# Ultrasonic Deicing Efficiency Prediction and Validation for a Flat Deicing System
## 2 Basic Theory and Model Establishment  
### Ultrasonic Deicing Principle Interpretation  
After the actuator bonded to the flat deicing system is turned on, Lamb waves and shear horizontal (SH) waves are generated in the structure. When the shear stress caused by these two kinds of waves exceed the ice adhesion strength, the ice layer is detached from the plate. The waves propagation in multiple layers of isotropic solid media can be investigated based on theory in Ref [27]. The coordinatesystem and ultrasonic waves propagating in the ice-plate layered structure for deicing is shown in Figure 1.  
The governing equation for Lamb waves and SH waves can be written as:  
\[\rho\frac{\partial^{2}u_{i}}{\partial t^{2}}=c_{ijkl}\frac{\partial^{2}u_{i}}{ \partial x_{j}\partial x_{k}}, \tag{1}\]  
where \(\rho\) is the density, \(C_{ijkl}\) is the stiffness tensor, and \(u_{i}\) is the displacement field. The waves are assumed to propagate along \(x_{1}\) direction. The solution of Equation (1) can be assumed as the following form:  
\[u_{i}=U_{i}\mathrm{e}^{\tilde{\mathrm{d}}(x_{1}+\alpha x_{3}-ct)}. \tag{2}\]  
In Equation (2), \(U_{i}=U\alpha_{i}\), \(\alpha_{i}\) is the cosine component in \(i\) direction. \(U_{i}\) is the polarization vector presenting the displacement in each direction, \(k\) stands for the wave number along \(x_{1}\) direction, c represents the phase velocity along \(x_{1}\) direction, and \(\alpha\) is the ratio of wave number in \(x_{3}\) with respect to the wave number in the \(x_{1}\) direction. By utilizing the inverse method [28], [PERSON]'s equation can be written as Equation (3):  
\[\left[\begin{array}{ccc}\lambda_{11}-\rho c^{2}&\lambda_{12}&\lambda_{13}\\ \lambda_{12}&\lambda_{22}-\rho c^{2}&\lambda_{23}\\ \lambda_{13}&\lambda_{23}&\lambda_{33}-\rho c^{2}\end{array}\right]\left\{ \begin{array}{c}U_{1}\\ U_{2}\\ U_{3}\end{array}\right\}=\left\{\begin{array}{c}0\\ 0\\ 0\end{array}\right\}. \tag{3}\]  
The values of \(\lambda_{im}\) (\(i\), \(m=1\), \(2\), \(3\)) in Equation (3) can be calculated from Equation (4):  
\[\lambda_{\mathrm{im}}=C_{iklm}n_{k}n_{l}, \tag{4}\]  
where \(n_{k}\), \(n_{l}\) are the direction cosines of the normal to the wave front. According to the boundary conditions, the displacements \(u_{1}\) and \(u_{3}\) are zero for SH waves and \(u_{2}\) is zero for Lamb waves. Thus, displacements and stresses can be expressed as Equation (5):  
\[u_{1}=\sum_{k=1}^{4}B_{k}\mathrm{e}^{\tilde{\mathrm{d}}(x_{1}+\alpha_{k}x_{3} -ct)}\]  
\[u_{2}=\sum_{k=1}^{2}B_{k}\mathrm{e}^{\tilde{\mathrm{d}}(x_{1}+\alpha_{k}x_{3} -ct)}\]  
Figure 1: The coordinate system and ultrasonic waves propagating in the ice-plate layered structure.  
\[\sigma_{31}=\sum_{k=1}^{4}B_{k}[\alpha_{k}+U_{3k}]\mu(ik)e^{ik(x_{1}+\alpha_{k}x_{3 }-ct)}\]  
\[\sigma_{32}=\sum_{k=1}^{2}B_{k}\alpha_{k}\mu(ik)e^{ik(x_{1}+\alpha_{k}x_{3}-ct)}\]  
\[\sigma_{33}=\sum_{k=1}^{4}B_{k}[\lambda+(\lambda+2\mu)\alpha_{k}U_{3k}](ik)e^{ ik(x_{1}+\alpha_{k}x_{3}-ct)}, \tag{5}\]  
where \(B_{k}\) is the weighting coefficient of the partial waves. \(U_{3k}\) is the relation of the polarization vectors between \(U_{3}\) and \(U_{1}\). The other variables are the same as mentioned before.