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.  
叶片1的分布式特性导致与叶片弹性、叶片阻尼、叶片重量及叶片气动力相关的广义惯性力和广义主动力。
$$
F_{r}^{*}\big|_{B1} = -\int_{0}^{BldFlexL} \mu^{B1}(r) \, {}^E \, \boldsymbol{v}_{r}^{S1}(r) \cdot {}^E \, \boldsymbol{a}^{S1}(r) \, dr - m^{BldTip} \, {}^E \, \boldsymbol{v}_{r}^{S1}(BldFlexL) \cdot {}^E \, \boldsymbol{a}^{S1}(BldFlexL) \quad (r = 1, 2, \ldots, 22) \tag{1}
$$

其中 
$$
\mu^{B1}(r) = AdjBlMs^{B1} \cdot BMassDen^{B1}(r) \tag{2}
$$

$$
m^{B1Tip} = TipMass(1) \tag{3}
$$
or, 

$$
F_{r}^{*}\big|_{B1} = -\int_{0}^{BldFlexL} \mu^{B1}(r) \, {}^E\boldsymbol{v}_{r}^{S1}(r) \cdot \left\{
\begin{array}{l}
\left( \sum_{i=1}^{14} {}^{E}\boldsymbol{v}_{i}^{S1}(r) \ddot{q}_{i} \right) + \left( \sum_{i=16}^{18} {}^{E}\boldsymbol{v}_{i}^{S1}(r) \ddot{q}_{i} \right) + {}^{E}\boldsymbol{v}_{Teet}^{S1}(r) \ddot{q}_{Teet} \\
+ \left[ \sum_{i=4}^{14} \frac{d}{dt} \left( {}^{E}\boldsymbol{v}_{i}^{S1}(r) \right) \dot{q}_{i} \right] + \left[ \sum_{i=16}^{18} \frac{d}{dt} \left( {}^{E}\boldsymbol{v}_{i}^{S1}(r) \right) \dot{q}_{i} \right] + \frac{d}{dt} \left( {}^{E}\boldsymbol{v}_{Teet}^{S1}(r) \right) \dot{q}_{Teet}
\end{array}
\right\} dr \\
- m^{B1Tip} {}^{E}\boldsymbol{v}_{r}^{S1}(BldFlexL) \cdot \left\{
\begin{array}{l}
\left( \sum_{i=1}^{14} {}^{E}\boldsymbol{v}_{i}^{S1}(BldFlexL) \ddot{q}_{i} \right) + \left( \sum_{i=16}^{18} {}^{E}\boldsymbol{v}_{i}^{S1}(BldFlexL) \ddot{q}_{i} \right) + {}^{E}\boldsymbol{v}_{Teet}^{S1}(BldFlexL) \ddot{q}_{Teet} \\
+ \left[ \sum_{i=4}^{14} \frac{d}{dt} \left( {}^{E}\boldsymbol{v}_{i}^{S1}(BldFlexL) \right) \dot{q}_{i} \right] + \left[ \sum_{i=16}^{18} \frac{d}{dt} \left( {}^{E}\boldsymbol{v}_{i}^{S1}(BldFlexL) \right) \dot{q}_{i} \right] \\
+ \frac{d}{dt} \left( {}^{E}\boldsymbol{v}_{Teet}^{S1}(BldFlexL) \right) \dot{q}_{Teet}
\end{array}
\right\} \quad (r = 1, 2, \ldots, 14; 16, 17, 18; Teet) \tag{4}
$$

因此，

$$
\left[C(q,t)\right]_{B1} (\text{Row,Col}) = \int_{0}^{BldFlexL} \mu^{B1}(r) \, {}^{E}\boldsymbol{v}_{\text{Row}}^{S1}(r) \cdot {}^{E}\boldsymbol{v}_{\text{Col}}^{S1}(r) \, dr + m^{B1Tip} {}^{E}\boldsymbol{v}_{\text{Row}}^{S1}(BldFlexL) \cdot {}^{E}\boldsymbol{v}_{\text{Col}}^{S1}(BldFlexL) \quad (\text{Row, Col} = 1,2,\ldots,14;16,17,18;22) \tag{5}
$$

$$
\left\{-f(\dot{q},q,t)\right\}_{B1} (\text{Row}) = -\int_{0}^{BldFlexL} \mu^{B1}(r) \, {}^{E}\boldsymbol{v}_{\text{Row}}^{S1}(r) \cdot \left\{ \left[ \sum_{i=4}^{14} \frac{d}{dt} \left( {}^{E}\boldsymbol{v}_{i}^{S1}(r) \right) \dot{q}_{i} \right] + \left[ \sum_{i=16}^{18} \frac{d}{dt} \left( {}^{E}\boldsymbol{v}_{i}^{S1}(r) \right) \dot{q}_{i} \right] + \frac{d}{dt} \left( {}^{E}\boldsymbol{v}_{Teet}^{S1}(r) \right) \dot{q}_{Teet} \right\} dr \\
- m^{B1Tip} {}^{E}\boldsymbol{v}_{\text{Row}}^{S1}(BldFlexL) \cdot \left\{ \left[ \sum_{i=4}^{14} \frac{d}{dt} \left( {}^{E}\boldsymbol{v}_{i}^{S1}(BldFlexL) \right) \dot{q}_{i} \right] + \left[ \sum_{i=16}^{18} \frac{d}{dt} \left( {}^{E}\boldsymbol{v}_{i}^{S1}(BldFlexL) \right) \dot{q}_{i} \right] + \frac{d}{dt} \left( {}^{E}\boldsymbol{v}_{Teet}^{S1}(BldFlexL) \right) \dot{q}_{Teet} \right\} \quad (\text{Row} = 1,2,\ldots,14;16,17,18;22) \tag{6}
$$


$$
F_{r}\big|_{ElasticB1} = -\frac{\partial V'^{B1}}{\partial q_r} \quad (r = 1, 2, \ldots, 22) \tag{7}
$$
$$
\left. F_{r} \right|_{ElasticB1} =
\begin{cases}
- {k'}_{11}^{B1F} q_{B1F1} - {k'}_{12}^{B1F} q_{B1F2}
 & \text{for } r = B1F1 \\
- {k'}_{11}^{B1E} q_{B1E1}
 & \text{for } r = B1E1 \\
- {k'}_{21}^{B1F} q_{B1F1} - {k'}_{22}^{B1F} q_{B1F2}
 & \text{for } r = B1F2 \\
0
 & \text{otherwise}
\end{cases} \tag{8}
$$

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:  
其中，$k_{ij}^{\,\prime BIF}$ 和 $\boldsymbol{k}_{II}^{\prime BIE}$ 分别为未考虑离心刚化效应时，叶片1在局部挥舞方向和局部摆振方向的广义刚度，具体如下：
$$
{k'}_{ij}^{B1F} = \sqrt{FlStTunr^{B1}(i) \, FlStTunr^{B1}(j)} \int_{0}^{BldFlexL} EI^{B1F}(r) \frac{d^2 \phi_i^{B1F}(r)}{dr^2} \frac{d^2 \phi_j^{B1F}(r)}{dr^2} dr \quad (i, j = 1, 2) \tag{9}
$$

其中，

$$
EI^{B1F}(r) = AdjFlSt^{B1} \cdot FlpStff^{B1}(r) \tag{10}
$$


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

其中，

$$
EI^{B1E}(r) = AdjEdSt^{B1} \cdot EdgStff^{B1}(r) \tag{12}
$$

Similarly, when using the Rayleigh damping technique where the damping is assumed proportional to the stiffness, then  
同样，在使用瑞利阻尼技术（假设阻尼与刚度成正比）时，则
$$
\left. F_r \right|_{DampB1} =
\begin{cases}
-\frac{\zeta_1^{B1F} {k'}_{11}^{B1F}}{\pi f^{\prime B 1 F}_1} \dot{q}_{B1F1} - \frac{\zeta_2^{B1F} {k'}_{12}^{B1F}}{\pi f^{\prime B 1 F}_2} \dot{q}_{B1F2} & \text{for } r = B1F1 \\
-\frac{\zeta_1^{B1E} {k'}_{11}^{B1E}}{\pi f^{\prime B 1 E}_1} \dot{q}_{B1E1} & \text{for } r = B1E1 \\
-\frac{\zeta_1^{B1F} {k'}_{21}^{B1F}}{\pi f^{\prime B 1 F}_1} \dot{q}_{B1F1} - \frac{\zeta_2^{B1F} {k'}_{22}^{B1F}}{\pi f^{\prime B 1 F}_2} \dot{q}_{B1F2} & \text{for } r = B1F2 \\
0 & \text{otherwise}
\end{cases} \tag{13}
$$
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,  
其中，$\zeta_{i}^{BIF}$ 和 $\zeta_{i}^{BIE}$ 分别代表叶片1在第 $i$ 阶模态下在**挥舞**方向和**摆振**方向的结构阻尼比，对应表达式为 $BldFlDmp^{B1}(i)/100$ 和 $BldEdDmp^{B1}(i)/100$。此外，${f^{\prime}}_{i}^{B I F}$ 和 $\boldsymbol{f}_{\ i}^{\prime B I E}$ 分别表示叶片1在第 $i$ 阶模态下，**不考虑离心刚度效应**时的**挥舞**方向和**摆振**方向的固有频率。也就是说，
$$
f_i^{\prime B1F} = \frac{1}{2\pi} \sqrt{\frac{k_{ii}^{\prime B1F}}{m_{ii}^{\prime B1F}}} \tag{14}
$$

$$
f_i^{\prime B1E} = \frac{1}{2\pi} \sqrt{\frac{k_{ii}^{\prime B1E}}{m_{ii}^{\prime B1E}}} \tag{15}
$$

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:  
其中，$m_{ii}^{\prime BIF}$ 和 $m_{ii}^{\prime BIE}$ 分别表示在未考虑离心刚化和叶尖质量效应的情况下，第 $i$ 阶模态下叶片1在**挥舞方向**和**摆振方向**的广义质量，具体定义如下：
$$
m_{ij}^{\prime B1F} = \int_{0}^{BldFlexL} \mu^{B1}(r) \phi_i^{B1F}(r) \phi_j^{B1F}(r) \, dr \quad (i, j = 1, 2) \tag{16}
$$
$$
m_{11}^{\prime B1E} = \int_{0}^{BldFlexL} \mu^{B1}(r) \left[ \phi_1^{B1E}(r) \right]^2 dr \tag{17}
$$

Thus
$$
[C(q,t)]|_{ElasticB1} = 0 \quad \text{and} \quad [C(q,t)]|_{DampB1} = 0 \tag{18}
$$

$$
\left. \left\{ -f(\dot{q},q,t) \right\} \right|_{ElasticB1} =
\begin{pmatrix}
\vdots \\
\vdots \\
\vdots \\
-k_{11}^{\prime B1F} q_{B1F1} - k_{12}^{\prime B1F} q_{B1F2} \\
-k_{11}^{\prime B1E} q_{B1E1} \\
-k_{21}^{\prime B1F} q_{B1F1} - k_{22}^{\prime B1F} q_{B1F2} \\
\vdots \\
\vdots \\
\vdots
\end{pmatrix} \tag{19}
$$

$$
\left. \left\{ -f(\dot{q},q,t) \right\} \right|_{DampB1} =
\begin{pmatrix}
\vdots \\
\vdots \\
\vdots \\
-\frac{\zeta_1^{B1F} k_{11}^{\prime B1F}}{\pi f_1^{\prime B1F}} \dot{q}_{B1F1} - \frac{\zeta_2^{B1F} k_{12}^{\prime B1F}}{\pi f_2^{\prime B1F}} \dot{q}_{B1F2} \\
-\frac{\zeta_1^{B1E} k_{11}^{\prime B1E}}{\pi f_1^{\prime B1E}} \dot{q}_{B1E1} \\
-\frac{\zeta_1^{B1F} k_{21}^{\prime B1F}}{\pi f_1^{\prime B1F}} \dot{q}_{B1F1} - \frac{\zeta_2^{B1F} k_{22}^{\prime B1F}}{\pi f_2^{\prime B1F}} \dot{q}_{B1F2} \\
\vdots \\
\vdots \\
\vdots
\end{pmatrix} \tag{20}
$$$$
F_{r}\big|_{GravB1} = -\int_{0}^{BldFlexL} \mu^{B1}(r) \, \boldsymbol{g}^{E}\boldsymbol{v}_{r}^{S1}(r) \cdot \boldsymbol{{z}}_2 \, dr - m^{B1Tip} \boldsymbol{g}^{E}\boldsymbol{v}_{r}^{S1}(BldFlexL) \cdot \boldsymbol{{z}}_2 \quad (r = 3,4,\ldots,14;16,17,18;Teet)\tag{21}
$$

Thus, 
$$
[C(q,t)]|_{GravB1} = 0\tag{22}
$$
$$
\left\{ -f(\dot{q}, q, t) \right\}|_{GravB1} (\text{Row}) = -\int_{0}^{BldFlexL} \mu^{B1}(r) \, g \, {}^E\boldsymbol{v}_{Row}^{S1}(r) \cdot \boldsymbol{z}_2 \, dr - m^{B1Tip} \, g \, {}^E\boldsymbol{v}_{Row}^{S1}(BldFlexL) \cdot \boldsymbol{z}_2 \quad (\text{Row} = 3, 4, \ldots, 14; 16, 17, 18; 22)\tag{23}
$$


$$
\boldsymbol{F}_r \big|_{AeroB1} = \int_{0}^{BldFlexL} \left[ {}^E\boldsymbol{v}_r^{S1}(r) \cdot \boldsymbol{F}_{AeroB1}^{S1}(r) + {}^E\boldsymbol{\omega}_r^{M1}(r) \cdot \boldsymbol{M}_{AeroB1}^{M1}(r) \right] dr + {}^E\boldsymbol{v}_r^{S1}(BldFlexL) \cdot \boldsymbol{F}_{TipDragB1}^{S1}(BldFlexL)(r = 1, 2, \ldots, 14; 16, 17, 18; Teet)\tag{24}
$$

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). 
其中，$F_{\text{AeroB1}}^{S1}(r)$ 和 $M_{\text{AeroB1}}^{M1}(r)$ 分别表示作用于叶片1上点S1处的单位展向长度上的气动力和气动俯仰力矩。需要注意的是，$M_{\text{AeroB1}}^{M1}(r)$ 可包含两部分气动俯仰力矩的影响：一是直接气动俯仰力矩（即 $\mathrm{Cm}$），二是由气动力（升力和阻力）在叶片元素气动弦线上相对于质心产生的偏移距离所引起的气动俯仰力矩。
$$
[C(q,t)]|_{AeroB1} = 0\tag{25}
$$

$$
\left\{ -f(\dot{q}, q, t) \right\}|_{AeroB1} (\text{Row}) = \int_{0}^{BldFlexL} \left[ {}^E\boldsymbol{v}_{Row}^{S1}(r) \cdot \boldsymbol{F}_{AeroB1}^{S1}(r) + {}^E\boldsymbol{\omega}_{Row}^{M1}(r) \cdot \boldsymbol{M}_{AeroB1}^{M1}(r) \right] dr + {}^E\boldsymbol{v}_{Row}^{S1}(BldFlexL) \cdot \boldsymbol{F}_{TipDragB1}^{S1}(BldFlexL) \quad (\text{Row} = 1, 2, \ldots, 14; 16, 17, 18; 22)\tag{26}
$$