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Michael Soukup 2019-05-11 15:18:34 +02:00
commit 283093ef2f
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{
"cells": [
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"%matplotlib notebook\n",
"\n",
"import numpy as np\n",
"from scipy.integrate import odeint\n",
"import matplotlib.pyplot as plt\n",
"from matplotlib import animation\n",
"plt.rcParams[\"animation.html\"] = \"jshtml\""
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"m1 = 1\n",
"m2 = 1\n",
"c1 = 0.1\n",
"c2 = 0.1\n",
"c3 = 0.1\n",
"a1 = 0.1\n",
"a2 = 0.1\n",
"a3 = 0.1\n",
"k1 = 0.05\n",
"k2 = 0.21\n",
"k3 = 0.11\n",
"C1 = 0.15\n",
"C2 = 0.30\n",
"omega = 0.1\n",
"\n",
"f1 = lambda t: C1*np.cos(omega*t)\n",
"f2 = lambda t: C2*np.cos(omega*t)\n",
"\n",
"def deriv(X, t):\n",
" \"\"\"Return the derivatives dx/dt and d2x/dt2.\"\"\"\n",
" x1, x2, xd1, xd2 = X\n",
" F1 = f1(t)\n",
" F2 = f2(t)\n",
" xdd1 = F1 - xd1*(c1 + c2) + xd2*c2 - x1*(k1 + k2) + x2*k2 - a1*x1**3 + a2*(x2 - x1)**3\n",
" xdd2 = F2 - xd2*(c2 + c3) + xd1*c1 - x2*(k2 + k3) + x1*k2 - a3*x2**3 + a2*(x2 - x1)**3\n",
" return xd1, xd2, xdd1/m1, xdd2/m2\n",
"\n",
"\n",
"def solve_duffing(tmax, dt_per_period, t_trans, x0, v0):\n",
" \"\"\"Solve the Duffing equation with the standard odeint solver.\n",
" \n",
" Find the numerical solution to the Duffing equation using a suitable\n",
" time grid: tmax is the maximum time (s) to integrate to; t_trans is\n",
" the initial time period of transient behaviour until the solution\n",
" settles down (if it does) to some kind of periodic motion (these data\n",
" points are dropped) and dt_per_period is the number of time samples\n",
" (of duration dt) to include per period of the driving motion (frequency\n",
" omega).\n",
" \n",
" Returns the time grid, t (after t_trans), position, x, and velocity,\n",
" xdot, dt, and step, the number of array points per period of the driving\n",
" motion.\n",
" \n",
" \"\"\"\n",
" # Time point spacings and the time grid\n",
"\n",
" period = 2*np.pi/omega\n",
" dt = 2*np.pi/omega / dt_per_period\n",
" step = int(period / dt)\n",
" t = np.arange(0, tmax, dt)\n",
" # Initial conditions: x, xdot\n",
" X0 = x0 + v0\n",
" X = odeint(deriv, X0, t)\n",
" idx = int(t_trans / dt)\n",
" return t[idx:], X[idx:], dt, step"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"# Set up the motion for a oscillator with initial positions and velocities\n",
"x0, v0 = (0.5, -0.5), (0.5, -0.1)\n",
"tmax, t_trans = 150, 0\n",
"dt_per_period = 100"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"# Solve the equation of motion.\n",
"t, X, dt, pstep = solve_duffing(tmax, dt_per_period, t_trans, x0, v0)\n",
"print(\"# samples: %d\" % len(t))\n",
"print(\"dt: %.4f\" % dt)\n",
"print(\"Steps per period: %d\" % pstep)\n",
"x1, x2, xd1, xd2 = X.T"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"fig1, ((ax11, ax12), (ax21, ax22)) = plt.subplots(nrows=2, ncols=2)\n",
"\n",
"ax11.plot(t, x1)\n",
"ax11.set_xlabel(r'$x_1$')\n",
"ax12.plot(t, x2)\n",
"ax12.set_xlabel(r'$x_2$')\n",
"ax21.plot(t, xd1)\n",
"ax21.set_xlabel(r'$v_1$')\n",
"ax22.plot(t, xd2)\n",
"ax22.set_xlabel(r'$v_2$')\n",
"plt.tight_layout()"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"scrolled": false
},
"outputs": [],
"source": [
"%%capture\n",
"fig2, ((ax11, ax12), (ax21, ax22)) = plt.subplots(nrows=2, ncols=2)\n",
"\n",
"# Positions\n",
"ax11.set_xlabel(r'$x_1$')\n",
"ax11.set_ylabel(r'$x_2$')\n",
"ln11, = ax11.plot([], [])\n",
"ax11.set_xlim([np.min(x1), np.max(x1)])\n",
"ax11.set_ylim([np.min(x2), np.max(x2)])\n",
"\n",
"# Velocities\n",
"ax12.set_xlabel(r'$v_1$')\n",
"ax12.set_ylabel(r'$v_2$')\n",
"ln12, = ax12.plot([], [])\n",
"ax12.set_xlim([np.min(xd1), np.max(xd1)])\n",
"ax12.set_ylim([np.min(xd2), np.max(xd2)])\n",
"\n",
"# Phase spaces\n",
"ax21.set_xlabel(r'$\\mathbf{x}$')\n",
"ax21.set_ylabel(r'$\\mathbf{v}$')\n",
"ln21a, = ax21.plot([], [])\n",
"ln21b, = ax21.plot([], [])\n",
"ax21.set_xlim([min(np.min(x1), np.min(x2)), max(np.max(x1), np.max(x2))])\n",
"ax21.set_ylim([min(np.min(xd1), np.min(xd2)), max(np.max(xd1), np.max(xd2))])\n",
"\n",
"# F(t)\n",
"F1 = f1(t)\n",
"F2 = f2(t)\n",
"ax22.set_xlabel(r'$t$')\n",
"ax22.set_ylabel(r'$\\mathbf{F}(t)$')\n",
"ln22a, = ax22.plot([], [])\n",
"ln22b, = ax22.plot([], [])\n",
"ax22.set_xlim([t[0], t[-1]])\n",
"ax21.set_ylim([min(np.min(F1), np.min(F2)), max(np.max(F1), np.max(F2))])\n",
"\n",
"plt.tight_layout()\n",
"\n",
"def animate(i):\n",
" ln11.set_data(x1[:i], x2[:i])\n",
" ln12.set_data(xd1[:i], xd2[:i])\n",
" ln21a.set_data(x1[:i], xd1[:i])\n",
" ln21b.set_data(x2[:i], xd2[:i])\n",
" ln22a.set_data(t[:i], F1[:i])\n",
" ln22b.set_data(t[:i], F2[:i])\n",
" \n",
"\n",
"anim = animation.FuncAnimation(fig2, animate, frames=len(t), interval=100, repeat_delay=1000)"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"anim"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"anaconda-cloud": {},
"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": "3.5.2"
}
},
"nbformat": 4,
"nbformat_minor": 1
}

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{
"cells": [
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"%matplotlib notebook\n",
"\n",
"import numpy as np\n",
"from scipy.integrate import odeint\n",
"import matplotlib.pyplot as plt\n",
"from matplotlib import animation\n",
"plt.rcParams[\"animation.html\"] = \"jshtml\""
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"m1 = 1\n",
"m2 = 1\n",
"c1 = 0.1\n",
"c2 = 0.1\n",
"c3 = 0.1\n",
"a1 = 0.1\n",
"a2 = 0.1\n",
"a3 = 0.1\n",
"k1 = 0.05\n",
"k2 = 0.21\n",
"k3 = 0.11\n",
"C1 = 0.15\n",
"C2 = 0.30\n",
"omega = 0.1\n",
"\n",
"f1 = lambda t: C1*np.cos(omega*t)\n",
"f2 = lambda t: C2*np.cos(omega*t)\n",
"\n",
"def deriv(X, t):\n",
" \"\"\"Return the derivatives dx/dt and d2x/dt2.\"\"\"\n",
" x1, x2, xd1, xd2 = X\n",
" F1 = f1(t)\n",
" F2 = f2(t)\n",
" xdd1 = F1 - xd1*(c1 + c2) + xd2*c2 - x1*(k1 + k2) + x2*k2 - a1*x1**3 + a2*(x2 - x1)**3\n",
" xdd2 = F2 - xd2*(c2 + c3) + xd1*c1 - x2*(k2 + k3) + x1*k2 - a3*x2**3 + a2*(x2 - x1)**3\n",
" return xd1, xd2, xdd1/m1, xdd2/m2\n",
"\n",
"\n",
"def solve_duffing(tmax, dt_per_period, t_trans, x0, v0):\n",
" \"\"\"Solve the Duffing equation with the standard odeint solver.\n",
" \n",
" Find the numerical solution to the Duffing equation using a suitable\n",
" time grid: tmax is the maximum time (s) to integrate to; t_trans is\n",
" the initial time period of transient behaviour until the solution\n",
" settles down (if it does) to some kind of periodic motion (these data\n",
" points are dropped) and dt_per_period is the number of time samples\n",
" (of duration dt) to include per period of the driving motion (frequency\n",
" omega).\n",
" \n",
" Returns the time grid, t (after t_trans), position, x, and velocity,\n",
" xdot, dt, and step, the number of array points per period of the driving\n",
" motion.\n",
" \n",
" \"\"\"\n",
" # Time point spacings and the time grid\n",
"\n",
" period = 2*np.pi/omega\n",
" dt = 2*np.pi/omega / dt_per_period\n",
" step = int(period / dt)\n",
" t = np.arange(0, tmax, dt)\n",
" # Initial conditions: x, xdot\n",
" X0 = x0 + v0\n",
" X = odeint(deriv, X0, t)\n",
" idx = int(t_trans / dt)\n",
" return t[idx:], X[idx:], dt, step"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"# Set up the motion for a oscillator with initial positions and velocities\n",
"x0, v0 = (0.5, -0.5), (0.5, -0.1)\n",
"tmax, t_trans = 150, 0\n",
"dt_per_period = 100"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"# Solve the equation of motion.\n",
"t, X, dt, pstep = solve_duffing(tmax, dt_per_period, t_trans, x0, v0)\n",
"print(\"# samples: %d\" % len(t))\n",
"print(\"dt: %.4f\" % dt)\n",
"print(\"Steps per period: %d\" % pstep)\n",
"x1, x2, xd1, xd2 = X.T"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"fig1, ((ax11, ax12), (ax21, ax22)) = plt.subplots(nrows=2, ncols=2)\n",
"\n",
"ax11.plot(t, x1)\n",
"ax11.set_xlabel(r'$x_1$')\n",
"ax12.plot(t, x2)\n",
"ax12.set_xlabel(r'$x_2$')\n",
"ax21.plot(t, xd1)\n",
"ax21.set_xlabel(r'$v_1$')\n",
"ax22.plot(t, xd2)\n",
"ax22.set_xlabel(r'$v_2$')\n",
"plt.tight_layout()"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"scrolled": false
},
"outputs": [],
"source": [
"%%capture\n",
"fig2, ((ax11, ax12), (ax21, ax22)) = plt.subplots(nrows=2, ncols=2)\n",
"\n",
"# Positions\n",
"ax11.set_xlabel(r'$x_1$')\n",
"ax11.set_ylabel(r'$x_2$')\n",
"ln11, = ax11.plot([], [])\n",
"ax11.set_xlim([np.min(x1), np.max(x1)])\n",
"ax11.set_ylim([np.min(x2), np.max(x2)])\n",
"\n",
"# Velocities\n",
"ax12.set_xlabel(r'$v_1$')\n",
"ax12.set_ylabel(r'$v_2$')\n",
"ln12, = ax12.plot([], [])\n",
"ax12.set_xlim([np.min(xd1), np.max(xd1)])\n",
"ax12.set_ylim([np.min(xd2), np.max(xd2)])\n",
"\n",
"# Phase spaces\n",
"ax21.set_xlabel(r'$\\mathbf{x}$')\n",
"ax21.set_ylabel(r'$\\mathbf{v}$')\n",
"ln21a, = ax21.plot([], [])\n",
"ln21b, = ax21.plot([], [])\n",
"ax21.set_xlim([min(np.min(x1), np.min(x2)), max(np.max(x1), np.max(x2))])\n",
"ax21.set_ylim([min(np.min(xd1), np.min(xd2)), max(np.max(xd1), np.max(xd2))])\n",
"\n",
"# F(t)\n",
"F1 = f1(t)\n",
"F2 = f2(t)\n",
"ax22.set_xlabel(r'$t$')\n",
"ax22.set_ylabel(r'$\\mathbf{F}(t)$')\n",
"ln22a, = ax22.plot([], [])\n",
"ln22b, = ax22.plot([], [])\n",
"ax22.set_xlim([t[0], t[-1]])\n",
"ax21.set_ylim([min(np.min(F1), np.min(F2)), max(np.max(F1), np.max(F2))])\n",
"\n",
"plt.tight_layout()\n",
"\n",
"def animate(i):\n",
" ln11.set_data(x1[:i], x2[:i])\n",
" ln12.set_data(xd1[:i], xd2[:i])\n",
" ln21a.set_data(x1[:i], xd1[:i])\n",
" ln21b.set_data(x2[:i], xd2[:i])\n",
" ln22a.set_data(t[:i], F1[:i])\n",
" ln22b.set_data(t[:i], F2[:i])\n",
" \n",
"\n",
"anim = animation.FuncAnimation(fig2, animate, frames=len(t), interval=100, repeat_delay=1000)"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"anim"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"anaconda-cloud": {},
"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": "3.5.2"
}
},
"nbformat": 4,
"nbformat_minor": 1
}