## 2 Equations of Motion  
The mathematical model for the cantilever is based on the nonlinear Euler-Bernoulli beam theory.The partial-differential equation of planar motion and that associated with an external excited elastic beam are as follows [54]:  
\[m\ddot{v}+c\dot{v}-EAp(t)v^{\prime\prime}+Elv^{\prime\prime\prime\prime}+kv \delta(x-l)-\frac{EA}{2l}v^{\prime\prime}\int_{0}^{l}v^{\prime 2}dx=q(x,t)+F_{n}(x) \cos\Omega t, \tag{1}\]  
\[v(0,t)=0,v^{\prime}(0,t)=0,v^{\prime\prime}(l,t)=0,Elv^{\prime\prime\prime}(l,t)+kv(l,t)=0, \tag{2}\]  
where \(m\) is the linear density; \(c\) is the coefficient of linear viscous damping per unit length; \(v\) denotes the displacement component along the \(y\)-axis; the primes and overdots indicate the derivatives with respect to the arc length \(x\) and time \(t\), respectively; \(E\) is [PERSON]'s modulus of elasticity; \(A\) is the cross-sectional area; \(l\) is the moment of inertia about the neutral axis of the beam; \(l\) is the length of the beam; and \(p(t)\) is the axial force; The distributed load \(q(x,t)\) of the piezoelectric actuator (see Figure 1), is given by:  
\[q(x,t)=\frac{\partial^{2}M}{\partial x^{2}}, \tag{3}\]  
where \(M\) is a uniformly-distributed bending moment expressed as:  
\[M=\frac{3 bdd_{31}E_{a}EIV_{a}(t)}{3 EI+6 bE_{a}t_{a}d^{2}+2 bE_{a}t_{a}^{3}}[H(x- x_{1})-H(x-x_{2})]. \tag{4}\]  
where \(b\) and \(t_{a}\) are the width and thickness of the piezoelectric actuator, respectively; \(d_{31}\) is a piezoelectric constant; \(E_{a}\) is the actuator Young's modulus; \(t_{b}\) is the thickness of the beam; \(V_{a}(t)\) is the control voltage; \(H(x)\) is the Heaviside step function; and \(x_{1}\) and \(x_{2}\) are the starting and ending coordinates of the piezoelectric strip. In this paper, the time delayed feedback control is used to suppress the large vibration of the beam. The block diagram is shown in Figure 2.  
We derive a reduced-model for the system under consideration by using the [PERSON] procedure in the form:  
\[\
u=\sum_{i=1}^{\infty}\phi_{i}(x)q_{i}(t), \tag{5}\]  
where the \(q_{i}(t)\) are generalized temporal coordinates and the \(\phi_{i}(x)\) are the linear mode shapes of a elastic beams and are given by:  
\[\phi_{i}(x)=A_{i}\left[\cos\delta_{i}x-\cosh\epsilon_{i}x-\sigma_{i}\left(\sin \delta_{i}x-\frac{\delta_{i}}{\epsilon_{i}}\sinh\epsilon_{i}x\right)\right] \tag{6}\]  
Figure 1: The theory model of the controlled beams. Figure 2: A block diagram of the time-delayed feedback control. where:  
\[\sigma_{i}=\frac{\delta_{i}^{2}\cos\delta_{i}l+\varepsilon_{i}^{2}\cosh\varepsilon _{i}l}{\delta_{i}^{2}\sin\delta_{i}l+\varepsilon_{i}\delta_{i}\sinh\varepsilon _{i}l},\delta_{i}=\sqrt{\left(r_{i}+\frac{g^{4}}{4}\right)^{1/2}+\frac{g^{2}}{4 }},\varepsilon_{i}=\sqrt{\left(r_{i}+\frac{g^{4}}{4}\right)^{1/2}-\frac{g^{2}} {4}},\]  
and \(g^{2}=p(t)/EI\), while \(r_{i}\) is calculated using the following transcendental equation:  
\[v(0,t)=0,v^{\prime}(0,t)=0,v^{\prime\prime}(l,t)=0,Elv^{\prime\prime\prime}(l,t)+kv(l,t)=0, \tag{2}\]0 \[v(0,t)=0,v^{\prime}(0,t)=0,v^{\prime\prime}(l,t)=0,Elv^{\prime\prime\prime}(l,t)+kv(l,t)=0, \tag{2}\]1  
Substituting Equation (5) into Equation (1) and using the [PERSON] method, we obtain the following set of nonlinear ordinary differential equations:  
\[ \begin{array}{l}\ddot{q}_{n}(t)+\mu_{n}\dot{q}_{n}(t)+\omega_{n}^{2}q_{n}(t )+\sum\limits_{i,j,k=1}^{\infty}\Gamma_{nijk}q_{i}(t)q_{j}(t)q_{k}(t)\\ \hskip 113.811024 pt=M_{n}V_{a}(t)+f_{n}(x)\cos\Omega t,\;n=1,2,...,\infty, \end{array}  \tag{8}\]  
where:  
\[ \begin{array}{l}\mu_{n}\quad=\frac{c}{m},\quad\Gamma_{nijk}=-\frac{EA}{2 ml} \int_{0}^{l}\phi_{i}^{\prime\prime}(x)\phi_{n}(x)\int_{0}^{l}\phi_{j}^{\prime }(x)\phi_{k}^{\prime}(x)dxdx,\\ M\quad=\frac{3 bdd_{3}E_{4}EI}{3E_{1}+6 bE_{4}tad^{2}+2 bE_{4}t^{2}}[\phi(x_{1})- \phi(x_{2})],\quad f_{n}=F_{n}\sum_{0}^{\infty}\phi_{n}dx.\end{array}  \tag{9}\]  
We nondimensionalize Equation (8) and obtain:  
\[