# Experiment and Analysis of Active Vibration Suppression via an Absorber with a Tunable Delay

Authors: Yixia Sun
Journal: Shock and Vibration
 (2020)

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## Abstract

A time-delayed absorber is utilized to suppress the vibration of a primary system excited by a simple harmonic force. The inherent and intentional time delays in the feedback control loop are taken into consideration. The value of the former is fixed, while the value of the latter is tunable in the controller. To begin with, the mechanical model of the system is established and the acceleration transfer functions of the system are derived. Consequently, the stability analysis of the coupled system is carried out. Finally, the experimental studies on the performance of the time-delayed absorber are conducted. Both experimental and theoretical results show that the time-delayed absorber with proper values of feedback gain coefficient and intentional time delay greatly suppresses the vibration of the primary system. The numerical results validate the correctness of the experimental and theoretical ones.


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## Body

## 1. Introduction

Time delay is inherent in the active control loop, which is derived from the signal acquisition and processing, filtering, the action of the actuator, etc [1]. In fact, time delay exists in various research fields, such as aerospace engineering [2], medicine [3], communication [4], and machining [5].Originally, the time delay was taken as a negative factor in the active control. It may result in the degradation of the control performance and the instability of the controlled system. Therefore, several methods were employed to compensate the adverse influence of time delay, such as phase-shift method [6], Smith predictor method [7], and Pade approximation method [8]. However, in the past three decades, lots of works have showed that intentional introduction of time delay in the feedback control loop benefits control effect. The time-delayed feedback control technique has been widely used in controlling chaos [9], improving system stability [10], and vibration control performance [11, 12].A time-delayed absorber is a new technique in the field of active vibration control. The key idea of the time-delayed absorber is the introduction of an actuator controlled via time-delayed feedback control. In 1994, Olgac and Holm-Hansen [13] firstly presented the concept of Delayed Resonator (DR). When the proportionality gain and time delay are properly selected, the resonator moves all vibration from a primary system at its point of attachment. After that, Olgac et al. conducted in-depth studies on the application of DR [14–16]. Zhao et al. [17] investigated the effect of a nonlinear time-delayed absorber on suppressing the vertical vibration of a primary system when the primary resonance and 1 : 1 internal resonance occurred simultaneously. Mohanty and Dwivedy [18] studied the vibration control performance of a piezoelectric-based nonlinear vibration absorber using time-delayed acceleration feedback when the nonlinear primary system is harmonically excited. Sun and Song [19] discussed the time-delayed active control of vibration absorbers attached to a continuous beam structure. It is worth noting that extensive studies on the time-delayed absorber focus on theoretical analysis and numerical simulation. Few efforts have been devoted to experiments [20, 21].In our previous work [22], it is demonstrated that the time-delayed absorber with proper feedback gain coefficient and inherent time delay greatly reduces the vibration amplitude of a primary system. Motivated by this finding, an intentional time delay is introduced in the controller and the vibration suppression effect of the time-delayed absorber with the inherent and the intentional time delays is studied in this paper.The present paper is organized as follows. The mechanical model and stability analysis are shown in Section2. In Sections 3, the experimental studies are conducted. The effects of the feedback gain coefficient and intentional time delay on vibration suppression effect of the time-delayed absorber are, respectively, investigated. Conclusions are presented in Section 4.


## 2. Modeling and Stability Analysis

### 2.1. Modeling

Figure1 illustrates the mechanical model of the 2-dof coupled system [23]. The coupled system consists of a time-delayed absorber and a primary system. m1 and m2 represent the mass of the absorber and the primary system, respectively. The motions of the absorber and the primary system are denoted by x1 and x2, respectively. k1 and k2 represent the linear stiffness coefficients. c1 and c2 represent the viscous damping coefficients. An actuator mounted between the primary mass and the absorber mass provides the time-delayed feedback control force.Figure 1
Mechanical model of the coupled system.Assuming that simply harmonic excitation is applied to the primary mass, the governing equations of the coupled system are given by(1)m1x¨1+k1x1−x2+c1x˙1−x˙2−kcxc=0,(2)m2x¨2+k2x2+c2x˙2+k1x2−x1+c1x˙2−x˙1+kcxc=fsinωt,where kcxc is the time-delayed feedback control force, kc is the stiffness coefficient of the actuator, and xc is the control signal with the following form:(3)xc=α1x1t−τ1−x2t−τ1+α2x1t−τ1−τ2−x1t−τ1,where τ1 and τ2, respectively, represent the inherent and the intentional time delays in the feedback control loop. The value of τ1 is constant, while the value of τ2 is tunable in the controller. Instead of eliminating, compensating for, and even ignoring the presence of τ1, it is thought that τ2 together with τ1 leads to the final control effect. α1 and α2 represent amplification factors of the control signal. The time-delayed feedback disappears and the time-delayed absorber degrades into the passive one when α1=α2=0.Substituting equation (3) into equations (1) and (2) gives(4)m1x¨1+k1x1−x2+c1x˙1−x˙2−g1x1t−τ1−x2t−τ1−g2x1t−τ1−τ2−x1t−τ1=0,(5)m2x¨2+k2x2+c2x˙2+k1x2−x1+c1x˙2−x˙1+g1x1t−τ1−x2t−τ1+g2x1t−τ1−τ2−x1t−τ1=fsinωt,where g1=α1kc and g2=α2kc represent feedback gain coefficients.The external excitation can be rewritten as(6)fsinωt=fejωt.The solutions of equations (4) and (5) are considered to be(7)x1x2=x¯1x¯2ejωt,where x¯1 and x¯2 are complex.Substituting equations (6) and (7) into equations (4) and (5), one gets(8)x¯1x¯2=fΔωk1+jc1ω−g1e−jωτ1−m1ω2+k1+jc1ω+g2−g1e−jωτ1−g2e−jωτ1+τ2,where(9)Δω=−m1ω2+k1+jc1ω+g2−g1e−jωτ1−g2e−jωτ1+τ2×−m2ω2+k2+k1+jc1+c2ω−g1e−jωτ1−k1+jc1ω−g1e−jωτ1k1+jc1ω−g1−g2e−jωτ1−g2e−jωτ1+τ2.To facilitate the following analysis, new variablesH1 and H2 are defined as follows:(10)Hi=x¨if,i=1,2,where H1 and H2 represent the acceleration transfer functions of the absorber and the primary system, respectively.It is seen from equation (10) that H1 and H2 describe the vibration intensity of the absorber and the primary system, respectively. Hence, for fixed values of physical parameters and inherent time delay, it is available to evaluate the vibration suppression effect of the time-delayed absorber by comparing the value of H2 under different values of g1, g2, and τ2.


### 2.2. Stability Analysis

It is known that the values of feedback gain coefficients and time delays determine the stability of the system when the physical parameters of the coupled system are fixed. Hence, it is necessary to analyze the stability of the system before the experiments are carried out.In Laplace domain, equations (4) and (5) become(11)AsXs=Fs,where(12)As=m1s2+k1+c1s+g2−g1e−τ1s−g2e−τ1+τ2s−k1−c1s+g1e−τ1s−k1−c1s−g2−g1e−τ1s+g2e−τ1+τ2sm2s2+k1+k2+c1+c2s−g1e−τ1s,Xs=X1s,X2sT,Fs=0,FsT.The characteristic equation of the coupled system is det (A) = 0; that is,(13)P0s+P1se−τ1s+P2se−τ1+τ2s=0,where(14)P0s=k1k2+c2k1+c1k2s+c1c2+k1m1+k2m1+k1m2s2+c1m1+c2m1+c1m2s3+m1m2s4,P1s=g2−g1k2+−g1m1−g1m2+g2m2s2+c2g2−g1s,P2s=−g2k2−c2g2s−g2m2s2.The coupled system is stable if and only if all characteristic roots of equation (13) have negative real parts. When equation (13) has a pure imaginary root, stability switch may occur. Let(15)s=ωci,ωc>0.Substituting equation (15) into equation (13) and separating the real and the imaginary parts, one obtains(16)g2−g1k2+−g2m2+g1m1+g1m2ωc2cosωcτ1−c2ωcg1−g2sinωcτ1−c2ωcg2sinωcτ1+τ2+g2m2ωc2−k2cosωcτ1+τ2+k1k2+m1m2ωc4−c1c2+k1m1+k2m1+k1m2ωc2=0,(17)g2m2ωc2−k2+g1k2−m1ωc2−m2ωc2sinωcτ1+c2g2−g1ωccosωcτ1−c2g2ωccosωcτ1+τ2+g2k2−m2ωc2sinωcτ1+τ2+c2k1+c1k2ωc−c2m1+c1m1+c1m2ωc3=0.Usingsin2ωcτ1+τ2+cos2ωcτ1+τ2=1, one has(18)d8ωc8+d6ωc6+d5ωc5+d4ωc4+d3ωc3+d2ωc2+d1ωc+d0=0,where the expressions of dll=1,2,3,4,5,6,8 are given in Appendix.Since equation (18) is a high-order equation with transcendental terms, the values of ωc are obtained by numerical calculation. For fixed values of τ1 and certain values of feedback gain coefficients g1=g10 and g2=g20, N is assigned to denote the number of the positive real roots of equation (18). When N=0, there is no stability switch. In other words, the stability of the system remains the same for all τ2∈ℜ+. When N≠0, we have ωc1,ωc2,…,ωcn. For each ωcjj=1,2,…,N, infinite τ2 values τ21,τ22,…,τ2∞ are determined by solving equations (16) and (17). The transition direction of the roots at ωcj as τ2 increases from τ2p−ε to τ2p+ε0<ε≪1,p=1,2,…,∞ is decided by the expression:(19)RD=sgnRe∂s∂τ2s=ωcji,and +1 and −1 values of RD represent destabilizing and stabilizing transitions, respectively.Through the above computation, the stable and unstable ranges ofτ2 for g1=g10 and g2=g20 are divided. Performing the same procedure for other values of g1 and g2, the stable and unstable regions in the space g1,g2,τ2 are plotted.


## 2.1. Modeling

Figure1 illustrates the mechanical model of the 2-dof coupled system [23]. The coupled system consists of a time-delayed absorber and a primary system. m1 and m2 represent the mass of the absorber and the primary system, respectively. The motions of the absorber and the primary system are denoted by x1 and x2, respectively. k1 and k2 represent the linear stiffness coefficients. c1 and c2 represent the viscous damping coefficients. An actuator mounted between the primary mass and the absorber mass provides the time-delayed feedback control force.Figure 1
Mechanical model of the coupled system.Assuming that simply harmonic excitation is applied to the primary mass, the governing equations of the coupled system are given by(1)m1x¨1+k1x1−x2+c1x˙1−x˙2−kcxc=0,(2)m2x¨2+k2x2+c2x˙2+k1x2−x1+c1x˙2−x˙1+kcxc=fsinωt,where kcxc is the time-delayed feedback control force, kc is the stiffness coefficient of the actuator, and xc is the control signal with the following form:(3)xc=α1x1t−τ1−x2t−τ1+α2x1t−τ1−τ2−x1t−τ1,where τ1 and τ2, respectively, represent the inherent and the intentional time delays in the feedback control loop. The value of τ1 is constant, while the value of τ2 is tunable in the controller. Instead of eliminating, compensating for, and even ignoring the presence of τ1, it is thought that τ2 together with τ1 leads to the final control effect. α1 and α2 represent amplification factors of the control signal. The time-delayed feedback disappears and the time-delayed absorber degrades into the passive one when α1=α2=0.Substituting equation (3) into equations (1) and (2) gives(4)m1x¨1+k1x1−x2+c1x˙1−x˙2−g1x1t−τ1−x2t−τ1−g2x1t−τ1−τ2−x1t−τ1=0,(5)m2x¨2+k2x2+c2x˙2+k1x2−x1+c1x˙2−x˙1+g1x1t−τ1−x2t−τ1+g2x1t−τ1−τ2−x1t−τ1=fsinωt,where g1=α1kc and g2=α2kc represent feedback gain coefficients.The external excitation can be rewritten as(6)fsinωt=fejωt.The solutions of equations (4) and (5) are considered to be(7)x1x2=x¯1x¯2ejωt,where x¯1 and x¯2 are complex.Substituting equations (6) and (7) into equations (4) and (5), one gets(8)x¯1x¯2=fΔωk1+jc1ω−g1e−jωτ1−m1ω2+k1+jc1ω+g2−g1e−jωτ1−g2e−jωτ1+τ2,where(9)Δω=−m1ω2+k1+jc1ω+g2−g1e−jωτ1−g2e−jωτ1+τ2×−m2ω2+k2+k1+jc1+c2ω−g1e−jωτ1−k1+jc1ω−g1e−jωτ1k1+jc1ω−g1−g2e−jωτ1−g2e−jωτ1+τ2.To facilitate the following analysis, new variablesH1 and H2 are defined as follows:(10)Hi=x¨if,i=1,2,where H1 and H2 represent the acceleration transfer functions of the absorber and the primary system, respectively.It is seen from equation (10) that H1 and H2 describe the vibration intensity of the absorber and the primary system, respectively. Hence, for fixed values of physical parameters and inherent time delay, it is available to evaluate the vibration suppression effect of the time-delayed absorber by comparing the value of H2 under different values of g1, g2, and τ2.


## 2.2. Stability Analysis

It is known that the values of feedback gain coefficients and time delays determine the stability of the system when the physical parameters of the coupled system are fixed. Hence, it is necessary to analyze the stability of the system before the experiments are carried out.In Laplace domain, equations (4) and (5) become(11)AsXs=Fs,where(12)As=m1s2+k1+c1s+g2−g1e−τ1s−g2e−τ1+τ2s−k1−c1s+g1e−τ1s−k1−c1s−g2−g1e−τ1s+g2e−τ1+τ2sm2s2+k1+k2+c1+c2s−g1e−τ1s,Xs=X1s,X2sT,Fs=0,FsT.The characteristic equation of the coupled system is det (A) = 0; that is,(13)P0s+P1se−τ1s+P2se−τ1+τ2s=0,where(14)P0s=k1k2+c2k1+c1k2s+c1c2+k1m1+k2m1+k1m2s2+c1m1+c2m1+c1m2s3+m1m2s4,P1s=g2−g1k2+−g1m1−g1m2+g2m2s2+c2g2−g1s,P2s=−g2k2−c2g2s−g2m2s2.The coupled system is stable if and only if all characteristic roots of equation (13) have negative real parts. When equation (13) has a pure imaginary root, stability switch may occur. Let(15)s=ωci,ωc>0.Substituting equation (15) into equation (13) and separating the real and the imaginary parts, one obtains(16)g2−g1k2+−g2m2+g1m1+g1m2ωc2cosωcτ1−c2ωcg1−g2sinωcτ1−c2ωcg2sinωcτ1+τ2+g2m2ωc2−k2cosωcτ1+τ2+k1k2+m1m2ωc4−c1c2+k1m1+k2m1+k1m2ωc2=0,(17)g2m2ωc2−k2+g1k2−m1ωc2−m2ωc2sinωcτ1+c2g2−g1ωccosωcτ1−c2g2ωccosωcτ1+τ2+g2k2−m2ωc2sinωcτ1+τ2+c2k1+c1k2ωc−c2m1+c1m1+c1m2ωc3=0.Usingsin2ωcτ1+τ2+cos2ωcτ1+τ2=1, one has(18)d8ωc8+d6ωc6+d5ωc5+d4ωc4+d3ωc3+d2ωc2+d1ωc+d0=0,where the expressions of dll=1,2,3,4,5,6,8 are given in Appendix.Since equation (18) is a high-order equation with transcendental terms, the values of ωc are obtained by numerical calculation. For fixed values of τ1 and certain values of feedback gain coefficients g1=g10 and g2=g20, N is assigned to denote the number of the positive real roots of equation (18). When N=0, there is no stability switch. In other words, the stability of the system remains the same for all τ2∈ℜ+. When N≠0, we have ωc1,ωc2,…,ωcn. For each ωcjj=1,2,…,N, infinite τ2 values τ21,τ22,…,τ2∞ are determined by solving equations (16) and (17). The transition direction of the roots at ωcj as τ2 increases from τ2p−ε to τ2p+ε0<ε≪1,p=1,2,…,∞ is decided by the expression:(19)RD=sgnRe∂s∂τ2s=ωcji,and +1 and −1 values of RD represent destabilizing and stabilizing transitions, respectively.Through the above computation, the stable and unstable ranges ofτ2 for g1=g10 and g2=g20 are divided. Performing the same procedure for other values of g1 and g2, the stable and unstable regions in the space g1,g2,τ2 are plotted.


## 3. Experimental Studies on the Time-Delayed Absorber

### 3.1. Experimental Setup

The photo of the experimental setup is shown in Figure2. The absorber mass (1) is attached to the primary mass (2) by five sheets of steel. A servo motor (3) is fixed on the primary mass. The primary mass is connected to the base (5) by four sheets of steel. The controlled steel sheet (4) acts as the actuator and exerts the control force, whose lower end is linked to the shaft of the servo motor by a wire rope. A shaker (6) provides a horizontal exciting force to the primary mass.Figure 2
Photo of experiment setup: (a) front view; (b) right view (1: absorber mass, 2: primary mass, 3: servo motor, 4: controlled steel sheet, 5: base, and 6: shaker).
(a)(b)Figure3 illustrates the schematic of time-delayed feedback control. A shaker (5) provides a sinusoidal excitation force to the primary mass (2). The amplitude and frequency of the excitation force can be set in M + p vib-pilot. A force sensor (6) and two acceleration sensors (7) and (8) are used to monitor the excitation force and the responses of the absorber and the primary mass, respectively. The time-delayed feedback control loop is described as follows:Step 1: the acceleration signals of the absorber mass (1) and the primary mass enter into signal conditioning instrument, in which the functions of signal amplification and low-pass filtering are achieved to improve the signal-to-noise ratio.Step 2: the processed signals go into the voltage lifting device, where the voltage of the input signal is raised 5 volts.Step 3: the raised voltage signal enters into Trio motion controller, where control commands are written. The values of feedback control feedback gains and intentional time delay can be adjusted in the control commands.Step 4: the control commands are transferred into the servo controller, which guides the shaft rotation of the servo motor (3).Step 5: driven by the rotation of the servo motor shaft, the lower end of the controlled steel sheet (4) realizes the horizontal reciprocating motion and applies the time-delayed feedback control force.Figure 3
Schematic of time-delayed feedback control (1: absorber mass, 2: primary mass, 3: servo motor, 4: controlled steel sheet, 5: shaker, 6: force sensor, and 7 and 8: acceleration sensors).As a preliminary, the values of the physical parameters of the 2-dof coupled system need to be identified when the time-delayed feedback control is absent (i.e.,α1=α2=0 in equations (4) and (5)). Appling a sine swept excitation to the primary mass, the acceleration transfer function curves of the coupled system are obtained. Using the least-squares method, the values of the physical parameters are identified, as shown in Table 1. In addition, the value of inherent time delay τ1 is identified to be 63 ms [22].Table 1
Physical parameters of the coupled system.
m1kgm2kgk1N/mk2N/mkcN/mc1Ns/mc2Ns/m0.6676.82431.7213502.4198.960.171Figure4 shows the comparison of the experimental and the theoretical results of the acceleration transfer functions, where Ω=ω/2π denotes the excitation frequency. It is seen that the passive absorber is most effective when Ω=9.63Hz. When the excitation frequency is disturbed and deviates from 9.63 Hz, vibration control performance of the passive absorber deteriorates. In this case, the time-delayed feedback control is introduced to transform the passive absorber into a time-delayed one. Proper values of intentional time delay and feedback gain coefficients are adapted to improve the vibration suppression effect of the absorber.Figure 4
Comparison of the experimental and theoretical results ofHii=1,2: (a) absorber and (b) primary system.
(a)(b)


### 3.2. Experimental Results

In this subsection, the effects ofα2 and τ2 on the vibration suppression effect of the time-delayed absorber are, respectively, discussed when Ω=9.75Hz and 10 Hz.Figure5 shows the stability charts of the coupled system for τ2=50ms, τ2=30ms, and α2=0.5, respectively. In Figures 5(a) and (b), the coupled system is stable in region II, while it is unstable in regions I and III. In Figure 5(c), the coupled system is stable in region I, while it is unstable in regions II-VI.Figure 5
Stability charts of the coupled system. (a)τ2=50ms. (b) τ2=30ms. (c) α2=0.5.
(a)(b)(c)


#### 3.2.1. Effect ofα2

Case 1.
Ω=9.75Hz, α1=−0.4, and τ2=50ms.
Figure6 shows how the acceleration transfer functions of the system Hii=1,2 change as a function of α2 for Ω=9.75Hz, α1=−0.4, and τ2=50ms. Obviously, the experimental results agree well with the theoretical ones. Figure 6(a) indicates that the value of H1 monotonically increases with the increase of α2. As shown in Figure 6(b), the value of the H2 decreases firstly and then increases as α2 increases. The minimum of H2 occurs for α2=−0.33.
Figure7 shows the measured time histories of excitation force and system accelerations. The total settling time is 144 s. The time-delayed feedback control is activated at t=29s. After a short transient, the amplitude of the excitation force and system acceleration responses reaches fixed values. It is found that although the amplitude of the excitation force increases, the acceleration amplitude of the primary system decreases sharply. The time-delayed feedback control is deactivated at t=113s and the coupled system returns to the initial uncontrolled state.
Table2 shows the effect of α2 on the vibration suppression effect for Ω=9.75Hz, α1=−0.4, and τ2=50ms. When the passive absorber works, the value of H2 is 0.037m/s2/N. When the time-delayed feedback control is activated, the value of H2 decreases 45.95% for α1=−0.4 and α2=−0.33 and increases 159.46% for α1=−0.4 and α2=−0.055. It is drawn from Table 2 and Figure 6 that the vibration suppression effect of the time-delayed absorber is superior to that of the passive one for α2∈−0.495,−0.275.
Hereinafter,D=H2α1≠0,α2≠0−H2α1=α2=0/H2α1=α2=0×100%.
To verify the experimental and theoretical results, numerical results are obtained by numerical integration of equations (4) and (5). Figure 8 shows the numerical simulations of system responses for Ω=9.75Hz and f=1N. From the comparison between Figures 6 and 8, the numerical results agree well with experimental and theoretical ones.Figure 6
Hii=1,2 versus α2 for Ω=9.75Hz, α1=−0.4, and τ2=50ms: (a) absorber and (b) primary system.
(a)(b)Figure 7
Measured time histories of excitation force and system accelerations whenΩ=9.75Hz (for t<29s and t>113s, α1=α2=0; for 29s≤t≤113s, α1=−0.4, α2=−0.33, and τ2=50ms): (a) excitation force; (d) absorber; (g) primary system; (b, c) zoom of (a); (e, f) zoom of (d); (h, i) zoom of (g).
(a)(b)(c)(d)(e)(f)(g)(h)(i)Table 2
Effect ofα2 on the vibration suppression effect for Ω=9.75Hz, α1=−0.4, and τ2=50ms.
ΩHzτ2msα1α2H2m/s2/ND%9.75—000.03709.7550−0.4−0.330.020−45.959.7550−0.4−0.0550.096159.46Figure 8
Numerical simulations of system responses forΩ=9.75Hz and f=1N: (a) absorber and (b) primary system.
(a)(b)Case 2.
Ω=10Hz, α1=0.25, and τ2=30ms.
The variations ofHii=1,2 versus α2 for Ω=10Hz, α1=0.25, and τ2=30ms are shown in Figure 9. It can be concluded from the experimental results that the value of H2 is maximal for α2=−0.065 and minimal for α2=−0.455.
Figure10 shows the measured time histories of excitation force and system accelerations before and after the time-delayed feedback control. It is seen that after the feedback control is exerted, the amplitude of the excitation force increases 46.3%, whereas the acceleration amplitudes of the absorber and the primary system, respectively, decrease 4.2% and 72.9%.
Table3 shows the values of H2 under different values of feedback control parameters. It is attractive that the value of H2 decreases 78.92% for α1=0.25 and α2=−0.455.
Figure11 shows the numerical simulations of system acceleration responses for Ω=10Hz and f=1N, which are in agreement with the results shown in Figure 9 and Table 3.Figure 9
Hii=1,2 versus α2 for Ω=10Hz, α1=0.25, and τ2=30ms: (a) absorber and (b) primary system.
(a)(b)Figure 10
Measured time histories of excitation force and system acceleration whenΩ=10Hz (for t<25s and t>125s, α1=α2=0; for 25s≤t≤125s, α1=0.25, α2=−0.455, and τ2=30ms): (a) excitation force; (d) absorber; (g) primary system; (b, c) zoom of (a); (e, f) zoom of (d); (h, i) zoom of (g).
(a)(b)(c)(d)(e)(f)(g)(h)(i)Table 3
Influence ofα2 on the vibration suppression effect for Ω=10Hz, α1=0.25, and τ2=30ms.
ΩHzτ2msα1α2H2m/s2/ND%10—000.166010300.25−0.4550.035−78.9210300.25−0.0650.126−24.10Figure 11
Numerical simulations of the time histories of system responses forΩ=10Hz and f=1N: (a) absorber and (b) primary system.
(a)(b)


#### 3.2.2. Effect ofτ2

The variations ofHii=1,2 versus τ2 for Ω=10.25Hz and α1=α2=0.5 are shown in Figure 12. It is clear that the values of Hii=1,2 fluctuate as τ2 increases. The value of H2 is maximal for τ2=36ms and minimal for τ2=84ms. Figure 13 shows the measured time histories of excitation force and system acceleration for Ω=10.25Hz.Figure 12
Hii=1,2 versus τ2 for Ω=10.25Hz and α1=α2=0.5: (a) absorber and (b) primary system.
(a)(b)Figure 13
Time history of excitation force and system responses whenΩ=10.25Hz (for t<11s and t>110s, α1=α2=0; for 11s≤t≤110s, α1=α2=0.5 and τ2=96ms): (a) excitation force; (d) absorber; (g) primary system; (b, c) zoom of (a); (e, f) zoom of (d); (h, i) zoom of (g).
(a)(b)(c)(d)(e)(f)(g)(h)(i)Table4 shows the influence of τ2 on the vibration suppression effect for Ω=10.25Hz and α1=α2=0.5. The value of H2 decreases 69.33% for τ2=84ms. However, it is undesirable that the time-delayed absorber greatly intensifies the vibration of the primary system for τ2=36ms.Table 4
Influence ofτ2 on the vibration suppression effect for Ω=10.25Hz and α1=α2=0.5.
ΩHzτ2msα1α2H2m/s2/ND%10.25—000.538010.25360.50.53.272508.1810.25840.50.50.165−69.33Figure14 shows the numerical simulations of system responses for Ω=10.25Hz and f=1N, which coincide with the experimental and the theoretical results shown in Figure 12.Figure 14
Numerical simulations of the time histories of system responses forΩ=10.25Hz and f=1N: (a) absorber and (b) primary system.
(a)(b)


## 3.1. Experimental Setup

The photo of the experimental setup is shown in Figure2. The absorber mass (1) is attached to the primary mass (2) by five sheets of steel. A servo motor (3) is fixed on the primary mass. The primary mass is connected to the base (5) by four sheets of steel. The controlled steel sheet (4) acts as the actuator and exerts the control force, whose lower end is linked to the shaft of the servo motor by a wire rope. A shaker (6) provides a horizontal exciting force to the primary mass.Figure 2
Photo of experiment setup: (a) front view; (b) right view (1: absorber mass, 2: primary mass, 3: servo motor, 4: controlled steel sheet, 5: base, and 6: shaker).
(a)(b)Figure3 illustrates the schematic of time-delayed feedback control. A shaker (5) provides a sinusoidal excitation force to the primary mass (2). The amplitude and frequency of the excitation force can be set in M + p vib-pilot. A force sensor (6) and two acceleration sensors (7) and (8) are used to monitor the excitation force and the responses of the absorber and the primary mass, respectively. The time-delayed feedback control loop is described as follows:Step 1: the acceleration signals of the absorber mass (1) and the primary mass enter into signal conditioning instrument, in which the functions of signal amplification and low-pass filtering are achieved to improve the signal-to-noise ratio.Step 2: the processed signals go into the voltage lifting device, where the voltage of the input signal is raised 5 volts.Step 3: the raised voltage signal enters into Trio motion controller, where control commands are written. The values of feedback control feedback gains and intentional time delay can be adjusted in the control commands.Step 4: the control commands are transferred into the servo controller, which guides the shaft rotation of the servo motor (3).Step 5: driven by the rotation of the servo motor shaft, the lower end of the controlled steel sheet (4) realizes the horizontal reciprocating motion and applies the time-delayed feedback control force.Figure 3
Schematic of time-delayed feedback control (1: absorber mass, 2: primary mass, 3: servo motor, 4: controlled steel sheet, 5: shaker, 6: force sensor, and 7 and 8: acceleration sensors).As a preliminary, the values of the physical parameters of the 2-dof coupled system need to be identified when the time-delayed feedback control is absent (i.e.,α1=α2=0 in equations (4) and (5)). Appling a sine swept excitation to the primary mass, the acceleration transfer function curves of the coupled system are obtained. Using the least-squares method, the values of the physical parameters are identified, as shown in Table 1. In addition, the value of inherent time delay τ1 is identified to be 63 ms [22].Table 1
Physical parameters of the coupled system.
m1kgm2kgk1N/mk2N/mkcN/mc1Ns/mc2Ns/m0.6676.82431.7213502.4198.960.171Figure4 shows the comparison of the experimental and the theoretical results of the acceleration transfer functions, where Ω=ω/2π denotes the excitation frequency. It is seen that the passive absorber is most effective when Ω=9.63Hz. When the excitation frequency is disturbed and deviates from 9.63 Hz, vibration control performance of the passive absorber deteriorates. In this case, the time-delayed feedback control is introduced to transform the passive absorber into a time-delayed one. Proper values of intentional time delay and feedback gain coefficients are adapted to improve the vibration suppression effect of the absorber.Figure 4
Comparison of the experimental and theoretical results ofHii=1,2: (a) absorber and (b) primary system.
(a)(b)


## 3.2. Experimental Results

In this subsection, the effects ofα2 and τ2 on the vibration suppression effect of the time-delayed absorber are, respectively, discussed when Ω=9.75Hz and 10 Hz.Figure5 shows the stability charts of the coupled system for τ2=50ms, τ2=30ms, and α2=0.5, respectively. In Figures 5(a) and (b), the coupled system is stable in region II, while it is unstable in regions I and III. In Figure 5(c), the coupled system is stable in region I, while it is unstable in regions II-VI.Figure 5
Stability charts of the coupled system. (a)τ2=50ms. (b) τ2=30ms. (c) α2=0.5.
(a)(b)(c)


### 3.2.1. Effect ofα2

Case 1.
Ω=9.75Hz, α1=−0.4, and τ2=50ms.
Figure6 shows how the acceleration transfer functions of the system Hii=1,2 change as a function of α2 for Ω=9.75Hz, α1=−0.4, and τ2=50ms. Obviously, the experimental results agree well with the theoretical ones. Figure 6(a) indicates that the value of H1 monotonically increases with the increase of α2. As shown in Figure 6(b), the value of the H2 decreases firstly and then increases as α2 increases. The minimum of H2 occurs for α2=−0.33.
Figure7 shows the measured time histories of excitation force and system accelerations. The total settling time is 144 s. The time-delayed feedback control is activated at t=29s. After a short transient, the amplitude of the excitation force and system acceleration responses reaches fixed values. It is found that although the amplitude of the excitation force increases, the acceleration amplitude of the primary system decreases sharply. The time-delayed feedback control is deactivated at t=113s and the coupled system returns to the initial uncontrolled state.
Table2 shows the effect of α2 on the vibration suppression effect for Ω=9.75Hz, α1=−0.4, and τ2=50ms. When the passive absorber works, the value of H2 is 0.037m/s2/N. When the time-delayed feedback control is activated, the value of H2 decreases 45.95% for α1=−0.4 and α2=−0.33 and increases 159.46% for α1=−0.4 and α2=−0.055. It is drawn from Table 2 and Figure 6 that the vibration suppression effect of the time-delayed absorber is superior to that of the passive one for α2∈−0.495,−0.275.
Hereinafter,D=H2α1≠0,α2≠0−H2α1=α2=0/H2α1=α2=0×100%.
To verify the experimental and theoretical results, numerical results are obtained by numerical integration of equations (4) and (5). Figure 8 shows the numerical simulations of system responses for Ω=9.75Hz and f=1N. From the comparison between Figures 6 and 8, the numerical results agree well with experimental and theoretical ones.Figure 6
Hii=1,2 versus α2 for Ω=9.75Hz, α1=−0.4, and τ2=50ms: (a) absorber and (b) primary system.
(a)(b)Figure 7
Measured time histories of excitation force and system accelerations whenΩ=9.75Hz (for t<29s and t>113s, α1=α2=0; for 29s≤t≤113s, α1=−0.4, α2=−0.33, and τ2=50ms): (a) excitation force; (d) absorber; (g) primary system; (b, c) zoom of (a); (e, f) zoom of (d); (h, i) zoom of (g).
(a)(b)(c)(d)(e)(f)(g)(h)(i)Table 2
Effect ofα2 on the vibration suppression effect for Ω=9.75Hz, α1=−0.4, and τ2=50ms.
ΩHzτ2msα1α2H2m/s2/ND%9.75—000.03709.7550−0.4−0.330.020−45.959.7550−0.4−0.0550.096159.46Figure 8
Numerical simulations of system responses forΩ=9.75Hz and f=1N: (a) absorber and (b) primary system.
(a)(b)Case 2.
Ω=10Hz, α1=0.25, and τ2=30ms.
The variations ofHii=1,2 versus α2 for Ω=10Hz, α1=0.25, and τ2=30ms are shown in Figure 9. It can be concluded from the experimental results that the value of H2 is maximal for α2=−0.065 and minimal for α2=−0.455.
Figure10 shows the measured time histories of excitation force and system accelerations before and after the time-delayed feedback control. It is seen that after the feedback control is exerted, the amplitude of the excitation force increases 46.3%, whereas the acceleration amplitudes of the absorber and the primary system, respectively, decrease 4.2% and 72.9%.
Table3 shows the values of H2 under different values of feedback control parameters. It is attractive that the value of H2 decreases 78.92% for α1=0.25 and α2=−0.455.
Figure11 shows the numerical simulations of system acceleration responses for Ω=10Hz and f=1N, which are in agreement with the results shown in Figure 9 and Table 3.Figure 9
Hii=1,2 versus α2 for Ω=10Hz, α1=0.25, and τ2=30ms: (a) absorber and (b) primary system.
(a)(b)Figure 10
Measured time histories of excitation force and system acceleration whenΩ=10Hz (for t<25s and t>125s, α1=α2=0; for 25s≤t≤125s, α1=0.25, α2=−0.455, and τ2=30ms): (a) excitation force; (d) absorber; (g) primary system; (b, c) zoom of (a); (e, f) zoom of (d); (h, i) zoom of (g).
(a)(b)(c)(d)(e)(f)(g)(h)(i)Table 3
Influence ofα2 on the vibration suppression effect for Ω=10Hz, α1=0.25, and τ2=30ms.
ΩHzτ2msα1α2H2m/s2/ND%10—000.166010300.25−0.4550.035−78.9210300.25−0.0650.126−24.10Figure 11
Numerical simulations of the time histories of system responses forΩ=10Hz and f=1N: (a) absorber and (b) primary system.
(a)(b)


### 3.2.2. Effect ofτ2

The variations ofHii=1,2 versus τ2 for Ω=10.25Hz and α1=α2=0.5 are shown in Figure 12. It is clear that the values of Hii=1,2 fluctuate as τ2 increases. The value of H2 is maximal for τ2=36ms and minimal for τ2=84ms. Figure 13 shows the measured time histories of excitation force and system acceleration for Ω=10.25Hz.Figure 12
Hii=1,2 versus τ2 for Ω=10.25Hz and α1=α2=0.5: (a) absorber and (b) primary system.
(a)(b)Figure 13
Time history of excitation force and system responses whenΩ=10.25Hz (for t<11s and t>110s, α1=α2=0; for 11s≤t≤110s, α1=α2=0.5 and τ2=96ms): (a) excitation force; (d) absorber; (g) primary system; (b, c) zoom of (a); (e, f) zoom of (d); (h, i) zoom of (g).
(a)(b)(c)(d)(e)(f)(g)(h)(i)Table4 shows the influence of τ2 on the vibration suppression effect for Ω=10.25Hz and α1=α2=0.5. The value of H2 decreases 69.33% for τ2=84ms. However, it is undesirable that the time-delayed absorber greatly intensifies the vibration of the primary system for τ2=36ms.Table 4
Influence ofτ2 on the vibration suppression effect for Ω=10.25Hz and α1=α2=0.5.
ΩHzτ2msα1α2H2m/s2/ND%10.25—000.538010.25360.50.53.272508.1810.25840.50.50.165−69.33Figure14 shows the numerical simulations of system responses for Ω=10.25Hz and f=1N, which coincide with the experimental and the theoretical results shown in Figure 12.Figure 14
Numerical simulations of the time histories of system responses forΩ=10.25Hz and f=1N: (a) absorber and (b) primary system.
(a)(b)


## 3.2.1. Effect ofα2

Case 1.
Ω=9.75Hz, α1=−0.4, and τ2=50ms.
Figure6 shows how the acceleration transfer functions of the system Hii=1,2 change as a function of α2 for Ω=9.75Hz, α1=−0.4, and τ2=50ms. Obviously, the experimental results agree well with the theoretical ones. Figure 6(a) indicates that the value of H1 monotonically increases with the increase of α2. As shown in Figure 6(b), the value of the H2 decreases firstly and then increases as α2 increases. The minimum of H2 occurs for α2=−0.33.
Figure7 shows the measured time histories of excitation force and system accelerations. The total settling time is 144 s. The time-delayed feedback control is activated at t=29s. After a short transient, the amplitude of the excitation force and system acceleration responses reaches fixed values. It is found that although the amplitude of the excitation force increases, the acceleration amplitude of the primary system decreases sharply. The time-delayed feedback control is deactivated at t=113s and the coupled system returns to the initial uncontrolled state.
Table2 shows the effect of α2 on the vibration suppression effect for Ω=9.75Hz, α1=−0.4, and τ2=50ms. When the passive absorber works, the value of H2 is 0.037m/s2/N. When the time-delayed feedback control is activated, the value of H2 decreases 45.95% for α1=−0.4 and α2=−0.33 and increases 159.46% for α1=−0.4 and α2=−0.055. It is drawn from Table 2 and Figure 6 that the vibration suppression effect of the time-delayed absorber is superior to that of the passive one for α2∈−0.495,−0.275.
Hereinafter,D=H2α1≠0,α2≠0−H2α1=α2=0/H2α1=α2=0×100%.
To verify the experimental and theoretical results, numerical results are obtained by numerical integration of equations (4) and (5). Figure 8 shows the numerical simulations of system responses for Ω=9.75Hz and f=1N. From the comparison between Figures 6 and 8, the numerical results agree well with experimental and theoretical ones.Figure 6
Hii=1,2 versus α2 for Ω=9.75Hz, α1=−0.4, and τ2=50ms: (a) absorber and (b) primary system.
(a)(b)Figure 7
Measured time histories of excitation force and system accelerations whenΩ=9.75Hz (for t<29s and t>113s, α1=α2=0; for 29s≤t≤113s, α1=−0.4, α2=−0.33, and τ2=50ms): (a) excitation force; (d) absorber; (g) primary system; (b, c) zoom of (a); (e, f) zoom of (d); (h, i) zoom of (g).
(a)(b)(c)(d)(e)(f)(g)(h)(i)Table 2
Effect ofα2 on the vibration suppression effect for Ω=9.75Hz, α1=−0.4, and τ2=50ms.
ΩHzτ2msα1α2H2m/s2/ND%9.75—000.03709.7550−0.4−0.330.020−45.959.7550−0.4−0.0550.096159.46Figure 8
Numerical simulations of system responses forΩ=9.75Hz and f=1N: (a) absorber and (b) primary system.
(a)(b)Case 2.
Ω=10Hz, α1=0.25, and τ2=30ms.
The variations ofHii=1,2 versus α2 for Ω=10Hz, α1=0.25, and τ2=30ms are shown in Figure 9. It can be concluded from the experimental results that the value of H2 is maximal for α2=−0.065 and minimal for α2=−0.455.
Figure10 shows the measured time histories of excitation force and system accelerations before and after the time-delayed feedback control. It is seen that after the feedback control is exerted, the amplitude of the excitation force increases 46.3%, whereas the acceleration amplitudes of the absorber and the primary system, respectively, decrease 4.2% and 72.9%.
Table3 shows the values of H2 under different values of feedback control parameters. It is attractive that the value of H2 decreases 78.92% for α1=0.25 and α2=−0.455.
Figure11 shows the numerical simulations of system acceleration responses for Ω=10Hz and f=1N, which are in agreement with the results shown in Figure 9 and Table 3.Figure 9
Hii=1,2 versus α2 for Ω=10Hz, α1=0.25, and τ2=30ms: (a) absorber and (b) primary system.
(a)(b)Figure 10
Measured time histories of excitation force and system acceleration whenΩ=10Hz (for t<25s and t>125s, α1=α2=0; for 25s≤t≤125s, α1=0.25, α2=−0.455, and τ2=30ms): (a) excitation force; (d) absorber; (g) primary system; (b, c) zoom of (a); (e, f) zoom of (d); (h, i) zoom of (g).
(a)(b)(c)(d)(e)(f)(g)(h)(i)Table 3
Influence ofα2 on the vibration suppression effect for Ω=10Hz, α1=0.25, and τ2=30ms.
ΩHzτ2msα1α2H2m/s2/ND%10—000.166010300.25−0.4550.035−78.9210300.25−0.0650.126−24.10Figure 11
Numerical simulations of the time histories of system responses forΩ=10Hz and f=1N: (a) absorber and (b) primary system.
(a)(b)


## 3.2.2. Effect ofτ2

The variations ofHii=1,2 versus τ2 for Ω=10.25Hz and α1=α2=0.5 are shown in Figure 12. It is clear that the values of Hii=1,2 fluctuate as τ2 increases. The value of H2 is maximal for τ2=36ms and minimal for τ2=84ms. Figure 13 shows the measured time histories of excitation force and system acceleration for Ω=10.25Hz.Figure 12
Hii=1,2 versus τ2 for Ω=10.25Hz and α1=α2=0.5: (a) absorber and (b) primary system.
(a)(b)Figure 13
Time history of excitation force and system responses whenΩ=10.25Hz (for t<11s and t>110s, α1=α2=0; for 11s≤t≤110s, α1=α2=0.5 and τ2=96ms): (a) excitation force; (d) absorber; (g) primary system; (b, c) zoom of (a); (e, f) zoom of (d); (h, i) zoom of (g).
(a)(b)(c)(d)(e)(f)(g)(h)(i)Table4 shows the influence of τ2 on the vibration suppression effect for Ω=10.25Hz and α1=α2=0.5. The value of H2 decreases 69.33% for τ2=84ms. However, it is undesirable that the time-delayed absorber greatly intensifies the vibration of the primary system for τ2=36ms.Table 4
Influence ofτ2 on the vibration suppression effect for Ω=10.25Hz and α1=α2=0.5.
ΩHzτ2msα1α2H2m/s2/ND%10.25—000.538010.25360.50.53.272508.1810.25840.50.50.165−69.33Figure14 shows the numerical simulations of system responses for Ω=10.25Hz and f=1N, which coincide with the experimental and the theoretical results shown in Figure 12.Figure 14
Numerical simulations of the time histories of system responses forΩ=10.25Hz and f=1N: (a) absorber and (b) primary system.
(a)(b)


## 4. Conclusions

An active vibration suppression via a time-delayed absorber is presented. Case studies are provided to demonstrate the effects of feedback gain coefficient and intentional time delay on the vibration suppression performance of the time-delayed absorber. The following points are concluded:(1)
From the viewpoint of vibration suppression, the time-delayed absorber has advantages and disadvantages over the passive one, which depend on the values of the feedback gain coefficient and intentional time delay. The time-delayed absorber with proper choices of the two parameters decreases the values of acceleration transfer function of the primary system by 45.95%, 78.92%, and 69.33% forΩ=9.75Hz, 10 Hz, and 10.25 Hz, respectively. However, the time-delayed absorber fails when the value of the two parameters is improperly selected, which leads to the sharp vibration of the primary system.(2)
When the values of feedback gain coefficients are fixed, the value of intentional time delay determines the vibration suppression effect of the time-delayed absorber. It acts as a double-edged sword. Reasonable values of intentional time delay effectively improve the vibration suppression effect without changing the mass or stiffness of the absorber. However, unreasonable values of intentional time delay greatly intensify the vibration of the primary system. This situation should be avoided in practical engineering application.



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