# A Novel Bearing Lubricating Device Based on the Piezoelectric Micro-Jet
## 4 The Velocity-Inlet Boundary Conditions  
In order to obtain the velocity-inlet boundary conditions for the injection simulations, we decided to get the volume change function of the cavity of the micro-jet first, which was caused by the vibration of the piezoelectric vibrator. As we mainly focused on the injection performances of the micro-jet for simplicity's sake, the backward flow occurred at the inlet of ball-race contact, and the thermal balance in the bearing _etc._, which had effects on the lubrication performances of the bearing, were not considered for this study. For details, please read the article by [PERSON] _et al._[15]. The first four mode shapes of the micro-jet are shown in Figure 4; the corresponding resonant frequencies are 1.83 kHz, 4.32 kHz, 7.01 kHz and 9.49 kHz, respectively. The frequency of the first mode shape was preferable as the working frequency due to the central symmetrical characteristic, which can reduce the difficulty of the positioning when being assembled. Figure 4: The first four modes of vibration. The displacement-time relation curves of particles with different radial positions on the fluid-structure interface were obtained, as shown in Figure 5. The square pulse voltages were selected, as the excitations with the frequency and amplitude were set to 1.83 kHz and 100 V, respectively. A numerical fitting model of a Fourier series function was adopted to fit the displacement-time relation curve of the particles with radius \(R\) = 15 mm, in accordance with the data of Figure 5, as shown in Equation (1). In the equation,\(w/2\pi\) is the vibration frequency and \(t\) represents the time. The fitting coefficients in the equation above are listed in Table 1.  
\[u_{z}(t)=a_{0}+\sum_{n=1}^{13}a_{n}\text{cos}\,nwt+\sum_{n=1}^{13}b_{n}\text{ sin}\,nwt,\quad 0\leqslant t\leqslant 684\text{$\upmu$s} \tag{1}\]  
The fitting curve, which was determined by Equation (1), gives a better fit for the simulation data, as shown in Figure 6. \begin{table}
\begin{tabular}{c c c c c c} \hline \(n\) & \(a_{n}\) & \(b_{n}\) & \(n\) & \(a_{n}\) & \(b_{n}\) \\ \hline
0 & 0.3456 & — & 7 & \(-\)0.0204 & \(-\)0.0282 \\
1 & \(-\)0.4588 & 1.5104 & 8 & \(-\)0.0180 & \(-\)0.0242 \\
2 & 0.2656 & \(-\)0.4576 & 9 & \(-\)0.0203 & \(-\)0.0145 \\
3 & 0.0208 & \(-\)0.0999 & 10 & \(-\)0.0211 & \(-\)0.0140 \\
4 & \(-\)0.0034 & \(-\)0.0718 & 11 & \(-\)0.0183 & \(-\)0.0099 \\
5 & \(-\)0.0069 & \(-\)0.0491 & 12 & \(-\)0.0217 & \(-\)0.0057 \\
6 & \(-\)0.0122 & \(-\)0.0310 & 13 & \(-\)0.0211 & \(-\)0.0047 \\ \hline \end{tabular}
\end{table} 
Table 1: The coefficients of Equation (1). Figure 5: The displacement-time relation curves of particles with different radial position on the fluid-structure interface (\(R\) is the radius of the particle’s position). Figure 6: The fitting curve, which was determined by Equation (1). The relation curve of vibration amplitude and radial position of particles was obtained from Figure 5, as shown in Figure 7. As can be seen, the curve is approximately axially symmetric, and the symmetry axis is \(R=15\) mm. Moreover, the vibration directions and frequencies of the particles are all the same. For convenience, we call the particles with radius \(R=15\) mm \"central line particles.\"  
Combined with Equation (1), the vibration displacement \(U_{Z}(t,r)\) of particles on the fluid-structure interface was deduced by fitting the curve in Figure 7, as shown in Equation (2). In this equation, \(r\) is the radial distance between particles and the central line particles, \(u_{zm0}\) is the vibration amplitude of the central line particles, and \(2r_{0}\) is the effective radial width of the vibrator (here \(r_{0}=~{}3 mm\)). The fitting coefficients in Equation (2) are listed in Table 2.  
\[U_{Z}(t,r)=(c_{0}+\sum_{n=1}^{6}c_{n}\text{cos}\,nwr+\sum_{n=1}^{6}d_{n}\text{ sin}\,nwr).u_{z}(t)/u_{zm0},~{}~{}-r_{0}\leqslant r\leqslant r_{0} \tag{2}\]  
The volume change of the cavity \(\Delta V(t)\) was deduced, as shown in Equation (3), where \(R_{0}\) is the radius of the central line particles.