# A Time-History Contact Force Model of the Dynamic Load of AERORail Structures
## 2 Data Fitting and Bezier Curves  
Data fitting is a commonly used method in research to summarize empirical laws, which can find trend rules from measured data or simulation results simply and effectively [8; 9]. For data that are discrete and random, the commonly used fitting methods include least squares, maximum likelihood estimation, etc. The least squares method refers to determining the best fit for the original data \(y(x_{i})\) by choosing a function \(\hat{y}(x)\) that minimizes the sum of the squares of the differences for a given set of data [10]. This process can be represented by Equation (1):  
\[e=\sum_{i=1}^{N}\lVert y(x_{i})-\hat{y}(x_{i})\rVert^{2},e\to 0 \tag{1}\]  
where \(e\) is the error function, and \(N\) is the total number of sample points. Generally speaking, directly obtaining the expression of the fitting function is difficult. Usually, a fitting function (Equation (2)) is determined using a set of basis functions and variable parameters:  
\[\hat{y}(x)=\sum_{i=1}^{N}\hat{y}_{i}(x)\hat{\beta}_{i} \tag{2}\]  
where \(\hat{y}_{i}(x)\) and \(\hat{\beta}_{i}\) are the basis functions and the variable parameters, respectively. Once the basis functions are selected, the actual problem becomes solving for \(\hat{\beta}_{i}\). With \(\hat{\beta}_{i}\), the expression of the fitting function can be obtained through Equation (2). There are various choices for the basis functions of the fitting function, including polynomial, sine function, power function, and exponential function, etc. For the contact force's time-history curve to be analyzed, due to its relatively uniform curve shape (double-valley curve or single-valley curve), traditional polynomial fitting is feasible [11; 12]. However, in polynomial fitting, the physical meanings of the coefficients are not very clear, and most of the time researchers cannot derive further rules or gain any intuitive understanding of it, which hinders further in-depth research. In addition, considering that the contact force's time-history curve contains many high-frequency low-amplitude vibrations, and these have little impact on the structure because the vibrational response of the structure is mainly caused by low-frequency vibrations within its fundamental frequency range, the chosen basis function for fitting should ideally disregard this part of the vibration. Besides polynomial and other basis functions, there are Bezier curves [13; 14], B-spline curves [15; 16], and non-uniform rational B-splines (NURBS) [17] available in the field of data analysis. Among these, Bezier curves have the advantages of simple form, smooth curvature, easy calculation, and easy implementation, and have been widely used in fields such as CAD, artistic design, and complex shape modeling. Bezier curves determine the shape and position of the curve through a finite number of control points, and the visually smooth curves determined by the control points are also an important method for vector graphics. Using Bezier curves as the basis function for fitting the contact force time-history curve can effectively address the issues mentioned earlier, and Bezier curves themselves have good parameterization capabilities, making it convenient for future research to start from the curve control points and explore deeper patterns [18; 19]. A Bezier curve is obtained by recursively inserting points to interpolation lines connecting consecutive control points and inserted points. For \(n+1\) control points, one connects these points and inserts \(n\) intermediate points at a local coordinate \(t\) on each interpolation line. These intermediate points are then treated as the new control points and the above step is repeated to get new intermediate points. After the recursion, there is only one intermediate point left. If the value of \(t\) is changed (e.g., from 0.5 to 0.75), the final intermediate point will also move accordingly, thereby generating a smooth trajectory in space. This trajectory is the Bezier curve.