vault backup: 2025-09-03 17:09:39

杂项/随礼清单.md

@@ -9,4 +9,4 @@
 | 唐世泽 | 结婚  | 600 |
 | 华瑞  | 结婚  | 800 |
 | 朱钰龙 | 结婚  | 800 |
-|     |     |     |
+| 许建强 | 结婚  | 800 |

线性化求解器/参考文献/Kallesøe-Equations of motion for a rotor blade/auto/Kallesøe-Equations of motion for a rotor blade.md

@@ -539,19 +539,81 @@ Coordinate Transformations
 The derivation of the transformation matrices follows the method used in Hodges and Dowell.3 The major difference between these matrices and those of Hodges and Dowell3 is the inclusion of the pitch angle $\beta$ . The transformation between the initial $\left(X,\,Y,\,Z\right)$ -frame and the $\left(\hat{x},\,\hat{y},\,\hat{z}\right)$ -frame is given by  
 
 $$
-\begin{array}{r}{\left[\hat{\mathbf{i}}\right]}\\ {\left[\hat{\mathbf{j}}\right]=\mathbf{T}_{\phi}\left[\mathbf{J}\right]=\left[\begin{array}{c c c}{\cos(\phi(t))}&{0}&{-\sin(\phi(t))}\\ {0}&{1}&{0}\\ {\sin(\phi(t))}&{0}&{\cos(\phi(t))}\end{array}\right]\left[\mathbf{J}\right]}\end{array}
+\begin{bmatrix}
+\hat{\mathbf{i}} \\
+\hat{\mathbf{j}} \\
+\hat{\mathbf{k}}
+\end{bmatrix} = \mathbf{T}_{\phi} \begin{bmatrix}
+\mathbf{I} \\
+\mathbf{J} \\
+\mathbf{K}
+\end{bmatrix} = \begin{bmatrix}
+\cos(\phi(t)) & 0 & -\sin(\phi(t)) \\
+0 & 1 & 0 \\
+\sin(\phi(t)) & 0 & \cos(\phi(t))
+\end{bmatrix} \begin{bmatrix}
+\mathbf{I} \\
+\mathbf{J} \\
+\mathbf{K}
+\end{bmatrix}
+\tag{43}
 $$  
 
 and between the $(\hat{x},\hat{y},\hat{z})$ -frame and the $(x,\,y,\,z)$ -frame is given by  
 
 $$
-\begin{array}{r l}&{\left[\!\!\begin{array}{c}{\mathbf{\hat{i}}}\\ {\mathbf{j}}\\ {\mathbf{k}}\end{array}\!\!\right]=\mathbf{T}_{\beta}\left[\!\!\begin{array}{c}{\hat{\mathbf{i}}}\\ {\hat{\mathbf{j}}}\\ {\hat{\mathbf{k}}}\end{array}\!\!\right]=\left[\!\!\begin{array}{c c c}{\cos(\beta(t))}&{\sin(\beta(t))}&{0\!\!\sqrt{\!\!\dot{\mathbf{i}}}}\\ {-\sin(\beta(t))}&{\cos(\beta(t))}&{0\!\!\sqrt{\!\!\dot{\mathbf{j}}}}\\ {0}&{0}&{1\!\!\sqrt{\!\!\dot{\mathbf{k}}}}\end{array}\!\!\right]\!\!\right]_{\mathbf{\hat{k}}}}\end{array}
+\begin{bmatrix}
+\mathbf{i} \\
+\mathbf{j} \\
+\mathbf{k}
+\end{bmatrix} = \mathbf{T}_{\beta} \begin{bmatrix}
+\hat{\mathbf{i}} \\
+\hat{\mathbf{j}} \\
+\hat{\mathbf{k}}
+\end{bmatrix} = \begin{bmatrix}
+\cos(\beta(t)) & \sin(\beta(t)) & 0 \\
+-\sin(\beta(t)) & \cos(\beta(t)) & 0 \\
+0 & 0 & 1
+\end{bmatrix} \begin{bmatrix}
+\hat{\mathbf{i}} \\
+\hat{\mathbf{j}} \\
+\hat{\mathbf{k}}
+\end{bmatrix}
+\tag{44}
 $$  
 
 The principle axis of each cross section of the blade is described by the $(\mathfrak{n},\xi,\zeta)$ -frame with origin at $e a$ , where $\eta$ and $\zeta$ are the principle axes of the cross section and the $\zeta.$ -axis points outward along the elastic axis of the deformed blade. This frame has the unit vectors $(\tilde{\mathrm{i}},\tilde{\mathrm{j}},\tilde{\mathrm{k}})$ given by the following transformation:  
 
 $$
-\begin{array}{r l}&{\left[\begin{array}{l}{\overline{{\mathbf{i}}}}\\ {\overline{{\mathbf{j}}}}\\ {\overline{{\mathbf{k}}}}\end{array}\right]=\mathbf{T}_{\mathrm{e}}\left[\begin{array}{l l l}{{\mathbf{i}}}\\ {{\bf{j}}}\\ {{\bf{k}}}\end{array}\right]=\left[\begin{array}{l l l}{\cos(\hat{\theta}(s,t))}&{\sin(\hat{\theta}(s,t))}&{0}\\ {-\sin(\hat{\theta}(s,t))}&{\cos(\hat{\theta}(s,t))}&{0}\\ {0}&{0}&{1}\end{array}\right]}\\ &{\begin{array}{r l}{{\mathbf{\theta}}_{1}}&{0}\\ {0}&{\sqrt{1-\nu^{\prime}(s,t)^{2}}}&{-\nu^{\prime}(s,t)}\\ {0}&{0}&{1}\end{array}\right]}\\ &{\begin{array}{r l}{\sqrt{\bigg[\frac{1-(l_{p}^{\prime}(s)+u^{\prime}(s,t))^{2}-\nu^{\prime}(s,t)^{2}}{1-\nu^{\prime}(s,t)^{2}}}}&{0}&{\cdots\frac{l_{p}^{\prime}(s)+u^{\prime}(s,t)}{\sqrt{1-\nu^{\prime}(s,t)^{2}}}}\\ {0}&{0}&{1}\end{array}\right]\times\left[\begin{array}{l}{1}\\ {1}\\ {\overline{{\mathbf{k}}}}\\ {\overline{{\mathbf{k}}}}\end{array}\right]}\\ &{\begin{array}{r l}{\frac{L_{p}^{\prime}(s)+u^{\prime}(s,t)}{\sqrt{1-\nu^{\prime}(s,t)^{2}}}}&{0}\\ {0}&{\sqrt{\frac{1-\big(l_{p}^{\prime}(s)+u^{\prime}(s,t)\big)^{2}-\nu^{\prime}(s,t)^{2}}{1-\nu^{\prime}(s,t)^{2}}}}\end{array}\right]\mathbf{k},}\end{array}}\end{array}
+\begin{bmatrix}
+\tilde{\mathbf{i}} \\
+\tilde{\mathbf{j}} \\
+\tilde{\mathbf{k}}
+\end{bmatrix} = \mathbf{T}_c \begin{bmatrix}
+\mathbf{i} \\
+\mathbf{j} \\
+\mathbf{k}
+\end{bmatrix} = \begin{bmatrix}
+\cos(\hat{\theta}(s,t)) & \sin(\hat{\theta}(s,t)) & 0 \\
+-\sin(\hat{\theta}(s,t)) & \cos(\hat{\theta}(s,t)) & 0 \\
+0 & 0 & 1
+\end{bmatrix}
+\begin{bmatrix}
+1 & 0 & 0 \\
+0 & \sqrt{1-v'(s,t)^2} & -v'(s,t) \\
+0 & v'(s,t) & \sqrt{1-v'(s,t)^2}
+\end{bmatrix}
+\begin{bmatrix}
+\sqrt{\frac{1-(l'_{pi}(s)+u'(s,t))^2 - v'(s,t)^2}{1-v'(s,t)^2}} & 0 & \frac{-(l'_{pi}(s)+u'(s,t))}{\sqrt{1-v'(s,t)^2}} \\
+0 & 1 & 0 \\
+\frac{l'_{pi}(s)+u'(s,t)}{\sqrt{1-v'(s,t)^2}} & 0 & \sqrt{\frac{1-(l'_{pi}(s)+u'(s,t))^2 - v'(s,t)^2}{1-v'(s,t)^2}}
+\end{bmatrix}
+\begin{bmatrix}
+\mathbf{i} \\
+\mathbf{j} \\
+\mathbf{k}
+\end{bmatrix}
+\tag{45}
 $$  
 
 where $\tilde{\theta}$ is the rotation of the blade around the elastic axis. The first matrix in equation (45) is the rotation about the $\hat{z}$ -axis, the next matrix is the rotation about the $x$ -axis and the last matrix is the rotation about the $y$ -axis.  
@@ -567,7 +629,7 @@ where ${{\bf{T}}_{c}}$ is equal to ${\bf{T}}_{e}$ except that the twist qˆ incl
 The rotation of the principle axis of the blade sections as a function of the $s$ coordinate is given by the differential equation:  
 
 $$
-\begin{array}{r}{\mathbf{T}_{e}^{\prime}\!=\!\!\left[\!\!\begin{array}{c c c}{0}&{\tilde{\omega}_{k}}&{-\tilde{\omega}_{j}}\\ {-\tilde{\omega}_{k}}&{0}&{\tilde{\omega}_{i}}\\ {\tilde{\omega}_{j}}&{-\tilde{\omega}_{i}}&{0}\end{array}\!\!\right]\!\mathbf{T}_{c}\!\Rightarrow\!\!\left[\!\!\begin{array}{c c c}{0}&{\tilde{\omega}_{k}}&{-\tilde{\omega}_{j}}\\ {-\tilde{\omega}_{k}}&{0}&{\tilde{\omega}_{i}}\\ {\tilde{\omega}_{j}}&{-\tilde{\omega}_{i}}&{0}\end{array}\!\!\right]\!=\!T_{\mathrm{c}}\mathbf{T}_{\mathrm{c}}^{-1}}\end{array}
+\begin{array}{r}{\mathbf{T}_{e}^{\prime}=\left[\!\!\begin{array}{c c c}{0}&{\tilde{\omega}_{k}}&{-\tilde{\omega}_{j}}\\ {-\tilde{\omega}_{k}}&{0}&{\tilde{\omega}_{i}}\\ {\tilde{\omega}_{j}}&{-\tilde{\omega}_{i}}&{0}\end{array}\right]\!\mathbf{T}_{c}\!\Rightarrow\left[\begin{array}{c c c}{0}&{\tilde{\omega}_{k}}&{-\tilde{\omega}_{j}}\\ {-\tilde{\omega}_{k}}&{0}&{\tilde{\omega}_{i}}\\ {\tilde{\omega}_{j}}&{-\tilde{\omega}_{i}}&{0}\end{array}\right]=T_{\mathrm{c}}\mathbf{T}_{\mathrm{c}}^{-1}}\end{array}
 $$  
 
 where $(\tilde{w}_{i},\,\tilde{w}_{j},\,\tilde{w}_{k})$ is the rotation about the $(\tilde{\mathbf{i}},\tilde{\mathbf{j}},\tilde{\mathbf{k}})$ -directions respectively, and it is utilized that $\mathbf{T}_{\phi}^{\prime}=\mathbf{T}_{\beta}^{\prime}=0$ . The rotation about the $\tilde{\mathbf{k}}$ -direction is also measured by changes in the twist coordinates of the blade, (the pre-twist $\tilde{\theta}=\tilde{\theta}(s)$ and the elastic twist $\theta_{e l a}=\theta_{e l a}(s,\,t))$ . Hence,  
@@ -595,7 +657,31 @@ Note that $\mathbf{T}^{\mathrm{T}}\mathbf{T}=\mathbf{I}$ holds for all the trans
 The individual terms in the assumed mode approximated blade model (equation (39)) are  
 
 $$
-\begin{array}{r l}&{\mathbb{D}(\boldsymbol{\phi},\boldsymbol{\beta},\boldsymbol{\beta})=\hat{\wp}(\mathbb{D}_{\boldsymbol{\ell}\times\boldsymbol{\sin}}(\boldsymbol{\beta})+\mathbb{D}_{\boldsymbol{\ell}\times\boldsymbol{\cos}}(\boldsymbol{\beta}))+\hat{\varrho}(\mathbb{D}_{\boldsymbol{\ell}\times\boldsymbol{\sin}}(\boldsymbol{\beta})+\mathbb{D}_{\boldsymbol{\ell}\times\boldsymbol{\cos}}\cos(\beta))+\hat{\jmath}\mathfrak{D}_{\boldsymbol{\mathcal{N}}_{\boldsymbol{\ell}}}(\boldsymbol{\beta})}\\ &{\mathrm{K}(\boldsymbol{\phi},\boldsymbol{\beta},\boldsymbol{\beta})=\mathbf{K}_{\boldsymbol{\ell}}+\boldsymbol{\mathrm{K}}+\boldsymbol{\beta}^{2}\mathbb{K}_{\boldsymbol{\ell}}+2\hat{\beta}\hat{\boldsymbol{\psi}}(\mathbb{R}_{\boldsymbol{\ell}\times\boldsymbol{\sin}}\sin(\beta)+\mathbb{K}_{\boldsymbol{\ell}\times\boldsymbol{\cos}}(\boldsymbol{\beta}))}\\ &{\qquad\qquad\quad+\vec{\varrho}^{2}\left(\mathbb{K}_{\boldsymbol{\ell}\times\boldsymbol{\sin}}+\mathbb{K}_{\boldsymbol{\ell}\times\boldsymbol{\sin}}\sin^{2}(\beta)+\mathbb{K}_{\boldsymbol{\ell}\times\boldsymbol{\cos}}\cos^{2}(\beta)+\mathbb{K}_{\boldsymbol{\ell}\times\boldsymbol{\sin}}\sin(\beta)\cos(\beta)\right)}\\ &{L(\vec{\beta},\boldsymbol{\phi},\boldsymbol{\beta})=\hat{\jmath}\mathcal{R}_{\boldsymbol{\ell}\times\boldsymbol{\sin}}(\boldsymbol{\phi})(\mathbb{D}_{\boldsymbol{\ell}\times\boldsymbol{\sin}},\sin(\beta)+\mathbb{F}_{\boldsymbol{\ell}\times\boldsymbol{\cos}}(\boldsymbol{\beta}))+\boldsymbol{\mathrm{g}}\sin(\phi)\boldsymbol{\mathrm{F}}_{\boldsymbol{\ell}},}\\ &{\mathrm{N}(\boldsymbol{\phi},\boldsymbol{\beta},\boldsymbol{\beta},\boldsymbol{\mathrm{q}})=\varrho,\mathbb{F}_{\boldsymbol{\ell}}(\boldsymbol{\mathrm{q}})+\mathbb{F}_{\boldsymbol{\ell}}\left[L_{\boldsymbol{\ell}\times\boldsymbol{\sin}}^{\prime}(\boldsymbol{\mu})^{2}\right]^{\dagger}+2\hat{\varrho}(\mathbb{F}_{\boldsymbol{\ell}},\sin(\beta)+\mathbb{F}_{\boldsymbol{\ell}\times\boldsymbol{\cos}}(\boldsymbol{\beta}))\boldsymbol{\mathrm{q}}}\\ &{\qquad 
+\begin{align*}
+\mathbf{D}(\phi, \dot{\beta}, \beta) &= \dot{\phi}(\mathbf{D}_{\phi,s} \sin(\beta) + \mathbf{D}_{\phi,c} \cos(\beta)) + \dot{\phi}(\mathbf{D}_{\phi,a,s} \sin(\beta) + \mathbf{D}_{\phi,a,c} \cos(\beta)) + \dot{\beta}\mathbf{D}_{\beta}(\beta) \\
+\mathbf{K}(\phi, \dot{\beta}, \beta) &= \mathbf{K}_s + \mathbf{K} + \dot{\beta}^2 \mathbf{K}_{\beta} + 2\dot{\phi}\dot{\beta}(\mathbf{K}_{\beta\phi,s} \sin(\beta) + \mathbf{K}_{\beta\phi,c} \cos(\beta)) \\
+&\quad + \dot{\phi}^2(\mathbf{K}_{\phi,0} + \mathbf{K}_{\phi,ss} \sin^2(\beta) + \mathbf{K}_{\phi,cc} \cos^2(\beta) + \mathbf{K}_{\phi,sc} \sin(\beta)\cos(\beta)) \\
+\mathbf{L}(\dot{\beta}, \phi, \beta) &= \ddot{\beta}\mathbf{F}_{\beta,1} + g \sin(\phi)(\mathbf{F}_{g,1,s} \sin(\beta) + \mathbf{F}_{g,1,c} \cos(\beta)) + g \sin(\phi)\mathbf{F}_{g,2} \\
+\mathbf{N}(\dot{\phi}, \dot{\beta}, \beta, q) &= \theta_t \mathbf{F}_1 q + \mathbf{F}_2[u_l^2 v_l^2 \theta_l^2]^T + 2\dot{\phi}\mathbf{Q}(\mathbf{F}_{3,s} \sin(\beta) + \mathbf{F}_{3,c} \cos(\beta))\dot{q} \\
+&\quad + 2\dot{\phi}\dot{\beta}\mathbf{Q}(\mathbf{F}_{4,s} \sin(\beta) + \mathbf{F}_{4,c} \cos(\beta))q + f_5 u_l v_l \\
+\mathbf{F}(\ddot{\beta}, \dot{\phi}, \dot{\beta}, \dot{\phi}, \beta, \phi) &= 2\dot{\phi}\dot{\beta}\mathbf{F}_{\phi\beta}(\beta) + \dot{\phi}^2 f_{\phi} + g(\mathbf{F}_{g,0} + \mathbf{F}_{g,s} \sin(\beta) + \mathbf{F}_{g,c} \cos(\beta))
+\begin{bmatrix}
+\sin(\phi) \\
+\cos(\phi)
+\end{bmatrix} \\
+&\quad + \mathbf{F}_{\beta,0}
+\begin{bmatrix}
+\beta \\
+\dot{\beta}^2
+\end{bmatrix}
++ (\mathbf{F}_{\phi,0} + \mathbf{F}_{\phi,s} \sin(\beta) + \mathbf{F}_{\phi,c} \cos(\beta) + \mathbf{F}_{\phi,ss} \sin^2(\beta) \\
+&\quad + \mathbf{F}_{\phi,cc} \cos^2(\beta) + \mathbf{F}_{\phi,sc} \sin(\beta)\cos(\beta))
+\begin{bmatrix}
+\ddot{\phi} \\
+\dot{\phi}^2
+\end{bmatrix}
+\tag{50}
+\end{align*}
 $$  
 
 where the constants for the linear terms are  
@@ -605,7 +691,13 @@ $$
 $$  
 
 $$
-\int_{r}^{R}\left[\frac{E\big(I_{\xi}\;\mathrm{cos}^{2}(\tilde{\theta})+I_{\eta}\;\mathrm{sin}^{2}(\tilde{\theta})\big)u_{*}^{\prime\prime}u_{*}^{\prime\prime}}{E\big(I_{\xi}\;I_{\eta}\big)\cos(\tilde{\theta})\sin(\tilde{\theta})u_{*}^{\prime\prime}v_{*}^{\prime\prime}}\right.\qquad E\big(I_{\xi}\;-\;I_{\eta}\big)\cos(\tilde{\theta})\sin(\tilde{\theta})u_{*}^{\prime\prime}v_{*}^{\prime\prime}\qquad-E\big(I_{\xi}\;I_{\eta}\big)\cos(\tilde{\theta})\sin(\tilde{\theta})\big)d\tilde{\eta}=0,\quad\mathrm{for~o~r~o~r~}\quad R\in\mathbb{R}^{3},
+\mathbf{K}_s = \int_r^R
+\begin{bmatrix}
+E(I_{\xi} \cos^2(\tilde{\theta}) + I_{\eta} \sin^2(\tilde{\theta}))u_s''u_s'' & E(I_{\xi} - I_{\eta})\cos(\tilde{\theta})\sin(\tilde{\theta})u_s''v_s'' & -E(I_{\xi} - I_{\eta})\cos(\tilde{\theta})\sin(\tilde{\theta})l_{pi}''u_s''\theta_s \\
+E(I_{\xi} - I_{\eta})\cos(\tilde{\theta})\sin(\tilde{\theta})u_s''v_s'' & E(I_{\xi} \sin^2(\tilde{\theta}) + I_{\eta} \cos^2(\tilde{\theta}))v_s''v_s'' & -E(I_{\xi} \sin^2(\tilde{\theta}) + I_{\eta} \cos^2(\tilde{\theta}))l_{pi}''v_s''\theta_s \\
+-E(I_{\xi} - I_{\eta})\cos(\tilde{\theta})\sin(\tilde{\theta})l_{pi}''u_s''\theta_s & -E(I_{\xi} \sin^2(\tilde{\theta}) + I_{\eta} \cos^2(\tilde{\theta}))l_{pi}''u_s''\theta_s & GJ\theta_s'\theta_s'
+\end{bmatrix}
+ds
 $$  
 
 $$
@@ -617,7 +709,7 @@ $$
 $$  
 
 $$
-\mathbf{K}_{\phi,s s}\!=\!\int_{r}^{R}\!\!\left[\!\!\begin{array}{c c c}{0}&{0}&{0}\\ {0}&{-m\nu_{s}^{2}}&{-l_{c g}m\cos(\overline{{\theta}})\nu_{s}\theta_{s}}\\ {0}&{-l_{c g}m\cos(\overline{{\theta}})\nu_{s}\theta_{s}}&{0}\end{array}\!\!\right]\!\mathrm{d}s
+\mathbf{K}_{\phi,s s}=\int_{r}^{R}\left[\begin{array}{c c c}{0}&{0}&{0}\\ {0}&{-m\nu_{s}^{2}}&{-l_{c g}m\cos(\overline{{\theta}})\nu_{s}\theta_{s}}\\ {0}&{-l_{c g}m\cos(\overline{{\theta}})\nu_{s}\theta_{s}}&{0}\end{array}\right]\!\mathrm{d}s
 $$  
 
 $$
@@ -625,11 +717,24 @@ $$
 $$  
 
 $$
-\mathbf{D}_{\phi,a,\mathrm{s}}\!=\!\int_{r}^{R}\!\left[m l_{c g}\cos(\overline{{\theta}})u_{s}^{\prime}\nu_{s}\right.\left.\right.-m l_{c g}\cos(\overline{{\theta}})\nu_{s}u_{s}^{\prime}\left.\right.\left.0\right]\!\!\!\!\mathrm{d}s
+\mathbf{D}_{\phi,a,s} = \int_r^R
+\begin{bmatrix}
+0 & -ml_{cg} \cos(\bar{\theta})v_s u_s' & 0 \\
+ml_{cg} \cos(\bar{\theta})u_s'v_s & -ml_{cg} \sin(\bar{\theta})v_s v_s' & 0 \\
+0 & +ml_{cg} \sin(\bar{\theta})v_s v_s' & 0
+\end{bmatrix}
+ds
 $$  
 
 $$
-\mathbf{D}_{\phi,a,\varsigma}=\int_{r}^{R}\left[\begin{array}{c c c}{m l_{c g}\cos(\overline{{\theta}})u_{s}u_{s}^{\prime}}&{-m l_{c g}\sin(\overline{{\theta}})\nu_{s}^{\prime}u_{s}}&{0}\\ {-m l_{c g}\cos(\overline{{\theta}})u_{s}u_{s}^{\prime}}&{0}&{0}\\ {m l_{c g}\sin(\overline{{\theta}})u_{s}\nu_{s}^{\prime}}&{0}&{0}\\ {0}&{0}&{0}\\ {0}&{0}&{0}\end{array}\right]\mathrm{d}s
+\mathbf{D}_{\phi,a,c} = \int_r^R
+\begin{bmatrix}
+ml_{cg} \cos(\bar{\theta})u_s u_s' & -ml_{cg} \sin(\bar{\theta})v_s' u_s & 0 \\
+-ml_{cg} \cos(\bar{\theta})u_s u_s' & 0 & 0 \\
+ml_{cg} \sin(\bar{\theta})u_s v_s' & 0 & 0 \\
+0 & 0 & 0
+\end{bmatrix}
+ds
 $$  
 
 $$
@@ -641,31 +746,91 @@ $$
 $$  
 
 $$
-\mathbf{K}_{\beta\phi,s}=\int_{r}^{R}\left[\begin{array}{c c c}{-m l_{c g}u_{s}u_{s}^{\prime}\cos(\overline{{\theta}})-l_{p i}^{\prime}u_{s}^{\prime}\biggr\int_{s}^{R}m u_{s}\mathrm{d}\rho}&{0}&{0}\\ {-u_{s}^{\prime2}\int_{s}^{R}m(l_{p i}+l_{c g}\cos(\overline{{\theta}}))m u_{s}\mathrm{d}\rho}&{0}&{0}\\ {-m l_{c g}u_{s}\nu_{s}^{\prime}\sin(\overline{{\theta}})}&{-u_{s}^{\prime2}\int_{s}^{R}m(l_{p i}+l_{c g}\cos(\overline{{\theta}}))\mathrm{d}\rho}&{0}\\ {0}&{0}&{0}\\ {0}&{0}&{0}\end{array}\right]\mathrm{d}s
+\mathbf{F}_{\beta,1} = \int_r^R
+\begin{bmatrix}
+0 & -mv_s u_s & -ml_{cg} \theta_s u_s \cos(\bar{\theta}) \\
+m u_s v_s & 0 & -ml_{cg} \theta_s v_s \sin(\bar{\theta}) \\
+ml_{cg} u_s \theta_s \cos(\bar{\theta}) & ml_{cg} \theta_s v_s \sin(\bar{\theta}) & 0
+\end{bmatrix}
+ds
 $$  
 
 $$
-\mathbf{K}_{\beta\phi,c}=\int_{r}^{R}\left[\begin{array}{c c c c}{-{u_{s}^{\prime}}^{2}\displaystyle\int_{s}^{R}m l_{c g}\sin(\overline{{\theta}}){\mathrm{d}}\rho}&{-{m l_{c g}}{\nu_{s}}{u_{s}^{\prime}}\cos(\overline{{\theta}})}&{0}\\ {0}&{-{l_{p}^{\prime}}{u_{s}^{\prime}}\displaystyle\int_{s}^{R}m{\nu_{s}}{\mathrm{d}}\rho}&{0}\\ {0}&{-{m l_{c g}}{\nu_{s}}{\nu_{s}^{\prime}}\sin(\overline{{\theta}}){\mathrm{d}}\rho}&{0}\\ {0}&{-{\nu_{s}^{\prime}}^{2}\displaystyle\int_{r}^{R}m l_{c g}\sin(\overline{{\theta}}){\mathrm{d}}\rho}&{0}\\ {0}&{0}&{0}\end{array}\right]{\mathrm{d}}s
+\mathbf{K}_{\beta \phi,s} = \int_r^R
+\begin{bmatrix}
+-ml_{cg}u_s u_s' \cos(\bar{\theta}) - l_{pi}'u_s' \int_s^R mu_s d\rho & 0 & 0 \\
+-u_s'^2 \int_s^R m(l_{pi} + l_{cg} \cos(\bar{\theta}))mu_s d\rho & 0 & 0 \\
+-ml_{cg}u_s v_s' \sin(\bar{\theta}) & -u_s'^2 \int_s^R m(l_{pi} + l_{cg} \cos(\bar{\theta}))d\rho & 0 \\
+0 & 0 & 0
+\end{bmatrix}
+ds
+ds
+$$  
+$$
+\mathbf{K}_{\beta \phi,c} = \int_r^R
+\begin{bmatrix}
+-u_s'^2 \int_s^R ml_{cg} \sin(\bar{\theta}) d\rho & -ml_{cg} v_s u_s' \cos(\bar{\theta}) & 0 \\
+0 & -l_{pi}'u_s' \int_s^R mv_s d\rho & 0 \\
+0 & -ml_{cg} v_s v_s' \sin(\bar{\theta}) d\rho & 0 \\
+0 & -v_s'^2 \int_r^R ml_{cg} \sin(\bar{\theta}) d\rho & 0
+\end{bmatrix}
+ds
+$$
+$$
+\mathbf{F}_{g,1,s} = \int_r^R
+\begin{bmatrix}
+0 & 0 & 0 \\
+0 & ml_{cg}v_s'^2 \sin(\bar{\theta}) & 0 \\
+0 & 0 & ml_{cg}\theta_s^2 \sin(\bar{\theta})
+\end{bmatrix}
+ds
+$$
+
+
+$$
+\mathbf{F}_{g,1,c} = \int_r^R
+\begin{bmatrix}
+-ml_{cg}u_s'^2 \cos(\bar{\theta}) & -ml_{cg}u_s u_s' \sin(\bar{\theta}) & 0 \\
+-ml_{cg}v_s'u_s' \sin(\bar{\theta}) & 0 & 0 \\
+0 & 0 & -ml_{cg}\theta_s^2 \cos(\bar{\theta})
+\end{bmatrix}
+ds
 $$  
 
 $$
-\mathbf{F}_{g,1,s}=\int_{r}^{R}\!\left[\begin{array}{c c c}{0}&{0}&{0}\\ {0}&{m l_{c g}\nu_{s}^{\prime2}\sin(\overline{{\theta}})}&{0}\\ {0}&{0}&{m l_{c g}\theta_{s}^{2}\sin(\overline{{\theta}})\right]\!\mathrm{d}s}\end{array}
+\mathbf{F}_{g,2} = \int_r^R
+\begin{bmatrix}
+-u_s'^2 \int_s^R md\rho & 0 & -ml_{cg} \theta_s u_s' \sin(\bar{\theta}) \\
+0 & -v_s'^2 \int_s^R md\rho & ml_{cg} \theta_s v_s' \cos(\bar{\theta}) \\
+-ml_{cg} \theta_s u_s' \sin(\bar{\theta}) & ml_{cg} \theta_s v_s' \cos(\bar{\theta}) & 0
+\end{bmatrix}
+ds
 $$  
 
 $$
-\mathbf{F}_{g,1,c}=\int_{r}^{R}\!\left[\!\!{\begin{array}{c c c}{-m l_{c g}u_{s}^{\prime\,2}\cos(\overline{{\theta}})}&{-m l_{c g}u_{s}^{\prime}u_{s}^{\prime}\sin(\overline{{\theta}})}&{0}\\ {-m l_{c g}{\nu}_{s}^{\prime}u_{s}^{\prime}\sin(\overline{{\theta}})}&{0}&{0}\\ {0}&{0}&{-m l_{c g}{\theta}_{s}^{2}\cos(\overline{{\theta}})}\end{array}}\!\!\right]\!\mathrm{d}s
+\mathbf{D}_{\phi,s} = \int_r^R
+\begin{bmatrix}
+0 & -l'_{pi}u'_{s}\int_s^R mv_s d\rho & 0 \\
+0 & 0 & 0 \\
+0 & 0 & 0
+\end{bmatrix}
+ds, \quad \mathbf{D}_{\phi,c} = \int_r^R
+\begin{bmatrix}
+l'_{pi}u'_{s}\int_s^R mu_s d\rho & 0 & 0 \\
+0 & 0 & 0 \\
+0 & 0 & 0
+\end{bmatrix}
+ds
 $$  
 
 $$
-\mathbf{F}_{g,2}=\int_{r}^{R}{\left[\begin{array}{c c c}{-{u_{s}^{\prime}}^{2}{\int_{s}^{R}}m\mathrm{d}\rho}&{0}&{-m l_{c g}\theta_{s}{u_{s}^{\prime}}\sin{(\overline{{\theta}})}}\\ {0}&{-{\nu_{s}^{\prime}}^{2}{\int_{s}^{R}}m\mathrm{d}\rho}&{m l_{c g}\theta_{s}{\nu_{s}^{\prime}}\cos(\overline{{\theta}})}\\ {-m l_{c g}\theta_{s}{u_{s}^{\prime}}\sin(\overline{{\theta}})}&{m l_{c g}\theta_{s}{\nu_{s}^{\prime}}\cos(\overline{{\theta}})}&{0}\end{array}\right]}\mathrm{d}s
+\mathbf{K} = \int_r^R
-$$  
+\begin{bmatrix}
-
+0 & 0 & 0 \\
-$$
+0 & 0 & 0 \\
-\mathbf{D}_{\phi,s}=\int_{r}^{R}\left[\begin{array}{c c c c}{0}&{-l_{p i}^{\prime}u_{s}^{\prime}\int_{s}^{R}m\nu_{s}\mathrm{d}\rho}&{0}\\ {0}&{0}&{0}\\ {0}&{0}&{0\Biggr]\mathrm{d}s,}&{\mathbf{D}_{\phi,c}=\int_{r}^{R}\left[\begin{array}{c c c c}{l_{p i}^{\prime}u_{s}^{\prime}\int_{s}^{R}m u_{s}\mathrm{d}\rho}&{0}&{0}\\ {0}&{0}&{0}\\ {0}&{0}&{0}\end{array}\right]\mathrm{d}s
+EI_{\eta\eta\zeta}\bar{\theta}'\theta_s'u_s''\sin(\bar{\theta}) - EI_{\eta\zeta\zeta}\bar{\theta}'\theta_s'u_s''\cos(\bar{\theta}) & -EI_{\eta\eta\zeta}\bar{\theta}'\theta_s''u_s'\cos(\tilde{\theta}) - EI_{\eta\zeta\zeta}\bar{\theta}'\theta_s'v_s''\sin(\tilde{\theta}) & 0
-$$  
+\end{bmatrix}
-
+ds
-$$
-\mathbf{K}=\int_{r}^{\kappa}\left[\begin{array}{c c c c}{0}&{0}&{0}&{0}\\ {0}&{0}&{0}\\ {E I_{\eta\eta\xi}\overline{{\theta}}^{\prime}\theta_{s}^{\prime}u_{s}^{\prime\prime}\mathrm{sin}(\overline{{\theta}})-E I_{\eta\xi\overline{{\xi}}}\overline{{\theta}}^{\prime}\theta_{s}^{\prime}u_{s}^{\prime\prime}\mathrm{cos}(\overline{{\theta}})}&{-E I_{\eta\eta\overline{{\xi}}}\overline{{\theta}}^{\prime}\theta_{s}^{\prime}u_{s}^{\prime\prime}\mathrm{cos}(\overline{{\theta}})-E I_{\eta\xi\overline{{\xi}}}\overline{{\theta}}^{\prime}\theta_{s}^{\prime}\nu_{s}^{\prime\prime}\mathrm{sin}(\widetilde{\theta})}&{0}\end{array}\right]^{0},
 $$  
 
 and for constants for the nonlinear terms:  
@@ -679,11 +844,23 @@ $$
 $$  
 
 $$
-\mathbf{F}_{3,s}=\int_{r}^{R}{\left[\begin{array}{l l l l}{0}&{-u_{s}^{\prime\,2}\displaystyle\int_{s}^{R}m\nu_{s}\mathrm{d}\rho}&{0}\\ {0}&{-\nu_{s}^{\prime\,2}\displaystyle\int_{s}^{R}m\nu_{s}\mathrm{d}\rho}&{0}\\ {0}&{0}&{0}\\ {0}&{\qquad0}&{0}\end{array}\right]}\mathrm{d}s,\quad\mathbf{F}_{3,c}=\int_{r}^{R}{\left[\begin{array}{l l l}{u_{s}^{\prime\,2}\displaystyle\int_{s}^{R}m u_{s}\mathrm{d}\rho}&{0}&{0}\\ {\nu_{s}^{\prime\,2}\displaystyle\int_{s}^{R}m u_{s}\mathrm{d}\rho}&{0}&{0}\\ {0}&{0}&{0}\\ {0}&{0}&{0}\end{array}\right]}\mathrm{d}s
+\mathbf{F}_{3,s}=\int_{r}^{R}{\left[\begin{array}{l l l}{0}&{-u_{s}^{\prime\,2}\displaystyle\int_{s}^{R}m\nu_{s}\mathrm{d}\rho}&{0}\\ {0}&{-\nu_{s}^{\prime\,2}\displaystyle\int_{s}^{R}m\nu_{s}\mathrm{d}\rho}&{0}\\ {0}&{\qquad0}&{0}\end{array}\right]}\mathrm{d}s,\quad\mathbf{F}_{3,c}=\int_{r}^{R}{\left[\begin{array}{l l l}{u_{s}^{\prime\,2}\displaystyle\int_{s}^{R}m u_{s}\mathrm{d}\rho}&{0}&{0}\\ {\nu_{s}^{\prime\,2}\displaystyle\int_{s}^{R}m u_{s}\mathrm{d}\rho}&{0}&{0}\\ {\qquad0}&{0}&{0}\\ \end{array}\right]}\mathrm{d}s
 $$  
 
 $$
-\mathbf{F}_{4,s}=\int_{r}^{R}\left[-\nu_{s}^{\prime2}\int_{s}^{R}m u_{s}\mathrm{d}\rho\quad0\quad0\right]\mathrm{d}s,\quad\mathbf{F}_{4,c}=\int_{r}^{R}\left[0\quad-u_{s}^{\prime2}\int_{s}^{R}m\nu_{s}\mathrm{d}\rho\quad0\right]}\\ {0\quad\qquad\qquad\quad0\quad0\quad0\quad0\right]\mathrm{d}s,\quad\mathbf{F}_{4,c}=\int_{r}^{R}\left[0\quad-\nu_{s}^{\prime2}\int_{s}^{R}m\nu_{s}\mathrm{d}\rho\quad0\right]\mathrm{d}s}\end{array}
+\mathbf{F}_{4,s} = \int_r^R
+\begin{bmatrix}
+-u_s'^2 \int_s^R mu_s d\rho & 0 & 0 \\
+-v_s'^2 \int_s^R mu_s d\rho & 0 & 0 \\
+0 & 0 & 0
+\end{bmatrix}
+ds, \quad \mathbf{F}_{4,c} = \int_r^R
+\begin{bmatrix}
+0 & -u_s'^2 \int_s^R mv_s d\rho & 0 \\
+0 & -v_s'^2 \int_s^R mv_s d\rho & 0 \\
+0 & 0 & 0
+\end{bmatrix}
+ds
 $$  
 
 $$