Beneficial performance of a quasi-zero-stiffness vibration isolator with displacement-velocity feedback control

A displacement-velocity feedback control method is proposed to enhance the isolation performance of a quasi-zero-stiffness vibration isolator (QZS-VI). Time delay is considered in the controlled QZS-VI system. First, the steady-state solutions are obtained using the averaging method and validated by a numerical method. The jump phenomenon and frequency island phenomenon can occur, and a stability analysis is implemented. Then, the effects of the time delay and feedback gain on the frequency response and stability of solutions are analyzed in detail. Then, the force transmissibility is defined to evaluate the isolation performance of the controlled QZS-VI system. The results show that the time delay mainly affects the stability of the controlled system and weakly influences the isolation performance. The proposed displacement-velocity feedback control method can effectively suppress the vibration in the resonant region without affecting the performance in the isolation region. Finally, the vibration control effect is illustrated by the concept of an equivalent damping ratio.


Introduction
Prolonged vibration and impact may make the human body uncomfortable and tired in vehicle engineering. Severe vibration can even cause structural damage in mechanical and architectural engineering. Thus, vibration control is important in engineering. Linear passive vibration isolators are widely applied in engineering because of their high cost performance. However, the linear vibration isolator can only attenuate vibration when the external excitation frequency is greater than ffiffi ffi 2 p times the natural frequency of a vibration system [1]. This shortcoming causes a dilemma: reducing the natural frequency of the vibration system decreases the bearing capacity. As a result, linear vibration isolators are not suitable for low-frequency vibration isolation. Therefore, quasizero-stiffness vibration isolators (QZS-VIs) have become a research hotspot because they have both high static and low dynamic stiffness [2]. The high static stiffness makes QZS-VI have a high bearing capacity, while the low dynamic stiffness makes QZS-VI possess a low natural frequency.
Various QZS-VIs have been developed by combining a linear vibration isolator with negative stiffness mechanisms in parallel [3]. Most of the negative stiffness mechanisms are based on geometric nonlinearity, such as inclined springs [4], connecting rods [5], and scissor-like structures [6]. Recently, Zhou et al. [7] developed an asymmetric QZS-VI using a cantilever plate spring and L-shaped levers as the negative stiffness mechanism. A long stroke and a high bearing capacity were achieved for this QZS-VI. Ye and Ji [8] developed a QZS-VI using a truss-springbased stack Miura-ori structure as the negative stiffness mechanism. Liu et al. [9] introduced a tunable nonlinear inerter into a QZS-VI to improve its vibration isolation performance. Zeng et al. [10] proposed a limb-inspired bionic structure to achieve a negative stiffness mechanism and developed a novel QZS-VI. Theoretical and experimental studies have proven that increasing the excitation amplitude can weaken the isolation performance of the QZS-VI because of inherent stiffness nonlinearity, which increases both peak amplitude and resonant frequency [11,12]. An unbounded response can occur if a large base excitation is applied to a QZS-VI, which is undesirable in engineering [13]. To attenuate excessive vibration in the resonant region, heavy damping can be applied. However, increasing the linear damping can inevitably weaken the isolation performance in the isolation region [14].
To improve the isolation performance of QZS-VI, various active control methods have been investigated by researchers. Sun et al. [15] introduced a timedelayed displacement feedback control method into a QZS-VI system. The displacement feedback control can effectively improve the transmissibility performance by designing an appropriate time delay. Then, this method was applied to a multi-directional QZS-VI system [16]. Although displacement feedback control does not require a high response speed for the control system, the transmissibility performance outside the resonant region deteriorates. A dual feedback control method that includes displacement and velocity feedback control was introduced into QZS-VI systems [17,18]. Its vibration control mechanism has been investigated by Hu et al. [19] from the perspective of equivalent damping. Although the dual feedback control method can effectively attenuate resonant vibration, the transmissibility performance in the isolation region remains affected. To keep the transmissibility performance in the isolation region unaffected, Gao and Chen [20] proposed a time-delayed cubic velocity feedback control method, which was further applied in a high-static-low-dynamic-stiffness (HSLDS) vibration isolator [21]. However, this control method can only achieve the best isolation performance when there is no time delay, which requires a high response speed for the control system. Cheng et al. [22] proposed a time-delayed cubic displacement feedback control method to improve the isolation performance of an HSLDS vibration isolator. When an appropriate time delay is designed, this feedback control can achieve the best isolation performance and hardly affects the transmissibility performance in the isolation region. These feedback control methods are sensitive to the time delay, which makes the isolation performance in the resonant region unstable.
This paper aims to present a feedback control method that can effectively attenuate vibration in the resonant region and has little effect on the transmissibility performance in the isolation region. Furthermore, this feedback control method should be as insensitive to the time delay as possible within a certain time delay range. To achieve these objectives, a displacement-velocity feedback control method using displacement and velocity signals is proposed in this paper. The averaging method is used to evaluate the isolation performance of the controlled system from the viewpoint of dynamics.
The remainder of this paper is organized as follows. The restoring force of the controlled QZS-VI system is obtained in Sect. 2. In Sect. 3, steady-state solutions of the controlled QZS-VI system are derived, and their stability is analyzed. Section 4 shows the effects of feedback parameters on the frequency response and stability of solutions. The vibration isolation performance of the controlled system is analyzed in Sect. 5. Section 6 illustrates the vibration control effect using the concept of an equivalent damping ratio. Conclusions are drawn in Sect. 7.

Modeling of a controlled QZS-VI system
A QZS-VI system with displacement-velocity feedback control is shown in Fig. 1. This QZS-VI is mainly composed of two identical vertical springs, a horizontal spring, and a scissor-like structure, which has been reported in our previous work [23]. The scissor-like structure is composed of four inner connecting rods and four outer connecting rods. In this QZS-VI, the horizontal spring and scissor-like structure construct a negative stiffness mechanism. It is assumed that the stiffness of a single vertical spring is K v =2, the stiffness of the horizontal spring is K h , and the length of all connecting rods is L. When subjected to a rated load m, the QZS-VI is at a static equilibrium position, where all connecting rods are in a plane. At the static equilibrium position, the stretch length of the horizontal spring is d, and the static deflection of the vertical springs is Dx. A linear damper is considered, and its damping coefficient is c. The control system is mainly composed of a controller, an actuator, and sensors. Sensors are used to collect displacement and velocity signals of the load. The collected signals are input into the controller to calculate the desired control force signal. The actuator outputs the control force to the load according to the control force signal.
For a QZS-VI carrying a load m, when the loading support is subjected to a static force f, it will deviate from the static equilibrium position by a displacement x. The restoring force has been given by [23], which yields For simplicity, the above restoring force is transformed into the following nondimensional form: where The restoring force of the QZS-VI is given by where d qzs is the nondimensional stretch length of the horizontal spring when the quasi-zero-stiffness condition is satisfied.
To simplify the subsequent dynamic analysis, Eq. (3) is approximated by a third-order Taylor series, which yields where Figure 2 shows the comparison between the exact restoring force and its approximation. The approximate restoring force is consistent with the exact force over most of the selected displacement range, which shows that the approximation is feasible.
Time delay is considered in the feedback control, since it is inevitable in the process from signal acquisition to output of the control force. Specifically, there is time delay in the collection and transmission of sensor signals, calculation of the controller, and response of the actuator. Thus, the control force of the displacement-velocity feedback control can be expressed as where t 0 is the time delay, and G is the feedback gain.

Frequency response
The differential equation of motion of the controlled QZS-VI system subjected to harmonic force excitation is given by where F e and x are the amplitude and circular frequency of the force excitation, respectively. Replacing the exact restoring force with its approximation and transforming Eq. (6) into a nondimensional form can yield The averaging method [24] is employed to obtain the analytical solutions. Thus, X 2 ¼ 1 þ er is introduced, where e and r denote a perturbation parameter and a detuning parameter, respectively. Then, Eq. (7) can be written in the following form: The steady-state solutions of Eq. (8) are expressed as where A and h are the amplitude and phase of a solution, respectively, and they are slowly varying functions related to T. By applying the averaging method, the following equations that govern A and h are obtained: where u ¼ XT þ h. Since A and h are slowly varying functions, averaging the right sides of Eqs. (12) and (13) over 0; 2p ð Þcan yield The following equations can be obtained by completing the integral calculation and eliminating the perturbation parameter e The steady-state solutions can be obtained by Combining Eqs. (18) and (19) can obtain the following amplitude-frequency equation: To validate the effectiveness of the averaging method, a numerical method based on the Runge-Kutta algorithm is used to obtain the first-order harmonic amplitude, and the results are shown in Fig. 3. The analytical solutions of the averaging method are consistent with the numerical solutions. The analytical solutions that cannot be simulated by the numerical method are unstable. Figure 3b shows that the amplitude-frequency curve is separated into two parts when s ¼ p, and a closed loop, which is also called a frequency island, is above the main curve. Since this closed loop is unstable, there is no numerical solution. Similarly, the amplitude-frequency curve confined by two dotted lines is also unstable, as shown in Fig. 3c. The stability of solutions is analyzed in the subsequent subsection.

Stability analysis
The perturbation method is used for stability analysis. It is assumed that the steady-state solution has a small disturbance. Thus, A and h can be disturbed as where DA and Dh are small disturbances. Substituting Eq. (21) into Eqs. (16) and (17) and using sinDh % Dh and cosDh % 1 yield the following linearized equations: where The characteristic equation of the coefficient matrix of Eqs. (22) and (23) is According to Routh-Hurwitz stability criterion [25], the steady-state solution is asymptotically stable only when both inequalities below are satisfied: The steady-state solution becomes unstable as long as one of the conditions is not satisfied. C 1 ¼ 0 and C 2 ¼ 0 can obtain two stability boundaries. Specifically, C 1 ¼ 0 indicates that the real parts of the eigenvalues are zero. As a result, Hopf bifurcation phenomenon occurs at the intersections of the amplitude-frequency curve and stability boundary. C 2 ¼ 0 indicates that the imaginary parts of the eigenvalues are zero. Thus, the saddle-node bifurcation phenomenon occurs at the intersections of the amplitude-frequency curve and stability boundary, so the amplitude jump phenomenon occurs. Figure 4 shows the stability boundaries of the controlled QZS-VI system with different time delays. The region within boundary C 1 ¼ 0 is the unstable region R 1 , and the region within boundary C 2 ¼ 0 is the unstable region R 2 . When s ¼ p=8, only the unstable region R 2 appears in the selected frequency range, as shown in Fig. 4a. The amplitude-frequency curve is partially covered by R 2 , which indicates that the steady-state solutions covered by R 2 are unstable. When s increases to p, a closed loop appears above the main amplitude-frequency curve, and an unstable region R 1 also appears, as shown in Fig. 4b. The closed loop is partially covered by R 2 and completely covered by R 1 . Therefore, this closed loop is unstable. When s increases to p ¼ 5p=4, the closed loop is partially covered by unstable regions R 1 and R 2 , as shown in Fig. 4c. Thus, only part of the closed loop is stable. When s increases to p ¼ 3p=2, the amplitude-frequency curve is partially covered by regions R 1 and R 2 , as shown in Fig. 4d. Thus, there are no corresponding numerical solutions for the amplitudefrequency curve confined by the two dotted lines in Fig. 3c. Figure 5 shows the effects of time delay on the frequency response. When there is no time delay, the amplitude of the controlled QZS-VI system in the resonant region is suppressed effectively compared with the uncontrolled one, as shown in Fig. 5a. When the time delay increases from 0 to p=2, the amplitudefrequency curve of the controlled QZS-VI system hardly changes. When the time delay increases to 3p=4, the frequency island phenomenon occurs, which indicates that a closed loop appears above the main amplitude-frequency curve, as shown in Fig. 5d. When the time delay increases from 3p=4 to 5p=4, the closed loop gradually approaches the main amplitude-frequency curve and becomes larger. When the time delay increases to 3p=2, the closed loop is fused with the main amplitude-frequency curve and disappears, as shown in Fig. 5g. The peak amplitude and resonant frequency of the controlled QZS-VI system greatly increase compared with those of the uncontrolled system. Interestingly, when the time delay increases to 2p, the closed loop emerges again, as shown in Fig. 5h. This loop is at a higher position than the first closed loop.

Effects of time delay
When the time delay is greater than p=2, the frequency response in the resonant region increases because time delay can cause extra phase delay. In addition, increasing the excitation frequency increases the phase delay. Thus, the feedback control force can be in the opposite direction to the external excitation force because of phase delay, which results in a poor control effect.
To understand the specific effects of time delay on the peak amplitude and resonant frequency of the controlled system, differentiating Eq. (20) with respect to X yields Letting dA=dX ¼ 0, we obtain the following condition related to the extreme point: Solving the combined Eqs. (20) and (36) can yield the resonant frequency and peak amplitude of the main amplitude-frequency curve, and the results are shown in Fig. 6. When the time delay increases within the range of [0, 2], the resonant frequency and peak amplitude slowly increase. When the time delay further increases, the resonant frequency and peak amplitude rapidly increase. When the time delay increases to 3.94, there is a jump in the resonant frequency and peak amplitude because the closed loop is fused with the main amplitude-frequency curve. When the time delay is greater than 3.94, the resonant frequency and peak amplitude first decrease and subsequently increase with increasing time delay. Figure 7 shows the effects of time delay on the response amplitude of X ¼ 0:3. When g ¼ À0:1, there are always three steady-state solutions, as shown in Fig. 7a. The middle solutions are always covered by the unstable region R 2 , and some of the largest solutions are covered by unstable region R 1 . Therefore, these solutions are unstable. When g ¼ À0:4, there is only one steady-state solution over the range of [0, 3.4], as shown in Fig. 7b. When the time delay is greater than 3.4, three solutions reappear, most of which are covered by R 1 and R 2 . The jump phenomenon occurs at s ¼ 3:4 because of saddle-node bifurcation. When g ¼ À0:7, a closed loop appears in the range of [3.77, 9.14], as shown in Fig. 7c. Most of the solutions in the closed loop are unstable, and the jump phenomenon occurs at s ¼ 3:77 and s ¼ 9:14.

Effects of feedback gain
The amplitude-frequency curves of the controlled QZS-VI system with different feedback gains are shown in Fig. 8. When the feedback gain is 0.1, the peak amplitude and resonant frequency of the controlled QZS-VI system are much larger than those of the uncontrolled system. Thus, only negative feedback gain is considered for the controlled system. Increasing the absolute value of the feedback gain can suppress the vibration amplitude in the resonant region and reduce the resonant frequency. When the feedback gain reaches À0:7, the jump phenomenon disappears; thus, the controlled system has only one steady-state solution at any frequency. The feedback gain has little effect on the response amplitude outside the resonant region. Figure 9 shows the effects of feedback gain on the response amplitude of X ¼ 0:3. When the feedback gain is 0, there are three steady-state solutions, and the middle solution is unstable. Two of the solutions approach each other when the absolute value of the feedback gain increases. The jump phenomenon  occurs when the feedback gain reaches À0:135. When the absolute value of the feedback gain further increases, there is only one steady-state solution, which hardly changes with the feedback gain. The aforementioned analysis shows that selecting an appropriate time delay and a feedback gain can obtain a small and stable solution.

Force transmissibility
Force transmissibility is used to evaluate the isolation performance of the controlled QZS-VI system. The force transmitted to the base is expressed as Thus, the amplitude of the transmitted force is The force transmissibility can be expressed in terms of decibel Figure 10 shows the force transmissibility curves of the controlled QZS-VI system with different time delays. The controlled QZS-VI system exhibits lower force transmissibility in the resonant region than the uncontrolled system when the time delay is less than tan p=2. When the time delay is very high, such as s ¼ 3p=2, the isolation performance worsens. Thus, a small time delay should be selected in the feedback control. Increasing the time delay leads to a small change in force transmissibility in the resonant region over a certain range. Thus, the feedback control effects are not sensitive to the time delay over a certain range, which does not require a high response speed of the controlled system. In addition, the transmissibility outside the resonant region hardly changes with the time delay, which implies that the time delay has little effect on the transmissibility outside the resonant region. Figure 11 shows the force transmissibility curves of the controlled QZS-VI system with different feedback gains. Increasing the absolute value of the feedback gain can suppress the force transmissibility in the resonant region and reduce the resonant frequency. The feedback gain has little effect on the transmissibility outside the resonant region. Specifically, the transmissibility in the high-frequency region hardly changes with the feedback gain. The aforementioned analysis shows that selecting an appropriate time delay and feedback gain can achieve excellent isolation performance.
The isolation performance of the proposed feedback is compared with that of a cubic velocity feedback developed in [20] and [21], as shown in Fig. 12. When the same feedback gain is selected, the proposed feedback can achieve lower peak transmissibility than the cubic velocity feedback.
6 Discussion on the displacement-velocity feedback control The effect of feedback control is equivalent to adding a damping force to the QZS-VI system. To obtain the equivalent damping ratio, Eq. (20) can be rewritten as Thus, the equivalent damping ratio can be expressed as Since only negative displacement-velocity feedback is considered, the maximum equivalent damping ratio is n max is a nonlinear function of vibration amplitude A when n and g are fixed. Figure 13 shows the maximum value of the equivalent damping ratio under different feedback gains. n max nonlinearly increases with increasing vibration amplitude. Thus, a high vibration amplitude leads to heavy damping, while a low amplitude leads to slight damping. n max hardly changes when the vibration amplitude varies at approximately 0, as shown in the pink dotted box. Thus, when the vibration amplitude is very small, increasing the absolute value of the feedback gain hardly changes the equivalent damping ratio, and n max is almost equal to the linear damping ratio n.
Substituting the steady-state solutions into Eq. (41) yields the change in equivalent damping ratio of the controlled QZS-VI system, as shown in Fig. 14. The controlled QZS-VI system exhibits a high equivalent damping ratio in the resonant region and a low Fig. 11 Force transmissibility curves of the controlled QZS-VI system with different feedback gains (d qzs ¼ 1, n ¼ 0:02, f e ¼ 0:02, s ¼ p=4) Fig. 12 Comparison of force transmissibility between displacement-velocity feedback and cubic velocity feedback (d qzs ¼ 1, n ¼ 0:02, f e ¼ 0:02, s ¼ p=4, g ¼ À0:1) Fig. 13 Maximum value of the equivalent damping ratio (n ¼ 0:02) equivalent damping ratio in the isolation region because the controlled QZS-VI system outputs a high response amplitude in the resonant region and a low response amplitude in the isolation region. In addition, the response amplitude trends to 0 with increasing excitation frequency in the isolation region. As a result, the introduction of displacement-velocity feedback control into the QZS-VI system can effectively suppress the vibration in the resonant region without affecting the high-frequency isolation performance.

Conclusions
A time-delayed displacement-velocity feedback control method is proposed to enhance the isolation performance of a QZS-VI system. Both time delay and feedback gain affect the frequency response and stability. When the time delay varies over a certain range, the frequency island phenomenon can occur. Selecting an appropriate time delay and a negative feedback gain can suppress the response amplitude effectively.
The time delay weakly affects the isolation performance, and increasing the feedback gain can enhance the control effect. The proposed displacement-velocity feedback control can suppress the vibration in the resonant region effectively without affecting the performance in the isolation region because the displacement-velocity feedback control offers heavy damping in the resonant region and slight damping in the isolation region. Compared with cubic velocity feedback control, the proposed feedback control can achieve lower peak transmissibility.
The proposed feedback control method can be applied in vehicle engineering and mechanical engineering. In future work, we will investigate and verify the practical control effects, and determine the best parameter combination.