# Vibration Energy Harvester Based on Bilateral Periodic One-Dimensional Acoustic Black Hole
## 2 VEH Integrated with Bilateral Periodic 1D ABH
### Dynamic Modeling of ABH Beam  
As shown in Figure 3, the left side of the beam is fix constrained and the right side is left free. It is assumed that the external excitation \(Y(x,t)=Y_{0}\sin Pt\) is applied at \(0.1\,x_{L}\) from the left end, where \(Y_{0}\) is the amplitude of the external excitation, and \(P\) is its frequency. The thickness of the constant thickness part of the beam is \(h_{m}\), \(x_{A}\) is the length of a single ABH unit, \(h_{t}\) is the truncated thickness of the ABH and \(x_{t}\) is the truncated extended platform length. The adverse effects of the truncated platform can be offset to some extent by properly adding damping material to the surface of the curved portion of the ABH. \(x_{E}\) is the length of the periodic joint in the middle and \(x_{L}\) is the total length of the beam. The shear deformation of the cantilever beam is neglected because the length of the ABH beam is much larger than its thickness, and the transverse vibration is dominant in practice. The undamped transverse vibration equation of the beam based on the Euler-Bernoulli beam theory is [18]:  
\[\rho T^{B}(x)\frac{\partial^{2}w(x,t)}{\partial t^{2}}+\frac{\partial^{2}M^{ T}(x,t)}{\partial x^{2}}=Y(x,t), \tag{1}\]  
where \(\rho\) is the material density of the ABH beam, \(T^{B}(x)\) is the cross-section area of the beam and \(M^{T}(x,t)\) is the bending moment of the beam. The ABH beam is a variable cross-section beam, which can be approximately equivalent to a linear combination of \(N\) equal cross-section beams as shown in Figure 4. When the \(N\) is large enough, the ABH beam can be approximated by the equivalent segmented beam, and then the vibration of the equivalent beam is analyzed by the transfer matrix method. The transfer matrix method is that any uniform beam in the segmented beam is  
Figure 3: Bilateral one-dimensional periodic ABH beam Model 1. Figure 2: Structure diagram of piezoelectric energy harvester. analyzed separately to determine its corresponding transfer matrix, and then the whole beam is combined and analyzed according to the continuity between the segmented beams to obtain the whole vibration response of the ABH beam. For the single-section uniform beam after discretization, if the damping layer is neglected, the transverse free vibration equation of the number \(i\) uniform beam can be expressed as [19]:  
\[\rho T_{i}^{B}(x)\frac{\partial^{2}w(x,t)}{\partial t^{2}}+\frac{\partial^{2}M _{i}^{\tau}(x,t)}{\partial x^{2}}=Y(x,t),x_{i}\leq x<x_{i+1},i=1,2,\ldots N. \tag{2}\]  
Let \(w(x,t)\) and \(j(x,t)\) be the components of the beam in the \(x\)-axis and \(z\)-axis directions, respectively, as shown in Figure 5. The shear and rotation of the beam are ignored since the thickness of the beam is much smaller than its length, where \(r\) and \(\theta\) are the beam's axial strain and rotation angle after flexural deformation. Then, the length of the deformed element can be expressed as [20]:  
\[dxw^{\prime}=\sqrt{\left(\frac{dw}{dx}\right)^{2}+\left(1+\frac{dj}{dx} \right)^{2}}dx, \tag{3}\]  
and the axial strain of the beam is:  
\[r=\frac{dx^{\prime}-dx}{dx}, \tag{4}\]  
the axial strain of the beam can be neglected when the excitation of the ABH beam is mainly transverse vibration, which means \(r=0\). At this time, the following can be obtained:  
\[\dot{f}^{\prime}=\sqrt{1-w^{\prime 2}}-1, \tag{5}\]  
\[\tan\theta=\frac{w^{\prime}}{1+\dot{f}^{\prime}}, \tag{6}\]  
by substituting Equation (5) into Equation (6), the rotation angle \(\theta\) can be expressed as:  
Figure 4.0  
The Taylor expansion of Equation (7) can be obtained:  
Figure 4.1  
the curvature \(\lambda_{\theta}\) of the beam element after deformation can be expressed as:  
Figure 4.2  
Figure 4.3 Sectional diagram of ABH beam. in summary, the transfer matrix method mechanical model of the ABH beam considering the nonlinear strain can be expressed as:  
\[