{ "cells": [ { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "import sympy as sm\n", "import sympy.physics.mechanics as me\n", "me.init_vprinting(use_latex='mathjax')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 建立坐标系" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "def smll_rot_trans(theta1, theta2, theta3):\n", " theta11 = theta1 * theta1\n", " theta22 = theta2 * theta2\n", " theta33 = theta3 * theta3\n", " sqrd_sum = theta11 + theta22 + theta33\n", " sqrt1_sqrd_sum = (1.0 + sqrd_sum)**0.5\n", " com_denom = sqrd_sum * sqrt1_sqrd_sum\n", " theta12s = theta1 * theta2 * (sqrt1_sqrd_sum - 1.0)\n", " theta13s = theta1 * theta3 * (sqrt1_sqrd_sum - 1.0)\n", " theta23s = theta2 * theta3 * (sqrt1_sqrd_sum - 1.0)\n", " \n", " if com_denom == 0.0:\n", " trans_mat = sm.Matrix([[1.0, 0.0, 0.0],\n", " [0.0, 1.0, 0.0],\n", " [0.0, 0.0, 1.0]])\n", " \n", " else:\n", " trans_mat = sm.Matrix([[(theta11 * sqrt1_sqrd_sum + theta22 + theta33) / com_denom, (theta3 * sqrd_sum + theta12s) / com_denom, (-theta2 * sqrd_sum + theta13s) / com_denom],\n", " [(-theta3 * sqrd_sum + theta12s) / com_denom, (theta11 + theta22 * sqrt1_sqrd_sum + theta33) / com_denom, (theta1 * sqrd_sum + theta23s) / com_denom],\n", " [(theta2 * sqrd_sum + theta13s) / com_denom, (-theta1 * sqrd_sum + theta23s) / com_denom, (theta11 + theta22 + theta33 * sqrt1_sqrd_sum) / com_denom]])\n", " return trans_mat" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}\\frac{\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5} \\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}}{\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5}} & \\frac{\\left(\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5} - 1.0\\right) \\theta_{1} \\theta_{2} + \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\theta_{3}}{\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5}} & \\frac{\\left(\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5} - 1.0\\right) \\theta_{1} \\theta_{3} - \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\theta_{2}}{\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5}}\\\\\\frac{\\left(\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5} - 1.0\\right) \\theta_{1} \\theta_{2} - \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\theta_{3}}{\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5}} & \\frac{\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5} \\theta_{2}^{2} + \\theta_{1}^{2} + \\theta_{3}^{2}}{\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5}} & \\frac{\\left(\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5} - 1.0\\right) \\theta_{2} \\theta_{3} + \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\theta_{1}}{\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5}}\\\\\\frac{\\left(\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5} - 1.0\\right) \\theta_{1} \\theta_{3} + \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\theta_{2}}{\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5}} & \\frac{\\left(\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5} - 1.0\\right) \\theta_{2} \\theta_{3} - \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\theta_{1}}{\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5}} & \\frac{\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5} \\theta_{3}^{2} + \\theta_{1}^{2} + \\theta_{2}^{2}}{\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5}}\\end{matrix}\\right]$" ], "text/plain": [ "⎡ ⎛ 0.5 ⎞ ↪\n", "⎢ ⎜⎛ 2 2 2 ⎞ 2 2 2⎟ ⎛ 2 2 2 ↪\n", "⎢ ⎝⎝θ₁ + θ₂ + θ₃ + 1.0⎠ ⋅θ₁ + θ₂ + θ₃ ⎠⋅⎝θ₁ + θ₂ + θ₃ + 1.0 ↪\n", "⎢ ─────────────────────────────────────────────────────────────────── ↪\n", "⎢ 2 2 2 ↪\n", "⎢ θ₁ + θ₂ + θ₃ ↪\n", "⎢ ↪\n", "⎢⎛⎛ 0.5 ⎞ ⎞ ↪\n", "⎢⎜⎜⎛ 2 2 2 ⎞ ⎟ ⎛ 2 2 2⎞ ⎟ ⎛ 2 2 ↪\n", "⎢⎝⎝⎝θ₁ + θ₂ + θ₃ + 1.0⎠ - 1.0⎠⋅θ₁⋅θ₂ - ⎝θ₁ + θ₂ + θ₃ ⎠⋅θ₃⎠⋅⎝θ₁ + θ₂ ↪\n", "⎢───────────────────────────────────────────────────────────────────────────── ↪\n", "⎢ 2 2 2 ↪\n", "⎢ θ₁ + θ₂ + θ₃ ↪\n", "⎢ ↪\n", "⎢⎛⎛ 0.5 ⎞ ⎞ ↪\n", "⎢⎜⎜⎛ 2 2 2 ⎞ ⎟ ⎛ 2 2 2⎞ ⎟ ⎛ 2 2 ↪\n", "⎢⎝⎝⎝θ₁ + θ₂ + θ₃ + 1.0⎠ - 1.0⎠⋅θ₁⋅θ₃ + ⎝θ₁ + θ₂ + θ₃ ⎠⋅θ₂⎠⋅⎝θ₁ + θ₂ ↪\n", "⎢───────────────────────────────────────────────────────────────────────────── ↪\n", "⎢ 2 2 2 ↪\n", "⎣ θ₁ + θ₂ + θ₃ ↪\n", "\n", "↪ -0.5 ⎛⎛ 0.5 ⎞ ↪\n", "↪ ⎞ ⎜⎜⎛ 2 2 2 ⎞ ⎟ ⎛ 2 2 ↪\n", "↪ ⎠ ⎝⎝⎝θ₁ + θ₂ + θ₃ + 1.0⎠ - 1.0⎠⋅θ₁⋅θ₂ + ⎝θ₁ + θ₂ + θ ↪\n", "↪ ───── ────────────────────────────────────────────────────────── ↪\n", "↪ 2 2 2 ↪\n", "↪ θ₁ + θ₂ + θ₃ ↪\n", "↪ ↪\n", "↪ -0.5 ⎛ 0.5 ⎞ ↪\n", "↪ 2 ⎞ ⎜⎛ 2 2 2 ⎞ 2 2 2⎟ ⎛ ↪\n", "↪ + θ₃ + 1.0⎠ ⎝⎝θ₁ + θ₂ + θ₃ + 1.0⎠ ⋅θ₂ + θ₁ + θ₃ ⎠⋅⎝θ₁ ↪\n", "↪ ──────────────── ──────────────────────────────────────────────── ↪\n", "↪ 2 2 2 ↪\n", "↪ θ₁ + θ₂ + θ₃ ↪\n", "↪ ↪\n", "↪ -0.5 ⎛⎛ 0.5 ⎞ ↪\n", "↪ 2 ⎞ ⎜⎜⎛ 2 2 2 ⎞ ⎟ ⎛ 2 2 ↪\n", "↪ + θ₃ + 1.0⎠ ⎝⎝⎝θ₁ + θ₂ + θ₃ + 1.0⎠ - 1.0⎠⋅θ₂⋅θ₃ - ⎝θ₁ + θ₂ + θ ↪\n", "↪ ──────────────── ────────────────────────────────────────────────────────── ↪\n", "↪ 2 2 2 ↪\n", "↪ θ₁ + θ₂ + θ₃ ↪\n", "\n", "↪ ⎞ -0.5 ⎛⎛ 0.5 ⎞ ↪\n", "↪ 2⎞ ⎟ ⎛ 2 2 2 ⎞ ⎜⎜⎛ 2 2 2 ⎞ ⎟ ↪\n", "↪ ₃ ⎠⋅θ₃⎠⋅⎝θ₁ + θ₂ + θ₃ + 1.0⎠ ⎝⎝⎝θ₁ + θ₂ + θ₃ + 1.0⎠ - 1.0⎠⋅θ₁⋅ ↪\n", "↪ ─────────────────────────────────── ─────────────────────────────────────── ↪\n", "↪ ↪\n", "↪ ↪\n", "↪ ↪\n", "↪ -0.5 ⎛⎛ 0.5 ⎞ ↪\n", "↪ 2 2 2 ⎞ ⎜⎜⎛ 2 2 2 ⎞ ⎟ ↪\n", "↪ + θ₂ + θ₃ + 1.0⎠ ⎝⎝⎝θ₁ + θ₂ + θ₃ + 1.0⎠ - 1.0⎠⋅θ₂⋅ ↪\n", "↪ ──────────────────────── ─────────────────────────────────────── ↪\n", "↪ ↪\n", "↪ ↪\n", "↪ ↪\n", "↪ ⎞ -0.5 ⎛ 0.5 ↪\n", "↪ 2⎞ ⎟ ⎛ 2 2 2 ⎞ ⎜⎛ 2 2 2 ⎞ ↪\n", "↪ ₃ ⎠⋅θ₁⎠⋅⎝θ₁ + θ₂ + θ₃ + 1.0⎠ ⎝⎝θ₁ + θ₂ + θ₃ + 1.0⎠ ⋅θ ↪\n", "↪ ─────────────────────────────────── ───────────────────────────── ↪\n", "↪ ↪\n", "↪ θ ↪\n", "\n", "↪ ⎞ -0.5⎤\n", "↪ ⎛ 2 2 2⎞ ⎟ ⎛ 2 2 2 ⎞ ⎥\n", "↪ θ₃ - ⎝θ₁ + θ₂ + θ₃ ⎠⋅θ₂⎠⋅⎝θ₁ + θ₂ + θ₃ + 1.0⎠ ⎥\n", "↪ ──────────────────────────────────────────────────────⎥\n", "↪ 2 2 2 ⎥\n", "↪ θ₁ + θ₂ + θ₃ ⎥\n", "↪ ⎥\n", "↪ ⎞ -0.5⎥\n", "↪ ⎛ 2 2 2⎞ ⎟ ⎛ 2 2 2 ⎞ ⎥\n", "↪ θ₃ + ⎝θ₁ + θ₂ + θ₃ ⎠⋅θ₁⎠⋅⎝θ₁ + θ₂ + θ₃ + 1.0⎠ ⎥\n", "↪ ──────────────────────────────────────────────────────⎥\n", "↪ 2 2 2 ⎥\n", "↪ θ₁ + θ₂ + θ₃ ⎥\n", "↪ ⎥\n", "↪ ⎞ -0.5 ⎥\n", "↪ 2 2 2⎟ ⎛ 2 2 2 ⎞ ⎥\n", "↪ ₃ + θ₁ + θ₂ ⎠⋅⎝θ₁ + θ₂ + θ₃ + 1.0⎠ ⎥\n", "↪ ─────────────────────────────────────────── ⎥\n", "↪ 2 2 2 ⎥\n", "↪ ₁ + θ₂ + θ₃ ⎦" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A = me.ReferenceFrame('A')\n", "B = me.ReferenceFrame('B')\n", "D = me.ReferenceFrame('D')\n", "C = me.ReferenceFrame('C')\n", "G = me.ReferenceFrame('G')\n", "G_prime1 = me.ReferenceFrame('G_prime1')\n", "G_prime2 = me.ReferenceFrame('G_prime2')\n", "G_prime3 = me.ReferenceFrame('G_prime3')\n", "# fast x-1, -y-3, z-2\n", "\n", "# 广义坐标\n", "q1, q2, q3, q4, q_yaw, q_drtr, q_geaz, q_1f1, q_1f2, q_1e1, q_2f1, q_2f2, q_2e1, q_3f1, q_3f2, q_3e1 = me.dynamicsymbols('q1, q2, q3, q4, q_yaw, q_drtr, q_geaz, q_1f1, q_1f2, q_1e1, q_2f1, q_2f2, q_2e1, q_3f1, q_3f2, q_3e1')\n", "# tower top\n", "theta1, theta2, theta3 = me.dynamicsymbols(\"theta1, theta2, theta3\")\n", "dcm = smll_rot_trans(theta1, theta2, theta3)\n", "B.orient_dcm(A, dcm)\n", "\n", "# nacelle\n", "D.orient_axis(B, q_yaw, B.y)\n", "\n", "# shaft C \n", "theta_tilt = sm.symbols(\"theta_tilt\")\n", "C.orient_axis(D, theta_tilt, D.z)\n", "\n", "# Hub G = Azimuth E\n", "Azimuth = q_drtr + q_geaz\n", "G.orient_axis(C, Azimuth, C.x)\n", "\n", "# 到每只叶片\n", "angle = 2*sm.pi/3\n", "G_prime1.orient_axis(G, angle * 0, G.x)\n", "G_prime2.orient_axis(G, angle * 1, G.x)\n", "G_prime3.orient_axis(G, angle * 2, G.x)\n", "\n", "B.dcm(A)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 速度" ] } ], "metadata": { "kernelspec": { "display_name": "MinerU", "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": "3.10.16" } }, "nbformat": 4, "nbformat_minor": 2 }