{ "cells": [ { "cell_type": "markdown", "metadata": { "toc": "true" }, "source": [ "# Table of Contents\n", "

0.1  現実的なマシンの加速性能
0.2  課題
0.2.1  ヒント
0.3  課題4
1  課題4: fitting
2  課題5:
2.1  空気抵抗なしの場合
2.1.1  課題4のfittingで求めた加速度
2.2  空気抵抗が小さい場合(cc=0.1)
2.3  空気抵抗が大きい場合(cc=0.5)
3  馬力換算,実データの再現
4  訂正した問題
4.1  コメント
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", " スーパーカーのシミュレーション\n", "
\n", "
\n", " cc by Shigeto R. Nishitani\n", "
\n", "\n", "\n", "* /Users/bob/Desktop/maple_ode/python_ode.ipynb\n", "* origin\tgit@github.com:daddygongon/maple_ode.git (fetch)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 現実的なマシンの加速性能\n", "## 課題\n", "課題4で計測したのと同じマシンを使って,Euler法による\n", "シミュレーションから空気抵抗と馬力を予測する.\n", "空気抵抗をいくつか現実的な程度に設定してシミュレーションを行え.\n", "馬力がどの程度変化するかを報告せよ.\n", "現実的なゼロヨンのタイムとその時の時速を報告せよ.\n", "\n", "### ヒント\n", "* 課題4と課題5のEuler法に空気抵抗を加えてシミュレーションを行い,\n", "* 図表にまとめてA4 2枚程度のレポートに仕上げよ.\n", "* 厳密にフィッティングするとおかしくなるので,目視確認でよい.\n", "\n", "## 課題4\n", "次のデータを示す自動車の加速度$a$を \n", "$d = a t^2 (y = a x^2)$にフィットして\n", "求めよ.\n", "\n", "|time[sec] | dist[m] \n", "|---:|---:\n", "|0 | 0 \n", "|0.751 | 10 \n", "|1.113 | 20 \n", "|1.504 | 40\n", "* 自動車の走行距離データ.\n", "\n", "100[m]何秒で通過するか? 小数点以下2桁程度で答えよ.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 課題4: fitting\n", "\n", "データにfitさせると次のようになる.\n" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 17.34679]\n" ] }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "import numpy as np\n", "from scipy.optimize import curve_fit\n", "\n", "np.set_printoptions(precision=5, suppress=True)\n", "\n", "def fitting_func(t,a):\n", " d =a*(t**2)\n", " return d\n", "\n", "t =np.array([0, 0.751, 1.113, 1.504])\n", "d = np.array([0, 10, 20, 40])\n", "\n", "param, cov = curve_fit(fitting_func, t, d)\n", "x = np.arange(0,5, 0.0001)\n", "print(param)\n", "fit_y = fitting_func(x, *param)\n", "plt.plot(x, fit_y)\n", "plt.scatter(t,d)\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "このマシンの加速度は17.35[m/sec^2]となっており,グラフの目視によると2.4秒ほどで100mを通過し,4.8秒ほどで400mを通過している.その時の速度は299km/hで,ゼロヨン時の速度は600km/hであることが次の通り確認できる." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[(-2.40099080206130,), (2.40099080206130,)] [(-4.80198160412261,), (4.80198160412261,)]\n", "[ 299.75245]\n", "[ 599.50491]\n" ] } ], "source": [ "from sympy import *\n", "t0=symbols('t0')\n", "a0 = param\n", "eq1=a0*t0*t0\n", "s0=solve(eq1-100, t0)\n", "s1=solve(eq1-400, t0)\n", "\n", "print(s0,s1)\n", "\n", "print(2*a0*2.4*60*60/1000)\n", "print(2*a0*4.8*60*60/1000)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 課題5: \n", "## 空気抵抗なしの場合\n", "### 課題4のfittingで求めた加速度\n", "まず空気抵抗なしでのマシンの運動を微分方程式で立ててみる.\n", "空気抵抗なしなので,cc=0である.また,物体の落下ではg=9.8としていたが,\n", "今は,加速度としてaaとし,向きを逆にしてeuler法に入れておく.\n", "\n", "加速度のaaは,先ほどの課題4から17.35を初期値にして様子をみる." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import matplotlib.pyplot as plt\n", "import numpy as np\n", "\n", "def euler(x0, v0):\n", " v1 = v0 + (-cc * v0 + aa) * dt\n", " x1 = x0 + v0 * dt\n", " return x1,v1\n", "\n", "def my_plot(xx, vv, tt):\n", " t =np.array([0, 0.751, 1.113, 1.504])\n", " d = np.array([0, 10, 20, 40])\n", "\n", " plt.plot(tt,xx,color='b')\n", " plt.plot(tt,vv,color='r')\n", " plt.scatter(t,d,color='k')\n", "\n", " plt.grid()\n", " plt.show()" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "aa, dt, cc = a0, 0.01, 0.0\n", "tt, xx, vv =[0.0],[0.0],[0.0]\n", "time = 0.0\n", "\n", "for i in range(0, 200):\n", " time += dt\n", " x, v = euler(xx[-1], vv[-1])\n", " tt.append(time)\n", " xx.append(x)\n", " vv.append(v)\n", "\n", "plt.scatter(t,d)\n", "my_plot(xx,vv,tt)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "全く合ってない.そこで,aaをいくつか変えてプロットしてみた.\n", "\n", "その結果,a0(課題4で求めたfitting parametersの値)を2倍すると,データに一致する.\n", "これは,\n", "$$\n", "dist = \\int v dt = \\int a t dt = \\frac{1}{2}at^2\n", "$$\n", "\n", "における係数から出てきたものと考えられる.\n", "\n", "すなわち,課題4のfittingでは実データに対して加速度がfittingされるが,\n", "euler法では微分方程式に基づいて逐次的に解かれていくため,\n", "方程式の係数に差異が出る." ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "def super_machine(r_max):\n", " tt, xx, vv =[0.0],[0.0],[0.0]\n", " time = 0.0\n", " for i in range(0, r_max):\n", " time += dt\n", " x, v = euler(xx[-1], vv[-1])\n", " tt.append(time)\n", " xx.append(x)\n", " vv.append(v)\n", " return(xx,vv,tt)\n", "\n", "aa, dt, cc = a0*2, 0.01, 0.0\n", "xx,vv,tt = super_machine(200)\n", "my_plot(xx,vv,tt)" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "aa, dt, cc = a0*2, 0.01, 0.0\n", "xx,vv,tt = super_machine(500)\n", "my_plot(xx,vv,tt)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "空気抵抗のない場合のfitting結果は上の通り.綺麗に加速して,約4.8秒で400mに到達する." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 空気抵抗が小さい場合(cc=0.1)\n", "\n", "空気抵抗が0.1ある場合では,馬力はそれほどあげなくてもほぼおなじfitが可能.ただし,高速での\n", "伸びは抑えられて,ゼロヨンまで5.1秒ほどかかる.\n" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "aa, dt, cc = a0*2, 0.01, 0.1\n", "xx,vv,tt = super_machine(550)\n", "my_plot(xx,vv,tt)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 空気抵抗が大きい場合(cc=0.5)\n", "さらに空気抵抗が0.5まで大きくすると,$aa=2a0$ではデータに合わず,\n", "より大きな加速度(2.4a0)が要求される.\n" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "image/png": 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lnFvknFvinOt+gMedc+71wOOznXONQ31uBDMlB7LMcc794Jw7p9Bj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\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "aa, dt, cc = a0*2.4, 0.01, 0.5\n", "xx,vv,tt = super_machine(200)\n", "my_plot(xx,vv,tt)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "高速での空気抵抗による速度の低下はより顕著となり,ゼロヨンまで6.5秒ほどもかかることになる." ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "aa, dt, cc = a0*2.4, 0.01, 0.5\n", "xx,vv,tt = super_machine(700)\n", "my_plot(xx,vv,tt)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 馬力換算,実データの再現\n", "\n", "馬力換算は難しいですね.単純な計算式があるのかと\n", "思ってたのですが,なさそうです.\n", "\n", "そこで実データの再現性を確認しておきます.\n", "\n", "今話題(2017/11発表)のテスラのロードスターは,\n", "\n", "|スペック| |\n", "|---:|---:\n", "|0-60マイル(0-96km/h)加速|1.9秒\n", "|最高速度|400km/h|\n", "|ゼロヨン|8.9秒|\n", "\n", "だそうです.でも,馬力のデータがなくって...\n", "\n", "その前の世代のロードスターは詳細なデータがあって,\n", "\n", "|スペック| |\n", "|---:|---:\n", "|0-60マイル(0-96km/h)加速|3.7秒\n", "|車両重量|1,238kg\n", "|最高出力|215kW(292馬力)\n", "\n", "となっています.これを換算したかったのですが,\n", "ちょっと無理みたい.ごめんなさい.\n", "\n", "我々のsuper machineの馬力を下げて,空気抵抗も下げると\n", "加速性能とゼロヨンは最新のロードスターのスペックを\n", "再現できそうです.\n" ] }, { "cell_type": "code", "execution_count": 122, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "108.67011841024657 402.2828535082075 233.1173749704914\n" ] } ], "source": [ "aa, dt, cc = a0*1.2, 0.01, 0.3\n", "xx,vv,tt = super_machine(1000)\n", "my_plot(xx,vv,tt)\n", "print(vv[190]*3.6,xx[890],vv[900]*3.6)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "上の設定(aa, cc = 1.2a0, 0.3)だと\n", "108km/hまでで1.9秒,8.9秒で402m\n", "となります.\n", "\n", "でも,その時の時速は233km/hで,最高速度が抑えられてスペック通りにはなりません.\n", "現実と理論との間にはまだギャップがあるようです.申し訳ない." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 訂正した問題\n", "課題4で計測したのと同じマシンを使って,Euler法による シミュレーションから空気抵抗と加速性能を予測する. 空気抵抗をいくつか現実的な程度に設定してシミュレーションを行え. 加速性能をどの程度に設定する必要があるかを推測せよ. その場合のゼロヨンのタイムとその時の時速を計算せよ.\n", "\n", "## コメント\n", "ここでやったような目視確認を,「現場合わせ」と呼んでいます.なにかを設計・制作する時には,ここで使った微分方程式のような経験的な式をたてて,部品の性能を示すパラメータをいじって,シミュレートを繰り返し,開発に必要となる部品性能や制御パラメータを割り出します.現場で必要となる知恵として心に留めておいてください." ] }, { "cell_type": "code", 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