{"id":26307,"date":"2025-08-27T23:00:24","date_gmt":"2025-08-27T15:00:24","guid":{"rendered":"http:\/\/www.jdcui.com\/?p=26307"},"modified":"2026-02-15T10:41:37","modified_gmt":"2026-02-15T02:41:37","slug":"%e5%8a%a8%e5%8a%9b%e5%ad%a6-%e5%bc%ba%e8%bf%ab%e6%8c%af%e5%8a%a8%e5%a4%8d%e6%95%b0%e8%a7%a3%e6%b3%95%e4%b8%8e%e5%ae%9e%e6%95%b0%e8%a7%a3%e6%b3%95%ef%bc%88%e4%b8%89%e8%a7%92%e5%87%bd%e6%95%b0","status":"publish","type":"post","link":"http:\/\/www.jdcui.com\/?p=26307","title":{"rendered":"[\u632f\u52a8\u63a7\u5236] \u7b80\u8c10\u6fc0\u52b1\u4e0b\u5f3a\u8feb\u632f\u52a8\u590d\u6570\u89e3\u6cd5\u4e0e\u5b9e\u6570\u89e3\u6cd5\u7684\u7b49\u6548\u6027 [Equivalence of Complex and Real Solutions for Harmonically Excited Forced Vibration]"},"content":{"rendered":"<p><script><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span>\nMathJax = {\n  tex: {\n    inlineMath: [['$', '$'], ['\\\\(', '\\\\)']]\n  }\n};<span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span>\n<\/script><br \/>\n<script id=\"MathJax-script\" src=\"http:\/\/www.jdcui.com\/wp-content\/MathJax_3_1_2\/MathJax-master\/es5\/tex-mml-chtml.js\" async=\"\"><\/script><\/p>\n<p><span style=\"color: #ff00ff; background-color: #ccffcc;\"><strong>\u5b9e\u5e72\u3001\u5b9e\u8df5\u3001\u79ef\u7d2f\u3001\u601d\u8003\u3001\u521b\u65b0\uff01<\/strong><\/span><\/p>\n<hr \/>\n<p>\u5728\u5b66\u4e60\u52a8\u529b\u5b66\u4e0e\u7ed3\u6784\u632f\u52a8\u63a7\u5236\u76f8\u5173\u5185\u5bb9\u65f6\uff0c\u590d\u6570\u89e3\u6cd5\u5e38\u7528\u4e8e\u5206\u6790\u7b80\u8c10\u6fc0\u52b1\u4e0b\u7684\u5f3a\u8feb\u632f\u52a8\u95ee\u9898\u3002\u56de\u60f3\u8d77\u5728\u6821\u521d\u5b66\u6b64\u90e8\u5206\u65f6\uff0c\u5e38\u611f\u5230\u56f0\u60d1\u3002\u968f\u7740\u540e\u7eed\u6df1\u5165\u7406\u89e3\uff0c\u624d\u8ba4\u8bc6\u5230\u5176\u80cc\u540e\u7684\u5b9e\u8d28\uff1a\u7b80\u8c10\u6fc0\u52b1\u4e0b\u7684\u5f3a\u8feb\u632f\u52a8\u95ee\u9898\uff0c\u590d\u6570\u89e3\u6cd5\u4e0e\u5b9e\u6570\u89e3\u6cd5\uff08\u4e09\u89d2\u51fd\u6570\u6cd5\uff09\u5728\u6570\u5b66\u4e0a\u662f\u5b8c\u5168\u7b49\u6548\u7684\u3002\u590d\u6570\u89e3\u6cd5\u7684\u4f18\u52bf\u5728\u4e8e\u63a8\u5bfc\u8fc7\u7a0b\u66f4\u4e3a\u7b80\u6d01\uff0c\u5c24\u5176\u5728\u5904\u7406\u9ad8\u9636\u7cfb\u7edf\u6216\u591a\u9891\u6fc0\u52b1\u65f6\u663e\u5f97\u66f4\u4e3a\u9ad8\u6548\u3002<\/p>\n<p>\u4ee5\u4e0b\u4ee5\u5355\u81ea\u7531\u5ea6\u7b80\u8c10\u6fc0\u52b1\u53d7\u8feb\u632f\u52a8\u4e3a\u4f8b\uff0c\u5bf9\u8fd9\u4e24\u79cd\u65b9\u6cd5\u7684\u7b49\u6548\u6027\u4f5c\u4e00\u7b80\u8981\u603b\u7ed3\u3002<\/p>\n<p>\u4e09\u89d2\u51fd\u6570\u6fc0\u52b1\u4f5c\u7528\u4e0b\uff0c\u5355\u81ea\u7531\u5ea6\u52a8\u529b\u4f53\u7cfb\u52a8\u529b\u5e73\u8861\u65b9\u7a0b\u5982\u4e0b\uff1a<\/p>\n<p>\\[m\\ddot x + c\\dot x + kx = F\\cos \\left( {\\omega t} \\right)\\]\n<p>\u4e5f\u53ef\u8868\u793a\u4e3a\u6b63\u5f26\u51fd\u6570\u6fc0\u52b1\u7684\u5f62\u5f0f<\/p>\n<p>\\[m\\ddot x + c\\dot x + kx = F\\sin \\left( {\\omega t} \\right)\\]\n<p>\u4e0a\u8ff0\u5c31\u662f\u5b9e\u6570\u89e3\u6cd5\uff0c\u5982\u679c\u6362\u6210\u590d\u6570\u89e3\u6cd5\uff0c\u5982\u4e0b<\/p>\n<p>\u9996\u5148\uff0c\u5c06\u5916\u529b\u53d8\u4e3a\u590d\u6307\u6570\u51fd\u6570\u5f62\u5f0f\uff0c\u5373<\/p>\n<p>\\[f = F{e^{i\\omega t}}\\]\n<p>\u6b64\u65f6\u7cfb\u7edf\u8fd0\u52a8\u65b9\u7a0b\u53d8\u4e3a\uff1a<\/p>\n<p>\\[m\\ddot z + c\\dot z + k{\\rm{z}} = F{e^{i\\omega t}}\\]\n<p>\u5229\u7528\u5f3a\u5927\u7684\u6b27\u62c9\u516c\u5f0f<em>Euler&#8217;s formula\uff1a<\/em><\/p>\n<p>\\[{e^{i\\omega t}} =\u00a0 {\\cos \\omega t + i\\sin \\omega t} \\]\n<p>\u7cfb\u7edf\u8fd0\u52a8\u65b9\u7a0b\u8fdb\u4e00\u6b65\u53d8\u4e3a\uff1a<\/p>\n<p>\\[m\\ddot z + c\\dot z + k{\\rm{z}} = F{e^{i\\omega t}} = F\\left( {\\cos \\omega t + i\\sin \\omega t} \\right)\\]\n<p>\u4e0d\u59a8\u8bbe\u590d\u6570\u54cd\u5e94\u4e3a\uff1a<\/p>\n<p>\\[\\left\\{ \\begin{array}{l}<br \/>\nz = x + iy\\\\<br \/>\n\\dot z = \\dot x + i\\dot y\\\\<br \/>\n\\ddot z = \\ddot x + i\\ddot y<br \/>\n\\end{array} \\right.\\]\n<p>\u5176\u4e2d\uff0c\\( x\\)\u662f\u590d\u4f4d\u79fb\u54cd\u5e94\u7684\u5b9e\u90e8\uff0c \\( y\\)\u4e3a\u590d\u4f4d\u79fb\u54cd\u5e94\u7684\u865a\u90e8\uff0c\u4e0a\u8ff0\u53d8\u91cf\u5747\u4e3a\u65f6\u7a0b\u53d8\u91cf\u3002<\/p>\n<p>\u5c06\u4e0a\u8ff0\u516c\u5f0f\u4ee3\u5165\u590d\u632f\u52a8\u65b9\u7a0b\uff0c\u5e76\u6839\u636e\u7b49\u53f7\u4e24\u4fa7\u5b9e\u90e8\u548c\u865a\u90e8\u5206\u522b\u76f8\u7b49\u7684\u539f\u5219\uff0c\u53ef\u4ee5\u5f97\u5230\uff1a<\/p>\n<p>\\[\\left\\{ \\begin{array}{l}<br \/>\nm\\ddot x + c\\dot x + kx = F\\cos \\left( {\\omega t} \\right)\\\\<br \/>\nm\\ddot y + c\\dot y + ky = F\\sin \\left( {\\omega t} \\right)<br \/>\n\\end{array} \\right.\\]\n<p>\u7531\u6b64\u53ef\u89c1\uff0c\u590d\u6570\u5f62\u5f0f\u632f\u52a8\u65b9\u7a0b\u7684\u5b9e\u90e8\u4e0e\u865a\u90e8\u5206\u522b\u5bf9\u5e94\u4e8e\u5b9e\u6570\u57df\u4e0b\u7684\u4e09\u89d2\u51fd\u6570\u89e3\u3002\u8fd9\u8868\u660e\u590d\u6570\u89e3\u6cd5\u4e0e\u5b9e\u6570\u89e3\u6cd5\uff08\u4e09\u89d2\u51fd\u6570\u6cd5\uff09\u5728\u6570\u5b66\u4e0a\u5b8c\u5168\u7b49\u6548\u3002\u4e00\u65e6\u6c42\u5f97\u7cfb\u7edf\u7684\u590d\u632f\u52a8\u54cd\u5e94\uff0c\u53ea\u9700\u53d6\u8be5\u89e3\u7684\u5b9e\u90e8\u6216\u865a\u90e8\uff0c\u5373\u53ef\u5f97\u5230\u76f8\u5e94\u7684\u5b9e\u6570\u89e3\u3002\u82e5\u4ec5\u9700\u83b7\u53d6\u54cd\u5e94\u7684\u632f\u5e45\uff08\u5373\u5b9e\u90e8\u6216\u865a\u90e8\u7684\u6700\u5927\u503c\uff09\uff0c\u53ef\u76f4\u63a5\u5bf9\u590d\u632f\u52a8\u54cd\u5e94\u53d6\u6a21\u3002\u56e0\u6b64\uff0c\u590d\u632f\u52a8\u54cd\u5e94\u7684\u6a21\u957f\u5373\u4e3a\u5b9e\u6570\u89e3\u7684\u632f\u5e45\u3002<\/p>\n<p>\u6b64\u5916\uff0c\u7531\u4e8e\u590d\u6307\u6570\u51fd\u6570\u5728\u5fae\u79ef\u5206\u8fd0\u7b97\u4e0a\u5177\u6709\u663e\u8457\u4f18\u52bf\uff0c\u4f7f\u5f97\u590d\u6570\u89e3\u6cd5\u5728\u6574\u4e2a\u63a8\u5bfc\u8fc7\u7a0b\u4e2d\u66f4\u4e3a\u7b80\u6d01\u9ad8\u6548\u3002\u901a\u8fc7\u7b80\u6d01\u7684\u4ee3\u6570\u8fd0\u7b97\u5373\u53ef\u6c42\u5f97\u590d\u6570\u89e3\uff0c\u5176\u89e3\u672c\u8eab\u4fbf\u76f4\u63a5\u8574\u542b\u4e86\u632f\u5e45\u4e0e\u76f8\u4f4d\u7b49\u5173\u952e\u54cd\u5e94\u4fe1\u606f\u3002\u8fd9\u4e00\u65b9\u6cd5\u5728\u5904\u7406\u591a\u9891\u7b80\u8c10\u6fc0\u52b1\u53e0\u52a0\u7b49\u590d\u6742\u95ee\u9898\u65f6\uff0c\u4f18\u52bf\u5c24\u4e3a\u7a81\u51fa\u3002\u56e0\u6b64\uff0c\u6df1\u5165\u7406\u89e3\u590d\u6570\u89e3\u6cd5\u662f\u638c\u63e1\u632f\u52a8\u63a7\u5236\u7406\u8bba\u7684\u91cd\u8981\u57fa\u7840\u3002<\/p>\n<ul style=\"list-style-type: square;\">\n<li><strong>\u76f8\u5173\u535a\u6587 (<span style=\"color: #000080;\"> Related Topics<\/span>)<\/strong><\/li>\n<\/ul>\n<p><strong>[00] <a title=\"[\u6570\u5b66][\u5730\u9707\u52a8][\u8f6f\u4ef6] FOUR_TRAN: Fourier Analysis Tool [\u5085\u91cc\u53f6\u5206\u6790\u5de5\u5177]\" href=\"http:\/\/www.jdcui.com\/?p=13874\" target=\"_blank\" rel=\"noopener\">[\u6570\u5b66][\u5730\u9707\u52a8][\u8f6f\u4ef6] FOUR_TRAN: Fourier Analysis Tool [\u5085\u91cc\u53f6\u5206\u6790\u5de5\u5177]<\/a><\/strong><\/p>\n<p><strong>[01] <a href=\"http:\/\/www.jdcui.com\/?p=13909\" target=\"_blank\" rel=\"noopener noreferrer\">[\u6570\u5b66][\u8f6f\u4ef6] FOUR_TRAN Example 1: Filtering [FOUR_TRAN\u5085\u91cc\u53f6\u5206\u6790\u5de5\u5177\u4f7f\u7528\u6848\u4f8b1: \u6ee4\u6ce2]<\/a><\/strong><\/p>\n<p><strong>[02] <a href=\"http:\/\/www.jdcui.com\/?p=13911\" target=\"_blank\" rel=\"noopener noreferrer\">[\u6570\u5b66][\u8f6f\u4ef6] FOUR_TRAN Example 2: Square Wave Signal Decomposition [FOUR_TRAN\u5085\u91cc\u53f6\u5206\u6790\u5de5\u5177\u4f7f\u7528\u6848\u4f8b2: \u65b9\u6ce2\u4fe1\u53f7\u5206\u89e3]<\/a><\/strong><\/p>\n<p><strong>[03] <a href=\"http:\/\/www.jdcui.com\/?p=13914\" target=\"_blank\" rel=\"noopener noreferrer\">[\u6570\u5b66][\u8f6f\u4ef6] FOUR_TRAN Example 3: Earthquake Ground Acceleration Frequency Spectrum Analysis [FOUR_TRAN\u5085\u91cc\u53f6\u5206\u6790\u5de5\u5177\u4f7f\u7528\u6848\u4f8b3: \u5730\u9707\u6ce2\u9891\u8c31\u5206\u6790]<\/a><\/strong><\/p>\n<p><strong>[04] <a title=\"[\u6570\u5b66][\u8f6f\u4ef6] FOUR_TRAN Example 4: Stock Periodicity and Volatility Analysis [FOUR_TRAN\u5085\u91cc\u53f6\u5206\u6790\u5de5\u5177 \u6848\u4f8b4: \u80a1\u7968\u5468\u671f\u6027\u548c\u6ce2\u52a8\u6027\u5206\u6790]\" href=\"http:\/\/www.jdcui.com\/?p=19570\" target=\"_blank\" rel=\"noopener\">[\u6570\u5b66][\u8f6f\u4ef6] FOUR_TRAN Example 4: Stock Periodicity and Volatility Analysis [FOUR_TRAN\u5085\u91cc\u53f6\u5206\u6790\u5de5\u5177 \u6848\u4f8b4: \u80a1\u7968\u5468\u671f\u6027\u548c\u6ce2\u52a8\u6027\u5206\u6790]<\/a><\/strong><\/p>\n<p><strong>[05] <a title=\"[\u8f6f\u4ef6] FOUR_TRAN \u6848\u4f8b5: \u98ce\u632f\u54cd\u5e94\u9891\u8c31\u5206\u6790 ( FOUR_TRAN Ex5: Wind induced vibration response spectrum analysis)\" href=\"http:\/\/www.jdcui.com\/?p=22674\" target=\"_blank\" rel=\"noopener\">[\u8f6f\u4ef6] FOUR_TRAN \u6848\u4f8b5: \u98ce\u632f\u54cd\u5e94\u9891\u8c31\u5206\u6790 ( FOUR_TRAN Ex5: Wind induced vibration response spectrum analysis)<\/a><\/strong><\/p>\n<p><strong>[06] <a title=\"[\u8f6f\u4ef6][\u632f\u52a8\u63a7\u5236] FOUR_TRAN \u6848\u4f8b6: \u94a2\u697c\u68af\u5b9e\u6d4b\u632f\u52a8\u6570\u636e\u9891\u8c31\u5206\u6790 (FOUR_TRAN Ex6: Frequency spectrum of measured vibration data of Steel stairs)\" href=\"http:\/\/www.jdcui.com\/?p=23673\" target=\"_blank\" rel=\"noopener\">[\u8f6f\u4ef6][\u632f\u52a8\u63a7\u5236] FOUR_TRAN \u6848\u4f8b6: \u94a2\u697c\u68af\u5b9e\u6d4b\u632f\u52a8\u6570\u636e\u9891\u8c31\u5206\u6790 (FOUR_TRAN Ex6: Frequency spectrum of measured vibration data of Steel stairs)<\/a><\/strong><\/p>\n<p><strong>[07] <a title=\"[\u8f6f\u4ef6][\u632f\u52a8\u63a7\u5236] FOUR_TRAN \u6848\u4f8b7: \u94a2\u697c\u68af\u5b9e\u6d4b\u632f\u52a8\u6570\u636e(\u4e0d\u540c\u9891\u7387\u6210\u5206\u5bf9\u52a0\u901f\u5ea6\u5cf0\u503c\u7684\u8d21\u732e) (FOUR_TRAN Ex7: Frequency spectrum of measured vibration data of Steel stairs)\" href=\"http:\/\/www.jdcui.com\/?p=23716\" target=\"_blank\" rel=\"noopener\">[\u8f6f\u4ef6][\u632f\u52a8\u63a7\u5236] FOUR_TRAN \u6848\u4f8b7: \u94a2\u697c\u68af\u5b9e\u6d4b\u632f\u52a8\u6570\u636e(\u4e0d\u540c\u9891\u7387\u6210\u5206\u5bf9\u52a0\u901f\u5ea6\u5cf0\u503c\u7684\u8d21\u732e) (FOUR_TRAN Ex7: Frequency spectrum of measured vibration data of Steel stairs)<\/a><\/strong><\/p>\n<p><strong>[08] <a title=\"[\u632f\u52a8\u63a7\u5236] \u62cd\u632f\u73b0\u8c61 (Beats Phenomenon)\" href=\"http:\/\/www.jdcui.com\/?p=23747\" target=\"_blank\" rel=\"noopener\">[\u632f\u52a8\u63a7\u5236] \u62cd\u632f\u73b0\u8c61 (Beats Phenomenon)<\/a><\/strong><\/p>\n<p><strong>[09] <a title=\"[\u632f\u52a8\u63a7\u5236] \u521a\u5ea6\u3001\u8d28\u91cf\u3001\u963b\u5c3c\u5bf9\u632f\u52a8\u8212\u9002\u5ea6\u7684\u5f71\u54cd\u603b\u7ed3 (Summary of the influence of stiffness, mass, and damping on vibration comfort)\" href=\"http:\/\/www.jdcui.com\/?p=24577\" target=\"_blank\" rel=\"noopener\">[\u632f\u52a8\u63a7\u5236] \u521a\u5ea6\u3001\u8d28\u91cf\u3001\u963b\u5c3c\u5bf9\u632f\u52a8\u8212\u9002\u5ea6\u7684\u5f71\u54cd\u603b\u7ed3 (Summary of the influence of stiffness, mass, and damping on vibration comfort)<\/a><\/strong><\/p>\n<p><strong>[10] <a title=\"[\u632f\u52a8\u63a7\u5236][\u7f16\u7a0b] \u8f66\u8f86-\u8f68\u9053\u8026\u5408\u52a8\u529b\u5b66\u6a21\u578b [Vehicle\u2013Track Coupled Dynamics Model]\" href=\"http:\/\/www.jdcui.com\/?p=24312\" target=\"_blank\" rel=\"noopener\">[\u632f\u52a8\u63a7\u5236][\u7f16\u7a0b] \u8f66\u8f86-\u8f68\u9053\u8026\u5408\u52a8\u529b\u5b66\u6a21\u578b [Vehicle\u2013Track Coupled Dynamics Model]<\/a><\/strong><\/p>\n<p><strong>[11] <a title=\"[\u632f\u52a8\u63a7\u5236] \u5e38\u89c1\u7684\u632f\u52a8\u91cf\u5316\u6307\u6807 [Vibration Quantity Indicators]\" href=\"http:\/\/www.jdcui.com\/?p=24089\" target=\"_blank\" rel=\"noopener\">[\u632f\u52a8\u63a7\u5236] \u5e38\u89c1\u7684\u632f\u52a8\u91cf\u5316\u6307\u6807 [Vibration Quantity Indicators]<\/a><\/strong><\/p>\n<p><strong>[12] <a title=\"[\u632f\u52a8\u63a7\u5236][\u7f16\u7a0b] OSA: Octave Spectra Analysis Program [\u500d\u9891\u7a0b\u8c31\u5206\u6790\u8f6f\u4ef6]\" href=\"http:\/\/www.jdcui.com\/?p=20164\" target=\"_blank\" rel=\"noopener\">[\u632f\u52a8\u63a7\u5236][\u7f16\u7a0b] OSA: Octave Spectra Analysis Program [\u500d\u9891\u7a0b\u8c31\u5206\u6790\u8f6f\u4ef6]<\/a><\/strong><\/p>\n<p><strong>[13] <a title=\"[\u632f\u52a8\u63a7\u5236] \u5e38\u89c1\u8d28\u91cf\u963b\u5c3c\u5668\u5206\u7c7b [Passive, semi-active, active and hybrid mass dampers]\" href=\"http:\/\/www.jdcui.com\/?p=22357\" target=\"_blank\" rel=\"noopener\">[\u632f\u52a8\u63a7\u5236] \u5e38\u89c1\u8d28\u91cf\u963b\u5c3c\u5668\u5206\u7c7b [Passive, semi-active, active and hybrid mass dampers]<\/a><\/strong><\/p>\n<p><strong>[14] <a title=\"[\u52a8\u529b\u5b66][\u632f\u52a8\u63a7\u5236][\u7f16\u7a0b] SDOF_FRE: Dynamic Response Analysis of SDOF System using Frequency Domain Analysis Method [\u5355\u81ea\u7531\u5ea6\u4f53\u7cfb\u52a8\u529b\u54cd\u5e94\u9891\u57df\u5206\u6790\u7a0b\u5e8f]\" href=\"http:\/\/www.jdcui.com\/?p=18955\" target=\"_blank\" rel=\"noopener\">[\u52a8\u529b\u5b66][\u632f\u52a8\u63a7\u5236][\u7f16\u7a0b] SDOF_FRE: Dynamic Response Analysis of SDOF System using Frequency Domain Analysis Method [\u5355\u81ea\u7531\u5ea6\u4f53\u7cfb\u52a8\u529b\u54cd\u5e94\u9891\u57df\u5206\u6790\u7a0b\u5e8f]<\/a><\/strong><\/p>\n<p><strong>[15] <a title=\"[\u8d44\u6599][\u632f\u52a8\u63a7\u5236] \u57fa\u4e8e\u73af\u5883\u6fc0\u52b1\u7684\u6a21\u6001\u53c2\u6570\u8bc6\u522b\u6cd5 (Modal parameter identification method based on environmental excitation)\" href=\"http:\/\/www.jdcui.com\/?p=18955\" target=\"_blank\" rel=\"noopener\">[\u8d44\u6599][\u632f\u52a8\u63a7\u5236] \u57fa\u4e8e\u73af\u5883\u6fc0\u52b1\u7684\u6a21\u6001\u53c2\u6570\u8bc6\u522b\u6cd5 (Modal parameter identification method based on environmental excitation)<\/a><\/strong><\/p>\n<p><strong>[16] <a title=\"[\u52a8\u529b\u5b66][\u632f\u52a8\u63a7\u5236][\u7f16\u7a0b] SDOF_FRE \u6848\u4f8b1 \u2014\u2014 \u52a8\u529b\u65f6\u7a0b\u54cd\u5e94\u5206\u6790 [SDOF_FRE Example 1: Dynamic Force Time History Analysis]\" href=\"http:\/\/www.jdcui.com\/?p=20087\" target=\"_blank\" rel=\"noopener\">[\u52a8\u529b\u5b66][\u632f\u52a8\u63a7\u5236][\u7f16\u7a0b] SDOF_FRE \u6848\u4f8b1 \u2014\u2014 \u52a8\u529b\u65f6\u7a0b\u54cd\u5e94\u5206\u6790 [SDOF_FRE Example 1: Dynamic Force Time History Analysis]<\/a><\/strong><\/p>\n<p><strong>[17] <a title=\"[\u52a8\u529b\u5b66][\u632f\u52a8\u63a7\u5236][\u8f6f\u4ef6] SDOF_FRE \u6848\u4f8b 2 \u2014\u2014 \u5730\u9707\u65f6\u7a0b\u54cd\u5e94\u5206\u6790 [SDOF_FRE Example 2: Earthquake Time History Analysis]\" href=\"http:\/\/www.jdcui.com\/?p=20090\" target=\"_blank\" rel=\"noopener\">[\u52a8\u529b\u5b66][\u632f\u52a8\u63a7\u5236][\u8f6f\u4ef6] SDOF_FRE \u6848\u4f8b 2 \u2014\u2014 \u5730\u9707\u65f6\u7a0b\u54cd\u5e94\u5206\u6790 [SDOF_FRE Example 2: Earthquake Time History Analysis]<\/a><\/strong><\/p>\n<p><strong>[18] <a title=\"[\u632f\u63a7][\u8f6f\u4ef6] OSA\u500d\u9891\u7a0b\u5206\u6790\u8f6f\u4ef6 \u6848\u4f8b1\uff1a\u632f\u52a8\u52a0\u901f\u5ea6 1\/3\u500d\u9891\u7a0b\u5206\u6790 [OSA octave band analysis software case 1: Vibration acceleration 1\/3 octave band analysis]\" href=\"http:\/\/www.jdcui.com\/?p=24156\" target=\"_blank\" rel=\"noopener\">[\u632f\u63a7][\u8f6f\u4ef6] OSA\u500d\u9891\u7a0b\u5206\u6790\u8f6f\u4ef6 \u6848\u4f8b1\uff1a\u632f\u52a8\u52a0\u901f\u5ea6 1\/3\u500d\u9891\u7a0b\u5206\u6790 [OSA octave band analysis software case 1: Vibration acceleration 1\/3 octave band analysis]<\/a><\/strong><\/p>\n<p><strong>[19] <a title=\"[\u632f\u63a7][\u8f6f\u4ef6] OSA\u500d\u9891\u7a0b\u5206\u6790\u8f6f\u4ef6 \u6848\u4f8b2\uff1a\u591a\u7ec4\u6570\u636e\u5bfc\u5165 [OSA octave band analysis software case 2: Support importing multiple sets of data]\" href=\"http:\/\/www.jdcui.com\/?p=24159\" target=\"_blank\" rel=\"noopener\">[\u632f\u63a7][\u8f6f\u4ef6] OSA\u500d\u9891\u7a0b\u5206\u6790\u8f6f\u4ef6 \u6848\u4f8b2\uff1a\u591a\u7ec4\u6570\u636e\u5bfc\u5165 [OSA octave band analysis software case 2: Support importing multiple sets of data]<\/a><\/strong><\/p>\n<p><strong>[20] <a title=\"[\u632f\u52a8\u63a7\u5236] \u4eba\u884c\u6865\u3001\u8fde\u5eca\u3001\u5929\u6865\u4eba\u81f4\u632f\u52a8\u8212\u9002\u5ea6\u8bc4\u4f30\u6b65\u9aa4 [Assessment steps for human induced vibration comfort of footbridges and corridors]\" href=\"http:\/\/www.jdcui.com\/?p=25414\" target=\"_blank\" rel=\"noopener\">[\u632f\u52a8\u63a7\u5236] \u4eba\u884c\u6865\u3001\u8fde\u5eca\u3001\u5929\u6865\u4eba\u81f4\u632f\u52a8\u8212\u9002\u5ea6\u8bc4\u4f30\u6b65\u9aa4 [Assessment steps for human induced vibration comfort of footbridges and corridors]<\/a><\/strong><\/p>\n<p><strong>[21] <a title=\"[\u632f\u52a8\u63a7\u5236][\u8f6f\u4ef6] VCAP: Vibration Comfort Assessment Program [\u632f\u52a8\u8212\u9002\u5ea6\u8bc4\u4f30\u8f6f\u4ef6]\" href=\"http:\/\/www.jdcui.com\/?p=24093\" target=\"_blank\" rel=\"noopener\">[\u632f\u52a8\u63a7\u5236][\u8f6f\u4ef6] VCAP: Vibration Comfort Assessment Program [\u632f\u52a8\u8212\u9002\u5ea6\u8bc4\u4f30\u8f6f\u4ef6]<\/a><\/strong><\/p>\n<p><strong>[22] <a title=\"[\u632f\u52a8\u63a7\u5236] \u9694\u632f\u5efa\u7b51\u7ad6\u5411\u632f\u52a8\u9891\u7387\u4e0e\u538b\u7f29\u91cf\u7684\u5173\u7cfb [The relationship between vertical vibration frequency and compression of isolated buildings]\" href=\"http:\/\/www.jdcui.com\/?p=25465\" target=\"_blank\" rel=\"noopener\">[\u632f\u52a8\u63a7\u5236] \u9694\u632f\u5efa\u7b51\u7ad6\u5411\u632f\u52a8\u9891\u7387\u4e0e\u538b\u7f29\u91cf\u7684\u5173\u7cfb [The relationship between vertical vibration frequency and compression of isolated buildings]<\/a><\/strong><\/p>\n<p><strong>[23] <a title=\"[\u632f\u52a8\u63a7\u5236] GenVibrationTool: A Toolbox for Vibration Comfort Analysis of Floor Structures Developed Based on MidasGen API [\u57fa\u4e8emidasGen API\u5f00\u53d1\u7684\u697c\u76d6\u7ed3\u6784\u632f\u52a8\u8212\u9002\u5ea6\u5206\u6790\u5de5\u5177\u7bb1]\" href=\"http:\/\/www.jdcui.com\/?p=24973\" target=\"_blank\" rel=\"noopener\">[\u632f\u52a8\u63a7\u5236] GenVibrationTool: A Toolbox for Vibration Comfort Analysis of Floor Structures Developed Based on MidasGen API [\u57fa\u4e8emidasGen API\u5f00\u53d1\u7684\u697c\u76d6\u7ed3\u6784\u632f\u52a8\u8212\u9002\u5ea6\u5206\u6790\u5de5\u5177\u7bb1]<\/a><\/strong><\/p>\n<p><strong>[24] <a title=\"[\u632f\u52a8\u63a7\u5236] \u652f\u5ea7\u7ad6\u5411\u9694\u632f\u4e00\u4e9b\u6ce8\u610f\u70b9\u5907\u5fd8 [Some precautions for vertical vibration isolation of supports]\" href=\"http:\/\/www.jdcui.com\/?p=25732\" target=\"_blank\" rel=\"noopener\">[\u632f\u52a8\u63a7\u5236] \u652f\u5ea7\u7ad6\u5411\u9694\u632f\u4e00\u4e9b\u6ce8\u610f\u70b9\u5907\u5fd8 [Some precautions for vertical vibration isolation of supports]<\/a><\/strong><\/p>\n<p><strong>[25] <a title=\"[\u52a8\u529b\u5b66] \u5f3a\u8feb\u632f\u52a8\u590d\u6570\u89e3\u6cd5\u4e0e\u5b9e\u6570\u89e3\u6cd5\uff08\u4e09\u89d2\u51fd\u6570\u6cd5\uff09\u7684\u7b49\u6548\u6027 [Equivalence of Complex and Real (Trigonometric) Methods for Forced Vibrations]\" href=\"http:\/\/www.jdcui.com\/?p=26307\" target=\"_blank\" rel=\"noopener\">[\u52a8\u529b\u5b66] \u5f3a\u8feb\u632f\u52a8\u590d\u6570\u89e3\u6cd5\u4e0e\u5b9e\u6570\u89e3\u6cd5\uff08\u4e09\u89d2\u51fd\u6570\u6cd5\uff09\u7684\u7b49\u6548\u6027 [Equivalence of Complex and Real (Trigonometric) Methods for Forced Vibrations]<\/a><\/strong><\/p>\n<p><strong>[26] <a title=\"[\u7a0b\u5e8f] PDDVA: Parameter design of dynamic vibration absorber [PDDVA: \u52a8\u529b\u5438\u632f\u5668\u53c2\u6570\u8bbe\u8ba1\u8f6f\u4ef6]\" href=\"http:\/\/www.jdcui.com\/?p=20799\" target=\"_blank\" rel=\"noopener\">[\u7a0b\u5e8f] PDDVA: Parameter design of dynamic vibration absorber [PDDVA: \u52a8\u529b\u5438\u632f\u5668\u53c2\u6570\u8bbe\u8ba1\u8f6f\u4ef6]<\/a><\/strong><\/p>\n<section>\n<section>\n<section>\n<section>\n<section><\/section>\n<\/section>\n<\/section>\n<\/section>\n<section>\n<section>\n<hr \/>\n<p style=\"text-align: center;\"><span style=\"font-size: 14pt; background-color: #999999; color: #000000;\"><strong>\u5173\u4e8e\u6211\u4eec<\/strong><\/span><\/p>\n<p style=\"text-align: center;\">\u8d85\u9650\u590d\u6742\u9ad8\u5c42\u7ed3\u6784\u8bbe\u8ba1 | \u00a0\u7f8e\u6807\u6b27\u6807\u7ed3\u6784\u8bbe\u8ba1| \u8f6f\u4ef6\u5b9a\u5236\u5f00\u53d1| \u73af\u8bc4\u51cf\u632f\u63a7\u5236 |\u4eba\u884c\u53ca\u98ce\u81f4\u632f\u52a8\u63a7\u5236 | \u51cf\u9694\u9707\u8bbe\u8ba1 | \u65bd\u5de5\u8fc7\u7a0b\u6a21\u62df | \u5c0f\u54c1\u94a2\u7ed3\u6784 | \u6709\u9650\u5143\u4eff\u771f\u5206\u6790 | BIM\u4e0eGH\u53c2\u6570\u5316 | \u5927\u9707\u5f39\u5851\u6027\u5206\u6790<\/p>\n<p style=\"text-align: center;\"><a href=\"http:\/\/www.jdcui.com\/wp-content\/uploads\/2017\/01\/QRCODE.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3636 alignnone\" src=\"http:\/\/www.jdcui.com\/wp-content\/uploads\/2017\/01\/QRCODE.jpg\" alt=\"WeChat_QRCode\" width=\"250\" height=\"255\" \/><\/a><\/p>\n<p style=\"text-align: center;\">https:\/\/www.jdcui.com<\/p>\n<p style=\"text-align: center;\">\u5408\u4f5c\u53ca\u6280\u672f\u54a8\u8be2<\/p>\n<p style=\"text-align: center;\">COOPERATION &amp; CONTACT<\/p>\n<p style=\"text-align: center;\">E-mail\uff1ajidong_cui@163.com<\/p>\n<p style=\"text-align: center;\">WeChat &amp; Tel: 13450468449<\/p>\n<\/section>\n<\/section>\n<\/section>\n","protected":false},"excerpt":{"rendered":"<p>\u5b9e\u5e72\u3001\u5b9e\u8df5\u3001\u79ef\u7d2f\u3001\u601d\u8003\u3001\u521b\u65b0\uff01 \u5728\u5b66\u4e60\u52a8\u529b\u5b66\u4e0e\u7ed3\u6784\u632f\u52a8\u63a7\u5236\u76f8\u5173\u5185\u5bb9\u65f6\uff0c\u590d\u6570\u89e3\u6cd5\u5e38\u7528\u4e8e\u5206\u6790\u7b80\u8c10\u6fc0\u52b1\u4e0b\u7684\u5f3a\u8feb\u632f\u52a8\u95ee\u9898\u3002\u56de\u60f3\u8d77\u5728\u6821\u521d\u5b66\u6b64\u90e8\u5206\u65f6\uff0c\u5e38\u611f\u5230\u56f0\u60d1\u3002\u968f\u7740\u540e\u7eed\u6df1\u5165\u7406\u89e3\uff0c\u624d\u8ba4\u8bc6\u5230\u5176\u80cc\u540e\u7684\u5b9e\u8d28\uff1a\u7b80\u8c10\u6fc0\u52b1\u4e0b\u7684\u5f3a\u8feb\u632f\u52a8\u95ee\u9898\uff0c\u590d\u6570\u89e3\u6cd5\u4e0e\u5b9e\u6570\u89e3\u6cd5\uff08\u4e09\u89d2\u51fd\u6570\u6cd5\uff09\u5728\u6570\u5b66\u4e0a\u662f\u5b8c\u5168\u7b49\u6548\u7684\u3002\u590d\u6570\u89e3\u6cd5\u7684\u4f18\u52bf\u5728\u4e8e\u63a8\u5bfc\u8fc7\u7a0b\u66f4\u4e3a\u7b80\u6d01\uff0c\u5c24\u5176\u5728\u5904\u7406\u9ad8\u9636\u7cfb\u7edf\u6216\u591a\u9891\u6fc0\u52b1\u65f6\u663e\u5f97\u66f4\u4e3a\u9ad8\u6548\u3002 \u4ee5\u4e0b\u4ee5\u5355\u81ea\u7531\u5ea6\u7b80\u8c10\u6fc0\u52b1\u53d7\u8feb\u632f\u52a8\u4e3a\u4f8b\uff0c\u5bf9\u8fd9\u4e24\u79cd\u65b9\u6cd5\u7684\u7b49\u6548\u6027\u4f5c\u4e00\u7b80\u8981\u603b\u7ed3\u3002 \u4e09\u89d2\u51fd\u6570\u6fc0\u52b1\u4f5c\u7528\u4e0b\uff0c\u5355\u81ea\u7531\u5ea6\u52a8\u529b\u4f53\u7cfb\u52a8\u529b\u5e73\u8861\u65b9\u7a0b\u5982\u4e0b\uff1a \\[m\\ddot x + c\\dot x + kx = F\\cos \\left( {\\omega t} \\right)\\] \u4e5f\u53ef\u8868\u793a\u4e3a\u6b63\u5f26\u51fd\u6570\u6fc0\u52b1\u7684\u5f62\u5f0f \\[m\\ddot x + c\\dot x + kx = F\\sin \\left( {\\omega t} 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\u5728\u5b66\u4e60\u52a8\u529b\u5b66\u4e0e\u7ed3\u6784\u632f\u52a8\u63a7\u5236\u76f8\u5173\u5185\u5bb9\u65f6\uff0c\u590d\u6570\u89e3\u6cd5\u5e38\u7528\u4e8e\u5206\u6790\u7b80\u8c10\u6fc0\u52b1\u4e0b\u7684\u5f3a\u8feb\u632f\u52a8\u95ee\u9898\u3002\u56de\u60f3\u8d77\u5728\u6821\u521d\u5b66\u6b64\u90e8\u5206\u65f6\uff0c\u5e38\u611f\u5230\u56f0\u60d1\u3002\u968f\u7740\u540e\u7eed\u6df1\u5165\u7406\u89e3\uff0c\u624d\u8ba4\u8bc6\u5230\u5176\u80cc\u540e\u7684\u5b9e\u8d28\uff1a\u7b80\u8c10\u6fc0\u52b1\u4e0b\u7684\u5f3a\u8feb\u632f\u52a8\u95ee\u9898\uff0c\u590d\u6570\u89e3\u6cd5\u4e0e\u5b9e\u6570\u89e3\u6cd5\uff08\u4e09\u89d2\u51fd\u6570\u6cd5\uff09\u5728\u6570\u5b66\u4e0a\u662f\u5b8c\u5168\u7b49\u6548\u7684\u3002\u590d\u6570\u89e3\u6cd5\u7684\u4f18\u52bf\u5728\u4e8e\u63a8\u5bfc\u8fc7\u7a0b\u66f4\u4e3a\u7b80\u6d01\uff0c\u5c24\u5176\u5728\u5904\u7406\u9ad8\u9636\u7cfb\u7edf\u6216\u591a\u9891\u6fc0\u52b1\u65f6\u663e\u5f97\u66f4\u4e3a\u9ad8\u6548\u3002 \u4ee5\u4e0b\u4ee5\u5355\u81ea\u7531\u5ea6\u7b80\u8c10\u6fc0\u52b1\u53d7\u8feb\u632f\u52a8\u4e3a\u4f8b\uff0c\u5bf9\u8fd9\u4e24\u79cd\u65b9\u6cd5\u7684\u7b49\u6548\u6027\u4f5c\u4e00\u7b80\u8981\u603b\u7ed3\u3002 \u4e09\u89d2\u51fd\u6570\u6fc0\u52b1\u4f5c\u7528\u4e0b\uff0c\u5355\u81ea\u7531\u5ea6\u52a8\u529b\u4f53\u7cfb\u52a8\u529b\u5e73\u8861\u65b9\u7a0b\u5982\u4e0b\uff1a \\[m\\ddot x + c\\dot x + kx = F\\cos \\left( {\\omega t} \\right)\\] \u4e5f\u53ef\u8868\u793a\u4e3a\u6b63\u5f26\u51fd\u6570\u6fc0\u52b1\u7684\u5f62\u5f0f \\[m\\ddot x + c\\dot x + kx = F\\sin \\left( {\\omega t} \\right)\\] \u4e0a\u8ff0\u5c31\u662f\u5b9e\u6570\u89e3\u6cd5\uff0c\u5982\u679c\u6362\u6210\u590d\u6570\u89e3\u6cd5\uff0c\u5982\u4e0b \u9996\u5148\uff0c\u5c06\u5916\u529b\u53d8\u4e3a\u590d\u6307\u6570\u51fd\u6570\u5f62\u5f0f\uff0c\u5373 \\[f = F{e^{i\\omega t}}\\] \u6b64\u65f6\u7cfb\u7edf\u8fd0\u52a8\u65b9\u7a0b\u53d8\u4e3a\uff1a \\[m\\ddot z + c\\dot z + k{\\rm{z}} = F{e^{i\\omega t}}\\] \u5229\u7528\u5f3a\u5927\u7684\u6b27\u62c9\u516c\u5f0fEuler's formula\uff1a \\[{e^{i\\omega t}} = {\\cos \\omega","canonical_url":"http:\/\/www.jdcui.com\/?p=26307","robots":"max-image-preview:large","keywords":"complex exponential method,dynamics,forced vibrations,harmonically excited,trigonometric function,\u52a8\u529b\u5b66,\u590d\u6307\u6570\u51fd\u6570,\u590d\u6570\u89e3\u6cd5,\u5b9e\u6570\u89e3\u6cd5,\u5f3a\u8feb\u632f\u52a8,\u632f\u52a8\u63a7\u5236,\u7a33\u6001\u54cd\u5e94,\u7b80\u8c10\u6fc0\u52b1","webmasterTools":{"miscellaneous":""},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"http:\/\/www.jdcui.com\/?p=26307#article","name":"[\u632f\u52a8\u63a7\u5236] \u7b80\u8c10\u6fc0\u52b1\u4e0b\u5f3a\u8feb\u632f\u52a8\u590d\u6570\u89e3\u6cd5\u4e0e\u5b9e\u6570\u89e3\u6cd5\u7684\u7b49\u6548\u6027 [Equivalence of Complex and Real Solutions for Harmonically Excited Forced Vibration] | \u5d14\u6d4e\u4e1c\u7684\u535a\u5ba2 - www.jdcui.com","headline":"[\u632f\u52a8\u63a7\u5236] \u7b80\u8c10\u6fc0\u52b1\u4e0b\u5f3a\u8feb\u632f\u52a8\u590d\u6570\u89e3\u6cd5\u4e0e\u5b9e\u6570\u89e3\u6cd5\u7684\u7b49\u6548\u6027 [Equivalence of Complex and Real Solutions for Harmonically Excited Forced Vibration]","author":{"@id":"http:\/\/www.jdcui.com\/?author=1#author"},"publisher":{"@id":"http:\/\/www.jdcui.com\/#organization"},"image":{"@type":"ImageObject","url":"http:\/\/www.jdcui.com\/wp-content\/uploads\/2025\/08\/Complex-and-Real-Solutions-min.png","width":676,"height":338},"datePublished":"2025-08-27T23:00:24+08:00","dateModified":"2026-02-15T10:41:37+08:00","inLanguage":"en-US","mainEntityOfPage":{"@id":"http:\/\/www.jdcui.com\/?p=26307#webpage"},"isPartOf":{"@id":"http:\/\/www.jdcui.com\/?p=26307#webpage"},"articleSection":"Dynamics [\u52a8\u529b\u5b66], Structural Engineering [\u7ed3\u6784\u5de5\u7a0b], Vibration Control [\u632f\u52a8\u63a7\u5236], Complex Exponential Method, Dynamics, Forced Vibrations, 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\u7b80\u8c10\u6fc0\u52b1\u4e0b\u5f3a\u8feb\u632f\u52a8\u590d\u6570\u89e3\u6cd5\u4e0e\u5b9e\u6570\u89e3\u6cd5\u7684\u7b49\u6548\u6027 [Equivalence of Complex and Real Solutions for Harmonically Excited Forced Vibration]"},"previousItem":{"@type":"ListItem","@id":"http:\/\/www.jdcui.com#listItem","name":"Home"}},{"@type":"ListItem","@id":"http:\/\/www.jdcui.com\/?p=26307#listItem","position":3,"name":"[\u632f\u52a8\u63a7\u5236] \u7b80\u8c10\u6fc0\u52b1\u4e0b\u5f3a\u8feb\u632f\u52a8\u590d\u6570\u89e3\u6cd5\u4e0e\u5b9e\u6570\u89e3\u6cd5\u7684\u7b49\u6548\u6027 [Equivalence of Complex and Real Solutions for Harmonically Excited Forced Vibration]","previousItem":{"@type":"ListItem","@id":"http:\/\/www.jdcui.com\/?cat=5#listItem","name":"Structural Engineering [\u7ed3\u6784\u5de5\u7a0b]"}}]},{"@type":"Organization","@id":"http:\/\/www.jdcui.com\/#organization","name":"\u5d14\u6d4e\u4e1c\u7684\u535a\u5ba2 - 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\u5927\u9707\u5f39\u5851\u6027","url":"http:\/\/www.jdcui.com\/"},{"@type":"Person","@id":"http:\/\/www.jdcui.com\/?author=1#author","url":"http:\/\/www.jdcui.com\/?author=1","name":"CJD","image":{"@type":"ImageObject","@id":"http:\/\/www.jdcui.com\/?p=26307#authorImage","url":"https:\/\/secure.gravatar.com\/avatar\/e2fdb66ee1ab31c12f692ce40f21c023cc9c65dfbcf39aa491fb3b6785061f54?s=96&d=blank&r=g","width":96,"height":96,"caption":"CJD"}},{"@type":"WebPage","@id":"http:\/\/www.jdcui.com\/?p=26307#webpage","url":"http:\/\/www.jdcui.com\/?p=26307","name":"[\u632f\u52a8\u63a7\u5236] \u7b80\u8c10\u6fc0\u52b1\u4e0b\u5f3a\u8feb\u632f\u52a8\u590d\u6570\u89e3\u6cd5\u4e0e\u5b9e\u6570\u89e3\u6cd5\u7684\u7b49\u6548\u6027 [Equivalence of Complex and Real Solutions for Harmonically Excited Forced Vibration] | \u5d14\u6d4e\u4e1c\u7684\u535a\u5ba2 - www.jdcui.com","description":"\u5b9e\u5e72\u3001\u5b9e\u8df5\u3001\u79ef\u7d2f\u3001\u601d\u8003\u3001\u521b\u65b0\uff01 \u5728\u5b66\u4e60\u52a8\u529b\u5b66\u4e0e\u7ed3\u6784\u632f\u52a8\u63a7\u5236\u76f8\u5173\u5185\u5bb9\u65f6\uff0c\u590d\u6570\u89e3\u6cd5\u5e38\u7528\u4e8e\u5206\u6790\u7b80\u8c10\u6fc0\u52b1\u4e0b\u7684\u5f3a\u8feb\u632f\u52a8\u95ee\u9898\u3002\u56de\u60f3\u8d77\u5728\u6821\u521d\u5b66\u6b64\u90e8\u5206\u65f6\uff0c\u5e38\u611f\u5230\u56f0\u60d1\u3002\u968f\u7740\u540e\u7eed\u6df1\u5165\u7406\u89e3\uff0c\u624d\u8ba4\u8bc6\u5230\u5176\u80cc\u540e\u7684\u5b9e\u8d28\uff1a\u7b80\u8c10\u6fc0\u52b1\u4e0b\u7684\u5f3a\u8feb\u632f\u52a8\u95ee\u9898\uff0c\u590d\u6570\u89e3\u6cd5\u4e0e\u5b9e\u6570\u89e3\u6cd5\uff08\u4e09\u89d2\u51fd\u6570\u6cd5\uff09\u5728\u6570\u5b66\u4e0a\u662f\u5b8c\u5168\u7b49\u6548\u7684\u3002\u590d\u6570\u89e3\u6cd5\u7684\u4f18\u52bf\u5728\u4e8e\u63a8\u5bfc\u8fc7\u7a0b\u66f4\u4e3a\u7b80\u6d01\uff0c\u5c24\u5176\u5728\u5904\u7406\u9ad8\u9636\u7cfb\u7edf\u6216\u591a\u9891\u6fc0\u52b1\u65f6\u663e\u5f97\u66f4\u4e3a\u9ad8\u6548\u3002 \u4ee5\u4e0b\u4ee5\u5355\u81ea\u7531\u5ea6\u7b80\u8c10\u6fc0\u52b1\u53d7\u8feb\u632f\u52a8\u4e3a\u4f8b\uff0c\u5bf9\u8fd9\u4e24\u79cd\u65b9\u6cd5\u7684\u7b49\u6548\u6027\u4f5c\u4e00\u7b80\u8981\u603b\u7ed3\u3002 \u4e09\u89d2\u51fd\u6570\u6fc0\u52b1\u4f5c\u7528\u4e0b\uff0c\u5355\u81ea\u7531\u5ea6\u52a8\u529b\u4f53\u7cfb\u52a8\u529b\u5e73\u8861\u65b9\u7a0b\u5982\u4e0b\uff1a \\[m\\ddot x + c\\dot x + kx = F\\cos \\left( {\\omega t} \\right)\\] \u4e5f\u53ef\u8868\u793a\u4e3a\u6b63\u5f26\u51fd\u6570\u6fc0\u52b1\u7684\u5f62\u5f0f \\[m\\ddot x + c\\dot x + kx = F\\sin \\left( {\\omega t} \\right)\\] \u4e0a\u8ff0\u5c31\u662f\u5b9e\u6570\u89e3\u6cd5\uff0c\u5982\u679c\u6362\u6210\u590d\u6570\u89e3\u6cd5\uff0c\u5982\u4e0b \u9996\u5148\uff0c\u5c06\u5916\u529b\u53d8\u4e3a\u590d\u6307\u6570\u51fd\u6570\u5f62\u5f0f\uff0c\u5373 \\[f = F{e^{i\\omega t}}\\] \u6b64\u65f6\u7cfb\u7edf\u8fd0\u52a8\u65b9\u7a0b\u53d8\u4e3a\uff1a \\[m\\ddot z + c\\dot z + k{\\rm{z}} = F{e^{i\\omega t}}\\] \u5229\u7528\u5f3a\u5927\u7684\u6b27\u62c9\u516c\u5f0fEuler's formula\uff1a \\[{e^{i\\omega t}} = {\\cos \\omega","inLanguage":"en-US","isPartOf":{"@id":"http:\/\/www.jdcui.com\/#website"},"breadcrumb":{"@id":"http:\/\/www.jdcui.com\/?p=26307#breadcrumblist"},"author":{"@id":"http:\/\/www.jdcui.com\/?author=1#author"},"creator":{"@id":"http:\/\/www.jdcui.com\/?author=1#author"},"image":{"@type":"ImageObject","url":"http:\/\/www.jdcui.com\/wp-content\/uploads\/2025\/08\/Complex-and-Real-Solutions-min.png","@id":"http:\/\/www.jdcui.com\/?p=26307\/#mainImage","width":676,"height":338},"primaryImageOfPage":{"@id":"http:\/\/www.jdcui.com\/?p=26307#mainImage"},"datePublished":"2025-08-27T23:00:24+08:00","dateModified":"2026-02-15T10:41:37+08:00"},{"@type":"WebSite","@id":"http:\/\/www.jdcui.com\/#website","url":"http:\/\/www.jdcui.com\/","name":"\u5d14\u6d4e\u4e1c\u7684\u535a\u5ba2 - 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