{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "*(Note: the US spelling is maneuver, but it took me years of muscle memory training to spell manoeuvre correctly, and I won't be losing it easily.)*\n", "\n", "# Acceleration, Manoeuvres, and Aircraft Loading\n", "\n", "The preceding analysis has been constrained to *steady* flight - that is, with zero acceleration. For the cases of climb, the climb rate was assumed to be steady.\n", "\n", "To understand unconstrained aircraft manoeuvres requires an understanding of accelerated flight.\n", "\n", "Manoeuvre will be broken down into horizontal (e.g., flat or banked turns) and vertical manoeuvres (e.g., loops, pull-ups), comprising curvilinear motion. Such manoeuvres are the result of a force *perpendicular* to the flight path, giving a normal acceleration.\n", "\n", "All of the manoevures discussed are the result of a **variation in lift**, which can be *large*. Consider that the dynamic pressure rises with the square of the forward speed, so a five-fold speed increase results in twenty-five times the aerodynamic forces.\n", "\n", "Before discussing manoeuvres, a means to represent the allowable amount of load on an aircraft will be introduced. \n", "\n", "## Load Factor\n", "\n", "The load that can be safely taken through an aircraft dictates the load limits on an aircraft - for this the **load factor** is introduced as a non-dimensional measure of the load variation.\n", "\n", "{math}\n", ":label: LoadFactor\n", "n=\\frac{L}{W}\n", "\n", "\n", "{admonition} So - *what is the load factor for steady level flight?*\n", ":class: dropdown\n", "In steady level flight, the equilibrium steady flight condition is $L=W$ so $n=1$\n", "\n", "\n", "There are two structural limits defined for aircraft:\n", "- **Limit Loads**, $n_{l}$ are the loads at which *plastic deformation* will occur. At flight with $1n_l$, the aircraft will require inspection and likely replacement of parts.\n", " \n", " \n", "- **Ultimate Loads**, $n_{u}$ are the loads at which *failure* will occur. At flight with $n>n_u$, *parts of the aircraft will break*\n", "\n", "The numerical value of the load factor is an exercise in structural analysis, whereby the loads and their paths are applied to a model of the aircraft, and a determination of $n_l$ and $n_u$ can be made.\n", "\n", "In general, $n_l$ and $n_u$ will be defined with a safety factor of 50% - such that plastic structural deformation may not actually occur until $n_l*1.5$. This should not be considered a margin to 'play with', however!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Values of the load factor\n", "\n", "FAR 23 (Federal Aviation Requirement) dictates the *minimum* load factor required for three categories of aircraft:\n", "- Commuter Aircraft\n", "- Utility Aircraft\n", "- Aerobatic Aircraft\n", "\n", "The minimum load factors are defined as (since this is from FAR 23, $W$ is defined in $lbf$)\n", "\n", "\n", "| Aircraft Category | Minimum positive load factor, $n+$ | Minimum negative load factor, $n-$ |\n", "|-------------------|---------------------------------------|------------------------------------|\n", "| Normal/Commuter | smaller of $2.1+\\frac{24000}{W+1000}$ or $3.8$ | $0.4n+$ |\n", "| Utility | $4.4$ | $0.4n+$ |\n", "| Aerobatic | $6.0$ | $0.5n+$ |\n", "\n", "Often pilots will talk about *load factor* in terms of \"g\"s - for straight and level flight, 1$g$ feels like regular earth gravity, whilst an $n=2$ or $2g$ manoeuvre makes the occupants feel *twice as heavy*.\n", "\n", "Since the load factor represents the amount of lift being produced, it is an easy and instructive means to define the structural limits of an aircraft - $e.g.,$ the loads for which the wings will break off." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### V-n diagram\n", "\n", "{admonition} Bear in mind...\n", ":class: dropdown\n", "\n", "In what's becoming a common phrase in this document - the following is a simplification of the theory, designed to give you a good insight into the physics of the problem. Construction of a full $V-n$ diagram is complicated and more of an exercise in understanding FAA certification than it is the flight physics.\n", "\n", "What follows is a suitable explanation for an undergraduate aerospace engineer.\n", "\n", "\n", "\n", "The values of the limit loads, and ultimate loads are presented on a graph vs. airspeed. For the following, the loads will be presented against *equivalent airspeed* because that enables a single graph to be plotted, but for real aircraft this is usually presented against *indicated airspeed*, with several lines representing different altitudes.\n", "\n", "There are many different means of showing how to construct a $V-n$ diagram in textbook and online, but most seem geared up to helping pilots understand them rather than aerospace engineers. For example - the following show good examples of how to formulate a $V-n$ diagram, [website one](http://www.aviationchief.com/operating-flight-strength-v-g--v-n-diagrams.html), [website two](https://code7700.com/g450_limitations.htm#weight).\n", "\n", "For the following example, a representative jet trainer aircraft will be used, with structural limit loads of $-3.0\n", " \n", " \n", " \n", " \n", " " ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import plotly.graph_objects as go\n", "\n", "# Limit loads\n", "n_l = [-3.0, 7.0]\n", "\n", "# Ultimate loads\n", "n_u = [-5.0, 11.0]\n", "\n", "# Make a figure\n", "fig = go.Figure()\n", "\n", "# Make a vector of equivalent airspeed (m/s)\n", "VE = np.linspace(0, 340, 1000)\n", "\n", "# Plot the Structural damage area\n", "x_structural_damage = [VE.min(), VE.max()]\n", "y_structural_damage = [n_l[1], n_l[1]]\n", "y_structural_failure = [n_u[1], n_u[1]]\n", "\n", "## Positive strucutral damage\n", "# Plot a line of the structural damage\n", "fig.add_trace(go.Scatter(x=VE, y=n_l[1] * np.ones(VE.shape),\n", " fill=None,\n", " mode='lines',\n", " line_color='indigo', name=\"+nl\",\n", " showlegend=False))\n", "\n", "# Then a line of the structural failure that is filled to the structural damage\n", "fig.add_trace(go.Scatter(\n", " x=x_structural_damage,\n", " y=y_structural_failure,\n", " fill='tonexty', # fill area between structural failure and structural Damage\n", " mode='none', fillcolor='rgba(255, 0, 0, 0.25)', name=\"Structural Damage\", showlegend=False))\n", "\n", "# Annotate the structural failure area\n", "fig.add_trace(go.Scatter(\n", " x=[np.mean(x_structural_damage)],\n", " y=[0.5 * (n_l[1] + n_u[1])],\n", " mode='text',\n", " text=\"Structural Damage\", showlegend=False))\n", "\n", "# Add the ultimate load factor\n", "fig.add_trace(go.Scatter(x=VE, y=n_u[1] * np.ones(VE.shape),\n", " mode='lines',\n", " line_color='red', name=\"+nu\",\n", " showlegend=False))\n", "\n", "# Then a line of the structural failure that is filled to the structural damage\n", "fig.add_trace(go.Scatter(\n", " x=VE, y=3*n_u[1] * np.ones(VE.shape),\n", " fill='tonexty', # fill area between structural failure and structural Damage\n", " mode='none', fillcolor='rgba(255, 0, 0, 0.55)', name=\"Structural Damage\", showlegend=False, hoverinfo=\"none\"))\n", "\n", "# Annotate the structural failure area\n", "fig.add_trace(go.Scatter(\n", " x=[np.mean(x_structural_damage)],\n", " y=[n_u[1] + 0.05 * (n_l[1] + n_u[1])],\n", " mode='text',\n", " text=\"Structural Failure\", showlegend=False, hoverinfo=\"none\"))\n", "\n", "\n", "\n", "\n", "### Negative side\n", "\n", "\n", "\n", "## Negative structural damage\n", "x_structural_damage = [VE.min(), VE.max()]\n", "y_structural_damage = [n_l[0], n_l[0]]\n", "y_structural_failure = [n_u[0], n_u[0]]\n", "# Plot a line of the structural damage\n", "fig.add_trace(go.Scatter(x=VE, y=n_l[0] * np.ones(VE.shape),\n", " fill=None,\n", " mode='lines',\n", " line_color='indigo',\n", " name=\"-nl\",\n", " showlegend=False))\n", "\n", "# Then a line of the structural failure that is filled to the structural damage\n", "fig.add_trace(go.Scatter(\n", " x=x_structural_damage,\n", " y=y_structural_failure,\n", " fill='tonexty', # fill area between trace0 and trace1\n", " mode='none', fillcolor='rgba(255, 0, 0, 0.25)', name=\"Structural Damage\", showlegend=False))\n", "\n", "# Annotate the structural damage area\n", "fig.add_trace(go.Scatter(\n", " x=[np.mean(x_structural_damage)],\n", " y=[0.5 * (n_l[0] + n_u[0])],\n", " mode='text',\n", " text=\"Structural Damage\", showlegend=False))\n", "\n", "# Add the ultimate load factor\n", "fig.add_trace(go.Scatter(x=VE, y=n_u[0] * np.ones(VE.shape),\n", " fill=None,\n", " mode='lines',\n", " line_color='red', name=\"-nu\",\n", " showlegend=False))\n", "\n", "# Then a line of the structural failure that is filled to the structural damage\n", "fig.add_trace(go.Scatter(\n", " x=VE, y=3*n_u[0] * np.ones(VE.shape),\n", " fill='tonexty', # fill area between structural failure and structural Damage\n", " mode='none', fillcolor='rgba(255, 0, 0, 0.55)', name=\"Structural Damage\", showlegend=False, hoverinfo=\"none\"))\n", "\n", "\n", "# Annotate the structural failure area\n", "fig.add_trace(go.Scatter(\n", " x=[np.mean(x_structural_damage)],\n", " y=[n_u[0] + 0.05 * (n_l[0] + n_u[0])],\n", " mode='text',\n", " text=\"Structural Failure\", showlegend=False, hoverinfo=\"none\"))\n", "\n", "# Change the limits\n", "fig.update_yaxes(range= [1.2 * n_u[0], 1.2 * n_u[1]])\n", "\n", "fig.update_layout(\n", " title=\"Representative Structural Damage and Failure Load Factors\",\n", " xaxis_title=\"$V_E/\\\\text{m/s}$\",\n", " yaxis_title=\"$n$\",\n", ")\n", "\n", "# Dick around with the hover behaviour\n", "fig.update_traces(hovertemplate=None)\n", "fig.update_layout(hovermode=\"x unified\")\n", "\n", "fig.update_yaxes(zeroline=True, zerolinewidth=2, zerolinecolor='black')\n", "fig.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Stall limits on V-n diagram\n", "\n", "At this stage, the$V-n$diagram isn't particularly useful. You can hover over the graph and see that values of allowable load is constant with airspeed. \n", "\n", "The graph will be adapted by inclusion of other limitations. The first of which is that **at certain airspeeds, the aircraft will stall prior to$n_l$being reached**, and therefore this influences the speed at which manoeuvres can be attempted safely.\n", "\n", "It can be shown from the definition of the load factor, and the definition of the lift coefficient that the load factor associated with *stall* is given by\n", "\n", "$$n_{stall}=\\frac{C_{L,max}\\,\\tfrac{1}{2}\\rho V^2\\,S}{W}$$\n", "\n", "If the aircraft is taken to have a wing area of 16m$^2$, and a weight of 53kN, and a$C_{L,max}$of 1.6 with a$C_{L,min}$of -1.0, then the variation of$n_{stall}$can be overlaid on the previous graph\n", "\n", "{margin}\n", "You might not have considered Clmin before - but it's the condition at which the flow around the wing will separate due to *negative* incidence\n", "`\n", "\n", "It can be appreciated that if the stall occurs for$n < 1$, then steady flight is not possible at this speed.\n" ] }, { "cell_type": "code", "execution_count": 89, 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