#### 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.

#### 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 Section 2. 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

Figure 1 illustrates the mechanical model of the 2-dof coupled system [23]. The coupled system consists of a time-delayed absorber and a primary system. and represent the mass of the absorber and the primary system, respectively. The motions of the absorber and the primary system are denoted by and , respectively. and represent the linear stiffness coefficients. and represent the viscous damping coefficients. An actuator mounted between the primary mass and the absorber mass provides the time-delayed feedback control force.

Assuming that simply harmonic excitation is applied to the primary mass, the governing equations of the coupled system are given bywhere is the time-delayed feedback control force, is the stiffness coefficient of the actuator, and is the control signal with the following form:where and , respectively, represent the inherent and the intentional time delays in the feedback control loop. The value of is constant, while the value of is tunable in the controller. Instead of eliminating, compensating for, and even ignoring the presence of , it is thought that together with leads to the final control effect. and represent amplification factors of the control signal. The time-delayed feedback disappears and the time-delayed absorber degrades into the passive one when .

Substituting equation (3) into equations (1) and (2) giveswhere and represent feedback gain coefficients.

The external excitation can be rewritten as

The solutions of equations (4) and (5) are considered to bewhere and are complex.

Substituting equations (6) and (7) into equations (4) and (5), one getswhere

To facilitate the following analysis, new variables and are defined as follows:where and represent the acceleration transfer functions of the absorber and the primary system, respectively.

It is seen from equation (10) that and 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 under different values of , , and .

##### 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) becomewhere

The characteristic equation of the coupled system is det (**A**) = 0; that is,where

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

Substituting equation (15) into equation (13) and separating the real and the imaginary parts, one obtains

Using , one haswhere the expressions of are given in Appendix.

Since equation (18) is a high-order equation with transcendental terms, the values of are obtained by numerical calculation. For fixed values of and certain values of feedback gain coefficients and , *N* is assigned to denote the number of the positive real roots of equation (18). When , there is no stability switch. In other words, the stability of the system remains the same for all . When , we have . For each , infinite values are determined by solving equations (16) and (17). The transition direction of the roots at as increases from to is decided by the expression:and and values of RD represent destabilizing and stabilizing transitions, respectively.

Through the above computation, the stable and unstable ranges of for and are divided. Performing the same procedure for other values of and , the stable and unstable regions in the space are plotted.

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

##### 3.1. Experimental Setup

The photo of the experimental setup is shown in Figure 2. 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.

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Figure 3 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.

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., 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 is identified to be 63 ms [22].

Figure 4 shows the comparison of the experimental and the theoretical results of the acceleration transfer functions, where denotes the excitation frequency. It is seen that the passive absorber is most effective when . 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.

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##### 3.2. Experimental Results

In this subsection, the effects of and on the vibration suppression effect of the time-delayed absorber are, respectively, discussed when and 10 Hz.

Figure 5 shows the stability charts of the coupled system for , , and , 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.

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###### 3.2.1. Effect of

*Case 1. *, , and .

Figure 6 shows how the acceleration transfer functions of the system change as a function of for , , and . Obviously, the experimental results agree well with the theoretical ones. Figure 6(a) indicates that the value of monotonically increases with the increase of . As shown in Figure 6(b), the value of the decreases firstly and then increases as increases. The minimum of occurs for .

Figure 7 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 . 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 and the coupled system returns to the initial uncontrolled state.

Table 2 shows the effect of on the vibration suppression effect for , , and . When the passive absorber works, the value of is . When the time-delayed feedback control is activated, the value of decreases 45.95% for and and increases 159.46% for and . 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 .

Hereinafter, .

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 and . From the comparison between Figures 6 and 8, the numerical results agree well with experimental and theoretical ones.

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*Case 2. *, , and .

The variations of versus for , , and are shown in Figure 9. It can be concluded from the experimental results that the value of is maximal for and minimal for .

Figure 10 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%.

Table 3 shows the values of under different values of feedback control parameters. It is attractive that the value of decreases 78.92% for and .

Figure 11 shows the numerical simulations of system acceleration responses for and , which are in agreement with the results shown in Figure 9 and Table 3.

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###### 3.2.2. Effect of

The variations of versus for and are shown in Figure 12. It is clear that the values of fluctuate as increases. The value of is maximal for and minimal for . Figure 13 shows the measured time histories of excitation force and system acceleration for .

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Table 4 shows the influence of on the vibration suppression effect for and . The value of decreases 69.33% for . However, it is undesirable that the time-delayed absorber greatly intensifies the vibration of the primary system for .

Figure 14 shows the numerical simulations of system responses for and , which coincide with the experimental and the theoretical results shown in Figure 12.

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#### 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% for , 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.

#### Appendix

One has

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Disclosure

This work was based on the manuscript presented in the 9th European Nonlinear Dynamics Conference.

#### Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant no. 11602135.