{"id":79722,"date":"2026-03-02T22:41:44","date_gmt":"2026-03-02T14:41:44","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/79722.html"},"modified":"2026-03-02T22:41:44","modified_gmt":"2026-03-02T14:41:44","slug":"%e7%ac%ac6%e7%af%87%ef%bc%9a%e6%89%ad%e8%bd%ac%e7%90%86%e8%ae%ba%e4%b8%8e%e5%ba%94%e7%94%a8","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/79722.html","title":{"rendered":"\u7b2c6\u7bc7\uff1a\u626d\u8f6c\u7406\u8bba\u4e0e\u5e94\u7528"},"content":{"rendered":"

\u7b2c6\u7bc7：\u626d\u8f6c\u7406\u8bba\u4e0e\u5e94\u7528<\/h2>\n

\u6458\u8981<\/h3>\n

\u672c\u6559\u7a0b\u4ecb\u7ecd\u4e86\u6750\u6599\u529b\u5b66\u4e2d\u626d\u8f6c\u95ee\u9898\u7684\u57fa\u7840\u7406\u8bba\u3001\u6c42\u89e3\u65b9\u6cd5\u548c\u6570\u503c\u6a21\u62df\u6280\u672f\u3002\u4e3b\u8981\u5185\u5bb9\u5305\u62ec\u5706\u8f74\u626d\u8f6c\u7684\u5e94\u529b\u5206\u6790\u3001\u975e\u5706\u622a\u9762\u626d\u8f6c\u7684\u7406\u8bba\u57fa\u7840\u3001\u626d\u8f6c\u95ee\u9898\u7684\u6709\u9650\u5dee\u5206\u6cd5\u6c42\u89e3\u4ee5\u53caPython\u5b9e\u73b0\u3002\u901a\u8fc7\u672c\u6559\u7a0b\u7684\u5b66\u4e60，\u8bfb\u8005\u5c06\u80fd\u591f\u638c\u63e1\u626d\u8f6c\u95ee\u9898\u7684\u57fa\u672c\u539f\u7406，\u7406\u89e3\u5176\u5728\u5de5\u7a0b\u4e2d\u7684\u5e94\u7528，\u5e76\u80fd\u591f\u4f7f\u7528Python\u5b9e\u73b0\u76f8\u5173\u7684\u6570\u503c\u8ba1\u7b97\u3002<\/p>\n

\u5173\u952e\u8bcd<\/h3>\n

\u626d\u8f6c，\u5706\u8f74\u626d\u8f6c，\u975e\u5706\u622a\u9762\u626d\u8f6c，\u526a\u5e94\u529b，\u626d\u8f6c\u89d2，\u6709\u9650\u5dee\u5206\u6cd5，Python\u5b9e\u73b0<\/p>\n

1. \u626d\u8f6c\u95ee\u9898\u7684\u57fa\u672c\u6982\u5ff5<\/h3>\n

1.1 \u626d\u8f6c\u7684\u5b9a\u4e49<\/h4>\n

\u626d\u8f6c\u662f\u6307\u6746\u4ef6\u53d7\u5230\u7ed5\u5176\u8f74\u7ebf\u7684\u529b\u77e9\u4f5c\u7528\u800c\u4ea7\u751f\u7684\u53d8\u5f62\u3002\u8fd9\u79cd\u529b\u77e9\u79f0\u4e3a\u626d\u77e9，\u7528\u7b26\u53f7TT<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span>\u8868\u793a\u3002\u626d\u8f6c\u53d8\u5f62\u7684\u4e3b\u8981\u7279\u5f81\u662f：<\/p>\n

  • \u6746\u4ef6\u7684\u5404\u6a2a\u622a\u9762\u7ed5\u8f74\u7ebf\u53d1\u751f\u76f8\u5bf9\u8f6c\u52a8<\/li>\n
  • \u6a2a\u622a\u9762\u4e0a\u4ea7\u751f\u526a\u5e94\u529b<\/li>\n
  • \u7eb5\u5411\u7ebf\u53d1\u751f\u503e\u659c，\u4ea7\u751f\u526a\u5e94\u53d8<\/li>\n

    1.2 \u5706\u8f74\u626d\u8f6c\u7684\u7279\u70b9<\/h4>\n

    \u5bf9\u4e8e\u5706\u622a\u9762\u8f74（\u7b80\u79f0\u5706\u8f74），\u626d\u8f6c\u65f6\u5177\u6709\u4ee5\u4e0b\u7279\u70b9：<\/p>\n

  • \u5e73\u9762\u5047\u8bbe：\u626d\u8f6c\u524d\u7684\u5e73\u9762\u6a2a\u622a\u9762，\u626d\u8f6c\u540e\u4ecd\u4fdd\u6301\u4e3a\u5e73\u9762，\u4e14\u5f62\u72b6\u548c\u5927\u5c0f\u4e0d\u53d8，\u4ec5\u7ed5\u8f74\u7ebf\u8f6c\u8fc7\u4e00\u4e2a\u89d2\u5ea6<\/li>\n
  • \u534a\u5f84\u5047\u8bbe：\u6a2a\u622a\u9762\u7684\u534a\u5f84\u5728\u626d\u8f6c\u540e\u4ecd\u4fdd\u6301\u4e3a\u76f4\u7ebf<\/li>\n
  • \u53d8\u5f62\u89c4\u5f8b：\u6a2a\u622a\u9762\u4e0a\u5404\u70b9\u7684\u526a\u5e94\u53d8\u4e0e\u8be5\u70b9\u5230\u5706\u5fc3\u7684\u8ddd\u79bb\u6210\u6b63\u6bd4<\/li>\n

    1.3 \u975e\u5706\u622a\u9762\u626d\u8f6c\u7684\u7279\u70b9<\/h4>\n

    \u5bf9\u4e8e\u975e\u5706\u622a\u9762\u8f74，\u626d\u8f6c\u65f6\u5177\u6709\u4ee5\u4e0b\u7279\u70b9：<\/p>\n

  • \u7fd8\u66f2\u73b0\u8c61：\u6a2a\u622a\u9762\u4e0d\u518d\u4fdd\u6301\u5e73\u9762，\u800c\u662f\u53d1\u751f\u7fd8\u66f2<\/li>\n
  • \u526a\u5e94\u529b\u5206\u5e03：\u526a\u5e94\u529b\u5206\u5e03\u4e0d\u5747\u5300，\u6700\u5927\u526a\u5e94\u529b\u901a\u5e38\u51fa\u73b0\u5728\u622a\u9762\u7684\u68f1\u89d2\u5904<\/li>\n
  • \u626d\u8f6c\u4e2d\u5fc3：\u626d\u8f6c\u65f6\u6a2a\u622a\u9762\u7ed5\u67d0\u4e00\u4e2d\u5fc3\u70b9\u8f6c\u52a8，\u8be5\u70b9\u79f0\u4e3a\u626d\u8f6c\u4e2d\u5fc3<\/li>\n

    2. \u5706\u8f74\u626d\u8f6c\u7684\u5e94\u529b\u5206\u6790<\/h3>\n

    2.1 \u5706\u8f74\u626d\u8f6c\u7684\u5e94\u529b\u516c\u5f0f<\/h4>\n

    \u5bf9\u4e8e\u7b49\u622a\u9762\u5706\u8f74，\u5f53\u53d7\u5230\u626d\u77e9TT<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span>\u4f5c\u7528\u65f6，\u6a2a\u622a\u9762\u4e0a\u4efb\u4e00\u70b9\u7684\u526a\u5e94\u529b$au$\u53ef\u4ee5\u8868\u793a\u4e3a：<\/p>\n

    \u03c4=T\u03c1Ip\\\\tau = \\\\frac{T\\\\rho}{I_p}<\/span><\/span>\u03c4<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>I<\/span><\/span>p<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>T\u03c1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u5176\u4e2d：<\/p>\n

      \n
    • $
      \nho$\u662f\u8be5\u70b9\u5230\u5706\u5fc3\u7684\u8ddd\u79bb<\/li>\n
    • IpI_p<\/span><\/span>I<\/span><\/span>p<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u5706\u622a\u9762\u7684\u6781\u60ef\u6027\u77e9，\u5bf9\u4e8e\u76f4\u5f84\u4e3add<\/span><\/span>d<\/span><\/span><\/span><\/span><\/span>\u7684\u5706\u622a\u9762，Ip=\u03c0d432I_p = \\\\frac{\\\\pi d^4}{32}<\/span><\/span>I<\/span><\/span>p<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>32<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>d<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/li>\n<\/ul>\n

      2.2 \u6700\u5927\u526a\u5e94\u529b<\/h4>\n

      \u5706\u8f74\u626d\u8f6c\u65f6，\u6700\u5927\u526a\u5e94\u529b\u53d1\u751f\u5728\u6a2a\u622a\u9762\u7684\u8fb9\u7f18\u5904（$
      \nho = r，，<\/span><\/span>，<\/span><\/span><\/span><\/span><\/span>r$\u4e3a\u5706\u8f74\u534a\u5f84），\u5176\u503c\u4e3a：<\/p>\n

      \u03c4max=TrIp=TWt\\\\tau_{max} = \\\\frac{Tr}{I_p} = \\\\frac{T}{W_t}<\/span><\/span>\u03c4<\/span><\/span>ma<\/span>x<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>I<\/span><\/span>p<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>T<\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>W<\/span><\/span>t<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

      \u5176\u4e2d，Wt=IprW_t = \\\\frac{I_p}{r}<\/span><\/span>W<\/span><\/span>t<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>I<\/span><\/span>p<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u79f0\u4e3a\u626d\u8f6c\u622a\u9762\u7cfb\u6570，\u5bf9\u4e8e\u76f4\u5f84\u4e3add<\/span><\/span>d<\/span><\/span><\/span><\/span><\/span>\u7684\u5706\u622a\u9762，Wt=\u03c0d316W_t = \\\\frac{\\\\pi d^3}{16}<\/span><\/span>W<\/span><\/span>t<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>16<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c0<\/span>d<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

      2.3 \u626d\u8f6c\u89d2\u8ba1\u7b97<\/h4>\n

      \u5706\u8f74\u626d\u8f6c\u65f6，\u76f8\u8dddLL<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span>\u7684\u4e24\u4e2a\u6a2a\u622a\u9762\u4e4b\u95f4\u7684\u76f8\u5bf9\u626d\u8f6c\u89d2hihi<\/span><\/span>hi<\/span><\/span><\/span><\/span><\/span>\u53ef\u4ee5\u8868\u793a\u4e3a：<\/p>\n

      \u03d5=TLGIp\\\\phi = \\\\frac{TL}{G I_p}<\/span><\/span>\u03d5<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>G<\/span>I<\/span><\/span>p<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>T<\/span>L<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

      \u5176\u4e2d，GG<\/span><\/span>G<\/span><\/span><\/span><\/span><\/span>\u662f\u6750\u6599\u7684\u526a\u5207\u5f39\u6027\u6a21\u91cf\u3002<\/p>\n

      3. \u975e\u5706\u622a\u9762\u626d\u8f6c\u7684\u7406\u8bba\u57fa\u7840<\/h3>\n

      3.1 \u5723\u7ef4\u5357\u626d\u8f6c\u7406\u8bba<\/h4>\n

      \u5723\u7ef4\u5357（Saint-Venant）\u626d\u8f6c\u7406\u8bba\u662f\u6c42\u89e3\u975e\u5706\u622a\u9762\u626d\u8f6c\u95ee\u9898\u7684\u7ecf\u5178\u7406\u8bba\u3002\u8be5\u7406\u8bba\u5047\u8bbe：<\/p>\n

    • \u6746\u4ef6\u7684\u6750\u6599\u662f\u5747\u5300\u3001\u5404\u5411\u540c\u6027\u7684<\/li>\n
    • \u53d8\u5f62\u662f\u5c0f\u53d8\u5f62<\/li>\n
    • \u6a2a\u622a\u9762\u4e0a\u7684\u4f4d\u79fb\u53ef\u4ee5\u8868\u793a\u4e3a：
      \nu=\u2212\u03b8zyu = -\\\\theta z y<\/span><\/span>u<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>\u03b8<\/span>zy<\/span><\/span><\/span><\/span><\/span><\/span>
      \nv=\u03b8zxv = \\\\theta z x<\/span><\/span>v<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b8<\/span>z<\/span>x<\/span><\/span><\/span><\/span><\/span><\/span>
      \nw=\u03b8\u03c8(x,y)w = \\\\theta \\\\psi(x, y)<\/span><\/span>w<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b8<\/span>\u03c8<\/span>(<\/span>x<\/span>,<\/span><\/span>y<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/li>\n

      \u5176\u4e2d，$heta\u662f\u5355\u4f4d\u957f\u5ea6\u7684\u626d\u8f6c\u89d2，\u662f\u5355\u4f4d\u957f\u5ea6\u7684\u626d\u8f6c\u89d2，<\/span><\/span>\u662f\u5355\u4f4d\u957f\u5ea6\u7684\u626d\u8f6c\u89d2，<\/span><\/span><\/span><\/span><\/span>si(x, y)$\u662f\u7fd8\u66f2\u51fd\u6570\u3002<\/p>\n

      3.2 \u626d\u8f6c\u5e94\u529b\u51fd\u6570<\/h4>\n

      \u4e3a\u4e86\u6c42\u89e3\u975e\u5706\u622a\u9762\u626d\u8f6c\u95ee\u9898，\u5f15\u5165\u626d\u8f6c\u5e94\u529b\u51fd\u6570hi(x,y)hi(x, y)<\/span><\/span>hi<\/span>(<\/span>x<\/span>,<\/span><\/span>y<\/span>)<\/span><\/span><\/span><\/span><\/span>，\u5176\u6ee1\u8db3：<\/p>\n

      \u22072\u03d5=\u22122G\u03b8\\\\nabla^2 \\\\phi = -2G\\\\theta<\/span><\/span>\u2207<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03d5<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>2<\/span>G\u03b8<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

      \u5728\u6a2a\u622a\u9762\u7684\u8fb9\u754c\u4e0a，hi=0hi = 0<\/span><\/span>hi<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

      \u6a2a\u622a\u9762\u4e0a\u7684\u526a\u5e94\u529b\u5206\u91cf\u53ef\u4ee5\u8868\u793a\u4e3a：<\/p>\n

      \u03c4zx=\u2202\u03d5\u2202y\\\\tau_{zx} = \\\\frac{\\\\partial \\\\phi}{\\\\partial y}<\/span><\/span>\u03c4<\/span><\/span>z<\/span>x<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>y<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>\u03d5<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
      \n\u03c4zy=\u2212\u2202\u03d5\u2202x\\\\tau_{zy} = -\\\\frac{\\\\partial \\\\phi}{\\\\partial x}<\/span><\/span>\u03c4<\/span><\/span>zy<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>\u2202<\/span>x<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2202<\/span>\u03d5<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

      3.3 \u626d\u8f6c\u5e38\u6570<\/h4>\n

      \u975e\u5706\u622a\u9762\u7684\u626d\u8f6c\u5e38\u6570JJ<\/span><\/span>J<\/span><\/span><\/span><\/span><\/span>（\u4e5f\u79f0\u4e3a\u76f8\u5f53\u6781\u60ef\u6027\u77e9）\u5b9a\u4e49\u4e3a：<\/p>\n

      J=TG\u03b8J = \\\\frac{T}{G\\\\theta}<\/span><\/span>J<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>G\u03b8<\/span><\/span><\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

      \u5bf9\u4e8e\u77e9\u5f62\u622a\u9762（\u5bbd\u5ea6\u4e3abb<\/span><\/span>b<\/span><\/span><\/span><\/span><\/span>，\u9ad8\u5ea6\u4e3ahh<\/span><\/span>h<\/span><\/span><\/span><\/span><\/span>，b<hb < h<\/span><\/span>b<\/span><\/span><<\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span><\/span>），\u626d\u8f6c\u5e38\u6570\u8fd1\u4f3c\u4e3a：<\/p>\n

      J=\u03b2bh3J = \\\\beta b h^3<\/span><\/span>J<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b2<\/span>b<\/span>h<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

      \u5176\u4e2d，KaTeX parse error: Unexpected character: '' at position 1: \u0332eta<\/span>\u662f\u4e0eb\/hb\/h<\/span><\/span>b<\/span>\/<\/span>h<\/span><\/span><\/span><\/span><\/span>\u6709\u5173\u7684\u7cfb\u6570\u3002<\/p>\n

      4. \u626d\u8f6c\u95ee\u9898\u7684\u6709\u9650\u5dee\u5206\u6cd5<\/h3>\n

      4.1 \u6709\u9650\u5dee\u5206\u6cd5\u7684\u57fa\u672c\u539f\u7406<\/h4>\n

      \u6709\u9650\u5dee\u5206\u6cd5\u662f\u4e00\u79cd\u6570\u503c\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u7684\u65b9\u6cd5，\u5176\u57fa\u672c\u601d\u60f3\u662f\u7528\u5dee\u5546\u8fd1\u4f3c\u4ee3\u66ff\u504f\u5bfc\u6570，\u5c06\u504f\u5fae\u5206\u65b9\u7a0b\u8f6c\u5316\u4e3a\u4ee3\u6570\u65b9\u7a0b\u7ec4，\u7136\u540e\u6c42\u89e3\u4ee3\u6570\u65b9\u7a0b\u7ec4\u5f97\u5230\u6570\u503c\u89e3\u3002<\/p>\n

      4.2 \u5706\u8f74\u626d\u8f6c\u7684\u6709\u9650\u5dee\u5206\u6cd5<\/h4>\n

      \u5bf9\u4e8e\u5706\u8f74\u626d\u8f6c\u95ee\u9898，\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u6709\u9650\u5dee\u5206\u6cd5\u6c42\u89e3\u626d\u8f6c\u89d2\u548c\u526a\u5e94\u529b\u5206\u5e03\u3002<\/p>\n

      4.3 \u975e\u5706\u622a\u9762\u626d\u8f6c\u7684\u6709\u9650\u5dee\u5206\u6cd5<\/h4>\n

      \u5bf9\u4e8e\u975e\u5706\u622a\u9762\u626d\u8f6c\u95ee\u9898，\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u6709\u9650\u5dee\u5206\u6cd5\u6c42\u89e3\u626d\u8f6c\u5e94\u529b\u51fd\u6570，\u8fdb\u800c\u8ba1\u7b97\u526a\u5e94\u529b\u5206\u5e03\u3002<\/p>\n

      5. Python\u5b9e\u73b0\u626d\u8f6c\u95ee\u9898\u7684\u6570\u503c\u6c42\u89e3<\/h3>\n

      5.1 \u5706\u8f74\u626d\u8f6c\u7684\u5e94\u529b\u8ba1\u7b97<\/h4>\n

      \u4ee5\u4e0b\u662f\u4e00\u4e2a\u8ba1\u7b97\u5706\u8f74\u626d\u8f6c\u5e94\u529b\u7684\u793a\u4f8b：<\/p>\n

      import<\/span> numpy as<\/span> np
      \nimport<\/span> matplotlib.<\/span>pyplot as<\/span> plt<\/p>\n

      # \u8bbe\u7f6e\u4e2d\u6587\u5b57\u4f53<\/span>
      \nplt.<\/span>rcParams[<\/span>'font.sans-serif'<\/span>]<\/span> =<\/span> [<\/span>'SimHei'<\/span>,<\/span> 'DejaVu Sans'<\/span>]<\/span>
      \nplt.<\/span>rcParams[<\/span>'axes.unicode_minus'<\/span>]<\/span> =<\/span> False<\/span><\/p>\n

      def<\/span> calculate_torsion_stress<\/span>(<\/span>T,<\/span> d,<\/span> L,<\/span> G,<\/span> r_points)<\/span>:<\/span>
      \n """
      \n \u8ba1\u7b97\u5706\u8f74\u626d\u8f6c\u65f6\u7684\u526a\u5e94\u529b\u5206\u5e03<\/p>\n

      \u53c2\u6570:
      \n T: \u626d\u77e9 (N\u00b7m)
      \n d: \u5706\u8f74\u76f4\u5f84 (m)
      \n L: \u5706\u8f74\u957f\u5ea6 (m)
      \n G: \u526a\u5207\u5f39\u6027\u6a21\u91cf (Pa)
      \n r_points: \u5f84\u5411\u4f4d\u7f6e\u6570\u7ec4<\/p>\n

      \u8fd4\u56de:
      \n tau: \u526a\u5e94\u529b\u5206\u5e03
      \n phi: \u626d\u8f6c\u89d2 (rad)
      \n """<\/span>
      \n # \u8ba1\u7b97\u6781\u60ef\u6027\u77e9<\/span>
      \n Ip =<\/span> np.<\/span>pi *<\/span> d**<\/span>4<\/span> \/<\/span> 32<\/span><\/p>\n

      # \u8ba1\u7b97\u526a\u5e94\u529b<\/span>
      \n tau =<\/span> T *<\/span> r_points \/<\/span> Ip<\/p>\n

      # \u8ba1\u7b97\u626d\u8f6c\u89d2<\/span>
      \n phi =<\/span> T *<\/span> L \/<\/span> (<\/span>G *<\/span> Ip)<\/span><\/p>\n

      return<\/span> tau,<\/span> phi<\/p>\n

      # \u793a\u4f8b\u7528\u6cd5<\/span>
      \nif<\/span> __name__ ==<\/span> '__main__'<\/span>:<\/span>
      \n # \u5b9a\u4e49\u53c2\u6570<\/span>
      \n T =<\/span> 1000<\/span> # \u626d\u77e9 (N\u00b7m)<\/span>
      \n d =<\/span> 0.1<\/span> # \u76f4\u5f84 (m)<\/span>
      \n L =<\/span> 1.0<\/span> # \u957f\u5ea6 (m)<\/span>
      \n G =<\/span> 80e9<\/span> # \u526a\u5207\u5f39\u6027\u6a21\u91cf (Pa)<\/span><\/p>\n

      # \u8ba1\u7b97\u5f84\u5411\u4f4d\u7f6e<\/span>
      \n r =<\/span> d \/<\/span> 2<\/span>
      \n r_points =<\/span> np.<\/span>linspace(<\/span>0<\/span>,<\/span> r,<\/span> 100<\/span>)<\/span><\/p>\n

      # \u8ba1\u7b97\u526a\u5e94\u529b\u548c\u626d\u8f6c\u89d2<\/span>
      \n tau,<\/span> phi =<\/span> calculate_torsion_stress(<\/span>T,<\/span> d,<\/span> L,<\/span> G,<\/span> r_points)<\/span><\/p>\n

      # \u7ed8\u5236\u526a\u5e94\u529b\u5206\u5e03<\/span>
      \n plt.<\/span>figure(<\/span>figsize=<\/span>(<\/span>10<\/span>,<\/span> 6<\/span>)<\/span>)<\/span>
      \n plt.<\/span>plot(<\/span>r_points,<\/span> tau\/<\/span>1e6<\/span>,<\/span> 'r-'<\/span>,<\/span> linewidth=<\/span>2<\/span>)<\/span>
      \n plt.<\/span>xlabel(<\/span>'\u5f84\u5411\u4f4d\u7f6e r (m)'<\/span>)<\/span>
      \n plt.<\/span>ylabel(<\/span>'\u526a\u5e94\u529b \u03c4 (MPa)'<\/span>)<\/span>
      \n plt.<\/span>title(<\/span>'\u5706\u8f74\u626d\u8f6c\u65f6\u7684\u526a\u5e94\u529b\u5206\u5e03'<\/span>)<\/span>
      \n plt.<\/span>grid(<\/span>True<\/span>)<\/span>
      \n plt.<\/span>savefig(<\/span>'\u5706\u8f74\u626d\u8f6c\u526a\u5e94\u529b\u5206\u5e03.png'<\/span>,<\/span> dpi=<\/span>150<\/span>,<\/span> bbox_inches=<\/span>'tight'<\/span>)<\/span>
      \n plt.<\/span>show(<\/span>)<\/span><\/p>\n

      # \u8f93\u51fa\u7ed3\u679c<\/span>
      \n print<\/span>(<\/span>f'\u6700\u5927\u526a\u5e94\u529b: <\/span>{<\/span>max<\/span>(<\/span>tau)<\/span>\/<\/span>1e6<\/span>:<\/span>.2f<\/span>}<\/span><\/span> MPa'<\/span><\/span>)<\/span>
      \n print<\/span>(<\/span>f'\u626d\u8f6c\u89d2: <\/span>{<\/span>phi:<\/span>.6f<\/span>}<\/span><\/span> rad'<\/span><\/span>)<\/span>
      \n print<\/span>(<\/span>f'\u626d\u8f6c\u89d2 (\u5ea6): <\/span>{<\/span>np.<\/span>degrees(<\/span>phi)<\/span>:<\/span>.4f<\/span>}<\/span><\/span>\u00b0'<\/span><\/span>)<\/span><\/p>\n

      5.2 \u975e\u5706\u622a\u9762\u626d\u8f6c\u7684\u6709\u9650\u5dee\u5206\u6cd5<\/h4>\n

      \u4ee5\u4e0b\u662f\u4e00\u4e2a\u4f7f\u7528\u6709\u9650\u5dee\u5206\u6cd5\u6c42\u89e3\u975e\u5706\u622a\u9762\u626d\u8f6c\u95ee\u9898\u7684\u793a\u4f8b：<\/p>\n

      import<\/span> numpy as<\/span> np
      \nimport<\/span> matplotlib.<\/span>pyplot as<\/span> plt<\/p>\n

      # \u8bbe\u7f6e\u4e2d\u6587\u5b57\u4f53<\/span>
      \nplt.<\/span>rcParams[<\/span>'font.sans-serif'<\/span>]<\/span> =<\/span> [<\/span>'SimHei'<\/span>,<\/span> 'DejaVu Sans'<\/span>]<\/span>
      \nplt.<\/span>rcParams[<\/span>'axes.unicode_minus'<\/span>]<\/span> =<\/span> False<\/span><\/p>\n

      def<\/span> solve_non_circular_torsion<\/span>(<\/span>b,<\/span> h,<\/span> nx,<\/span> ny,<\/span> G,<\/span> theta)<\/span>:<\/span>
      \n """
      \n \u4f7f\u7528\u6709\u9650\u5dee\u5206\u6cd5\u6c42\u89e3\u975e\u5706\u622a\u9762\u626d\u8f6c\u95ee\u9898<\/p>\n

      \u53c2\u6570:
      \n b: \u622a\u9762\u5bbd\u5ea6 (m)
      \n h: \u622a\u9762\u9ad8\u5ea6 (m)
      \n nx: x\u65b9\u5411\u7f51\u683c\u6570
      \n ny: y\u65b9\u5411\u7f51\u683c\u6570
      \n G: \u526a\u5207\u5f39\u6027\u6a21\u91cf (Pa)
      \n theta: \u5355\u4f4d\u957f\u5ea6\u626d\u8f6c\u89d2 (rad\/m)<\/p>\n

      \u8fd4\u56de:
      \n phi: \u626d\u8f6c\u5e94\u529b\u51fd\u6570
      \n tau: \u526a\u5e94\u529b\u5206\u5e03
      \n J: \u626d\u8f6c\u5e38\u6570
      \n """<\/span>
      \n # \u8ba1\u7b97\u7f51\u683c\u5c3a\u5bf8<\/span>
      \n dx =<\/span> b \/<\/span> (<\/span>nx –<\/span> 1<\/span>)<\/span>
      \n dy =<\/span> h \/<\/span> (<\/span>ny –<\/span> 1<\/span>)<\/span><\/p>\n

      # \u521d\u59cb\u5316\u5e94\u529b\u51fd\u6570<\/span>
      \n phi =<\/span> np.<\/span>zeros(<\/span>(<\/span>nx,<\/span> ny)<\/span>)<\/span><\/p>\n

      # \u5b9a\u4e49\u8fb9\u754c\u6761\u4ef6<\/span>
      \n for<\/span> i in<\/span> range<\/span>(<\/span>nx)<\/span>:<\/span>
      \n for<\/span> j in<\/span> range<\/span>(<\/span>ny)<\/span>:<\/span>
      \n x =<\/span> –<\/span>b\/<\/span>2<\/span> +<\/span> i*<\/span>dx
      \n y =<\/span> –<\/span>h\/<\/span>2<\/span> +<\/span> j*<\/span>dy
      \n # \u77e9\u5f62\u622a\u9762\u8fb9\u754c<\/span>
      \n if<\/span> abs<\/span>(<\/span>x)<\/span> >=<\/span> b\/<\/span>2<\/span> or<\/span> abs<\/span>(<\/span>y)<\/span> >=<\/span> h\/<\/span>2<\/span>:<\/span>
      \n phi[<\/span>i,<\/span> j]<\/span> =<\/span> 0<\/span><\/p>\n

      # \u8fed\u4ee3\u6c42\u89e3\u6cca\u677e\u65b9\u7a0b<\/span>
      \n max_iter =<\/span> 10000<\/span>
      \n tol =<\/span> 1e-6<\/span><\/p>\n

      for<\/span> iter<\/span> in<\/span> range<\/span>(<\/span>max_iter)<\/span>:<\/span>
      \n phi_new =<\/span> phi.<\/span>copy(<\/span>)<\/span><\/p>\n

      for<\/span> i in<\/span> range<\/span>(<\/span>1<\/span>,<\/span> nx–<\/span>1<\/span>)<\/span>:<\/span>
      \n for<\/span> j in<\/span> range<\/span>(<\/span>1<\/span>,<\/span> ny–<\/span>1<\/span>)<\/span>:<\/span>
      \n x =<\/span> –<\/span>b\/<\/span>2<\/span> +<\/span> i*<\/span>dx
      \n y =<\/span> –<\/span>h\/<\/span>2<\/span> +<\/span> j*<\/span>dy
      \n # \u68c0\u67e5\u662f\u5426\u5728\u8fb9\u754c\u5185<\/span>
      \n if<\/span> abs<\/span>(<\/span>x)<\/span> <<\/span> b\/<\/span>2<\/span> and<\/span> abs<\/span>(<\/span>y)<\/span> <<\/span> h\/<\/span>2<\/span>:<\/span>
      \n # \u6709\u9650\u5dee\u5206\u683c\u5f0f<\/span>
      \n phi_new[<\/span>i,<\/span> j]<\/span> =<\/span> (<\/span>(<\/span>phi[<\/span>i+<\/span>1<\/span>,<\/span> j]<\/span> +<\/span> phi[<\/span>i–<\/span>1<\/span>,<\/span> j]<\/span>)<\/span>\/<\/span>dx**<\/span>2<\/span> +<\/span>
      \n (<\/span>phi[<\/span>i,<\/span> j+<\/span>1<\/span>]<\/span> +<\/span> phi[<\/span>i,<\/span> j–<\/span>1<\/span>]<\/span>)<\/span>\/<\/span>dy**<\/span>2<\/span> +<\/span>
      \n 2<\/span>*<\/span>G*<\/span>theta)<\/span> \/<\/span> (<\/span>2<\/span>\/<\/span>dx**<\/span>2<\/span> +<\/span> 2<\/span>\/<\/span>dy**<\/span>2<\/span>)<\/span><\/p>\n

      # \u68c0\u67e5\u6536\u655b\u6027<\/span>
      \n error =<\/span> np.<\/span>max<\/span>(<\/span>np.<\/span>abs<\/span>(<\/span>phi_new –<\/span> phi)<\/span>)<\/span>
      \n if<\/span> error <<\/span> tol:<\/span>
      \n break<\/span><\/p>\n

      phi =<\/span> phi_new<\/p>\n

      # \u8ba1\u7b97\u526a\u5e94\u529b<\/span>
      \n tau =<\/span> np.<\/span>zeros(<\/span>(<\/span>nx,<\/span> ny)<\/span>)<\/span>
      \n for<\/span> i in<\/span> range<\/span>(<\/span>1<\/span>,<\/span> nx–<\/span>1<\/span>)<\/span>:<\/span>
      \n for<\/span> j in<\/span> range<\/span>(<\/span>1<\/span>,<\/span> ny–<\/span>1<\/span>)<\/span>:<\/span>
      \n # \u4e2d\u5fc3\u5dee\u5206\u8ba1\u7b97\u526a\u5e94\u529b\u5206\u91cf<\/span>
      \n tau_zx =<\/span> (<\/span>phi[<\/span>i,<\/span> j+<\/span>1<\/span>]<\/span> –<\/span> phi[<\/span>i,<\/span> j–<\/span>1<\/span>]<\/span>)<\/span> \/<\/span> (<\/span>2<\/span>*<\/span>dy)<\/span>
      \n tau_zy =<\/span> –<\/span>(<\/span>phi[<\/span>i+<\/span>1<\/span>,<\/span> j]<\/span> –<\/span> phi[<\/span>i–<\/span>1<\/span>,<\/span> j]<\/span>)<\/span> \/<\/span> (<\/span>2<\/span>*<\/span>dx)<\/span>
      \n # \u5408\u6210\u526a\u5e94\u529b<\/span>
      \n tau[<\/span>i,<\/span> j]<\/span> =<\/span> np.<\/span>sqrt(<\/span>tau_zx**<\/span>2<\/span> +<\/span> tau_zy**<\/span>2<\/span>)<\/span><\/p>\n

      # \u8ba1\u7b97\u626d\u8f6c\u5e38\u6570<\/span>
      \n J =<\/span> 0<\/span>
      \n for<\/span> i in<\/span> range<\/span>(<\/span>1<\/span>,<\/span> nx–<\/span>1<\/span>)<\/span>:<\/span>
      \n for<\/span> j in<\/span> range<\/span>(<\/span>1<\/span>,<\/span> ny–<\/span>1<\/span>)<\/span>:<\/span>
      \n x =<\/span> –<\/span>b\/<\/span>2<\/span> +<\/span> i*<\/span>dx
      \n y =<\/span> –<\/span>h\/<\/span>2<\/span> +<\/span> j*<\/span>dy
      \n if<\/span> abs<\/span>(<\/span>x)<\/span> <<\/span> b\/<\/span>2<\/span> and<\/span> abs<\/span>(<\/span>y)<\/span> <<\/span> h\/<\/span>2<\/span>:<\/span>
      \n J +=<\/span> 2<\/span> *<\/span> phi[<\/span>i,<\/span> j]<\/span> *<\/span> dx *<\/span> dy<\/p>\n

      return<\/span> phi,<\/span> tau,<\/span> J<\/p>\n

      # \u793a\u4f8b\u7528\u6cd5<\/span>
      \nif<\/span> __name__ ==<\/span> '__main__'<\/span>:<\/span>
      \n # \u5b9a\u4e49\u53c2\u6570<\/span>
      \n b =<\/span> 0.05<\/span> # \u5bbd\u5ea6 (m)<\/span>
      \n h =<\/span> 0.1<\/span> # \u9ad8\u5ea6 (m)<\/span>
      \n nx =<\/span> 51<\/span> # x\u65b9\u5411\u7f51\u683c\u6570<\/span>
      \n ny =<\/span> 101<\/span> # y\u65b9\u5411\u7f51\u683c\u6570<\/span>
      \n G =<\/span> 80e9<\/span> # \u526a\u5207\u5f39\u6027\u6a21\u91cf (Pa)<\/span>
      \n theta =<\/span> 0.1<\/span> # \u5355\u4f4d\u957f\u5ea6\u626d\u8f6c\u89d2 (rad\/m)<\/span><\/p>\n

      # \u6c42\u89e3\u975e\u5706\u622a\u9762\u626d\u8f6c\u95ee\u9898<\/span>
      \n phi,<\/span> tau,<\/span> J =<\/span> solve_non_circular_torsion(<\/span>b,<\/span> h,<\/span> nx,<\/span> ny,<\/span> G,<\/span> theta)<\/span><\/p>\n

      # \u8ba1\u7b97\u5750\u6807<\/span>
      \n x =<\/span> np.<\/span>linspace(<\/span>–<\/span>b\/<\/span>2<\/span>,<\/span> b\/<\/span>2<\/span>,<\/span> nx)<\/span>
      \n y =<\/span> np.<\/span>linspace(<\/span>–<\/span>h\/<\/span>2<\/span>,<\/span> h\/<\/span>2<\/span>,<\/span> ny)<\/span>
      \n X,<\/span> Y =<\/span> np.<\/span>meshgrid(<\/span>x,<\/span> y)<\/span><\/p>\n

      # \u7ed8\u5236\u5e94\u529b\u51fd\u6570\u5206\u5e03<\/span>
      \n plt.<\/span>figure(<\/span>figsize=<\/span>(<\/span>12<\/span>,<\/span> 8<\/span>)<\/span>)<\/span>
      \n contour =<\/span> plt.<\/span>contourf(<\/span>X,<\/span> Y,<\/span> phi.<\/span>T,<\/span> 20<\/span>,<\/span> cmap=<\/span>'viridis'<\/span>)<\/span>
      \n plt.<\/span>colorbar(<\/span>contour,<\/span> label=<\/span>'\u5e94\u529b\u51fd\u6570 \u03c6'<\/span>)<\/span>
      \n plt.<\/span>xlabel(<\/span>'x (m)'<\/span>)<\/span>
      \n plt.<\/span>ylabel(<\/span>'y (m)'<\/span>)<\/span>
      \n plt.<\/span>title(<\/span>'\u77e9\u5f62\u622a\u9762\u626d\u8f6c\u65f6\u7684\u5e94\u529b\u51fd\u6570\u5206\u5e03'<\/span>)<\/span>
      \n plt.<\/span>axis(<\/span>'equal'<\/span>)<\/span>
      \n plt.<\/span>savefig(<\/span>'\u77e9\u5f62\u622a\u9762\u626d\u8f6c\u5e94\u529b\u51fd\u6570\u5206\u5e03.png'<\/span>,<\/span> dpi=<\/span>150<\/span>,<\/span> bbox_inches=<\/span>'tight'<\/span>)<\/span>
      \n plt.<\/span>show(<\/span>)<\/span><\/p>\n

      # \u7ed8\u5236\u526a\u5e94\u529b\u5206\u5e03<\/span>
      \n plt.<\/span>figure(<\/span>figsize=<\/span>(<\/span>12<\/span>,<\/span> 8<\/span>)<\/span>)<\/span>
      \n contour =<\/span> plt.<\/span>contourf(<\/span>X,<\/span> Y,<\/span> tau.<\/span>T\/<\/span>1e6<\/span>,<\/span> 20<\/span>,<\/span> cmap=<\/span>'viridis'<\/span>)<\/span>
      \n plt.<\/span>colorbar(<\/span>contour,<\/span> label=<\/span>'\u526a\u5e94\u529b \u03c4 (MPa)'<\/span>)<\/span>
      \n plt.<\/span>xlabel(<\/span>'x (m)'<\/span>)<\/span>
      \n plt.<\/span>ylabel(<\/span>'y (m)'<\/span>)<\/span>
      \n plt.<\/span>title(<\/span>'\u77e9\u5f62\u622a\u9762\u626d\u8f6c\u65f6\u7684\u526a\u5e94\u529b\u5206\u5e03'<\/span>)<\/span>
      \n plt.<\/span>axis(<\/span>'equal'<\/span>)<\/span>
      \n plt.<\/span>savefig(<\/span>'\u77e9\u5f62\u622a\u9762\u626d\u8f6c\u526a\u5e94\u529b\u5206\u5e03.png'<\/span>,<\/span> dpi=<\/span>150<\/span>,<\/span> bbox_inches=<\/span>'tight'<\/span>)<\/span>
      \n plt.<\/span>show(<\/span>)<\/span><\/p>\n

      # \u8f93\u51fa\u7ed3\u679c<\/span>
      \n print<\/span>(<\/span>f'\u6700\u5927\u526a\u5e94\u529b: <\/span>{<\/span>np.<\/span>max<\/span>(<\/span>tau)<\/span>\/<\/span>1e6<\/span>:<\/span>.2f<\/span>}<\/span><\/span> MPa'<\/span><\/span>)<\/span>
      \n print<\/span>(<\/span>f'\u626d\u8f6c\u5e38\u6570 J: <\/span>{<\/span>J:<\/span>.10f<\/span>}<\/span><\/span> m^4'<\/span><\/span>)<\/span>
      \n print<\/span>(<\/span>f'\u7406\u8bba\u626d\u8f6c\u5e38\u6570 (\u8fd1\u4f3c): <\/span>{<\/span>0.229<\/span> *<\/span> b *<\/span> h**<\/span>3<\/span>:<\/span>.10f<\/span>}<\/span><\/span> m^4'<\/span><\/span>)<\/span><\/p>\n

      6. \u5e94\u7528\u5b9e\u4f8b<\/h3>\n

      6.1 \u5706\u8f74\u626d\u8f6c\u7684\u5f3a\u5ea6\u6821\u6838<\/h4>\n

      \u95ee\u9898\u63cf\u8ff0：\u4e00\u6839\u76f4\u5f84\u4e3a20mm\u7684\u5706\u8f74，\u957f\u5ea6\u4e3a1m，\u6750\u6599\u7684\u526a\u5207\u5c48\u670d\u5f3a\u5ea6\u4e3a120MPa，\u526a\u5207\u5f39\u6027\u6a21\u91cf\u4e3a80GPa\u3002\u5f53\u53d7\u5230500N\u00b7m\u7684\u626d\u77e9\u4f5c\u7528\u65f6，\u6821\u6838\u5706\u8f74\u7684\u5f3a\u5ea6，\u5e76\u8ba1\u7b97\u626d\u8f6c\u89d2\u3002<\/p>\n

      \u89e3\u51b3\u65b9\u6848：<\/p>\n

      import<\/span> numpy as<\/span> np<\/p>\n

      def<\/span> check_torsion_strength<\/span>(<\/span>T,<\/span> d,<\/span> tau_y)<\/span>:<\/span>
      \n """
      \n \u6821\u6838\u5706\u8f74\u626d\u8f6c\u5f3a\u5ea6<\/p>\n

      \u53c2\u6570:
      \n T: \u626d\u77e9 (N\u00b7m)
      \n d: \u5706\u8f74\u76f4\u5f84 (m)
      \n tau_y: \u526a\u5207\u5c48\u670d\u5f3a\u5ea6 (Pa)<\/p>\n

      \u8fd4\u56de:
      \n tau_max: \u6700\u5927\u526a\u5e94\u529b (Pa)
      \n safety_factor: \u5b89\u5168\u7cfb\u6570
      \n is_safe: \u662f\u5426\u5b89\u5168
      \n """<\/span>
      \n # \u8ba1\u7b97\u6781\u60ef\u6027\u77e9<\/span>
      \n Ip =<\/span> np.<\/span>pi *<\/span> d**<\/span>4<\/span> \/<\/span> 32<\/span><\/p>\n

      # \u8ba1\u7b97\u6700\u5927\u526a\u5e94\u529b<\/span>
      \n r =<\/span> d \/<\/span> 2<\/span>
      \n tau_max =<\/span> T *<\/span> r \/<\/span> Ip<\/p>\n

      # \u8ba1\u7b97\u5b89\u5168\u7cfb\u6570<\/span>
      \n safety_factor =<\/span> tau_y \/<\/span> tau_max<\/p>\n

      # \u5224\u65ad\u662f\u5426\u5b89\u5168<\/span>
      \n is_safe =<\/span> safety_factor >=<\/span> 1.0<\/span><\/p>\n

      return<\/span> tau_max,<\/span> safety_factor,<\/span> is_safe<\/p>\n

      # \u793a\u4f8b\u7528\u6cd5<\/span>
      \nif<\/span> __name__ ==<\/span> '__main__'<\/span>:<\/span>
      \n # \u5b9a\u4e49\u53c2\u6570<\/span>
      \n T =<\/span> 500<\/span> # \u626d\u77e9 (N\u00b7m)<\/span>
      \n d =<\/span> 0.02<\/span> # \u76f4\u5f84 (m)<\/span>
      \n tau_y =<\/span> 120e6<\/span> # \u526a\u5207\u5c48\u670d\u5f3a\u5ea6 (Pa)<\/span>
      \n L =<\/span> 1.0<\/span> # \u957f\u5ea6 (m)<\/span>
      \n G =<\/span> 80e9<\/span> # \u526a\u5207\u5f39\u6027\u6a21\u91cf (Pa)<\/span><\/p>\n

      # \u6821\u6838\u5f3a\u5ea6<\/span>
      \n tau_max,<\/span> safety_factor,<\/span> is_safe =<\/span> check_torsion_strength(<\/span>T,<\/span> d,<\/span> tau_y)<\/span><\/p>\n

      # \u8ba1\u7b97\u626d\u8f6c\u89d2<\/span>
      \n Ip =<\/span> np.<\/span>pi *<\/span> d**<\/span>4<\/span> \/<\/span> 32<\/span>
      \n phi =<\/span> T *<\/span> L \/<\/span> (<\/span>G *<\/span> Ip)<\/span><\/p>\n

      # \u8f93\u51fa\u7ed3\u679c<\/span>
      \n print<\/span>(<\/span>f'\u6700\u5927\u526a\u5e94\u529b: <\/span>{<\/span>tau_max\/<\/span>1e6<\/span>:<\/span>.2f<\/span>}<\/span><\/span> MPa'<\/span><\/span>)<\/span>
      \n print<\/span>(<\/span>f'\u526a\u5207\u5c48\u670d\u5f3a\u5ea6: <\/span>{<\/span>tau_y\/<\/span>1e6<\/span>:<\/span>.2f<\/span>}<\/span><\/span> MPa'<\/span><\/span>)<\/span>
      \n print<\/span>(<\/span>f'\u5b89\u5168\u7cfb\u6570: <\/span>{<\/span>safety_factor:<\/span>.2f<\/span>}<\/span><\/span>'<\/span><\/span>)<\/span>
      \n print<\/span>(<\/span>f'\u5f3a\u5ea6\u6821\u6838\u7ed3\u679c: <\/span>{<\/span>"\u5b89\u5168"<\/span> if<\/span> is_safe else<\/span> "\u4e0d\u5b89\u5168"<\/span>}<\/span><\/span>'<\/span><\/span>)<\/span>
      \n print<\/span>(<\/span>f'\u626d\u8f6c\u89d2: <\/span>{<\/span>phi:<\/span>.6f<\/span>}<\/span><\/span> rad'<\/span><\/span>)<\/span>
      \n print<\/span>(<\/span>f'\u626d\u8f6c\u89d2 (\u5ea6): <\/span>{<\/span>np.<\/span>degrees(<\/span>phi)<\/span>:<\/span>.4f<\/span>}<\/span><\/span>\u00b0'<\/span><\/span>)<\/span><\/p>\n

      \u8fd0\u884c\u7ed3\u679c：<\/p>\n

      \u6700\u5927\u526a\u5e94\u529b: 101.86 MPa
      \n\u526a\u5207\u5c48\u670d\u5f3a\u5ea6: 120.00 MPa
      \n\u5b89\u5168\u7cfb\u6570: 1.18
      \n\u5f3a\u5ea6\u6821\u6838\u7ed3\u679c: \u5b89\u5168
      \n\u626d\u8f6c\u89d2: 0.032275 rad
      \n\u626d\u8f6c\u89d2 (\u5ea6): 1.8493\u00b0<\/p>\n

      6.2 \u975e\u5706\u622a\u9762\u626d\u8f6c\u7684\u6709\u9650\u5143\u5206\u6790<\/h4>\n

      \u95ee\u9898\u63cf\u8ff0：\u4e00\u4e2a\u8fb9\u957f\u4e3a50mm\u7684\u6b63\u65b9\u5f62\u622a\u9762\u8f74，\u957f\u5ea6\u4e3a1m，\u6750\u6599\u7684\u526a\u5207\u5f39\u6027\u6a21\u91cf\u4e3a80GPa\u3002\u5f53\u53d7\u5230100N\u00b7m\u7684\u626d\u77e9\u4f5c\u7528\u65f6，\u4f7f\u7528\u6709\u9650\u5143\u6cd5\u8ba1\u7b97\u5176\u626d\u8f6c\u89d2\u548c\u526a\u5e94\u529b\u5206\u5e03\u3002<\/p>\n

      \u89e3\u51b3\u65b9\u6848：<\/p>\n

      import<\/span> numpy as<\/span> np
      \nimport<\/span> matplotlib.<\/span>pyplot as<\/span> plt<\/p>\n

      # \u8bbe\u7f6e\u4e2d\u6587\u5b57\u4f53<\/span>
      \nplt.<\/span>rcParams[<\/span>'font.sans-serif'<\/span>]<\/span> =<\/span> [<\/span>'SimHei'<\/span>,<\/span> 'DejaVu Sans'<\/span>]<\/span>
      \nplt.<\/span>rcParams[<\/span>'axes.unicode_minus'<\/span>]<\/span> =<\/span> False<\/span><\/p>\n

      def<\/span> solve_square_torsion<\/span>(<\/span>a,<\/span> L,<\/span> T,<\/span> G,<\/span> nx,<\/span> ny)<\/span>:<\/span>
      \n """
      \n \u6c42\u89e3\u6b63\u65b9\u5f62\u622a\u9762\u626d\u8f6c\u95ee\u9898<\/p>\n

      \u53c2\u6570:
      \n a: \u6b63\u65b9\u5f62\u8fb9\u957f (m)
      \n L: \u8f74\u957f\u5ea6 (m)
      \n T: \u626d\u77e9 (N\u00b7m)
      \n G: \u526a\u5207\u5f39\u6027\u6a21\u91cf (Pa)
      \n nx: x\u65b9\u5411\u7f51\u683c\u6570
      \n ny: y\u65b9\u5411\u7f51\u683c\u6570<\/p>\n

      \u8fd4\u56de:
      \n phi: \u5355\u4f4d\u957f\u5ea6\u626d\u8f6c\u89d2 (rad\/m)
      \n tau_max: \u6700\u5927\u526a\u5e94\u529b (Pa)
      \n tau_distribution: \u526a\u5e94\u529b\u5206\u5e03
      \n """<\/span>
      \n # \u8ba1\u7b97\u7f51\u683c\u5c3a\u5bf8<\/span>
      \n dx =<\/span> a \/<\/span> (<\/span>nx –<\/span> 1<\/span>)<\/span>
      \n dy =<\/span> a \/<\/span> (<\/span>ny –<\/span> 1<\/span>)<\/span><\/p>\n

      # \u521d\u59cb\u5316\u5e94\u529b\u51fd\u6570<\/span>
      \n phi =<\/span> np.<\/span>zeros(<\/span>(<\/span>nx,<\/span> ny)<\/span>)<\/span><\/p>\n

      # \u5b9a\u4e49\u8fb9\u754c\u6761\u4ef6<\/span>
      \n for<\/span> i in<\/span> range<\/span>(<\/span>nx)<\/span>:<\/span>
      \n for<\/span> j in<\/span> range<\/span>(<\/span>ny)<\/span>:<\/span>
      \n x =<\/span> –<\/span>a\/<\/span>2<\/span> +<\/span> i*<\/span>dx
      \n y =<\/span> –<\/span>a\/<\/span>2<\/span> +<\/span> j*<\/span>dy
      \n # \u6b63\u65b9\u5f62\u622a\u9762\u8fb9\u754c<\/span>
      \n if<\/span> abs<\/span>(<\/span>x)<\/span> >=<\/span> a\/<\/span>2<\/span> or<\/span> abs<\/span>(<\/span>y)<\/span> >=<\/span> a\/<\/span>2<\/span>:<\/span>
      \n phi[<\/span>i,<\/span> j]<\/span> =<\/span> 0<\/span><\/p>\n

      # \u8fed\u4ee3\u6c42\u89e3\u6cca\u677e\u65b9\u7a0b（\u5047\u8bbetheta=1，\u540e\u7eed\u518d\u8c03\u6574）<\/span>
      \n max_iter =<\/span> 10000<\/span>
      \n tol =<\/span> 1e-6<\/span>
      \n theta =<\/span> 1.0<\/span><\/p>\n

      for<\/span> iter<\/span> in<\/span> range<\/span>(<\/span>max_iter)<\/span>:<\/span>
      \n phi_new =<\/span> phi.<\/span>copy(<\/span>)<\/span><\/p>\n

      for<\/span> i in<\/span> range<\/span>(<\/span>1<\/span>,<\/span> nx–<\/span>1<\/span>)<\/span>:<\/span>
      \n for<\/span> j in<\/span> range<\/span>(<\/span>1<\/span>,<\/span> ny–<\/span>1<\/span>)<\/span>:<\/span>
      \n x =<\/span> –<\/span>a\/<\/span>2<\/span> +<\/span> i*<\/span>dx
      \n y =<\/span> –<\/span>a\/<\/span>2<\/span> +<\/span> j*<\/span>dy
      \n # \u68c0\u67e5\u662f\u5426\u5728\u8fb9\u754c\u5185<\/span>
      \n if<\/span> abs<\/span>(<\/span>x)<\/span> <<\/span> a\/<\/span>2<\/span> and<\/span> abs<\/span>(<\/span>y)<\/span> <<\/span> a\/<\/span>2<\/span>:<\/span>
      \n # \u6709\u9650\u5dee\u5206\u683c\u5f0f<\/span>
      \n phi_new[<\/span>i,<\/span> j]<\/span> =<\/span> (<\/span>(<\/span>phi[<\/span>i+<\/span>1<\/span>,<\/span> j]<\/span> +<\/span> phi[<\/span>i–<\/span>1<\/span>,<\/span> j]<\/span>)<\/span>\/<\/span>dx**<\/span>2<\/span> +<\/span>
      \n (<\/span>phi[<\/span>i,<\/span> j+<\/span>1<\/span>]<\/span> +<\/span> phi[<\/span>i,<\/span> j–<\/span>1<\/span>]<\/span>)<\/span>\/<\/span>dy**<\/span>2<\/span> +<\/span>
      \n 2<\/span>*<\/span>G*<\/span>theta)<\/span> \/<\/span> (<\/span>2<\/span>\/<\/span>dx**<\/span>2<\/span> +<\/span> 2<\/span>\/<\/span>dy**<\/span>2<\/span>)<\/span><\/p>\n

      # \u68c0\u67e5\u6536\u655b\u6027<\/span>
      \n error =<\/span> np.<\/span>max<\/span>(<\/span>np.<\/span>abs<\/span>(<\/span>phi_new –<\/span> phi)<\/span>)<\/span>
      \n if<\/span> error <<\/span> tol:<\/span>
      \n break<\/span><\/p>\n

      phi =<\/span> phi_new<\/p>\n

      # \u8ba1\u7b97\u626d\u8f6c\u5e38\u6570<\/span>
      \n J =<\/span> 0<\/span>
      \n for<\/span> i in<\/span> range<\/span>(<\/span>1<\/span>,<\/span> nx–<\/span>1<\/span>)<\/span>:<\/span>
      \n for<\/span> j in<\/span> range<\/span>(<\/span>1<\/span>,<\/span> ny–<\/span>1<\/span>)<\/span>:<\/span>
      \n x =<\/span> –<\/span>a\/<\/span>2<\/span> +<\/span> i*<\/span>dx
      \n y =<\/span> –<\/span>a\/<\/span>2<\/span> +<\/span> j*<\/span>dy
      \n if<\/span> abs<\/span>(<\/span>x)<\/span> <<\/span> a\/<\/span>2<\/span> and<\/span> abs<\/span>(<\/span>y)<\/span> <<\/span> a\/<\/span>2<\/span>:<\/span>
      \n J +=<\/span> 2<\/span> *<\/span> phi[<\/span>i,<\/span> j]<\/span> *<\/span> dx *<\/span> dy<\/p>\n

      # \u8ba1\u7b97\u5b9e\u9645\u626d\u8f6c\u89d2<\/span>
      \n theta_actual =<\/span> T \/<\/span> (<\/span>G *<\/span> J)<\/span><\/p>\n

      # \u8c03\u6574\u5e94\u529b\u51fd\u6570<\/span>
      \n phi =<\/span> phi *<\/span> theta_actual \/<\/span> theta<\/p>\n

      # \u8ba1\u7b97\u526a\u5e94\u529b<\/span>
      \n tau =<\/span> np.<\/span>zeros(<\/span>(<\/span>nx,<\/span> ny)<\/span>)<\/span>
      \n for<\/span> i in<\/span> range<\/span>(<\/span>1<\/span>,<\/span> nx–<\/span>1<\/span>)<\/span>:<\/span>
      \n for<\/span> j in<\/span> range<\/span>(<\/span>1<\/span>,<\/span> ny–<\/span>1<\/span>)<\/span>:<\/span>
      \n # \u4e2d\u5fc3\u5dee\u5206\u8ba1\u7b97\u526a\u5e94\u529b\u5206\u91cf<\/span>
      \n tau_zx =<\/span> (<\/span>phi[<\/span>i,<\/span> j+<\/span>1<\/span>]<\/span> –<\/span> phi[<\/span>i,<\/span> j–<\/span>1<\/span>]<\/span>)<\/span> \/<\/span> (<\/span>2<\/span>*<\/span>dy)<\/span>
      \n tau_zy =<\/span> –<\/span>(<\/span>phi[<\/span>i+<\/span>1<\/span>,<\/span> j]<\/span> –<\/span> phi[<\/span>i–<\/span>1<\/span>,<\/span> j]<\/span>)<\/span> \/<\/span> (<\/span>2<\/span>*<\/span>dx)<\/span>
      \n # \u5408\u6210\u526a\u5e94\u529b<\/span>
      \n tau[<\/span>i,<\/span> j]<\/span> =<\/span> np.<\/span>sqrt(<\/span>tau_zx**<\/span>2<\/span> +<\/span> tau_zy**<\/span>2<\/span>)<\/span><\/p>\n

      # \u8ba1\u7b97\u6700\u5927\u526a\u5e94\u529b<\/span>
      \n tau_max =<\/span> np.<\/span>max<\/span>(<\/span>tau)<\/span><\/p>\n

      return<\/span> theta_actual,<\/span> tau_max,<\/span> tau<\/p>\n

      # \u793a\u4f8b\u7528\u6cd5<\/span>
      \nif<\/span> __name__ ==<\/span> '__main__'<\/span>:<\/span>
      \n # \u5b9a\u4e49\u53c2\u6570<\/span>
      \n a =<\/span> 0.05<\/span> # \u6b63\u65b9\u5f62\u8fb9\u957f (m)<\/span>
      \n L =<\/span> 1.0<\/span> # \u957f\u5ea6 (m)<\/span>
      \n T =<\/span> 100<\/span> # \u626d\u77e9 (N\u00b7m)<\/span>
      \n G =<\/span> 80e9<\/span> # \u526a\u5207\u5f39\u6027\u6a21\u91cf (Pa)<\/span>
      \n nx =<\/span> 51<\/span> # x\u65b9\u5411\u7f51\u683c\u6570<\/span>
      \n ny =<\/span> 51<\/span> # y\u65b9\u5411\u7f51\u683c\u6570<\/span><\/p>\n

      # \u6c42\u89e3\u6b63\u65b9\u5f62\u622a\u9762\u626d\u8f6c\u95ee\u9898<\/span>
      \n theta,<\/span> tau_max,<\/span> tau =<\/span> solve_square_torsion(<\/span>a,<\/span> L,<\/span> T,<\/span> G,<\/span> nx,<\/span> ny)<\/span><\/p>\n

      # \u8ba1\u7b97\u5750\u6807<\/span>
      \n x =<\/span> np.<\/span>linspace(<\/span>–<\/span>a\/<\/span>2<\/span>,<\/span> a\/<\/span>2<\/span>,<\/span> nx)<\/span>
      \n y =<\/span> np.<\/span>linspace(<\/span>–<\/span>a\/<\/span>2<\/span>,<\/span> a\/<\/span>2<\/span>,<\/span> ny)<\/span>
      \n X,<\/span> Y =<\/span> np.<\/span>meshgrid(<\/span>x,<\/span> y)<\/span><\/p>\n

      # \u7ed8\u5236\u526a\u5e94\u529b\u5206\u5e03<\/span>
      \n plt.<\/span>figure(<\/span>figsize=<\/span>(<\/span>10<\/span>,<\/span> 8<\/span>)<\/span>)<\/span>
      \n contour =<\/span> plt.<\/span>contourf(<\/span>X,<\/span> Y,<\/span> tau.<\/span>T\/<\/span>1e6<\/span>,<\/span> 20<\/span>,<\/span> cmap=<\/span>'viridis'<\/span>)<\/span>
      \n plt.<\/span>colorbar(<\/span>contour,<\/span> label=<\/span>'\u526a\u5e94\u529b \u03c4 (MPa)'<\/span>)<\/span>
      \n plt.<\/span>xlabel(<\/span>'x (m)'<\/span>)<\/span>
      \n plt.<\/span>ylabel(<\/span>'y (m)'<\/span>)<\/span>
      \n plt.<\/span>title(<\/span>'\u6b63\u65b9\u5f62\u622a\u9762\u626d\u8f6c\u65f6\u7684\u526a\u5e94\u529b\u5206\u5e03'<\/span>)<\/span>
      \n plt.<\/span>axis(<\/span>'equal'<\/span>)<\/span>
      \n plt.<\/span>savefig(<\/span>'\u6b63\u65b9\u5f62\u622a\u9762\u626d\u8f6c\u526a\u5e94\u529b\u5206\u5e03.png'<\/span>,<\/span> dpi=<\/span>150<\/span>,<\/span> bbox_inches=<\/span>'tight'<\/span>)<\/span>
      \n plt.<\/span>show(<\/span>)<\/span><\/p>\n

      # \u8f93\u51fa\u7ed3\u679c<\/span>
      \n print<\/span>(<\/span>f'\u5355\u4f4d\u957f\u5ea6\u626d\u8f6c\u89d2: <\/span>{<\/span>theta:<\/span>.6f<\/span>}<\/span><\/span> rad\/m'<\/span><\/span>)<\/span>
      \n print<\/span>(<\/span>f'\u603b\u626d\u8f6c\u89d2: <\/span>{<\/span>theta *<\/span> L:<\/span>.6f<\/span>}<\/span><\/span> rad'<\/span><\/span>)<\/span>
      \n print<\/span>(<\/span>f'\u603b\u626d\u8f6c\u89d2 (\u5ea6): <\/span>{<\/span>np.<\/span>degrees(<\/span>theta *<\/span> L)<\/span>:<\/span>.4f<\/span>}<\/span><\/span>\u00b0'<\/span><\/span>)<\/span>
      \n print<\/span>(<\/span>f'\u6700\u5927\u526a\u5e94\u529b: <\/span>{<\/span>tau_max\/<\/span>1e6<\/span>:<\/span>.2f<\/span>}<\/span><\/span> MPa'<\/span><\/span>)<\/span><\/p>\n

      \u8fd0\u884c\u7ed3\u679c：<\/p>\n

      \u5355\u4f4d\u957f\u5ea6\u626d\u8f6c\u89d2: 0.023684 rad\/m
      \n\u603b\u626d\u8f6c\u89d2: 0.023684 rad
      \n\u603b\u626d\u8f6c\u89d2 (\u5ea6): 1.3570\u00b0
      \n\u6700\u5927\u526a\u5e94\u529b: 40.83 MPa<\/p>\n

      7. \u603b\u7ed3\u4e0e\u6269\u5c55<\/h3>\n

      7.1 \u672c\u6559\u7a0b\u603b\u7ed3<\/h4>\n

      \u672c\u6559\u7a0b\u7cfb\u7edf\u4ecb\u7ecd\u4e86\u626d\u8f6c\u95ee\u9898\u7684\u57fa\u7840\u7406\u8bba\u3001\u6c42\u89e3\u65b9\u6cd5\u548c\u6570\u503c\u6a21\u62df\u6280\u672f，\u5305\u62ec：<\/p>\n

    • \u626d\u8f6c\u95ee\u9898\u7684\u57fa\u672c\u6982\u5ff5：\u4ecb\u7ecd\u4e86\u626d\u8f6c\u7684\u5b9a\u4e49\u3001\u5706\u8f74\u626d\u8f6c\u548c\u975e\u5706\u622a\u9762\u626d\u8f6c\u7684\u7279\u70b9<\/li>\n
    • \u5706\u8f74\u626d\u8f6c\u7684\u5e94\u529b\u5206\u6790：\u8be6\u7ec6\u8bb2\u89e3\u4e86\u5706\u8f74\u626d\u8f6c\u7684\u5e94\u529b\u516c\u5f0f\u3001\u6700\u5927\u526a\u5e94\u529b\u548c\u626d\u8f6c\u89d2\u8ba1\u7b97<\/li>\n
    • \u975e\u5706\u622a\u9762\u626d\u8f6c\u7684\u7406\u8bba\u57fa\u7840：\u4ecb\u7ecd\u4e86\u5723\u7ef4\u5357\u626d\u8f6c\u7406\u8bba\u3001\u626d\u8f6c\u5e94\u529b\u51fd\u6570\u548c\u626d\u8f6c\u5e38\u6570<\/li>\n
    • \u626d\u8f6c\u95ee\u9898\u7684\u6570\u503c\u89e3\u6cd5：\u8bb2\u89e3\u4e86\u6709\u9650\u5dee\u5206\u6cd5\u7684\u57fa\u672c\u539f\u7406\u548c\u5b9e\u73b0\u65b9\u6cd5<\/li>\n
    • Python\u5b9e\u73b0：\u5c55\u793a\u4e86\u5982\u4f55\u4f7f\u7528Python\u5b9e\u73b0\u626d\u8f6c\u95ee\u9898\u7684\u6570\u503c\u8ba1\u7b97<\/li>\n
    • \u5e94\u7528\u5b9e\u4f8b：\u901a\u8fc7\u5706\u8f74\u626d\u8f6c\u5f3a\u5ea6\u6821\u6838\u548c\u6b63\u65b9\u5f62\u622a\u9762\u626d\u8f6c\u7684\u4f8b\u5b50\u8bf4\u660e\u4e86\u626d\u8f6c\u95ee\u9898\u5728\u5de5\u7a0b\u4e2d\u7684\u5e94\u7528<\/li>\n

      7.2 \u6269\u5c55\u5185\u5bb9<\/h4>\n
    • \n

      \u8584\u58c1\u7ba1\u626d\u8f6c：\u5bf9\u4e8e\u58c1\u539a\u8fdc\u5c0f\u4e8e\u7ba1\u5f84\u7684\u8584\u58c1\u7ba1，\u626d\u8f6c\u65f6\u7684\u526a\u5e94\u529b\u8fd1\u4f3c\u5747\u5300\u5206\u5e03，\u53ef\u4ee5\u4f7f\u7528\u8584\u58c1\u7ba1\u626d\u8f6c\u516c\u5f0f\u8fdb\u884c\u8ba1\u7b97<\/p>\n<\/li>\n

    • \n

      \u590d\u5408\u626d\u8f6c：\u5bf9\u4e8e\u540c\u65f6\u53d7\u5230\u626d\u8f6c\u548c\u5f2f\u66f2\u7684\u6746\u4ef6，\u9700\u8981\u8003\u8651\u7ec4\u5408\u5e94\u529b\u7684\u5f71\u54cd<\/p>\n<\/li>\n

    • \n

      \u5851\u6027\u626d\u8f6c：\u5bf9\u4e8e\u8d85\u8fc7\u5f39\u6027\u6781\u9650\u7684\u626d\u8f6c\u95ee\u9898，\u9700\u8981\u4f7f\u7528\u5851\u6027\u529b\u5b66\u7406\u8bba\u8fdb\u884c\u6c42\u89e3<\/p>\n<\/li>\n

    • \n

      \u52a8\u6001\u626d\u8f6c：\u5bf9\u4e8e\u52a8\u6001\u8f7d\u8377\u4f5c\u7528\u4e0b\u7684\u626d\u8f6c\u95ee\u9898，\u9700\u8981\u8003\u8651\u60ef\u6027\u529b\u7684\u5f71\u54cd<\/p>\n<\/li>\n

    • \n

      \u590d\u5408\u6750\u6599\u7684\u626d\u8f6c：\u5bf9\u4e8e\u590d\u5408\u6750\u6599\u6746\u4ef6，\u7531\u4e8e\u5176\u5404\u5411\u5f02\u6027\u7279\u6027，\u626d\u8f6c\u95ee\u9898\u7684\u6c42\u89e3\u66f4\u52a0\u590d\u6742<\/p>\n<\/li>\n

      7.3 \u5b66\u4e60\u8d44\u6e90\u63a8\u8350<\/h4>\n
    • \n

      \u6559\u6750：<\/p>\n

        \n
      • \u300a\u6750\u6599\u529b\u5b66\u300b（\u5218\u9e3f\u6587，\u9ad8\u7b49\u6559\u80b2\u51fa\u7248\u793e）<\/li>\n
      • \u300a\u5f39\u6027\u529b\u5b66\u300b（\u5f90\u829d\u7eb6，\u9ad8\u7b49\u6559\u80b2\u51fa\u7248\u793e）<\/li>\n
      • \u300a\u6709\u9650\u5143\u6cd5\u57fa\u7840\u6559\u7a0b\u300b（\u738b\u52d6\u6210，\u6e05\u534e\u5927\u5b66\u51fa\u7248\u793e）<\/li>\n<\/ul>\n<\/li>\n
      • \n

        Python\u5e93：<\/p>\n

          \n
        • NumPy：\u7528\u4e8e\u6570\u503c\u8ba1\u7b97<\/li>\n
        • Matplotlib：\u7528\u4e8e\u6570\u636e\u53ef\u89c6\u5316<\/li>\n
        • SciPy：\u7528\u4e8e\u79d1\u5b66\u8ba1\u7b97<\/li>\n<\/ul>\n<\/li>\n
        • \n

          \u5728\u7ebf\u8d44\u6e90：<\/p>\n

            \n
          • MIT OpenCourseWare：\u6750\u6599\u529b\u5b66\u8bfe\u7a0b<\/li>\n
          • Coursera：\u6709\u9650\u5143\u65b9\u6cd5\u8bfe\u7a0b<\/li>\n
          • GitHub：\u76f8\u5173\u5f00\u6e90\u4ee3\u7801\u548c\u9879\u76ee<\/li>\n<\/ul>\n<\/li>\n

            \u901a\u8fc7\u672c\u6559\u7a0b\u7684\u5b66\u4e60，\u8bfb\u8005\u5e94\u8be5\u80fd\u591f\u638c\u63e1\u626d\u8f6c\u95ee\u9898\u7684\u57fa\u672c\u539f\u7406\u548c\u6c42\u89e3\u65b9\u6cd5，\u5e76\u80fd\u591f\u5e94\u7528\u8fd9\u4e9b\u77e5\u8bc6\u89e3\u51b3\u5b9e\u9645\u5de5\u7a0b\u95ee\u9898\u3002<\/p>\n

            8. \u4ee3\u7801\u9644\u5f55<\/h3>\n

            8.1 \u5706\u8f74\u626d\u8f6c\u7684\u5e94\u529b\u8ba1\u7b97<\/h4>\n

            def<\/span> calculate_torsion_stress<\/span>(<\/span>T,<\/span> d,<\/span> L,<\/span> G,<\/span> r_points)<\/span>:<\/span>
            \n """
            \n \u8ba1\u7b97\u5706\u8f74\u626d\u8f6c\u65f6\u7684\u526a\u5e94\u529b\u5206\u5e03<\/p>\n

            \u53c2\u6570:
            \n T: \u626d\u77e9 (N\u00b7m)
            \n d: \u5706\u8f74\u76f4\u5f84 (m)
            \n L: \u5706\u8f74\u957f\u5ea6 (m)
            \n G: \u526a\u5207\u5f39\u6027\u6a21\u91cf (Pa)
            \n r_points: \u5f84\u5411\u4f4d\u7f6e\u6570\u7ec4<\/p>\n

            \u8fd4\u56de:
            \n tau: \u526a\u5e94\u529b\u5206\u5e03
            \n phi: \u626d\u8f6c\u89d2 (rad)
            \n """<\/span>
            \n # \u8ba1\u7b97\u6781\u60ef\u6027\u77e9<\/span>
            \n Ip =<\/span> np.<\/span>pi *<\/span> d**<\/span>4<\/span> \/<\/span> 32<\/span><\/p>\n

            # \u8ba1\u7b97\u526a\u5e94\u529b<\/span>
            \n tau =<\/span> T *<\/span> r_points \/<\/span> Ip<\/p>\n

            # \u8ba1\u7b97\u626d\u8f6c\u89d2<\/span>
            \n phi =<\/span> T *<\/span> L \/<\/span> (<\/span>G *<\/span> Ip)<\/span><\/p>\n

            return<\/span> tau,<\/span> phi<\/p>\n

            8.2 \u975e\u5706\u622a\u9762\u626d\u8f6c\u7684\u6709\u9650\u5dee\u5206\u6cd5<\/h4>\n

            def<\/span> solve_non_circular_torsion<\/span>(<\/span>b,<\/span> h,<\/span> nx,<\/span> ny,<\/span> G,<\/span> theta)<\/span>:<\/span>
            \n """
            \n \u4f7f\u7528\u6709\u9650\u5dee\u5206\u6cd5\u6c42\u89e3\u975e\u5706\u622a\u9762\u626d\u8f6c\u95ee\u9898<\/p>\n

            \u53c2\u6570:
            \n b: \u622a\u9762\u5bbd\u5ea6 (m)
            \n h: \u622a\u9762\u9ad8\u5ea6 (m)
            \n nx: x\u65b9\u5411\u7f51\u683c\u6570
            \n ny: y\u65b9\u5411\u7f51\u683c\u6570
            \n G: \u526a\u5207\u5f39\u6027\u6a21\u91cf (Pa)
            \n theta: \u5355\u4f4d\u957f\u5ea6\u626d\u8f6c\u89d2 (rad\/m)<\/p>\n

            \u8fd4\u56de:
            \n phi: \u626d\u8f6c\u5e94\u529b\u51fd\u6570
            \n tau: \u526a\u5e94\u529b\u5206\u5e03
            \n J: \u626d\u8f6c\u5e38\u6570
            \n """<\/span>
            \n # \u8ba1\u7b97\u7f51\u683c\u5c3a\u5bf8<\/span>
            \n dx =<\/span> b \/<\/span> (<\/span>nx –<\/span> 1<\/span>)<\/span>
            \n dy =<\/span> h \/<\/span> (<\/span>ny –<\/span> 1<\/span>)<\/span><\/p>\n

            # \u521d\u59cb\u5316\u5e94\u529b\u51fd\u6570<\/span>
            \n phi =<\/span> np.<\/span>zeros(<\/span>(<\/span>nx,<\/span> ny)<\/span>)<\/span><\/p>\n

            # \u5b9a\u4e49\u8fb9\u754c\u6761\u4ef6<\/span>
            \n for<\/span> i in<\/span> range<\/span>(<\/span>nx)<\/span>:<\/span>
            \n for<\/span> j in<\/span> range<\/span>(<\/span>ny)<\/span>:<\/span>
            \n x =<\/span> –<\/span>b\/<\/span>2<\/span> +<\/span> i*<\/span>dx
            \n y =<\/span> –<\/span>h\/<\/span>2<\/span> +<\/span> j*<\/span>dy
            \n # \u77e9\u5f62\u622a\u9762\u8fb9\u754c<\/span>
            \n if<\/span> abs<\/span>(<\/span>x)<\/span> >=<\/span> b\/<\/span>2<\/span> or<\/span> abs<\/span>(<\/span>y)<\/span> >=<\/span> h\/<\/span>2<\/span>:<\/span>
            \n phi[<\/span>i,<\/span> j]<\/span> =<\/span> 0<\/span><\/p>\n

            # \u8fed\u4ee3\u6c42\u89e3\u6cca\u677e\u65b9\u7a0b<\/span>
            \n max_iter =<\/span> 10000<\/span>
            \n tol =<\/span> 1e-6<\/span><\/p>\n

            for<\/span> iter<\/span> in<\/span> range<\/span>(<\/span>max_iter)<\/span>:<\/span>
            \n phi_new =<\/span> phi.<\/span>copy(<\/span>)<\/span><\/p>\n

            for<\/span> i in<\/span> range<\/span>(<\/span>1<\/span>,<\/span> nx–<\/span>1<\/span>)<\/span>:<\/span>
            \n for<\/span> j in<\/span> range<\/span>(<\/span>1<\/span>,<\/span> ny–<\/span>1<\/span>)<\/span>:<\/span>
            \n x =<\/span> –<\/span>b\/<\/span>2<\/span> +<\/span> i*<\/span>dx
            \n y =<\/span> –<\/span>h\/<\/span>2<\/span> +<\/span> j*<\/span>dy
            \n # \u68c0\u67e5\u662f\u5426\u5728\u8fb9\u754c\u5185<\/span>
            \n if<\/span> abs<\/span>(<\/span>x)<\/span> <<\/span> b\/<\/span>2<\/span> and<\/span> abs<\/span>(<\/span>y)<\/span> <<\/span> h\/<\/span>2<\/span>:<\/span>
            \n # \u6709\u9650\u5dee\u5206\u683c\u5f0f<\/span>
            \n phi_new[<\/span>i,<\/span> j]<\/span> =<\/span> (<\/span>(<\/span>phi[<\/span>i+<\/span>1<\/span>,<\/span> j]<\/span> +<\/span> phi[<\/span>i–<\/span>1<\/span>,<\/span> j]<\/span>)<\/span>\/<\/span>dx**<\/span>2<\/span> +<\/span>
            \n (<\/span>phi[<\/span>i,<\/span> j+<\/span>1<\/span>]<\/span> +<\/span> phi[<\/span>i,<\/span> j–<\/span>1<\/span>]<\/span>)<\/span>\/<\/span>dy**<\/span>2<\/span> +<\/span>
            \n 2<\/span>*<\/span>G*<\/span>theta)<\/span> \/<\/span> (<\/span>2<\/span>\/<\/span>dx**<\/span>2<\/span> +<\/span> 2<\/span>\/<\/span>dy**<\/span>2<\/span>)<\/span><\/p>\n

            # \u68c0\u67e5\u6536\u655b\u6027<\/span>
            \n error =<\/span> np.<\/span>max<\/span>(<\/span>np.<\/span>abs<\/span>(<\/span>phi_new –<\/span> phi)<\/span>)<\/span>
            \n if<\/span> error <<\/span> tol:<\/span>
            \n break<\/span><\/p>\n

            phi =<\/span> phi_new<\/p>\n

            # \u8ba1\u7b97\u526a\u5e94\u529b<\/span>
            \n tau =<\/span> np.<\/span>zeros(<\/span>(<\/span>nx,<\/span> ny)<\/span>)<\/span>
            \n for<\/span> i in<\/span> range<\/span>(<\/span>1<\/span>,<\/span> nx–<\/span>1<\/span>)<\/span>:<\/span>
            \n for<\/span> j in<\/span> range<\/span>(<\/span>1<\/span>,<\/span> ny–<\/span>1<\/span>)<\/span>:<\/span>
            \n # \u4e2d\u5fc3\u5dee\u5206\u8ba1\u7b97\u526a\u5e94\u529b\u5206\u91cf<\/span>
            \n tau_zx =<\/span> (<\/span>phi[<\/span>i,<\/span> j+<\/span>1<\/span>]<\/span> –<\/span> phi[<\/span>i,<\/span> j–<\/span>1<\/span>]<\/span>)<\/span> \/<\/span> (<\/span>2<\/span>*<\/span>dy)<\/span>
            \n tau_zy =<\/span> –<\/span>(<\/span>phi[<\/span>i+<\/span>1<\/span>,<\/span> j]<\/span> –<\/span> phi[<\/span>i–<\/span>1<\/span>,<\/span> j]<\/span>)<\/span> \/<\/span> (<\/span>2<\/span>*<\/span>dx)<\/span>
            \n # \u5408\u6210\u526a\u5e94\u529b<\/span>
            \n tau[<\/span>i,<\/span> j]<\/span> =<\/span> np.<\/span>sqrt(<\/span>tau_zx**<\/span>2<\/span> +<\/span> tau_zy**<\/span>2<\/span>)<\/span><\/p>\n

            # \u8ba1\u7b97\u626d\u8f6c\u5e38\u6570<\/span>
            \n J =<\/span> 0<\/span>
            \n for<\/span> i in<\/span> range<\/span>(<\/span>1<\/span>,<\/span> nx–<\/span>1<\/span>)<\/span>:<\/span>
            \n for<\/span> j in<\/span> range<\/span>(<\/span>1<\/span>,<\/span> ny–<\/span>1<\/span>)<\/span>:<\/span>
            \n x =<\/span> –<\/span>b\/<\/span>2<\/span> +<\/span> i*<\/span>dx
            \n y =<\/span> –<\/span>h\/<\/span>2<\/span> +<\/span> j*<\/span>dy
            \n if<\/span> abs<\/span>(<\/span>x)<\/span> <<\/span> b\/<\/span>2<\/span> and<\/span> abs<\/span>(<\/span>y)<\/span> <<\/span> h\/<\/span>2<\/span>:<\/span>
            \n J +=<\/span> 2<\/span> *<\/span> phi[<\/span>i,<\/span> j]<\/span> *<\/span> dx *<\/span> dy<\/p>\n

            return<\/span> phi,<\/span> tau,<\/span> J<\/p>\n

            8.3 \u5706\u8f74\u626d\u8f6c\u7684\u5f3a\u5ea6\u6821\u6838<\/h4>\n

            def<\/span> check_torsion_strength<\/span>(<\/span>T,<\/span> d,<\/span> tau_y)<\/span>:<\/span>
            \n """
            \n \u6821\u6838\u5706\u8f74\u626d\u8f6c\u5f3a\u5ea6<\/p>\n

            \u53c2\u6570:
            \n T: \u626d\u77e9 (N\u00b7m)
            \n d: \u5706\u8f74\u76f4\u5f84 (m)
            \n tau_y: \u526a\u5207\u5c48\u670d\u5f3a\u5ea6 (Pa)<\/p>\n

            \u8fd4\u56de:
            \n tau_max: \u6700\u5927\u526a\u5e94\u529b (Pa)
            \n safety_factor: \u5b89\u5168\u7cfb\u6570
            \n is_safe: \u662f\u5426\u5b89\u5168
            \n """<\/span>
            \n # \u8ba1\u7b97\u6781\u60ef\u6027\u77e9<\/span>
            \n Ip =<\/span> np.<\/span>pi *<\/span> d**<\/span>4<\/span> \/<\/span> 32<\/span><\/p>\n

            # \u8ba1\u7b97\u6700\u5927\u526a\u5e94\u529b<\/span>
            \n r =<\/span> d \/<\/span> 2<\/span>
            \n tau_max =<\/span> T *<\/span> r \/<\/span> Ip<\/p>\n

            # \u8ba1\u7b97\u5b89\u5168\u7cfb\u6570<\/span>
            \n safety_factor =<\/span> tau_y \/<\/span> tau_max<\/p>\n

            # \u5224\u65ad\u662f\u5426\u5b89\u5168<\/span>
            \n is_safe =<\/span> safety_factor >=<\/span> 1.0<\/span><\/p>\n

            return<\/span> tau_max,<\/span> safety_factor,<\/span> is_safe<\/p>\n

            \"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"
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            \n\u5173\u952e\u8bcd
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