# Novelty in Intelligent Controlled Oscillations in Smart Structures
## 2 Motion Formula of the Smart Structure
### Modelling  
In this work, the Eyler-Bernoulli beam equation (Equation (1)) is used, by integrating this equation and using a Hermite multivariable, the local matrices are derived for the mass, stiffness, and damping electric charge matrices of the structure. Assembling is then performed on the global stiffness model. A reference related to the finite element method is given [1; 33]. In our analysis, we use two degrees of freedom in the finite element method--the transport deflection \(\psi\) and the rotation. In addition, we use two different disturbances, one static and one dynamic. The goal of this study was to reduce oscillations using intricate control techniques and piezoelectric materials. In particular, the locations of piezoelectric actuators were defined. Figure 4 shows the actuators positioned at each of the four points (labels 1, 2, 3, and 4) along the beam. The dynamical description of the system is given by [1; 35]  
\[\mathrm{M}\ddot{q}(t)+\mathrm{D}\dot{q}(t)+\mathrm{K}\mathrm{q}(t)=\mathrm{fm}( t)+\mathrm{fe}(t) \tag{10}\]  
where fe represents the results of the electromechanical coupling affecting the global control force vector, D is the viscous damping matrix, K is the global stiffness matrix, and M is  
Figure 3: An intelligent beam that incorporates piezoelectric actuators and sensors. the global mass matrix. fm is the mechanical vector of the global external loading and fe is the electrical vector. The independent variable is a vector q(t) composed of transversal deflections \(\psi_{\mathrm{i}}\) and rotations \(w_{\mathrm{i}}\), or  
\[q(t)= \begin{bmatrix}w_{1}\\ \psi_{1}\\ \vdots\\ w_{n}\\ \psi_{n}\end{bmatrix}  \tag{11}\]  
where n denotes the finite element number employed in the analysis [7]. As is common practice, we switch the representation to a state-space control form [9; 10; 11; 12; 13; 14; 15]. \[ \begin{array}{l}x(t)=\begin{bmatrix}q(t)\\ \dot{q}(t)\end{bmatrix}\\ =\begin{bmatrix}0_{2 nxn}\\ \mathrm{M}^{-1}(f_{m}(t)+f_{e}(t)\end{bmatrix}+\begin{bmatrix}\dot{q}(t)\\ -\mathrm{M}^{-1}D\dot{q}(t)-\mathrm{M}^{-1}Kq(t)\end{bmatrix}\\ =\begin{bmatrix}0_{2 nxn}\\ \mathrm{M}^{-1}(f_{m}+f_{e})(t)\end{bmatrix}+\begin{bmatrix}0_{2 nx2n}&I_{2 nx2n} \\ -\mathrm{M}^{-1}K&-\mathrm{M}^{-1}D\end{bmatrix}\begin{bmatrix}q(t)\\ \dot{q}(t)\end{bmatrix}\\ =\begin{bmatrix}0_{2 nxn}\\ \mathrm{M}^{-1}f_{m}(t)\end{bmatrix}+\begin{bmatrix}0_{2 nxn}\\ \mathrm{M}^{-1}f_{e}(t)\end{bmatrix}+\begin{bmatrix}0_{2 nx2n}&I_{2 nx2n}\\ -\mathrm{M}^{-1}K&-\mathrm{M}^{-1}D\end{bmatrix}\begin{bmatrix}q(t)\\ \dot{q}(t)\end{bmatrix}\end{array}  \tag{12}\]  
Additionally, the value \(f_{e}(t)\) is denoted as where the unit's piezoelectric force (of size 2n \(\times\) n) is when it is placed on the proper actuator. Figure 4: The actuators were positioned over the entire smart structure. where u denotes the actuator voltage. The disturbance vector is denoted as d(t) = f\({}_{\text{m}}\)(t). Then,  
\[ \begin{array}{ll}\dot{\text{x}}(t)=\begin{bmatrix}0_{2n\times 2n}&I_{2n\times 2n}\\ -M^{-1}K&-M^{-1}D\end{bmatrix}\text{x}(t)+\begin{bmatrix}0_{2n\times n}\\ M^{-1}F_{\text{e}}^{-1}\end{bmatrix}\text{u}(t)+\begin{bmatrix}0_{2n\times 2n} \\ M^{-1}\end{bmatrix}\text{d}(t)\\ &=\text{Ax}(t)+\text{Bu}(t)+\text{Gd}(t)\\ &=\text{Ax}(t)+[\text{BG}]\begin{bmatrix}\text{u}(t)\\ \text{d}(t)\end{bmatrix}\\ &=\text{Ax}(t)+\widetilde{\text{Bu}}\widetilde{\text{u}}(t),\end{array}  \tag{14}\]  
The output function (displacement measured alone) can be used to enhance this result. \[\text{y}(t)=[\text{x}1(t)\text{ x}3(t)\ldots\text{xn }-1(t)]T=\text{C x}(t)\]  
where  
\[\text{C}=[1\ 0\ 0\ldots 0;\ -1\ 0\ 1\ 0\ldots 0;\ 0\ 0\ -1\ 0\ 1\ldots 0 ;\ 0\ 0\ 0\ -1\ 0\ 1\ldots 0]\]  
The ability to convert mechanical stress into strain, and vice versa, is a feature of the piezoelectric effect. The reduction in oscillations attained in this study is based on this. Actuator pairs co-localized with piezoceramics (PZT G-1195) were employed in our simulation for laminated composite beams made of graphite/epoxy, aluminum, and glass/epoxy (Figure 5).