obsidian_backup/书籍/力学书籍/OpenFast/FASTKinetics/auto/Blade.md

The distributed properties of blade 1 bring about generalized inertia forces and generalized active forces associated with blade elasticity, blade damping, blade weight, and blade aerodynamics.  

$$
F_{r}^{*}\big|_{Bl} = -\int_{0}^{Bl \, dFlexL} \mu^{Bl}(r) \, {}^E \, \boldsymbol{v}_{r}^{SI}(r) \cdot E \, \boldsymbol{a}^{SI}(r) \, dr - m^{BldTip} \, {}^E \, \boldsymbol{v}_{r}^{SI}(Bl \, dFlexL) \cdot {}^E \, \boldsymbol{a}^{SI}(Bl \, dFlexL) \quad (r = 1, 2, \ldots, 22)
$$

其中 
$$
\mu^{B1}(r) = AdjBlMs^{B1} \cdot BMassDen^{B1}(r)
$$

$$
m^{B1Tip} = TipMass(1)
$$
or, 

$$
F_{r}^{*}\big|_{B1} = -\int_{0}^{BldFlexL} \mu^{B1}(r) \, {}^E\boldsymbol{v}_{r}^{SI}(r) \cdot \left\{
\begin{array}{l}
\left( \sum_{i=1}^{14} {}^{E}\boldsymbol{v}_{i}^{SI}(r) \ddot{q}_{i} \right) + \left( \sum_{i=16}^{18} {}^{E}\boldsymbol{v}_{i}^{SI}(r) \ddot{q}_{i} \right) + {}^{E}\boldsymbol{v}_{Teet}^{SI}(r) \ddot{q}_{Teet} \\
+ \left[ \sum_{i=4}^{14} \frac{d}{dt} \left( {}^{E}\boldsymbol{v}_{i}^{SI}(r) \right) \dot{q}_{i} \right] + \left[ \sum_{i=16}^{18} \frac{d}{dt} \left( {}^{E}\boldsymbol{v}_{i}^{SI}(r) \right) \dot{q}_{i} \right] + \frac{d}{dt} \left( {}^{E}\boldsymbol{v}_{Teet}^{SI}(r) \right) \dot{q}_{Teet}
\end{array}
\right\} dr \\
- m^{B1Tip} {}^{E}\boldsymbol{v}_{r}^{SI}(BldFlexL) \cdot \left\{
\begin{array}{l}
\left( \sum_{i=1}^{14} {}^{E}\boldsymbol{v}_{i}^{SI}(BldFlexL) \ddot{q}_{i} \right) + \left( \sum_{i=16}^{18} {}^{E}\boldsymbol{v}_{i}^{SI}(BldFlexL) \ddot{q}_{i} \right) + {}^{E}\boldsymbol{v}_{Teet}^{SI}(BldFlexL) \ddot{q}_{Teet} \\
+ \left[ \sum_{i=4}^{14} \frac{d}{dt} \left( {}^{E}\boldsymbol{v}_{i}^{SI}(BldFlexL) \right) \dot{q}_{i} \right] + \left[ \sum_{i=16}^{18} \frac{d}{dt} \left( {}^{E}\boldsymbol{v}_{i}^{SI}(BldFlexL) \right) \dot{q}_{i} \right] \\
+ \frac{d}{dt} \left( {}^{E}\boldsymbol{v}_{Teet}^{SI}(BldFlexL) \right) \dot{q}_{Teet}
\end{array}
\right\} \quad (r = 1, 2, \ldots, 14; 16, 17, 18; Teet)
$$

因此，

$$
\left[C(q,t)\right]_{B1} (\text{Row,Col}) = \int_{0}^{BldFlexL} \mu^{B1}(r) \, {}^{E}\boldsymbol{v}_{\text{Row}}^{S1}(r) \cdot {}^{E}\boldsymbol{v}_{\text{Col}}^{SI}(r) \, dr + m^{B1Tip} {}^{E}\boldsymbol{v}_{\text{Row}}^{SI}(BldFlexL) \cdot {}^{E}\boldsymbol{v}_{\text{Col}}^{SI}(BldFlexL) \quad (\text{Row, Col} = 1,2,\ldots,14;16,17,18;22)
$$

$$
\left\{-f(\dot{q},q,t)\right\}_{B1} (\text{Row}) = -\int_{0}^{BldFlexL} \mu^{B1}(r) \, {}^{E}\boldsymbol{v}_{\text{Row}}^{SI}(r) \cdot \left\{ \left[ \sum_{i=4}^{14} \frac{d}{dt} \left( {}^{E}\boldsymbol{v}_{i}^{SI}(r) \right) \dot{q}_{i} \right] + \left[ \sum_{i=16}^{18} \frac{d}{dt} \left( {}^{E}\boldsymbol{v}_{i}^{SI}(r) \right) \dot{q}_{i} \right] + \frac{d}{dt} \left( {}^{E}\boldsymbol{v}_{Teet}^{SI}(r) \right) \dot{q}_{Teet} \right\} dr \\
- m^{B1Tip} {}^{E}\boldsymbol{v}_{\text{Row}}^{SI}(BldFlexL) \cdot \left\{ \left[ \sum_{i=4}^{14} \frac{d}{dt} \left( {}^{E}\boldsymbol{v}_{i}^{SI}(BldFlexL) \right) \dot{q}_{i} \right] + \left[ \sum_{i=16}^{18} \frac{d}{dt} \left( {}^{E}\boldsymbol{v}_{i}^{SI}(BldFlexL) \right) \dot{q}_{i} \right] + \frac{d}{dt} \left( {}^{E}\boldsymbol{v}_{Teet}^{SI}(BldFlexL) \right) \dot{q}_{Teet} \right\} \quad (\text{Row} = 1,2,\ldots,14;16,17,18;22)
$$


$$
F_{r}\big|_{ElasticB1} = -\frac{\partial V'^{B1}}{\partial q_r} \quad (r = 1, 2, \ldots, 22)
$$
$$
\left. F_{r} \right|_{ElasticB1} =
\begin{cases}
- {k'}_{11}^{B1F} q_{B1F1} - {k'}_{12}^{B1F} q_{B1F2}
 & \text{for } r = B1F1 \\
- {k'}_{11}^{B1E} q_{B1E1}
 & \text{for } r = B1E1 \\
- {k'}_{21}^{B1F} q_{B1F1} - {k'}_{22}^{B1F} q_{B1F2}
 & \text{for } r = B1F2 \\
0
 & \text{otherwise}
\end{cases}
$$

where $k_{\ i j}^{\,\prime B I F}$ and $\boldsymbol{k\,}_{I I}^{\prime B I E}$ are the generalized stiffnesses of blade 1 in the local flap and local edge directions respectively when centrifugal-stiffening effects are not included as follows:  

$$
{k'}_{ij}^{B1F} = \sqrt{FlStTunr^{B1}(i) \, FlStTunr^{B1}(j)} \int_{0}^{BldFlexL} EI^{B1F}(r) \frac{d^2 \phi_i^{B1F}(r)}{dr^2} \frac{d^2 \phi_j^{B1F}(r)}{dr^2} dr \quad (i, j = 1, 2)
$$

其中，

$$
EI^{B1F}(r) = AdjFlSt^{B1} \cdot FlpStff^{B1}(r)
$$


$$
{k'}_{11}^{B1E} = \int_{0}^{BldFlexL} EI^{B1E}(r) \left[ \frac{d^2 \phi_{1}^{B1E}(r)}{dr^2} \right]^2 dr
$$

其中，

$$
EI^{B1E}(r) = AdjEdSt^{B1} \cdot EdgStff^{B1}(r)
$$

Similarly, when using the Rayleigh damping technique where the damping is assumed proportional to the stiffness, then  

$$
\left. F_r \right|_{DampB1} =
\begin{cases}
-\frac{\zeta_1^{B1F} {k'}_{11}^{B1F}}{\pi f^{\prime,B 1 F}_1} \dot{q}_{B1F1} - \frac{\zeta_2^{B1F} {k'}_{12}^{B1F}}{\pi f^{\prime,B 1 F}_2} \dot{q}_{B1F2} & \text{for } r = B1F1 \\
-\frac{\zeta_1^{B1E} {k'}_{11}^{B1E}}{\pi f^{\prime,B 1 E}_1} \dot{q}_{B1E1} & \text{for } r = B1E1 \\
-\frac{\zeta_1^{B1F} {k'}_{21}^{B1F}}{\pi f^{\prime,B 1 F}_1} \dot{q}_{B1F1} - \frac{\zeta_2^{B1F} {k'}_{22}^{B1F}}{\pi f^{\prime,B 1 F}_2} \dot{q}_{B1F2} & \text{for } r = B1F2 \\
0 & \text{otherwise}
\end{cases}
$$
where $\zeta_{i}^{B I F}$ and $\zeta_{i}^{B I E}$ represent the structural damping ratio of blade 1 for the $i^{\mathrm{th}}$ mode in the local flap and edge directions, $B l d F l D m p^{B 1}(i)/100$ and $B l d E d D m p^{B 1}\left(i\right)/100$ respectively.  Also, ${f^{\prime}}_{i}^{B I F}$ and $\boldsymbol{f}_{\ i}^{\prime B I E}$ represent the natural frequency of blade 1 for the $i^{\mathrm{th}}$ mode in the local flap and edge directions respectively without centrifugal-stiffening effects.  That is,  

$$
f_i^{\prime B1F} = \frac{1}{2\pi} \sqrt{\frac{k_{ii}^{\prime B1F}}{m_{ii}^{\prime B1F}}}
$$

$$
f_i^{\prime B1E} = \frac{1}{2\pi} \sqrt{\frac{k_{ii}^{\prime B1E}}{m_{ii}^{\prime B1E}}}
$$

where $m_{\ i i}^{\prime B I F}$ and $m_{\ i i}^{\prime B I E}$ represent the generalized mass of blade 1 for the $i^{\mathrm{th}}$ mode in the local flap and edge directions respectively without centrifugal-stiffening and tip mass effects as follows:  

$$
m_{ij}^{\prime B1F} = \int_{0}^{BldFlexL} \mu^{B1}(r) \phi_i^{B1F}(r) \phi_j^{B1F}(r) \, dr \quad (i, j = 1, 2)
$$
$$
m_{11}^{\prime B1E} = \int_{0}^{BldFlexL} \mu^{B1}(r) \left[ \phi_1^{B1E}(r) \right]^2 dr
$$

Thus
$$
[C(q,t)]_{ElasticB1} = 0 \quad \text{and} \quad [C(q,t)]_{DampB1} = 0
$$

$$
\left. \left\{ -f(\dot{q},q,t) \right\} \right|_{ElasticB1} =
\begin{pmatrix}
\vdots \\
\vdots \\
\vdots \\
-k_{11}^{\prime B1F} q_{B1F1} - k_{12}^{\prime B1F} q_{B1F2} \\
-k_{11}^{\prime B1E} q_{B1E1} \\
-k_{21}^{\prime B1F} q_{B1F1} - k_{22}^{\prime B1F} q_{B1F2} \\
\vdots \\
\vdots \\
\vdots
\end{pmatrix}
$$

$$
\left. \left\{ -f(\dot{q},q,t) \right\} \right|_{DampB1} =
\begin{pmatrix}
\vdots \\
\vdots \\
\vdots \\
-\frac{\zeta_1^{B1F} k_{11}^{\prime B1F}}{\pi f_1^{\prime B1F}} \dot{q}_{B1F1} - \frac{\zeta_2^{B1F} k_{12}^{\prime B1F}}{\pi f_2^{\prime B1F}} \dot{q}_{B1F2} \\
-\frac{\zeta_1^{B1E} k_{11}^{\prime B1E}}{\pi f_1^{\prime B1E}} \dot{q}_{B1E1} \\
-\frac{\zeta_1^{B1F} k_{21}^{\prime B1F}}{\pi f_1^{\prime B1F}} \dot{q}_{B1F1} - \frac{\zeta_2^{B1F} k_{22}^{\prime B1F}}{\pi f_2^{\prime B1F}} \dot{q}_{B1F2} \\
\vdots \\
\vdots \\
\vdots
\end{pmatrix}
$$$$
F_{r}\big|_{GravB1} = -\int_{0}^{BldFlexL} \mu^{B1}(r) \, \boldsymbol{g}^{E}\boldsymbol{v}_{r}^{SI}(r) \cdot \boldsymbol{{z}}_2 \, dr - m^{B1Tip} \boldsymbol{g}^{E}\boldsymbol{v}_{r}^{S1}(BldFlexL) \cdot \boldsymbol{{z}}_2 \quad (r = 3,4,\ldots,14;16,17,18;Teet)
$$

where $F_{A e r o B 1}^{S 1}(r)$ and $M_{A e r o B 1}^{M 1}\left(r\right)$ are aerodynamic forces and pitching moments applied to point S1 on blade 1 respectively expressed per unit span. Note that $M_{A e r o B 1}^{M 1}\left(r\right)$ can include effects of both direct aerodynamic pitching moments (i.e., $\mathrm{Cm})$ ) and aerodynamic pitching moments caused by an aerodynamic offset (i.e., moments due to aerodynamic lift and drag forces acting at a distance away from the center of mass of the blade element along the aerodynamic chord).