105 lines
17 KiB
Plaintext
105 lines
17 KiB
Plaintext
{
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"nodes":[
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{"id":"2144fa1ef85968d0","type":"text","text":"气弹模型","x":-140,"y":-400,"width":250,"height":60},
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{"id":"bde0f2550987a8cc","type":"text","text":"线性系统","x":-140,"y":-240,"width":250,"height":60},
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{"id":"d7b8720eb69baf7d","type":"text","text":"线性系统方程组以状态空间方程形式表示","x":-265,"y":-40,"width":250,"height":60},
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{"id":"62dcdf23e738af0d","type":"text","text":"$$\n\\underline{{\\mathbf{x}}}=\\mathbf{x}-\\mathbf{x_{0}},\\quad\\underline{{\\mathbf{y}}}=\\mathbf{y}-\\mathbf{y_{0}},\\quad\\mathrm{~and~}\\quad\\underline{{\\mathbf{u}}}=\\mathbf{u}-\\mathbf{u_{0}}\n$$","x":85,"y":80,"width":435,"height":60},
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{"id":"5797579dbe07ce53","type":"text","text":" $\\mathbf{x}$ 是状态向量，代表系统状态；$\\mathbf{u}$ 是系统输入向量；$\\mathbf{y}$ 是系统输出向量。","x":85,"y":180,"width":435,"height":60},
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{"id":"58c8fd4b2b79d5c4","type":"text","text":"归一化向量 $\\underline{{\\mathbf{x}}}、\\underline{{\\mathbf{y}}}$ 和 $\\underline{{\\mathbf{u}}}$ 代表偏离平衡态的量。","x":85,"y":280,"width":435,"height":60},
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{"id":"75f03285489bc502","type":"text","text":"矩阵 $\\mathbf{A}$、$\\mathbf{B}$、$\\mathbf{C}$ 和 $\\mathbf{D}$ 代表这些向量之间的线性化关系","x":85,"y":380,"width":435,"height":60},
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{"id":"172345b7932e4879","type":"text","text":"$$\n\\begin{array}{r}{\\underline{{\\dot{\\mathbf{x}}}}=\\mathbf{A}\\underline{{\\mathbf{x}}}+\\mathbf{B}\\underline{{\\mathbf{u}}}}\\\\ {\\underline{{\\mathbf{y}}}=\\mathbf{C}\\underline{{\\mathbf{x}}}+\\mathbf{D}\\underline{{\\mathbf{u}}}}\\end{array}\n$$","x":178,"y":-60,"width":250,"height":100},
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{"id":"7f80af6dea4a5833","type":"text","text":"$$\n\\begin{array}{r}{\\dot{\\mathbf{x}}=f(t,\\mathbf{x},\\mathbf{u})}\\\\ {\\mathbf{y}=h(t,\\mathbf{x},\\mathbf{u}).}\\end{array}\n$$","x":178,"y":480,"width":250,"height":100},
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{"id":"a1a2472e2f6a21f7","type":"text","text":"弹性动力学 (Elastodynamic)：这些状态代表系统的结构模态。**弹性动力学模态由二阶运动方程控制**。因此，为了在状态空间形式中表示，每个模态由两个状态表示——位移和速度。这还包括主要的风轮刚体自由度。","x":660,"y":160,"width":283,"height":256},
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{"id":"fe565c4907cb3e77","type":"text","text":"状态分为两大类","x":840,"y":-40,"width":250,"height":60},
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{"id":"8d024129778a2f9a","type":"text","text":"气动 (Aerodynamic)：这些主要用于模拟动态失速和动态尾流。**这些状态通常是一阶的**，因为它们与时滞有关。","x":1000,"y":160,"width":250,"height":256},
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{"id":"cfe15ef44610d3d4","type":"text","text":"Bladed线性化分析过程","x":-264,"y":760,"width":250,"height":60,"color":"1"},
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{"id":"f9c41a29bddeb8cc","type":"text","text":"依次取每个工作点","x":120,"y":760,"width":250,"height":60},
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{"id":"fb99badcdb47fb3c","type":"text","text":"找到风电机组的稳态条件，这意味着风轮没有加速，模态变形使得弹性载荷平衡外部载荷。","x":480,"y":730,"width":250,"height":120},
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{"id":"7d322544cd4c0e9e","type":"text","text":"对于每个输入或状态，Bladed然后在平衡点两侧进行一系列幅度**逐渐增加的扰动**。**人为地增加或减少状态或输入的数值，用这些修改后的数值求解系统**，并记录状态导数和输出。扰动的数量和最大扰动幅度可以由用户定义。","x":880,"y":650,"width":250,"height":280},
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{"id":"7798ef385eff4f6c","type":"text","text":"通过对状态导数与所有扰动值及其平衡值之间的关系进行线性回归，来**推导矩阵A、B、C和D的元素**。线性回归的梯度给出元素的值。如果相关系数小于最小相关系数，则认为该关系无效，并给该元素赋予零值。","x":1260,"y":650,"width":250,"height":280},
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{"id":"d528580e06b6b8a1","type":"text","text":"这定义了$\\mathbf{x_{0}},\\,\\mathbf{y_{0}}$和$\\mathbf{u_{0}}$的值，即所有扰动围绕的主要平衡点。","x":465,"y":960,"width":280,"height":90},
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{"id":"bc2d79101ffc2c67","type":"text","text":"- 需移除方位角依赖性，包括风切变、偏航、风轮不平衡等。\n- 需移除无法线性化的物理效应，例如风湍流、黏滞滑动等。","x":-302,"y":180,"width":325,"height":140},
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{"id":"e7f67865a05e0be3","type":"text","text":"多叶片坐标系转化非旋转坐标系","x":-264,"y":1140,"width":250,"height":60},
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{"id":"d9a4691cf95d96c3","type":"text","text":"基于 (Bir, 2008) and (Hansen, 2003)","x":-264,"y":1260,"width":250,"height":60},
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{"id":"8d7c49775076894a","type":"text","text":"旋转坐标系到非旋转坐标系变换关系","x":465,"y":1140,"width":250,"height":60},
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{"id":"081fcfd3683a8ce4","type":"text","text":"$$\n\\begin{align}\n\\begin{bmatrix}\nq_{0} \\\\\nq_{c} \\\\\nq_{s} \\\\\n\\end{bmatrix} &= \n\\frac{1}{3}\\begin{bmatrix}\n1 & 1 & 1 \\\\\n2\\cos\\psi_{1} & 2\\cos\\psi_{2} & 2\\cos\\psi_{3} \\\\\n2\\sin\\psi_{1} & 2\\sin\\psi_{2} & 2\\sin\\psi_{3} \\\\\n\\end{bmatrix}\n\n\\begin{bmatrix}\nq_{1} \\\\\nq_{2} \\\\\nq_{3} \\\\\n\\end{bmatrix},\n\\end{align}\n$$","x":805,"y":1108,"width":400,"height":125},
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{"id":"975926818ccaa396","type":"text","text":"方位角表示：$\\psi_{i}=\\Omega t+\\Psi_{i}$","x":1260,"y":1141,"width":250,"height":59},
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{"id":"c1c28df14fb3bfbb","type":"text","text":"变换矩阵求时间导数","x":1560,"y":1140,"width":250,"height":60},
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{"id":"edd4cdf7e36d0cfc","type":"text","text":"$$\n\\begin{align*}\n\\dot{{t}}_{R \\rightarrow NR} =\n\\frac{\\Omega}{3}\n\\begin{bmatrix}\n0 & 0 & 0 \\\\\n- 2\\sin\\psi_{1} & - 2\\sin\\psi_{2} & - 2\\sin\\psi_{3} \\\\\n2\\cos\\psi_{1} & 2\\cos\\psi_{2} & 2\\cos\\psi_{3} \\\\\n\\end{bmatrix}\n\\end{align*}\n$$","x":1880,"y":1005,"width":368,"height":135},
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{"id":"8622b2d75c26d18b","type":"text","text":"$$\n\\begin{align*}\n\\ddot{{t}}_{R \\rightarrow NR} =\n- \\frac{\\Omega^{2}}{3}\n\\begin{bmatrix}\n0 & 0 & 0 \\\\\n2\\cos\\psi_{1} & 2\\cos\\psi_{2} & 2\\cos\\psi_{3} \\\\\n2\\sin\\psi_{1} & 2\\sin\\psi_{2} & 2\\sin\\psi_{3} \\\\\n\\end{bmatrix}\n\\end{align*}\n$$","x":1880,"y":1200,"width":368,"height":137},
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{"id":"90ebf789028488ba","type":"text","text":"方位角带入到变换矩阵中了","x":1555,"y":1269,"width":260,"height":60},
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{"id":"2e0560ed3e0b3d1e","type":"text","text":"系统变换矩阵","x":-264,"y":1440,"width":250,"height":60},
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{"id":"9e5fe38ea39cb6a4","type":"text","text":"$$\n{\\bf q}_{N R}={\\bf t}_{R\\rightarrow N R}{\\bf q}_{R}\n$$","x":880,"y":1260,"width":250,"height":60},
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{"id":"b4199d2560e40041","type":"text","text":"a common transformation matrix $\\mathbf{T}$","x":880,"y":1440,"width":250,"height":60},
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{"id":"6980197894c4c44e","type":"text","text":"$$\n\\begin{array}{r}\n{\\bf q}_{N R}={\\bf t}_{R\\rightarrow N R}{\\bf q}_{R}\\\\\n\\dot{\\bf q}_{N R}={\\bf t}_{R\\rightarrow N R}\\dot{\\bf q}_{R}+\\dot{\\bf t}_{R\\rightarrow N R}{\\bf q}_{R}\n\\end{array}\n$$","x":408,"y":1420,"width":395,"height":100},
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{"id":"d2862b9b5b4f20bd","type":"text","text":"$$\n\\begin{align}\n\n{T} &:= \\begin{bmatrix}\n\n{t}_{R \\rightarrow NR} & 0 \\\\\n\n\\dot{{t}}_{R \\rightarrow NR} & {t}_{R \\rightarrow NR} \\\\\n\n\\end{bmatrix}\n\n\\end{align}\n$$","x":1205,"y":1420,"width":250,"height":100},
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{"id":"3bc2446646bd3d8b","type":"text","text":"$$\n\\begin{align}\n\n\\dot{{T}}=\n\n\\begin{bmatrix}\n\n\\dot{{t}}_{R \\rightarrow NR} & 0 \\\\\n\n\\ddot{{t}}_{R \\rightarrow NR} & \\dot{{t}}_{R \\rightarrow NR} \\\\\n\n\\end{bmatrix}.\n\n\\end{align}\n$$","x":1555,"y":1420,"width":250,"height":100},
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{"id":"eaa37292565b4f12","type":"text","text":"$$\n\\begin{align}\n\\begin{bmatrix}\n q_{NR} \\\\\n \\dot{q}_{NR}\n\\end{bmatrix}\n&=\n\\begin{bmatrix}\n t_{R \\rightarrow NR} & 0 \\\\\n \\dot{t}_{R \\rightarrow NR} & t_{R \\rightarrow NR}\n\\end{bmatrix}\n\\begin{bmatrix}\n q_{R} \\\\\n \\dot{q}_{R}\n\\end{bmatrix}.\n\\end{align}\n$$","x":1880,"y":1420,"width":286,"height":100},
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{"id":"9165677a568e0e8c","type":"text","text":"计算耦合模态","x":-264,"y":1800,"width":250,"height":60},
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{"id":"1d1483f33284f3fb","type":"text","text":"一个包含位移和速度状态的整个状态列表的变换矩阵","x":80,"y":1420,"width":250,"height":100},
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{"id":"9152c2d34d6164f1","type":"text","text":"坎贝尔图和叶片稳定性分析都是对**指定工作点的矩阵A**的分析","x":80,"y":1780,"width":250,"height":100},
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{"id":"67680eb011c14d65","type":"text","text":"得到基于非旋转坐标系的模态","x":80,"y":1140,"width":250,"height":60},
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{"id":"fad223ea1c18fca6","type":"text","text":"三叶片模型消除方位角的影响","x":80,"y":1260,"width":250,"height":60},
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{"id":"1f219abcb5ce249f","type":"text","text":"**每个耦合模态对应一个特征值及其特征向量**","x":465,"y":1800,"width":250,"height":60},
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{"id":"6115769fc1310020","type":"text","text":"**给定矩阵 A 的一个（复数）特征值 λ**","x":880,"y":1800,"width":250,"height":60},
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{"id":"2203401c5015c9b8","type":"text","text":"Bladed 会根据Argand Diagram得到无阻尼频率、阻尼频率、阻尼比","x":1205,"y":1780,"width":250,"height":100},
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{"id":"57346d11b4b1a3d0","type":"text","text":"考虑旋转坐标系下的线性模型方程，定义$\\begin{align}{x}_{R} :=\\begin{bmatrix}{q}_{R} \\\\ \\dot{{q}}_{R}\\end{bmatrix}\\end{align}$","x":1205,"y":1600,"width":305,"height":100},
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{"id":"a680b9a9a3275ad0","type":"text","text":"矩阵A从旋转坐标系转换到非旋转坐标系","x":880,"y":1620,"width":250,"height":60},
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{"id":"c644b36ef5f053f2","type":"text","text":"$$\n\\begin{array}{r}{\\dot{\\mathbf{x}}_{R}=\\mathbf{A}_{R}\\mathbf{x}_{R}+\\mathbf{B}_{R}\\mathbf{u}}\\\\ {\\mathbf{y}=\\mathbf{C}_{R}\\mathbf{x}_{R}+\\mathbf{D}_{R}\\mathbf{u}}\\end{array}\n$$","x":1555,"y":1595,"width":250,"height":110},
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{"id":"32a0144ad0800646","type":"text","text":"状态向量从旋转坐标系变换到非旋转坐标系，$\\mathbf{x}_{N R}=\\mathbf{T}\\mathbf{x}_{R}$","x":1880,"y":1593,"width":250,"height":115},
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{"id":"00f636edd7d9bb41","type":"text","text":"$$\n\\dot{{\\bf x}}_{N R}={\\bf T}\\dot{\\bf x}_{R}+\\dot{\\bf T}{\\bf x}_{R}.\n$$","x":2200,"y":1607,"width":250,"height":88},
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{"id":"cdda60f3f9d6cdf7","type":"text","text":"$$\n\\begin{array}{r l}&{\\dot{\\mathbf{x}}_{N R}=\\mathbf{T}\\left(\\mathbf{A}_{R}\\mathbf{x}_{R}+\\mathbf{B}_{R}\\mathbf{u}\\right)+\\dot{\\mathbf{T}}\\mathbf{x}_{R}}\\\\ &{\\qquad=\\Big(\\mathbf{T}\\mathbf{A}_{R}+\\dot{\\mathbf{T}}\\Big)\\mathbf{x}_{R}+\\mathbf{T}\\mathbf{B}_{R}\\mathbf{u}}\\\\ &{\\qquad=\\Big(\\mathbf{T}\\mathbf{A}_{R}+\\dot{\\mathbf{T}}\\Big)\\mathbf{T}^{-1}\\mathbf{x}_{N R}+\\mathbf{T}\\mathbf{B}_{R}\\mathbf{u}}\\end{array}\n$$","x":2520,"y":1574,"width":334,"height":154},
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{"id":"74b2f2439c0340f6","type":"text","text":"$$\n\\mathbf{C}_{N R}:=\\mathbf{C}_{R}\\mathbf{T}^{-1},\\quad\\mathrm{~and~}\\mathbf{D}_{N R}:=\\mathbf{D}_{R}\n$$","x":2940,"y":1678,"width":317,"height":82},
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{"id":"e707ffb73e63911e","type":"text","text":"$$\n\\begin{array}{r l}&{\\mathbf{A}_{N R}=\\Big(\\mathbf{TA}_{R}+\\dot{\\mathbf{T}}\\Big)\\mathbf{T}^{-1}}\\\\ &{\\mathbf{B}_{N R}=\\mathbf{TB}_{R}}\\end{array}\n$$","x":2940,"y":1525,"width":250,"height":126},
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{"id":"e1785c07b66dee47","type":"text","text":"命名，连接","x":-264,"y":2000,"width":250,"height":60},
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{"id":"5d7b2c4818b9df8a","type":"text","text":"耦合模态如何由未耦合模态得到","x":1560,"y":1840,"width":250,"height":60,"color":"2"},
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{"id":"849dc73be675038b","type":"text","text":"跟耦合、未耦合模态有什么关系","x":1560,"y":1760,"width":250,"height":60,"color":"2"}
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