## 4 Optimization of the Damping Layer's Position and Thickness
Considering that the output value of the Sigmoid function ranges from 0 to 1, and the derivative ranges from 0 to 0.25, it can conveniently control the magnitude of weight updates during the backpropagation process and avoid the problem of gradient vanishing. The choice of activation function is the Sigmoid function. When seeking the minimum value of the error function, the Levenberg-Marquardt algorithm [40] is used for iteration. The algorithm is an improved method that lies between the Gauss-Newton method and the gradient descent method [41]. The algorithm incorporates a step-size adjustment parameter. In situations where the error function decreases sharply in the direction of the gradient, a reduced step-size factor is applied, thereby aligning the method more closely with the Gauss-Newton approach. Conversely, when the descent is slower, a larger step-size factor is used, making the algorithm more similar to the gradient descent method. Figure 10: A BP neural network structure diagram with one hidden layer. We set the following range for the six input parameters separately,  
\[c_{0}\in[0,0.2],\quad c_{1}\in[-2,2],\quad c_{2}\in[-0.06,0]\] \[c_{3}\in[0,0.005],\quad n\in[0,3],\quad[x_{4J},x_{4J}]\subseteq[0.1,0.3],\]  
and use the larger mean of output data and the smaller variance as the evaluation criteria. The training samples and validation samples both originate from theoretical computational results. Figure 11 illustrates the degree of fit of the BP network after being trained with Matlab software. It is evident that the linear correlation between the output data of the BP model and the training, validation, and test datasets exceeds 0.84 in each case. The overall fit rate of the entire model is 0.98657, indicating a high degree of fit, making the model acceptable and suitable as a predictive network model. The optimal damping layer thickness variation function predicted by the BP network is  
\[h_{dp}(x)=\frac{4}{3}(|x-0.2|-0.03)^{2},\quad x\in[0.17,0.23] \tag{30}\]  
Figure 12 displays the graphical representations of the functions \(h_{dp}\), as defined by Equation (30), and \(h_{dP}\), as defined by Equation (27). The comparison of the effect of different damping layer thicknesses (varying according to the functions \(h_{dp}\) and \(h_{dP}\)) on the ABH effect is shown in Figure 13. It is clear that as the resonance order increases, the modal loss factors of the system with damping layer thickness set according to the function \(h_{dP}\) gradually stabilize around 0.029, whereas with the damping layer thickness set according to the function \(h_{dp}\), the modal loss factors gradually stabilize around 0.035. The average value of the modal loss factors for the first fifteen modes of the system with damping layer thickness set according to the function \(h_{dP}\) is 0.0252, while for the function \(h_{dp}\) it is 0.0293,  
Figure 11: The BP neural network regression analysis chart. representing an approximate 16.2% increase compared to the \(h_{d\tau}\) scenario. Figure 13 shows the effectiveness of optimization using the BP neural network. The analysis based on BP neural network optimization indicates that to achieve a good ABH effect, it is not necessary to apply the damping layer over the entire ABH section. The thickness of the damping layer should be maximized at the ABH tip, with a swift reduction as one moves away from the tip. Additionally, it is unnecessary to apply a damping layer in regions that are distant from the tip. Given that the thickness of the damping layer significantly affects the vibration reduction performance of the system, precise control over the damping layer thickness on the ABH beam is essential in practical applications. To further elucidate the impact of the damping layer thickness tolerances on the ABH effect, we consider the damping layer thickness varying according to the following two functions:  
\[ \begin{split} h_{dpa}(x)&=\frac{4}{3}(|x-0.2|-0.031)^ {2},\\ h_{dps}(x)&=\frac{4}{3}(|x-0.2|-0.029)^{2},\qquad x \in[0.17,0.23].\end{split}