2=1+34βa2+32ζ3Ω3a2sinΩτ≤1+34βa2+32ζ3Ω3a2.Obviously, compared to the controlled system without time delay, when time delayτ exists, it can decrease the equivalent damping ratio; then the peak amplitude increases; the equivalent resonance frequency can also increase which leads to an undesirable isolation performance. In the primary resonance region, assuming that Ω≈1, the equivalent damping ratio and equivalent resonance frequency can be approximated as(21a)ζeq=ζ1+34ζ3a2cos⁡τ,(21b)weq2=1+34βa2+32ζ3a2sinτ.Then the equivalent damping ratio and equivalent resonance frequency can be approximately considered as the periodic functions in2π periods. Based on this assumption, the effect of the time delay τ on the primary resonance can be clearly seen in Figure 4. When the time delay τ lies in the ranges of 0,π/2 and 3π/2,2π, the equivalent damping ratio is larger than the linear damping ratio; then the peak amplitude and resonance frequency are smaller than the uncontrolled system which can achieve a better isolation performance in the primary resonance region; the equivalent damping ratio takes a major effect on the peak amplitude and resonance frequency in these two ranges. When the time delay τ lies in the range of π/2,π, the equivalent damping ratio is smaller than the linear damping ratio and the equivalent resonance frequency is larger resulting in a poorer isolation performance in the primary resonance region than the uncontrolled system. When the time delay τ lies in the range of π,3π/2, although the equivalent damping ratio is smaller than the linear damping ratio, the equivalent resonance frequency is smaller and takes a major effect on the peak amplitude and resonance frequency; the peak amplitude and resonance frequency are smaller which can achieve a better isolation performance in the primary resonance region compared to the uncontrolled system.Figure 4
Amplitude-frequency relationships of the primary resonance for the controlled system with different time delayτ (ζ1=0.025, ζ3=0.05, f1=0.05, k=1, and λ^=0.4).
(a)
τ
=
0
(b)
τ
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π
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4
(c)
τ
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π
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(d)
τ
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3
π
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(e)
τ
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(f)
τ
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(g)
τ
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(h)
τ
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(i)
τ
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πFigure4 also shows the numerical results obtained by solving (7) using the fourth order Runge-Kutta method; the analytic method gives very well results compared with the numerical results. But when the time delay τ takes some special values, a closed-loop response curve [23] emerges above the primary resonance because multiple solutions exist in these frequency bands; some closed-loop response curves can be estimated by numerical results and some cannot be which are due to their stability. In Figures 4(c), 4(d), 4(e), and 4(i), some parts of the upper branch of the primary resonance cannot also be estimated by numerical results which are also due to their stability. Stability analysis of the controlled system will be investigated in the stability analysis part.As can be clearly seen in Figures3 and 4, the feedback gain ζ3 mostly influences the vibration amplitude of the controlled system around the resonance frequency band. When the feedback gain ζ3 changes, the vibration amplitudes of the controlled system in the low and high frequency bands are almost the same as the uncontrolled system. Figure 5 shows the effect of the time delay τ on the primary resonance for the controlled system with different feedback gain ζ3 around the resonance frequency band. In this case only one single vibration amplitude is considered, the frequency Ω is chosen as 1 and 1.1, respectively, and the stability analysis of the vibration amplitude is also investigated in the stability analysis part.Figure 5
Effect of the time delayτ on the primary resonance for the controlled system with different feedback gain ζ3 (ζ1=0.025, f1=0.05, k=1, and λ^=0.4).
(a)
Ω
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1
(b)
Ω
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1.1The vibration amplitude is a periodic function of time delayτ which can be seen in (17a). In the first half period, the vibration amplitude decreases with the increase of feedback gain ζ3, while in the latter half period the vibration amplitude first increases and then decreases with the increase of feedback gain ζ3. Figure 6 shows the corresponding amplitude-frequency relationships of the primary resonance for the controlled system with different feedback gain ζ3. When τ=π/4 lies in the range of the first half period, with the increase of feedback gain ζ3, the vibration amplitude around the resonance frequency band decreases and jump phenomenon can be avoided. When τ=5π/4 lies in the range of the latter half period, with the increase of feedback gain ζ3, the vibration amplitude around the resonance frequency band first increases and then decreases, and jump phenomenon can also be avoided when feedback gain ζ3 increases to a larger value.Figure 6
Amplitude-frequency relationships of the primary resonance for the controlled system with different feedback gainζ3 (ζ1=0.025, f1=0.05, k=1, and λ^=0.4).
(a)
τ
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π
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4
(b)
τ
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### 3.2. Jump Phenomenon Analysis

From Figures3 and 4, it can be seen that jump phenomenon can occur in the primary resonance of the controlled system when the structural parameters and excitation force amplitude are chosen as some specific values. The primary response amplitude decreases at jump-down frequency and increases at jump-up frequency. Jump phenomenon can be detrimental and can probably cause extensive physical damage to the controlled system due to the large amplitude changes occurring at jump-down or jump-up frequency. So it is beneficial to determine the critical surface of the structural parameters which lead to no jump phenomenon, and the jump frequencies should be determined when jump phenomenon occurs. The jump phenomenon analysis of the controlled system without and with time delay is investigated, respectively, in this part.The amplitude-frequency function of the controlled system without time delay can be simplified into(22)2Ωζ1a+32ζ3Ω3a32+Ω2-1a-34βa32=f12.Equation (22) can be written as a polynomial function in amplitude a or frequency Ω as follows:(23a)HΩ,a=p1a6+q1a4+r1a2+s1,(23b)p1=916β2+94ζ32Ω6,(23c)q1=6ζ1ζ3Ω4-32βΩ2-1,(23d)r1=4ζ12Ω2+Ω2-12,(23e)s1=-f12,(23f)HΩ,a=p2Ω6+q2Ω4+r2Ω2+s2,(23g)p2=94ζ32a6,(23h)q2=6ζ1ζ3a4+a2,(23i)r2=-32βa4+4ζ12-2a2,(23j)s2=916β2a6+32βa4+a2-f12.It is inconvenient to solve the amplitudea or frequency Ω in an analytic expression since (23a)–(23j) can be treated as a cubic function of amplitude a2 or frequency Ω2; it is difficult to determine the critical surface of the structural parameters and jump frequencies based on these analytic expressions. So the method of Sylvester resultant [24] is used in this paper. Assuming two polynomials fx and gx as(24a)fx=∑i=0naixi,an≠0,(24b)gx=∑i=0mbixi,bm≠0.The Sylvester resultant offx and gx can be obtained [25]:(25)Rf,g=anan-1⋯⋯a1a00⋯00anan-1⋯⋯a1a0⋯0⋯⋯⋯⋯⋯⋯0⋯0anan-1⋯⋯⋯a0bmbm-1⋯b1b00⋯⋯00bmbm-1⋯b1b00⋯0⋯⋯⋯⋯⋯⋯0⋯⋯0bmbm-1⋯⋯b0.A necessary and sufficient condition for two polynomialsfx and gx to have a common root is that the Sylvester resultant Rf,g=0 [26].When the excitation force amplitude takes a smaller value for chosen structural parameter, jump phenomenon does not occur and the amplitude-frequency curve is single-valued which is similar to the linear harmonic excited system; that is, there exists a unique value of amplitudea for each value of frequency Ω. When the excitation force amplitude increases to a larger value, jump phenomenon can occur due to the existence of cubic nonlinear terms; there exist multiple values of amplitude a for each value of frequency Ω in a specific frequency band. Denote f1cr as the critical value of the excitation force amplitude, that is the maximum excitation force amplitude which leads to no jump phenomenon. Denoting Ωcr,acr as the infection point in the amplitude-frequency curve when f1=f1cr, then the function HΩcr,a has three positive real roots equal to acr. Assuming the functions H′Ω,a and H′′Ω,a as the derivative and second derivative of the function HΩ,a with respect to the amplitude a, respectively, then the function H′Ωcr,a has two positive real roots equal to acr and the function H′′Ωcr,a has one positive real root equal to acr. According to the definition of Sylvester resultant, the resultant RH′,H′′ should be equal to zero at the infection point Ωcr,acr.Combining (23a)–(23j) and (25), the following equation can be obtained:(26)Gacr=RH′,H′′a=acr=∑i=010biacri=0,where the definitions of the coefficients bi are given in Appendix. Using (26), Ωcr can be eliminated and can obtain a polynomial equation in acr. Then the value of acr can be obtained numerically. Of the ten roots of GΩcr=0, there are two real and positive roots and the value of acr should be taking the smaller value. The critical excitation frequency Ωcr can be given by substituting acr into the function H′′Ωcr,a. Once the critical excitation frequency Ωcr and critical amplitude acr are known, the critical excitation force amplitude f1cr can be obtained by using (23a) or (23f).When the excitation force amplitudef1>f1cr, jump phenomenon occurs; it is necessary to determine the jump-up and jump-down frequencies. Denoting Ω∗,a∗ as the jump points in the amplitude-frequency curve, then the function HΩ∗,a has two positive real roots equal to a∗, which requires that the resultant RH,H′ should be equal to zero at the jump points Ω∗,a∗.Combining (23a)–(23j) and (25), the following equation can be obtained:(27)Ga∗=RH,H′a=a∗=∑i=018cia∗i=0,where the definitions of the coefficients ci are given in Appendix. Using (27), Ω∗ can be eliminated and can obtain a polynomial equation in a∗. Then the value of a∗ can be obtained numerically. Of the eighteen roots of GΩcr=0, there are two real and positive roots; the smaller value corresponds to the amplitude of jump-up frequency and the larger value corresponds to the amplitude of jump-down frequency. The jump frequencies can be obtained by substituting a∗ into the function H′Ωcr,a. But for each value of a∗, there are two real and positive values of Ω∗, one of which is spurious. To investigate which one is spurious, substitute Ω∗ into the function HΩ∗,a and check whether there are two real and positive values equal to a∗ or not. If there are none, then this particular value of Ω∗ is spurious and should be discarded.Amplitude-frequency relationship of the primary resonance for the controlled system under different excitation force amplitude without time delay is shown in Figure7. Using the method of Sylvester resultant, the critical excitation force amplitude f1cr=0.01102. The infection point Ωcr,acr can be obtained as 1.0485,0.1698 which can be seen in Figure 7 denoted by black asterisk point. As discussed previously, when the excitation force amplitude f1=0.03>f1cr, jump phenomenon occurs; the jump points Ω∗,a∗ can also be determined by the method of Sylvester resultant. The jump-up points and jump-down points can be obtained as 1.108,0.1945 and 1.163,0.3891, respectively, which can be seen in Figure 7 denoted by black dot points. Figure 8 shows the critical surface of the structural parameters which lead to no jump phenomenon. It is apparent that the reasonable structural parameter combinations should lie under the critical surface. Figure 8 also shows the partial contours of the critical surface; the direction of the arrow indicates the reasonable structural parameter combinations which lead to no jump phenomenon.Figure 7
Amplitude-frequency relationship of the primary resonance for the controlled system under different excitation force amplitude without time delay (ζ1=0.025, ζ3=0.05, k=1, and λ^=0.4).Figure 8
Critical surface and contours of the critical surface of the controlled system without time delay (k=1 and λ^=0.4).The jump phenomenon analysis of the controlled system with time delay is then considered. Because the trigonometric functions exist in the amplitude-frequency function of the primary resonance, the method of Sylvester resultant cannot be used in this case. The critical boundary which leads to no jump phenomenon and jump frequencies will be investigated by using the optimization method.For the controlled system with time delay, when there are no points in the amplitude-frequency curves satisfying the conditiondΩ/da=0, jump phenomenon does not occur [27]. But when there are at least two points in the amplitude-frequency curves satisfying the condition dΩ/da=0, the jump phenomenon can occur. So the critical condition is that there is only one point in the amplitude-frequency curves satisfying the condition dΩ/da=0. Differentiating both sides of (17a) with respect to a and substituting the condition dΩ/da=0 in it, the critical function can be given as(28)2Ωζ1+32ζ3Ω3a2cos⁡Ωτ2Ωζ1+92ζ3Ω3a2cos⁡Ωτ+Ω2-1-34βa2-32ζ3Ω3a2sinΩτ·Ω2-1-94βa2-92ζ3Ω3a2sinΩτ=0.Combining the critical function equation (28) and the amplitude-frequency function equation (17a), the critical boundary and jump frequencies can be achieved.An example of how to determine the critical excitation force amplitudef1cr by using the optimization method is presented when the other structural parameters are chosen. The Levenberg-Marquardt (LM) optimization method is considered in this paper. Denoting the critical function equation (28) and the amplitude-frequency function equation (17a) as the objective functions and defining frequency Ω and amplitude a as the independent variable. Set up the convergence index η as 1e-10 and the maximum number of iterations N as 2000. For a given critical excitation force amplitude f1cr, the frequency Ω and amplitude a can be determined by using the LM optimization method to minimize the objective functions; then the value of objective functions can be obtained.Substituting the initial guess value of critical excitation force amplitudef1icr into the objective functions and using the LM optimization method, when the initial value of f1icr is chosen as a larger one, the value of the objective functions is smaller than the convergence index η which indicates the value of f1icr is larger and jump phenomenon occurs; the frequency Ω and amplitude a can be obtained as the jump frequency and the amplitude of jump frequency, respectively, decreasing the value of f1icr slowly until the value of the objective functions is larger than the convergence index η for the first time; then the critical excitation force amplitude f1cr can be obtained; when the initial value of f1icr is chosen as a smaller one, the value of the objective functions is larger than the convergence index η which indicates the value of f1icr is smaller and jump phenomenon does not occur; the obtained frequency Ω and amplitude a are spurious and should be discarded, increasing the value of f1icr slowly until the value of the objective functions is smaller than the convergence index η for the first time; then the critical excitation force amplitude f1cr can be obtained. The solution process for determining the critical excitation force amplitude f1icr is an iterative process.The critical border of the controlled system with time delayτ=π/4 is shown in Figure 9. Using the LM optimization method, the critical excitation force amplitude f1cr=0.01046. When the excitation force amplitude f1=f1cr, there is only one point in the amplitude-frequency curves satisfying the condition dΩ/da=0 which corresponds to the boundary. When the excitation force amplitude f1>f1cr, there are two points in the amplitude-frequency curves satisfying the condition dΩ/da=0 which implies the jump phenomenon. When the jump phenomenon occurs, the jump points can be determined conveniently by using the LM optimization method. Figure 10 shows the amplitude-frequency relationship of the primary resonance for the controlled system under different excitation force amplitude with time delay τ=π/4. When the excitation force amplitude f1=0.03>f1cr, the jump-up points and jump-down points can be obtained as 1.1095,0.1914 and 1.1973,0.4129, respectively, which can be seen in Figure 10 denoted by black dot points.Figure 9
Critical border of the controlled system with time delayτ=π/4 (ζ1=0.025, ζ3=0.05, k=1, and λ^=0.4).Figure 10
Amplitude-frequency relationship of the primary resonance for the controlled system under different excitation force amplitude with time delayτ=π/4 (ζ1=0.025, ζ3=0.05, k=1, and λ^=0.4).The other critical values of structural parameters, for example, damping ratioζ1 and feedback gain ζ3, can be determined just as the critical excitation force amplitude case, but the solution process is complicated which can be seen in the above part, so it is no longer discussed here.


### 3.3. Stability Analysis

To analyze the stability of the primary response of the controlled system, linearizing (15a) and (15b) with respect to a and Φ gives(29a)D1Δa=-ζ^1+94ζ^3Ω2a2cos⁡ΩτΔa-f2cos⁡ΦΔΦ,(29b)D1ΔΦ=34Ωβ^a+32ζ^3Ω2asinΩτ+f2Ωa2cos⁡ΦΔa+f2ΩasinΦΔΦ.The characteristic equation of (29a) and (29b) is given as(30)det⁡s+ζ^1+94ζ^3Ω2a2cos⁡Ωτf2cos⁡Φ-34Ωβ^a+32ζ^3Ω2asinΩτ+f2Ωa2cos⁡Φs-f2ΩasinΦ=0.Equation (30) can be simplified using (16a) and (16b):(31a)s2-G1s+G2=0,(31b)G1=-2ζ^1-3ζ^3Ω2a2cos⁡Ωτ,(31c)G2=ζ^1+34ζ^3Ω2a2cos⁡Ωτζ^1+94ζ^3Ω2a2cos⁡Ωτ+1Ω2σ2-38β^a2+34ζ^3Ω3a2sinΩτ·σ2-98β^a2+94ζ^3Ω3a2sinΩτ.According to the Routh-Hurwitz criterion, the controlled system has stable primary response corresponding to stable node whenG1<0 and G2>0, has unstable primary response corresponding to unstable node when G1>0 and G2>0, and has unstable primary response corresponding to saddle point when G2<0. Actually, the critical condition G1=0 indicates a pair of purely imaginary eigenvalues exist and hence a Hopf bifurcation occurs [16]. For the uncontrolled system G1=-2ζ^1<0 and controlled system without time delay G1=-2ζ^1-3ζ^3Ω2a2<0, the Hopf bifurcation can be avoided.The stability boundary of the primary resonance for the controlled system with different time delay is shown in Figure11. The gray shaded unstable regions denoted by  USR1 are determined by G1 and the green shaded unstable regions denoted by  USR2 are determined by G2. When time delay τ=π/4, G1>0 in the given frequency band, and the unstable regions are determined by G2. When time delay τ=π/2, the unstable regions are determined by both G1 and G2. The upper branch of the primary resonance is cut off by the boundary of USR1 determined by G1; then these unstable responses cannot be obtained by numerical method which can be clearly seen in Figure 4(c). When time delay τ=7π/4, a closed-loop response curve emerges above the primary resonance. It is observed that the lower branch of the closed-loop response curve is all covered by USR1 and USR2 and the upper branch of the closed-loop response curve is partially covered by USR1 which can also be clearly seen in Figure 4(h) with using numerical method checked.Figure 11
Stability boundary of the primary resonance for the controlled system with different time delayτ (ζ1=0.025, ζ3=0.05, f1=0.05, k=1, and λ^=0.4).
(a)
τ
=
π
/
4
(b)
τ
=
π
/
2
(c)
τ
=
7
π
/
4From (31a)–(31c), it can be seen that the stability boundary of the primary resonance for the controlled system is greatly influenced by the nondimensional feedback gain ζ3 and time delay τ when the other structural parameters and excitation force amplitude are chosen. Two cases are considered when investigating the effects of these two parameters on the stability of vibration amplitude for the controlled system. The first case is when only one single vibration amplitude exists and the second case is when three vibration amplitudes may exist in the controlled system.Figure12 shows the effect of the time delay τ on the stability of vibration amplitude for the controlled system with different feedback gain ζ3. The dashed line indicates the unstable vibration amplitude. When Ω=1.1, only one single vibration amplitude exists in the controlled system. When feedback gain ζ3 takes a small value, the vibration amplitude is always stable regardless of the value of time delay τ; with the increase of feedback gain ζ3, the vibration amplitude can be unstable for some specific regions of time delay τ; the unstable solutions where a Hopf bifurcation occurs are determined by G1. Behaviour is more complicated when Ω=1.3 because three vibration amplitudes exist for some specific regions of time delay τ. The lowest branch is always stable regardless of the value of time delay τ and remains the same for different feedback gain ζ3 which can be seen in Figure 4. The vibration amplitude of the lowest branch is smaller compared to the upper two branches. When feedback gain ζ3 takes a small value, the upper branch is stable and the lower branch is unstable which is determined by G2; with the increase of feedback gain ζ3, the upper branch can be unstable for some specific regions of time delay τ; the unstable solutions where a Hopf bifurcation occurs are determined by G1.Figure 12
Effect of the time delayτ on the stability of vibration amplitude for the controlled system with different feedback gain ζ3 (ζ1=0.025, f1=0.05, k=1, and λ^=0.4).
(a)
Ω
=
1.1
(b)
Ω
=
1.3


## 3.1. Steady State Resonance

The dynamic equation of the HSLDS vibration isolator with time-delayed cubic velocity feedback control under harmonic force excitation using third-order Taylor series expansion is given as(6)mx¨+Cx˙+C3x˙3t-δ+Kvαx+γKvx3L2=Fecos⁡wt,where C3 and δ are the feedback gain and time delay, respectively; the dots denote derivatives with respect to time t. Equation (6) can be written in nondimensional form as(7)x^′′+2ζ1x^′+2ζ3x^′3T-τ+x^+βx^3=f1cos⁡ΩT,where x^=x/L, ζ1=C/2mwn, ζ3=C3L2wn/2m, β=γ/α, wn=αKv/m, Ω=w/wn, T=wnt, f1=Fe/αKvL, and τ=wnδ. The primes denote derivatives with respect to T. To analyze the primary resonance of the controlled system under force excitation by using the multiple scales method, it is assumed that(8a)ζ1=εζ^1,(8b)ζ3=εζ^3,(8c)β=εβ^,(8d)f1=εf,(8e)Ω2=1+εσ,where ε is a formal small parameter and σ is the detuning frequency. Then it is convenient to rewrite (7) as(9)x^′′+2εζ^1x^′+2εζ^3x^′3T-τ+Ω2-εσx^+εβ^x^3=εfcos⁡ΩT.For simplicity, assuming a two-scale expansion of (9) as(10)x^τ=x0T0,T1+εx1T0,T1+Oε2,Tn=εnT,n=0,1and using the following differential operators [22](11)ddT=∂∂T0+ε∂∂T1+Oε2=D0+εD1+Oε2,d2dT2=D02+2εD0D1+Oε2.Substituting (10) and (11) into (9) and equating the coefficients of the same power of ε, the following equations can be obtained:(12a)D02x0+Ω2x0=0,(12b)D02x1+Ω2x1=-2D0D1x0-2ξ^1D0x0-2ξ^3D0x0T0-τ3-β^x03+σx0+fcos⁡ΩT0.The solution of (12a) can be written in the form(13a)x0T0,T1=AT1ejΩT0+cc,(13b)AT1=12aT1ejΦT1,where cc denotes the conjugate term and the amplitude a and phase Φ are functions of the slow time scale T1. Substituting (13a) into (12b) and removing secular terms from (12b) require(14)-j2ΩD1A-j2ξ^1ΩA-3β^A2A--j6ζ^3Ω3A2A-e-jΩτ+σA+f2=0.Substituting (13b) into (14) and separating the real and imaginary terms, the following equations can be obtained:(15a)a′=-1ΩΩζ^1a+34ζ^3Ω3a3cos⁡Ωτ+f2sinΦ,(15b)aΦ′=1Ω-12σa+38β^a3+34ζ^3Ω3a3sinΩτ-f2cos⁡Φ.The steady state solutions of (9) for the primary resonance corresponding to the fixed points are given as(16a)Ωζ^1a+34ζ^3Ω3a3cos⁡Ωτ=-f2sinΦ,(16b)-12σa+38β^a3+34ζ^3Ω3a3sinΩτ=f2cos⁡Φ.Then the amplitude-frequency and phase-frequency relationships using the original structural parameter can be given by(17a)2Ωζ1a+32ζ3Ω3a3cos⁡Ωτ2+Ω2-1a-34βa3-32ζ3Ω3a3sinΩτ2=f12,(17b)tanΦ=Ω2-1a-3/4βa3-3/2ζ3Ω3a3sinΩτ2Ωζ1a+3/2ζ3Ω3a3cos⁡Ωτ.From (17a), it can be seen that the primary resonance of the controlled system is greatly influenced by the nondimensional feedback gain ζ3 and time delay τ when the other structural parameters and excitation force amplitude are chosen. The effects of these two parameters on the primary resonance can be seen by defining the equivalent damping ratio and equivalent resonance frequency as follows:(18a)ζeq=ζ1+34ζ3Ω2a2cos⁡Ωτ,(18b)weq2=1+34βa2+32ζ3Ω3a2sinΩτ.Figure3 shows the amplitude-frequency and phase-frequency relationships of the primary resonance for the controlled system under different feedback gain ζ3 without time delay. When the time delay τ=0, the feedback gain ζ3 only appears in the equivalent damping ratio term and (18a) and (18b) simplify into(19a)ζeq=ζ1+34ζ3Ω2a2,(19b)weq2=1+34βa2.Figure 3
Amplitude-frequency and phase-frequency relationships of the primary resonance for the controlled system under different feedback gainζ3 without time delay (ζ1=0.025, f1=0.05, k=1, and λ^=0.4).The effect of the feedback gainζ3 on the primary resonance can be clearly seen in Figure 3. When the linear damping ratio ζ1 is small and the system is uncontrolled, the peak amplitude of the primary resonance takes a larger value and the jump phenomenon occurs. When the feedback gain ζ3 increases, the equivalent damping ratio becomes larger, the peak amplitude of the primary resonance decreases, the equivalent resonance frequency decreases, and the jump phenomenon can be avoided.When the time delayτ exists in the controlled system, the following inequalities of the equivalent damping ratio and equivalent resonance frequency can be obtained:(20a)ζ1-34ζ3Ω2a2≤ζeq=ζ1+34ζ3Ω2a2cos⁡Ωτ≤ζ1+34ζ3Ω2a2,(20b)1+34βa2-32ζ3Ω3a2≤weq2=1+34βa2+32ζ3Ω3a2sinΩτ≤1+34βa2+32ζ3Ω3a2.Obviously, compared to the controlled system without time delay, when time delayτ exists, it can decrease the equivalent damping ratio; then the peak amplitude increases; the equivalent resonance frequency can also increase which leads to an undesirable isolation performance. In the primary resonance region, assuming that Ω≈1, the equivalent damping ratio and equivalent resonance frequency can be approximated as(21a)ζeq=ζ1+34ζ3a2cos⁡τ,(21b)weq2=1+34βa2+32ζ3a2sinτ.Then the equivalent damping ratio and equivalent resonance frequency can be approximately considered as the periodic functions in2π periods. Based on this assumption, the effect of the time delay τ on the primary resonance can be clearly seen in Figure 4. When the time delay τ lies in the ranges of 0,π/2 and 3π/2,2π, the equivalent damping ratio is larger than the linear damping ratio; then the peak amplitude and resonance frequency are smaller than the uncontrolled system which can achieve a better isolation performance in the primary resonance region; the equivalent damping ratio takes a major effect on the peak amplitude and resonance frequency in these two ranges. When the time delay τ lies in the range of π/2,π, the equivalent damping ratio is smaller than the linear damping ratio and the equivalent resonance frequency is larger resulting in a poorer isolation performance in the primary resonance region than the uncontrolled system. When the time delay τ lies in the range of π,3π/2, although the equivalent damping ratio is smaller than the linear damping ratio, the equivalent resonance frequency is smaller and takes a major effect on the peak amplitude and resonance frequency; the peak amplitude and resonance frequency are smaller which can achieve a better isolation performance in the primary resonance region compared to the uncontrolled system.Figure 4
Amplitude-frequency relationships of the primary resonance for the controlled system with different time delayτ (ζ1=0.025, ζ3=0.05, f1=0.05, k=1, and λ^=0.4).
(a)
τ
=
0
(b)
τ
=
π
/
4
(c)
τ
=
π
/
2
(d)
τ
=
3
π
/
4
(e)
τ
=
π
(f)
τ
=
5
π
/
4
(g)
τ
=
3
π
/
2
(h)
τ
=
7
π
/
4
(i)
τ
=
2
πFigure4 also shows the numerical results obtained by solving (7) using the fourth order Runge-Kutta method; the analytic method gives very well results compared with the numerical results. But when the time delay τ takes some special values, a closed-loop response curve [23] emerges above the primary resonance because multiple solutions exist in these frequency bands; some closed-loop response curves can be estimated by numerical results and some cannot be which are due to their stability. In Figures 4(c), 4(d), 4(e), and 4(i), some parts of the upper branch of the primary resonance cannot also be estimated by numerical results which are also due to their stability. Stability analysis of the controlled system will be investigated in the stability analysis part.As can be clearly seen in Figures3 and 4, the feedback gain ζ3 mostly influences the vibration amplitude of the controlled system around the resonance frequency band. When the feedback gain ζ3 changes, the vibration amplitudes of the controlled system in the low and high frequency bands are almost the same as the uncontrolled system. Figure 5 shows the effect of the time delay τ on the primary resonance for the controlled system with different feedback gain ζ3 around the resonance frequency band. In this case only one single vibration amplitude is considered, the frequency Ω is chosen as 1 and 1.1, respectively, and the stability analysis of the vibration amplitude is also investigated in the stability analysis part.Figure 5
Effect of the time delayτ on the primary resonance for the controlled system with different feedback gain ζ3 (ζ1=0.025, f1=0.05, k=1, and λ^=0.4).
(a)
Ω
=
1
(b)
Ω
=
1.1The vibration amplitude is a periodic function of time delayτ which can be seen in (17a). In the first half period, the vibration amplitude decreases with the increase of feedback gain ζ3, while in the latter half period the vibration amplitude first increases and then decreases with the increase of feedback gain ζ3. Figure 6 shows the corresponding amplitude-frequency relationships of the primary resonance for the controlled system with different feedback gain ζ3. When τ=π/4 lies in the range of the first half period, with the increase of feedback gain ζ3, the vibration amplitude around the resonance frequency band decreases and jump phenomenon can be avoided. When τ=5π/4 lies in the range of the latter half period, with the increase of feedback gain ζ3, the vibration amplitude around the resonance frequency band first increases and then decreases, and jump phenomenon can also be avoided when feedback gain ζ3 increases to a larger value.Figure 6
Amplitude-frequency relationships of the primary resonance for the controlled system with different feedback gainζ3 (ζ1=0.025, f1=0.05, k=1, and λ^=0.4).
(a)
τ
=
π
/
4
(b)
τ
=
5
π
/
4


## 3.2. Jump Phenomenon Analysis

From Figures3 and 4, it can be seen that jump phenomenon can occur in the primary resonance of the controlled system when the structural parameters and excitation force amplitude are chosen as some specific values. The primary response amplitude decreases at jump-down frequency and increases at jump-up frequency. Jump phenomenon can be detrimental and can probably cause extensive physical damage to the controlled system due to the large amplitude changes occurring at jump-down or jump-up frequency. So it is beneficial to determine the critical surface of the structural parameters which lead to no jump phenomenon, and the jump frequencies should be determined when jump phenomenon occurs. The jump phenomenon analysis of the controlled system without and with time delay is investigated, respectively, in this part.The amplitude-frequency function of the controlled system without time delay can be simplified into(22)2Ωζ1a+32ζ3Ω3a32+Ω2-1a-34βa32=f12.Equation (22) can be written as a polynomial function in amplitude a or frequency Ω as follows:(23a)HΩ,a=p1a6+q1a4+r1a2+s1,(23b)p1=916β2+94ζ32Ω6,(23c)q1=6ζ1ζ3Ω4-32βΩ2-1,(23d)r1=4ζ12Ω2+Ω2-12,(23e)s1=-f12,(23f)HΩ,a=p2Ω6+q2Ω4+r2Ω2+s2,(23g)p2=94ζ32a6,(23h)q2=6ζ1ζ3a4+a2,(23i)r2=-32βa4+4ζ12-2a2,(23j)s2=916β2a6+32βa4+a2-f12.It is inconvenient to solve the amplitudea or frequency Ω in an analytic expression since (23a)–(23j) can be treated as a cubic function of amplitude a2 or frequency Ω2; it is difficult to determine the critical surface of the structural parameters and jump frequencies based on these analytic expressions. So the method of Sylvester resultant [24] is used in this paper. Assuming two polynomials fx and gx as(24a)fx=∑i=0naixi,an≠0,(24b)gx=∑i=0mbixi,bm≠0.The Sylvester resultant offx and gx can be obtained [25]:(25)Rf,g=anan-1⋯⋯a1a00⋯00anan-1⋯⋯a1a0⋯0⋯⋯⋯⋯⋯⋯0⋯0anan-1⋯⋯⋯a0bmbm-1⋯b1b00⋯⋯00bmbm-1⋯b1b00⋯0⋯⋯⋯⋯⋯⋯0⋯⋯0bmbm-1⋯⋯b0.A necessary and sufficient condition for two polynomialsfx and gx to have a common root is that the Sylvester resultant Rf,g=0 [26].When the excitation force amplitude takes a smaller value for chosen structural parameter, jump phenomenon does not occur and the amplitude-frequency curve is single-valued which is similar to the linear harmonic excited system; that is, there exists a unique value of amplitudea for each value of frequency Ω. When the excitation force amplitude increases to a larger value, jump phenomenon can occur due to the existence of cubic nonlinear terms; there exist multiple values of amplitude a for each value of frequency Ω in a specific frequency band. Denote f1cr as the critical value of the excitation force amplitude, that is the maximum excitation force amplitude which leads to no jump phenomenon. Denoting Ωcr,acr as the infection point in the amplitude-frequency curve when f1=f1cr, then the function HΩcr,a has three positive real roots equal to acr. Assuming the functions H′Ω,a and H′′Ω,a as the derivative and second derivative of the function HΩ,a with respect to the amplitude a, respectively, then the function H′Ωcr,a has two positive real roots equal to acr and the function H′′Ωcr,a has one positive real root equal to acr. According to the definition of Sylvester resultant, the resultant RH′,H′′ should be equal to zero at the infection point Ωcr,acr.Combining (23a)–(23j) and (25), the following equation can be obtained:(26)Gacr=RH′,H′′a=acr=∑i=010biacri=0,where the definitions of the coefficients bi are given in Appendix. Using (26), Ωcr can be eliminated and can obtain a polynomial equation in acr. Then the value of acr can be obtained numerically. Of the ten roots of GΩcr=0, there are two real and positive roots and the value of acr should be taking the smaller value. The critical excitation frequency Ωcr can be given by substituting acr into the function H′′Ωcr,a. Once the critical excitation frequency Ωcr and critical amplitude acr are known, the critical excitation force amplitude f1cr can be obtained by using (23a) or (23f).When the excitation force amplitudef1>f1cr, jump phenomenon occurs; it is necessary to determine the jump-up and jump-down frequencies. Denoting Ω∗,a∗ as the jump points in the amplitude-frequency curve, then the function HΩ∗,a has two positive real roots equal to a∗, which requires that the resultant RH,H′ should be equal to zero at the jump points Ω∗,a∗.Combining (23a)–(23j) and (25), the following equation can be obtained:(27)Ga∗=RH,H′a=a∗=∑i=018cia∗i=0,where the definitions of the coefficients ci are given in Appendix. Using (27), Ω∗ can be eliminated and can obtain a polynomial equation in a∗. Then the value of a∗ can be obtained numerically. Of the eighteen roots of GΩcr=0, there are two real and positive roots; the smaller value corresponds to the amplitude of jump-up frequency and the larger value corresponds to the amplitude of jump-down frequency. The jump frequencies can be obtained by substituting a∗ into the function H′Ωcr,a. But for each value of a∗, there are two real and positive values of Ω∗, one of which is spurious. To investigate which one is spurious, substitute Ω∗ into the function HΩ∗,a and check whether there are two real and positive values equal to a∗ or not. If there are none, then this particular value of Ω∗ is spurious and should be discarded.Amplitude-frequency relationship of the primary resonance for the controlled system under different excitation force amplitude without time delay is shown in Figure7. Using the method of Sylvester resultant, the critical excitation force amplitude f1cr=0.01102. The infection point Ωcr,acr can be obtained as 1.0485,0.1698 which can be seen in Figure 7 denoted by black asterisk point. As discussed previously, when the excitation force amplitude f1=0.03>f1cr, jump phenomenon occurs; the jump points Ω∗,a∗ can also be determined by the method of Sylvester resultant. The jump-up points and jump-down points can be obtained as 1.108,0.1945 and 1.163,0.3891, respectively, which can be seen in Figure 7 denoted by black dot points. Figure 8 shows the critical surface of the structural parameters which lead to no jump phenomenon. It is apparent that the reasonable structural parameter combinations should lie under the critical surface. Figure 8 also shows the partial contours of the critical surface; the direction of the arrow indicates the reasonable structural parameter combinations which lead to no jump phenomenon.Figure 7
Amplitude-frequency relationship of the primary resonance for the controlled system under different excitation force amplitude without time delay (ζ1=0.025, ζ3=0.05, k=1, and λ^=0.4).Figure 8
Critical surface and contours of the critical surface of the controlled system without time delay (k=1 and λ^=0.4).The jump phenomenon analysis of the controlled system with time delay is then considered. Because the trigonometric functions exist in the amplitude-frequency function of the primary resonance, the method of Sylvester resultant cannot be used in this case. The critical boundary which leads to no jump phenomenon and jump frequencies will be investigated by using the optimization method.For the controlled system with time delay, when there are no points in the amplitude-frequency curves satisfying the conditiondΩ/da=0, jump phenomenon does not occur [27]. But when there are at least two points in the amplitude-frequency curves satisfying the condition dΩ/da=0, the jump phenomenon can occur. So the critical condition is that there is only one point in the amplitude-frequency curves satisfying the condition dΩ/da=0. Differentiating both sides of (17a) with respect to a and substituting the condition dΩ/da=0 in it, the critical function can be given as(28)2Ωζ1+32ζ3Ω3a2cos⁡Ωτ2Ωζ1+92ζ3Ω3a2cos⁡Ωτ+Ω2-1-34βa2-32ζ3Ω3a2sinΩτ·Ω2-1-94βa2-92ζ3Ω3a2sinΩτ=0.Combining the critical function equation (28) and the amplitude-frequency function equation (17a), the critical boundary and jump frequencies can be achieved.An example of how to determine the critical excitation force amplitudef1cr by using the optimization method is presented when the other structural parameters are chosen. The Levenberg-Marquardt (LM) optimization method is considered in this paper. Denoting the critical function equation (28) and the amplitude-frequency function equation (17a) as the objective functions and defining frequency Ω and amplitude a as the independent variable. Set up the convergence index η as 1e-10 and the maximum number of iterations N as 2000. For a given critical excitation force amplitude f1cr, the frequency Ω and amplitude a can be determined by using the LM optimization method to minimize the objective functions; then the value of objective functions can be obtained.Substituting the initial guess value of critical excitation force amplitudef1icr into the objective functions and using the LM optimization method, when the initial value of f1icr is chosen as a larger one, the value of the objective functions is smaller than the convergence index η which indicates the value of f1icr is larger and jump phenomenon occurs; the frequency Ω and amplitude a can be obtained as the jump frequency and the amplitude of jump frequency, respectively, decreasing the value of f1icr slowly until the value of the objective functions is larger than the convergence index η for the first time; then the critical excitation force amplitude f1cr can be obtained; when the initial value of f1icr is chosen as a smaller one, the value of the objective functions is larger than the convergence index η which indicates the value of f1icr is smaller and jump phenomenon does not occur; the obtained frequency Ω and amplitude a are spurious and should be discarded, increasing the value of f1icr slowly until the value of the objective functions is smaller than the convergence index η for the first time; then the critical excitation force amplitude f1cr can be obtained. The solution process for determining the critical excitation force amplitude f1icr is an iterative process.The critical border of the controlled system with time delayτ=π/4 is shown in Figure 9. Using the LM optimization method, the critical excitation force amplitude f1cr=0.01046. When the excitation force amplitude f1=f1cr, there is only one point in the amplitude-frequency curves satisfying the condition dΩ/da=0 which corresponds to the boundary. When the excitation force amplitude f1>f1cr, there are two points in the amplitude-frequency curves satisfying the condition dΩ/da=0 which implies the jump phenomenon. When the jump phenomenon occurs, the jump points can be determined conveniently by using the LM optimization method. Figure 10 shows the amplitude-frequency relationship of the primary resonance for the controlled system under different excitation force amplitude with time delay τ=π/4. When the excitation force amplitude f1=0.03>f1cr, the jump-up points and jump-down points can be obtained as 1.1095,0.1914 and 1.1973,0.4129, respectively, which can be seen in Figure 10 denoted by black dot points.Figure 9
Critical border of the controlled system with time delayτ=π/4 (ζ1=0.025, ζ3=0.05, k=1, and λ^=0.4).Figure 10
Amplitude-frequency relationship of the primary resonance for the controlled system under different excitation force amplitude with time delayτ=π/4 (ζ1=0.025, ζ3=0.05, k=1, and λ^=0.4).The other critical values of structural parameters, for example, damping ratioζ1 and feedback gain ζ3, can be determined just as the critical excitation force amplitude case, but the solution process is complicated which can be seen in the above part, so it is no longer discussed here.


## 3.3. Stability Analysis

To analyze the stability of the primary response of the controlled system, linearizing (15a) and (15b) with respect to a and Φ gives(29a)D1Δa=-ζ^1+94ζ^3Ω2a2cos⁡ΩτΔa-f2cos⁡ΦΔΦ,(29b)D1ΔΦ=34Ωβ^a+32ζ^3Ω2asinΩτ+f2Ωa2cos⁡ΦΔa+f2ΩasinΦΔΦ.The characteristic equation of (29a) and (29b) is given as(30)det⁡s+ζ^1+94ζ^3Ω2a2cos⁡Ωτf2cos⁡Φ-34Ωβ^a+32ζ^3Ω2asinΩτ+f2Ωa2cos⁡Φs-f2ΩasinΦ=0.Equation (30) can be simplified using (16a) and (16b):(31a)s2-G1s+G2=0,(31b)G1=-2ζ^1-3ζ^3Ω2a2cos⁡Ωτ,(31c)G2=ζ^1+34ζ^3Ω2a2cos⁡Ωτζ^1+94ζ^3Ω2a2cos⁡Ωτ+1Ω2σ2-38β^a2+34ζ^3Ω3a2sinΩτ·σ2-98β^a2+94ζ^3Ω3a2sinΩτ.According to the Routh-Hurwitz criterion, the controlled system has stable primary response corresponding to stable node whenG1<0 and G2>0, has unstable primary response corresponding to unstable node when G1>0 and G2>0, and has unstable primary response corresponding to saddle point when G2<0. Actually, the critical condition G1=0 indicates a pair of purely imaginary eigenvalues exist and hence a Hopf bifurcation occurs [16]. For the uncontrolled system G1=-2ζ^1<0 and controlled system without time delay G1=-2ζ^1-3ζ^3Ω2a2<0, the Hopf bifurcation can be avoided.The stability boundary of the primary resonance for the controlled system with different time delay is shown in Figure11. The gray shaded unstable regions denoted by  USR1 are determined by G1 and the green shaded unstable regions denoted by  USR2 are determined by G2. When time delay τ=π/4, G1>0 in the given frequency band, and the unstable regions are determined by G2. When time delay τ=π/2, the unstable regions are determined by both G1 and G2. The upper branch of the primary resonance is cut off by the boundary of USR1 determined by G1; then these unstable responses cannot be obtained by numerical method which can be clearly seen in Figure 4(c). When time delay τ=7π/4, a closed-loop response curve emerges above the primary resonance. It is observed that the lower branch of the closed-loop response curve is all covered by USR1 and USR2 and the upper branch of the closed-loop response curve is partially covered by USR1 which can also be clearly seen in Figure 4(h) with using numerical method checked.Figure 11
Stability boundary of the primary resonance for the controlled system with different time delayτ (ζ1=0.025, ζ3=0.05, f1=0.05, k=1, and λ^=0.4).
(a)
τ
=
π
/
4
(b)
τ
=
π
/
2
(c)
τ
=
7
π
/
4From (31a)–(31c), it can be seen that the stability boundary of the primary resonance for the controlled system is greatly influenced by the nondimensional feedback gain ζ3 and time delay τ when the other structural parameters and excitation force amplitude are chosen. Two cases are considered when investigating the effects of these two parameters on the stability of vibration amplitude for the controlled system. The first case is when only one single vibration amplitude exists and the second case is when three vibration amplitudes may exist in the controlled system.Figure12 shows the effect of the time delay τ on the stability of vibration amplitude for the controlled system with different feedback gain ζ3. The dashed line indicates the unstable vibration amplitude. When Ω=1.1, only one single vibration amplitude exists in the controlled system. When feedback gain ζ3 takes a small value, the vibration amplitude is always stable regardless of the value of time delay τ; with the increase of feedback gain ζ3, the vibration amplitude can be unstable for some specific regions of time delay τ; the unstable solutions where a Hopf bifurcation occurs are determined by G1. Behaviour is more complicated when Ω=1.3 because three vibration amplitudes exist for some specific regions of time delay τ. The lowest branch is always stable regardless of the value of time delay τ and remains the same for different feedback gain ζ3 which can be seen in Figure 4. The vibration amplitude of the lowest branch is smaller compared to the upper two branches. When feedback gain ζ3 takes a small value, the upper branch is stable and the lower branch is unstable which is determined by G2; with the increase of feedback gain ζ3, the upper branch can be unstable for some specific regions of time delay τ; the unstable solutions where a Hopf bifurcation occurs are determined by G1.Figure 12
Effect of the time delayτ on the stability of vibration amplitude for the controlled system with different feedback gain ζ3 (ζ1=0.025, f1=0.05, k=1, and λ^=0.4).
(a)
Ω
=
1.1
(b)
Ω
=
1.3


## 4. 1/3 Subharmonic Resonance

Since there exist cubic nonlinear terms in the controlled system, when the excitation force amplitude increases to a larger value, 1/3 subharmonic resonance can occur which results in a worse isolation performance. So 1/3 subharmonic resonance of the controlled system is studied in this section. For the case of 1/3 subharmonic resonance, the excitation force amplitude is larger; it is assumed that(32a)ζ1=εζ^1,(32b)ζ3=εζ^3,(32c)β=εβ^,(32d)Ω2=9+εσ.Then (7) can be rewritten as(33)x^′′+2εζ^1x^′+2εζ^3x^′3T-τ+Ω2-εσ9x^+εβ^x^3=f1cos⁡ΩT.Substituting (10) and (11) into (33) and equating the coefficients of the same power of ε, the following equations can be obtained:(34a)D02x0+Ω29x0=f1cos⁡ΩT0,(34b)D02x1+Ω29x1=-2D0D1x0-2ξ^1D0x0-2ξ^3D0x0T0-τ3-β^x03+σ9x0.The solution of (32a) can be written in the form(35a)x0T0,T1=AT1ejΩ/3T0+ΛejΩT0+cc,(35b)AT1=12aT1ejΦT1,(35c)Λ=-9f116Ω2.Substituting (35a) into (34b) and removing secular terms from (34b) require(36)-j23ΩD1A-j23ξ^1ΩA-3β^A2A--3β^A-2Λ-6β^AΛ2+19σA-j29ζ^3Ω3A2A-e-jΩ/3τ-j4ζ^3Ω3AΛ2e-jΩ/3τ+j23ζ^3Ω3A-2Λe-jΩ/3τ=0.Substituting (35b) into (36) and separating the real and imaginary terms, the following equations can be obtained:(37a)a′=1Ω-Ωζ^1a-112ζ^3Ω3a3cos⁡Ω3τ-6ζ^3Ω3Λ2acos⁡Ω3τ+94β^Λa2sin3Φ+12ζ^3Ω3Λa2cos⁡Ω3τ+3Φ,(37b)aΦ′=1Ω-16σa+98β^a3+9β^aΛ2+112ζ^3Ω3a3sinΩ3τ+6ζ^3Ω3Λ2asinΩ3τ+94β^Λa2cos⁡3Φ-12ζ^3Ω3Λa2sinΩ3τ+3Φ.Then the amplitude-frequency and phase-frequency relationships of 1/3 subharmonic resonance using the original structural parameter can be given by(38a)ζ1Ω+112ζ3Ω3a2cos⁡Ω3τ+6ζ3Ω3Λ2cos⁡Ω3τ2+9-Ω26+98βa2+9βΛ2+112ζ3Ω3a2sinΩ3τ9-Ω26+6ζ3Ω3Λ2sinΩ3τ2=94βΛa-12ζ3Ω3ΛasinΩ3τ2+12ζ3Ω3Λacos⁡Ω3τ2,(38b)tan3Φ=A1C1-A2C2A1C2+A2C1,(38c)C1=94βΛa-12ζ3Ω3ΛasinΩ3τ,(38d)C2=12ζ3Ω3Λacos⁡Ω3τ,(38e)A1=ζ1Ω+112ζ3Ω3a2cos⁡Ω3τ+6ζ3Ω3Λ2cos⁡Ω3τ,(38f)A2=9-Ω26+98βa2+9βΛ2+112ζ3Ω3a2sinΩ3τ+6ζ3Ω3Λ2sinΩ3τ.Equation (38a) can be written as a polynomial function in a as(39a)a4-2p3a2+q3=0,(39b)p3=-38ζ32Ω6-24332β2-698βζ3Ω3sinΩ3τΛ2+172ζ3Ω5sinΩ3τ-112ζ1ζ3Ω4cos⁡Ω3τ-38ζ32Ω6-24332β2-698βζ3Ω3sinΩ3τ-18ζ3Ω3sinΩ3τ+316βΩ2-2716β·1144ζ32Ω6+8164β2+316βζ3Ω3sinΩ3τ-1,(39c)q3=ζ1Ω+6ζ3Ω3Λ2cos⁡Ω3τ232+9βΛ2+6ζ3Ω3Λ2sinΩ3τ-Ω262+32+9βΛ2+6ζ3Ω3Λ2sinΩ3τ-Ω262·1144ζ32Ω6+8164β2+316βζ3Ω3sinΩ3τ-1.It can be seen from (39a) that there exists amplitude for a given frequency if p3>0 and p32≥q3 since q3>0. Substituting ((39b)-(39c)) into these inequalities, the existence condition of the 1/3 subharmonic resonance can be obtained.From (38a)–(38f) and (39a)–(39c), it can be seen that the amplitude-frequency, phase-frequency relationships, and existence regions of the 1/3 subharmonic resonance for the controlled system are greatly influenced by the nondimensional feedback gain ζ3 and time delay τ when the other structural parameters are taking the same values as in the primary resonance part. The effect of the feedback gain ζ3 on the 1/3 subharmonic resonance of the controlled system without time delay can be clearly seen in Figure 13. When the system is uncontrolled, the existence regions and amplitude of the 1/3 subharmonic resonance are relatively large. When the feedback gain ζ3 increases, the existence regions of the 1/3 subharmonic resonance decrease rapidly. So by choosing appreciate value of feedback gain ζ3, the 1/3 subharmonic resonance of the controlled system without time delay can be avoided.Figure 13
Existence regions, amplitude-frequency, and phase-frequency relationships of the 1/3 subharmonic resonance for the controlled system under different feedback gainζ3 without time delay (ζ1=0.025, k=1, λ^=0.4, and f1=2).The effect of the time delayτ on the 1/3 subharmonic resonance of the controlled system can be seen in Figure 14. Compared to the uncontrolled system, the existence regions of the 1/3 subharmonic resonance for the controlled system with time delay can also become smaller but are larger than the controlled system without time delay. Because when time delay exists, the trigonometric functions appear in both the amplitude-frequency function and existence region function and cannot be eliminated, which can be clearly seen in (37a) and  (37b) and (38a)–(38f). So there exist multiple existence regions for the 1/3 subharmonic resonance; the existence regions for larger frequency are not shown in Figure 14 for simplicity. When the time delay exists in the controlled system, the 1/3 subharmonic resonance can occur for a smaller excitation force amplitude and increasing the feedback gain cannot eliminate the 1/3 subharmonic resonance which is undesirable in practical engineering. So in the meaning of controlling 1/3 subharmonic resonance, the controlled system without time delay is better.Figure 14
Existence regions and amplitude-frequency relationships of the 1/3 subharmonic resonance for the controlled system with different time delayτ (ζ1=0.025, ζ3=0.025, k=1, λ^=0.4, and f1=2).
(a)
τ
=
π
/
4
(b)
τ
=
2
π
/
4
(c)
τ
=
3
π
/
4The stability analysis of the 1/3 subharmonic resonance for the controlled system can be investigated as the primary resonance case, so it is no longer described here. It can be proven that the upper branch of the 1/3 subharmonic resonance is asymptotically stable, while the lower branch is unstable.


## 5. Force Transmissibility

Force transmissibility is an important index to describe the performance of the vibration isolator. It is defined as the ratio between the force transmitted to the base and the excitation force. In this section, force transmissibility is investigated to evaluate the performance of the controlled HSLDS vibration isolator and compared with an equivalent linear passive vibration isolator. The nondimensional force transmitted to the base through the controlled HSLDS vibration isolator is defined as the following form:(40)ft=2ζ1x^′+2ζ3x^′3T-τ+x^+β^x^3.Using (13a) and (13b), the nondimensional force is obtained as(41)ft=2ζ1Ωa+32ζ3Ω3a3cos⁡Ωτ2+a+34βa3+32ζ3Ω3a3sinΩτ21/2.The force transmissibility of the controlled system can be expressed as(42)TNF=ftf1=2ζ1Ωa+32ζ3Ω3a3cos⁡Ωτ2+a+34βa3+32ζ3Ω3a3sinΩτ21/2·f1-1.It is also of interest to compare the force transmissibility of the controlled system with an equivalent linear one with the same load bearing capacity. The force transmissibility of the equivalent linear vibration isolator is given as(43)TLF=1+4ζl2Ωl21-Ωl22+4ζl2Ωl2,where ξl, Ωl are the damping ratio and frequency ratio of the equivalent linear vibration isolator, respectively. Since the HSLDS vibration isolator is comprised of load bearing element and stiffness correctors, the equivalent linear one is the HSLDS vibration isolator with the stiffness correctors removed; then ξl and Ωl should satisfy the following conditions:(44a)ζl=αζ1,(44b)Ωl=Ωα.For the sake of consistency, when plotting the force transmissibility of the controlled system and an equivalent linear one on the same figure, the values on the frequency axis have to comply with (44b). The force transmissibility of these two vibration isolators is expressed in dB scale as 20log⁡10TNF and 20log⁡10TLF, respectively. As can be seen in (42), the force transmissibility of the controlled system is greatly influenced by the nondimensional damping ratio ζ1, feedback gain ζ3, and time delay τ when the other structural parameters are chosen. The following discussion investigates the effects of these parameters on the force transmissibility.Force transmissibility of the uncontrolled system with different damping ratioζ1 is shown in Figure 15; the force transmissibility of the equivalent linear one is also plotted in the same figure for comparison. From Figure 15, it can be seen that when the damping ratio is smaller, the jump phenomenon occurs and the smaller the damping ratio is, the larger the force transmissibility curve bends to the right; then the HSLDS vibration isolator can lead to an undesirable isolation performance compared to the linear one. When the damping ratio increases, it can significantly reduce the vibration amplitude around the resonance frequency region and the isolation frequency band is larger than the linear one; then the performance of the HSLDS vibration isolator is superior to the linear one, but it results in a poor performance in the high frequency band where the isolation is required.Figure 15
Force transmissibility of the uncontrolled system with different damping ratioζ1 (f1=0.05, k=1, and λ^=0.4).Force transmissibility of the controlled system under different feedback gainζ3 without time delay is shown in Figure 16. The effect of feedback gain ζ3 on the force transmissibility can be clearly seen in this figure. When the feedback gain ζ3 increases, the vibration amplitude around the resonance frequency region decreases; the jump phenomenon can be eliminated and can achieve a better isolation performance in the high frequency band where the isolation is required. Thus the cubic velocity feedback control strategy can solve the dilemma existing in the passive vibration isolators with linear damping.Figure 16
Force transmissibility of the controlled system under different feedback gainζ3 without time delay (ζ1=0.025, f1=0.05, k=1, and λ^=0.4).Force transmissibility of the controlled system with different time delayτ is shown in Figure 17. The effect of time delay τ on the force transmissibility can also be interpreted by the equivalent damping ratio and equivalent resonance frequency. Then the force transmissibility of the controlled system can be obtained as(45)TNF=ftf1=2ζeqΩa2+aweq22f1.Figure 17
Force transmissibility of the controlled system with different time delayτ (ζ1=0.025, ζ3=0.025, f1=0.05, k=1, and λ^=0.4).The time delayτ can influence the peak amplitude and resonance frequency of the force transmissibility just as the primary resonance case which can achieve a better or poorer isolation performance than the uncontrolled system, so it is no longer described in detail here. Another important feature that should be addressed is that the time delay τ does not influence the isolation performance in the high frequency band where the isolation is required.


## 6. Conclusion

This paper proposes the time-delayed cubic velocity feedback control strategy to improve the isolation performance of HSLDS vibration isolator. Both the resonance and performance of the controlled HSLDS vibration isolator are considered. The primary resonance of the controlled system is obtained by using the multiple scales method. The equivalent damping ratio and equivalent resonance frequency are defined to study the effects of feedback gainζ3 and time delay τ on the primary resonance. When time delay τ=0, with the increase of feedback gain ζ3, the equivalent damping ratio becomes larger; the peak amplitude of the primary resonance decreases and is smaller than the uncontrolled system. When time delay τ exists in the controlled system, the equivalent damping ratio and equivalent resonance frequency can be considered as the periodic functions of time delay τ; the peak amplitude of the primary resonance can be smaller or larger than the uncontrolled system based on the value of time delay τ. A closed-loop response curve is found above the primary resonance because multiple solutions exist in these frequency bands when time delay τ takes some special values.The jump phenomenon analysis of the controlled system is then investigated. When time delayτ=0, it is convenient to use the method of Sylvester resultant to determine the critical surface of the structural parameters which lead to no jump phenomenon and the jump frequencies when jump phenomenon occurs. When time delay τ exists in the controlled system, the method of Sylvester resultant cannot be used because the trigonometric functions exist in the amplitude-frequency function of the primary resonance; the critical boundary which leads to no jump phenomenon and jump frequencies can be obtained by using the optimization method but is more complicated. The stability analysis of the controlled system is also considered; it is found that the stability boundary of the primary resonance is greatly influenced by the feedback gain ζ3 and time delay τ.The 1/3 subharmonic resonance of the controlled system is studied by using the multiple scales method. When time delayτ=0, with the increase of feedback gain ζ3, the existence regions of the 1/3 subharmonic resonance decrease rapidly and the 1/3 subharmonic resonance can be avoided by choosing appreciate value of feedback gain  ζ3. When time delay τ exists in the controlled system, the 1/3 subharmonic resonance can occur for a smaller excitation force amplitude and increasing the feedback gain cannot eliminate the 1/3 subharmonic resonance. So in the meaning of controlling 1/3 subharmonic resonance, the controlled system without time delay is better.Last, the force transmissibility of the controlled system is considered. When the feedback gainζ3 increases, the vibration amplitude around the resonance frequency region decreases and the isolation frequency band is larger which is superior to the equivalent linear passive vibration isolator. It can also achieve a better isolation performance in the high frequency band where the isolation is required. Thus the cubic velocity feedback control strategy can solve the dilemma existing in the passive vibration isolators with linear damping.



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*Source: 712851-2015-03-31.xml*