## 3 Time Delayed Velocity Feedback Control
### Nonlinear Primary Resonance Response  
By using the method of multiple scales, the perturbation solution of Equation (3) is assumed as follows:  
\[q=q_{0}(t_{0},t_{1})+\varepsilon q_{1}(t_{0},t_{1})+\cdots, \tag{4}\]  
where \(T_{0}=t\), \(T_{1}=\varepsilon t\). In the case of primary resonance, we let  
\[\omega=\omega_{0}+\varepsilon\sigma, \tag{5}\]  
where \(\sigma\) is the detuning parameter. Substituting Equations (4) and (5) into Equation (3) and equating coefficients of similar powers of \(\varepsilon\) yield the following equations:  
\[D_{0}^{2}q_{0}+\omega_{0}^{2}q_{0}=0. \tag{6}\]  
\[D_{0}^{2}q_{1}+\omega_{0}^{2}q_{1}= -2D_{0}D_{1}q_{0}-\mu D_{0}q_{0}+\alpha q_{0}^{3}+f\sin(\omega_{0} T_{0}+\sigma T_{1})+pD_{0}q_{0}(t-\tau). \tag{7}\]  
The solution of Equation (6) is written as follows:  
\[q_{0}=A(T_{1})e^{i\omega T_{0}}+\overline{A}(T_{1})e^{-i\omega T_{0}}. \tag{8}\]  
Figure 1: The simple controlled beam structure model. Substituting Equation (8) into Equation (7) we obtain  
\[ \begin{split}& D_{0}^{2}q_{1}+\omega_{0}^{2}q_{1}=[-2i\omega D_{1}A- i\mu\omega_{0}A+3\alpha A^{2}\overline{A}+\frac{f}{2}e^{i\sigma T_{1}}\\ &+i\omega p(\cos\omega\tau-i\sin\omega\tau)A]e^{i\omega_{0}T_{0}} +\alpha A^{3}e^{3i\omega\tau_{0}}+cc,\end{split}  \tag{9}\]  
where \(cc\) denotes the conjugate term. Eliminating secular terms from Equation (9), we have  
\[-2i\omega_{0}D_{1}A-i\mu\omega_{0}A+3\alpha A^{2}\overline{A}+\frac{f}{2}e^{i \sigma T_{1}}+i\omega_{0}p(\cos\omega\tau-i\sin\omega\tau)A=0. \tag{10}\]  
Let \(A=\frac{1}{2}a\exp(i\beta)\) and substituting it into Equation (10) and separating the real and imaginary part yield the averaged equation as follows:  
\[ \begin{split} a^{\prime}&=\frac{-\mu a}{2}-\frac{f} {2\omega}\sin\gamma+\frac{ap}{2}\cos\omega_{0}\tau,\\ a\gamma^{\prime}&=\sigma a+\frac{3\alpha a^{3}}{8 \omega_{0}}+\frac{f}{2\omega}\cos\gamma-\frac{ap}{2}\sin\omega_{0}\tau,\end{split}  \tag{11}\]  
where \(\gamma=\sigma T_{1}-\beta\). Then the frequency response relations are obtained by:  
\[ \begin{split}&\frac{f^{2}}{4\omega_{0}^{2}}=\frac{1}{4}(\mu-p \cos\omega_{0}\tau)^{2}a^{2}+[(\sigma-\frac{p}{2}\sin\omega_{0}\tau)-\frac{3 \alpha a^{2}}{8\omega_{0}}]^{2}a^{2},\\ &\tan\gamma=\frac{4\omega_{0}(\mu-p\cos\omega_{0}\tau)}{8\omega _{0}(\sigma-\frac{p}{2}\sin\omega_{0}\tau)+3\alpha a^{2}}.\end{split}  \tag{12}\]  
For a special case, the parameters \(\mu,\omega,\alpha\) are confirmed as \(\mu=0.02,\omega=1,\alpha=0.4\). The frequency-response curves (\(a\) versus \(\sigma\)) for the primary response are depicted in Figure 2. Selecting the control gain value \(p=-0.05\), it can be seen from the frequency amplitude curve that by adjusting different time delay, a better control effect can be obtained, and the peak value of the response can be significantly reduced. Figure 3 shows the variation of the response amplitude with the excitation amplitude for several values of \(\sigma\) and \(\tau\). These curves were obtained directly from Equation (12). We note that, depending on the value of \(\sigma\), some curves are multivalued while others are single-valued. Figure 2: Frequency response curves showing the primary resonances response of the nonlinear beam with time delayed velocity feedback. It is seen that the saddle-node (SN) bifurcations located on the primary branch. Then, to determine the lowest excitation amplitude, the locus of the SN bifurcation points in the \(f-\sigma\) plane is obtained in Figure 4 for \(p=-0.05\) and different values of \(\tau\). It is shown that the minimum point determines the minimum excitation level below and the excitation frequency band can also be determined. As the time delay increases, the amplitude of the minimum excitation also decreases. The trajectories of the damped motions of the nonlinear beam are plotted in Figure 5. Points \(P_{1}\) and \(P_{2}\) are stable foci, and point \(P_{3}\) is a saddle point.