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书籍/力学书籍/BladedTheoryManual/Structural Dynamics/auto/Structural Dynamics.md

@@ -323,18 +323,43 @@ An equilibrium-based method is available for calculating internal forces, which
 
 For blades, the equilibrium-based method is always used to calculate internal forces. For the tower, either method can be used.  
 
-# Conventional Displacement-Based Finite Element Method  
+柔性体部件的关键功能是计算内力（也称为应力结果或截面力），**在动态分析的每个时间步末或稳态分析的计算稳态解时进行计算**。计算的一个基本假设是，外力（包括重力和惯性力）施加于柔性体部件的弯曲状态，因为外力的二阶效应对最终内力有显著影响。
+
+在Bladed中，计算内力的传统方法是一种displacement-based的有限元方法，它使用梁单元的刚度矩阵来计算单元端节点的内力。
+
+还提供了一种基于平衡的方法来计算内力，该方法是通过使内力与外力达到平衡而推导出来的。对于具有静不定性的柔性体复杂分析，必须包含刚度特性，但对于具有静定性的柔性体，可以在不使用任何刚度特性的情况下进行计算。
+
+**对于叶片，始终使用基于平衡的方法来计算内力。对于塔架，可以使用任一种方法。**
+
+## Conventional Displacement-Based Finite Element Method  
 
 With the modal model, the deformed shape of the flexible body components like the tower at any instant is a linear combination of the selected mode shape functions. With a reduced number of modes, the resulting deformation may therefore not be accurately predicted, which means that it is not possible to calculate the internal forces directly from the deformations as done by standard finite element technique.  
 
 In order to calculate the internal forces of flexible body components, the deformation at all stations is therefore calculated from a static equilibrium analysis, where the applied force is calculated as the sum of all external forces including the inertial loads. In case that some fundamental degrees of freedom are constrained the system is solved with respect to a reduced set of independent degrees of freedom, and the Lagrange multipliers associated with the constraints are calculated. Finally the internal forces of all beam elements at both ends are calculated from the fundamental equilibrium equation of the beam element. The second order effects of the external loads on the calculated internal forces are accounted for through the geometric stiffness model as described in (Przemieniecki, 1968).  
 
+借助模态模型，塔架等柔性构件在任何时刻的变形都是选定的模态函数线性组合。如果采用较少的模态，则可能无法准确预测变形，这意味着无法像标准有限元技术那样直接从变形计算内部力。
+
+为了计算柔性构件的内部力，因此需要从静态平衡分析计算所有位置的变形，其中施加力被计算为所有外部力的总和，包括惯性载荷。如果一些基本自由度受到约束，则系统将针对一组减少的独立自由度进行求解，并计算与约束相关的拉格朗日乘子。最后，**从梁单元的基本平衡方程计算所有梁单元两端的内部力**。外部载荷对计算出的内部力的二阶效应通过几何刚度模型进行考虑，如 (Przemieniecki, 1968) 中所述。
+
+
 # Equilibrium-Based Method  
 
 The method employs the multibody_approach used by Bladed for modelling of the complete wind turbine structure on a flexible body component level. With this approach described in (Nim et al.,. 2024), a flexible body is modelled as an assembly of rigid or deformable elements, interconnected at $N$ nodes each including six nodal DOFs, i.e., three translation components and three rotation components. According to the multibody approach, the deformation state of a flexible element is described by $N_{\epsilon}^{\epsilon}$ generalised strains that are collected in the vector $\epsilon^{\mathrm{{e}}}$ . The external element loading including inertial loads is represented by the vector $\mathbf{p}_{\mathrm{r}}^{\mathrm{e}}$ conjugate to nodal motion and the vector $\mathbf{p}_{\epsilon}^{\mathrm{e}}$ conjugate to generalised strains, defining the deformation state of a deformable element with elastic properties defined by the stiffness matrix ${\bf K}_{\epsilon\epsilon}^{\mathrm{e}}$ . The kinematical relations between nodal displacement and the generalised strains motion are described by $N_{\mathrm{c}}^{\mathrm{e}}$ geometric constraint relations in terms of the constraint matrices ${\bf C}_{\mathrm{r}}^{\mathrm{e}}$ and ${\bf C}_{\epsilon}^{\mathrm{e}},$ which associate a set of unknown Lagrange multipliers $\lambda^{\mathrm{~e~}}$ . By application of the principle of virtual work, it appears that the resulting equilibrium equations for an element can be written in a simplified form as  
 
+该方法采用Bladed软件使用的多体法，用于在柔性体组件层级上对整个风轮机结构的建模。这种方法如(Nim et al.,. 2024)中所述，将柔性体建模为由刚性或变形元件的组合，这些元件在$N$个节点处相互连接，每个节点包含六个节点自由度 (DOF)，即三个平动分量和三个转动分量。根据多体法，柔性元件的变形状态由$N_{\epsilon}^e$个广义应变描述，并被收集在向量 $\epsilon^{\mathrm{{e}}}$ 中。包括惯性载荷在内的外部元件载荷由与节点运动共轭的向量 $\mathbf{p}_{\mathrm{r}}^{\mathrm{e}}$ 和与广义应变共轭的向量 $\mathbf{p}_{\epsilon}^{\mathrm{e}}$ 表示，它们定义了具有由刚度矩阵 ${\bf K}_{\epsilon\epsilon}^{\mathrm{e}}$ 定义的弹性特性的变形元件的变形状态。节点位移与广义应变运动之间的运动学关系由$N_{\mathrm{c}}^{\mathrm{e}}$个几何约束关系描述，这些关系通过约束矩阵 ${\bf C}_{\mathrm{r}}^{\mathrm{e}}$ 和 ${\bf C}_{\epsilon}^{\mathrm{e}}$ 关联，并关联一组未知的拉格朗日乘子 $\lambda^{\mathrm{~e~}}$。通过应用虚拟功原理，可以发现对于一个元件，得到的平衡方程可以写成简化的形式，如下所示：
+
+
 $$
-\left[\mathbf{C}_{\mathrm{r}}^{\mathrm{e}T}\right]\boldsymbol{\lambda}^{\mathrm{e}}=\left[\begin{array}{l}{\mathbf{p}_{\mathrm{r}}^{\mathrm{e}}+\mathbf{f}_{0}^{\mathrm{e}}}\\ {\mathbf{p}_{\epsilon}^{\mathrm{e}}-\mathbf{K}_{\epsilon\epsilon}^{\mathrm{e}}\epsilon^{\mathrm{e}}}\end{array}\right]
+\begin{equation}
+\left[\begin{array}{c}
+{\bmatrix{C}_{\mathrm{r}}^{\mathrm{e}}}^T \\
+{\bmatrix{C}_{\epsilon}^{\mathrm{e}}}^T
+\end{array}\right] \bvector{\uplambda}^{\mathrm{e}}=\left[\begin{array}{c}
+\bvector{p}_{\mathrm{r}}^{\mathrm{e}}+\bvector{f}_0^{\mathrm{e}} \\
+\bvector{p}_{\epsilon}^{\mathrm{e}}-\bmatrix{K}_{\epsilon \epsilon}^{\mathrm{e}} \bvector{\epsilon}^{\mathrm{e}}
+\end{array}\right]
+\label{eq:motionforaflexelement}
+\end{equation}
 $$  
 
 Thevector $\mathbf{f}_{0}^{\mathrm{{e}}}$ contains the resulting element internal forces, which originate from the connection to neighbouring elements. This vector cancels out in the assembly process, which means that the resulting equilibrium equations for the flexible body take the form  

书籍/力学书籍/力学/Flexible Multibody Dynamics (O. A. Bauchau) (Z-Library)/auto/Flexible Multibody Dynamics (O. A. Bauchau) (Z-Library).md

@@ -2869,15 +2869,16 @@ Principle 1 (Principle of work and energy for a particle) The work done by the e
 Figure 3.2 depicts a particle of mass $m$ whose position is described by position vector ${\underline{{r}}}(t)$ with respect to an inertial frame, $\mathcal{F}^{I}=[\mathbf{O},\mathcal{T}=(\bar{\iota}_{1},\bar{\iota}_{2},\bar{\iota}_{3})]$ . Conservative forces are a class of forces that depend only upon the position of the particles on which they act, $\underline{{F}}=\underline{{F}}(\underline{{r}})$ . Although these forces may vary with time as the particle moves, they do not depend explicitly on time or velocity. Figure 3.3 shows two arbitrary paths, denoted ACB and ADB, along which the particle moves in space from point A to point $\mathbf{B}$ .  
 图 3.2 描绘了一个质量为 $m$ 的粒子，其位置由相对于惯性系 $\mathcal{F}^{I}=[\mathbf{O},\mathcal{T}=(\bar{\iota}_{1},\bar{\iota}_{2},\bar{\iota}_{3})]$ 的位置矢量 ${\underline{{r}}}(t)$ 描述。保守力是一类仅取决于它们作用的粒子位置的力，$\underline{{F}}=\underline{{F}}(\underline{{r}})$ 。 尽管这些力在粒子运动时可能随时间变化，但它们不显式地依赖于时间或速度。图 3.3 显示了两个任意路径，分别标记为 ACB 和 ADB，粒子沿这些路径在空间中从点 A 移动到点 $\mathbf{B}$ 。
 
-# Definition  
+### Definition  
 
 By definition, force $\underline{{F}}$ is conservative if and only if the work it performs along any path joining the same initial and final points is identical. This is expressed by the following equation  
-
+根据定义，力 $\underline{{F}}$ 仅当它在连接相同起始点和终点的任意路径上所做的功相同，才被认为是保守力。这由以下方程表达：
 $$
 W_{A\rightarrow B}=\int_{\mathrm{Path~ACB}}\underline{{F}}^{T}\mathrm{d}\underline{{r}}=\int_{\mathrm{Path~ADB}}\underline{{F}}^{T}\mathrm{d}\underline{{r}}.
 $$  
 
 Since reversing the limits of integration simply changes the sign of the integral, the work done by the force along path ADB is equal in magnitude and opposite in sign to that along path BDA. Equation (3.13) then implies the vanishing of the work done by the force over the closed path ACBDA. Because path ACB and ADB are arbitrary paths joining points $\mathbf{A}$ and $\mathbf{B}$ , it follows that a force is conservative if and only if the work it performs vanishes over any arbitrary closed path,  
+由于反转积分上下限仅仅改变积分的符号，因此力沿路径ADB所做的功，其大小等于沿路径BDA所做的功，但符号相反。由此，方程(3.13)推导出，力在闭合路径ACBDA上所做的功为零。因为路径ACB和ADB是连接点$\mathbf{A}$和$\mathbf{B}$的任意路径，因此可以得出，如果力所做的功在任意闭合路径上消失，则该力是保守力。
 
 $$
 W=\oint_{\mathrm{Any~path}}\underline{{F}}^{T}\mathrm{d}\underline{{r}}=\oint_{\mathbb{C}}\underline{{F}}^{T}\mathrm{d}\underline{{r}}=0,
@@ -2891,25 +2892,30 @@ Fig. 3.3. Paths ACB and ADB join the same two points, A and $\mathbf{B}$ .
 ![](fbafde9173ed8f0a72b92314bccc2b8d199d27f3890b3e572f09d40effd4cece.jpg)  
 Fig. 3.4. Path enclosing a surface of area $\mathbb{S}$ with a normal $\bar{n}$ .  
 
-# Potential of a conservative force  
+### Potential of a conservative force  
 
 Based on the definition of conservative forces, eq. (3.14), Stokes’ theorem [2] then implies that  
-
+基于保守力的定义，公式（3.14），斯托克斯定理[2]由此推导出，
 $$
 \oint_{\mathbb{C}}\underline{{F}}^{T}\mathrm{d}\underline{{r}}=\int_{\mathbb{S}}\bar{n}^{T}\widetilde{\nabla}\underline{{F}}\,\mathrm{d}\mathbb{S}=0,
 $$  
 
 where $\mathbb{S}$ is a surface bounded by curve $\mathbb{C},\,\bar{n}$ the outward normal to surface $\mathbb{S}$ , as shown in fig. 3.4, and $\widetilde\nabla\underline{{F}}=\mathrm{curl}(\underline{{F}}).$ If the force is conservative, the surface integral must vanish for any  s urface, $\mathbb{S}$ , and this can only occur if the integrand vanishes, leading to $\widetilde\nabla\underline{{F}}=0$ for any curve, $\mathbb{C}$ , and surface, $\mathbb{S}$ . Textbooks on vector algebra [2], prove the  fo llowing identity: $\widetilde\nabla\underline{{\nabla}}V=0$ , where $V$ is an arbitrary scalar function and $\Sigma V\,=\,\operatorname{grad}(V)$ . It can the n  be shown that the solution of equation $\widetilde\nabla\underline{{F}}\,=\,0$ is simply  
+其中 $\mathbb{S}$ 为由曲线 $\mathbb{C}$ 围成的曲面，$\bar{n}$ 为曲面 $\mathbb{S}$ 的外法向量，如图 3.4 所示，且 $\widetilde\nabla\underline{{F}}=\mathrm{curl}(\underline{{F}})$. 如果力是保守力，则对于任意曲面 $\mathbb{S}$，表面积分必须为零，这只能在积分表达式为零时发生，从而导致 $\widetilde\nabla\underline{{F}}=0$ 对于任意曲线 $\mathbb{C}$ 和曲面 $\mathbb{S}$。 矢量代数教材 [2] 证明了以下恒等式：$\widetilde\nabla\underline{{\nabla}}V=0$ ，其中 $V$ 是任意标量函数，且 $\Sigma V\,=\,\operatorname{grad}(V)$。 随后可以证明，方程 $\widetilde\nabla\underline{{F}}\,=\,0$ 的解是简单的
 
 $$
 \underline{{F}}=-\underline{{\nabla}}V,
 $$  
 
-where $\boldsymbol{\Sigma}$ is the gradient operator.  
+where $\underline{{\nabla}}$ is the gradient operator.  
 
 If a vector field, $\underline{{F}}$ , can be derived from a scalar function, $V$ , this function is called a potential, and the vector function is said to “be derived from a potential.” Because the potential is an arbitrary scalar function, the minus sign is redundant, but is, however, a convention that will be justified later.  
 
 It has now been established that if a force is conservative, it can be “derived from a potential.” In more mathematical terms, a conservative force must be the gradient a scalar function, called the potential of the force. If $\mathcal{T}=(\bar{\iota}_{1},\bar{\iota}_{2},\bar{\iota}_{3})$ is an orthonormal basis, conservative forces can be expressed as  
+如果一个向量场 $\underline{{F}}$ 可以由一个标量函数 $V$ 推导出来，则称该函数为势函数，并且称该向量函数“是由势函数推导出来的”。由于势函数是一个任意标量函数，因此负号是冗余的，但后续会对其进行合理的解释。
+
+目前已经确定，如果一个力是保守力，那么它可以“由势函数推导出来”。用更数学化的语言来说，保守力必须是标量函数（称为该力的势函数）的梯度。如果 $\mathcal{T}=(\bar{\iota}_{1},\bar{\iota}_{2},\bar{\iota}_{3})$ 是一个正交基，那么保守力可以表示为：
+
 
 $$
 \underline{{F}}=-\underline{{\nabla}}V=-\frac{\partial V}{\partial x_{1}}\bar{\iota}_{1}-\frac{\partial V}{\partial x_{2}}\bar{\iota}_{2}-\frac{\partial V}{\partial x_{3}}\bar{\iota}_{3}.
@@ -2918,26 +2924,27 @@ $$
 The work done by a conservative force over an arbitrary path joining point 1 to point 2, with position vectors $\underline{{r}}_{1}$ and $\underline{{r}}_{2}$ , respectively, is then  
 
 $$
-\begin{array}{l}{{W_{1\to2}=\displaystyle\int_{\frac{T_{1}}{\underline{{r}}_{1}}}^{\frac{T_{2}}{\underline{{r}}}}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! 
+\begin{array}{l}{{W_{1\to2}=\displaystyle\int_{\frac{T_{1}}{\underline{{r}}_{1}}}^{\frac{T_{2}}{\underline{{r}}}} 
 $$  
 
 Thus the work done by a conservative force along any path joining point 1 to point 2 depends only on the positions of these points and can be evaluated as the difference between the values of the potential function expressed at these two points,  
-
+因此，保守力沿连接点 1 到点 2 的任意路径所做的功仅取决于这两个点的位置，并且可以评估为在这些点处势函数值的差。
 $$
 W_{1\rightarrow2}=V(\underline{{r}}_{1})-V(\underline{{r}}_{2})=-\varDelta V.
 $$  
 
 If point 1 and 2 are an infinitesimal distance apart,  
-
+如果点 1 和 2 相隔一个无穷小的距离，
 $$
 \mathrm{d}W=V(\underline{{r}}_{1})-V(\underline{{r}}_{1}+\mathrm{d}\underline{{r}})=-\mathrm{d}(V).
 $$  
 
 The differential work is now the true derivative of the potential function.  
-
+现在，增量工作是势函数真正的导数。
-# Summary  
+### Summary  
 
 Conservative forces enjoy a number of remarkable properties. Initially, conservative forces are defined as forces that perform the same work along any path joining the same initial and final points, as expressed by eq. (3.13). Simple calculus reasoning is then used to prove that a force is conservative if and only if the work it performs vanishes over any arbitrary closed path, see eq. (3.14). Finally, conservative forces are shown to be derivable from a potential, as expressed by eq. (3.16). Consequently, the work done by a conservative force along any path joining two points can be evaluated as the difference between the potential function evaluated at these two points, see eq. (3.18).  
+保守力具有一些显著的特性。首先，保守力被定义为连接相同始点和终点的任意路径上所做的功相同，如公式(3.13)所示。随后，利用简单的微积分推理证明，一个力是保守力，当且仅当它在任意闭合路径上所做的功为零，参见公式(3.14)。最后，证明保守力可以从势函数导出，如公式(3.16)所示。因此，保守力沿连接两个点的任意路径所做的功，可以评估为在这些点处势函数之差，参见公式(3.18)。
 
 # Examples of conservative forces  
 
@@ -2950,7 +2957,17 @@ As another example, consider the restoring force of an elastic spring of stiffne
 Quantity $V(u)$ is called the strain energy and it can be viewed as a “potential of the elastic forces” in the spring. Hence, the strain energy function implicitly defines the constitutive behavior of the component. Finally, the work done by the elastic restoring force as the spring stretches from $u_{a}$ to $u_{b}$ is $\begin{array}{r}{W\,=\,\int_{u_{a}}^{u_{b}}F_{s}\;\mathrm{d}u\,=\,}\end{array}$ $\begin{array}{r}{-\int_{u_{a}}^{u_{b}}\partial V/\partial u~\mathrm{d}u\,=\,V(u_{a})\,-\,V(u_{b})}\end{array}$ . Here again, the work depends only on the initial and final positions.  
 
 At first glance, the potential of a gravity force and the strain energy of an elastic spring seem to be distinct, unrelated concepts. Both quantities, however, share a common property: forces can be derived from these scalar potentials. Consider a particle of mass $m$ connected to an elastic spring of stiffness constant $k$ and subjected to a gravity force acting in the direction of the spring. The downward displacement, $u$ , of the mass measures both the spring stretch and the elevation of the particle. The externally applied gravity force can be derived from the potential, $V\,=\,m g u$ , as $F_{g}\,=\,-\partial V/\partial u\,=\,-m g$ ; the restoring force in the spring can be derived from the strain energy, $V=1/2\,k u^{2}$ , which can also be viewed as the potential of the internal forces, as $F_{s}=-\partial V/\partial u=-k u.$ . The two forces acting on the particle can therefore be derived from a potential.  
+为了阐述这些概念，考虑地球表面上质量为 $m$ 的粒子的重力。很容易证明这种力是保守力。因此，重力势能 $V$ 为 $V=m g\,\underline{{{r}}}{\cdot}\bar{\imath}_{3}=m g x_{3}$ ，其中 ${\underline{{r}}}=x_{1}{\bar{\imath}}_{1}+x_{2}{\bar{\imath}}_{2}+x_{3}{\bar{\imath}}_{3}$ 是粒子的位置矢量。作用在粒子上的重力，$\underline{{F}}_{g}$ ，可以通过使用公式 (3.17) 从这个势能中获得，得到 $\underline{{\tilde{F_{g}}}}=-\nabla V=-\partial V/\partial x_{3}\,\bar{\iota}_{3}=-m g\bar{\iota}_{3}$ ，并且重力被认为是“从势能中推导出来的”。
 
+当粒子从高度 $x_{3a}$ 移动到 $x_{3b}$ 时，重力所做的功变为 $\begin{array}{r}{W=\int_{x_{3a}}^{x_{3b}}\underline{{F}}_{g}\cdot\mathrm{d}\underline{{r}}=-\int_{x_{3a}}^{x_{3b}}\partial V/\partial x_{3}\;\mathrm{d}x_{3}=V(x_{3a})-V(x_{3b})}\end{array}$ 。显然，这项功仅取决于初始和最终高度，而不取决于粒子在初始高度到最终高度之间移动所遵循的特定路径。如果粒子沿一条起始和结束于同一高度的闭合路径移动，则重力所做的功为零。
+
+作为另一个例子，考虑弹性弹簧的回复力，其刚度常数为 $k$ 。如果弹簧被拉伸 $u$ 的量，回复力为 $-k u$ ，并且可以从形式为 $V(u)\,=\,1/2\,\,k u^{2}$ 的势能中推导出来。事实上，使用公式 (3.17)，弹簧中的弹性力变为 $F_{s}\;=\;-\partial V/\partial u\;=\;-k u$ 。这种关系是弹簧的本构律，因为它将弹簧中的力与弹簧的伸长量相关联。
+
+量 $V(u)$ 被称为应变能，可以将其视为弹簧中的“弹性力的势能”。因此，应变能函数隐式地定义了组件的本构行为。最后，当弹簧从 $u_{a}$ 伸长到 $u_{b}$ 时，弹性回复力所做的功为 $\begin{array}{r}{W\,=\,\int_{u_{a}}^{u_{b}}F_{s}\;\mathrm{d}u\,=\,}\end{array}$ $\begin{array}{r}{-\int_{u_{a}}^{u_{b}}\partial V/\partial u~\mathrm{d}u\,=\,V(u_{a})\,-\,V(u_{b})}\end{array}$ 。这里再次，这项功仅取决于初始和最终位置。
+
+乍一看，重力势能和弹性弹簧的应变能似乎是截然不同、无关的概念。然而，这两个量都共享一个共同的属性：力可以从这些标量势能中推导出来。考虑一个质量为 $m$ 的粒子，该粒子连接到刚度常数为 $k$ 的弹性弹簧，并且受到一个在弹簧方向上的重力作用。粒子的向下位移 $u$ 测量了弹簧的伸长量和粒子的高度。施加的外部重力可以从势能 $V\,=\,m g u$ 中推导出来，得到 $F_{g}\,=\,-\partial V/\partial u\,=\,-m g$；弹簧中的回复力可以从应变能 $V=1/2\,k u^{2}$ 中推导出来，后者也可以被视为内部力的势能，得到 $F_{s}=-\partial V/\partial u=-k u.$ 。因此，作用在粒子上的两个力可以从势能中推导出来。
+
+Blade翻译为叶片，flapwise翻译为挥舞，edgewise翻译为摆振，pitch angle翻译成变桨角度，twist angle翻译为扭角，rotor翻译为风轮，span翻译为展向。
 # 3.2.1 Principle of conservation of energy  
 
 The forces applied to a particle can be divided into two categories: the conservative forces, which can be derived from a potential, and the non-conservative forces, for which no potential function exists. The principle of work and energy, eq. (3.11), now becomes  

学术讲座-交流-面试/Bladed交流.md

@@ -1,4 +1,8 @@
 
+
+
+
+
 问题
 1 steady operational loads 
 叶片变形如何稳态求解的？
@@ -8,5 +12,75 @@
 BEM在90°攻角如何计算的？
 
 3 分段模态有什么用，与单段模态相比，有没有相应的数据支持？
+考虑叶片的非线性变形，把整个叶片看成一个整体的话，只能考虑线性变形，分段后可以考虑非线性变形
+
+$VE 文件 coupled mode uncoupled mode 计算变形时依旧使用组合后的模态计算?
+
+4 中性轴的概念，定义，跟主轴的关系，station中性轴xy坐标如何得到？
+主轴 中性轴 弹性轴都是一个东西
+
+5 6 $\times$ 6 刚度矩阵影响模态，影响载荷
+
+
+Bladed 5 今年9月份发布
+
+bladed使用fvm算出来的载荷比bem小，能小17%
+fvm诱导速度波动更大，随着自由风速趋势变化，更加精确？
+推荐用于极限载荷工况
+
+坎贝尔图结果变得更易于理解
+显示结构模块，耦合模态的动画效果显示出来
+改进了模态的命名
+
+22mw yaml给出了参考轴，每个轴上的刚度矩阵
+windio_to_bladed.py
+
+积分算法 newmark-β 隐式积分算法比较快
+
+问题
+1 地震
+地震的谱生成加速度，得到地震的文件，频谱是拿过来的，bladed把谱生成时序加速度数据。
+
+2 舱室破损、叶片断裂、锚链断裂
+external loads dll 可以把自己的模型加载进来
+叶片断裂 v4不行，bladed5有一个试验模型，只能从叶根地方断裂
+锚链断裂 v4不行，moordyn-link某个时刻不施加锚链拉力
+
+3 多风轮机组
+4.14已经支持，rotor里选2个，目前两个风轮只能相同，可以用两个控制器控制它，
+
+4、 6 6 刚度矩阵
+更符合结构分析的流程，有限元模型
+windio叶片格式，
+从叶片刚度矩阵，了解基于哪些假设，耦合项可以定义在bladed里，结构模型更精确
+bladed的所有参数都是基于弹性轴，
+
+5、分段模态有什么用，与单段模态相比，有没有相应的数据支持？
+考虑叶片的非线性变形，把整个叶片看成一个整体的话，只能考虑线性变形，分段后可以考虑非线性变形
+
+收敛性试验，dlc工况里比较危险的工况挑出来算一算，作为收敛性评判的依据
+段上模态叠加，段之间可以传递变形量
+
+$VE 文件 coupled mode uncoupled mode 计算变形时依旧使用组合后的模态计算?
+
+
+中性轴的概念，定义，跟主轴的关系，station中性轴xy坐标如何得到？
+主轴 中性轴 弹性轴都是一个东西
+中性轴xy，基于root坐标系，定义弹性中心的位置
+弹性中心xy，基于前后缘位置定义弹性中心
+
+载荷，主弹性轴 z沿着中性轴，xy方向沿着，弹性轴方向，原点在弹性中心
+叶根坐标系 xyz都与叶根坐标系相同，原点在弹性中心
+
+1 steady operational loads 
+叶片变形如何稳态求解的？
+变桨角度、转速如何求解的？
+
+外力 F = k x   刚度矩阵 * 变形量 模态的刚度矩阵  广义刚度
+不用模态？
+
+2 steady parked loads
+BEM在90°攻角如何计算的？
+
+
 
-$VE 文件 coupled mode uncoupled mode