# Millimeter Wave Attenuation Due to Wind and Heavy Rain in a Tropical Region
## 3 Theory and Algorithms
### Inclination Attenuation Due to Wind
#### 3.3.1 Dynamic Pole Angle Estimation via Wind-Induced Mechanical Vibration  
Pole inclination, \(\theta\), in the direction of the antenna front is equal to the angular misalignment of the beam, as shown in Figure 6. The estimation of the pole inclination is based on the theory presented in the APT-AWG-REP-81 report ([PERSON] et al., 2017) and a previous study ([PERSON] et al., 2018):  
\[\theta=\theta_{0}+\theta_{s}+\theta_{d}, \tag{14}\]  
where \(\theta_{0}\) is the initial axis misalignment, \(\theta_{s}\) is the static inclination angle and \(\theta_{d}\) is the dynamic inclination angle. The static inclination angle is caused by the assumed constant wind load and is given using  
\[\theta_{s}=(C_{1}A_{1}+3C_{2}A_{2})\frac{\rho|^{2}}{12 EI}v^{2}=C_{s}v^{2}. \tag{15}\]  
The dynamic inclination is caused by the natural vibration frequency of the pole and is given using  
\[\theta_{d}=C_{d}v^{2}. \tag{16}\]  
Figure 5: Radiation pattern envelope as a function of the misalignment angle using Equation (12) compared to the manufacturer’s data.  
The wind load coefficients for static and dynamic inclination, \(C_{s}\) and \(C_{d}\), depend on the type and dimensions of the pole, respectively. \(v\) is the effective wind speed (m/s), \(C_{1}\) and \(A_{1}\) are the drag coefficient and the wind receiving area of the pole, respectively, \(C_{2}\) and \(A_{2}\) are the drag coefficient and the wind receiving area of the antenna, respectively, \(\rho\) is the air density (kg/m\({}^{3}\)), \(l\) is the length of the pole (m), \(E\) is the Young's modulus and \(l\) is the second moment of area (m\({}^{4}\)).  
For this setup in Figure 1, the parameters and their values used for angle calculation are given in Table 2. The pole length and diameter are used to calculate the wind-receiving area of the pole (\(A_{1}\)). The antenna diameter is used to calculate the wind-receiving area of the antenna (\(A_{2}\)). The drag coefficients \(C_{1}\) and \(C_{2}\) are determined by \(A_{1}\) and \(A_{2}\), respectively, using [34] and taking into account the cylindrical and flat shapes of the pole and antenna. The value of air density, \(\rho=1.124\) kg/m\({}^{3}\), is calculated from the average temperature and pressure using the ideal gas law. The Young's modulus, \(E=200\times 10^{9}\) Pa, is typical of stainless steel SS400. The second moment of area can be calculated to be \(1.8\times 10^{-4}\) m\({}^{4}\) for the annulus cross-section of the pole [35].  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Quantity** & **Value** & **Unit** \\ \hline \(C_{1}\) & Drag coefficient of the pole & 0.98 & - \\ \(A_{1}\) & Wind receiving area of the pole & 0.36 & m\({}^{2}\) \\ \(C_{2}\) & Drag coefficient of the antenna & 1.42 & - \\ \(A_{2}\) & Wind receiving area of the antenna & 0.118 & m\({}^{2}\) \\ \(\rho\) & Air density & 1.124 & kg/m\({}^{3}\) \\ \(l\) & Pole length & 4 & m \\ \(d\) & Pole diameter & 101.6 (4) & mm (inch) \\ \(D\) & Antenna diameter & 0.3 (1) & m (ft) \\ \(E\) & [PERSON]’s modulus & \(200\times 10^{9}\) & Pa \\ \(l\) & Second moment of area & \(1.8\times 10^{-6}\) & m\({}^{4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Values of parameters for the calculation of inclination angle due to wind speed.  
Figure 6: 3D inclination angle measurement diagram.