# Dynamic Performance Optimization of Circular Sawing Machine Gearbox
## 2 Theoretical Basis
### Sensitivity Analysis Theory  
Sensitivity analysis refers to the degree to which a change of structural design parameters affects the objective performance function. The expression for sensitivity analysis is as follows:  
\[\text{S}=\frac{\partial F(x)}{\partial x_{i}}\text{ or }\text{S}=\frac{\Delta F(x)}{ \Delta x_{i}}. \tag{3}\]  
In Equation (3), the former is the differential sensitivity of a continuous system and the latter is the differential sensitivity of a discrete system. Sensitivity quantifies the influence of a parameter change on the change of the objective function [29; 30; 31; 32]. In the process of sensitivity analysis optimization, the designer selects effective design parameters quickly and purposefully according to actual needs to improve the optimization efficiency [33; 34; 35; 36]. In the sensitivity analysis of the gearbox structure, the objective function is the modal frequency, and the variable parameters are the thickness of the panels and stiffeners. The mathematical models for the modal frequencies and the dimension parameters of the structures can be expressed as follows:  
\[f=g(x_{1},x_{2},\cdots x_{n}). \tag{4}\]  
In Equation (4), \(f\) is the modal frequency, \(g\) represents the function of the modal frequency to the dimension thickness, and \(x_{1}\), \(x_{2}\),..., \(x_{n}\) is a dimension variable of the structure. By calculating the partial derivative of the modal frequency to the variables, the corresponding sensitivity can be obtained. The mathematical expression is as follows:  
\[S_{fi}=\frac{\partial_{f}}{\partial x_{i}}=\frac{\partial g(x_{1},x_{2},\cdots,x_{n})}{\partial x_{i}}. \tag{5}\]  
In Equation (5), \(\text{S}_{fi}\) is the sensitivity of the structure size to the modal frequency. The larger the absolute value of the sensitivity, the greater the influence of the size variable on the objective function. It is also evident from the formula that the value of the sensitivity can have a positive or a negative sign. Therefore, the influence of the size variable on the objective function may decrease with the increase of the size variable, while the sensitivity response represents the speed of the change. In structural modal analysis, the low-order mode has a great influence on the dynamic characteristics of the structure. Therefore, the weighted average \(F\) of the first five modal frequencies is extracted as the objective function, as follows:  
\[F=0.3\times f_{1}+0.25\times f_{2}+0.2\times f_{3}+0.15\times f_{4}+0.1\times f _{5}. \tag{6}\]  
In Equation (6), \(f_{1}\), \(f_{2}\), \(f_{3}\), \(f_{4}\), and \(f_{5}\) correspond to frequencies 1 to 5 respectively, and 0.3, 0.25, 0.2, 0.15, and 0.1 are the respective weighting factors that correspond to the first to the 5 th modal frequencies. If the first five order modal frequencies of the gearbox have a linear relationship with respect to the change of the size parameters, the fitting equation is given as follows:  
\[\hat{f_{i}}=\alpha_{k}+\beta_{k}x_{i}. \tag{7}\]  
In Equation (7), \(\hat{f_{i}}\) represents the regression value; \(\alpha_{k}\), \(\beta_{k}\) represents the regression coefficient; \(x_{i}\) represents the amount of change in the design variable; and \(k\) represents the \(k\)-\(th\) design variable. The residual is defined as the difference between the regression and experimental values. To achieve a convenient solution, the difference between the regression and experimental values is squared. If the result satisfies the minimum value, the fitting effect is optimal. The expression used to perform this calculation is as follows:\[SS_{e}=\sum_{i=1}^{n}e_{i}^{2}=\sum_{i=1}^{n}\left(f_{i}-f_{i}\right)^{2}=\sum_{i=1 }^{n}\left[f_{i}-\left(\alpha_{k}+\beta_{k}x_{i}\right)\right]^{2}. \tag{8}\]  
In Equation (8), \(f_{i}\) represents the experimental value. To achieve the best fitting effect, the least square method is applied. The partial derivatives of Equation (8) are calculated for \(\alpha_{k}\) and \(\beta_{k}\) respectively, and the result of these equations is 0.