# Dynamic Analysis on the Parametric Resonance of the Tower-Multicable-Beam Coupled System
## Appendix B  
\[\mathbf{\alpha_{1}}=\frac{\mathbf{\pi^{5}}\cdot\mathbf{E_{t}}\mathbf{l_{t}}}{16(3\mathbf{\pi}-8 )m_{t}\mathbf{L_{t}}^{4}}+\sum_{i=1}^{6}\frac{\mathbf{\pi^{2}}\mathbf{E_{ci}}^{*}\mathbf{A_{ ci}}_{\mathbf{ci}}sin2\theta_{i}}{(3\mathbf{\pi}-8)(-2+\mathbf{\pi})m_{t}\mathbf{L_{ci}}\mathbf{L_{t}} ^{4}};\mathbf{\kappa_{2}}=\frac{\mathbf{c_{t}}}{\mathbf{m_{t}}};\]  
\[\mathbf{\alpha_{3}}=-\sum_{i=1}^{6}\frac{\mathbf{\pi^{2}}\mathbf{E_{ci}}\mathbf{A_{ci}}}{(-2+ \mathbf{\pi})(3\mathbf{\pi}-8)m_{t}\mathbf{L_{t}}\mathbf{L_{ci}}^{2}}\cdot sin\theta_{i}\cdot \mathbf{\alpha_{4}}=\sum_{i=1}^{6}\frac{\mathbf{\pi}\mathbf{E_{ci}}^{*}\mathbf{A_{ci}}}{(3\mathbf{ \pi}-8)m_{t}\mathbf{L_{t}}\mathbf{L_{ci}}^{2}}\cdot sin2\theta_{i};\]  
\[\mathbf{\alpha_{5}}=-\sum_{i=1}^{6}\frac{\mathbf{\pi}\mathbf{E_{ci}}\mathbf{A_{ci}}}{(-2+\mathbf{ \pi})(3\mathbf{\pi}-8)m_{t}\mathbf{L_{t}}\mathbf{L_{ci}}^{2}}\cdot sin\theta_{i}\cdot cos^{ 2}\theta_{i};\ \mathbf{\alpha_{6}}=-\sum_{i=1}^{6}\frac{32\mathbf{\pi}\mathbf{D_{ci}}\mathbf{E_{ci}}\mathbf{A_{ci} }}{(-2+\mathbf{\pi})(3\mathbf{\pi}-8)m_{t}\mathbf{L_{t}}\mathbf{L_{t}}^{2}}\cdot sin\theta_{i};\]  
\[\mathbf{\alpha_{7}}=-\sum_{i=1}^{6}\frac{\mathbf{\pi^{4}}\mathbf{E_{ci}}\mathbf{A_{ci}}}{2(-2+ \mathbf{\pi})(3\mathbf{\pi}-8)m_{t}\mathbf{L_{t}}\mathbf{L_{ci}}^{2}}\cdot sin\theta_{i};\ \mathbf{\alpha_{8}}=\sum_{i=1}^{6}\frac{\mathbf{\pi}\mathbf{E_{ci}}\mathbf{A_{ci}}}{(-2+\mathbf{ \pi})(3\mathbf{\pi}-8)m_{t}\mathbf{L_{t}}\mathbf{L_{ci}}^{2}}\cdot sin\theta_{i}\sin 2 \theta_{i};\]  
\[\mathbf{\alpha_{9}}=-\sum_{i=1}^{6}\frac{2\mathbf{\pi^{2}}\mathbf{H_{i}}sin\theta_{i}}{(-2+ \mathbf{\pi})(3\mathbf{\pi}-8)m_{t}\mathbf{L_{t}}};\] (A11)  
\[\beta_{1}^{i}=\frac{512\mathbf{D_{ci}}\mathbf{E_{ci}}^{*}\mathbf{A_{ci}}}{\pi^{2}m_{ci} \cdot\mathbf{L_{t}}^{4}}+\frac{\mathbf{\pi^{2}}\mathbf{H_{ci}}}{m_{ci}\cdot\mathbf{L_{ci}}^{ 2}};\beta_{2}^{i}=-\frac{2}{\pi}\cdot cos\theta_{i};\ \beta_{3}^{i}=-\frac{2}{\pi}\cdot sin\theta_{i};\ \beta_{4}^{i}= \frac{1}{m_{ci}}\cdot c_{ci};\]  
\[\beta_{5}^{i}=-\frac{2}{m_{ci}\cdot\pi}\cdot cos\theta_{i}\cdot c_{ci};\beta_ {6}^{i}=-\frac{2}{m_{ci}\cdot\pi}\cdot sin\theta_{i}\cdot c_{ci};\ \beta_{7}^{i}=\frac{16\mathbf{\pi}\mathbf{D_{ci}}\mathbf{E_{ci}}^{*}\mathbf{A_{ci}}}{m_{ci} \cdot\mathbf{L_{t}}^{2}\cdot\mathbf{L_{t}}^{2}};\beta_{8}^{i}=\frac{\mathbf{\pi^{4}}\mathbf{L_{ ci}}\mathbf{A_{ci}}}{4\cdot m_{ci}\cdot\mathbf{L_{ci}}^{4}};\]  
\[\beta_{9}^{i}=-\frac{32\mathbf{D_{ci}}\mathbf{E_{ci}}^{*}\mathbf{A_{ci}}}{m_{ci}\cdot\mathbf{L_{ t}}\cdot\mathbf{L_{t}}^{2}}\cdot sin\theta_{i};\ \beta_{10}^{i}=-\frac{32\mathbf{D_{ci}}\mathbf{E_{ci}}^{*}\mathbf{A_{ci}}}{m_{ci}\cdot\mathbf{ \pi}\mathbf{L_{ci}}\cdot\mathbf{L_{t}}^{2}}\cdot cos\theta_{i};\ \beta_{11}^{i}=\frac{16\mathbf{D_{ci}}\mathbf{E_{ci}}^{*}\mathbf{A_{ci}}}{m_{ci}\cdot\mathbf{ \pi}\mathbf{L_{ci}}^{2}\cdot\mathbf{L_{t}}^{2}}\cdot sin^{2}\theta_{i};\] (A12)  
\[\beta_{12}^{i}=\frac{16\mathbf{D_{ci}}\mathbf{E_{ci}}^{*}\mathbf{A_{ci}}}{m_{ci}\cdot\mathbf{ \pi}\mathbf{L_{ci}}\cdot\mathbf{L_{t}}^{2}}\cdot cos^{2}\theta_{i};\ \beta_{13}^{i}=-\frac{16\mathbf{D_{ci}}\mathbf{E_{ci}}^{*}\mathbf{A_{ci}}}{m_{ci}\cdot\mathbf{ \pi}\mathbf{L_{ci}}^{2}\cdot\mathbf{L_{t}}^{2}}\cdot sin\theta_{i};\ \beta_{14}^{i}=-\frac{\mathbf{\pi^{2}}\mathbf{E_{ci}}^{*}\mathbf{A_{ci}}}{m_{ci\[\beta_{18}^{i}=-\frac{E_{G}^{*}A_{ci}}{2\cdot M_{bi}\cdot L_{ci}^{2}}sin\theta_{i} \text{;}\]  
\[\gamma_{1}^{i}=\frac{E_{G}^{*}A_{ci}}{M_{bi}L_{ci}}sin^{2}\theta_{i}+\mu\cdot \frac{E_{bi}I_{bi}}{M_{bi}d^{3}}\text{; }\mu=\left\{ \begin{array}{ll}-5,\text{ }i=1 \text{or }N\\ -6,\text{ }1<i<N\end{array}