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@ -1550,115 +1550,129 @@ The kinematics expressions for the entire wind turbine structure found in sectio
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3.3节中可用于构建动力学表达式的风轮整体结构运动学表达式，可用于构建动力学表达式。Kane运动方程（见3.5节）使用两组标量量，分别称为广义惯性力，$F_{r}{}^{*}$'s，和广义主动力，$F_{r}$ ’s：
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$$
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F_{\,r}^{\;\,*}\!=\!\sum_{i=I}^{\nu}{^{E}{\nu}_{\!r}^{X_{i}}\cdot\left(-{m_{\,x_{i}}}\,^{E}{\bf{a}}^{X_{i}}\right)}\quad\left(r=I,\!2,\!...,\!I\,\bar{\bf{\var S}}\right)
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F_{\,r}^{\;\,*}\!=\!\sum_{i=1}^{\nu}{^{E}{\nu}_{\!r}^{X_{i}}\cdot\left(-{m_{\,x_{i}}}\,^{E}{{a}}^{X_{i}}\right)}\quad\left(r=1,\!2,\!...,\!15\right)
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$$
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$$
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F_{r}=\sum_{i=I}^{\nu}\d^{\scriptscriptstyle E}\nu_{r}^{\scriptscriptstyle X_{i}}\cdot\left(F^{\scriptscriptstyle X_{i}}\right)\;\;\left(r=I,\!2,...,\!I\,\bar{\!}\,\right)
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F_{r}=\sum_{i=1}^{\nu}{}^{ E}\nu_{r}^{ X_{i}}\cdot\left(F^{ X_{i}}\right)\;\;\left(r=1,\!2,...,\!15\,\right)
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$$
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where $\nu$ is the number of particles with mass in the system, $E_{\nu}{x}i_{r}$ is the $r^{\mathrm{th}}$ partial velocity associated with particle $X_{i}$ , $m_{X i}$ is the mass of particle $X_{i}$ , $E_{\pmb{q}}X^{i}$ is the acceleration of particle $X_{i}$ in the inertial frame, and $F^{X i}$ is the resultant of all applied forces acting on particle $X_{i}$ . If the system is composed of a number of rigid bodies instead of particles, the generalized inertia forces can be simplified:
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where $\nu$ is the number of particles with mass in the system, $^{ E}\nu_{r}^{ X_{i}}$ is the $r^{\mathrm{th}}$ partial velocity associated with particle $X_{i}$ , $m_{X i}$ is the mass of particle $X_{i}$ , $^{E}{{a}}^{X_{i}}$ is the acceleration of particle $X_{i}$ in the inertial frame, and $F^{X i}$ is the resultant of all applied forces acting on particle $X_{i}$ . If the system is composed of a number of rigid bodies instead of particles, the generalized inertia forces can be simplified:
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其中，$\nu$ 为系统中的粒子数量，$^{ E}\nu_{r}^{ X_{i}}$ 为与粒子 $X_{i}$ 相关的第 $r^{\mathrm{th}}$ 偏速度，$m_{X i}$ 为粒子 $X_{i}$ 的质量，$^{E}{{a}}^{X_{i}}$ 为粒子 $X_{i}$ 在惯性系中的加速度，且 $F^{X i}$ 为作用于粒子 $X_{i}$ 上的所有合力。如果系统由若干刚体组成，而不是粒子，则广义惯性力可以简化为：
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$$
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{F_{r}}^{*}\!=\sum_{i=l}^{w}{^{E}\nu_{{r}}^{{X}_{i}}\cdot\left(-{m_{{X}_{i}}}^{{E}}a^{{X}_{i}}\right)\!\!+^{{E}}\!\omega_{{r}}^{{X}_{i}}\cdot\left(\!\!-^{{E}}\!\!\dot{H}^{{X}_{i}}\right)\;\;\left(r={I},\!2,...,{I}\boldsymbol{\mathcal{S}}\right)
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F_{r}^{*} = \sum_{i=l}^{w} \left( ^{E}\nu_{{r}}^{{X}_{i}} \cdot \left(-m_{{X}_{i}}{}^{E} a^{{X}_{i}}\right) + {}^{E}\omega_{{r}}^{{X}_{i}} \cdot \left(-^{E}\dot{H}^{{X}_{i}}\right) \right) \quad \text{for } r = 1, 2, \ldots, 15
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$$
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where $\boldsymbol{w}$ is the number of rigid bodies with mass in the system, $E_{\nu}{x^{i}}_{r}$ is the $r^{\mathrm{th}}$ partial velocity associated with the center of mass of rigid body $X_{i},m_{X i}$ is the mass of rigid body $X_{i},{}^{E}\pmb{a}^{X i}$ is the acceleration of the center of mass of rigid body $X_{i}$ in the inertial frame, $\breve{\varepsilon}_{\omega_{\quad r}^{{X i}}}$ is the $r^{\mathrm{th}}$ partial
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angular velocity associated with rigid body $X_{i},$ and ${\pmb E}\hat{\pmb H}^{X i}$ is the time derivative of the angular momentum of rigid body $X_{i}$ about its center of mass, in the inertial frame. For ease in computation, ${^E}\bar{H^{X i}}$ can be written in terms of a body-fixed coordinate system22:
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where $\boldsymbol{w}$ is the number of rigid bodies with mass in the system, $^{E}\nu_{{r}}^{{X}_{i}}$ is the $r^{\mathrm{th}}$ partial velocity associated with the center of mass of rigid body $X_{i},m_{X i}$ is the mass of rigid body $X_{i},{}^{E}\pmb{a}^{X i}$ is the acceleration of the center of mass of rigid body $X_{i}$ in the inertial frame, $^{E}\omega_{{r}}^{{X}_{i}}$ is the $r^{\mathrm{th}}$ partial angular velocity associated with rigid body $X_{i},$ and $^{E}\dot{H}^{{X}_{i}}$ is the time derivative of the angular momentum of rigid body $X_{i}$ about its center of mass, in the inertial frame. For ease in computation, $^{E}\dot{H}^{{X}_{i}}$ can be written in terms of a body-fixed coordinate system:
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其中 $\boldsymbol{w}$ 为系统中的刚体数量，$^{E}\nu_{{r}}^{{X}_{i}}$ 为刚体 $X_{i}$ 的质心相关的第 $r$ 个偏速度，$m_{X i}$ 为刚体 $X_{i}$ 的质量，$^{E}\pmb{a}^{X i}$ 为惯性系中刚体 $X_{i}$ 的质心加速度，$^{E}\omega_{{r}}^{{X}_{i}}$ 为刚体 $X_{i}$ 相关的第 $r$ 个偏角速度，$^{E}\dot{H}^{{X}_{i}}$ 为刚体 $X_{i}$ 关于其质心的角动量的时间导数，均在惯性系中。为了便于计算，$^{E}\dot{H}^{{X}_{i}}$ 可以用固定于刚体的坐标系来表示：
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$$
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{}^{E}\dot{H}^{X_{i}}=\left(\dot{H}^{X_{i}}\right)^{\prime}\!\!+^{E}\!\omega^{X_{i}}\!\times^{E}\!H^{X_{i}}
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{}^{E}\dot{H}^{X_{i}}=\left(\dot{H}^{X_{i}}\right)^{\prime}\!\!+{}^{E}\!\omega^{X_{i}}\!\times^{E}\!H^{X_{i}}
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$$
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where $(\hat{H}^{X i})\,^{,}$ is the time derivative of the angular momentum of rigid body $X_{i}$ about its center of mass, relative to the body-fixed coordinate system $[(\hat{H}^{X i})\,^{,}$ contains time derivatives of the angular velocity of the body but the moments and products of inertia are all constant], $\varepsilon_{\pmb{\omega}}^{\phantom{\dagger}}x^{i}$ is the angular velocity associated with rigid body $X_{i}$ in the inertial frame (equal to the angular velocity of the body-fixed coordinate system in the inertial frame), and ${}^{E}{H}^{X\dot{\imath}}$ is the angular momentum of rigid body $X_{i}$ about its center of mass, in the inertial frame.
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where $(\hat{H}^{X i})\,^{,}$ is the time derivative of the angular momentum of rigid body $X_{i}$ about its center of mass, relative to the body-fixed coordinate system $[(\hat{H}^{X i})\,^{,}$ contains time derivatives of the angular velocity of the body but the moments and products of inertia are all constant], ${}^{E}\!\omega^{X_{i}}$ is the angular velocity associated with rigid body $X_{i}$ in the inertial frame (equal to the angular velocity of the body-fixed coordinate system in the inertial frame), and ${}^{E}{H}^{X\dot{\imath}}$ is the angular momentum of rigid body $X_{i}$ about its center of mass, in the inertial frame.
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其中，$(\hat{H}^{X i})\,^{,}$是刚体 $X_{i}$ 关于其质心的时间导数，相对于固定于体的坐标系 $[(\hat{H}^{X i})\,^{,}$包含刚体的角速度的时间导数，但所有力矩和惯性积均保持不变]，${}^{E}\!\omega^{X_{i}}$ 是惯性系中与刚体 $X_{i}$ 相关的角速度（等于惯性系中固定于体的坐标系的角速度），并且 ${}^{E}{H}^{X\dot{\imath}}$ 是刚体 $X_{i}$ 关于其质心的角动量，在惯性系中。
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For the wind turbine modeled in FAST_AD, the mass of the tower, nacelle, hub, and blades contribute to the total generalized inertia forces:
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对于在 FAST_AD 中建模的风力发电机，塔架、舱盖、轮毂和叶片的质量共同贡献于总的广义惯性力：
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$$
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F_{\scriptscriptstyle r}*=F_{\scriptscriptstyle r}*\big|_{\scriptscriptstyle T}+F_{\scriptscriptstyle r}*\big|_{\scriptscriptstyle N}+F_{\scriptscriptstyle r}*\big|_{\scriptscriptstyle H}+F_{\scriptscriptstyle r}*\big|_{\scriptscriptstyle B}\quad\big(r=I,2,...,I\,5\big)
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{F_{\scriptscriptstyle r}}^*=F_{\scriptscriptstyle r}^*\big|_{\scriptscriptstyle T}+F_{\scriptscriptstyle r}^*\big|_{\scriptscriptstyle N}+F_{\scriptscriptstyle r}^*\big|_{\scriptscriptstyle H}+F_{\scriptscriptstyle r}^*\big|_{\scriptscriptstyle B}\quad\big(r=1,2,...,15\big)
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$$
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The generalized inertia forces associated with the tower, $F_{r}{}^{*}\!|_{T},$ result from the tower’s distributed lineal density, $\mu_{T}(h)$ :
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The generalized inertia forces associated with the tower, $F_{r}^{*}|_{T},$ result from the tower’s distributed lineal density, $\mu_{T}(h)$ :
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与塔相关的广义惯性力，$F_{r}^{*}|_{T},$ 源于塔的分布式线密度，$\mu_{T}(h)$：
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$$
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F_{r}\left.^{*}\right|_{T}=-{\int_{0}^{H}\mu_{T}\big(h\big)^{E}\pmb{\nu}_{r}^{T}\cdot^{E}\pmb{a}^{T}}d h\quad\left(r=I,2,...,I\,\pmb{\zeta}\right)
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F_{r}\left.^{*}\right|_{T}=-{\int_{0}^{H}\mu_{T}\big(h\big){}^{E}{\nu}_{r}^{T}\cdot{}^{E}\pmb{a}^{T}}d h\quad\left(r=1,2,...,15\right)
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$$
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where $\varepsilon_{\nu_{r}^{T}}$ is the $r^{\mathrm{th}}$ partial velocity associated with point $\mathrm{T}$ in the tower and ${\pmb{\varepsilon}}_{\pmb{a}}{\pmb{T}}$ is the acceleration of the same point in the inertial frame.
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where ${}^{E}{\nu}_{r}^{T}$ is the $r^{\mathrm{th}}$ partial velocity associated with point $\mathrm{T}$ in the tower and ${}^{E}\pmb{a}^{T}$ is the acceleration of the same point in the inertial frame.
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其中 ${}^{E}{\nu}_{r}^{T}$ 为塔架上点 $\mathrm{T}$ 对应的第 $r$ 个偏导速度，${}^{E}\pmb{a}^{T}$ 为该点在惯性坐标系下的加速度。**------基于质点的广义惯性力公式**
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Tower deflections, yaw rates, and tilt rates contribute to the generalized inertia forces associated with the nacelle, $F_{r}{^\ast}|_{D}$ :
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塔架挠度、偏航速率和倾覆速率共同作用，导致机舱的广义惯性力 $F_{r}{^\ast}|_{D}$ ：
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$$
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F_{r}*\!\!\mid_{_{N}}=^{E}\!\nu_{r}^{D}\cdot\left(-m_{\scriptscriptstyle N}^{\phantom{~}E}a^{D}\right)\!\!+^{E}\!\omega_{r}^{^{N}}\cdot\left(-^{E}\!\dot{H}^{D}\right)\;\;\left(r=I,\!2,...,I\bar{S}\right)
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F_{r}^*\mid_{_{N}}={}^{E}\nu_{r}^{D}\cdot\left(-m_{ N}{}^{E}a^{D}\right)\!\!+{}^{E}\omega_{r}^{N}\cdot\left(-{}^{E}\dot{H}^{D}\right)\;\;\left(r=1,\!2,...,15\right)
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$$
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where $\varepsilon_{\boldsymbol{\nu}_{r}^{D}_{r}}$ is the $r^{\mathrm{th}}$ partial velocity associated with the center of mass (point D) of the nacelle, $m_{N}$ is the mass of the nacelle, $\varepsilon_{\pmb{a}}^{~~D}$ is the acceleration of the center of mass of the nacelle in the inertial frame, $^E_{\pmb{\omega}_{\ r}^{N}}$ is the $r^{\mathrm{th}}$ partial angular velocity associated with the nacelle, and ${\boldsymbol{E}}\hat{\boldsymbol{H}}^{D}$ is the time derivative of angular momentum of the nacelle about its center of mass, in the inertial frame.
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Similarly, the generalized inertia forces associated with the hub, $F_{r}*|_{H},$ are:
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where ${}^{E}\nu_{r}^{D}$ is the $r^{\mathrm{th}}$ partial velocity associated with the center of mass (point D) of the nacelle, $m_{N}$ is the mass of the nacelle, ${}^{E}a^{D}$ is the acceleration of the center of mass of the nacelle in the inertial frame, ${}^{E}\omega_{r}^{N}$ is the $r^{\mathrm{th}}$ partial angular velocity associated with the nacelle, and ${}^{E}\dot{H}^{D}$ is the time derivative of angular momentum of the nacelle about its center of mass, in the inertial frame.
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其中 ${}^{E}\nu_{r}^{D}$ 是与机舱质量中心（点D）相关的第 $r^{\mathrm{th}}$ 分向速度，$m_{N}$ 是机舱的质量，${}^{E}a^{D}$ 是机舱质量中心在惯性坐标系中的加速度，${}^{E}\omega_{r}^{N}$ 是与机舱相关的第 $r^{\mathrm{th}}$ 分角速度，而 ${}^{E}\dot{H}^{D}$ 是机舱绕其质量中心的角动量在惯性坐标系中的时间导数。
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Similarly, the generalized inertia forces associated with the hub, $F_{r}^*|_{H},$ are:
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同样地，与枢轴相关的广义惯性力，$F_{r}^*|_{H}$ 为：
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$$
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\boldsymbol{F}_{r}*\big|_{H}\!=\!^{E}\boldsymbol{\nu}_{r}^{C}\cdot\left(-\boldsymbol{m}_{H}^{\phantom{\dagger}E}\boldsymbol{a}^{C}\right)\!\!+\!^{E}\boldsymbol{\omega}_{r}^{H}\cdot\left(\!-^{E}\dot{H}^{C}\right)\;\;\left(r=I,\!2,...,I\boldsymbol{S}\!\right)
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F_{r}^*|_{H}={}^{E}\boldsymbol{\nu}_{r}^{C}\cdot\left(-\boldsymbol{m}_{H}{}^{E}\boldsymbol{a}^{C}\right)\!\!+\!{}^{E}\boldsymbol{\omega}_{r}^{H}\cdot\left(\!-{}^{E}\dot{H}^{C}\right)\;\;\left(r=1,\!2,...,15\right)
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$$
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where $\varepsilon_{\boldsymbol{\nu}_{r}^{c}}$ is the $r^{\mathrm{th}}$ partial velocity associated with the center of mass (point C) of the hub, $m_{H}$ is the hub mass, $\varepsilon_{\pmb{a}}c$ is the acceleration of the center of mass of the hub in the inertial frame, $\varepsilon_{\omega_{\ r}}^{}$ is the $r^{\mathrm{th}}$ partial angular velocity associated with the hub, and ${}^{E}\hat{H}^{C}$ is the time derivative of angular momentum of the hub about its center of mass, in the inertial frame.
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where ${}^{E}\boldsymbol{\nu}_{r}^{C}$ is the $r^{\mathrm{th}}$ partial velocity associated with the center of mass (point C) of the hub, $m_{H}$ is the hub mass, ${}^{E}\boldsymbol{a}^{C}$ is the acceleration of the center of mass of the hub in the inertial frame, ${}^{E}\boldsymbol{\omega}_{r}^{H}$ is the $r^{\mathrm{th}}$ partial angular velocity associated with the hub, and ${}^{E}\hat{H}^{C}$ is the time derivative of angular momentum of the hub about its center of mass, in the inertial frame.
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其中 ${}^{E}\boldsymbol{\nu}_{r}^{C}$ 是与轮毂质量心（点 C）相关的第 $r$ 个偏速度，$m_{H}$ 是轮毂质量，${}^{E}\boldsymbol{a}^{C}$ 是轮毂在惯性系中的质量心加速度，${}^{E}\boldsymbol{\omega}_{r}^{H}$ 是与轮毂相关的第 $r$ 个偏角速度，而 ${}^{E}\hat{H}^{C}$ 是轮毂绕其质量心的角动量时变率，在惯性系中。
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Finally, the distributed lineal density of the blades, $\mu_{B}(r)$ , contribute to the generalized inertia forces associated with the blades, $F_{r}{^{*}\!|}_{B}$ , in the same way the distributed lineal density of the tower contributed to its corresponding generalized inertia forces:
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Finally, the distributed lineal density of the blades, $\mu_{B}(r)$ , contribute to the generalized inertia forces associated with the blades, $F_{r}^{*}|_{B}$ , in the same way the distributed lineal density of the tower contributed to its corresponding generalized inertia forces:
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最终，叶片的分布式线密度 $\mu_{B}(r)$ 同样会像塔的分布式线密度对塔的广义惯性力贡献一样，对叶片的广义惯性力 $F_{r}^{*}|_{B}$ 产生影响。
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$$
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\begin{array}{l}{{\left.F_{r}\mathbf{\Phi}^{*}\right|_{B}=-\!\!\int_{0}^{R-R_{H}}\mu_{B}\bigl(r_{l}\bigr)^{E}\nu_{r}^{S_{l}}\mathbf{\cdot}^{E}a^{S_{l}}d r_{l}\,-}}\\ {{\qquad\,\,\,\int_{0}^{R-R_{H}}\mu_{B}\bigl(r_{2}\bigr)^{E}\nu_{r}^{S_{2}}\mathbf{\cdot}^{E}a^{S_{2}}d r_{2}\quad\left(r=l,2,\ldots,\!l\,5\right)}}\end{array}
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\begin{array}{l}{{\left.F_{r}^{*}\right|_{B}=-\!\!\int_{0}^{R-R_{H}}\mu_{B}\bigl(r_{1}\bigr){}^{E}\nu_{r}^{S_{1}}\mathbf{\cdot}{}^{E}a^{S_{1}}d r_{1}\,-}}\\ {{\qquad\,\,\,\int_{0}^{R-R_{H}}\mu_{B}\bigl(r_{2}\bigr){}^{E}\nu_{r}^{S_{2}}\mathbf{\cdot}{}^{E}a^{S_{2}}d r_{2}\quad\left(r=1,2,\ldots,15\right)}}\end{array}
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$$
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where $E_{\nu_{~r}^{s i}}$ is the $r^{\mathrm{th}}$ partial velocity associated with point $S$ in blade $i$ ${\mathrm{\Delta}i=1}$ and 2) and $\pmb{{\cal E}}_{\pmb{q}}s i$ is the acceleration of the same point in the inertial frame.
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where ${}^{E}\nu_{r}^{S_{1}}$ is the $r^{\mathrm{th}}$ partial velocity associated with point $S$ in blade $i$ ($i=1$ and 2) and ${}^{E}a^{S_{i}}$ is the acceleration of the same point in the inertial frame.
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其中 ${}^{E}\nu_{r}^{S_{1}}$ 是与叶片 $i$ ( $i=1$ 和 2) 中的点 $S$ 相关的第 $r$ 偏导速度，而 ${}^{E}a^{S_{i}}$ 是该点在惯性坐标系下的加速度。
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The resultant of all applied forces acting on elements of the wind turbine contributes to the total generalized active forces that govern the equations of motion. These forces include aerodynamic forces (see Chapter 2); elastic forces from the tower, blade, and drive train flexibility; elastic forces from the springs inherent in the yaw drive (yaw and tilt motion) and teetering device; gravitational forces; generator forces; and damping forces:
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所有作用于风轮叶片的载荷合力，共同构成控制运动方程的总体广义主动力。这些力包括气动力（见第二章）；塔架、叶片和传动系统柔性引起的弹性力；偏航驱动（偏航和俯仰运动）和摆振装置固有弹簧引起的弹性力；重力；发电机力；以及阻尼力：
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$$
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F_{r}=F_{r}\big|_{A e r o}+F_{r}\big|_{E l a s t i c}+F_{r}\big|_{G r a v}+F_{r}\big|_{G e n e r a t o r}+F_{r}\big|_{D a m p}\quad(r=l,2,...,l\,5)
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F_{r}=F_{r}\big|_{A e r o}+F_{r}\big|_{E l a s t i c}+F_{r}\big|_{G r a v}+F_{r}\big|_{G e n e r a t o r}+F_{r}\big|_{D a m p}\quad(r=1,2,...,15)
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$$
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An exhaustive development of all these local generalized active forces is beyond the scope of this work. Brief descriptions of a few follow.
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对所有这些局部广义主动力的详尽推导超出了本工作的范围。以下是对此进行简要描述。
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If ${F_{A e r o}}^{s i}$ is the resultant of all aerodynamic forces acting on point $S$ in blade $i$ ${\mathrm{[}}i=1$ and 2), then the generalized active aerodynamic forces, $F_{r\mid A e r o}$ , are:
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If ${F_{A e r o}}^{s i}$ is the resultant of all aerodynamic forces acting on point $S$ in blade $i$ (i=1 and 2), then the generalized active aerodynamic forces, $F_{r\mid A e r o}$ , are:
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如果 ${F_{A e r o}}^{s i}$ 是作用于叶片 $i$ (i=1 和 2) 的点 $S$ 上的所有气动力合力，那么广义主动气动力，$F_{r\mid A e r o}$ ，是：
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$$
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F_{r}\big|_{A e r o}=\int_{0}^{R-R_{H}}\,E_{\,r}^{\,\,S_{I}}\cdot F_{\,A e r o}^{\,\,s_{I}}d r_{I}+\int_{0}^{R-R_{H}}\,E_{\,r}^{\,\,S_{2}}\cdot F_{\,A e r o}^{\,\,s_{2}}d r_{2}\quad(r=I,2,...,I\,S)
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F_{r}\big|_{A e r o}=\int_{0}^{R-R_{H}}\,{}^{E}\nu_{r}^{S_{1}}\cdot F_{\,A e r o}^{\,\,s_{1}}d r_{1}+\int_{0}^{R-R_{H}}\,{}^{E}\nu_{r}^{S_{2}}\cdot F_{\,A e r o}^{\,\,s_{2}}d r_{2}\quad(r=1,2,...,15)
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$$
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FAST_AD assumes aerodynamic forces act only on the rotor blades. FAST_AD neglects any aerodynamic forces that are imparted on the tower or the rest of the wind turbine structure, though FAST_AD does assume that the tower forms a wake (tower shadow) that affects the aerodynamics of downwind turbines (see section 2.6).
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FAST_AD 假设气动力仅作用于风轮叶片。FAST_AD 忽略了传递到塔架或其他风力发电机结构的气动力，尽管 FAST_AD 假设塔架会形成一个尾流（塔架阴影），从而影响下游风轮的空气动力学（见 2.6 节）。
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Instead of defining the generalized active elastic restoring forces of the tower and blades with Eq. (3.101), FAST_AD uses the equivalent form:
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Instead of defining the generalized **active elastic restoring forces** of the tower and blades with Eq. (3.101), FAST_AD uses the equivalent form:
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不使用公式 (3.101) 定义塔架和叶片的**广义主动弹性回复力**，FAST_AD 采用等效形式：
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$$
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F_{r}={\frac{\partial V}{\partial q_{r}}}\quad(r=l,\!2,...,\!l\,5)
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F_{r}={\frac{\partial V}{\partial q_{r}}}\quad(r=1,\!2,...,15)
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$$
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where the potential energy, $V$ :
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其中，势能，$V$：
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$$
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V\!=\!\frac{I}{2}\!\sum_{i=I}^{N}\!\sum_{j=I}^{N}\!k_{i j}q_{i}(t)q_{j}(t)
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V=\frac{1}{2}\sum_{i=1}^{N}\sum_{j=1}^{N}k_{i j}q_{i}(t)q_{j}(t)
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$$
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where $N$ is the number of DOFs, $k_{i j}$ is the generalized stiffness, and the $q_{i}(t)$ ’s are the generalized coordinates pertaining to the flexible body. There is a subtle difference between Eqs. (3.112) and (3.44). Since Eq. (3.112) contains $q_{i}(t)$ ’s and Eq. (3.44) contains $c_{i}(t)^{\circ}\mathbf{s}.$ , the generalized stiffness inherent in Eq. (3.112) is related to the integrals of the partial derivatives of the normal mode shapes [the $\phi_{i}(z)^{\circ}\mathbf{s}]$ , whereas the generalized stiffness inherent in Eq. (3.44) is related to the integrals of the partial derivatives of the shape functions [the $\varphi_{i}(z)\,{\mathrm{'s]}}$ .
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where $N$ is the number of DOFs, $k_{i j}$ is the generalized stiffness, and the $q_{i}(t)$ ’s are the generalized coordinates pertaining to the flexible body. There is a subtle difference between Eqs. (3.112) and (3.44). Since Eq. (3.112) contains $q_{i}(t)$ ’s and Eq. (3.44) contains $c_{i}(t)'s$ , the generalized stiffness inherent in Eq. (3.112) is related to the integrals of the partial derivatives of the normal mode shapes (the $\phi_{i}(z)'{s}$) , whereas the generalized stiffness inherent in Eq. (3.44) is related to the integrals of the partial derivatives of the shape functions (the $\varphi_{i}(z)\,{\mathrm{'s}}$ ).
|
||||
其中，$N$ 为自由度数，$k_{i j}$ 为广义刚度，而 $q_{i}(t)$ 对应于柔性体的广义坐标。 方程 (3.112) 和 (3.44) 之间存在微妙的差异。 由于方程 (3.112) 包含 $q_{i}(t)$，而方程 (3.44) 包含 $c_{i}(t)$，因此方程 (3.112) 中的广义刚度与正模态形状normal mode shapes的偏导数积分有关（即 $\phi_{i}(z)$），而方程 (3.44) 中的广义刚度与形状函数shape functions的偏导数积分有关（即 $\varphi_{i}(z)$）。
|
||||
|
||||
Combining the elastic restoring forces and the gravitational forces of the tower into a single set of forces, the generalized active forces attributed to the tower deflection are found by replacing the $\varphi_{i}(h)$ ’s with $\phi_{i}(h)$ ’s in the generalized stiffness of the tower [Eq. (3.69)], substituting the result into Eq. (3.112), and then substituting the combined result into Eq. (3.111). Since the normal mode shapes are orthogonal to each other, the cross-coupled terms will drop out of this process (the generalized stiffness terms, $k_{i j},$ , involving $i\not=j$ will be zero).
|
||||
将塔的弹性回复力和重力合为一体，通过在塔的广义刚度中[公式 (3.69)]用 $\phi_{i}(h)$ 替换 $\varphi_{i}(h)$，将结果代入公式 (3.112)，再将合并后的结果代入公式 (3.111)，即可得到归因于塔的挠度所产生的广义主动力。由于正交模态normal mode shapes彼此正交，此过程中的耦合项将消失（涉及 $i\not=j$ 的广义刚度项，$k_{i j},$ 将为零）。
|
||||
|
||||
A similar process is performed when computing the generalized active elastic restoring forces of the blades. However, since the centrifugal stiffening effects are accounted for in the generalized inertia force of the blades [see Eq. (3.108)], the centrifugal stiffening terms must be dropped from the generalized stiffness of the blades given in Eq. (3.77). The potential energy of the flexing beam is the only potential energy term substituted into Eq. (3.111) when finding the generalized active elastic restoring forces of the blades. Since the centrifugal force terms are included when calculating the natural mode shapes but not in the immediate discussion of the generalized blade stiffness, the cross-coupled generalized stiffness terms will not drop out.
|
||||
在计算叶片（blade）的广义主动弹性回复力时，也会执行类似的过程。然而，由于离心加固效应已包含在叶片的广义惯性力中[见公式(3.108)]，因此必须从公式(3.77)给出的叶片的广义刚度中去除离心加固项。在寻找叶片的广义主动弹性回复力时，仅将弯曲梁的势能项代入公式(3.111)。由于离心力项在计算固有振型时包含，但在广义叶片刚度讨论中未直接考虑，因此交叉耦合的广义刚度项将不会消去。
|
||||
|
||||
Structural damping inherent in the tower and blades is modeled in FAST_AD using the Rayleigh damping technique. To characterize the magnitude of this damping, modal damping ratios are defined in the FAST_AD input file.
|
||||
FAST_AD 使用瑞利阻尼技术模拟塔架和叶片固有的结构阻尼。为了表征这种阻尼的强度，在 FAST_AD 输入文件中定义了模态阻尼比。
|
||||
|
||||
The flexibility of the drive shaft is modeled with a linear torsional spring and a linear viscous damper. The flexibility of the yaw drive, which allows everything atop the tower-top base plate to yaw and tilt, is modeled with a linear torsional spring and a linear torsional viscous damper in both the yaw and tilt directions. The teetering device is modeled with a constant coulomb damping moment (that is, constant in magnitude, but always opposing the teeter motion), a linear torsional viscous damper, and a nonlinear, cubic torsional spring. A teeter stop, which prevents excessive teeter motion, is modeled with a linear spring.
|
||||
|
||||
驱动轴的挠曲被模拟为线性扭转弹簧和一个线性粘性阻尼器。允许塔顶底板上所有部件进行偏航和倾斜的偏航驱动的挠曲，被模拟为在偏航和倾斜方向上都具有线性扭转弹簧和一个线性扭转粘性阻尼器。摆振机构被模拟为具有一个恒定库仑阻尼力矩（即大小恒定，但始终与摆振运动相反），一个线性扭转粘性阻尼器，以及一个非线性、三次扭转弹簧。一个摆振限位器，用于防止过度摆振，被模拟为线性弹簧。
|
||||
# 3.5 Kane’s Equations of Motion
|
||||
|
||||
FAST_AD uses Kane’s method to set up equations of motion, which are then solved by numerical integration (see Chapter 4). By a direct result of Newton’s laws of motion, Kane’s equations of motion for a simple holonomic system can be stated as (Kane and Levinson 1985):
|
||||
FAST_AD 使用 Kane 方法建立运动方程，然后通过数值积分求解（见第四章）。根据牛顿运动定律的直接结果，对于一个简单的holonomic system，Kane 的运动方程可以表示为（Kane and Levinson 1985）：
|
||||
|
||||
$$
|
||||
F_{r}+F_{r}^{~*}\,{=}\,0\quad\left(r\,{=}\,I,2,...,I\,5\right)
|
||||
$$
|
||||
|
||||
Substituting the equations presented in the previous sections of this chapter into Eq. (3.113) results in a set of 15 coupled, dynamic equations that prescribe the motion of the entire wind turbine structure as a function of time.
|
||||
|
||||
将本章前几节中给出的方程代入公式(3.113)，得到一组由15个耦合动态方程，这些方程描述了整个风轮结构的随时间变化的运动。
|
||||
# 4. FAST_AD Design Code Overview and Limitations
|
||||
|
||||
The FAST_AD design code is a medium-complexity code used to (1) model a wind turbine structurally given the turbine layout and aerodynamic and mechanical properties of its members and (2) simulate the wind turbine’s aerodynamic and structural response by imposing complex virtual wind-inflow conditions. Outputs of these simulations include time-series data on the loads and responses of the structural members of the wind turbine. Post-processing codes can than be used to analyze these data, enabling researchers and designers to efficiently and safely design, analyze, and improve wind energy systems and lower the cost of wind-generated electricity. Wind energy researchers and designers rely heavily on computer modeling tools such as FAST_AD to accurately predict, quantify, and understand the complex physical interactions that characterize component loads and overall wind turbine performance.
|
||||
|
||||
@ -42,3 +42,50 @@ AD 15
|
||||
|
||||
- 采用v3.5气动数据情况下，多体计算结果与FAST一致
|
||||
- 采用v4.0气动数据情况下，多体计算结果与FAST一致
|
||||
|
||||
|
||||
|
||||
fast bladed 载荷对比
|
||||
叶片载荷
|
||||
|
||||
| fast | | | |
|
||||
| ------------- | ---------------------------------------------------- | --------------------------------- | ------ |
|
||||
| Spn1MLxb1 | Blade 1 local edgewise moment at span station 1 | About the local xb1-axis | (kN-m) |
|
||||
| Spn1MLyb1 | Blade 1 local flapwise moment at span station 1 | About the local yb1-axis | (kN-m) |
|
||||
| Spn1MLzb1 | Blade 1 local pitching moment at span station 1 | About the local zb1-axis | (kN-m) |
|
||||
| | | | |
|
||||
| Spn1FLxb1 | Blade 1 local flapwise shear force at span station 1 | Directed along the local xb1-axis | (kN) |
|
||||
| Spn1FLyb1 | Blade 1 local edgewise shear force at span station 1 | Directed along the local yb1-axis | (kN) |
|
||||
| Spn1FLzb1 | Blade 1 local axial force at span station 1 | Directed along the local zb1-axis | (kN) |
|
||||
| | | | |
|
||||
| bladed | | | |
|
||||
| Bladed 1 Mx | Principal axes | | (N-m) |
|
||||
| Bladed 1 My | Principal axes | | (N-m) |
|
||||
| Bladed 1 Mxyz | Principal axes | | (N-m) |
|
||||
| | | | |
|
||||
| Bladed 1 Fx | Principal axes | | (kN) |
|
||||
| Bladed 1 Fy | Principal axes | | (kN) |
|
||||
| Bladed 1 Fxyz | Principal axes | | (kN) |
|
||||
|
||||
Principal axes: The positive z-axis follows the local deflected neutral axis at each blade station towards the blade tip. The positive y axis is defined by the principal axis orientation. The positive x axis is orthogonal to the y and z and follows the right hand rule. For output loads, the origin of the axes is on the neutral axis at each local deflected blade station. (see diagram below)
|
||||
|
||||
主轴：正向z轴沿每个叶片位置向叶片末端的挠曲中性轴延伸。正向y轴由主要轴系方向定义。正向x轴垂直于y轴和z轴，并遵循右手螺旋定则。对于输出载荷，坐标原点位于每个挠曲中性轴上。（见下方的示意图）
|
||||
|
||||
![[Pasted image 20250610111507.png]]Blade principal axes coordinate system
|
||||
|
||||
Note that there is a subtle difference between the “principal axis” frame and the “blade local element frame”.
|
||||
The “blade local element frame” is orientated so that the X vector in this coordinate system points directly between adjacent nodes on the blade. The other two coordinate system vector directions are determined by the “principal axis twist” angle. This is explained in more detail in DNV GL technical note UKBR-110052-T-31-A
|
||||
The “principal axis” frame is used for load output in Bladed. The principal axis orientation is calculated by taking the average orientation of the two blade elements at the node where the elements join. This is illustrated below. The two adjoining local element frames are shown in green and red. The principal axes output frame is shown in blue.
|
||||
Note that the element local coordinate system has its x direction along the element, unlike the “principal axes” coordinate system which has z along the element axis.
|
||||
需要注意的是，“主轴系”和“叶片局部单元系”之间存在细微差别。
|
||||
“叶片局部单元系”的定向方式是，该坐标系中的X向量直接指向叶片上相邻节点之间的连线。另外两个坐标系向量方向由“主轴扭转”角度决定。 详情请参见DNV GL技术备忘录UKBR-110052-T-31-A。
|
||||
**Bladed中用于输出载荷的是“主轴系”**。主轴方向的计算是通过取叶片节点处两个叶片单元的平均方向来实现的。下图所示。两个相邻的局部单元系分别用绿色和红色表示。主轴输出系用蓝色表示。
|
||||
需要注意的是，局部单元坐标系的x方向沿着单元方向，而“主轴系”坐标系的z方向沿着单元轴方向。
|
||||
|
||||
![[Pasted image 20250610113014.png]]
|
||||
Relationship between blade "local element axes" and "principal axes coordinates"
|
||||
|
||||
|
||||
Root axes: The orientation of the axes is fixed to the blade root and does not rotate with either twist or blade deflection. The axis set does rotate about the z axis with pitch. For output loads, the origin of the axes is on the neutral axis at each local deflected blade station.
|
||||
根部坐标轴：坐标轴的方向固定于叶片根部，不会随扭角或叶片挠曲而旋转。坐标轴系会绕z轴旋转，以适应变桨角度。对于输出载荷，坐标轴原点位于每个局部挠曲叶片位置的受力中性轴上。
|
||||
|
||||
|
||||
BIN
多体+耦合求解器/images/Pasted image 20250610111507.png
Normal file
BIN
多体+耦合求解器/images/Pasted image 20250610111507.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 82 KiB |
@ -15,10 +15,24 @@
|
||||
{"id":"d405163cb9ecd804","type":"text","text":"叶片多段拆分，小段做模态叠加？","x":60,"y":670,"width":250,"height":60},
|
||||
{"id":"4a08e20366911d68","type":"text","text":"v1pt_theory","x":290,"y":140,"width":250,"height":60},
|
||||
{"id":"6e5a6a3cdd47bd52","type":"text","text":"v2pt_theory","x":290,"y":207,"width":250,"height":60},
|
||||
{"id":"4bfacdf3ddedbdec","x":-340,"y":960,"width":250,"height":60,"type":"text","text":"柔性体 / 连续体"},
|
||||
{"id":"ae48e80ccd92bffa","x":60,"y":960,"width":250,"height":60,"type":"text","text":"连续体振动"},
|
||||
{"id":"ed5a265dc4b72aaa","x":420,"y":960,"width":250,"height":60,"type":"text","text":"连续体动力学"},
|
||||
{"id":"fc13b731983ac384","x":-340,"y":1181,"width":250,"height":60,"type":"text","text":"浮动坐标系"}
|
||||
{"id":"4bfacdf3ddedbdec","type":"text","text":"柔性体 / 连续体","x":-340,"y":960,"width":250,"height":60},
|
||||
{"id":"ae48e80ccd92bffa","type":"text","text":"连续体振动","x":60,"y":960,"width":250,"height":60},
|
||||
{"id":"ed5a265dc4b72aaa","type":"text","text":"连续体动力学","x":420,"y":960,"width":250,"height":60},
|
||||
{"id":"fc13b731983ac384","type":"text","text":"浮动坐标系","x":-340,"y":1181,"width":250,"height":60},
|
||||
{"id":"df8e8a93ea4af203","x":984,"y":-235,"width":250,"height":60,"type":"text","text":"叶片"},
|
||||
{"id":"7dc483819d8b527b","x":1280,"y":-235,"width":250,"height":60,"type":"text","text":"轮毂"},
|
||||
{"id":"c02c22667637367f","x":1580,"y":-235,"width":250,"height":60,"type":"text","text":"机舱"},
|
||||
{"id":"623ac90ce5bad160","x":1580,"y":-60,"width":250,"height":60,"type":"text","text":"塔架"},
|
||||
{"id":"0324f23e7dc78d68","x":1580,"y":55,"width":250,"height":60,"type":"text","text":"机舱"},
|
||||
{"id":"ba5c15a27611709c","x":1580,"y":-150,"width":250,"height":60,"type":"text","text":"偏航轴承"},
|
||||
{"id":"6c3a6982a53e7df9","x":984,"y":177,"width":250,"height":60,"type":"text","text":"刚体部件："},
|
||||
{"id":"f3c36900dd3ed9d1","x":1280,"y":177,"width":250,"height":60,"type":"text","text":"刚体的广义主动力、惯性力公式"},
|
||||
{"id":"b624e8c6302b9a1c","x":984,"y":310,"width":250,"height":60,"type":"text","text":"叶片、塔架"},
|
||||
{"id":"1054909d642ce071","x":1280,"y":310,"width":250,"height":60,"type":"text","text":"广义惯性力：质点广义惯性力公式 积分"},
|
||||
{"id":"1e1a1a0d67920443","x":1280,"y":420,"width":250,"height":60,"type":"text","text":"广义主动力：由势能dV/dq_r"},
|
||||
{"id":"092cf6719d47d01d","x":1280,"y":530,"width":250,"height":60,"type":"text","text":"弹性恢复力、阻尼、重力、气动力"},
|
||||
{"id":"32b762bd2a4b0d66","x":984,"y":640,"width":250,"height":60,"type":"text","text":"传动链、偏航"},
|
||||
{"id":"14a4450243ca9953","x":1620,"y":530,"width":250,"height":60,"type":"text","text":"原理呢？"}
|
||||
],
|
||||
"edges":[
|
||||
{"id":"647c1b45edc92b02","fromNode":"9461f7dd96103316","fromSide":"bottom","toNode":"0c8534c8ba68c9a6","toSide":"top"},
|
||||
@ -31,6 +45,11 @@
|
||||
{"id":"5d5f4fef281a656e","fromNode":"6094c53caf966263","fromSide":"right","toNode":"da500b2b12ed0901","toSide":"left"},
|
||||
{"id":"5573fa12a3a02ee0","fromNode":"da500b2b12ed0901","fromSide":"right","toNode":"20ce8d75f0f35588","toSide":"left"},
|
||||
{"id":"c4828432b71fd99a","fromNode":"4bfacdf3ddedbdec","fromSide":"right","toNode":"ae48e80ccd92bffa","toSide":"left"},
|
||||
{"id":"8ded09c7902a30b6","fromNode":"ae48e80ccd92bffa","fromSide":"right","toNode":"ed5a265dc4b72aaa","toSide":"left"}
|
||||
{"id":"8ded09c7902a30b6","fromNode":"ae48e80ccd92bffa","fromSide":"right","toNode":"ed5a265dc4b72aaa","toSide":"left"},
|
||||
{"id":"4818c7bc8aa99b57","fromNode":"b624e8c6302b9a1c","fromSide":"right","toNode":"1054909d642ce071","toSide":"left"},
|
||||
{"id":"d58b296db32dc932","fromNode":"1054909d642ce071","fromSide":"bottom","toNode":"1e1a1a0d67920443","toSide":"top"},
|
||||
{"id":"f88001eedfed69b8","fromNode":"1e1a1a0d67920443","fromSide":"bottom","toNode":"092cf6719d47d01d","toSide":"top"},
|
||||
{"id":"d0bfe0faa41e4e2f","fromNode":"6c3a6982a53e7df9","fromSide":"right","toNode":"f3c36900dd3ed9d1","toSide":"left"},
|
||||
{"id":"14130329273bf6fc","fromNode":"092cf6719d47d01d","fromSide":"right","toNode":"14a4450243ca9953","toSide":"left"}
|
||||
]
|
||||
}
|
||||
10
学术讲座-交流-面试/images/Bladed交流.md
Normal file
10
学术讲座-交流-面试/images/Bladed交流.md
Normal file
@ -0,0 +1,10 @@
|
||||
|
||||
问题
|
||||
1 steady operational loads
|
||||
叶片变形如何稳态求解的？
|
||||
变桨角度、转速如何求解的？
|
||||
|
||||
2 steady parked loads
|
||||
BEM在90°攻角如何计算的？
|
||||
|
||||
3 分段模态有什么用，与单段模态相比，有没有相应的数据支持？
|
||||
@ -5,7 +5,7 @@
|
||||
{"id":"d2c5e076ba6cf7d7","type":"text","text":"# 推进计划\n未来四周计划推进的重要事情\n\n文献调研启动\n\n建模重新推导\n\n\n","x":-600,"y":-306,"width":456,"height":347},
|
||||
{"id":"82708a439812fdc7","type":"text","text":"# 7月已完成\n\n","x":-220,"y":134,"width":440,"height":560},
|
||||
{"id":"505acb3e6b119076","type":"text","text":"# 6月已完成\n\n\nP1 结果对比\n- Herowind 带3.5气动与fast3.5对比 相同\n- Herowind 带4.0气动与fast4.0对比 相同\n- Herowind 带hrl气动与fast对比 需气动支持15MW\n\nP1 Bladed交流问题汇总","x":-700,"y":134,"width":440,"height":560},
|
||||
{"id":"c18d25521d773705","type":"text","text":"# 计划\n这周要做的3~5件重要的事情，这些事情能有效推进实现OKR。\n\nP1 必须做。P2 应该做\n\n\nP1 柔性部件 叶片、塔架主动力惯性力算法 主线\n- 变形体动力学 简略看看ing\n- 浮动坐标系方法 如何用于梁模型 \n\t- Q 问孟航 不用浮动坐标系\n- 柔性梁弯曲变形振动学习，主线 \n\t- 广义质量 刚度矩阵及含义\n- 如何静力学求解\n\t- 基于本构方程 读孟的论文\n\t\nP1 结果对比\n- Herowind 带hrl气动与fast对比 需气动支持15MW\n- Bladed与FAST之间的对比\n\nP1 如何优雅的存储、输出结果。\nP1 国产化适配交给甲子营，对接\nP1 推进气动、控制、多体、水动 耦合计算\nP2 yaw 自由度再bug确认 已知原理了\n\nP1 模型线性化调研\n \n","x":-594,"y":-693,"width":450,"height":347},
|
||||
{"id":"c18d25521d773705","type":"text","text":"# 计划\n这周要做的3~5件重要的事情，这些事情能有效推进实现OKR。\n\nP1 必须做。P2 应该做\n\n\nP1 柔性部件 叶片、塔架主动力惯性力算法 主线\n- 变形体动力学 简略看看ing\n- 柔性梁弯曲变形振动学习，主线 \n\t- 广义质量 刚度矩阵及含义\n- 如何静力学求解\n\t- 基于本构方程 读孟的论文\n\t- normal mode shape 能否使用？\n\nP1 模型线性化原理\nP1 结果对比\n- Bladed与FAST之间的对比 去掉角度\n\nP1 如何优雅的存储、输出结果。\nP1 推进气动、控制、多体、水动 耦合计算\nP2 yaw 自由度再bug确认 已知原理了\n\nP1 模型线性化调研\n \n","x":-594,"y":-693,"width":450,"height":347},
|
||||
{"id":"30cb7486dc4e224c","type":"text","text":"# 8月已完成\n\n","x":260,"y":134,"width":440,"height":560}
|
||||
],
|
||||
"edges":[]
|
||||
|
||||
Loading…
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Reference in New Issue
Block a user