## 2 Materials and Methods
The electric circuit can be found by using the following equations, which are based on Kirchhoff's law and the study of alternating current circuits:  
\[\frac{d^{2}\eta_{k}(t)}{dt^{2}}+2\xi\omega_{k}\frac{d\eta_{k}(t)}{dt}+\omega_{k }^{2}\eta_{k}(t)=f_{k}(t). \tag{15}\]0  
where \(R_{z}\) is the resistance connected for the purpose of measuring voltage drop due to vibrational energy (shown in Figure 1), \(C_{p}\) represents the electric capacity of the MFC transducer, and \(V_{c}\) represents the voltage on the electrodes of the capacitor. Using Equation (22) through (24) we can obtain:  
\[\frac{d^{2}\eta_{k}(t)}{dt^{2}}+2\xi\omega_{k}\frac{d\eta_{k}(t)}{dt}+\omega_{k }^{2}\eta_{k}(t)=f_{k}(t). \tag{15}\]1  
where:  
\[\frac{d^{2}\eta_{k}(t)}{dt^{2}}+2\xi\omega_{k}\frac{d\eta_{k}(t)}{dt}+\omega_{k }^{2}\eta_{k}(t)=f_{k}(t). \tag{15}\]2  
* \(\gamma\) is the surface area of the electrodes in the considered transducer and was defined as an area of the piezo-element active part multiplied by a copper fiber volume fraction, and \(l_{b},b_{p},t_{p}\) are active length, width, and thickness of the transducer, respectively. The value of the effective power can be computed based on the relationship: \[\frac{d^{2}\eta_{k}(t)}{dt^{2}}+2\xi\omega_{k}\frac{d\eta_{k}(t)}{dt}+\omega_{k }^{2}\eta_{k}(t)=f_{k}(t). \tag{15}\]3 (27)  
where \(V_{a}\) represents the voltage's root mean square value that is produced by the transducer. Commercially available MFC transducers are ready-made piezoelectric transducers in the form of thin, flexible films. The transducer is made of rectangular piezo-ceramic bars sandwiched between layers of electrodes and polyimide foil. The transducer's electrodes are connected in such a way that they form an interdigitated system. Because of this design, the transducer is a ready-to-use, long-lasting element that can be used in systems for energy recovery from mechanical vibrations, among other things. There are numerous piezoelectric transducer solutions that are attached to the surfaces of vibrating mechanical systems or integrated with them by laminating them in composite structures. The harvester in this paper is made up of three MFC transducers of type M8514P2, laminated between layers of a composite panel. The system is made of a glass fabric with a twill weave and a weight of 600 g/m\({}^{2}\). The matrix consisted of Ig700 epoxy resin and hg700 hardener mixed in a mass ratio of 100/30. The detailed procedure of the system built with material properties can be found in a separate paper [29]. Figure 2 depicts a ready-to-use composite panel with MFC piezoelectric transducers. The developed solution is an intelligent structure that can function as both a system for generating electricity from mechanical vibrations and a structure which dynamic parameters can be controlled using piezoelectric transducers laminated within it. The tested piezoelectric transducer's terminals were connected to an external resistor (positioned outside of the thermal chamber), and an electric voltage drop was noted on it with a National Instruments measuring card NI9215 with a monitoring range of \(-10\) V, \(+10\) V. The measurement card was installed in the cDAQ-9191 CompactDAQ Chassis and connected to a computer, where the results were noted using LabVIEW software. During measurements, the number of cycles was set to 500. Data was collected during the tests using a voltage trigger released by the analogue output of the Instron controller only during the excitation of the test samples. Figure 2 depicts a view of a laminated MFC test sample. The MFC piezoelectric transducer was evaluated using a temperature chamber equipped with a universal electromechanical testing machine Instron ElectroPuls E10,000 (Instron, High Wycombe, UK) with variable strain-controlled dynamic forcing function. The test was conducted with displacement control. The strain was measured using an extensometer designed to record under dynamic loading conditions.