## 2 Statement of the Energy Harvester Problem  
In this section, we present a formulation of the initial-boundary value problem describing the dynamics of a specific model of an energy harvester. The electromechanical equations of motion are presented in transverse and rotational vibrations using Timoshenko beam theory. The beam is assumed to be excited by small (not necessary harmonic) transverse motion of the base. The schematic diagram of the piezoelectric, vibration-based energy harvester considered in this study is shown on Figure 1.  
It is a bimorph configuration consisting of a piezoelectric material layer bonded to both surfaces of a supporting, inactive core. Electrodes are assumed to cover the upper and lower surfaces of each layer and they are wired together in a parallel configuration. The voltage across each layer is assumed to be equal, and the charge displaced by each layer is additive. Because each layer experiences opposite strains (i.e., one layer is in tension while the other is in compression), they must be in the same direction to avoid charge cancellation. It is assumed that the electrodes and connecting wires have negligible resistance and that the resistivity of the piezoelectric material is significantly higher than that of the external circuitry (i.e., an insulator). In what follows, the equations of motion for the electromechanical system are derived through force, moment, and charge balances adopting the Timoshenko beam assumptions (see [PERSON] [23], [PERSON] et al. [24], [PERSON] [25]). Now, we are in a position to present mathematical statement of the problem. The model is governed by a system of three coupled partial differential equations for the following unknown functions: \(W(x,t)\): the relative transverse deflection of the beam with respect to its base; \(\Phi(x,t)\): the rotation angle of the beam cross-section; and \(v(t)\): the voltage across the energy harvester. To describe the dynamics of the non-homogeneous Timoshenko beam with the piezoelectric patches, we need the following notations: \(EI(x)\) is the flexural rigidity of a beam, \(\rho(x)\) is the mass density, \(S\) is the cross-sectional area of beam, and \(K(x)\) is the shear stiffness of a cross-section (as in [PERSON] [26], [PERSON] [27], [PERSON] [28], and [PERSON] et al. [29]). For the electrical circuit part, we denote by \(C\) the net clamped (i.e., constant strain) capacitance of the piezoelectric material. The electrical load attached to the energy harvester is assumed to be presented by a simple resistor with resistance \(R\); and \(\theta\) is the electromechanical coupling coefficient. The following boundary-value problem for the electromechanical system is valid:  
\[\rho(x)W_{tt}(x,t)+\frac{d}{dx}\bigg{(}K(x)\Big{(}\Phi(x,t)-W_{x}(x,t)\Big{)} \bigg{)}=-\rho(x)y_{tt}(t), \tag{1}\]  
\[ \begin{split}& S\rho\Phi_{tt}(x,t)-\frac{d}{dx}\Big{(}EI(x) \Phi_{x}(x,t)\Big{)}+\\ & K(x)\Big{(}\Phi(x,t)-W_{x}(x,t)\Big{)}+\theta v(t)\Big{(}\delta (x)-\delta(x-L)\Big{)}=0,\end{split}  \tag{2}\]  
Figure 1: Cantilevered vibration energy harvester in parallel bimorph configuration. \[v_{t}(t)+\frac{\theta}{C}\Phi_{t}(L,t)+\frac{1}{CR}v(t)=0. \tag{3}\]  
In Equation (1), \(y(t)\) is the absolute transverse base displacement, and \(\delta(\cdot)\) is the delta function in Equation (2). Since we consider the clamped-free model, the boundary conditions are (see [18; 30]):  
\[W(0,t)=\Phi(0,t)=0, \tag{4}\]  
\[\Phi(L,t)-W_{x}(L,t)=0,\qquad\Phi_{x}(L,t)=0. \tag{5}\]  
Note that the base excitation only directly contributes to the transverse dynamics of Equation (1); it does not appear in the rotational dynamics equation of Equation (2). Furthermore, the transverse dynamics is coupled to the rotational dynamics, and the rotational dynamics is coupled to the electrical dynamics, but the electrical dynamics is not directly coupled to the transverse dynamics.