## 3 Numerical Analysis of Dynamic Contact Characteristics
### Experiment  
In order to verify the numeric results of the dynamic contact model and sensorless stiffness estimating method, an experiment is conducted to measure the axial stiffness of a SNBSP in the Z-axis feed system of a CNC lathe (Shenyang Machine Tool Group, Shenyang, China) with Fanuc Series (Figure 25). The experimental lathe parameters are exhibited in Table 2. The parameters of SNBSP(Best Precision, Jining, China) are exhibited in Table 1.  
In order to verify this, an external sensor is needed to measure the response signals during the operation of the machine tool. Fast Fourier transform (FFT) is used to process the signals, and the frequency components obtained are compared with the theoretical values to verification. Hence an acceleration sensor, B&K4520-004 (Bruel & Kjaer, Naerum, Denmark), of which the maximum of sampling frequency is 25.6 kHz, was mounted on the worktable and moves together with it. The acceleration sensor was connected to NI-DAQ (National Instruments, Austin, TX, USA) which was connected to Notebook PC. The max travel of the worktable is 300 mm. However, considering the safety of high-speed operation, the travel was set to 270 mm with two end point coordinates 15 mm (denoted by Z0) and 285 mm (denoted by Z1), respectively. The experimental process is as follows. 1. Locate the worktable at the given position Z1 by using a NC program. 2. Run acceleration sensor data acquisition program. Run servo axis data acquisition program to capture the TCSM. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Value** & **Unit** \\ \hline I. Geometric parameters of feed system & & \\ \multicolumn{2}{c}{orktable mass \(m_{t}\)} & 325 & Kg \\ Axial stiffness of fixed end bearing \(K_{b}\) & \(2\times 10^{8}\) & N/m \\ \multicolumn{2}{c}{The length of the screw} & 500 & mm \\ \multicolumn{2}{c}{The travel of the feed system} & 300 & mm \\ \multicolumn{2}{c}{Motor torque coefficient} & 1.85 & Nm/A \\ \multicolumn{2}{c}{Elastic modulus of screw shaft \(Es\)} & 210 & GPa \\ \multicolumn{2}{c}{II. Operating conditions} & & \\ \multicolumn{2}{c}{Feed speed} & 10,000 & mm/min \\ \multicolumn{2}{c}{Two end point coordinates} & & \\ \multicolumn{2}{c}{Z0} & 15 & mm \\ \multicolumn{2}{c}{Z1} & 285 & mm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Specific parameters of experimental CNC lathe. Figure 25: Experimental system. Step 3: Run the NC program. The worktable moves from Z1 to Z0 at the specified speed. Step 4: All the application programs are stopped and the data are saved. Because the maximum feed speed of the Z feed shaft of the NC lathe is locked to 10,000 mm/min, and the lead of the Z axis screw shaft is 10 mm, that means that the screw speed is 1000 rpm. Hence, the high speed of the screw is selected as 1000 rpm, and 50 rpm is selected as the low speed. Under each rotating speed, the worktable is run three times according to the above test steps, in the forward (Z1-Z0) and reverse (Z0-Z1) direction. Then, the acceleration and TCSM signals are obtained. Figure 26 shows the acceleration and TCSM signals of the first positive feed at the screw speed 1000 rpm. It can be seen from Figure 26a that there are three states in the process: a state of acceleration, stable running and deceleration. The acceleration signals in the stability state can be used for FFT analysis. It can be seen from Figure 26b that there are also three operation states in TCSM signals. Calculate the average value of the TCSM in the stability state. By substituting the average value of the current into Equations (47) and (48), the axial force of the Z axis screw of the feed system can be obtained as 637 N under the screw speed of 1000 rpm. According to the model presented in this paper, the dynamic axial contact stiffness of the SNBSP can be calculated. When the screw speed is 50 rpm, the axial force is about 684.8 N. Because the speed is very low, the axial stiffness of the SNBSP can be solved by static model, and the result is \(3.89\times 10^{5}\) N/mm.