fixing Ct description to constant

586f49aa · Jenni Rinker · 3b8ff546 · 586f49aa
Commit 586f49aa authored 3 years ago by Jenni Rinker
--- a/docs/source/turbie/1-definition.ipynb
+++ b/docs/source/turbie/1-definition.ipynb
@@ -105,21 +105,21 @@
    "\n",
    "A wind turbine is, as you might expect, forced by the wind. To accurately model aerodynamics, you should include extra time-dependent variables to include phenomena such as dynamic stall, tower shadow, variable turbine speed, etc. For simplicity, we make the following assumptions:  \n",
    "* No dynamic inflow, dynamic stall, or tower shadow;  \n",
-    "* Turbie's thrust coefficient can be directly calculated from the current wind speed;  \n",
+    "* Turbie's thrust coefficient is constant for a simulation and can be calculated from the mean wind speed;  \n",
    "* The only aerodynamic forcing is on the blades;  \n",
    "* No spatial variation of turbulence.\n",
    "\n",
    "With these assumptions, the aerodynamic forcing on the blades is given by  \n",
    "\\begin{equation}\n",
-    "    f_{aero}(t) = \\frac12\\, \\rho\\, C_T(t)\\, A\\, (u(t) - \\dot{x}_1)\\,|u(t) - \\dot{x}_1|,\n",
+    "    f_{aero}(t) = \\frac12\\, \\rho\\, C_T\\, A\\, (u(t) - \\dot{x}_1)\\,|u(t) - \\dot{x}_1|,\n",
    "\\end{equation}\n",
-    "where $A$ is the rotor area and $u(t)$ is the wind speed at time $t$. The thrust coefficient $C_T(t)$ is determined from $u(t)$ using the look-up table defined above.\n",
+    "where $A$ is the rotor area and $u(t)$ is the wind speed at time $t$. The thrust coefficient $C_T$ is determined from the mean wind speed $U = \\overline{u(t)}$ using the look-up table defined above.\n",
    "\n",
    "The full dynamical equations for Turbie are then given by\n",
    "\\begin{equation}\n",
    "[M]\\ddot{\\overline{x}}(t) + [C]\\dot{\\overline{x}}(t) + [K]\\overline{x}(t) = \\overline{F}(t)\n",
    "\\end{equation}\n",
-    "where $\\overline{x}(t)=[x_1(t), x_2(t)]^T$ is the state vector and the forcing vector is given by\n",
+    "where $\\overline{x}(t)=[x_1(t), x_2(t)]^T$ is the state vector and the forcing vector is given by  \n",
    "\\begin{equation}\n",
    "\\overline{F}(t) = \\left[ \\begin{array}{c} f_{aero}(t)  \\\\ 0\\end{array} \\right].\n",
    "\\end{equation}"

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 # Definition

 Turbie is a simple, two-degree-of-freedom (2DOF) system based of the DTU 10 MW Reference Wind Turbine. She is equivalent to the forced 2DOF mass-spring-damper system shown below, which is defined by a mass matrix, stiffness matrix, and damping matrix.

 <img src="figures/1-diagram.png" alt="Diagram of Turbie" style="width: 600px;"/>

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 Here are two files that you can download with parameters for Turbie: :download:`download turbie_parameters.txt <turbie_parameters.txt>` and :download:`download CT.txt <CT.txt>`

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 ## Mass, stiffness and damping matrices

 To derive Turbie's mass, stiffness and damping matrices, we make a few assumptions about our dynamical system:
 * The turbine can only move in the fore-aft direction;
 * The 3 blade deflections in the fore-aft direction are syncronized, i.e., only collective flapwise deflections.

 With these assumptions, we have reduced Turbie to two degrees of freedom: the deflection of the blades in the fore-aft direction and the deflection of the nacelle in the fore-aft direction. The 2 DOFs of the system are therefore defined as
 * $x_1(t)$: the deflection of the blades from their undeflected position in the global coordinate system;
 * $x_2(t)$: the deflection of the nacelle from its undeflected position in the global coordinate system.

 With these DOFs, Turbie is equivalent to a 2DOF mass-spring-damper system, as shown above, where Mass 1 represents the 3 blades and Mass 2 represents the combined effects of the nacelle, hub and tower.

 **Exercise for the reader!** Given the diagram of the 2DOF mass, spring and damper system, derive the equations of motion and give the mass, stiffness and damping matrices.

 **Answer**.
 The 2DOF mass-spring-damper system has the following system matrices:
 \begin{equation}
 [M] = \left[\begin{array}{cc}m_1 & 0\\0 & m_2\end{array} \right]
 \end{equation}
 \begin{equation}
 [C] = \left[\begin{array}{cc}c_1 & -c_1\\-c_1 & c_1+c_2\end{array} \right]
 \end{equation}
 \begin{equation}
 [K] = \left[\begin{array}{cc}k_1 & -k_1\\-k_1 & k_1+k_2\end{array} \right]
 \end{equation}

 Here is a table of relevant parameter values for Turbie,

 Symbol |Decription |Value
 -----|-----|-----
 $m_b$|Mass of a single blade|41 metric tons
 $m_n$|Mass of the nacelle|446 metric tons
 $m_t$|Mass of the tower|628 metric tons
 $m_h$|Mass of the hub|105 metric tons
 $c_1$|Equivalent damping of the blades|4208 N/(m/s)
 $c_2$|Equivalent damping of the nacelle, hub and tower|12730 N/(m/s)
 $k_1$|Equivalent stiffness of the blades|1711000 N/m
 $k_2$|Equivalent stiffness of the nacelle, hub and tower|3278000 N/m
 $D_{rotor}$|Rotor diameter|180 m
 $\rho$|Air density|1.22 kg/m^3

 and here is the thrust-coefficient look-up table:

 Wind speed (m/s) | CT (-)
 ------|-----
 4.0  | 0.923
 5.0  | 0.919
 6.0  | 0.904
 7.0  | 0.858
 8.0  | 0.814
 9.0  | 0.814
 10.0 | 0.814
 11.0 | 0.814
 12.0 | 0.577
 13.0 | 0.419
 14.0 | 0.323
 15.0 | 0.259
 16.0 | 0.211
 17.0 | 0.175
 18.0 | 0.148
 19.0 | 0.126
 20.0 | 0.109
 21.0 | 0.095
 22.0 | 0.084
 23.0 | 0.074
 24.0 | 0.066
 25.0 | 0.059

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 ## Dynamical equations

 A wind turbine is, as you might expect, forced by the wind. To accurately model aerodynamics, you should include extra time-dependent variables to include phenomena such as dynamic stall, tower shadow, variable turbine speed, etc. For simplicity, we make the following assumptions:
 * No dynamic inflow, dynamic stall, or tower shadow;
-* Turbie's thrust coefficient can be directly calculated from the current wind speed;
+* Turbie's thrust coefficient is constant for a simulation and can be calculated from the mean wind speed;
 * The only aerodynamic forcing is on the blades;
 * No spatial variation of turbulence.

 With these assumptions, the aerodynamic forcing on the blades is given by
 \begin{equation}
-    f_{aero}(t) = \frac12\, \rho\, C_T(t)\, A\, (u(t) - \dot{x}_1)\,|u(t) - \dot{x}_1|,
+    f_{aero}(t) = \frac12\, \rho\, C_T\, A\, (u(t) - \dot{x}_1)\,|u(t) - \dot{x}_1|,
 \end{equation}
-where $A$ is the rotor area and $u(t)$ is the wind speed at time $t$. The thrust coefficient $C_T(t)$ is determined from $u(t)$ using the look-up table defined above.
+where $A$ is the rotor area and $u(t)$ is the wind speed at time $t$. The thrust coefficient $C_T$ is determined from the mean wind speed $U = \overline{u(t)}$ using the look-up table defined above.

 The full dynamical equations for Turbie are then given by
 \begin{equation}
 [M]\ddot{\overline{x}}(t) + [C]\dot{\overline{x}}(t) + [K]\overline{x}(t) = \overline{F}(t)
 \end{equation}
 where $\overline{x}(t)=[x_1(t), x_2(t)]^T$ is the state vector and the forcing vector is given by
 \begin{equation}
 \overline{F}(t) = \left[ \begin{array}{c} f_{aero}(t)  \\ 0\end{array} \right].
 \end{equation}

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