## 4 Optimization of the Damping Layer's Position and Thickness  
From the analysis in the previous section, we consider that the thickness of the damping layer varies as follows:  
\[h_{d}(x)=c_{1}(|x-c_{0}|+c_{2})^{n}+c_{3}, \tag{28}\]  
where \(c_{0,1,2,3}\) and \(n\) are the parameters that need to be optimized for determining the best damping layer thickness variation function under conditions (24) and (25). In this section, we will find the optimal form of the damping layer thickness variation function based on BP neural networks. BP neural networks are a class of artificial neural networks that are widely used for various tasks such as pattern recognition, prediction, and data classification. A BP neural network typically consists of an input layer, one or more hidden layers, and an output layer. Each layer is composed of a set of neurons, and each neuron is connected to the neurons in the next layer through weights. A typical BP neural network structure is shown in Figure 10.  
When a training example is presented, the input is passed through the network. Each neuron in the hidden layers computes a weighted sum of its inputs and applies an activation function to it. The activation function introduces nonlinearity into the model, allowing it to learn complex relationships. Common activation functions used in BP neural networks include the Sigmoid function, hyperbolic tangent (tanh), and the rectified linear unit (ReLU). The choice of activation function can have a significant impact on the network's performance. After the forward pass, the network's prediction is compared to the actual target value, and the error is calculated. The choice of loss function depends on the specific task. Common loss functions include mean squared error for regression tasks and cross-entropy loss for classification tasks. This error is then propagated back through the network, adjusting the weights to minimize the error. The process involves the calculation of gradients using the chain rule of calculus. The learning in a BP neural network is achieved through iterative training. During each iteration, the network processes a batch of training data, calculates the error, and  
The parameters \(c_{0,1,2,3}\) and \(n\) in Equation (28) and the position of the damping layer \([x_{41},x_{44}]\) are used as six different indicator inputs for the BP neural network. The values of the output layer represent the desired results of the prediction, signifying the functional purpose that the network aims to achieve. In this model, the damping coefficients of the system at different vibration orders need to be predicted. Considering the practical needs of engineering, this paper predicts the damping coefficients for the first ten orders of the system; hence, the number of nodes in the output layer is 10. We set up two hidden layers through the design of network architecture, cross-validation, and performance evaluation. Determining the appropriate number of neurons in each hidden layer is crucial for neural network design. Insufficient neurons may result in underfitting, preventing the network from capturing the full complexity of the training data and thus weakening its predictive capabilities. Conversely, an excess of neurons can cause overfitting, where the network performs well on training data but poorly on unseen data, and also prolongs the training process. For networks with multiple hidden layers, the number of nodes in the hidden layers can be determined using the following empirical formula:  
\[m=\sqrt{n+l}+\alpha,\quad m=\sqrt{nl}, \tag{29}\]  
where the number of nodes in the hidden layer is denoted by \(m\), the number of nodes in the input layer by \(n\), and the number of nodes in the output layer by \(l\). The constant \(\alpha\) is greater than 1 and less than 10. According to the empirical formula, this paper sets the number of hidden layer nodes to be eight and nine, respectively.