diff --git a/copilot-conversations/20250905_092042__[[Kallesøe-Equations_of_motion_for_a_rotor_blade]]方程9如何得到.md b/copilot-conversations/20250905_092042__[[Kallesøe-Equations_of_motion_for_a_rotor_blade]]方程9如何得到.md index 263c150..c2efb6c 100644 --- a/copilot-conversations/20250905_092042__[[Kallesøe-Equations_of_motion_for_a_rotor_blade]]方程9如何得到.md +++ b/copilot-conversations/20250905_092042__[[Kallesøe-Equations_of_motion_for_a_rotor_blade]]方程9如何得到.md @@ -212,7 +212,7 @@ $m\big(\ddot{u}-\ddot{\theta}l_{c g}\sin(\overline{{\theta}})\big)+F_{u,1}+F_{u, * 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。 + * 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。 diff --git a/copilot-conversations/20250908_092205__[[Kallesøe-Equations_of_motion_for_a_rotor_blade]]方程13-14如何得到？以及解释方程各项的意义，.md b/copilot-conversations/20250908_092205__[[Kallesøe-Equations_of_motion_for_a_rotor_blade]]方程13-14如何得到？以及解释方程各项的意义，.md new file mode 100644 index 0000000..25579fc --- /dev/null +++ b/copilot-conversations/20250908_092205__[[Kallesøe-Equations_of_motion_for_a_rotor_blade]]方程13-14如何得到？以及解释方程各项的意义，.md @@ -0,0 +1,245 @@ +--- +epoch: 1757294525893 +modelKey: gemini-2.5-pro|google +tags: + - copilot-conversation +--- + +**user**: [[Kallesøe-Equations of motion for a rotor blade]]方程13-14如何得到？以及解释方程各项的意义， +[Timestamp: 2025/09/08 09:22:05] + +**ai**: 好的，我们来分析一下您在笔记 [[Kallesøe-Equations of motion for a rotor blade]] 中提到的方程(13)和(14)的推导和物理意义。 + +### **方程的推导方法** + +根据笔记中的“Derivation of the Equations of Motion”章节，这些运动方程并不是孤立推导的，而是通过一个统一的、基于能量的方法——**扩展的哈密顿原理 (extended Hamilton’s principle)** 推导出来的。 + +具体步骤如下： +1. **建立能量表达式**：首先，为整个系统（包括叶片、轮毂、传动链等）建立动能 $T$ 和势能 $V$ 的表达式。势能包括弹性应变能 $V_{ela}$ 和重力势能 $V_{gra}$。 +2. **考虑非保守力**：通过虚功原理，引入气动力、发电机转矩等非保守力所做的功 $\delta Q$。 +3. **应用哈密顿原理**：哈密顿原理指出，在任意时间段内，系统的真实运动路径使得作用量积分 $H = \int (T - V) dt$ 的变分为零。扩展的哈密顿原理将非保守力也包含进来，即要求作用量积分的变分 $\delta H\equiv\int_{t_{1}}^{t_{2}}(\delta T-\delta V_{e l a}-\delta V_{g r a}+\delta Q)d t$ 对任何容许的变分都为零。 +4. **导出运动方程**：通过对系统的广义坐标（如 $u, \nu, \theta, \beta, \phi$）进行变分，从上述积分方程中就可以导出一系列偏微分运动方程和相应的边界条件。 + +方程(12)是叶片弯曲运动的总方程。而**方程(13)和(14)是方程(12)中的具体项**，它们分别代表了**变桨运动**和**风轮旋转**对叶片弯曲运动产生的各种惯性力（包括离心力、科里奥利力等）。这些项是在对总动能表达式（方程(9)）进行变分和求导的过程中自然产生的。 + +--- + +### **方程 (13) 各项的物理意义 (变桨作用的影响)** + +方程(13)描述了由于变桨运动（即桨距角 $\beta$ 的变化）而在叶片局部坐标系 $(x,y,z)$ 中产生的惯性力。我们以 $x$ 方向的力 $F_{u,1}$ (方程13a) 为例进行解释： + +$F_{u,1}\!=\!-\ddot{\beta}m\nu_{c g}-\dot{\beta}^{2}m u_{c g}-2\dot{\beta}m\dot{\nu}_{c g}+\left(T_{1}l_{c g}\cos(\overline{{{\theta}}})\right)^{\prime}+\left((u^{\prime}+l_{p i}^{\prime})\!\int_{s}^{R}T_{1}\!\mathrm{d}\rho\right)^{\prime}$ + +* **第一项: $-\ddot{\beta}m\nu_{c g}$** + * **物理意义**: 这是与变桨角加速度 $\ddot{\beta}$ 相关的**虚拟力 (fictitious force)**。当叶片绕其纵轴（z轴）加速转动时，对于叶片上每个质量微元 $m$，如果其在 $y$ 方向有偏移 $\nu_{cg}$，就会在 $x$ 方向产生一个惯性力。 + +* **第二项: $-\dot{\beta}^{2}m u_{c g}$** + * **物理意义**: 这是与变桨角速度 $\dot{\beta}$ 相关的**离心力 (centrifugal force)**。由于叶片绕 z 轴以角速度 $\dot{\beta}$ 旋转，位于 $x$ 方向偏移量为 $u_{cg}$ 的质量微元会受到一个指向旋转中心的离心力，该力在 $x$ 方向的分量即为此项。 + +* **第三项: $-2\dot{\beta}m\dot{\nu}_{c g}$** + * **物理意义**: 这是**科里奥利力 (Coriolis force)**。它产生于变桨旋转（角速度 $\dot{\beta}$）和质心在 $y$ 方向的相对速度 $\dot{\nu}_{cg}$ 的相互作用。 + +* **第四项: $\left(T_{1}l_{c g}\cos(\overline{{{\theta}}})\right)^{\prime}$** + * **物理意义**: 这是一个由力矩产生的力。$T_1$ 是一个与变桨和风轮旋转耦合的科里奥利力（在z方向）。这个力作用在偏离弹性轴的重心上（距离为 $l_{cg}$），产生了一个力矩。该项是这个力矩沿叶片展向（s方向）的**空间导数**，代表了力矩变化引起的剪力。 + +* **第五项: $\left((u^{\prime}+l_{p i}^{\prime})\!\int_{s}^{R}T_{1}\!\mathrm{d}\rho\right)^{\prime}$** + * **物理意义**: 这是一个由**弯矩**产生的力。积分项 $\int_{s}^{R}T_{1}\!\mathrm{d}\rho$ 表示从当前截面 $s$ 到叶尖 $R$ 的所有 $z$ 方向科里奥利力 $T_1$ 的总和。这个合力作用在变形后的叶片上（其斜率为 $u^{\prime}+l_{p i}^{\prime}$），产生了一个弯矩。最外层的导数 `(')` 将这个弯矩的变化转换为了作用在截面上的剪力。 + +--- + +### **方程 (14) 各项的物理意义 (风轮转速的影响)** + +方程(14)描述了由于整个风轮的旋转（即方位角 $\phi$ 的变化）而在叶片局部坐标系中产生的惯性力。我们同样以 $x$ 方向的力 $F_{u,2}$ (方程14a) 为例进行解释： + +$F_{u,2} = -\dot{\phi}^{2}m\hat{u}_{cg}\cos(\beta) - \left[\dot{\phi}^{2}\left(m l_{c g}w_{0}\cos(\overline{\theta}) - \theta\sin(\overline{\theta})\right)\right]' - \left(l_{cg}T_{2}\right)' \cos(\overline{\theta}) - 2\dot{\phi}m l_{c g}\left(\dot{u}'\cos(\overline{\theta}) + \dot{\nu}'\sin(\overline{\theta})\right)\cos(\beta) - \left[\Big(u' + l_{pi}'\Big)\int_{s}^{R}\left(\dot{\phi}^{2}m w_{0} + T_{2}\right)\mathrm{d}\rho\right]'$ + +* **第一项: $-\dot{\phi}^{2}m\hat{u}_{cg}\cos(\beta)$** + * **物理意义**: 这是风轮旋转产生的最主要的**离心力**。$\dot{\phi}^2$ 是风轮旋转角速度的平方，$\hat{u}_{cg}$ 是质心在轮毂坐标系 $(\hat{x},\hat{y},\hat{z})$ 中的 $\hat{x}$ 坐标。这一项表示由于风轮旋转，叶片质量微元受到的离心力在叶片局部坐标系 $x$ 方向上的投影。 + +* **第二项和第三项**: + * **物理意义**: 这两项与方程(13)的第四项类似，是由于离心力和科里奥利力 $T_2$ 作用在偏离弹性轴的重心上而产生的力矩的空间导数。 + +* **第四项: $- 2\dot{\phi}m l_{c g}\left(\dot{u}'\cos(\overline{\theta}) + \dot{\nu}'\sin(\overline{\theta})\right)\cos(\beta)$** + * **物理意义**: 这是**科里奥利力**。它产生于风轮旋转（角速度 $\dot{\phi}$）和质心沿叶片轴向（$\hat{z}$ 方向）的相对速度的相互作用。 + +* **第五项: $- \left[\Big(u' + l_{pi}'\Big)\int_{s}^{R}\left(\dot{\phi}^{2}m w_{0} + T_{2}\right)\mathrm{d}\rho\right]'$** + * **物理意义**: 这是一个由**弯矩**产生的力，结构上与方程(13)最后一项非常相似。积分项代表了从当前截面 $s$ 到叶尖 $R$ 的所有离心力（$\dot{\phi}^{2}m w_{0}$）和科里奥利力（$T_2$）的总和。这个合力作用在变形叶片上产生弯矩，其空间导数表现为剪力。其中，$\dot{\phi}^{2}m w_{0}$ 这一部分是著名的**离心刚度 (centrifugal stiffness)**效应的来源，它会使旋转的叶片变得更“硬”，抵抗弯曲变形。 + +总结来说，方程(13)和(14)详细地分解了变桨运动和风轮旋转所引入的各种复杂的惯性力项，这些力是导致叶片挥舞、摆振和扭转运动之间耦合的重要原因。 +[Timestamp: 2025/09/08 09:22:53] + +**user**: 方程12由$\delta H\equiv\int_{\mathbf{t_{1}}}^{\mathbf{t_{2}}}(\delta T-\delta V_{e l a}-\delta V_{g r a}+\delta Q)d t$ 推导得出？ +[Timestamp: 2025/09/08 09:27:59] + +**ai**: 是的，完全正确。 + +根据您提供的笔记 [[Kallesøe-Equations of motion for a rotor blade]]，**方程(12)正是通过扩展的哈密顿原理 (extended Hamilton’s principle)，即令作用量积分的变分 $\delta H = 0$ 推导出来的**。 + +笔记中的 **"Equations of Motion"** 章节明确描述了这一过程： + +> By demanding that any admissible variation of the action integral $\delta H\equiv\int_{\mathbf{t_{1}}}^{\mathbf{t_{2}}}(\delta T-\delta V_{e l a}-\delta V_{g r a}+\delta Q)d t$ is zero, a set of partial differential equations of motion and a set of boundary condition equations are derived (extended Hamilton’s principle11). + +这段话的核心思想是：系统的真实运动路径必须使作用量积分 $H$ 的一阶变分为零。这是一个功能强大的变分原理，可以将整个系统的动力学问题归结为一个单一的积分方程。 + +### 推导过程简述如下： + +1. **建立能量和虚功的表达式**： + * **动能 T**: 由方程(9)给出，包含了轮毂、传动链以及叶片各截面平动和转动的动能。它是广义坐标 $(u, \nu, \theta, \beta, \phi)$ 及其对时间导数 $(\dot{u}, \dot{\nu}, ...)$ 的复杂函数。 + * **弹性势能 $V_{ela}$**: 由方程(6)的积分形式给出，主要与叶片的弯曲和扭转变形有关，是广义坐标对空间导数 $(u', u'', \nu', \nu'', \theta')$ 的函数。 + * **重力势能 $V_{gra}$**: 由方程(7)给出，与叶片质心的空间位置有关。 + * **非保守力虚功 $\delta Q$**: 由方程(10)给出，包含了外部气动力、发电机转矩和变桨力矩所做的功。 + +2. **进行变分运算**： + * 对作用量积分 $H$ 中的所有广义坐标 $(u, \nu, \theta, \beta, \phi)$ 进行变分（例如，用 $u + \delta u$ 替换 $u$）。 + * 这个过程会涉及到大量的链式法则求导和**分部积分 (integration by parts)**。例如，对 $\delta \dot{u}$ 进行时间上的分部积分，会得到包含 $\delta u$ 的项和边界项；对 $\delta u'$ 进行空间上的分部积分，会得到包含 $\delta u$ 的项和边界项。 + +3. **分离并应用变分基本引理**： + * 经过整理，整个变分方程 $\delta H = 0$ 会变成类似如下的形式： + $\int_{t_1}^{t_2} \int_{r}^{R} \left[ (\text{...}) \delta u + (\text{...}) \delta \nu + (\text{...}) \delta \theta \right] ds dt + \text{...其他项...} = 0$ + * 根据**变分法基本引理 (fundamental lemma of calculus of variations)**，因为变分 $\delta u, \delta \nu, \delta \theta$ 等是任意的，所以要使整个积分为零，必须保证每一项变分前的系数（即括号内的表达式）恒等于零。 + +4. **得到运动方程**： + * 令 $\delta u$ 的系数等于零，就得到了关于 $u$ 方向运动的偏微分方程，即**方程(12a)**。 + * 令 $\delta \nu$ 的系数等于零，就得到了关于 $\nu$ 方向运动的偏微分方程，即**方程(12b)**。 + * 同理，令 $\delta \theta$ 的系数等于零，会得到扭转运动方程(18)；令 $\delta \beta$ 和 $\delta \phi$ 的系数等于零，则分别得到变桨和风轮旋转的运动方程(27)和(33)。 + +因此，方程(12)是哈密顿原理在叶片弯曲自由度 $u$ 和 $\nu$ 上的具体体现，它系统地包含了所有惯性力、弹性恢复力、重力以及外部载荷对这两个方向运动的影响。 +[Timestamp: 2025/09/08 09:28:29] + +**user**: 展示详细的推导过程 +[Timestamp: 2025/09/08 09:29:53] + +**ai**: 好的，展示从扩展的哈密顿原理推导方程(12)的详细过程是一项非常复杂的任务，因为它涉及到大量的符号运算、坐标变换和变分法。实际上，这篇论文本身也省略了完整的代数推导，因为它会占据几十页的篇幅。 + +然而，我可以为您**分步展示这个推导的逻辑框架和关键步骤**，让您清楚地看到方程(12)中每一组项是如何从哈密顿原理中的能量项和功的表达式中产生的。 + +### **推导的起点：扩展的哈密顿原理** + +我们的出发点是作用量积分的变分为零： +$$ +\delta H = \int_{t_1}^{t_2} (\delta T - \delta V_{ela} - \delta V_{gra} + \delta Q) dt = 0 +$$ +为了得到关于叶片弯曲位移 $u$ 和 $\nu$ 的运动方程，我们需要计算出每一项的变分，然后收集所有与 $\delta u$ 和 $\delta \nu$ 相关的项，并令它们的系数为零。 + +--- + +### **第一步：动能的变分 ($\delta T$) → 惯性项** + +动能表达式为 (方程9): +$$ +T = \frac{1}{2}J_{gen}\dot{\phi}^2 + \int_r^R \left( \frac{1}{2}m (\dot{\mathbf{r}}_{cg} \cdot \dot{\mathbf{r}}_{cg}) + \frac{1}{2}I_{cg}(\dot{\beta}+\dot{\theta})^2 \right) ds +$$ +我们关注与弯曲相关的项，主要是质心速度平方项 $\frac{1}{2}m (\dot{\mathbf{r}}_{cg} \cdot \dot{\mathbf{r}}_{cg})$。 + +1. **计算变分 $\delta T$**: + 对 $T$ 进行变分，主要部分是： + $$ + \delta T = \int_r^R m (\dot{\mathbf{r}}_{cg} \cdot \delta\dot{\mathbf{r}}_{cg}) ds + \text{... (其他项)} + $$ + +2. **对时间进行分部积分**: + 将 $\delta T$ 代入哈密顿原理的积分中，并对时间进行分部积分，以消去变分的速度项 $\delta\dot{\mathbf{r}}_{cg}$，从而得到位移的变分项 $\delta\mathbf{r}_{cg}$： + $$ + \int_{t_1}^{t_2} \left( \int_r^R m \dot{\mathbf{r}}_{cg} \cdot \delta\dot{\mathbf{r}}_{cg} ds \right) dt = \int_r^R \left( [m \dot{\mathbf{r}}_{cg} \cdot \delta\mathbf{r}_{cg}]_{t_1}^{t_2} - \int_{t_1}^{t_2} m \ddot{\mathbf{r}}_{cg} \cdot \delta\mathbf{r}_{cg} dt \right) ds + $$ + 由于在时间边界 $t_1$ 和 $t_2$ 上的变分为零，第一项消失。于是，动能变分在作用量积分中最终贡献的项是： + $$ + \int_{t_1}^{t_2} \delta T dt = - \int_{t_1}^{t_2} \int_r^R (m \ddot{\mathbf{r}}_{cg} \cdot \delta\mathbf{r}_{cg}) ds dt + \text{...} + $$ + 这里的核心是得到了**质心加速度 $\ddot{\mathbf{r}}_{cg}$**。 + +3. **展开加速度项**: + 质心位置矢量 $\mathbf{r}_{cg}$ (方程8) 是一个关于所有广义坐标 $(u, \nu, \theta, \beta, \phi)$ 的极其复杂的函数。对其求二次时间导数 $\ddot{\mathbf{r}}_{cg}$ 会产生大量的项，包括： + * **平动加速度**: $\ddot{u}, \ddot{\nu}$ 等。 + * **离心加速度**: 与 $\dot{\phi}^2$ 和 $\dot{\beta}^2$ 成正比的项。 + * **科里奥利加速度**: 与 $\dot{\phi}\dot{u}$, $\dot{\phi}\dot{\nu}$, $\dot{\beta}\dot{u}$, $\dot{\beta}\dot{\nu}$ 等成正比的项。 + * **切向加速度**: 与 $\ddot{\phi}$ 和 $\ddot{\beta}$ 成正比的项。 + +4. **投影到 $\delta u$ 和 $\delta \nu$**: + 最后，将内积 $m \ddot{\mathbf{r}}_{cg} \cdot \delta\mathbf{r}_{cg}$ 展开。$\delta\mathbf{r}_{cg}$ 可以表示为 $\frac{\partial \mathbf{r}_{cg}}{\partial u}\delta u + \frac{\partial \mathbf{r}_{cg}}{\partial \nu}\delta \nu + ...$。收集所有 $\delta u$ 和 $\delta \nu$ 的系数。 + * **结果**: 这些系数构成了方程(12)左侧所有的**惯性力项**，包括： + * $m(\ddot{u} - \ddot{\theta}l_{cg}\sin(\overline{\theta}))$ 和 $m(\ddot{\nu} + \ddot{\theta}l_{cg}\cos(\overline{\theta}))$ + * $F_{u,1}, F_{\nu,1}$ (来自变桨运动的惯性力) + * $F_{u,2}, F_{\nu,2}$ (来自风轮旋转的惯性力) + * $F_{u,5}, F_{\nu,5}$ (来自风轮角加速度的惯性力) + +--- + +### **第二步：弹性势能的变分 ($-\delta V_{ela}$) → 弹性恢复力** + +弹性势能的变分由方程(6)给出： +$$ +\delta V_{ela} = \int_{r}^{R}\iint_{A}{(\sigma_{s s}\delta\varepsilon_{s s}+\sigma_{s\eta}\delta\varepsilon_{s\eta}+\sigma_{s\xi}\delta\varepsilon_{s\xi})\mathrm{d}\eta\mathrm{d}\xi\mathrm{d}s} +$$ +1. **关联应变和位移**: + 根据方程(5)，工程应变 $\varepsilon_{ss}$ 与曲率 $u''$ 和 $\nu''$ 直接相关。因此，应变的变分 $\delta\varepsilon_{ss}$ 与曲率的变分 $\delta u''$ 和 $\delta \nu''$ 相关。 + +2. **引入弯矩**: + 将应力-应变关系代入，并对截面进行积分，可以将变分写成弯矩 $M$ 和曲率变分的形式： + $$ + \delta V_{ela} = \int_r^R (M_{\eta} \delta u'' + M_{\xi} \delta \nu'' + \text{... coupling terms ...}) ds + $$ + 其中 $M_{\eta}$ 和 $M_{\xi}$ 是关于主轴的弯矩，它们是弯曲刚度 (如 $EI_{\xi}$) 和曲率 $(u'', \nu'')$ 的函数。 + +3. **对空间进行两次分部积分**: + 为了从 $\delta u''$ 得到 $\delta u$，我们需要对空间变量 $s$ 进行两次分部积分。以 $u$ 项为例： + $$ + \int_r^R M_{\eta} \delta u'' ds = [M_{\eta} \delta u']_r^R - \int_r^R M_{\eta}' \delta u' ds + $$ + 再次分部积分： + $$ + = [M_{\eta} \delta u']_r^R - [M_{\eta}' \delta u]_r^R + \int_r^R M_{\eta}'' \delta u ds + $$ + 边界项将用于推导边界条件(方程26)，而积分项则进入运动方程。 + +4. **结果**: + 经过整理，$-\delta V_{ela}$ 在作用量积分中贡献的项为 $-\int_r^R (M_{\eta}'' \delta u + M_{\xi}'' \delta \nu + ...) ds$。这些项就是方程(12)中的**弹性恢复力** $F_{u,4}$ 和 $F_{\nu,4}$。 + +--- + +### **第三步：重力势能的变分 ($-\delta V_{gra}$) → 重力项** + +重力势能为 (方程7): +$$ +V_{gra} = \int_r^R m \mathbf{g} \cdot \mathbf{r}_{cg} ds +$$ +1. **计算变分**: + $$ + \delta V_{gra} = \int_r^R m \mathbf{g} \cdot \delta\mathbf{r}_{cg} ds + $$ +2. **展开并收集系数**: + 与动能部分类似，将 $\delta\mathbf{r}_{cg}$ 展开为 $\frac{\partial \mathbf{r}_{cg}}{\partial u}\delta u + \frac{\partial \mathbf{r}_{cg}}{\partial \nu}\delta \nu + ...$，然后收集 $\delta u$ 和 $\delta \nu$ 的系数。 +3. **结果**: + 项 $-\delta V_{gra}$ 贡献了方程(12)中的**重力项** $F_{u,3}$ 和 $F_{\nu,3}$。 + +--- + +### **第四步：非保守力的虚功 ($\delta Q$) → 外力项** + +虚功表达式为 (方程10): +$$ +\delta Q = ... + \int_r^R (\mathbf{f}^T \cdot \delta\mathbf{r}_{ea} + M\delta(\theta+\beta)) ds +$$ +1. **展开虚位移**: + 弹性轴位置的变分 $\delta\mathbf{r}_{ea}$ 同样可以展开为关于 $\delta u, \delta \nu$ 等的表达式。 +2. **结果**: + $\delta Q$ 直接给出了方程(12)的**右侧项**，即广义外力。 + * $f_u$ 和 $f_\nu$ 直接来自于 $\mathbf{f}^T \cdot \delta\mathbf{r}_{ea}$ 的展开。 + * 右侧的积分项，如 $\Big((u^{\prime}+l_{p i}^{\prime})\int_{s}^{R}f_{w}\mathrm{d}\rho\Big)^{\prime}$，来自于一个更微妙的效应：轴向位移 $w_1$ 的变分。因为 $w_1$ 依赖于 $u'$ (见"Model Description"部分)，$\delta w_1$ 会产生 $\delta u'$ 项。当这个项与轴向力 $f_w$ 产生的虚功结合，并通过分部积分后，就形成了这个看起来复杂的弯矩项。 + +### **总结：组装方程** + +将以上所有步骤的结果代入 $\delta H = 0$，并分别收集所有 $\delta u$ 和 $\delta \nu$ 的系数： + +* **$\delta u$ 的系数**: + $$(-\text{惯性项}) - (\text{弹性项}) - (\text{重力项}) + (\text{外力项}) = 0$$ + 整理后即为方程(12a): + $$ + m\big(\ddot{u}...\big)+F_{u,1}+F_{u,2}+F_{u,3}+F_{u,4}+F_{u,5} = f_{u}+\Big((u^{\prime}...)\Big)^{\prime} + $$ + +* **$\delta \nu$ 的系数**: + 同理，收集所有 $\delta \nu$ 的系数并令其为零，得到方程(12b)。 + +这个过程虽然复杂，但逻辑清晰：哈密顿原理作为一个统一的框架，系统地将所有物理效应（惯性、弹性、重力、外力）自动地分配到最终的运动方程中。 +[Timestamp: 2025/09/08 09:30:50] \ No newline at end of file diff --git a/书籍/力学书籍/GarradAD-Dynamics of wind turbines/auto/GarradAD-Dynamics of wind turbines.md b/书籍/力学书籍/GarradAD-Dynamics of wind turbines/auto/GarradAD-Dynamics of wind turbines.md new file mode 100644 index 0000000..637508c --- /dev/null +++ b/书籍/力学书籍/GarradAD-Dynamics of wind turbines/auto/GarradAD-Dynamics of wind turbines.md @@ -0,0 +1,249 @@ +# Dynamics of wind turbines + +A.D. Garrad, B.A., Ph.D + +Indexing terms: Wind power, Turbogenerators + +Abstract: Three principal aspects of the dynamics of wind-turbine behaviour are discussed: forced response to deterministic loads, forced response to stochastic loads and stability. In each case, an introduction to the physics of the problem is presented and a means of analysis is described. A simple analytical model is derived to help illustrate some of the analytical techniques commonly employed. + +# 1 Introduction + +The most noticeable change in wind-turbine design, during the last ten years, is in the increased flexibility of the structure as a whole. This change is well illustrated by comparison of the NASA MOD-O and the Boeing MOD-2 machines. The most apparent outward sign of this evolution is the change in tower structure, the stiff-truss tower being replaced by a considerably softer cylinderical structure. The tower structure, although the most visible, is not the only component to evolve in this way. The same procedure has taken place with rotors (in some cases, fibre glass and wood are replacing steel) and with transmission systems where stiff, rugged versions are being replaced by flexible ones of various types, both mechanical and electrical. Almost all large two-bladed machines now have teetered hubs rather than rigid ones. There is, at present, no concensus of opinion about the ideal combinationof structural components. The solution adopted depends on the size, duty and location of each machine. Opting for a flexible tower, for instance, does not necessarily imply the use of a flexible transmission system or rotor. + +The increasing structural flexibility of wind turbines means that their dynamic behaviour, and our ability to predict it, becomes more important. Inherent with increased fexibility are large displacements which may give rise to large inertial loads and, in some cases, instabilities. + +This paper reviews some problems that arise in windturbine dynamics and describes methods for their analysis. Only horizontal-axis machines will be treated in any detail. Vertical-axis machines have many similarities, but also many important differences which put their · treatment outside the scope of this review. + +# 2 Basic analysis + +The dynamic problems that are encountered in windturbine systems may be conveniently divided into three separate sections, by means of identifying the different types of forcing functions that are involved. These categoriesare: + +(a) stability $(b)$ forced response to deterministic loads (c) forced response to stochastic loads. + +Stability is a property of the system and may be determined by analysis of the homogeneous equations of motion. Stability analysis is, therefore, characterised by the absence of a forcing function. This is not strictly true, but this definition serves as a means of categorising the nature of theproblem. + +The rotor of a wind turbine operating in the atmosphere will encounter wind velocities that are continuously changing. These changes come from a number of sources. The wind turbine will be controlled to yaw itself into the wind, but there will inevitably be some error giving yaw misalignment, which results in one blade moving into the wind while the others move out of it. A similar effect results from shaft tilt and wind pitch. The presence of the earth's boundary layer means that the air near the ground has a lower velocity than that higher up, resulting in a velocity gradient across the rotor disc. When the wind fows round the tower, it is declerated so that, in the neighbourhood of the tower, there is an area of retarded air. All these sources combine together to give a fairly complicated cyclic variation in wind speed and, hence, load, as the blade rotates; although these loads may be complicated, they are well defined and, hence, are termed \*deterministic'. In addition to this cyclic variation, there will be changes in wind speed resulting from wind turbulence which produces stochastic loads. It is useful to make a distinction between the cyclic, deterministic and the turbulent, stochastic variations. + +The deterministic loads may be calculated relatively easily. Detailed discussion of the process is, however, outside the scope of this article. Most authors adopt a pseudo-steady-state approach to the aerodynamic calculations, the basic principles of which are given in Wilson and Lissaman [1]. This approach is not strictly applicable to cyclic conditions and some consideration is given to proper means of dealing with the periodic nature of the loads by Miller et al. [2]. The cyclic nature of the loads also suggests that unsteady effects such as stall hysteresis may be important; so far this has received little attention for HAWTs, although some work is presently under way; see, for example, Reference 3.A recent review of horizontal-axis wind-turbine aerodynamics may be found in de Vries [4, 5]. + +Assumingthatadequatemeansareavailableforthe determination of the cyclic aerodynamic loads, we may consider the most basic type of dynamic analysis that is available for any form of rotating machinery. The underlying principle involved in designing a turbine against the deterministic loads is the avoidance of resonance. This topic was the subject of a recent review paper by Sullivan [6]. The basic, and indeed essential, means of avoiding the coincidence of a frequency corresponding to the natural mode of the structure and that of a forcing function is the ability to perform reliable calculations of thenatural frequencies of the important modes of vibration of the structure. Nowadays, most designers have finite-element packages available to them for such calculations, and no pioneering work is required for their application to stationary wind-turbine systems. The most convenient way of presenting and interpreting these results is in at interference diagram, sometimes called a Campbell diagram, an example of which is shown in Fig. 1. The finite-element calculations are usually performed for a stationary system, although the frequencies of interest are those corresponding to the natural modes of vibration of the rotating system. Proper calculation of these is fairly complicated and will be dealt with in the following text; however, a reasonably accurate estimate may be obtained by simply allowing for the centrifugal stiffening effects on the rotating blades, which manifest themselves as a change in natural frequency with rotational speed, as shown in Fig. 1. The interference diagram shows the natural frequencies, together with the rotational speed and its harmonics. It is considered prudent to arrange these so that the harmonic rays do not coincide or come near to coinciding with the natural frequencies at the operating speed, thus avoiding possible resonances. + +![](images/24e849e6361b1c876e19f6ccfb1a0040da10e5b432dc5b272cf0bf39c22da3b6.jpg) +Fig.1 Typicalinterference diagram + +In general, each blade will see the cyclic aerodynamic loads at a frequency of once per revolution (or $1P_{,}$ ,and may also be affected by the harmonics. The importance of the harmonics depends on the particular load case considered. The support structure will experience the sum of the blade loads at any particular instant, and, if all the blades are identical, these loads will be modulated so that an $_n$ -bladed rotor will give rise to loads at a frequency of $n P$ and its harmonics on the support structure. Small amplitude loads, at frequencies other than these, will arise because of slight imbalance between the blades. This may be due to manufacturing imperfections or difference in blade setting due to operation of the control system. + +# 3 Dynamic modelling + +Despite the fact that wind-turbine technology is relatively new, there are, in fact, quite a few mathematical models available which attempt to compute the aeroelastic behaviour of HAWTs. These vary from the very simple, such as the closed-form solution described by Stoddard [7] through limited degrees of freedom models, Garrad [8], to very complex systems, of which there are now quite a number; see, for example, Hoffmann [9], Friedmann and Warmbrodt [10], Vollan [11], Fabian [12] and Thurgood.t + +An excellent introduction to this subject may be found in the substantial text [13] developed by the Massachusetts Institute of Technology, which considers aerodynamics and transmission systems as well as general aeroelastic problems. A very good idea of the state of the art may be obtained from Reference 14. An overview of the subject is also presented by Thresher [15]. + +Any dynamic analysis of a wind turbine that addresses the problems of stability or forced response will rapidly encounter some common difficulties. Many engineers, faced with problems in structural dynamics, will immediately consider the use of one of the finite-element packages that are now very widely available. However, a particularly interesting aspect of dynamic modelling of a complete wind-turbine system is the fact that gross movement of one part of the structure occurs relative to another part. This precludes the use of standard finite-element packages that normallyconsider structures in which motion occurs about a mean undisplaced position. For this reason, all dynamic analysis packages used for wind turbines have had to be specially constructed. Some have opted for finite-element approaches which are developed from first principles [12], some for lumped parameter methods,f but most have opted for a modal description. + +Whatever approach is used, similar problems and results are encountered. It is useful to consider a specific, but highly idealised, model to demonstrate some fairly general points. As an example, let us take the three degrees of freedom system shown in Fig. 2. This model has been used in various other papers, see for example Dugundji et al. [13]; it is intended to represent a two-bladed rotor mounted on a flexible tower. The blades are allowed to exercise independent in-plane “lead-lag’ motion, and the tower is permittedtomove laterally in the plane of rotation. If $\xi_{i}$ represents the lead-lag angle of blade $i,\;\psi$ the azimuthand $\pmb q$ the linear motion of the tower head, then the kinetic energy of the system may be easily calculated to give: + +![](images/040c9461464b4240d15e4dc19fbeabe39bba35cc93d3f4e89a7848b4d12a23e1.jpg) +Fig. 2Simple three degrees of freedom model + +$$ +\begin{array}{l}{{T=\displaystyle\frac{1}{2}\left[I\sum_{i=1}^{2}{(\dot{\psi}_{i}+\dot{\xi}_{i})^{2}}+2M\dot{q}^{2}\right.}}\\ {{\displaystyle\left.\qquad\qquad+\left.2\dot{q}\sum_{i=1}^{2}{(\dot{\psi}_{i}+\dot{\xi}_{i})}\cos{(\psi_{i}+\xi_{i})}+M_{e}\dot{q}^{2}\right]}}\end{array} +$$ + +$$ +\begin{array}{r l r}&{\displaystyle\psi_{i}=\psi+(i-1)}&\\ &{\displaystyle I=\int_{0}^{R}m r^{2}\ d r}&\\ &{\displaystyle S=\int_{0}^{R}m r\ d r}&\\ &{\displaystyle M=\int_{0}^{R}m\ d r}&\end{array} +$$ + +$M_{e}$ is the mass of the tower, $\pmb R$ is the radius of the blades and $m$ is their mass per unit length. + +Assuming further that the blade and tower flexibilities may be expressed in terms of a simple spring, the potential energy may also be formulated as: + +$$ +U=\textstyle{\frac{1}{2}}K_{\xi}\sum_{i\,=\,1}^{2}\xi_{i}^{2}\,+\frac{1}{2}K_{\t}q^{2} +$$ + +where $K_{\xi}$ represents an equivalent spring for the blades and $K_{\tau}$ is a similar parameter for the tower. + +Equations of motion for wind-turbine systems tend to be very clumsy, as they involve rotating and nonrotating components, and, hence, a well organised approach to their derivation is essential. Virtually all authors adopt the well known Lagrangian method that permits the derivation of equations of motion to be performed mechanically, after expressions for the kinetic and potential energy have been formed. The equation of motion for the generalised co-ordinate $q_{i}$ is given by: + +$$ +{\frac{d}{d t}}\,{\frac{\partial T}{\partial{\dot{q}}_{i}}}-{\frac{\partial T}{\partial q_{i}}}+{\frac{\partial U}{\partial q_{i}}}=Q_{i} +$$ + +where $Q_{i}$ is the generalised force. In this particular example, we intend to neglect all infuences except the mechanical coupling in the system, so the generalised forces are zero and, hence, the equations of motion are: + +$$ +\begin{array}{l}{{M_{t}\ddot{q}+S\displaystyle\frac{d^{2}}{d t^{2}}\left[(\xi_{1}-\xi_{2})\cos\psi\right]+K_{t}q=0}}\\ {{\displaystyle I\ddot{\xi}_{1}+S\ddot{q}\cos\left(\psi+\xi_{1}\right)+K_{\xi}\xi_{1}=0}}\\ {{\displaystyle I\ddot{\xi}_{2}-S\ddot{q}\cos\left(\psi+\xi_{2}\right)+K_{\xi}\xi_{2}=0}}\end{array} +$$ + +where $M_{\nu}=M_{e}+2M$ , the total linear mass moving at the top of the tower. + +# 4 Stability + +These equations may be linearised, which allows the term cOs $(\psi+\xi_{i})$ to be reduced to cos $\psi$ , but the periodic terms cannot be removed. Thus, for a two-bladed rotor mounted on a flexible tower, the equations of motion written in matrix notation are always of the form: + +$$ +[M(t)]\ddot{y}+[C(t)]\dot{y}+[K(t)]y=0\quad t=\psi/\Omega +$$ + +For rotors with three or more blades, co-ordinate transformations exist that allow periodicity to be removed. However, as most large wind turbines have two blades, and a two-bladed rotor is more difficult to analyse, the discussion will be limited to that case. + +There is, of course, no trouble in solving a linear differential equation of this sort, many numerical algorithms exist for that purposes. Indeed, in principle, the problem of solution would be no more complex for a realistic system than for the trivial example described here. It is highly desirable that the system is checked for stability. This can be done in the time domain by simply supplying a set of initial conditions and allowing the equations to be integrated over a long period of time. The solution may then be observed, to see if it is convergent or divergent. Such an approach is, however, unreliable, and a more direct method is desirable. Had the matrices in eqn. 5 been constant with time, standard eigenvalue analysis could be conducted to check for stability, and to determine the natural frequencies of the rotating system. The presence of the periodic coefficients do, however, preclude such an approach. + +The problem of predicting the stability of differential equations with periodic coefficients is by no means new. A great deal of work has been done on the problem, largely inspired by the helicopter industry, where very similar dynamic problems occur. The wind-turbine community is, therefore, fortunate that it may apply the existing helicopter technology to wind-turbine problems. To illustrate the type of analysis that is required and to demonstrate the existence of at least one type of mechanical instability, we shall continue the analysis of the simple example described here. + +The most widely used method is known as FloquetLiapunov theory. Rather than describing the theory mathematically, it is perhaps useful to attempt to understand, in a general way, how the theory works. The first step of the approach is to transform the $\pmb{n}$ equations of motion for an n degrees of freedom model into $2n$ first-order equations. Each state is individually perturbed by assigning it a unit initial value while allthe other states have zeroinitial values. The system equations are then integrated around one revolution of the rotor, and the final solution vector ${\pmb v}_{1}(T)$ is stored. This process is repeated for each state until $_{2n}$ solution vectors ${\pmb v}_{1}(T),\ {\pmb v}_{2}(T),\ \ldots,\ {\pmb v}_{2n}(T)$ have been obtained. These are assembled column by column into a matrix, termed the transition matrix $[Q]$ It seems sensible that, because such a matrix contains information about the transient behaviour, it should also be useful in analysing the stability of the system. It is outside the scope of this paper to demonstrate this fact, but knowledge of $[Q]$ is indeed the key to ascertaining the stability information about the system it describes. + +$\lambda_{\kappa}$ are the eigenvalues of $[Q]$ suchthat $\lambda_{K}=e^{p\kappa t}$ and $p_{\kappa}(=\alpha_{\kappa}+i\omega_{\kappa})$ are the stability exponents of the system which may conveniently be determined by the relation: + +$$ +\alpha_{\kappa}=\frac{1}{T}\ln\ |\lambda_{\kappa}| +$$ + +and + +$$ +\omega_{K}=\frac{1}{T}\tan^{-1}\!\left\{\frac{\mathrm{Im}\ (\lambda_{K})}{\mathrm{Re}\ (\lambda_{K})}\right\} +$$ + +The damping in the system $\alpha_{\kappa}$ is, therefore, uniquely determined. Unfortunately, $\omega_{\kappa}$ is indeterminate by integer multiples of $2\pi/T=\Omega$ . The actual frequency can either be determined by inspection or by obtaining a solution using the eigenvector corresponding to $\omega_{\kappa}$ as an initial solution vector. The resulting solution will be periodic with frequency $\omega_{\kappa}$ + +It should be apparent that use of Floquet theory is very cumbersome and expensive. It is necessary, when considering stability, to construct a root-locus plot, which means calculating the stability parameters for a range of rotational speeds. For the simple 3 dof system used here as an example, six rotations of the rotor must be performed to provide the transition matrix for a single eigenvalue calculation. For a more representative system with, say, twelve modal degrees of freedom, $2\,\times\,12=24$ rotationsare required for each eigenvalue. Repeating this process for, say, ten rotational speeds, therefore, requires 240 revolutions. Some of the natural frequencies may be quite high and, hence, place considerable demands on the integration algorithm. It is, therefore, useful to consider more elegant approaches. + +Physically it seems fairly obvious that the equations of motion themselves must contain the stability information, and it ought not, therefore, be necessary to solve them to deduce the system frequencies and damping. This approach has been persued by Kaza and Hammond [16], who have produced a considerably more economical scheme. Further information about Floquet analysis, in general, and wind-turbine stability analysis, in particular, may be found in References 10, 11, 13, 17 and 18. + +Floquet analysis has been used to determine the system frequencies and damping of the model described by eqn. 4. The results of this analysis are presented in Fig. 3, which is typical of a stability plot for a wind-turbine system. Normally, the machine would operate at the low-frequency end of the plot. The frequencies here have been nondimensionalised with respect to $\sqrt{(K_{\xi}/I)}=\omega_{\xi}$ . The important characteristic demonstrated by this plot is the coalescence of the tower mode and the blade lead-lag cyclic mode. In helicopter parlance, this coalescence represents an instability known as 'ground resonance' that can be very violent. Note also that, as the two modes coalesce, the damping which had hitherto been identically zero becomes finite, shown in the Figure as positive, for convenience, but is in fact destabilising. A model that included aerodynamics and structural damping would have had some finite but stabilising damping at all stable rotational speeds. It is customary to conduct stability analyses in the absence of structural damping, as its infuence is very powerful, and it is reassuring to know that a system is stable without it. + +![](images/3afdcddbaa0bb7c43ab0db3d0208277fcb17e7e651ca55efc23c39ed108b90b8.jpg) +Fig. 3Typical stability plot rotorcollectivemode —rotor cyclicmode O—towerlateralmode + +In conclusion to this discussion of stability analysis, some comment should be made about the physical characteristics of mechanical instabilities. It is important to appreciate the difference between instabilities and resonances. To excite a resonance, the system must be forced at a certain frequency that coincides with a natural frequency of the structure. The presence of an instability, on the other hand, will result in some arbitary perturbation of a linear system growing without limit in the absence of any forcing. The above illustration of mechanical instability used ^ground resonance' as an example. Ground resonance involves the in-plane motion of the hub and blades. It occurs when the in-plane motion of the blades generates inertial loads which react with the tower, in such a way as toproducehubmotionthatfurtherexcitestheblade lagging motion. For the present generation of wind turbines, this particular instability cannot occur. At present, wind-turbine instabilities are not categorised as rigorously as helicopter instabilities; although they do not have particular names, they can occur and should be checked for during the design process. Finally, it should be stressed that the example used for illustrative purposes here is a drastic oversimplification. Typical analyses of real systems have many degrees of freedom, and may exhibit not only pure mechanical instabilities such as described here, but also true aeroelastic instabilities akin to the well known 'futter' problems encountered in aircraft. To predict aeroelastic instabilities, aerodynamic effects must of course be included. + +# 5 Forced response to deterministic loads + +A true aeroelastic model of a wind turbine must contain a structuraldynamic modelof the systemincludingthe power train and control system, as well as the blades and tower. All of these elements play their part in determining the behaviour of the system as a whole when it is excited by aerodynamic loads. There is, in principle, no difference in deriving the mathematical description of such a model from the simple derivation described in the preceding text; precisely the same steps would be followed. However, the analysiswould nowhave toinclude thegeneralisedforces, which would be derived from considering the aerodynamic loads. + +The basis of the aerodynamic calculations is exactly that used for performance and static load prediction. It is, however, important to appreciate that the structural velocities of the turbine are now superimposed onto the wind speed and rotational velocities. The additional velocities modify the angle of attack of the blade section and, hence, change the blade loads as illustrated for fatwise motion in Fig. 4. It is the change in angle of attack that results in so called‘aerodynamic damping'. This effect is essential to the aeroelastic behaviour of the system; it implies that the formulation of the generalised forces is fairly complex algebraically, although, in principle, it is straightforward. Unsteady effects may also be included at this stage. + +It is easy to discuss such a task in a few words, but experience of such a problem soon shows that the algebraic manipulation involved in the formulation of a good structural model can be truly formidable and run to hundreds of pages of calculations. The accuracy of such a model is limited only by the analyst's stamina and care! Short of providing a detailed derivation and presentation of an aeroelastic model, little more need be said about structural response predictions. Interested readers can consult References 7 and 8, for relatively simple examples, and References 10, 11 and 12, for more complex analyses. + +![](images/d8542de124ce52a8ea8c7356efeca554fad80ab5c8d7c32df297905dfb20d61b.jpg) +Fig.4Impactofstructuralmotiononangleofattack $\pmb\theta=$ built-in twist, $\pmb{\alpha}=$ angleofattack + +Fig. 5 shows the blade root loads for the NASA MOD-o, which has been a workhorse for much experimental testing and validation of computer codes. Fig. 5a shows the flatwise bending moment as a function of azimuth.Thedominantfeatureofthiswaveformisthe large peak that occurs about $50^{\circ}$ after bottom dead centre, which is a result of passing through the tower shadow. The MOD-0 rotor is downwind of the tower and hence this load is very large. For a HAwT, the dominant blade load for the inner portion, at least, is the gravitational bending moment which is shown clearly as a steady 1P oscillation in the edgewise loads of Fig. 5b. It is interesting to note the higher frequency oscillation in both of these loads. These are more prevalent in the edgewise than in the fatwise direction. The flatwise motion is heavily damped by virtue of the aerodynamics, whereas, even in a quite highly twisted blade such as this, the edgewise direction must rely mostly on structural damping, with a little help from the drive train. Upwind rotors will have more sinusoidal waveforms than these, but, otherwise, the characteristics will be similar. Any rotor which is subjected to an appreciable tower shadow, which is present in upwind as well as downwind rotors, will exhibit some high harmonic loads which result from the impulsive nature of the shadow. Fig. 5 also shows curves predicted by the author and reported by Garrad [8]. + +![](images/18db1740611a25857e0d723c1029a3dda70e0e358a991b203cd8ea4cf6481833.jpg) +Fig. 5 Mod-0 rigid hub loads—comparison of predicted and measured bladeloads - predicted,Z measured aFlatwise bEdgewise + +The example cited was an early version of the MOD-0 machine which had a rigid hub. Later versions, and indeed most large two-bladed machines, have teetered rotors. For a rigid-hub machine, the dominant cyclic load on the shaft, nacelle and tower is the out-of-plane bending moment. Removal or reduction of this load helps considerably in the design of the components downwind of the rotor. This is achieved in two distinct ways, either by the addition of a third blade or by use of a teeter hinge. The addition of the third blade increases the rotor's symmetry so that the large periodic hub loads are greatly reduced by the spatial averaging of the three-bladed rotor. The addition of the teeter hinge, see Fig. 6, which permits the rotor to move as a rigid body out of the plane of rotation, allows the reaction to the out-of-plane moments to be derived from the considerable inertia of the rotor. The introduction of the teeter pin at the end of the shaft completely eliminates the troublesome moment loads. + +![](images/bf51ee4012405f26e13b43c10985288d776dda599a20a36e0e5b6c5f71493a63.jpg) +Fig.6 Teeter hinge + +Many different methods are available for predicting the types of loads described by Fig. 5; they differ quite radically in their complexity. In the USA and Europe, experimental data from large machines has been collected for about one year. Owing to the commercial nature of this information, little has been published. In the UK we are just entering the phase of comparing measurements with predicted dynamic loads, a process that should allow more reliable judgments about the efficiency of the various analytical procedures to be made. + +# 6 Forced response to stochastic loads + +To keep the analytical methods described in the preceding text in perspective, it is important to be sure that the various elements that make them up are of comparable accuracy. There seems little point in developing the structural or aerodynamic models too far, in the absence of an adequate representation of the behaviour of the wind itseif; whose fluctuation is, after all, responsible for a large proportion of the load variation. Compared with the structural models, this aspect of wind-turbine analysis is very much in its infancy; although well established descriptions of the wind are available. + +We have so far dealt with steady-state loading of rotors and their support structures. In addition to this cyclic loading, there will be transient loads that result from wind gusts, or, more accurately, from turbulent variations in the wind velocity. Wind turbines must be designed against these as well as steady-state loads.A fast acting control system may be able to alleviate these loads to some degree, but, as they result from a continuously varying source, such a course of action may result in considerable wear in the control system itself, and possibly in large loads in the rotor, if the control surfaces are made to move very fast. Whether the turbulent loads are removed or accommodated, it is necessary to be able to predict their nature. Such a step is significantly more complicated than the prediction of the deterministic loads. + +The structural dynamic models used for prediction of stochastic loads are similar to those required for deterministic work, although it will normally be necessary to make some simplifications. The description of the wind does, however, require some discussion. The idea of a gust of wind is a familiar one. It is easy to understand and has the advantage of being relatively easy to analyse. Given the existence of a structural dynamic model of the system, using a time-domain integration scheme and bearing in mind the validity of any aerodynamic models used for local calculations, it is a fairly trivial step to perturb the wind input and, hence, model a gust. The transient behaviour of the system may be clearly observed with only minimal changes to any mathematical models adopted. There are plenty of data available on the modelling of these discrete gusts from the meteorological point of view. Frost [19] has compiled a fairly comprehensive set of data intended specifically for wind modelling for wind-turbine applications. The use of such a wind model has the advantage of analytical simplicity, but does not provide a realistic representation of the wind itself. + +An important aspect of the natural wind is the distribution of turbulent energy at different frequencies, a characteristic that is easily described in the frequency domain. The modelling of the coherence, or rather incoherence, of the wind turbulence is also vital in providing a realistic input to load prediction procedures. The fact that, as the wind velocity increases on one part of the rotor disc, a corresponding increase does not necessarily occur elsewhere will obviously play an important part in determining differential loads. Both of these characteristics are difficult to model accurately using discrete gust methods. + +It has long been recognised in the field of wind loading of stationary structures that frequency-domain methods are superior to discrete gust methods. There are wellestablished spectral representations of the wind: again these are conveniently collected by Frost [19] and a good state-of-the-art review is to be found in the CIRIA proceedings [20]. Hitherto, these methods have found little application in wind-turbine analysis. The reason for this is the added complexity introduced into the structure by the rotation of the blades. To understand the importance of this difference, a short digression is required. + +It is useful to consider a turbulent eddy being convected past a structure. Consider, first, a stationary structure, as in Fig. $_{7a}$ ; if the eddy has a length l and the mean wind speed is $\pmb{u}$ .a useful parameter is the time taken for this eddy to pass the building, which may be derived as $t_{1}=$ $\boldsymbol{l}/\boldsymbol{u}.$ . Now consider the same situation, except that the stationary structure is replaced by a rotating blade, Fig. 7b. For a point on the blade moving at a speed $r\Omega$ , the time for the passage of the eddy is now the time taken for the blade to cut through it $t_{2}=l/r\Omega.$ The time is therefore considerably smaller for the turbine blade than for the stationary structure, the ratio being $u/r\Omega_{:}$ . or the inverse of the local speed ratio $1/\lambda$ . This process has the effect of moving the turbulent energy to higher frequencies and, in particular, to harmonics of the rotor speed; this is shown graphically in Fig. 8. + +![](images/be439160b09221d395c790fd6d33fb274c5a34546134ded6114e41759690e68d.jpg) +Fig.7 Turbulent loads a Stationary structure b Rotating turbine blade + +![](images/fb4605e04207e2d7f638c3509c36b76879b9cdc3f28e5ad27bf5b305e88b7c29.jpg) +Fig.8Turbulent velocity spectra seen from stationary and rotating framesofreference aRotatingframe, $\boldsymbol{b}$ stationaryframe + +The ‘slicing’ process may be conveniently described mathematically in terms of correlation functions. Many such functions exist and only one, the von Karman, will be used here. Assuming that the turbulence is homogeneous and that Taylor's ‘frozen turbulence’ hypothesis which relates spatial and temporal separation is valid, the crosscorrelation of the velocity, at two points separated by the vector r, may be expressed using the standard expression + +$$ +\begin{array}{c}{{\displaystyle\rho_{u u}(r,\,\tau)=\frac{2}{\Gamma(\frac{1}{3})}\left\{\left(\frac{\eta}{2}\right)^{1/3}\!K_{1/3}(\eta)\right.}}\\ {{\displaystyle-\left.\frac{\pi}{4}\left(\frac{\Gamma(\frac{5}{6})}{\Gamma(\frac{1}{3})}\right)^{2}\frac{r^{2}}{l^{2}}\left(\frac{\eta}{2}\right)^{-2/3}\!K_{-\,2/3}(\eta)\right\}}}\end{array} +$$ + +a detailed description of which may be found in Reference 20. $K_{\scriptscriptstyle{1/3}}$ and $K_{-2/3}$ are Bessel functions of the second kind, $\Gamma$ is the Gamma function, $l$ is the turbulence length scale, + +$$ +\eta=\sqrt{\pi}\;\frac{\Gamma(\frac{5}{6})}{\Gamma(\frac{1}{3})}\;\frac{1}{l}\;(r^{2}+u^{2}\tau^{2})^{1/2} +$$ + +and $\pmb{u}$ is the mean wind speed. Simple physical reasoning shows that the wind that comes into contact with a point on a rotating wind-turbine blade occupies a spiral shape stretching upwind of the turbine, at any instant in time. This is in contrast to the straight line that would characterise the wind incident on a stationary building. By considering this spiral, eqn. 7 may be transformed to provide an expression for the crosscorrelation of the wind velocity fuctuations between two points on a wind-turbine rotor. This is achieved by replacing eqn. 8 with + +$$ +\begin{array}{r l}&{\eta=\sqrt{\pi}\,\frac{\Gamma(\frac{5}{6})}{\Gamma(\frac{1}{3})}\,\frac{1}{l}}\\ &{\qquad\qquad\times\,\left\{r_{1}^{2}+r_{2}^{2}-2r_{1}\,r_{2}\,\cos\,(\theta_{s}+\Omega\tau)+(u\tau)^{2}\right\}^{1/2}}\end{array} +$$ + +where $r_{1}$ and $r_{2}$ are the radii of the two points considered, $\pmb{\Omega}$ is the speed of rotation and $\theta_{s}$ is an angle: $\theta_{s}=\pi$ if the points are on different blades and $\theta_{s}=0$ , if they are on the same blade. This particular transformation only applies for two-bladed machines. + +This approach was first suggested by Rosenbrock [21] and has since been rediscovered by Kristensen and Frandsen [22] and Anderson [23]. It was validated experimentally by Connell [24]. A similar study is presented for helicopter rotors in Reference 25. + +It is perhaps more familiar to consider turbulent fluctuations in spectral terms, rather than as correlation functions. Eqns. 7 and 9 lay the foundations for the use of a proper description of the wind, together with a structural model of the turbine. Such a calculation has been undertaken by Garrad and Hassan [26] and Madsen [27]. To do this in the frequency domain, the equations of motion must be transformed and generalised forces calculated by combining eqn. 7 with some suitable aerodynamic model of the blades. A description of this process is given by Garrad and Hassan [26]. In addition, some suitable usually fairly simple structural model must be incorporated. Using this approach, a response spectrum may be computed which has the rather clumsy form: + +$$ +S_{j j}(p)=\mathrm{constant}\times\frac{S_{j j}^{F}(p)}{|(P_{j}^{2}-p^{2}+i D_{j}\,p)|^{2}} +$$ + +where + +$$ +\begin{array}{r l r}{\lefteqn{S_{j j}^{F}(p)=\int_{-1}^{1}\int_{-1}^{1}|x_{K}|\ |\ x_{l}|}}\\ &{}&{\times\ \phi_{j}(x_{K})\phi_{j}(x_{l})S^{u}(x_{K},\ x_{l},\ p)\ d x_{K}\ d x_{l}}\end{array} +$$ + +$p$ is nondimensional frequency $\mathbf{\nabla}=\omega/\Omega,\ \boldsymbol{D}_{j}$ is a damping parameter dependent on the blade aerodynamics and mode shape, $\phi_{j}$ is the blade mode shape, $x_{K},\,x_{t}$ are nondimensionalbiaderadi $=r/R$ and $S^{u}(x_{K},\ x_{l},\ p)$ is the Fourier transform of eqn. 7 with $\eta$ taken from eqn. 9. Some typical response spectra for blade teetering and two-blade vibrational modes taken from Garrad and Hassan [26] are shown in Fig. 9. This Figure clearly demonstrates both the blade resonances and the peaks in the turbulent loads. + +Use of a spectral model of the wind turbulence allows the coherence of the wind to be included in a realistic way. It also permits a proper description of frequency content of the wind to be included in the aeroelastic model. Use of such a model brings with it considerable analytical complexity. As the turbine rotor is by its very nature sensitive to changes in flow around it, coupling between the various modes of vibration of the rotor may occur via the aerodynamic loads. Application of spectral methods to stationary structures such as buildings usually assumes that the modes of vibration are uncoupled, because there is no feedback between the applied loads and resulting displacements. This will necessarily be the case, the modal independence being a basic building block of modal analysis. + +![](images/4aa78ab227616160f9d9fab6234735e4b61f9941588c3afd2e1f7d3d9078ff4a.jpg) +Fig.9Structural response of a rotor to turbulent loads expressed as modalresponseinteeter,symmetric and asymmetricfatwisemodes asymmetric,---symmetric,——·—teeter + +The presence of coupling significantly complicates the analysis. The main aim of an analysis that includes wind turbulence will probably be the prediction of fatigue damage. It is not valid to combine the stochastic and deterministic loads by simple addition, and some further work is required to provide a suitable analytical basis for such a combination. The foundation for this work already exists in the field of communication engineering, see for example Bendat [28]. For rotors mounted on flexible support structures, it is important to include tower models in any dynamic analysis. For two-bladed rotors, the inclusion of a tower model and the consequent presence of periodic terms in the equations of motion, combined with the use of spectral methods, presents further complications. + +Reference to Fig. 8 demonstrates that proper analysis of turbulent wind is required to estimate the stochastic loads. The Figure clearly demonstrates that, under certain conditions, the higher harmonic loads may assume considerable importance. The spatial distribution of the loads which can be accurately modelled by such a system is also of obvious importance. + +It is evident from the experience of design teams, who have been operating large wind turbines, that the ability to predict fatigue life may rely very heavily on the availability of a dynamic model that contains a realistic wind description. No doubt in the near future the analytical effort in this field will increase substantially. Only spectral methods have been described here; there is also some research effort underway that attempts to simulate realistic turbulence in the time domain, using considerably more sophisticated methods than those usually described by the term ‘discrete gust'. + +This paper has attempted to outline some of the problems and means for their solution that occur in the dynamics of wind-turbine design. It was not intended to make a critical review of existing methods, but rather to enable an interestedreadertoobtaininsightintothenature ofthe problems and to lead him to more detailed works where required. + +The dynamics of rotating machines is a complex subject and, in the case of a wind turbine where the main motive force is so difficult to characterise, additional complications arise. The absence of any concensus of opinion about design choices, even over such a basic characteristic as the number of blades, demonstrates that there is still much tobe learned about wind-turbine behaviour.This review has demonstrated that the dynamics of the turbine play a central role in design and, consequently, this aspect of wind-turbine technology will no doubt evolve considerably in the near future. + +# 8 Acknowledgments + +The author wishes to thank the directors of Taylor Woodrow Construction for their permission to publish this paper. + +# 9 References + +1 WILSON, R.E., and LISSAMAN, P.: “Aerodynamic performance of wind turbines'(Oregon State University,June 1976) +2 MILLER, R.H., DUGUNDJI, J., and MARTINEZ-SANCHEZ, M.: ‘Aerodynamics of horizontal axis wind turbines—Vol.2'.Wind Energy Conversion, Massachusetts Institute of Technology Aeroelastic and Structures Laboratory, Report ASRL-TR-184-8, 1978 +3 GARSIDE, A.J., HALES, R.L., and GREENHOUGH, V.W.: 'Design andanalysis technique for wind turbinesinunsteadyfow environments'. Cranfield Institute of Technology, Cranfield, Bedford, UK +4 DE VRIES, O.: \*On the theory of the horizontal axis wind turbine', Annu.Rev.Fluid Mech.,1983,15,pp.77-96 +5 DE VRIES, O.: “Fluid dynamic aspects of wind energy conversion'. Agardograph 243, 1979,AGARD-AG-243 +6 SULLIVAN, T.L.: ^A review of resonance response in large horizontal axis wind turbines', Sol. Energy, 1982, 29, pp. 377-384 +7 STODDARD, F.S.: 'Dynamic rotor loads of a wind turbine via handheld Calculators', J. Energy, 1981, 5, pp. 99-103 +8 GARRAD, A.D.: \*An approximate method for the dynamic analysis of a 2-bladed horizontal axis wind turbine system'. Proceedings of the 4th international symposium on wind energy systems, Stockholm, September 1982 +9 HOFFMAN, JA: "Coupled dynamic analysis of wind energy systems'.NASA CR-135152,Feb.1977 +10 FRIEDMANN, P., and WARMBRODT: 'Coupled rotortower aeroelastic analysis of large horizontal axis wind turbines', AlAA J., 18, pp. 1118-1124 +11 VOLLAN, A.: "Aeroelastic stability and dynamic response calculations for wind energy converters. Proceedings of the 4th international conference on wind energy systems,Stockholm,1982 +12 FABIAN, O.: \*A new method for aeroelastic analysis of wind turbines. Technical University of Denmark, Internal report AFM: 81-06 +13 DUNGUNDJI, J. et al.: 10 volume series on wind energy conversion, MIT Aeroelastic and Structures Research Laboratory, 1978 +14 Proceedings of the second DoE/NASA wind turbine dynamics workshop, Cleveland State University, Cleveland, Ohio, 24th-26th February NASA CONF-810226, 1982 +15 THRESHER, R.W.: ‘Structural dynamic analysis of wind turbine systems', J. Sol. Energy Eng., 1982, 104, pp. 89-95 +16 KAZA, K.R.V., and HAMMOND, C.E.: An investigation of flap-lag stability of wind turbine rotors in the presence of velocity gradients' +17 OTTENS, H., and ZWAAN: Description of a method to calculate the aeroelastic stability of a 2-bladed horizontal axis wind turbine'. NLR National Aerospace Laboratory, Report NLR TR 78115L, The Netherlands, 1978 +18 HULTGREN, L.S., and DUGUNDJI, J.: Dynamics of flexible rotortower system'. FFA technical note AV-1499, Part 3, The Aeronautical Research Institute of Sweden, Stockholm, 1979 +19 FROST, W.:Enginering handbook on the atmospheric environmental guidelines for use in wind turbine generator development. NASA technical paper 1359, Dec. 1978 +20 Proceedings of the CIRIA conference on wind engineering in the eighties, London, Nov. 1980 +21 ROSENBROCK, H.H.: 'Vibration and stability problems in large wind turbines having hinged blades'. ERA report C/T 113, 1955, ERA, Leatherhead, Surrey +22 KRISTENSEN, L., and FRANDSEN, S.: Model for power spectra of the blade of a wind turbine measured from the moving frame of reference', Wind Eng & Ind. Aerodyn., 1982, 10, (2) +23 ANDERSON, M.B.: The interaction of turbulence with a horizontal axis wind turbine.Proceedings of the 4th BWEA wind energy conference, Cranfeld, UK, 1982, pp. 104-118 +24 CONNELL, J.R.: The spectrum of wind speed fluctuations encountered by a rotating blade of a wind energy conversion system', Sol. Energy, 1982, 29, pp. 363-376 +25 BARLOW, J.B.: Theory of propellor forces in a turbulent atmosphere'. Institute for Aerospace Studies, University of Toronto, UTIAS report 155, 1970 +26 GARRAD, A.D, and HASSAN, U.: Turbulence induced loads in a wind turbine rotor. Proceedings of the 5th BWEA workshop, Reading, 1983 +27 MADSEN, P.: Dynamic analysis of wind turbine rotors for life prediction'. Presented at Expert Meeting on structural design criteria for LS WECS, Greenford, Middlesex, March 1983 +28 BENDAT, J.S.: “Principles and applications of random noise theory' (John Wiley & Son, New York, 1958) \ No newline at end of file diff --git a/书籍/力学书籍/GarradAD-Dynamics of wind turbines/auto/GarradAD-Dynamics of wind turbines_origin.pdf b/书籍/力学书籍/GarradAD-Dynamics of wind turbines/auto/GarradAD-Dynamics of wind turbines_origin.pdf new file mode 100644 index 0000000..c0e6b25 Binary files /dev/null and b/书籍/力学书籍/GarradAD-Dynamics of wind turbines/auto/GarradAD-Dynamics of wind turbines_origin.pdf differ diff --git a/书籍/力学书籍/GarradAD-Dynamics of wind turbines/auto/images/040c9461464b4240d15e4dc19fbeabe39bba35cc93d3f4e89a7848b4d12a23e1.jpg b/书籍/力学书籍/GarradAD-Dynamics of wind turbines/auto/images/040c9461464b4240d15e4dc19fbeabe39bba35cc93d3f4e89a7848b4d12a23e1.jpg new file mode 100644 index 0000000..5666432 Binary files /dev/null and b/书籍/力学书籍/GarradAD-Dynamics of wind 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D. GARRADt AND D. C. QUARTON + +Wind Energy Group, Taylor Woodrow Construction, 345 Ruislip Road, Southall UB1 2QX, England (Received 26 October 1984, and in revised form 25 September 1985) + +The problem of deriving equations of motion that describe the coupled tower-rotor system of a wind turbine is discussed. The mathematical formulation of these equations is described first in a form suitable for manual derivation and then as a step by step process suitable for automation. Reasons for and experience in using a symbolic computing system to undertake this work for large models are described. The general approach is illustrated by means of a very simple model. Finally, some results of the stability analysis of the simple example and an eleven degree of freedom three-bladed model are presented together with a set of measured and predicted blade loads for a $250\,\mathbf{kW}$ wind turbine. + +# 1. INTRODUCTION + +We recently encountered a problem of algebraic manipulation in wind turbine dynamics which was conceptually very straightforward but immensely tedious to perform. We required a solution to this problem and were interested in finding a means of performing it automatically, first as a check on the manual calculation, and then as a method in its own right. The power and possible application'of computers as “"equation crunchers" rather'than "number crunchers" was a revelation to us. In this paper we shall not seek to break any new ground, but rather describe the analysis which we conducted as an illustration of the possible applications of symbolic or algebraic computing as a tool in engineering analysis. Our particular problem was concerned with formulating the equations of motion of a wind turbine system. This problem will be used here as an example, but it will become obvious that many problems in engineering dynamics are of a similar nature. + +Symbolic computing systems have been in existence for some 20 years. References were made to this application of digital computers as early as 1953 [1, 2]. The systems seem to have been developed for high energy physics applications and, indeed, the system used by us (REDUCE) bears definite signs of its origins in this respect. Their application to engineering in general and engineering mechanics in particular seems to have been very limited. Some work on helicopter dynamics has been reported by Nagabhushanam et al. [3]; a review of various systems applied to structural dynamics has been given by Noor and Anderson [4]. A layman's guide to computer algebra and its applications was given by Pavelle et al. [5]. + +There are quite a number of systems available, notably FORMAC, MACSYMA, REDUCE, SCRATCHPAD and most recently SMP. Some of these were reviewed by Jensen and Niordsen [6]. The present authors have first hand experience only of REDUCE and our limited knowledge of symbolic computing indicates that this system is the most widely used in engineering applications. Some recent papers have appeared in which problems very similar to our own have been treated. Kiessling [7, 8] discussed REDUCE in the context of wind turbine problems, but unfortunately both papers appeared in rather obscure publications. Reference [8] is a yery thorough description of the use of symbolic computing in this type of work. Koppens [9] has considered some rotorcraft problems, less complex than those considered by Kiessling or in the present paper, but he retained the non-linear terms which, of course, is no hardship when using the computer to perform the algebra. + +The growing interest in symbolic computing is aptly demonstrated by 'the advent of a new technical journal: Journal of Symbolic Computation. In a paper published there Fitch [1o] has summed up its position in the engineering field: “"(We) ... undertook a programme for the education of ... companies in both Sweden and Britain in computer algebra, and we discovered that once these techniques had been shown to them they immediately recognized that they would be able to ask questions that they had been suppressing as incapable of solution. Many of these companies are now acquiring algebra systems. At present it is unfortunately still the case that much algebra is being done by hand'". That is the view of a computer scientist. We hope that the present paper will reinforce his view and encourage many other engineers to use this powerful aid to analysis. + +# 2. WIND TURBINE DYNAMICS + +The main problem in producing a mathematical model of a wind turbine is the fact that the rotor exercises gross angular movements with respect to the support structure. In.standard structural analysis techniques, such as finite element packages, it is assumed that the structure has a mean position about which deflections, albeit large deflections, occur. This is evidently not the case for a wind turbine. The analyst is therefore forced to develop the equations of motion from first principles. The goal of the analysis is to be able to predict the stability of the system and to be able to calculate the forced response. + +# 3. DERIVATION OF THE EQUATIONS OF MOTION + +3.1. DEFINITION OF THE CO-ORDINATE SYSTEMS + +It is common practice when faced with the derivation of the equations of motion of a complex dynamic system to use the well-known approach due to Lagrange. This requires the calculation of an expression for the various forms of energy in the system, the kinetic energy $\pmb{T,}$ the strain energy $\pmb{U}$ and the dissipation energy $U_{D}$ . After having calculated these quantities the Lagrangian may be derived and equated with an expression for the generalized force $Q.$ + +For a system such as we are considering the computation of the strain and dissipation energies is relatively straightforward compared with the kinetic energy and generalized force. + +For the wind turbine problems we considered standard methods were used to set up the description of an arbitrary point on one of the blades. A series of transformations (rotations and translations) were calculated that allowed the position vector of an arbitrary point on the blades to be expressed in inertially fixed axes. These transformations are listed below and sketched in Figure 1. + +The inertially fixed co-ordinate system is defined such that it is aligned along the unperturbed tower. The remaining co-ordinate systems are defined as follows: $\begin{array}{r}{T_{0}(x_{0},y_{0},}\end{array}$ $\scriptstyle z_{0})$ is the inertially fixed system; $T_{1}(x_{1},\,y_{1},\,z_{1})$ is obtained by translation of $\boldsymbol{T_{0}}$ by the tower head displacement vector, $\mathfrak{n}_{1};\;T_{2}(x_{2},\;y_{2},\;z_{2})$ is obtained by torsional rotation, $\theta_{y},$ about $y_{1};\,T_{3}(x_{3},\,y_{3},\,z_{3})$ is obtained by “"nodding" rotation, $\pmb{\theta_{x}}$ , about $x_{2};\,T_{4}(x_{4},\,y_{4},\,z_{4})$ is obtained by lateral rotation, $\pmb{\theta}_{\pmb{z}}$ , about $z_{3};\,I_{5}(x_{5},\,y_{5},\,z_{5})$ is obtained by translation of $\pmb{T_{4}}$ by the overhang vector $\mathbf{n_{2}}$ $T_{6}(x_{6},y_{6},z_{6})$ is obtained by azimuthal rotation, $(\psi+\phi_{z})$ ,about $z_{5};\,I_{7}(x_{7},\,y_{7},\,z_{7})$ is obtained by teeter rotation, $\pmb{\gamma},$ about $x_{6}$ + +当面临复杂动力系统运动方程的推导时，通常采用著名的拉格朗日方法。这需要计算系统中各种形式的能量表达式，即动能 $\pmb{T}$、应变能 $\pmb{U}$ 和耗散能 $U_{D}$。在计算出这些量之后，可以推导出拉格朗日量，并将其与广义力 $Q$ 的表达式相等。 + +对于我们所考虑的系统，与动能和广义力相比，应变能和耗散能的计算相对简单。 + +对于我们考虑的风电机组问题，采用标准方法来建立叶片上任意点的描述。计算了一系列变换（旋转和平移），使得叶片上任意点的位矢能够在惯性固定坐标系中表示。这些变换列举如下，并在图1中示意。 + +惯性固定坐标系定义为沿未受扰动的塔架对齐。其余坐标系定义如下：$T_{0}(x_{0},y_{0},z_{0})$ 是惯性固定坐标系；$T_{1}(x_{1},\,y_{1},\,z_{1})$ 是通过将 $\boldsymbol{T_{0}}$ 沿塔顶位移矢量 $\mathfrak{n}_{1}$ 平移得到；$T_{2}(x_{2},\;y_{2},\;z_{2})$ 是通过绕 $y_{1}$ 轴进行扭转旋转 $\theta_{y}$ 得到；$T_{3}(x_{3},\,y_{3},\,z_{3})$ 是通过绕 $x_{2}$ 轴进行“点头”旋转 $\pmb{\theta_{x}}$ 得到；$T_{4}(x_{4},\,y_{4},\,z_{4})$ 是通过绕 $z_{3}$ 轴进行侧向旋转 $\pmb{\theta}_{\pmb{z}}$ 得到；$I_{5}(x_{5},\,y_{5},\,z_{5})$ 是通过将 $\pmb{T_{4}}$ 沿悬伸矢量 $\mathbf{n_{2}}$ 平移得到；$T_{6}(x_{6},y_{6},z_{6})$ 是通过绕 $z_{5}$ 轴进行方位角旋转 $(\psi+\phi_{z})$ 得到；$I_{7}(x_{7},\,y_{7},\,z_{7})$ 是通过绕 $x_{6}$ 轴进行摆动旋转 $\pmb{\gamma}$ 得到。 +![](images/3cc981e37406bc5eafd73cf77240b5fcb6041716321a2bd522f65b4209f5b9d1.jpg) +Figure 1. (a) System degrees of freedom; $\pmb{\mathrm{\vec{z}_{1}}}$ ,tower head displacement; $\theta_{y}$ , tower torsion; $\theta_{\mathbf{x}}$ ,tower nodding; $\pmb{\theta}_{\pmb{z}}$ , tower lateral rotation; $\mathfrak{n}_{2}$ nacelle overhang; $\psi+\phi_{z}$ , azimuthal rotation; $\pmb{\gamma},$ rotor teeter; $\mathbb{W}_{1}$ ,rotor flapwise bending; $w_{2}$ rotor edgewise bending; (b) co-ordinate systems. + +The azimuthal rotation, $(\psi+\phi_{z})$ is the sum of the steady state and power train perturbation components. + +The rotational matrices defining the co-ordinate transformations are standard and may be easily determined from the definition of the co-ordinate systems given above. + +Generalized co-ordinates were used that corresponded to the following physical degrees of freedom; rotor teeter; rotor azimuth and power train rotation; flapwise and edgewise displacement including arbitrarily large initial deflections; tower torsion; tower fore-aft and side-to-side displacement. +方位角旋转 $(\psi+\phi_{z})$ 是稳态分量和传动链扰动分量之和。 + +定义坐标变换的旋转矩阵是标准的，并且可以从上面给出的坐标系定义中很容易地确定。 + +采用了与以下物理自由度相对应的广义坐标：风轮跷跷板运动；风轮方位角和传动链旋转；包括任意大初始变形的挥舞和摆振位移；塔架扭转；塔架前后和侧向位移。 +# 3.2. DERIVATION OF KINETIC ENERGY + +The kinetic energy of the wind turbine structure may be written as + +$$ +T\!=\!\frac{1}{2}\int_{{\mathrm{rotor}}}\mathbf{V}_{6}^{\mathsf{T}}M_{R}\mathbf{V}_{6}\,\mathrm{d}r\!+\!\frac{1}{2}\!\!\int_{{\mathrm{tower}}}\mathbf{V}_{0_{T}}^{\mathsf{T}}[M_{T}]\mathbf{V}_{0_{T}}\,\mathrm{d}h, +$$ + +where $\mathbf{v}_{\mathfrak{o}_{\tau}}$ and $\mathbf{y}_{6}$ are the absolute velocity vectors of a point on the tower and on the rotor, expressed in $\scriptstyle{T_{0}}$ and $\boldsymbol{T_{6}}$ co-ordinate systems respectively. The tower distributed mass and inertia properties are contained in the matrix $M_{T}$ + +After considerable manipulation the velocity of a point on the rotor can be expressed as + +$$ +\begin{array}{l}{{{\bf{V}}_{6}\!=\![\,T_{\bar{\nu}}]^{\top}[\,T_{\theta_{c}}]^{\top}[\,T_{\theta_{s}}]^{\top}[\,T_{\theta_{\nu}}]^{\top}\bar{\bf{n}}_{1}\!+\![\,T_{\bar{\nu}}]^{\top}[\,T_{\theta_{c}}]^{\top}[\,T_{\theta_{s}}]^{\top}(\dot{\bf{0}}_{y}\times([\,T_{\nu}]{\bf{R}}_{\bar{\nu}}\!+\![\,T_{\theta_{x}}][\,T_{\theta_{z}}]{\bf{n}}_{2}))}}\\ {{\mathrm{~}\qquad+[\,T_{\bar{\nu}}\,]^{\top}[\,T_{\theta_{z}}]^{\top}(\dot{\bf{0}}_{x}\times([\,T_{\gamma}]{\bf{R}}_{\bar{\nu}}\!+\![\,T_{\theta_{c}}]{\bf{n}}_{2}))\!+\![\,T_{\bar{\nu}}]^{\top}[\,\dot{\bf{0}}_{z}\times([\,T_{\gamma}]{\bf{R}}_{\bar{\nu}}\!+\!{\bf{n}}_{2}))}}\\ {{\mathrm{~}\qquad+\dot{\bar{\Psi}}\times[\,T_{\gamma}]{\bf{R}}_{7}\!+\![\,T_{\gamma}](\dot{\bf{R}}_{7}\!+\!\dot{\gamma}\times{\bf{R}}_{7}),}}\end{array} +$$ + +where the rotational matrices $[T]$ are standard and may easily be determined from the definition of the co-ordinate systems. + +After the rotor absolute velocity has been calculated in this way, the kinetic energy is derived by substitution of the generalized co-ordinates into equation (1). The resultant expression is not included in this text since it is extremely cumbersome and of no great physical significance. + +# 3.3. DERIVATION OF THE GENERALIZED FORCE + +The generalized force corresponding to the arbitrary generalized co-ordinate $\pmb q$ is given by， + +$$ +Q=\int\mathbf{F}_{0}\cdot{\frac{\partial\mathbf{R}_{0}}{\partial q}}\,\mathrm{d}r, +$$ + +where $\mathbf{F_{0}}$ is the aerodynamic force vector acting at a point on the rotor defined by the displacement vector $\mathbf{R_{0}}$ + +The rotor-fixed force vector $\mathbf{F}_{7}$ is more convenient to work with and is derived from consideration of the local aerodynamic lift and drag loads resolved into the blade edgewise and flapwise directions and is given by + +$$ +\begin{array}{r}{\mathrm{d}\mathbf{F}_{7}\!=\!\left(\!\frac{1}{2}\rho U c(C_{L}U_{P_{7}}\!-\!C_{D}U_{T_{7}})\:\mathrm{d}r\!\right),}\\ {\mathrm{d}\mathbf{F}_{7}\!=\!\left(\!\frac{1}{2}\rho U c(C_{L}U_{T_{7}}\!+\!C_{D}U_{P_{7}})\:\mathrm{d}r\!\right),}\end{array} +$$ + +where $U_{T_{7}}$ and $U_{P_{7}}$ are the edgewise and flapwise components of the relative wind velocity vector $\mathbf{V}_{\mathtt{R e I}_{7}}$ expressed in $\scriptstyle{T_{7}}$ rotor-fixed axes, and $\bar{U^{=}}(U_{T_{7}}^{2}\!+U_{P_{7}}^{2})^{1/2}$ + +The relative wind velocity vector defines the velocity of the wind with respect to the structural velocity of the rotor and is computed from + +$$ +\Psi_{\mathrm{Rel}_{7}}\!=\![\,T_{\gamma}][\,T_{\bar{\psi}}][\,T_{\theta_{\varepsilon}}][\,T_{\theta_{x}}][\,T_{\theta_{y}}](\Psi_{w_{0}}\!-\!\dot{\bf R}_{0}), +$$ + +where $\mathbf{v}_{\mathsf{w}_{0}}$ is the absolute velocity of the wind expressed in inertial axes and the $\dot{\mathbf{R}}_{0}$ is the structural velocity. + +The aerodynamic force vector $\mathbf{F_{0}}$ is derived by transforming $\mathbf{F}_{7}$ Thegeneralizedforce may then be computed from equation (2) after having first substituted into it the generalized co-ordinates.· + +# 3.4. EQUATIONS FOR THREE-BLADED MACHINES + +It is interesting to consider the slightly different but strongly related problem of a three-bladed rotor. The discussion above described the derivation of the equations of motion of a two-bladed rotor starting with an arbitrary point on the blade. The teeter freedom requires the two-bladed rotor to be treated as a single entity rather than as two independent blades. In the basic steps involved an arbitrary point was considered and it is only the specification of the mode shapes themselves that narrows the analysis to a two-bladed case. One may, therefore, take the kinetic energy derivation of a blade on a flexible tower as a starting point for the analysis of a three-bladed machine. It was mentioned above that two-bladed rotors pose considerable numerical problems in stability calculations. A three-bladed rotor is much simpler since the co-ordinates may be transformed to remove the periodic terms. Dugundji and Wendell [11] have given a good review of this aspect of wind turbine dynamics. In our case we were able to make uise of the single-blade analysis that we had derived for the two-bladed work. Like the derivation of the equations of motion themselves, the transformation from physical co-ordinates into multi-blade co-ordinates can be very simply specified and was therefore suitable for symboliccomputing. + +To illustrate this transformation consider a state vector that describes a three-bladed rotor supported by a flexible tower. The blades are modelled in terms of their first flapping and lead-lag modes. $\pmb q_{1,i}$ denotes the generalized co-ordinate of the first flapping mode of bladei and $\pmb{p}_{1,i}$ the first lead-lag co-ordinate. + +$\phi_{z}$ is the power train angular deflection, $\pmb{n},\ \pmb{s},\ \pmb{\theta}_{\pmb{y}}$ are the fore-aft, side-to-side and torsional generalized co-ordinates of the tower. The state vector of the system is $\pmb{\mathrm{x}}$ where + +$$ +\mathbf{x}^{\mathsf{T}}\!=\![q_{1,1}\quad q_{1,2}\quad q_{1,3}\quad p_{1,1}\quad p_{1,2}\quad p_{1,3}\quad\phi_{z}\quad n\quad s\quad\theta_{y}]. +$$ + +The system can be transformed into multi-blade co-ordinates by making standard substitutions and defining a new state vector ${\bf x^{i}}$ + +$$ +\mathbf{x}^{1\!\mathrm{T}}\!=\![q_{0}\quad q_{\!}\quad q_{\!}\quad p_{0}\quad p_{s}\quad p_{c}\quad\phi_{z}\quad n\quad s\quad\theta_{y}]. +$$ + +If the untransformed equation of motion was + +$$ +[M]{\ddot{\mathbf{x}}}+[C]{\dot{\mathbf{x}}}+[K]\mathbf{x}=\mathbf{0}, +$$ + +where $[M],[C]$ and $[\kappa]$ are periodic, then the new equation of motion is + +$$ +[M^{1}]{\ddot{\bf x}}^{1}\!+\![C^{1}]{\dot{\bf x}}^{1}\!+\![K^{1}]{\bf x}^{1}\!=\!{\bf0}, +$$ + +where + +$$ +\begin{array}{r}{[M^{1}]\!=\![P]^{\top}\![M][P],\qquad[C^{1}]\!=\![P]^{\top}\!(2[M][\dot{P}]\!+\![C][P]),}\\ {[K^{1}]\!=\![P]^{\top}\!([M][\ddot{P}]\!+\![C][\dot{P}]\!+\![K][P]),~~~~~~~~~~~~}\end{array} +$$ + +which contain only constant terms due to the resulting trigonometric summations. Here + +$$ +[P]\!=\!\!{\left[\begin{array}{l l l}{[Q]}&{0}&{0}\\ {0}&{[Q]}&{0}\\ {0}&{0}&{[I]}\end{array}\right]},\qquad[Q]\!=\!{\left[\begin{array}{l l l}{1}&{\sin\psi_{1}}&{\cos\psi_{1}}\\ {1}&{\sin\psi_{2}}&{\cos\psi_{2}}\\ {1}&{\sin\psi_{3}}&{\cos\psi_{3}}\end{array}\right]}, +$$ + +Where $[I]$ is the $\pmb{4\times4}$ identity matrix and the variable $\psi_{i}$ is the azimuth of the ith blade. + +3.5. SYSTEMATIC DERIVATION OF THE LAGRANGIAN AND GENERALIZED FORCE + +Equation (1) appears fairly innocent, but reference to equation (2) which shows the algebraicformof $\mathbf{v_{6}}$ demonstrates that calculation of ${\cal T}_{R}$ is formidable indeed. The complexity and tedium of the algebra involved becomes still worse when the derivation of the equations of motion is considered and the evaluation of the Lagrangian is attempted. The approach adopted to derive the expression for $T_{R}$ is fairly standard and follows' very closely that described by Ottens and Zwaan [12]. One attempts to minimize the algebraic manipulation by avoiding the calculation of the position vector and then taking its time derivative to calculate the velocity which would be the most direct method. The complexity is reduced by calculating the inertial velocity in an intermediate set of axes, the result being $\mathbf{y}_{6}$ . Nevertheless, the algebra involved is extremely tedious and, in the view of the present authors, approaches the limits of human endurance or at least reliability. This sort of manipulation can, of course, be performed, but examination of the resulting expressions, which must be checked, suggested that short of performing a wholly independent analysis and comparing the results there was no possible, reliable method of checking. + +The derivation of the kinetic energy and the steps involved in the derivation of the equations of motion are very well defined. It is possible to outline a general scheme for calculation of the Lagrangian in a step by step method: (1) specify the position vector $(\mathbf{\mathbf{R}}_{7})$ of an arbitrary point of the blade in blade axes; (2) specify the series of transformations that are required to express the position vector in inertial axes, ${\bf R}_{0}\!=\![T]{\bf R}_{7}$ (3) take the derivativeof $\scriptstyle\mathbf{R_{0}}$ with respect to time in order to calculate the inertial velocity, $\dot{\mathbf{R}}_{0}$ (4) calculate the kinetic energy of the point mass and integrate over the rotor, $\int m\dot{\bf r}_{i}^{2}\,{\bf d}r=T_{R}$ (5) form the Lagrangian for each of the generalized co-ordinates. + +By following the five steps specified above it is possible to calculate the equation of motion for each of the generalized co-ordinates. Thus, after having completed step 4, step 5 may be repeated for each of the co-ordinates so that the complete system equations are produced. In the list above a very inelegant method has been adopted to calculate the velocity--the position vector is simply differentiated with respect to time. This approach makes the algebra still more tedious but makes the specification of the steps considerably simpler. Faced with the problem of checking the derivation of the equations of motion, and realizing the mechanical way in which they were derived, suggested to us that our problem was ideally suited to solution by means of an algebraic computing system. + +As with the Lagrangian, the derivation of the generalized force vector appears to be a straightforward mathematical procedure. However, the algebraic complexity of the derivation is again a great obstacle to a manual solution. The problem is therefore well-suited to the application of symbolic computing techniques. It is possible to define a series of mathematical steps for calculation of the generalized force: (1) specify the wind velocity vector $\mathbf{V}_{\ast_{0}}$ in inertial axes and subtract the inertial rotor velocity, $\dot{\mathbf{R_{0}}}\mathbf{;}$ (2) compute the relative wind velocity in blade axes by means of the rotational transformations, $v_{\mathtt{R e l},}\!=\![T]^{\mathtt{T}}$ $(\bar{\mathbf{V}}_{\mathbf{w}_{0}}\!-\!\dot{\mathbf{R}}_{0})$ (see equation (5));(3) compute the aerodynamic force vector in blade axes by meansof $\mathbf{v}_{\mathtt{p e l},}$ and the lift and drag coefficients; (4) express the aerodynamic force vector in inertial axes, $\mathbf{F}_{0}\!=\![\,T]\mathbf{F}_{7};$ (5) take the partial derivative of $\mathbf{R_{0}}$ with respect to the generalized co-ordinate to give the vector, $\partial\mathbb{R}_{0}/\partial q$ ; (6) form the dot product of $\mathbf{F_{0}}$ with $\partial\mathbb{R}_{0}/\partial q$ and integrate over the rotor to give the generalized force. + +The algebra involved in the derivation of the generalized force vector is clearly complex and yet the mathematical procedure remains relatively straightforward. It is true to say, however, that the procedure requires a greater degree of careful consideration at each step compared to the easily automated derivation of the Lagrangian. + +方程(1)看起来相当简单，但参考显示 $\mathbf{v_{6}}$ 代数形式的方程(2)表明 ${\cal T}_{R}$ 的计算确实非常艰巨。在考虑运动方程的推导以及尝试评估拉格朗日量时，所涉及代数的复杂性和繁琐性变得更糟。采用的推导 $T_{R}$ 表达式的方法相当标准，并与 Ottens 和 Zwaan [12] 所述的方法非常相似。人们试图通过避免计算位矢，然后对其进行时间求导来计算速度，从而最小化代数运算，尽管这将是最直接的方法。通过在中间坐标系中计算惯性速度来降低复杂性，结果为 $\mathbf{y}_{6}$。尽管如此，所涉及的代数运算极其繁琐，在本文作者看来，这接近了人类耐力或至少是可靠性的极限。这种操作当然可以执行，但对必须检查的所得表达式进行检查，表明除了进行完全独立的分析并比较结果之外，没有其他可能可靠的检查方法。 + +动能的推导以及运动方程推导中涉及的步骤定义得非常清楚。可以概述一种分步计算拉格朗日量的通用方案：(1) 在叶片坐标系中指定叶片任意点的位矢 $(\mathbf{\mathbf{R}}_{7})$；(2) 指定将位矢表示为惯性坐标系所需的系列变换，${\bf R}_{0}\!=\![T]{\bf R}_{7}$；(3) 对 $\scriptstyle\mathbf{R_{0}}$ 进行时间求导以计算惯性速度 $\dot{\mathbf{R}}_{0}$；(4) 计算质点的动能并在风轮上积分，$\int m\dot{\bf r}_{i}^{2}\,{\bf d}r=T_{R}$；(5) 为每个广义坐标形成拉格朗日量。 + +通过遵循上述五个步骤，可以计算每个广义坐标的运动方程。因此，在完成步骤4后，可以对每个坐标重复步骤5，以便生成完整的系统方程。在上述列表中，采用了一种非常不优雅的方法来计算速度——即简单地对位矢进行时间求导。这种方法使代数运算更加繁琐，但使步骤的规定大大简化。面对检查运动方程推导的问题，并意识到它们的机械推导方式，使我们认为我们的问题非常适合通过代数计算系统来解决。 + +与拉格朗日量一样，广义力矢量的推导似乎是一个直接的数学过程。然而，推导的代数复杂性再次成为手动解决的巨大障碍。因此，该问题非常适合应用符号计算技术。可以定义一系列计算广义力的数学步骤：(1) 在惯性坐标系中指定风速矢量 $\mathbf{V}_{\ast_{0}}$ 并减去惯性风轮速度 $\dot{\mathbf{R_{0}}}$；(2) 通过旋转变换计算叶片坐标系中的相对风速，$v_{\mathtt{R e l},}\!=\![T]^{\mathtt{T}}$ $(\bar{\mathbf{V}}_{\mathbf{w}_{0}}\!-\!\dot{\mathbf{R}}_{0})$ (参见方程(5))；(3) 通过 $\mathbf{v}_{\mathtt{p e l},}$ 以及升力系数和阻力系数计算叶片坐标系中的气动力矢量；(4) 将气动力矢量表示为惯性坐标系，$\mathbf{F}_{0}\!=\![\,T]\mathbf{F}_{7}$；(5) 对 $\mathbf{R_{0}}$ 关于广义坐标求偏导，得到矢量 $\partial\mathbb{R}_{0}/\partial q$；(6) 将 $\mathbf{F_{0}}$ 与 $\partial\mathbb{R}_{0}/\partial q$ 进行点积，并在风轮上积分以得到广义力。 + +广义力矢量推导中涉及的代数运算显然很复杂，但数学过程仍然相对直接。然而，可以说，与易于自动化的拉格朗日量推导相比，该过程在每个步骤都需要更仔细的考虑。 +# 4.SOME EXAMPLES OF SYMBOLIC MANIPULATIONS + +Symbolic computing comes into its own when tackling complicated and tedious prob. lems. The type of problem for which we have found it useful is described in section 3. It will come as no surprise to workers in the field of dynamics that the matrices involved in these problems are very large and, although it is vital that they are correct, it is difficult, if not impossible, to glean any physical understanding of the system they describe by scrutinizing them term by term. + +The problems described above have been analyzed by using symbolic computing techniques but, rather than tackle a large problem in detail here, which will result in large and complex algebraic expressions, it is more illustrative to consider a very simple example which may be followed through from beginning to end. + +![](images/a2cd117bf0a8fcb5957e734db7f3e18f6c3f3d4c0b0c9f1daba042d2d36f4a6e.jpg) +Figure 2. Three degree of freedom model. $\psi=\pmb{\Omega}t.$ + +Consider the wind turbine model shown in Figure 2. This is a simple three degree of freedom model first described by Kaza et al. [13]. It bears little physical resemblance to a real wind turbine, in fact its dynamic characteristics correspond rather more closely to a helicopter; however, it does do well as an illustrative model. The degrees of freedom contained in the model are tower head lateral motion and blade lead-lag motion. The derivation of the equations of motion of this system follows exactly the same steps as for the larger system described in section 3. The equivalent REDUCE steps, together with the inertial parts of the equations themselves, are given in Table 1. + +TABLE 1 A simple example of REDUCE + +$\%$ The following REDUCE program calculates the Lagrangians of +$\%$ the simple 3 d.o.f system described in Figure 2. +$\%$ +MATRIX MPSI(3,3),VTOW(3,1),VR1(3,1),VR2(3,1),VR3(3,1),VR3DOT(3,1)S $\%$ +$\%$ Algebraic procedure for differentiation w.r.t. time: +$\%$ +PROCEDURE DIFFRT(A)S +BEGIN +RETURN QDOT\*DF(A,Q) ++PSIDOT\*DF(A,PSI)S +ENDS +$\%$ +$\%$ Algebraic procedure for determination of a Lagrangian: +$\%$ +PROCEDURE LAG(B,C)S +BEGIN +$\begin{array}{r}{\mathsf{D D}\!:=\!\mathsf{D F}(\mathsf{T F},\!\mathbf{B})\mathbb{S}}\\ {\mathsf{A A}\!:=\!\mathsf{D F}(\mathsf{T F},\!\mathbf{C})\mathbb{S}}\end{array}$ +RETURN OMEGA\*DF(AA,PSI0) ++QDOT\*DF(AA,Q) ++QDOTDOT\*DF(AA,QDOT) ++Z1DOT\*DF(AA,Z1) ++Z1DOTDOT\*DF(AA,Z1DOT) ++Z2DOT\*DF(AA,Z2) ++Z2DOTDOT\*DF(AA,ZDOT) +-DDS +ENDS +$\%$ +$\%$ MPSI is the azimuthal rotation matrix: +$\%$ +MPSI := MAT(COS(PSI),-SIN(PSI),0),(SIN(PSI),COS(PSI),0),(0,0,1))S +$\%$ +$\%$ VTOw is the tower head displacement defined in inertial axes: +$\%$ +$\mathbf{VIOW}{:=}\,\mathbf{MAT}((\mathbf{Q}),(\varnothing),(\varnothing))\mathbb{S}$ +$\%$ +$\%$ VR1 is the position vector of an arbitrary point on the blade in blade axes: $\%$ +$\mathbf{VR1}\!:=\mathbf{MAT}((\boldsymbol{\mathfrak{G}}),(\mathbf{RR}),(\boldsymbol{\mathfrak{G}}))\mathbf{S}$ +VR2:= MPSI\*VR1S +$\begin{array}{r}{\mathbf{V}\mathbf{R}3:=\mathbf{V}\mathbf{R}2\mathbf{+}\mathbf{V}\mathbf{T}0\mathbf{W}\S}\end{array}$ +CLEAR MPSI,VTOW,VR1, VR2S +$\%$ +$\%$ VR3DOT is the general blade velocity vector defined in inertial axes: +$\%$ +VR3DOT $:=$ MAT(DIFFRT(VR3(1,1)),(DIFFRT(VR3(2,1)),(DIFFRT(VR3(3,1)s CLEAR VR3S +$\%$ +$\%$ VSQ is the velocity squared expression for an arbitrary blade station: +$\%$ +VSQ := VR3D0T(1,1)\*VR3D0T(1,1) $^{1+}$ VR3DOT(2,1)\*VR3DOT(2,1)+ +VR3DOT(3,1)\*VR3D0T(3,1)S +LET SIN(PSI) $\lvert=$ SINPSIS +LET COS(PS $\scriptstyle\mathtt{I}\mathtt{J}=\mathtt{I}$ COSPSIS +LET $\mathrm{COSPSI^{**}2}\!\!=\!1\!-\!\mathrm{SINPSI^{**}2S}$ +$\mathbf{VSQ:=VSQS}$ + +TABLE 1 (cont.) + +LET SINPSI\*\*2=1-COSPSI\*\*2S +$\mathbf{VSQ:=VSQ5}$ +$\%$ Z1 and ${\bf z}{\bf2}$ are the perturbation co-ordinates for blades I and 2: +$\%$ +PSIDC $)\mathbf{T}{:=}\,\mathbf{OMEGA}$ +Z1DOTS +LET $\mathbf{SIN}(\mathbf{ZI})\!=\!\mathbf{Z1}\mathfrak{S}$ +LET $\mathsf{C O S}(\mathbf{Z1})\!=\!1\!-\!z1^{**}\!2/2\mathfrak{I}$ +SINPSI := SIN(PSI0)\*COS(Z1)+COS(PSI0)\*SIN(Z1)\$ +$\mathbf{COSPSI:=COS(PSI/\beta)^{\ast}C O S(Z1)\mathbf{-SIN(PSI/\beta)^{\ast}S I N(Z1)\mathfrak{H}}}$ +$\%$ +$\%$ VSQ1 and VSQ2 are the velocity squared expressions for blades 1 and 2: $\%$ +$\mathbf{VSQ1:=VSQ95}$ +$\mathbf{VSO2:=SUB}$ (Z1=Z2,Z1DOT $\ddots$ Z2DOT,SIN(PSI0) $=$ -SIN(PSI0,COS(PSI0)= $-\mathbf{COS}(\mathbf{PSI\emptyset}),\mathbf{vSQ1})\mathbb{S}$ +VSQTOT $:=$ VSQ1+VSQ2\$ +CLEAR VSQ,VSQ1,VSQ2\$ +$\%$ +$\%$ TT is the total kinetic energy of the rotor: +$\%$ +TT:= M\*VSQTOT/2\$ +$\%$ +$\%$ Definition of blade inertia IB and first mass moment integral SB: +$\%$ +LET M\*RR\*RR $\cdot\overline{{-}}$ IB\$ +$\mathbf{T}\mathbf{T}:=\mathbf{T}\mathbf{T}\mathbf{\mathbb{S}}$ +LET $\Delta A^{*}\mathbb{R}\mathbb{R}{=}\mathbb{S}\mathbb{B}\mathbb{S}$ +$\mathbf{T}:=\mathbf{T}\mathbf{T}\mathbf{S}$ +CLEAR Q,QDOT,Z1,Z1DOT,Z2,Z2DOT\$ +$\%$ +$\%$ The Lagrangians are now computed: +$\%$ +$\mathbf{LAGQ}:=\mathbf{LAG}(0,\mathbf{QDOT})\mathbf{\hat{S}}$ +$\mathbf{LAGZ1\!:=\!LAG(Z1,\!Z1DOT)\!\!\!\S}$ +$\mathbf{LAGZ2}:=\mathbf{LAG}(\mathbf{Z}2,\mathbf{Z2DOT})\hat{\mathbf{S}}$ +$\%$ +$\%$ A weight level is defined for linearization of the Lagrangians. +$\%$ The linearized expressions are then output; +$\%$ +WTLEVEL 1\$ +WEIGHT $\mathbf{Q}\mathbf{=}1,\mathbf{Q}\mathbf{D}\mathbf{O}\mathbf{T}\mathbf{=}1,\mathbf{Q}\mathbf{D}\mathbf{O}\mathbf{T}\mathbf{D}\mathbf{O}\mathbf{T}\mathbf{=}1,\mathbf{Z}1\mathbf{=}1,\mathbf{Z}1\mathbf{D}\mathbf{O}\mathbf{T}\mathbf{=}1,\mathbf{Z}1\mathbf{D}\mathbf{O}\mathbf{T}\mathbf{D}\mathbf{O}\mathbf{T}\mathbf{=}$ +$1,\!Z\!2\!=\!1,\!Z\!2\mathrm{DOT}\!\!=\!1,\!Z\!2\mathrm{DOTDOT}\!\!=\!1\mathfrak{F}$ +$\mathbf{LAGQ}:=\mathbf{LAGQ};$ +$\mathbf{LAGZ1}:=\mathbf{LAGZ1}:$ +$\mathbf{LAGZ2}:=\mathbf{LAGZ2}$ +ENDS +$\%$ Lagrangian for 'q generalized coordinate; +$\mathrm{LAGQ}:=\mathrm{COS}(\mathrm{PSI}\emptyset)^{\ast}\mathrm{Z2DOTDOT}^{\ast}\mathrm{SB}-\mathrm{COS}(\mathrm{PSI}\emptyset)^{\ast}\mathrm{Z2}^{\ast}\mathrm{OMEGA}^{\ast}\mathrm{SB}$ +-COS(PSI0) +$\mathrm{^{*}Z1D O T D O T^{*}S B+C O S(P S I/6)^{*}Z1^{*}O M E G A^{*}S B-2^{*}S I N(P S I/6)^{*}}$ +$\mathrm{Z2DOT^{+}O M E G A^{*}S B+2^{*}S I N(P S I\beta)^{*}Z1D O T^{*}O M E G A^{*}S B+2^{*}M^{*}Q D O T D O T^{-}}$ $\%$ Lagrangian for Z1' generalized coordinate; +$\mathrm{LAGZ1}:=-\mathrm{COS}(\mathrm{PSI}\emptyset)^{\ast}\mathrm{QDOTDOT^{\ast}S B}+\mathrm{Z1DOTDOT^{\ast}I B}$ + +$\%$ Lagrangianfor $\mathbf{\mathcal{Z}}2^{\bullet}$ generalized coordinate: LAGZ2 := COS(PSI0)\*QDOTDOT\*SB+ Z2DOTDOT\*1B + +The steps described in the table are fairly straightforward for any reader familiar with FORTRAN and dynamics. Some general comments may, however, be made. REDUCE has the capability of implementing "Procedures" that are very much akin to FORTRAN subroutines. In the example here we have used two procedures, one for differentiation with respect to time and one to form the Lagrangian as defined in equation (1). Note that REDUCE can only perform partial differentiation. The derivation of the equations of motion involves considerable manipulation of matrices. The simple example cited here shows how REDUCE can perform these operations. It is often necessary, even when manipulation is being performed automatically, to linearize expressions. This may simply be done by REDUCE by using the “"WEIGHT' statement. This statement is used at the end of the example to assign weights to the individual variables and, given a general weight level of 1, all quadratic and higher order terms are discarded. The resulting Lagrangians for each of the three degrees of freedom are then calculated and printed at the end of the example. + +The WEIGHT command may also be used to reduce the size of the kinetic energy expression. When considering a particular generalized co-ordinate many of the terms are redundant since they disappear when the expression is differentiated. Judicious use of the WEIGHT and WTLEVEL statements avoids the computation of the redundant terms which brings about significant reductions in the size of the expressions used. Failure to carry out this procedure does, in our experience, lead to unacceptably large expressions which brings the calculation to a halt when storage is exhausted. This problem of mid-calculation “expression swell is very common and has been discussed by Nagabhushanam et al. [3]. + +One further, and very useful, facility that REDUCE possesses is the ability to write expressions to files in FORTRAN format thus enabling the computer to both derive the equation of motion and transcribe them into a FORTRAN program for numerical solution. This final step helps to eliminate further possible sources of careless mistakes, since a manual transcription process is a very likely place for such problems to arise. + +# 5. SOME COMMENTS ABOUT THE USE OF REDUCE + +The automatic derivation of the equations of motion of large dynamical systems brings with it two main advantages: increased confidence in the results and increased scope in the scale of problems that may be tackled. Some limitations are still imposed either by financial constraints or limitations in computer storage capacity, but, in principle, the size and complexity of a mathematical model need not be limited by the size and complexity of the algebraic expressions which it produces. + +The analyses described here exercized only a small part of the REDUCE system. We have used matrix manipulation and differential calculus. The system contains a great many more analytical facilities particularly in the field of high energy physics. For the engineering user the integration and factorization facilities are probably the most relevant. It is interesting to note that the integrator does not simply run through a “"look-up table" of known integrals but tackles each one on a rigorous and systematic basis. There is still some considerable work to be done on this aspect of the system; for example, integrals containing square roots present great difficulties and REDUCE often fails to produce an answer. In view of the indirect nature of integration this is not surprising. Much work is evidently underway on this subject at present. + +We have found REDUCE very useful. We have also found it to be difficult to implement and sometimes very frustrating to use. It would be erroneous to give the impression that problems such as those described in this paper have been solved without difficulty. The highly sophisticated nature of REDUCE which gives it its wide applicability and power brings with it much scope for strange errors and obtuse messages. However the new release, REDUCE 3·2 is evidently more robust and has overcome many of these problems. + +We approached symbolic computing in a cautious frame of mind. We first attempted to do small problems with solutions that could also be obtained manually with reasonable ease. We then proceeded to use REDUCE to check a very long and complicated analysis as described above. Only when we had achieved term for term agreement with this calculation did we feel happy to use the technique alone. Although there are certain features of our present system (REDUCE 3·0 implemented on an IBM 3033 operating CMS running under VM/SP release 2) that do not work, we have not discovered any way of obtaining erroneous results. Discussions with other users have also failed to bring any such problems to light. Our careful approach was therefore unnecessary which is no surprise since REDUCE has been in use in high energy physics for many years. + +# 6. RESULTS FROM THE WIND TURBINE CALCULATIONS + +It is hoped that readers of this paper will be interested in both the application of symbolic computing to engineering problems and in the problems themselves. The original reason for embarking on the analyses described here was to ascertain the stability characteristics of wind turbine designs. It is well known that helicopters, which have some similarity with wind turbines, are prone to severe mechanical and aeroelastic instabilities. Wind turbines seem less susceptible to these problems, but it is nevertheless prudent to check for possible unstable regions within the operational envelope. + +It was pointed out earlier that the simple three degree-of-freedom system used as an example above had dynamic characteristics which had more in common with helicopters than wind turbines. This comment is confirmed by consideration of the results presented in Figure 3. These were computed by using standard Floquet techniques together with some more economical methods; these calculations have been described by Quarton and Garrad [14]. They clearly show a region of instability. Investigation of the instability reveals it to be a “ground resonance" type. Such an instability requires the blades to be very soft in the lead-lag direction. The present generation of wind turbines tend to have blades that are very stiff in this direction and hence this type of instability cannot occur. + +![](images/c8d820d96f90b393f0c6fe9aa78bd14a03824908d65c486af171f72eae40e405.jpg) +Figure 3. System frequencies of three degree of freedom model. -——, Rotor collective mode; -- -, rotor cyclicmode;·..,tower lateral mode. + +![](images/c3a00899a6eb2ce0606a1aba87996943bfbfb45ce4b41d11362028aa0bdae146.jpg) +Figure 4. System WEG MS-2 frequencies. + +![](images/f1ce29a078c5953544017d5a415330c0c1322ee43b8a0abd49e258cc8e23f18b.jpg) +Figure 5. Predicted and measured WEG MS-1 blade bending moments at $16\,\Pi/\mathfrak{s}$ windspeed.-, Predicted; -- -, measured. (a) Edgewise bending moment at $3{\cdot}28\,\mathrm{m}$ ; (b) fapwise bending moment at $\mathbf{3\cdot28}\,\mathbf{m}$ + +A more typical set of stability results is shown in Figure 4. These were obtained for the WEG MS-2 wind turbine (a $25\,\uppi$ diameter,3-bladed, $200\,\mathbf{kW}$ machine). This model contains the fundamental fapping and lagging modes of each blade, a rigid body drive train freedom, and the first fore-aft, side-to-side and torsional modes of the tower. The eigenvalues of the system were obtained from a constant coefficient set of equations which resulted from performing a Coleman multi-blade transformation on the set of periodic coefficient equations as described above in section 3.4. The equations of motion of the system were derived entirely by REDUCE which also wrote a considerable part of the FORTRAN program used to calculate the numerical results. + +In addition to using REDUCE for the stability analysis of wind turbine systems it has also been used for the derivation of mathematical models for the calculation of forced response. The procedure for deriving the equations of motion for a forced response model is essentially the same as for stability analysis except it then becomes important to retain all steady forcing and defection terms as well as those which are proportional to the system generalized co-ordinates. + +A forced response model for the WEG $_{20\,\mathfrak{m}}$ diameter,2-bladed, $250\,\mathbf{kW}$ MS-1wind turbine has been derived by using REDUCE and typical response predictions are presented in Figure 5. These predictions of blade bending moment have been compared with equivalent measured results and their satisfactory agreement provides validation of the mathematicalmodel. + +# 7. CONCLUDING REMARKS + +The purpose of this paper was to describe one area of engineering analysis in which symbolic computing has played a useful role. It is hoped that the examples described here together with the specific comments on the use of the REDUCE system will enable other analysts to consider the possibility of using symbolic computing to help tackle complicated algebraic problems. + +# ACKNOWLEDGMENTS + +The authors wish to thank the U.K. Department of Energy and the Directors of Taylor Woodrow Construction, both of whom have supported parts of this work, for their permission to publish this paper. We have received invaluable help over our problems with REDUCE from Dr Pat Pearce of Kingston Polytechnic and Professor John Fitch of the University of Bath. We are also very much in debt to Rob Warren of Taylor Woodrow Construction and Dr Larry Seward of the Rand Corporation, who have spent a lot of time and energy implementing the REDUCE system on Taylor Woodrow's IBM3033. + +# REFERENCES + +1. H.G. KAHRIMANIAN 1953 MS Thesis, Temple University, Philadelphia, Pennsylvania, Analytical differentiation by digital computer. +2. J. NOLAN 1953 SM Thesis, MIT Cambridge, Massachusetts. Analytical differentiation on a digital computer. +3. J. NAGABHUSHANAM, G. H. GAONKAR and T. S. R. REDDY 1981 Proceedings of the Seventh European Rotorcraft and Powered Lift Aircraft Forum, Garmisch-Partenkirchen,Federal Republic of Germany, Paper No 37. Automatic generation of equations for rotor body systems with dynamic inflow for a priori ordering schemes. +4. A. K. NoOR and C. M. ANDERsON 1979 Computers and Structures 10, 95-118. Computerised symbolic manipulation in structural mechanics progress and potential. +5. R. PAVELLE, M. ROTHSTEIN and J. FITCH 1981 Scientific American, 101-113. Computer algebra. +6. J. JENSEN and F. N1ORDsEN 1977 in Structural Mechanics Software Series (Editors N. Perrone and W. Pilkey) Charlottesville: University Press of Virginia. Symbolic and algebraic manipulation languages and their applications in mechanics. +7. F. KIEsSLING 1982 ICAS Proceedings, 13th Congress of the International Council of the Aeronautical Sciences/ AIAA Aircraft Systems and Technology Conference, Seattle, Washington. Computer-aided derivation of equations of motion for rotary-wing aeroelastic problems. +8. F. KIEssLING 1984 DFVLR Forschunsbevicht 84-10, 187 pages. Modelling of the overall aeroelastic system of a wind turbine aided by symbolic programming. (In German.) +9. W. P. KoPpENS 1983 Technische Hogeschool Delft, Luchtuaart en Ruimtevaarttechnik, Memorandum M477. Utilization of the symbolic manipulation system REDUCE 2 for deriving equations of motion. +10. J. FITCH 1985 Journal of Symbolic Computation 1, 211-227. Solving algebraic problems with REDUCE. +11. J. DUGUNDJ1 and J. H. WENDELL 1983 American Institute of Aeronautics and Astronautics Journal 21, 890-897. Some analysis methods for rotating systems with periodic coeffcients. +12. H. H. OTTENS and R. J. ZWAAN 1978 NLR TR 78115 L. Description of a method to calculate the aeroelastic stability of a two-bladed horizontal axis wind turbine. +13. K. R. V. KAZA, D. C. JANETZKE and T. L. SULLIVAN 1980 Journal of Energy 4, 162-169. MOSTAS computer code evaluation for dynamic analysis of two-bladed wind turbines. (Also NASA TM 79101.) +14. D. C. QUARTON and A. D. GARRAD 1984 Proceedings of the 6th British Wind Energy Association Conference, pp. 197-209. Some comments on the stability analysis of horizontal axis wind turbines. \ No newline at end of file diff --git a/书籍/力学书籍/GarradAD-SYMBOHC COMPUTING AS A TOOL/auto/GarradAD-SYMBOHC COMPUTING AS A TOOL_origin.pdf b/书籍/力学书籍/GarradAD-SYMBOHC COMPUTING AS A TOOL/auto/GarradAD-SYMBOHC COMPUTING AS A TOOL_origin.pdf new file mode 100644 index 0000000..fd03a30 Binary files /dev/null and b/书籍/力学书籍/GarradAD-SYMBOHC COMPUTING AS A TOOL/auto/GarradAD-SYMBOHC COMPUTING AS A TOOL_origin.pdf differ diff --git a/书籍/力学书籍/GarradAD-SYMBOHC COMPUTING AS A TOOL/auto/images/3cc981e37406bc5eafd73cf77240b5fcb6041716321a2bd522f65b4209f5b9d1.jpg b/书籍/力学书籍/GarradAD-SYMBOHC COMPUTING AS A TOOL/auto/images/3cc981e37406bc5eafd73cf77240b5fcb6041716321a2bd522f65b4209f5b9d1.jpg new file mode 100644 index 0000000..1f6f47d Binary files /dev/null and b/书籍/力学书籍/GarradAD-SYMBOHC COMPUTING AS A TOOL/auto/images/3cc981e37406bc5eafd73cf77240b5fcb6041716321a2bd522f65b4209f5b9d1.jpg differ diff --git a/书籍/力学书籍/GarradAD-SYMBOHC COMPUTING AS A TOOL/auto/images/a2cd117bf0a8fcb5957e734db7f3e18f6c3f3d4c0b0c9f1daba042d2d36f4a6e.jpg b/书籍/力学书籍/GarradAD-SYMBOHC COMPUTING AS A TOOL/auto/images/a2cd117bf0a8fcb5957e734db7f3e18f6c3f3d4c0b0c9f1daba042d2d36f4a6e.jpg new file mode 100644 index 0000000..94ce8ec Binary files /dev/null and b/书籍/力学书籍/GarradAD-SYMBOHC COMPUTING AS A TOOL/auto/images/a2cd117bf0a8fcb5957e734db7f3e18f6c3f3d4c0b0c9f1daba042d2d36f4a6e.jpg differ diff --git a/书籍/力学书籍/GarradAD-SYMBOHC COMPUTING AS A TOOL/auto/images/c3a00899a6eb2ce0606a1aba87996943bfbfb45ce4b41d11362028aa0bdae146.jpg b/书籍/力学书籍/GarradAD-SYMBOHC COMPUTING AS A TOOL/auto/images/c3a00899a6eb2ce0606a1aba87996943bfbfb45ce4b41d11362028aa0bdae146.jpg new file mode 100644 index 0000000..f8ceb17 Binary files /dev/null and b/书籍/力学书籍/GarradAD-SYMBOHC COMPUTING AS A TOOL/auto/images/c3a00899a6eb2ce0606a1aba87996943bfbfb45ce4b41d11362028aa0bdae146.jpg differ diff --git a/书籍/力学书籍/GarradAD-SYMBOHC COMPUTING AS A TOOL/auto/images/c8d820d96f90b393f0c6fe9aa78bd14a03824908d65c486af171f72eae40e405.jpg b/书籍/力学书籍/GarradAD-SYMBOHC COMPUTING AS A TOOL/auto/images/c8d820d96f90b393f0c6fe9aa78bd14a03824908d65c486af171f72eae40e405.jpg new file mode 100644 index 0000000..9c44e70 Binary files /dev/null and b/书籍/力学书籍/GarradAD-SYMBOHC COMPUTING AS A TOOL/auto/images/c8d820d96f90b393f0c6fe9aa78bd14a03824908d65c486af171f72eae40e405.jpg differ diff --git a/书籍/力学书籍/GarradAD-SYMBOHC COMPUTING AS A TOOL/auto/images/f1ce29a078c5953544017d5a415330c0c1322ee43b8a0abd49e258cc8e23f18b.jpg b/书籍/力学书籍/GarradAD-SYMBOHC COMPUTING AS A TOOL/auto/images/f1ce29a078c5953544017d5a415330c0c1322ee43b8a0abd49e258cc8e23f18b.jpg new file mode 100644 index 0000000..d68e132 Binary files /dev/null and b/书籍/力学书籍/GarradAD-SYMBOHC COMPUTING AS A TOOL/auto/images/f1ce29a078c5953544017d5a415330c0c1322ee43b8a0abd49e258cc8e23f18b.jpg differ diff --git a/书籍/力学书籍/input/CASEstab_theory_manual.pdf b/书籍/力学书籍/input/CASEstab_theory_manual.pdf deleted file mode 100644 index b91c9a5..0000000 Binary files a/书籍/力学书籍/input/CASEstab_theory_manual.pdf and /dev/null differ diff --git a/工作OKRs/25.9-11 OKR.canvas b/工作OKRs/25.9-11 OKR.canvas index b960152..cb2bd0d 100644 --- a/工作OKRs/25.9-11 OKR.canvas +++ b/工作OKRs/25.9-11 OKR.canvas @@ -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 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":"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\n\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，等待气动完成后开始\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} ], diff --git a/生活OKRs/25.6-8 OKR.canvas b/生活OKRs/25.6-8 OKR.canvas index 7784a9d..fbaeced 100644 --- a/生活OKRs/25.6-8 OKR.canvas +++ b/生活OKRs/25.6-8 OKR.canvas @@ -5,7 +5,7 @@ {"id":"52c483d4870680c3","type":"text","text":"# 推进计划\nRag系统、agent系统调研\n新闻 公众号\n小说推广系统 可能容易实现一些\n","x":-490,"y":-186,"width":456,"height":347}, {"id":"0b25ceb1c28f6da1","type":"text","text":"# 六月已完成\n\nP1 海龟系统测试\n- 代理测试，5分钟间隔，全天监控成功1次\n- 代理池增加，但是没用\n- yiy.one.config会变 检测是否变化，重新修改以及重新加载 done\n- 邮件服务器连接使用代理 done\n","x":-482,"y":240,"width":440,"height":340}, {"id":"b79f3a0f35402ec1","type":"text","text":"# 七月已完成\n\nP1 短剧使用AI生成解说视频功能跑通\n\nP1 短剧使用AI生成解说视频\n- 提示词更新，流程缩减 done\n- 考虑增加解说词长度 完成\n\nP1 交易系统 pass 也要增加冷却时间 done\n\nP1 公众号系统工作流 done\n- 每天的文章改名字\n- 控制一批处理新闻的数量加个loop，加一个wait\n- 加个ai控制主题别重复\n- 做一个封面图，以后就用一个\n\n","x":34,"y":240,"width":440,"height":340}, - {"id":"5aac58c184e57887","type":"text","text":"# 计划\n这周要做的3~5件重要的事情，这些事情能有效推进实现OKR。\n\nP1 必须做。P2 应该做\n\n\nP1 suno youtube视频\n- 生成音频\n- 动画 done\n- 音频有什么特征\n- 控制音乐时长\n\nP1 海龟交易法 \n\n- 选股部分\n\n\n","x":-490,"y":-573,"width":450,"height":347}, + {"id":"5aac58c184e57887","type":"text","text":"# 计划\n这周要做的3~5件重要的事情，这些事情能有效推进实现OKR。\n\nP1 必须做。P2 应该做\n\n\nP1 suno youtube视频\n- 生成音频\n- 动画 done\n- 音频有什么特征\n- 控制音乐时长\n\nP1 海龟交易法 \n\n- 选股部分\n- 9点28-29分开始检测价格，做出预备\n- 突破价格似乎不对-以及突破但是没有提示\n\n\n","x":-490,"y":-573,"width":450,"height":347}, {"id":"1597ad15730745ea","type":"text","text":"# 八月已完成\n\nP1 suno youtube视频\n- 创建场景形象 - 听音乐观海 done\n- logo形象 重绘看看 done\n- 修复手 done\n- 名字 done\n- 横幅 done\n- 动画 done\n\nP1 海龟交易法 \n\n- 止盈卖出总份额 done\n- 分仓位加仓的止损价格 done\n","x":540,"y":240,"width":440,"height":340} ], "edges":[] diff --git a/线性化求解器/参考文献/Kallesøe-Equations of motion for a rotor blade/auto/Kallesøe-Equations of motion for a rotor blade.md b/线性化求解器/参考文献/Kallesøe-Equations of motion for a rotor blade/auto/Kallesøe-Equations of motion for a rotor blade.md index 24a6fd1..18014e3 100644 --- a/线性化求解器/参考文献/Kallesøe-Equations of motion for a rotor blade/auto/Kallesøe-Equations of motion for a rotor blade.md +++ b/线性化求解器/参考文献/Kallesøe-Equations of motion for a rotor blade/auto/Kallesøe-Equations of motion for a rotor blade.md @@ -189,7 +189,7 @@ is a position vector describing the elastic axis. # Equations of Motion -By demanding that any admissible variation of the action integral $\delta H\equiv\int_{\mathbf{\Omega}_{t_{1}}}^{\mathbf{\Omega}_{t_{2}}}(\delta T-\delta V_{e l a}-\delta V_{g r a}+\delta Q)d t$ is zero, a set of partial differential equations of motion and a set of boundary condition equations are derived (extended Hamilton’s principle11). The variation of the $w_{1}$ term leads to integral terms in the equations of motion, while the $w_{1}$ itself does not appear because it is relatively small. First, the partial differential equations of blade bending and torsional motion are presented, followed by the corresponding boundary conditions. Second, the equations of motion for the rotor azimuth angle and the pitch angle are presented. +By demanding that any admissible variation of the action integral $\delta H\equiv\int_{\mathbf{t_{1}}}^{\mathbf{t_{2}}}(\delta T-\delta V_{e l a}-\delta V_{g r a}+\delta Q)d t$ is zero, a set of partial differential equations of motion and a set of boundary condition equations are derived (extended Hamilton’s principle11). The variation of the $w_{1}$ term leads to integral terms in the equations of motion, while the $w_{1}$ itself does not appear because it is relatively small. First, the partial differential equations of blade bending and torsional motion are presented, followed by the corresponding boundary conditions. Second, the equations of motion for the rotor azimuth angle and the pitch angle are presented. 通过要求作用量积分 $\delta H\equiv\int_{\mathbf{t_{1}}}^{\mathbf{t_{2}}}(\delta T-\delta V_{e l a}-\delta V_{g r a}+\delta Q)d t$ 的任何可容许变分为零，导出了一组运动偏微分方程和一组边界条件方程（广义哈密顿原理11）。$w_{1}$项的变分导致运动方程中出现积分项，而$w_{1}$本身不出现，因为它相对较小。首先，介绍了叶片弯曲和扭转运动的偏微分方程，接着是相应的边界条件。其次，介绍了风轮方位角和变桨角度的运动方程。 @@ -1082,4 +1082,13 @@ $$ 11. Thomsen JJ. Vibrations and Stability: Advanced Theory, Analysis, and Tools. Springer-Verlag: Berlin-Heidelberg-New York, 2003. 12. Jonkman J. Nreloffshrbsline5mw. Technical report, NREL/NWTC, 1617 Cole Boulevard; Golden, CO 80401-3393, USA, 2005. 13. Hansen MH. Aeroelastic stability analysis of wind turbines using an eigenvalue approach. Wind Energy 2004; 7: 133–143. -14. Meirovitvh L. Computational Methods in Structural Dynamics. Sijthoff & Noordhoff: Alphen aan den Rijn, The Netherlands, 1980. \ No newline at end of file +14. Meirovitvh L. Computational Methods in Structural Dynamics. Sijthoff & Noordhoff: Alphen aan den Rijn, The Netherlands, 1980. + + + +# 点评 + +基于哈密顿原理推导动力学方程 + +弹性势能、重力势能方程列起来容易，好理解， +但是由哈密顿原理作用量积分的变分为零，推导出运动方程的具体形式，极其复杂，有变分，有积分 \ No newline at end of file diff --git a/线性化求解器/计算流程框架.canvas b/线性化求解器/计算流程框架.canvas index 68dcd50..1ec8e05 100644 --- a/线性化求解器/计算流程框架.canvas +++ b/线性化求解器/计算流程框架.canvas @@ -26,7 +26,10 @@ {"id":"45421c5911e8d893","type":"text","text":"能不能算？应该可以","x":1540,"y":98,"width":250,"height":60}, {"id":"bbcf3043979eac49","type":"text","text":"平衡状态平衡值x：---q","x":-720,"y":0,"width":250,"height":60}, {"id":"efcf091d8aaf324d","type":"text","text":"仅考虑求A矩阵","x":-720,"y":140,"width":250,"height":60}, - {"id":"db651e5e09444ed4","type":"text","text":"Δq 求 $Δ \\dot{q}$ 再线性回归","x":-720,"y":280,"width":250,"height":60} + {"id":"db651e5e09444ed4","type":"text","text":"Δq 求 $Δ \\dot{q}$ 再线性回归","x":-720,"y":280,"width":250,"height":60}, + {"id":"c86002941101b4b0","x":160,"y":420,"width":380,"height":120,"type":"text","text":"自由度上，其中一个做一组扰动，求瞬态的其他自由度的加速度，线性拟合，Aij"}, + {"id":"f264418318fe13a2","x":225,"y":620,"width":250,"height":60,"type":"text","text":"组装A矩阵"}, + {"id":"84420c9c180de186","x":225,"y":800,"width":250,"height":80,"type":"text","text":"A矩阵求特征值和特征向量"} ], "edges":[ {"id":"83d84c5b21257d2f","fromNode":"c03f206d2e22c014","fromSide":"right","toNode":"9effe93fe812b3d5","toSide":"left"}, @@ -53,6 +56,9 @@ {"id":"83cd6a37db2f73c9","fromNode":"e9c01c636e40c0b3","fromSide":"bottom","toNode":"45421c5911e8d893","toSide":"top"}, {"id":"820254294add6d7b","fromNode":"bbcf3043979eac49","fromSide":"bottom","toNode":"efcf091d8aaf324d","toSide":"top"}, {"id":"382a228fe50bb9e2","fromNode":"efcf091d8aaf324d","fromSide":"bottom","toNode":"db651e5e09444ed4","toSide":"top"}, - {"id":"e7df3158c1adbec6","fromNode":"d3aa69200118cea0","fromSide":"left","toNode":"bbcf3043979eac49","toSide":"right"} + {"id":"e7df3158c1adbec6","fromNode":"d3aa69200118cea0","fromSide":"left","toNode":"bbcf3043979eac49","toSide":"right"}, + {"id":"8ec58bd022f131da","fromNode":"d3aa69200118cea0","fromSide":"right","toNode":"c86002941101b4b0","toSide":"left"}, + {"id":"8afc7d5b908382ee","fromNode":"c86002941101b4b0","fromSide":"bottom","toNode":"f264418318fe13a2","toSide":"top"}, + {"id":"1b3efa1ca44739ca","fromNode":"f264418318fe13a2","fromSide":"bottom","toNode":"84420c9c180de186","toSide":"top"} ] } \ No newline at end of file