## 4 The Simplified Theoretical Model of the LPCB with One PCS
For the cantilever, the boundaries are as bellow:  
\[W(0)=0\ \ ;\ \ \ W^{\prime}(0)=0\ \ \ ;\ \ E{I}W^{\prime\prime}(L)=0\ \ ;\ \ E{I}W^{ \prime\prime\prime}(L)=0\]  
According to the shape function and the natural frequencies of the cantilever (force arm segment), the generalized coordinate can be expressed as follows:  
\[q_{n}(t)=\frac{1}{\overline{m}B\omega_{n}}\int_{0}^{t}Q_{n}(\tau)\sin{\omega_ {n}}(t-\tau)d\tau \tag{4}\]  
In which:  
\[\omega_{n}=(sL)_{n}^{2}\sqrt{\frac{EI}{\overline{m}L^{4}}},B=\int_{0}^{L}W_{n }^{2}(x)dx,Q(t)=\int_{0}^{L}F(t)W_{n}(x)\delta(x-L)dx=F(t)W_{n}(L)\]  
Based on the shape function, the generalized coordinates and principle of force amplifier shown in Figure 2a, the horizontal displacement (\(w(L,t)\)) and the bending moment (\(M(0,t)\)) of the cantilever (force arm segment) can be expressed as follows:  
\[w(L,t)=\sum_{n=1}^{\infty}W_{n}(L)q_{n}(t) \tag{5}\]  
\[M(0,t)=F_{h}(t)L=\frac{F(t)L}{2\tan{\theta(t)}} \tag{6}\]  
\[\tan{\theta(t)}=\frac{\sqrt{L_{c}^{2}-\left(L_{h}(t)\right)^{2}}}{L_{h}(t)} \tag{7}\]  
\[L_{h}(t)=w(L,t)+L_{h0} \tag{8}\]  
where \(w(L,t)\) is the horizontal displacement of the cantilever at \(x_{2}=L\); \(M(0,t)\) is the bending moment of the cantilever (force arm segment) at \(x_{2}=0\); \(F_{h}(t)\) is the horizontal force exerted on the force arm segment at time \(t\); \(L=l+na\) denotes the force arm length and \(n\) equal to 0, 1, or 2; \(F(t)\) is the vertical force from footstep at time \(t\); \(\theta(t)\) denoted the angle of the force amplifier at time \(t\); \(L_{c}\) is the length of \(BC/BD\); \(L_{h}\) and \(L_{h0}\) are the length and the initial length of horizontal projection of \(BC/BD\). Due to the bending moment exerted on the PCS is equal to \(M\left(0,t\right)\), the stress \(\sigma(t)\) and the strain \(\varepsilon(t)\) of the piezoelectric patch in PCS can be obtained as follows:  
Figure 5:0  
Figure 5:1  
Figure 5:2  
where \(I\) denotes the inertia moments of the PCS of LPCB; \(t_{1}\) and \(t_{p}\) express the substrate thickness and the piezoelectric patch thickness; \(l_{p}\) denotes the PCS length; \(E\) and \(E_{p}\) are the Young's modulus of the substrate and the piezoelectric patch. Figure 5:3  
Figure 5:4Figure 5:5  
Figure 5:6  
where \(Q_{g}\) (\(t\)) and \(V_{g}\) (\(t\)) are the generated electric charge and voltage for an open-circuit at time \(t\) from one piezoelectric patch of the brick; \(d_{31}\) is the piezoelectric constant; \(A_{p}\) and \(C\) are the surface area and the capacitance of the piezoelectric patch; \(V_{rms}\), \(V_{gmax}\), \(V_{gmin}\), and \(V_{d}\) are the root mean square of open-circuit voltage, the maximum of \(V_{g}\) (\(t\)), the minimum of \(V_{g}\) (\(t\)), and the peak-to-peak voltage density from one piezoelectric patch of the brick; \(\varepsilon_{0}\) is the permittivity of air and \(\varepsilon_{r}\) is the dielectric constant of the piezoelectric patch; \(T\) is a footstep period. Figure 5:7  
Figure 5:8  
Figure 5:9  
\[W(x_{2})=C_{1}\cos{sx_{2}}+C_{2}\sin{sx_{2}}+C_{3}\cosh{sx_{2}}+C_{4}\text{sinhs}x _{2} \tag{3}\]0  
where \(V_{rms}\) are the root mean square of close-circuit voltage; \(R\) and \(R_{c}\) means the load resistance and the internal resistance of PZT4 patch in the circuit; \(\omega\) is the excitation frequency; \(\overline{P}\), \(\overline{P}_{d}\), and \(\overline{P}_{\text{max}}\) represent the average power, the average power density, and the maximum average power from a PZT4 patch of an L-shaped beam respectively; \(R_{opt}\) is the optimum load resistance in the circuit and can be solved by \(R=R_{c}\) and \(\partial\overline{P}/\partial R=0\).