# Effect of Ultrasonic Vibration on the Surface Adhesive Characteristic of Iced Aluminum Alloy Plate
## 4 Experiment on the Adhesive Force of Ice
### Experimental Results and Analysis  
To comparatively analyze the experimental results with theoretical ones, the adhesive shear stress of ice, which is its shear force in unit area, must be calculated. Therefore, based on the measuring adhesive torque, an analytical model for calculating the adhesive torsional shear stress of ice was established. The principle of modeling is shown in Figure 14.  
As shown in Figure 14a, because the shape of the iced aluminum alloy is square, for simplifying the process of modeling, the diagonals of the iced aluminum alloy plate coincide with the X-axis (abscissa) and the Y-axis (ordinate) of the Cartesian coordinates, respectively. Considering the axial symmetry of the square shape along the diagonals, one-fourth of it was selected as the modeling object, which is shown in Figure 14b. For simplifying the processes of modeling and calculation, the equation of infinitesimal adhesive torque \(dM\) is expressed in the form of the polar coordinate system, which is expressed as Equation (5). \[dM=\tau_{\mathrm{a}}\rho^{2}d\rho d\theta \tag{5}\]  
Figure 14: Principle of the analytical model: (**a**) position of the icing sample in the Cartesian coordinates; (**b**) a quarter of the icing sample. Figure 13: Measuring device for the adhesive torque of ice: (**a**) sketch map of the measuring device; (**b**) prototype of the measuring device. where \(dM\) is the infinitesimal adhesive torque of the ice; \(\tau_{\rm a}\) is the adhesive torsional shear stress of the ice; \(\rho\) is the polar radius of the infinitesimal ice; \(\theta\) is the polar angle of the infinitesimal ice. According to the range of the polar angle and the polar radius in Figure 13b, the adhesive torque \(M\) of the whole ice covering on an aluminum alloy plate surface can be calculated by Equation (5), which is expressed as Equation (6). \[M=4\tau_{\rm a}\int_{0}^{\frac{2}{2}}\int_{0}^{\frac{L}{\sqrt{2}(\ln\theta+ \cos\theta)}}\rho^{2}\rmd\rho\rmd\theta \tag{6}\]  
where \(M\) is the adhesive torque of ice covering on the whole aluminum alloy plate surface; \(L\) is the side length of the iced aluminum alloy plate. According to Equation (6), the adhesive torsional shear stress of ice is calculated as Equation (7). \[\tau_{\rm a}=\frac{12M}{\left[2\sqrt{2}+\ln\left(3+2\sqrt{2}\right)\right]L^{3}} \tag{7}\]  
For calculating the adhesive torsional shear stress \(\tau_{\rm a}\), the torque \(M\) in Equation (7) is substituted by the adhesive torque measured in the experiment, and the calculation results under different excitation frequencies are shown in Figure 15.  
As shown in Figure 15, when the iced aluminum alloy plate is not excited by ultrasonic vibration, the adhesive torsional shear stress of ice is about 0.21 MPa. In contrast, when the size of the PZT patch is 14 mm \(\times\) 14 mm \(\times\) 2 mm, and the vibration frequency is 79 kHz, the adhesive torsional shear stress of ice is lowest, which is about 0.014 MPa. It is only approximately 7% of that without ultrasonic vibration. When the vibration frequency is 81 kHz, the adhesive torsional shear stress is approximate to the one without ultrasonic vibration. In this case, the ultrasonic vibration does not have a deicing effect. The experimental results show that high shear stresses exist at the adhesive interface when the vibration frequency is 79 kHz, which leads to a decrease in the adhesive force of ice. On the contrary, the shear stress is small when the vibration frequency is 81 kHz. These experimental results agree with the theoretical analysis results listed in Table 2.  
In addition, when the size of the PZT patch is 16 mm\(\times\)16 mm\(\times\)2 mm, and the excitation frequencies are 85 kHz and 87 kHz, the adhesive torsional shear stresses of ice are also low. They are 0.037 MPa and 0.021 MPa, respectively. From Figure 15, it can be derived that the adhesive torsional shear stresses are lower than those of the iced aluminum alloy plate with a PZT size of 14 mm \(\times\) 14 mm \(\times\) 2 mm in general.