{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Ressort hélicoïdal - formules Bresse - visu 3D" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Paramètrisation" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\theta_{A} \\theta_{B}$ " ], "text/plain": [ "theta_A ' ' theta_B" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|Ressort|\\phantom{\\verb!x!}\\verb|à| 3 \\phantom{\\verb!x!}\\verb|spires,|\\phantom{\\verb!x!}\\verb|theta_B|\\phantom{\\verb!x!}\\verb|=| 6 \\, \\pi$ " ], "text/plain": [ "'Ressort à ' 3 ' spires, theta_B = ' 6*pi" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Données géométriques fixes\n", "var('R r theta_A theta_B beta n S S_1 I J'); \n", "# Variables géométriques\n", "var('rho theta phi alpha')\n", "assume(theta>0); assume(alpha>0); assume(phi>0)\n", "S_1 = S_1.subs(S_1=S)\n", "# Intervalle de l'abscisse curviligne\n", "var('theta_A theta_B'); show(theta_A, ' ',theta_B); theta_A = 0;\n", "# combien de spires\n", "n = 3; theta_B = 2*n*pi; show('Ressort à ', n, ' spires, theta_B = ',theta_B)\n", "#beta = pi/8; show('Pas hélicoïdal: beta =',beta)" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|Force|\\phantom{\\verb!x!}\\verb|résultante|\\phantom{\\verb!x!}\\verb|axiale,|\\phantom{\\verb!x!}\\verb|F|\\phantom{\\verb!x!}\\verb|=| \\left(0,\\,0,\\,F_{z}\\right)$ " ], "text/plain": [ "'Force résultante axiale, F =' (0, 0, F_z)" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|Moment|\\phantom{\\verb!x!}\\verb|résultant|\\phantom{\\verb!x!}\\verb|M|\\phantom{\\verb!x!}\\verb|=| \\left(0,\\,0,\\,0\\right)$ " ], "text/plain": [ "'Moment résultant M =' (0, 0, 0)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Données sthéniques et matériau\n", "var('E G F_x F_y F_z M_x M_y M_z'); #show(E,G,F_x, F_y, F_z, M_x, M_y, M_z)\n", "#G = E/2;\n", "F_x = 0; F_y = 0; M_x = 0; M_y = 0; M_z = 0;\n", "F = vector(SR,3,[F_x, F_y, F_z]); show('Force résultante axiale, F =', F)\n", "M = vector(SR,3,[M_x, M_y, M_z]); show('Moment résultant M =', M)" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|Base|\\phantom{\\verb!x!}\\verb|cylindrique:|$ " ], "text/plain": [ "'Base cylindrique: '" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|e_R|\\phantom{\\verb!x!}\\verb|=| \\left(\\cos\\left(\\theta\\right),\\,\\sin\\left(\\theta\\right),\\,0\\right) \\phantom{\\verb!xxx!}\\verb|e_t|\\phantom{\\verb!x!}\\verb|=| \\left(-\\sin\\left(\\theta\\right),\\,\\cos\\left(\\theta\\right),\\,0\\right) \\phantom{\\verb!xxx!}\\verb|e_Z|\\phantom{\\verb!x!}\\verb|=| \\left(0,\\,0,\\,1\\right)$ " ], "text/plain": [ "'e_R =' (cos(theta), sin(theta), 0) ' e_t =' (-sin(theta), cos(theta), 0) ' e_Z =' (0, 0, 1)" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|Base|\\phantom{\\verb!x!}\\verb|locale|\\phantom{\\verb!x!}\\verb|en|\\phantom{\\verb!x!}\\verb|G:|$ " ], "text/plain": [ "'Base locale en G: '" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|e_R|\\phantom{\\verb!x!}\\verb|=| \\left(\\cos\\left(\\theta\\right),\\,\\sin\\left(\\theta\\right),\\,0\\right) \\phantom{\\verb!xxx!}\\verb|e_tau|\\phantom{\\verb!x!}\\verb|=| \\left(-\\cos\\left(\\beta\\right) \\sin\\left(\\theta\\right),\\,\\cos\\left(\\beta\\right) \\cos\\left(\\theta\\right),\\,\\sin\\left(\\beta\\right)\\right) \\phantom{\\verb!xx!}\\verb|e_n|\\phantom{\\verb!x!}\\verb|=| \\left(\\sin\\left(\\beta\\right) \\sin\\left(\\theta\\right),\\,-\\cos\\left(\\theta\\right) \\sin\\left(\\beta\\right),\\,\\cos\\left(\\beta\\right)\\right)$ " ], "text/plain": [ "'e_R =' (cos(theta), sin(theta), 0) ' e_tau =' (-cos(beta)*sin(theta), cos(beta)*cos(theta), sin(beta)) ' e_n =' (sin(beta)*sin(theta), -cos(theta)*sin(beta), cos(beta))" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Base cylindrique en G (d'abscisse theta): e_R, e_t, e_Z\n", "e_R = vector(SR,3,[cos(theta),sin(theta),0]); e_t = vector(SR,3,[-sin(theta),cos(theta),0]);\n", "e_Z = vector(SR,3,[0,0,1]); \n", "show('Base cylindrique: ')\n", "show('e_R =',e_R,' e_t =', e_t,' e_Z =',e_Z); \n", "# base locale en G (theta): e_R, e_tau, e_n\n", "e_tau = cos(beta)*e_t + sin(beta)*e_Z; e_n = -sin(beta)*e_t + cos(beta)*e_Z;\n", "show('Base locale en G: ')\n", "show('e_R =',e_R,' e_tau =', e_tau, ' e_n =',e_n);" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|Couple|\\phantom{\\verb!x!}\\verb|induit|\\phantom{\\verb!x!}\\verb|en|\\phantom{\\verb!x!}\\verb|B:|\\phantom{\\verb!x!}\\verb|M|\\phantom{\\verb!x!}\\verb|=| \\left(0,\\,F_{z} R,\\,0\\right)$ " ], "text/plain": [ "'Couple induit en B: M =' (0, F_z*R, 0)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Couple induit par la force axiale à l'extrémité B\n", "M = -R*e_R.cross_product(F); M = M.subs(theta=theta_B); show('Couple induit en B: M =',M)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|Coordonées|\\phantom{\\verb!x!}\\verb|de|\\phantom{\\verb!x!}\\verb|G,|\\phantom{\\verb!x!}\\verb|X|\\phantom{\\verb!x!}\\verb|=| \\left(R \\cos\\left(\\theta\\right),\\,R \\sin\\left(\\theta\\right),\\,R \\theta \\sin\\left(\\beta\\right)\\right)$ " ], "text/plain": [ "'Coordonées de G, X =' (R*cos(theta), R*sin(theta), R*theta*sin(beta))" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Positions des points G\n", "X = vector(SR,3,[R*cos(theta),R*sin(theta),R*theta*sin(beta)]); show('Coordonées de G, X =',X);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Visualisation 3D" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "# Valeurs numériques (pour affichage)\n", "RR = 1; rr = RR/5; bbeta = pi/32; EE = 1000; GG = EE/2;\n", "SS = pi*rr^2; SS_1 = SS; II = pi*rr^4/4; JJ = pi*rr^4/2; \n", "FF_x = 0; FF_y = 0; FF_z = 1/20; MM_x = 0; MM_y = 0; MM_z = 0;# ....." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|XX|\\phantom{\\verb!x!}\\verb|=| \\left(\\cos\\left(\\theta\\right),\\,\\sin\\left(\\theta\\right),\\,\\theta \\sin\\left(\\frac{1}{32} \\, \\pi\\right)\\right)$ " ], "text/plain": [ "'XX = ' (cos(theta), sin(theta), theta*sin(1/32*pi))" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Ligne neutre initiale\n", "XX = X.subs(R=RR,beta=bbeta); show('XX = ',XX)\n", "Init = parametric_plot3d((XX[0],XX[1],XX[2]), (theta,0,theta_B),color='black')\n", "show(Init)" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|e_r|\\phantom{\\verb!x!}\\verb|=| \\left(\\sin\\left(\\beta\\right) \\sin\\left(\\phi\\right) \\sin\\left(\\theta\\right) + \\cos\\left(\\phi\\right) \\cos\\left(\\theta\\right),\\,-\\cos\\left(\\theta\\right) \\sin\\left(\\beta\\right) \\sin\\left(\\phi\\right) + \\cos\\left(\\phi\\right) \\sin\\left(\\theta\\right),\\,\\cos\\left(\\beta\\right) \\sin\\left(\\phi\\right)\\right)$ " ], "text/plain": [ "'e_r = ' (sin(beta)*sin(phi)*sin(theta) + cos(phi)*cos(theta), -cos(theta)*sin(beta)*sin(phi) + cos(phi)*sin(theta), cos(beta)*sin(phi))" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|e_r|\\phantom{\\verb!x!}\\verb|=| \\left(\\sin\\left(\\frac{1}{32} \\, \\pi\\right) \\sin\\left(\\phi\\right) \\sin\\left(\\theta\\right) + \\cos\\left(\\phi\\right) \\cos\\left(\\theta\\right),\\,-\\cos\\left(\\theta\\right) \\sin\\left(\\frac{1}{32} \\, \\pi\\right) \\sin\\left(\\phi\\right) + \\cos\\left(\\phi\\right) \\sin\\left(\\theta\\right),\\,\\cos\\left(\\frac{1}{32} \\, \\pi\\right) \\sin\\left(\\phi\\right)\\right)$ " ], "text/plain": [ "'e_r = ' (sin(1/32*pi)*sin(phi)*sin(theta) + cos(phi)*cos(theta), -cos(theta)*sin(1/32*pi)*sin(phi) + cos(phi)*sin(theta), cos(1/32*pi)*sin(phi))" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|e_z|\\phantom{\\verb!x!}\\verb|=| \\left(\\cos\\left(\\phi\\right) \\sin\\left(\\beta\\right) \\sin\\left(\\theta\\right) - \\cos\\left(\\theta\\right) \\sin\\left(\\phi\\right),\\,-\\cos\\left(\\phi\\right) \\cos\\left(\\theta\\right) \\sin\\left(\\beta\\right) - \\sin\\left(\\phi\\right) \\sin\\left(\\theta\\right),\\,\\cos\\left(\\beta\\right) \\cos\\left(\\phi\\right)\\right)$ " ], "text/plain": [ "'e_z = ' (cos(phi)*sin(beta)*sin(theta) - cos(theta)*sin(phi), -cos(phi)*cos(theta)*sin(beta) - sin(phi)*sin(theta), cos(beta)*cos(phi))" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Bases en M, point du domaine 3D\n", "e_r = cos(phi)*e_R + sin(phi)*e_n; show('e_r = ',e_r)\n", "e_r = e_r.subs(beta=bbeta); show('e_r = ',e_r)\n", "e_z = -sin(phi)*e_R + cos(phi)*e_n; show('e_z = ',e_z)" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|X_A|\\phantom{\\verb!x!}\\verb|=| \\left(\\rho \\cos\\left(\\phi\\right) + 1,\\,-\\rho \\sin\\left(\\frac{1}{32} \\, \\pi\\right) \\sin\\left(\\phi\\right),\\,\\rho \\cos\\left(\\frac{1}{32} \\, \\pi\\right) \\sin\\left(\\phi\\right)\\right)$ " ], "text/plain": [ "'X_A =' (rho*cos(phi) + 1, -rho*sin(1/32*pi)*sin(phi), rho*cos(1/32*pi)*sin(phi))" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|X_B|\\phantom{\\verb!x!}\\verb|=| \\left(\\rho \\cos\\left(\\phi\\right) + 1,\\,-\\rho \\sin\\left(\\frac{1}{32} \\, \\pi\\right) \\sin\\left(\\phi\\right),\\,\\rho \\cos\\left(\\frac{1}{32} \\, \\pi\\right) \\sin\\left(\\phi\\right) + 6 \\, \\pi \\sin\\left(\\frac{1}{32} \\, \\pi\\right)\\right)$ " ], "text/plain": [ "'X_B =' (rho*cos(phi) + 1, -rho*sin(1/32*pi)*sin(phi), rho*cos(1/32*pi)*sin(phi) + 6*pi*sin(1/32*pi))" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Sections extrémales\n", "var('rho z'); OPA = 0.5;\n", "X_A = (XX + rho*e_r).subs(theta=theta_A); show('X_A =',X_A)\n", "S_A = parametric_plot3d((X_A[0],X_A[1],X_A[2]),(rho,0,rr),(phi,0,2*pi),opacity=OPA,color='orange');\n", "X_B = (XX + rho*e_r).subs(theta=theta_B); show('X_B =',X_B)\n", "S_B = parametric_plot3d((X_B[0],X_B[1],X_B[2]),(rho,0,rr),(phi,0,2*pi),opacity=OPA,color='orange');\n", "#show(Init + S_A + S_B)" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "scrolled": false }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|X_lat|\\phantom{\\verb!x!}\\verb|=| \\left(\\frac{1}{5} \\, \\sin\\left(\\frac{1}{32} \\, \\pi\\right) \\sin\\left(\\phi\\right) \\sin\\left(\\theta\\right) + \\frac{1}{5} \\, \\cos\\left(\\phi\\right) \\cos\\left(\\theta\\right) + \\cos\\left(\\theta\\right),\\,-\\frac{1}{5} \\, \\cos\\left(\\theta\\right) \\sin\\left(\\frac{1}{32} \\, \\pi\\right) \\sin\\left(\\phi\\right) + \\frac{1}{5} \\, \\cos\\left(\\phi\\right) \\sin\\left(\\theta\\right) + \\sin\\left(\\theta\\right),\\,\\theta \\sin\\left(\\frac{1}{32} \\, \\pi\\right) + \\frac{1}{5} \\, \\cos\\left(\\frac{1}{32} \\, \\pi\\right) \\sin\\left(\\phi\\right)\\right)$ " ], "text/plain": [ "'X_lat = ' (1/5*sin(1/32*pi)*sin(phi)*sin(theta) + 1/5*cos(phi)*cos(theta) + cos(theta), -1/5*cos(theta)*sin(1/32*pi)*sin(phi) + 1/5*cos(phi)*sin(theta) + sin(theta), theta*sin(1/32*pi) + 1/5*cos(1/32*pi)*sin(phi))" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Visualisation de la configuration initiale\n", "X_lat = XX + rr*e_r; show('X_lat = ',X_lat)\n", "S_lat = parametric_plot3d((X_lat[0],X_lat[1],X_lat[2]),(theta,theta_A,theta_B),(phi,0,2*pi),opacity=OPA,color='green',\n", " plot_points=[100,100]);\n", "INIT = Init + S_A + S_B + S_lat; show(INIT)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Efforts généralisés" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Raideur par Castigliano" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Formules de Bresse" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Visualisation 3D" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "SageMath 9.1", "language": "sage", "name": "sagemath" }, "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 }