{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "The following is a direct adaptation of an example given in my own undergraduate notes - reproduced with permission. Thanks, Dougie.\n", "\n", "# Comparison of the Linear and Nonlinear Equations of Motion: Aircraft Simulation\n", "\n", "The nonlinear equations of motion have been derived and will be repeated here:\n", "\n", "$$m\\begin{bmatrix} \\dot{U} + QW - UR \\\\ \\dot{V} + RU-PW\\\\\\dot{W}+PV-QU\\end{bmatrix}=\\begin{matrix} -mg\\sin\\Theta - D\\cos\\alpha + L\\sin\\alpha + T\\cos\\Theta_T\\\\mg\\sin\\Phi\\cos\\Theta + F_{A_Y} + F_{T_Y}\\\\mg\\cos\\Phi\\cos\\Theta - D\\sin\\alpha - L\\cos\\alpha - T\\sin\\Theta_T\\end{matrix}$$ \n", "\n", "$$\\left[{\\begin{matrix} \\dot{P}\\cdot I_{xx} \\\\ \\dot{Q}\\cdot I_{yy} \\\\\\dot{R}\\cdot I_{zz}\\end{matrix}} \\hspace{.5cm} \\begin{matrix} + \\\\+\\\\+\\end{matrix} \\hspace{.5cm} {\\begin{matrix} Q\\cdot r\\left(I_{zz} - I_{yy}\\right) \\\\ P\\cdot R\\left(I_{xx} - I_{yy}\\right) \\\\ P\\cdot Q\\left(I_{yy} - I_{xx}\\right)\\end{matrix}} \\hspace{.5cm} \\begin{matrix} - \\\\+\\\\+\\end{matrix} \\hspace{.5cm} {\\begin{matrix}\\left(\\dot{R} + P\\cdot Q\\right) I_{xz} \\\\ \\left(P^2 - R^2\\right) I_{xz} \\\\ \\left(Q\\cdot R - \\dot{P}\\right) I_{xz}\\end{matrix}} \\right] = \\begin{bmatrix} L \\\\ M \\\\ N\\end{bmatrix}$$ \n", "\n", "$$\\begin{aligned}\n", " \\begin{bmatrix} P\\\\Q\\\\R\\end{bmatrix} &= \\begin{bmatrix}\n", " 1 & 0 & -\\sin\\Theta\\\\\n", " 0 & \\cos\\Phi & \\sin\\Phi\\cos\\Theta\\\\\n", " 0 & -\\sin\\Phi & \\cos\\Phi\\cos\\Theta\n", " \\end{bmatrix}\\begin{bmatrix} \\dot{\\Phi}\\\\\\dot{\\Theta}\\\\\\dot{\\Psi}\\end{bmatrix}\\end{aligned}$$ \n", " \n", " $$\\begin{aligned}\n", "\\begin{bmatrix} \\dot{\\Phi}\\\\\\dot{\\Theta}\\\\\\dot{\\Psi}\\end{bmatrix}&= \\begin{bmatrix}\n", " 1 & \\sin\\Phi\\tan\\Theta & \\cos\\Phi\\tan\\Theta\\\\\n", " 0 & \\cos\\Phi & -\\sin\\Phi \\\\\n", " 0 & \\frac{\\sin\\Phi}{\\cos\\Theta} & \\frac{\\cos\\Phi}{\\cos\\Theta}\n", " \\end{bmatrix}\\begin{bmatrix} P\\\\Q\\\\R\\end{bmatrix}\\end{aligned}$$\n", "\n", "a simulation of this aircraft in the longitudinal plane will be presented in this section. By constraining to longitudinal motion only, the solution of the nonlinear equations is heavily simplified, but the analysis can be expanded for the unconstrained case and if we assume the thrust inclination is zero. The simplifying assumption is: \n", "\n", "$$V=P=R=\\Phi=\\Psi=\\Theta_0 = 0$$ \n", "\n", "and the equations describing longitudinal flight may now be we written: \n", "\n", "$$\\begin{aligned}\n", " m\\left(\\dot{U} + Q\\,W\\right) &= T - D\\cos\\alpha + L\\cos\\alpha - mg\\,\\sin\\Theta\\\\\n", " m\\left(\\dot{W} - Q\\,U\\right) &= -L\\cos\\alpha - D\\sin\\alpha+ mg\\,\\cos\\Theta\\\\\n", " I_{yy}\\,\\dot{Q} &= M_A - T\\,\\Delta_{zT}\\\\\n", " \\dot{\\Theta}&=Q\\end{aligned}$$\n", " \n", "## HS125 (Hawker 800) Business Jet)\n", "\n", "```{image} /Images/Hawker800.jpg\n", ":alt: Hawker800\n", ":class: bg-primary\n", ":width: 300px\n", ":align: center\n", "```\n", "\n", "The data for the HS125 Business jet are given in the table below. The aerodynamic model gives the lift, drag, and pitching moment coefficients as:\n", "\n", "$$\\begin{aligned}\n", " C_L &= C_{L_0} + C_{L_\\alpha}\\,\\alpha + C_{L_{\\delta_e}}\\,\\delta_e\\\\\n", " C_D &= C_{D_0} + C_{D_\\alpha}\\,\\alpha + C_{D_{\\alpha^2}}\\,\\alpha^2\\\\\n", " C_M &= C_{M_0} + C_{M_\\alpha}\\,\\alpha + C_{M_{\\delta_e}}\\,\\delta_e + C_{M_q}\\,\\hat{q}\\\\\n", " \\hat{q}&\\triangleq \\frac{Q\\,\\bar{c}}{V_f}\\end{aligned}$$\n", "\n", "\n", "\n", "\n", "| **Parameter** | **Symbol** | **Value** |\n", "|:--------------------------------:|:-----------------------------------------------------:|:-----------------------------:|\n", "| Mass | $m$ | 7500kg |\n", "| Wing Area | $S$ | 32.8$\\text{m}^2$ |\n", "| Pitching Moment of Inertia | $I_{yy}$ | 84,309$\\text{kg m}^2$ |\n", "| MAC | $\\bar{c}$ | 2.29m |\n", "| Thrustline vertical displacement | $\\Delta_{zT}$ | -0.378m |\n", "| Lift Coefficients | $C_{L_0},\\, C_{L_\\alpha},\\, C_{L_{\\delta_e}}$ | 0.895, 5.01, 0.722 |\n", "| Drag Coefficients | $C_{D_0},\\, C_{D_\\alpha},\\, C_{D_{\\alpha^2}}$ | 0.177, 0.232, 1.393 |\n", "| PM Coefficients | $C_{M_0},\\, C_{M_\\alpha},\\, C_{M_{\\alpha^2},\\, C_{M_q}}$| -0.046, -1.087, -1.88, -7.055 |\n", "\n", "The reduced nonlinear equations of motion can be constructed into a function `eqnofmotion800`:" ] }, { "cell_type": "code", "execution_count": 163, "metadata": {}, "outputs": [], "source": [ "def eqnofmotion800(y, t=[], Thrust=13878):\n", " # The reduced state vector is a 7 x 1 vector\n", " # y = [U, W, Q, Theta, X_E, Z_E, dE]\n", "\n", " \n", " # Aircraft Parameters (all in SI)\n", " m=7484.4 # mass = 7484.4kg etc.\n", " s=32.8 #\n", " Iyy=84309\n", " cbar=2.29 \n", " dZt=-0.378\n", " rho=1.225 \n", " CD0=0.177 \n", " CDa=0.232 \n", " CDa2= 1.393\n", " CL0=0.895\n", " CLa=5.01 \n", " CLde=0.722\n", " CM0=-0.046 \n", " CMa=-1.087 \n", " CMde=-1.88 \n", " CMq=-7.055 \n", "\n", " \n", " # Get the current values of aircraft states\n", " dE = y[6] # Elevator input\n", "\n", " U=y[0] # Forward speed\n", " W=y[1] # Heave velocity\n", " Q=y[2] # Pitch rate\n", " Theta=y[3] # Pitch attitude\n", " \n", " # Determine some aerodynamic terms\n", " Vf=np.sqrt(U**2 + W**2) # Flight speed\n", " alpha = np.arctan(W/U) # Angle of attack\n", " qh = Q*cbar/Vf # Nondimensional pitch rate\n", " \n", " # Aerodynamic coefficients from Hawker model:\n", " CL = CL0 + CLa*alpha + CLde*dE\n", " CD = CD0 + CDa*alpha + CDa2*alpha*alpha\n", " CM = CM0 + CMa*alpha + CMde * dE + CMq*qh\n", "\n", " q_inf = .5*rho*Vf**2*s\n", " \n", " # Dimensional aero terms\n", " lift = q_inf*CL\n", " drag = q_inf*CD\n", " pm = q_inf*cbar*CM\n", "\n", " # Simplified nonlinear equations of motion\n", " ydot = np.zeros(7, dtype='float64')\n", " ydot[0] = -Q*W + (lift*np.sin(alpha) - drag*np.cos(alpha) + Thrust)/m - 9.80665*np.sin(Theta) # Udot\n", " ydot[1] = Q*U + (-lift*np.cos(alpha) - drag*np.sin(alpha))/m + 9.80665*np.cos(Theta) # Wdot\n", " ydot[2] = (pm - Thrust*dZt)/Iyy # M\n", " ydot[3] = Q # ThetaDot\n", " ydot[4] = U*np.cos(Theta) - W*np.sin(Theta) # X_Edot (earth axes)\n", " ydot[5] = U*np.sin(Theta) + W*np.cos(Theta) # Z_Edot (earth axes)\n", " ydot[6] = 0 # No change to elevator\n", "\n", " return ydot\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To determine the transient aircraft response, first the trim state must be determined.\n", "\n", "## Trim State Determination\n", "\n", "By definition, the trim case is found by setting the accelerations and angular velocities to zero. Hence the equations of motion become:\n", "\n", "$$\\begin{aligned}\n", " 0 &= T - D\\cos\\alpha + L\\cos\\alpha - mg\\,\\sin\\Theta\\\\\n", " 0 &= -L\\cos\\alpha - D\\sin\\alpha+ mg\\,\\cos\\Theta\\\\\n", " 0 &= M_A - T\\Delta_{zT}\\end{aligned}$$\n", " \n", "The knowns in the above are flightspeed, $V_f$, density (from altitude), $\\rho$, and climb angle/flight path angle $\\gamma$ (setting $U_E$ and $W_E$). These may be solved via any means you like to get the reference trim state. An example of a Newton-Raphson solver[^4] is included below as `TrimState` which finds trim using `TotalForces`:\n", "\n", "[^4]: You will not be tested on things like this in an examination in this course, but it's really in your interest to ensure that you can a) understand how this solver works and b) be confident you could write something that does the same job yourself." ] }, { "cell_type": "code", "execution_count": 106, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Converged after 1390 iterations\n", "For inputs of Vf = 61.73m/s, h = 0.00km, gamma = 0.00deg\n", "\n", "Thrust = 13.84kN, de = -0.98deg, Theta = 0.84deg\n", "\n" ] } ], "source": [ "from ambiance import Atmosphere\n", "import numpy as np\n", "\n", "def TotalForces(T, de, Theta, Vf, h, gamma):\n", " '''This gives the total forces for the HS 125 simplified longitudinal\n", " aircraft\n", " Inputs: Thrust/N, elevator deflection/deg, theta/rad, flightspeed/m/s,\n", " flightpath/rad\n", " altitude/m\n", " '''\n", "\n", "\n", " # Aircraft Parameters (all in SI)\n", " m=7484.4 # mass = 7484.4kg etc.\n", " s=32.8 #\n", " Iyy=84309\n", " cbar=2.29 \n", " dZt=-0.378\n", " rho=1.225 \n", " CD0=0.177 \n", " CDa=0.232 \n", " CDa2= 1.393\n", " CL0=0.895\n", " CLa=5.01 \n", " CLde=0.722\n", " CM0=-0.046 \n", " CMa=-1.087 \n", " CMde=-1.88 \n", " CMq=-7.055 \n", " Thrust=13878\n", " g = 9.80665\n", "\n", " qh = 0 # Trim definition\n", " \n", " rho = Atmosphere(h).density\n", "\n", " # Get the Ue and Ve\n", " alpha = Theta - gamma\n", " CL = CL0 + CLa*alpha + CLde*de\n", " CD = CD0 + CDa*alpha + CDa2*alpha*alpha\n", " CM = CM0 + CMa*alpha + CMde * de + CMq*qh\n", "\n", " q_infs = .5*rho*Vf**2*s\n", "\n", " # Dimensional lift\n", " lift = q_infs*CL\n", " drag = q_infs*CD\n", " pm = q_infs*cbar*CM\n", "\n", "\n", " # Determine lift and drag\n", " F = np.zeros((3, 1))\n", "\n", " F[0] = T - drag*np.cos(alpha) + lift*np.sin(alpha) - m*g*np.sin(Theta)\n", " F[1] = -lift*np.cos(alpha) -drag*np.sin(alpha) + m*g*np.cos(Theta)\n", " F[2] = pm - T*dZt\n", " return F\n", "\n", "def TrimState(Vf = 120 * 0.5144444, h = 0, gamma = 0):\n", " '''This function solves for the longitudinal trim of the HS125 using a Newton-Raphson method'''\n", "\n", "\n", "\n", " # First guess and increments for the jacobian <---- may need to adjust these\n", " T = 15000\n", " dT = 1\n", " de = 0*np.pi/180\n", " dde = .01*np.pi/180\n", " Theta = 2*np.pi/180\n", " dTheta = dde;\n", "\n", " # Gets the value of the functions at the initial guess\n", " trim = TotalForces(T, de, Theta, Vf, h, gamma);\n", "\n", " Trimstate = np.array([[T], [de], [Theta]])\n", "\n", " itercount = 0\n", " while max(abs(trim)) > 1e-5: \n", " itercount = itercount + 1\n", " # Get value of the function\n", " trim = TotalForces(T, de, Theta, Vf, h, gamma);\n", "\n", "\n", " # Get the Jacobian approximation (3 x 3)\n", " JT = np.squeeze(TotalForces(T + dT, de, Theta, Vf, h, gamma)/dT)\n", " Jde = np.squeeze(TotalForces(T, de + dde, Theta, Vf, h, gamma)/dde)\n", " JTheta = np.squeeze(TotalForces(T, de, Theta + dTheta, Vf, h, gamma)/dTheta)\n", " Jac = np.transpose(np.array([JT, Jde, JTheta]))\n", "\n", " # Get the next iteration\n", " Trimstate = Trimstate - np.dot(np.linalg.inv(Jac), trim)\n", "\n", "\n", " T = Trimstate[0]\n", " de = Trimstate[1]\n", " Theta = Trimstate[2]\n", " \n", " print(f\"Converged after {itercount} iterations\")\n", " print(f'For inputs of Vf = {Vf:1.2f}m/s, h = {h/1e3:1.2f}km, gamma = {gamma:1.2f}deg\\n')\n", " print(f'Thrust = {T[0]/1e3:1.2f}kN, de = {np.degrees(de[0]):1.2f}deg, Theta = {np.degrees(Theta[0]):1.2f}deg\\n')\n", "\n", " return Trimstate\n", "\n", "TrimState();\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For a trim input of sea-level and 120kn, the trim is found to be - $T = 13.84$kN, $\\delta_e = -0.98^\\circ$, $\\Theta = 0.84^\\circ$. A quick sense check shows that the aircraft is slightly nose up (less than a degree) so the aerodynamic pitching moment from the wing/fuselage will be nose-up, and the elevator is deflected trailing edge *up* to balance the pitching moment.\n", "\n", "## Transient Simulation\n", "\n", "For level flight (trim), the initial conditions are given from from the previous step - if the aircraft remains undisturbed, then there will be no variation in any of the parameters. The nonlinear equations cannot be solved analytically, so if the aircraft response to control inputs is desired, then a numerical scheme must be used.\n", "\n", "The case of a positive elevator deflection corresponding to a stick-forward displacement will be explored. First, the response to a $\\delta_e^\\prime=1^\\circ$ will be explored - this is equivalent to the pilot pushing the stick forward and holding it, keeping other controls constant. \n", "\n", "The equations are solved numerically using `scipy`'s `odeint` (which is very similar to ODE45 in MATLAB)" ] }, { "cell_type": "code", "execution_count": 171, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Converged after 1390 iterations\n", "For inputs of Vf = 61.73m/s, h = 0.00km, gamma = 0.00deg\n", "\n", "Thrust = 13.84kN, de = -0.98deg, Theta = 0.84deg\n", "\n" ] }, { "data": { "application/vnd.plotly.v1+json": { "config": { "plotlyServerURL": "https://plot.ly" }, "data": [ { "showlegend": false, "type": "scatter", "x": [ 0, 0.1001001001001001, 0.2002002002002002, 0.3003003003003003, 0.4004004004004004, 0.5005005005005005, 0.6006006006006006, 0.7007007007007008, 0.8008008008008008, 0.9009009009009009, 1.001001001001001, 1.1011011011011012, 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Set the inputs\n", "de = -1\n", "de = np.radians(de)\n", "h = 0\n", "Vf = 120*.51444\n", "gamma = 0\n", "\n", "# Determine the trim state \n", "trimstate = TrimState(Vf, h, gamma)\n", "alpha = trimstate[2] - gamma\n", "Ue = Vf*np.cos(alpha)\n", "We = Vf*np.sin(alpha)\n", "Qe = 0\n", "ThetaE = trimstate[2]\n", "Xe = 0\n", "Ze = h; \n", " \n", "# Initial conditions\n", "from scipy.integrate import odeint\n", "y0 = np.array([Ue[0], We[0], Qe, ThetaE[0], Xe, Ze, (de + trimstate[1])[0]])\n", "\n", "t = np.linspace(0, 100, 1000)\n", "output = odeint(eqnofmotion800, y0, t, args=(trimstate[0],))\n", "\n", "U = output[:, 0]\n", "W = output[:, 1]\n", "Q = output[:, 2]\n", "Theta = output[:, 3]\n", "Alt = output[:, 5]\n", "alpha = np.arctan2(W, U)\n", "FS = np.sqrt(U**2 + W**2)\n", "\n", "# Plot 'em\n", "from plotly.subplots import make_subplots\n", "import plotly.graph_objects as go\n", "\n", "fig = make_subplots(rows=3, cols=2, subplot_titles=(\"Forward Speed\", \"Heave Velocity\", \"Pitch Rate\", \"Pitch Attitude\", \"Altitude\", \"AoA\"))\n", "\n", "fig.add_trace(\n", " go.Scatter(x = t, y = U, showlegend=False), row=1, col=1)\n", "\n", "fig.add_trace(\n", " go.Scatter(x = t, y = W, showlegend=False), row=1, col=2)\n", "\n", "fig.add_trace(\n", " go.Scatter(x = t, y = Q, showlegend=False), row=2, col=1)\n", "\n", "fig.add_trace(\n", " go.Scatter(x = t, y = Theta, showlegend=False), row=2, col=2)\n", "\n", "fig.add_trace(\n", " go.Scatter(x = output[:, 4], y = Alt, showlegend=False), row=3, col=1)\n", "\n", "fig.add_trace(\n", " go.Scatter(x = t, y = alpha, showlegend=False), row=3, col=2)\n", "\n", "\n", "\n", "fig.update_xaxes(title_text=\"Time\", row=1, col=1)\n", "fig.update_yaxes(title_text=f\"U / (m/s)\", row=1, col=1)\n", "fig.update_xaxes(title_text=\"Time\", row=1, col=2)\n", "fig.update_yaxes(title_text=f\"W / (m/s)\", row=1, col=2)\n", "fig.update_xaxes(title_text=\"Time\", row=2, col=1)\n", "fig.update_yaxes(title_text=\"Q / (deg/s)\", row=2, col=1)\n", "fig.update_xaxes(title_text=\"Time\", row=2, col=2)\n", "fig.update_yaxes(title_text=\"θ / deg\", row=2, col=2)\n", "fig.update_xaxes(title_text=\"Horizontal Distance\", row=3, col=1)\n", "fig.update_yaxes(title_text=\"Alt / m\", row=3, col=1)\n", "fig.update_xaxes(title_text=\"Time\", row=3, col=2)\n", "fig.update_yaxes(title_text=\"AoA / deg\", row=3, col=2)\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It should be noted that the values above are *total* values and NOT perturbational values. We can compare a linear model of the HS125 with the reduced nonlinear model, above - but rather than sourcing HS 125 data of unknown legacy, it makes _more_ sense (but more work) to use the HS125 model to produce numerical derivatives based upon small perturbation theory using the model itself." ] }, { "cell_type": "markdown", "metadata": { "tags": [ "remove-input", "remove-output" ] }, "source": [ "$$\\newcommand{\\pd}[2]{\\frac{\\partial#1}{\\partial#2}}$$\n", "$$\\newcommand{\\ppd}[2]{\\frac{\\partial^2#1}{\\partial#2^2}}$$\n", "$$\\newcommand{\\pppd}[2]{\\frac{\\partial^3#1}{\\partial#2^3}}$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Numerical Linearisation of the Equations of Motion\n", "\n", "This section shall stick with the HS 125 business jet. For the longitudinal equations of motion:\n", "\n", "$$\\begin{aligned}\n", " \\begin{bmatrix} \\dot{u}\\\\\\dot{w}\\\\\\dot{q}\\\\\\dot{\\theta}\\end{bmatrix} &= \\begin{bmatrix}\n", " X_u & X_w & 0 & -g\\cdot\\cos\\theta_0\\\\\n", " Z_u & Z_w & U_0 & -g\\cdot\\sin\\theta_0\\\\\n", " M_u^* & M_w^* & M_q^* & M_\\theta^*\\\\\n", " 0 & 0 & 1 & 0 \n", " \\end{bmatrix}\\begin{bmatrix}\n", " {u}\\\\{w}\\\\{q}\\\\{\\theta^\\prime} \n", " \\end{bmatrix} + \\begin{bmatrix}\n", " 0\\\\Z_{\\delta_0}\\\\M_{\\delta_0}^*\\\\0 \n", " \\end{bmatrix}\\left[\\delta_0\\right]\\\\\n", " \\dot{\\vec{x}} &= \\hspace{2cm} A\\vec{x} + B\\vec{u}\n", "\\end{aligned}$$\n", "\n", "$$\\begin{aligned}\n", " M_u^*&\\triangleq M_u + M_{\\dot{w}}Z_u\\,\\,\\,\\,\\,\\,\\,\\,M_w^*\\triangleq M_w+M_{\\dot{w}}Z_w\\,\\,\\,\\,\\,\\,\\,\\,M_q^*\\triangleq M_q+M_{\\dot{w}}U_0\\nonumber\\\\\n", " M_{\\delta_0}^*&\\triangleq M_{\\delta_0}+M_{\\dot{w}}Z_{\\delta_0}\\,\\,\\,\\,\\,\\,\\,\\,M_{\\theta}^*\\triangleq-M_{\\dot{w}}g\\sin\\theta_0 \\nonumber \\end{aligned}$$\n", "\n", "For each of the derivatives contained above, a central-difference approach may be utilised. That is, if the longitudinal force is a function of the aircraft states:\n", "\n", "$$X = f\\left(U, W, Q,\\Theta_0,\\delta_{e_0}\\right)$$\n", "\n", "then each of the stability derivatives may be expressed as a numerical partial derivative. For example\n", "\n", "$$X_u = \\frac{1}{m}\\left.\\pd{X}{u}\\right|_0=\\frac{1}{m}\\left[\\frac{f\\left(U_0+\\delta U,W_0,Q_0,\\theta_0,\\delta_{e_0}\\right)-f\\left(U_0- \\delta U,W_0,Q_0,\\theta_0,\\delta_{e_0}\\right)}{2\\delta U}\\right]$$\n", "\n", "Note that the variable that the derivative is with respect to is altered, and the others are held constant - this is a partial derivative. This can be repeated for the remaining derivatives. This procedure has been used below to create stability derivatives.\n", "\n", "`ForcesandMoments` gives values of the $X$, $Z$, and $M$ force and moments, and then `NumerialDerivatives` uses a central difference approach _with_ `ForcesandMoments` to create the s " ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "celltoolbar": "Tags", "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.5" } }, "nbformat": 4, "nbformat_minor": 4 }