{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Worked Example\n", "\n", "Data for several aircraft are presented in {cite}`Heffley:1972wb` and, honestly, this is a good exercise in showing you what a tremendous amount of being an engineer is - _extracting data from old reports and doing stuff with it_.\n", "\n", "## Data for Boeing 747\n", "\n", "Data are given on Page 217 of the aforementioned report for the B-747 in power approach configuration.\n", "\n", "### Configuration Information\n", "\n", "$$h=SL$$\n", "$$V_\\infty=165\\text{KTAS}$$\n", "$$S=5500\\text{ft}^2$$\n", "$$b=195.68\\text{ft}$$\n", "$$\\bar{c}=27.31\\text{ft}$$\n", "$$\\theta_0=0$$\n", "\n", "_note that the trim theta is not actually given, but assumed to be zero_\n", "\n", "### Nondimensional Stability Derivatives\n", "\n", "| Longitudinal \t| Lateral-Directional \t|\n", "|------------------------------------------\t|---------------------------------------------\t|\n", "| $C_L=1.11$ \t| $C_{y_\\beta}=-0.96\\text{rad}^{-1}$ \t|\n", "| $C_D=0.102$ \t| $C_{\\ell_\\beta}=-0.221\\text{rad}^{-1}$ \t|\n", "| $C_{L_\\alpha}=5.70\\text{rad}^{-1}$ \t| $C_{n_\\beta}=0.150\\text{rad}^{-1}$ \t|\n", "| $C_{D_\\alpha}=0.66\\text{rad}^{-1}$ \t| $C_{\\ell_\\hat{p}}=-0.45$ \t|\n", "| $C_{m_\\alpha}=-1.26\\text{rad}^{-1}$ \t| $C_{n_\\hat{p}}=-0.121$ \t|\n", "| $C_{L_\\dot{\\alpha}}=-6.7\\text{rad}^{-1}$ \t| $C_{\\ell_\\hat{r}}=0.101$ \t|\n", "| $C_{m_\\dot{\\alpha}}=-3.2\\text{rad}^{-1}$ \t| $C_{n_\\hat{r}}=-0.30$ \t|\n", "| $C_{L_\\hat{q}}=5.40$ \t| $C_{\\ell_{\\delta_a}}=0.0461\\text{rad}^{-1}$ \t|\n", "| $C_{m_\\hat{q}}=-20.8$ \t| $C_{n_{\\delta_a}}=0.0064\\text{rad}^{-1}$ \t|\n", "| $C_{L_\\text{M}}=-0.81$ \t| $C_{y_{\\delta_r}}=0.175\\text{rad}^{-1}$ \t|\n", "| $C_{m_\\text{M}}=0.27$ \t| $C_{\\ell_{\\delta_r}}=0.007\\text{rad}^{-1}$ \t|\n", "| $C_{L_{\\delta_e}}=0.338\\text{rad}^{-1}$ \t| $C_{n_{\\delta_r}}=-0.109\\text{rad}^{-1}$ \t|\n", "| $C_{m_{\\delta_e}}=-1.34\\text{rad}^{-1}$ \t| \t|\n", "\n", "Note that in {cite}`Heffley:1972wb`, the derivatives with respect to angular rates are erroneously given units of /rad, when they clearly are non-dimensional rates. The derivatives with respect to aerodynamic angles correctly are given units of /rad." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "scrolled": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Xu = -0.0212\n", "Xw = 0.0467\n", "Zu = -0.2098\n", "Zw = -0.6027\n", "Zdw = -0.0341\n", "Zq = -7.6596\n", "Mu = 0.0001\n", "Mw = -0.0019\n", "Mw = -0.0019\n", "Mdw = -0.0002\n", "Mq = -0.4373\n", "Zde = -9.7779\n", "Mde = -0.5746\n" ] } ], "source": [ "import numpy as np\n", "from IPython.display import display, Math, Latex, Markdown\n", "\n", "\n", "# Condition and geometry information\n", "rho = 0.002377 # slugs/cubic feet\n", "# rho = rho * 32.174 # to lb / cubic feet\n", "rho = 0.0765 # lb/ft^3\n", "U0 = 165 # Knots\n", "U0 = U0 * 1.68781 # to feet/second\n", "S = 5500 # square feet\n", "b = 195.68 # feet\n", "c = 27.31 # feet\n", "theta_0 = 0\n", "m = 564000 # lb\n", "\n", "\n", "Iyy = 32.3e6 # slug-ft^2\n", "Ixx = 14.3e6\n", "Izz = 45.3e6\n", "Ixz = -2.23e6\n", "\n", "SIunits = False\n", "\n", "if SIunits:\n", " # Conversion factors\n", " ft_to_metre = 0.3048\n", " lb_to_kg = 0.453592\n", " slug_to_kg = 14.5939\n", " \n", " MOIconvert = slug_to_kg*ft_to_metre**2\n", "\n", " \n", " # Constants\n", " a = 340 # sonic velocity in m/s\n", " g = 9.80665 # acceleration due to gravity\n", " rho = 1.225 # density in kg/m^3\n", " \n", " # Convert\n", " U0 = U0 * ft_to_metre\n", " S = S * ft_to_metre**2\n", " b = b * ft_to_metre\n", " c = c * ft_to_metre\n", " m = m * lb_to_kg\n", " Ixx = Ixx * MOIconvert\n", " Iyy = Iyy * MOIconvert\n", " Izz = Izz * MOIconvert\n", " Ixz = Ixz * MOIconvert \n", " \n", "else:\n", " a = 1125.33 # sonic velocity in ft/s\n", " g = 32.174\n", " \n", " # Convert mass moments of inertia to consistent units --> INTO lb-ft^2 FROM slug-ft^2\n", " Ixx = Ixx * g\n", " Iyy = Iyy * g\n", " Izz = Izz * g\n", " Ixz = Ixz * g\n", "\n", "# Store the derivatives in two dictionaries\n", "B747_lon_ders = {'C_L': 1.11, 'C_D': 0.102, 'C_L_a' : 5.7, 'C_D_a' : 0.66, 'C_m_a' : -1.26,\n", " 'C_L_da' : -6.7, 'C_m_da' : -3.2, 'C_L_hq' : 5.4, 'C_m_hq' : -20.8, 'C_L_M' : -0.81,\n", " 'C_m_M' : 0.27, 'C_L_de' : 0.338, 'C_m_de' : -1.34}\n", "\n", "B747_lat_ders = {'C_y_b' : -0.96, 'C_l_b' : -0.221, 'C_n_b' : 0.150, 'C_l_hp' : -0.45, 'C_n_hp' : -0.121,\n", " 'C_l_hr' : 0.101, 'C_n_hr': -0.30, 'C_l_da' : 0.0461, 'C_n_da' : 0.0064, 'C_y_dr' : 0.175,\n", " 'C_l_dr' : 0.007, 'C_n_dr' : -0.109}\n", "\n", "# Put the dictionaries into the local namespace\n", "# This might seem a bit convoluted but it enables us to store values of derivatives in the dicionary, above,\n", "# and then put them all into the global namespace.\n", "#\n", "# It's fairly easy to get (key: value) pairs into a dictionary from a text file or xls so this will be handy\n", "# later.\n", "locals().update(B747_lon_ders)\n", "\n", "# Note that without doing the dictionary --> local namespace, the we'd have to write:\n", "# Xu = q * S / m / U0 * (2 * B747_lon_ders[\"C_D\"] + M * B747_lon_ders[\"C_D_M\"])\n", "\n", "# Convert to dimensional form\n", "q = 0.5 * rho * U0**2\n", "M = U0 / a\n", "\n", "Xu = -q * S / m / U0 * (2 * C_D) # No C_D_M term so assumed zero\n", "Xw = q * S / m / U0 * (C_L - C_D_a)\n", "Xq = 0 # No CDq term given in the table so assumed zero\n", "Zu = -q * S / m / U0 * (2 * C_L + M * C_L_M)\n", "Zw = -q * S / m / U0 * (C_D + C_L_a)\n", "Zdw = q * S * c / m / 2 / U0**2 * C_L_da # This is a NEW term for us, but since it was given as C_L_da, must be included\n", "Zq = -q * S * c / 2 / m / U0 * C_L_hq\n", "Mu = q * S * c / Iyy / U0 * M * C_m_M\n", "Mw = q * S * c / Iyy / U0 * C_m_a\n", "Mdw = q * S * c**2 / 2 / Iyy / U0**2 * C_m_da\n", "Mq = q * S * c**2 / 2 / Iyy / U0 * C_m_hq\n", "Mq = q * S * c**2 / 2 / Iyy / U0 * C_m_hq\n", "Zde = -q * S / m * C_L_de\n", "Mde = q * S * c / Iyy * C_m_de\n", "\n", "print(f\"Xu = {Xu:1.4f}\")\n", "print(f\"Xw = {Xw:1.4f}\")\n", "print(f\"Zu = {Zu:1.4f}\")\n", "print(f\"Zw = {Zw:1.4f}\")\n", "print(f\"Zdw = {Zdw:1.4f}\")\n", "print(f\"Zq = {Zq:1.4f}\")\n", "print(f\"Mu = {Mu:1.4f}\")\n", "print(f\"Mw = {Mw:1.4f}\")\n", "print(f\"Mw = {Mw:1.4f}\")\n", "print(f\"Mdw = {Mdw:1.4f}\")\n", "print(f\"Mq = {Mq:1.4f}\")\n", "\n", "print(f\"Zde = {Zde:1.4f}\")\n", "print(f\"Mde = {Mde:1.4f}\")\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Some units have dimensions of $[\\text{T}^{-1}]$ and hence have the same value in either unit system. Expand the below to see this. Other derivatives do not." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Using US Customary Units:\n", "Dynamic pressure, q = 2966.5143 lbs/ft^2\n", "Wing area, S = 5500 ft^2\n", "Mass, m = 564000 lbs\n", "Trim forward speed, U0 = 278.5 ft/s\n", "Gives a value for Xu = -0.2098, and a value for Zq of -7.6596\n", "\n", "Using SI Units:\n", "Dynamic pressure, q = 4414.9750 Pa\n", "Wing area, S = 511 m^2\n", "Mass, m = 255826 kg\n", "Trim forward speed, U0 = 84.9 m/s\n", "Gives a value for Xu = -0.2098, and a value for Zq of -2.3348\n", "\n" ] } ], "source": [ "# Demonstration of the equality of some derivatives in both unit systems:\n", "SIunits = {\"Pressure\": \"Pa\", \"Mass\": \"kg\", \"Area\" : \"m^2\", \"Speed\" : \"m/s\"}\n", "USunits = {\"Pressure\": \"lbs/ft^2\", \"Mass\": \"lbs\", \"Area\" : \"ft^2\", \"Speed\" : \"ft/s\"}\n", "\n", "# Conversion factors\n", "ft_to_metre = 0.3048\n", "lb_to_kg = 0.453592\n", "slug_to_kg = 14.5939\n", "\n", "for SIUNIT in [False, True]:\n", " if SIUNIT:\n", " UNITS = SIunits\n", " else:\n", " UNITS = USunits\n", " \n", " # Convert to some temporary variables to not overwrite the original values\n", " if SIUNIT:\n", " u0 = U0 * ft_to_metre\n", " s = S * ft_to_metre**2\n", " c2 = c * ft_to_metre\n", " mass = m * lb_to_kg\n", " rho = 1.225\n", " qinf = 0.5 * 1.2255 * u0**2\n", " else:\n", " qinf = q\n", " u0 = U0\n", " s = S\n", " c2 = c\n", " mass = m\n", "\n", " if SIUNIT: print(\"Using SI Units:\") \n", " else: print(\"Using US Customary Units:\")\n", " print(f\"Dynamic pressure, q = {qinf:1.4f} {UNITS['Pressure']}\")\n", " print(f\"Wing area, S = {s:1.0f} {UNITS['Area']}\")\n", " print(f\"Mass, m = {mass:1.0f} {UNITS['Mass']}\")\n", " print(f\"Trim forward speed, U0 = {u0:1.1f} {UNITS['Speed']}\")\n", " \n", " testXu = -qinf * s / mass / u0 * (C_L * 2 + C_L_M * M) \n", " testZq = -qinf * s * c2 / 2 / mass / u0 * C_L_hq\n", "\n", " print(f\"Gives a value for Xu = {testXu:1.4f}, and a value for Zq of {testZq:1.4f}\\n\")\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now the system matrix can be constructed:\n", "\n", "$$\\begin{aligned}\n", " \\begin{bmatrix} \\dot{u}\\\\\\dot{w}\\\\\\dot{q}\\\\\\dot{\\theta}\\end{bmatrix} &= \\begin{bmatrix}\n", " X_u & X_w & 0 & -g\\cdot\\cos\\Theta_e\\\\\n", " Z_u & Z_w & U_0 & -g\\cdot\\sin\\Theta_e\\\\\n", " M_u^* & M_w^* & M_q^* & M_\\theta^*\\\\\n", " 0 & 0 & 1 & 0 \n", " \\end{bmatrix}\\begin{bmatrix}\n", " {u}\\\\{w}\\\\{q}\\\\{\\theta} \n", " \\end{bmatrix} + \\begin{bmatrix}\n", " 0\\\\Z_{\\delta_e}\\\\M_{\\delta_e}^*\\\\0 \n", " \\end{bmatrix}\\left[\\delta_e\\right]\\end{aligned}$$\n", " \n", "well...almost - you'll see two things that need to be performed:\n", "1. The starred terms need to be made from the products of the stability derivatives.\n", "2. We have a $Z_q$ term from conversion of the stability derivatives, but no corresponding place to put it.\n", "\n", "The first problem is easy. From Equations {eq}`eq:conciseqterms`" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "scrolled": true }, "outputs": [], "source": [ "Mustar = Mu + Mdw * Zu\n", "Mwstar = Mw + Mdw * Zw\n", "Mqstar = Mq + Mdw * Zq\n", "Mthetastar = -Mdw * g * np.sin(theta_0)\n", "Mdestar = Mde + Mdw * Zde" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we've got all the terms, we don't know where to put $Z_q$. The answer should be obvious - it goes where the $U_0$ term is - it doesn't _replace_ it, but is _added to it_. So:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "scrolled": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The system matrix for longitudinal motion is \n" ] }, { "data": { "text/latex": [ "$\\displaystyle [A_{lon}] = \\left[\\begin{matrix}-0.021191 & 0.046745 & 0 & -32.174\\\\-0.20979 & -0.6027 & 270.83 & 0\\\\0.00015355 & -0.0017943 & -0.43544 & 0\\\\0 & 0 & 1.0 & 0\\end{matrix}\\right]$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "The eigenvalues are: [-0.52829455+0.69158467j -0.52829455-0.69158467j -0.00137077+0.14113918j\n", " -0.00137077-0.14113918j]\n", "The coefficients of the CE are: [1. 1.05933064 0.7802033 0.02312592 0.01508872]\n" ] } ], "source": [ "import sympy as sp\n", "\n", "Alon = np.matrix([[Xu, Xw, 0, -g*np.cos(theta_0)],\n", " [Zu, Zw, U0 + Zq, -g*np.sin(theta_0)],\n", " [Mustar, Mwstar, Mqstar, Mthetastar],\n", " [0, 0, 1, 0]])\n", "\n", "print(\"The system matrix for longitudinal motion is \")\n", "display(Math('[A_{lon}] = ' + sp.latex(sp.Matrix(Alon).evalf(5))))\n", "\n", "# Get the eigenvalues - this is my own checking I got the matrix correct. It'll make sense later.\n", "eigs, _ = np.linalg.eig(Alon)\n", "\n", "print(\"The eigenvalues are:\", eigs)\n", "\n", "print(\"The coefficients of the CE are:\", np.poly(Alon))\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The same can be performed for the lateral-directional matrix:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "scrolled": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Yv = -0.0997\n", "Yp = 0.0000\n", "Yr = 0.0000\n", "Lv = -0.0055\n", "Lp = -1.0971\n", "Lr = 0.2462\n", "Nv = 0.0012\n", "Np = -0.0931\n", "Nr = -0.2309\n" ] } ], "source": [ "# Put the non-dimensional derivatives into the local namespace\n", "locals().update(B747_lat_ders)\n", "\n", "Yv = q * S / m / U0 * C_y_b\n", "Yp = 0\n", "Yr = 0\n", "Lv = q * S * b / Ixx / U0 * C_l_b\n", "Lp = q * S * b**2 / 2/ Ixx / U0 * C_l_hp\n", "Lr = q * S * b**2 / 2/ Ixx / U0 * C_l_hr\n", "Nv = q * S * b / Izz / U0 * C_n_b\n", "Np = q * S * b**2 / 2 / Izz / U0 * C_n_hp\n", "Nr = q * S * b**2 / 2 / Izz / U0 * C_n_hr\n", "\n", "\n", "print(f\"Yv = {Yv:1.4f}\")\n", "print(f\"Yp = {Yp:1.4f}\")\n", "print(f\"Yr = {Yr:1.4f}\")\n", "print(f\"Lv = {Lv:1.4f}\")\n", "print(f\"Lp = {Lp:1.4f}\")\n", "print(f\"Lr = {Lr:1.4f}\")\n", "print(f\"Nv = {Nv:1.4f}\")\n", "print(f\"Np = {Np:1.4f}\")\n", "print(f\"Nr = {Nr:1.4f}\")\n", "\n", "# Do the sadme with the control terms \n", "# 'C_l_da' : 0.0461, 'C_n_da' : 0.0064, 'C_y_dr' : 0.175,\n", "# 'C_l_dr' : 0.007, 'C_n_dr' : -0.109}\n", "Yda = 0\n", "Ydr = q * S / m * C_y_dr\n", "Lda = q * S * b / Ixx * C_l_da\n", "Ldr = q * S * b / Ixx * C_l_dr\n", "Nda = q * S * b / Izz * C_n_da\n", "Ndr = q * S * b / Izz * C_n_dr\n", "\n", "\n", "# \n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "And put into the system matrix for lateral-directional motion\n", "\n", "$$\\boldsymbol{A}=\\begin{bmatrix}Y_v & 0 & -U_0 & g\\cdot\\cos\\theta_0 & 0\\\\L_v^* & L_p^* & L_r^* & 0 & 0\\\\N_v^* & N_p^* & N_r^* & 0 & 0\\\\0 & 1 & \\tan\\theta_0 & 0 & 0 \\\\ 0 & 0 & \\sec\\theta_0 & 0 & 0\\end{bmatrix}$$" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "scrolled": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The system matrix for lateral-directional motion is \n" ] }, { "data": { "text/latex": [ "$\\displaystyle [A_{lat}] = \\left[\\begin{matrix}-0.099723 & 0 & -278.49 & 32.174 & 0\\\\-0.0057348 & -1.0909 & 0.28442 & 0 & 0\\\\0.0014622 & -0.039417 & -0.24488 & 0 & 0\\\\0 & 1.0 & 0 & 0 & 0\\\\0 & 0 & 1.0 & 0 & 0\\end{matrix}\\right]$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "The eigenvalues are: [ 0. +0.j -1.2285872 +0.j -0.08023692+0.74204291j\n", " -0.08023692-0.74204291j -0.04646794+0.j ]\n", "The coefficients of the CE are: [1. 1.43552897 0.81876855 0.71945085 0.03180283 0. ]\n" ] } ], "source": [ "# Making the starred terms\n", "Imess = Ixx * Izz / (Ixx * Izz - Ixz**2)\n", "I2 = Ixz / Ixx\n", "\n", "# Lstarred terms\n", "Lvstar = Imess * (Lv + I2 * Nv)\n", "Lpstar = Imess * (Lp + I2 * Np)\n", "Lrstar = Imess * (Lr + I2 * Nr)\n", "Ldrstar = Imess * (Ldr + I2 * Ndr)\n", "Ldastar = Imess * (Lda + I2 * Nda)\n", "\n", "# Nstarred terms\n", "I2 = Ixz / Izz\n", "Nvstar = Imess * (Nv + I2 * Lv)\n", "Npstar = Imess * (Np + I2 * Lp)\n", "Nrstar = Imess * (Nr + I2 * Lr)\n", "Ndrstar = Imess * (Ndr + I2 * Ldr)\n", "Ndastar = Imess * (Nda + I2 * Lda)\n", "\n", "Alat = np.matrix([[Yv, 0, -U0, g*np.cos(theta_0), 0],\n", " [Lvstar, Lpstar, Lrstar, 0, 0],\n", " [Nvstar, Npstar, Nrstar, 0, 0],\n", " [0, 1, np.tan(theta_0), 0, 0],\n", " [0, 0, 1/np.cos(theta_0), 0, 0]])\n", "\n", "print(\"The system matrix for lateral-directional motion is \")\n", "display(Math('[A_{lat}] = ' + sp.latex(sp.Matrix(Alat).evalf(5))))\n", "\n", "print(\"The eigenvalues are:\", np.linalg.eig(Alat)[0])\n", "\n", "print(\"The coefficients of the CE are:\", np.poly(Alat))" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "8.467904726657125" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.pi*2/.7420" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Transient Longitudinal Response\n", "\n", "Note that we've not made either of the control matrices yet - and there's a good reason for that. If we consider \n", "the linearised model of the aircraft created, the _stability_ of the aircraft (_i.e.,_ its response to a perturbation) will be governed by the system matrix.\n", "\n", "In the first instance, it is desired to understand the _stick fixed stability_ and understand what will happen to the aircraft if no inputs are made.\n", "\n", "Consider what the two matrices developed actually show us; if the aircraft is disturbed from a given condition, then if the disturbance is a _symmetric_ state variable ($u, w, q, \\theta$), then the output is a time rate of change of _all_ other symmetric variables. \n", "\n", "The control matrices tell us how the aircraft responds due to control input. Therefore, if we're interested in aircraft response in the absense of pilot or control-system input, then we're only interested in the A matrices themselves. The A matrices give the _open loop response_ of the aircraft.\n", "\n", "Since these are ODEs, we could write a means to time-march through the solution - but there are some handy inbuilt tools in Python to look at the responses of state-space systems to different inputs.\n", "\n", "You will probably need to install these to work on your system - [look here](https://python-control.readthedocs.io/en/0.8.3/intro.html#overview-of-the-toolbox).\n", "\n", "But if we create the $B$ matrix, it can be used to excite the aircraft with unity impulse input. For this reason, it makes sense to make the two control matrices with derivatives that have been scaled by a factor of $\\frac{\\pi}{180}$ to allow a unity _degree_ control input." ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "application/vnd.plotly.v1+json": { "config": { "plotlyServerURL": "https://plot.ly" }, "data": [ { "showlegend": false, "type": "scatter", "x": [ 0, 0.020002000200020003, 0.040004000400040006, 0.060006000600060005, 0.08000800080008001, 0.10001000100010002, 0.12001200120012001, 0.14001400140014003, 0.16001600160016002, 0.18001800180018002, 0.20002000200020004, 0.22002200220022003, 0.24002400240024002, 0.26002600260026004, 0.28002800280028006, 0.3000300030003, 0.32003200320032005, 0.34003400340034007, 0.36003600360036003, 0.38003800380038005, 0.4000400040004001, 0.42004200420042004, 0.44004400440044006, 0.4600460046004601, 0.48004800480048004, 0.5000500050005001, 0.5200520052005201, 0.54005400540054, 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\n", " \n", " \n", "
\n", " \n", "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import control\n", "import control.matlab\n", "\n", "# We will make a B matrix to enable us to use the control system toolbox by _exciting the aircraft_ through\n", "# Elevator input. Turn from Zde in 1/radians to 1/degree to put a useful input in.\n", "Blon = np.matrix([[0], [np.radians(Zde)], [np.radians(Mdestar)], [0]])\n", "\n", "# Turn the matrices into a state space object\n", "LonSS = control.StateSpace(Alon, Blon, np.eye(Alon.shape[0]), np.zeros(Blon.shape))\n", "\n", "# Look at the first 100 seconds response to a unit impulse in the only\n", "Time, [u, w, q, theta] = control.impulse_response(LonSS, T=np.linspace(0, 200, 10000))\n", "\n", "u, w, q, theta = u[0], w[0], q[0], theta[0]\n", "\n", "# Convert q and theta\n", "q = np.degrees(q)\n", "theta = np.degrees(theta)\n", "\n", "from plotly.subplots import make_subplots\n", "import plotly.graph_objects as go\n", "\n", "fig = make_subplots(rows=2, cols=2, subplot_titles=(\"Forward Speed\", \"Heave Velocity\", \"Pitch Rate\", \"Pitch Attitude\"))\n", "\n", "fig.add_trace(\n", " go.Scatter(x = Time, y = u, showlegend=False), row=1, col=1)\n", "\n", "fig.add_trace(\n", " go.Scatter(x = Time, y = w, showlegend=False), row=1, col=2)\n", "\n", "fig.add_trace(\n", " go.Scatter(x = Time, y = q, showlegend=False), row=2, col=1)\n", "\n", "fig.add_trace(\n", " go.Scatter(x = Time, y = theta, showlegend=False), row=2, col=2)\n", "\n", "# Make a label based upon the units\n", "if SIunits:\n", " speedlabel = \"m/s\"\n", "else:\n", " speedlabel = \"ft/s\"\n", "\n", "fig.update_xaxes(title_text=\"Time\", row=1, col=1)\n", "fig.update_yaxes(title_text=f\"u / ({speedlabel})\", row=1, col=1)\n", "fig.update_xaxes(title_text=\"Time\", row=1, col=2)\n", "fig.update_yaxes(title_text=f\"w / ({speedlabel})\", 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", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As much as a pain as the non-dimensional to dimensional procedure is, it is useful in that you can take fully non-dimensional data and then use whatever unit system you prefer, and you'll get results in those units.\n", "\n", "Note that there is a flag to convert between US Customary (ugh) and SI units in the code above. You can see that there's some very annoying fiddly bits that have to be performed with the US Customary units to ensure consistency - that is, the mass moments of inertia are given in slugs-ft^2, but the masses etc. are presented in lbs. This can cause some dimensional moments of inertia to be about thirty-two times too large, which has a large effect on the end result.\n", "\n", "My strong advice is if you're required to do anything in US Customary units, to convert to SI at the start, and then convert to US Customary at the end. There's just too many pitfalls in the US Customary system.\n", "\n", "```{admonition} Other inputs?\n", "If you wanted to look at the transient response of the aircraft to control inputs other than one degree, what could we do with the _output_ to explore this?\n", "\n", "Discuss this on the slack space.\n", "```\n", "#### Response characteristics\n", "\n", "We'll spend the next module understanding the actual characteristics of the aircraft in terms of dynamic response, but we can start to intuit some of this now:\n", "- There are two phases to the aircraft response - a short-term response that is heavily damped, affecting mainly heave (_i.e._, angle of attack) and pitch rate, and a long-term response that is lightly-damped, affecting mainly speed and pitch attitude, but also showing in the other state variables.\n", "\n", "Non-state variables can be explored - we *could* go back and change the state variable $w$ for $\\alpha$, for example, and then explore the change in AoA. This would require calculating a new set of stability derivatives (many sources use $\\alpha$ in place of $u$). Instead, perturbational angle of attack can be determined from the linear relationship" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "application/vnd.plotly.v1+json": { "config": { "plotlyServerURL": "https://plot.ly" }, "data": [ { "showlegend": false, "type": "scatter", "x": [ 0, 0.020002000200020003, 0.040004000400040006, 0.060006000600060005, 0.08000800080008001, 0.10001000100010002, 0.12001200120012001, 0.14001400140014003, 0.16001600160016002, 0.18001800180018002, 0.20002000200020004, 0.22002200220022003, 0.24002400240024002, 0.26002600260026004, 0.28002800280028006, 0.3000300030003, 0.32003200320032005, 0.34003400340034007, 0.36003600360036003, 0.38003800380038005, 0.4000400040004001, 0.42004200420042004, 0.44004400440044006, 0.4600460046004601, 0.48004800480048004, 0.5000500050005001, 0.5200520052005201, 0.54005400540054, 0.5600560056005601, 0.5800580058005801, 0.6000600060006, 0.6200620062006201, 0.6400640064006401, 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "alpha = np.degrees(w/(U0+u))\n", "\n", "fig = go.Figure()\n", "fig.add_trace(\n", " go.Scatter(x = Time, y = alpha, showlegend=False))\n", "\n", "fig.update_xaxes(title_text=\"Time\")\n", "fig.update_yaxes(title_text=\"$\\\\alpha/\\\\text{deg}$\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You will see that the angle of attack is affected much more by the short-term response than the long-term response.\n", "\n", "### Transient Lateral Response\n", "\n", "A similar exercise can be performed for the lateral response\n" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "application/vnd.plotly.v1+json": { "config": { "plotlyServerURL": "https://plot.ly" }, "data": [ { "showlegend": false, "type": "scatter", "x": [ 0, 0.020002000200020003, 0.040004000400040006, 0.060006000600060005, 0.08000800080008001, 0.10001000100010002, 0.12001200120012001, 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# We will make a B matrix to enable us to use the control system toolbox by _exciting the aircraft_ through\n", "# Elevator input. Turn from Zde in 1/radians to 1/degree to put a useful input in.\n", "Blat = np.matrix([[Ydr, Yda], [Ldrstar, Ldastar], [Ndrstar, Ndastar], [0, 0], [0, 0]])\n", "Blat = np.radians(Blat)\n", "\n", "# Turn the matrices into a state space object\n", "LatSS = control.StateSpace(Alat, Blat, np.eye(Alat.shape[0]), np.zeros(Blat.shape))\n", "\n", "# do this twice, for rudder and then aileron\n", "Time, [v, p, r, phi, psi] = control.impulse_response(LatSS, T=np.linspace(0, 200, 10000), input=0)\n", "\n", "v, p, r, phi, psi = v[0], p[0], r[0], phi[0], psi[0]\n", "\n", "# Convert p, r, and phi\n", "p = np.degrees(p)\n", "r = np.degrees(r)\n", "phi = np.degrees(phi)\n", "\n", "from plotly.subplots import make_subplots\n", "import plotly.graph_objects as go\n", "\n", "fig = make_subplots(rows=2, cols=2, subplot_titles=(\"Sideslip Velocity\", \"Roll Rate\", \"Yaw Rate\", \"Roll Attitude\"))\n", "\n", "fig.update_layout(title=f\"Unit Rudder Input\", title_x=0.5)\n", "\n", "fig.add_trace(\n", " go.Scatter(x = Time, y = v, showlegend=False), row=1, col=1)\n", "\n", "fig.add_trace(\n", " go.Scatter(x = Time, y = p, showlegend=False), row=1, col=2)\n", "\n", "fig.add_trace(\n", " go.Scatter(x = Time, y = r, showlegend=False), row=2, col=1)\n", "\n", "fig.add_trace(\n", " go.Scatter(x = Time, y = phi, showlegend=False), row=2, col=2)\n", "\n", "# Make a label based upon the units\n", "if SIunits:\n", " speedlabel = \"m/s\"\n", "else:\n", " speedlabel = \"ft/s\"\n", "\n", "fig.update_xaxes(title_text=\"Time\", row=1, col=1)\n", "fig.update_yaxes(title_text=f\"v / ({speedlabel})\", row=1, col=1)\n", "fig.update_xaxes(title_text=\"Time\", row=1, col=2)\n", "fig.update_yaxes(title_text=f\"p / (deg/s)\", row=1, col=2)\n", "fig.update_xaxes(title_text=\"Time\", row=2, col=1)\n", "fig.update_yaxes(title_text=\"r / (deg/s)\", row=2, col=1)\n", "fig.update_xaxes(title_text=\"Time\", row=2, col=2)\n", "fig.update_yaxes(title_text=\"φ / deg\", row=2, col=2)\n", "fig.show()\n" ] }, { 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\n", " \n", " \n", "
\n", " \n", "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "\n", "# Second time - aileron\n", "Time, [v, p, r, phi, psi] = control.impulse_response(LatSS, T=np.linspace(0, 200, 10000), input=1)\n", "v, p, r, phi, psi = v[0], p[0], r[0], phi[0], psi[0]\n", "\n", "\n", "# Convert p, r, and phi\n", "p = np.degrees(p)\n", "r = np.degrees(r)\n", "phi = np.degrees(phi)\n", "\n", "from plotly.subplots import make_subplots\n", "import plotly.graph_objects as go\n", "\n", "fig = make_subplots(rows=2, cols=2, subplot_titles=(\"Sideslip Velocity\", \"Roll Rate\", \"Yaw Rate\", \"Roll Attitude\"))\n", "fig.update_layout(title=\"Unit Aileron Input\", title_x=0.5)\n", "\n", "fig.add_trace(\n", " go.Scatter(x = Time, y = v, showlegend=False), row=1, col=1)\n", "\n", "fig.add_trace(\n", " go.Scatter(x = Time, y = p, showlegend=False), row=1, col=2)\n", "\n", "fig.add_trace(\n", " go.Scatter(x = Time, y = r, showlegend=False), row=2, col=1)\n", "\n", "fig.add_trace(\n", " go.Scatter(x = Time, y = phi, showlegend=False), row=2, col=2)\n", "\n", "# Make a label based upon the units\n", "if SIunits:\n", " speedlabel = \"m/s\"\n", "else:\n", " speedlabel = \"ft/s\"\n", "\n", "\n", "fig.update_xaxes(title_text=\"Time\", row=1, col=1)\n", "fig.update_yaxes(title_text=f\"v / ({speedlabel})\", row=1, col=1)\n", "fig.update_xaxes(title_text=\"Time\", row=1, col=2)\n", "fig.update_yaxes(title_text=f\"p / (deg/s)\", row=1, col=2)\n", "fig.update_xaxes(title_text=\"Time\", row=2, col=1)\n", "fig.update_yaxes(title_text=\"r / (deg/s)\", row=2, col=1)\n", "fig.update_xaxes(title_text=\"Time\", row=2, col=2)\n", "fig.update_yaxes(title_text=\"φ / deg\", row=2, col=2)\n", "fig.show()\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Getting eigenvalues from the graph data\n", "\n", "This bit wont make much sense until you watch the corresponding lecture..." ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Estimated spiral eigenvalue = -0.04503\n" ] }, { "data": { "application/vnd.plotly.v1+json": { "config": { "plotlyServerURL": "https://plot.ly" }, "data": [ { "showlegend": false, "type": "scatter", "x": [ 0, 0.020002000200020003, 0.040004000400040006, 0.060006000600060005, 0.08000800080008001, 0.10001000100010002, 0.12001200120012001, 0.14001400140014003, 0.16001600160016002, 0.18001800180018002, 0.20002000200020004, 0.22002200220022003, 0.24002400240024002, 0.26002600260026004, 0.28002800280028006, 0.3000300030003, 0.32003200320032005, 0.34003400340034007, 0.36003600360036003, 0.38003800380038005, 0.4000400040004001, 0.42004200420042004, 0.44004400440044006, 0.4600460046004601, 0.48004800480048004, 0.5000500050005001, 0.5200520052005201, 0.54005400540054, 0.5600560056005601, 0.5800580058005801, 0.6000600060006, 0.6200620062006201, 0.6400640064006401, 0.6600660066006601, 0.6800680068006801, 0.7000700070007001, 0.7200720072007201, 0.7400740074007401, 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" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Estimated spiral eigenvalue = -0.04503\n", "Estimated DR eigenvalue -0.08413±0.7515j\n", "\n", "Compared with those from the A matix - spiral = -0.0465\n", "Compared with those from the A matrix - Dutch Roll = -0.0802+0.7420j\n" ] } ], "source": [ "# Just plot the roll attitude\n", "fig = go.Figure()\n", "fig.update_layout(title=\"Roll Attitude\", title_x=0.5)\n", "\n", "fig.add_trace(\n", " go.Scatter(x = Time, y = phi, showlegend=False))\n", "\n", "# Estimate the spiral mode\n", "lam_sp_est = 1/50 * np.log(0.02/0.19)\n", "print(f\"Estimated spiral eigenvalue = {lam_sp_est:1.4}\")\n", "\n", "dphi_est = 0.19\n", "sp_est = dphi_est * np.exp(lam_sp_est*Time)\n", "\n", "fig.add_trace(\n", " go.Scatter(x = Time, y = sp_est, name=\"Estimated Spiral Mode\", showlegend=True))\n", "\n", "# Now estimate the dutch roll\n", "import scipy.signal as sig\n", "peaks_all = sig.find_peaks(phi)[0]\n", "\n", "phi_dr = np.zeros(peaks_all.shape)\n", "\n", "for i, p in enumerate(peaks_all[:4]):\n", " fig.add_trace(go.Scatter(x = [Time[p]], y = [phi[p]], mode='markers+text', text=f\"{Time[p]:1.3f},{phi[p]:1.3f}\", showlegend=False))\n", "\n", " # Get the value without the spiral mode\n", " phi_dr[i] = phi[p] - dphi_est * np.exp(lam_sp_est * Time[p])\n", " \n", "fig.show()\n", "fig.update_traces(textposition='top right')\n", "\n", "# Estimate the damping in the Dutch Roll\n", "lam_real_DR = 1/(Time[peaks_all[2]] - Time[peaks_all[1]]) * np.log(phi_dr[2]/phi_dr[1])\n", "\n", "\n", "# remove the spiral completely from the data\n", "just_dr = phi - sp_est\n", "fig.add_trace(\n", " go.Scatter(x = Time, y = just_dr, name=\"Estimated Dutch Mode\", showlegend=True))\n", "\n", "# And the imaginary part is simply the damped natural frequency\n", "lam_imag_DR = 2 * np.pi / ((Time[peaks_all[2]] - Time[peaks_all[1]]))\n", "print(f\"Estimated spiral eigenvalue = {lam_sp_est:1.4}\")\n", "print(f\"Estimated DR eigenvalue {lam_real_DR:1.4}±{lam_imag_DR:1.4}j\\n\")\n", "\n", "# Compare to those from the data - first get the spiral\n", "lat_eigs = np.linalg.eig(Alat)[0]\n", "real_eigs = lat_eigs[lat_eigs.imag == 0] # Gets just the real eigenvalues but they're still returned as complex numbers\n", "real_eigs = real_eigs[real_eigs != 0] \n", "the_spiral_index = np.argmin(np.abs(real_eigs)) # Finds which has the least damping - this must be the spiral\n", "spiral_eigenvalue = real_eigs[the_spiral_index].real # Returns it without the zero imaginary part\n", "print(f\"Compared with those from the A matix - spiral = {spiral_eigenvalue:1.4f}\")\n", "\n", "# Now get the DR\n", "DR_eigs = lat_eigs[lat_eigs.imag != 0]\n", "print(f\"Compared with those from the A matrix - Dutch Roll = {DR_eigs[0]:1.4f}\")\n", "\n" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "tags": [ "remove-input", "remove_output" ] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0.04223124601721076\n", "0.020431824451614247\n", "-0.08683969221285175\n", "Help on ndarray object:\n", "\n", "class ndarray(builtins.object)\n", " | ndarray(shape, dtype=float, buffer=None, offset=0,\n", " | strides=None, order=None)\n", " | \n", " | An array object represents a multidimensional, homogeneous array\n", " | of fixed-size items. An associated data-type object describes the\n", " | format of each element in the array (its byte-order, how many bytes it\n", " | occupies in memory, whether it is an integer, a floating point number,\n", " | or something else, etc.)\n", " | \n", " | Arrays should be constructed using `array`, `zeros` or `empty` (refer\n", " | to the See Also section below). The parameters given here refer to\n", " | a low-level method (`ndarray(...)`) for instantiating an array.\n", " | \n", " | For more information, refer to the `numpy` module and examine the\n", " | methods and attributes of an array.\n", " | \n", " | Parameters\n", " | ----------\n", " | (for the __new__ method; see Notes below)\n", " | \n", " | shape : tuple of ints\n", " | Shape of created array.\n", " | dtype : data-type, optional\n", " | Any object that can be interpreted as a numpy data type.\n", " | buffer : object exposing buffer interface, optional\n", " | Used to fill the array with data.\n", " | offset : int, optional\n", " | Offset of array data in buffer.\n", " | strides : tuple of ints, optional\n", " | Strides of data in memory.\n", " | order : {'C', 'F'}, optional\n", " | Row-major (C-style) or column-major (Fortran-style) order.\n", " | \n", " | Attributes\n", " | ----------\n", " | T : ndarray\n", " | Transpose of the array.\n", " | data : buffer\n", " | The array's elements, in memory.\n", " | dtype : dtype object\n", " | Describes the format of the elements in the array.\n", " | flags : dict\n", " | Dictionary containing information related to memory use, e.g.,\n", " | 'C_CONTIGUOUS', 'OWNDATA', 'WRITEABLE', etc.\n", " | flat : numpy.flatiter object\n", " | Flattened version of the array as an iterator. The iterator\n", " | allows assignments, e.g., ``x.flat = 3`` (See `ndarray.flat` for\n", " | assignment examples; TODO).\n", " | imag : ndarray\n", " | Imaginary part of the array.\n", " | real : ndarray\n", " | Real part of the array.\n", " | size : int\n", " | Number of elements in the array.\n", " | itemsize : int\n", " | The memory use of each array element in bytes.\n", " | nbytes : int\n", " | The total number of bytes required to store the array data,\n", " | i.e., ``itemsize * size``.\n", " | ndim : int\n", " | The array's number of dimensions.\n", " | shape : tuple of ints\n", " | Shape of the array.\n", " | strides : tuple of ints\n", " | The step-size required to move from one element to the next in\n", " | memory. For example, a contiguous ``(3, 4)`` array of type\n", " | ``int16`` in C-order has strides ``(8, 2)``. This implies that\n", " | to move from element to element in memory requires jumps of 2 bytes.\n", " | To move from row-to-row, one needs to jump 8 bytes at a time\n", " | (``2 * 4``).\n", " | ctypes : ctypes object\n", " | Class containing properties of the array needed for interaction\n", " | with ctypes.\n", " | base : ndarray\n", " | If the array is a view into another array, that array is its `base`\n", " | (unless that array is also a view). The `base` array is where the\n", " | array data is actually stored.\n", " | \n", " | See Also\n", " | --------\n", " | array : Construct an array.\n", " | zeros : Create an array, each element of which is zero.\n", " | empty : Create an array, but leave its allocated memory unchanged (i.e.,\n", " | it contains \"garbage\").\n", " | dtype : Create a data-type.\n", " | \n", " | Notes\n", " | -----\n", " | There are two modes of creating an array using ``__new__``:\n", " | \n", " | 1. If `buffer` is None, then only `shape`, `dtype`, and `order`\n", " | are used.\n", " | 2. If `buffer` is an object exposing the buffer interface, then\n", " | all keywords are interpreted.\n", " | \n", " | No ``__init__`` method is needed because the array is fully initialized\n", " | after the ``__new__`` method.\n", " | \n", " | Examples\n", " | --------\n", " | These examples illustrate the low-level `ndarray` constructor. Refer\n", " | to the `See Also` section above for easier ways of constructing an\n", " | ndarray.\n", " | \n", " | First mode, `buffer` is None:\n", " | \n", " | >>> np.ndarray(shape=(2,2), dtype=float, order='F')\n", " | array([[0.0e+000, 0.0e+000], # random\n", " | [ nan, 2.5e-323]])\n", " | \n", " | Second mode:\n", " | \n", " | >>> np.ndarray((2,), buffer=np.array([1,2,3]),\n", " | ... offset=np.int_().itemsize,\n", " | ... dtype=int) # offset = 1*itemsize, i.e. skip first element\n", " | array([2, 3])\n", " | \n", " | Methods defined here:\n", " | \n", " | __abs__(self, /)\n", " | abs(self)\n", " | \n", " | __add__(self, value, /)\n", " | Return self+value.\n", " | \n", " | __and__(self, value, /)\n", " | Return self&value.\n", " | \n", " | __array__(...)\n", " | a.__array__(|dtype) -> reference if type unchanged, copy otherwise.\n", " | \n", " | Returns either a new reference to self if dtype is not given or a new array\n", " | of provided data type if dtype is different from the current dtype of the\n", " | array.\n", " | \n", " | __array_function__(...)\n", " | \n", " | __array_prepare__(...)\n", " | a.__array_prepare__(obj) -> Object of same type as ndarray object obj.\n", " | \n", " | __array_ufunc__(...)\n", " | \n", " | __array_wrap__(...)\n", " | a.__array_wrap__(obj) -> Object of same type as ndarray object a.\n", " | \n", " | __bool__(self, /)\n", " | self != 0\n", " | \n", " | __complex__(...)\n", " | \n", " | __contains__(self, key, /)\n", " | Return key in self.\n", " | \n", " | __copy__(...)\n", " | a.__copy__()\n", " | \n", " | Used if :func:`copy.copy` is called on an array. Returns a copy of the array.\n", " | \n", " | Equivalent to ``a.copy(order='K')``.\n", " | \n", " | __deepcopy__(...)\n", " | a.__deepcopy__(memo, /) -> Deep copy of array.\n", " | \n", " | Used if :func:`copy.deepcopy` is called on an array.\n", " | \n", " | __delitem__(self, key, /)\n", " | Delete self[key].\n", " | \n", " | __divmod__(self, value, /)\n", " | Return divmod(self, value).\n", " | \n", " | __eq__(self, value, /)\n", " | Return self==value.\n", " | \n", " | __float__(self, /)\n", " | float(self)\n", " | \n", " | __floordiv__(self, value, /)\n", " | Return self//value.\n", " | \n", " | __format__(...)\n", " | Default object formatter.\n", " | \n", " | __ge__(self, value, /)\n", " | Return self>=value.\n", " | \n", " | __getitem__(self, key, /)\n", " | Return self[key].\n", " | \n", " | __gt__(self, value, /)\n", " | Return self>value.\n", " | \n", " | __iadd__(self, value, /)\n", " | Return self+=value.\n", " | \n", " | __iand__(self, value, /)\n", " | Return self&=value.\n", " | \n", " | __ifloordiv__(self, value, /)\n", " | Return self//=value.\n", " | \n", " | __ilshift__(self, value, /)\n", " | Return self<<=value.\n", " | \n", " | __imatmul__(self, value, /)\n", " | Return self@=value.\n", " | \n", " | __imod__(self, value, /)\n", " | Return self%=value.\n", " | \n", " | __imul__(self, value, /)\n", " | Return self*=value.\n", " | \n", " | __index__(self, /)\n", " | Return self converted to an integer, if self is suitable for use as an index into a list.\n", " | \n", " | __int__(self, /)\n", " | int(self)\n", " | \n", " | __invert__(self, /)\n", " | ~self\n", " | \n", " | __ior__(self, value, /)\n", " | Return self|=value.\n", " | \n", " | __ipow__(self, value, /)\n", " | Return self**=value.\n", " | \n", " | __irshift__(self, value, /)\n", " | Return self>>=value.\n", " | \n", " | __isub__(self, value, /)\n", " | Return self-=value.\n", " | \n", " | __iter__(self, /)\n", " | Implement iter(self).\n", " | \n", " | __itruediv__(self, value, /)\n", " | Return self/=value.\n", " | \n", " | __ixor__(self, value, /)\n", " | Return self^=value.\n", " | \n", " | __le__(self, value, /)\n", " | Return self<=value.\n", " | \n", " | __len__(self, /)\n", " | Return len(self).\n", " | \n", " | __lshift__(self, value, /)\n", " | Return self<>self.\n", " | \n", " | __rshift__(self, value, /)\n", " | Return self>>value.\n", " | \n", " | __rsub__(self, value, /)\n", " | Return value-self.\n", " | \n", " | __rtruediv__(self, value, /)\n", " | Return value/self.\n", " | \n", " | __rxor__(self, value, /)\n", " | Return value^self.\n", " | \n", " | __setitem__(self, key, value, /)\n", " | Set self[key] to value.\n", " | \n", " | __setstate__(...)\n", " | a.__setstate__(state, /)\n", " | \n", " | For unpickling.\n", " | \n", " | The `state` argument must be a sequence that contains the following\n", " | elements:\n", " | \n", " | Parameters\n", " | ----------\n", " | version : int\n", " | optional pickle version. If omitted defaults to 0.\n", " | shape : tuple\n", " | dtype : data-type\n", " | isFortran : bool\n", " | rawdata : string or list\n", " | a binary string with the data (or a list if 'a' is an object array)\n", " | \n", " | __sizeof__(...)\n", " | Size of object in memory, in bytes.\n", " | \n", " | __str__(self, /)\n", " | Return str(self).\n", " | \n", " | __sub__(self, value, /)\n", " | Return self-value.\n", " | \n", " | __truediv__(self, value, /)\n", " | Return self/value.\n", " | \n", " | __xor__(self, value, /)\n", " | Return self^value.\n", " | \n", " | all(...)\n", " | a.all(axis=None, out=None, keepdims=False)\n", " | \n", " | Returns True if all elements evaluate to True.\n", " | \n", " | Refer to `numpy.all` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.all : equivalent function\n", " | \n", " | any(...)\n", " | a.any(axis=None, out=None, keepdims=False)\n", " | \n", " | Returns True if any of the elements of `a` evaluate to True.\n", " | \n", " | Refer to `numpy.any` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.any : equivalent function\n", " | \n", " | argmax(...)\n", " | a.argmax(axis=None, out=None)\n", " | \n", " | Return indices of the maximum values along the given axis.\n", " | \n", " | Refer to `numpy.argmax` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.argmax : equivalent function\n", " | \n", " | argmin(...)\n", " | a.argmin(axis=None, out=None)\n", " | \n", " | Return indices of the minimum values along the given axis of `a`.\n", " | \n", " | Refer to `numpy.argmin` for detailed documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.argmin : equivalent function\n", " | \n", " | argpartition(...)\n", " | a.argpartition(kth, axis=-1, kind='introselect', order=None)\n", " | \n", " | Returns the indices that would partition this array.\n", " | \n", " | Refer to `numpy.argpartition` for full documentation.\n", " | \n", " | .. versionadded:: 1.8.0\n", " | \n", " | See Also\n", " | --------\n", " | numpy.argpartition : equivalent function\n", " | \n", " | argsort(...)\n", " | a.argsort(axis=-1, kind=None, order=None)\n", " | \n", " | Returns the indices that would sort this array.\n", " | \n", " | Refer to `numpy.argsort` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.argsort : equivalent function\n", " | \n", " | astype(...)\n", " | a.astype(dtype, order='K', casting='unsafe', subok=True, copy=True)\n", " | \n", " | Copy of the array, cast to a specified type.\n", " | \n", " | Parameters\n", " | ----------\n", " | dtype : str or dtype\n", " | Typecode or data-type to which the array is cast.\n", " | order : {'C', 'F', 'A', 'K'}, optional\n", " | Controls the memory layout order of the result.\n", " | 'C' means C order, 'F' means Fortran order, 'A'\n", " | means 'F' order if all the arrays are Fortran contiguous,\n", " | 'C' order otherwise, and 'K' means as close to the\n", " | order the array elements appear in memory as possible.\n", " | Default is 'K'.\n", " | casting : {'no', 'equiv', 'safe', 'same_kind', 'unsafe'}, optional\n", " | Controls what kind of data casting may occur. Defaults to 'unsafe'\n", " | for backwards compatibility.\n", " | \n", " | * 'no' means the data types should not be cast at all.\n", " | * 'equiv' means only byte-order changes are allowed.\n", " | * 'safe' means only casts which can preserve values are allowed.\n", " | * 'same_kind' means only safe casts or casts within a kind,\n", " | like float64 to float32, are allowed.\n", " | * 'unsafe' means any data conversions may be done.\n", " | subok : bool, optional\n", " | If True, then sub-classes will be passed-through (default), otherwise\n", " | the returned array will be forced to be a base-class array.\n", " | copy : bool, optional\n", " | By default, astype always returns a newly allocated array. If this\n", " | is set to false, and the `dtype`, `order`, and `subok`\n", " | requirements are satisfied, the input array is returned instead\n", " | of a copy.\n", " | \n", " | Returns\n", " | -------\n", " | arr_t : ndarray\n", " | Unless `copy` is False and the other conditions for returning the input\n", " | array are satisfied (see description for `copy` input parameter), `arr_t`\n", " | is a new array of the same shape as the input array, with dtype, order\n", " | given by `dtype`, `order`.\n", " | \n", " | Notes\n", " | -----\n", " | .. versionchanged:: 1.17.0\n", " | Casting between a simple data type and a structured one is possible only\n", " | for \"unsafe\" casting. Casting to multiple fields is allowed, but\n", " | casting from multiple fields is not.\n", " | \n", " | .. versionchanged:: 1.9.0\n", " | Casting from numeric to string types in 'safe' casting mode requires\n", " | that the string dtype length is long enough to store the max\n", " | integer/float value converted.\n", " | \n", " | Raises\n", " | ------\n", " | ComplexWarning\n", " | When casting from complex to float or int. To avoid this,\n", " | one should use ``a.real.astype(t)``.\n", " | \n", " | Examples\n", " | --------\n", " | >>> x = np.array([1, 2, 2.5])\n", " | >>> x\n", " | array([1. , 2. , 2.5])\n", " | \n", " | >>> x.astype(int)\n", " | array([1, 2, 2])\n", " | \n", " | byteswap(...)\n", " | a.byteswap(inplace=False)\n", " | \n", " | Swap the bytes of the array elements\n", " | \n", " | Toggle between low-endian and big-endian data representation by\n", " | returning a byteswapped array, optionally swapped in-place.\n", " | Arrays of byte-strings are not swapped. The real and imaginary\n", " | parts of a complex number are swapped individually.\n", " | \n", " | Parameters\n", " | ----------\n", " | inplace : bool, optional\n", " | If ``True``, swap bytes in-place, default is ``False``.\n", " | \n", " | Returns\n", " | -------\n", " | out : ndarray\n", " | The byteswapped array. If `inplace` is ``True``, this is\n", " | a view to self.\n", " | \n", " | Examples\n", " | --------\n", " | >>> A = np.array([1, 256, 8755], dtype=np.int16)\n", " | >>> list(map(hex, A))\n", " | ['0x1', '0x100', '0x2233']\n", " | >>> A.byteswap(inplace=True)\n", " | array([ 256, 1, 13090], dtype=int16)\n", " | >>> list(map(hex, A))\n", " | ['0x100', '0x1', '0x3322']\n", " | \n", " | Arrays of byte-strings are not swapped\n", " | \n", " | >>> A = np.array([b'ceg', b'fac'])\n", " | >>> A.byteswap()\n", " | array([b'ceg', b'fac'], dtype='|S3')\n", " | \n", " | ``A.newbyteorder().byteswap()`` produces an array with the same values\n", " | but different representation in memory\n", " | \n", " | >>> A = np.array([1, 2, 3])\n", " | >>> A.view(np.uint8)\n", " | array([1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0,\n", " | 0, 0], dtype=uint8)\n", " | >>> A.newbyteorder().byteswap(inplace=True)\n", " | array([1, 2, 3])\n", " | >>> A.view(np.uint8)\n", " | array([0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0,\n", " | 0, 3], dtype=uint8)\n", " | \n", " | choose(...)\n", " | a.choose(choices, out=None, mode='raise')\n", " | \n", " | Use an index array to construct a new array from a set of choices.\n", " | \n", " | Refer to `numpy.choose` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.choose : equivalent function\n", " | \n", " | clip(...)\n", " | a.clip(min=None, max=None, out=None, **kwargs)\n", " | \n", " | Return an array whose values are limited to ``[min, max]``.\n", " | One of max or min must be given.\n", " | \n", " | Refer to `numpy.clip` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.clip : equivalent function\n", " | \n", " | compress(...)\n", " | a.compress(condition, axis=None, out=None)\n", " | \n", " | Return selected slices of this array along given axis.\n", " | \n", " | Refer to `numpy.compress` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.compress : equivalent function\n", " | \n", " | conj(...)\n", " | a.conj()\n", " | \n", " | Complex-conjugate all elements.\n", " | \n", " | Refer to `numpy.conjugate` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.conjugate : equivalent function\n", " | \n", " | conjugate(...)\n", " | a.conjugate()\n", " | \n", " | Return the complex conjugate, element-wise.\n", " | \n", " | Refer to `numpy.conjugate` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.conjugate : equivalent function\n", " | \n", " | copy(...)\n", " | a.copy(order='C')\n", " | \n", " | Return a copy of the array.\n", " | \n", " | Parameters\n", " | ----------\n", " | order : {'C', 'F', 'A', 'K'}, optional\n", " | Controls the memory layout of the copy. 'C' means C-order,\n", " | 'F' means F-order, 'A' means 'F' if `a` is Fortran contiguous,\n", " | 'C' otherwise. 'K' means match the layout of `a` as closely\n", " | as possible. (Note that this function and :func:`numpy.copy` are very\n", " | similar, but have different default values for their order=\n", " | arguments.)\n", " | \n", " | See also\n", " | --------\n", " | numpy.copy\n", " | numpy.copyto\n", " | \n", " | Examples\n", " | --------\n", " | >>> x = np.array([[1,2,3],[4,5,6]], order='F')\n", " | \n", " | >>> y = x.copy()\n", " | \n", " | >>> x.fill(0)\n", " | \n", " | >>> x\n", " | array([[0, 0, 0],\n", " | [0, 0, 0]])\n", " | \n", " | >>> y\n", " | array([[1, 2, 3],\n", " | [4, 5, 6]])\n", " | \n", " | >>> y.flags['C_CONTIGUOUS']\n", " | True\n", " | \n", " | cumprod(...)\n", " | a.cumprod(axis=None, dtype=None, out=None)\n", " | \n", " | Return the cumulative product of the elements along the given axis.\n", " | \n", " | Refer to `numpy.cumprod` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.cumprod : equivalent function\n", " | \n", " | cumsum(...)\n", " | a.cumsum(axis=None, dtype=None, out=None)\n", " | \n", " | Return the cumulative sum of the elements along the given axis.\n", " | \n", " | Refer to `numpy.cumsum` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.cumsum : equivalent function\n", " | \n", " | diagonal(...)\n", " | a.diagonal(offset=0, axis1=0, axis2=1)\n", " | \n", " | Return specified diagonals. In NumPy 1.9 the returned array is a\n", " | read-only view instead of a copy as in previous NumPy versions. In\n", " | a future version the read-only restriction will be removed.\n", " | \n", " | Refer to :func:`numpy.diagonal` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.diagonal : equivalent function\n", " | \n", " | dot(...)\n", " | a.dot(b, out=None)\n", " | \n", " | Dot product of two arrays.\n", " | \n", " | Refer to `numpy.dot` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.dot : equivalent function\n", " | \n", " | Examples\n", " | --------\n", " | >>> a = np.eye(2)\n", " | >>> b = np.ones((2, 2)) * 2\n", " | >>> a.dot(b)\n", " | array([[2., 2.],\n", " | [2., 2.]])\n", " | \n", " | This array method can be conveniently chained:\n", " | \n", " | >>> a.dot(b).dot(b)\n", " | array([[8., 8.],\n", " | [8., 8.]])\n", " | \n", " | dump(...)\n", " | a.dump(file)\n", " | \n", " | Dump a pickle of the array to the specified file.\n", " | The array can be read back with pickle.load or numpy.load.\n", " | \n", " | Parameters\n", " | ----------\n", " | file : str or Path\n", " | A string naming the dump file.\n", " | \n", " | .. versionchanged:: 1.17.0\n", " | `pathlib.Path` objects are now accepted.\n", " | \n", " | dumps(...)\n", " | a.dumps()\n", " | \n", " | Returns the pickle of the array as a string.\n", " | pickle.loads or numpy.loads will convert the string back to an array.\n", " | \n", " | Parameters\n", " | ----------\n", " | None\n", " | \n", " | fill(...)\n", " | a.fill(value)\n", " | \n", " | Fill the array with a scalar value.\n", " | \n", " | Parameters\n", " | ----------\n", " | value : scalar\n", " | All elements of `a` will be assigned this value.\n", " | \n", " | Examples\n", " | --------\n", " | >>> a = np.array([1, 2])\n", " | >>> a.fill(0)\n", " | >>> a\n", " | array([0, 0])\n", " | >>> a = np.empty(2)\n", " | >>> a.fill(1)\n", " | >>> a\n", " | array([1., 1.])\n", " | \n", " | flatten(...)\n", " | a.flatten(order='C')\n", " | \n", " | Return a copy of the array collapsed into one dimension.\n", " | \n", " | Parameters\n", " | ----------\n", " | order : {'C', 'F', 'A', 'K'}, optional\n", " | 'C' means to flatten in row-major (C-style) order.\n", " | 'F' means to flatten in column-major (Fortran-\n", " | style) order. 'A' means to flatten in column-major\n", " | order if `a` is Fortran *contiguous* in memory,\n", " | row-major order otherwise. 'K' means to flatten\n", " | `a` in the order the elements occur in memory.\n", " | The default is 'C'.\n", " | \n", " | Returns\n", " | -------\n", " | y : ndarray\n", " | A copy of the input array, flattened to one dimension.\n", " | \n", " | See Also\n", " | --------\n", " | ravel : Return a flattened array.\n", " | flat : A 1-D flat iterator over the array.\n", " | \n", " | Examples\n", " | --------\n", " | >>> a = np.array([[1,2], [3,4]])\n", " | >>> a.flatten()\n", " | array([1, 2, 3, 4])\n", " | >>> a.flatten('F')\n", " | array([1, 3, 2, 4])\n", " | \n", " | getfield(...)\n", " | a.getfield(dtype, offset=0)\n", " | \n", " | Returns a field of the given array as a certain type.\n", " | \n", " | A field is a view of the array data with a given data-type. The values in\n", " | the view are determined by the given type and the offset into the current\n", " | array in bytes. The offset needs to be such that the view dtype fits in the\n", " | array dtype; for example an array of dtype complex128 has 16-byte elements.\n", " | If taking a view with a 32-bit integer (4 bytes), the offset needs to be\n", " | between 0 and 12 bytes.\n", " | \n", " | Parameters\n", " | ----------\n", " | dtype : str or dtype\n", " | The data type of the view. The dtype size of the view can not be larger\n", " | than that of the array itself.\n", " | offset : int\n", " | Number of bytes to skip before beginning the element view.\n", " | \n", " | Examples\n", " | --------\n", " | >>> x = np.diag([1.+1.j]*2)\n", " | >>> x[1, 1] = 2 + 4.j\n", " | >>> x\n", " | array([[1.+1.j, 0.+0.j],\n", " | [0.+0.j, 2.+4.j]])\n", " | >>> x.getfield(np.float64)\n", " | array([[1., 0.],\n", " | [0., 2.]])\n", " | \n", " | By choosing an offset of 8 bytes we can select the complex part of the\n", " | array for our view:\n", " | \n", " | >>> x.getfield(np.float64, offset=8)\n", " | array([[1., 0.],\n", " | [0., 4.]])\n", " | \n", " | item(...)\n", " | a.item(*args)\n", " | \n", " | Copy an element of an array to a standard Python scalar and return it.\n", " | \n", " | Parameters\n", " | ----------\n", " | \\*args : Arguments (variable number and type)\n", " | \n", " | * none: in this case, the method only works for arrays\n", " | with one element (`a.size == 1`), which element is\n", " | copied into a standard Python scalar object and returned.\n", " | \n", " | * int_type: this argument is interpreted as a flat index into\n", " | the array, specifying which element to copy and return.\n", " | \n", " | * tuple of int_types: functions as does a single int_type argument,\n", " | except that the argument is interpreted as an nd-index into the\n", " | array.\n", " | \n", " | Returns\n", " | -------\n", " | z : Standard Python scalar object\n", " | A copy of the specified element of the array as a suitable\n", " | Python scalar\n", " | \n", " | Notes\n", " | -----\n", " | When the data type of `a` is longdouble or clongdouble, item() returns\n", " | a scalar array object because there is no available Python scalar that\n", " | would not lose information. Void arrays return a buffer object for item(),\n", " | unless fields are defined, in which case a tuple is returned.\n", " | \n", " | `item` is very similar to a[args], except, instead of an array scalar,\n", " | a standard Python scalar is returned. This can be useful for speeding up\n", " | access to elements of the array and doing arithmetic on elements of the\n", " | array using Python's optimized math.\n", " | \n", " | Examples\n", " | --------\n", " | >>> np.random.seed(123)\n", " | >>> x = np.random.randint(9, size=(3, 3))\n", " | >>> x\n", " | array([[2, 2, 6],\n", " | [1, 3, 6],\n", " | [1, 0, 1]])\n", " | >>> x.item(3)\n", " | 1\n", " | >>> x.item(7)\n", " | 0\n", " | >>> x.item((0, 1))\n", " | 2\n", " | >>> x.item((2, 2))\n", " | 1\n", " | \n", " | itemset(...)\n", " | a.itemset(*args)\n", " | \n", " | Insert scalar into an array (scalar is cast to array's dtype, if possible)\n", " | \n", " | There must be at least 1 argument, and define the last argument\n", " | as *item*. Then, ``a.itemset(*args)`` is equivalent to but faster\n", " | than ``a[args] = item``. The item should be a scalar value and `args`\n", " | must select a single item in the array `a`.\n", " | \n", " | Parameters\n", " | ----------\n", " | \\*args : Arguments\n", " | If one argument: a scalar, only used in case `a` is of size 1.\n", " | If two arguments: the last argument is the value to be set\n", " | and must be a scalar, the first argument specifies a single array\n", " | element location. It is either an int or a tuple.\n", " | \n", " | Notes\n", " | -----\n", " | Compared to indexing syntax, `itemset` provides some speed increase\n", " | for placing a scalar into a particular location in an `ndarray`,\n", " | if you must do this. However, generally this is discouraged:\n", " | among other problems, it complicates the appearance of the code.\n", " | Also, when using `itemset` (and `item`) inside a loop, be sure\n", " | to assign the methods to a local variable to avoid the attribute\n", " | look-up at each loop iteration.\n", " | \n", " | Examples\n", " | --------\n", " | >>> np.random.seed(123)\n", " | >>> x = np.random.randint(9, size=(3, 3))\n", " | >>> x\n", " | array([[2, 2, 6],\n", " | [1, 3, 6],\n", " | [1, 0, 1]])\n", " | >>> x.itemset(4, 0)\n", " | >>> x.itemset((2, 2), 9)\n", " | >>> x\n", " | array([[2, 2, 6],\n", " | [1, 0, 6],\n", " | [1, 0, 9]])\n", " | \n", " | max(...)\n", " | a.max(axis=None, out=None, keepdims=False, initial=, where=True)\n", " | \n", " | Return the maximum along a given axis.\n", " | \n", " | Refer to `numpy.amax` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.amax : equivalent function\n", " | \n", " | mean(...)\n", " | a.mean(axis=None, dtype=None, out=None, keepdims=False)\n", " | \n", " | Returns the average of the array elements along given axis.\n", " | \n", " | Refer to `numpy.mean` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.mean : equivalent function\n", " | \n", " | min(...)\n", " | a.min(axis=None, out=None, keepdims=False, initial=, where=True)\n", " | \n", " | Return the minimum along a given axis.\n", " | \n", " | Refer to `numpy.amin` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.amin : equivalent function\n", " | \n", " | newbyteorder(...)\n", " | arr.newbyteorder(new_order='S')\n", " | \n", " | Return the array with the same data viewed with a different byte order.\n", " | \n", " | Equivalent to::\n", " | \n", " | arr.view(arr.dtype.newbytorder(new_order))\n", " | \n", " | Changes are also made in all fields and sub-arrays of the array data\n", " | type.\n", " | \n", " | \n", " | \n", " | Parameters\n", " | ----------\n", " | new_order : string, optional\n", " | Byte order to force; a value from the byte order specifications\n", " | below. `new_order` codes can be any of:\n", " | \n", " | * 'S' - swap dtype from current to opposite endian\n", " | * {'<', 'L'} - little endian\n", " | * {'>', 'B'} - big endian\n", " | * {'=', 'N'} - native order\n", " | * {'|', 'I'} - ignore (no change to byte order)\n", " | \n", " | The default value ('S') results in swapping the current\n", " | byte order. The code does a case-insensitive check on the first\n", " | letter of `new_order` for the alternatives above. For example,\n", " | any of 'B' or 'b' or 'biggish' are valid to specify big-endian.\n", " | \n", " | \n", " | Returns\n", " | -------\n", " | new_arr : array\n", " | New array object with the dtype reflecting given change to the\n", " | byte order.\n", " | \n", " | nonzero(...)\n", " | a.nonzero()\n", " | \n", " | Return the indices of the elements that are non-zero.\n", " | \n", " | Refer to `numpy.nonzero` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.nonzero : equivalent function\n", " | \n", " | partition(...)\n", " | a.partition(kth, axis=-1, kind='introselect', order=None)\n", " | \n", " | Rearranges the elements in the array in such a way that the value of the\n", " | element in kth position is in the position it would be in a sorted array.\n", " | All elements smaller than the kth element are moved before this element and\n", " | all equal or greater are moved behind it. The ordering of the elements in\n", " | the two partitions is undefined.\n", " | \n", " | .. versionadded:: 1.8.0\n", " | \n", " | Parameters\n", " | ----------\n", " | kth : int or sequence of ints\n", " | Element index to partition by. The kth element value will be in its\n", " | final sorted position and all smaller elements will be moved before it\n", " | and all equal or greater elements behind it.\n", " | The order of all elements in the partitions is undefined.\n", " | If provided with a sequence of kth it will partition all elements\n", " | indexed by kth of them into their sorted position at once.\n", " | axis : int, optional\n", " | Axis along which to sort. Default is -1, which means sort along the\n", " | last axis.\n", " | kind : {'introselect'}, optional\n", " | Selection algorithm. Default is 'introselect'.\n", " | order : str or list of str, optional\n", " | When `a` is an array with fields defined, this argument specifies\n", " | which fields to compare first, second, etc. A single field can\n", " | be specified as a string, and not all fields need to be specified,\n", " | but unspecified fields will still be used, in the order in which\n", " | they come up in the dtype, to break ties.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.partition : Return a parititioned copy of an array.\n", " | argpartition : Indirect partition.\n", " | sort : Full sort.\n", " | \n", " | Notes\n", " | -----\n", " | See ``np.partition`` for notes on the different algorithms.\n", " | \n", " | Examples\n", " | --------\n", " | >>> a = np.array([3, 4, 2, 1])\n", " | >>> a.partition(3)\n", " | >>> a\n", " | array([2, 1, 3, 4])\n", " | \n", " | >>> a.partition((1, 3))\n", " | >>> a\n", " | array([1, 2, 3, 4])\n", " | \n", " | prod(...)\n", " | a.prod(axis=None, dtype=None, out=None, keepdims=False, initial=1, where=True)\n", " | \n", " | Return the product of the array elements over the given axis\n", " | \n", " | Refer to `numpy.prod` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.prod : equivalent function\n", " | \n", " | ptp(...)\n", " | a.ptp(axis=None, out=None, keepdims=False)\n", " | \n", " | Peak to peak (maximum - minimum) value along a given axis.\n", " | \n", " | Refer to `numpy.ptp` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.ptp : equivalent function\n", " | \n", " | put(...)\n", " | a.put(indices, values, mode='raise')\n", " | \n", " | Set ``a.flat[n] = values[n]`` for all `n` in indices.\n", " | \n", " | Refer to `numpy.put` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.put : equivalent function\n", " | \n", " | ravel(...)\n", " | a.ravel([order])\n", " | \n", " | Return a flattened array.\n", " | \n", " | Refer to `numpy.ravel` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.ravel : equivalent function\n", " | \n", " | ndarray.flat : a flat iterator on the array.\n", " | \n", " | repeat(...)\n", " | a.repeat(repeats, axis=None)\n", " | \n", " | Repeat elements of an array.\n", " | \n", " | Refer to `numpy.repeat` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.repeat : equivalent function\n", " | \n", " | reshape(...)\n", " | a.reshape(shape, order='C')\n", " | \n", " | Returns an array containing the same data with a new shape.\n", " | \n", " | Refer to `numpy.reshape` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.reshape : equivalent function\n", " | \n", " | Notes\n", " | -----\n", " | Unlike the free function `numpy.reshape`, this method on `ndarray` allows\n", " | the elements of the shape parameter to be passed in as separate arguments.\n", " | For example, ``a.reshape(10, 11)`` is equivalent to\n", " | ``a.reshape((10, 11))``.\n", " | \n", " | resize(...)\n", " | a.resize(new_shape, refcheck=True)\n", " | \n", " | Change shape and size of array in-place.\n", " | \n", " | Parameters\n", " | ----------\n", " | new_shape : tuple of ints, or `n` ints\n", " | Shape of resized array.\n", " | refcheck : bool, optional\n", " | If False, reference count will not be checked. Default is True.\n", " | \n", " | Returns\n", " | -------\n", " | None\n", " | \n", " | Raises\n", " | ------\n", " | ValueError\n", " | If `a` does not own its own data or references or views to it exist,\n", " | and the data memory must be changed.\n", " | PyPy only: will always raise if the data memory must be changed, since\n", " | there is no reliable way to determine if references or views to it\n", " | exist.\n", " | \n", " | SystemError\n", " | If the `order` keyword argument is specified. This behaviour is a\n", " | bug in NumPy.\n", " | \n", " | See Also\n", " | --------\n", " | resize : Return a new array with the specified shape.\n", " | \n", " | Notes\n", " | -----\n", " | This reallocates space for the data area if necessary.\n", " | \n", " | Only contiguous arrays (data elements consecutive in memory) can be\n", " | resized.\n", " | \n", " | The purpose of the reference count check is to make sure you\n", " | do not use this array as a buffer for another Python object and then\n", " | reallocate the memory. However, reference counts can increase in\n", " | other ways so if you are sure that you have not shared the memory\n", " | for this array with another Python object, then you may safely set\n", " | `refcheck` to False.\n", " | \n", " | Examples\n", " | --------\n", " | Shrinking an array: array is flattened (in the order that the data are\n", " | stored in memory), resized, and reshaped:\n", " | \n", " | >>> a = np.array([[0, 1], [2, 3]], order='C')\n", " | >>> a.resize((2, 1))\n", " | >>> a\n", " | array([[0],\n", " | [1]])\n", " | \n", " | >>> a = np.array([[0, 1], [2, 3]], order='F')\n", " | >>> a.resize((2, 1))\n", " | >>> a\n", " | array([[0],\n", " | [2]])\n", " | \n", " | Enlarging an array: as above, but missing entries are filled with zeros:\n", " | \n", " | >>> b = np.array([[0, 1], [2, 3]])\n", " | >>> b.resize(2, 3) # new_shape parameter doesn't have to be a tuple\n", " | >>> b\n", " | array([[0, 1, 2],\n", " | [3, 0, 0]])\n", " | \n", " | Referencing an array prevents resizing...\n", " | \n", " | >>> c = a\n", " | >>> a.resize((1, 1))\n", " | Traceback (most recent call last):\n", " | ...\n", " | ValueError: cannot resize an array that references or is referenced ...\n", " | \n", " | Unless `refcheck` is False:\n", " | \n", " | >>> a.resize((1, 1), refcheck=False)\n", " | >>> a\n", " | array([[0]])\n", " | >>> c\n", " | array([[0]])\n", " | \n", " | round(...)\n", " | a.round(decimals=0, out=None)\n", " | \n", " | Return `a` with each element rounded to the given number of decimals.\n", " | \n", " | Refer to `numpy.around` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.around : equivalent function\n", " | \n", " | searchsorted(...)\n", " | a.searchsorted(v, side='left', sorter=None)\n", " | \n", " | Find indices where elements of v should be inserted in a to maintain order.\n", " | \n", " | For full documentation, see `numpy.searchsorted`\n", " | \n", " | See Also\n", " | --------\n", " | numpy.searchsorted : equivalent function\n", " | \n", " | setfield(...)\n", " | a.setfield(val, dtype, offset=0)\n", " | \n", " | Put a value into a specified place in a field defined by a data-type.\n", " | \n", " | Place `val` into `a`'s field defined by `dtype` and beginning `offset`\n", " | bytes into the field.\n", " | \n", " | Parameters\n", " | ----------\n", " | val : object\n", " | Value to be placed in field.\n", " | dtype : dtype object\n", " | Data-type of the field in which to place `val`.\n", " | offset : int, optional\n", " | The number of bytes into the field at which to place `val`.\n", " | \n", " | Returns\n", " | -------\n", " | None\n", " | \n", " | See Also\n", " | --------\n", " | getfield\n", " | \n", " | Examples\n", " | --------\n", " | >>> x = np.eye(3)\n", " | >>> x.getfield(np.float64)\n", " | array([[1., 0., 0.],\n", " | [0., 1., 0.],\n", " | [0., 0., 1.]])\n", " | >>> x.setfield(3, np.int32)\n", " | >>> x.getfield(np.int32)\n", " | array([[3, 3, 3],\n", " | [3, 3, 3],\n", " | [3, 3, 3]], dtype=int32)\n", " | >>> x\n", " | array([[1.0e+000, 1.5e-323, 1.5e-323],\n", " | [1.5e-323, 1.0e+000, 1.5e-323],\n", " | [1.5e-323, 1.5e-323, 1.0e+000]])\n", " | >>> x.setfield(np.eye(3), np.int32)\n", " | >>> x\n", " | array([[1., 0., 0.],\n", " | [0., 1., 0.],\n", " | [0., 0., 1.]])\n", " | \n", " | setflags(...)\n", " | a.setflags(write=None, align=None, uic=None)\n", " | \n", " | Set array flags WRITEABLE, ALIGNED, (WRITEBACKIFCOPY and UPDATEIFCOPY),\n", " | respectively.\n", " | \n", " | These Boolean-valued flags affect how numpy interprets the memory\n", " | area used by `a` (see Notes below). The ALIGNED flag can only\n", " | be set to True if the data is actually aligned according to the type.\n", " | The WRITEBACKIFCOPY and (deprecated) UPDATEIFCOPY flags can never be set\n", " | to True. The flag WRITEABLE can only be set to True if the array owns its\n", " | own memory, or the ultimate owner of the memory exposes a writeable buffer\n", " | interface, or is a string. (The exception for string is made so that\n", " | unpickling can be done without copying memory.)\n", " | \n", " | Parameters\n", " | ----------\n", " | write : bool, optional\n", " | Describes whether or not `a` can be written to.\n", " | align : bool, optional\n", " | Describes whether or not `a` is aligned properly for its type.\n", " | uic : bool, optional\n", " | Describes whether or not `a` is a copy of another \"base\" array.\n", " | \n", " | Notes\n", " | -----\n", " | Array flags provide information about how the memory area used\n", " | for the array is to be interpreted. There are 7 Boolean flags\n", " | in use, only four of which can be changed by the user:\n", " | WRITEBACKIFCOPY, UPDATEIFCOPY, WRITEABLE, and ALIGNED.\n", " | \n", " | WRITEABLE (W) the data area can be written to;\n", " | \n", " | ALIGNED (A) the data and strides are aligned appropriately for the hardware\n", " | (as determined by the compiler);\n", " | \n", " | UPDATEIFCOPY (U) (deprecated), replaced by WRITEBACKIFCOPY;\n", " | \n", " | WRITEBACKIFCOPY (X) this array is a copy of some other array (referenced\n", " | by .base). When the C-API function PyArray_ResolveWritebackIfCopy is\n", " | called, the base array will be updated with the contents of this array.\n", " | \n", " | All flags can be accessed using the single (upper case) letter as well\n", " | as the full name.\n", " | \n", " | Examples\n", " | --------\n", " | >>> y = np.array([[3, 1, 7],\n", " | ... [2, 0, 0],\n", " | ... [8, 5, 9]])\n", " | >>> y\n", " | array([[3, 1, 7],\n", " | [2, 0, 0],\n", " | [8, 5, 9]])\n", " | >>> y.flags\n", " | C_CONTIGUOUS : True\n", " | F_CONTIGUOUS : False\n", " | OWNDATA : True\n", " | WRITEABLE : True\n", " | ALIGNED : True\n", " | WRITEBACKIFCOPY : False\n", " | UPDATEIFCOPY : False\n", " | >>> y.setflags(write=0, align=0)\n", " | >>> y.flags\n", " | C_CONTIGUOUS : True\n", " | F_CONTIGUOUS : False\n", " | OWNDATA : True\n", " | WRITEABLE : False\n", " | ALIGNED : False\n", " | WRITEBACKIFCOPY : False\n", " | UPDATEIFCOPY : False\n", " | >>> y.setflags(uic=1)\n", " | Traceback (most recent call last):\n", " | File \"\", line 1, in \n", " | ValueError: cannot set WRITEBACKIFCOPY flag to True\n", " | \n", " | sort(...)\n", " | a.sort(axis=-1, kind=None, order=None)\n", " | \n", " | Sort an array in-place. Refer to `numpy.sort` for full documentation.\n", " | \n", " | Parameters\n", " | ----------\n", " | axis : int, optional\n", " | Axis along which to sort. Default is -1, which means sort along the\n", " | last axis.\n", " | kind : {'quicksort', 'mergesort', 'heapsort', 'stable'}, optional\n", " | Sorting algorithm. The default is 'quicksort'. Note that both 'stable'\n", " | and 'mergesort' use timsort under the covers and, in general, the\n", " | actual implementation will vary with datatype. The 'mergesort' option\n", " | is retained for backwards compatibility.\n", " | \n", " | .. versionchanged:: 1.15.0.\n", " | The 'stable' option was added.\n", " | \n", " | order : str or list of str, optional\n", " | When `a` is an array with fields defined, this argument specifies\n", " | which fields to compare first, second, etc. A single field can\n", " | be specified as a string, and not all fields need be specified,\n", " | but unspecified fields will still be used, in the order in which\n", " | they come up in the dtype, to break ties.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.sort : Return a sorted copy of an array.\n", " | numpy.argsort : Indirect sort.\n", " | numpy.lexsort : Indirect stable sort on multiple keys.\n", " | numpy.searchsorted : Find elements in sorted array.\n", " | numpy.partition: Partial sort.\n", " | \n", " | Notes\n", " | -----\n", " | See `numpy.sort` for notes on the different sorting algorithms.\n", " | \n", " | Examples\n", " | --------\n", " | >>> a = np.array([[1,4], [3,1]])\n", " | >>> a.sort(axis=1)\n", " | >>> a\n", " | array([[1, 4],\n", " | [1, 3]])\n", " | >>> a.sort(axis=0)\n", " | >>> a\n", " | array([[1, 3],\n", " | [1, 4]])\n", " | \n", " | Use the `order` keyword to specify a field to use when sorting a\n", " | structured array:\n", " | \n", " | >>> a = np.array([('a', 2), ('c', 1)], dtype=[('x', 'S1'), ('y', int)])\n", " | >>> a.sort(order='y')\n", " | >>> a\n", " | array([(b'c', 1), (b'a', 2)],\n", " | dtype=[('x', 'S1'), ('y', '>> x = np.array([[0, 1], [2, 3]], dtype='>> x.tobytes()\n", " | b'\\x00\\x00\\x01\\x00\\x02\\x00\\x03\\x00'\n", " | >>> x.tobytes('C') == x.tobytes()\n", " | True\n", " | >>> x.tobytes('F')\n", " | b'\\x00\\x00\\x02\\x00\\x01\\x00\\x03\\x00'\n", " | \n", " | tofile(...)\n", " | a.tofile(fid, sep=\"\", format=\"%s\")\n", " | \n", " | Write array to a file as text or binary (default).\n", " | \n", " | Data is always written in 'C' order, independent of the order of `a`.\n", " | The data produced by this method can be recovered using the function\n", " | fromfile().\n", " | \n", " | Parameters\n", " | ----------\n", " | fid : file or str or Path\n", " | An open file object, or a string containing a filename.\n", " | \n", " | .. versionchanged:: 1.17.0\n", " | `pathlib.Path` objects are now accepted.\n", " | \n", " | sep : str\n", " | Separator between array items for text output.\n", " | If \"\" (empty), a binary file is written, equivalent to\n", " | ``file.write(a.tobytes())``.\n", " | format : str\n", " | Format string for text file output.\n", " | Each entry in the array is formatted to text by first converting\n", " | it to the closest Python type, and then using \"format\" % item.\n", " | \n", " | Notes\n", " | -----\n", " | This is a convenience function for quick storage of array data.\n", " | Information on endianness and precision is lost, so this method is not a\n", " | good choice for files intended to archive data or transport data between\n", " | machines with different endianness. Some of these problems can be overcome\n", " | by outputting the data as text files, at the expense of speed and file\n", " | size.\n", " | \n", " | When fid is a file object, array contents are directly written to the\n", " | file, bypassing the file object's ``write`` method. As a result, tofile\n", " | cannot be used with files objects supporting compression (e.g., GzipFile)\n", " | or file-like objects that do not support ``fileno()`` (e.g., BytesIO).\n", " | \n", " | tolist(...)\n", " | a.tolist()\n", " | \n", " | Return the array as an ``a.ndim``-levels deep nested list of Python scalars.\n", " | \n", " | Return a copy of the array data as a (nested) Python list.\n", " | Data items are converted to the nearest compatible builtin Python type, via\n", " | the `~numpy.ndarray.item` function.\n", " | \n", " | If ``a.ndim`` is 0, then since the depth of the nested list is 0, it will\n", " | not be a list at all, but a simple Python scalar.\n", " | \n", " | Parameters\n", " | ----------\n", " | none\n", " | \n", " | Returns\n", " | -------\n", " | y : object, or list of object, or list of list of object, or ...\n", " | The possibly nested list of array elements.\n", " | \n", " | Notes\n", " | -----\n", " | The array may be recreated via ``a = np.array(a.tolist())``, although this\n", " | may sometimes lose precision.\n", " | \n", " | Examples\n", " | --------\n", " | For a 1D array, ``a.tolist()`` is almost the same as ``list(a)``, \n", " | except that ``tolist`` changes numpy scalars to Python scalars:\n", " | \n", " | >>> a = np.uint32([1, 2])\n", " | >>> a_list = list(a)\n", " | >>> a_list\n", " | [1, 2]\n", " | >>> type(a_list[0])\n", " | \n", " | >>> a_tolist = a.tolist()\n", " | >>> a_tolist\n", " | [1, 2]\n", " | >>> type(a_tolist[0])\n", " | \n", " | \n", " | Additionally, for a 2D array, ``tolist`` applies recursively:\n", " | \n", " | >>> a = np.array([[1, 2], [3, 4]])\n", " | >>> list(a)\n", " | [array([1, 2]), array([3, 4])]\n", " | >>> a.tolist()\n", " | [[1, 2], [3, 4]]\n", " | \n", " | The base case for this recursion is a 0D array:\n", " | \n", " | >>> a = np.array(1)\n", " | >>> list(a)\n", " | Traceback (most recent call last):\n", " | ...\n", " | TypeError: iteration over a 0-d array\n", " | >>> a.tolist()\n", " | 1\n", " | \n", " | tostring(...)\n", " | a.tostring(order='C')\n", " | \n", " | Construct Python bytes containing the raw data bytes in the array.\n", " | \n", " | Constructs Python bytes showing a copy of the raw contents of\n", " | data memory. The bytes object can be produced in either 'C' or 'Fortran',\n", " | or 'Any' order (the default is 'C'-order). 'Any' order means C-order\n", " | unless the F_CONTIGUOUS flag in the array is set, in which case it\n", " | means 'Fortran' order.\n", " | \n", " | This function is a compatibility alias for tobytes. Despite its name it returns bytes not strings.\n", " | \n", " | Parameters\n", " | ----------\n", " | order : {'C', 'F', None}, optional\n", " | Order of the data for multidimensional arrays:\n", " | C, Fortran, or the same as for the original array.\n", " | \n", " | Returns\n", " | -------\n", " | s : bytes\n", " | Python bytes exhibiting a copy of `a`'s raw data.\n", " | \n", " | Examples\n", " | --------\n", " | >>> x = np.array([[0, 1], [2, 3]], dtype='>> x.tobytes()\n", " | b'\\x00\\x00\\x01\\x00\\x02\\x00\\x03\\x00'\n", " | >>> x.tobytes('C') == x.tobytes()\n", " | True\n", " | >>> x.tobytes('F')\n", " | b'\\x00\\x00\\x02\\x00\\x01\\x00\\x03\\x00'\n", " | \n", " | trace(...)\n", " | a.trace(offset=0, axis1=0, axis2=1, dtype=None, out=None)\n", " | \n", " | Return the sum along diagonals of the array.\n", " | \n", " | Refer to `numpy.trace` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.trace : equivalent function\n", " | \n", " | transpose(...)\n", " | a.transpose(*axes)\n", " | \n", " | Returns a view of the array with axes transposed.\n", " | \n", " | For a 1-D array this has no effect, as a transposed vector is simply the\n", " | same vector. To convert a 1-D array into a 2D column vector, an additional\n", " | dimension must be added. `np.atleast2d(a).T` achieves this, as does\n", " | `a[:, np.newaxis]`.\n", " | For a 2-D array, this is a standard matrix transpose.\n", " | For an n-D array, if axes are given, their order indicates how the\n", " | axes are permuted (see Examples). If axes are not provided and\n", " | ``a.shape = (i[0], i[1], ... i[n-2], i[n-1])``, then\n", " | ``a.transpose().shape = (i[n-1], i[n-2], ... i[1], i[0])``.\n", " | \n", " | Parameters\n", " | ----------\n", " | axes : None, tuple of ints, or `n` ints\n", " | \n", " | * None or no argument: reverses the order of the axes.\n", " | \n", " | * tuple of ints: `i` in the `j`-th place in the tuple means `a`'s\n", " | `i`-th axis becomes `a.transpose()`'s `j`-th axis.\n", " | \n", " | * `n` ints: same as an n-tuple of the same ints (this form is\n", " | intended simply as a \"convenience\" alternative to the tuple form)\n", " | \n", " | Returns\n", " | -------\n", " | out : ndarray\n", " | View of `a`, with axes suitably permuted.\n", " | \n", " | See Also\n", " | --------\n", " | ndarray.T : Array property returning the array transposed.\n", " | ndarray.reshape : Give a new shape to an array without changing its data.\n", " | \n", " | Examples\n", " | --------\n", " | >>> a = np.array([[1, 2], [3, 4]])\n", " | >>> a\n", " | array([[1, 2],\n", " | [3, 4]])\n", " | >>> a.transpose()\n", " | array([[1, 3],\n", " | [2, 4]])\n", " | >>> a.transpose((1, 0))\n", " | array([[1, 3],\n", " | [2, 4]])\n", " | >>> a.transpose(1, 0)\n", " | array([[1, 3],\n", " | [2, 4]])\n", " | \n", " | var(...)\n", " | a.var(axis=None, dtype=None, out=None, ddof=0, keepdims=False)\n", " | \n", " | Returns the variance of the array elements, along given axis.\n", " | \n", " | Refer to `numpy.var` for full documentation.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.var : equivalent function\n", " | \n", " | view(...)\n", " | a.view(dtype=None, type=None)\n", " | \n", " | New view of array with the same data.\n", " | \n", " | Parameters\n", " | ----------\n", " | dtype : data-type or ndarray sub-class, optional\n", " | Data-type descriptor of the returned view, e.g., float32 or int16. The\n", " | default, None, results in the view having the same data-type as `a`.\n", " | This argument can also be specified as an ndarray sub-class, which\n", " | then specifies the type of the returned object (this is equivalent to\n", " | setting the ``type`` parameter).\n", " | type : Python type, optional\n", " | Type of the returned view, e.g., ndarray or matrix. Again, the\n", " | default None results in type preservation.\n", " | \n", " | Notes\n", " | -----\n", " | ``a.view()`` is used two different ways:\n", " | \n", " | ``a.view(some_dtype)`` or ``a.view(dtype=some_dtype)`` constructs a view\n", " | of the array's memory with a different data-type. This can cause a\n", " | reinterpretation of the bytes of memory.\n", " | \n", " | ``a.view(ndarray_subclass)`` or ``a.view(type=ndarray_subclass)`` just\n", " | returns an instance of `ndarray_subclass` that looks at the same array\n", " | (same shape, dtype, etc.) This does not cause a reinterpretation of the\n", " | memory.\n", " | \n", " | For ``a.view(some_dtype)``, if ``some_dtype`` has a different number of\n", " | bytes per entry than the previous dtype (for example, converting a\n", " | regular array to a structured array), then the behavior of the view\n", " | cannot be predicted just from the superficial appearance of ``a`` (shown\n", " | by ``print(a)``). It also depends on exactly how ``a`` is stored in\n", " | memory. Therefore if ``a`` is C-ordered versus fortran-ordered, versus\n", " | defined as a slice or transpose, etc., the view may give different\n", " | results.\n", " | \n", " | \n", " | Examples\n", " | --------\n", " | >>> x = np.array([(1, 2)], dtype=[('a', np.int8), ('b', np.int8)])\n", " | \n", " | Viewing array data using a different type and dtype:\n", " | \n", " | >>> y = x.view(dtype=np.int16, type=np.matrix)\n", " | >>> y\n", " | matrix([[513]], dtype=int16)\n", " | >>> print(type(y))\n", " | \n", " | \n", " | Creating a view on a structured array so it can be used in calculations\n", " | \n", " | >>> x = np.array([(1, 2),(3,4)], dtype=[('a', np.int8), ('b', np.int8)])\n", " | >>> xv = x.view(dtype=np.int8).reshape(-1,2)\n", " | >>> xv\n", " | array([[1, 2],\n", " | [3, 4]], dtype=int8)\n", " | >>> xv.mean(0)\n", " | array([2., 3.])\n", " | \n", " | Making changes to the view changes the underlying array\n", " | \n", " | >>> xv[0,1] = 20\n", " | >>> x\n", " | array([(1, 20), (3, 4)], dtype=[('a', 'i1'), ('b', 'i1')])\n", " | \n", " | Using a view to convert an array to a recarray:\n", " | \n", " | >>> z = x.view(np.recarray)\n", " | >>> z.a\n", " | array([1, 3], dtype=int8)\n", " | \n", " | Views share data:\n", " | \n", " | >>> x[0] = (9, 10)\n", " | >>> z[0]\n", " | (9, 10)\n", " | \n", " | Views that change the dtype size (bytes per entry) should normally be\n", " | avoided on arrays defined by slices, transposes, fortran-ordering, etc.:\n", " | \n", " | >>> x = np.array([[1,2,3],[4,5,6]], dtype=np.int16)\n", " | >>> y = x[:, 0:2]\n", " | >>> y\n", " | array([[1, 2],\n", " | [4, 5]], dtype=int16)\n", " | >>> y.view(dtype=[('width', np.int16), ('length', np.int16)])\n", " | Traceback (most recent call last):\n", " | ...\n", " | ValueError: To change to a dtype of a different size, the array must be C-contiguous\n", " | >>> z = y.copy()\n", " | >>> z.view(dtype=[('width', np.int16), ('length', np.int16)])\n", " | array([[(1, 2)],\n", " | [(4, 5)]], dtype=[('width', '>> x = np.array([[1.,2.],[3.,4.]])\n", " | >>> x\n", " | array([[ 1., 2.],\n", " | [ 3., 4.]])\n", " | >>> x.T\n", " | array([[ 1., 3.],\n", " | [ 2., 4.]])\n", " | >>> x = np.array([1.,2.,3.,4.])\n", " | >>> x\n", " | array([ 1., 2., 3., 4.])\n", " | >>> x.T\n", " | array([ 1., 2., 3., 4.])\n", " | \n", " | See Also\n", " | --------\n", " | transpose\n", " | \n", " | __array_finalize__\n", " | None.\n", " | \n", " | __array_interface__\n", " | Array protocol: Python side.\n", " | \n", " | __array_priority__\n", " | Array priority.\n", " | \n", " | __array_struct__\n", " | Array protocol: C-struct side.\n", " | \n", " | base\n", " | Base object if memory is from some other object.\n", " | \n", " | Examples\n", " | --------\n", " | The base of an array that owns its memory is None:\n", " | \n", " | >>> x = np.array([1,2,3,4])\n", " | >>> x.base is None\n", " | True\n", " | \n", " | Slicing creates a view, whose memory is shared with x:\n", " | \n", " | >>> y = x[2:]\n", " | >>> y.base is x\n", " | True\n", " | \n", " | ctypes\n", " | An object to simplify the interaction of the array with the ctypes\n", " | module.\n", " | \n", " | This attribute creates an object that makes it easier to use arrays\n", " | when calling shared libraries with the ctypes module. The returned\n", " | object has, among others, data, shape, and strides attributes (see\n", " | Notes below) which themselves return ctypes objects that can be used\n", " | as arguments to a shared library.\n", " | \n", " | Parameters\n", " | ----------\n", " | None\n", " | \n", " | Returns\n", " | -------\n", " | c : Python object\n", " | Possessing attributes data, shape, strides, etc.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.ctypeslib\n", " | \n", " | Notes\n", " | -----\n", " | Below are the public attributes of this object which were documented\n", " | in \"Guide to NumPy\" (we have omitted undocumented public attributes,\n", " | as well as documented private attributes):\n", " | \n", " | .. autoattribute:: numpy.core._internal._ctypes.data\n", " | :noindex:\n", " | \n", " | .. autoattribute:: numpy.core._internal._ctypes.shape\n", " | :noindex:\n", " | \n", " | .. autoattribute:: numpy.core._internal._ctypes.strides\n", " | :noindex:\n", " | \n", " | .. automethod:: numpy.core._internal._ctypes.data_as\n", " | :noindex:\n", " | \n", " | .. automethod:: numpy.core._internal._ctypes.shape_as\n", " | :noindex:\n", " | \n", " | .. automethod:: numpy.core._internal._ctypes.strides_as\n", " | :noindex:\n", " | \n", " | If the ctypes module is not available, then the ctypes attribute\n", " | of array objects still returns something useful, but ctypes objects\n", " | are not returned and errors may be raised instead. In particular,\n", " | the object will still have the ``as_parameter`` attribute which will\n", " | return an integer equal to the data attribute.\n", " | \n", " | Examples\n", " | --------\n", " | >>> import ctypes\n", " | >>> x\n", " | array([[0, 1],\n", " | [2, 3]])\n", " | >>> x.ctypes.data\n", " | 30439712\n", " | >>> x.ctypes.data_as(ctypes.POINTER(ctypes.c_long))\n", " | \n", " | >>> x.ctypes.data_as(ctypes.POINTER(ctypes.c_long)).contents\n", " | c_long(0)\n", " | >>> x.ctypes.data_as(ctypes.POINTER(ctypes.c_longlong)).contents\n", " | c_longlong(4294967296L)\n", " | >>> x.ctypes.shape\n", " | \n", " | >>> x.ctypes.shape_as(ctypes.c_long)\n", " | \n", " | >>> x.ctypes.strides\n", " | \n", " | >>> x.ctypes.strides_as(ctypes.c_longlong)\n", " | \n", " | \n", " | data\n", " | Python buffer object pointing to the start of the array's data.\n", " | \n", " | dtype\n", " | Data-type of the array's elements.\n", " | \n", " | Parameters\n", " | ----------\n", " | None\n", " | \n", " | Returns\n", " | -------\n", " | d : numpy dtype object\n", " | \n", " | See Also\n", " | --------\n", " | numpy.dtype\n", " | \n", " | Examples\n", " | --------\n", " | >>> x\n", " | array([[0, 1],\n", " | [2, 3]])\n", " | >>> x.dtype\n", " | dtype('int32')\n", " | >>> type(x.dtype)\n", " | \n", " | \n", " | flags\n", " | Information about the memory layout of the array.\n", " | \n", " | Attributes\n", " | ----------\n", " | C_CONTIGUOUS (C)\n", " | The data is in a single, C-style contiguous segment.\n", " | F_CONTIGUOUS (F)\n", " | The data is in a single, Fortran-style contiguous segment.\n", " | OWNDATA (O)\n", " | The array owns the memory it uses or borrows it from another object.\n", " | WRITEABLE (W)\n", " | The data area can be written to. Setting this to False locks\n", " | the data, making it read-only. A view (slice, etc.) inherits WRITEABLE\n", " | from its base array at creation time, but a view of a writeable\n", " | array may be subsequently locked while the base array remains writeable.\n", " | (The opposite is not true, in that a view of a locked array may not\n", " | be made writeable. However, currently, locking a base object does not\n", " | lock any views that already reference it, so under that circumstance it\n", " | is possible to alter the contents of a locked array via a previously\n", " | created writeable view onto it.) Attempting to change a non-writeable\n", " | array raises a RuntimeError exception.\n", " | ALIGNED (A)\n", " | The data and all elements are aligned appropriately for the hardware.\n", " | WRITEBACKIFCOPY (X)\n", " | This array is a copy of some other array. The C-API function\n", " | PyArray_ResolveWritebackIfCopy must be called before deallocating\n", " | to the base array will be updated with the contents of this array.\n", " | UPDATEIFCOPY (U)\n", " | (Deprecated, use WRITEBACKIFCOPY) This array is a copy of some other array.\n", " | When this array is\n", " | deallocated, the base array will be updated with the contents of\n", " | this array.\n", " | FNC\n", " | F_CONTIGUOUS and not C_CONTIGUOUS.\n", " | FORC\n", " | F_CONTIGUOUS or C_CONTIGUOUS (one-segment test).\n", " | BEHAVED (B)\n", " | ALIGNED and WRITEABLE.\n", " | CARRAY (CA)\n", " | BEHAVED and C_CONTIGUOUS.\n", " | FARRAY (FA)\n", " | BEHAVED and F_CONTIGUOUS and not C_CONTIGUOUS.\n", " | \n", " | Notes\n", " | -----\n", " | The `flags` object can be accessed dictionary-like (as in ``a.flags['WRITEABLE']``),\n", " | or by using lowercased attribute names (as in ``a.flags.writeable``). Short flag\n", " | names are only supported in dictionary access.\n", " | \n", " | Only the WRITEBACKIFCOPY, UPDATEIFCOPY, WRITEABLE, and ALIGNED flags can be\n", " | changed by the user, via direct assignment to the attribute or dictionary\n", " | entry, or by calling `ndarray.setflags`.\n", " | \n", " | The array flags cannot be set arbitrarily:\n", " | \n", " | - UPDATEIFCOPY can only be set ``False``.\n", " | - WRITEBACKIFCOPY can only be set ``False``.\n", " | - ALIGNED can only be set ``True`` if the data is truly aligned.\n", " | - WRITEABLE can only be set ``True`` if the array owns its own memory\n", " | or the ultimate owner of the memory exposes a writeable buffer\n", " | interface or is a string.\n", " | \n", " | Arrays can be both C-style and Fortran-style contiguous simultaneously.\n", " | This is clear for 1-dimensional arrays, but can also be true for higher\n", " | dimensional arrays.\n", " | \n", " | Even for contiguous arrays a stride for a given dimension\n", " | ``arr.strides[dim]`` may be *arbitrary* if ``arr.shape[dim] == 1``\n", " | or the array has no elements.\n", " | It does *not* generally hold that ``self.strides[-1] == self.itemsize``\n", " | for C-style contiguous arrays or ``self.strides[0] == self.itemsize`` for\n", " | Fortran-style contiguous arrays is true.\n", " | \n", " | flat\n", " | A 1-D iterator over the array.\n", " | \n", " | This is a `numpy.flatiter` instance, which acts similarly to, but is not\n", " | a subclass of, Python's built-in iterator object.\n", " | \n", " | See Also\n", " | --------\n", " | flatten : Return a copy of the array collapsed into one dimension.\n", " | \n", " | flatiter\n", " | \n", " | Examples\n", " | --------\n", " | >>> x = np.arange(1, 7).reshape(2, 3)\n", " | >>> x\n", " | array([[1, 2, 3],\n", " | [4, 5, 6]])\n", " | >>> x.flat[3]\n", " | 4\n", " | >>> x.T\n", " | array([[1, 4],\n", " | [2, 5],\n", " | [3, 6]])\n", " | >>> x.T.flat[3]\n", " | 5\n", " | >>> type(x.flat)\n", " | \n", " | \n", " | An assignment example:\n", " | \n", " | >>> x.flat = 3; x\n", " | array([[3, 3, 3],\n", " | [3, 3, 3]])\n", " | >>> x.flat[[1,4]] = 1; x\n", " | array([[3, 1, 3],\n", " | [3, 1, 3]])\n", " | \n", " | imag\n", " | The imaginary part of the array.\n", " | \n", " | Examples\n", " | --------\n", " | >>> x = np.sqrt([1+0j, 0+1j])\n", " | >>> x.imag\n", " | array([ 0. , 0.70710678])\n", " | >>> x.imag.dtype\n", " | dtype('float64')\n", " | \n", " | itemsize\n", " | Length of one array element in bytes.\n", " | \n", " | Examples\n", " | --------\n", " | >>> x = np.array([1,2,3], dtype=np.float64)\n", " | >>> x.itemsize\n", " | 8\n", " | >>> x = np.array([1,2,3], dtype=np.complex128)\n", " | >>> x.itemsize\n", " | 16\n", " | \n", " | nbytes\n", " | Total bytes consumed by the elements of the array.\n", " | \n", " | Notes\n", " | -----\n", " | Does not include memory consumed by non-element attributes of the\n", " | array object.\n", " | \n", " | Examples\n", " | --------\n", " | >>> x = np.zeros((3,5,2), dtype=np.complex128)\n", " | >>> x.nbytes\n", " | 480\n", " | >>> np.prod(x.shape) * x.itemsize\n", " | 480\n", " | \n", " | ndim\n", " | Number of array dimensions.\n", " | \n", " | Examples\n", " | --------\n", " | >>> x = np.array([1, 2, 3])\n", " | >>> x.ndim\n", " | 1\n", " | >>> y = np.zeros((2, 3, 4))\n", " | >>> y.ndim\n", " | 3\n", " | \n", " | real\n", " | The real part of the array.\n", " | \n", " | Examples\n", " | --------\n", " | >>> x = np.sqrt([1+0j, 0+1j])\n", " | >>> x.real\n", " | array([ 1. , 0.70710678])\n", " | >>> x.real.dtype\n", " | dtype('float64')\n", " | \n", " | See Also\n", " | --------\n", " | numpy.real : equivalent function\n", " | \n", " | shape\n", " | Tuple of array dimensions.\n", " | \n", " | The shape property is usually used to get the current shape of an array,\n", " | but may also be used to reshape the array in-place by assigning a tuple of\n", " | array dimensions to it. As with `numpy.reshape`, one of the new shape\n", " | dimensions can be -1, in which case its value is inferred from the size of\n", " | the array and the remaining dimensions. Reshaping an array in-place will\n", " | fail if a copy is required.\n", " | \n", " | Examples\n", " | --------\n", " | >>> x = np.array([1, 2, 3, 4])\n", " | >>> x.shape\n", " | (4,)\n", " | >>> y = np.zeros((2, 3, 4))\n", " | >>> y.shape\n", " | (2, 3, 4)\n", " | >>> y.shape = (3, 8)\n", " | >>> y\n", " | array([[ 0., 0., 0., 0., 0., 0., 0., 0.],\n", " | [ 0., 0., 0., 0., 0., 0., 0., 0.],\n", " | [ 0., 0., 0., 0., 0., 0., 0., 0.]])\n", " | >>> y.shape = (3, 6)\n", " | Traceback (most recent call last):\n", " | File \"\", line 1, in \n", " | ValueError: total size of new array must be unchanged\n", " | >>> np.zeros((4,2))[::2].shape = (-1,)\n", " | Traceback (most recent call last):\n", " | File \"\", line 1, in \n", " | AttributeError: incompatible shape for a non-contiguous array\n", " | \n", " | See Also\n", " | --------\n", " | numpy.reshape : similar function\n", " | ndarray.reshape : similar method\n", " | \n", " | size\n", " | Number of elements in the array.\n", " | \n", " | Equal to ``np.prod(a.shape)``, i.e., the product of the array's\n", " | dimensions.\n", " | \n", " | Notes\n", " | -----\n", " | `a.size` returns a standard arbitrary precision Python integer. This\n", " | may not be the case with other methods of obtaining the same value\n", " | (like the suggested ``np.prod(a.shape)``, which returns an instance\n", " | of ``np.int_``), and may be relevant if the value is used further in\n", " | calculations that may overflow a fixed size integer type.\n", " | \n", " | Examples\n", " | --------\n", " | >>> x = np.zeros((3, 5, 2), dtype=np.complex128)\n", " | >>> x.size\n", " | 30\n", " | >>> np.prod(x.shape)\n", " | 30\n", " | \n", " | strides\n", " | Tuple of bytes to step in each dimension when traversing an array.\n", " | \n", " | The byte offset of element ``(i[0], i[1], ..., i[n])`` in an array `a`\n", " | is::\n", " | \n", " | offset = sum(np.array(i) * a.strides)\n", " | \n", " | A more detailed explanation of strides can be found in the\n", " | \"ndarray.rst\" file in the NumPy reference guide.\n", " | \n", " | Notes\n", " | -----\n", " | Imagine an array of 32-bit integers (each 4 bytes)::\n", " | \n", " | x = np.array([[0, 1, 2, 3, 4],\n", " | [5, 6, 7, 8, 9]], dtype=np.int32)\n", " | \n", " | This array is stored in memory as 40 bytes, one after the other\n", " | (known as a contiguous block of memory). The strides of an array tell\n", " | us how many bytes we have to skip in memory to move to the next position\n", " | along a certain axis. For example, we have to skip 4 bytes (1 value) to\n", " | move to the next column, but 20 bytes (5 values) to get to the same\n", " | position in the next row. As such, the strides for the array `x` will be\n", " | ``(20, 4)``.\n", " | \n", " | See Also\n", " | --------\n", " | numpy.lib.stride_tricks.as_strided\n", " | \n", " | Examples\n", " | --------\n", " | >>> y = np.reshape(np.arange(2*3*4), (2,3,4))\n", " | >>> y\n", " | array([[[ 0, 1, 2, 3],\n", " | [ 4, 5, 6, 7],\n", " | [ 8, 9, 10, 11]],\n", " | [[12, 13, 14, 15],\n", " | [16, 17, 18, 19],\n", " | [20, 21, 22, 23]]])\n", " | >>> y.strides\n", " | (48, 16, 4)\n", " | >>> y[1,1,1]\n", " | 17\n", " | >>> offset=sum(y.strides * np.array((1,1,1)))\n", " | >>> offset/y.itemsize\n", " | 17\n", " | \n", " | >>> x = np.reshape(np.arange(5*6*7*8), (5,6,7,8)).transpose(2,3,1,0)\n", " | >>> x.strides\n", " | (32, 4, 224, 1344)\n", " | >>> i = np.array([3,5,2,2])\n", " | >>> offset = sum(i * x.strides)\n", " | >>> x[3,5,2,2]\n", " | 813\n", " | >>> offset / x.itemsize\n", " | 813\n", " | \n", " | ----------------------------------------------------------------------\n", " | Data and other attributes defined here:\n", " | \n", " | __hash__ = None\n", "\n" ] } ], "source": [ "phi_dr1 = 0.164 - 0.19 * np.exp(lam_sp_est * 9.881)\n", "phi_dr2 = 0.104 - 0.19 * np.exp(lam_sp_est * 18.242)\n", "\n", "print(phi_dr1)\n", "print(phi_dr2)\n", "\n", "sp_dr = 1/(18.242 - 9.881)*np.log(phi_dr2/phi_dr1)\n", "print(sp_dr)\n", "\n", "np.sqrt(0.025**2 + 0.025**2)\n", "\n", "help(np.random.randint(10, size=(10, 10)))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Other Condition Data\n", "\n", "The data as presented in the original report are only tabulated for a couple of aircraft conditions for the Boeing 747, with the remainder of the data presented in graphical form. Thankfully, someone has already done the hard work of extracting those data from the graph and presenting in tabular form (it wasn't me, the following is taken from [Caughey's online course notes](https://courses.cit.cornell.edu/mae5070/Caughey_2011_04.pdf), borrowed with thanks).\n", "\n", "The condition number confers to the Mach/SL diagram on Page 212 of the original report.\n", "\n", "| Derivative | Condition 2 | Condition 5 | Condition 7 | Condition 9 | Condition 10 |\n", "|-----------------------------|--------------------|--------------------|-------------------|--------------------|--------------------|\n", "| $h$/ft | 0 | 20,000 | 20,000 | 40,000 | 40,000 |\n", "| $M$ | 0.25 | 0.50 | 0.80 | 0.80 | 0.90 |\n", "| $\\alpha$/$^\\circ$ | 5.70 | 6.80 | 0.0 | 4.60 | 2.40 |\n", "| $W$/lb | 564,032 | 636,636 | 636,636 | 636,636 | 636,636 |\n", "| $I_{xx}$ /(slug-ft$^2$) | 14.3 x $10^6$ | 18.4 x $10^6$ | 18.2 x $10^6$ | 18.2 x $10^6$ | 18.2 x $10^6$ |\n", "| $I_{yy}$ /(slug-ft$^2$) | 32.3 x $10^6$ | 33.1 x $10^6$ | 33.1 x $10^6$ | 33.1 x $10^6$ | 33.1 x $10^6$ |\n", "| $I_{zz}$ /(slug-ft$^2$) | 45.3 x $10^6$ | 49.5 x $10^6$ | 49.7 x $10^6$ | 49.7 x $10^6$ | 49.7 x $10^6$ |\n", "| $I_{xz}$ /(slug-ft$^2$) | -2.23 x $10^6$ | -2.76 x $10^6$ | 0.97 x $10^6$ | -1.56 x $10^6$ | -0.35 x $10^6$ |\n", "| $C_L$ | 1.11 | 0.68 | 0.266 | 0.66 | 0.521 |\n", "| $C_D$ | 0.102 | 0.0393 | 0.0174 | 0.0415 | 0.0415 |\n", "| $C_{L_\\alpha}$ | 5.7 | 4.67 | 4.24 | 4.92 | 5.57 |\n", "| $C_{D_\\alpha}$ | 0.66 | 0.366 | 0.084 | 0.425 | 0.527 |\n", "| $C_{m_\\alpha}$ | -1.26 | -1.146 | -629 | -1.033 | -1.613 |\n", "| $C_{L_\\dot{\\alpha}}$ | 6.7 | 6.53 | 5.99 | 5.91 | 5.53 |\n", "| $C_{m_\\dot{\\alpha}}$ | -3.2 | -3.35 | -5.4 | -6.41 | -8.82 |\n", "| $C_{L_q}$ | 5.4 | 5.13 | 5.01 | 6 | 6.94 |\n", "| $C_{m_q}$ | -20.8 | -20.7 | -20.5 | -24 | -25.1 |\n", "| $C_{L_M}$ | 0 | -0.0875 | 0.105 | 0.205 | -0.278 |\n", "| $C_{D_M}$ | 0 | 0 | 0.008 | 0.0275 | 0.242 |\n", "| $C_{m_M}$ | 0 | 0.121 | -0.116 | 0.166 | -0.114 |\n", "| $C_{L_{\\delta_e}}$ | 0.338 | 0.356 | 0.27 | 0.367 | 0.3 |\n", "| $C_{m_{\\delta_e}}$ | -1.34 | -1.43 | -1.06 | -1.45 | -1.2 |\n", "| $C_{y_\\beta}$ | -0.96 | -0.9 | -0.81 | -0.88 | -0.92 |\n", "| $C_{l_\\beta}$ | -0.221 | -193 | -0.164 | -0.277 | -0.095 |\n", "| $C_{n_\\beta}$ | 0.15 | 0.147 | 0.179 | 0.195 | 0.207 |\n", "| $C_{l_p}$ | -0.45 | -0.323 | -0.315 | -0.334 | -0.296 |\n", "| $C_{n_p}$ | -0.121 | -0.069 | 0.0028 | -0.042 | 0.023 |\n", "| $C_{l_r}$ | 0.101 | 0.212 | 0.0979 | 0.3 | 0.193 |\n", "| $C_{n_r}$ | -0.3 | -0.278 | -0.265 | -0.327 | -333 |\n", "| $C_{l_{\\delta_a}}$ | 0.0461 | 0.0129 | 0.012 | 0.0137 | 0.0139 |\n", "| $C_{n_{\\delta_a}}$ | 0.0064 | 0.0015 | 0.0008 | 0.0002 | -0.003 |\n", "| $C_{y_{\\delta_r}}$ | 0.175 | 0.1448 | 0.0841 | 0.1157 | 0.062 |\n", "| $C_{l_{\\delta_r}}$ | 0.007 | 0.0039 | 0.009 | 0.007 | 0.0052 |\n", "| $C_{n_{\\delta_r}}$ | -0.109 | -0.1081 | -0.099 | -0.1256 | -0.091 |" ] }, { "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": "3.8.3" } }, "nbformat": 4, "nbformat_minor": 4 }