## 2 Electromechanical Model  
In this study, we investigated the mathematical model of a slitted cantilever beam energy harvester with a PZT-5 layer on top, as shown in Figure 1. The beam is modelled using Euler-Bernoulli beam theory, considering tip masses on both free ends and incorporating damping effects. It is assumed that plane sections remain plane and perpendicular to the neutral axis after deformation. The figure is for illustrative purposes, but it is important to note that the PZT material is brittle and requires substantial support from the substrate and intermediate layers to prevent cracking during testing, as seen in commercially available designs. For a cantilever beam with a piezoelectric layer, the strain \(S\) due to bending can be expressed as [46]  
\[S(x,z,t)=-z\frac{\partial^{2}w(x,t)}{\partial x^{2}} \tag{1}\]  
where \(z\) is the distance from the neutral axis of the beam, and \(w(x,t)\) is the transverse displacement of the beam at a distance \(x\) from the fixed end and at time \(t\). This displacement \(w(x,t)\) represents the deflection of the beam in the direction perpendicular to its length due to bending. The piezoelectric constitutive equations relating the mechanical stress \(T\) and electric displacement \(D\) to the mechanical strain \(S\) and electric field \(E\) are [47]  
\[ \begin{cases}T=c^{E}S-eE\\ D=eS+\varepsilon^{S}E\end{cases}  \tag{2}\]  
where \(c^{E}\) is the stiffness at a constant electric field, \(e\) is the piezoelectric stress constant, and \(\varepsilon^{S}\) is the permittivity at constant strain. The electric field in the piezoelectric layer is related to the voltage \(V(t)\) across the thickness \(h_{p}\) of the piezoelectric layer as follows [48]:  
\[E_{3}=\frac{V(t)}{h_{p}} \tag{3}\]  
Figure 1: Schematic of a slitted beam of overall width \(\mathrm{b}_{\mathrm{b}}\), thickness \(\mathrm{h}_{\mathrm{s}}\), PZT layer thickness \(\mathrm{h}_{\mathrm{p}}\), and tip mass configuration; beam rigidly fixed at non-slit end. The bending moment \(M_{p}(x,t)\) due to the piezoelectric effect in a slitted beam can be derived by integrating the stress over the piezoelectric layer:  
\[M_{p}(x,t)=\int_{-h_{p}/2}^{h_{p}/2}T(x,z,t)\mathrm{b_{p}}dz \tag{4}\]  
where \(\mathrm{b_{p}}\) is the width of the piezoelectric layer, while \(h_{p}\) is the thickness of the piezoelectric layer. Substituting for \(T(x,z,t)\) and integrating, we obtain  
\[Mp(x,t)=\int_{-h_{p}/2}^{h_{p}/2}\bigg{[}c^{E}\bigg{(}-z\frac{\partial^{2}w(x, t)}{\partial x^{2}}\bigg{)}-e\ E_{3}\bigg{]}b_{p}dz \tag{5}\]  
Considering the piezoelectric coupling term \(\theta\)[49],  
\[\theta=-e\int_{-h_{p}/2}^{h_{p}/2}b_{p}dz=-d_{31}E_{p}b_{p}h_{p} \tag{6}\]  
where \(d_{31}\) is the piezoelectric strain constant, and \(E_{p}\) is [PERSON]'s modulus of the piezoelectric material. For a slitted beam, the coupling term must consider the reduced cross-sectional area due to the slit(s). The effective width \(b_{eff}\) can be approximated by subtracting the width of the slits from the total width:  
\[b_{eff}=b_{p}-nw_{slit} \tag{7}\]  
where \(n\) is the number of slits, and \(w_{slit}\) is the width of each slit. Thus, the modified coupling term \(\theta_{\mathit{eff}}\) is  
\[\theta_{eff}=-d_{31}E_{p}b_{eff}h_{p}\frac{\partial^{2}w(x,t)}{\partial x^{2} }=-d_{31}E_{p}(b_{p}-nw_{slit})h_{p} \tag{8}\]  
The introduction of slits in a cantilever beam alters the stiffness and mode shapes, leading to mode coupling, which results in two resonance peaks and anti-resonance. The slit changes the effective stiffness, splitting a single resonance into two distinct peaks. The transverse displacement \(w(x,t)\) of the beam can be expressed as a superposition of mode shapes:  
\[w(x,t)=\sum_{r=1}^{2}\phi_{r}(x)\eta_{r}(t) \tag{9}\]  
where \(\phi_{r}(x)\) represents the mode shapes, and \(\eta_{r}(t)\) denotes the modal coordinates.