# A Novel Noncontact Ultrasonic Levitating Bearing Excited by Piezoelectric Ceramics
## 4 Load-Bearing Capacity of a Single Transducer
### Modeling of Radical Radiation Force  
Figure 5 shows the cross section of a radical levitation structure including a journal and a radiator excited by single peizoelctric transducer. The concave surface with a radius \(R\) and width \(L\) vibrates at a resonance frequency \(f\) and a journal with a radius \(r\) is placed above at a height \(h_{0}\). The layer of air between the concave surface and journal surface is conventionally called gas squeeze film. The order of the air squeeze film thickness is \(10^{-5}\) m, which is three orders smaller than the dimensions of the vibrating surface \(R\) and \(L\) (with order of \(10^{-2}\) m). Thus, an acoustic wave generated by vibrating surface can be considered an approximate plane wave, and the air film pressure gradient along the thickness direction could be ignored. Since the suspended journal is a rigid body, the deformation will not occur, and the journal will exert no impact on the sound field. In return, air squeeze film pressure cannot change the modes of the vibrating surface. The wave equation of a one-dimensional sound wave can be written as  
\[\frac{\partial^{2}p}{\partial t^{2}}=c^{2}\
abla^{2}p \tag{1}\]  
where \(p\) represents the acoustic pressure, \(t\) the time, \(c\) acoustic velocity, and \(\
abla\) Laplace operator. Considering the multiple reflection between transducer's radiation surface and the journal's cylindrical surface, the sound pressure \(p\) and velocity \(v\) of waves in \(z\)-direction are calculated as  
\[\left\{ \begin{array}{l}p=2A\mathrm{cos}kz\,\mathrm{cos}\omega t\\ v=\frac{2A}{\rho_{0}c}\mathrm{sin}kz\,\mathrm{sin}\omega t\end{array} \right. \tag{2}\]  
where \(A\) is the amplitude of acoustic pressure before stacking of incident sound waves and reflected sound waves, \(k\) the wave number, \(\rho_{0}\) the density of air, and \(\omega\) the angular frequency. Since the heat  
Figure 5: Cross-section diagram of radical levitation structure driven by a single transducer. propagation velocity in the air is much lower than the acoustic velocity, the ultrasonic levitating process can be considered adiabatic. In addition, there is little acoustic energy dissipation caused by thermal conduction and viscous force among air molecules in the air squeeze film, so acoustic absorption in levitating process is also neglected. Based on the theory of thermodynamics, kinetic energy density \(\overline{K}\), potential energy density \(\overline{V}\) and total acoustic energy density \(\overline{E}\) after time-averaging in the bearing's air squeeze film are derived as  
\[\left\{ \begin{array}{l}\overline{K}=\frac{1}{2}\rho_{0}(\frac{1}{T}\int \limits_{0}^{T}{vdt})^{2}=A^{2}\text{sin}^{2}kz/(\rho_{0}c^{2})\\ \overline{V}=(\frac{1}{T}\int\limits_{0}^{T}{pdt})^{2}\text{cos}^{2}kz/(2\rho_ {0}c^{2})=A^{2}\text{cos}^{2}kz/(\rho_{0}c^{2})\\ \overline{E}=\overline{K}+\overline{V}=A^{2}/\rho_{0}c_{0}^{2}\end{array} \right. \tag{3}\]  
According to a study by [PERSON] and [PERSON] [31], the mean radiation pressure in an ideal gas is described as  
\[p_{\text{ra}}=(p-p_{0})_{avg}=\overline{K}-\overline{V}+\frac{\gamma-1}{2} \overline{E}=(\frac{\gamma-1}{2}+\text{cos}2 kz)\frac{v_{0}^{2}\rho_{0}}{4\text {sin}^{2}kh} \tag{4}\]  
Given cos\(2 kz\approx 1\), \(\text{sin}kh\approx kh\), \(a_{0}=\frac{v_{0}}{\omega}\), \(k=\frac{\omega}{c}\), Equation (4) can be simplified as  
\[p_{\text{ra}}=(p-p_{0})_{avg}=\frac{\gamma+1}{8}\frac{a_{0}^{2}\rho_{0}c^{2}}{ h^{2}} \tag{5}\]  
During the bearing's levitation process, when the journal is suspended at different heights \(h_{0}\) in \(z\)-direction, the thickness \(h\) of air film varies apparently along the circumferential direction. As shown in Figure 6, if levitation height \(h\) is equal to the radius difference \(\delta\) between the journal's cylindrical surface and the bearing's radiation surface, the thickness \(h\) of air squeeze-film is uniform.