{"id":23747,"date":"2024-05-10T17:15:49","date_gmt":"2024-05-10T09:15:49","guid":{"rendered":"http:\/\/www.jdcui.com\/?p=23747"},"modified":"2026-02-15T19:05:59","modified_gmt":"2026-02-15T11:05:59","slug":"%e6%8c%af%e5%8a%a8%e6%8e%a7%e5%88%b6-%e6%8b%8d%e6%8c%af%e7%8e%b0%e8%b1%a1-beat-vibration","status":"publish","type":"post","link":"http:\/\/www.jdcui.com\/?p=23747","title":{"rendered":"[\u632f\u52a8\u63a7\u5236] \u62cd\u632f\u73b0\u8c61 (Beats Phenomenon)"},"content":{"rendered":"<p><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\u3002<\/strong><\/span><\/p>\n<hr \/>\n<p>\u62cd\u632f\u73b0\u8c61\u6307\u7684\u662f\u4e24\u4e2a\u9891\u7387\u63a5\u8fd1\u7684\u529b\uff08\u6fc0\u52b1\u6e90\uff09\u4ea7\u751f\u7684\u632f\u52a8\u53e0\u52a0\u5728\u4e00\u8d77\uff0c\u7531\u4e8e\u9891\u7387\u63a5\u8fd1\uff0c\u5468\u671f\u4e5f\u63a5\u8fd1\uff0c\u6bcf\u8fc7\u4e00\u4e2a\u5468\u671f\u4e24\u4e2a\u4fe1\u53f7\u7684\u76f8\u5bf9\u76f8\u4f4d\u5c31\u4f1a\u6709\u4e00\u70b9\u53d8\u5316\uff0c\u63a5\u8fd1\u540c\u76f8\u7684\u65f6\u5019\u4e24\u4e2a\u4fe1\u53f7\u53e0\u52a0\uff0c\u5e45\u503c\u53d8\u5927\uff1b\u63a5\u8fd1\u53cd\u76f8\u7684\u65f6\u5019\u4e24\u4e2a\u4fe1\u53f7\u76f8\u4e92\u62b5\u6d88\uff0c\u5e45\u503c\u53d8\u5c0f\uff0c\u9020\u6210\u6ce2\u5f62\u603b\u5e45\u503c\u7684\u5468\u671f\u6027\u6ce2\u52a8\u3002<\/p>\n<p>\u8fd9\u91cc\u6709\u4e00\u4e9b\u4e0d\u9519\u7684\u4ecb\u7ecd\u8d44\u6599\uff0c\u4e00\u5e76\u5217\u51fa\uff1a<\/p>\n[1] <strong>Beats\uff1a<a href=\"https:\/\/www.physicsbootcamp.org\/beat-phenomenon.html\">https:\/\/www.physicsbootcamp.org\/beat-phenomenon.html<\/a><\/strong><\/p>\n[2] <strong><a href=\"https:\/\/mp.weixin.qq.com\/s?__biz=MzA4Nzg4MDY1Mw==&amp;mid=2652474491&amp;idx=2&amp;sn=3c9cdede97c0aa54a075f348f024f670&amp;chksm=8bdf8ad9bca803cf4e0462dbf32b1e6299a12f33e137ac1a68f5c377271875863438029f6d85&amp;scene=27\">\u822a\u5929\u5668\u706b\u7bad\u53d1\u5c04\u62cd\u632f\u73b0\u8c61<\/a><\/strong><\/p>\n<p>\u5047\u5b9a\u4e24\u7ec4\u6ce2\uff0c\u5706\u9891\u7387\u5206\u522b\u4e3a\u03c81\u548c\u03c82\u7684\u4e24\u7ec4\u6ce2\uff0c\u8868\u8fbe\u5f0f\u5982\u4e0b\uff1a<\/p>\n<p>$${\\psi _1}(0,t) = Acos({\\omega _1}t) = Acos(2\\pi {f_1}t)$$<\/p>\n<p>$${\\psi _2}(0,t) = Acos({\\omega _2}t) = Acos(2\\pi {f_2}t)$$<\/p>\n<p>\u53e0\u52a0\u540e\u7684\u6ce2\u8868\u8fbe\u5f0f\u5982\u4e0b<\/p>\n<p>$$\\psi = {\\psi _1} + {\\psi _2} = A[cos({\\omega _1}t) + cos({\\omega _2}t)]$$<\/p>\n<p>\u6839\u636e\u4e09\u89d2\u6052\u7b49\u5f0f\uff08trigonometric identity\uff09\uff0c\u8fdb\u4e00\u6b65\u8f6c\u5316\u4e3a\u5982\u4e0b\u7684\u5f62\u5f0f<\/p>\n<p>$$\\psi = \\{ 2Acos[({{{\\omega _1} &#8211; {\\omega _2}} \\over 2})t]\\} cos[({{{\\omega _1} + {\\omega _2}} \\over 2})t]$$<\/p>\n<p>\u8fdb\u4e00\u6b65\u5c06\u9891\u7387\u5199\u6210\u5747\u503c\\({\\omega _0}\\)\u4e0e\u504f\u5dee\\({\\Delta \\omega }\\)\u7684\u5f62\u5f0f\uff0c\u5982\u4e0b<\/p>\n<p>$${\\omega _1} = {\\omega _0} &#8211; {{\\Delta \\omega } \\over 2}$$<\/p>\n<p>$${\\omega _2} = {\\omega _0} + {{\\Delta \\omega } \\over 2}$$<\/p>\n<p>\u4e8e\u662f\uff0c\u53ef\u8fdb\u4e00\u6b65\u5c06\u5c06\u53e0\u52a0\u540e\u7684\u632f\u52a8\u5199\u6210\u5982\u4e0b\u5f62\u5f0f<\/p>\n<p>$$\\psi = \\{ 2Acos({{\\Delta \\omega } \\over 2}t)\\} cos({\\omega _0}t)$$<\/p>\n<p>\u4e0a\u5f0f\u662f\u4e24\u4e2acos\u51fd\u6570\u76f8\u4e58\uff0c\u7531\u4e8e\u5047\u5b9a\u4e24\u4e2a\u9891\u7387\u76f8\u5dee\u7279\u522b\u5c0f\uff0c\u56e0\u6b64\u5747\u503c\\({\\omega _0}\\)\u8fdc\u5927\u4e8e\u504f\u5dee\\({\\Delta \\omega }\\)\uff0c\u4e8e\u662f\u7b2c\u4e00\u9879\u4e09\u89d2\u51fd\u6570\u53ef\u4ee5\u8ba4\u4e3a\u662f\u7b2c\u4e8c\u9879\u4e09\u89d2\u51fd\u6570\u7684\u4e00\u4e2a\u5e45\u5ea6\u503c\uff0c\u4e14\u662f\u4e00\u4e2a\u5468\u671f\u53d8\u5316\u7684\u5e45\u5ea6\u503c\uff0c\u5f62\u6210\u8fde\u7eed\u7684\u6ce2\u5cf0\u548c\u6ce2\u8c37\u3002<\/p>\n<p>\u7531\u4e8e\\(cos({{\\Delta \\omega } \\over 2}t)\\)\u7684\u6ce2\u5cf0\u548c\u6ce2\u8c37\uff0c\u5747\u662f\u5bf9\u5e94\u6781\u5927\u503c\uff0c\u56e0\u6b64\u62cd\u7684\u5468\u671f\u662f\\(cos({{\\Delta \\omega } \\over 2}t)\\)\u51fd\u6570\u5468\u671f\u7684\u4e00\u534a\uff1a<\/p>\n<p>$${T_{beat}} = {1 \\over 2}({{2\\pi } \\over {\\Delta \\omega \/2}}) = {{2\\pi } \\over {\\Delta \\omega }} = {1 \\over {|f1 &#8211; f2|}}$$<\/p>\n<p>\u62cd\u7684\u9891\u7387\u5219\u4e3a\u4e24\u4e2a\u9891\u7387\u4e4b\u5dee\uff1a<\/p>\n<p>$${f_{beat}} = |f1 &#8211; f2|$$<\/p>\n<p>\u6839\u636e\u6587\u732e2\u7684\u89e3\u91ca\uff0c\u62cd\u632f\u6709\u4ee5\u4e0b\u7279\u6027\uff0c\u6211\u4eec\u76f4\u63a5\u6458\u6284\u4e0b\u6765\uff1a<\/p>\n<ul class=\"list-paddingleft-2\">\n<li>\u5f53\u9891\u7387\u6bd4\u6ee1\u8db30.85\u2264<em>\u03be<\/em>\u22641.18\uff0c\u632f\u5e45\u6bd4\u6ee1\u8db30.33\u2264<em>\u03b2<\/em>\u22643\uff0c\u4e24\u4e2a\u5206\u632f\u52a8\u53ef\u5408\u6210\u62cd\u9891\u632f\u52a8\uff0c\u521d\u59cb\u76f8\u4f4d\u5dee\u5bf9\u62cd\u9891\u632f\u52a8\u5e76\u65e0\u5b9e\u8d28\u6027\u5f71\u54cd\uff1b<\/li>\n<\/ul>\n<ul class=\"list-paddingleft-2\">\n<li>\u9891\u7387\u6bd4\u5f71\u54cd\u5408\u6210\u632f\u52a8\u632f\u5e45\u5468\u671f\uff1b<\/li>\n<li>\u632f\u5e45\u6bd4\u8d8a\u5c0f\uff0c\u632f\u52a8\u7684\u5bf9\u6bd4\u5ea6\u8d8a\u5c0f\uff0c\u62cd\u632f\u73b0\u8c61\u8d8a\u4e0d\u660e\u663e\u3002\u5f53\u632f\u5e45\u6bd4\u4e3a1\u65f6\uff0c\u62cd\u632f\u73b0\u8c61\u6700\u660e\u663e\u3002<\/li>\n<\/ul>\n<p>\u6211\u4eec\u8fd9\u91cc\u4e0d\u518d\u8fdb\u884c\u7ec6\u81f4\u7684\u7406\u8bba\u5206\u6790\uff0c\u6211\u4eec\u901a\u8fc7\u7f16\u5199\u5b9e\u9645\u6848\u4f8b\u6765\u770b\u4e00\u4e0b\u6548\u679c\u3002<\/p>\n<p>\u4f8b\u5b501\uff1aA1=A2 =1,\u00a0 f1=10Hz, f2 = 10.5Hz\uff0c\u9891\u7387\u6bd4f2\/f1= 1.05\uff0c\u632f\u5e45\u6bd41\uff0c\u6839\u636e\u4e0a\u9762\u7684\u5206\u6790\u53ef\u77e5\uff0c\u53ef\u5f62\u6210\u62cd\u632f\uff0c\u62cd\u632f\u5468\u671f\u662f2s\u3002<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-23777\" src=\"http:\/\/www.jdcui.com\/wp-content\/uploads\/2024\/05\/Beat_1-min.png\" alt=\"\" width=\"1013\" height=\"667\" srcset=\"http:\/\/www.jdcui.com\/wp-content\/uploads\/2024\/05\/Beat_1-min.png 1013w, http:\/\/www.jdcui.com\/wp-content\/uploads\/2024\/05\/Beat_1-min-768x506.png 768w\" sizes=\"auto, (max-width: 1013px) 100vw, 1013px\" \/><\/p>\n<p>\u4f8b\u5b502\uff1aA1=A2 =1,\u00a0 f1=10Hz, f2 = 11Hz\uff0c\u9891\u7387\u6bd4f2\/f1= 1.1\uff0c\u632f\u5e45\u6bd41\uff0c\u6839\u636e\u4e0a\u9762\u7684\u5206\u6790\u53ef\u77e5\uff0c\u53ef\u5f62\u6210\u62cd\u632f\uff0c\u62cd\u632f\u5468\u671f\u662f1s\u3002<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-23778\" src=\"http:\/\/www.jdcui.com\/wp-content\/uploads\/2024\/05\/Beat_2-min.png\" alt=\"\" width=\"1013\" height=\"667\" srcset=\"http:\/\/www.jdcui.com\/wp-content\/uploads\/2024\/05\/Beat_2-min.png 1013w, http:\/\/www.jdcui.com\/wp-content\/uploads\/2024\/05\/Beat_2-min-768x506.png 768w\" sizes=\"auto, (max-width: 1013px) 100vw, 1013px\" \/><\/p>\n<p>\u4f8b\u5b503\uff1aA1=1, A2 =3,\u00a0 f1=10Hz, f2 = 11Hz\uff0c\u9891\u7387\u6bd4f2\/f1= 1.1\uff0c\u632f\u5e45\u6bd43\uff0c\u6839\u636e\u4e0a\u9762\u7684\u5206\u6790\u53ef\u77e5\uff0c\u53ef\u5f62\u6210\u62cd\u632f\uff0c\u62cd\u632f\u5468\u671f\u662f1s\uff0c\u540c\u65f6\u4e0e\u4f8b\u5b502\u6bd4\uff0c\u62cd\u632f\u73b0\u8c61\u6ca1\u4f8b\u5b502\u5f3a\u3002<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-23779\" src=\"http:\/\/www.jdcui.com\/wp-content\/uploads\/2024\/05\/Beat_3-min.png\" alt=\"\" width=\"1013\" height=\"667\" srcset=\"http:\/\/www.jdcui.com\/wp-content\/uploads\/2024\/05\/Beat_3-min.png 1013w, http:\/\/www.jdcui.com\/wp-content\/uploads\/2024\/05\/Beat_3-min-768x506.png 768w\" sizes=\"auto, (max-width: 1013px) 100vw, 1013px\" \/><\/p>\n<p>\u4f8b\u5b504\uff1aA1=A2 =1,\u00a0 f1=10Hz, f2 = 13Hz\uff0c\u9891\u7387\u6bd4f2\/f1= 1.3\uff0c\u632f\u5e45\u6bd41\uff0c\u6309\u516c\u5f0f\u62cd\u632f\u5468\u671f\u662f3.3s\uff0c\u4f46\u6839\u636e\u4e0a\u9762\u7684\u5206\u6790\u53ef\u77e5\uff0c\u53ef\u77e5\u5df2\u65e0\u660e\u663e\u62cd\u632f\u73b0\u8c61\u3002<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-23780\" src=\"http:\/\/www.jdcui.com\/wp-content\/uploads\/2024\/05\/Beat_4-min.png\" alt=\"\" width=\"1013\" height=\"667\" srcset=\"http:\/\/www.jdcui.com\/wp-content\/uploads\/2024\/05\/Beat_4-min.png 1013w, http:\/\/www.jdcui.com\/wp-content\/uploads\/2024\/05\/Beat_4-min-768x506.png 768w\" sizes=\"auto, (max-width: 1013px) 100vw, 1013px\" \/><\/p>\n<p><strong>Video \u89c6\u9891\u6f14\u793a<\/strong><\/p>\n<p><iframe loading=\"lazy\" src=\"\/\/player.bilibili.com\/player.html?isOutside=true&amp;aid=1254824087&amp;bvid=BV1nJ4m1P7Bg&amp;cid=1550925632&amp;p=1\" width=\"800\" height=\"600\" frameborder=\"no\" scrolling=\"no\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<hr \/>\n<p style=\"text-align: center;\"><strong><div  class=\"thumbs-rating-container\" id=\"thumbs-rating-23747\" data-content-id=\"23747\"><button class=\"thumbs-rating-up thumbs-rating-voted\" onclick=\"thumbs_rating_vote(23747, 1);\">Vote Up +10<\/button> <button class=\"thumbs-rating-down thumbs-rating-voted\" onclick=\"thumbs_rating_vote(23747, 2);\">Vote Down -2<\/button><span class=\"thumbs-rating-already-voted\">You already voted!<\/span><\/div><\/strong><\/p>\n<ul style=\"list-style-type: square;\">\n<li><strong>\u6848\u4f8b (<span style=\"color: #000080;\"> Examples <\/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<hr \/>\n<p style=\"text-align: center;\"><strong>\u7ed3\u6784|\u8d85\u9650|\u8f6f\u4ef6\u5f00\u53d1|\u7f16\u7a0b|\u53c2\u6570\u5316|\u4f18\u5316|\u7b97\u6cd5|\u632f\u52a8\u63a7\u5236|\u51cf\u9694\u9707|\u6709\u9650\u5143|\u6280\u672f\u57f9\u8bad<\/strong><\/p>\n<p style=\"text-align: center;\">\u8ffd\u6c42\u5353\u8d8a \u811a\u8e0f\u5b9e\u5730 \u81f4\u529b\u4e8e\u63a2\u7d22\u548c\u62d3\u5c55\u884c\u4e1a\u8bbe\u8ba1\u524d\u6cbf<\/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","protected":false},"excerpt":{"rendered":"<p>\u5b9e\u5e72\u3001\u5b9e\u8df5\u3001\u79ef\u7d2f\u3001\u601d\u8003\u3001\u521b\u65b0\u3002 \u62cd\u632f\u73b0\u8c61\u6307\u7684\u662f\u4e24\u4e2a\u9891\u7387\u63a5\u8fd1\u7684\u529b\uff08\u6fc0\u52b1\u6e90\uff09\u4ea7\u751f\u7684\u632f\u52a8\u53e0\u52a0\u5728\u4e00\u8d77\uff0c\u7531\u4e8e\u9891\u7387\u63a5\u8fd1\uff0c\u5468\u671f\u4e5f\u63a5\u8fd1\uff0c\u6bcf\u8fc7\u4e00\u4e2a\u5468\u671f\u4e24\u4e2a\u4fe1\u53f7\u7684\u76f8\u5bf9\u76f8\u4f4d\u5c31\u4f1a\u6709\u4e00\u70b9\u53d8\u5316\uff0c\u63a5\u8fd1\u540c\u76f8\u7684\u65f6\u5019\u4e24\u4e2a\u4fe1\u53f7\u53e0\u52a0\uff0c\u5e45\u503c\u53d8\u5927\uff1b\u63a5\u8fd1\u53cd\u76f8\u7684\u65f6\u5019\u4e24\u4e2a\u4fe1\u53f7\u76f8\u4e92\u62b5\u6d88\uff0c\u5e45\u503c\u53d8\u5c0f\uff0c\u9020\u6210\u6ce2\u5f62\u603b\u5e45\u503c\u7684\u5468\u671f\u6027\u6ce2\u52a8\u3002 \u8fd9\u91cc\u6709\u4e00\u4e9b\u4e0d\u9519\u7684\u4ecb\u7ecd\u8d44\u6599\uff0c\u4e00\u5e76\u5217\u51fa\uff1a [1] Beats\uff1ahttps:\/\/www.physicsbootcamp.org\/beat-phenomenon.html [2] \u822a\u5929\u5668\u706b\u7bad\u53d1\u5c04\u62cd\u632f\u73b0\u8c61 \u5047\u5b9a\u4e24\u7ec4\u6ce2\uff0c\u5706\u9891\u7387\u5206\u522b\u4e3a\u03c81\u548c\u03c82\u7684\u4e24\u7ec4\u6ce2\uff0c\u8868\u8fbe\u5f0f\u5982\u4e0b\uff1a $${\\psi _1}(0,t) = Acos({\\omega _1}t) = Acos(2\\pi {f_1}t)$$ $${\\psi _2}(0,t) = Acos({\\omega _2}t) = Acos(2\\pi {f_2}t)$$ \u53e0\u52a0\u540e\u7684\u6ce2\u8868\u8fbe\u5f0f\u5982\u4e0b $$\\psi = {\\psi _1} + &#8230;<\/p>\n","protected":false},"author":1,"featured_media":23782,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5,3164],"tags":[4170,4168,1291,4171,4172,4169],"class_list":["post-23747","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-structural-engineering","category--vibration-control","tag-angular-frequency","tag-beat-vibration","tag-frequency","tag-trigonometric-identity","tag-4172","tag-4169"],"aioseo_notices":[],"views":5569,"_links":{"self":[{"href":"http:\/\/www.jdcui.com\/index.php?rest_route=\/wp\/v2\/posts\/23747","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.jdcui.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.jdcui.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.jdcui.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.jdcui.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=23747"}],"version-history":[{"count":0,"href":"http:\/\/www.jdcui.com\/index.php?rest_route=\/wp\/v2\/posts\/23747\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/www.jdcui.com\/index.php?rest_route=\/wp\/v2\/media\/23782"}],"wp:attachment":[{"href":"http:\/\/www.jdcui.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=23747"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.jdcui.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=23747"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.jdcui.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=23747"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}