{ "cells": [ { "cell_type": "markdown", "id": "6cf173b5-b29a-432e-88bf-6d729c1407a3", "metadata": {}, "source": [ "# Damped and Driven Harmonic Oscillator" ] }, { "cell_type": "markdown", "id": "71dc82c1-a87e-41fc-87aa-028fc01acba4", "metadata": {}, "source": [ "A simple harmonic oscillator is described by the equation of motion:\n", "\\begin{equation}\n", "\\ddot{x} = - \\omega_0^2 \\, x\n", "\\end{equation}\n", "where $\\omega_0$ is the natural frequency of the oscillator. For example, a mass attached to a spring has $\\omega_0^2 = k/m$, whereas a simple pendulum has $\\omega_0^2 = g/l$. The solution to the equation is a sinusoidal function of time:\n", "\\begin{equation}\n", "x(t) = A \\cos(\\omega_0 t + \\theta_0)\n", "\\end{equation}\n", "where $A$ is the amplitude of the oscillation and $\\theta_0$ is the initial phase." ] }, { "cell_type": "markdown", "id": "7ec5639e-6c57-4818-a7a9-9cea1347aff5", "metadata": {}, "source": [ "## Damped harmonic oscillator" ] }, { "cell_type": "markdown", "id": "af681f6e-a343-470e-a2b4-bfec579f855f", "metadata": {}, "source": [ "Now consider a *damped* harmonic oscillator, which is subject to a damping force that is proportional to the velocity. The equation of motion becomes:\n", "\\begin{equation}\n", "\\ddot{x} = - \\omega_0^2 \\, x - \\gamma \\, \\dot{x}\n", "\\end{equation}\n", "This equation can be solved by using the ansatz $x \\sim \\mathrm{e}^{i \\omega t}$, with the understanding that $x$ is the real part of the solution. Plugging that into the differential equation leads to an algebraic equation for $\\omega$:\n", "\\begin{equation}\n", "\\omega^2 - i \\gamma \\omega - \\omega_0^2 = 0\n", "\\end{equation}\n", "The solutions are:\n", "\\begin{equation}\n", "\\omega = i \\frac{\\gamma}{2} \\pm \\sqrt{\\omega_0^2 - \\frac{\\gamma^2}{4}}\n", "\\end{equation}\n", "\n", "- For weak damping ($\\gamma < 2 \\omega_0$), the square-root is real, and let us denote it by $\\omega_1 \\equiv \\sqrt{\\omega_0^2 - \\gamma^2 /4}$. Then the solution looks like:\n", "\\begin{equation}\n", "x \\sim \\mathrm{e}^{i \\omega t} = \\mathrm{e}^{-(\\gamma/2)\\, t} \\, \\mathrm{e}^{i \\omega_1 t}\n", "\\end{equation}\n", "This describes an oscillation with frequency $\\omega_1$ and an amplitude that decays with time.\n", "\n", "- For strong damping ($\\gamma > 2 \\omega_0$), the square-root becomes imaginary, so we can write $\\omega = i \\Big( \\frac{\\gamma}{2} \\pm \\sqrt{\\frac{\\gamma^2}{4} - \\omega_0^2} \\Big) \\equiv i \\lambda_\\pm$. Then the solution will behave like:\n", "\\begin{equation}\n", "x \\sim A \\, \\mathrm{e}^{-\\lambda_+ t} + B \\, \\mathrm{e}^{-\\lambda_- t}\n", "\\end{equation}\n", "Both terms are exponentially decaying with time, without any oscillatory motion." ] }, { "cell_type": "markdown", "id": "a2b21463-90cb-45ad-85ce-ed4d92724055", "metadata": {}, "source": [ "We can numerically solve the equation for the damped harmonic oscillator to verify these behaviors." ] }, { "cell_type": "code", "execution_count": 1, "id": "0b63c7c6-a27c-4ac1-a1fd-2bb2ef787c9e", "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "import scipy.integrate as intgr" ] }, { "cell_type": "code", "execution_count": 2, "id": "742f929c-c91c-4979-9c77-0175e2e5887a", "metadata": {}, "outputs": [], "source": [ "def damped(X, t, omega0, gamma):\n", " x, v = X # unpack variables\n", " dxdt = v\n", " dvdt = - omega0**2 * x - gamma * v\n", " dXdt = [dxdt, dvdt] # pack derivatives\n", " return dXdt" ] }, { "cell_type": "markdown", "id": "4cd48ac5-5c82-4e24-936d-32b8b14c1785", "metadata": {}, "source": [ "Let us first try a small damping strength, in which case we expect the oscillation to decay in time." ] }, { "cell_type": "code", "execution_count": 3, "id": "be28f24a-d7a4-4f54-b4d4-71f2482c21a7", "metadata": {}, "outputs": [], "source": [ "# set parameters\n", "omega0 = 1 # can always set this to 1 by rescaling time\n", "gamma = 0.2\n", "\n", "# specify initial values\n", "x0 = 1\n", "v0 = 0\n", "\n", "time = np.arange(0, 50, 0.1) # time points to evaluate solutions at\n", "sol = intgr.odeint(damped, [x0, v0], time, args=(omega0, gamma))\n", "xt = sol[:,0] # x(t) is the first component" ] }, { "cell_type": "code", "execution_count": 4, "id": "766c0134-48e3-438d-b561-c85bcf5fbb4b", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.figure()\n", "plt.axhline(0, lw=1, color='k')\n", "plt.plot(time, xt)\n", "plt.plot(time, np.exp(-gamma/2*time), 'k--', lw=1) # expected amplitude that decays with time\n", "plt.plot(time, -np.exp(-gamma/2*time), 'k--', lw=1) # expected amplitude that decays with time\n", "plt.ylim(-1.2, 1.2)\n", "plt.xlabel(r'$t$')\n", "plt.ylabel(r'$x$')\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "f79e0e63-ec64-40de-9e54-23f77af09789", "metadata": {}, "source": [ "The dashed lines show the analytic expression for the decaying amplitude. It matches well with the numerical solution.\n", "\n", "Let us now try a wide range of damping strengths, including below and above the threshold $\\gamma = 2 \\omega_0$." ] }, { "cell_type": "code", "execution_count": 5, "id": "fe57ba46-43ac-4f1d-a610-8ae3fd56ffb5", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "gamma_list = [0.2, 1.0, 2.0, 3.0]\n", "\n", "plt.figure()\n", "plt.axhline(0, lw=1, color='k')\n", "for gamma in gamma_list:\n", " sol = intgr.odeint(damped, [x0, v0], time, args=(omega0, gamma))\n", " xt = sol[:,0]\n", " plt.plot(time, xt, label=f'$\\gamma={gamma}$')\n", "plt.xlim(0, 10)\n", "plt.ylim(-1.2, 1.2)\n", "plt.xlabel(r'$t$')\n", "plt.ylabel(r'$x$')\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "89da3dc5-742e-401e-9bbc-59d915add7c4", "metadata": {}, "source": [ "As can be seen from the plot, once $\\gamma$ is equal to or larger than the threshold, the solution no longer oscillates, but decays monotonically instead." ] }, { "cell_type": "markdown", "id": "57523717-7ef7-4930-96ff-598516922fe4", "metadata": {}, "source": [ "## Driven harmonic oscillator (with damping)" ] }, { "cell_type": "markdown", "id": "b223c1f5-94fe-475a-ac15-79748a1f74f2", "metadata": {}, "source": [ "We have seen that the damped harmonic oscillator will stop moving eventually. That is because the energy of the system is being dissipated by the damping force. In order to sustain motion, we need to pump energy into the system." ] }, { "cell_type": "markdown", "id": "9081f926-f531-4e95-9294-312a87cba3e4", "metadata": {}, "source": [ "Consider a damped harmonic oscillator that is *driven* by an external force. We will consider a sinusoidal force, such that the equation of motion is given by:\n", "\\begin{equation}\n", "\\ddot{x} + \\gamma \\, \\dot{x} + \\omega_0^2 \\, x = \\xi \\, \\cos(\\omega t)\n", "\\end{equation}\n", "Here $\\xi$ and $\\omega$ are the amplitude and frequency of the driving force.\n", "\n", "To solve this equation, we may again use the ansatz $x \\sim \\mathrm{e}^{i \\omega t}$. However, this time we will assume that the oscillation frequency is the same as the driving force, $\\omega$. Intuitively, the damped motion we saw above will vanish after a transient period of time, and only the mode driven by the external force will remain. To proceed analytically, we will also include an amplitude in the ansatz, so that $x = \\tilde{A} \\, \\mathrm{e}^{i \\omega t}$. Note that $\\tilde{A}$ can be a complex number, with the understanding that the solution will be the real part of the whole expression. Similarly, we will modify the equation by replacing the driving term with $\\xi \\, \\mathrm{e}^{i \\omega t}$, so that the original equation is the real part of the modified equation:\n", "\\begin{equation}\n", "\\ddot{x} + \\gamma \\, \\dot{x} + \\omega_0^2 \\, x = \\xi \\, \\mathrm{e}^{i \\omega t}\n", "\\end{equation}\n", "Inserting the ansatz into the equation yields:\n", "\\begin{equation}\n", "\\big( (-\\omega^2 + i \\gamma \\omega + \\omega_0^2) \\tilde{A} - \\xi \\big) \\, \\mathrm{e}^{i \\omega t} = 0\n", "\\end{equation}\n", "Solving for $\\tilde{A}$ gives:\n", "\\begin{equation}\n", "\\tilde{A} = \\frac{\\xi}{\\omega_0^2 - \\omega^2 + i \\gamma \\omega}\n", "\\end{equation}\n", "\n", "We are interested in the amplitude of the driven motion. Write $\\tilde{A}$ in the polar form $\\tilde{A} = A \\, \\mathrm{e}^{i \\theta_0}$, then we have:\n", "\\begin{equation}\n", "x = A \\, \\mathrm{e}^{i (\\omega t + \\theta_0)}\n", "\\end{equation}\n", "which is exactly the form of a harmonic oscillation with an amplitude\n", "\\begin{equation}\n", "A = \\big| \\tilde{A} \\big| = \\bigg( \\frac{\\xi^2}{(\\omega_0^2 - \\omega^2)^2 + \\gamma^2 \\omega^2} \\bigg)^{\\frac12}\n", "\\end{equation}\n", "and a phase shift\n", "\\begin{equation}\n", "\\theta_0 = \\arg \\big(\\tilde{A} \\big) = - \\arctan \\bigg( \\frac{\\gamma \\omega}{\\omega_0^2 - \\omega^2} \\bigg)\n", "\\end{equation}\n", "As we will see below, this amplitude depends on how close the driving frequency $\\omega$ is to the natural frequency $\\omega_0$ of the oscillator. Note that, even when these frequencies are very different, the system is still forced to oscillate at the *driving* frequency, not its natural frequency." ] }, { "cell_type": "markdown", "id": "23d53ab2-f688-484c-ab2e-c93231825fc6", "metadata": {}, "source": [ "Let us now solve the equation numerically to see how the system settles into the synchronous motion with the driving force." ] }, { "cell_type": "code", "execution_count": 6, "id": "37e66ee8-bd9c-4418-99b3-bef21976d9a7", "metadata": {}, "outputs": [], "source": [ "def driven(X, t, omega0, gamma, xi, omega):\n", " x, v = X # unpack variables\n", " dxdt = v\n", " dvdt = - omega0**2 * x - gamma * v + xi * np.cos(omega*t)\n", " dXdt = [dxdt, dvdt] # pack derivatives\n", " return dXdt" ] }, { "cell_type": "code", "execution_count": 7, "id": "3e5382af-f1dd-48c8-89c2-9eeca8ca76c3", "metadata": {}, "outputs": [], "source": [ "# set parameters\n", "omega0 = 1\n", "gamma = 0.2\n", "xi = 1\n", "omega = 1.5\n", "\n", "# specify initial values\n", "x0 = 1\n", "v0 = 0\n", "\n", "time = np.arange(0, 50, 0.1) # time points to evaluate solutions at\n", "sol = intgr.odeint(driven, [x0, v0], time, args=(omega0, gamma, xi, omega))\n", "xt = sol[:,0] # x(t) is the first component" ] }, { "cell_type": "code", "execution_count": 8, "id": "3fd1fbb7-28a8-48a2-8929-757a333a4e01", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "amp = xi / np.sqrt((omega0**2 - omega**2)**2 + gamma**2 * omega**2) # theoretical amplitude\n", "\n", "plt.figure()\n", "plt.axhline(0, lw=1, color='k')\n", "plt.plot(time, xt, label='oscillator')\n", "plt.plot(time, xi*np.cos(omega*time), label='driving')\n", "plt.axhline(amp, lw=1, ls='--', color='k')\n", "plt.axhline(-amp, lw=1, ls='--', color='k')\n", "plt.ylim(-2, 2)\n", "plt.xlabel(r'$t$')\n", "plt.ylabel(r'$x$')\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "fcc11298-2fa1-410f-ae9e-a1628b5333ab", "metadata": {}, "source": [ "We can see that there is a transient period, which has to do with the vanishing of the homogeneous solution due to damping alone. After that, the solution settles to oscillating with the driving frequency. Notice that there is a phase shift from the driving force, and the amplitude of the driven oscillation matches our analytical result." ] }, { "cell_type": "markdown", "id": "fc2a8708-2514-4a60-b8d3-558dc59317f3", "metadata": {}, "source": [ "Importantly, the amplitude varies with the frequency of the driving force. Here is a plot of the analytical result." ] }, { "cell_type": "code", "execution_count": 9, "id": "1fb55fe5-52dc-408c-b777-b92b8afd659a", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "gamma_list = [0.2, 0.3, 0.5, 1]\n", "omega_list = np.arange(0.01, 2, 0.01)\n", "\n", "plt.figure()\n", "for gamma in gamma_list:\n", " amp_list = xi / np.sqrt((omega0**2 - omega_list**2)**2 + gamma**2 * omega_list**2)\n", " plt.plot(omega_list, amp_list, label=f'$\\gamma={gamma}$')\n", "plt.xlabel(r'$\\omega$')\n", "plt.ylabel(r'$A$')\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "3ba35099-a9d9-4a67-8e14-b27e172e53c1", "metadata": {}, "source": [ "One can see that, for $\\gamma \\ll \\omega_0$, the amplitude peaks around the natural frequency of the harmonic oscillator, $\\omega_0$ (set to 1). Near this frequency, the oscillator can be driven to a large amplitude by just a small driving force, which is known as the phenomenon of \"resonance\". As the driving frequency deviates from the natural frequency, the amplitude drops rapidly, and the width of the peak is roughly given by the damping coefficient, $\\gamma$. For larger $\\gamma$, the peak widens and the maximum amplitude also decreases, until there is no visible peak when $\\gamma > \\omega_0$." ] }, { "cell_type": "markdown", "id": "fd5f22a4-4079-4d3f-b32d-305f850480e7", "metadata": {}, "source": [ "```{admonition} Exercise\n", ":class: tip\n", "\n", "**Driven harmonic oscillator without damping**\n", "\n", "Without the damping term, the equation of a driven harmonic oscillator becomes:\n", "\\begin{equation}\n", "\\ddot{x} + \\omega_0^2 \\, x = \\xi \\, \\cos(\\omega t)\n", "\\end{equation}\n", "Use the ansatz $x = A \\mathrm{e}^{i \\omega t}$ to solve the equation and obtain an expression for the amplitude $A$. Then solve the equation numerically and compare the solution to the ansatz. Do they match? If not, what is missing?\n", "```" ] }, { "cell_type": "markdown", "id": "04907e76-daaf-44d7-9f15-170680f92556", "metadata": {}, "source": [ "```{admonition} Hint\n", ":class: note, dropdown\n", "\n", "The ansatz gives the special solution to the ODE. You need to also include the general solution to the homogeneous equation.\n", "```" ] }, { "cell_type": "code", "execution_count": null, "id": "bf206671-9379-473f-bbcc-20f7643b1da8", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "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": "3.8.10" } }, "nbformat": 4, "nbformat_minor": 5 }