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---
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epoch: 1757035242142
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modelKey: gemini-2.5-flash|google
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tags:
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- copilot-conversation
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---
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**user**: [[Kallesøe-Equations of motion for a rotor blade]]方程9如何得到
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[Timestamp: 2025/09/05 09:20:42]
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**ai**: 根据提供的笔记 [[Kallesøe-Equations of motion for a rotor blade]]，方程(9)描述了整个系统的总动能（Kinetic Energy）。
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其表达式为：
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$T\!=\!\frac{1}{2}J_{g c n}\dot{\phi}^{2}+\int_{r}^{R}\!\Big(\frac{1}{2}m\mathbf{r}_{c g}^{\mathrm{T}}\cdot\dot{\mathbf{r}}_{c g}+\frac{1}{2}I_{c g}\big(\dot{\beta}+\dot{\theta}\big)^{2}\Big)\mathrm{d}s\tag 9$
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这个方程是通过将系统中所有运动部件的动能相加得到的，具体可以分解为以下三个部分：
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1. **轮毂、齿轮箱和发电机的转动动能**:
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* 第一项是 $\frac{1}{2}J_{g c n}\dot{\phi}^{2}$。
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* 根据文中的“Kinetic Energy”一节，$J_{g c n}$ 是描述轮毂、齿轮箱和发电机的总转动惯量。
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* $\dot{\phi}$ 是风轮的角速度（$\phi$ 是方位角）。
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* 因此，这一项是整个驱动链（不包括叶片）绕旋转轴的转动动能，符合转动动能公式 $\frac{1}{2}I\omega^2$。
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2. **叶片质量的平动动能**:
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* 第二项是 $\int_{r}^{R}\!\Big(\frac{1}{2}m\mathbf{r}_{c g}^{\mathrm{T}}\cdot\dot{\mathbf{r}}_{c g}\Big)\mathrm{d}s$。
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* 这是一个沿着叶片长度（从根部 $r$ 到叶尖 $R$）的积分。
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* $m$ 是叶片单位长度的质量。
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* $\mathbf{r}_{c g}$ 是叶片截面重心的位置矢量，$\dot{\mathbf{r}}_{c g}$ 则是其速度矢量。
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* $\frac{1}{2}m(\dot{\mathbf{r}}_{c g})^2$ 表示单位长度叶片的平动动能。
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* 通过对整个叶片进行积分，这一项计算了整个叶片因平动（包括挥舞和摆振）而产生的总动能。
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3. **叶片截面绕自身弹性轴的转动动能**:
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* 第三项是 $\int_{r}^{R}\!\Big(\frac{1}{2}I_{c g}\big(\dot{\beta}+\dot{\theta}\big)^{2}\Big)\mathrm{d}s$。
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* 这也是一个沿着叶片长度的积分。
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* $I_{c g}$ 是叶片截面绕其重心的转动惯量。
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* 文中明确指出，${\dot{\beta}}+{\dot{\theta}}$ 是叶片截面绕弹性轴的角速度，其中 $\dot{\beta}$ 是变桨角速度，$\dot{\theta}$ 是叶片弹性扭转的角速度。
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* 因此，这一项代表了叶片截面因扭转运动而产生的转动动能，并沿叶片全长进行积分。
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综上所述，方程(9)是通过线性叠加原理，将驱动链的转动动能、叶片的平动动能以及叶片自身的扭转动能这三个部分相加，从而得到了整个系统的总动能表达式。
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[Timestamp: 2025/09/05 09:21:10]
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**user**: \frac{1}{2}m\mathbf{r}_{c g}^{\mathrm{T}}\cdot\dot{\mathbf{r}}_{c g}与1/2 m \dot {r_{cg}}^2不同吧
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[Timestamp: 2025/09/05 09:23:57]
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**ai**: 您提了一个非常好的问题，这涉及到矢量和标量表示法的区别，以及物理学中常见的符号简写。
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简单来说：**在这篇论文的语境下，这两个表达式代表的是同一个物理量——动能，但它们的数学写法不同。**
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下面是详细的解释：
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### 1. 论文中的写法：$\frac{1}{2}m\dot{\mathbf{r}}_{c g}^{\mathrm{T}}\cdot\dot{\mathbf{r}}_{c g}$
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首先，请注意在论文的方程(9)中，正确的表达式是 $\frac{1}{2}m\dot{\mathbf{r}}_{c g}^{\mathrm{T}}\cdot\dot{\mathbf{r}}_{c g}$，这里用的是速度矢量 $\dot{\mathbf{r}}_{c g}$ 而不是位置矢量 $\mathbf{r}_{c g}$。
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这是一种严谨的**矢量/矩阵表示法**：
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* $\mathbf{r}_{c g}$ 是描述质心的**位置矢量**。由于运动是三维的，它可以写成一个列向量：
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$\mathbf{r}_{c g} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$
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* $\dot{\mathbf{r}}_{c g}$ 是对时间求导得到的**速度矢量**，通常也表示为列向量：
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$\dot{\mathbf{r}}_{c g} = \begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix}$
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* $\dot{\mathbf{r}}_{c g}^{\mathrm{T}}$ 是速度矢量的**转置**，即一个行向量：
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$\dot{\mathbf{r}}_{c g}^{\mathrm{T}} = [v_x, v_y, v_z]$
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* $\dot{\mathbf{r}}_{c g}^{\mathrm{T}}\cdot\dot{\mathbf{r}}_{c g}$ 是行向量与列向量的**矩阵乘法**（或者理解为矢量的点积）：
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$[v_x, v_y, v_z] \begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix} = v_x^2 + v_y^2 + v_z^2$
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这个结果 $v_x^2 + v_y^2 + v_z^2$ 正是速度矢量大小（即速率 $v$）的平方，也就是 $v^2$。
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所以，$\frac{1}{2}m\dot{\mathbf{r}}_{c g}^{\mathrm{T}}\cdot\dot{\mathbf{r}}_{c g} = \frac{1}{2}m(v_x^2 + v_y^2 + v_z^2) = \frac{1}{2}mv^2$。
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### 2. 您提到的写法：$\frac{1}{2} m \dot{r_{cg}}^2$
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这是一种更常见的**标量表示法**，通常在不涉及复杂坐标变换的物理学入门教材中看到。这里的 $\dot{r_{cg}}^2$ 通常被用作**速率平方 $v^2$ 的简写**。
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* **重要区别**：严格来说，如果 $r_{cg}$ 表示位置矢量 $\mathbf{r}_{c g}$ 的大小（即到原点的距离），那么 $\dot{r}_{cg}$ 是这个距离的变化率，它**不等于**速率 $|\dot{\mathbf{r}}_{c g}|$。例如，一个物体做匀速圆周运动，它的 $r_{cg}$ 是恒定的，所以 $\dot{r}_{cg}=0$，但它的速率 $v=|\dot{\mathbf{r}}_{c g}|$ 显然不为零。
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* **常见约定**：然而，在物理学中，为了书写方便，人们常常用 $\dot{r}^2$ 来代表速率的平方 $v^2 = |\dot{\mathbf{r}}|^2$。
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### 结论
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* **物理意义相同**：两者都表示质点的平动动能 $\frac{1}{2}mv^2$。
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* **数学严谨性不同**：
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* $\frac{1}{2}m\dot{\mathbf{r}}_{c g}^{\mathrm{T}}\cdot\dot{\mathbf{r}}_{c g}$ 是一个无歧义的、严谨的矢量表示，它明确地表示了速度矢量与自身的点积。
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* $\frac{1}{2} m \dot{r_{cg}}^2$ 是一种简写，依赖于上下文约定它代表速率的平方。
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**为什么论文要用更复杂的矢量写法？**
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因为这篇论文涉及多个旋转坐标系之间的复杂变换（如方程(1)、(3)、(8)所示）。在进行这些变换时，使用矩阵和矢量的形式进行推导会更加清晰、系统和不容易出错。这种表示法在高等动力学和连续介质力学中是标准做法。
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[Timestamp: 2025/09/05 09:24:30]
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**user**: 原文中\mathbf{r}_{c g}^{\mathrm{T}，上面没有\dot，是否是笔误
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[Timestamp: 2025/09/05 16:24:59]
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**ai**: 您观察得非常仔细，这是一个非常关键的问题。
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根据动能的物理定义，原文方程(9)中的表达式 $\frac{1}{2}m\mathbf{r}_{c g}^{\mathrm{T}}\cdot\dot{\mathbf{r}}_{c g}$ **极有可能是一个笔误 (typographical error)**。
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下面是详细的分析：
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1. **动能的正确物理定义**
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* 一个质量为 $m$ 的质点的平动动能是 $T = \frac{1}{2}mv^2$，其中 $v$ 是质点的速率（速度的大小）。
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* 在矢量表示中，速度是位置矢量对时间的导数，即 $\mathbf{v} = \dot{\mathbf{r}}_{c g}$。
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* 速率的平方 $v^2$ 等于速度矢量与其自身的点积：$v^2 = |\dot{\mathbf{r}}_{c g}|^2 = \dot{\mathbf{r}}_{c g} \cdot \dot{\mathbf{r}}_{c g}$。
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* 使用矩阵/向量的转置写法，这个点积就表示为 $\dot{\mathbf{r}}_{c g}^{\mathrm{T}}\cdot\dot{\mathbf{r}}_{c g}$。
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* 因此，正确的动能项应该是：$\frac{1}{2}m\dot{\mathbf{r}}_{c g}^{\mathrm{T}}\cdot\dot{\mathbf{r}}_{c g}$。
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2. **原文写法的物理意义**
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* 原文写的是 $\mathbf{r}_{c g}^{\mathrm{T}}\cdot\dot{\mathbf{r}}_{c g}$，这是**位置矢量**与**速度矢量**的点积。
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* 这个量在物理上并不代表动能。实际上，$\mathbf{r} \cdot \dot{\mathbf{r}} = \frac{1}{2}\frac{d}{dt}(\mathbf{r} \cdot \mathbf{r}) = \frac{1}{2}\frac{d}{dt}(r^2) = r\dot{r}$。它描述的是质点到坐标原点距离平方变化率的一半，这与动能是完全不同的物理量。
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* 从**单位**上分析也可以看出问题：
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* **动能**的单位是：质量 × (长度/时间)² = $M L^2 T^{-2}$。
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* **原文项**的单位是：质量 × (长度) × (长度/时间) = $M L^2 T^{-1}$。
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* 两者单位不匹配，这从根本上说明了原文的写法是错误的。
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### 结论
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您是对的，**原文中第一个 $\mathbf{r}_{c g}$ 上方缺少一个点 ($\dot{}$) 是一个笔误**。
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这篇论文的后续推导（例如推导出的离心力、科里奥利力等项）都是基于正确的动能表达式进行的。作者在进行复杂的推导时，很可能在撰写最终稿件时，在公式(9)这个中间步骤出现了一个小的排版错误。这是一个在学术文献中偶尔会发生的情况。
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[Timestamp: 2025/09/05 16:25:29]
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**user**: 公式10是如何得到的
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[Timestamp: 2025/09/05 16:32:09]
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**ai**: 好的，我们来分析一下公式(10)是如何得到的。
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公式(10)的表达式为：
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$$
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\delta Q=T_{g c n}\delta\phi+M_{p i t c h}\delta\beta+\int_{r}^{R}(\mathbf{f}^{\mathrm{{T}}}\cdot\delta\mathbf{r}_{e a}+M\delta(\theta+\beta))\mathrm{d}s\tag {10}
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$$
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这个公式并不是通过数学推导得出的，而是基于**虚功原理 (Principle of Virtual Work)** 对系统中所有**非保守力 (Nonconservative Forces)** 所做的功进行定义的。在分析力学中，特别是使用哈密顿原理（Hamilton's Principle）时，需要将力的作用以“功”的形式表述。
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$\delta Q$ 代表的是所有非保守的外部广义力所做的**虚功 (Variational Work)**。虚功的基本定义是：**力**与它作用点上**虚位移**的点积，或者是**力矩**与它作用物体上**虚转角**的乘积。
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公式(10)就是将系统中所有外部非保守力/力矩所做的虚功加起来得到的总和。我们来逐项分解：
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1. **$T_{g c n}\delta\phi$**:
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* $T_{g c n}$：根据文中的定义，这是**发电机扭矩 (generator torque)**。这是一个作用在整个风轮上的集中力矩。
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* $\phi$：这是风轮的**方位角 (azimuth angle)**。
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* $\delta\phi$：这是风轮方位角的**虚转角 (admissible variation or virtual rotation)**。
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* **物理意义**：这一项代表了发电机扭矩 $T_{g c n}$ 在风轮发生一个微小的虚转动 $\delta\phi$ 时所做的虚功。
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2. **$M_{p i t c h}\delta\beta$**:
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* $M_{p i t c h}$：这是**变桨力矩 (pitch moment)**，由变桨系统施加。
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* $\beta$：这是叶片的**变桨角 (pitch angle)**。
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* $\delta\beta$：这是变桨角的**虚转角**。
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* **物理意义**：这一项代表了变桨力矩 $M_{p i t c h}$ 在叶片发生一个微小的虚变桨转动 $\delta\beta$ 时所做的虚功。
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3. **$\int_{r}^{R}\mathbf{f}^{\mathrm{{T}}}\cdot\delta\mathbf{r}_{e a}\mathrm{d}s$**:
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* $\mathbf{f}$：这是作用在叶片弹性轴上的**分布外力**（例如气动力），文中定义为 $\mathbf{f}=\mathbf{T}_{\phi}^{\mathrm{T}}\mathbf{T}_{\beta}^{\mathrm{T}}[f_{u},f_{\nu},f_{w}]^{\mathrm{T}}$，即转换到惯性坐标系下的力矢量。
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* $\mathbf{r}_{e a}$：这是叶片**弹性轴的位置矢量**（见公式11）。
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* $\delta\mathbf{r}_{e a}$：这是弹性轴上一点的**虚位移 (virtual displacement)**。
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* **物理意义**：$\mathbf{f}^{\mathrm{{T}}}\cdot\delta\mathbf{r}_{e a}$ 是作用在叶片微元段 $ds$ 上的外力所做的虚功。通过从叶根 $r$ 到叶尖 $R$ 进行积分，就得到了整个叶片上所有分布外力所做的总虚功。
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4. **$\int_{r}^{R}M\delta(\theta+\beta)\mathrm{d}s$**:
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* $M$：这是作用在叶片上的**分布扭转力矩 (twisting moment)**。
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* $\theta$：这是叶片的**弹性扭转角**。
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* $\theta+\beta$：这是叶片截面相对于 $(\hat{x},\hat{y},\hat{z})$ 坐标系的**总扭转角**。
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* $\delta(\theta+\beta)$：这是叶片截面总扭转角的**虚转角**。
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* **物理意义**：$M\delta(\theta+\beta)$ 是作用在叶片微元段 $ds$ 上的扭转力矩所做的虚功。沿叶片全长积分，就得到了总的扭转虚功。
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**总结来说**，公式(10)是依据虚功原理，将系统中所有外部的、非保守的广义力（包括集中力矩 $T_{g c n}$ 和 $M_{p i t c h}$，以及分布力 $\mathbf{f}$ 和分布力矩 $M$）与它们对应的广义坐标的虚位移（$\delta\phi$, $\delta\beta$, $\delta\mathbf{r}_{e a}$, $\delta(\theta+\beta)$）相乘，然后求和（或积分），从而构建出总的虚功表达式 $\delta Q$。这个表达式是应用哈密顿原理推导运动方程的必要组成部分。
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[Timestamp: 2025/09/05 16:32:59]
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**user**: 解释方程12
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[Timestamp: 2025/09/05 16:52:46]
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**ai**: 好的，我们来详细解释一下方程(12)。
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方程(12)是描述风力机叶片**弯曲运动**的偏微分方程组，它包含了两个耦合的方程：
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* **方程(12a)** 描述了叶片在 $x$ 方向（通常称为**摆振 (edgewise)** 方向）的运动。
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* **方程(12b)** 描述了叶片在 $y$ 方向（通常称为**挥舞 (flapwise)** 方向）的运动。
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这两个方程是根据**扩展哈密顿原理 (extended Hamilton’s principle)** 推导出来的，其基本形式是：
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**惯性力 + 变桨作用引起的力 + 风轮转速变化引起的力 + 重力 + 弹性恢复力 + 风轮角加速度引起的力 = 外部作用力**
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由于论文中提到“方程(12a)和(12b)中的各项具有相似性，因此只讨论方程(12a)中的各项”，所以我们主要关注方程(12a)的物理意义，并理解其各项如何影响叶片的摆振运动。
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### 方程(12a)的结构：
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$m\big(\ddot{u}-\ddot{\theta}l_{c g}\sin(\overline{{\theta}})\big)+F_{u,1}+F_{u,2}+F_{u,3}+F_{u,4}+F_{u,5}=f_{u}+\Big((u^{\prime}+l_{p i}^{\prime})\int_{s}^{R}f_{w}\mathrm{d}\rho\Big)^{\prime}$
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其中：
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* $u(s,t)$ 是叶片在 $x$ 方向（摆振方向）的挠度（变形）。
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* $s$ 是沿叶片长度方向的距离。
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* $t$ 是时间。
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* $(\mathbf{})\equiv{\frac{\mathrm{d}}{\mathrm{d}t}}\,$ 表示对时间求导（速度或加速度）。
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* $(\mathbf{})^{'}\equiv\frac{\mathrm{d}\mathbf{\omega}}{\mathrm{d}s}$ 表示对空间位置 $s$ 求导。
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现在我们逐项解释：
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#### 左侧（LHS）- 惯性力、内部力及虚拟力：
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1. **$m\big(\ddot{u}-\ddot{\theta}l_{c g}\sin(\overline{{\theta}})\big)$ - 惯性力项**
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* $m$: 叶片单位长度的质量。
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* $\ddot{u}$: 叶片在 $x$ 方向的加速度，这是最直接的惯性力项（质量乘以加速度）。
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* $-\ddot{\theta}l_{c g}\sin(\overline{{\theta}})$: 这是一个**惯性耦合项**。
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* $\ddot{\theta}$: 叶片的扭转加速度。
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* $l_{c g}$: 质心到弹性轴的距离。
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* $\overline{{\theta}}$: 弦线的预扭角。
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* 这一项表示由于叶片扭转加速度，以及质心偏离弹性轴的距离，在 $x$ 方向上产生的惯性力。它体现了叶片弯曲运动与扭转运动之间的耦合。
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2. **$F_{u,1}\big(\ddot{\beta},\dot{\beta},\dot{\phi},\dot{\nu},\dot{\theta},u^{\prime},u,\nu,\theta,\beta\big)$ - 变桨作用的影响 (Pitch Action Influence)**
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* 这一项的详细表达式在**方程(13a)**中给出。
|
||||
* 它包含了由**变桨运动**（变桨角 $\beta$ 及其导数 $\dot{\beta}, \ddot{\beta}$）引起的各种**虚拟力 (fictitious forces)**，例如：
|
||||
* 与变桨角加速度 $\ddot{\beta}$ 相关的虚拟力。
|
||||
* 与变桨角速度 $\dot{\beta}$ 相关的虚拟离心力。
|
||||
* 与变桨角速度 $\dot{\beta}$ 和质心速度相关的科里奥利力。
|
||||
* 这些力是由于叶片所在的 $(x,y,z)$ 坐标系在变桨过程中绕其 $z$ 轴旋转而产生的非惯性效应。
|
||||
|
||||
3. **$F_{u,2}\big(\dot{\phi},\dot{u},\dot{\nu},u^{\prime},\nu,\theta,\beta\big)$ - 风轮转速的影响 (Rotor Speed Influence)**
|
||||
* 这一项的详细表达式在**方程(14a)**中给出。
|
||||
* 它包含了由**风轮旋转**（方位角 $\phi$ 的角速度 $\dot{\phi}$）引起的各种**虚拟力**，例如：
|
||||
* 与风轮角速度 $\dot{\phi}$ 相关的虚拟离心力（离心刚度）。
|
||||
* 与风轮角速度 $\dot{\phi}$ 和叶片变形速度相关的科里奥利力。
|
||||
* 这些力是由于叶片所在的 $(\hat{x},\hat{y},\hat{z})$ 坐标系（以及其后的 $(x,y,z)$ 坐标系）绕 $Y$ 轴旋转而产生的非惯性效应。
|
||||
|
||||
4. **$F_{u,3}(\phi,\beta,\theta,u^{\prime},\nu^{\prime})$ - 重力的影响 (Gravity Influence)**
|
||||
* 这一项的详细表达式在**方程(15a)**中给出。
|
||||
* 它包含了**重力**在 $x$ 方向的分量，以及重力与质心偏移、叶片变形等因素耦合产生的弯矩项。
|
||||
* 重力效应会随着风轮方位角 $\phi$ 和变桨角 $\beta$ 的变化而变化。
|
||||
|
||||
5. **$F_{u,4}(u^{\prime\prime},\nu^{\prime\prime},\theta)$ - 弹性恢复力 (Elastic Restoring Force)**
|
||||
* 这一项的详细表达式在**方程(16a)**中给出。
|
||||
* 它代表了叶片弯曲刚度产生的恢复力，试图将叶片拉回其未变形位置。
|
||||
* 它包含了叶片在 $x$ 方向的弯曲刚度项，以及与 $y$ 方向弯曲和扭转运动的耦合项。这表明叶片的弯曲和扭转运动是相互关联的。
|
||||
|
||||
6. **$F_{u,5}\big(\ddot{\phi},\beta\big)$ - 风轮角加速度的影响 (Rotor Angular Acceleration Influence)**
|
||||
* 这一项的详细表达式在**方程(17)**中给出。
|
||||
* 它表示由于风轮的角加速度 $\ddot{\phi}$ 引起的虚拟力，该力也取决于变桨角 $\beta$。
|
||||
|
||||
#### 右侧（RHS）- 外部作用力：
|
||||
|
||||
1. **$f_{u}$ - 直接外部力**
|
||||
* 这是直接作用在叶片弹性轴上，并沿 $x$ 方向的外部力，例如气动力的 $x$ 分量。
|
||||
|
||||
2. **$\Big((u^{\prime}+l_{p i}^{\prime})\int_{s}^{R}f_{w}\mathrm{d}\rho\Big)^{\prime}$ - $z$ 方向外部力引起的弯矩**
|
||||
* $f_{w}$: 作用在叶片弹性轴上，并沿 $z$ 方向的外部力。
|
||||
* $\int_{s}^{R}f_{w}\mathrm{d}\rho$: 表示从当前位置 $s$ 到叶尖 $R$ 的所有 $z$ 方向外部力的合力。
|
||||
* $(u^{\prime}+l_{p i}^{\prime})$: 与叶片在 $x-z$ 平面内的斜率有关。
|
||||
* 整个表达式的导数 `$(\cdot)'$` 表示这是一个由 $z$ 方向外部力在叶片剩余部分上产生的弯矩对 $s$ 的空间导数。
|
||||
|
||||
### 方程(12b)的解释：
|
||||
方程(12b)的各项与方程(12a)完全对称，只是将 $x$ 方向的挠度 $u$ 替换为 $y$ 方向的挠度 $\nu$，并将各项的投影方向从 $x$ 轴变为 $y$ 轴。例如，惯性项变为 $m\big(\ddot{\nu}+\ddot{\theta}l_{c g}\cos(\overline{{\theta}})\big)$，其中 $\ddot{\nu}$ 是 $y$ 方向的加速度，而扭转耦合项现在通过 $\cos(\overline{{\theta}})$ 投影到 $y$ 方向。
|
||||
|
||||
### 总结：
|
||||
方程(12)是一个复杂的偏微分方程组，它综合考虑了叶片在旋转环境中（包括变桨和风轮转速变化）的弯曲运动。它不仅包含了基本的惯性力和弹性恢复力，还引入了由于坐标系旋转（非惯性系）产生的虚拟力（如离心力和科里奥利力），以及重力、外部气动力等多种因素对叶片弯曲变形的影响。这些耦合项使得叶片的摆振和挥舞运动与扭转运动、变桨运动以及风轮整体运动紧密关联。
|
||||
[Timestamp: 2025/09/05 16:53:08]
|
||||
@ -6,7 +6,7 @@
|
||||
{"id":"82708a439812fdc7","type":"text","text":"# 10月已完成\n\n","x":-220,"y":134,"width":440,"height":560},
|
||||
{"id":"505acb3e6b119076","type":"text","text":"# 9月已完成\n","x":-700,"y":134,"width":440,"height":560},
|
||||
{"id":"30cb7486dc4e224c","type":"text","text":"# 11月已完成\n\n\n\n","x":260,"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\nP2 柔性部件 叶片、塔架变形算法 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\t\n- 梳理bladed动力学框架\n\t- 子结构文献阅读\n\t- 叶片模型建模 done\n- 共旋方法学习\n- DTU 变形量计算方法学习\n\n\nP1 线性化方法编写 搁置\nP1 气动、多体、控制、水动联调\nP1 湍流 气动 多体 控制联调\nP2 停机工况等调试\n\nP1 bladed对比--稳态，产出报告\n- 模态对比 两种描述方法不同，bladed方向更多，x y z deflection, x y z rotation，不好对比\n- 气动对比 aerodynamic info 轴向切向诱导因子，根部，尖部差距较大\n- 稳态变形量对比\n- 稳态变形量对比 -- steady power production loading、steady parked loading\n\nP1 稳态工况前端对接\n- 是否拆分成单独的bin\n- 如何接收参数 配置文件\n\n\nP2 如何优雅的存储、输出结果。\nP2 yaw 自由度再bug确认 已知原理了\n","x":-597,"y":-803,"width":453,"height":457},
|
||||
{"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\nP2 柔性部件 叶片、塔架变形算法 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\t\n- 梳理bladed动力学框架\n\t- 子结构文献阅读\n\t- 叶片模型建模 done\n- 共旋方法学习\n- DTU 变形量计算方法学习\n\n\nP1 线性化方法编写 搁置\nP1 气动、多体、控制、水动联调\nP1 湍流 气动 多体 控制联调\nP2 停机工况等调试\n\nP1 bladed对比--稳态，产出报告\n- 模态对比 两种描述方法不同，bladed方向更多，x y z deflection, x y z rotation，不好对比\n- 气动对比 aerodynamic info 轴向切向诱导因子，根部，尖部差距较大\n- 稳态变形量对比\n- 稳态变形量对比 -- steady power production loading、steady parked loading\n\nP1 稳态工况前端对接\n- 是否拆分成单独的bin no\n- 如何接收参数 配置文件 \n\n\nP2 如何优雅的存储、输出结果。\nP2 yaw 自由度再bug确认 已知原理了\n","x":-597,"y":-803,"width":453,"height":457},
|
||||
{"id":"86ab96a25a3bf82e","type":"text","text":" 湍流风+ 控制的联调，bladed也算一个算例\n- 加水动的联调\n- 8月份底完成这两个\n- 9月份完成停机等工况测试\n- 10月份明阳实际机型测试","x":580,"y":-803,"width":480,"height":220},
|
||||
{"id":"e355f33c92cf18ea","type":"text","text":"9月份定常计算对接前端\n非定常测试完也对接前端","x":580,"y":-500,"width":480,"height":100}
|
||||
],
|
||||
|
||||
BIN
工作总结/周报/周报93-郭翼泽.docx
Normal file
BIN
工作总结/周报/周报93-郭翼泽.docx
Normal file
Binary file not shown.
@ -166,7 +166,7 @@ The inertia of the system is described by a mass pr. length $m$ , a moment of ro
|
||||
系统的惯性由单位长度质量 $m$、叶片的单位长度转动惯量 $I_{c g}$ 以及描述轮毂、齿轮箱和发电机的转动惯量 $J_{g e n}$ 描述。使用叶片惯性的集中质量描述，而不是更一般的截面积分描述，大大降低了推导的复杂性。一般描述会引入额外的项，例如绕 $x$ 轴和 $y$ 轴的转动惯量，但这些项无论如何都相对较小。系统的动能由下式给出
|
||||
$$
|
||||
T\!=\!\frac{1}{2}J_{g c n}\dot{\phi}^{2}+\int_{r}^{R}\!\Big(\frac{1}{2}m\mathbf{r}_{c g}^{\mathrm{T}}\cdot\dot{\mathbf{r}}_{c g}+\frac{1}{2}I_{c g}\big(\dot{\beta}+\dot{\theta}\big)^{2}\Big)\mathrm{d}s\tag 9
|
||||
$$
|
||||
$$ 有笔误，应为$\dot{\mathbf{r}_{c g}}^{\mathrm{T}}$
|
||||
|
||||
where ${\dot{\beta}}+{\dot{\theta}}$ is the angular velocity of the blade section around the elastic axis.
|
||||
其中 ${\dot{\beta}}+{\dot{\theta}}$ 是叶片在弹性轴周围的角速度。
|
||||
@ -176,13 +176,13 @@ where ${\dot{\beta}}+{\dot{\theta}}$ is the angular velocity of the blade sectio
|
||||
The nonconservative forces are taken into account by describing the variational work done by them for any admissible variation:
|
||||
非保守力通过描述它们对任何容许变分所做的变分功来加以考虑：
|
||||
$$
|
||||
\delta Q=T_{g c n}\delta\phi+M_{p i t c h}\delta\beta+\int_{r}^{R}(\mathbf{f}^{\mathrm{{T}}}\cdot\delta\mathbf{r}_{e a}+M\delta(\theta+\beta))\mathrm{d}s\tag 10
|
||||
\delta Q=T_{g c n}\delta\phi+M_{p i t c h}\delta\beta+\int_{r}^{R}(\mathbf{f}^{\mathrm{{T}}}\cdot\delta\mathbf{r}_{e a}+M\delta(\theta+\beta))\mathrm{d}s\tag {10}
|
||||
$$
|
||||
|
||||
where $\mathbf{f}=\mathbf{T}_{\phi}^{\mathrm{T}}\mathbf{T}_{\beta}^{\mathrm{T}}[f_{u},f_{\nu},f_{w}]^{\mathrm{T}}$ and
|
||||
|
||||
$$
|
||||
\mathbf{r}_{e a}\!=\![\mathbf{I},\mathbf{J},\mathbf{K}]\mathbf{T}_{\phi}^{\mathrm{T}}\mathbf{T}_{\beta}^{\mathrm{T}}[l_{p i}+u,\nu,w_{0}+w_{1}]^{\mathrm{T}}\tag 11
|
||||
\mathbf{r}_{e a}\!=\![\mathbf{I},\mathbf{J},\mathbf{K}]\mathbf{T}_{\phi}^{\mathrm{T}}\mathbf{T}_{\beta}^{\mathrm{T}}[l_{p i}+u,\nu,w_{0}+w_{1}]^{\mathrm{T}}\tag {11}
|
||||
$$
|
||||
|
||||
is a position vector describing the elastic axis.
|
||||
@ -198,7 +198,7 @@ By demanding that any admissible variation of the action integral $\delta H\equi
|
||||
The equation of motion of the $x$ - and $y$ -directions becomes
|
||||
$x$ 和 $y$ 方向的运动方程变为
|
||||
$$
|
||||
\begin{array}{r l}&{m\big(\ddot{u}-\ddot{\theta}l_{c g}\sin(\overline{{\theta}})\big)+F_{u,1}\big(\ddot{\beta},\dot{\beta},\dot{\phi},\dot{\nu},\dot{\theta},u^{\prime},u,\nu,\theta,\beta\big)+F_{u,2}\big(\dot{\phi},\dot{u},\dot{\nu},u^{\prime},\nu,\theta,\beta\big)}\\ &{\quad+F_{u,3}(\phi,\beta,\theta,u^{\prime},\nu^{\prime})+F_{u,4}(u^{\prime\prime},\nu^{\prime\prime},\theta)+F_{u,5}\big(\ddot{\phi},\beta\big)=f_{u}+\Big((u^{\prime}+l_{p i}^{\prime})\int_{s}^{R}f_{w}\mathrm{d}\rho\Big)^{\prime}}\\ &{\quad m\big(\ddot{\nu}+\ddot{\theta}l_{c g}\cos(\overline{{\theta}})\big)+F_{\nu,1}\big(\ddot{\beta},\dot{\beta},\dot{\phi},\dot{\theta},\nu^{\prime},u,\nu,\theta,\beta\big)+F_{\nu,2}\big(\dot{\phi},\dot{u},\dot{\nu},u^{\prime},\nu,\theta,\beta\big)}\\ &{\quad+F_{\nu,3}\big(\phi,\beta,\theta,u^{\prime},\nu^{\prime}\big)+F_{\nu,4}\big(u^{\prime\prime},\nu^{\prime\prime},\theta\big)+F_{\nu,5}\big(\ddot{\phi},\beta\big)=f_{\nu}+\Big(\nu^{\prime}\int_{s}^{R}f_{w}\mathrm{d}\rho\Big)^{\prime}}\end{array}\tag 12
|
||||
\begin{array}{r l}&{m\big(\ddot{u}-\ddot{\theta}l_{c g}\sin(\overline{{\theta}})\big)+F_{u,1}\big(\ddot{\beta},\dot{\beta},\dot{\phi},\dot{\nu},\dot{\theta},u^{\prime},u,\nu,\theta,\beta\big)+F_{u,2}\big(\dot{\phi},\dot{u},\dot{\nu},u^{\prime},\nu,\theta,\beta\big)}\\ &{\quad+F_{u,3}(\phi,\beta,\theta,u^{\prime},\nu^{\prime})+F_{u,4}(u^{\prime\prime},\nu^{\prime\prime},\theta)+F_{u,5}\big(\ddot{\phi},\beta\big)=f_{u}+\Big((u^{\prime}+l_{p i}^{\prime})\int_{s}^{R}f_{w}\mathrm{d}\rho\Big)^{\prime}}\\ &{\quad m\big(\ddot{\nu}+\ddot{\theta}l_{c g}\cos(\overline{{\theta}})\big)+F_{\nu,1}\big(\ddot{\beta},\dot{\beta},\dot{\phi},\dot{\theta},\nu^{\prime},u,\nu,\theta,\beta\big)+F_{\nu,2}\big(\dot{\phi},\dot{u},\dot{\nu},u^{\prime},\nu,\theta,\beta\big)}\\ &{\quad+F_{\nu,3}\big(\phi,\beta,\theta,u^{\prime},\nu^{\prime}\big)+F_{\nu,4}\big(u^{\prime\prime},\nu^{\prime\prime},\theta\big)+F_{\nu,5}\big(\ddot{\phi},\beta\big)=f_{\nu}+\Big(\nu^{\prime}\int_{s}^{R}f_{w}\mathrm{d}\rho\Big)^{\prime}}\end{array}\tag {12}
|
||||
$$
|
||||
|
||||
The direction of the $x_{\mathrm{{}}}$ - and $y$ -axis can be swapped by changing the $\beta$ angle, hence the only differences between the terms in equations (12a) and (12b) are the directions of projection of the forces. In the following, the individual terms in equation (12) are shown and the physical interpretation of them is discussed. Because of the similarity between the terms from equations (12a) and (12b), only the terms from equation (12a) will be discussed. The influence of pitch action is described by
|
||||
@ -207,11 +207,11 @@ The direction of the $x_{\mathrm{{}}}$ - and $y$ -axis can be swapped by changin
|
||||
|
||||
|
||||
$$
|
||||
F_{u,1}\!=\!-\ddot{\beta}m\nu_{c g}-\dot{\beta}^{2}m u_{c g}-2\dot{\beta}m\dot{\nu}_{c g}+\left(T_{1}l_{c g}\cos(\overline{{{\theta}}})\right)^{\prime}+\left((u^{\prime}+l_{p i}^{\prime})\!\int_{s}^{R}T_{1}\!\mathrm{d}\rho\right)^{\prime}
|
||||
F_{u,1}\!=\!-\ddot{\beta}m\nu_{c g}-\dot{\beta}^{2}m u_{c g}-2\dot{\beta}m\dot{\nu}_{c g}+\left(T_{1}l_{c g}\cos(\overline{{{\theta}}})\right)^{\prime}+\left((u^{\prime}+l_{p i}^{\prime})\!\int_{s}^{R}T_{1}\!\mathrm{d}\rho\right)^{\prime}\tag {13a}
|
||||
$$
|
||||
|
||||
$$
|
||||
F_{u,1}\!=\!\ddot{\beta}m u_{c g}-\dot{\beta}^{2}m\nu_{c g}+2\dot{\beta}m\dot{u}_{c g}+\left(T_{1}l_{c g}\sin(\overline{{{\theta}}}\,)\right)^{\prime}+\left(\nu^{\prime}\!\int_{s}^{R}T_{1}\mathrm{d}\rho\right)^{\prime}
|
||||
F_{u,1}\!=\!\ddot{\beta}m u_{c g}-\dot{\beta}^{2}m\nu_{c g}+2\dot{\beta}m\dot{u}_{c g}+\left(T_{1}l_{c g}\sin(\overline{{{\theta}}}\,)\right)^{\prime}+\left(\nu^{\prime}\!\int_{s}^{R}T_{1}\mathrm{d}\rho\right)^{\prime}\tag {13b}
|
||||
$$
|
||||
|
||||
Where $u_{c g}=u+l_{p i}+l_{c g}\mathrm{cos}(\overline{{\theta}})-l_{c g}\theta\mathrm{sin}(\overline{{\theta}})$ and $\nu_{c g}=\nu+l_{c g}\mathrm{sin}(\overline{{\theta}})+l_{c g}\theta\mathrm{cos}(\overline{{\theta}})$ are the $x$ and $y$ coordinates of the center of gravity in the $(x,y,z)$ -frame, respectively, $T_{1}=2m\dot{\beta}\dot{\phi}((u+l_{p i})\mathrm{sin}(\beta)+\nu\mathrm{cos}(\beta)+l_{c g}\mathrm{sin}(\overline{{{\theta}}}+\beta))$ is the Coriolis force in the $z-$ -direction, associated with the angular velocity of the $(x,y,z)$ -frame about the $z$ -axis and about the $\hat{y}$ -axis. The first term in equation (13a) is the fictitious force\* in the $x$ -direction associated with the angular acceleration of the $(x,\,y,\,z)$ -frame about the $z-$ -axis. The second term is the fictitious centrifugal force associated with the angular velocity of the $(x,y,z)$ -frame about the $z$ -axis and the offset of $c g$ in the $x$ - direction. The third term is the Coriolis force associated with the angular velocity of the $(x,y,z)$ -frame about the $z$ -axis and the velocity of $c g$ in the $y$ -direction. The fourth term in equation (13a) is the spatial derivative of the moment caused by the offset of $c g$ and the Coriolis force $T_{1}$ . The last term is the bending moment caused by the Coriolis force $T_{1}$ on the remaining part of the blade, from this point to the tip. The influence from the rotor speed is described by
|
||||
@ -224,12 +224,11 @@ F_{u,2} &= -\dot{\phi}^{2}m\hat{u}_{cg}\cos(\beta) - \left[\dot{\phi}^{2}\left(m
|
||||
&\quad - \left(l_{cg}T_{2}\right)' \cos(\overline{\theta}) \\
|
||||
&\quad - 2\dot{\phi}m l_{cg}\left(\dot{u}'\cos(\overline{\theta}) + \dot{\nu}'\sin(\overline{\theta})\right)\cos(\beta) \\
|
||||
&\quad - \left[\Big(u' + l_{pi}'\Big)\int_{s}^{R}\left(\dot{\phi}^{2}m w_{0} + T_{2}\right)\mathrm{d}\rho\right]' \\
|
||||
|
||||
F_{\nu,2} &= \dot{\phi}^{2}m\hat{u}_{cg}\sin(\beta) - \left[\dot{\phi}^{2}\left(m l_{cg}w_{0}\sin(\overline{\theta}) + \theta\cos(\overline{\theta})\right)\right]' \\
|
||||
&\quad - \left(l_{cg}T_{2}\right)' \sin(\overline{\theta}) \\
|
||||
&\quad + 2\dot{\phi}m l_{cg}\left(\dot{u}'\cos(\overline{\theta}) + \dot{\nu}'\sin(\overline{\theta})\right)\sin(\beta) \\
|
||||
&\quad - \left[\nu'\int_{s}^{R}\left(\dot{\phi}^{2}m w_{0} + T_{2}\right)\mathrm{d}\rho\right]'
|
||||
\end{align}
|
||||
\end{align}\tag {14}
|
||||
$$
|
||||
|
||||
where $\hat{u}_{c g}=(u+l_{p i})\mathrm{cos}(\beta)-\nu\mathrm{sin}(\beta)+l_{c g}\mathrm{cos}(\overline{{{\theta}}}+\beta)-l_{c g}\theta\mathrm{sin}(\overline{{{\theta}}}+\beta)$ is the $\hat{x}$ coordinate of the center of gravity given in the $(\hat{x},\hat{y},\hat{z})$ -frame, $T_{2}=2m\dot{\phi}(\dot{u}\cos(\upbeta)-\dot{\nu}\sin(\beta))$ is the Coriolis force in the $z$ -direction associated with the rotation in the rotor plane and the velocity of $c g$ in the $\boldsymbol{\hat{x}}$ -direction The first term in equation (14a) is the fictitious centrifugal force associated with the rotation in the rotor plane and the offset of $c g$ in the $x_{\mathrm{{}}}$ -direction projected onto the $x_{\mathrm{{}}}$ -direction. The second and third terms in equation (14a) are the spatial derivative of the moment caused by the distance from $c g$ to $e a$ in the $x$ -direction and the fictitious centrifugal and the Coriolis force $T_{2}$ , respectively. The centrifugal force is associated with the rotation in the rotor plane and the offset of $c g$ from the center of rotation. The fourth term is the fictitious Coriolis force associated with the rotation of the blade in the rotor plane and the velocity of $c g$ in the $\hat{z}$ -direction The last term in equation (14a) is the bending moment from the fictitious centrifugal and the Coriolis force $T_{2}$ on the remaining part of the blade from this point to the tip. The influence from gravity is described by
|
||||
@ -237,21 +236,21 @@ where $\hat{u}_{c g}=(u+l_{p i})\mathrm{cos}(\beta)-\nu\mathrm{sin}(\beta)+l_{c
|
||||
其中 $\hat{u}_{c g}=(u+l_{p i})\mathrm{cos}(\beta)-\nu\mathrm{sin}(\beta)+l_{c g}\mathrm{cos}(\overline{{{\theta}}}+\beta)-l_{c g}\theta\mathrm{sin}(\overline{{{\theta}}}+\beta)$ 是在 $(\hat{x},\hat{y},\hat{z})$ 坐标系中给出的质心的 $\hat{x}$ 坐标，$T_{2}=2m\dot{\phi}(\dot{u}\cos(\upbeta)-\dot{\nu}\sin(\beta))$ 是与风轮平面内的旋转以及质心在 ${\hat{x}}$ 方向的速度相关联的 $z$ 方向的科里奥利力。方程 (14a) 中的第一项是与风轮平面内的旋转以及质心在 $x_{\mathrm{{}}}$ 方向的偏移投影到 $x_{\mathrm{{}}}$ 方向相关联的虚拟离心力。方程 (14a) 中的第二项和第三项分别是质心到 $e a$ 在 $x$ 方向的距离引起的力矩的空间导数，以及虚拟离心力和科里奥利力 $T_{2}$。离心力与风轮平面内的旋转以及质心偏离旋转中心的偏移相关联。第四项是与叶片在风轮平面内的旋转以及质心在 $\hat{z}$ 方向的速度相关联的虚拟科里奥利力。方程 (14a) 中的最后一项是虚拟离心力和科里奥利力 $T_{2}$ 作用在叶片从该点到叶尖的剩余部分上的弯矩。重力影响由下式描述：
|
||||
|
||||
$$
|
||||
\begin{array}{r l}&{F_{u,3}\!=\!m g\sin(\phi)\cos(\beta)\!+\!\left(\!\left(l_{c g}(u^{\prime}\!+\!l_{p i}^{\prime})\right)^{\prime}\cos(\overline{{\theta}})\cos(\beta)\!+\!\left(l_{c g}\nu^{\prime}\right)^{\prime}\sin(\overline{{\theta}})\cos(\beta)m g\sin(\phi)\!\right.}\\ &{\qquad\left.-\left(m l_{c g}\!\left(\cos(\overline{{\theta}})-\theta\sin(\overline{{\theta}})\right)\right)^{\prime}g\cos(\phi)\!+\!\left((u^{\prime}\!+\!l_{p i}^{\prime})\right)\int_{s}^{R}m g\cos(\phi)\mathrm{d}\rho\right)^{\prime}}\\ &{F_{u,3}\!=\!-\!m g\sin(\phi)\sin(\beta)\!+\!\left(\!\left(l_{c g}\left(u^{\prime}\!+\!l_{p i}^{\prime}\right)\right)^{\prime}\sin(\overline{{\theta}})\cos(\beta)\!-\!\left(l_{c g}\nu^{\prime}\right)^{\prime}\sin(\overline{{\theta}})\sin(\beta)\!\right)\!m g\sin(\phi)}\\ &{\qquad-\left(m l_{c g}\left(\sin(\overline{{\theta}})-\theta\cos(\overline{{\theta}})\right)\right)^{\prime}g\cos(\phi)\!+\!\left(\nu^{\prime}\!\int_{s}^{R}m g\cos(\phi)\mathrm{d}\rho\right)^{\prime}}\end{array}
|
||||
\begin{array}{r l}&{F_{u,3}\!=\!m g\sin(\phi)\cos(\beta)\!+\!\left(\!\left(l_{c g}(u^{\prime}\!+\!l_{p i}^{\prime})\right)^{\prime}\cos(\overline{{\theta}})\cos(\beta)\!+\!\left(l_{c g}\nu^{\prime}\right)^{\prime}\sin(\overline{{\theta}})\cos(\beta)m g\sin(\phi)\!\right.}\\ &{\qquad\left.-\left(m l_{c g}\!\left(\cos(\overline{{\theta}})-\theta\sin(\overline{{\theta}})\right)\right)^{\prime}g\cos(\phi)\!+\!\left((u^{\prime}\!+\!l_{p i}^{\prime})\right)\int_{s}^{R}m g\cos(\phi)\mathrm{d}\rho\right)^{\prime}}\\ &{F_{u,3}\!=\!-\!m g\sin(\phi)\sin(\beta)\!+\!\left(\!\left(l_{c g}\left(u^{\prime}\!+\!l_{p i}^{\prime}\right)\right)^{\prime}\sin(\overline{{\theta}})\cos(\beta)\!-\!\left(l_{c g}\nu^{\prime}\right)^{\prime}\sin(\overline{{\theta}})\sin(\beta)\!\right)\!m g\sin(\phi)}\\ &{\qquad-\left(m l_{c g}\left(\sin(\overline{{\theta}})-\theta\cos(\overline{{\theta}})\right)\right)^{\prime}g\cos(\phi)\!+\!\left(\nu^{\prime}\!\int_{s}^{R}m g\cos(\phi)\mathrm{d}\rho\right)^{\prime}}\end{array}\tag {15}
|
||||
$$
|
||||
|
||||
where the first term in equation (15a) is the $x$ -component of the gravity force. The second term is the spatial derivative of the moment caused by the $\hat{x}$ -component of the gravity force and the offset of $c g$ in the $z$ -direction. The third term is the spatial derivative of the moment caused by the distance between $c g$ and $e a$ in the $x$ -direction and the $z$ -component of the gravity force. The last term in equation (15a) is the bending moment from the $z-$ -component of the gravity force on the remaining part of the blade, from this point to the tip. The restoring force caused by the bending stiffness of the blade is described by
|
||||
其中，方程(15a)中的第一项是重力在$x$方向的分量。第二项是由重力在$\hat{x}$方向的分量和质心($cg$)在$z$方向的偏移引起的力矩的空间导数。第三项是由质心($cg$)和$ea$在$x$方向的距离以及重力在$z$方向的分量引起的力矩的空间导数。方程(15a)中的最后一项是重力在$z$方向的分量作用在叶片剩余部分（从该点到叶尖）上的弯矩。由叶片的弯曲刚度引起的恢复力由下式描述：
|
||||
|
||||
$$
|
||||
\begin{array}{r l}&{F_{u,4}=\big(E\big(I_{\xi}\cos^{2}\big(\tilde{\theta}\big)+I_{\eta}\sin^{2}\big(\tilde{\theta}\big)\big)u^{\prime\prime}\big)^{\prime\prime}+\big(E(I_{\xi}-I_{\eta})\cos\big(\tilde{\theta}\big)\sin\big(\tilde{\theta}\big)\nu^{\prime\prime}\big)^{\prime\prime}}\\ &{\qquad\qquad-\big(E(I_{\xi}-I_{\eta})\theta\big(u^{\prime\prime}\sin\big(2\tilde{\theta}\big)-\nu^{\prime\prime}\cos\big(2\tilde{\theta}\big)+I_{p i}^{\prime\prime}\sin\big(\tilde{\theta}\big)\cos\big(\tilde{\theta}\big)\big)\big)^{\prime\prime}}\\ &{F_{v,4}=\big(E\big(I_{\xi}\sin^{2}\big(\tilde{\theta}\big)+I_{\eta}\cos^{2}\big(\tilde{\theta}\big)\big)\nu^{\prime\prime}\big)^{\prime\prime}+\big(E(I_{\xi}-I_{\eta})\cos\big(\tilde{\theta}\big)\sin\big(\tilde{\theta}\big)u^{\prime\prime}\big)^{\prime\prime}}\\ &{\qquad+\left(E(I_{\xi}-I_{\eta})\theta\big(u^{\prime\prime}\cos\big(2\tilde{\theta}\big)+\nu^{\prime\prime}\sin\big(2\tilde{\theta}\big)\big)\right)^{\prime\prime}-\big(l_{p i}^{\prime\prime}\theta E(I_{\xi}\sin^{2}(\theta)+I_{\eta}\cos^{2}(\theta))\big)^{\prime\prime}}\end{array}
|
||||
\begin{array}{r l}&{F_{u,4}=\big(E\big(I_{\xi}\cos^{2}\big(\tilde{\theta}\big)+I_{\eta}\sin^{2}\big(\tilde{\theta}\big)\big)u^{\prime\prime}\big)^{\prime\prime}+\big(E(I_{\xi}-I_{\eta})\cos\big(\tilde{\theta}\big)\sin\big(\tilde{\theta}\big)\nu^{\prime\prime}\big)^{\prime\prime}}\\ &{\qquad\qquad-\big(E(I_{\xi}-I_{\eta})\theta\big(u^{\prime\prime}\sin\big(2\tilde{\theta}\big)-\nu^{\prime\prime}\cos\big(2\tilde{\theta}\big)+I_{p i}^{\prime\prime}\sin\big(\tilde{\theta}\big)\cos\big(\tilde{\theta}\big)\big)\big)^{\prime\prime}}\\ &{F_{v,4}=\big(E\big(I_{\xi}\sin^{2}\big(\tilde{\theta}\big)+I_{\eta}\cos^{2}\big(\tilde{\theta}\big)\big)\nu^{\prime\prime}\big)^{\prime\prime}+\big(E(I_{\xi}-I_{\eta})\cos\big(\tilde{\theta}\big)\sin\big(\tilde{\theta}\big)u^{\prime\prime}\big)^{\prime\prime}}\\ &{\qquad+\left(E(I_{\xi}-I_{\eta})\theta\big(u^{\prime\prime}\cos\big(2\tilde{\theta}\big)+\nu^{\prime\prime}\sin\big(2\tilde{\theta}\big)\big)\right)^{\prime\prime}-\big(l_{p i}^{\prime\prime}\theta E(I_{\xi}\sin^{2}(\theta)+I_{\eta}\cos^{2}(\theta))\big)^{\prime\prime}}\end{array}\tag {16}
|
||||
$$
|
||||
|
||||
where the first term is the bending stiffness in the $x$ -direction, the second term is the coupling to the $\nu$ -direction, and the last term is the coupling to the twist. The principle moments of inertia are given by $I_{\xi}=\int\!\int_{A}\!\eta^{2}\mathrm{d}\eta\mathrm{d}\xi$ and $I_{\eta}=\int\int_{A}\!\xi^{2}\mathrm{d}\eta\mathrm{d}\xi.$ . The effect of an angular acceleration of the rotor is described by
|
||||
其中第一项是 $x$ 方向的弯曲刚度，第二项是与 $\nu$ 方向的耦合，最后一项是与扭转的耦合。主惯性矩由 $I_{\xi}=\int\!\int_{A}\!\eta^{2}\mathrm{d}\eta\mathrm{d}\xi$ 和 $I_{\eta}=\int\int_{A}\!\xi^{2}\mathrm{d}\eta\mathrm{d}\xi$ 给出。风轮角加速度的影响由
|
||||
|
||||
$$
|
||||
F_{u,5}\!={m\ddot{\phi}w_{0}\cos(\beta)},\ \ \ F_{v,5}\!=\!-m{\ddot{\phi}w_{0}}\sin(\beta)
|
||||
F_{u,5}\!={m\ddot{\phi}w_{0}\cos(\beta)},\ \ \ F_{v,5}\!=\!-m{\ddot{\phi}w_{0}}\sin(\beta)\tag {17}
|
||||
$$
|
||||
|
||||
which is the fictitious angular acceleration of $c g$ associated with the angular acceleration of the $(x,y,z)$ -frame about the $Y.$ -axis. The right hand side of equations (12a) and (12b) describes the external forces, $f_{u}$ and $f_{\nu}$ are the forces in the $x\cdot$ - and $y$ -directions, respectively. The last term is the bending moment from the external force in the $z$ -direction on the remaining part of the blade, from this point to the tip.
|
||||
@ -262,45 +261,45 @@ which is the fictitious angular acceleration of $c g$ associated with the angula
|
||||
The equation of torsional motion is
|
||||
|
||||
$$
|
||||
\begin{array}{r l}&{\bigl(I_{c g}+m l_{c g}^{2}\bigr)\ddot{\theta}-m l_{c g}\bigl(\ddot{u}\sin(\overline{{\theta}})-\ddot{\nu}\cos(\overline{{\theta}})\bigr)+F_{\theta,1}\bigl(\dot{\phi},u^{\prime},\nu^{\prime},u,\nu,\beta\bigr)+F_{\theta,2}\bigl(\ddot{\beta},\dot{\beta},\dot{u},\dot{\nu},u,\nu\bigr)}\\ &{\quad+\,F_{\theta,3}\bigl(\ddot{\phi},\beta\bigr)+F_{\theta,4}\bigl(\phi,u^{\prime},\nu^{\prime},\theta,\beta\bigr)+F_{\theta,5}\bigl(u^{\prime\prime},\nu^{\prime\prime},\theta^{\prime}\bigr)+F_{\theta,6}\bigl(\theta^{\prime}\bigr)=M}\end{array}
|
||||
\begin{array}{r l}&{\bigl(I_{c g}+m l_{c g}^{2}\bigr)\ddot{\theta}-m l_{c g}\bigl(\ddot{u}\sin(\overline{{\theta}})-\ddot{\nu}\cos(\overline{{\theta}})\bigr)+F_{\theta,1}\bigl(\dot{\phi},u^{\prime},\nu^{\prime},u,\nu,\beta\bigr)+F_{\theta,2}\bigl(\ddot{\beta},\dot{\beta},\dot{u},\dot{\nu},u,\nu\bigr)}\\ &{\quad+\,F_{\theta,3}\bigl(\ddot{\phi},\beta\bigr)+F_{\theta,4}\bigl(\phi,u^{\prime},\nu^{\prime},\theta,\beta\bigr)+F_{\theta,5}\bigl(u^{\prime\prime},\nu^{\prime\prime},\theta^{\prime}\bigr)+F_{\theta,6}\bigl(\theta^{\prime}\bigr)=M}\end{array}\tag {18}
|
||||
$$
|
||||
|
||||
where the rotor speed leads to the fictitious centrifugal forces:
|
||||
在风轮转速导致虚拟离心力的情况下：
|
||||
|
||||
$$
|
||||
F_{\theta,1}\!=\!m l_{c g}{\dot{\phi}}^{2}{\hat{u}}_{c g}\sin({\overline{{\theta}}}+\beta)+m l_{c g}w_{0}\phi^{2}\big(\nu^{\prime}\cos({\overline{{\theta}}})\!-\!(u^{\prime}\!+\!l_{p i}^{\prime})\sin({\overline{{\theta}}})\big)
|
||||
F_{\theta,1}\!=\!m l_{c g}{\dot{\phi}}^{2}{\hat{u}}_{c g}\sin({\overline{{\theta}}}+\beta)+m l_{c g}w_{0}\phi^{2}\big(\nu^{\prime}\cos({\overline{{\theta}}})\!-\!(u^{\prime}\!+\!l_{p i}^{\prime})\sin({\overline{{\theta}}})\big)\tag {19}
|
||||
$$
|
||||
where the first term is associated with the offset of $c g$ in the $\hat{x}$ -direction and the second is associated with the distance from the center of rotation to $c g$ . The influence from the pitch action is described by
|
||||
其中第一项与 $c g$ 在 $\hat{x}$ 方向上的偏移量相关联，而第二项与从旋转中心到 $c g$ 的距离相关联。变桨作用的影响由下式描述：
|
||||
|
||||
$$
|
||||
\begin{array}{l}{{F_{\theta,2}=(I_{c g}+m l_{c g}^{2})\ddot{\beta}-\ddot{\beta}m l_{c g}(u\cos(\overline{{{\theta}}})+\nu\sin(\overline{{{\theta}}}))+m l_{c g}\dot{\beta}^{2}\big((u+l_{p i})\sin(\overline{{{\theta}}})-\nu\cos(\overline{{{\theta}}})\big)}}\\ {{\mathrm{}}}\\ {{\mathrm{}\qquad+\,2m l_{c g}\dot{\beta}(\dot{u}\cos(\overline{{{\theta}}})+\dot{\nu}\sin(\overline{{{\theta}}}))}}\end{array}
|
||||
\begin{array}{l}{{F_{\theta,2}=(I_{c g}+m l_{c g}^{2})\ddot{\beta}-\ddot{\beta}m l_{c g}(u\cos(\overline{{{\theta}}})+\nu\sin(\overline{{{\theta}}}))+m l_{c g}\dot{\beta}^{2}\big((u+l_{p i})\sin(\overline{{{\theta}}})-\nu\cos(\overline{{{\theta}}})\big)}}\\ {{\mathrm{}}}\\ {{\mathrm{}\qquad+\,2m l_{c g}\dot{\beta}(\dot{u}\cos(\overline{{{\theta}}})+\dot{\nu}\sin(\overline{{{\theta}}}))}}\end{array}\tag {20}
|
||||
$$
|
||||
|
||||
where the first term is the fictitious angular acceleration associated with the angular acceleration of the $(x,y,$ , $z)$ -frame about the $z$ -axis. The second term is the fictitious centrifugal force associated with the rotation of the $(x,y,z)$ -frame about the $z-$ -axis. The last term is the fictitious Coriolis force associated with the rotation of the $(x,y,z)$ -frame about the $z$ -axis and the velocity of $c g$ in the chord direction. The acceleration of the rotor leads to the following term:
|
||||
其中第一项是与 $(x,y,z)$ 坐标系绕 $z$ 轴的角加速度相关的虚拟角加速度。第二项是与 $(x,y,z)$ 坐标系绕 $z$ 轴旋转相关的虚拟离心力。最后一项是与 $(x,y,z)$ 坐标系绕 $z$ 轴的旋转以及 $cg$ 在弦向的速度相关的虚拟科里奥利力。风轮的加速度导致以下项：
|
||||
$$
|
||||
F_{\theta,3}\,{=}\,{-}m\ddot{\phi}w_{0}l_{c g}\sin(\overline{{\theta}}+\beta)
|
||||
F_{\theta,3}\,{=}\,{-}m\ddot{\phi}w_{0}l_{c g}\sin(\overline{{\theta}}+\beta)\tag {21}
|
||||
$$
|
||||
|
||||
which is the fictitious angular acceleration of $c g$ associated with the angular acceleration of the $(x,y,z)$ -frame about the $Y\cdot$ -axis. The effect of gravity is described by
|
||||
这是与 $(x,y,z)$ 坐标系绕 $Y$ 轴的角加速度相关的重心 $cg$ 的虚拟角加速度。重力效应由
|
||||
$$
|
||||
F_{\theta,4}=-l_{c g}\big(\mathrm{sin}\big(\beta+\overline{{\theta}}\big)+\theta\cos\big(\beta+\overline{{\theta}}\big)m g\sin(\phi)+l_{c g}\big(\nu^{\prime}\cos\big(\overline{{\theta}}\big)-(u^{\prime}+l_{p i}^{\prime})\sin(\overline{{\theta}}\big)\big)m g\cos(\phi)
|
||||
F_{\theta,4}=-l_{c g}\big(\mathrm{sin}\big(\beta+\overline{{\theta}}\big)+\theta\cos\big(\beta+\overline{{\theta}}\big)m g\sin(\phi)+l_{c g}\big(\nu^{\prime}\cos\big(\overline{{\theta}}\big)-(u^{\prime}+l_{p i}^{\prime})\sin(\overline{{\theta}}\big)\big)m g\cos(\phi)\tag {22}
|
||||
$$
|
||||
|
||||
where the first term is the twisting moment caused by the $\hat{x}$ -component of the gravity force and the distance between $c g$ and $e a$ in the ${\hat{y}}.$ -direction. The last term is the twisting moment caused by the distance between $c g$ and $e a$ and the $z-$ -component of the gravity force projected onto the cross section of the deformed blade. The elastic coupling between the bending and twisting of the blade is described by
|
||||
其中第一项是由重力在$\hat{x}$方向的分量以及$cg$和$ea$在${\hat{y}}$方向上的距离引起的扭转力矩。最后一项是由$cg$和$ea$之间的距离以及投影到变形叶片截面上的重力在$z$方向的分量引起的扭转力矩。叶片的弯曲和扭转之间的弹性耦合由下式描述。
|
||||
$$
|
||||
\begin{array}{r l}&{F_{\theta,5}\!=\!-\!\bigg(E I_{\eta\eta\xi}\big(\tilde{\theta}+\theta\big)^{\prime}\big(u^{\prime\prime}\sin(\overline{{\theta}})\!-\!\nu^{\prime\prime}\cos(\theta)\big)\bigg)^{\prime}\!+\!\bigg(E I_{\eta\xi\xi}\big(\tilde{\theta}+\theta\big)^{\prime}\big(u^{\prime\prime}\cos(\tilde{\theta})\!+\nu^{\prime\prime}\sin(\tilde{\theta})\big)\bigg)^{\prime}}\\ &{\qquad-\left(E I_{\xi}-E I_{\eta}\right)\!\big((u^{\prime\prime2}-\nu^{\prime\prime2})\cos(\tilde{\theta})\sin(\tilde{\theta})\!-u^{\prime\prime}\nu^{\prime\prime}\cos(2\tilde{\theta})\big)}\\ &{\qquad-\left.E I_{\xi}l_{p i}^{\prime\prime}\big(u^{\prime\prime}\cos(\tilde{\theta})\!+\nu^{\prime\prime}\sin(\tilde{\theta})\big)\sin(\tilde{\theta})\!+E I_{\eta}l_{p i}^{\prime\prime}\big(u^{\prime\prime}\sin(\tilde{\theta})\!-\!\nu^{\prime\prime}\cos(\tilde{\theta})\big)\right)\!\cos(\tilde{\theta})}\end{array}
|
||||
\begin{array}{r l}&{F_{\theta,5}\!=\!-\!\bigg(E I_{\eta\eta\xi}\big(\tilde{\theta}+\theta\big)^{\prime}\big(u^{\prime\prime}\sin(\overline{{\theta}})\!-\!\nu^{\prime\prime}\cos(\theta)\big)\bigg)^{\prime}\!+\!\bigg(E I_{\eta\xi\xi}\big(\tilde{\theta}+\theta\big)^{\prime}\big(u^{\prime\prime}\cos(\tilde{\theta})\!+\nu^{\prime\prime}\sin(\tilde{\theta})\big)\bigg)^{\prime}}\\ &{\qquad-\left(E I_{\xi}-E I_{\eta}\right)\!\big((u^{\prime\prime2}-\nu^{\prime\prime2})\cos(\tilde{\theta})\sin(\tilde{\theta})\!-u^{\prime\prime}\nu^{\prime\prime}\cos(2\tilde{\theta})\big)}\\ &{\qquad-\left.E I_{\xi}l_{p i}^{\prime\prime}\big(u^{\prime\prime}\cos(\tilde{\theta})\!+\nu^{\prime\prime}\sin(\tilde{\theta})\big)\sin(\tilde{\theta})\!+E I_{\eta}l_{p i}^{\prime\prime}\big(u^{\prime\prime}\sin(\tilde{\theta})\!-\!\nu^{\prime\prime}\cos(\tilde{\theta})\big)\right)\!\cos(\tilde{\theta})}\end{array}\tag {23}
|
||||
$$
|
||||
|
||||
where $I_{\eta\eta\xi}=\ \int\!\int_{A}\!\eta(\eta^{2}\,+\,\xi^{2})\mathrm{d}\eta\mathrm{d}\xi$ and $I_{\eta\xi\xi}=\ \int\int_{A}^{}\!\xi(\eta^{2}\,+\,\xi^{2})\mathrm{d}\eta\mathrm{d}\xi$ . The restoring force caused by torsional stiffness is given by
|
||||
其中，$I_{\eta\eta\xi}=\ \int\!\int_{A}\!\eta(\eta^{2}\,+\,\xi^{2})\mathrm{d}\eta\mathrm{d}\xi$ 且 $I_{\eta\xi\xi}=\ \int\int_{A}^{}\!\xi(\eta^{2}\,+\,\xi^{2})\mathrm{d}\eta\mathrm{d}\xi$ 。由扭转刚度引起的回复力为：
|
||||
|
||||
$$
|
||||
F_{\theta,6}\,{=}\,{-}(G J(\theta^{\prime}\,{+}\,\nu^{\prime}(u^{\prime\prime}\,{+}\,l_{p i}^{\prime\prime})))^{\prime}
|
||||
F_{\theta,6}\,{=}\,{-}(G J(\theta^{\prime}\,{+}\,\nu^{\prime}(u^{\prime\prime}\,{+}\,l_{p i}^{\prime\prime})))^{\prime}\tag {24}
|
||||
$$
|
||||
|
||||
where the polar moment of inertia is $J=\int\int_{\cal A}{\left(\eta^{2}+\xi^{2}\right)}d\eta d\xi$ . The right hand side describes the external moment on the blade $M$ .
|
||||
@ -311,7 +310,7 @@ where the polar moment of inertia is $J=\int\int_{\cal A}{\left(\eta^{2}+\xi^{2}
|
||||
The boundary conditions for the root of the blade are given by the geometric constraints:
|
||||
叶片根部的边界条件由几何约束给出：
|
||||
$$
|
||||
u(0,t)\!=\!u^{\prime}(0,t)\!=\!\nu(0,t)\!=\!\nu^{\prime}(0,t)\!=\!\theta(0,t)\!=\!0
|
||||
u(0,t)\!=\!u^{\prime}(0,t)\!=\!\nu(0,t)\!=\!\nu^{\prime}(0,t)\!=\!\theta(0,t)\!=\!0\tag {25}
|
||||
$$
|
||||
|
||||
because the coordinate frame used to describe the blade follows the root of the blade.
|
||||
@ -321,7 +320,7 @@ The boundary conditions for the tip of the blade are determined by the boundary
|
||||
|
||||
叶尖的边界条件由要求作用量积分的任何容许变分为零所推导出的边界条件方程确定。边界条件为：
|
||||
$$
|
||||
\begin{array}{l}{{u^{\prime\prime}(R,t)=\nu^{\prime\prime}(R,t)\,{=}\,\theta^{\prime}(R,t)=0}}\\ {{E I_{\xi}I_{\eta}u^{\prime\prime\prime}(R,t)=m l_{c g}\big(\dot{\phi}^{2}w_{0}-g\cos(\phi)\big)\big(I_{\eta}\sin\!\big(\tilde{\theta}-\overline{{\theta}}\big)\!\sin\!\big(\overline{{\theta}}\big)+I_{\xi}\cos\!\big(\tilde{\theta}-\overline{{\theta}}\big)\!\cos\!\big(\tilde{\theta}\big)\big)}}\\ {{E I_{\xi}I_{\eta}\nu^{\prime\prime\prime}(R,t)=m l_{c g}\big(\dot{\phi}^{2}w_{0}-g\cos(\phi)\big)\big(I_{\eta}\cos\!\big(\tilde{\theta}-\overline{{\theta}}\big)\!\sin\!\big(\tilde{\theta}\big)-I_{\xi}\sin\!\big(\tilde{\theta}-\overline{{\theta}}\big)\!\cos\!\big(\tilde{\theta}\big)\big)}}\end{array}
|
||||
\begin{array}{l}{{u^{\prime\prime}(R,t)=\nu^{\prime\prime}(R,t)\,{=}\,\theta^{\prime}(R,t)=0}}\\ {{E I_{\xi}I_{\eta}u^{\prime\prime\prime}(R,t)=m l_{c g}\big(\dot{\phi}^{2}w_{0}-g\cos(\phi)\big)\big(I_{\eta}\sin\!\big(\tilde{\theta}-\overline{{\theta}}\big)\!\sin\!\big(\overline{{\theta}}\big)+I_{\xi}\cos\!\big(\tilde{\theta}-\overline{{\theta}}\big)\!\cos\!\big(\tilde{\theta}\big)\big)}}\\ {{E I_{\xi}I_{\eta}\nu^{\prime\prime\prime}(R,t)=m l_{c g}\big(\dot{\phi}^{2}w_{0}-g\cos(\phi)\big)\big(I_{\eta}\cos\!\big(\tilde{\theta}-\overline{{\theta}}\big)\!\sin\!\big(\tilde{\theta}\big)-I_{\xi}\sin\!\big(\tilde{\theta}-\overline{{\theta}}\big)\!\cos\!\big(\tilde{\theta}\big)\big)}}\end{array}\tag {26}
|
||||
$$
|
||||
|
||||
If $l_{c g}(R)\neq0$ , the boundary conditions for the tip are functions of rotor speed $\dot{\phi}$ and rotor position $\phi$ and therefore time-varying. This is because an offset of the center of gravity from the elastic axis at the blade tip leads to a bending moment at the tip, caused by the gravity and centrifugal force. Most modern wind turbine blades, however, are tapered at the tip, leading to $l_{c g}(R)/R<<\varepsilon_{*}$ , making the time variation of the boundary conditions negligible.
|
||||
@ -331,37 +330,37 @@ If $l_{c g}(R)\neq0$ , the boundary conditions for the tip are functions of roto
|
||||
The equation of motion for the pitch angle is
|
||||
|
||||
$$
|
||||
\begin{array}{l}{{\displaystyle\int_{r}^{R}\big(\big(I_{c g}+m I_{c g}^{2}\big)\big(\ddot{\theta}+\ddot{\beta}\big)+m\big(l_{p i}^{2}+2l_{c g}l_{p i}\cos(\overline{{\theta}})\big)\ddot{\beta}-m\ddot{u}l_{c g}\sin(\overline{{\theta}})+m\ddot{\nu}\big(l_{p i}+l_{c g}\cos(\overline{{\theta}})\big)\big)\mathrm{d}s+F_{\beta,1}\big(\ddot{\beta},\dot{u},u\big)}\ ~}\\ {{\displaystyle~~~+F_{\beta,2}\big(\dot{\beta},\dot{\nu},\nu\big)+F_{\beta,3}\big(\dot{\phi},u,\nu,\beta\big)+F_{\beta,4}\big(\ddot{\phi},u,\nu,\beta\big)+F_{\beta,5}\big(u,\nu,\theta,\beta,\phi\big)+F_{\beta,6}\big(\ddot{\beta},\ddot{u},\ddot{\nu},u,\nu\big)}\ ~}\\ {{\displaystyle=M_{p i r c h}+\int_{r}^{R}\big(M+f_{\nu}\big(u+l_{p i}\big)-f_{u}\nu\big)\mathrm{d}s}}\end{array}
|
||||
\begin{array}{l}{{\displaystyle\int_{r}^{R}\big(\big(I_{c g}+m I_{c g}^{2}\big)\big(\ddot{\theta}+\ddot{\beta}\big)+m\big(l_{p i}^{2}+2l_{c g}l_{p i}\cos(\overline{{\theta}})\big)\ddot{\beta}-m\ddot{u}l_{c g}\sin(\overline{{\theta}})+m\ddot{\nu}\big(l_{p i}+l_{c g}\cos(\overline{{\theta}})\big)\big)\mathrm{d}s+F_{\beta,1}\big(\ddot{\beta},\dot{u},u\big)}\ ~}\\ {{\displaystyle~~~+F_{\beta,2}\big(\dot{\beta},\dot{\nu},\nu\big)+F_{\beta,3}\big(\dot{\phi},u,\nu,\beta\big)+F_{\beta,4}\big(\ddot{\phi},u,\nu,\beta\big)+F_{\beta,5}\big(u,\nu,\theta,\beta,\phi\big)+F_{\beta,6}\big(\ddot{\beta},\ddot{u},\ddot{\nu},u,\nu\big)}\ ~}\\ {{\displaystyle=M_{p i r c h}+\int_{r}^{R}\big(M+f_{\nu}\big(u+l_{p i}\big)-f_{u}\nu\big)\mathrm{d}s}}\end{array}\tag {27}
|
||||
$$
|
||||
|
||||
where
|
||||
|
||||
$$
|
||||
F_{\beta,1}\big(\dot{\beta},\dot{\nu},u\big)=2\dot{\beta}\int_{r}^{R}m\dot{u}u_{c g}{d}s,\quad F_{\beta,2}\big(\dot{\beta},\dot{\nu},u\big)=2\dot{\beta}\int_{r}^{R}m\dot{\nu}\nu_{c g}{d}s
|
||||
F_{\beta,1}\big(\dot{\beta},\dot{\nu},u\big)=2\dot{\beta}\int_{r}^{R}m\dot{u}u_{c g}{d}s,\quad F_{\beta,2}\big(\dot{\beta},\dot{\nu},u\big)=2\dot{\beta}\int_{r}^{R}m\dot{\nu}\nu_{c g}{d}s\tag {28}
|
||||
$$
|
||||
|
||||
are the moments caused by the fictitious Coriolis force associated with the relative velocity of the blade and rotation of the $(x,y,z)$ -frame about the $z_{i}$ -axis. The effect of the fictitious centrifugal force associated with the rotation of the $(x,y,z)$ -frame about the $\hat{y}_{}$ -axis is described by
|
||||
这些是由于与叶片相对速度和 $(x,y,z)$ -坐标系绕 $z_{i}$ -轴旋转相关的虚构科里奥利力所造成的。与 $(x,y,z)$ -坐标系绕 $\hat{y}_{}$ -轴旋转相关的虚构离心力效应由以下内容描述：
|
||||
$$
|
||||
F_{\beta,3}\big(\dot{\phi},u,\nu,\beta\big)\!=\dot{\phi}^{2}\int_{r}^{R}m\hat{u}_{c g}\hat{\nu}_{c g}\mathrm{d}s
|
||||
F_{\beta,3}\big(\dot{\phi},u,\nu,\beta\big)\!=\dot{\phi}^{2}\int_{r}^{R}m\hat{u}_{c g}\hat{\nu}_{c g}\mathrm{d}s\tag {29}
|
||||
$$
|
||||
|
||||
where $\hat{\nu}_{c g}=(u+l_{p i})\mathrm{sin}(\beta)+\nu\mathrm{cos}(\beta)+l_{c g}\mathrm{sin}(\overline{{{\theta}}}+\beta)$ is the $\hat{y}$ coordinate of the center of gravity in the (ˆx,ˆy,ˆz)- frame. The effect of an angular acceleration of the $(x,y,z)$ -frame about the $\hat{y}$ -axis is described by
|
||||
其中 $\hat{\nu}_{c g}=(u+l_{p i})\mathrm{sin}(\beta)+\nu\mathrm{cos}(\beta)+l_{c g}\mathrm{sin}(\overline{{{\theta}}}+\beta)$ 是重心在 (ˆx,ˆy,ˆz) 坐标系中的 $\hat{y}$ 坐标。 $(x,y,z)$ 坐标系绕 $\hat{y}$ 轴的角加速度的影响由下式描述。
|
||||
$$
|
||||
F_{\beta,4}\big(\ddot{\phi},u,\nu,\beta\big)\!=-\ddot{\phi}\!\!\int_{r}^{R}m w_{0}\hat{\nu}_{c g}\mathrm{d}s
|
||||
F_{\beta,4}\big(\ddot{\phi},u,\nu,\beta\big)\!=-\ddot{\phi}\!\!\int_{r}^{R}m w_{0}\hat{\nu}_{c g}\mathrm{d}s\tag {30}
|
||||
$$
|
||||
|
||||
The gravity force is described by
|
||||
|
||||
$$
|
||||
F_{\beta,5}(u,\nu,\beta,\phi)=g\sin(\phi)\!\!\int_{r}^{R}m\hat{\nu}_{c g}\mathrm{d}s
|
||||
F_{\beta,5}(u,\nu,\beta,\phi)=g\sin(\phi)\!\!\int_{r}^{R}m\hat{\nu}_{c g}\mathrm{d}s\tag {31}
|
||||
$$
|
||||
|
||||
and
|
||||
|
||||
$$
|
||||
F_{\beta,6}(\vec{\beta},\ddot{u},\ddot{\nu},u,\nu)=\ddot{\beta}\int_{r}^{R}m(u^{2}+\nu^{2}+2l_{c g}(u\cos(\overline{{\theta}})+\nu\sin(\overline{{\theta}}))+2l_{p i}u)\mathrm{d}s+\int_{r}^{R}m(\ddot{\nu}u-\ddot{u}\nu)\mathrm{d}s
|
||||
F_{\beta,6}(\vec{\beta},\ddot{u},\ddot{\nu},u,\nu)=\ddot{\beta}\int_{r}^{R}m(u^{2}+\nu^{2}+2l_{c g}(u\cos(\overline{{\theta}})+\nu\sin(\overline{{\theta}}))+2l_{p i}u)\mathrm{d}s+\int_{r}^{R}m(\ddot{\nu}u-\ddot{u}\nu)\mathrm{d}s\tag {32}
|
||||
$$
|
||||
|
||||
is nonlinear inertia.
|
||||
@ -375,31 +374,31 @@ If the pitch angle is prescribed or given by an external model, equation (27) ca
|
||||
Assuming a rigid drive train and no gearing, the rotor position is described by
|
||||
假设传动链为刚性且无齿轮传动，风轮位置由
|
||||
$$
|
||||
\begin{array}{l}{{\displaystyle J_{g c n}{\ddot{\phi}}+\int_{r}^{R}m w_{0}\big(w_{0}{\ddot{\phi}}+u\cos(\beta)-{\ddot{\nu}}\sin(\beta)\big)\mathrm{d}s}\ ~}\\ {{\displaystyle\quad+\;F_{\phi,1}\big({\dot{\beta}},u,\nu,\beta\big)+F_{\phi,2}\big({\dot{\beta}},{\dot{u}},{\dot{\nu}},{\dot{\theta}},\beta\big)+F_{\phi,3}\big(u,\phi\big)+F_{\phi,4}\big({\ddot{\beta}},u,\nu,\beta\big)\ ~}}\\ {{\displaystyle=T_{g c n}+\int_{r}^{R}\big(\big(f_{u}\cos(\beta)-f_{\nu}\sin(\beta)\big)w_{0}+f_{w}\big(\nu\sin(\beta)-\big(u+l_{p i}\big)\cos(\beta)\big)\big)\mathrm{d}s}\ ~}\end{array}
|
||||
\begin{array}{l}{{\displaystyle J_{g c n}{\ddot{\phi}}+\int_{r}^{R}m w_{0}\big(w_{0}{\ddot{\phi}}+u\cos(\beta)-{\ddot{\nu}}\sin(\beta)\big)\mathrm{d}s}\ ~}\\ {{\displaystyle\quad+\;F_{\phi,1}\big({\dot{\beta}},u,\nu,\beta\big)+F_{\phi,2}\big({\dot{\beta}},{\dot{u}},{\dot{\nu}},{\dot{\theta}},\beta\big)+F_{\phi,3}\big(u,\phi\big)+F_{\phi,4}\big({\ddot{\beta}},u,\nu,\beta\big)\ ~}}\\ {{\displaystyle=T_{g c n}+\int_{r}^{R}\big(\big(f_{u}\cos(\beta)-f_{\nu}\sin(\beta)\big)w_{0}+f_{w}\big(\nu\sin(\beta)-\big(u+l_{p i}\big)\cos(\beta)\big)\big)\mathrm{d}s}\ ~}\end{array}\tag {33}
|
||||
$$
|
||||
|
||||
The effect of the fictitious centrifugal force associated with rotation of the $(x,\,y,\,z)$ -frame about the $z$ -axis is described by
|
||||
与 $(x,\,y,\,z)$ 坐标系绕 $z$ 轴旋转相关的虚构离心力的影响由下式描述：
|
||||
$$
|
||||
F_{\phi,1}\big(\dot{\beta},u,\nu,\beta\big)\!=-\dot{\beta}^{2}\!\int_{r}^{R}m w_{0}\hat{u}_{c g}\mathrm{d}s
|
||||
F_{\phi,1}\big(\dot{\beta},u,\nu,\beta\big)\!=-\dot{\beta}^{2}\!\int_{r}^{R}m w_{0}\hat{u}_{c g}\mathrm{d}s\tag {34}
|
||||
$$
|
||||
|
||||
and
|
||||
|
||||
$$
|
||||
F_{\phi,2}\big(\dot{\beta},\dot{u},\dot{\nu},\dot{\theta},\beta\big)\!=-2\dot{\beta}\!\!\int_{r}^{R}\!m w_{0}(\dot{u}\sin(\beta)\!+\!\dot{\nu}\cos(\beta))\mathrm{d}s
|
||||
F_{\phi,2}\big(\dot{\beta},\dot{u},\dot{\nu},\dot{\theta},\beta\big)\!=-2\dot{\beta}\!\!\int_{r}^{R}\!m w_{0}(\dot{u}\sin(\beta)\!+\!\dot{\nu}\cos(\beta))\mathrm{d}s\tag {35}
|
||||
$$
|
||||
|
||||
describes the fictitious Coriolis force associated with the rotation of the $(x,y,z)$ -frame about the $z$ -axis and the relative velocity of the blade. The effect of gravity is described by
|
||||
描述了由 $(x,y,z)$ 坐标系绕 $z$ 轴的旋转以及叶片的相对速度所产生的虚构科里奥利力。重力效应由
|
||||
$$
|
||||
F_{\phi,3}(u,\phi)=g\sin(\phi)\!\!\int_{r}^{R}m w_{0}\mathrm{d}s+g\cos(\phi)\!\!\int_{r}^{R}m{\hat{u}}_{c g}\mathrm{d}s
|
||||
F_{\phi,3}(u,\phi)=g\sin(\phi)\!\!\int_{r}^{R}m w_{0}\mathrm{d}s+g\cos(\phi)\!\!\int_{r}^{R}m{\hat{u}}_{c g}\mathrm{d}s\tag {36}
|
||||
$$
|
||||
|
||||
and
|
||||
|
||||
$$
|
||||
F_{\phi,4}\big(\ddot{\beta},u,\nu,\beta\big)\!=-\ddot{\beta}\!\!\int_{r}^{R}m w_{0}\hat{\nu}_{c g}\mathrm{d}s
|
||||
F_{\phi,4}\big(\ddot{\beta},u,\nu,\beta\big)\!=-\ddot{\beta}\!\!\int_{r}^{R}m w_{0}\hat{\nu}_{c g}\mathrm{d}s\tag {37}
|
||||
$$
|
||||
|
||||
describes the fictitious acceleration associated with an angular acceleration of the $\left(x,\,y,\,z\right)$ -frame about the $z$ -axis.
|
||||
@ -443,7 +442,7 @@ The spatial derivatives of an unforced and linearized version of the partial dif
|
||||
运动偏微分方程（方程 (12) 和 (18)）的无强迫和线性化版本的空间导数，采用二阶有限差分近似。得到的近似常微分方程可以写成
|
||||
|
||||
$$
|
||||
\hat{\mathbf{M}}\ddot{\tilde{\mathbf{q}}}+\tilde{\mathbf{D}}\ddot{\tilde{\mathbf{q}}}+\tilde{\mathbf{K}}\tilde{\mathbf{q}}=\mathbf{0}
|
||||
\hat{\mathbf{M}}\ddot{\tilde{\mathbf{q}}}+\tilde{\mathbf{D}}\ddot{\tilde{\mathbf{q}}}+\tilde{\mathbf{K}}\tilde{\mathbf{q}}=\mathbf{0}\tag {38}
|
||||
$$
|
||||
|
||||
where $\tilde{\textbf{M}}$ , $\tilde{\mathbf{D}}$ and $\tilde{\bf K}$ hold the constant coefficients from the discretization and $\tilde{\mathbf{q}}=[u_{I},\nu_{I},\theta_{I}$ , … $,u_{n},\nu_{n},\theta_{n}]$ holds the deformations at the $n$ discretization points. Equation (38) is a differential eigenvalue problem where the eigenvalues give the frequency and damping of natural vibrations of the blade and the corresponding eigenvectors give the shape of the natural vibrations.
|
||||
@ -474,27 +473,27 @@ The ordinary differential equation of blade motion becomes
|
||||
叶片运动的常微分方程变为：
|
||||
|
||||
$$
|
||||
\begin{array}{r l}&{\mathbf{M}\ddot{\mathbf{q}}+2\mathbf{D}\big(\dot{\phi},\dot{\beta},\beta\big)\dot{\mathbf{q}}+\mathbf{K}\big(\dot{\phi},\dot{\beta},\beta\big)\mathbf{q}+\mathbf{L}\big(\ddot{\beta},\phi,\beta\big)\mathbf{q}+\mathbf{N}\big(\dot{\phi},\dot{\beta},\beta,\mathbf{q}\big)+\mathbf{F}_{e x t,1}\mathbf{q}f_{w,t}}\\ &{\quad=\mathbf{F}\big(\ddot{\beta},\ddot{\phi},\dot{\beta},\dot{\phi},\beta,\phi\big)+\mathbf{F}_{e x t,0}[f_{u,t}f_{\nu,t}f_{w,t}M_{t}]^{\mathrm{T}}}\end{array}
|
||||
\begin{array}{r l}&{\mathbf{M}\ddot{\mathbf{q}}+2\mathbf{D}\big(\dot{\phi},\dot{\beta},\beta\big)\dot{\mathbf{q}}+\mathbf{K}\big(\dot{\phi},\dot{\beta},\beta\big)\mathbf{q}+\mathbf{L}\big(\ddot{\beta},\phi,\beta\big)\mathbf{q}+\mathbf{N}\big(\dot{\phi},\dot{\beta},\beta,\mathbf{q}\big)+\mathbf{F}_{e x t,1}\mathbf{q}f_{w,t}}\\ &{\quad=\mathbf{F}\big(\ddot{\beta},\ddot{\phi},\dot{\beta},\dot{\phi},\beta,\phi\big)+\mathbf{F}_{e x t,0}[f_{u,t}f_{\nu,t}f_{w,t}M_{t}]^{\mathrm{T}}}\end{array}\tag {39}
|
||||
$$
|
||||
|
||||
where $\mathbf{q}=[u_{t},\,\nu_{t},\,\theta_{t}]^{\mathrm{T}}$ and the rest of the terms are given in equation (50) in Appendix B. Inserting the expansions into equation (27), the integrals can be computed and the equation of pitch action becomes
|
||||
其中 $\mathbf{q}=[u_{t},\,\nu_{t},\,\theta_{t}]^{\mathrm{T}}$，其余项在附录B的公式(50)中给出。将展开式代入公式(27)，可以计算积分，变桨作用方程变为
|
||||
$$
|
||||
\begin{array}{r l}&{\big(I_{\beta,0}+I_{\beta,1}(\mathbf{q})\big)\ddot{\beta}+D_{\beta}(\dot{\mathbf{q}},\mathbf{q})\dot{\beta}=M_{p i t c h}+\big(\mathbf{M}_{e x t,0}+\mathbf{q}^{\mathrm{T}}\mathbf{M}_{e x t,1}\big)[f_{u,t}f_{\nu,t}M_{t}]^{\mathrm{T}}}\\ &{\quad+\,\mathbf{I}_{\beta,\mathbf{q}}\ddot{\mathbf{q}}+f_{\beta,\ddot{\mathbf{q}}}(\ddot{\mathbf{q}},\mathbf{q})+f_{\beta,\phi}\big(\ddot{\phi},\dot{\phi},\mathbf{q}\big)+f_{\beta,g r a v}(\mathbf{q})\sin(\phi)}\end{array}
|
||||
\begin{array}{r l}&{\big(I_{\beta,0}+I_{\beta,1}(\mathbf{q})\big)\ddot{\beta}+D_{\beta}(\dot{\mathbf{q}},\mathbf{q})\dot{\beta}=M_{p i t c h}+\big(\mathbf{M}_{e x t,0}+\mathbf{q}^{\mathrm{T}}\mathbf{M}_{e x t,1}\big)[f_{u,t}f_{\nu,t}M_{t}]^{\mathrm{T}}}\\ &{\quad+\,\mathbf{I}_{\beta,\mathbf{q}}\ddot{\mathbf{q}}+f_{\beta,\ddot{\mathbf{q}}}(\ddot{\mathbf{q}},\mathbf{q})+f_{\beta,\phi}\big(\ddot{\phi},\dot{\phi},\mathbf{q}\big)+f_{\beta,g r a v}(\mathbf{q})\sin(\phi)}\end{array}\tag {40}
|
||||
$$
|
||||
|
||||
The individual terms are given in equation (51) in Appendix B. Inserting the expansions into equation (33) and computing the integrals, the equation of rotor position becomes
|
||||
各项见附录 B 的方程 (51)，将展开式代入方程 (33) 并计算积分后，风轮位置方程变为
|
||||
|
||||
$$
|
||||
\begin{array}{r l}&{(J_{g c n}+I_{\phi})\ddot{\phi}+f_{\phi,g}(\phi,\beta,\mathbf{q})\!=T_{g c n}+f_{\phi,\mathbf{q}}(\ddot{\mathbf{q}},\beta)\!+I_{\phi,\beta}(\mathbf{q},\beta)\ddot{\beta}+f_{\phi,\beta}\big(\dot{\beta},\dot{\mathbf{q}},\beta,\mathbf{q}\big)}\\ &{\quad+\,\mathbf{f}_{e x t,0}(\beta)[f_{u,t}f_{\nu,t}f_{w,t}]^{\mathrm{T}}+f_{e x t,1}(\mathbf{q},\beta)f_{w,t}}\end{array}
|
||||
\begin{array}{r l}&{(J_{g c n}+I_{\phi})\ddot{\phi}+f_{\phi,g}(\phi,\beta,\mathbf{q})\!=T_{g c n}+f_{\phi,\mathbf{q}}(\ddot{\mathbf{q}},\beta)\!+I_{\phi,\beta}(\mathbf{q},\beta)\ddot{\beta}+f_{\phi,\beta}\big(\dot{\beta},\dot{\mathbf{q}},\beta,\mathbf{q}\big)}\\ &{\quad+\,\mathbf{f}_{e x t,0}(\beta)[f_{u,t}f_{\nu,t}f_{w,t}]^{\mathrm{T}}+f_{e x t,1}(\mathbf{q},\beta)f_{w,t}}\end{array}\tag {41}
|
||||
$$
|
||||
|
||||
The individual terms are given in equation (52) in Appendix B. An unforced and linearized version of equation (39) gives a differential eigenvalue problem:
|
||||
各项在附录 B 的方程 (52) 中给出。方程 (39) 的无强迫和线性化版本得到一个微分特征值问题：
|
||||
|
||||
$$
|
||||
\mathbf{M}_{l i n}{\ddot{\mathbf{q}}}+\mathbf{D}_{l i n}{\dot{\mathbf{q}}}+\mathbf{K}_{l i n}\mathbf{q}=\mathbf{0}
|
||||
\mathbf{M}_{l i n}{\ddot{\mathbf{q}}}+\mathbf{D}_{l i n}{\dot{\mathbf{q}}}+\mathbf{K}_{l i n}\mathbf{q}=\mathbf{0}\tag {42}
|
||||
$$
|
||||
|
||||
where the eigenvalue gives the frequency of natural vibrations of the assumed mode approximated model, and the eigenvectors give the coupling of the assumed modes in the natural vibrations. The found frequencies are compared with the previously found frequencies in Table I showing a good agreement. Figure 2 shows the natural mode shapes together with the previously found mode shapes. The edgewise and flapwise contents of the first and second modes are seen to agree very well with previous results. The torsional contents of the first mode are seen to disagree slightly from the previous results. The torsional contents of the second mode are seen to disagree with the previous result, but the value of the torsional contents is small compared to the edgewise and flapwise contents, hence the error is acceptable. The edgewise and flapwise contents of the sixth mode (first torsional mode) are seen to disagree quite a lot with the previous results. This is because the edgewise and flapwise contents are dominated by higher order edgewise and flapwise motion, which cannot be captured by this low order model. The value of the edgewise and flapwise contents is, however, small compared to the torsional contents, hence the error is acceptable.
|
||||
@ -621,7 +620,7 @@ where $\tilde{\theta}$ is the rotation of the blade around the elastic axis. The
|
||||
The chord is described by the $(\overline{{\mathbf{i}}},\overline{{\mathbf{j}}},\overline{{\mathbf{k}}})$ unit vectors parallel to the chord, normal upward from the chord and parallel to the elastic axis, respectively. This set of unit vectors is given by
|
||||
|
||||
$$
|
||||
[\bar{\mathbf{i}}\quad\bar{\mathbf{j}}\quad\overline{{\mathbf{k}}}]^{\mathrm{T}}=\mathbf{T}_{\mathrm{c}}[\mathbf{i}\quad\mathbf{j}\quad\mathbf{k}]^{\mathrm{T}}
|
||||
[\bar{\mathbf{i}}\quad\bar{\mathbf{j}}\quad\overline{{\mathbf{k}}}]^{\mathrm{T}}=\mathbf{T}_{\mathrm{c}}[\mathbf{i}\quad\mathbf{j}\quad\mathbf{k}]^{\mathrm{T}}\tag {46}
|
||||
$$
|
||||
|
||||
where ${{\bf{T}}_{c}}$ is equal to ${\bf{T}}_{e}$ except that the twist qˆ includes the aerodynamic pre-twist instead of the elastic pre-twist.
|
||||
@ -629,13 +628,13 @@ 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:
|
||||
|
||||
$$
|
||||
\begin{array}{r}{\mathbf{T}_{e}^{\prime}=\left[\!\!\begin{array}{c c c}{0}&{\tilde{\omega}_{k}}&{-\tilde{\omega}_{j}}\\ {-\tilde{\omega}_{k}}&{0}&{\tilde{\omega}_{i}}\\ {\tilde{\omega}_{j}}&{-\tilde{\omega}_{i}}&{0}\end{array}\right]\!\mathbf{T}_{c}\!\Rightarrow\left[\begin{array}{c c c}{0}&{\tilde{\omega}_{k}}&{-\tilde{\omega}_{j}}\\ {-\tilde{\omega}_{k}}&{0}&{\tilde{\omega}_{i}}\\ {\tilde{\omega}_{j}}&{-\tilde{\omega}_{i}}&{0}\end{array}\right]=T_{\mathrm{c}}\mathbf{T}_{\mathrm{c}}^{-1}}\end{array}
|
||||
\begin{array}{r}{\mathbf{T}_{e}^{\prime}=\left[\!\!\begin{array}{c c c}{0}&{\tilde{\omega}_{k}}&{-\tilde{\omega}_{j}}\\ {-\tilde{\omega}_{k}}&{0}&{\tilde{\omega}_{i}}\\ {\tilde{\omega}_{j}}&{-\tilde{\omega}_{i}}&{0}\end{array}\right]\!\mathbf{T}_{c}\!\Rightarrow\left[\begin{array}{c c c}{0}&{\tilde{\omega}_{k}}&{-\tilde{\omega}_{j}}\\ {-\tilde{\omega}_{k}}&{0}&{\tilde{\omega}_{i}}\\ {\tilde{\omega}_{j}}&{-\tilde{\omega}_{i}}&{0}\end{array}\right]=T_{\mathrm{c}}\mathbf{T}_{\mathrm{c}}^{-1}}\end{array}\tag {47}
|
||||
$$
|
||||
|
||||
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,
|
||||
|
||||
$$
|
||||
\left(\tilde{\theta}+\theta_{e l a}\right)^{\prime}=\tilde{\omega}_{k}=\hat{\theta}+\nu^{\prime}(u^{\prime\prime}+l_{p i}^{\prime\prime})+O(\varepsilon^{3})
|
||||
\left(\tilde{\theta}+\theta_{e l a}\right)^{\prime}=\tilde{\omega}_{k}=\hat{\theta}+\nu^{\prime}(u^{\prime\prime}+l_{p i}^{\prime\prime})+O(\varepsilon^{3})\tag {48}
|
||||
$$
|
||||
|
||||
using the order scheme (see previous discussion).
|
||||
@ -643,7 +642,7 @@ using the order scheme (see previous discussion).
|
||||
Rearranging and intergrading equation (48) lead to an expression for the rotation of each blade section around the elastic axis:
|
||||
|
||||
$$
|
||||
\hat{\theta}=\tilde{\theta}+\theta_{e l a}-\int_{0}^{s}\nu^{\prime}\big(\boldsymbol{u}^{\prime\prime}+\boldsymbol{l}_{p i}^{\prime}\big)\mathrm{d}\rho=\tilde{\theta}+\theta,\quad0=\theta\big(\boldsymbol{s},t\big)=\theta_{e l a}-\int_{0}^{s}\nu^{\prime}\big(\boldsymbol{u}^{\prime\prime}+\boldsymbol{l}_{p i}^{\prime\prime}\big)\mathrm{d}\rho
|
||||
\hat{\theta}=\tilde{\theta}+\theta_{e l a}-\int_{0}^{s}\nu^{\prime}\big(\boldsymbol{u}^{\prime\prime}+\boldsymbol{l}_{p i}^{\prime}\big)\mathrm{d}\rho=\tilde{\theta}+\theta,\quad0=\theta\big(\boldsymbol{s},t\big)=\theta_{e l a}-\int_{0}^{s}\nu^{\prime}\big(\boldsymbol{u}^{\prime\prime}+\boldsymbol{l}_{p i}^{\prime\prime}\big)\mathrm{d}\rho\tag {49}
|
||||
$$
|
||||
|
||||
where $\theta$ is the time-dependent twist of the blade relative to the $\left(x,\,y,\,z\right)$ -frame. Inserting equation (49) into the expression for ${\bf{T}}_{e}$ leads to the transformation matrix of the elastic properties. Replacing $\tilde{\theta}$ with $\bar{\theta}$ in ${\bf\delta T}_{e}$ gives the transformation matrix ${{\bf{T}}_{c}}$ of the chord.
|
||||
@ -687,7 +686,7 @@ $$
|
||||
where the constants for the linear terms are
|
||||
|
||||
$$
|
||||
{\bf M}=\int_{\nu}^{R}{\left[\begin{array}{c c c}{m u_{s}^{2}}&{0}&{-u_{s}\theta_{s}m l_{c g}\sin(\overline{{\theta}})}\\ {0}&{m\nu_{s}^{2}}&{\nu_{s}\theta_{s}m l_{c g}\cos(\overline{{\theta}})}\\ {-u_{s}\theta_{s}m l_{c g}\sin(\overline{{\theta}})}&{\nu_{s}\theta_{s}m l_{c g}\cos(\overline{{\theta}})}&{(I_{c g}+m l_{c g}^{2})\theta_{s}^{2}}\end{array}\right]}\mathrm{d}s
|
||||
{\bf M}=\int_{\nu}^{R}{\left[\begin{array}{c c c}{m u_{s}^{2}}&{0}&{-u_{s}\theta_{s}m l_{c g}\sin(\overline{{\theta}})}\\ {0}&{m\nu_{s}^{2}}&{\nu_{s}\theta_{s}m l_{c g}\cos(\overline{{\theta}})}\\ {-u_{s}\theta_{s}m l_{c g}\sin(\overline{{\theta}})}&{\nu_{s}\theta_{s}m l_{c g}\cos(\overline{{\theta}})}&{(I_{c g}+m l_{c g}^{2})\theta_{s}^{2}}\end{array}\right]}\mathrm{d}s\tag {51}
|
||||
$$
|
||||
|
||||
$$
|
||||
@ -697,23 +696,23 @@ 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} \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
|
||||
ds\tag {52}
|
||||
$$
|
||||
|
||||
$$
|
||||
\mathbf{K}_{\phi,0}=\int_{r}^{R}\left[\begin{array}{c c c}{\left(u_{s}^{\prime}\right)^{2}\int_{s}^{R}m w_{0}\mathrm{d}\rho}&{0}&{-m l_{c g}w_{0}\sin(\overline{{\theta}})u_{s}^{\prime}\theta_{s}}\\ {0}&{\left(\nu_{s}^{\prime}\right)^{2}\int_{s}^{R}m w_{0}\mathrm{d}\rho}&{m l_{c g}w_{0}\cos(\overline{{\theta}})\nu_{s}^{\prime}\theta_{s}}\\ {-m l_{c g}w_{0}\sin(\overline{{\theta}})u_{s}^{\prime}\theta_{s}}&{m l_{c g}w_{0}\cos(\overline{{\theta}})\nu_{s}^{\prime}\theta_{s}}&{0}\end{array}\right]\mathrm{d}s
|
||||
\mathbf{K}_{\phi,0}=\int_{r}^{R}\left[\begin{array}{c c c}{\left(u_{s}^{\prime}\right)^{2}\int_{s}^{R}m w_{0}\mathrm{d}\rho}&{0}&{-m l_{c g}w_{0}\sin(\overline{{\theta}})u_{s}^{\prime}\theta_{s}}\\ {0}&{\left(\nu_{s}^{\prime}\right)^{2}\int_{s}^{R}m w_{0}\mathrm{d}\rho}&{m l_{c g}w_{0}\cos(\overline{{\theta}})\nu_{s}^{\prime}\theta_{s}}\\ {-m l_{c g}w_{0}\sin(\overline{{\theta}})u_{s}^{\prime}\theta_{s}}&{m l_{c g}w_{0}\cos(\overline{{\theta}})\nu_{s}^{\prime}\theta_{s}}&{0}\end{array}\right]\mathrm{d}s\tag {53}
|
||||
$$
|
||||
|
||||
$$
|
||||
\mathbf{K}_{\phi,c c}=\int_{r}^{R}\!\left[\begin{array}{c c c c}{-m u_{s}^{2}}&{0}&{m l_{c g}\sin(\overline{{\theta}})u_{s}\theta_{s}}\\ {0}&{0}&{0}\\ {m l_{c g}\sin(\overline{{\theta}})u_{s}\theta_{s}}&{0}&{0}\end{array}\right]\!\mathrm{d}s
|
||||
\mathbf{K}_{\phi,c c}=\int_{r}^{R}\!\left[\begin{array}{c c c c}{-m u_{s}^{2}}&{0}&{m l_{c g}\sin(\overline{{\theta}})u_{s}\theta_{s}}\\ {0}&{0}&{0}\\ {m l_{c g}\sin(\overline{{\theta}})u_{s}\theta_{s}}&{0}&{0}\end{array}\right]\!\mathrm{d}s\tag {54}
|
||||
$$
|
||||
|
||||
$$
|
||||
\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\tag {55}
|
||||
$$
|
||||
|
||||
$$
|
||||
{\bf K}_{\dot{\phi},s c}=\int_{r}^{R}\!\left[\begin{array}{c c c}{0}&{m u_{s}\nu_{s}}&{m l_{c g}\cos(\overline{{\theta}})u_{s}\theta_{s}}\\ {m u_{s}\nu_{s}}&{0}&{-l_{c g}m\sin(\overline{{\theta}})\nu_{s}\theta_{s}}\\ {m l_{c g}\cos(\overline{{\theta}})u_{s}\theta_{s}}&{-l_{c g}m\sin(\overline{{\theta}})\nu_{s}\theta_{s}}&{0}\end{array}\right]\mathrm{d}s
|
||||
{\bf K}_{\dot{\phi},s c}=\int_{r}^{R}\!\left[\begin{array}{c c c}{0}&{m u_{s}\nu_{s}}&{m l_{c g}\cos(\overline{{\theta}})u_{s}\theta_{s}}\\ {m u_{s}\nu_{s}}&{0}&{-l_{c g}m\sin(\overline{{\theta}})\nu_{s}\theta_{s}}\\ {m l_{c g}\cos(\overline{{\theta}})u_{s}\theta_{s}}&{-l_{c g}m\sin(\overline{{\theta}})\nu_{s}\theta_{s}}&{0}\end{array}\right]\mathrm{d}s\tag {56}
|
||||
$$
|
||||
|
||||
$$
|
||||
@ -723,7 +722,7 @@ $$
|
||||
ml_{cg} \cos(\bar{\theta})u_s'v_s & -ml_{cg} \sin(\bar{\theta})v_s v_s' & 0 \\
|
||||
0 & +ml_{cg} \sin(\bar{\theta})v_s v_s' & 0
|
||||
\end{bmatrix}
|
||||
ds
|
||||
ds\tag {57}
|
||||
$$
|
||||
|
||||
$$
|
||||
@ -734,15 +733,15 @@ ml_{cg} \cos(\bar{\theta})u_s u_s' & -ml_{cg} \sin(\bar{\theta})v_s' u_s & 0 \\
|
||||
ml_{cg} \sin(\bar{\theta})u_s v_s' & 0 & 0 \\
|
||||
0 & 0 & 0
|
||||
\end{bmatrix}
|
||||
ds
|
||||
ds\tag {58}
|
||||
$$
|
||||
|
||||
$$
|
||||
\mathbf{k}_{\beta}=\int_{r}^{R}{\left[\begin{array}{c c c c}{-m u_{s}^{2}}&{0}&{m l_{c g}\sin(\overline{{\theta}})u_{s}\theta_{s}}\\ {0}&{-m\nu_{s}^{2}}&{-m l_{c g}\cos(\overline{{\theta}})\nu_{s}\theta_{s}}\\ {m l_{c g}\sin(\overline{{\theta}})u_{s}\theta_{s}}&{-m l_{c g}\cos(\overline{{\theta}})\nu_{s}\theta_{s}}&{0}\end{array}\right]}\mathrm{d}s
|
||||
\mathbf{k}_{\beta}=\int_{r}^{R}{\left[\begin{array}{c c c c}{-m u_{s}^{2}}&{0}&{m l_{c g}\sin(\overline{{\theta}})u_{s}\theta_{s}}\\ {0}&{-m\nu_{s}^{2}}&{-m l_{c g}\cos(\overline{{\theta}})\nu_{s}\theta_{s}}\\ {m l_{c g}\sin(\overline{{\theta}})u_{s}\theta_{s}}&{-m l_{c g}\cos(\overline{{\theta}})\nu_{s}\theta_{s}}&{0}\end{array}\right]}\mathrm{d}s\tag {59}
|
||||
$$
|
||||
|
||||
$$
|
||||
\mathbf{D}_{\beta}\!=\!\int_{r}^{R}\!\left[\begin{array}{c c c}{0}&{-m\nu_{s}u_{s}}&{-m l_{c g}\cos(\overline{{\theta}})u_{s}\theta_{s}}\\ {m\nu_{s}u_{s}}&{0}&{-m l_{c g}\sin(\overline{{\theta}})\nu_{s}\theta_{s}}\\ {m l_{c g}\cos(\overline{{\theta}})u_{s}\theta_{s}}&{m l_{c g}\sin(\overline{{\theta}})\nu_{s}\theta_{s}}&{0}\end{array}\right]\!\mathrm{d}s
|
||||
\mathbf{D}_{\beta}\!=\!\int_{r}^{R}\!\left[\begin{array}{c c c}{0}&{-m\nu_{s}u_{s}}&{-m l_{c g}\cos(\overline{{\theta}})u_{s}\theta_{s}}\\ {m\nu_{s}u_{s}}&{0}&{-m l_{c g}\sin(\overline{{\theta}})\nu_{s}\theta_{s}}\\ {m l_{c g}\cos(\overline{{\theta}})u_{s}\theta_{s}}&{m l_{c g}\sin(\overline{{\theta}})\nu_{s}\theta_{s}}&{0}\end{array}\right]\!\mathrm{d}s\tag {60}
|
||||
$$
|
||||
|
||||
$$
|
||||
@ -752,7 +751,7 @@ $$
|
||||
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
|
||||
ds\tag {61}
|
||||
$$
|
||||
|
||||
$$
|
||||
@ -763,8 +762,7 @@ $$
|
||||
-ml_{cg}u_s v_s' \sin(\bar{\theta}) & -u_s'^2 \int_s^R m(l_{pi} + l_{cg} \cos(\bar{\theta}))d\rho & 0 \\
|
||||
0 & 0 & 0
|
||||
\end{bmatrix}
|
||||
ds
|
||||
ds
|
||||
ds\tag {62}
|
||||
$$
|
||||
$$
|
||||
\mathbf{K}_{\beta \phi,c} = \int_r^R
|
||||
@ -774,7 +772,7 @@ $$
|
||||
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
|
||||
ds\tag {63}
|
||||
$$
|
||||
$$
|
||||
\mathbf{F}_{g,1,s} = \int_r^R
|
||||
@ -794,7 +792,7 @@ $$
|
||||
-ml_{cg}v_s'u_s' \sin(\bar{\theta}) & 0 & 0 \\
|
||||
0 & 0 & -ml_{cg}\theta_s^2 \cos(\bar{\theta})
|
||||
\end{bmatrix}
|
||||
ds
|
||||
ds\tag {64}
|
||||
$$
|
||||
|
||||
$$
|
||||
@ -804,7 +802,7 @@ $$
|
||||
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
|
||||
ds\tag {65}
|
||||
$$
|
||||
|
||||
$$
|
||||
@ -820,7 +818,7 @@ l'_{pi}u'_{s}\int_s^R mu_s d\rho & 0 & 0 \\
|
||||
0 & 0 & 0 \\
|
||||
0 & 0 & 0
|
||||
\end{bmatrix}
|
||||
ds
|
||||
ds\tag {66}
|
||||
$$
|
||||
|
||||
$$
|
||||
@ -830,21 +828,21 @@ $$
|
||||
0 & 0 & 0 \\
|
||||
EI_{\eta\eta\zeta}\bar{\theta}'\theta_s'u_s''\sin(\bar{\theta}) - EI_{\eta\zeta\zeta}\bar{\theta}'\theta_s'u_s''\cos(\bar{\theta}) & -EI_{\eta\eta\zeta}\bar{\theta}'\theta_s''u_s'\cos(\tilde{\theta}) - EI_{\eta\zeta\zeta}\bar{\theta}'\theta_s'v_s''\sin(\tilde{\theta}) & 0
|
||||
\end{bmatrix}
|
||||
ds
|
||||
ds\tag {67}
|
||||
$$
|
||||
|
||||
and for constants for the nonlinear terms:
|
||||
|
||||
$$
|
||||
\mathbf{F}_{1}=\int_{r}^{R}{\left[\begin{array}{c c c}{-E(I_{\xi}-I_{\eta})u_{s}^{\prime\prime}u_{s}^{\prime}\theta_{s}\sin(2\tilde{\theta})}&{E(I_{\xi}-I_{\eta})\nu_{s}^{\prime\prime}u_{s}^{\prime\prime}\theta_{s}\cos(2\tilde{\theta})}&{0}\\ {E(I_{\xi}-I_{\eta})u_{s}^{\prime\prime}\nu_{s}^{\prime\prime}\theta_{s}\cos(2\tilde{\theta})}&{E(I_{\xi}-I_{\eta})\nu_{s}^{\prime\prime}\nu_{s}^{\prime\prime}\theta_{s}\sin(2\tilde{\theta})}&{0}\\ {\theta_{s}^{\prime2}u_{s}^{\prime\prime}E(I_{\eta\eta\xi}\sin(\overline{{\theta}})-E I_{\eta\xi\xi}\cos(\overline{{\theta}}))}&{-\theta_{s}^{\prime2}\nu_{s}^{\prime\prime}E(I_{\eta\eta\xi}\cos(\tilde{\theta})+I_{\eta\xi\xi}\sin(\tilde{\theta}))}&{0}\end{array}\right]}\mathrm{d}s
|
||||
\mathbf{F}_{1}=\int_{r}^{R}{\left[\begin{array}{c c c}{-E(I_{\xi}-I_{\eta})u_{s}^{\prime\prime}u_{s}^{\prime}\theta_{s}\sin(2\tilde{\theta})}&{E(I_{\xi}-I_{\eta})\nu_{s}^{\prime\prime}u_{s}^{\prime\prime}\theta_{s}\cos(2\tilde{\theta})}&{0}\\ {E(I_{\xi}-I_{\eta})u_{s}^{\prime\prime}\nu_{s}^{\prime\prime}\theta_{s}\cos(2\tilde{\theta})}&{E(I_{\xi}-I_{\eta})\nu_{s}^{\prime\prime}\nu_{s}^{\prime\prime}\theta_{s}\sin(2\tilde{\theta})}&{0}\\ {\theta_{s}^{\prime2}u_{s}^{\prime\prime}E(I_{\eta\eta\xi}\sin(\overline{{\theta}})-E I_{\eta\xi\xi}\cos(\overline{{\theta}}))}&{-\theta_{s}^{\prime2}\nu_{s}^{\prime\prime}E(I_{\eta\eta\xi}\cos(\tilde{\theta})+I_{\eta\xi\xi}\sin(\tilde{\theta}))}&{0}\end{array}\right]}\mathrm{d}s\tag {68}
|
||||
$$
|
||||
|
||||
$$
|
||||
\mathbf{F}_{2}=\int_{r}^{R}\left[\begin{array}{c c c}{0}&{0}&{0}\\ {0}&{0}&{0}\\ {-E(I_{\xi}-I_{\eta})\theta_{s}u_{s}^{\prime\prime}u_{s}^{\prime\prime}\mathrm{cos}(\tilde{\theta})\mathrm{sin}(\tilde{\theta})}&{E(I_{\xi}-I_{\eta})\theta_{s}\nu_{s}^{\prime\prime}\nu_{s}^{\prime\prime}\mathrm{cos}(\tilde{\theta})\mathrm{sin}(\tilde{\theta})}&{0}\end{array}\right]\mathrm{d}s
|
||||
\mathbf{F}_{2}=\int_{r}^{R}\left[\begin{array}{c c c}{0}&{0}&{0}\\ {0}&{0}&{0}\\ {-E(I_{\xi}-I_{\eta})\theta_{s}u_{s}^{\prime\prime}u_{s}^{\prime\prime}\mathrm{cos}(\tilde{\theta})\mathrm{sin}(\tilde{\theta})}&{E(I_{\xi}-I_{\eta})\theta_{s}\nu_{s}^{\prime\prime}\nu_{s}^{\prime\prime}\mathrm{cos}(\tilde{\theta})\mathrm{sin}(\tilde{\theta})}&{0}\end{array}\right]\mathrm{d}s\tag {69}
|
||||
$$
|
||||
|
||||
$$
|
||||
\mathbf{F}_{3,s}=\int_{r}^{R}{\left[\begin{array}{l l l}{0}&{-u_{s}^{\prime\,2}\displaystyle\int_{s}^{R}m\nu_{s}\mathrm{d}\rho}&{0}\\ {0}&{-\nu_{s}^{\prime\,2}\displaystyle\int_{s}^{R}m\nu_{s}\mathrm{d}\rho}&{0}\\ {0}&{\qquad0}&{0}\end{array}\right]}\mathrm{d}s,\quad\mathbf{F}_{3,c}=\int_{r}^{R}{\left[\begin{array}{l l l}{u_{s}^{\prime\,2}\displaystyle\int_{s}^{R}m u_{s}\mathrm{d}\rho}&{0}&{0}\\ {\nu_{s}^{\prime\,2}\displaystyle\int_{s}^{R}m u_{s}\mathrm{d}\rho}&{0}&{0}\\ {\qquad0}&{0}&{0}\\ \end{array}\right]}\mathrm{d}s
|
||||
\mathbf{F}_{3,s}=\int_{r}^{R}{\left[\begin{array}{l l l}{0}&{-u_{s}^{\prime\,2}\displaystyle\int_{s}^{R}m\nu_{s}\mathrm{d}\rho}&{0}\\ {0}&{-\nu_{s}^{\prime\,2}\displaystyle\int_{s}^{R}m\nu_{s}\mathrm{d}\rho}&{0}\\ {0}&{\qquad0}&{0}\end{array}\right]}\mathrm{d}s,\quad\mathbf{F}_{3,c}=\int_{r}^{R}{\left[\begin{array}{l l l}{u_{s}^{\prime\,2}\displaystyle\int_{s}^{R}m u_{s}\mathrm{d}\rho}&{0}&{0}\\ {\nu_{s}^{\prime\,2}\displaystyle\int_{s}^{R}m u_{s}\mathrm{d}\rho}&{0}&{0}\\ {\qquad0}&{0}&{0}\\ \end{array}\right]}\mathrm{d}s\tag {69}
|
||||
$$
|
||||
|
||||
$$
|
||||
@ -860,11 +858,11 @@ ds, \quad \mathbf{F}_{4,c} = \int_r^R
|
||||
0 & -v_s'^2 \int_s^R mv_s d\rho & 0 \\
|
||||
0 & 0 & 0
|
||||
\end{bmatrix}
|
||||
ds
|
||||
ds\tag {70}
|
||||
$$
|
||||
|
||||
$$
|
||||
\mathbf{f}_{5}=\int_{r}^{R}[0\quad0\quad E(I_{\xi}-I_{\eta})u_{s}^{\prime\prime}\nu_{s}^{\prime\prime}\mathrm{cos}\big(2\tilde{\theta}\big)]^{\mathrm{T}}\mathrm{d}s
|
||||
\mathbf{f}_{5}=\int_{r}^{R}[0\quad0\quad E(I_{\xi}-I_{\eta})u_{s}^{\prime\prime}\nu_{s}^{\prime\prime}\mathrm{cos}\big(2\tilde{\theta}\big)]^{\mathrm{T}}\mathrm{d}s\tag {71}
|
||||
$$
|
||||
|
||||
and for the forcing terms:
|
||||
@ -876,7 +874,7 @@ 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
|
||||
ds\tag {72}
|
||||
$$
|
||||
|
||||
$$
|
||||
@ -886,7 +884,7 @@ ml_{cg}(l_{pi} + l_{cg} \cos(\bar{\theta}))u_s' \cos(\bar{\theta}) + l'_{pi}u'_{
|
||||
ml_{cg}(l_{pi} + l_{cg} \cos(\bar{\theta}))v_s' \sin(\bar{\theta}) \\
|
||||
0
|
||||
\end{bmatrix}
|
||||
ds
|
||||
ds\tag {73}
|
||||
$$
|
||||
|
||||
$$
|
||||
@ -896,7 +894,7 @@ ml_{cg}l_{cg} \cos(\beta)\sin(\bar{\theta})u_s' \cos(\bar{\theta}) + l'_{pi}u'_{
|
||||
ml_{cg}l_{cg} \sin^2(\bar{\theta})v_s' \\
|
||||
0
|
||||
\end{bmatrix}
|
||||
ds
|
||||
ds\tag {74}
|
||||
$$
|
||||
|
||||
$$
|
||||
@ -906,11 +904,11 @@ $$
|
||||
0 & -ml_{cg}w_0 v_s' \sin(\bar{\theta}) \\
|
||||
0 & -ml_{cg}l_{pi}' w_0 \theta_s \sin(\bar{\theta})
|
||||
\end{bmatrix}
|
||||
ds
|
||||
ds\tag {75}
|
||||
$$
|
||||
|
||||
$$
|
||||
\mathbf{F}_{\phi,s}=\int_{r}^{R}{\left[\begin{array}{c c c c}{0}&{0}\\ {m w_{0}u_{s}}&{0}\\ {m w_{0}l_{c g}\theta_{s}\cos({\overline{{\theta}}})}&{0}\end{array}\right]}\mathrm{d}s,\ \ \ \mathbf{F}_{\phi,c}=\int_{r}^{R}{\left[\begin{array}{c c c c}{-m w_{0}u_{s}}&{0}\\ {0}&{0}\\ {m w_{0}l_{c g}\theta_{s}\sin({\overline{{\theta}}})}&{0}\end{array}\right]}\mathrm{d}s
|
||||
\mathbf{F}_{\phi,s}=\int_{r}^{R}{\left[\begin{array}{c c c c}{0}&{0}\\ {m w_{0}u_{s}}&{0}\\ {m w_{0}l_{c g}\theta_{s}\cos({\overline{{\theta}}})}&{0}\end{array}\right]}\mathrm{d}s,\ \ \ \mathbf{F}_{\phi,c}=\int_{r}^{R}{\left[\begin{array}{c c c c}{-m w_{0}u_{s}}&{0}\\ {0}&{0}\\ {m w_{0}l_{c g}\theta_{s}\sin({\overline{{\theta}}})}&{0}\end{array}\right]}\mathrm{d}s\tag {76}
|
||||
$$
|
||||
|
||||
$$
|
||||
@ -920,7 +918,7 @@ $$
|
||||
0 & ml_{cg}\sin(\bar{\theta})v_s \\
|
||||
0 & ml_{cg}l_{cg} \sin(\bar{\theta})\theta_s \cos(\bar{\theta})
|
||||
\end{bmatrix}
|
||||
ds
|
||||
ds\tag {77}
|
||||
$$
|
||||
|
||||
$$
|
||||
@ -930,11 +928,11 @@ $$
|
||||
0 & 0 \\
|
||||
0 & -ml_{cg}(l_{pi} + l_{cg} \cos(\bar{\theta}))\theta_s \sin(\bar{\theta})
|
||||
\end{bmatrix}
|
||||
ds
|
||||
ds\tag {78}
|
||||
$$
|
||||
|
||||
$$
|
||||
\mathbf{F}_{\phi,\mathrm{sc}}=\int_{r}^{R}\left[\begin{array}{c c}{0}&{-m l_{c g}\sin(\overline{{\theta}})u_{s}}\\ {0}&{-m\big(l_{p i}+l_{c g}\cos(\overline{{\theta}})\big)\nu_{s}}\\ {0}&{-m l_{c g}\big(l_{p i}\cos(\theta)+l_{c g}\cos(\overline{{\theta}})-l_{c g}\sin^{2}(\overline{{\theta}})\big)\theta_{s}\,}\end{array}\!\!\right]\mathrm{d}s
|
||||
\mathbf{F}_{\phi,\mathrm{sc}}=\int_{r}^{R}\left[\begin{array}{c c}{0}&{-m l_{c g}\sin(\overline{{\theta}})u_{s}}\\ {0}&{-m\big(l_{p i}+l_{c g}\cos(\overline{{\theta}})\big)\nu_{s}}\\ {0}&{-m l_{c g}\big(l_{p i}\cos(\theta)+l_{c g}\cos(\overline{{\theta}})-l_{c g}\sin^{2}(\overline{{\theta}})\big)\theta_{s}\,}\end{array}\!\!\right]\mathrm{d}s\tag {79}
|
||||
$$
|
||||
|
||||
$$
|
||||
@ -944,7 +942,7 @@ $$
|
||||
0 & -ml_{cg}v_s'\sin(\bar{\theta}) \\
|
||||
0 & ml_{cg}l_{pi}'\theta_s\sin(\bar{\theta})
|
||||
\end{bmatrix}
|
||||
ds
|
||||
ds\tag {80}
|
||||
$$
|
||||
|
||||
$$
|
||||
@ -961,19 +959,19 @@ ds, \quad
|
||||
ml_{cg}l_{pi}'v_s'\sin(\bar{\theta}) & 0 \\
|
||||
ml_{cg}\theta_s \sin(\bar{\theta}) & 0
|
||||
\end{bmatrix}
|
||||
ds
|
||||
ds\tag {81}
|
||||
$$
|
||||
|
||||
$$
|
||||
\mathbf{f}_{\phi}=\int_{r}^{R}\Bigl[-l_{p i}^{\prime}u_{s}^{\prime}\Bigr]_{s}^{R}m w_{0}\mathrm{d}\rho\quad0\quad0\Bigr]^{\mathrm{T}}\mathrm{d}s
|
||||
\mathbf{f}_{\phi}=\int_{r}^{R}\Bigl[-l_{p i}^{\prime}u_{s}^{\prime}\Bigr]_{s}^{R}m w_{0}\mathrm{d}\rho\quad0\quad0\Bigr]^{\mathrm{T}}\mathrm{d}s\tag {82}
|
||||
$$
|
||||
|
||||
$$
|
||||
\mathbf{F}_{e x t,0}=\int_{r}^{R}{\left[\begin{array}{c c c c}{f_{u,s}u_{s}}&{0}&{-l_{p i}^{\prime}u_{s}^{\prime}{\int_{s}^{R}}f_{w,s}\mathrm{d}\rho}&{0}\\ {0}&{f_{v,s}\nu_{s}}&{0}&{0}\\ {0}&{0}&{0}&{M_{s}\theta_{s}}\end{array}\right]}^{\mathrm{T}}\mathrm{d}s
|
||||
\mathbf{F}_{e x t,0}=\int_{r}^{R}{\left[\begin{array}{c c c c}{f_{u,s}u_{s}}&{0}&{-l_{p i}^{\prime}u_{s}^{\prime}{\int_{s}^{R}}f_{w,s}\mathrm{d}\rho}&{0}\\ {0}&{f_{v,s}\nu_{s}}&{0}&{0}\\ {0}&{0}&{0}&{M_{s}\theta_{s}}\end{array}\right]}^{\mathrm{T}}\mathrm{d}s\tag {83}
|
||||
$$
|
||||
|
||||
$$
|
||||
\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
|
||||
\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\tag {84}
|
||||
$$
|
||||
|
||||
# Pitch Model
|
||||
@ -987,49 +985,49 @@ f_{\beta,\ddot{\mathbf{q}}}(\ddot{\mathbf{q}}, \mathbf{q}) = m_{uv}(\ddot{u}_t v
|
||||
f_{\beta,\phi}(\ddot{\phi}, \dot{\phi}, \mathbf{q}) = \ddot{\phi}(f_{\beta,\ddot{\phi},s}u_t \sin(\beta) + f_{\beta,\ddot{\phi},c}V_t \cos(\beta) + I_{\beta,\phi,s} \sin(\beta) + I_{\beta,\phi,c} \cos(\beta)) \\
|
||||
- 2\dot{\phi}^2(f_{\beta,\phi,s} \sin(2\beta) + f_{\beta,\phi,c} \cos(2\beta) + (f_{\beta,\phi,s} \sin(2\beta) + f_{\beta,\phi,c} \cos(2\beta))\mathbf{q} \\
|
||||
+ \sin(2\beta)\mathbf{f}_{\beta,\phi,2}[u_t^2v_t^2]^T + \cos(2\beta)f_{\beta,\phi,2}u_tv_t) \\
|
||||
f_{\beta,grav}(\mathbf{q}) = -g((f_{\beta,g,2,s}\sin(\beta) + \mathbf{f}_{\beta,g,2,c}\cos(\beta))\mathbf{q} + f_{\beta,g,s}\sin(\beta) + f_{\beta,g,c}\cos(\beta)) \tag{51}
|
||||
f_{\beta,grav}(\mathbf{q}) = -g((f_{\beta,g,2,s}\sin(\beta) + \mathbf{f}_{\beta,g,2,c}\cos(\beta))\mathbf{q} + f_{\beta,g,s}\sin(\beta) + f_{\beta,g,c}\cos(\beta)) \tag{85}
|
||||
$$
|
||||
|
||||
where the constants are
|
||||
|
||||
$$
|
||||
I_{\beta,0}=\int_{r}^{R}\big(l_{c g}+m\big(l_{c g}^{2}+l_{p i}^{2}+2l_{p i}l_{c g}\cos(\overline{{\theta}})\big)\big)\mathrm{d}s,\quad\mathbf{I}_{\beta,1}=\int_{r}^{R}[2m\big(l_{p i}+l_{c g}\cos(\theta)\big)\quad2m l_{c g}\sin(\overline{{\theta}})\quad0]\mathrm{d}s,
|
||||
I_{\beta,0}=\int_{r}^{R}\big(l_{c g}+m\big(l_{c g}^{2}+l_{p i}^{2}+2l_{p i}l_{c g}\cos(\overline{{\theta}})\big)\big)\mathrm{d}s,\quad\mathbf{I}_{\beta,1}=\int_{r}^{R}[2m\big(l_{p i}+l_{c g}\cos(\theta)\big)\quad2m l_{c g}\sin(\overline{{\theta}})\quad0]\mathrm{d}s\tag {86}
|
||||
$$
|
||||
|
||||
$$
|
||||
\mathbf{I}_{\beta,2}=\int_{r}^{R}[m u_{s}^{2}\quad m\nu_{s}^{2}]\mathrm{d}s,\quad\mathbf{I}_{\beta,\mathbf{q}}=\int_{r}^{R}[m u_{s}l_{c g}\sin(\overline{{\theta}})\quad-m\nu_{s}(l_{p i}+l_{c g}\sin(\overline{{\theta}}))\quad-I_{c g}-m l_{c g}^{2}]\mathrm{d}s
|
||||
\mathbf{I}_{\beta,2}=\int_{r}^{R}[m u_{s}^{2}\quad m\nu_{s}^{2}]\mathrm{d}s,\quad\mathbf{I}_{\beta,\mathbf{q}}=\int_{r}^{R}[m u_{s}l_{c g}\sin(\overline{{\theta}})\quad-m\nu_{s}(l_{p i}+l_{c g}\sin(\overline{{\theta}}))\quad-I_{c g}-m l_{c g}^{2}]\mathrm{d}s\tag {87}
|
||||
$$
|
||||
|
||||
$$
|
||||
I_{\beta,\phi,s}=\int_{r}^{R}m w_{0}(l_{p i}+l_{c g}\cos(\overline{{\theta}}))\mathrm{d}s,\quad I_{\beta,\phi,c}=\int_{r}^{R}m w_{0}l_{c g}\sin(\overline{{\theta}})\mathrm{d}s,\quad m_{u\nu}=\int_{r}^{R}m u_{s}\nu_{s}\mathrm{d}s
|
||||
I_{\beta,\phi,s}=\int_{r}^{R}m w_{0}(l_{p i}+l_{c g}\cos(\overline{{\theta}}))\mathrm{d}s,\quad I_{\beta,\phi,c}=\int_{r}^{R}m w_{0}l_{c g}\sin(\overline{{\theta}})\mathrm{d}s,\quad m_{u\nu}=\int_{r}^{R}m u_{s}\nu_{s}\mathrm{d}s\tag {88}
|
||||
$$
|
||||
|
||||
$$
|
||||
\mathbf{f}_{\beta,\mathbf{q}}=\int_{r}^{R}[m u_{s}(l_{p i}+l_{c g}\cos(\overline{{\theta}}))\quad m v_{s}l_{c g}\sin(\overline{{\theta}})\quad0]\mathrm{d}s,\quad f_{\beta,\ddot{\phi},s} = \int_r^R mu_s w_0 ds, \quad f_{\beta,\ddot{\phi},c} = \int_r^R mv_s w_0 ds
|
||||
\mathbf{f}_{\beta,\mathbf{q}}=\int_{r}^{R}[m u_{s}(l_{p i}+l_{c g}\cos(\overline{{\theta}}))\quad m v_{s}l_{c g}\sin(\overline{{\theta}})\quad0]\mathrm{d}s,\quad f_{\beta,\ddot{\phi},s} = \int_r^R mu_s w_0 ds, \quad f_{\beta,\ddot{\phi},c} = \int_r^R mv_s w_0 ds\tag {89}
|
||||
$$
|
||||
|
||||
$$
|
||||
\mathbf{M}_{ext,0} = \int_r^R [0 \quad l_{pi}f_{v,s} \quad M_s]ds, \quad \mathbf{M}_{ext,1} = \int_r^R \begin{bmatrix} 0 & u_s f_{v,s} & 0 \\ -v_s f_{u,s} & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} ds
|
||||
\mathbf{M}_{ext,0} = \int_r^R [0 \quad l_{pi}f_{v,s} \quad M_s]ds, \quad \mathbf{M}_{ext,1} = \int_r^R \begin{bmatrix} 0 & u_s f_{v,s} & 0 \\ -v_s f_{u,s} & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} ds\tag {90}
|
||||
$$
|
||||
|
||||
$$
|
||||
f_{\beta,\phi,s}=\int_{r}^{R}m\Big(\frac{1}{2}(l_{p i}^{2}-l_{c g}^{2})+l_{c g}^{2}\cos^{2}(\overline{{\theta}}\big)+l_{c g}l_{p i}\cos(\overline{{\theta}})\Big)\mathrm{d}s
|
||||
f_{\beta,\phi,s}=\int_{r}^{R}m\Big(\frac{1}{2}(l_{p i}^{2}-l_{c g}^{2})+l_{c g}^{2}\cos^{2}(\overline{{\theta}}\big)+l_{c g}l_{p i}\cos(\overline{{\theta}})\Big)\mathrm{d}s\tag {91}
|
||||
$$
|
||||
|
||||
$$
|
||||
f_{\beta,\phi,c}=\int_{r}^{R}m(l_{c g}^{2}\cos(\overline{{\theta}})\sin(\overline{{\theta}}\,)\,+l_{c g}l_{p i}\sin(\overline{{\theta}}\,))\mathrm{d}s
|
||||
f_{\beta,\phi,c}=\int_{r}^{R}m(l_{c g}^{2}\cos(\overline{{\theta}})\sin(\overline{{\theta}}\,)\,+l_{c g}l_{p i}\sin(\overline{{\theta}}\,))\mathrm{d}s\tag {92}
|
||||
$$
|
||||
|
||||
$$
|
||||
\mathbf{f}_{\beta,\phi,s}=\int_{r}^{R}{\left[\begin{array}{c}{m u_{s}(l_{p i}+l_{e g}\cos(\overline{{\theta}}))}\\ {-m\nu_{s}l_{c g}\sin(\overline{{\theta}})}\\ {0}\end{array}\right]}^{\mathrm{T}}{\mathrm{d}}s,\quad\mathbf{f}_{\beta,\phi,c}=\int_{r}^{R}{\left[\begin{array}{c}{m u_{s}l_{c g}\sin(\overline{{\theta}})}\\ {m\nu_{s}(l_{p i}+l_{c g}\cos(\overline{{\theta}}))}\\ {0}\end{array}\right]}^{\mathrm{T}}{\mathrm{d}}s
|
||||
\mathbf{f}_{\beta,\phi,s}=\int_{r}^{R}{\left[\begin{array}{c}{m u_{s}(l_{p i}+l_{e g}\cos(\overline{{\theta}}))}\\ {-m\nu_{s}l_{c g}\sin(\overline{{\theta}})}\\ {0}\end{array}\right]}^{\mathrm{T}}{\mathrm{d}}s,\quad\mathbf{f}_{\beta,\phi,c}=\int_{r}^{R}{\left[\begin{array}{c}{m u_{s}l_{c g}\sin(\overline{{\theta}})}\\ {m\nu_{s}(l_{p i}+l_{c g}\cos(\overline{{\theta}}))}\\ {0}\end{array}\right]}^{\mathrm{T}}{\mathrm{d}}s\tag {93}
|
||||
$$
|
||||
|
||||
$$
|
||||
\mathbf{f}_{\beta,\phi,2}=\int_{r}^{R}{\left[\begin{array}{l}{1/2m u_{s}^{2}}\\ {-1/2m v_{s}^{2}}\end{array}\right]}^{\mathrm{T}}\mathrm{d}s,\quad\mathbf{f}_{\beta,g,2,s}=\int_{r}^{R}{\left[\begin{array}{c}{m u_{s}}\\ {0}\\ {-m\theta_{s}l_{c g}\sin(\overline{{\theta}})\mathrm{.}}\end{array}\right]}^{\mathrm{T}}\mathrm{d}s,\quad\mathbf{f}_{\beta,g,2,c}=\int_{r}^{R}{\left[\begin{array}{c}{0}\\ {m v_{s}}\\ {m\theta_{s}l_{c g}\cos(\overline{{\theta}})}\end{array}\right]}^{\mathrm{T}}\mathrm{d}s
|
||||
\mathbf{f}_{\beta,\phi,2}=\int_{r}^{R}{\left[\begin{array}{l}{1/2m u_{s}^{2}}\\ {-1/2m v_{s}^{2}}\end{array}\right]}^{\mathrm{T}}\mathrm{d}s,\quad\mathbf{f}_{\beta,g,2,s}=\int_{r}^{R}{\left[\begin{array}{c}{m u_{s}}\\ {0}\\ {-m\theta_{s}l_{c g}\sin(\overline{{\theta}})\mathrm{.}}\end{array}\right]}^{\mathrm{T}}\mathrm{d}s,\quad\mathbf{f}_{\beta,g,2,c}=\int_{r}^{R}{\left[\begin{array}{c}{0}\\ {m v_{s}}\\ {m\theta_{s}l_{c g}\cos(\overline{{\theta}})}\end{array}\right]}^{\mathrm{T}}\mathrm{d}s\tag {94}
|
||||
$$
|
||||
|
||||
$$
|
||||
f_{\beta,\phi,2}\!=\!\int_{r}^{R}m u_{s}\nu_{s}\mathrm{d}s,\quad f_{\beta,g,s}\!=\!\int_{r}^{R}m(l_{p i}+l_{c g}\cos(\overline{{\theta}}))\mathrm{d}s,\quad f_{\beta,g,c}\!=\!\int_{r}^{R}m l_{c g}\sin(\overline{{\theta}})\mathrm{d}s
|
||||
f_{\beta,\phi,2}\!=\!\int_{r}^{R}m u_{s}\nu_{s}\mathrm{d}s,\quad f_{\beta,g,s}\!=\!\int_{r}^{R}m(l_{p i}+l_{c g}\cos(\overline{{\theta}}))\mathrm{d}s,\quad f_{\beta,g,c}\!=\!\int_{r}^{R}m l_{c g}\sin(\overline{{\theta}})\mathrm{d}s\tag {95}
|
||||
$$
|
||||
|
||||
# Rotor Speed Model
|
||||
@ -1044,29 +1042,29 @@ f_{\phi,\ddot{\mathbf{q}}}(\ddot{\mathbf{q}}, \beta) = I_{\phi,u,1}\ddot{u}_t \c
|
||||
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}
|
||||
\mathbf{f}_{ext,1}(\mathbf{q}, \beta) = f_{ext,1,v}v_t \sin(\beta) - f_{ext,1,u}u_t \cos(\beta) \tag {96}
|
||||
$$
|
||||
|
||||
where the constants are
|
||||
|
||||
$$
|
||||
I_{\phi}=\int_{r}^{R}m w_{0}^{2},\quad I_{\phi,u,0}=\int_{r}^{R}m w_{0}\,(l_{p i}+l_{c g}\cos(\overline{{\theta}}))\mathrm{d}s,\quad I_{\phi,v,0}=\int_{r}^{R}m w_{0}l_{c g}\sin(\overline{{\theta}})\mathrm{d}s
|
||||
I_{\phi}=\int_{r}^{R}m w_{0}^{2},\quad I_{\phi,u,0}=\int_{r}^{R}m w_{0}\,(l_{p i}+l_{c g}\cos(\overline{{\theta}}))\mathrm{d}s,\quad I_{\phi,v,0}=\int_{r}^{R}m w_{0}l_{c g}\sin(\overline{{\theta}})\mathrm{d}s\tag {97}
|
||||
$$
|
||||
|
||||
$$
|
||||
I_{\phi,u,1}\!=\!\int_{r}^{R}m w_{0}u_{s}\mathrm{d}s,\quad I_{\phi,v,1}\!=\!\int_{r}^{R}m w_{0}\nu_{s}\mathrm{d}s,\quad f_{\phi,g,u,1}\!=\!\int_{r}^{R}g m u_{s}\mathrm{d}s,\quad f_{\phi,g,v,1}\!=\!\int_{r}^{R}g m\nu_{s}\mathrm{d}s
|
||||
I_{\phi,u,1}\!=\!\int_{r}^{R}m w_{0}u_{s}\mathrm{d}s,\quad I_{\phi,v,1}\!=\!\int_{r}^{R}m w_{0}\nu_{s}\mathrm{d}s,\quad f_{\phi,g,u,1}\!=\!\int_{r}^{R}g m u_{s}\mathrm{d}s,\quad f_{\phi,g,v,1}\!=\!\int_{r}^{R}g m\nu_{s}\mathrm{d}s\tag {98}
|
||||
$$
|
||||
|
||||
$$
|
||||
f_{\phi,g,0}=\int_{r}^{R}g m w_{0}\mathrm{d}s,\quad f_{\phi,g,u,0}=\int_{r}^{R}g m(l_{p i}+l_{c g}\cos(\overline{{\theta}}))\mathrm{d}s,\quad f_{\phi,g,\nu,0}=\int_{r}^{R}g m l_{c g}\sin(\overline{{\theta}})\mathrm{d}s
|
||||
f_{\phi,g,0}=\int_{r}^{R}g m w_{0}\mathrm{d}s,\quad f_{\phi,g,u,0}=\int_{r}^{R}g m(l_{p i}+l_{c g}\cos(\overline{{\theta}}))\mathrm{d}s,\quad f_{\phi,g,\nu,0}=\int_{r}^{R}g m l_{c g}\sin(\overline{{\theta}})\mathrm{d}s\tag {99}
|
||||
$$
|
||||
|
||||
$$
|
||||
f_{e x t,0,s}=\int_{r}^{R}[w_{0}\,f_{u,s}\quad0\quad-l_{p i}\,f_{w,s}]\mathrm{d}s,\quad f_{e x t,0,c}=\int_{r}^{R}[0\quad-w_{0}\,f_{v,s}\quad0]\mathrm{d}s
|
||||
f_{e x t,0,s}=\int_{r}^{R}[w_{0}\,f_{u,s}\quad0\quad-l_{p i}\,f_{w,s}]\mathrm{d}s,\quad f_{e x t,0,c}=\int_{r}^{R}[0\quad-w_{0}\,f_{v,s}\quad0]\mathrm{d}s\tag {100}
|
||||
$$
|
||||
|
||||
$$
|
||||
f_{e x t,1,u}=\int_{r}^{R}u_{s}\,f_{w,s}\mathrm{d}s,\quad f_{e x t,1,v}=\int_{r}^{R}u_{s}\,f_{w,s}d s,\quad f_{e x t,1,u}=\int_{r}^{R}u_{s}\,f_{w,s}\mathrm{d}s,\quad f_{e x t,1,v}=\int_{r}^{R}\nu_{s}\,f_{w,s}\mathrm{d}s
|
||||
f_{e x t,1,u}=\int_{r}^{R}u_{s}\,f_{w,s}\mathrm{d}s,\quad f_{e x t,1,v}=\int_{r}^{R}u_{s}\,f_{w,s}d s,\quad f_{e x t,1,u}=\int_{r}^{R}u_{s}\,f_{w,s}\mathrm{d}s,\quad f_{e x t,1,v}=\int_{r}^{R}\nu_{s}\,f_{w,s}\mathrm{d}s\tag {101}
|
||||
$$
|
||||
|
||||
# References
|
||||
|
||||
Loading…
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Reference in New Issue
Block a user