# A Novel Piezoelectric Energy Harvester Using a Multi-Stepped Beam with Rectangular Cavities
## 3 Analytical Modeling  
The modeling of the harvester was carried out using the Euler Bernoulli beam theory. The harvester was made to vibrate at its natural frequency by applying a point tip force \(f(x,t)=F\sin(\omega t)\delta(x-L)\)  
Figure 1: Harvester; (**a**) structure, (**b**) sections II and IV, (**c**) section VII. with angular frequency \(\omega\) and magnitude \(F\). \(x\) representing the position on the beam between the fixed and free end \((0\leq x\leq L)\). The Dirac delta function \(\delta(x-L)\) was used to model a point force [31]. The electric charge (\(Q\)) generated on the surface of the piezoelectric patch 1, 2 and 3 bonded between \(L_{1}\) and \(L_{2}\), \(L_{3}\) and \(L_{4}\) and \(L_{6}\) and \(L_{7}\) and were found as follows [32]:  
\[Q_{1}^{(r)}(t) = -\int_{L_{1}}^{L_{2}}\Biggl{(}d_{31}y_{2}E_{p}b_{2}\frac{\partial ^{2}w_{2}^{(r)}(x,t)}{\partial x^{2}}\Biggr{)}dx \tag{1}\] \[Q_{2}^{(r)}(t) = -\int_{L_{3}}^{L_{4}}\left(d_{31}y_{4}E_{p}b_{4}\frac{\partial^{2 }w_{4}^{(r)}(x,t)}{\partial x^{2}}\right)dx\] (2) \[Q_{3}^{(r)}(t) = -\int_{L_{6}}^{L_{7}}\left(d_{31}y_{7}E_{p}b_{7}\frac{\partial^{2 }w_{7}^{(r)}(x,t)}{\partial x^{2}}\right)dx \tag{3}\]  
where \(d_{31}\) is the piezoelectric charge coefficient, \(E_{p}\) is the [PERSON]'s modulus of piezoelectric patches, \(b_{2}\), \(b_{4}\) and \(b_{7}\) are the width of patch 1, 2 and 3, \(y_{2}\), \(y_{4}\) and \(y_{7}\) are the distance between midpoints of the patches to the neutral axis, \(w_{2}^{(r)}(x,t)\), \(w_{4}^{(r)}(x,t)\) and \(w_{7}^{(r)}(x,t)\) are the \(r\)th mode transverse displacement of the beam at sections II, IV and VII. The transverse displacement \(w(x,t)\) of the beam was solved using the method of separation of variables [23]. \[w(x,t)=\sum_{r=1}^{\infty}W^{(r)}(x)T^{(r)}(t) \tag{4}\]  
where \(W^{(r)}(x)\) denotes the spatial mode shape function, \(T^{(r)}(t)\) represents the generalized time-dependent coordinate for the \(r\)th mode. Substituting Equation (4) into Equations (1)-(3), the charge can be rewritten as follows:  
\[Q_{1}^{(r)}(t) = -d_{31}y_{2}E_{p}b_{2}\Biggl{[}\left.\frac{dW_{2}^{(r)}(x)}{dx} \right|_{x=L_{2}}-\left.\frac{dW_{2}^{(r)}(x)}{dx}\right|_{x=L_{1}}\Biggr{]}T _{1}^{(r)}(t) \tag{5}\] \[Q_{2}^{(r)}(t) = -d_{31}y_{4}E_{p}b_{4}\[PERSON]{[}\left.\frac{dW_{4}^{(r)}(x)}{dx} \right|_{x=L_{4}}-\left.\frac{dW_{4}^{(r)}(x)}{dx}\right|_{x=L_{3}}\Biggr{]}T _{2}^{(r)}(t)\] (6) \[Q_{3}^{(r)}(t) = -d_{31}y_{7}E_{p}b_{7}\[PERSON]{[}\left.\frac{dW_{7}^{(r)}(x)}{dx} \right|_{x=L_{7}}-\left.\frac{dW_{7}^{(r)}(x)}{dx}\right|_{x=L_{6}}\[PERSON]{]}T _{3}^{(r)}(t) \tag{7}\]  
The spatial mode shape function \(W^{(r)}(x)\) was solved by using the free undamped vibration governing equation [33]:  
\[(EI)_{i}\frac{d^{4}W_{i}^{(r)}(x)}{dx^{4}}=\omega_{r}{}^{2}m_{i}W_{i}^{(r)}(x ),\ L_{i-1}\leq x\leq L_{i} \tag{8}\]  
where \(i=1,2,\ldots 8\)\(L_{0}=0\), \(L_{8}=L\)  
The free vibration solution of Equation (8) was given by:  
\[W_{i}^{(r)}(x)=A_{i}^{(r)}\sin\beta_{i}^{(r)}x+B_{i}^{(r)}\cos\beta_{i}^{(r)}x +C_{i}^{(r)}\mbox{sinh}\beta_{i}^{(r)}x+D_{i}^{(r)}\cosh\beta_{i}^{(r)}x \tag{9}\]  
where \(\beta_{i}^{(r)4}=\frac{\omega_{r}^{2}m_{i}}{(EI)_{i}}\), \(i=1,2,\ldots 8\)\(A_{i}^{(r)}\), \(B_{i}^{(r)}\), \(C_{i}^{(r)}\), \(D_{i}^{(r)}\) are unknown constants for the \(r\)th mode. \(\beta_{i}^{(r)}\)'s of each section can be replaced with a single parameter \(\beta^{(r)}\). \[Q_{1}^{(r)}(t) = -\int_{L_{1}}^{L_{2}}\Biggl{(}d_{31}y_{2}E_{p}b_{2}\frac{\partial ^{2}w_{2}^{(r)}(x,t)}{\partial x^{2}}\Biggr{)}dx \tag{1}\]0  
where \(\alpha_{i}=\left(\frac{m_{i}(EI)_{i}}{m_{1}(EI)_{i}}\right)^{\frac{1}{4}}\), \(\alpha_{1}=1\) and \(\beta^{(r)}=\beta_{1}^{(r)}\).