For the two degrees of freedom system with an amplification mechanism, the effect of vibration control using passive vibration reduction methods is extremely limited. To further widen its vibration reduction frequency band, time delay feedback control is considered based on reference [25] . The mechanical model of time delay feedback control is shown in Figure 1 . There is a lever with fixed support between the main system and the dynamic vibration absorber. The main system and the vibration absorber are connected via sliders at both ends of the lever, forming a complete system. Here, $r_1$ and $r_2$ respectively represent the distances from the lever support point $O$ to the fixed points $M$ and $N$ of the sliders. $m_1$, $c_1$, and $k_1$ respectively represent the mass, damping, and stiffness of the main system; $m_2$, $c_2$, and $k_2$ respectively represent the mass, damping, and stiffness of the vibration absorber; $k$ represents the grounded stiffness; $x_1$ and $x_2$ respectively represent the displacements of the main system and the vibration absorber; $\tau ,g_1$ respectively represent the time delay and the feedback gain coefficient; $b$ represents the inerter; $F_0,\omega$ respectively represent the amplitude and frequency of the external excitation. Assume that $L=r_1/r_2$ is the amplification ratio of the system. According to the similarity theorem of triangles, the force acting on the main system is $L$ times that on the vibration absorber, and the displacement is $1/L$ of the vibration absorber.

Give the vibration differential equations of the system according to Newton’s second law:

$\{\begin{array}{l}{m}_1{\ddot{x}}_1+k_{2}L\left(L{x}_1-x_2\right)+c_1{\stackrel{\dot{}}{x}}_1+k_1{x}_1+c_{2}L\left(L{\stackrel{\dot{}}{x}}_1-{\stackrel{\dot{}}{x}}_2\right)\\ -g_{1}L{x}_2\left(t-\tau \right)+bL\left(L{\ddot{x}}_1-{\ddot{x}}_2\right)=F_0\cos \left(\omega t\right)\\ m_2{\ddot{x}}_2-b\left(L{\ddot{x}}_1-{\ddot{x}}_2\right)-c_2\left(L{\stackrel{\dot{}}{x}}_1-{\stackrel{\dot{}}{x}}_2\right)-k_2\left(L{x}_1-x_2\right)+g_1{x}_2\left(t-\tau \right)+k{x}_2=0\end{array}$(1)

In order to simplify the calculation process, the following dimensionless quantities are introduced:

$\begin{array}{l}{\omega }_1=\sqrt{\frac{k_1}{m_1}},{\omega }_2=\sqrt{\frac{k_2}{m_2}},\mu =\frac{m_2}{m_1},{\xi }_1=\frac{c_1}{2m_1{\omega }_1},{\xi }_2=\frac{c_2}{2m_2{\omega }_2},p=\frac{{\omega }_2}{{\omega }_1},\\ \Omega =\frac{\omega }{{\omega }_1},g=\frac{g_1}{k_2},{\tilde{x}}_1=\frac{k_1{x}_1}{F_0},{\tilde{x}}_2=\frac{k_1{x}_2}{F_0},\tilde{t}={\omega }_{1}t,\tilde{\tau }={\omega }_1\tau ,\beta =\frac{b}{m_2},\alpha =\frac{k}{k_2},\end{array}$(2)

For the convenience of writing, the above parameters are expressed as: $x_1={\tilde{x}}_1$, $x_2={\tilde{x}}_2,t=\tilde{t},\tau =\tilde{\tau }$, where ${(\cdot )}^{\prime }=\text{d}(\cdot )/\text{d}t,{(\cdot )}^{\prime \text{}\prime }={\text{d}}^2(\cdot )/\text{d}{t}^2$. Substitute the above dimensionless quantities, and transform Equation (1) into the following dimensionless form:

$\{\begin{array}{l}{x^{″}}_1+\left(1+\mu L^2{p}^2\right)x_1+2\left({\xi }_1+L^2\mu p{\xi }_2\right){x^{\prime }}_1-L\mu p^2{x}_2-2L\mu p{\xi }_2{x^{\prime }}_2\\ +\beta \mu L\left(L{\ddot{x}}_1-{\ddot{x}}_2\right)-L{p}^2\mu g{x}_2\left(t-\tau \right)=\cos \left(\Omega t\right)\\ {x^{″}}_2+\left(1+\alpha \right)p^2{x}_2+2p{\xi }_2{x^{\prime }}_2-L{p}^2{x}_1-2Lp{\xi }_2{x^{\prime }}_1-\beta \left(L{\ddot{x}}_1-{\ddot{x}}_2\right)+g{p}^2{x}_2\left(t-\tau \right)=0\end{array}$(3)

Assume that the solution of (3) is as follows:

$\left[\begin{array}{l}{x}_1\\ x_2\end{array}\right]=\left[\begin{array}{l}{\bar{x}}_1\\ {\bar{x}}_2\end{array}\right]{\text{e}}^{j\Omega t}$(4)

In the formula, $j$ is the imaginary unit. Substitute Equation (4) into Equation (3), and the expression of its analytical solution is obtained as follows:

$\left[\begin{array}{l}{\bar{x}}_1\\ {\bar{x}}_2\end{array}\right]=\frac{1}{\Delta \left(\Omega \right)}\left[\begin{array}{c}-{\Omega }^2+2p{\xi }_{2}j\Omega +\left(1+\alpha \right)p^2+p^{2}g{\text{e}}^{-j\Omega \tau }-\beta {\Omega }^2\\ 2Lp{\xi }_{2}j\Omega +L{p}^2-\beta L{\Omega }^2\end{array}\right]$(5)

$\begin{array}{c}\Delta \left(\Omega \right)=\left(-{\Omega }^2+1+L^2{p}^2\mu +2{\xi }_{1}j\Omega +2L^2\mu p{\xi }_{2}j\Omega -\beta \mu L^2{\Omega }^2\right)\\ \times \left(-{\Omega }^2+2p{\xi }_{2}j\Omega +\left(1+\alpha \right)p^2+p^{2}g{\text{e}}^{-j\Omega \tau }-\beta {\Omega }^2\right)\\ -\left(2L\mu p{\xi }_{2}j\Omega +L{p}^2\mu +L\mu p^{2}g{\text{e}}^{-j\Omega \tau }-\beta \mu L{\Omega }^2\right)\\ \times \left(2Lp{\xi }_{2}j\Omega +L{p}^2-\beta L{\Omega }^2\right)\end{array}$ (6)

Separate the expressions of ${\bar{x}}_1$ and ${\bar{x}}_2$ into real and imaginary parts and find their moduli. Define the expressions of the variables $A_1$ and $A_2$ as follows:

$A_i={\Vert x_i\Vert }_{\left(i=1,2\right)}$(7)

where, ${\bar{x}}_1$ and ${\bar{x}}_2$ represent the amplitude amplification factors of the main system and the vibration absorber respectively.

For the above time-delay vibration reduction system, if it is assumed that the damping of the main system ${\xi }_1=0$, and the time-delay feedback gain coefficient $g=0$, then the above system will degenerate into a passive dynamic vibration absorber. Substitute the above system parameters into Equation (3), and the motion differential equation of the passive vibration reduction device is obtained as follows:

$\{\begin{array}{l}{x^{″}}_1+\left(1+\mu L^2{p}^2\right)x_1+2L^2\mu p{\xi }_2{x^{\prime }}_1-L\mu p^2{x}_2-2L\mu p{\xi }_2{x^{\prime }}_2+\beta \mu L\left(L{\ddot{x}}_1-{\ddot{x}}_2\right)=\cos \left(\Omega t\right)\\ {x^{″}}_2+\left(1+\alpha \right)p^2{x}_2+2p{\xi }_2{x^{\prime }}_2-L{p}^2{x}_1-2Lp{\xi }_2{x^{\prime }}_1-\beta \left(L{\ddot{x}}_1-{\ddot{x}}_2\right)=0\end{array}$(8)

Similarly, assume that the form of the solution of the equation is as shown in Equation (4). After substitution, the form of the solution of the passive vibration reduction device is obtained as follows:

$\left[\begin{array}{l}{\bar{x}}_1\\ {\bar{x}}_2\end{array}\right]=\frac{1}{\Delta \left(\Omega \right)}\left[\begin{array}{c}-{\Omega }^2+2p{\xi }_{2}j\Omega +\left(1+\alpha \right)p^2-\beta {\Omega }^2\\ 2Lp{\xi }_{2}j\Omega +L{p}^2-\beta L{\Omega }^2\end{array}\right]$(9)

$\begin{array}{c}\Delta \left(\Omega \right)=\left(-{\Omega }^2+1+L^2{p}^2\mu +2L^2\mu p{\xi }_{2}j\Omega -\beta \mu L^2{\Omega }^2\right)\\ \times \left(-{\Omega }^2+2p{\xi }_{2}j\Omega +(1+\alpha )p^2-\beta {\Omega }^2\right)\\ -\left(2L\mu p{\xi }_{2}j\Omega +L{p}^2\mu -\beta \mu L{\Omega }^2\right)\left(2Lp{\xi }_{2}j\Omega +L{p}^2-\beta L{\Omega }^2\right)\end{array}$(10)

Separate the real and imaginary parts of the above expression and calculate its modulus. The expression for the amplitude amplification factor $A_1$ of the passive vibration reduction system is obtained as follows:

$A_1=\Vert x_1\Vert =\sqrt{\frac{C^2+D^2}{A^2+B^2}}$(11)

In the formula:

$\{\begin{array}{l}A=2p\Omega {\xi }_2\left[1+L^2\mu \alpha p^2-{\Omega }^2\left(1+L^2\mu \right)\right]\\ B=\left(1-{\Omega }^2\right)\left[\left(1+\beta \right){\Omega }^2-\left(1+\alpha \right)p^2\right]-L^2\beta \mu {\Omega }^4-L^2\mu p^2\left[\alpha p^2-\left(1+\alpha \beta \right){\Omega }^2\right]\\ C=-{\Omega }^2+\left(1+\alpha \right)p^2-\beta {\Omega }^2\\ D=2p{\xi }_2\Omega \end{array}$(12)

For the PDVA system with the main system having no damping, through the fixed point theory, it is easy to prove that the normalized amplitude-frequency response curves will all pass through two points that are independent of the damping ratio. Figure 2 shows the normalized curves when the damping ratios ${\xi }_2$ are 0.3, 0.5, and 0.7 respectively. It can be seen from the figure that the three curves all intersect at point P and point Q.

For the grounded stiffness system with an inerter, the reference [25] has calculated the optimal parameters of the system through the fixed-point theory, enabling the two fixed points P and Q to be adjusted to approximately the same height. However, as can be observed from Figure 3 , for the passive dynamic vibration absorber with the main system having no damping, after optimizing its system parameters using the fixed-point theory, issues such as a narrow vibration reduction frequency band and the amplitude of the main system being highly sensitive to the changes in the excitation frequency near the anti-resonant frequency exist. There is still considerable room for optimizing the amplitude of the main system. Subsequently, time-delay feedback control will be introduced to further reduce the amplitude of the main system and widen the vibration reduction frequency band.

Under the condition of ensuring the system’s stable operation without sacrificing data accuracy, it is necessary to reasonably set the value ranges of the feedback gain coefficient and the time delay. Considering the actual situation of the experimental equipment, the value ranges of $g$ and $\tau$ are limited as follows:

$C_0=\{\left(g,\tau \right)|0\le g\le 2,0\le \tau \le 2\}$(13)

At the same time, define the form of the optimal parameter $D_{\text{opt}}^i$ of the system as follows. The parameter $D_{\text{opt}}^i$ is the optimal system structural parameter at the $i$th step, and its value range is $i=1,2,3,4$. Similar notations that appear subsequently will not be repeated for explanation.

$D_{\text{opt}}^i=\{\mu ,{\xi }_{\text{1opt}}^i,{\xi }_{\text{2opt}},p_{\text{opt}},{\alpha }_{\text{opt}}^i,{\beta }_{\text{opt}}^i,L_{\text{opt}}\}$(14)

Assume that the mass ratio $\mu =m_2/m_1=0.1$, and the damping ratio $c_1/c_2=10$. According to the relationship in the dimensionless expression (2), the calculation expression for the initial value of the system parameter ${\xi }_{1ori}^1$ is 0.2505. By combining with the optimal values of the passive system provided in the reference [25] , the specific values of the optimal parameters in the first step are as follows:

$\begin{array}{c}{D}_{\text{opt}}^1=\{\mu ,{\xi }_{\text{1opt}}^1,{\xi }_{\text{2opt}},p_{\text{opt}},{\alpha }_{\text{opt}}^1,{\beta }_{\text{opt}}^1,L_{\text{opt}}\}\\ =\{0.1,0.2505,1.0279,0.7302,8.923,0.8,2.5\}\end{array}$(15)

In the dynamic model of the coupled system established in the previous section, the introduction of the time-delay feedback control system will affect the stability of the system. When solving for the range of the optimal parameters, it is necessary to conduct a stability analysis of the feedback gain coefficient $g$ and the time delay $\tau$, so as to obtain the value ranges of $g$ and $\tau$. First, perform a Laplace transform on Equation (3), and represent it in the following matrix form in the Laplace domain:

$E\left(s\right)X\left(s\right)=F\left(s\right)$(16)

where:

$\{\begin{array}{l}X\left(s\right)={\left[{\bar{x}}_1\left(s\right),{\bar{x}}_2\left(s\right)\right]}^{\text{T}}\\ E\left(s\right)=\left|\begin{array}{cc}{s}^2+1+\mu L^2{p}^2+2s\left({\xi }_1+L^2\mu p{\xi }_2\right)+\beta \mu L^2{s}^2& \mu Lp\left(p+2{\xi }_{2}s+pg{\text{e}}^{-s\tau }\right)+\beta \mu L{s}^2\\ Lp\left(p+2{\xi }_{2}s\right)+\beta L{s}^2& s^2+p^2+\alpha p^2+2p{\xi }_{2}s+\beta {\text{s}}^2+p^{2}g{\text{e}}^{-s\tau }\end{array}\right|\\ F\left(s\right)={\left[f,0\right]}^{\text{T}}\end{array}$

Let the characteristic equation of the system be $\left|E\left(s\right)\right|=0$, and then the characteristic equation can be obtained as follows:

$\begin{array}{c}CE\left(g,\tau ,D_{opt}^i\right)=p^2+p^2\alpha +L^2{p}^4\alpha \mu +s^4\left(1+\beta +L^2\beta \mu \right)\\ +s^3\left(2{\xi }_1+2\beta {\xi }_1+2p{\xi }_2+2L^{2}p\mu {\xi }_2\right)\\ +s\left(2p^2{\xi }_1+2p^2\alpha {\xi }_1+2p{\xi }_2+2L^2{p}^3\alpha \mu {\xi }_2\right)\\ +s^2\left(1+p^2+p^2\alpha +\beta +L^2{p}^2\mu +L^2{p}^2\alpha \beta \mu +4p{\xi }_1{\xi }_2\right)\\ +{\text{e}}^{-s\tau }\left(g{p}^2+g{p}^2{s}^2+2g{p}^{2}s{\xi }_1\right)\end{array}$ (17)

Since the established system model contains a time delay term, the expression of the obtained characteristic equation is a transcendental equation with an exponential term. The solutions of the equation are infinite-dimensional, and there will be an infinite number of characteristic roots. To address this issue, the CTCR [26] method is introduced for analysis. The specific calculation steps are as follows:

1) Define Kernel Hyper curves: When the structural parameters of the system are fixed at a certain specific value, the two-dimensional plane of $\left(g,\tau \right)$ will give rise to an eigenvalue $s=j{\Omega }_c,{\Omega }_c\in R^{+},0\le {\Omega }_c\tau \le 2\pi ,0\le g\le 2$, with its real part being equal to zero. All the curves composed of points that exist within the two-dimensional plane $\left(g,\tau \right)$ and satisfy the given conditions are referred to as KH.

2) Define Offspring Hyper curves: The representation of OH can be obtained from Definition 1) through the following non-linear transformation:

$\langle \tau \pm \frac{2\pi }{{\Omega }_c}i,g\rangle ,i=0,1,2,3,\cdots .$(18)

3) Define the variation trend of the characteristic root Root Tendency as:

$RT=\mathrm{sgn}\left[\mathrm{Re}\left({\frac{\partial s}{\partial \tau }|}_{s=j{\Omega }_c}\right)\right]$(19)

4) Replace the exponential term using Euler’s formula:

${\text{e}}^{-j{\Omega }_c\tau }=\cos \left(h_c\right)-j\sin \left(h_c\right),h_c={\Omega }_c\tau$(20)

Among them, the trigonometric functions $\text{cos}$ and $\text{sin}$ can be replaced by the half-angle formulas when the tangent function $\text{tan}$:

$\cos \left(h_c\right)=\frac{1-z_1^2}{1+z_1^2},\sin \left(h_c\right)=\frac{2z_1}{1+z_1^2},z_1=\tan \left(\frac{h_c}{2}\right)$(21)

Continue to use the CTCR method to determine the stability of the system. Substitute the results obtained from Equations (19) and (20) into Equation (17), and the characteristic equation of the system can be rewritten in the form of a polynomial function of ${\Omega }_c$:

$q\left({\Omega }_c,g,z_1\right)={\displaystyle \underset{k=0}{\overset{4}{\sum }}{c}_k\left(g,z_1\right)}{\left({\Omega }_{c}j\right)}^k=0$(22)

Among them, $c_k$ is the coefficient expression of ${\Omega }_c$ and is a function of $g,z_1$. By separating the real and imaginary parts of Equation (22), we can obtain:

$\{\begin{array}{l}\mathrm{Re}\left[q\left({\Omega }_c,g,z_1\right)\right]={\displaystyle \underset{k=0}{\overset{4}{\sum }}{Q}_k}\left(g,z_1\right){\left({\Omega }_c\right)}^k=0\\ \mathrm{Im}\left[q\left({\Omega }_c,g,z_1\right)\right]={\displaystyle \underset{k=0}{\overset{4}{\sum }}{P}_k}\left(g,z_1\right){\left({\Omega }_c\right)}^k=0\end{array}$(23)

Among them, both $P_k$ and $Q_k$ are coefficient expressions of ${\Omega }_c$, and they are also functions of $g,z_1$. The specific results are as follows:

$\{\begin{array}{l}{Q}_4=\left(1+L^2\beta \mu +\beta +z_1^2+\beta z_1^2+L^2\beta \mu z_1^2\right)\\ Q_3=0\\ Q_2=-1-p^2-g{p}^2-p^2\alpha -\beta -L^2{p}^2\mu -z_1^2-L^2{p}^2\alpha \beta \mu -p^2{z}_1^2\\ +g{p}^2{z}_1^2-p^2\alpha z_1^2-\beta z_1^2-L^2{p}^2\mu z_1^2-L^2{p}^2\alpha \beta \mu z_1^2-4p{\xi }_1{\xi }_2-4p{z}_1^2{\xi }_1{\xi }_2\\ Q_1=4g{p}^2{z}_1{\xi }_1\\ Q_0=p^2+g{p}^2+p^2\alpha +L^2{p}^4\alpha \mu +p^2{z}_1^2-g{p}^2{z}_1^2+p^2\alpha z_1^2+L^2{p}^4\alpha \mu z_1^2\\ P_4=0\\ P_3=-2{\xi }_1-2\beta {\xi }_1-2z_1^2{\xi }_1-2\beta z_1^2{\xi }_1-2p{\xi }_2-2L^{2}p\mu {\xi }_2-2p{z}_1^2{\xi }_2-2L^{2}p\mu z_1^2{\xi }_2\\ P_2=2g{p}^2{z}_1\\ P_1=2p^2{\xi }_1\left(1+g+\alpha +z_1^2-g{z}_1^2+\alpha z_1^2\right)+2p{\xi }_2\left(1+L^2{p}^2\alpha \mu +z_1^2+L^2{p}^2\alpha \mu z_1^2\right)\\ P_0=-2g{p}^2{z}_1\end{array}$(24)

By combining Equation (23), ${\Omega }_c$ can be obtained. Its necessary and sufficient condition needs to be derived through the discriminant and the Dixon resultant. According to the Dixon resultant, assume that:

$\{\begin{array}{l}{q}_1\left({\Omega }_c\right)\equiv \mathrm{Re}\left[q\left({\Omega }_c,g,z_1\right)\right]\\ q_2\left({\Omega }_c\right)\equiv \mathrm{Im}\left[q\left({\Omega }_c,g,z_1\right)\right]\\ \delta \left({\Omega }_c,\varphi \right)\equiv \frac{1}{\left({\Omega }_c,\varphi \right)}=\left|\begin{array}{cc}{q}_1\left({\Omega }_c\right)& q_2\left({\Omega }_c\right)\\ q_1\left(\varphi \right)& q_2\left(\varphi \right)\end{array}\right|\end{array}$(25)

Among them, $q_1\left(\varphi \right)$ and $q_2\left(\varphi \right)$ are obtained by replacing $\delta$ with the imaginary variable ${\Omega }_c$. The expressions of $q_1\left(\varphi \right)$ and $q_2\left(\varphi \right)$ are as follows:

$\{\begin{array}{l}{q}_1=\left(1+L^2\beta \mu +\beta +z_1^2+\beta z_1^2+L^2\beta \mu z_1^2\right){\Omega }_c^4+(-1-p^2\\ -g{p}^2-p^2\alpha -\beta -L^2{p}^2\mu -z_1^2-L^2{p}^2\alpha \beta \mu -p^2{z}_1^2+g{p}^2{z}_1^2-p^2\alpha z_1^2\\ -\beta z_1^2-L^2{p}^2\mu z_1^2-L^2{p}^2\alpha \beta \mu z_1^2-4p{\xi }_1{\xi }_2-4p{z}_1^2{\xi }_1{\xi }_2){\Omega }_c^2+L^2{p}^4\alpha \mu z_1^2\\ +4g{p}^2{z}_1{\xi }_1{\Omega }_c+p^2+g{p}^2+p^2\alpha +L^2{p}^4\alpha \mu +p^2{z}_1^2-g{p}^2{z}_1^2+p^2\alpha z_1^2\\ q_2=(-2{\xi }_1-2\beta {\xi }_1-2z_1^2{\xi }_1-2\beta z_1^2{\xi }_1-2p{\xi }_2-2L^{2}p\mu {\xi }_2-2p{z}_1^2{\xi }_2\\ -2L^{2}p\mu z_1^2{\xi }_2){\Omega }_c^3+2g{p}^2{z}_1{\Omega }_c^2+[2p^2{\xi }_1\left(1+g+\alpha +z_1^2-g{z}_1^2+\alpha z_1^2\right)\\ +2p{\xi }_2\left(1+L^2{p}^2\alpha \mu +z_1^2+L^2{p}^2\alpha \mu z_1^2\right)]{\Omega }_c-2g{p}^2{z}_1\end{array}$(26)

Through observation, it can be found that Equation (25) is a polynomial of $deg-1$ that is symmetric about $\sigma$ and ${\Omega }_c$, and it can also be understood as a Dixon polynomial about $q_1\left({\Omega }_c\right)$ and $q_2\left({\Omega }_c\right)$. Due to its symmetric property, changing the positions of the elements within the expression will not change its result, that is, $\delta \left({\Omega }_c,\varphi \right)=\delta \left(\varphi ,{\Omega }_c\right)$. The above-mentioned $deg-1$ represents the highest order of ${\Omega }_c$ in $q_1\left({\Omega }_c\right)$ and $q_2\left({\Omega }_c\right)$. Among them, $deg=\max \left(deg\left(q_1\left({\Omega }_c\right)\right),deg\left(q_2\left({\Omega }_c\right)\right)\right)$, $deg\left(q_1\left({\Omega }_c\right)\right)$ and $deg\left(q_2\left({\Omega }_c\right)\right)$ also represent the highest order terms about ${\Omega }_c$. Any common zero point of $q_1\left({\Omega }_c\right)$ and $q_2\left({\Omega }_c\right)$ is also a zero point of $\delta \left({\Omega }_c,\varphi \right)$ corresponding to $\varphi$. Therefore, under the condition of the common zero point, the coefficient of the coefficient $\delta \left({\Omega }_c,\varphi \right)$ in ${\varphi }^i$ is zero, which can be expressed in matrix form as follows:

$D\left(g,z_1\right)=\left(\begin{array}{c}1\\ {\Omega }_c\\ ⋮\\ {\Omega }_c^{deg-1}\end{array}\right)=\left(\begin{array}{c}0\\ 0\\ ⋮\\ 0\end{array}\right)$(27)

Among them, $i=0,1,2,3,\cdots ,deg-1$. The coefficient matrix $D\left(g,z_1\right)\in R^d$ is called the Dixon matrix, and $d=deg\times deg$.

$D\left(g,z_1\right)=\left(\begin{array}{cccc}{D}_1& D_2& D_3& D_4\\ D_5& D_6& D_7& D_8\\ D_9& D_{10}& D_{11}& D_{12}\\ D_{13}& D_{14}& D_{15}& D_{16}\end{array}\right)$(28)

When the matrix $D\left(g,z_1\right)$ is a singular matrix, a non-trivial solution of Equation (26) can be obtained, that is: $deg=\left[D\left(g,z_1\right)\right]=0$. When $g$ is within a specific value range, Equation (28) will transform into a 16^th-degree polynomial about $z_1$:

$R_z={\displaystyle \underset{l=0}{\overset{16}{\sum }}{R}_l}\left(g\right)z_1^l=0$(29)

Among them, $l=0,1,2,\cdots ,16$. The specific expression of the coefficient $R_l\left(g\right)$ is quite complex and will not be presented in detail. To obtain the stable interval ranges of the feedback gain coefficient and the time delay amount, the above algebraic expressions will be numerically solved according to the following steps:

1) Given the initial value of the parameter $g$, find the roots of the equation $deg=\left[D\left(g,z_1\right)\right]=0$, and substitute the real roots into Equation (21) to obtain the value of $h_c$.

2) Substitute $h_c$ into Equation (20) to obtain the value of ${\text{e}}^{-j{\Omega }_c\tau }$. Then substitute this value into the characteristic Equation (17), and all the characteristic roots can be obtained.

3) Among the obtained characteristic roots, filter out the pure imaginary roots and mark them as ${\Omega }_c$. Substitute the obtained ${\Omega }_c$ and $h_c$ into the characteristic Equation (17) to obtain the time delay amount ${\tau }_i,i=1,2,3,4,\cdots$

4) Change the value of the feedback gain coefficient $g$ to make it linearly increase within a certain range until the value of $g$ reaches the set value, and repeat the above steps to obtain $C_1=\left\{\left(g,\tau \right)|\mathrm{Re}\left(CE\left(D,g,\tau \right)<0\right)\right\}\cap C_0$. The parameter ranges are as follows:

$D=\left\{\mu ,{\xi }_1,{\xi }_2,p,\alpha ,\beta ,L\right\}=\left\{0.1,0.1~0.9,0~2,0.7302,-2~9,0.4923~2.5846,2.5\right\}$;

When optimizing system parameters, the selection of optimization algorithm is particularly important. In order to achieve accurate vibration suppression of key frequencies and broaden the frequency band of vibration reduction, the anti-formant optimization criterion can solve the vibration problem of specific frequencies more directly than its optimization method, and achieve the purpose of wide-band vibration reduction through reasonable parameter design.

Next, regarding the system’s system of differential equations of motion (3), the amplitude optimization problem of the anti-resonance points is studied by adopting time-delay feedback active control. The time delay is used to control the amplitude of the anti-resonance points and the position of the anti-resonance frequencies, and a criterion for minimizing the amplitude of the anti-resonance points is designed. To achieve this goal, it is first necessary to obtain the frequencies and amplitudes of the anti-resonance points, and then use the minimum value of the function to derive the conditions for the minimum amplitude of the main system as follows:

$\begin{array}{l}\frac{\text{d}{A}_1}{\text{d}\Omega }=0\\ \frac{{\text{d}}^2{A}_1}{\text{d}{\Omega }^2}>0\end{array}$(30)

Among them, $\text{d}{A}_1/\text{d}\Omega$ and ${\text{d}}^2{A}_1/\text{d}{\Omega }^2$ are obtained according to the amplitude magnification factor expression (7). For a two-degree-of-freedom linear system, the conditions for finding the anti-resonance points are the same as the conditions for the minimum value of the amplitude of the main system. According to the above conditions for the minimum value of the system amplitude, the frequencies of the anti-resonance points can be obtained. Substituting these frequencies into the expression (7) of the amplitude magnification factor can yield the amplitudes of the anti-resonance points. For the given system structure parameter $D=\left\{\mu ,{\xi }_1,{\xi }_2,p,\alpha ,\beta ,L\right\}$, subsequently, by changing the feedback gain coefficient and the time-delay amount, the amplitudes of the anti-resonance points can be adjusted to make them within a sufficiently small range, which is defined as: $\min \Vert A_1\left(D,{\Omega }_c,g,\tau \right)\Vert <{10}^{-1}$. According to the above, the optimization criterion for satisfying the minimum of the anti-resonance points can be obtained as:

$\begin{array}{l}{\frac{\text{d}{A}_1}{\text{d}\Omega }|}_{\Omega ={\Omega }_{anti}}=0\\ {\frac{{\text{d}}^2{A}_1}{\text{d}{\Omega }^2}|}_{\Omega ={\Omega }_{anti}}>0\\ \min \Vert A_1\left(D,{\Omega }_{anti},g,\tau \right)\Vert <{10}^{-1}\end{array}$(31)

Among them, ${\Omega }_{anti}$ represents the anti-resonance frequency, and $A_1\left(D,{\Omega }_c,g,\tau \right)$ is the amplitude of the anti-resonance peak of the system. It is assumed that $C_2$ is not only the interval of $g$ and $\tau$ that enables the system to operate stably, but also the stable operating interval of $g$ and $\tau$ under the condition of satisfying the minimum criterion of the anti-resonance points of the system. Its specific form is:

$C_2=\left\{\left(g,\tau \right)|\min \Vert A_1\left(D,{\Omega }_{anti},g,\tau \right)\Vert <{10}^{-1}\right\}\cap C_1$(32)

In order to study the vibration reduction effect of the time-delay feedback active control on the vibration system at the anti-resonance points, taking the optimal structural parameter $D_{\text{opt}}^1$ obtained in the passive system in reference [25] as an example, the anti-resonance frequency ${\Omega }_a\in \left(0,2\right)$ is restricted, and by changing the time-delay amount and the feedback gain coefficient, the amplitude $A_1$ of the main system at the anti-resonance frequency can be obtained. The amplitude-frequency response curve of the time-delay feedback control is shown in Figure 4 .

Within the optimization criterion that the anti-resonance amplitude is less than 0.01, there is an adjustable interval of the feedback gain coefficient $g$ and the time-delay amount $\tau$ within the range of ${\Omega }_a\in \left(0,2\right)$. However, it can be found from the figure that the time-delay feedback active control is mainly concentrated in the high-frequency region, and when the anti-resonance frequency is small, the time-delay feedback control will fail. Therefore, it is necessary to optimize the design of the system structural parameters $D$ in combination with the time-delay feedback control, so that it can achieve the best optimization effect while having a wide vibration reduction frequency band and frequency band symmetry.

According to the above optimization requirements, the initial values for the optimization of the vibration system are given as follows:

$D_0=\left\{\mu ,{\xi }_1,{\xi }_2,p,{\alpha }^0,\beta ,L\right\}=\left\{0.1,0.2505,1.0279,0.7302,0,0.8,2.5\right\}$(33)

Then, within the range of structural parameter optimization, the difference between the initial parameter $D_0$ and the final parameter $D$ is divided into $N$ equal parts, and incremental searches are carried out step by step to obtain the optimal parameters. In order to ensure the accuracy of the data and at the same time shorten the search time, $N=100$ is taken, and the calculation step size is ${\Delta }_D=\left(D_0-D\right)/N$. The structural parameter of the ${\left(h+1\right)}^{th}$ step is set as $D_{h+1}=D_h+{\Delta }_D$, where $D_h$ is the structural parameter of the $h^{th}$ step and $h=0,1,2,\cdots ,N$.

Finally, by substituting the structural parameter $D_{h+1}$ into Expressions (31) and (32), the feedback gain coefficient $g$ and the time delay $\tau$ corresponding to the condition that satisfies the minimum amplitude criterion of the anti-resonance points can be obtained. The specific process is shown in Figure 5 .

The ground negative stiffness absorber uses a special structure to generate negative stiffness force, which presents a nonlinear relationship between force and displacement, and can more effectively offset or reduce the exciting force under specific vibration conditions, and rapidly suppress vibration to achieve better vibration reduction effect. Its implementation principle is shown in Figure 6 .

The two springs in the horizontal plane are preloaded springs, one end of the spring is fixed on the support, and the other end is connected with the linear bearing slider. The cylindrical guide rail plays a role in fixing and limiting the slider and the spring, and the slider can slide freely along the cylindrical guide rail. Among them, the stiffness of the two horizontal springs is $k_a$, the stiffness of the vertical spring is $k_b$, the angle between the horizontal spring and the connecting rod is $\theta$, the original length of the horizontal spring is $L_{\text{0}}$, the spring compression of the horizontal spring in the equilibrium position is $h$, and the length of the two connecting rods is $a$. When the connecting rod is in the horizontal position, the vertical spring does not deform.

When the displacement in the vertical direction is $x$, the following expression is obtained:

$F=k_{b}x-2\frac{x}{\sqrt{a^2-x^2}}\left(h-a+\sqrt{a^2-x^2}\right)k_a$(34)

After derivation of displacement by formula (34), and then dimensionless quantization simplification, the relationship between dimensionless stiffness and dimensionless displacement can be obtained as follows:

$\overline{K_3}=1-2d\left(\frac{d_1-d_2+\sqrt{d_2^2-\overline{x^2}}+\overline{x^2}\left(d_1-d_2\right)}{d_2^2-\overline{x^2}}\right)$ (35)

In formula (35), the force-displacement relationship of the horizontal negative stiffness device is completely derived. According to Taylor expansion, it can be seen that under the condition of small displacement, the higher-order nonlinear term can be ignored, and the device has good robustness.

According to the above calculation process of the anti-resonance points, the optimization design of the system parameter $\alpha$ is carried out first. The values of other system parameters are all those in $D_{\text{opt}}^1$. A value interval is given for the stiffness coefficient $\alpha$ so that the optimal value of the stiffness coefficient $\alpha$ in $D_{\text{opt}}^1$ lies within this interval. The specific structural parameter $D_{\text{opt}}^1$ is as follows:

$D=\left\{\mu ,{\xi }_1,{\xi }_2,p,\alpha ,\beta ,L\right\}=\left\{0.1,0.2505,1.0279,0.7302,-2~9,0.8,2.5\right\}$ (36)

After determining the values of the above structural parameters, the stability region diagram of the stiffness coefficient, the feedback gain coefficient and the time delay is drawn as follows:

From Figure 7 , it can be found that when $\alpha$ is within the range of $\left(-2,0\right)$, there exists an adjustable interval for the feedback gain coefficient $g$ and the time delay $\tau$. Moreover, as the stiffness coefficient $\alpha$ continuously increases, the adjustable interval for the feedback gain coefficient and the time-delay amount also keeps expanding. When $\alpha \in \left(0,9\right)$, since a stable $g,\tau$ interval has not been found, the stability diagram of this region is not presented in the figure.

Figure 8(a) and Figure 8(b) represent the relationship diagrams between the anti-resonance frequency range of $g$ and $\tau$, the amplitude $A_1$ of the anti-resonance point under different stiffness coefficients $\alpha$. It can be seen from Figure 8(b) that when $\alpha \in \left[-1.616,-0.8081\right]$, as the stiffness coefficient $\alpha$ continuously increases, the adjustable width of the anti-resonance frequency gradually increases; when , as the stiffness coefficient $\alpha$ continuously increases, the adjustable width of the anti-resonance frequency gradually decreases. From the perspective of controlling the symmetry of the anti-resonance frequency band, there exists an optimal stiffness coefficient $\alpha$ in Figure 8(c) . While ensuring that both $g$ and $\tau$ are within the stable range, the frequency band width of its anti-resonance point also maintains a certain symmetry with the main resonance point.

Record the stiffness coefficient ${\alpha }_{\text{opt}}^2=-1.4444$ marked by the purple triangle in Figure 8(c) as the optimal value. The specific range of the adjustable anti-resonance frequency band corresponding to this point is $\left({\Omega }_{aL},{\Omega }_{aH}\right)=\left(0.5606,1.4434\right)$. Take the value of the above optimal stiffness coefficient $\alpha$ as the initial value for the second-step optimization. At this time, the optimal structural parameters of the system are as follows:

$D_{\text{opt}}^2=\left\{\mu ,{\xi }_1,{\xi }_2,p,{\alpha }^2,\beta ,L\right\}=\left\{0.1,0.2505,1.0279,0.7302,-1.4444,0.8,2.5\right\}$(37)

Under these structural parameters, the requirement for the symmetry of the adjustable anti-resonance point frequency band still has a large gap compared with the expected value, and it is necessary to further optimize other parameters.

Next, carry out the optimization design of the system parameter ${\xi }_1$. Similarly,

follow the above calculation process of the anti-resonance points. The values of other system parameters are all those in $D_{\text{opt}}^2$. Give a value interval for the stiffness coefficient ${\xi }_1$ so that the optimal value of the damping coefficient ${\xi }_1$ in $D_{\text{opt}}^2$ is within this range. The specific structural parameter $D_{\text{opt}}^2$ is as follows:

$D=\left\{\mu ,{\xi }_1,{\xi }_2,p,\alpha ,\beta ,L\right\}=\left\{0.1,0.1~0.9,1.0279,0.7302,-1.4444,0.8,2.5\right\}$(38)

After determining the values of the above structural parameters, the stability region diagram of the main system damping, the feedback gain coefficient and the time delay is drawn as shown in Figure 9 .

It can be found from Figure 9 that when ${\xi }_1$ is within the range of $\left(0.1,0.9\right)$, there exists an adjustable interval for the feedback gain coefficient $g$ and the time delay amount $\tau$. Moreover, as the main system damping ${\xi }_1$ continuously increases, the adjustable interval for the feedback gain coefficient and the time delay amount also keeps expanding.

Figure 10(a) represents the relationship diagram between the anti-resonance frequency range of the time-delay quantity $g$ and the feedback gain coefficient $\tau$ and the amplitude $A_1$ of the anti-resonance point under the main system damping ${\xi }_1$. It can be seen from Figure 10(b) that there exists an optimal main system damping ${\xi }_1$. While ensuring that both $g$ and $\tau$ are within the stable range, the frequency band width of its anti-resonance point also maintains good symmetry with the main resonance point.

Record the main system damping ${\xi }_{1\text{opt}}=0.4797$ marked by the blue five-pointed star in Figure 10(b) as the optimal value. The specific range of the adjustable anti-resonance frequency band corresponding to this point is $\left({\Omega }_{aL},{\Omega }_{aH}\right)=\left(0.4445,1.6196\right)$. Take the value of the above optimal main system damping ${\xi }_1$ as the initial value for the third-step optimization. At this time, the optimal structural parameters of the system are as follows:

$D_{\text{opt}}^3=\left\{\mu ,{\xi }_{{}_1}^2,{\xi }_2,p,{\alpha }^2,\beta ,L\right\}=\left\{0.1,0.4797,1.0279,0.7302,-1.4444,0.8,2.5\right\}$(39)

Finally, optimize the inerter ratio. Within the optimal working range of the inerter, select the inerter ratio $\beta$ that simultaneously meets the best adjustable intervals of $g$ and $\tau$ for optimization design. The values of other system parameters are all those in $D_{\text{opt}}^3$. Give a value interval for the inerter ratio $\beta$ so that the optimal value of the inerter ratio $\beta$ in $D_{\text{opt}}^1$ is within this range. The specific structural parameter $D$ is shown in Equation (40).

$D=\left\{\mu ,{\xi }_1,{\xi }_2,p,\alpha ,\beta ,L\right\}=\left\{0.1,0.4797,1.0279,0.7302,-1.4444,0.4~2.5,2.5\right\}$(40)

After determining the values of the above structural parameters according to the cyclic process, the stability region diagram of the main system damping, the feedback gain coefficient and the time delay is drawn as follows:

It can be found from Figure 11 that when $\beta$ is within the range of $\left(0,2.48\right)$, there exists an adjustable interval for $g$ and $\tau$. Moreover, as the inerter ratio $\beta$ continuously decreases, the adjustable interval for the feedback gain coefficient $g$ and the time-delay quantity $\tau$ keeps expanding.

Figure 12(a) shows the relationship diagram between the anti-resonance frequency range of $g$ and $\tau$. and the amplitude $A_1$ of the anti-resonance point under the inerter ratio $\beta$. As can be seen from Figure 12(b) , there exists an optimal inerter ratio $\beta$. While ensuring that both $g$ and $\tau$. are within the stable range, the frequency-band width of its anti-resonance point maintains good symmetry with the main resonance point.

Record the inerter ratio $\beta =1.0222$ marked by the triangle in Figure 12(b) as the optimal value. The specific range of the adjustable anti-resonance frequency band corresponding to this point is $\left({\Omega }_{aL},{\Omega }_{aH}\right)=\left(0.2965,1.7809\right)$. Take the value of the above optimal inerter ratio $\beta$ as the initial value for the fourth-step optimization. At this time, the optimal structural parameters of the system are as follows:

$D_{opt}^4=\left\{\mu ,{\xi }_1^2,{\xi }_2,p,{\alpha }^2,{\beta }^2,L\right\}=\left\{0.1,0.7390,1.0279,0.7302,-1.444,1.0222,2.5\right\}$(41)

Now that the optimal structural parameters $D_4$ of the system have been obtained, the CTCR method [26] is continuously used to conduct stability verification on the parameters $D_4$. Replace the structural parameters $D$ in equation $C_1$ in Subsection 3.2 with the optimal system parameters $D_4$ to obtain $C_3$ under the optimal parameter range, that is: ADSDA; Similarly, replace the structural parameters $D$ in equation $C_2$ in Subsection 3.3 with the optimal system parameters $D_4$, and replace $C_1$ with $C_3$ to obtain the stable interval $C_4$ of the feedback gain coefficient and the time delay quantity that satisfy the criterion of the minimum amplitude of the anti-resonance point and are stable, $C_4=\left\{\left(g,\tau \right)|\min \Vert A_1\left(D_4,{\Omega }_{anti},g,\tau \right)\Vert <{10}^{-2}\right\}\cap C_3$;

The range of its stable interval is shown in Figure 14 . Below the solid blue square line in the figure represents the stable range of $C_3$, and below the red dotted line represents the stable range of $C_4$. Select five points of $T_1,T_2,T_3,T_4,T_5$ at the same interval on $C_4$. Only $T_1,T_3$ and $T_5$ are marked in Figure 14(a) , and their coordinates are $\left(1.541,0.873\right)$, $\left(1.749,0.703\right)$ and $\left(1.749,0.703\right)$ respectively.

In order to verify that the gain coefficient and time delay corresponding to the points on the $C_4$ curve can make the system stable, first, substitute the complex number $s=a+ib$ into the characteristic Equation (17) and expand all polynomials; then use Euler’s formula ${\text{e}}^{-s\tau }={\text{e}}^{-a\tau }\left[\cos \left(b\tau \right)-i\sin \left(b\tau \right)\right]$ to deal with the exponential terms; finally, organize the characteristic equation into the real part $\mathrm{Re}\left(CE\right)$ and the imaginary part $\mathrm{Im}\left(CE\right)$. Substitute the values of $g$ and $\tau$ corresponding to the above five measurement points into the expressions of the real part and the imaginary part, and the structural parameter values are still the optimized structural parameters $D_4$ in the fourth step. By solving the simultaneous equations of $\mathrm{Re}\left(CE\right)=0$ and $\mathrm{Im}\left(CE\right)=0$ numerically, the values of $a$ and $b$ can be obtained.

According to the above steps, the characteristic root with the maximum real part of the characteristic Equation (17) is obtained, and the results are shown in Figure 14(b) . As can be seen from Figure 14(b) , the real parts of the characteristic roots of the five measuring points are all less than 0. This indicates that $g$ and $\tau$ on the $C_4$ curve can make the system stable. Subsequently, the anti-resonance amplitude verification will be carried out by taking the three measuring points marked in Figure 14(a) as examples.

According to Figure 14(a) , the values of the corresponding gain coefficient and time delay can be determined. By substituting the coordinates of these three measuring points into Figure 13 , the anti-resonance point frequencies ${\Omega }_a$ corresponding to these three points can be determined as: ${\Omega }_{a1}=0.5135$, ${\Omega }_{a3}=0.9069$, ${\Omega }_{a5}=1.2762$.

Substitute $g$ and $\tau$ corresponding to the three points of $T_1,T_3,T_5$, as well as the optimal structural parameters $D_4$ into the amplitude Equation (11), and draw the amplitude-frequency response curves corresponding to these three points as shown in Figure 15 . It can be seen from the figure that the anti-resonance amplitudes corresponding to the three measuring points all meet the requirements of the anti-resonance optimization criterion (that is $A_1<{10}^{-1}$), and their corresponding anti-resonance frequencies are the same as the values in Figure 13 .

Finally, a time-domain verification is carried out for the three points of $T_1,T_3,T_5$. The anti-resonance frequencies ${\Omega }_a$ corresponding to these three measuring points are selected as the external excitation frequencies. After substituting the known parameters into the dimensionless Equation (3-3), the dde23 function in Matlab is used to solve Equation (3). The time t = 500S is selected, and the time history curves of the three measuring points are drawn as shown in Figure 16 . It can be found from the figure that the amplitudes of the three measuring points gradually decrease and converge over time, indicating that the system tends to be stable. From the locally enlarged view, it can be found that the amplitudes of the three measuring points are all within 0.1, which meets the conditions set by the optimization criterion.

In order to further analyze the excellent vibration reduction performance of the dynamic vibration absorber with time-delay feedback control, the optimized results are quantitatively compared with the vibration reduction effects of the other two types of dynamic vibration absorbers. Taking the measuring point $T_1$ as an example for the optimization results in this paper, the structural parameters used are represented by $D_4$, and the optimal parameters of Xing and Yang are represented by $P_X$ and $P_Y$ respectively. The parameters are shown in Table 2 , and the amplitude-frequency response curves of the three different dynamic vibration absorbers are drawn as shown in Figure 17 .

It can be seen from Figure 17 that when the external excitation frequency is within 3, the time-delay negative stiffness dynamic vibration absorber with an inerter designed in this chapter has achieved a very good vibration reduction effect. Compared with Xing’s vibration absorber model, it has significantly broadened the vibration reduction frequency band. When compared with Yang’s vibration absorber model, it has excellent vibration reduction characteristics throughout the entire frequency range. Moreover, at the anti-resonance point, the dynamic vibration absorber designed in this paper can almost completely absorb the vibration. This can effectively prevent the main system from generating large-amplitude vibrations at specific frequencies, ensuring its operational stability and safety, and preventing problems such as structural damage and performance degradation caused by resonance.