{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Circular membrane vibrations. Plotly animation " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The study of circular membrane vibration is of interest in acoustical applications. Circular membranes are components of drums, microphones and other devices.\n", "\n", "The free vibrations of a circular membrane of radius $a$, are modeled by the two dimensional wave equation\n", "\n", "$$ u_{tt}=c^2\\Delta u$$\n", "\n", "expressed in polar coordinates $(r, \\theta)$:\n", " $$\n", " u_{tt}=c^2(u_{rr}+\\displaystyle\\frac{1}{r}u_r+\\displaystyle\\frac{1}{r^2}u_{\\theta\\theta}),\n", " $$ \n", " with the boundary condition\n", " $$u(r=a, \\theta, t)=0, $$\n", " that imposes zero membrane deflection on the circular boundary. \n", "\n", "Usually to get the analytical solution $u(r, \\theta, t)$, we should\n", " set the initial displacement, $u(r, \\theta,0)=d(r,\\theta)$, and the initial velocity, $u_t(r,\\theta,0)=v(r, \\theta)$ of the membrane. \n", "But our goal here is to derive and animate the vibrational modes (eigenfunctions) of\n", "the circular membrane, and in this case these initial conditions are not involved." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Using the method of separation of variables, i.e. looking for a solution of the form $u(r,\\theta, t)=R(r)f(\\theta)T(t)$, we get the vibrational modes (eigenfunctions):\n", " $$ u_{nm}(r, \\theta, t)=J_n(r\\cdot z_{nm}/a)\\cos(n\\theta)\\cos(c\\cdot z_{mn}t/a),$$\n", "where $J_n$ is the Bessel function of the first kind and integer order, $n\\geq 0$, and $z_{nm}$ is the $m^{th}$ positive zero\n", "of the function $J_n$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Below are illustrated the graphs of the Bessel functions $J_0, J_1, J_2$, colored respectively with red, blue and green, as well as the first three positive zeros, $z_{11}, z_{12}, z_{13}$, of the function $J_1(x)$.\n" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "" ], "text/plain": [ "" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from IPython.display import SVG\n", "SVG(filename=\"data/Bessel.svg\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In a vibrating membrane the points $(r, \\theta)$, that don't move, i.e. $u(r,\\theta, t)=u(r, \\theta, 0)$,\n", "for any $t>0$, are called nodes. A vibrating circular membrane can exhibit diametrical nodes, respectively circular nodes. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The integers $n=0,1,2, \\ldots, m=1,2,\\ldots$ in the eigenfunction notation, $u_{nm}$, defined above, give \n", "the number of diametrical nodes, respectively circular nodes, i.e. the number of diameters, respectively circles in the disk of radius $a$ that do not move during the vibration. By the boundary condition, the membrane boundary is a node line. \n", "\n", "\n", "For example $u_{12}(r, \\theta, t)$ has one diameter and two circles consisting entirely in nodes. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Across the WEB there are a lot of animations of vibrational modes of the circular membrane, including the \n", "[Wikipedia page](https://en.wikipedia.org/wiki/Vibrations_of_a_circular_membrane), \n", "but usually they do not point out the node lines for each illustrated mode.\n", "\n", "Here we define a function to animate the modes $u_{nm}(r,\\theta, t)$, via Python Plotly animation, and plot each surface\n", "$z=u_{nm}(r, \\theta, t)$ (where t is running through the points in a grid over the animation time interval) with a diverging colorscale.\n", " So we illlustrate both the membrane vibration and its node lines, as lines colored with the central color of the diverging colormap." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/html": [ " \n", " " ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "from plotly.offline import download_plotlyjs, init_notebook_mode, iplot, plot\n", "init_notebook_mode(connected=True)\n", "import plotly.graph_objects as go" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "from scipy.special import jn, jn_zeros #Bessel functions of the first kind, and their zeros" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "`jn(n, x)` evaluates $J_n(x)$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The first `nz` positive zeros of the Bessel function $J_n$ are returned by the function `scipy.special.jn_zeros(n,nz)`:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([ 3.83170597, 7.01558667, 10.17346814])" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "n=1\n", "nz=3\n", "\n", "jn_zeros(n,nz)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "def vibrate(membrane_radius=1.0, n=0, m=1, N=50, pl_colorscale='RdBu', reverse=False):\n", " \n", " # n is the index of the Bessel function, J_n, n=0,1,2, ...; it gives the number of diametric nodes\n", " # m points out the m^th positive zero of the the Bessel function J_n, m=1, 2, 3, ...\n", " # N is the number of points in a grid along the radial and circular direction\n", " r = np.linspace(0, membrane_radius, N)\n", " theta = np.linspace(0, 2*np.pi, N)\n", " r, theta = np.meshgrid(r, theta)\n", "\n", " x = r*np.cos(theta)\n", " y = r*np.sin(theta)\n", "\n", " time = np.linspace(0, np.pi, N)\n", " frames = []\n", " zmin = 100\n", " zmax = 0\n", " \n", " bz_nm=jn_zeros(n,m)[-1] #the m^th positive zero of J_n\n", "\n", " for t in time:\n", " # z=u_nm(r, theta, t):\n", " z = jn(n, bz_nm*r/membrane_radius)*np.cos(n*theta)*np.cos(t* bz_nm/membrane_radius) \n", " zmin = min([zmin, z.min()])\n", " zmax = max([zmax, z.max()])\n", " frames.append(go.Frame(data= [dict(type='surface',\n", " z= z)], \n", " ) \n", " )\n", " fig = go.Figure(go.Surface( \n", " x=x, y=y, \n", " z=np.zeros(x.shape),# the initial position of the mebrane is 0 at each point\n", " colorscale=pl_colorscale,\n", " reversescale=reverse,\n", " cmin=zmin, \n", " cmax=zmax,\n", " colorbar=dict(thickness=23, \n", " len=0.5)))\n", " fig.update(frames=frames)\n", " title = f'Vibrational mode ({m}, {n})
with {n} diametric and {m} circular node lines'\n", "\n", " return fig, title, zmin, zmax " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us animate the mode (1,2). To test any other mode just change the values for n and m in the call of the vibrate function, below:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "pride= [[0.0, '#160d27'],\n", " [0.1, '#2a3472'],\n", " [0.2, '#24687e'],\n", " [0.3, '#57927d'],\n", " [0.4, '#9aba6b'],\n", " [0.5, '#f8dd28'],\n", " [0.6, '#f1940d'],\n", " [0.7, '#d5532b'],\n", " [0.8, '#a71b47'],\n", " [0.9, '#650650'],\n", " [1.0, '#1f0a1a']]" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "fig, title, zmin, zmax = vibrate(n=1, m=2, pl_colorscale=pride)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Set the plot layout:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [], "source": [ "fig.update_layout(\n", " title_text=title, title_x=0.5,\n", " width=700,\n", " height=700,\n", " scene=dict(camera = dict(eye=dict(x=1, y=1.25, z=0.6)),\n", " #aspectmode='data',\n", " aspectratio=dict(x=1, y=1, z=0.35),\n", " xaxis_visible=False,\n", " yaxis_visible=False, \n", " zaxis_visible=False,\n", " zaxis_range=[zmin, zmax]),\n", " \n", " updatemenus=[dict(type='buttons', \n", " y=0.9,\n", " x=1.275,\n", " xanchor='right',\n", " yanchor='top',\n", " pad=dict(t=0, r=10),\n", " buttons=[dict(label='Play',\n", " method='animate',\n", " args=[None, \n", " dict(frame=dict(duration=100, \n", " redraw=True),\n", " transition=dict(duration=0),\n", " fromcurrent=True,\n", " mode='immediate')])])]);\n" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "application/vnd.plotly.v1+json": { "config": { "linkText": "Export to plotly.com/", "plotlyServerURL": "https://plotly.com/", "showLink": false }, "data": [ { "cmax": 0.5817824557832924, "cmin": -0.5817682213824885, "colorbar": { "len": 0.5, "thickness": 23 }, "colorscale": [ [ 0, "#160d27" ], [ 0.1, "#2a3472" ], [ 0.2, "#24687e" ], [ 0.3, "#57927d" ], [ 0.4, "#9aba6b" ], [ 0.5, 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true, "subunitcolor": "white" }, "hoverlabel": { "align": "left" }, "hovermode": "closest", "mapbox": { "style": "light" }, "paper_bgcolor": "white", "plot_bgcolor": "#E5ECF6", "polar": { "angularaxis": { "gridcolor": "white", "linecolor": "white", "ticks": "" }, "bgcolor": "#E5ECF6", "radialaxis": { "gridcolor": "white", "linecolor": "white", "ticks": "" } }, "scene": { "xaxis": { "backgroundcolor": "#E5ECF6", "gridcolor": "white", "gridwidth": 2, "linecolor": "white", "showbackground": true, "ticks": "", "zerolinecolor": "white" }, "yaxis": { "backgroundcolor": "#E5ECF6", "gridcolor": "white", "gridwidth": 2, "linecolor": "white", "showbackground": true, "ticks": "", "zerolinecolor": "white" }, "zaxis": { "backgroundcolor": "#E5ECF6", "gridcolor": "white", "gridwidth": 2, "linecolor": "white", "showbackground": true, "ticks": "", "zerolinecolor": "white" } }, "shapedefaults": { "line": { "color": "#2a3f5f" } }, "ternary": { "aaxis": { "gridcolor": "white", "linecolor": "white", "ticks": "" }, "baxis": { "gridcolor": "white", "linecolor": "white", "ticks": "" }, "bgcolor": "#E5ECF6", "caxis": { "gridcolor": "white", "linecolor": "white", "ticks": "" } }, "title": { "x": 0.05 }, "xaxis": { "automargin": true, "gridcolor": "white", "linecolor": "white", "ticks": "", "title": { "standoff": 15 }, "zerolinecolor": "white", "zerolinewidth": 2 }, "yaxis": { "automargin": true, "gridcolor": "white", "linecolor": "white", "ticks": "", "title": { "standoff": 15 }, "zerolinecolor": "white", "zerolinewidth": 2 } } }, "title": { "text": "Vibrational mode (2, 1)
with 1 diametric and 2 circular node lines", "x": 0.5 }, "updatemenus": [ { "buttons": [ { "args": [ null, { "frame": { "duration": 100, "redraw": true }, "fromcurrent": true, "mode": "immediate", "transition": { "duration": 0 } } ], "label": "Play", "method": "animate" } ], "pad": { "r": 10, "t": 0 }, "type": "buttons", "x": 1.275, "xanchor": "right", "y": 0.9, "yanchor": "top" } ], "width": 700 } }, "text/html": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "iplot(fig)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "At the end of animation we can rotate the surface to place its circular range in the vertical plane, and look at its heatmap, that points out better the node lines." ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from IPython.core.display import HTML\n", "def css_styling():\n", " styles = open(\"./custom.css\", \"r\").read()\n", " return HTML(styles)\n", "css_styling()" ] } ], "metadata": { "@webio": { "lastCommId": null, "lastKernelId": null }, "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": "3.7.3" } }, "nbformat": 4, "nbformat_minor": 2 }