## 3 Superharmonic Resonance
The steady-state motions correspond to \(a_{n}^{\prime}=\gamma_{n}^{\prime}=0\); that is, they correspond to the solutions of:  
\[-\frac{1}{2}\mu_{n}a_{n}+\frac{k_{amn}a_{n}}{2\omega_{0}}\sin(\omega_{0}\tau)= \frac{\Gamma_{mmn}\Lambda_{n}^{3}}{\omega_{0}}\sin\gamma_{n} \tag{24}\]  
\[\left(\sigma-\frac{3\Gamma_{mmn}\Lambda_{n}^{2}}{\omega_{0}}\right)a_{n}- \frac{3\Gamma_{mmn}a_{n}}{8\omega_{0}}a_{n}^{3}-\frac{k_{amn}a_{n}}{2\omega_{0 }}\cos(\omega_{0}\tau)=\frac{\Gamma_{mnm}\Lambda_{n}^{3}}{\omega_{0}}\cos \gamma_{n} \tag{25}\]  
Squaring and adding these equations leads to the frequency-response equation:  
\[\left[\left(-\frac{1}{2}\mu_{n}+\frac{k_{amn}}{2\omega_{0}}\sin (\omega_{0}\tau)\right)^{2}+\left(\sigma-\frac{3\Gamma_{mmn}\Lambda_{n}^{2}}{ \omega_{0}}-\frac{3\Gamma_{mmn}}{8\omega_{0}}a_{n}^{2}-\frac{k_{amn}}{2 \omega_{0}}\cos(\omega_{0}\tau)\right)^{2}\right]a_{n}^{2} \tag{26}\] \[=\frac{\Gamma_{nmm}^{2}\Lambda_{n}^{6}}{\omega_{0}^{2}}.\]  
Here, we study the stability of the steady-state motion, setting:  
\[a=a_{n0}+a_{n1},\gamma_{n}=\gamma_{n0}+\gamma_{n1}, \tag{27}\]  
Substituting Equation (27) into Equations (21) and (22), expanding for small \(a_{n1}\) and \(\gamma_{n1}\), noting that \(a_{n0}\) and \(\gamma_{n0}\) satisfy Equation (24), and keeping linear terms in \(a_{n1}\) and \(\gamma_{n1}\), we obtain:  
\[a_{n1}^{\prime}=-\frac{1}{2}\mu_{ne}a_{n1}-a_{n0}(\sigma_{e}-\frac{3\Gamma_{ mnm}\Lambda_{n}^{2}}{\omega_{0}}-\frac{3\Gamma_{mnm}a_{n0}^{2}}{8\omega_{0}}) \gamma_{n1}, \tag{28}\]  
\[a_{n}^{\prime}=-\frac{1}{2}\mu_{ne}a_{n}-\frac{\Gamma_{mmn}\Lambda_{n}^{3}}{ \omega_{0}}\sin\gamma_{n}, \tag{21}\]0  
Using Equations (28) and (29), one can obtain the following eigenvalue equation:  
\[\left| \begin{array}{cc}-\frac{1}{2}\mu_{ne}-\lambda&-a_{n0}(\sigma_{e}- \frac{3\Gamma_{mnm}\Lambda_{n}^{2}}{\omega_{0}}-\frac{3\Gamma_{mnm}a_{n0}^{2} }{8\omega_{0}})\\ \frac{1}{a_{n0}}(\sigma_{e}-\frac{3\Gamma_{mmn}\Lambda_{n}^{2}}{\omega_{0}}- \frac{9\Gamma_{mmm}}{8\omega_{0}}a_{n0}^{2})&-\frac{1}{2}\mu_{ne}-\lambda \end{array} \right|=0\]  
Expanding this determinant yields:  
\[\lambda^{2}+\mu_{ne}\lambda+\rho=0, \tag{30}\]  
where:  
\[\rho=\frac{1}{4}\mu_{ne}^{2}+(\sigma_{e}-\frac{3\Gamma_{mnm}\Lambda_{n}^{2}}{ \omega_{0}}-\frac{3\Gamma_{mnm}a_{n0}^{2}}{8\omega_{0}})(\sigma_{e}-\frac{3 \Gamma_{mnm}\Lambda_{n}^{2}}{\omega_{0}}-\frac{9\Gamma_{mmn}}{8\omega_{0}}a_{ n0}^{2}).\]  
Hence, the steady-state motions are stable when \(\mu_{ne}>0\) and \(\rho>0\), and are otherwise unstable. Through concrete examples, we carry out the numerical analysis and discussion of the superharmonic resonance response of the first order modal of the controlled beam. Geometric dimensions and material characteristic parameters of the beam and piezoelectric actuator are as follows. Beam: \(l=99.62\times 10^{-2}\) m, \(A=15.36\times 10^{-4}\) m\({}^{2},E=34.5\) GPa, \(l=9.8662\times 10^{-8}\) m\({}^{4},k=6.872\times 10^{4}\) N/m, \(p=2.574\times 10^{-1}\) kN, \(m=4.4\) kg/m.Piezoelectric actuator: \(d_{31}=-270\times 10^{-12}\) m/V, \(E_{a}=108\) GPa, \(b=0.2\times 10^{-2}\) m, \(2t_{a}=0.04\times 10^{-2}\) m, \(d=0.5\times 10^{-2}\) m, \(x_{1}=12\times 10^{-2}\) m, \(x_{2}=18\times 10^{-2}\) m, \(x_{3}=80\times 10^{-2}\) m. For such an elastic beam, the first four non-dimensional natural frequencies and eigenfunctions are shown in Figure 3.  
In order to more intuitively display the suppression effect of the delay feedback control, Figure 4a shows the response of the system with no control, active control, and time delay feedback control. It shows that the time delay feedback control can achieve significant vibration suppression effects, and the effect is better than active control. On the other hand, the delay feedback control depends on two important parameters, the control gain and the time lag value. If the parameters are not properly selected, the system response will increase, as shown in Figure 4b.