## 4 Optimizations for Structural Parameters
### Method and Process  
A complex nonlinear relationship exists between the resonance frequency \(f_{s}\) and the structural parameters. The BP-network model is more advantageous in expressing the nonlinear relationship than the orthogonal experiment and the response surface method. In the first stage, in obtaining the fitness function of the structural parameters concerning the resonance frequency \(f_{s}\), the BP neural network method containing 10 hidden layers is employed. The dataset used for the BP network was obtained from the analytical model with a total of 3000 samples, of which 80% were used for training, 10% for validation, and the rest for testing. The overall optimization framework is shown in Figure 11, where the fitness function of the BP neural network output is set as an equation constraint in the second stage of the improved sparrow search algorithm (ISSA). The ISSA, which has good global search and local exploitation capabilities, is an algorithm that improves on SSA by introducing logistic chaotic mapping to avoid oscillating near the global optimal solution during the iteration process [36; 37]. As a population optimization algorithm inspired by the foraging and anti-predation behaviors of sparrows, sparrows are classified into discoverers, joiners, and watchers. The calculation process is shown in Figure 12. The discoverer sparrows are responsible for finding food for the entire population and providing directions to joiner sparrows for food. The watchers are responsible for monitoring the foraging area to avoid predation. When the optimization procedure starts, logistic chaotic mapping is introduced to improve the initial population diversity and algorithm stability, which is denoted as [38]  
\[\mu=\beta\mu_{0}(1-\mu_{0}) \tag{16}\]  
where \(\beta\in[0,\,4]\) and \(\mu_{0}\in[0,\,1]\). The closer the value of \(\beta\) is to 4, the more uniform the distribution of values of \(\mu\) is, and when \(\beta\) = 4, the system is in a completely chaotic state. Figure 11: Overall optimization framework. Figure 10: Variation in resonant frequency and \(\Omega\) with \(l_{1}\) and \(l_{2}\). (**a**) Resonant frequency and (**b**) \(\Omega\). The initialized individual values associated with sparrow locations are calculated as [39; 40]  
\[x_{*,j}=\mu\big{(}ub_{j}-lb_{j}\big{)}+lb_{j} \tag{17}\]  
Individual values are constantly updated during the optimization process. The population can be represented as a data collection, assuming the number of sparrows in the population is \(n\), and the population dimension is \(m\). The population is expressed as  
\[X= \begin{pmatrix}x_{1,1}&x_{1,2}&\cdots&x_{1,m}\\ x_{2,1}&x_{2,2}&\cdots&x_{2,m}\\ \vdots&\vdots&\ddots&\vdots\\ x_{n,1}&x_{n,2}&\cdots&x_{n,m}\end{pmatrix}  \tag{18}\]  
The fitness values for all sparrows can be expressed as  
\[F_{X}= \begin{bmatrix}f\big{(}\begin{bmatrix}X_{1,1}&X_{1,2}&\cdots&X_{1,m}\\ f\big{(}\begin{bmatrix}X_{2,1}&X_{2,2}&\cdots&X_{2,m}\end{bmatrix}\end{bmatrix} \big{)}\\ &\vdots&\\ f\big{(}\begin{bmatrix}X_{n,1}&X_{n,2}&\cdots&X_{n,m}\end{bmatrix}\big{)} \end{bmatrix}  \tag{19}\]  
where \(f\) is the fitness function. In the population search process, the goals of discoverers are used as the optimization goals. The discoverers are given more food to improve search ability and range, and the position is transformed as follows:  
\[x_{i,m}^{t+1}=\left\{ \begin{array}{ll}x_{i,m}^{t}\exp\Bigl{(}\frac{-i}{T} \Bigr{)}\,&R_{2}<ST\\ x_{i,m}^{t}+Q\,&R_{2}\geq ST\end{array} \right. \tag{20}\]  
where \(i\) is the current number of iterations, \(T\) is the maximum number of iterations, and \(\delta\) is a random value between 0 and 1. \(Q\) is a random number obeying a normal distribution. Figure 12: Calculation process of ISSA model. By behavioral guidelines of sparrows, the location update description for joiners is as follows:  
\[x_{i,m}^{t+1}=\left\{