## 3 Stability Analysis
### Time-Delay Independent Stability
We can obtain  
\[D(i\omega_{0},\tau)=P_{R}(\omega_{0})+iP_{I}(\omega_{0})+[Q_{R}(\omega_{0})+i \mathbb{Q}_{I}(\omega_{0})][\cos(\omega_{0}\tau)-i\sin(\omega_{0}\tau)]=0, \tag{13}\]  
where \(P_{R}(\omega_{0})=-\omega_{0}{}^{6}+p_{4}\omega_{0}{}^{4}-p_{2}\omega_{0}{}^{2 }+p_{0}\), \(Q_{R}(\omega_{0})=q_{4}\omega_{0}{}^{4}-q_{2}\omega_{0}{}^{2}+q_{0}\), \(Q_{I}(\omega_{0})=-q_{3}\omega_{0}{}^{3}+q_{1}\omega_{0}\), \(P_{I}(\omega_{0})=p_{5}\omega_{0}{}^{5}-p_{3}\omega_{0}{}^{3}+p_{1}\omega_{0}\). If Equation (13) is equal to zero, then the real and imaginary parts of the polynomial are zero, respectively. That is,  
\[\left\{ \begin{array}{l}P_{R}(\omega_{0})+Q_{R}(\omega_{0})\cos(\omega_{0} \tau^{*})+Q_{I}(\omega_{0})\sin(\omega_{0}\tau^{*})=0\\ P_{I}(\omega_{0})-Q_{R}(\omega_{0})\sin(\omega_{0}\tau^{*})+Q_{I}(\omega_{0}) \cos(\omega_{0}\tau^{*})=0\end{array} \right.. \tag{14}\]  
By solving Equation (14) and eliminating the trigonometric function terms, a polynomial without time-delay \(\tau^{*}\) can be obtained:  
\[P_{R}(\omega_{0})^{2}+P_{I}(\omega_{0})^{2}-[Q_{R}(\omega_{0})^{2}+Q_{I}( \omega_{0})^{2}]=0. \tag{15}\]  
Let \(F(\omega_{0})=P_{R}(\omega_{0})^{2}+P_{I}(\omega_{0})^{2}-[Q_{R}(\omega_{0})^ {2}+Q_{I}(\omega_{0})^{2}]\). If Equation (10) has a pair of purely imaginary roots \(\pm i\omega_{0}\) under the condition of \(\tau^{*}\geq 0\), it must be satisfied that \(F(\omega_{0})=0\) has a positive real root. The polynomial \(F(\omega_{0})=0\) does not include the time-delay \(\tau^{*}\). Through the above description, we can simplify the solution of whether the polynomial \(D(i\omega_{0},\tau^{*})=0\) has real roots or not to the solution of whether the polynomial \(F(\omega_{0})=0\) has real roots or not. By this method, the judgment of condition 2 can be completed. Equation (10) can be expressed as  
\[F(\omega_{0})=\omega_{0}{}^{12}+b_{1}\omega_{0}{}^{10}+b_{2}\omega_{0}{}^{8} +b_{3}\omega_{0}{}^{6}+b_{4}\omega_{0}{}^{4}+b_{5}\omega_{0}{}^{2}+b_{6}, \tag{16}\]  
where  
\[ \begin{array}{l}b_{1}=p_{5}{}^{2}-2p_{4}\\ b_{2}=2p_{2}-2p_{5}p_{3}+{p_{4}}^{2}-{q_{4}}^{2}\\ b_{3}={p_{3}}^{2}-{q_{3}}^{2}+2p_{1}p_{5}-2p_{2}p_{4}+2q_{4}q_{2}-2p_{0}\\ b_{4}={p_{2}}^{2}-2p_{1}p_{3}-{q_{2}}^{2}-2q_{0}q_{4}+2p_{0}p_{4}+2q_{1}q_{3} \\ b_{5}={p_{1}}^{2}+2q_{0}q_{2}-{q_{1}}^{2}-2p_{0}p_{2}\\ b_{6}=-{q_{0}}^{2}+{p_{0}}^{2}\end{array} \]  
Equation (16) is a 12 th order polynomial, and the number of real roots of the polynomial equation \(F(\omega_{0})=0\) can be determined according to the generalized Sturm criterion. Based on the theory of time-delay independent stability of two-degree-of-freedom vibration system, this paper extends it to the three-degree-of-freedom vibration system. We define the discriminant sequence \(F(\omega_{0})\) as \([D_{1}(F),D_{2}(F),\ldots,D_{12}(F)]\), i.e., \([1,d_{0},d_{0}d_{1},d_{1}d_{2},\cdots,d_{8}d_{9},\)\({d_{9}}^{2}d_{10}]\). The formula \(d_{j}(j=1,\ 2,\ \ldots,\ 10)\) is shown in Appendix A. To draw the time-delay independent stable region, The parameters of Table 3 are substituted into Equation (16); then, the coefficients \(b_{1}\), \(b_{2}\), \(b_{3}\), \(b_{4}\), \(b_{5}\), and \(b_{6}\) in Equation (16) are substituted into the Sturm criterion. We can draw the curve of the discriminant sequence coefficient \(d_{j}=0\,(j=0,\ 1,\ \ldots,\ 10)\). Taking the intersection with the curve satisfying condition 1, we can see that the selected parameter range is divided into 15 regions, as shown in Figure 3.  
Sign tables of the discrimination sequence of the polynomial \(F(\omega_{0})\) are shown in Table 4. \(l\) is the number of non-zero terms of the discriminant sequence and \(k\) denotes the number of discriminant sequence number changes, then \(F(\omega_{0})\) has \(l-2k\) distinct real roots. If \(l-2k=0\), the polynomial \(F(\omega_{0})\) has no real roots, which means that the system increases with time delay, and the characteristic roots do not cross the imaginary axis.