# Novelty in Intelligent Controlled Oscillations in Smart Structures
## 2 Motion Formula of the Smart Structure  
The beam formula for both mechanical and electrical loads is calculated by [7; 17; 18; 19; 28; 33]:  
\[EI\frac{\partial^{4}y(t,x)}{\partial x^{4}}+\rho bAb\frac{\partial^{2}y(t,x)}{ \partial t^{2}}=f_{\text{m}}(t,x)+f_{e}(t,x) \tag{1}\]  
\(E\) is [PERSON]'s modulus, \(I\) is the moment of inertia, \(p\) is density, \(A\) is area, \(f_{\text{e}}\) is the electrical force, and \(f_{\text{m}}\) is the disturbance (mechanical force) [33]. Figures 1 and 2 depict a smart beam incorporating an embedded piezoelectric actuator that produces a mechanical load in the form of a force when subjected to an electrical current [6; 7]. Figure 1 shows a smart beam with embedded actuators. Figure 1: Piezoelectric patch attached to a beam. Figure 2: One pair of actuator patches. Figure 2 shows the actuator integrated into the beam. This figure shows the ends of the actuator used in further equations, which are very important for the electrical matrices of our system. All modeling is based on Equation (1) [33]; from the solution of this equation, the results are given in the time domain. The results without control are given by applying the differential, whereas the results with control are given after the application of the infinity control. To determine the electric force generated by the piezoelectric actuator f\({}_{e}\)(t,x), the following equation was used:  
\[f_{e}(t,x)=\frac{\partial^{2}M_{px}(t,x)}{\partial x^{2}} \tag{2}\]  
where M\({}_{\text{px}}\) is the piezoelectric actuator's bending moment. Where Pzt (piezoelectric patch) placed on the beam is indicated by the step function \(H\). The bending moment M\({}_{\text{px}}\) is given by  
\[M_{px}(t,x)=C_{0}e_{pe}(t)\big{[}H\big{(}x-x_{1j}\big{)}-H\big{(}x-x_{2j}\big{)} \big{]}u_{j}(t) \tag{3}\]  
This equation is derived from the theory of piezoelectric materials, where [16; 17]  
\[C_{0}=EI\cdot K_{f} \tag{4}\]  
\[K_{f}=\frac{12 EE_{p}hh_{p}\Big{(}2h+h_{p}\Big{)}}{16E^{2}h^{4}+EE_{p}\Big{(}32 h^{3}h_{p}+24h^{2}h_{p}^{2}+8 hh_{p}^{3}\Big{)}+E_{p}^{2}h_{p}^{4}} \tag{5}\]  
The piezoelectric patch mechanical tension e\({}_{\text{pe}}\)(t) is derived by  
\[e_{pe}(t)=\frac{d_{31}}{h_{p}}u_{j}(t) \tag{6}\]  
The constant d\({}_{31}\) combines the electric intensity e\({}_{\text{pe}}\)(t) with the electric voltage u\({}_{\text{j}}\)(t), which is generated in an actuator j (Figure 2). Where d\({}_{31}\) is the piezoelectric constant d31 = 280 \(\times\) 10\({}^{-12}\) m/V (Table 1). Thus, Function (3) can be expressed as a flexural moment as follows:  
\[M_{px}(t,x)=C_{p}\big{[}H\big{(}x-x_{1j}\big{)}-H\big{(}x-x_{2j}\big{)} \big{]}u_{j}(t) \tag{7}\]  
where  
\[C_{p}=EIK_{f}\frac{d_{31}}{h_{p}}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Specifications of the Beam** & **Value** \\ \hline L stands for length & 1.40 m \\ \hline W stands for the width & 0.07 m \\ \hline h is height & 0.02 m \\ \hline \(\rho\) represents the density & 1700 kg/m\({}^{3}\) \\ \hline E is the [PERSON]’s modulus & 1.8 \(\times\) 10\({}^{11}\) N/m\({}^{2}\) \\ \hline Pzt thickness is bs and ba & 0.003 m \\ \hline d31 is the electrical conductivity & 280 \(\times\) 10\({}^{-12}\) m/V \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Smart beam specifications (graphite/epoxy T300/976).