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April 22, 2015 02:49
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Stackexchange answer for optimal path for fuel efficiency
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{ | |
"cells": [ | |
{ | |
"cell_type": "markdown", | |
"metadata": {}, | |
"source": [ | |
"# Most Fuel Efficient Road\n", | |
"\n", | |
"This notebook contains code and some discussion created to address the [stackexchange question](http://physics.stackexchange.com/questions/6220/can-a-car-get-better-mileage-driving-over-hills)\n", | |
"\n", | |
">##Can a car get better mileage driving over hills?\n", | |
">Two towns are at the same elevation and are connected by two roads of the same length. One road is flat, the other road goes up and down some hills. Will an automobile always get the best mileage driving between the two towns on the flat road versus the hilly one, if operated by a perfect driver with knowledge of the route? Is there any hilly road on which a better mileage can be achieved?" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 1, | |
"metadata": { | |
"collapsed": true | |
}, | |
"outputs": [], | |
"source": [ | |
"import numpy as np\n", | |
"import scipy as sp\n", | |
"import matplotlib.pyplot as plt\n", | |
"%matplotlib inline" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"metadata": {}, | |
"source": [ | |
"# Car Model" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"metadata": {}, | |
"source": [ | |
"We begin by defining some parameters for our car model, from an [EPA report](http://www.epa.gov/oms/models/ngm/420p05001.pdf) for a 2004 Honda Civic." | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 2, | |
"metadata": { | |
"collapsed": true | |
}, | |
"outputs": [], | |
"source": [ | |
"A = 105.47 #N\n", | |
"B = 5.4276 #N/mps\n", | |
"C = 0.267 #N/mps^2\n", | |
"m = 1239 #kg\n", | |
"g = 9.8 #m/s^2\n", | |
"a = 3.19 #0.15 * g\n", | |
"b = 7.193 #0.6 * g\n", | |
"P0 = 6000 # Watts\n", | |
"\n", | |
"mphfact = 2.237\n", | |
"gallonperj = 9.478e-4 / 114000.0 / 0.34" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"metadata": {}, | |
"source": [ | |
"This hypothetical vehicle feels a friction force, a rotating friction, and air drag\n", | |
"\n", | |
"$$ f = A (v > 0) + B v + C v^2 $$\n", | |
"at constant velocity, the power consumed by the vehicle will be\n", | |
"$$ P = P_0 + Fv = P_0 + A v + B v^2 + C v^3 $$\n", | |
"if we are concerned with the energy to go some constant distance, we have that\n", | |
"$$ U = \\int dt\\, P = \\int dx\\, \\frac{dt}{dx} P = \\Delta x \\frac{P}{v} $$\n", | |
"so that our fuel efficiency is given by\n", | |
"$$ \\frac{\\Delta x}{U} = \\frac{v}{ P } $$\n", | |
"\n", | |
"using the numbers from the EPA report, we can see the fuel efficiency of this vehicle" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 90, | |
"metadata": { | |
"collapsed": false | |
}, | |
"outputs": [ | |
{ | |
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], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7f0ba69b16d0>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
} | |
], | |
"source": [ | |
"vs = np.linspace(1, 90/mphfact, 300)\n", | |
"mpg = vs/( A*vs + B*vs**2 + C*vs**3 + P0 ) / 1609 / gallonperj\n", | |
"plt.plot( vs * mphfact, mpg, lw=3, alpha=0.8 );\n", | |
"plt.xlabel(\"v (mph)\");\n", | |
"plt.ylabel(\"mpg\");" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"metadata": {}, | |
"source": [ | |
"I've actually cheated a bit, and picked $P_0$ to give a maximum fuel efficiency near 50 mph. In either case, we now have a decent model of a car's fuel consumption.\n", | |
"\n", | |
"Or in the more useful gallons/100 miles" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 91, | |
"metadata": { | |
"collapsed": false | |
}, | |
"outputs": [ | |
{ | |
"data": { | |
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"YII=\n" | |
], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7f0ba68ac590>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
} | |
], | |
"source": [ | |
"plt.plot( vs * mphfact, 100./mpg, lw=3, alpha=0.8 );\n", | |
"plt.xlabel(\"v (mph)\");\n", | |
"plt.ylabel(\"gallons/(100 miles)\");\n", | |
"plt.ylim((0,5));" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"metadata": {}, | |
"source": [ | |
"The peak efficiency of our model car is:" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 97, | |
"metadata": { | |
"collapsed": false | |
}, | |
"outputs": [ | |
{ | |
"data": { | |
"text/plain": [ | |
"41.421056713604862" | |
] | |
}, | |
"execution_count": 97, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"from scipy.optimize import minimize\n", | |
"minimize( lambda v: -v/( A*v + B*v**2 + C*v**3 + P0 ), 25 ).x[0] * mphfact" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"metadata": {}, | |
"source": [ | |
"# Dynamic Programming for Flat Terrain" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"metadata": {}, | |
"source": [ | |
"We need to define the acceleration and fuel consumption of our car" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 229, | |
"metadata": { | |
"collapsed": true | |
}, | |
"outputs": [], | |
"source": [ | |
"# the acceleration of our car\n", | |
"def f(v,u):\n", | |
" return u - A/m*(v>0) - B/m*v - C/m*v**2 \n", | |
"\n", | |
"#the power consumption of the car\n", | |
"def r(v,u):\n", | |
" return m*v*u*(u>0) + P0" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"metadata": {}, | |
"source": [ | |
"Now we use dynamic programming to find an optimal control trajectory for our vehicle, on a grid of positions and possible velocities." | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 230, | |
"metadata": { | |
"collapsed": false | |
}, | |
"outputs": [ | |
{ | |
"name": "stderr", | |
"output_type": "stream", | |
"text": [ | |
"/home/alemi/anaconda/lib/python2.7/site-packages/IPython/kernel/__main__.py:26: RuntimeWarning: divide by zero encountered in divide\n" | |
] | |
} | |
], | |
"source": [ | |
"# define our grid\n", | |
"XNUM = 161\n", | |
"VNUM = 100\n", | |
"VMAX = 50/mphfact\n", | |
"dx = 10\n", | |
"\n", | |
"# create our dynamic programming variables\n", | |
"costs = np.zeros((XNUM, VNUM))\n", | |
"uvals = np.zeros((XNUM, VNUM))\n", | |
"args = np.zeros((XNUM, VNUM))\n", | |
"vs = np.linspace(0, VMAX, VNUM)\n", | |
"\n", | |
"# set the end conditions\n", | |
"costs[-1,:] = np.inf\n", | |
"costs[-1,0] = 0\n", | |
"\n", | |
"# fill in all of the other values\n", | |
"for i in xrange(costs.shape[0]-2,-1,-1):\n", | |
" # for each position going backwards\n", | |
" for j,vim1 in enumerate(vs):\n", | |
" # for each of our discritized velocities, compute the optimal cost\n", | |
" v = 0.5*(vs + vim1) #candidate average velocities\n", | |
" dv = vs - vim1 # candidate differences in velocities\n", | |
" u = ( v * dv + 0.5 * (dv)**2 )/dx + A/m * (v>0) + B/m * v + C/m * v**2 # candidate values of u\n", | |
" u[np.isnan(u)] = 0 # fix broken values at 0\n", | |
" dcost = r(v, u)/v # compute all of the candidate costs\n", | |
" # apply our physical constraints on u\n", | |
" constraint = np.piecewise( u, [ u < -b, (u <= a)*(u >= -b), u > a ], [ np.inf, 0, np.inf ])\n", | |
" # all possible new costs\n", | |
" newcost = costs[i+1,:] + dcost*dx + constraint\n", | |
" # take the best one\n", | |
" costs[i,j] = newcost.min()\n", | |
" uvals[i,j] = u[newcost.argmin()]\n", | |
" args[i,j] = newcost.argmin()" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"metadata": {}, | |
"source": [ | |
"Having computed all of the possible costs for all possible $x$s and $v$s," | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 231, | |
"metadata": { | |
"collapsed": false | |
}, | |
"outputs": [ | |
{ | |
"data": { | |
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], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7f0ba4b95990>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
} | |
], | |
"source": [ | |
"xgrid,vgrid = np.mgrid[0:dx*XNUM:XNUM*1j,0:VMAX:VNUM*1j]\n", | |
"galloncosts = costs*gallonperj\n", | |
"plt.pcolor( xgrid/1609.0, vgrid*mphfact, galloncosts, vmin=0, vmax=galloncosts[np.isfinite(costs)].max(), cmap='cubehelix' );\n", | |
"plt.xlim((0,dx*XNUM/1609));\n", | |
"plt.ylim((0,VMAX*mphfact));\n", | |
"plt.ylabel(\"v (mph)\");\n", | |
"plt.xlabel(\"x (miles)\");\n", | |
"plt.title(\"Fuel cost for optimal path (gallons)\");\n", | |
"plt.colorbar();\n", | |
"plt.tight_layout();" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"metadata": {}, | |
"source": [ | |
"Now we want to extract the path we are interested in, namely $C(0,0)$. For this we need to trace through all of the optimal arguments at each step" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 232, | |
"metadata": { | |
"collapsed": true | |
}, | |
"outputs": [], | |
"source": [ | |
"traceargs = np.zeros(XNUM)\n", | |
"traceargs[0] = 0\n", | |
"for i in xrange(1,XNUM):\n", | |
" traceargs[i] = args[i-1,traceargs[i-1]]\n", | |
" \n", | |
"traceu = np.array([ uvals[i,arg] for i,arg in enumerate(traceargs) ])\n", | |
"tracev = np.array([ vs[arg] for i,arg in enumerate(traceargs) ])\n", | |
"tracecost = np.array([ costs[i,arg] for i,arg in enumerate(traceargs) ])" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 233, | |
"metadata": { | |
"collapsed": false | |
}, | |
"outputs": [ | |
{ | |
"name": "stdout", | |
"output_type": "stream", | |
"text": [ | |
"108.905442186\n" | |
] | |
} | |
], | |
"source": [ | |
"v = 0.5*(tracev[:-1] + tracev[1:])\n", | |
"dv = tracev[1:] - tracev[:-1]\n", | |
"vdot = f(v, traceu[:-1])\n", | |
"dt = dx / v\n", | |
"ts = dt.cumsum()\n", | |
"print ts.max()\n", | |
"xs = np.linspace(0, XNUM*dx, XNUM)" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 234, | |
"metadata": { | |
"collapsed": true | |
}, | |
"outputs": [], | |
"source": [ | |
"# Convience routines for a nice plot \n", | |
"# FROM: http://stackoverflow.com/questions/10481990/matplotlib-axis-with-two-scales-shared-origin\n", | |
"def adjust_yaxis(ax,ydif,v):\n", | |
" \"\"\"shift axis ax by ydiff, maintaining point v at the same location\"\"\"\n", | |
" inv = ax.transData.inverted()\n", | |
" _, dy = inv.transform((0, 0)) - inv.transform((0, ydif))\n", | |
" miny, maxy = ax.get_ylim()\n", | |
" miny, maxy = miny - v, maxy - v\n", | |
" if -miny>maxy or (-miny==maxy and dy > 0):\n", | |
" nminy = miny\n", | |
" nmaxy = miny*(maxy+dy)/(miny+dy)\n", | |
" else:\n", | |
" nmaxy = maxy\n", | |
" nminy = maxy*(miny+dy)/(maxy+dy)\n", | |
" ax.set_ylim(nminy+v, nmaxy+v)\n", | |
" \n", | |
"def align_yaxis(ax1, v1, ax2, v2):\n", | |
" \"\"\"adjust ax2 ylimit so that v2 in ax2 is aligned to v1 in ax1\"\"\"\n", | |
" _, y1 = ax1.transData.transform((0, v1))\n", | |
" _, y2 = ax2.transData.transform((0, v2))\n", | |
" adjust_yaxis(ax2,(y1-y2)/2,v2)\n", | |
" adjust_yaxis(ax1,(y2-y1)/2,v1)" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 235, | |
"metadata": { | |
"collapsed": false | |
}, | |
"outputs": [ | |
{ | |
"data": { | |
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], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7f0b97d50450>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
} | |
], | |
"source": [ | |
"# plot v(x), u(x)\n", | |
"fig,ax = plt.subplots()\n", | |
"c1 = \"#e41a1c\"\n", | |
"c2 = \"#377eb8\"\n", | |
"ax.plot( xs, tracev*mphfact, lw=2, alpha=0.8, c=c1 );\n", | |
"ax.set_xlabel(r\"$x$ (mph)\");\n", | |
"for tl in ax.get_yticklabels():\n", | |
" tl.set_color(c1)\n", | |
"ax.set_ylabel(r\"$v(x)$ (m/s)\", color=c1);\n", | |
"ax2 = ax.twinx()\n", | |
"ax2.plot( xs, traceu, lw=2, alpha=0.8, c=c2 );\n", | |
"for tl in ax2.get_yticklabels():\n", | |
" tl.set_color(c2)\n", | |
"ax2.set_ylabel(r\"$u(x)$ (m/s$^2$)\", color=c2);\n", | |
"ax.set_xlim((0,1609));\n", | |
"ax2.grid('off');\n", | |
"align_yaxis(ax,0,ax2,0);" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 236, | |
"metadata": { | |
"collapsed": false | |
}, | |
"outputs": [ | |
{ | |
"data": { | |
"image/png": [ | |
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"rkJggg==\n" | |
], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7f0ba453c450>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
} | |
], | |
"source": [ | |
"# plot v(t) and x(t)\n", | |
"fig,ax = plt.subplots()\n", | |
"c1 = \"#e41a1c\"\n", | |
"c3 = \"#984ea3\"\n", | |
"tts = np.r_[0,ts]\n", | |
"ax.plot( tts, tracev*mphfact, lw=2, alpha=0.8, c=c1 );\n", | |
"ax.set_xlabel(r\"$t$ (s)\");\n", | |
"for tl in ax.get_yticklabels():\n", | |
" tl.set_color(c1)\n", | |
"ax.set_ylabel(r\"$v(t)$ (m)\", color=c1);\n", | |
"ax2 = ax.twinx()\n", | |
"ax2.plot( tts, xs, lw=2, alpha=0.8, c=c3 );\n", | |
"for tl in ax2.get_yticklabels():\n", | |
" tl.set_color(c3)\n", | |
"ax2.set_ylabel(r\"$x(t)$ (m/s)\", color=c3);\n", | |
"ax.set_xlim((0,ts.max()));\n", | |
"ax2.grid('off');" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 237, | |
"metadata": { | |
"collapsed": false | |
}, | |
"outputs": [ | |
{ | |
"data": { | |
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], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7f0ba45469d0>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
} | |
], | |
"source": [ | |
"# plot fuel consumption\n", | |
"fig,ax = plt.subplots()\n", | |
"c4 = \"#4daf4a\"\n", | |
"consump = r(v, traceu[:-1])/v * dx * 100 * gallonperj\n", | |
"ax.plot( xs[:-1]/1609, consump, lw=3, alpha=0.8, c=c4 );\n", | |
"ax.set_ylabel(\"fuel use (gallons/100 mi)\");\n", | |
"ax.set_xlabel(\"x (mi)\");\n", | |
"gallons = consump.sum()/100.0\n", | |
"mpg = 100./consump.sum()\n", | |
"ax.set_title(\"tot={:0.2g} ga={:.1f} mpg\".format(gallons, mpg));" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 250, | |
"metadata": { | |
"collapsed": false | |
}, | |
"outputs": [ | |
{ | |
"data": { | |
"image/png": [ | |
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], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7f0babd21e10>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
} | |
], | |
"source": [ | |
"#plot -a/g\n", | |
"fig,ax = plt.subplots();\n", | |
"c5=\"#a65628\"\n", | |
"ax.plot( xs/1609, np.r_[0,(-f(v,traceu[:-1])*dx/g).cumsum()], lw=3, alpha=0.8, c=c5 );\n", | |
"ax.set_xlim((0,1));\n", | |
"ax.set_xlabel(\"x (mi)\");\n", | |
"ax.set_ylabel(r\"$\\int\\, dx\\, \\left(-\\frac{a}{g}\\right)$ (m)\");" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"metadata": {}, | |
"source": [ | |
"## Dynamic Program for Arbitrary Hill" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"metadata": {}, | |
"source": [ | |
"Now we want to solve for the optimal hill, this just means adding another dimension to our grid to optimize over" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 203, | |
"metadata": { | |
"collapsed": true | |
}, | |
"outputs": [], | |
"source": [ | |
"# new acceleration\n", | |
"def f(v,u,hp):\n", | |
" return u - A/m*(v>0) - B/m*v - C/m*v**2 - g*hp\n", | |
"\n", | |
"# new fuel use\n", | |
"def r(v,u,hp):\n", | |
" return m*v*u*(u>0) + P0" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 205, | |
"metadata": { | |
"collapsed": false | |
}, | |
"outputs": [ | |
{ | |
"name": "stderr", | |
"output_type": "stream", | |
"text": [ | |
"/home/alemi/anaconda/lib/python2.7/site-packages/IPython/kernel/__main__.py:37: RuntimeWarning: divide by zero encountered in divide\n" | |
] | |
} | |
], | |
"source": [ | |
"# define our grid\n", | |
"XNUM = 161\n", | |
"VNUM = 100\n", | |
"VMAX = 50/mphfact\n", | |
"dx = 10\n", | |
"HMAX = 20\n", | |
"HMIN = -20\n", | |
"HNUM = 51\n", | |
"HZERO = HNUM//2\n", | |
"\n", | |
"# set up our matrix\n", | |
"costs = np.zeros((XNUM, VNUM, HNUM))\n", | |
"uvals = np.zeros((XNUM, VNUM, HNUM))\n", | |
"args = np.zeros((XNUM, VNUM, HNUM), dtype='int')\n", | |
"vs = np.linspace(0, VMAX, VNUM)[:,None]\n", | |
"hs = np.linspace(HMIN, HMAX, HNUM)[None,:]\n", | |
"\n", | |
"#set the endpoint\n", | |
"costs[-1,:,:] = np.inf\n", | |
"costs[-1,0,HZERO] = 0\n", | |
"args[:,:,:] = -1\n", | |
"uvals[:,:,:] = np.nan\n", | |
"\n", | |
"# fill in all of the other values\n", | |
"for i in xrange(costs.shape[0]-2,-1,-1):\n", | |
" # for each row going backward, fill in all of the costs\n", | |
" for j,vim1 in enumerate(vs):\n", | |
" # for each of our discritized velocities, compute the optimal cost\n", | |
" for k,him1 in enumerate(hs.flat):\n", | |
" # for each of our candidate heights\n", | |
" v = 0.5*(vs + vim1) #average velocity\n", | |
" dv = vs - vim1 # candidate differences\n", | |
" hp = (hs - him1)/dx # candidate slopes\n", | |
" # compute the candidate u values\n", | |
" u = ( v * dv + 0.5 * (dv)**2 )/dx + A/m * (v>0) + B/m * v + C/m * v**2 + g*hp\n", | |
" u[np.isnan(u)] = 0 # fix issues with zeros\n", | |
" dcost = r(v, u, hp)/v # new costs\n", | |
" # put in our constraint on the us\n", | |
" constraint = np.piecewise( u, [ u < -b, (u <= a)*(u >= -b), u > a ], [ np.inf, 0, np.inf ])\n", | |
" # all new candidate costs\n", | |
" newcost = costs[i+1,:,:] + dcost*dx + constraint\n", | |
" # pick the best\n", | |
" mincost = newcost.min()\n", | |
" costs[i,j,k] = mincost # record it\n", | |
" if np.isfinite(mincost): \n", | |
" pk = newcost.argmin() # record what choice we made\n", | |
" args[i,j,k] = pk \n", | |
" uvals[i,j,k] = u.flat[pk]" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"metadata": {}, | |
"source": [ | |
"Now trace out our optimal path" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 206, | |
"metadata": { | |
"collapsed": true | |
}, | |
"outputs": [], | |
"source": [ | |
"traceargs = np.zeros(XNUM, dtype='int')\n", | |
"traceargs[0] = args[0,0,HZERO]\n", | |
"for i in xrange(1,XNUM):\n", | |
" traceargs[i] = args[i-1].flat[traceargs[i-1]]\n", | |
" \n", | |
"traceu = np.array([ uvals[i].flat[arg] for i,arg in enumerate(traceargs) ])\n", | |
"tracev = np.array([ vs[np.unravel_index(arg, (VNUM, HNUM))[0],0] for i,arg in enumerate(traceargs) ])\n", | |
"traceh = np.array([ hs[0,np.unravel_index(arg, (VNUM, HNUM))[1]] for i,arg in enumerate(traceargs) ])\n", | |
"tracecost = np.array([ costs[i].flat[arg] for i,arg in enumerate(traceargs) ])" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 207, | |
"metadata": { | |
"collapsed": false | |
}, | |
"outputs": [ | |
{ | |
"name": "stdout", | |
"output_type": "stream", | |
"text": [ | |
"92.9024849766\n" | |
] | |
} | |
], | |
"source": [ | |
"v = 0.5*(tracev[:-1] + tracev[1:])\n", | |
"dv = tracev[1:] - tracev[:-1]\n", | |
"dh = (traceh[1:] - traceh[:-1])/dx\n", | |
"vdot = f(v, traceu[:-1], dh)\n", | |
"dt = dx / v\n", | |
"ts = dt.cumsum()\n", | |
"print ts.max()\n", | |
"xs = np.linspace(0, XNUM*dx, XNUM)" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 227, | |
"metadata": { | |
"collapsed": false | |
}, | |
"outputs": [ | |
{ | |
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], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7f0b97d50110>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
} | |
], | |
"source": [ | |
"#plot h(x)\n", | |
"fig, ax = plt.subplots()\n", | |
"c5=\"#a65628\"\n", | |
"ax.plot(xs/1609, np.r_[0,traceh[:-1]], lw=3, alpha=0.8, c=c5);\n", | |
"ax.set_xlim((0,1));\n", | |
"ax.set_xlabel(\"x (mi)\");\n", | |
"ax.set_ylabel(\"h(x) (m)\");" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 208, | |
"metadata": { | |
"collapsed": false | |
}, | |
"outputs": [ | |
{ | |
"data": { | |
"image/png": [ | |
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], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7f0ba478d110>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
} | |
], | |
"source": [ | |
"# plot v(x) and u(x)\n", | |
"fig,ax = plt.subplots()\n", | |
"c1 = \"#e41a1c\"\n", | |
"c2 = \"#377eb8\"\n", | |
"ax.plot( xs, tracev*mphfact, lw=2, alpha=0.8, c=c1 );\n", | |
"ax.set_xlabel(r\"$x$ (mph)\");\n", | |
"for tl in ax.get_yticklabels():\n", | |
" tl.set_color(c1)\n", | |
"ax.set_ylabel(r\"$v(x)$ (m/s)\", color=c1);\n", | |
"ax2 = ax.twinx()\n", | |
"ax2.plot( xs, traceu, lw=2, alpha=0.8, c=c2 );\n", | |
"for tl in ax2.get_yticklabels():\n", | |
" tl.set_color(c2)\n", | |
"ax2.set_ylabel(r\"$u(x)$ (m/s$^2$)\", color=c2);\n", | |
"ax.set_xlim((0,1609));\n", | |
"ax2.grid('off');\n", | |
"align_yaxis(ax,0,ax2,0);" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 209, | |
"metadata": { | |
"collapsed": false | |
}, | |
"outputs": [ | |
{ | |
"data": { | |
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], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7f0ba476de10>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
} | |
], | |
"source": [ | |
"# plot v(t) and x(t)\n", | |
"fig,ax = plt.subplots()\n", | |
"c1 = \"#e41a1c\"\n", | |
"c3 = \"#984ea3\"\n", | |
"tts = np.r_[0,ts]\n", | |
"ax.plot( tts, tracev*mphfact, lw=2, alpha=0.8, c=c1 );\n", | |
"ax.set_xlabel(r\"$t$ (s)\");\n", | |
"for tl in ax.get_yticklabels():\n", | |
" tl.set_color(c1)\n", | |
"ax.set_ylabel(r\"$v(t)$ (m)\", color=c1);\n", | |
"ax2 = ax.twinx()\n", | |
"ax2.plot( tts, xs, lw=2, alpha=0.8, c=c3 );\n", | |
"for tl in ax2.get_yticklabels():\n", | |
" tl.set_color(c3)\n", | |
"ax2.set_ylabel(r\"$x(t)$ (m/s)\", color=c3);\n", | |
"ax.set_xlim((0,ts.max()));\n", | |
"ax2.grid('off');" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 219, | |
"metadata": { | |
"collapsed": false | |
}, | |
"outputs": [ | |
{ | |
"data": { | |
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], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7f0ba4c47fd0>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
} | |
], | |
"source": [ | |
"# plot fuel consumption\n", | |
"fig,ax = plt.subplots()\n", | |
"c4 = \"#4daf4a\"\n", | |
"consump = r(v, traceu[:-1], traceh[:-1])/v * dx * 100 * gallonperj\n", | |
"ax.plot( xs[:-1]/1609, consump, lw=3, alpha=0.8, c=c4 );\n", | |
"ax.set_ylabel(\"fuel use (gallons/100 mi)\");\n", | |
"ax.set_xlabel(\"x (mi)\");\n", | |
"gallons = consump.sum()/100.0\n", | |
"mpg = 100./consump.sum()\n", | |
"ax.set_title(\"tot={:0.2g} ga={:.1f} mpg\".format(gallons, mpg));" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": null, | |
"metadata": { | |
"collapsed": true | |
}, | |
"outputs": [], | |
"source": [] | |
} | |
], | |
"metadata": { | |
"kernelspec": { | |
"display_name": "Python 2", | |
"language": "python", | |
"name": "python2" | |
}, | |
"language_info": { | |
"codemirror_mode": { | |
"name": "ipython", | |
"version": 2 | |
}, | |
"file_extension": ".py", | |
"mimetype": "text/x-python", | |
"name": "python", | |
"nbconvert_exporter": "python", | |
"pygments_lexer": "ipython2", | |
"version": "2.7.9" | |
} | |
}, | |
"nbformat": 4, | |
"nbformat_minor": 0 | |
} |
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