## 2 Stiffness Identification Method Based on Joint Simulation
### Method Verification  
The method is applied to identify the stiffness of the bed-column joint of the precision grinding machine, as shown in Figure 3.  
The accelerometer is fixed on the column, and the vibration signal of the joint is obtained by rotating the spindle. The signal is first processed by Kalman filtering, and then the natural frequencies of the joint are obtained by the least squares complex frequency domain method, as shown in Figure 4. Taking the first three orders as experimental natural frequencies from Figure 4, the first-order, second-order, and third-order experimental natural frequencies are 48.7 Hz, 166.5 Hz, and 310.7 Hz, respectively.  
Figure 3: Joint interfaces to be measured.  
Figure 2: Flowchart of the stiffness identification method.  
The joint stiffness is used as the design variable, and the calculated natural frequency _fc_ is used as the state variable. Equation (6) is the objective function, and the stiffness value is outputted when the y-value converges.  
\[y=\sqrt{w_{1}(f_{1c}-f_{1 test})^{2}+w_{2}(f_{2c}-f_{2 test})^{2}+\ldots+w_{n}(f_{ nc}-f_{ntest})^{2}} \tag{6}\]  
where \(\omega_{1}\) is the weights (\(0<\omega_{1}<1\)), \(f_{1c}\) is the calculated natural frequency, and \(f_{1 test}\) is the experimental natural frequency.  
The joint simulation process and simulation results are shown in Figure 5. Figure 5a indicates the relationship between the best fitness value and the mean fitness value in each generation. The best fitness is the minimum y value in the current generation, and the mean fitness is the mean value of y in the current generation. As can be seen from Figure 5a, the difference between the best fitness and the mean fitness gradually decreases, and the two values are equal in the tenth generation, at which time the y value converges, and the corresponding solution is the optimal solution. The optimal solution is shown in Figure 5b, and from Figure 5b, the identified stiffnesses are 1.68 \(\times\) 10\({}^{10}\) N/m in the normal direction and 2.46 \(\times\) 10\({}^{7}\) N/m and 6.16 \(\times\) 10\({}^{8}\) N/m in the tangential direction, respectively.  
The identified stiffness parameters are substituted into the model for modal analysis to obtain the natural frequencies considering the joint. The glue operation in Ansys is applied to bond the two objects together to obtain the natural frequencies without considering the joint. By comparing natural frequencies considering the joint with natural frequencies without considering the joint, as shown in Table 1, it can be seen that the simulation considering the influence of the joint is closer to the experimental results, which verifies the accuracy of the method of identifying the joint stiffness. It can also be seen that the presence of the joint reduces the overall natural frequencies of the structure significantly, proving that the presence of the joint needs to be taken into account when analyzing machine tools.  
Figure 4: Experimental natural frequencies.  
Figure 5: Joint simulation. (a) Simulation process; (b) Simulation results.  
The method of identifying the stiffness of the joint is based on experimental natural frequencies, and the accuracy of natural frequencies obtained from the modal experiment directly determines the accuracy of the method. However, the modal experiment is generally carried out on the machine tool. Although sensors are placed on the joint to be measured, the response will still be affected by the influence of other components. The response obtained by the sensor is the result of the vibration of the joint to be measured and other components. Natural frequencies obtained by the modal experiment will have errors, which will affect the accuracy of the stiffness identified by the joint simulation method. Therefore, this paper corrects the fractal model of the normal stiffness of the joint so as to make the theoretical model better characterize the actual surface.