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书籍/力学书籍/力学/Kane-Dynamics-Theory-Applications/auto/Kane-Dynamics-Theory-Applications.md

@@ -5087,22 +5087,25 @@ $$
 # ENERGY FUNCTIONS  
 
 The use of potential energy functions and kinetic energy functions sometimes enables one to construct integrals of equations of motion (see Secs. 7.1 and 7.2). In addition, potential energy functions can be helpful when one seeks to form expressions for generalized active forces, and expressions for generalized inertia forces can be formed with the aid of kinetic energy functions. Hence, familiarity with these functions is certainly desirable. However, since one can readily formulate equations of motion and extract information from such equations without ininvoking energy concepts, one need not master the material in the present chapter before moving on to Chapter 6.  
+使用势能函数和动能函数有时可以构造运动方程的积分（见第7.1和7.2节）。此外，势能函数在寻求广义主动力表达式时可以提供帮助，而动能函数则可以辅助形成广义惯性力表达式。因此，熟悉这些函数无疑是值得的。然而，由于可以无需调用能量概念，即可直接构建运动方程并从中提取信息，因此在进入第6章之前，不必完全掌握本章的内容。
 
 # 5.1 POTENTIAL ENERGY  
 
-If $\boldsymbol{s}$ is a holonomic system (see Sec. 2.13) possessing generalized coordinates $q_{1},\ldots,q_{n}$ (see Sec. 2.10) and generalized speeds $u_{1},\ldots,u_{n}$ (see Sec. 2.12) in a reference frame $\pmb{A}$ , and the generalized speeds are defined as  
+If $\boldsymbol{S}$ is a holonomic system (see Sec. 2.13) possessing generalized coordinates $q_{1},\ldots,q_{n}$ (see Sec. 2.10) and generalized speeds $u_{1},\ldots,u_{n}$ (see Sec. 2.12) in a reference frame $\pmb{A}$ , and the generalized speeds are defined as  
-
+如果 $\boldsymbol{S}$ 是一个holonomic系统（见第 2.13 节），拥有广义坐标 $q_{1},\ldots,q_{n}$（见第 2.10 节）和广义速度 $u_{1},\ldots,u_{n}$（见第 2.12 节）在一个参考系 $\pmb{A}$ 中，且广义速度被定义为：
 $$
 u_{r}\triangleq{\dot{q}}_{r}\qquad(r=1,\dots,n)
 $$  
 
 then there may exist functions $V$ of $q_{1},\ldots,q_{n}$ and the time $\boldsymbol{t}$ that satisfy all of the equations  
-
+那么，可能存在函数 $V$，它依赖于 $q_{1},\ldots,q_{n}$ 和时间 $\boldsymbol{t}$，并且能够满足所有方程。
 $$
 F_{r}=\,-\,{\frac{\partial V}{\partial q_{r}}}\qquad(r=1,\,.\,.\,.\,,n)
 $$  
 
-where $F_{1},\ldots,F_{n}$ are generalized active forces for $s$ in $\pmb{A}$ (see Sec. 4.4) associated with $u_{1},\ldots,u_{n}$ , respectively. Any such function $V$ is called a potential energy of $s$ in $A$ [One speaks of $^{\pmb{a}}$ potential energy, rather than the potential energy because, $V$ satisfies Eqs. (2), then $V+C$ ,where $C$ is any function of t, also satisfies Eqs. (2) and is, therefore, a potential energy of $s$ in A.]  
+where $F_{1},\ldots,F_{n}$ are generalized active forces for $\boldsymbol{S}$ in $\pmb{A}$ (see Sec. 4.4) associated with $u_{1},\ldots,u_{n}$ , respectively. Any such function $V$ is called a potential energy of $s$ in $A$ （One speaks of $^{\pmb{a}}$ potential energy, rather than the potential energy because, $V$ satisfies Eqs. (2), then $V+C$ ,where $C$ is any function of t, also satisfies Eqs. (2) and is, therefore, a potential energy of $s$ in A.）
+
+其中，$F_{1},\ldots,F_{n}$ 是与 $u_{1},\ldots,u_{n}$ 分别相关的 $\boldsymbol{S}$ 在 $\pmb{A}$ 中的广义主动力（见第 4.4 节）。任何这样的函数 $V$ 被称为 $\boldsymbol{S}$ 在 $A$ 中的势能（人们说的是 ${\pmb{a}}$ 势能，而不是简单的势能，因为 $V$ 满足方程 (2)，那么 $V+C$，其中 $C$ 是 $t$ 的任意函数，也满足方程 (2)，因此是 $s$ 在 $A$ 中的势能。）
 
 When a potential energy $V$ of $s$ satisfies the equation  
 
@@ -5111,28 +5114,29 @@ $$
 $$  
 
 then $\dot{V},$ the total time-derivative of $V.$ is given by  
-
+then $\dot{V},$ $V$ 的总时间导数，由
 $$
 {\dot{V}}=\,-\,\sum_{r\,=\,1}^{n}\,F_{r}{\dot{q}}_{r}
 $$  
 
 It is by virtue of this fact that potential energy plays an important part in the construction of integrals of equations of motion, as will be shown in Sec. 7.2.  
+正因如此，势能在运动方程积分的构建中扮演着重要的角色，正如将在第7.2节中所示。
 
 Given generalized active forces $F$ $(r=1,\ldots,n)$ all of which can be regarded as functions of $q_{1},\ldots,q_{n}$ , and $t$ (but not of $u_{1},\dotsc,u_{n})$ . one can either prove that $V$ does not exist, or find $V(q_{1},\dots,q_{n};t)$ explicitly, as follows: Determine whether or not all of the equations  
-
+给定广义主动力 $F$ ($r=1,\ldots,n$) ，它们都可以被视为 $q_{1},\ldots,q_{n}$ 的函数，以及 $t$ 的函数（但不是 $u_{1},\dotsc,u_{n}$ 的函数）。 可以证明 $V$ 不存在，或者按以下方式显式地找到 $V(q_{1},\dots,q_{n};t)$：确定以下所有方程是否成立：
 $$
 \frac{\partial F_{r}}{\partial q_{s}}=\frac{\partial F_{s}}{\partial q_{r}}\qquad(r,s=1,\ldots,n)
 $$  
 
 are satisfied. If one or more of Eqs. (5) are violated, then $V$ does not exist; if all of Eqs. (5) are satisfied, then $V$ exists and is given by  
-
+满足。如果 Eqs. (5) 中的一个或多个不满足，则 $V$ 不存在；如果 Eqs. (5) 全部满足，则 $V$ 存在，且由
 $$
 \begin{array}{l}{{V=\displaystyle\int_{\alpha_{1}}^{q_{1}}\!\frac{\partial}{\partial q_{1}}\,V(\zeta,\alpha_{2},\ldots,\alpha_{n};\,t)\,d\zeta\,+\,\displaystyle\int_{\alpha_{2}}^{q_{2}}\!\frac{\partial}{\partial q_{2}}\,V(q_{1},\zeta,\alpha_{3},\ldots,\alpha_{n};\,t)\,d\zeta}}\\ {{\mathrm{}+\ldots+\,\displaystyle\int_{\alpha_{n}}^{q_{n}}\!\frac{\partial}{\partial q_{n}}\,V(q_{1},\ldots,q_{n-1},\zeta;\,t)\,d\zeta\,+\,C}}\end{array}
 $$  
 
-Wwhere $\alpha_{1},\ldots,\alpha_{n}$ and $C$ are any functions of $t$ [It is advantageous to set as many of $\alpha_{1},\ldots,\alpha_{n}$ equal to zero as is possible without rendering any of the integrals in Eq.(6) improper.]  
+Wwhere $\alpha_{1},\ldots,\alpha_{n}$ and $C$ are any functions of $t$ (It is advantageous to set as many of $\alpha_{1},\ldots,\alpha_{n}$ equal to zero as is possible without rendering any of the integrals in Eq.(6) improper.) 
-
+其中 $\alpha_{1},\ldots,\alpha_{n}$ 和 $C$ 是任意的 $t$ 的函数（为了避免使公式(6)中的任何积分失效，应尽可能将 $\alpha_{1},\ldots,\alpha_{n}$ 设置为零）。
-When $s$ is holonomic and $u_{1},\ldots,u_{n}$ are defined as  
+When $\boldsymbol{S}$ is holonomic and $u_{1},\ldots,u_{n}$ are defined as  
 
 $$
 u_{r}\underset{(2.12.1)}{\triangleq}\sum_{1}^{n}\,Y_{r s}{\dot{q}}_{s}+Z_{r}\quad\quad(r=1,\ldots,n)
@@ -5161,15 +5165,16 @@ $$
 $$  
 
 respectively. Under these circumstances, one can either prove that $V$ doesnot exist, or find $V(q_{1},\dots,q_{n};t)$ explicitly, as follows: Solve Eqs. (9) for $\partial V/\partial q_{s}$ $(s=1,\ldots,n)$ and determine whether or not all of the equations  
-
+在这些情况下，要么可以证明 $V$ 不存在，或者像下面这样显式地找到 $V(q_{1},\dots,q_{n};t)$：解 Eqs. (9) 得到 $\partial V/\partial q_{s}$ $(s=1,\ldots,n)$，并确定是否所有方程
 $$
 {\frac{\partial}{\partial q_{s}}}\left({\frac{\partial V}{\partial q_{r}}}\right)={\frac{\partial}{\partial q_{r}}}\left({\frac{\partial V}{\partial q_{s}}}\right)\qquad(r,s=1,\ldots,n)
 $$  
 
 are satisfied. If one or more of Eqs. (12) are violated, then $V$ does not exist ; if all of Eqs. (12) are satisfied, then $V$ exists and can be found by using Eq. (6).  
+如果 Eqs. (12) 中的一个或多个不满足，则 $V$ 不存在；如果 Eqs. (12) 中的所有条件都满足，则 $V$ 存在，可以通过 Eq. (6) 找到。
 
-When $s$ is a simple nonholonomic system possessing $\pmb{p}$ degrees of freedom in $A$ (see Sec. 2.13), $u_{1},\ldots,u_{n}$ are defined as in Eqs. (1), and the motion constraint equations relating ${\dot{q}}_{p+1},\dots,{\dot{q}}_{n}$ to ${\dot{q}}_{1},\dots,{\dot{q}}_{p}$ are [this is a special case of Eqs. (2.13.1)]  
+When $\boldsymbol{S}$ is a simple nonholonomic system possessing $\pmb{p}$ degrees of freedom in $A$ (see Sec. 2.13), $u_{1},\ldots,u_{n}$ are defined as in Eqs. (1), and the motion constraint equations relating ${\dot{q}}_{p+1},\dots,{\dot{q}}_{n}$ to ${\dot{q}}_{1},\dots,{\dot{q}}_{p}$ are (this is a special case of Eqs. (2.13.1))
-
+当 $\boldsymbol{S}$ 是一个简单的nonholonomic系统，在 $A$ 中具有 $\pmb{p}$ 个自由度（见第 2.13 节），$u_{1},\ldots,u_{n}$ 的定义如 Eqs. (1) 所示，且将 ${\dot{q}}_{p+1},\dots,{\dot{q}}_{n}$ 与 ${\dot{q}}_{1},\dots,{\dot{q}}_{p}$ 关联的运动约束方程为（这是 Eqs. (2.13.1) 的一个特例）
 $$
 \dot{q}_{k}=\sum_{r=1}^{p}C_{k r}\dot{q}_{r}+D_{k}\qquad(k=p+1,\ldots,n)
 $$  
@@ -5191,7 +5196,7 @@ $$
 $$  
 
 respectively, whereas, when $u_{1},\ldots,u_{n}$ are defined as in Eqs. (7), so that Eqs. (8) apply, while the motion constraint equations relating $u_{p+1},\dotsc,u_{n}$ $u_{1},\dotsc,u_{p}$ are  
-
+鉴于，当 $u_{1},\ldots,u_{n}$ 按照公式 (7) 定义，使得公式 (8) 适用，而将 $u_{p+1},\dotsc,u_{n}$ 与 $u_{1},\dotsc,u_{p}$ 之间的运动约束方程为
 $$
 u_{k}\mathop{=}_{(2.13.1)}\sum_{r\,=\,1}^{p}A_{k r}u_{r}+B_{k}\qquad(k=p+1,\dots,n)
 $$  
@@ -5213,26 +5218,47 @@ $$
 $$  
 
 respectively. In both cases, the procedure for either proving that $V$ does not exist or finding $V$ explicitly is more complicated than in the two cases considered previously, the underlying reason for this being that the $\pmb{n}$ partial derivatives $\partial V/\partial q_{1},\ldots,\partial V/\partial q_{n}$ needed in Eqs. (6) appear in only $\pmb{p}$ equations, namely, Eqs. (14) or (18). What follows is a seven-step procedure for surmounting this hurdle.  
+在两种情况下，无论是证明 $V$ 不存在还是显式地找到 $V$，其过程都比先前考虑的两种情况更为复杂，其根本原因是 Eqs. (6) 中需要的 $\pmb{n}$ 个偏导数 $\partial V/\partial q_{1},\ldots,\partial V/\partial q_{n}$ 只出现在 $\pmb{p}$ 个方程中，即 Eqs. (14) 或 (18)。 以下是一个七步程序，用于克服这一难点。
 
-Step $^{\,\prime}$ Introduce $m\triangleq n-p$ quantities $f_{1},\ldots,f_{m}$ as  
+Step 1  Introduce $m\triangleq n-p$ quantities $f_{1},\ldots,f_{m}$ as  
 
 $$
 f_{s-p}\triangleq{\frac{\partial V}{\partial q_{s}}}\qquad(s=p+1,\ldots,n)
 $$  
 
-and regard each of these as a function of $q_{1},\ldots,q_{n}$ , and $t$ , except when both of the following conditions are fulfilled for some value of $r$ say, $r=i;$ (1) the generalized active force $\widetilde{\boldsymbol{F}}_{i}$ is a function of $q_{i}$ only ; (2) the right-hand members of Eqs. (14) or (18) reduce to $-\partial V/\partial q_{i}$ . In that event, regard each of $f_{1},\ldots,f_{m}$ as a function of $t$ and all of $q_{1},\ldots,q_{n}$ except $q_{i}$ [Unless this is done,Eqs. (21) and the now applicable relationship $\tilde{F}_{i}=\,-\partial V/\partial q_{i}$ lead to conflicting expressions for $\partial^{2}V/\partial q_{s}\partial q_{i}$ $(s=$ $p+1,\ldots,n;s\neq i)$ , namely, $\partial f_{s-p}/\partial q_{i}\neq0$ and $\partial\tilde{F}_{i}/\partial q_{s}=0,$ , respectively.]  
+and regard each of these as a function of $q_{1},\ldots,q_{n}$ , and $t$ , except when both of the following conditions are fulfilled for some value of $r$ say, $r=i;$ (1) the generalized active force $\widetilde{\boldsymbol{F}}_{i}$ is a function of $q_{i}$ only ; (2) the right-hand members of Eqs. (14) or (18) reduce to $-\partial V/\partial q_{i}$ . In that event, regard each of $f_{1},\ldots,f_{m}$ as a function of $t$ and all of $q_{1},\ldots,q_{n}$ except $q_{i}$ (Unless this is done,Eqs. (21) and the now applicable relationship $\tilde{F}_{i}=\,-\partial V/\partial q_{i}$ lead to conflicting expressions for $\partial^{2}V/\partial q_{s}\partial q_{i}$ $(s=$ $p+1,\ldots,n;s\neq i)$ , namely, $\partial f_{s-p}/\partial q_{i}\neq0$ and $\partial\tilde{F}_{i}/\partial q_{s}=0,$ , respectively.)
+
+并将这些力视为是 $q_{1},\ldots,q_{n}$ 和 $t$ 的函数，除非在某个特定的 $r$ 值（设为 $r=i;$）下，满足以下两个条件：
+
+1. 广义主动力 $\widetilde{\boldsymbol{F}}_{i}$ 仅是 $q_{i}$​ 的函数；
+    
+2. 方程 (14) 或 (18) 的右端项简化为 $-\partial V/\partial q_{i}$。
+    
+
+在这种情况下，应将每个$f_{1},\ldots,f_{m}$​ 看作是 $t$以及所有 $q_{1},\ldots,q_{n}$ 中除了 $q_{i}$ 之外变量的函数。（除非这样处理，否则方程 (21) 和此时适用的关系 $\tilde{F}_{i}=\,-\partial V/\partial q_{i}$会导致关于 $\partial^{2}V/\partial q_{s}\partial q_{i}$ （其中  $(s=$ $p+1,\ldots,n;s\neq i)$ 的矛盾表达式，也就是一方面有 $\partial f_{s-p}/\partial q_{i}\neq0$，另一方面又有 $\partial\tilde{F}_{i}/\partial q_{s}=0,$
 
 Step 2 In accordance with Eqs. (21),replace $\partial V/\partial q_{s}$ with $f_{s-p}\,(s=p\,+\,1,\,.\,.\,.\,,n)$ in Eqs. (14) or (18), and solve the resulting $\pmb{p}$ equations for $\partial V/\partial{\boldsymbol{q}}_{i}$ $(r=1,\ldots,p)$  
 
+第2步 根据公式(21)，将$\partial V/\partial q_{s}$ 替换为 $f_{s-p}\,(s=p\,+\,1,\,.\,.\,.\,,n)$，代入公式(14)或(18)并求解得到的$\pmb{p}$方程，以获得 $\partial V/\partial{\boldsymbol{q}}_{i}$ $(r=1,\ldots,p)$
+
 Step 3 Using the expressions obtained in Step 2 for ${\hat{c}}V/{\hat{c}}q,$ $(r=1,\hdots,p)$ , form $p(n-1)$ expressions for $\partial(\partial V/\partial q_{r})/\partial q_{j}$ $(r=1,\dots,p$ .， $j=1,\,.\,.\,.\,,n;\,j\neq r)$ . Referring to Eqs. (21), form the $m(n-1)$ equations $\partial(\partial V/\partial q_{s})/\partial q_{j}=\partial f_{s-p}/\partial q_{j}$ $(s=$ $p+1,\ldots,n;\,j=1,\ldots,n;\,j\neq s)$ . Substitute into Eqs. (12) to obtain $n(n-1)/2$ linear algebraic equations in the mn quantities ${\partial f_{i}}/{\partial q_{j}}\,({i=1,\dots,m;j=1,\dots,n})$  
+第3步 利用第2步获得表达式 ${\hat{c}}V/{\hat{c}}q,$ $(r=1,\hdots,p)$ ， 形成 $p(n-1)$ 个关于 $\partial(\partial V/\partial q_{r})/\partial q_{j}$ 的表达式 $(r=1,\dots,p$ .， $j=1,\,.\,.\,.\,,n;\,j\neq r)$ 。 参考公式 (21)， 形成 $m(n-1)$ 个方程 $\partial(\partial V/\partial q_{s})/\partial q_{j}=\partial f_{s-p}/\partial q_{j}$ $(s=$ $p+1,\ldots,n;\,j=1,\ldots,n;\,j\neq s)$ 。 将其代入公式 (12) 得到 $n(n-1)/2$ 个关于 $mn$ 个量 ${\partial f_{i}}/{\partial q_{j}}\,({i=1,\dots,m;j=1,\dots,n})$ 的线性代数方程。
 
-Step $\pmb{\mathscr{d}}$ Identify an $n(n\,-\,1)/2\,\times\,m n$ matrix $[Z]$ and an $n(n\mathrm{~-~}1)/2\,\times\,1$ matrix $\{Y\}$ such that the set of equations written in Step 3 is equivalent to the matrix equation $[Z]\ \{X\}=\{Y\}$ where $\{X\}$ is an $m n\times1$ matrix having $\partial f_{1}/\partial q_{1},\dots$ $\partial f_{1}/\partial q_{n},\ldots,\partial f_{m}/\partial q_{1},\ldots,\partial f_{m}/\partial q_{n}$ as successive elements.  
+Step 4 Identify an $n(n\,-\,1)/2\,\times\,m n$ matrix $[Z]$ and an $n(n\mathrm{~-~}1)/2\,\times\,1$ matrix $\{Y\}$ such that the set of equations written in Step 3 is equivalent to the matrix equation $[Z]\ \{X\}=\{Y\}$ where $\{X\}$ is an $m n\times1$ matrix having $\partial f_{1}/\partial q_{1},\dots$ $\partial f_{1}/\partial q_{n},\ldots,\partial f_{m}/\partial q_{1},\ldots,\partial f_{m}/\partial q_{n}$ as successive elements.  
 
-Step 5Determine the rank $\rho$ of $[Z]$ . If $\rho\,=\,n(n\,-\,1)/2$ , then $V$ may exist, but cannot be found by the application of a straightforward procedure. If $\rho\neq n(n-1)/2,$ use any $\rho$ rows of $[Z]$ , hereafter called independent rows, to express each of the remaining rows of $[Z]$ , hereafter called the dependent rows, as a weighted, linear combination of the $\rho$ independent rows; and solve the resulting set of equations simultaneously to determine the weighting factors.  
+Step 5 Determine the rank $\rho$ of $[Z]$ . If $\rho\,=\,n(n\,-\,1)/2$ , then $V$ may exist, but cannot be found by the application of a straightforward procedure. If $\rho\neq n(n-1)/2,$ use any $\rho$ rows of $[Z]$ , hereafter called independent rows, to express each of the remaining rows of $[Z]$ , hereafter called the dependent rows, as a weighted, linear combination of the $\rho$ independent rows; and solve the resulting set of equations simultaneously to determine the weighting factors.  
 
 Step 6 Express each element of $\{\cal Y\}$ corresponding to a dependent row of $[Z]$ as a weighted, linear combination of the $\rho$ elements of $\{Y\}$ corresponding to the independent rows of $[Z]$ , using the weighting factors found in Step 5, and solve the resulting set of equations for $f_{1},\ldots,f_{m}$ . If this cannot be done uniquely, or if one or more of $f_{1},\ldots,f_{m}$ turn out to be functions of a generalized coordinate of which they should be independent in accordance with Step 1, then a potential energy $V$ of $_s$ in $A$ does not exist.  
 
 Step 7 Substitute the functions $f_{1},\ldots,f_{m}$ found in Step 6 into Eqs. (21) and into the expressions for $\partial V/\partial q_{1},\dots,\partial V/\partial q_{s}$ formed in Step 2, thus obtaining expressions for $\partial V/\partial q_{1},\dots,\partial V/\partial q_{n}$ as explicit functions of $q_{1},\ldots,q_{n}$ , and $t.$ Finally, form $V$ in accordance with Eq. (6).  
+第4步 确定一个 $n(n\,-\,1)/2\,\times\,m n$ 矩阵 $[Z]$ 和一个 $n(n\mathrm{~-~}1)/2\,\times\,1$ 矩阵 $\{Y\}$，使得第3步中列出的方程组等价于矩阵方程 $[Z]\ \{X\}=\{Y\}$，其中 $\{X\}$ 是一个 $m n\times1$ 矩阵，其元素依次为 $\partial f_{1}/\partial q_{1},\dots$ $\partial f_{1}/\partial q_{n},\ldots,\partial f_{m}/\partial q_{1},\ldots,\partial f_{m}/\partial q_{n}$。
+
+第5步 确定矩阵 $[Z]$ 的秩 $\rho$。如果 $\rho\,=\,n(n\,-\,1)/2$ ，则 $V$ 可能存在，但无法通过简单的程序找到。如果 $\rho\neq n(n-1)/2$，则使用 $[Z]$ 的任意 $\rho$ 行（今后称为独立行）来表达 $[Z]$ 的其余行（今后称为相关行），将其表示为这 $\rho$ 个独立行的加权线性组合；并同时求解由此产生的方程组，以确定权重因子。
+
+第6步 使用第5步中找到的权重因子，将对应于 $[Z]$ 中相关行的每个元素 $\{\cal Y\}$ 表示为这 $\rho$ 个对应于 $[Z]$ 中独立行的 $\{Y\}$ 元素的加权线性组合；并求解由此产生的方程组，以确定 $f_{1},\ldots,f_{m}$。如果无法唯一地进行此操作，或者如果 $f_{1},\ldots,f_{m}$ 中的一个或多个实际上是泛坐标的函数，而这些函数应该根据第1步是独立的，那么$_s$ 在 $A$ 中的势能 $V$ 不存在。
+
+第7步 将第6步中找到的函数 $f_{1},\ldots,f_{m}$ 代入 Eqs. (21) 以及在第2步中形成的 $\partial V/\partial q_{1},\dots,\partial V/\partial q_{s}$ 表达式，从而获得 $\partial V/\partial q_{1},\dots,\partial V/\partial q_{n}$ 关于 $q_{1},\ldots,q_{n}$ 和 $t$ 的显式函数；最后，根据 Eq. (6) 构造 $V$。
+
 
 Derivations Multiplication of both sides of Eqs. (18) with $u_{r}$ and subsequent summation yields  
 
@@ -6642,7 +6668,15 @@ When nonlinear kinematical and/or dynamical equations are in hand, one forms the
 Develop fully nonlinear expressions for angular velocities of rigid bodies belonging to S, for velocities of mass centers of such bodies, and for velocities of particles of $s$ to which contact and/or distance forces contributing to generalized active forces are applied. Use these nonlinear expressions to determine partial angular velocities and partial velocities by inspection. Linearize all angular velocities of rigid bodies and velocities of particles, and use the linearized forms to construct linearized angular accelerations and accelerations. Linearize all partial angular velocities and partial velocities. Form linearized generalized active forces and linearized generalized inertia forces, and substitute into Eqs. (6.1.1) or (6.1.2).  
 
 Examples As was shown in the example in Sec. 6.1, all motions of the Foucault pendulum are governed by the equations  
+经常，人们可以通过系统 $s$ 的线性化运动学和/或动力学方程，获得关于其行为的许多有用的信息，即从非线性方程中省略所有关于某些（或全部）广义速度 $u_{1},\ldots,u_{n}$ 和广义坐标 $q_{1},\ldots;q_{n}$ 的扰动二阶或更高阶项而导出的方程。这主要是因为线性微分方程通常比非线性微分方程更容易求解。当然，这些线性化方程的解只能导出对应完整非线性方程解的近似值，而且这些近似值可能相当粗略。无论如何，随着线性化中涉及的扰动值越来越小，这些近似值会变得越来越好。
 
+当手头有非线性运动学和/或动力学方程时，通过将线性化中涉及的所有函数按这些扰动展开成幂级数，并舍弃所有非线性项，来形成它们的线性化对应物。要直接制定线性化动力学方程，即在没有首先写出精确动力学方程的情况下，请按以下步骤进行：
+
+充分展开属于 S 的刚体角速度、这些刚体的质量中心速度以及应用于贡献于广义主动力的接触和/或距离力的 $s$ 中粒子的速度。使用这些非线性表达式来确定部分角速度和部分速度。线性化所有刚体的角速度和粒子的速度，并使用线性化形式构造线性化的角加速度和加速度。线性化所有部分角速度和部分速度。形成线性化的广义主动力和线性化的广义惯性力，并代入 Eqs. (6.1.1) 或 (6.1.2)。
+
+
+
+示例：正如 6.1 节中的示例所示，福柯摆的所有运动都由方程控制。
 $$
 \begin{array}{r l}&{\dot{u}_{1\,\,\,\,\,=\,\,\,\,2\omega u_{2}}(\mathtt{c}_{1}\mathtt{c}_{2}\mathtt{c}\phi\,-\,\mathtt{s}_{2}\,\mathtt{s}\phi)+g\mathtt{c}_{1}\mathtt{s}_{2}}\\ &{\qquad\overset{(6.1.29)}{\underbrace{(6.1.29)}}-2\omega u_{1}(\mathtt{c}_{1}\mathtt{c}_{2}\mathtt{c}\phi-\mathtt{s}_{2}\,\mathtt{s}\phi)-g\mathtt{s}_{1}}\end{array}
 $$  
@@ -7137,7 +7171,9 @@ With $r=2\AA$ ,Eqs. (10) yield $\widetilde{F}_{2}=0$ , so that Eqs. (1) are sati
 
 # 6.6 STEADY MOTION  
 
-A simple nonholonomic system S possessing $\pmb{p}$ degrees of freedom in a Newtonian reference frame $N$ is said to be in a state of steady motion in $N$ when the generalized speeds $u_{1},\dotsc,u_{p}$ have constant values, say, $\bar{u}_{1},...,\bar{u}_{p}$ , respectively. To determine the conditions under which steady motions can occur, use Eqs. (6.1.1) or (6.1.2), proceeding as follows: Form expressions for angular velocities of rigid bodies belonging to $\pmb{S},$ velocities of mass centers of these bodies, and so forth, without regard to the fact that $u_{1},\dotsc,u_{p}$ are to remain constant, and use these expressions to construct partial angular velocities and partial velocities. Set $u_{r}=\bar{u}_{r}(r=1,\ldots,p)$ in angular velocity and velocity expressions, then differentiate with respect to time to generate needed angular accelerations of rigid bodies and accelerations of various points.Formulate expressions for ${\widetilde{\boldsymbol{F}}},$ and $\tilde{F}_{r}^{\mathrm{~*~}}(r=1,\dotsc,p)$ in the case of Eqs. (6.1.1), or $F_{r}$ and $F_{r}^{\ast}\left(r=1,\ldots,n\right)$ in the case of Eqs. (6.1.2), and substitute into Eqs. (6.1.1) or (6.1.2).  
+A simple nonholonomic system S possessing $\pmb{p}$ degrees of freedom in a Newtonian reference frame $N$ is said to be in a state of steady motion in $N$ when the generalized speeds $u_{1},\dotsc,u_{p}$ have constant values, say, $\bar{u}_{1},...,\bar{u}_{p}$ , respectively. To determine the conditions under which steady motions can occur, use Eqs. (6.1.1) or (6.1.2), proceeding as follows: Form expressions for angular velocities of rigid bodies belonging to $\pmb{S},$ velocities of mass centers of these bodies, and so forth, without regard to the fact that $u_{1},\dotsc,u_{p}$ are to remain constant, and use these expressions to construct partial angular velocities and partial velocities. Set $u_{r}=\bar{u}_{r}(r=1,\ldots,p)$ in angular velocity and velocity expressions, then differentiate with respect to time to generate needed angular accelerations of rigid bodies and accelerations of various points. Formulate expressions for ${\widetilde{\boldsymbol{F}}},$ and $\tilde{F}_{r}^{\mathrm{~*~}}(r=1,\dotsc,p)$ in the case of Eqs. (6.1.1), or $F_{r}$ and $F_{r}^{\ast}\left(r=1,\ldots,n\right)$ in the case of Eqs. (6.1.2), and substitute into Eqs. (6.1.1) or (6.1.2).  
+一个简单的非完整系统 S 拥有 $\pmb{p}$ 个自由度，在牛顿参考系 $N$ 中，当广义速度 $u_{1},\dotsc,u_{p}$ 分别具有常数值，例如 $\bar{u}_{1},...,\bar{u}_{p}$ 时，则认为该系统在 $N$ 中处于稳速运动状态。为了确定稳速运动发生的条件，使用公式 (6.1.1) 或 (6.1.2)，按以下步骤进行：首先，不考虑 $u_{1},\dotsc,u_{p}$ 需要保持常数的前提，分别建立属于 $\pmb{S}$ 的刚体的角速度表达式、这些刚体的质心速度表达式等等。然后，利用这些表达式构造偏角速度和偏速度。将 $u_{r}=\bar{u}_{r}(r=1,\ldots,p)$ 代入角速度和速度表达式中，然后对时间求导，以生成所需的刚体角加速度和各种点的加速度。建立 ${\widetilde{\boldsymbol{F}}}$ 和 $\tilde{F}_{r}^{\mathrm{~*~}}(r=1,\dotsc,p)$ 的表达式（对于公式 (6.1.1) 的情况），或者建立 $F_{r}$ 和 $F_{r}^{\ast}\left(r=1,\ldots,n\right)$ 的表达式（对于公式 (6.1.2) 的情况），然后代入公式 (6.1.1) 或 (6.1.2) 中。
+
 
 Example Figure 6.6.1 shows a right-circular, uniform, solid cone $C$ in contact with a fixed, horizontal plane $P$ The motion that $C$ performs-when $C$ rolls on $P$ in such a way that the mass center $C^{*}$ of $C$ (see Fig. 6.6.1) remains fixed while theplane determinedby the axis of $C$ and a verticai line passing through $C^{*}$ has an angular velocity $-\Omega\mathbf{k}$ $\Omega$ constant)--is a steady motion, as will be shown presently. But this motion can take place only if $\Omega$ , the radius $R$ of the base of $C$ ,theheight $4h$ of $C$ ,and the inclination angle $\theta$ (see Fig. 6.6.1) are related to each other suitably.To determine the conditionsunderwhich the motion is possible, we begin by noting that $C$ has three degrees of freedom, and introduce generalized speeds $u_{1},u_{2}$ , and $u_{3}$ as  
 

多体+耦合求解器/稳态平衡点求解.canvas

@@ -0,0 +1,22 @@
+{
+	"nodes":[
+		{"id":"4cabdcd7e0d502fd","x":80,"y":-100,"width":250,"height":60,"type":"text","text":"x是广义坐标\nq几到 q几   9个"},
+		{"id":"d823cc290bdbbdbc","x":-280,"y":-100,"width":250,"height":100,"type":"text","text":"F\n每个叶素的力、力矩\n重力"},
+		{"id":"adba3d94be07f144","x":80,"y":84,"width":250,"height":60,"type":"text","text":"变形量= "},
+		{"id":"83f0241674e9e5cb","x":420,"y":-100,"width":250,"height":60,"type":"text","text":"k？"},
+		{"id":"d80813d7cb9408cb","x":80,"y":-260,"width":250,"height":60,"type":"text","text":"F = kx"},
+		{"id":"a86cab933bc80a24","x":-280,"y":240,"width":250,"height":60,"type":"text","text":"受力分析"},
+		{"id":"93f1b7fd6fa3224c","x":80,"y":220,"width":250,"height":100,"type":"text","text":"叶片上气动力、力矩、重力\n是否还有别的力"},
+		{"id":"18bb8e4ab0f60e38","x":420,"y":240,"width":250,"height":60,"type":"text","text":"造成什么结果"},
+		{"id":"6703fb1c685819a1","x":-280,"y":460,"width":250,"height":60,"type":"text","text":"虚功原理/达朗贝尔原理"},
+		{"id":"9f72751890c22622","x":80,"y":460,"width":250,"height":60,"type":"text","text":"是什么，是否是依据"}
+	],
+	"edges":[
+		{"id":"c906e0b3a77f40a3","fromNode":"d80813d7cb9408cb","fromSide":"bottom","toNode":"d823cc290bdbbdbc","toSide":"top"},
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+		{"id":"0c09663a87eec93d","fromNode":"4cabdcd7e0d502fd","fromSide":"bottom","toNode":"adba3d94be07f144","toSide":"top"},
+		{"id":"adee06cd6c2acaef","fromNode":"d80813d7cb9408cb","fromSide":"bottom","toNode":"83f0241674e9e5cb","toSide":"top"},
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+		{"id":"a0bdb8c360ee84c5","fromNode":"6703fb1c685819a1","fromSide":"right","toNode":"9f72751890c22622","toSide":"left"}
+	]
+}

线性化求解器/参考文献/González-Assessment of linearization approache/auto/González-Assessment of linearization approache.md

@@ -124,22 +124,22 @@ $$
 
 Equation (6) has $2\left(n-m\right)$ eigenvalues that represent the exact spectrum of the problem. Moreover, its leading matrix $\mathbf{M}_{r}$ is symmetric and positive-definite. As a consequence, Eq. (6) can be used in the solution of forward-dynamics problems with explicit integration schemes.  
 方程 (6) 具有 $2\left(n-m\right)$ 个特征值，它们代表问题的精确谱。 此外，其首导矩阵 $\mathbf{M}_{r}$ 是对称正定矩阵。 因此，方程 (6) 可用于带有显式积分方案的逆动力学问题的求解。
-# 3.2 RCS Formulation: Generalized Eigenanalysis  
+# 3.2 RCS Formulation: Generalized Eigenanalysis广义特征分析
 
 Following a Lagrangian approach, the dynamics equations (1) can be expressed in the form  
-
+采用拉格朗日方法，动力学方程 (1) 可以表达为：
 $$
 \mathbf{H}_{2}=\left\{\begin{array}{c}{\mathbf{M}\ddot{\mathbf{q}}+\boldsymbol{\Phi}_{\mathbf{q}}^{\mathrm{T}}\boldsymbol{\lambda}-\mathbf{f}}\\ {\boldsymbol{\Phi}\left(\mathbf{q},t\right)}\end{array}\right\}=\mathbf{0}
 $$  
 
 which can be linearized directly and cast in descriptor form  
-
+可以被直接线性化并转化为描述符形式。
 $$
 \mathbf{E}_{q}\delta\dot{\mathbf{y}}=\mathbf{A}_{q}\delta\mathbf{y}+\mathbf{B}_{q}\delta\mathbf{u}
 $$  
 
 $$
-\mathbf{E}_{q}=\left[\begin{array}{c c}{\mathbf{I}}&{\mathbf{0}}&{\mathbf{0}}\\ {\mathbf{0}}&{\mathbf{M}_{q}\mathbf{\Lambda}\mathbf{0}}\\ {\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}\end{array}\right];\qquad\mathbf{A}_{q}=\left[\begin{array}{c c}{\mathbf{0}}&{\mathbf{I}}&{\mathbf{0}}\\ {-\mathbf{K}_{q}-\mathbf{C}_{q}\mathbf{\Lambda}-\mathbf{\Phi}\mathbf{\Phi}\mathbf{\Phi}\mathbf{0}}\\ {-\mathbf{\Phi}\mathbf{\Phi}\mathbf{q}}&{\mathbf{0}}&{\mathbf{0}}\end{array}\right]
+\mathbf{E}_{q}=\left[\begin{array}{c c}{\mathbf{I}}&{\mathbf{0}}&{\mathbf{0}}\\ {\mathbf{0}}&{\mathbf{M}_{q}}&{\mathbf{0}}\\ {\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}\end{array}\right];\qquad\mathbf{A}_{q}=\left[\begin{array}{c c}{\mathbf{0}}&{\mathbf{I}}&{\mathbf{0}}\\ {-\mathbf{K}_{q}-\mathbf{C}_{q}\mathbf{\Lambda}-\mathbf{\Phi}\mathbf{\Phi}\mathbf{\Phi}\mathbf{0}}\\ {-\mathbf{\Phi}\mathbf{\Phi}\mathbf{q}}&{\mathbf{0}}&{\mathbf{0}}\end{array}\right]
 $$  
 
 with  
@@ -150,6 +150,8 @@ $$
 
 It must be noted that the evaluation of $\mathbf{K}_{q}$ requires the computation of the Lagrange multipliers $\lambda$ at equilibrium. Besides the $2\left(n-m\right)$ eigenvalues in the spectrum of the constrained problem, the generalized eigenanalysis of the $(2n+m)$ linearized equations (8) introduces $3m$ spurious eigenvalues, related to the constrained kinematic variables and the Lagrange multipliers. These are easily identifiable because their value is either infinity [6] or zero, if either or both $\Phi_{\mathbf{q}}$ and $\Phi_{\mathbf{q}}^{\mathrm{T}}$ are transferred from $\mathbf{A}_{q}$ to $\mathbf{E}_{q}$ . Because Eq. (7) represents a differential-algebraic problem, matrix $\mathbf{E}_{q}$ is structurally singular. Therefore, Eq. (8) cannot be used in forward-dynamics problems with explicit numerical integrators. It is possible though to overcome this issue applying a QZ decomposition to matrices $\mathbf{E}_{q}$ and $\mathbf{A}_{q}$ [30]. The details of this method are discussed in [6].  
 
+需要注意的是，评估 $\mathbf{K}_{q}$ 需要计算在平衡状态下的拉格朗日乘子 $\lambda$。 除了受约束问题谱中的 $2\left(n-m\right)$ 个特征值外，对 $(2n+m)$ 个线性化方程 (8) 的广义特征分析会引入 $3m$ 个虚假特征值，它们与受约束的运动学变量和拉格朗日乘子相关。 这些虚假特征值很容易识别，因为它们的值要么是无穷大 [6]，要么是零，如果 $\Phi_{\mathbf{q}}$ 和 $\Phi_{\mathbf{q}}^{\mathrm{T}}$ 中的一个或两个从 $\mathbf{A}_{q}$ 传递到 $\mathbf{E}_{q}$。 由于方程 (7) 代表一个微分代数问题，矩阵 $\mathbf{E}_{q}$ 具有结构奇异性。 因此，方程 (8) 不能用于带有显式数值积分器的正向动力学问题。 然而，可以通过对矩阵 $\mathbf{E}_{q}$ 和 $\mathbf{A}_{q}$ 进行 QZ 分解来克服这个问题 [30]。 该方法的细节见 [6]。
+
 # 3.3 UCS Formulation: Penalty Method  
 
 Penalty-based relaxation of the constraints can be used to transform the system of DAEs in Eq. (1) into a set of $n$  
@@ -184,7 +186,15 @@ $$
 \begin{array}{l}{\displaystyle\frac{{\partial{\bf{H}}_{3}}}{{\partial{\bf{q}}}}={\bf{K}}_{p}=\frac{{\partial{\bf{M}}}}{{\partial{\bf{q}}}}{\dot{\bf{q}}}+\frac{{\partial\Phi_{\bf{q}}^{\mathrm{T}}}}{{\partial{\bf{q}}}}\Xi\left({\ddot{\Phi}}+\Theta{\dot{\Phi}}+\Omega{\Phi}\right)\quad}\\ {\displaystyle\quad\quad+\ \Phi_{\bf{q}}^{\mathrm{T}}\Xi\left({\Omega{\Phi_{\bf{q}}}+\frac{{\partial{\Phi_{\bf{q}}}}}{{\partial{\bf{q}}}}\left({\ddot{\bf{q}}}+{\bf{\Theta}}{\dot{\bf{q}}}\right)+\frac{{\partial{\dot{\Phi}}_{\bf{q}}}}{{\partial{\bf{q}}}}{\dot{\bf{q}}}\right)\quad}\\ {\displaystyle\quad\quad+\ \Phi_{\bf{q}}^{\mathrm{T}}\Xi\left({\Theta\frac{{\partial{\Phi_{t}}}}{{\partial{\bf{q}}}}+\frac{{\partial{\dot{\Phi}}_{t}}}{{\partial{\bf{q}}}}}\right)-\frac{{\partial{\bf{f}}}}{{\partial{\bf{q}}}}}\end{array}
 $$  
 
+$$
+\begin{array}{r l}&{\frac{\partial{\bf H}_{3}}{\partial{\bf\dot{q}}}={\bf C}_{p}}\\ &{\quad\quad=\Phi_{{\bf q}}^{\mathrm{T}}\Xi\left(\frac{\partial{\bf\dot{\Phi}}_{{\bf q}}}{\partial{\bf\dot{q}}}{\bf\dot{q}}+\frac{\partial{\bf\dot{\Phi}}_{t}}{\partial{\bf\dot{q}}}+\dot{\bf\Phi}_{{\bf q}}+\Theta{\bf\Phi}_{{\bf q}}\right)-\frac{\partial{\bf f}}{\partial{\bf q}}}\\ &{\quad\quad\frac{\partial{\bf H}_{3}}{\partial{\bf\dot{q}}}={\bf M}_{p}={\bf M}+\Phi_{{\bf q}}^{\mathrm{T}}\Xi\Phi_{{\bf q}}}\\ &{\quad\quad\frac{\partial{\bf H}_{3}}{\partial{\bf u}}=-{\bf F}_{p}=-\frac{\partial{\bf f}}{\partial{\bf u}}}\end{array}
+$$
+
+
+Equation (14) is a system of $n$ ODEs. The method delivers an approximation of the $2\left(n-m\right)$ true system eigenvalues,  
+
 ![](images/4dbaa92b67aa376a9c626b28d98d1f4d1a3a9ee5c377f60bc1a0d6a3d7c22fac.jpg)  
+
 Fig. 1: An $N$ -loop four-bar linkage with spring elements  
 
 together with $2m$ spurious eigenvalues related to the constrained coordinates. True and spurious eigenvalues can be told apart by checking if their associated eigenvectors v verify the velocity-level kinematic constraints Φ˙ΦΦ = 0. The violation of such constraints remains close to zero for true eigenvalues, while it is not negligible for the spurious ones.  
@@ -194,18 +204,20 @@ As it happened in Section 3.1, if the penalty matrix $\Xi$ is correctly chosen,
 # 4 Numerical Examples  
 
 The methods described in Section 3 were applied to the linearization of mechanical systems about static equilibrium configurations. Two examples were selected: the first was an $N$ -loop four-bar linkage with spring elements along the diagonals. The second was a flexible double pendulum. The four-bar linkage is heavily constrained and only has one degree of freedom; it is representative of mechanical systems in which the number of kinematic constraints is similar to the number of generalized coordinates $\left(n-m\ll n\right)$ . The double pendulum, on the other hand, includes degrees of freedom associated with flexibility among the generalized coordinates $\mathbf{q}$ . In this case, $n-m\approx n$ .  
+第3节中描述的方法被应用于机械系统在静态平衡构型附近的线性化。选择了两个例子：第一个是一个带有弹簧元件的 $N$ 环四杆机构；第二个是一个柔性双摆。四杆机构受到大量约束，只有一个自由度；它代表了在运动学约束的数量与广义坐标数 $\left(n-m\ll n\right)$ 相似的机械系统。另一方面，双摆包括与广义坐标 $\mathbf{q}$ 之间的柔性相关的自由度。在这种情况下，$n-m\approx n$。
+
+
   
-$$
-\begin{array}{r l}&{\frac{\partial{\bf H}_{3}}{\partial{\bf\dot{q}}}={\bf C}_{p}}\\ &{\quad\quad=\Phi_{{\bf q}}^{\mathrm{T}}\Xi\left(\frac{\partial{\bf\dot{\Phi}}_{{\bf q}}}{\partial{\bf\dot{q}}}{\bf\dot{q}}+\frac{\partial{\bf\dot{\Phi}}_{t}}{\partial{\bf\dot{q}}}+\dot{\bf\Phi}_{{\bf q}}+\Theta{\bf\Phi}_{{\bf q}}\right)-\frac{\partial{\bf f}}{\partial{\bf q}}}\\ &{\quad\quad\frac{\partial{\bf H}_{3}}{\partial{\bf\dot{q}}}={\bf M}_{p}={\bf M}+\Phi_{{\bf q}}^{\mathrm{T}}\Xi\Phi_{{\bf q}}}\\ &{\quad\quad\frac{\partial{\bf H}_{3}}{\partial{\bf u}}=-{\bf F}_{p}=-\frac{\partial{\bf f}}{\partial{\bf u}}}\end{array}
-$$  
 
 # 4.1 Multiple Loop Four-Bar Linkage  
 
 Figure 1 shows an $N$ -loop four-bar linkage made up of equal rods of length $l_{f}=1\;\mathrm{m}$ and uniformly distributed mass $m_{f}=1~\mathrm{kg}$ . It moves under gravity effects with $g=9.81$ $\mathrm{m}/\mathrm{s}^{2}$ . Each loop $i$ in the linkage has a spring connecting points $B_{i}$ and $A_{i-1}$ of stiffness $k_{f}\,{=}\,25\,\mathrm{N/m}$ and natural length $l_{f0}=\sqrt{2}\,\mathrm{m}$ . If $N_{\mathrm{}}=1$ , the system is equivalent to the test case discussed in [14].  
 
-The linkage is modeled with the $x$ and $y$ coordinates of points $B_{0}–B_{N}$ as variables plus the $\boldsymbol{\upvarphi}$ angle from the global $x_{\mathrm{{}}}$ - axis to the rod that connects points $A_{N}$ and $B_{N}$ $\begin{array}{r}{{n=2}N+3}\end{array}$ ). The kinematic constraints term $\Phi$ is composed by equations that enforce that the distances between the tips of each rod remain constant during motion, plus one extra equation that relates the value of $\boldsymbol{\upvarphi}$ to $x_{N}$ and $y_{N}$ $\!\!\!/m=2N+2\!\!\!$ ).  
+The linkage is modeled with the $x$ and $y$ coordinates of points $B_{0}–B_{N}$ as variables plus the $\boldsymbol{\upvarphi}$ angle from the global $x_{\mathrm{{}}}$ - axis to the rod that connects points $A_{N}$ and $B_{N}$ $\begin{array}{r}{{n=2}N+3}\end{array}$ ). The kinematic constraints term $\Phi$ is composed by equations that enforce that the distances between the tips of each rod remain constant during motion, plus one extra equation that relates the value of $\boldsymbol{\upvarphi}$ to $x_{N}$ and $y_{N}$ $(m=2N+2)$.  
 
-Equation (14) is a system of $n$ ODEs. The method delivers an approximation of the $2\left(n-m\right)$ true system eigenvalues,  
+图1所示为由长度为$l_{f}=1\;\mathrm{m}$且质量均匀分布为$m_{f}=1~\mathrm{kg}$的等长杆组成的$N$环四杆机构。其在重力作用下运动，重力加速度为$g=9.81$ $\mathrm{m}/\mathrm{s}^{2}$。机构中每个环 $i$ 都有一个刚度为$k_{f}\,{=}\,25\,\mathrm{N/m}$且自然长度为$l_{f0}=\sqrt{2}\,\mathrm{m}$的弹簧连接点 $B_{i}$ 和 $A_{i-1}$。当 $N_{\mathrm{}}=1$ 时，该系统等效于[14]中讨论的测试案例。
+
+该机构以点 $B_{0}–B_{N}$ 的 $x$ 和 $y$ 坐标以及连接点 $A_{N}$ 和 $B_{N}$ 的杆与全局 $x_{\mathrm{{}}}$ 轴之间的角度 $\boldsymbol{\upvarphi}$ 作为变量进行建模（共${n=2}N+3$个变量）。运动学约束项 $\Phi$ 由方程组成，这些方程强制每个杆的末端之间的距离在运动过程中保持恒定，外加一个方程，将 $\boldsymbol{\upvarphi}$ 与 $x_{N}$ 和 $y_{N}$ 相关联（$m=2N+2$）。
 
 # 4.2 Flexible Double Pendulum  
 

线性化求解器/参考文献/Kallesøe-Effect of steady deflections on the a/auto/Kallesøe-Effect of steady deflections on the a.md

@@ -0,0 +1,277 @@
+RESEARCH ARTICLE  
+
+# Effect of steady def lections on the aeroelastic stability of a turbine blade  
+
+B. S. Kallesøe  
+
+Wind Energy Department, Risø-DTU, Technical University of Denmark, DK-4000 Roskilde, Denmark  
+
+# ABSTRACT  
+
+This paper deals with effects of geometric non-linearities on the aeroelastic stability of a steady-state deflected blade. Today, wind turbine blades are long and slender structures that can have a considerable steady-state def lection which affects the dynamic behaviour of the blade. The f lapwise blade def lection causes the edgewise blade motion to couple to torsional blade motion and thereby to the aerodynamics through the angle of attack. The analysis shows that in the worst case for this particular blade, the edgewise damping can be decreased by half. 
+本文研究了几何非线性对稳态变桨叶片气弹振稳性的影响。如今，风电机组叶片是长而细的结构，可能存在相当大的稳态变形，这会影响叶片的动力学行为。挥舞方向的叶片变形会导致摆振方向的叶片运动与扭转叶片运动耦合，进而通过攻角影响气动特性。分析表明，对于特定叶片的最坏情况下，摆振阻尼可能会减小一半。
+Copyright $\circled{\mathrm{C}}\ 2010$ John Wiley & Sons, Ltd.  
+
+# KEYWORDS  
+
+stability analysis; aeroelasticity  
+
+# Correspondence  
+
+B. S. Kallesøe, Wind Energy Division, Risø DTU, Frederiksborgvej 399, P.O. Box 49, DK-4000 Roskilde, Denmark. E-mail: bska@risoe.dtu.dk  
+
+Received 5 October 2009; Revised 17 May 2010; Accepted; 31 May 2010  
+
+# 1. INTRODUCTION  
+
+A second-order non-linear beam model is used for aeroelastic stability analysis of a wind turbine blade. The importance of including the effects of non-linear geometric couplings in the stability analysis is considered and the aeroelastic mechanisms driving the aeroelastic response are described in detail.  
+
+The effect of non-linear geometric couplings in a curved rotating blade on the stability has been investigated in the helicopter society for decades1–4 and state-of-the-art comprehensive helicopter stability codes of today, like Hodges et al.,5 include both material and geometric non-linearities. However, most aeroelastic stability tools for wind turbines are based on linear beam theory and do not include the non-linear geometric coupling caused by, for instance, steady-state blade def lection, pre-bend or swept blade.  
+
+In the late 1970s, the oil crisis stimulated many MW size turbine projects. In a review of research on aeroelastic stability Friedmann6 concluded that ‘Reliable aeroelastic stability analyses should be based on non-linear formulations which account for both moderately large deformations (i.e. finite slopes) and non-linear aerodynamic effects, such as stall’. All these MW size turbine projects however ended without any commercial success. Later, the wind turbine followed a development starting at small $30\;\mathrm{kW}$ units gradually growing to today’s MW size commercial turbine. During this period, wind turbines have been relatively stiff constructions with only limited geometric couplings. Chaviaropoulos7 examines the influence of non-linear effects on the aeroelastic stability of a $19\;\mathrm{m}$ blade. It was found that the most important effect to include is the unsteady aerodynamics and that the structural def lection is unimportant. Modern wind turbine blades are longer (up to $60\;\mathrm{m}$ ) and more slender, thus increasing the blade def lection under normal operation and thereby reintroducing stability issues concerning geometric couplings. Steady-state blade def lection will result in geometrically non-linear couplings between the different blade modes. For instance, a large flapwise blade def lection will enhance the coupling between edgewise and torsional blade motion and consequently affect the aerodynamics through the angle of attack. Therefore, it can be important to include the non-linear geometric coupling between for example edgewise and torsional motion of a flapwise deflected blade. 
+
+为了风电机组叶片的气动弹性稳定性分析，采用二阶非线性梁模型。考虑在稳定性分析中包含非线性几何耦合效应的重要性，并详细描述驱动空载气动响应的气动弹性机制。
+
+在直升机领域，人们已经研究了几十年关于弯曲旋转叶片中非线性几何耦合对稳定性的影响<sup>1–4</sup>，如今最先进的直升机稳定性综合计算代码，如 Hodges 等人<sup>5</sup>，都包括了材料和几何非线性。然而，大多数风电机组的气动弹性稳定性工具仍然基于线性梁理论，并未包含由例如稳态叶片变形、预弯或掠角等引起的非线性几何耦合。
+
+在20世纪70年代末，石油危机刺激了许多兆瓦级风力发电机项目。在对气动弹性稳定性研究的回顾中，Friedmann<sup>6</sup> 结论是：“可靠的气动弹性稳定性分析应基于能够考虑中等幅度的变形（即有限斜率）和非线性气动效应（如失速）的非线性公式。” 然而，这些兆瓦级风力发电机项目最终都未能获得商业成功。 之后，风力发电机的发展始于小型30 kW机组，逐渐发展到如今的兆瓦级商业风电机组。在此期间，风力发电机结构相对刚性，几何耦合效应有限。 Chaviaropoulos<sup>7</sup> 考察了非线性效应对19 m叶片气动弹性稳定性的影响。研究发现，最重要的是包含非稳态气动效应，而结构变形的影响不重要。 现代风力发电机叶片更长（高达60 m），更细长，从而在正常运行期间增加了叶片变形，重新引入了关于几何耦合的稳定性问题。 稳态叶片变形会导致不同叶片模态之间的几何非线性耦合。 例如，较大的挥舞叶片变形会增强摆振和扭角叶片运动之间的耦合，从而通过迎角影响气动特性。 因此，在例如摆振和扭角运动之间包含非线性几何耦合，对于挥舞变形的叶片来说可能很重要。
+
+
+Research in utilizing sweep and pre-bend blades is ongoing. The European Union founded project UPWIND $^{8-10}$ deals, among other issues, with non-linear modelling of blades and the effects of including such non-linearities. Some stateof-the-art stability codes, such as TURBU,11 include the effect of geometric non-linearities. Riziotis et al.12 include these effects in a stability analysis of a turbine in closed-loop operation. There is also focus on utilizing the geometric couplings to reduce fatigue and/or ultimate loads, for instance Ashwill et al.,13 where a blade is swept to introduce a flapwise—torsion coupling.  
+
+Wind turbine stability can be analysed by a variety of different model types. The most detailed description of the turbine response is given by numerical non-linear time simulation tools.14–18 These tools show instabilities as well as non-linear effects limiting the response to for instance limit cycle oscillations. They can also be used to analyse the effect of, for instance, turbulence and wind shear’s effects on turbine stability. The referenced tools use different models and different model complexity. For instance, $\mathrm{FAST^{18}}$ is a modal-based code which on the one hand does not include a torsional degree of freedom of the blade and non-linear geometric couplings, but on the other hand is relatively computationally inexpensive. A code like $\mathrm{HAWC}2^{14,15}$ has a more complex model with a structural model based on a multi-body formulation where each body is a Timoshenko beam element including torsion. The drawback of these time-simulation tools is that they are computationally intensive and they can make it difficult to extract the important aeroelastic mechanisms from the large volume of results. Another approach is to use eigenvalue analysis of a linear (or linearized) model of the turbine.11,12,19–21 The HAWCStab code19,21 offers a platform for linearization of the undeflected turbine structure, while the code TURBU11 offers a platform for aeroservoelastic stability analysis based on linearization around the def lected/curved blade state. The structural model in TURBU is based on a simple co-rotational beam element approach. Each beam element consists of a rigid body with springs and dampers in its entry point; average strains in the springs and torque-free rotation offsets between the beam elements embody the average deflected/curved blade state. Riziotis et al.12 offers a multi-body platform which finds a reference state by time integration and linearizes the aeroservoelastic equations around this reference state to provide a stability tool including closed loop control. This type of tool can give both structural eigenfrequencies and eigenmodes that describe the basic structural dynamics of the turbine and aeroelastic frequencies, damping and modes of the aeroelastic motion. The aeroelastic damping reveals any stability problems for the turbine. However, since it is linear tools, they do not give any information concerning non-linear mechanism that limit the amplitude of a linear negative damped mode. The knowledge of structural and aeroelastic frequencies and mode shapes is very useful in the analysis and in the interpretation of results from aeroelastic time simulations. However, the modes of the aeroelastic response of the whole turbine can still be complex to analyse. To reduce the complexity, and thus make the results more transparent, a blade-only analysis is used.22 This allows a clear physical interpretation and insight into the mechanisms that govern the dynamic response of the blade and many basic characteristic of turbine stability can be extract from a blade-only analysis.  
+
+This paper uses a non-linear blade model23 which includes the effect of large blade def lections, pitch action and rotor speed variations. This blade model is strongly inspired by the work of Hodges and Dowell1 First, the structural model is combined with a steady-state aerodynamic model based on beam element momentum (BEM) theory and discritized by a f inite difference scheme. The resulting algebraic non-linear aeroelastic model is employed to compute steady-state blade def lections and induced velocities of a blade from the 5 MW Reference Wind Turbine (RWT) by National Renewable Energy Laboratory (NREL)24 at normal power production conditions. The steady-state def lections are compared with the results from HAWC2 simulations, showing good agreement. Throughout this paper, the 5 MW RWT by NREL is used as an example blade. The reference turbine is an artif icial turbine based on state-of-the-art turbines on the market. The blade is strongly inspired by the $61.5\mathrm{~m~LM}$ glasf iber blade (LM Wind Power, Kolding, Denmark). This blade belongs to the mid-region of f lexible designs of state-of-the-art blades, and hence, the geometric couplings can be more pronounced for other blade designs. The big advantage of this blade however is that all data is publicly available and it has been widely used in other research work and therefore a good reference with realistic f lexibility compared with most state-of-the-art blades. A non-linear structural blade model23 and an unsteady aerodynamic model25 are then linearized about the steadystate def lected blade, preserving the main effects of the geometric non-linearities. The linear model is discritized by the f inite difference scheme which along with boundary conditions form a differential eigenvalue problem. The solution to this eigenvalue problem gives the aeroelastic frequencies and damping, but also information concerning the fundamental aeroelastic behaviour of the blade. The analysis shows that the aeroelastic damping of the edgewise modes changes when the steady-state def lection is included. The aeroelastic motion is analysed in detail for three different operation conditions in which there is large differences in the damping when including or excluding steady-state blade def lections.  
+
+# 2. STRUCTURAL BLADE MODEL  
+
+The structural blade model described in Kallesøe23 is based on the work by Hodges and Dowell1 using secondorder Bernoulli–Euler beam theory to describe the blade motion by a non-linear partial integral-differential equation of motion  
+
+$$
+\vec{\bf M}\ddot{\bf u}+\overline{{{\bf F}}}\left(\dot{\bf u},\overline{{{\bf u}}}^{\prime\prime},\overline{{{\bf u}}}^{\prime},\overline{{{\bf u}}},\ddot{\beta},\dot{\beta},\beta,\ddot{\phi},\dot{\phi},\phi\right)\!=\overline{{{\bf f}}}\left({\bf f}_{a e r o},M_{a e r o},u^{\prime},\nu^{\prime}\right)
+$$  
+
+where $\bar{\bf M}$ is the mass matrix, $\bar{\mathbf{F}}$ is a non-linear function that includes stiffness, damping, gyroscopic terms together with centrifugal force-based integral terms. The state vector $\bar{\mathbf{u}}=[u\left(t,s\right),\nu\left(t,s\right),\theta\left(t,s\right)]$ holds edgewise, f lapwise and torsional deformations, respectively.  
+
+Flapwise is def ined as the direction normal to the rotor plane (positive downwind) and edgewise as in the rotor plane (positive towards leading edge) for a blade at zero pitch. When the blade pitches, the $(u,\,\nu)$ frame follows the blade. The position along the blades elastic axis is denoted $s$ , $t$ is the time, $\beta=\beta(t)$ is the global pitch of the blade, $\phi=\phi(t)$ is the azimuth angle of the rotor and the right hand side force function $\bar{\mathbf{f}}$ holds the effect of the aerodynamic forces $\mathbf{f}_{a e r o}$ and aerodynamic moment $M_{a e r o}$ on the blade. The dots denote time derivatives and the primes denote derivatives with respect to the longitudinal coordinate $s$ . As an example, the equation of motion for edgewise blade bending is given by  
+
+$$
+\begin{array}{r l}&{m\big(\ddot{u}-\ddot{\theta}l_{c g}\sin\big(\overline{{\theta}}\big)\big)+F_{u,1}\big(\ddot{\beta},\dot{\beta},\dot{\phi},\dot{\nu},\dot{\theta},u^{\prime},u,\theta,\beta\big)+F_{u,2}\big(\dot{\phi},\dot{u},\dot{\nu},u^{\prime},\nu,\theta,\beta\big)}\\ &{\quad+\,F_{u,3}(\phi,\beta,\theta,u^{\prime},\nu^{\prime})+F_{u,4}(u^{\prime\prime},\nu^{\prime\prime},\theta)+F_{u,5}\big(\ddot{\phi},\beta\big)}\\ &{\quad=f_{u}+\Big((u^{\prime}+l_{p i}^{\prime})\int_{s}^{R}f_{w}\mathrm{d}\rho\Big)^{\prime}}\end{array}
+$$  
+
+where the f irst term is the inertia forces, the second term $F_{u,1}$ describes the inf luence of pitch action, which will not be used in this work. The third term $\boldsymbol{F}_{u,2}$ describes centrifugal and Coriolis forces caused by the rotor speed. The fourth term $F_{u,3}$ describes the unsteady inf luence form gravity, which is neglected in this work. The f ifth term describes the restoring forces  
+
+$$
+\begin{array}{r l}&{F_{u,4}\!=\!\big(E\big(I_{\xi}\!\cos^{2}\!(\tilde{\theta})\!+I_{\eta}\sin^{2}\!(\tilde{\theta})\big)u^{\prime\prime}\big)^{\prime\prime}+\!\big(E(I_{\xi}\!-I_{\eta})\!\cos\!(\tilde{\theta})\!\sin\!(\tilde{\theta})\nu^{\prime\prime}\big)^{\prime\prime}}\\ &{\qquad-\big(E(I_{\xi}\!-I_{\eta})\theta\big(u^{\prime\prime}\!\sin\!\big(2\tilde{\theta}\big)\!-\nu^{\prime\prime}\!\cos\!\big(2\tilde{\theta}\big)\!+I_{p i}^{\prime\prime}\sin\!\big(\tilde{\theta}\big)\!\cos\!(\tilde{\theta})\!\big)\big)^{\prime\prime}}\end{array}
+$$  
+
+where the f irst term is the bending stiffness in the $x$ -direction, the second term is the coupling to the $y$ -direction and the last term in equation (3) is the coupling to the twist. The sixth term in equation (2) describes the inf luence of rotor speed variations, which is assumed constant in this work, so the term is not active. The right hand side holds the external forces, which in this case will be aerodynamic forces. Longitudinal forces on and in the blade, for example the centrifugal force, lead to integral terms in the equations of motion. A detailed description of all terms are found in Kallesøe.23  
+
+The boundary conditions employed in this paper are for simplif ication derived for blades without pre-curvature. The boundary conditions for the root of the blade are given by the geometric constraints  
+
+$$
+u(0,t)=u^{\prime}(0,t)=\nu(0,t)=\nu^{\prime}(0,t)=\theta(0,t)=0
+$$  
+
+because the frame used to describe the blade follows the root of the blade. The boundary conditions for the tip of the blade are23  
+
+$$
+\begin{array}{l}{{{u^{\prime\prime}}(R,t)=\nu^{\prime\prime}(R,t)\,{=}\,\theta^{\prime}(R,t)=0}}\\ {{u^{\prime\prime\prime}(R,t)\,{=}\,\displaystyle\frac{m l_{c g}}{E I_{\xi}I_{\eta}}\,\big(\dot{\phi}^{2}w-g\cos{(\phi)}\big)\big(I_{\eta}\sin{\big(\tilde{\theta}-\overline{{\theta}}\big)}\sin{\big(\tilde{\theta}\big)}+I_{\xi}\cos{\big(\tilde{\theta}-\overline{{\theta}}\big)}\cos{\big(\tilde{\theta}\big)}\big)}}\\ {{{\nu^{\prime\prime\prime}}(R,t)\,{=}\,\displaystyle\frac{m l_{c g}}{E I_{\xi}I_{\eta}}\big(\dot{\phi}^{2}w-g\cos{(\phi)}\big)\big(I_{\eta}\cos{\big(\tilde{\theta}-\overline{{\theta}}\big)}\sin{\big(\tilde{\theta}\big)}+I_{\xi}\sin{\big(\tilde{\theta}-\overline{{\theta}}\big)}\cos{\big(\tilde{\theta}\big)}\big)}}\end{array}
+$$  
+
+where $s=R$ is the tip of the blade, $m=m(s)$ is the mass per length of the blade, $l_{c g}=l_{c g}(s)$ is the offset of centre of gravity from the elastic axis, $E=E(s)$ is the Young’s modulus, $I=I\left(s\right)$ and $I_{\eta}=I_{\eta}(s)$ is the principle moments of inertia, $w=$ $w(s,t)$ is the radius to the position $s$ on the elastic axis, $g$ denotes gravity, $\tilde{\theta}=\tilde{\theta}\left(s\right)$ is the angle between chord and principle axis of elasticity and $\tilde{\theta}=\tilde{\theta}\left(s\right)$ is the angle between the chord and a line between elastic centre and centre of gravity along which $l_{c g}$ is measured. In the case that $l_{c g}(R)\neq0$ the boundary conditions for the tip are functions of the rotor speed $\dot{\phi}$ and the azimuth angle of the rotor $\phi$ and therefore time varying. This is because an offset of the centre of gravity from the elastic axis at the blade tip leads to a bending moment at the tip caused by gravity and centrifugal force. Most modern wind turbine blades are tapered at the tip, whereby $l_{c g}(s)\longrightarrow{\cal0}$ and $E I_{\xi}I_{\eta}\longrightarrow0$ . Hence, it depends on the individual blade design if this azimuth angle-dependent boundary conditions can be neglected or not. In this work, the blade is constructed such that $l_{c g}(R)=0$ and $E I_{\xi}I_{\eta}|_{s=R}\neq0$ , thus making the boundary conditions azimuth angel independent and hence all right hand sides of equation (5) become zero.  
+
+# 3. STEADY–STATE AEROELASTIC MODEL  
+
+To determine the steady-state def lection for the blade, a non-linear steady-state aeroelastic model i.s derived. Steady-state conditions are def ined as uniform inf low, zero gravity, constant rotor speed and pitch an.gle $\ddot{\phi}=\dot{\beta}=0$ whereby all time derivatives in the structural equations of motion (1) become zero $\ddot{u}=\ddot{\nu}=\ddot{\theta}=\dot{u}=\dot{\nu}=\dot{\theta}=0$ . These uniform conditions remove the periodicity of the system. The steady-state aerodynamic model is based on blade element momentum (BEM) theory including Prendtl’s tip loss correction.26 The BEM theory computes a balance between the forces on the blade and the momentum change in the wind. The aerodynamic model is coupled to the structural model through the local wind speed and angle of attack and the structural model is coupled to the aerodynamic model through the aerodynamic forces acting on the blade.  
+
+# 3.1. Discretization of structural model  
+
+The equations of motion (equation (1)) are discretized on an equidistant grid along the elastic axis with step size $h$ and $N$ computation points. The spatial derivatives of the partial differential equation of motion (1) are approximated by the f inite difference scheme given in Table I. The derivatives of parameters (such as mass, stiffness, etc.) are approximated by the same f inite difference scheme. The integral terms in the equation of motion are approximated by sums using the trapezoid rule.  
+
+The boundary conditions for the f inite difference formulation are derived by inserting the f inite difference approximations into the boundary conditions (equations (4) and (5)). It is assumed that the offset of the centre of gravity is zero at the blade tip, thus making the boundary condition independent of rotor position.  
+
+The discretized version of the partial differential equations of motion implemented on the $N$ discretization points forms a set of non-linear algebraic equations:  
+
+$$
+\mathbf{F}_{s t}\big(\mathbf{u}_{0},\dot{\phi}_{0},\beta_{0}\big)\!=\!\mathbf{f}_{0}
+$$  
+
+where $\mathbf{F}_{s t}\;(\mathbf{u}_{0},\;\dot{\phi}_{0},\;\beta_{0})$ holds the terms from the discretization of the structural equation and $\mathbf{u}_{0}=[u_{0,1},\,\nu_{0,1},\,\theta_{0,1},\,\dots\,,\,u_{0},$ $u_{0,N},$ $\nu_{0,N}$ , ${\theta_{0,N}}\mathrm{]}^{\mathrm{T}}$ holds the steady-state deformation at each discretization point. The f irst subscript 0 denotes that it is the steadystate solution (zero order) and the second subscript denotes the discretization point, counting from the root of the blade. The right hand side $\mathbf{f}_{0}$ holds the steady-state aerodynamic forces computed at each discretization point using BEM theory.  
+
+# 3.2. Solution scheme  
+
+The f inite difference discretized steady-state equation (equation (6)) has 3N unknown blade def lections (f lapwise, edgewise and torsional def lections of the $_\mathrm{N}$ discretization points) and 2N unknown induction factors (longitudinal and tangential induction factor at the $_\mathrm{N}$ discretization points). This system of non-linear equations is solved using the following iterative scheme: i) Operational conditions are chosen: steady-state wind speed $\left(U_{0}\right)$ , the corresponding rotor speed $\dot{\phi}_{0}=\dot{\phi}_{0}(U_{0}))$ and pitch setting $\begin{array}{r}{\beta_{0}=\beta_{0}(U_{0}))}\end{array}$ ; ii) apparent wind speed and angle of attack based on inf low conditions, blade def lections and induction factors are computed; iii) the aerodynamic forces using BEM theory are computed; iv) equation (6) is solved for the deformations $\mathbf{u}_{0}$ ; v) new induction factors are computed; and vi) if no convergence return to step 2. This gives the steady-state deformations $\mathbf{u}_{0}=\mathbf{u}_{0}\left(U_{0},\,\dot{\phi}_{0},\,\beta_{0}\right)$ and the induction factors for the given operational condition.  
+
+<html><body><table><tr><td colspan="2">Tablel.Second-orderfinitedifferenceformulationforuniformstepsize. fi+(t)-f-{(t)</td></tr><tr><td rowspan="2">f’(s, t) ds</td><td>af(s, t)</td></tr><tr><td>2h</td></tr><tr><td>f"(s, t) ²f(s, t)</td><td>fi+1(t)-2f(t)+f-(t)</td></tr><tr><td rowspan="2">ds2</td><td>h2</td></tr><tr><td>-f-2(t)+2f_(t)-2fi+1(t)+fi+2(t)</td></tr><tr><td>f"(s, t) a²f(s, t) ds3</td><td>2h3</td></tr><tr><td rowspan="3">f"(s, t)</td><td>a4f(s, t)</td><td></td></tr><tr><td>ds4</td><td>f-2(t)-4f-(t)+6f;(t)-4f+1(t)+f+2(t)</td></tr><tr><td></td><td>h4</td></tr></table></body></html>  
+
+![](ab42554a7d5e39cc6f7a030cf6653410bfdc8ab71835958c84d16e0ccaa7f153.jpg)  
+Figure 1. Edgewise and f lapwise def lection and angle of attack at $55.5~\mathsf{m}$ radius $(88\%)$ vs. wind speed for the present second-order Bernoulli–Euler blade model (equation (6)) and the non-linear aeroelastic time simulation code HAWC2.  
+
+# 3.3. Steady–state blade def lection at power production conditions  
+
+The steady-state model (equation (6)) is used to compute steady-state blade def lection and induction factors for the NREL 5 MW RWT24 blade at normal power production operation. The results are compared with results from the non-linear aeroelastic time simulation code HAWC2.14,15 The HAWC2 code is a multi-body formulation where each body is a linear Timoshenko beam element with a torsional degree of freedom. The geometric non-linearities are captured by the multibody formulation, in which the blades for example are modelled by 10 bodies each. If only one body per blade is used the code will become as a linear code since the beam model in each body is linear, whereas a convergence study has shown that with 10 bodies the geometric non-linearities are captured. In the present model, only one blade is considered and modelled as a f lexible beam. For f irst and second modes of blade motions, as considered in this paper, the rotary and shear effects are negligible, so the Bernoulli–Euler beam model in the present mode is comparable with the Timoshenko beam model in HAWC2. As for higher order modes of motion and other turbine components, the rotary and shear effects are of higher relevance. Figure 1 shows the blade f lapwise and edgewise def lections and angle of attack at radius $55.5~\mathrm{m}$ $88\%$ blade length) at different wind speeds. The angle of attack indicates how well the torsional deformation from the two models agrees. It is seen that there is good agreement between the present second-order Bernoulli–Euler blade model and HAWC2 for all operational conditions. The kink at rated wind speed $(\approx\!11\ensuremath{\mathrm{~m~s~}^{-1}})$ at the blade tip def lection curve is caused by the shift from variable speed, constant pitch to constant speed, variable pitch operation.  
+
+# 4. AEROELASTIC MODES OF BLADE MOTION  
+
+In this section, the aeroelastic modes of blade motion are analysed with particular emphasis on effects of steady-state f lapwise blade def lection. The stability of a specif ic blade at normal operation will be analysed in detail and differences including and excluding geometric couplings will be discussed. The effect of pre-bend is similar to the effects of steadystate blade def lection which is investigated in this analysis. The effect of sweep (edgewise curved blades) is different since it couples f lapwise and torsional motion instead of edgewise and torsion as characterized by the f lapwise def lection.  
+
+# 4.1. Linear aeroelastic model  
+
+The non-linear partial differential equations of motion is linearized by inserting ${\mathbf{u}}(s,\,t)={\mathbf{u}}_{0}(s)+\varepsilon{\mathbf{u}}_{1}(s,\,t)$ into equation (1), where ${\bf u}_{0}(s)$ is the steady-state def lected blade position including any pre-bend and sweep, ${\mathbf{u}}_{1}(s,t)$ is time-dependent variations around this position and $\varepsilon$ is a bookkeeping parameter denoting smallness of terms. The external inf luences, such as wind speed, pitch setting, etc. are split into a steady part and a time-varying part (denoted by the subscript 0 and 1, respectively) with the bookkeeping parameter $\varepsilon$ . The equation of motion (equation (1)) is Taylor expanded assuming $\varepsilon<<1$ . Balancing terms of order $\varepsilon^{\mathrm{l}}$ give the linear approximation around the def lected blade position $\mathbf{u}_{0}$ . By linearizing the equations of motion about the def lected blade the main effects for the geometric non-linearities are preserved. For example, the non-linear stiffness term in the edgewise equation  
+
+$$
+\left(\left(E I_{\xi}-E I_{\eta}\right)\cos\left(\tilde{\theta}+\theta\right)\sin\left(\tilde{\theta}+\theta\right)\nu^{\prime\prime}\right)^{\prime\prime}
+$$  
+
+becomes  
+
+$$
+\left(\left(E I_{\xi}-E I_{\eta}\right)\cos\left(2\left(\tilde{\theta}+\theta_{0}\right)\right)\nu_{0}^{\prime\prime}\theta_{1}\right)^{\prime\prime}+\dots
+$$  
+
+when linearized about the def lected blade (using $\theta=\theta_{0}+\theta_{1}$ and $\nu=\nu_{0}+\nu_{1}$ ), whereby the important coupling between edgewise and torsional blade motion of a f lapwise def lected blade is preserved. The subscript 1 denotes the linear variation around the linearization point $\mathbf{u}_{0}$ . Likewise the non-linear term in the torsional equation  
+
+$$
+(E I_{\xi}-E I_{\eta})\cos(2\big(\tilde{\theta}+\theta\big)\big)u^{\prime\prime}\nu^{\prime\prime}
+$$  
+
+becomes  
+
+$$
+\begin{array}{r}{\big(E I_{\xi}-E I_{\eta}\big)\mathrm{cos}\big(2\big(\widetilde{\theta}+\theta_{0}\big)\big)u_{0}^{\prime\prime}u_{1}^{\prime\prime}+...\,.}\end{array}
+$$  
+
+when linearized about the def lected blade. The major effect of the important geometric coupling in the stiffness terms (equations (7) and (9)) between edgewise and torsional motion of a f lapwise def lected blade is preserved when linearized about the steady-state def lected blade (equations (8) and (10)).  
+
+The linearized equations of motion are combined with a linearized Beddoes–Leishman27 type of unsteady aerodynamic model.25 The unsteady aerodynamic model is formulated in a state space formulation with four states; two states are second-order approximations to Thoedorsen’s function28 and two states describe the dynamics of the trailing edge separation point. Periodic effects, such as gravity, can be included in the linear model by considering $\sin(\phi_{1}\;+\;t\dot{\phi}_{0})$ and $\cos(\phi_{\mathrm{l}}\;+\;t\dot{\phi}_{\mathrm{0}})$ as independent variables, which subsequently can be obtained by a non-linear transformation, but are neglected in this work. The linear partial differential equation and the unsteady aerodynamic model are given by  
+
+$$
+\begin{array}{r}{\tilde{\mathbf{M}}\ddot{\mathbf{u}}+\tilde{\mathbf{D}}\dot{\mathbf{u}}+\left(\tilde{\mathbf{K}}_{s s}\mathbf{u}^{\prime\prime}\right)^{\prime\prime}+\left(\tilde{\mathbf{K}}_{s}\mathbf{u}^{\prime}\right)^{\prime}+\tilde{\mathbf{K}}\mathbf{u}+\tilde{\mathbf{C}}\mathbf{z}=\tilde{\mathbf{F}}_{s}\tilde{\mathbf{f}}\ }\\ {\dot{\mathbf{z}}+\tilde{\mathbf{T}}\mathbf{z}+\tilde{\mathbf{G}}\ddot{\mathbf{u}}+\tilde{\mathbf{H}}\dot{\mathbf{u}}+\tilde{\mathbf{J}}\mathbf{u}=\tilde{\mathbf{F}}_{a}\tilde{\mathbf{f}}}\end{array}
+$$  
+
+where $\mathbf{u}=\mathbf{u}(s,\,t)=[u_{1}(s,\,t)$ , $\nu_{1}(s,\,t),\,\theta_{1}(s,\,t)]$ are the linear def lections around the linearization point $\mathbf{u}_{0}$ , $\begin{array}{r}{\tilde{\mathbf{M}}{=}\tilde{\mathbf{M}}(\mathbf{u}_{0},\,\dot{\phi}_{0},\,\beta_{0},}\end{array}$ $U_{n,0})$ , $\tilde{\mathbf{D}}=\tilde{\mathbf{D}}$ $(\mathbf{u}_{0},\,\dot{\phi}_{0},\,\beta_{0},\,U_{n,0})$ , $\tilde{\mathbf{K}}_{s s}=\tilde{\mathbf{K}}_{s s}\;(\mathbf{u}_{0},\;\dot{\phi}_{0},\;\beta_{0}),\;\tilde{\mathbf{K}}_{s}=\tilde{\mathbf{K}}_{s}\;(\dot{\phi}_{0},\;\beta_{0}),\;\tilde{\mathbf{K}}=\tilde{\mathbf{K}}\;(\mathbf{u}_{0},\;\dot{\phi}_{0},\;\beta_{0},\;U_{0})$ are collections of the linear coeff icients, where $U_{0}$ is the mean wind speed, $\tilde{\mathbf{C}}=\tilde{\mathbf{C}}\left(\mathbf{u}_{0},\;\dot{\phi}_{0},\;\beta_{0}\right)$ is the unsteady aerodynamic’s effect on the structure, where $U_{1}$ is the variation of the wind speed. The coupling to external inf luences such as pitch action and wind speed variations is described on t.he right hand side, where $\tilde{\mathbf{F}_{s}}=\tilde{\mathbf{F}}_{s}\left(\mathbf{u}_{0},\,\dot{\phi}_{0},\,\beta_{0},\,U_{n,0}\right)$ holds the linear gains on the external inf luences given by f˜ $=[\beta(t),\ \dot{\beta}(t),\ \ddot{\beta}(t),\ \sin(\phi_{1}(t)+t\dot{\phi}_{0}),\cos(\phi_{1}(t)+t\dot{\phi}_{0}),\ \dot{\phi}_{1}(t),\ \ddot{\phi}_{1}(t),\ U_{1}(s,\ t),\ \dot{U}_{1}(s,\ t)]^{\mathrm{T}}$ . The four aerodynamic states in $\mathbf{z}$ are modelled by steady-state wind speed-dependent time constants $\tilde{\mathbf{T}}$ and affected by the linear blade def lection, speed and acceleration through time-varying angle of attack and local wind speed described by the matrices $\tilde{\bf G}$ , H˜ and $\tilde{\mathbf{J}}$ .25 The linear gains on external inf luences are given by $\tilde{\mathbf{F}}_{a}$ .  
+
+# 4.2. Aeroelastic modes of motion  
+
+The spatial derivatives in the linear equations of aeroelastic motion (equation (11)) are approximated by the f inite difference scheme (Table I) with N discretization points. The f inite difference implementation includes the spatial boundary conditions (equations (4) and (5)). The second-order differential equation is then rewritten into f irst-order form by introducing the f irst-order time derivatives as states and combining it with the unsteady aerodynamic model. The spatial discretized f irst-order equation of aeroelastic motion becomes  
+
+# $\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{f}$  
+
+where $\dot{\mathbf{x}}$ includes the linear deformation around the linearization point, speed and the aerodynamic states for each discretization point, giving $3N+3N+4N=10N$ degrees of freedom, A is the linear coeff icients, $\mathbf{B}$ is the linear gains on the external inf luences and f is the linear variation of the external inf luences. The unforced version of equation (12) forms a differential eigenvalue problem.29 The differential eigenvalue problem is casted into an algebraic eigenvalue problem by assuming a complex exponential solution. The eigenvalues and corresponding eigenvectors can be grouped into two sets: real valued and complex valued eigenvalues. Generally, the real valued eigenvalues are related to the aerodynamic states and correspond to the aerodynamic time lags. However, overdamped aeroelastic modes will also have real valued eigenvalues. The complex valued eigenvalues are related to the aeroelastic states and give the aeroelastic frequencies and damping. The corresponding eigenvectors give the aeroelastic mode shapes of the particular mode.  
+
+It is noted that since aerodynamic forces are included, the eigenvalue problem12 is not self-adjoint, and therefore, the eigenvectors are not orthogonal.  
+
+# 4.3. Frequency and damping of a blade at normal power production conditions  
+
+The model described above is used to analyse the effect of geometric non-linearities caused by steady-state blade def lections under normal operational conditions. The aeroelastic frequencies, damping and mode shapes of the NREL RWT blade are computed for different wind speeds in the power production region. The aeroelastic results are computed in two versions: one in which the model is linearized about the steady-state def lected blade, and another in which it is linearized about the undef lected blade, hereby including and excluding the effect of the geometric non-linearities, respectively.  
+
+The results from the present model are compared with the results from the non-linear aeroelastic time simulation code HAWC2. Since each body in this code is a linear beam model and the non-linearities are only included by the multi-body formulation, this model will produce linear results if only one body per blade is used and non-linear results if more bodies are used. Hence, a one body per blade model will correspond to the present model without geometric couplings and a model with more bodies will correspond to the present model with geometric couplings. Two versions of the HAWC2 model are used in this work: one with one body in the blade and one with 10 bodies in the blade. In both models, only the blade is considered as a f lexible beam. The frequencies and damping from the time simulation code are estimated by f itting the frequency, phase and damping of a number of exponentially decaying sinusoidal functions to the decay of the blade motion after an initial excitation at the expected aeroelastic frequency. In the simulations, the pitch angle is set to a prescribed value dependent on wind speed only.  
+
+Figure 2 shows the aeroelastic frequencies and damping of the two f irst f lapwise blade-bending modes. In the variable speed operation range (5 to $12\;\mathrm{m}\;\mathrm{s}^{-1}$ ), the aeroelastic frequency increases because of increased centrifugal stiffness. The disagreement between the undef lected and def lected blade case in aeroelastic frequency of the f irst f lapwise mode around $20\ \mathrm{m}\ \mathrm{s}^{-1}$ is caused by the increased steady-state blade twist, which changes the angle of attack and thereby the aerodynamic stiffness. The damping of the f irst f lapwise bending mode is almost the same for the undef lected and def lected blade case, there are only some minor differences at the same wind speeds that are also caused by the small change in steady angle of attack. For the second f lapwise bending mode, neither the frequency nor the damping are changed by the inclusion of the geometric non-linearities. The results for the second f lapwise mode from HAWC2 are seen to follow the same trend as the results from the present model. Because of the high damping of this mode, the decay of initial excited oscillations is very fast and the noise from other lower damped modes becomes relatively large, resulting in a large uncertainty on the f titing of damping to this short decay time. The geometric non-linearities do not have a large effect on the f lapwise bending modes since the edgewise steady-state def lection is relatively small, giving only a weak coupling from f lapwise motion to the other directions.  
+
+Figure 3(a) shows the aeroelastic frequencies and damping for the f irst edgewise blade-bending mode. There is an offset of the frequency of the two different models (HAWC2 and the present model). The reason for this offset is that the present model only includes the blade whereas the HAWC2 model includes the whole turbine. The turbine’s effect on the blade dynamics is minimized by making all other turbine components very stiff in the HAWC2 computations, but nonetheless there will always be a small effect. This effect is more pronounced for the edgewise mode since the coupling is more direct through the drive train and the other blades than it is for the f lapwise mode. The change in frequency caused by the blade def lection is also seen to have a minor difference in offset for the two models. This is due to the fact that in the  
+
+![](c5e8d2df7eb5d140fc387d95da939e17b18bb6b2fee037b01febff77ea7f2b1b.jpg)  
+Figure 2. Aeroelastic frequency and damping for the (a) f irst and (b) second f lapwise blade-bending modes. There are no HAWC2 results for f irst f lapwise mode because it is too highly damped for measuring the decay.  
+
+![](78d9e5ad9d9dd276825be7489b13bc961f04c7e720d1534c97397fe427cd65d0.jpg)  
+Figure 3. Aeroelastic frequency and damping ratio for (a) the f irst and (b) the second edgewise blade-bending modes. Damping ratio refers to the exponential damping rate.  
+
+HAWC2 model the aerodynamic forces are applied to the deformed blade position even if the blade is assumed linear whereas in the present model the forces are applied to the undef lected blade position. Regardless of these differences, the damping of the two models is qualitatively similar, and since the focus of this work is the qualitative effect of geometric couplings on the blade stability, the present model is well suited for this purpose. The aeroelastic damping around $14\;\mathrm{m}\;\mathrm{s}^{-1}$ decreases when the geometric non-linear couplings are included (def lected blade case). At the higher wind speeds, the damping of the model including the geometric non-linearities increase and becomes the highest. The reason for these differences will be analysed in the next section. Figure 3(b) shows the aeroelastic frequency and damping of the second edgewise bending mode. The frequency and damping from the present model differ from the results from HAWC2 at a wind speed around 11 m s−1, where the f lapwise def lection is largest. This case will be analysed in the next section.  
+
+# 4.4. Aeroelastic analysis of specif ic cases  
+
+The aeroelastic damping of the edgewise mode is a caused by both f lapwise and edgewise aerodynamic force variations, which results from angle off attack variations due to edge-torsion coupling of the f lapwise def lected blade and from f lap and edgewise blade motion. On the one hand, modal aerodynamic force variation that occurs in counter phase with the blade speed enhances the damping. On the other, when it is in-phase with speed, the damping decreases or even becomes negative. When modal aerodynamic force variations are in counter phase with the blade def lection, aerodynamic stiffening occurs and vice versa. The following discussion is clarif ied through phase-space plots of f lapwise and edgewise def lections; these phase-space plots also include distinct values of the belonging aerodynamic f lapping force variation through a scaled stem-like plot (vertical bars with an o-mark; sign from up/down orientation relative to trajectory). Furthermore, the elastic twist of the blade is included in a distinct number of points of the trajectory in the phase-space plot through a straight, mainly horizontally directed bar. The torsion will increase the angle of attack when the bar is decreasing from left to right and vice versa. The plots are included to clarify the aeroelastic damping mechanisms and to illustrate the difference for an undef lected and a def lected blade. The three cases where there are large differences between the def lected and undef lected blade cases are analysed in detail; the f irst edgewise bending mode at 14 and $25\ \mathrm{m\s^{-1}}$ and the second edgewise bending mode at $11~\mathrm{m~s}^{-1}$ . Summary: In the f irst cases (f irst edgewise bending mode at $14~\mathrm{m~s}^{-1}$ ), the damping of the def lected blade is lower than the damping of the undef lected blade. The damping decreases because the inclusion of geometric non-linearities reduces the f lapwise motion and the phase between f lapwise motion and f lapwise forces is changed. In the second case (f irst edgewise bending mode at $14~\mathrm{m~s}^{-1}$ ), the damping of the def lected blade is highest. The increased damping is due to the fact that the geometric non-linearities increase the torsional motion, and thereby the changes in angle of attack and thus the aerodynamic forces. The change in phase and amplitude of the aerodynamic forces relative to the edgewise motion increase the negative aerodynamic work, increasing damping. In the last case (second edgewise bending mode at $11~\mathrm{m~s^{-1}}$ ) relative large increase in damping is seen when the def lections are included. The increase is caused by an increased amount of torsional motion and negative aerodynamic work on the torsional motion.  
+
+![](4c2a7b6a4f10e19d5b9aafdcf62507e4b2a803746ee66d37af44ab1256dd3e93.jpg)  
+Figure 4. Traces of cross-sectional blade motion at $90\%$ radius in the f irst structural edgewise eigenmode at $14\;\mathrm{m}\;\mathrm{s}^{-1}$ for the blade with exaggeration of the torsional component. Arrows denote the direction of motion and the bars denote the torsional component. (a) Steady-state blade def lection terms are excluded and (b) steady-state blade def lection terms are included. Note that the relative wind comes from right to left in the displayed cross-sectional coordinate system.  
+
+The f irst case to be analysed is the aeroelastic response of the f irst edgewise blade-bending mode at $14~\mathrm{m~s^{-1}}$ , where the def lected blade cases are less damped than the undef lected blade case (Figure 3(a)). First, looking at the case without steady-state blade def lections, which for this blade without pre-bend and sweep will mean a straight blade removing geometric non-linearities: Figure 4 shows the normalized cross-sectional blade def lection at $90\%$ radius for the f irst edgewise bending structural eigenmodes for the undef lected blade and the steady-state def lected blade at $14~\mathrm{m~s}^{-1}$ . When the blade moves forward (left to right) in the structural eigenmode, the local wind speed increases, consequently increasing the aerodynamic forces, and vice versa when the blade moves backwards. The extremes of this variation of aerodynamic forces appear at the points with the largest blade speed, i.e. the midpoint of the edgewise blade motion. The f lapwise motion in the structural eigenmode also affects the aerodynamic force, increasing the angle of attack when the blade moves downwards and thereby increasing the aerodynamic force. Since the edgewise and f lapwise motion are in counter phase (blade moves forward and downwards) in this structural eigenmode, both effects described above give the highest aerodynamic forces when the blade moves forward and lowest when the blade moves backwards. In this case, without steady-state deformations, there is only a very limited and insignif icant torsional motion. The variations in aerodynamic f lapwise forces affect the f lapwise motion in the aeroelastic mode of motion. The frequency of the f irst edgewise mode $(1.1\ \mathrm{Hz})$ is higher than the resonance frequency of f irst f lapwise bending mode $(0.79\ \mathrm{Hz})$ . Hence, the f lapwise def lection lags approximately 180 degrees after the f lapwise force according to basic dynamic considerations. The f lapwise force is highest at the midpoint of the forward edgewise motion, increasing the f lapwise def lection around the midpoint of the backward edgewise motion. This increased f lapwise motion at the midpoint of the edgewise motion will increase the f lapwise speed at the edgewise turning points, affecting the angle of attack and thereby the aerodynamic force. The increased f lapwise forces will increase the f lapwise def lection ${\approx}180$ degrees later, which is the other edgewise turning point. Summing up, the f lapwise motion in the f irst edgewise aeroelastic bending mode is an equilibrium between the f lapwise motion caused by the structural coupling (eigenmode motion) and the variations in f lapwise aerodynamic force caused by the structural eigenmode and the f lapwise motion itself. Figure 5(b) shows the unsteady aerodynamic f lapwise force for the cross-sectional motion of f irst edgewise aeroelastic bending mode. The resulting aerodynamic f lapwise force variation is seen to be largest around the edgewise turning points, indicating that it is dominated by the force variation caused by the f lapwise motion itself. The black dot denotes the point with the largest f lapwise force.  
+
+Figure 6(b) shows the change in cross-sectional motion caused by the aerodynamic forces. It is seen that the largest f lapwise def lection caused by the aerodynamic forces is ${\approx}180$ degree offset from the largest f lapwise force.  
+
+When the steady-state def lections are included in the model, the geometric non-linear couplings between edgewise and torsional motion of a f lapwise def lected blade (equations (8) and (10)) become active and increase the torsional motion in the f irst f lapwise structural eigenmode (Figure 4(a)). The torsional motion is seen to decrease the angle of attack, and thereby the aerodynamic force, at the forward position of the edgewise motion so this torsional motion counteracts the angle of attack changes caused by the f lapwise speed at the edgewise turning points. The reduced effect of the f lapwise motion on the aerodynamic forces changes the phase between f lapwise and edgewise motion. The f lapwise motion relative to the local wind becomes smaller but only looking at the change in f lapwise motion caused by aerodynamic forces (Figure 6) the def lections are similar, so it is mainly the phase between f lapwise and edgewise motion that has changed.  
+
+![](24114c0ae04deca215e81b5e54bdf341ce99a342abd52bbf21767e23c9604ffc.jpg)  
+Figure 5. Traces of cross-sectional blade motion at $90\%$ radius in the f irst aeroelastic edgewise mode at $14~\mathsf{m}~\mathsf{S}^{-1}$ for the blade with exaggeration of the torsional component. Arrows denote the direction of motion, the bars denote the torsional component and the vertical lines the unsteady f lapwise aerodynamic force with the black dot at the point with highest force. (a) Steady-state blade def lection terms are excluded and (b) steady-state blade def lection terms are included. Note that the relative wind comes from right to left in the displayed cross-sectional coordinate system.  
+
+![](3881427e6d206bb657a228bf7f25604e8ec690a02246ec74bba12b5def7c604b.jpg)  
+Figure 6. Change in cross-sectional blade motion at $90\%$ radius of the f irst aeroelastic edgewise mode at $14\;\mathrm{m}\;\mathrm{s}^{-1}$ caused by aerodynamic forces. The f igure shows the difference between the structural eigenmode (Figure 4) and the aeroelastic mode (Figure 5) showing that the maximum f lapwise def lection caused by aerodynamic forces are 90 degrees phase shifted from the maximum force. Arrows denote the direction of motion and the vertical lines the unsteady f lapwise aerodynamic force with the black dot at the point with highest force. (a) Steady-state blade def lection terms are excluded and (b) steady-state blade def lection terms are included. Note that the relative wind comes from right to left in the displayed cross-sectional coordinate system.  
+
+Table II shows the aerodynamic sectional work for the two cases in Figure 5. Both the f lapwise and edgewise aerodynamic works are seen to be negative, thus extracting energy from the motion (adding damping to the mode). For the undef lected blade case, the total work is dominated by the f lapwise work. The relatively high f lapwise work is due to the fact that the f lapwise force is ${\approx}90\$ degrees phase shifted from the f lapwise motion, so for this reason the largest forces counteract at the highest velocities. The f lapwise force is mainly caused by the f lapwise component of the lift force on the airfoil. This lift force will also have an edgewise component pointing forward (the component driving the wind turbine) so the point with the highest f lapwise force also has a relatively large edgewise force component pointing forward. For the undef lected blade case (Figure 5(b)), the blade moves forward at the point with the highest forces. Consequently at this point, the edgewise component of the lift will add energy to the system, reducing the damping. This is the reason for the low damping value for the edgewise motion of the undef lected blade (Table II). In the def lected blade case, two effects reduce the f lapwise damping: f irst, the reduced f lapwise motion relative to the local wind, reduces the amount of work. Second, the f lapwise force and motion are almost in counter phase, so the maximum forces act at a low f lapwise velocity, extracting less energy from the system. The edgewise work is increased since the point of maximum force is moved towards the edgewise turning point compared with the undef lected blade case, which reduces the amount of energy that the lift force component on the edgewise motion adds to the system, leading to a higher edgewise damping contribution (Table II).  
+
+Table II. Aerodynamic sectional work for the sectional motion shown in Figure 5. The work is normalized with respect to the total work of the undef lected blade, except for the sign.   
+
+
+<html><body><table><tr><td></td><td>Edgewise</td><td>Flapwise</td><td>Total</td></tr><tr><td>Undeflectedblade</td><td>-0.03</td><td>-0.97</td><td>-1.00</td></tr><tr><td>Deflectedblade</td><td>-0.25</td><td>-0.27</td><td>-0.52</td></tr></table></body></html>  
+
+![](e36db93dd8609b19a020ef2ad5c5a0135ec309c2daca60e1a6c42d73393b59ff.jpg)  
+Figure 7. Traces of cross-sectional blade motion at $90\%$ radius in the f irst structural edgewise eigenmode at $25~\mathsf{m}~\mathsf{s}^{-1}$ for the blade with exaggeration of the torsional component. Arrows denote the direction of motion and the bars denote the torsional component. (a) Steady-state blade def lection terms are excluded and (b) steady-state blade def lection terms are included. Note that the relative wind comes from right to left in the displayed cross-sectional coordinate system.  
+
+The next case to be analysed is the f irst edgewise blade-bending mode at $25\ \mathrm{m\s^{-1}}$ where the damping of the def lected blade is higher than the damping of the undef lected blade case (Figure 3(a)). At this higher wind speed, the f lapwise tip def lection shifts sign (Figure 1) changing the sign of the coupling between edgewise and torsional motion for the f lapwise def lected blade (equations (8) and (10)). Figure 7 shows how the torsional def lection in the f irst edgewise structural eigenmode at $25\mathrm{~m~s~}^{-1}$ has changed sign compared with the results for $14~\mathrm{m~s^{-1}}$ (Figure 4). Figure 8 shows the crosssectional def lection of the f irst edgewise aeroelastic mode and the unsteady aerodynamic f lapwise forces at $25\mathrm{~m~s~}^{-1}$ . At this wind speed, the average angle of attack at the shown cross-section is ${\approx}{-4}$ degrees. At this negative angle of attack, the lift force is negative, so the effect of edgewise vibration change, since the forward motion, which gives larger local wind speed, increases the absolute value of the negative lift force. Hence, the forward motion decreases the lift and the backward motion increases the lift, opposite the case at $14~\mathrm{m~s^{-1}}$ . The effect of f lapwise motion is the same as before since this affects the angle of attack. So the two effects counteract each other, resulting in smaller unsteady aerodynamic forces in this mode at $25\ \mathrm{m\s^{-1}}$ than at $14~\mathrm{m~s}^{-1}$ (Figure 8). The phase between the f lapwise and edgewise motion determines how well the forces from the two effects cancel each other out and thereby also where the highest force appears. Because of the reduced aerodynamic forces, the aeroelastic mode is less affected by the aerodynamic forces and the direction of motion is similar to the structural eigenmode when compared with the previous case at $14~\mathrm{m~s}^{-1}$ . The edgewise force is mainly caused by the lift force on the blade, and since the angle of attack in this $25\ \mathrm{m}\ \mathrm{s}^{-1}$ case is negative $\approx\!-4$ degrees), a lift force giving a positive f lapwise force will give a negative edgewise force. Thus, for the f irst ${\approx}2/3$ for the forward and backward edgewise motion, the aerodynamics will contribute with negative work (Figure 8(b)). For the f lapwise motion, the f lapwise force is almost constantly in the opposite direction than the f lapwise motion, extracting energy from the motion. Table III shows that the f lapwise and edgewise works contribute equally to the damping of the undef lected blade case at $25\ \mathrm{m\s^{-1}}$ . The changes in blade twist, and thereby angle of attack, caused by the geometric nonlinearities increase the aerodynamic force at the forward edgewise position of the blade and decreases the forces at the backward position. Adding this extra effect to the effects of f lapwise and edgewise motion moves the point of highest f lapwise force towards the forward position and places it almost at the midpoint for both the f lapwise and edgewise motion. Having the highest f lapwise force (indicating high negative edgewise force at this negative angle of attack) close to the highest f lapwise and edgewise speed, results in high damping even though the force level is relatively low.  
+
+![](86a22c06cb1c9c13f4e6525b1ebeb66ed05c6c57bc20202cc866c4ec655e5773.jpg)  
+Figure 8. Traces of cross-sectional blade motion at $90\%$ radius in the f irst aeroelastic edgewise mode at $25~\mathsf{m}~\mathsf{s}^{-1}$ for the blade with exaggeration of the torsional component. Arrows denote the direction of motion, the bars denote the torsional component and the vertical lines the unsteady f lapwise aerodynamic force with the black dot at the point with highest force. (a) Steady-state blade def lection terms are excluded and (b) steady-state blade def lection terms are included. Note that the relative wind comes from right to left in the displayed cross-sectional coordinate system.  
+
+Table III. Aerodynamic sectional work for the sectional motion shown on Figure 8. The work is normalized with respect to the total work of the undef lected blade, except for the sign.   
+
+
+<html><body><table><tr><td></td><td>Edgewise</td><td>Flapwise</td><td>Total</td></tr><tr><td>Undeflectedblade</td><td>-0.56</td><td>-0.44</td><td>-1.00</td></tr><tr><td>Deflectedblade</td><td>-1.13</td><td>-0.43</td><td>-1.57</td></tr></table></body></html>  
+
+The last case to be analysed is the second edgewise blade-bending mode at $11~\mathrm{m~s}^{-1}$ , where the damping of the def lected blade case is much higher than the damping of the undef lected blade case (Figure 3(b)). On a pitch-regulated wind turbine, as the present one, the f lapwise tip def lection is largest around rated wind speed since the pitch regulation of the turbine relieves the aerodynamic loads at higher wind speeds. The large f lapwise steady-state def lection (indicating large curvature $\nu_{0}^{\prime\prime}\propto\nu_{0})$ together with the relatively large edgewise curvature $u_{1}^{\prime\prime}$ gives a large torsional component in the second edgewise bending mode (equation (10)). Figure 9 shows the content of f lapwise, edgewise and torsional motion in the second edgewise bending mode and it is seen how the inclusion of the non-linearities increases the torsional motion. Figure 10 shows the distribution of aerodynamic work done by the edgewise, f lapwise and torsional aerodynamic forces along the blade. It is on the outer $10\%$ of the blade, beyond the node of the second bending mode, that the majority of the aerodynamic work is done and the difference between the two blade def lection cases arises. The main differences in aerodynamic work between undef lected and def lected blade cases are in the torsional motions, which increase when the geometric non-linearities are included. Figure 11 shows the cross-sectional motion for the second aeroelastic edgewise bending mode for the undef lected and the def lected blade at $95\%$ blade radius. The modal aeroelastic cross-sectional motion of the undef lected blade is very similar to the structural eigenmode: this is because the unsteady aerodynamic forces are smaller relative to the higher inertia and structural restoring forces in this higher bending mode compared with the f irst edgewise mode. Figure 10 shows that the edgewise motion is slightly negatively damped for the outer part of the blade. This is because the edgewise component of the unsteady lift force acts in the direction of edgewise motion adding energy to the system. This results in minor negative damping because the drag force on the edgewise motion always adds damping. The f lapwise motion is positively damped since the unsteady f lapwise force works against the direction of f lapwise motion.  
+
+![](82eae939d1766cd56bf37d7d4af5c5b856ca4aebcb420f460f09675b60fd715d.jpg)  
+Figure 9. Edgewise, f lapwise and torsional components of the second edgewise aeroelastic bending mode at $11\ m\ s^{-1}$ for the undef lected and the def lected blade.  
+
+![](842002e9969b3a4782b7a69d9544ba74318e4f2f7a9a35a98086fc30744887a1.jpg)  
+Figure 10. Edgewise, f lapwise and torsional cross-sectional work in the second edgewise aeroelastic vibration mode at $11\ m\ s^{-1}$ for the undef lected and the def lected blade. Negative aerodynamic work corresponds to positive aeroelastic damping.  
+
+When the steady-state def lections are included, the large torsional component caused by the geometric non-linearities (equation (10)) has a large effect on the unsteady aerodynamic forces. Note that the direction of the loop has changed compared with the undef lected blade case. The edgewise force adds energy to the system, since the force acts in the same direction and the motion for the f irst ${\approx}2/3$ of the edgewise motion. The f lapwise forces in the def lected blade case add energy to the system (Figure 10) since they act in the same direction as the f lapwise motion. The amount of work is relatively small because the f lapwise amplitude normal to the local wind direction is relatively small. The large increase in aeroelastic damping of the def lected blade case compared with the undef lected blade case is caused by negative aerodynamic work of the torsional motion (Figure 10). The aerodynamic lift force acts at the aerodynamic centre, which is located in front of the elastic centre, where the blade twists. Thus, an increased lift results in an increased rotational moment on the cross-section. The cross-sectional motion of the undef lected blade (Figure 11(b)) has almost no torsional motion, resulting in small aerodynamic work (Figure 10). The def lected blade case, on the other hand, has much more torsional motion (Figure 11(b)). The cross-section has a nose down motion on its way forward to the lift force and thereby also the torsional moment is high and a nose up motion on its way back where the lift is low, resulting in negative aerodynamic work, increasing the damping.  
+
+![](32cb348a3666869d7206d3e6ee45313cc433826dfa892d18d1844f1842105d76.jpg)  
+Figure 11. Traces of cross-sectional blade motion at $95\%$ radius in the second aeroelastic edgewise mode at $11m\ s^{-1}$ for the blade with exaggeration of the torsional component. Arrows denote the direction of motion, the bars denote the torsional component and the vertical lines the unsteady f lapwise aerodynamic force with the black dot at the point with highest force. The dotted line shows the structural eigenmode. (a) Steady-state blade def lection is excluded and (b) steady-state blade def lection is included. Note that the relative wind comes from right to left in the displayed cross-sectional coordinate system.  
+
+# 5. CONCLUSION  
+
+In this paper, a second-order non-linear beam model is used for aeroelastic stability analysis of a turbine blade. The aeroelastic mechanisms of the different modes and the difference between including and excluding non-linear geometric couplings caused by steady-state def lection at normal operation are discussed in detail. The methodology can also be used to analyse the effects of pre-bend or swept blades.  
+
+The analysis is based on the non-linear structural blade model from Kallesøe,23 which in this work is extended to include an aerodynamic model. The resulting non-linear aeroelastic blade model is linearized about a curved blade position, caused by e.g. sweep, pre-bend or steady-state def lections. The linearized model is used to perform stability analysis of a steady-state def lected blade and to examine the effects of the linearized geometric non-linearities.  
+
+First, the derived non-linear aeroelastic model is used to compute steady-state blade def lections. The steady-state def lections are validated against results from a non-linear aeroelastic time simulation code, showing good agreement. Next, the non-linear aeroelastic model is linearized about the steady-state def lected blade. By linearizing about the def lected blade, the main effects of geometric non-linearities are preserved and the results show how the relative large f lapwise blade def lection introduces a coupling between edgewise and torsional blade motion.  
+
+Two versions of the linearized model are used to compute the aeroelastic stability of the blade: one linearized about the def lected blade, preserving the non-linearities and one linearized about an undef lected blade excluding the nonlinearities. The stability results from the two versions are compared and the differences discussed. It is found that the f lapwise modes are not as affected by the steady-state blade def lection as the edgewise modes. The damping of f irst edgewise bending mode of the steady-state def lected blade decreases around $14~\mathrm{m~s}^{-1}$ but increases around $25\ \mathrm{m\s^{-1}}$ compared with the undef lected blade. The reason for this change of the effect of the blade def lection on the aeroelastic damping is caused by the steady-state f lapwise def lection shifting sign around $20\ \mathrm{m}\ \mathrm{s}^{-1}$ . When the f lapwise def lection shifts sign, the coupling between the edgewise and torsional motion also shifts, and thereby changing the non-linear geometric couplings effect on the aeroelastic damping contribution. The damping of second edgewise bending mode is high around $11~\mathrm{m~s}^{-1}$ for the steady-state def lected blade compared with the undef lected blade. This is because the f lapwise steady-state def lection is largest around $11~\mathrm{m~s^{-1}}$ giving the largest effect of the geometric non-linear coupling between edgewise and torsional motion.  
+
+This work shows that the blade def lection under normal operation conditions affects the aeroelastic stability properties of the blades. In the worst case for this particular blade, the edgewise damping can be decreased by half.  
+
+# REFERENCES  
+
+1. Hodges DH, Dowell EH. Nonlinear equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades. Technical Report TN D-7818, NASA, 1974.   
+2. Friedmann PP, Kottapalli SBR. Coupled f lap-lag-torsional dynamics of hingeless rotor blades in forward f light. Journal of the American Helicopter Society 1982; 4: 28–36.   
+3. Panda B, Chopra I. Flap-lag-torsion in forward f ilght helicopter. Journal of the American Helicopter Society 1985; 30: 30–39.   
+4. Crespo da Silva MRM, Hodges DH. Nonlinear f lexure and torsion of rotating beams, with application to helicopter rotor blades—II. Response and stability results. Vertica 1986; 10: 171–186.   
+5. Hodges DH, Saberi H, Ormiston RA. Development of nonlinear beam elements for rotorcraft comprehensive analyses. Journal of the American Helicopter Society 2007; 52: 36–48.   
+6. Friedmann PP. Aeroelastic stability and response analysis of large horizontal-axis wind turbines. Journal of Wind Engineering and Industrial Aerodynamics 1980; 5: 373–401.   
+7. Chaviaropoulos PK. Flap/lead-lag aeroelastic stability of wind turbine blades. Wind Energy—Bognor Regis 2001; 4: 183–200.   
+8. Politis E, Riziotis V. The importance of nonlinear effects identif ied by aerodynamic and aero-elastic simulations on the 5mw reference wind turbine. Technical Report, Center for Renewable Energy Sources, Pikermi, Greece, 2007.   
+9. Kallesøe BS, Hansen MH. Effects of large bending def lections on blade f lutter limits. Technical Report, Risøe DTU, Roskilde, Denmark, 2008.   
+10. Riziotis VA, Voutsinas SG, Politis ES, Chaviaropoulos PK, Hansen AM, Madsen HA, Rasmussen F. Identif ication of structural non-linearities due to large def lections on a 5mw wind turbine blade. Proceedings of the EWEC ’08, Scientif ic Track, Brussels, 2008.   
+11. van Engelen TG. Control design based on aero-hydro-servo-elastic linear models from turbu (ecn). Proceedings of European Wind Conference, Milan, 2007.   
+12. Riziotis VA, Politis ES, Voutsinas SG, Chaviaropoulos PK. Stability analysis of pitch-regulated, variable-speed wind turbines in closed loop operation using a linear eigenvalue approach. Wind Energy 2008; 11: 517–535.   
+13. Ashwill TD, Kanaby G, Jackson K, Zuteck M. Development of the swept twist adaptive rotor (star) blade. Proceedings of the 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, Florida, 2010.   
+14. Larsen TJ, Hansen A, Buhl T. Aeroelastic effects of large blade def lections for wind turbines. Proceedings of the Special Topic Conference ‘The Science of making Torque from Wind’, Copenhagen, Denmark, 2004; 238–246.   
+15. Larsen TJ, Madsen HA, Hansen MA, Thomsen K. Investigations of stability effects of an offshore wind turbine using the new aeroelastic code hawc2. Proceedings of the Conference ‘Copenhagen Offshore Wind 2005’, Copenhagen, Denmark, 2005.   
+16. NWTC Design Codes (ADAMS2AD by LainoDJ, Jonkman J). [Online]. Available: http://wind.nrel.gov/designcodes/ simulators/adams2ad/ (Accessed 12 August 2005).   
+17. NWTC Design Codes (AeroDyn by Laino DJ). [Online]. Available: http://wind.nrel.gov/designcodes/simulators/ aerodyn/ (Accessed 5 July 2005).   
+18. NWTC Design Codes (FAST by Jonkman J). [Online]. Available: http://wind.nrel.gov/designcodes/simulators/fast/ (Accessed 12 August 2005).   
+19. Hansen MH. Improved modal dynamics of wind turbines to avoid stall-induced vibrations. Wind Energy 2003; 6: 179–195.   
+20. Riziotis VA, Voutsinas SG, Politis ES, Chaviaropoulos PK. Aeroelastic stability of wind turbines: the problem, the methods and the issues. Wind Energy 2004; 7: 373–392.   
+21. Hansen MH. Aeroelastic stability analysis of wind turbines using an eigenvalue approach. Wind Energy 2004; 7: 133–143.   
+22. Hansen MH. Aeroelastic instability problems for wind turbines. Wind Energy 2007; 10: 551–577.   
+23. Kallesøe BS. Equations of motion for a rotor blade, including gravity, pitch action and rotor speed variations. Wind Energy 2007; 10: 209–230.   
+24. Jonkman J. NREL 5MW baseline wind turbine. Technical Report, NREL/NWTC, Golden, Colorado, 2005.   
+25. Hansen MH, Gaunaa M, Madsen HA. A Beddoes-Leishman type dynamic stall model in state-space and indicial formulation. Technical Report Risø-R-1354(EN), Risø National Laboratory, Roskilde, 2004. [Online]. Available: http://www.risoe.dk (Accessed August 2004).   
+26. Hansen MOL. Aerodynamics of Wind Turbines. James & James: Earthscan, London, 2001.   
+27. Leishman JG, Beddoes TS. A semi-empirical model for dynamic stall. Journal of the American Helicopter Society 1989; 34: 3–17.   
+28. Thoedorsen T. General theory of aerodynamic instability and the mechanism of f lutter. NACA Report 496, 1935.   
+29. Thomsen JJ. Vibrations and Stability: Advanced Theory, Analysis, and Tools. Springer-Verlag: Berlin—Heidelberg— New York, 2003.  

线性化求解器/参考文献/Skjoldan-Implicit Floquet analysis of wind tur/auto/Skjoldan-Implicit Floquet analysis of wind tur.md

@@ -1,4 +1,5 @@
 # Implicit Floquet analysis of wind turbines using tangent matrices of a non-linear aeroelastic code  
+利用非线性气弹耦合代码的切线矩阵进行风电机组的隐式Floquet分析
 
 P. F. Skjoldan1 and M. H. Hansen2  
 
@@ -7,7 +8,10 @@ P. F. Skjoldan1 and M. H. Hansen2
 
 # ABSTRACT  
 
-The aeroelastic code BHawC for calculation of the dynamic response of a wind turbine uses a non-linear finite element formulation. Most wind turbine stability tools for calculation of the aeroelastic modes are, however, based on separate linearized models. This paper presents an approach to modal analysis where the linear structural model is extracted directly from BHawC using the tangent system matrices when the turbine is in a steady state. A purely structural modal analysis of the periodic system for an isotropic rotor operating at a stationary steady state was performed by eigenvalue analysis after describing the rotor degrees of freedom in the inertial frame with the Coleman transformation. For general anisotropic systems, implicit Floquet analysis, which is less computationally intensive than classical Floquet analysis, was used to extract the least damped modes. Both methods were applied to a model of a three-bladed $2.3\;\mathrm{MW}$ Siemens wind turbine model. Frequencies matched individually and with a modal identification on time simulations with the non-linear model. The implicit Floquet analysis performed for an anisotropic system in a periodic steady state showed that the response of a single mode contains multiple harmonic components differing in frequency by the rotor speed. Copyright $\copyright$ 2011 John Wiley & Sons, Ltd.  
+The aeroelastic code BHawC for calculation of the dynamic response of a wind turbine uses a non-linear finite element formulation. Most wind turbine stability tools for calculation of the aeroelastic modes are, however, based on separate linearized models. This paper presents an approach to modal analysis where the linear structural model is extracted directly from BHawC using the tangent system matrices when the turbine is in a steady state. A purely structural modal analysis of the periodic system for an isotropic rotor operating at a stationary steady state was performed by eigenvalue analysis after describing the rotor degrees of freedom in the inertial frame with the Coleman transformation. For general anisotropic systems, implicit Floquet analysis, which is less computationally intensive than classical Floquet analysis, was used to extract the least damped modes. Both methods were applied to a model of a three-bladed $2.3\;\mathrm{MW}$ Siemens wind turbine model. Frequencies matched individually and with a modal identification on time simulations with the non-linear model. The implicit Floquet analysis performed for an anisotropic system in a periodic steady state showed that the response of a single mode contains multiple harmonic components differing in frequency by the rotor speed. 
+用于计算风电机组动态响应的空气弹性代码 BHawC 采用非线性有限元公式。然而，大多数用于计算空气弹性模态的风电机组稳定性工具仍然基于单独的线性化模型。本文提出了一种模态分析方法，该方法利用风电机组处于稳态时的切线系统矩阵，直接从 BHawC 中提取线性结构模型。通过描述惯性坐标系下的风轮自由度，并使用科尔曼变换，对各向同性风轮在固定稳态下的周期系统进行纯结构模态分析，采用特征值分析实现。对于一般的各向异性系统，采用隐式Floquet分析，其计算强度小于传统的Floquet分析，用于提取阻尼最小的模态。这两种方法都应用于一个三叶片 2.3 MW 西门子风电机组模型。特征频率与非线性模型的时间模拟模态识别结果相符。隐式Floquet分析表明，对于周期稳态下的各向异性系统，单个模态的响应包含多个频率不同的谐波分量，这些频率之差为风轮转速。
+
+Copyright $\copyright$ 2011 John Wiley & Sons, Ltd.  
 
 # KEYWORDS  
 
@@ -33,29 +37,48 @@ In this paper, tangent matrices for mass, damping and stiffness are extracted fr
 
 Section 2 of this paper describes the BHawC model, and Section 3 explains the methods for modal analysis, the Coleman transformation approach, the implicit Floquet analysis and also the partial Floquet analysis, a system identification technique. In Section 4, the methods are applied to a model of a wind turbine. Section 5 discusses the approaches, and Section 6 concludes the paper.  
 
+今天，先进的非线性有限元代码1–3被常规地用于风电机组的载荷计算。然而，大多数用于计算气弹振模态的风电机组稳定性工具，仍然基于独立的线性化模型。**稳定性分析可以分为三个步骤：首先，计算稳态；然后，对稳态运动方程进行线性化；最后，进行模态分析以提取模态频率、阻尼和模态形状**。本文提出了一种适用于任何周期稳态的结构模态分析方法，该方法直接从非线性风电机组气弹振代码BHawC.3中获得线性化结果。
+
+在恒定平均风轮转速下运行的风电机组的运动方程包含周期系数，这阻止了对系统的直接特征值分析。大多数最近的风电机组稳定性工具$4\mathrm{-}7$采用了科尔曼变换，也称为多叶坐标变换，它在惯性坐标系中描述了风轮的自由度。如果系统是各向同性的，即风轮由对称安装的相同叶片组成，并且环境条件对称，则该变换可以消除周期系数。然而，Floquet分析适用于各向异性系统和任何周期稳态。它需要对运动方程在风轮旋转一个周期内进行积分，积分次数等于系统状态变量的数量。由于这种方法的计算负担，它仅被用于减少或简化具有有限自由度的风电机组模型。8–10 一种减少计算时间的方法是使用快速Floquet理论11，对于三叶各向同性风轮，只需要进行三分之一的积分。另一种方法是使用隐式Floquet分析12，可以在有限次数的积分后提取最弱阻尼的模态。
+
+Stol等人13将Floquet分析与应用于周期稳态的科尔曼变换方法进行比较，通过平均消除剩余的周期系数，发现模态频率和阻尼存在微小差异，得出结论：不需要使用Floquet分析。
+
+模态分析的另一种方法是系统辨识14–16，它基于数值模拟或测量结果，无需了解系统方程即可提取模态特性。然而，这些方法的精度有限，并且取决于所选的激励。
+
+在本文中，质量、阻尼和刚度的切线矩阵从气弹振代码BHawC中提取。如果系统是各向同性的，稳态是静态的，则在提取模态参数并通过特征值分析之前，应用科尔曼变换。对于各向异性系统，使用隐式Floquet分析进行模态分析。当系统是各向同性的时，单个模态的响应包含单个谐波分量，用于塔架自由度，对于叶片则包含多达三个分量。各向异性系统中的单个模态响应，对于叶片和塔架都包含多个谐波分量，这些分量在频率上不同，相差风轮转速。
+
+本文第2节描述了BHawC模型，第3节解释了模态分析方法，科尔曼变换方法、隐式Floquet分析以及部分Floquet分析（一种系统辨识技术）。第4节将这些方法应用于风电机组模型。第5节讨论这些方法，第6节总结了本文。
+
+
+
 # 2. STRUCTURAL MODEL  
 
 The BHawC wind turbine aeroelastic code3 is based on a structural finite element model sketched in Figure 1, where the main structural parts, tower, nacelle, shaft, hub and blades, are modelled as two-node 12-degrees of freedom Timoshenko beam elements. The code uses a corotational formulation, where each element has its own coordinate system that rotates with the element. The elastic deformation is described in the element frame, whereas the movement of the element coordinate system accounts for rigid body motion. In this way, a geometrically non-linear model is obtained using linear finite elements.  
 
 The configuration of the system, defined by nodal positions $\pmb{p}$ and orientations $\pmb q$ , nodal velocities $\dot{\pmb u}$ (of both positions and orientations) and nodal accelerations $\ddot{u}$ , must satisfy the equilibrium equation given in global coordinates as  
 
+BHawC风电机组气弹振代码3基于图1所示的结构有限元模型，其中主要结构部件，塔架、机舱、主轴、轮毂和叶片，被建模为两节点12自由度Timoshenko梁单元。该代码采用corotational公式，其中每个单元拥有自己的坐标系，该坐标系随单元旋转。弹性变形在单元坐标系中描述，而单元坐标系的运动则考虑了刚体运动。 这样，就使用线性有限元获得了几何非线性模型。
+
+系统的配置，由节点位置 $\pmb{p}$ 和姿态 $\pmb q$ ，节点速度 $\dot{\pmb u}$ （位置和姿态均包含）和节点加速度 $\ddot{u}$ 定义，必须满足以全局坐标表示的平衡方程。
+
+
 $$
 f_{\mathrm{iner}}(\boldsymbol{p},\boldsymbol{q},\dot{\boldsymbol{u}},\ddot{\boldsymbol{u}})+f_{\mathrm{damp}}(\boldsymbol{q},\dot{\boldsymbol{u}})+f_{\mathrm{int}}(\boldsymbol{p},\boldsymbol{q})=f_{\mathrm{ext}}
 $$  
 
 where $f_{\mathrm{iner}},f_{\mathrm{damp}},f_{\mathrm{int}}$ and $\pmb{f}_{\mathrm{ext}}$ are the inertial, damping, internal and external force vectors, respectively, and $\dot{\mathrm{O}}=\mathrm{d}/\mathrm{d}t$ denotes a time derivative. The inertial forces depend on the acceleration of the masses, the damping forces are given by viscous damping, the internal forces are due to elastic forces and the external forces contain the aerodynamic forces.17 To find  
-
+其中，$f_{\mathrm{iner}}$、 $f_{\mathrm{damp}}$、 $f_{\mathrm{int}}$ 和 $\pmb{f}_{\mathrm{ext}}$ 分别为惯性力、阻尼力、内力以及外力矢量，且 $\dot{\mathrm{O}}=\mathrm{d}/\mathrm{d}t$ 表示时间导数。惯性力取决于质量的加速度，阻尼力由粘性阻尼给出，内力由弹性力引起，外力包含气动力。<sup>17</sup> 为了找到
 ![](images/8c6afcabcd9b46b013d0b9eaf1928f2d43001587902377b212d0d820be50f7d8.jpg)  
 Figure 1. Sketch of the BHawC model substructures.  
 
 this equilibrium configuration, increments of the positions and the orientations $\delta\pmb{u}$ , the velocities $\delta\dot{\pmb{u}}$ and the accelerations $\delta\ddot{\pmb{u}}$ are obtained using Newton–Raphson iteration with the tangent relation obtained from the variation of Equation (1) as  
-
+这种平衡构型下，位置和姿态的增量 $\delta\pmb{u}$ 、速度 $\delta\dot{\pmb{u}}$ 和加速度 $\delta\ddot{\pmb{u}}$ 采用牛顿-拉夫逊迭代法获得，其切线关系式由方程 (1) 的变分推导得到，作为
 $$
 \mathbf{M}(q)\delta{\ddot{u}}+\mathbf{C}(q,{\dot{u}})\delta{\dot{u}}+\mathbf{K}(p,q,{\dot{u}},{\ddot{u}})\delta u=r
 $$  
 
 where M, C and $\mathbf{K}$ are the tangent mass, damping/gyroscopic and stiffness matrices, respectively, and $r=f_{\mathrm{ext}}+f_{\mathrm{iner}}+$ $f_{\mathrm{damp}}\!-\!f_{\mathrm{int}}$ is the residual. The stiffness matrix is composed of constitutive, geometric and inertial stiffness. The orientation $\pmb q$ of the nodes is described by quaternions, also known as the Euler parameters,18 a general four-parameter representation equivalent to a triad, which for node number $i$ is updated as  
-
+其中 M、C 和 $\mathbf{K}$ 分别为切向质量、阻尼/陀螺和刚度矩阵，且 $r=f_{\mathrm{ext}}+f_{\mathrm{iner}}+$ $f_{\mathrm{damp}}\!-\!f_{\mathrm{int}}$ 为残余量。刚度矩阵由本构刚度、几何刚度和惯性刚度组成。节点方向 $\pmb q$，由四元数描述，也称为欧拉参数<sup>18</sup>，这是一种与三维坐标系等效的通用四参数表示，对于节点编号 $i$ 而言，其更新方式为：
 $$
 \pmb q_{i}:=q u a t(\delta\pmb u_{i,\mathrm{rot}})*\pmb q_{i}
 $$  
@@ -63,20 +86,26 @@ $$
 where $\delta\mathbf{\boldsymbol{u}}_{i,\mathrm{rot}}$ contains three rotations that are assumed infinitesimal and thus commute and where this rotation pseudovector is transformed by the function termed quat into a quaternion, which is used to update the nodal quaternion $\pmb q_{i}$ employing the special quaternion product denoted by $^*$ , which maintains the unity of the quaternion. The nodal positions $\pmb{p}$ , the nodal velocities $\dot{\pmb u}$ and the accelerations $\ddot{u}$ are updated by regular addition of the positional part of $\delta\pmb{u},\,\delta\dot{\pmb{u}}$ and $\delta\ddot{\pmb{u}}$ , respectively. All components in $\pmb{p}$ , $\pmb q$ and $\delta\pmb{u}$ are absolute and described in a global frame.  
 
 The present work considers small perturbations in position and orientation ${\bf\delta y}$ , velocity $\dot{\mathbf{y}}$ and acceleration $\ddot{\mathbf{y}}$ to a steady state with constant mean rotor speed $\varOmega$ defined by $(p_{\mathrm{ss}},q_{\mathrm{ss}},\dot{\pmb u}_{\mathrm{ss}},\ddot{\pmb u}_{\mathrm{ss}})$ , the steady state positions, orientations, velocities and accelerations, respectively, all periodic with the rotor period $T=2\pi/\varOmega$ . The linearized equations of motion are obtained from equation (2) at $r\approx\theta$ as  
+其中 $\delta\mathbf{\boldsymbol{u}}_{i,\mathrm{rot}}$ 包含三个假设为无穷小的旋转，因此可以交换，并且这个旋转伪矢量通过称为“quat”的函数转换成一个四元数，用于更新节点四元数 $\pmb q_{i}$，采用特殊的四元数乘积（用 $^*$ 表示），该乘积保持四元数的模为一。节点位置 $\pmb{p}$ 、节点速度 $\dot{\pmb u}$ 和加速度 $\ddot{u}$ 分别通过正规地加回 $\delta\pmb{u}$、$\delta\dot{\pmb{u}}$ 和 $\delta\ddot{\pmb{u}}$ 的位置部分来更新。$\pmb{p}$、$\pmb q$ 和 $\delta\pmb{u}$ 中的所有分量都是绝对的，并且描述在全局坐标系中。
+
+本工作考虑了位置和姿态 ${\bf\delta y}$ 、速度 $\dot{\mathbf{y}}$ 和加速度 $\ddot{\mathbf{y}}$ 在稳态下发生的微小扰动，稳态具有恒定的平均风轮转速 $\varOmega$，由 $(p_{\mathrm{ss}},q_{\mathrm{ss}},\dot{\pmb u}_{\mathrm{ss}},\ddot{\pmb u}_{\mathrm{ss}})$ 定义，分别代表稳态位置、姿态、速度和加速度，它们都具有风轮周期 $T=2\pi/\varOmega$ 。运动方程的线性化是通过在 $r\approx\theta$ 时从方程 (2) 获得的。
+
 
 $$
 {\bf M}({q}_{\mathrm{ss}})\ddot{\boldsymbol{y}}+{\bf C}({q}_{\mathrm{ss}},\dot{\boldsymbol{u}}_{\mathrm{ss}})\dot{\boldsymbol{y}}+{\bf K}({p}_{\mathrm{ss}},{q}_{\mathrm{ss}},\dot{\boldsymbol{u}}_{\mathrm{ss}},\ddot{\boldsymbol{u}}_{\mathrm{ss}}){\boldsymbol{y}}=\boldsymbol{\theta}
 $$  
 
 where the matrices $\mathbf{M}$ , $\mathbf{C}$ and $\mathbf{K}$ are the $T$ -periodic tangent system matrices that are employed in the modal analysis described in the next section.  
-
+其中，矩阵 $\mathbf{M}$ 、 $\mathbf{C}$ 和 $\mathbf{K}$ 是在下一节中描述的模态分析中使用的 $T$ 周期切线系统矩阵。
 # 3. METHODS  
 
 In this section, the four methods for modal analysis of structures with rotors are presented.  
-
+在本节中，将介绍四种带有风轮结构的模态分析方法。
 # 3.1. Coleman approach  
 
 The Coleman transformation requires identical degrees of freedom on each blade, and therefore, the equations of motion (equation (4)) in global coordinates were first transformed into substructure coordinates $y_{\mathrm{T}}$ . The transformation is  
+科尔曼变换要求每个叶片具有相同的自由度，因此，首先将全局坐标系下的运动方程（方程(4)）转换到次结构坐标 $y_{\mathrm{T}}$ 。该变换是：
+
 
 $$
 \begin{array}{r l}&{\boldsymbol{y}=\mathrm{\mathbf{T}}\boldsymbol{y}_{\mathrm{T}}}\\ &{\mathbf{T}=\mathbf{diag}(\mathbf{I}_{N_{s}},\mathbf{T}_{\mathrm{r}},\mathbf{T}_{\mathrm{b1}},\mathbf{T}_{\mathrm{b2}},\mathbf{T}_{\mathrm{b3}})}\end{array}
@@ -86,18 +115,21 @@ where $\mathbf{T}$ is a block diagonal time-variant matrix composed of the ident
 
 The time-variant transformation into inertial frame coordinates $z$ is  
 
+其中，$\mathbf{T}$ 是一个块对角时间变动矩阵，由塔、机舱和传动系统自由度数量定义的单位矩阵 $\mathbf{I}_{N_{\mathrm{s}}}$ 组成，$\mathbf{T_{r}}$ 将主轴和轮毂的自由度变换到轮毂中心系，$\mathrm{T}_{\mathfrak{b}j}$ 将叶片编号 $j=1,2,3$ 的自由度变换到叶片 $j$ 的局部系。这些坐标系是在周期稳态下获得的，因此，$\mathbf{T}$ 是 $T$ 周期性的。
+
+时间变动变换到惯性系坐标 $z$ 是
 $$
 \begin{array}{r l}&{{\mathbf y}_{\mathrm{T}}={\mathbf B}\,z}\\ &{{\mathbf B}={\textbf d i a g}({\mathbf I}_{N_{\mathrm{s}}},{\mathbf B}_{\mathrm{r}},{\mathbf B}_{\mathrm{b}})}\end{array}
 $$  
 
 where $\mathbf{B}_{\mathrm{r}}$ is a simple rotational transformation of the shaft and the hub and $\mathbf{B}_{\mathrm{b}}$ is the Coleman transformation introducing multiblade coordinates for a three-bladed rotor11,19 as  
-
+其中 $\mathbf{B}_{\mathrm{r}}$ 是主轴和轮毂的一个简单旋转变换，而 $\mathbf{B}_{\mathrm{b}}$ 是科尔曼变换，它引入了三叶片风轮的多叶片坐标系<sup>11,19</sup>，如下所示：
 $$
 \mathbf{B}_{\mathrm{b}}=\left[\begin{array}{l l l}{\mathbf{I}_{N_{\mathrm{b}}}}&{\mathbf{I}_{N_{\mathrm{b}}}\cos\psi_{1}}&{\mathbf{I}_{N_{\mathrm{b}}}\sin\psi_{1}}\\ {\mathbf{I}_{N_{\mathrm{b}}}}&{\mathbf{I}_{N_{\mathrm{b}}}\cos\psi_{2}}&{\mathbf{I}_{N_{\mathrm{b}}}\sin\psi_{2}}\\ {\mathbf{I}_{N_{\mathrm{b}}}}&{\mathbf{I}_{N_{\mathrm{b}}}\cos\psi_{3}}&{\mathbf{I}_{N_{\mathrm{b}}}\sin\psi_{3}}\end{array}\right]
 $$  
 
 where $\psi_{j}=\varOmega t+2\pi(j-1)/3$ is the mean azimuth angle to blade number $j$ and $N_{\mathrm{b}}$ is the number of degrees of freedom on each blade. The inertial frame coordinate vector  
-
+其中 $\psi_{j}=\varOmega t+2\pi(j-1)/3$ 是叶片序号 $j$ 的平均方位角，$N_{\mathrm{b}}$ 是每个叶片上的自由度数量。惯性坐标系向量
 $$
 \boldsymbol{z}=\{y_{\mathrm{s}}^{\mathrm{T}}\,z_{\mathrm{r}}^{\mathrm{T}}\,a_{0}^{\mathrm{T}}\,a_{1}^{\mathrm{T}}\,b_{1}^{\mathrm{T}}\}^{\mathrm{T}}
 $$  
@@ -105,86 +137,93 @@ $$
 contains the untransformed coordinates for tower, nacelle and drivetrain ${\mathfrak{y}}_{\mathrm{s}}$ , the coordinates for shaft and hub $z_{\mathrm{r}}$ measured in a non-rotating frame aligned with the hub and the multiblade symmetric coordinates $\pmb{a}_{0}$ , cosine coordinates $\pmb{a}_{1}$ and sine coordinates $\pmb{b}_{1}$ . The details on how multiblade coordinates describe the motion of a wind turbine rotor in the inertial frame are discussed by Hansen.20,21  
 
 The Coleman transformed equations were obtained by first inserting equation (5) into equation (4), then converting it to first order form and lastly introducing the inertial frame transformation in equation (6) as ${\displaystyle y_{\mathrm{T}2}=\mathrm{diag}({\bf B},{\bf B})z_{2}}$ where ${y}_{\mathrm{T}2}=\{{y}^{\mathrm{T}}\,{\dot{{y}}}^{\mathrm{T}}\}^{\mathrm{T}}$ and $z_{2}=\dot{\{z^{\mathrm{T}}\,\tilde{z}^{\mathrm{T}}\}}^{\mathrm{T}}$ are the state v ectors in substructure and ine rtial frames, respectively, with $\tilde{z}=\dot{z}+\bar{\omega}z$ and the constant matrix $\bar{\boldsymbol{\omega}}=\mathbf{B}^{-1}\mathbf{B}$ . The result is  
+包含塔架、机舱和传动系统 ${\mathfrak{y}}_{\mathrm{s}}$ 的未转换坐标，主轴和轮毂 $z_{\mathrm{r}}$ 在与轮毂对齐的非旋转参考系中测量，以及多叶对称坐标 $\pmb{a}_{0}$，余弦坐标 $\pmb{a}_{1}$ 和正弦坐标 $\pmb{b}_{1}$。Hansen.20,21 讨论了多叶坐标如何描述风轮叶片在惯性参考系中的运动。
 
+通过首先将方程(5)代入方程(4)，然后将其转换为一阶形式，最后引入方程(6)中的惯性参考系变换，获得了 Coleman 变换后的方程，形式为 ${\displaystyle y_{\mathrm{T}2}=\mathrm{diag}({\bf B},{\bf B})z_{2}}$，其中 ${y}_{\mathrm{T}2}=\{{y}^{\mathrm{T}}\,{\dot{{y}}}^{\mathrm{T}}\}^{\mathrm{T}}$ 和 $z_{2}=\dot{\{z^{\mathrm{T}}\,\tilde{z}^{\mathrm{T}}\}}^{\mathrm{T}}$ 分别是子结构和惯性参考系中的状态向量，且 $\tilde{z}=\dot{z}+\bar{\omega}z$，$\bar{\boldsymbol{\omega}}=\mathbf{B}^{-1}\mathbf{B}$ 为常数矩阵。结果为：
 $$
 \begin{array}{r l}&{\dot{z}_{2}=\mathbf{A}_{\mathrm{B}}z_{2}}\\ &{\mathbf{A}_{\mathrm{B}}=\left[\mathbf{-}\mathbf{\bar{\omega}}-\mathbf{\bar{\omega}}\mathbf{\bar{\omega}}_{\mathrm{KB}}\quad\mathbf{-M}_{\mathrm{B}}^{-1}\mathbf{C}_{\mathrm{B}}-\mathbf{\bar{\omega}}\right]}\end{array}
 $$  
 
 where $\mathbf{A}_{\mathrm{B}}$ is the Coleman transformed system matrix and  
-
+其中 $\mathbf{A}_{\mathrm{B}}$ 是科尔曼变换后的系统矩阵，并且
 $$
 \begin{array}{r l}&{\mathbf{M}_{\mathrm{B}}=\mathbf{B}^{-1}\mathbf{T}^{\mathrm{T}}\mathbf{M}\mathbf{T}\,\mathbf{B}}\\ &{\mathbf{C}_{\mathrm{B}}=\mathbf{B}^{-1}\mathbf{T}^{\mathrm{T}}(\mathbf{C}\,\mathbf{T}+2\,\mathbf{M}\,\dot{\mathbf{T}})\mathbf{B}}\\ &{\mathbf{K}_{\mathrm{B}}=\mathbf{B}^{-1}\mathbf{T}^{\mathrm{T}}(\mathbf{K}\,\mathbf{T}+\mathbf{C}\,\dot{\mathbf{T}}+\mathbf{M}\,\ddot{\mathbf{T}})\mathbf{B}}\end{array}
 $$  
 
 are the Coleman transformed mass, damping/gyroscopic and stiffness matrices, respectively. If the system is isotropic, then $\mathbf{A}_{\mathrm{B}}$ is time-invariant, and a transient solution of equation (9) is  
-
+分别是柯尔曼变换的质量、阻尼/陀螺和刚度矩阵。如果系统是各向同性，则 $\mathbf{A}_{\mathrm{B}}$ 是时不变的，方程 (9) 的瞬态解是
 $$
-z_{2}=\mathrm{e}^{\mathbf{A}_{\mathrm{B}}t}z_{2}(0)=\mathbf{V}\mathrm{e}^{\mathbf{A}t}q(0)
+z_{2}=\mathrm{e}^{\mathbf{A}_{\mathrm{B}}t}z_{2}(0)=\mathbf{V}\mathrm{e}^{\Lambda t}q(0)
 $$  
 
-where $\mathbf{A}$ is a diagonal matrix containing the eigenvalues of $\mathbf{A}_{\mathrm{B}}$ , $\mathbf{V}$ contains the corresponding eigenvectors as columns and $\pmb q(0)=\mathbf V^{-1}z_{2}(0)$ are the initial conditions in modal c oordinates. It is assumed th at all eigenvect ors are linearly independen t .  
+where $\Lambda$ is a diagonal matrix containing the eigenvalues of $\mathbf{A}_{\mathrm{B}}$ , $\mathbf{V}$ contains the corresponding eigenvectors as columns and $\pmb q(0)=\mathbf V^{-1}z_{2}(0)$ are the initial conditions in modal coordinates. It is assumed that all eigenvectors are linearly independent .  
 
-The blade motion given in the inertial frame in equation (11) can be transformed back into the rotating frame using equation (6) as21  
+The blade motion given in the inertial frame in equation (11) can be transformed back into the rotating frame using equation (6) as $^{21}$  
+其中 $\mathbf{A}$ 是包含 $\mathbf{A}_{\mathrm{B}}$ 特征值的对角矩阵，$\mathbf{V}$ 包含相应的特征向量作为列，且 $\pmb q(0)=\mathbf V^{-1}z_{2}(0)$ 是模态坐标下的初始条件。假设所有特征向量线性无关。
 
+惯性坐标系中的叶片运动（见公式(11)）可以使用公式(6)转换回旋转坐标系，表示为 $^{21}$
 $$
-\Gamma,i k=\mathrm{e}^{\sigma_{k}t}\left(A_{0,i k}\cos(\omega_{k}t+\varphi_{0,i k})+A_{\mathrm{BW},i k}\cos\left((\omega_{k}+\Omega)t+\varphi_{j}+\varphi_{\mathrm{BW},i k}\right)+A_{\mathrm{FW},i k}\cos\left((\omega_{k}-\Omega)t-\varphi_{j}+\varphi_{\mathrm{GW},i k}\right)\right),
+y_T,i k=\mathrm{e}^{\sigma_{k}t}\left(A_{0,i k}\cos(\omega_{k}t+\varphi_{0,i k})+A_{\mathrm{BW},i k}\cos\left((\omega_{k}+\Omega)t+\varphi_{j}+\varphi_{\mathrm{BW},i k}\right)+A_{\mathrm{FW},i k}\cos\left((\omega_{k}-\Omega)t-\varphi_{j}+\varphi_{\mathrm{GW},i k}\right)\right),
 $$  
 
 where $\varphi_{j}=2\pi(j-1)/3$ and $\sigma_{k}$ and $\omega_{k}$ pare the modal damping and frequency of mode number $k$ , respectively, given by the eigenvalue $\lambda_{k}=\sigma_{k}+\mathrm{i}\omega_{k}$ with $\mathrm{i}=\sqrt{-1}$ . The amplitudes for degree of freedom number $i$ were determined from the components of the eigenvector $\nu_{k}$ gi ven in multiblade coordinates of equation (8) as $A_{0,i k}=|a_{0,i k}|$ and  
-
+其中 $\varphi_{j}=2\pi(j-1)/3$ ，$\sigma_{k}$ 和 $\omega_{k}$ 分别为模态阻尼和第 $k$ 个模态的频率，由特征值 $\lambda_{k}=\sigma_{k}+\mathrm{i}\omega_{k}$ 给出，其中 $\mathrm{i}=\sqrt{-1}$ 。自由度编号 $i$ 的振幅由方程 (8) 中多叶片坐标的特征向量 $\nu_{k}$ 的分量确定，为 $A_{0,i k}=|a_{0,i k}|$ 并且
 $$
 \begin{array}{r l}&{A_{\mathrm{BW},i k}=\frac{1}{2}\big((\mathrm{Re}\,(a_{1,i k})+\mathrm{Im}\,(b_{1,i k}))^{2}+(\mathrm{Re}\,(b_{1,i k})-\mathrm{Im}\,(a_{1,i k}))^{2}\big)^{1/2}}\\ &{A_{\mathrm{FW},i k}=\frac{1}{2}\big((\mathrm{Re}\,(a_{1,i k})-\mathrm{Im}\,(b_{1,i k}))^{2}+(\mathrm{Re}\,(b_{1,i k})+\mathrm{Im}\,(a_{1,i k}))^{2}\big)^{1/2}}\end{array}
 $$  
 
 where the subscripts 0, BW and FW denote symmetric, backward whirling and forward whirling motion, respectively.  
-
+其中，下标 0、BW 和 FW 分别表示对称、后旋和前旋运动。
 # 3.2. Classical Floquet analysis  
 
 Floquet analysis enables the solution of the periodic equations of motion directly without an explicit transformation. Equation (4) is written in first order form  
-
+Floquet 分析能够直接求解周期运动方程，无需显式变换。方程 (4) 以一阶形式写出。
 $$
 \begin{array}{r l}&{\dot{\boldsymbol{y}}_{2}=\mathbf{A}\boldsymbol{y}_{2}}\\ &{\mathbf{A}=\left[\mathbf{-M}^{-1}\mathbf{K}\quad\mathbf{-M}^{-1}\mathbf{C}\right]}\end{array}
 $$  
 
 where ${\mathbf y}_{2}=\{{\mathbf y}^{{\mathrm T}}\,\dot{{\mathbf y}}^{{\mathrm T}}\}^{{\mathrm T}}$ is the state vector and $\mathbf{A}$ is the $T$ -periodic system matrix.  
 
-Floquet theory22 states that the solution to equation (15) is of the form  
+Floquet theory$^{22}$ states that the solution to equation (15) is of the form  
+其中 ${\mathbf y}_{2}=\{{\mathbf y}^{{\mathrm T}}\,\dot{{\mathbf y}}^{{\mathrm T}}\}^{{\mathrm T}}$ 是状态向量，$\mathbf{A}$ 是 $T$ -周期系统矩阵。
 
+Floquet理论$^{22}$ 指出，方程 (15) 的解具有如下形式：
 $$
 \mathbf{\boldsymbol{y}}_{2}=\mathbf{\boldsymbol{U}}\mathbf{\boldsymbol{e}}^{\mathbf{\boldsymbol{\Lambda}}t}\mathbf{\boldsymbol{U}}^{-1}(0)\mathbf{\boldsymbol{y}}_{2}(0)
 $$  
 
 where $\mathbf{U}$ is a $T$ -periodic matrix and $\mathbf{A}$ is a diagonal matrix. One way to construct this solution is to form a fundamental solution to equation (15) as  
-
+其中 $\mathbf{U}$ 是一个 $T$ 周期矩阵，而 $\mathbf{A}$ 是一个对角矩阵。一种构造该解的方法是构造方程 (15) 的基本解，如下所示：
 $$
 \displaystyle\varphi=\bigl[\varphi_{1}\quad\varphi_{2}\quad.\ .\quad\varphi_{N}\bigr]
 $$  
 
 over one period, $t\ \in\ [0;T]$ , where $N$ is the number of state variables, such that $\dot{\varphi}\;=\;{\bf A}\varphi$ . The monodromy matrix defined as  
-
+在周期内，$t\ \in\ [0;T]$ ，其中 $N$ 为状态变量的数量，且 $\dot{\varphi}\;=\;{\bf A}\varphi$ 。单值性矩阵定义为：
 $$
 \mathbf{C}=\boldsymbol{\varphi}^{-1}(0)\boldsymbol{\varphi}(T)
 $$  
 
 contains all modal properties, which can be extracted from the eigenvalue decomposition  
-
+包含所有模态特性，可通过特征值分解提取。
 $$
 \mathbf{C}=\mathbf{V}\mathbf{J}\mathbf{V}^{-1}
 $$  
 
 where $\mathbf{V}$ contains the column eigenvectors $\nu_{k}$ of $\mathbf{C}$ , which are all assumed to be linearly independent and $\mathbf{J}$ is a diagonal matrix containing the eigenvalues $\rho_{k}$ of $\mathbf{C}$ , called the characteristic multipliers. The characteristic exponents $\lambda_{k}=\sigma_{k}+\mathrm{i}\omega_{k}$ contain the frequency $\omega_{k}$ and damping $\sigma_{k}$ and are related to the characteristic multipliers as $\rho_{k}=\exp(\lambda_{k}T)$ . Because the complex logarithm is not unique, the frequency is not determined uniquely, and the principal frequency $\omega_{\mathrm{p},k}$ and the damping $\sigma_{k}$ are defined from the characteristic multipliers as  
-
+其中 $\mathbf{V}$ 包含矩阵 $\mathbf{C}$ 的列特征向量 $\nu_{k}$，假设它们线性无关，而 $\mathbf{J}$ 是一个对角矩阵，包含矩阵 $\mathbf{C}$ 的特征值 $\rho_{k}$，称为特征乘数。特征指数 $\lambda_{k}=\sigma_{k}+\mathrm{i}\omega_{k}$ 包含频率 $\omega_{k}$ 和阻尼 $\sigma_{k}$，并且与特征乘数相关，关系为 $\rho_{k}=\exp(\lambda_{k}T)$。由于复数对数不唯一，频率不能唯一确定，因此从特征乘数定义了主频率 $\omega_{\mathrm{p},k}$ 和阻尼 $\sigma_{k}$。
 $$
 \begin{array}{c}{\displaystyle\sigma_{k}=\frac{1}{T}\ln(\vert\rho_{k}\vert)}\\ {\displaystyle\omega_{\mathrm{p},k}=\frac{1}{T}\arg(\rho_{k})}\end{array}
 $$  
 
-where $\arg(\rho_{k})\in]-\pi;\pi]$ is implied, resulting in $\omega_{\mathrm{p},k}\in\big[-\Omega/2;\Omega/2\big]$ . Any integer multiple of the rotor speed can be added to the principal frequency to obtain a more physically meaningful frequency23,24  
+where $\arg(\rho_{k})\in\big[-\pi;\pi]$ is implied, resulting in $\omega_{\mathrm{p},k}\in\big[-\Omega/2;\Omega/2\big]$ . Any integer multiple of the rotor speed can be added to the principal frequency to obtain a more physically meaningful frequency23,24  
-
+其中隐含条件为 $\arg(\rho_{k})\in\big[-\pi;\pi]$，由此得出 $\omega_{\mathrm{p},k}\in\big[-\Omega/2;\Omega/2\big]$。 可以将风轮转速的任意整数倍加到主频上，以获得更具物理意义的频率²³，²⁴
 $$
 \omega_{k}=\omega_{\mathrm{p},k}+j_{k}\Omega
 $$  
 
 a choice that also affects the periodic modal matrix $\mathbf{U}$ in equation (16). This matrix $\mathbf{U}$ contains the periodic mode shapes uk and is given as24  
+一个也会影响方程(16)中的周期模态矩阵 $\mathbf{U}$ 的选择。该矩阵 $\mathbf{U}$ 包含周期模态形状 uk，其表达式为24。
 
 $$
 \pmb{u}_{k}=\varphi\nu_{k}\mathrm{e}^{-\lambda_{k}t}
@@ -193,39 +232,45 @@ $$
 where the real part of $\lambda_{k}$ is given by equation (20) and the imaginary part of $\lambda_{k}$ is defined by equation (21) by selecting $j_{k}$ such that $\pmb{u}_{k}$ is as constant as possible for degrees of freedXom measured in the inertial frame.  
 
 Introducing the Fourier transform of the periodic mode shape  
+其中，$\lambda_{k}$ 的实部由公式(20)给出，虚部由公式(21)定义，通过选择 $j_{k}$ 使得在惯性坐标系测量的自由度方向上，$\pmb{u}_{k}$ 尽可能恒定。
 
+引入周期模态的傅里叶变换
 $$
 {\pmb u}_{k}=\sum_{j=-\infty}^{\infty}u_{j k}\mathrm{e}^{\mathrm{i}j\Omega t}
 $$  
 
 the transient solution in equation (16) can be written as a sum of harmonic components  
-
+方程 (16) 中的瞬态解可以写成谐波分量的和。
 $$
 y_{2}=\sum_{k=1}^{N}\sum_{j=-\infty}^{\infty}\mathcal{U}_{j k}\mathrm{e}^{(\sigma_{k}+\mathrm{i}(\omega_{k}+j\Omega))t}q_{k}(0)
 $$  
 
 where $\begin{array}{r}{\pmb q(0)=\mathbf{U}^{-1}(0)\mathbf{y}_{2}(0).}\end{array}$ . Note that equation (12) is a special case of this expression for $j=-1,0,1$  
-
+其中 $\begin{array}{r}{\pmb q(0)=\mathbf{U}^{-1}(0)\mathbf{y}_{2}(0).}\end{array}$ 。 注意，当 $j=-1,0,1$ 时，方程 (12) 是此表达式的一个特例。
 # 3.3. Implicit Floquet analysis  
 
 The implicit Floquet method is here described based on the detailed description in Bauchau and Nikishkov,12 which focuses on computation of the characteristic multipliers from the state transition matrix $\Phi(T,0)$ . It can be defined in classical Floquet theory as  
+基于Bauchau和Nikishkov的详细描述，本文介绍隐式Floquet方法，重点在于从状态转移矩阵$\Phi(T,0)$计算特征乘数。它可在经典Floquet理论中定义为：
+
 
 $$
 \boldsymbol{\varphi}(T)=\boldsymbol{\Phi}(T,0)\,\boldsymbol{\varphi}(0)
 $$  
 
 Using equation (18), the relationship between the state transition and monodromy matrices is derived as  
-
+使用公式(18)，推导了状态转移矩阵与单值性矩阵之间的关系。
 $$
 \Phi(T,0)=\varphi(0){\bf C}\,\varphi^{-1}(0)
 $$  
 
 showing that $\Phi(T,0)$ and C have identical eigenvalues (characteristic multipliers), and their eigenvectors are related as $\pmb{\nu}_{k}=\pmb{\varphi}^{-1}(0)\pmb{w}_{k}$ , where $w_{k}$ represents the eigenvectors of $\Phi(T,0)$ .  
-
+表明 $\Phi(T,0)$ 和 C 具有相同的特征值（特征乘数），且它们的特征向量之间存在如下关系：$\pmb{\nu}_{k}=\pmb{\varphi}^{-1}(0)\pmb{w}_{k}$ ，其中 $\pmb{w}_{k}$ 代表 $\Phi(T,0)$ 的特征向量。
 The key feature of the state transition matrix is that it defines the solution $y_{2}(T)=\Phi(T,0)y_{2}(0)$ for a time integration of the system equations (equation (15)) over one period $T$ with initial conditions ${\mathfrak{y}}_{2}(0)$ . Hence, without knowing the state transition matrix, it is possible to obtain the product of it with an arbitrary vector (the initial state vector) by integration of  
+关键在于状态转移矩阵定义了系统方程（式(15)）在周期 $T$ 内的时间积分解 $y_{2}(T)=\Phi(T,0)y_{2}(0)$，初始条件为 ${\mathfrak{y}}_{2}(0)$。因此，在不知道状态转移矩阵的情况下，可以通过积分来获得它与任意向量（初始状态向量）的乘积。
+
 
 equation (15) over one period. The Arnoldi algorithm25 is a method to approximate the eigenvalues and the eigenvectors of a matrix, say $\Phi(T,0)$ , using only the matrix multiplication with $\Phi(T,0)$ to construct an $m$ -sized subspace  
-
+方程 (15) 描述了一个周期内的状态。Arnoldi算法²⁵是一种近似矩阵（例如 $\Phi(T,0)$ ）的特征值和特征向量的方法，仅通过与 $\Phi(T,0)$ 的矩阵乘法构建一个 $m$ 维子空间。
 $$
 \mathbf{P}=\left[p_{1}\quad p_{2}\quad\ldots\quad p_{m}\right]
 $$  
@@ -244,18 +289,28 @@ $$
 
 converge towards the eigenvalues $\rho_{k}$ of $\Phi(T,0)$ with the largest modulus as the size $m$ of the subspace increases. The subspace eigenvectors $\tilde{w}_{k}$ of $\mathbf{H}$ projected back to the full state space converge towards the eigenvectors $w_{k}$ of $\Phi(T,0)$ , i.e. $w_{k}\approx\mathbf{P}\tilde{w}_{k}$ . The Arnoldi algorithm proceeds as follows:  
 
-Choose an arbitrary vector $\pmb{p}_{1}$ with $|p_{1}|=1$ for $n=1,2,\ldots,m$ $\pmb{a}:=\Phi(T,0)p_{n}$ (integration of equation (15) over $t\in[0;T])$ $\begin{array}{l}{b:=a}\\ {\mathrm{for~}j=1,2,\dotsc,n}\\ {\quad h_{j,n}:=p_{j}^{\operatorname{T}}a}\\ {\quad b:=b-h_{j,n}p_{j}}\end{array}$ end if $n<m$ $\begin{array}{l}{{h_{n+1,n}:=|b|}}\\ {{p_{n+1}:=b/h_{n+1,n}}}\end{array}$ end $\begin{array}{r}{p_{n+1}:=\!p_{n+1}-\!\sum_{j=1}^{n}\!(p_{j}^{\mathrm{T}}\!p_{n+1})p_{j}}\end{array}$ end  
+随着子空间维数 $m$ 的增大，子空间特征向量 $\tilde{w}_{k}$ 投影回完整状态空间，会收敛于 $\Phi(T,0)$ 的特征向量 $w_{k}$，即 $w_{k}\approx\mathbf{P}\tilde{w}_{k}$。阿诺尔迪算法的步骤如下：
+Choose an arbitrary vector $\pmb{p}_{1}$ with $|p_{1}|=1$ for $n=1,2,\ldots,m$ 
+$\pmb{a}:=\Phi(T,0)p_{n}$ (integration of equation (15) over $t\in[0;T])$ 
+$\begin{array}{l}{b:=a}\\ {\mathrm{for~}j=1,2,\dotsc,n}\\ {\quad h_{j,n}:=p_{j}^{\operatorname{T}}a}\\ {\quad b:=b-h_{j,n}p_{j}}\end{array}$ 
+end 
+if $n<m$ $\begin{array}{l}{{h_{n+1,n}:=|b|}}\\ {{p_{n+1}:=b/h_{n+1,n}}}\end{array}$ 
+end 
+$\begin{array}{r}{p_{n+1}:=\!p_{n+1}-\!\sum_{j=1}^{n}\!(p_{j}^{\mathrm{T}}\!p_{n+1})p_{j}}\end{array}$ 
+end  
 
 The last step in the $n$ -loop is an explicit re-orthogonalization to eliminate an otherwise progressing skewness of the subspace basis and thereby ensure convergence of the algorithm.12 Note that $\mathbf{H}$ with components $h_{j,n}$ , $n\,=\,1,\dots,m$ , $j=1,\dots,n$ , is an upper Hessenberg matrix for which there exist efficient eigenvalue solvers. In practice, the Arnoldi algorithm is continued until a desired number of eigenvalues $\tilde{\lambda}_{k}$ with the largest modulus and their corresponding eigenvectors $\mathbf{P}\tilde{{\boldsymbol{w}}}_{k}$ of the state transition matrix $\Phi(T,0)$ are converged to within a specific tolerance.  
 
 To construct the approximations to the periodic mo de shapes (equation (22)), the $m\times m$ fundamental solution matrix $\tilde{\varphi}$ to the subspace projected system equations is written as  
+最后一步是显式地重新正交化，以消除否则会逐渐累积的子空间基底的偏斜，从而确保算法的收敛。12 注意，$\mathbf{H}$ 具有分量 $h_{j,n}$ ，$n\,=\,1,\dots,m$ ，$j=1,\dots,n$ ，是一个Hessenberg矩阵，存在高效的特征值求解器。在实践中，阿诺尔迪算法会持续进行，直到收敛到期望数量的具有最大模值的特征值 $\tilde{\lambda}_{k}$ 及其对应于状态转移矩阵 $\Phi(T,0)$ 的特征向量 $\mathbf{P}\tilde{{\boldsymbol{w}}}_{k}$，并且在特定容差范围内。
 
+为了构造周期模态形状（方程 (22)）的近似值，将投影到子空间的系统方程的 $m\times m$ 基础解矩阵 $\tilde{\varphi}$ 写作：
 $$
 {\tilde{\boldsymbol{\varphi}}}=\mathbf{P}^{\mathrm{T}}\left[\varphi_{1}\quad\varphi_{2}\quad\ldots\quad\varphi_{m}\right]
 $$  
 
 where $\varphi_{j}$ is the solution of the full system (equation (15)) integrated over $t\in[0;T]$ for each initial condition $p_{j}$ , whereby ${\tilde{\boldsymbol{\varphi}}}(0)=\mathbf{I}$ because of equation (28). The eigenvectors $\tilde{\nu}_{k}$ of the subspace projected monodromy matrix $\tilde{\mathbf{C}}=\tilde{\boldsymbol{\Phi}}^{-1}(0)\tilde{\boldsymbol{\Phi}}(T)$ are therefore identical to the eigenvectors $\tilde{w}_{k}$ of the subspace projected state transition matrix (equation (29)). The periodic mode shapes in the subspace are therefore similar to equation (22) given by  
-
+其中 $\varphi_{j}$ 是对每个初始条件 $p_{j}$ 在 $t\in[0;T]$ 积分得到的完整系统（方程 (15)）的解，由于方程 (28)，${\tilde{\boldsymbol{\varphi}}}(0)=\mathbf{I}$。因此，子空间投影单值性矩阵 $\tilde{\mathbf{C}}=\tilde{\boldsymbol{\Phi}}^{-1}(0)\tilde{\boldsymbol{\Phi}}(T)$ 的特征向量 $\tilde{\nu}_{k}$ 与子空间投影状态转移矩阵（方程 (29)）的特征向量 $\tilde{w}_{k}$ 相同。因此，子空间中的周期模态形状与方程 (22) 相似，由...
 $$
 \tilde{\b u}_{k}=\tilde{\b\varphi}\tilde{\b w}_{k}\mathrm{e}^{-\tilde{\lambda}_{k}t}
 $$  
@@ -274,30 +329,44 @@ Partial Floquet analysis23 is a system identification technique that operates on
 
 Singular value decomposition is used to eliminate noise and extract the frequency and the damping of the most dominant modes from a matrix similar to the monodromy matrix assembled from a limited number of signals spanning several periods. The entries in this matrix can only be sampled once per period for periodic systems, which limits the accuracy because the signal damps away, decreasing the signal to noise ratio. Time-invariant systems can, however, be sampled once per time step. Therefore, partial Floquet analysis is combined with Coleman transformation of the signals,26 such that the response resembles that of a time-invariant system. This approach increases the accuracy and the number of modes that can be extracted from a given signal. However, a careful choice of forcing that excites all modes of interest to a sufficient level is necessary to extract these modes accurately.  
 
+部分Floquet分析23是一种系统识别技术，它基于系统的自由响应信号进行操作，因此无需了解系统方程。这些信号可以通过数值模拟或测量获得。
+
+利用奇异值分解来消除噪声，并从一个类似于由有限数量的信号组装而成的单值矩阵中提取最主要的模态的频率和阻尼。对于周期系统，该矩阵中的元素每周期只需采样一次，这会限制精度，因为信号会衰减，降低信噪比。然而，时不变系统可以每时间步采样一次。因此，部分Floquet分析与信号的科尔曼变换相结合26，使得响应类似于时不变系统的响应。这种方法提高了精度，并增加了可以从给定信号中提取的模态数量。然而，为了准确提取这些模态，需要仔细选择能够以足够的水平激发所有感兴趣模态的激励。
+
+
+
 # 4. NUMERICAL RESULTS  
 
 The modal analysis methods described in the previous sections are applied to a BHawC model of a $2.3\;\mathrm{MW}$ wind turbine with three $45\;\mathrm{m}$ blades, hub height $80\;\mathrm{m}$ and nominal speed $16\,\mathrm{rpm}$ . The model has 381 structural degrees of freedom.  
-
+前几节描述的模态分析方法应用于一个$2.3\;\mathrm{MW}$风电机组的BHawC模型，该风电机组具有三片$45\;\mathrm{m}$叶片，塔顶高度$80\;\mathrm{m}$，标称转速$16\,\mathrm{rpm}$。该模型包含381个结构自由度。
 # 4.1. Isotropic system  
 
 The turbine is mounted with identical blades and runs in a vacuum neglecting gravity forces, so the system is isotropic. The deflection of the blades because of centrifugal forces is therefore constant in the blade frame. The constant steady state is found at a given azimuth position by solving equation (1) statically, including centrifugal forces from the constant rotor speed. In this way, a steady state with no transients is obtained, and the system matrices become exactly periodic.  
-
+机组安装有完全相同的叶片，并在忽略重力作用的真空环境中运行，因此系统各向同性。由于离心力引起的叶片变形在叶片坐标系中是恒定的。通过静态求解方程（1），包括恒定风轮转速引起的离心力，可以在给定的方位角位置找到稳定的稳态。 这样可以获得无瞬态的稳态，并且系统矩阵变得精确的周期性。
 # 4.1.1. Coleman transformation approach  
 
 Because the system is isotropic, a modal analysis can be performed on the Coleman transformed system matrix. The system matrices M, C and $\mathbf{K}$ from equation (4) were extracted at a single azimuth angle and combined into the Coleman transformed system matrix of equation (9) from which the modal frequencies, damping and eigenvectors given in the inertial frame were extracted. The time-invariance of the system matrix was checked by calculation for several azimuth angles.  
 
 Figure 2(a) shows the lowest modal frequencies as a function of rotor speed where the frequency is normalized with the lowest modal frequency at $0\;\mathrm{rpm}$ . The modes were named according to their dominant motion determined from the eigenvector and the whirling amplitudes calculated from equations (13) and (14). The mode labels in Figure 2 first contain the index of that particular mode, then ‘T’ for tower, ‘F’ for blade flapwise, ‘E’ for blade edgewise or ‘DRV’ for drivetrain and ‘LO’ for longitudinal, ‘LA’ for lateral, ‘BW’ for backward whirling, ‘FW’ for forward whirling or $\mathbf{\nabla}^{\bullet}\mathbf{S}^{\bullet}$ for symmetric. For comparison, the frequencies extracted from time simulations with the non-linear BHawC model using the partial Floquet method26 are also shown. The agreement is within $0.4\%$ except for modes coupling to the drivetrain, i.e. the drivetrain, edgewise and lateral tower modes, where the discrepancy is up to $2\%$ at the highest rotor speed, which is caused by a difficulty with keeping the rotor speed exactly constant in the non-linear simulation because of the energy dissipated in the oscillation.  
 
+由于系统具有各向同性，因此可以在科尔曼变换后的系统矩阵上进行模态分析。从方程(4)中提取的系统矩阵M、C和$\mathbf{K}$，在单个方位角下组合成方程(9)中的科尔曼变换后的系统矩阵，从中提取了在惯性坐标系下的模态频率、阻尼和特征向量。通过计算，验证了系统矩阵的时间不变性，在多个方位角下进行验证。
+
+图2(a)显示了最低模态频率随风轮转速的变化曲线，频率以$0\;\mathrm{rpm}$时的最低模态频率为归一化值。模态的命名根据特征向量确定的主要运动和从方程(13)和(14)计算出的旋摆振幅来确定。图2中的模态标签首先包含该特定模态的索引，然后是‘T’代表塔架，‘F’代表叶片挥舞，‘E’代表叶片摆振，或‘DRV’代表机组，以及‘LO’代表纵向，‘LA’代表横向，‘BW’代表反向旋摆，‘FW’代表正向旋摆，或$\mathbf{\nabla}^{\bullet}\mathbf{S}^{\bullet}$代表对称模态。为了比较，还显示了使用部分Floquet方法26，从非线性BHawC模型的时间模拟中提取的频率。除了耦合到机组的模态（即机组、摆振和横向塔架模态）之外，一致性在0.4%以内，在最高风轮转速下，差异可达2%，这是由于在非线性模拟中，由于振荡过程中耗散的能量，难以保持风轮转速完全恒定。
 ![](images/995c2b6f3ee06ebcc2b34049a2242af9692ede73f9a2353ed42d793443f8df71.jpg)  
 Figure 2. Frequency (a) and damping (b) as a function of rotor speed. Standstill eigenvalue analysis (squares), Coleman approach (lines), partial Floquet analysis (circles). Legend entries are ordered after the sequence at 0 rpm.  
+图2. 转轮转速函数下的频率 (a) 和阻尼 (b)。静止特征值分析（方块），科尔曼法（线），偏分福克分析（圆）。图例条目按0 rpm时的顺序排列。
 
 Figure 2(b) shows the damping as a function of rotor speed where the logarithmic decrement is normalized with the value for the first tower longitudinal mode at $0\;\mathrm{rpm}$ . The agreement in damping between the results from the linear model and the partial Floquet analysis applied to the non-linear model is within $6\%$ , except for a discrepancy of up to $20\%$ for modes coupling to the drivetrain. It must be noted that the purely structural damping of the modes is small, and thus, a small absolute difference leads to a high relative difference. The results also show that damping is more difficult to estimate than frequency using system identification.  
-
+图 2(b) 显示了阻尼随风轮转速的变化，其中对第一塔纵向简正模态在 $0\;\mathrm{rpm}$ 时的对数衰减量进行了归一化。线性模型的结果与应用于非线性模型的偏Floquet分析结果在阻尼方面的吻合度在 $6\%$ 以内，除了与驱动系耦合的模态，存在高达 $20\%$ 的偏差。需要注意的是，这些模态的纯结构阻尼较小，因此，小的绝对差异会导致大的相对差异。结果还表明，与频率相比，系统辨识法更难估计阻尼。
 # 4.1.2. Implicit Floquet analysis  
 
 For the implicit Floquet analysis, the system matrices in global coordinates in equation (4) were extracted from the steady state at 16 azimuth angles equally spaced over a rotor rotation. For interpolation to other azimuth angles, a least squares fit of a truncated Fourier series with eight terms was used. The fundamental solutions in equation (30) were integrated with a Newmark-type solver from initial conditions determined by the Arnoldi algorithm. The principal frequencies and damping were found from equation (20) where $\rho_{k}$ are taken as the eigenvalues of the approximated state transition matrix. Figure 3 shows the real part $\sigma_{k}$ of the characteristic exponents calculated at each Arnoldi step for a steady state at $12\;\mathrm{rpm}$ using a time step of $\Delta t=T/1024=0.0049\;\mathrm{s}$ . The scattering of the highest damping values shows that the highest damped modes are spurious and do not represent actual eigenmodes of the system because of the approximate nature of the implicit Floquet analysis. To exclude these modes from the results, only modes satisfying a strict convergence criterion, where the absolute change of both damping $\sigma_{k}$ and principal frequency $\omega_{\mathrm{p},k}$ is less than $10^{-10}$ between three successive steps, were retained. After 50 Arnoldi steps, 19 modes were converged. The modal frequencies were determined using equation (21) by adding $j_{k}\Omega$ to the principal frequency, where $j_{k}\Omega$ is the single non-vanishing harmonic component in a Fourier transform of the periodic mode shape for degrees of freedom on the tower calculated from equation (32) using the principal frequency $\omega_{\mathrm{p},k}$ . The periodic mode shape components for degrees of freedom on the tower and the nacelle calculated with the modal frequency $\omega_{k}$ are thus constant. A detailed description of the process of frequency identification is given by Skjoldan and Hansen.24  
 
 Figure 4 shows the difference in frequency calculated with the Coleman transformation approach and the implicit Floquet analysis with different integration time steps. The implicit Floquet results converge towards the Coleman transformation results for decreasing time steps, the error being roughly proportional to $\varDelta t^{2}$ . Predominantly, the error increases with the modal frequency. A similar trend is seen for the damping.  
+为了进行隐式Floquet分析，方程(4)中的全局坐标系矩阵从风轮在16个方位角上均匀分布的稳态解中提取。为了插值到其他方位角，使用了截断的傅里叶级数，包含八个项，并采用最小二乘法拟合。方程(30)中的基本解使用Newmark型求解器，初始条件由Arnoldi算法确定。主频率和阻尼可以通过方程(20)得到，其中$\rho_{k}$被认为是近似状态转移矩阵的特征值。图3显示了在每个Arnoldi步计算得到的特征指数的实部$\sigma_{k}$，稳态转速为$12\;\mathrm{rpm}$，时间步长为$\Delta t=T/1024=0.0049\;\mathrm{s}$。最高阻尼值的散布表明，最高的阻尼模态是虚假的，由于隐式Floquet分析的近似性，它们不代表系统的实际特征模态。为了将这些模态从结果中排除，仅保留满足严格收敛判据的模态，即三个连续步长之间阻尼$\sigma_{k}$和主频率$\omega_{\mathrm{p},k}$的绝对变化均小于$10^{-10}$。经过50个Arnoldi步，19个模态收敛。模态频率使用方程(21)确定，通过将$j_{k}\Omega$加到主频率上，其中$j_{k}\Omega$是塔架自由度上周期模态的傅里叶变换中的单个非零谐波分量，该分量由方程(32)计算，并使用主频率$\omega_{\mathrm{p},k}$。因此，使用模态频率$\omega_{k}$计算出的塔架和机舱自由度上的周期模态分量是恒定的。Skjoldan和Hansen.24 详细描述了频率识别的过程。
+
+图4显示了使用Coleman变换方法和隐式Floquet分析计算出的频率差，使用了不同的积分时间步长。随着时间步长的减小，隐式Floquet结果趋近于Coleman变换结果，误差大致与$\varDelta t^{2}$成正比。主要趋势是误差随着模态频率的增加而增加。阻尼也呈现出类似的趋势。
+
 
 ![](images/d300c31577c207a813cae8b5b1a3dc8e8c0459cc32696c8195e9ce09628530b1.jpg)  
 Figure 3. Magnitude of implicit Floquet characteristic multipliers as function of steps in Arnoldi algorithm.   non-converged eigenvalues, $^{\circ}$ converged eigenvalues.  
@@ -306,7 +375,7 @@ Figure 3. Magnitude of implicit Floquet characteristic multipliers as function o
 Figure 4. Relative difference in implicit Floquet frequency compared with Coleman approach frequency for selected modes as a function of implicit Floquet integration time step.  
 
 Figure 5 shows the dominant harmonic components $\boldsymbol{u}_{j k}$ in equation (24) for the first flapwise forward whirling mode shape. The blade mode shape was transformed into substructure coordinates using equation (5) and contains the rigid body motion of the hub. The zoom factor in the lower right corner indicates how much each component has been enlarged. The ground fixed components in the mode shape are constant, consistent with the solution from the Coleman transformation approach. The mode shape for the blade has harmonic components at $j=-1,0,1$ , corresponding to the forward, symmetric and backward whirling components, respectively, in the Coleman transformation approach. Thus, in a pure excitation of this mode at $12{\mathrm{~rpm}}$ , according to equation (24), the tower vibrates with the normalized modal frequency $\omega^{\prime}=2.8$ , and the blades dominantly vibrate with $\omega^{\prime}-\Omega^{\prime}=2.2$ (FW) and to a lesser extent with $\omega^{\prime}+\Omega^{\prime}=3.3$ (BW) and $\omega^{\prime}=2.8$ (S) (see Figure 2(a)).  
-
+图 5 显示了方程 (24) 中第一简正挥舞前摆振模态的优势谐波分量 $\boldsymbol{u}_{j k}$。叶片模态已使用方程 (5) 转换到次结构坐标系，并包含塔轮刚体运动。图下角放大的倍数指示每个分量放大了多少。模态中的地面固定分量是恒定的，与科尔曼变换方法得到的解一致。叶片的模态具有 $j=-1,0,1$ 的谐波分量，分别对应于科尔曼变换方法中的前摆振、对称和后摆振分量。因此，在 $12{\mathrm{~rpm}}$ 的纯激励下，根据方程 (24)，塔按照归一化模态频率 $\omega^{\prime}=2.8$ 振动，而叶片主要以 $\omega^{\prime}-\Omega^{\prime}=2.2$ (FW) 振动，并在较小的程度上以 $\omega^{\prime}+\Omega^{\prime}=3.3$ (BW) 和 $\omega^{\prime}=2.8$ (S) 振动（见图 2(a)）。
 # 4.2. Anisotropic system  
 
 To investigate the effects of an anisotropic rotor on the modal properties, a mass of $485\;\mathrm{kg}$ because of ice coverage defined by DIN-1055- $5^{27}$ is added along the length of blade 1. Figure 6 shows the resulting steady state when running the turbine at $16~\mathrm{rpm}$ with a $10~\mathrm{m~s}^{-1}$ uniform wind field perpendicular to the rotor plane. Note that the wind is used only to drive the rotor, and the modal analysis is still purely structural. The steady state varies periodically both for the tower and the blades, and the blade motion for blade 1 is different from that of blades 2 and 3. The steady state was determined from a time simulation until transients have damped away, and system matrices were then extracted at each time step of the steady state simulation and interpolated onto integration time points using a truncated Fourier series with eight terms. The implicit Floquet analysis was carried out with an integration time step of $T/1024=0.0037$ s as described for the isotropic case. The frequencies were up to $4\%$ lower than in the isotropic case because of the added mass on one blade. The change in damping was slightly more pronounced, up to a $17\%$ decrease for the second flapwise forward whirling mode.  
@@ -317,6 +386,15 @@ For the mode shape of blade 1, the harmonic components at $j=-1,0,1$ are similar
 
 The identification of the first flapwise forward whirling modal frequency was not done by making the tower mode shape as constant as possible, as in the isotropic case. Rather, the modal frequency was chosen to be close to the one for the similar mode in the isotropic case. A more suitable criterion to give this result is to require that the mode shape with the rotor degrees of freedom in multiblade coordinates be as constant as possible.28  
 
+为了研究各向异性风轮对模态特性的影响，根据DIN-1055- $5^{27}$ 标准定义的冰覆盖量，在叶片1的长度方向上增加了$485\;\mathrm{kg}$的质量。图6显示了机组以$16~\mathrm{rpm}$转速，在$10~\mathrm{m~s}^{-1}$均匀风场（垂直于风轮平面）作用下的稳态响应。需要注意的是，风仅用于驱动风轮，模态分析仍然是纯粹的结构分析。稳态响应在塔架和叶片上都呈周期性变化，叶片1的运动与叶片2和3的运动不同。稳态响应是通过时间模拟，直到瞬态衰减后确定，然后从稳态模拟的每个时间步长提取系统矩阵，并使用截断傅里叶级数（八项）进行插值。隐式Floquet分析采用时间步长$T/1024=0.0037$ s，方法与各向异性情况描述相同。由于一个叶片增加了质量，频率比各向异性情况低高达$4\%$。阻尼的变化更为明显，对于第二简正挥舞前摆振模态，阻尼降低高达$17\%$。
+
+图7显示了塔架和叶片1的第一个简正挥舞前摆振模态的谐波分量 $\boldsymbol{u}_{j k}$，其频率为 $j\,\Omega$。与各向异性情况相比，塔架的模态形状现在具有多个谐波分量，而各向异性情况只有一种。$j\,=\,0$ 处的谐波分量形状与各向异性情况的对应分量相似，但现在主导分量位于 $j=-2$ ，并且 $j=-1$ 处也有显著分量。
+
+对于叶片1的模态形状，$j=-1,0,1$ 处的谐波分量与各向异性情况的对应分量相似。然而，现在叶片1的主导挥舞分量在 $j\,=\,-1$ 处的振幅是叶片2和3的3倍，叶片2和3与叶片1几乎同相位或反相位运动，如图8所示。因此，在纯激励下，塔架现在以归一化频率 $\omega^{\prime}\!-\!2\varOmega^{\prime}=1.6$ 和 $\omega^{\prime}=2.8$ 的分量主导振动。叶片1以与各向异性情况相同的 $\omega^{\prime}\,{-}\,\Omega^{\prime}=2.2$ 主导振动，并且在 $\omega^{\prime}-2\varOmega^{\prime}=1.6,\omega^{\prime}-3\varOmega^{\prime}=1.0$ 和 $\omega^{\prime}+3\Omega^{\prime}=4.5$ 处也显著振动，此外还有 $\omega^{\prime}+\Omega^{\prime}=3.3$ 和 $\omega^{\prime}=2.8$ 分量，与各向异性情况相同。
+
+第一个简正挥舞前摆振模态频率的识别不是通过使塔架模态形状尽可能恒定来实现，这与各向异性情况相同。相反，模态频率被选择为接近各向异性情况中相似模态的频率。获得此结果更合适的标准是要求具有风轮自由度的多叶坐标下的模态形状尽可能恒定。28
+
+
 ![](images/7e93824c056fe69263ae0ceee0705b5fbe122ce32c71bb9052a0133dc9d49790.jpg)  
 Figure 5. Amplitudes of harmonic components of the first flapwise forward whirling periodic mode shape for the isotropic rotor. Blades (top) flapwise and edgewise, and tower (bottom) longitudinal and lateral.  
 
@@ -325,6 +403,8 @@ The rotor with one ice-covered blade is an example of how an isotropic rotor can
 # 5. DISCUSSION  
 
 This paper has presented several different methods for structural modal analysis of wind turbines. The Coleman approach is simple and fast, and its basis in a physical coordinate transformation means that the results are easily interpreted. Its speed makes it useful for doing parameter studies early in the design process. But it is only applicable to isotropic systems. Floquet analysis can be applied to examine special cases where anisotropic effects are suspected to change the modal parameters. The implicit Floquet analysis is an efficient implementation of Floquet analysis for systems with many degrees of freedom. In the example given, the most important modes are extracted after 50 integrations of the system over a rotor period, whereas 762 integrations would be needed for a classical Floquet analysis. Finally, the partial Floquet analysis, or another means of system identification, is useful to check the validity of the linearization.  
+本文介绍了几种风电机组结构模态分析的不同方法。Coleman 方法简单快速，其基于物理坐标变换的原理使得结果易于解释。其速度使其在设计过程早期进行参数研究非常有用。但它仅适用于各向同性系统。Floquet 分析可用于检查各向异性效应可能改变模态参数的特殊情况。隐式 Floquet 分析是针对具有许多自由度的系统，Floquet 分析的一种高效实现。在给定的例子中，最重要的模态在对系统进行 50 次积分（积分周期为风轮周期）后提取，而经典 Floquet 分析需要 762 次积分。最后，偏 Floquet 分析，或另一种系统识别方法，可用于检查线性化的有效性。
+
 
 ![](images/3fe519c4b369da4448413f5b3cd0d0aa591fdf47cc6494f2e5e84863dd20c455.jpg)  
 Figure 6. Steady state over one rotor period for the anisotropic rotor at 16 rpm. Blade tips flapwise (top) 1, 2 and $\--3$ and edgewise (middle) 1, 2 and $-\cdot-3$ , and blade tips, tower top (bottom) longitudinal and lateral.  
@@ -336,10 +416,14 @@ Figure 7. Amplitudes of harmonic components of the first flapwise forward whirli
 Figure 8. Amplitudes and phases of the harmonic component at $j=-1$ of the first flapwise forward whirling periodic mode shape for the isotropic rotor (a) and the anisotropic rotor (b). Blades 1, 2 and $\--3$ .  
 
 The work presented in this paper is part of an ongoing effort to obtain a full aeroelastic linear model of the non-linear code BHawC. The approach presented in this paper is readily extendable to a linear aeroelastic model. The linear model will aid in the understanding of the loads obtained from a non-linear response, of which many features can be explained from the linear modes.  
+本文所呈现的工作是持续努力的一部分，旨在获得非线性代码BHawC的完整气弹道线性模型。本文所介绍的方法易于扩展至线性气弹道模型。该线性模型将有助于理解从非线性响应中获得载荷，其中许多特征可以通过线性模态进行解释。
+
 
 # 6. CONCLUSION  
 
 Tangent matrices for structural modal analysis are extracted directly from the non-linear model of a wind turbine in a steady state. When the system is isotropic, the preferred approach is to use the Coleman transformation for describing the equations of motion in the inertial frame allowing direct eigenvalue analysis to extract the modal frequencies, damping and mode shapes. When the system is anisotropic, implicit Floquet analysis, reduces the computational burden associated with classical Floquet analysis, is applied to yield the lowest damped eigenmodes. The linearized model is validated from numerical results for a three-bladed turbine, showing a reasonable agreement for the frequencies and the damping between the Coleman approach and the partial Floquet analysis on the response of the non-linear model for modes not related to the drivetrain. The implicit Floquet results converge to the results from the Coleman approach with the deviation in frequency and damping roughly proportional to the square of the integration time step and increasing with the modal frequency. This finding shows the importance of precise time integration in implicit Floquet analysis. An analysis applied to an anisotropic system with one blade covered with ice shows a decrease in frequency up to $3\%$ and changes in damping within $17\%$ . It also reveals multiple harmonic components in the response of a single mode that will show up in measurements.  
+在稳态条件下，直接从风电机组的非线性模型中提取结构模态分析的切线矩阵。当系统各向同性时，首选方法是使用科尔曼变换来描述惯性系中的运动方程，从而可以直接进行特征值分析，提取模态频率、阻尼和模态形状。当系统非各向同性时，采用隐式Floquet分析（Implicit Floquet analysis），以减轻与经典Floquet分析相关的计算负担，从而获得阻尼最低的简正模态。对三叶片风轮的线性化模型进行数值验证，结果表明科尔曼方法和部分Floquet分析在与驱动系无关的模态频率和阻尼方面具有合理的吻合度。隐式Floquet分析结果与科尔曼方法的结果收敛，频率和阻尼的偏差大致与积分时间步长的平方成正比，并且随着模态频率的增加而增加。这一发现表明了在隐式Floquet分析中精确时间积分的重要性。对一个带有冰层的叶片的非各向同性系统进行的分析表明，频率降低高达$3\%$，阻尼变化在$17\%$以内。它还揭示了单个模态响应中的多个谐波分量，这些分量将在测量中显现。
+
 
 # ACKNOWLEDGEMENT