# A Novel Noncontact Ultrasonic Levitating Bearing Excited by Piezoelectric Ceramics
## 4 Load-Bearing Capacity of a Single Transducer
### Modeling of Lateral Radiation Force  
Different from radical levitation process above, the acoustic field excited by flexural vibration in lateral radiation process is hard to be described accurately. Both vertical and horizontal forces acting on an object that levitated on the vibrating surface were theoretically and experimentally studied by [PERSON] et al. [32; 33; 34]. In their studies, the levitation force was generated by longitudinal vibration which was considered to be piston motion, while the levitation force is excited by flexural vibration in our study. For this complicated case, the thermodynamic theory of ideal gas is adopted to approximately estimate the sound pressure in air film between the supported object and the radiation head's lateral surface.  
Figure 8 shows different states of the lateral levitation process. The process includes two main stages: direct extrusion (see Figure 8b) and reverse extrusion (see Figure 8c). When the radiator's lateral surface vibrates at a high frequency, the air between the radiator and supported object will also be squeezed at a high frequency. Because of the viscosity of the gas, it is difficult for the air to flow in or out of the gap rapidly, so that the time-averaging pressure of air in the gap is higher than that of the ambient environment. Due to the existence of the pressure difference, an air squeeze film with a certain stiffness is generated.  
An assumption is made that the gas between two surfaces is perfectly ideal, and the heat that gas in the gap absorbs from or radiates to surroundings is neglected as the process of the air's condensation and expansion in the gap is very fast. Thus, according to the state equation of ideal gas in an adiabatic process, the relationship between ambient pressure \(p_{0}\) and air film pressure \(p\) can be expressed as  
\[\frac{p}{p_{0}}=\left(\frac{V_{0}}{V}\right)^{\gamma} \tag{9}\]  
where \(V_{0}\) is the initial volume of air in the gap and \(\gamma\) is a specific heat ratio (\(\gamma=1.4\) for air).  
The lateral surface of the radiator is subjected to the high-frequency excitation, and flexural vibration deformation occurs. The expression of flexural vibration is  
\[a=a_{0}\mathrm{cos}(\omega t) \tag{10}\]  
Figure 8: Lateral levitation process: (**a**) initial state; (**b**) direct extrusion; (**c**) reverse extrusion.  
where \(a_{0}\) and \(a\) respectively represent the peak value and instantaneous value of amplitude and \(\omega\) is angular frequency. If the thickness of air squeeze film is defined as \(\overline{h}_{0}\), the overlapping area of radiator and levitated object is \(S_{0}\), and the length and width of the flexural deformation area on the lateral surface are \(l\) and \(b\), then the approximate volume \(V\) of air in the gap can be expressed as  
\[V\approx S_{0}\overline{h}_{0}-\frac{1}{2}a_{0}bl\text{cos}(\omega t) \tag{11}\]  
So the mean pressure difference in a time period is derived as follows  
\[\overline{p}=\frac{1}{2\pi}\underset{0}{\overset{2\pi}{\int}}pd(\omega t)-p_{0} \tag{12}\]  
where \(p=\dfrac{p_{0}}{(1-\dfrac{a_{0}bl\text{cos}(\omega t)}{2S_{0}\overline{h}_{0}} )^{\gamma}}\).  
The lateral levitation force generated by the radiator is calculated by  
\[F_{\text{L}}=\underset{S}{\iint}\overline{p}dS=\frac{1}{16}\frac{p_{0}\gamma( \gamma+1)a_{0}^{2}b^{2}l^{2}}{S_{0}\overline{h}_{0}^{2}} \tag{13}\]