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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

@@ -870,19 +870,43 @@ $$
 and for the forcing terms:  
 
 $$
-\mathbf{F}_{\beta,0}\!=\!\int_{r}^{R}\!\left[-m(l_{r g}u_{s}\sin(\overline{{\theta}}))\!\!\begin{array}{c c c}{m(l_{p i}+l_{c g}\cos(\overline{{\theta}}))u_{s}}\\ {m{l}_{c g}\nu_{s}\sin(\overline{{\theta}})}\\ {-(I_{c g}+m l_{c g}^{2})\theta_{s}}&{-m l_{c g}l_{p i}\theta_{s}\sin(\overline{{\theta}})}\end{array}\right]\!\mathrm{d}s
+\mathbf{F}_{\beta,0} = \int_r^R
+\begin{bmatrix}
+ml_{cg}u_s \sin(\bar{\theta}) & m(l_{pi} + l_{cg} \cos(\bar{\theta}))u_s \\
+-m(l_{pi} + l_{cg} \cos(\bar{\theta}))v_s & ml_{cg}v_s \sin(\bar{\theta}) \\
+-(I_{cg} + ml_{cg}^2)\ddot{\theta}_s & -ml_{cg}l_{pi}\ddot{\theta}_s \sin(\bar{\theta})
+\end{bmatrix}
+ds
 $$  
 
 $$
-\mathbf{f}_{\beta{_{\phi,s}}}\!=\!\int_{r}^{R}\!\left[\!\!{\begin{array}{c}{\!\!\!{m l_{{c}g}}(l_{p i}+l_{{c}g}\cos(\overline{{\theta}}))u_{s}^{\prime}\cos(\overline{{\theta}})\!+\!l_{p i}^{\prime}u_{s}^{\prime}\displaystyle\int_{s}^{R}\!\!{m(l_{p i}+l_{{c}g}\cos(\overline{{\theta}}))}\mathrm{d}\rho}\\ {\!\!{m l_{{c}g}}(l_{p i}+l_{{c}g}\cos(\overline{{\theta}}))\nu_{s}^{\prime}\sin(\overline{{\theta}})}\\ {0}\end{array}}\!\!\right]\!\mathrm{d}s
+\mathbf{f}_{\beta\phi,s} = \int_r^R
+\begin{bmatrix}
+ml_{cg}(l_{pi} + l_{cg} \cos(\bar{\theta}))u_s' \cos(\bar{\theta}) + l'_{pi}u'_{s}\int_s^R m(l_{pi} + l_{cg} \cos(\bar{\theta}))d\rho \\
+ml_{cg}(l_{pi} + l_{cg} \cos(\bar{\theta}))v_s' \sin(\bar{\theta}) \\
+0
+\end{bmatrix}
+ds
 $$  
 
 $$
-\mathbf{f}_{\beta\phi,c}=\int_{r}^{R}\!\!\left[{m}l_{c g}l_{c g}\cos(\beta)\sin(\overline{{\theta}})u_{s}^{\prime}\cos(\overline{{\theta}})+l_{p i}^{\prime}u_{s}^{\prime}\!\int_{s}^{R}{m}l_{c g}\sin(\overline{{\theta}})\mathrm{d}\rho\right]_{\mathrm{d}s}}\\ {\quad{m}l_{c g}l_{c g}\sin^{2}(\overline{{\theta}})\nu_{s}^{\prime}}\\ {0}\end{array}\!\!\!\!
+\mathbf{f}_{\beta\phi,c} = \int_r^R
+\begin{bmatrix}
+ml_{cg}l_{cg} \cos(\beta)\sin(\bar{\theta})u_s' \cos(\bar{\theta}) + l'_{pi}u'_{s}\int_s^R ml_{cg} \sin(\bar{\theta})d\rho \\
+ml_{cg}l_{cg} \sin^2(\bar{\theta})v_s' \\
+0
+\end{bmatrix}
+ds
 $$  
 
 $$
-{\bf F}_{\phi,0}={\int_{r}^{R}}\Bigg[0\quad-m l_{c g}w_{0}u_{s}^{\prime}\cos(\overline{{\theta}})\\ {0\quad-m l_{c g}w_{0}\nu_{s}^{\prime}\sin(\overline{{\theta}})}\\ {0\quad-m l_{c g}l_{p i}^{\prime}w_{0}\theta_{s}\sin(\overline{{\theta}})\Bigg]\mathrm{d}s
+\mathbf{F}_{\phi,0} = \int_r^R
+\begin{bmatrix}
+0 & -ml_{cg}w_0 u_s' \cos(\bar{\theta}) \\
+0 & -ml_{cg}w_0 v_s' \sin(\bar{\theta}) \\
+0 & -ml_{cg}l_{pi}' w_0 \theta_s \sin(\bar{\theta})
+\end{bmatrix}
+ds
 $$  
 
 $$
@@ -890,23 +914,54 @@ $$
 $$  
 
 $$
-\mathbf{F}_{\phi,s s}=\int_{r}^{R}\left[\overset{\displaystyle0}{\underset{\displaystyle0}{0}}\right.\qquad m l_{c g}\sin(\overline{{\theta}})\nu_{s}\quad\quad\quad\Biggl]\mathrm{d}s
+\mathbf{F}_{\phi,ss} = \int_r^R
+\begin{bmatrix}
+0 & 0 \\
+0 & ml_{cg}\sin(\bar{\theta})v_s \\
+0 & ml_{cg}l_{cg} \sin(\bar{\theta})\theta_s \cos(\bar{\theta})
+\end{bmatrix}
+ds
 $$  
 
 $$
-\mathbf{F}_{\phi,c c}=\int_{r}^{R}\!\!\left[\begin{array}{c c c}{0}&{m(l_{p i}+l_{c g}\cos(\overline{{\theta}}))u_{s}}\\ {0}&{0}\\ {0}&{-m l_{c g}(l_{p i}+l_{c g}\cos(\overline{{\theta}}))\theta_{s}\sin(\overline{{\theta}})\!\right]\!\mathrm{d}s}\end{array}
+\mathbf{F}_{\phi,cc} = \int_r^R
+\begin{bmatrix}
+0 & m(l_{pi} + l_{cg} \cos(\bar{\theta}))u_s \\
+0 & 0 \\
+0 & -ml_{cg}(l_{pi} + l_{cg} \cos(\bar{\theta}))\theta_s \sin(\bar{\theta})
+\end{bmatrix}
+ds
 $$  
 
 $$
-\mathbf{F}_{\phi,\mathrm{sc}}=\int_{r}^{R}\!\left[\!\!\begin{array}{c c}{0}&{-m l_{c g}\sin(\overline{{\theta}})u_{s}}\\ {0}&{-m\big(l_{p i}+l_{c g}\cos(\overline{{\theta}})\big)\nu_{s}}\\ {0}&{-m l_{c g}\big(l_{p i}\cos(\theta)+l_{c g}\cos(\overline{{\theta}})-l_{c g}\sin^{2}(\overline{{\theta}})\big)\theta_{s}\,\!}\end{array}\!\!\right]\!\mathrm{d}s
+\mathbf{F}_{\phi,\mathrm{sc}}=\int_{r}^{R}\left[\begin{array}{c c}{0}&{-m l_{c g}\sin(\overline{{\theta}})u_{s}}\\ {0}&{-m\big(l_{p i}+l_{c g}\cos(\overline{{\theta}})\big)\nu_{s}}\\ {0}&{-m l_{c g}\big(l_{p i}\cos(\theta)+l_{c g}\cos(\overline{{\theta}})-l_{c g}\sin^{2}(\overline{{\theta}})\big)\theta_{s}\,}\end{array}\!\!\right]\mathrm{d}s
 $$  
 
 $$
-\mathbf{F}_{g,0}=\int_{r}^{R}{\left[\begin{array}{l l}{0}&{-m l_{c g}u_{s}^{\prime}\cos(\overline{{\theta}})\!-\!l_{p i}^{\prime}u_{s}^{\prime}\displaystyle\int_{s}^{R}m\mathrm{d}\rho}\\ {0}&{-m l_{c g}\nu_{s}^{\prime}\sin(\overline{{\theta}})}\\ {0}&{m l_{c g}l_{p i}^{\prime}\theta_{s}\sin(\overline{{\theta}})}\end{array}\right]}\mathrm{d}s
+\mathbf{F}_{g,0} = \int_r^R
+\begin{bmatrix}
+0 & -ml_{cg}u_s'\cos(\bar{\theta}) - l_{pi}'u_s' \int_s^R md\rho \\
+0 & -ml_{cg}v_s'\sin(\bar{\theta}) \\
+0 & ml_{cg}l_{pi}'\theta_s\sin(\bar{\theta})
+\end{bmatrix}
+ds
 $$  
 
 $$
-\mathbf{F}_{g,s}(\beta)\!=\!\int_{r}^{R}\!\left[\begin{array}{c c c}{0}&{0}\\ {m\nu_{s}(0}&{0\Biggr]\mathrm{d}s,}&{\mathbf{F}_{g,c}(\beta)\!=\!\int_{r}^{R}\!\left[\begin{array}{c c c}{-m u_{s}+m l_{c g}l_{p i}^{\prime}u_{s}^{\prime}\cos(\overline{{\theta}})}&{0}\\ {m l_{c g}l_{p i}^{\prime}\nu_{s}^{\prime}\sin(\overline{{\theta}})}&{0}\\ {m l_{c g}\theta_{s}\sin(\overline{{\theta}})}&{0}\end{array}\right]\mathrm{d}s
+\mathbf{F}_{g,s}(\beta) = \int_r^R
+\begin{bmatrix}
+0 & 0  \\
+mv_s & 0  \\
+ml_{cg}\theta_s \cos(\bar{\theta}) & 0 
+\end{bmatrix}
+ds, \quad
+\mathbf{F}_{g,c}(\beta) = \int_r^R
+\begin{bmatrix}
+-mu_s + ml_{cg}l_{pi}'u_s' \cos(\bar{\theta}) & 0  \\
+ml_{cg}l_{pi}'v_s'\sin(\bar{\theta}) & 0  \\
+ml_{cg}\theta_s \sin(\bar{\theta}) & 0 
+\end{bmatrix}
+ds
 $$  
 
 $$
@@ -918,7 +973,7 @@ $$
 $$  
 
 $$
-\mathbf{F}_{e x t,1}\!=\!\int_{r}^{R}\!\left[\begin{array}{c c c}{\!\!\!u_{s}^{\prime2}\!\int_{s}^{R}f_{w,s}\mathrm{d}\rho}&{\!\!\!0}&{\!\!\!0}\\ {\!\!\!0}&{\!\!\!u_{s}^{\prime2}\!\int_{s}^{R}f_{w,s}\mathrm{d}\rho}&{\!\!\!0}\\ {\!\!\!0}&{\!\!\!0}&{\!\!\!0}\end{array}\right]^{\mathrm{T}}\!
+\mathbf{F}_{e x t,1}=\int_{r}^{R}\!\left[\begin{array}{c c c}{u_{s}^{\prime2}\int_{s}^{R}f_{w,s}\mathrm{d}\rho}&{0}&{0}\\ {0}&{u_{s}^{\prime2}\int_{s}^{R}f_{w,s}\mathrm{d}\rho}&{0}\\ {0}&{0}&{0}\end{array}\right]^{\mathrm{T}}ds
 $$  
 
 # Pitch Model  
@@ -926,13 +981,19 @@ $$
 The individual terms in the assumed mode approximated pitch model (equation (40)) are  
 
 $$
-\begin{array}{r l}&{I_{\beta,1}(\mathbf{q})=\mathbf{I}_{\beta,1}\mathbf{q}+\mathbf{I}_{\beta,2}{[u_{t}^{2}{\nu}_{t}^{2}]}^{\top}}\\ &{D_{\beta}(\mathbf{\dot{q}},\mathbf{q})=2\mathbf{f}_{\beta,0}\mathbf{\dot{q}}+2\mathbf{I}_{\beta,2}{[\dot{u}_{i}{u_{t}}{\dot{\nu}}_{t}{]}^{\top}}}\\ &{f_{\beta,4}(\mathbf{\dot{q}},\mathbf{q})=m_{w}(\dot{u_{t}}{\nu}_{t}-\dot{\nu}_{t}{u_{t}})}\\ &{f_{\beta,\phi}(\vec{\phi},\vec{\phi},\mathbf{q})=\vec{\phi}(f_{\beta,\phi}{u_{t}}\sin(\beta)+f_{\beta,\vec{\phi},C}{V_{t}}\cos(\beta)+I_{\beta,\phi},\sin(\beta)+I_{\beta,\phi,c}\cos(\beta))}\\ &{\qquad\qquad\qquad-2\dot{\phi}^{2}(f_{\beta,\phi,\sin}(2\beta)+f_{\beta,\phi,c}\cos(2\beta)+(\mathbf{f}_{\beta,\phi,\sin}(2\beta)+f_{\beta,\phi,c}\cos(2\beta))\mathbf{q}}\\ &{\qquad\qquad\qquad+\sin(2\beta)\mathbf{f}_{\beta,\phi,2}\big[u_{t}^{2}{\nu}_{t}^{2}\big]^{\top}+\cos(2\beta)f_{\beta,\phi,2}u_{t}{\nu}_{t}\big)}\\ &{f_{\beta,\varepsilon\mu\nu}(\mathbf{q})=-g((\mathbf{f}_{\beta,\varepsilon\mathrm{2},s}\sin(\beta)+\mathbf{f}_{\beta,\varepsilon\mathrm{2},c}\cos(\beta))\mathbf{q}+f_{\beta,\varepsilon\mathrm{3}}\sin(\beta)+f_{\beta,\varepsilon\mathrm{c}}\cos(\beta)))}\end{array}
+I_{\beta,1}(\mathbf{q}) = \mathbf{I}_{\beta,1}\mathbf{q} + \mathbf{I}_{\beta,2}[u_t^2 v_t^2]^T \\
+D_\beta(\dot{\mathbf{q}}, \mathbf{q}) = 2\mathbf{f}_{\beta,\mathbf{q}}\dot{\mathbf{q}} + 2\mathbf{I}_{\beta,2}[\dot{u}_t u_t \dot{v}_t v_t]^T \\
+f_{\beta,\ddot{\mathbf{q}}}(\ddot{\mathbf{q}}, \mathbf{q}) = m_{uv}(\ddot{u}_t v_t - \ddot{v}_t u_t) \\
+f_{\beta,\phi}(\ddot{\phi}, \dot{\phi}, \mathbf{q}) = \ddot{\phi}(f_{\beta,\ddot{\phi},s}u_t \sin(\beta) + f_{\beta,\ddot{\phi},c}V_t \cos(\beta) + I_{\beta,\phi,s} \sin(\beta) + I_{\beta,\phi,c} \cos(\beta)) \\
+- 2\dot{\phi}^2(f_{\beta,\phi,s} \sin(2\beta) + f_{\beta,\phi,c} \cos(2\beta) + (f_{\beta,\phi,s} \sin(2\beta) + f_{\beta,\phi,c} \cos(2\beta))\mathbf{q} \\
++ \sin(2\beta)\mathbf{f}_{\beta,\phi,2}[u_t^2v_t^2]^T + \cos(2\beta)f_{\beta,\phi,2}u_tv_t) \\
+f_{\beta,grav}(\mathbf{q}) = -g((f_{\beta,g,2,s}\sin(\beta) + \mathbf{f}_{\beta,g,2,c}\cos(\beta))\mathbf{q} + f_{\beta,g,s}\sin(\beta) + f_{\beta,g,c}\cos(\beta)) \tag{51}
 $$  
 
 where the constants are  
 
 $$
-I_{\beta,0}=\int_{r}^{R}\big(I_{c g}+m\big(I_{c g}^{2}+I_{p i}^{2}+2I_{p i}I_{c g}\cos(\overline{{\theta}})\big)\big)\mathrm{d}s,\quad\mathbf{I}_{\beta,1}=\int_{r}^{R}[2m\big(I_{p i}+l_{c g}\cos(\theta)\big)\quad2m l_{c g}\sin(\overline{{\theta}})\quad0]\mathrm{d}s,
+I_{\beta,0}=\int_{r}^{R}\big(l_{c g}+m\big(l_{c g}^{2}+l_{p i}^{2}+2l_{p i}l_{c g}\cos(\overline{{\theta}})\big)\big)\mathrm{d}s,\quad\mathbf{I}_{\beta,1}=\int_{r}^{R}[2m\big(l_{p i}+l_{c g}\cos(\theta)\big)\quad2m l_{c g}\sin(\overline{{\theta}})\quad0]\mathrm{d}s,
 $$  
 
 $$
@@ -940,15 +1001,15 @@ $$
 $$  
 
 $$
-I_{\beta,\phi,s}=\int_{r}^{R}m w_{0}(I_{p i}+l_{c g}\cos(\overline{{\theta}}))\mathrm{d}s,\quad I_{\beta,\phi,c}=\int_{r}^{R}m w_{0}l_{c g}\sin(\overline{{\theta}})\mathrm{d}s,\quad m_{u\nu}=\int_{r}^{R}m u_{s}\nu_{s}\mathrm{d}s
+I_{\beta,\phi,s}=\int_{r}^{R}m w_{0}(l_{p i}+l_{c g}\cos(\overline{{\theta}}))\mathrm{d}s,\quad I_{\beta,\phi,c}=\int_{r}^{R}m w_{0}l_{c g}\sin(\overline{{\theta}})\mathrm{d}s,\quad m_{u\nu}=\int_{r}^{R}m u_{s}\nu_{s}\mathrm{d}s
 $$  
 
 $$
-\mathbf{f}_{\beta,\mathbf{q}}=\int_{r}^{R}[m u_{s}(l_{p i}+l_{c g}\cos(\overline{{\theta}}))\quad m v_{s}l_{c g}\sin(\overline{{\theta}})\quad0]\mathrm{d}s,\quad f_{\beta,\vec{\circ},\vec{\circ}}=\int_{r}^{R}m u_{s}w_{0}\mathrm{d}s,\quad f_{\beta,\vec{\circ},\vec{\circ}}=\int_{r}^{R}m v_{s}w_{0}\mathrm{d}s
+\mathbf{f}_{\beta,\mathbf{q}}=\int_{r}^{R}[m u_{s}(l_{p i}+l_{c g}\cos(\overline{{\theta}}))\quad m v_{s}l_{c g}\sin(\overline{{\theta}})\quad0]\mathrm{d}s,\quad f_{\beta,\ddot{\phi},s} = \int_r^R mu_s w_0 ds, \quad f_{\beta,\ddot{\phi},c} = \int_r^R mv_s w_0 ds
 $$  
 
 $$
-\mathbf{M}_{e x t,0}\!=\!\int_{r}^{R}[0\quad l_{p i}f_{\nu,s}\quad M_{s}]\mathrm{d}s,\quad\mathbf{M}_{e x t,1}\!=\!\int_{r}^{R}\!\left[{\!\!\begin{array}{c c c}{0}&{u_{s}f_{\nu,s}}&{0}\\ {-\nu_{s}f_{u,s}}&{0}&{0}\\ {0}&{0}&{0}\end{array}\!\!\right]\!\mathrm{d}s
+\mathbf{M}_{ext,0} = \int_r^R [0 \quad l_{pi}f_{v,s} \quad M_s]ds, \quad \mathbf{M}_{ext,1} = \int_r^R \begin{bmatrix} 0 & u_s f_{v,s} & 0 \\ -v_s f_{u,s} & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} ds
 $$  
 
 $$
@@ -964,7 +1025,7 @@ $$
 $$  
 
 $$
-\mathbf{f}_{\beta,\phi,2}=\int_{r}^{R}{\left[\begin{array}{l}{1/2m u_{s}^{2}}\\ {-1/2m v_{s}^{2}}\end{array}\right]}^{\mathrm{T}}\mathrm{d}s,\quad\mathbf{f}_{\beta,g,2,s}=\int_{r}^{R}{\left[\begin{array}{c}{m u_{s}}\\ {0}\\ {-m\theta_{s}l_{c g}\sin(\overline{{\theta}})\mathrm{.}}\end{array}\right]}^{\mathrm{T}}\mathrm{d}s,\quad\mathbf{f}_{\beta,g,2,c}=\int_{r}^{R}{\left[\begin{array}{c}{0}\\ {m v_{s}}\\ {m\theta_{s}l_{c g}\cos(\overline{{\theta}})\mathrm{.}}\end{array}\right]}^{\mathrm{T}}\mathrm{d}s
+\mathbf{f}_{\beta,\phi,2}=\int_{r}^{R}{\left[\begin{array}{l}{1/2m u_{s}^{2}}\\ {-1/2m v_{s}^{2}}\end{array}\right]}^{\mathrm{T}}\mathrm{d}s,\quad\mathbf{f}_{\beta,g,2,s}=\int_{r}^{R}{\left[\begin{array}{c}{m u_{s}}\\ {0}\\ {-m\theta_{s}l_{c g}\sin(\overline{{\theta}})\mathrm{.}}\end{array}\right]}^{\mathrm{T}}\mathrm{d}s,\quad\mathbf{f}_{\beta,g,2,c}=\int_{r}^{R}{\left[\begin{array}{c}{0}\\ {m v_{s}}\\ {m\theta_{s}l_{c g}\cos(\overline{{\theta}})}\end{array}\right]}^{\mathrm{T}}\mathrm{d}s
 $$  
 
 $$
@@ -976,7 +1037,14 @@ $$
 The individual terms in the assumed mode approximated pitch model (equation (41)) are  
 
 $$
-\begin{array}{r l}&{f_{\phi,z}(\phi,\beta,\mathbf{q})=f_{\phi,z,0}\sin(\phi)+(f_{\phi,z,\mu,0}\cos(\beta)-f_{\phi,z,\nu,0}\sin(\beta)}\\ &{\qquad\qquad\qquad+f_{\phi,z,\mu,1}u_{t}\cos(\beta)-f_{\phi,z,\nu,1}\nu_{t}\sin(\beta))\cos(\phi)}\\ &{I_{\phi,\beta}(\mathbf{q},\beta)=I_{\phi,z,0}\sin(\beta)+I_{\phi,z,0}\cos(\beta)+I_{\phi,u,1}u_{t}\sin(\beta)+I_{\phi,z,1}\nu_{t}\cos(\beta)}\\ &{f_{\phi,\alpha_{1}}(\vec{\mathbf{q}},\beta)=I_{\phi,u,1}\vec{u}_{t}\cos(\beta)-I_{\phi,x,1}\vec{\nu}_{t}\sin(\beta)}\\ &{f_{\phi,\beta}(\vec{\mathbf{\alpha}},\mathbf{i},\beta,\mathbf{q})=(I_{\phi,u,0}\cos(\beta)-I_{\phi,\nu,0}\sin(\beta)+I_{\phi,u,1}u_{t}\cos(\beta)-I_{\phi,z,1}\nu_{t}\sin(\beta))\dot{\beta}^{2}}\\ &{\qquad\qquad\qquad\qquad\qquad+2\dot{\beta}(I_{\phi,u,1}\dot{u}_{t}\sin(\beta)+I_{\phi,v,1}\dot{u}_{t}\cos(\beta))}\\ &{\mathbf{f}_{\mathrm{ext},0}(\beta)=\mathbf{f}_{\mathrm{ext},0,x}\cos(\beta)+\mathbf{f}_{\mathrm{ext},0,x}\sin(\beta)}\\ &{\mathbf{f}_{\mathrm{ext},\left(\mathbf{q},\beta\right)}(\mathbf{q},\beta)=f_{\mathrm{ext},1,\nu_{t},\sin}(\beta)-f_{\mathrm{ext},\left\mathbf{u},\mu_{t}\right.\cos(\beta)}}\end{array}
+f_{\phi,g}(\phi, \beta, \mathbf{q}) = f_{\phi,g,0} \sin(\phi) + (f_{\phi,g,u,0} \cos(\beta) - f_{\phi,g,v,0} \sin(\beta) \\
++ f_{\phi,g,u,1}u_t \cos(\beta) - f_{\phi,g,v,1}v_t \sin(\beta))\cos(\phi) \\
+I_{\phi,\beta}(\mathbf{q}, \beta) = I_{\phi,u,0} \sin(\beta) + I_{\phi,v,0} \cos(\beta) + I_{\phi,u,1}u_t \sin(\beta) + I_{\phi,v,1}v_t \cos(\beta) \\
+f_{\phi,\ddot{\mathbf{q}}}(\ddot{\mathbf{q}}, \beta) = I_{\phi,u,1}\ddot{u}_t \cos(\beta) - I_{\phi,v,1}\ddot{v}_t \sin(\beta) \\
+f_{\phi,\beta}(\dot{\beta}, \dot{\mathbf{q}}, \beta, \mathbf{q}) = (I_{\phi,u,0} \cos(\beta) - I_{\phi,v,0} \sin(\beta) + I_{\phi,u,1}u_t \cos(\beta) - I_{\phi,v,1}v_t \sin(\beta))\dot{\beta}^2 \\
++ 2\dot{\beta}(I_{\phi,u,1}\dot{u}_t \sin(\beta) + I_{\phi,v,1}\dot{v}_t \cos(\beta)) \\
+\mathbf{f}_{ext,0}(\beta) = \mathbf{f}_{ext,0,s} \cos(\beta) + \mathbf{f}_{ext,0,c} \sin(\beta) \\
+\mathbf{f}_{ext,1}(\mathbf{q}, \beta) = f_{ext,1,v}v_t \sin(\beta) - f_{ext,1,u}u_t \cos(\beta) \tag{52}
 $$  
 
 where the constants are