# Normalized Study of Three-Parameter System in the Time Domain and Frequency Domain

Authors: Xiao-Lei Jiao; Yang Zhao; Wen-Lai Ma
Journal: Shock and Vibration
 (2017)

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## Abstract

Three-parameter isolation system can be used to isolate microvibration for control moment gyroscopes. Normalized analytical model for three-parameter system in the time domain and frequency domain is proposed by using analytical method. Dynamic behavior of three-parameter system in the time domain and frequency domain is studied. Response in the time domain under different types of excitations is analyzed. In this paper, a regulatory factor is defined in order to analyze dynamic behavior in the frequency domain. For harmonic excitation, a comparison study is made on isolation performance between the case when the system has optimal damping and the case when regulatory factor is 1. Besides, phase margin of three-parameter system is obtained. Results show that dynamic behavior in the time domain and frequency domain changes with regulatory factor. Phase margin has the largest value when the value of regulatory factor is 1. System under impulse excitation and step excitation has the shortest settling time for the response in the time domain when the value of regulatory factor is 1. When stiffness ratio is small, isolation performances of two cases are nearly the same; when system has a large stiffness ratio, isolation performance of the first case is better.


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## Body

## 1. Introduction

The classic mass-spring-damper system includes spring and parallel viscous damper; it is two-parameter isolation system. Three-parameter system includes spring and an elastically supported damper. The system can be tuned by selecting parameter values that provide maximum damping at the fundamental frequency and reduced damping at higher frequencies [1]. Isolator based on three-parameter system is widely used in order to isolate microvibration in the aircraft. Microvibration is produced by control moment gyroscopes and reaction wheel assemblies with a frequency range of 0.1–300 Hz. Microvibration in a satellite can take a long time to attenuate because of special space environment. It is necessary to protect high precision payloads from the influence of microvibration [2–12]. Isolator based on three-parameter system can be used to tackle this problem.Early studies focused on two-parameter isolation system. References [13, 14] had a simple study on three-parameter system. They showed that the system had considerable advantages to prevent impact. Honeywell Inc. developed a series of isolators based on three-parameter system [1, 15–17]. The isolator had a pair of bellows which can be used to provide main stiffness; additional stiffness was volume stiffness produced by silicone oil. It was installed as a part of the complete reaction wheel isolation system; after being tested in Hubble Space Telescope, isolation performance was very good, with the maximum reduction in vibration, occurring at frequency as low as 50 Hz, and the maximum attenuation rate was approximately 60 dB. These isolators were assembled into isolation platform in order to prevent microvibration in all directions [18, 19]. Then, a lot of isolators based on three-parameter system were developed [20–27]; isolation performance became better. Dynamic behavior of three-parameter system was studied by Brennan [28]; he pointed out that the system had critical damping if additional stiffness was at least eight times that of the main stiffness; the isolator afforded no advantages if the system was excited by white noise. Liu et al. [29] developed a fluid viscous damper based on three-parameter system; it was used to isolate vibration of the whole satellite; the performance was better than traditional isolator. Zhang et al. [30, 31] had a research on the performance of isolation platform based on three-parameter system. If parameters were selected reasonably, the system not only satisfied the requirement for vibration isolation but also guaranteed the stability of closed-loop attitude control system. Wang et al. [32, 33] proposed a method to test stiffness and damping of the isolator based on three-parameter system; an equivalent two-parameter physical system was obtained based on mechanical impedance theory. Wang et al. [34] proposed a method to design isolator based on three-parameter system; optimal damping of the isolator was obtained. Wang et al. [35] had a research on nonlinear three-parameter system. An optimization method called the generalized pattern search (GPS) algorithm was proposed and applied to identify the nonlinear model parameters. Wang et al. [36] pointed out that the usage of nonlinear secondary spring had improved high frequency isolation performance and meanwhile maintained or even reduced its already low resonance amplitude. Shi et al. [37] developed a fluid viscous damper based on three-parameter system; they proposed equivalent stiffness and equivalent damping.This study is concerned with dynamic behavior of three-parameter system in the time domain and frequency domain. By defining normalized parametersα and β, analytical normalized model of three-parameter system in the time domain and frequency domain is derived. After that, dynamic behavior in the frequency domain is obtained. System’s response in the time domain under different types of excitation is analyzed. Then, we make a comparison research on isolation performance between the case when system has optimal damping and the case when the value of regulatory factor is 1. The results reveal that the second case has better isolation performance; this is a correction of previous studies.


## 2. Normalized Model of Three-Parameter System

### 2.1. Modeling of Three-Parameter System

Three-parameter system is shown in Figure1. M is mass of payload. KA is main stiffness of the system. KB is additional stiffness. CA is damping of the system. Figure 2 describes the mechanical impedance model of three-parameter system. F is total force. xp is displacement of the payload. xb is displacement of the base. xd is displacement of damper node. x˙p is velocity of the payload. x˙b is velocity of the base. x˙d is velocity of damper node.Figure 1
Three-parameter system (Zener model).Figure 2
Mechanical impedance model.The equations of motion for this system are(1)Mx¨p+KAxp-xb+KBxp-xd=0,KBxd-xp+CAx˙d-x˙b=0.Taking the Laplace transform and rearranging,(2)Ms2+KA+KB-KB-KBKB+CAsxpxd=KACAsxb.δ is the stroke, δ=xb-xp, FS is the static force, FS=KAδ, FD is the dynamic force, FD=KBxd-xp, and F is the total force, F=FS+FD.


### 2.2. Complex Mechanical Impedance

Figure2 is schematic diagram of mechanical impedance model, one end of the model connected to the ground; therefore xp=0.The equations of motion for this system can be written as(3)KAxb+CAx˙b-x˙d=F,CAx˙b-x˙d-KBxd=0.Taking the Laplace transform and rearranging,(4)KA+CAs-CAs-CAsKB+CAsxbxd=10F.The complex mechanical impedance Zs can be written as(5)Zs=KAKB+KA+KBCAsKB+CAs.ω1 is zero (lead) frequency in impedance, ω1=KAKB/KA+KBCA. ω2 is pole (lag) frequency in impedance, ω2=KB/CA.Equation (5) becomes(6)Zs=KAs/ω1+1s/ω2+1.For sinusoidal excitation, the complex mechanical impedance can be written as(7)Zω=KA1+ω2/ω1ω21+ω2/ω22+ω/ω1-ω/ω21+ω2/ω22.ω0 is maximum phase frequency, ω0=ω1ω2, α is lead/lag separation ratio, α=ω2/ω1, Ω is normalized frequency, Ω=ω/ω0, transformation between KA, KB, CA and KA, α, ω0 can be written as(8)α=1+KBKA,ω0=KBCAKAKA+KB,KB=KA1-α2,CA=KAα2-1αω0.


### 2.3. Normalized Open-Loop Transfer Function

Figure3 is general isolator block diagram for three-parameter system; the open-loop transfer function can be written as(9)Gols=KAMs21+s/ω11+s/ω2.Gols is open-loop transfer function.Figure 3
General isolator block diagram.Figure4 is Bode plot of the open-loop transfer function. Asymptotic gain is KA/Ms2; the asymptote intersects with the lateral axis at frequency ωs, ωs=KA/M; (9) can be written as(10)Gols=ωs2s21+αs/ω01+1/αs/ω0.ωs2=ω0/αωco, ωco is cross frequency in Bode plot of the open-loop transfer function, and it is marked in Figure 4. β is regulatory factor, β=ωco/ω0. ωs can be written as ωs=ω0β/α; Gols can be written as(11)Gols=βαω02s21+αs/ω01+1/αs/ω0.ω0,α,β can be written as(12)ω0=KAKB2KA+KBCA2,α=KA+KBKA,β=KA+KBCA2MKB2KA+KBKA.KA,KB,CA can be written as(13)KA=Mω02βα,KB=Mω02α2-1βα,CA=Mω0α2-1βα2.When β=1, KA,KB,CA can be written as(14)KA=Mω02α,KB=Mω02α2-1α,CA=Mω0α2-1α2.If parameters of physical model KA,KB,CA or normalized parameters α,β are known, transmissibility curve can be obtained; it is convenient to design three-parameter system if we use normalized model.Figure 4
Bode plot of open-loop transfer function.


### 2.4. Normalized Close-Loop Transfer Function

According to block diagram of open-loop transfer function, isolation transfer function can be written as(15)Gxp_xbs=xpsxbs.Gxp_xbs is open-loop transfer function. Equation (15) can be written as(16)Gxp_xbs=Gols1+Gols.Defining ψ=s/ω0, (16) can be written as(17)Gxp_xbψ=CψRψ=1+αβψ1+αβψ+αψ2+βψ3.Cψ is numerator, Cψ=1+αβψ, and Rψ is denominator, Rψ=1+αβψ+αψ2+βψ3.Equation (17) can be written as(18)Gxp_xbs=1+αβ/ω0s1+αβ/ω0s+α/ω02s2+β/ω03s3.When β=1, (17) can be written as(19)Gxp_xbψ=CψRψ=1+αψ1+αψ+αψ2+ψ3.


## 2.1. Modeling of Three-Parameter System

Three-parameter system is shown in Figure1. M is mass of payload. KA is main stiffness of the system. KB is additional stiffness. CA is damping of the system. Figure 2 describes the mechanical impedance model of three-parameter system. F is total force. xp is displacement of the payload. xb is displacement of the base. xd is displacement of damper node. x˙p is velocity of the payload. x˙b is velocity of the base. x˙d is velocity of damper node.Figure 1
Three-parameter system (Zener model).Figure 2
Mechanical impedance model.The equations of motion for this system are(1)Mx¨p+KAxp-xb+KBxp-xd=0,KBxd-xp+CAx˙d-x˙b=0.Taking the Laplace transform and rearranging,(2)Ms2+KA+KB-KB-KBKB+CAsxpxd=KACAsxb.δ is the stroke, δ=xb-xp, FS is the static force, FS=KAδ, FD is the dynamic force, FD=KBxd-xp, and F is the total force, F=FS+FD.


## 2.2. Complex Mechanical Impedance

Figure2 is schematic diagram of mechanical impedance model, one end of the model connected to the ground; therefore xp=0.The equations of motion for this system can be written as(3)KAxb+CAx˙b-x˙d=F,CAx˙b-x˙d-KBxd=0.Taking the Laplace transform and rearranging,(4)KA+CAs-CAs-CAsKB+CAsxbxd=10F.The complex mechanical impedance Zs can be written as(5)Zs=KAKB+KA+KBCAsKB+CAs.ω1 is zero (lead) frequency in impedance, ω1=KAKB/KA+KBCA. ω2 is pole (lag) frequency in impedance, ω2=KB/CA.Equation (5) becomes(6)Zs=KAs/ω1+1s/ω2+1.For sinusoidal excitation, the complex mechanical impedance can be written as(7)Zω=KA1+ω2/ω1ω21+ω2/ω22+ω/ω1-ω/ω21+ω2/ω22.ω0 is maximum phase frequency, ω0=ω1ω2, α is lead/lag separation ratio, α=ω2/ω1, Ω is normalized frequency, Ω=ω/ω0, transformation between KA, KB, CA and KA, α, ω0 can be written as(8)α=1+KBKA,ω0=KBCAKAKA+KB,KB=KA1-α2,CA=KAα2-1αω0.


## 2.3. Normalized Open-Loop Transfer Function

Figure3 is general isolator block diagram for three-parameter system; the open-loop transfer function can be written as(9)Gols=KAMs21+s/ω11+s/ω2.Gols is open-loop transfer function.Figure 3
General isolator block diagram.Figure4 is Bode plot of the open-loop transfer function. Asymptotic gain is KA/Ms2; the asymptote intersects with the lateral axis at frequency ωs, ωs=KA/M; (9) can be written as(10)Gols=ωs2s21+αs/ω01+1/αs/ω0.ωs2=ω0/αωco, ωco is cross frequency in Bode plot of the open-loop transfer function, and it is marked in Figure 4. β is regulatory factor, β=ωco/ω0. ωs can be written as ωs=ω0β/α; Gols can be written as(11)Gols=βαω02s21+αs/ω01+1/αs/ω0.ω0,α,β can be written as(12)ω0=KAKB2KA+KBCA2,α=KA+KBKA,β=KA+KBCA2MKB2KA+KBKA.KA,KB,CA can be written as(13)KA=Mω02βα,KB=Mω02α2-1βα,CA=Mω0α2-1βα2.When β=1, KA,KB,CA can be written as(14)KA=Mω02α,KB=Mω02α2-1α,CA=Mω0α2-1α2.If parameters of physical model KA,KB,CA or normalized parameters α,β are known, transmissibility curve can be obtained; it is convenient to design three-parameter system if we use normalized model.Figure 4
Bode plot of open-loop transfer function.


## 2.4. Normalized Close-Loop Transfer Function

According to block diagram of open-loop transfer function, isolation transfer function can be written as(15)Gxp_xbs=xpsxbs.Gxp_xbs is open-loop transfer function. Equation (15) can be written as(16)Gxp_xbs=Gols1+Gols.Defining ψ=s/ω0, (16) can be written as(17)Gxp_xbψ=CψRψ=1+αβψ1+αβψ+αψ2+βψ3.Cψ is numerator, Cψ=1+αβψ, and Rψ is denominator, Rψ=1+αβψ+αψ2+βψ3.Equation (17) can be written as(18)Gxp_xbs=1+αβ/ω0s1+αβ/ω0s+α/ω02s2+β/ω03s3.When β=1, (17) can be written as(19)Gxp_xbψ=CψRψ=1+αψ1+αψ+αψ2+ψ3.


## 3. Dynamic Behavior in the Frequency Domain

### 3.1. Root Locus

According to (17), characteristic equation can be written as (20)1+αβψ+αψ2+βψ3=0,α=KA+KBKA>1.The shape of root locus will be diverse because of different values of α.Equation (20) can be written as(21)1β1+αψ2ψ3+αψ=-1.Figure5 is root locus for α<3. Figure 6 is root locus for α=3. Figure 7 is root locus for α>3. Root locus is located on the left side of lateral axis; it means that the system is stable. Each root locus has three branches corresponding to three characteristic roots.Figure 5
Root locus (α<3).Figure 6
Root locus (α=3).Figure 7
Root locus (α>3).(1) Ifα<3, the system has real poles and conjugate poles. When the system has real poles, it means that the system has critical damping. Free vibration mode of the system is shown in Figure 8 (red line), which has obvious oscillation.Figure 8
Free motion modes for different values ofα.(2) Ifα=3, root locus intersects with lateral axis at point (-1,0). Real poles mean that the system has critical damping. Free vibration mode is shown in Figure 8 (green line); the oscillation is not obvious.(3) Ifα>3, the system has real poles and conjugate poles. Real poles mean that the system has critical damping. Free vibration mode of the system is shown in Figure 8 (blue line); oscillation decays with time quickly.Whenβ=1, denominator of close-loop transfer function can be written as(22)Rψ=ψ3+αψ2+aψ+1.Equation (22) can be written as(23)Rψ=ψ+1ψ2+ψα-1+1.For β=1, free vibration mode of the system will be analyzed for different values of α.(1) Ifα>3, Δ>0, and Δ=α-12-4, there are three different real poles(24)r1=-1,r2=-α-1+α-12-42,r3=-α-1-α-12-42.These three real poles mean that the system has three different free vibration modes(25)c1t=e-t,c2t=e-α-1+α-12-4/2t,c3t=e-α-1-α-12-4/2t.For unit pulse excitation, response in the time domain can be written as(26)ct=α1/α+r1r1-r2r1-r3er1t+1/α+r2r2-r1r2-r3er2t+1/α+r3r3-r1r3-r2er3t.(2) If α=3 and Δ=0, there is a single real pole(27)r=-1.It means that the system has a single free vibration mode; for unit pulse excitation, response in the time domain can be written as(28)ct=α1-α2t2+αte-t.(3) If 1<α<3 and Δ<0, there is a real pole and a pair of conjugate poles(29)r1=-1,r2=-α-1+j4-α-122,r3=-α-1-j4-α-122.Free vibration modes can be written as(30)c1t=e-t,c2t=e-α-1/2tcos⁡α-12-42t+sin⁡α-12-42t.For unit pulse excitation, response in the time domain can be written as(31)ct=α1/α-1α-1/2-12+1-α-1/22e-t+11-α-1/221/α-α-1/22+1-α-1/221-α-1/22+1-α-1/221/2e-α-1/2tsin⁡1-α-122t+φ.φ is phase angle.According to (23), zero-pole plots for different values of α can be obtained.Figures9, 10, and 11 are zero-pole plots for different values of α.Figure 9
Zero-pole plot (α=2,β=1).Figure 10
Zero-pole plot (α=3,β=1).Figure 11
Zero-pole plot (α=4,β=1).Ifα=2, there is a real pole and a pair of conjugate poles. The real pole means that the system has critical damping. It can be seen from Figure 9 that the system has dominant poles; it means that free vibration modes corresponding to conjugate poles are main modes. If α=3, there is a real pole and a pair of conjugate poles; it can be seen from Figure 10 that imaginary part of conjugate poles is very small. If α=4, there are three different real poles which means the system has critical damping.


### 3.2. Nyquist Diagram

Open-loop transfer function can be written as(32)Golψ=1+αβψβψ3+αψ2.Nyquist diagram can be obtained according to the open-loop transfer function.Figures12, 13, 14, and 15 are the Nyquist diagrams for different values of α. It can be concluded from Nyquist diagram that phase margin has the maximum value when β=1; phase margin increases with the value of α, which means that stability of system increases too.Figure 12
Nyquist diagram (α=1.2).Figure 13
Nyquist diagram (α=1.5).Figure 14
Nyquist diagram (α=2).Figure 15
Nyquist diagram (α=3).


### 3.3. Amplitude-Frequency Characteristic and Phase-Frequency Characteristic

Transmissibility is an important index for isolation performance of the isolator. In this section, amplitude-frequency characteristic and phase-frequency characteristic of three-parameter system will be analyzed.Figures16–19 have described three-parameter system’s amplitude-frequency characteristic and phase-frequency characteristic for different values of α and β. It can be concluded that when β=1, system has a minimum resonance peak compared with the case when β<1 and β>1; resonance peak decreases with the value of α. The case when β=1 is very important; it can be used to design an isolator which has minimum resonance peak.Figure 16
Amplitude-frequency characteristic and phase-frequency characteristic (α=1.2,β=0.16,1,6.25).Figure 17
Amplitude-frequency characteristic and phase-frequency characteristic (α=1.5,β=0.16,1,6.25).Figure 18
Amplitude-frequency characteristic and phase-frequency characteristic (α=2,β=0.16,1,6.25).Figure 19
Amplitude-frequency characteristic and phase-frequency characteristic (α=3,β=0.16,1,6.25).


## 3.1. Root Locus

According to (17), characteristic equation can be written as (20)1+αβψ+αψ2+βψ3=0,α=KA+KBKA>1.The shape of root locus will be diverse because of different values of α.Equation (20) can be written as(21)1β1+αψ2ψ3+αψ=-1.Figure5 is root locus for α<3. Figure 6 is root locus for α=3. Figure 7 is root locus for α>3. Root locus is located on the left side of lateral axis; it means that the system is stable. Each root locus has three branches corresponding to three characteristic roots.Figure 5
Root locus (α<3).Figure 6
Root locus (α=3).Figure 7
Root locus (α>3).(1) Ifα<3, the system has real poles and conjugate poles. When the system has real poles, it means that the system has critical damping. Free vibration mode of the system is shown in Figure 8 (red line), which has obvious oscillation.Figure 8
Free motion modes for different values ofα.(2) Ifα=3, root locus intersects with lateral axis at point (-1,0). Real poles mean that the system has critical damping. Free vibration mode is shown in Figure 8 (green line); the oscillation is not obvious.(3) Ifα>3, the system has real poles and conjugate poles. Real poles mean that the system has critical damping. Free vibration mode of the system is shown in Figure 8 (blue line); oscillation decays with time quickly.Whenβ=1, denominator of close-loop transfer function can be written as(22)Rψ=ψ3+αψ2+aψ+1.Equation (22) can be written as(23)Rψ=ψ+1ψ2+ψα-1+1.For β=1, free vibration mode of the system will be analyzed for different values of α.(1) Ifα>3, Δ>0, and Δ=α-12-4, there are three different real poles(24)r1=-1,r2=-α-1+α-12-42,r3=-α-1-α-12-42.These three real poles mean that the system has three different free vibration modes(25)c1t=e-t,c2t=e-α-1+α-12-4/2t,c3t=e-α-1-α-12-4/2t.For unit pulse excitation, response in the time domain can be written as(26)ct=α1/α+r1r1-r2r1-r3er1t+1/α+r2r2-r1r2-r3er2t+1/α+r3r3-r1r3-r2er3t.(2) If α=3 and Δ=0, there is a single real pole(27)r=-1.It means that the system has a single free vibration mode; for unit pulse excitation, response in the time domain can be written as(28)ct=α1-α2t2+αte-t.(3) If 1<α<3 and Δ<0, there is a real pole and a pair of conjugate poles(29)r1=-1,r2=-α-1+j4-α-122,r3=-α-1-j4-α-122.Free vibration modes can be written as(30)c1t=e-t,c2t=e-α-1/2tcos⁡α-12-42t+sin⁡α-12-42t.For unit pulse excitation, response in the time domain can be written as(31)ct=α1/α-1α-1/2-12+1-α-1/22e-t+11-α-1/221/α-α-1/22+1-α-1/221-α-1/22+1-α-1/221/2e-α-1/2tsin⁡1-α-122t+φ.φ is phase angle.According to (23), zero-pole plots for different values of α can be obtained.Figures9, 10, and 11 are zero-pole plots for different values of α.Figure 9
Zero-pole plot (α=2,β=1).Figure 10
Zero-pole plot (α=3,β=1).Figure 11
Zero-pole plot (α=4,β=1).Ifα=2, there is a real pole and a pair of conjugate poles. The real pole means that the system has critical damping. It can be seen from Figure 9 that the system has dominant poles; it means that free vibration modes corresponding to conjugate poles are main modes. If α=3, there is a real pole and a pair of conjugate poles; it can be seen from Figure 10 that imaginary part of conjugate poles is very small. If α=4, there are three different real poles which means the system has critical damping.


## 3.2. Nyquist Diagram

Open-loop transfer function can be written as(32)Golψ=1+αβψβψ3+αψ2.Nyquist diagram can be obtained according to the open-loop transfer function.Figures12, 13, 14, and 15 are the Nyquist diagrams for different values of α. It can be concluded from Nyquist diagram that phase margin has the maximum value when β=1; phase margin increases with the value of α, which means that stability of system increases too.Figure 12
Nyquist diagram (α=1.2).Figure 13
Nyquist diagram (α=1.5).Figure 14
Nyquist diagram (α=2).Figure 15
Nyquist diagram (α=3).


## 3.3. Amplitude-Frequency Characteristic and Phase-Frequency Characteristic

Transmissibility is an important index for isolation performance of the isolator. In this section, amplitude-frequency characteristic and phase-frequency characteristic of three-parameter system will be analyzed.Figures16–19 have described three-parameter system’s amplitude-frequency characteristic and phase-frequency characteristic for different values of α and β. It can be concluded that when β=1, system has a minimum resonance peak compared with the case when β<1 and β>1; resonance peak decreases with the value of α. The case when β=1 is very important; it can be used to design an isolator which has minimum resonance peak.Figure 16
Amplitude-frequency characteristic and phase-frequency characteristic (α=1.2,β=0.16,1,6.25).Figure 17
Amplitude-frequency characteristic and phase-frequency characteristic (α=1.5,β=0.16,1,6.25).Figure 18
Amplitude-frequency characteristic and phase-frequency characteristic (α=2,β=0.16,1,6.25).Figure 19
Amplitude-frequency characteristic and phase-frequency characteristic (α=3,β=0.16,1,6.25).


## 4. Dynamic Behavior in the Time Domain

### 4.1. Response in the Time Domain under Different Types of Excitations

When system is applied to different types of excitation, response in the time domain will be different. In this section, response in the time domain for different types of excitation will be analyzed.(
1) Step Excitation. As has been analyzed in Section 3.1, characteristic root has relationship with the value of Δ. In this section, only the case when system has a real root and conjugate roots will be analyzed.For step excitation, output function can be written as(33)Cψ=1βαβψ+1ψψ-ψ1ψ-ψ2ψ-ψ3,ψ=sω0.Equation (33) has three characteristic roots ψ1, ψ2, ψ3 which can be written as(34)ψ1=c,ψ2=a+bi,ψ3=a-bi.c is real root, a is real part of complex root, and b is the imaginary part of complex root. Response in the time domain can be derived by using inverse Laplace transform method; it can be written as(35)ct=1βA0+A1ect+2Ueatcos⁡bt-2Veatsin⁡bt,A0=-1ca2+b2,A1=1/c+αβc2-a2+b2,U=p1p3+p2p4p32+p42,V=p1p4-p2p3p32+p42,p1=ma+1,p2=mb,p3=-4ab2+2cb2,p4=2aba-c-2b3,m=αβ.(
2) Slope Excitation. For slope excitation, output function can be written as(36)Cψ=1βαβψ+1ψ2ψ-ψ1ψ-ψ2ψ-ψ3.Three characteristic roots are ψ1,ψ2,ψ3, which can be written as(37)ψ1=c,ψ2=a+bi,ψ3=a-bi.c is real root, a is real part of complex root, and b is the imaginary part of complex root. Response in the time domain can be derived by using inverse Laplace transform method; it can be written as(38)ct=1βA0t+A1+A2e-ct+2eat-Fsin⁡bt+Ecos⁡bt,A0=-1ca2+b2,A1=-mca2+b2-a2+b2+2acc2a2+b22,A2=mc+1c2c-a2+b2,p1=ma+1,p2=mb,p3=-2b2a2-b2-4ab2a-c,p4=2ba2-b2a-c-4ab3,E=p1p3+p2p4p32+p42,F=p2p3-p1p4p32+p42,m=αβ.(
3) Acceleration Excitation. For acceleration excitation, output function can be written as(39)Cψ=1βαβψ+1ψ3ψ-ψ1ψ-ψ2ψ-ψ3.Three characteristic roots are ψ1,ψ2,ψ3, which can be written as(40)ψ1=c,ψ2=a+bi,ψ3=a-bi.c is real root, a is real part of complex root, and b is the imaginary part of complex root.Response in the time domain can be derived by using inverse Laplace transform method; it can be written as(41)ct=1β12A0t2+A1t+A2+A3ect+2eat-Fsin⁡bt+Ecos⁡bt,A0=-1ca2+b2,A1=-mca2+b2-a2+b2+2acc2a2+b22,A2=c+2a-ca2+b2-a2+b2+2acm-ca2+b2-a2+b2+2ac-ca2+b23,E=p1p3+p2p4p32+p42,F=p2p3-p1p4p32+p42,p1=-ma+12b,p2=-m2,p3=ba-c3a2-b2-ab3b2-a2,p4=b23a2-b2+aa-c3b2-a2,m=αβ.(
4) Impulse Excitation. For impulse excitation, output function can be written as(42)Cψ=1βαβψ+1ψ-ψ1ψ-ψ2ψ-ψ3.Three characteristic roots are ψ1,ψ2,ψ3, which can be written as(43)ψ1=c,ψ2=a+bi,ψ3=a-bi.c is real root, a is real part of complex root, and b is the imaginary part of complex root. Response in the time domain can be derived by using inverse Laplace transform method; it can be written as(44)ct=1β-2Eect+2eat-Fsin⁡bt+Ecos⁡bt,p1=-a+c,p2=b,p3=-a2-b2+ac,p4=bc,E=mp4-p22p1p4-p2p3,F=mp3-p12p1p4-p2p3,m=αβ.(
5) Harmonic Excitation. For harmonic excitation, output function can be written as(45)Cψ=1βαβψ+1ψ2+ω2ψ-ψ1ψ-ψ2ψ-ψ3.Three characteristic roots are ψ1, ψ2, ψ3, which can be written as(46)ψ1=c,ψ2=a+bi,ψ3=a-bi.c is real root, a is real part of complex root, and b is the imaginary part of complex root. Response in the time domain can be derived by using inverse Laplace transform method; it can be written as(47)ct=Aωβmc+1c-a2+b2c2+ω2ect+2eat-Fsin⁡bt+Ecos⁡bt+2Nsin⁡ωt+Mcos⁡ωt,E=p1p3+p2p4p32+p42,F=p2p3-p1p4p32+p42,M=p5p7+p6p8p72+p82,N=p6p7-p5p8p72+p82,p1=ma+1,p2=mb,p3=-2b2a2-b2+ω2,p4=-4ab3+2ba-ca2-b2+ω2,p5=1,p6=-mω,p7=-2ω2a2+b2-ω2-4acω2,p8=-4aω3+2ωca2+b2-ω2,m=αβ.


### 4.2. Simulation Results

(
1) Step Excitation. Figures 20–23 are responses in the time domain for step excitation. Lateral axis represents normalized time. It can be concluded that the system has the shortest settling time when β=1 compared with other values of β. Settling time decreases with the value of α.Figure 20
Response in the time domain(α=1.2,β=0.16,1,6.25).Figure 21
Response in the time domain(α=1.5,β=0.16,1,6.25).Figure 22
Response in the time domain(α=2,β=0.16,1,6.25).Figure 23
Response in the time domain(α=3,β=0.16,1,6.25).(
2) Slop Excitation. Figures 24–27 are the responses in the time domain for slop excitation; lateral axis represents normalized time. It can be seen from these figures that response in the time domain has a oscillation at the beginning. Duration of oscillation decreases with the value of α; then, response is consistent with the input function.Figure 24
Response in the time domain (α=1.2,β=0.16,1,6.25).Figure 25
Response in the time domain (α=1.5,β=0.16,1,6.25).Figure 26
Response in the time domain (α=2,β=0.16,1,6.25).Figure 27
Response in the time domain (α=3,β=0.16,1,6.25).(
3) Acceleration Excitation. Figures 28–31 are the responses in the time domain for acceleration excitation. Lateral axis represents normalized time. It can be seen from these figures that response in the time domain is consistent with the input function; this trend has no relationship with the values of α and β.Figure 28
Response in the time domain (α=1.2,β=0.16,1,6.25).Figure 29
Response in the time domain (α=1.5,β=0.16,1,6.25).Figure 30
Response in the time domain (α=2,β=0.16,1,6.25).Figure 31
Response in the time domain (α=3,β=0.16,1,6.25).(
4) Impulse Excitation. Figures 32–35 are the responses in the time domain for impulse excitation. Lateral axis represents normalized time. When the value of α is small, response in the time domain has an obvious oscillation. Oscillation decays with the value of α. The system has the shortest settling time when β=1 compared with the case when β<1 and β>1.Figure 32
Response in the time domain (α=1.2,β=0.16,1,6.25).Figure 33
Response in the time domain (α=1.5,β=0.16,1,6.25).Figure 34
Response in the time domain (α=2,β=0.16,1,6.25).Figure 35
Response in the time domain (α=3,β=0.16,1,6.25).


### 4.3. A Comparison Study on the Case When the System Has Optimal Damping and the Case Whenβ=1

The system has minimum resonance peak when system has optimal damping. It can be proved that all the transmissibility curves will intersect at one point; this point is resonance peak when the system has optimal damping. In this section, a comparison study on the responses in the time domain at two frequencies (ω/ω0=2 and the resonance peak frequency) is made. According to (17), transmissibility can be written as(48)GjΩ=1+α2βΩ21-αΩ22+βΩ2α-Ω22.ψ=s/ω0, s=jω, and, defining Ω=ω/ω0, Ω is normalized frequency.G
j
Ω
1
=
G
j
Ω
2. Resonance frequency when system has optimal damping can be obtained(49)Ω0=2α1+α2.Taking the derivative of (48) and taking result into (49), the value of β can be obtained.(50)β=1+α22α.Figures38–41 are the responses in the time domain for sinusoidal excitation. Two cases are analyzed; the first case is when β=1 and the second case is when system has optimal damping.Figure36 is response in the time domain when ω/ω0=2 and α=2; there are three curves; the black line represents input function, red line represents response in the time domain when β=1, and blue line represents response in the time domain when system has optimal damping. According to (50), β=1.25. Longitudinal axis represents displacement transmissibility. It can be seen from the figure that transmissibility when β=1 is nearly the same as the case when β=1.25. Figure 37 is response at resonance frequency. It can be seen from the figure that the two cases (β=1 and β=1.25) nearly have the same transmissibility.Figure 36
Response in the time domain (α=2,β=1,1.25,ω/ω0=2).Figure 37
Response in the time domain (α=2,β=1,1.25, at resonance frequency).Figure 38
Response in the time domain (α=3,β=1,1.6667,ω/ω0=2).Figure 39
Response in the time domain (α=3,β=1,1.6667, at resonance frequency).Figure 40
Response in the time domain (α=4,β=1,2.125,ω/ω0=2).Figure 41
Response in the time domain (α=4,β=1,2.125, at resonance frequency).Figure38 is response in the time domain when ω/ω0=2 and α=3. According to (50), β=1.6667. It can be seen from the figure that the system has better isolation performance when β=1 than the case when β=1.6667. Figure 39 is response at resonance frequency. It can be seen from the figure that the two cases (β=1 and β=1.6667) nearly have the same transmissibility.Figure40 is response in the time domain when ω/ω0=2 and α=4. According to (50), β=2.215. It can be seen from the figure that the system has better isolation performance when β=1 than the case when β=2.215. Figure 41 is response at resonance frequency. It can be seen from the figure that the two cases (β=1 and β=2.215) nearly have the same transmissibility.With the increasing values ofα, isolation performance at ω/ω0=2 becomes better, while the two cases (the case when β=1 and the case when the system has the optimal damping) nearly have the same transmissibility at resonance frequency.


## 4.1. Response in the Time Domain under Different Types of Excitations

When system is applied to different types of excitation, response in the time domain will be different. In this section, response in the time domain for different types of excitation will be analyzed.(
1) Step Excitation. As has been analyzed in Section 3.1, characteristic root has relationship with the value of Δ. In this section, only the case when system has a real root and conjugate roots will be analyzed.For step excitation, output function can be written as(33)Cψ=1βαβψ+1ψψ-ψ1ψ-ψ2ψ-ψ3,ψ=sω0.Equation (33) has three characteristic roots ψ1, ψ2, ψ3 which can be written as(34)ψ1=c,ψ2=a+bi,ψ3=a-bi.c is real root, a is real part of complex root, and b is the imaginary part of complex root. Response in the time domain can be derived by using inverse Laplace transform method; it can be written as(35)ct=1βA0+A1ect+2Ueatcos⁡bt-2Veatsin⁡bt,A0=-1ca2+b2,A1=1/c+αβc2-a2+b2,U=p1p3+p2p4p32+p42,V=p1p4-p2p3p32+p42,p1=ma+1,p2=mb,p3=-4ab2+2cb2,p4=2aba-c-2b3,m=αβ.(
2) Slope Excitation. For slope excitation, output function can be written as(36)Cψ=1βαβψ+1ψ2ψ-ψ1ψ-ψ2ψ-ψ3.Three characteristic roots are ψ1,ψ2,ψ3, which can be written as(37)ψ1=c,ψ2=a+bi,ψ3=a-bi.c is real root, a is real part of complex root, and b is the imaginary part of complex root. Response in the time domain can be derived by using inverse Laplace transform method; it can be written as(38)ct=1βA0t+A1+A2e-ct+2eat-Fsin⁡bt+Ecos⁡bt,A0=-1ca2+b2,A1=-mca2+b2-a2+b2+2acc2a2+b22,A2=mc+1c2c-a2+b2,p1=ma+1,p2=mb,p3=-2b2a2-b2-4ab2a-c,p4=2ba2-b2a-c-4ab3,E=p1p3+p2p4p32+p42,F=p2p3-p1p4p32+p42,m=αβ.(
3) Acceleration Excitation. For acceleration excitation, output function can be written as(39)Cψ=1βαβψ+1ψ3ψ-ψ1ψ-ψ2ψ-ψ3.Three characteristic roots are ψ1,ψ2,ψ3, which can be written as(40)ψ1=c,ψ2=a+bi,ψ3=a-bi.c is real root, a is real part of complex root, and b is the imaginary part of complex root.Response in the time domain can be derived by using inverse Laplace transform method; it can be written as(41)ct=1β12A0t2+A1t+A2+A3ect+2eat-Fsin⁡bt+Ecos⁡bt,A0=-1ca2+b2,A1=-mca2+b2-a2+b2+2acc2a2+b22,A2=c+2a-ca2+b2-a2+b2+2acm-ca2+b2-a2+b2+2ac-ca2+b23,E=p1p3+p2p4p32+p42,F=p2p3-p1p4p32+p42,p1=-ma+12b,p2=-m2,p3=ba-c3a2-b2-ab3b2-a2,p4=b23a2-b2+aa-c3b2-a2,m=αβ.(
4) Impulse Excitation. For impulse excitation, output function can be written as(42)Cψ=1βαβψ+1ψ-ψ1ψ-ψ2ψ-ψ3.Three characteristic roots are ψ1,ψ2,ψ3, which can be written as(43)ψ1=c,ψ2=a+bi,ψ3=a-bi.c is real root, a is real part of complex root, and b is the imaginary part of complex root. Response in the time domain can be derived by using inverse Laplace transform method; it can be written as(44)ct=1β-2Eect+2eat-Fsin⁡bt+Ecos⁡bt,p1=-a+c,p2=b,p3=-a2-b2+ac,p4=bc,E=mp4-p22p1p4-p2p3,F=mp3-p12p1p4-p2p3,m=αβ.(
5) Harmonic Excitation. For harmonic excitation, output function can be written as(45)Cψ=1βαβψ+1ψ2+ω2ψ-ψ1ψ-ψ2ψ-ψ3.Three characteristic roots are ψ1, ψ2, ψ3, which can be written as(46)ψ1=c,ψ2=a+bi,ψ3=a-bi.c is real root, a is real part of complex root, and b is the imaginary part of complex root. Response in the time domain can be derived by using inverse Laplace transform method; it can be written as(47)ct=Aωβmc+1c-a2+b2c2+ω2ect+2eat-Fsin⁡bt+Ecos⁡bt+2Nsin⁡ωt+Mcos⁡ωt,E=p1p3+p2p4p32+p42,F=p2p3-p1p4p32+p42,M=p5p7+p6p8p72+p82,N=p6p7-p5p8p72+p82,p1=ma+1,p2=mb,p3=-2b2a2-b2+ω2,p4=-4ab3+2ba-ca2-b2+ω2,p5=1,p6=-mω,p7=-2ω2a2+b2-ω2-4acω2,p8=-4aω3+2ωca2+b2-ω2,m=αβ.


## 4.2. Simulation Results

(
1) Step Excitation. Figures 20–23 are responses in the time domain for step excitation. Lateral axis represents normalized time. It can be concluded that the system has the shortest settling time when β=1 compared with other values of β. Settling time decreases with the value of α.Figure 20
Response in the time domain(α=1.2,β=0.16,1,6.25).Figure 21
Response in the time domain(α=1.5,β=0.16,1,6.25).Figure 22
Response in the time domain(α=2,β=0.16,1,6.25).Figure 23
Response in the time domain(α=3,β=0.16,1,6.25).(
2) Slop Excitation. Figures 24–27 are the responses in the time domain for slop excitation; lateral axis represents normalized time. It can be seen from these figures that response in the time domain has a oscillation at the beginning. Duration of oscillation decreases with the value of α; then, response is consistent with the input function.Figure 24
Response in the time domain (α=1.2,β=0.16,1,6.25).Figure 25
Response in the time domain (α=1.5,β=0.16,1,6.25).Figure 26
Response in the time domain (α=2,β=0.16,1,6.25).Figure 27
Response in the time domain (α=3,β=0.16,1,6.25).(
3) Acceleration Excitation. Figures 28–31 are the responses in the time domain for acceleration excitation. Lateral axis represents normalized time. It can be seen from these figures that response in the time domain is consistent with the input function; this trend has no relationship with the values of α and β.Figure 28
Response in the time domain (α=1.2,β=0.16,1,6.25).Figure 29
Response in the time domain (α=1.5,β=0.16,1,6.25).Figure 30
Response in the time domain (α=2,β=0.16,1,6.25).Figure 31
Response in the time domain (α=3,β=0.16,1,6.25).(
4) Impulse Excitation. Figures 32–35 are the responses in the time domain for impulse excitation. Lateral axis represents normalized time. When the value of α is small, response in the time domain has an obvious oscillation. Oscillation decays with the value of α. The system has the shortest settling time when β=1 compared with the case when β<1 and β>1.Figure 32
Response in the time domain (α=1.2,β=0.16,1,6.25).Figure 33
Response in the time domain (α=1.5,β=0.16,1,6.25).Figure 34
Response in the time domain (α=2,β=0.16,1,6.25).Figure 35
Response in the time domain (α=3,β=0.16,1,6.25).


## 4.3. A Comparison Study on the Case When the System Has Optimal Damping and the Case Whenβ=1

The system has minimum resonance peak when system has optimal damping. It can be proved that all the transmissibility curves will intersect at one point; this point is resonance peak when the system has optimal damping. In this section, a comparison study on the responses in the time domain at two frequencies (ω/ω0=2 and the resonance peak frequency) is made. According to (17), transmissibility can be written as(48)GjΩ=1+α2βΩ21-αΩ22+βΩ2α-Ω22.ψ=s/ω0, s=jω, and, defining Ω=ω/ω0, Ω is normalized frequency.G
j
Ω
1
=
G
j
Ω
2. Resonance frequency when system has optimal damping can be obtained(49)Ω0=2α1+α2.Taking the derivative of (48) and taking result into (49), the value of β can be obtained.(50)β=1+α22α.Figures38–41 are the responses in the time domain for sinusoidal excitation. Two cases are analyzed; the first case is when β=1 and the second case is when system has optimal damping.Figure36 is response in the time domain when ω/ω0=2 and α=2; there are three curves; the black line represents input function, red line represents response in the time domain when β=1, and blue line represents response in the time domain when system has optimal damping. According to (50), β=1.25. Longitudinal axis represents displacement transmissibility. It can be seen from the figure that transmissibility when β=1 is nearly the same as the case when β=1.25. Figure 37 is response at resonance frequency. It can be seen from the figure that the two cases (β=1 and β=1.25) nearly have the same transmissibility.Figure 36
Response in the time domain (α=2,β=1,1.25,ω/ω0=2).Figure 37
Response in the time domain (α=2,β=1,1.25, at resonance frequency).Figure 38
Response in the time domain (α=3,β=1,1.6667,ω/ω0=2).Figure 39
Response in the time domain (α=3,β=1,1.6667, at resonance frequency).Figure 40
Response in the time domain (α=4,β=1,2.125,ω/ω0=2).Figure 41
Response in the time domain (α=4,β=1,2.125, at resonance frequency).Figure38 is response in the time domain when ω/ω0=2 and α=3. According to (50), β=1.6667. It can be seen from the figure that the system has better isolation performance when β=1 than the case when β=1.6667. Figure 39 is response at resonance frequency. It can be seen from the figure that the two cases (β=1 and β=1.6667) nearly have the same transmissibility.Figure40 is response in the time domain when ω/ω0=2 and α=4. According to (50), β=2.215. It can be seen from the figure that the system has better isolation performance when β=1 than the case when β=2.215. Figure 41 is response at resonance frequency. It can be seen from the figure that the two cases (β=1 and β=2.215) nearly have the same transmissibility.With the increasing values ofα, isolation performance at ω/ω0=2 becomes better, while the two cases (the case when β=1 and the case when the system has the optimal damping) nearly have the same transmissibility at resonance frequency.


## 5. Transmissibility of Three-Parameter System

Transmissibility is an important index for isolation performance. In this section, transmissibility of three-parameter system will be discussed. Two cases are analyzed; the first case is whenβ=1; the second case is when the system has optimal damping.Figure42 is transmissibility of three-parameter system (α=2). Lateral axis represents normalized frequency. Black line is transmissibility for the case when β=1, resonance frequency is 0.83, and amplification factor of resonance peak is 1.6823. Red line is transmissibility for the case when system has optimal damping, resonance frequency is 0.89, and amplification factor of resonance peak is 1.6667. It is shown that amplification factors of two cases are nearly the same (1.6823 and 1.6667). Besides, the two cases nearly have the same transmissibility in the high frequency domain.Figure 42
Transmissibility (α=2,β=1,1.25).Figure43 is transmissibility of three-parameter system (α=3). Lateral axis represents normalized frequency. Black line is transmissibility for the case when β=1, resonance frequency is 0.58, and amplification factor of resonance peak is 1.299. Red line is transmissibility for the case when system has optimal damping, resonance frequency is 0.77, and amplification factor of resonance peak is 1.25. It is shown that amplification factors of two cases are nearly the same (1.299 and 1.25), while transmissibility characteristics of two cases are different in the high frequency domain. Isolation performance when β=1 is better than the case when system has optimal damping.Figure 43
Transmissibility (α=3,β=1,1.6667).Figure44 is transmissibility of three-parameter system (α=4). Lateral axis represents normalized frequency. Black line is transmissibility for the case when β=1, resonance frequency is 0.44, and amplification factor of resonance peak is 1.1978. Red line is transmissibility for the case when system has optimal damping, resonance frequency is 0.69, and amplification factor of resonance peak is 1.3333. It is shown that amplification factors of two cases are nearly the same (1.1978 and 1.3333), while transmissibility characteristics of two cases are different in the high frequency domain. Isolation performance when β=1 is better than the case when system has optimal damping.Figure 44
Transmissibility (α=4,β=1,2.215).It can be concluded that the case whenβ=1 has better isolation performance in the high frequency domain, but amplification factors of two cases are nearly the same.Figure45 is a comparison of resonance factors between the case when β=1 and the case when system has optimal damping. Lateral axis represents the value of α. If the value of α is small, there is no difference between these two cases. If the value of α is large, difference between two cases is obvious; the second case has smaller resonance factor than the first case.Figure 45
Comparison of resonance peaks.Figures46–50 are the transmissibility surfaces for different values of α and β. The first axis represents normalized frequency (ω/ω0), the second axis represents the value of α, and the third axis represents transmissibility. Figure 46 is transmissibility surface when β=1. It can be seen from the figure that the largest amplification factor is less than 10, and amplification factor decreases with the value of α. Figure 47 is transmissibility surface when α=1.2. It can be seen from the figure that the largest amplification factor is less than 10; with the increasing values of β, amplification factor decreases, but this trend is not obvious. Figure 48 is transmissibility surface when α=1.5. It can be seen from the figure that the largest amplification factor is less than 4; amplification factor decreases obviously compared with the case when α=1.2. Figure 49 is transmissibility surface when α=2. It can be seen from the figure that the largest amplification factor is less than 3; amplification factor decreases obviously compared with the case when α=1.5. Figure 50 is transmissibility surface when α=3. It can be seen from the figure that the largest amplification factor is less than 3; amplification factor decreases obviously compared with the case when α=2. It can be concluded that amplification factor decreases with the value of α.Figure 46
Transmissibility surface (β=1).Figure 47
Transmissibility surface (α = 1.2, β = 0.2–7).Figure 48
Transmissibility surface (α = 1.5, β = 0.2–7).Figure 49
Transmissibility surface (α = 2, β = 0.2–7).Figure 50
Transmissibility surface (α = 3, β = 0.2–7).


## 6. Conclusion

In this paper, dynamic behavior of three-parameter system has been studied by using normalized model. Different types of excitation have been considered in order to analyze response in the time domain. In order to analyze dynamic behavior in the frequency domain, root locus and phase margin of the system have been analyzed. The results reveal the following:(1)
System under impulse excitation and step excitation has the shortest settling time for the response in the time domain when the value of regulatory factor is 1; meanwhile, the system has the largest phase margin.(2)
A comparison study for isolation performance at resonance frequency is made between the case when system has optimal damping and the case whenβ=1. If stiffness ratio is small, isolation performances of two cases are nearly the same; if stiffness ratio is large, isolation performance of the first case is much better. Previous researches showed that isolation performance of the second case was much better.(3)
Amplification factor decreases with the value ofα. Because parameter α is the function of stiffness ratio, it means that the system has small amplification factor when stiffness ratio is large.(4)
The normalized model has two dimensionless parametersα and β; it can make the dynamic model of three-parameter system more concise than the physical parameter model; therefore, dynamic behavior can be analyzed easily.



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