{"id":5297,"date":"2013-04-23T02:55:07","date_gmt":"2013-04-22T18:55:07","guid":{"rendered":"http:\/\/www.andrewsun.net\/panta_rhei\/?p=5297"},"modified":"2020-02-04T23:45:12","modified_gmt":"2020-02-04T15:45:12","slug":"fourier%e5%8f%98%e6%8d%a2%ef%bc%88%e4%ba%8c%ef%bc%89","status":"publish","type":"post","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/5297","title":{"rendered":"Fourier\u53d8\u6362\uff08\u4e8c\uff09"},"content":{"rendered":"
\n

[mathjax]
\n\u4e0a\u4e00\u7bc7\u4ecb\u7ecd\u4e86\u7528Fourier\u53d8\u6362\u6765\u91cd\u6784\u4e00\u4e2a\u672a\u77e5\u6b63\u5f26\u4fe1\u53f7\u7684\u65b9\u6cd5\u3002\u672c\u7bc7\u4ecb\u7ecd\u5bf9\u4e8e\u672a\u77e5\u975e<\/em>\u6b63\u5f26\u4fe1\u53f7\u7684\u60c5\u51b5\u3002<\/p>\n

\u5927\u5e45\u632f\u8361\u526a\u5207\u7684\u53c2\u6570<\/h3>\n

\u5927\u5e45\u632f\u8361\u526a\u5207\u5b9e\u9a8c\u4ea7\u751f\u7684\u6ce2\u5f62\u662f\u975e\u6b63\u5f26\u4fe1\u53f7\uff0c\u975e\u6b63\u5f26\u662f\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7684\u7279\u70b9\u3002\u6240\u4ee5\u5bf9\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7a0b\u5ea6\u7684\u8868\u5f81\uff0c\u5c31\u662f\u5bf9\u6ce2\u5f62\u6b63\u5f26\u5f62\u72b6\u504f\u79bb\u7a0b\u5ea6\u7684\u8868\u5f81\u3002\u5173\u4e8e\u5927\u5e45\u632f\u8361\u526a\u5207\u548c\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7684\u539f\u7406\u4e0e\u5b9e\u8df5\uff0c\u53ef\u89c1Hyhn\u7b49\u7684\u7efc\u8ff0[1]<\/a><\/span>\uff0c\u6216\u8005\u53c2\u9605\u6211\u7684\u535a\u58eb\u5b66\u4f4d\u8bba\u6587\u7b2c1.2.1\u8282\u3001\u7b2c1.3\u8282\u548c\u7b2c2\u7ae0\u3002\u672c\u6587\u53ea\u89e3\u51b3MATLAB\u7f16\u7a0b\u5b9e\u73b0\u95ee\u9898\u3002<\/p>\n

\u5927\u5e45\u632f\u8361\u526a\u5207\uff08LAOS\uff09\u5b9e\u9a8c\u4e2d\uff0c\u6211\u4eec\u4ece\u6d41\u53d8\u4eea\u540c\u65f6\u83b7\u53d6\u4e24\u4e2a\u4fe1\u53f7\u3002\u4e00\u4e2a\u662f\u5e94\u53d8\u7684\u4fe1\u53f7\uff0c\u4e00\u4e2a\u662f\u5e94\u529b\u7684\u4fe1\u53f7\uff08\u4e8b\u5b9e\u4e0a\uff0c\u4e00\u4e2a\u662f\u9a6c\u8fbe\u4f4d\u79fb\u4fe1\u53f7\uff0c\u53e6\u4e00\u4e2a\u662f\u529b\u77e9\u4fe1\u53f7\uff0c\u5b83\u4eec\u90fd\u8981\u4e58\u4ee5\u4e00\u4e2a\u7cfb\u6570\uff0c\u624d\u662f\u5e94\u53d8\u548c\u5e94\u529b\u3002\u6b64\u6587\u5148\u7565\u53bb\u8fd9\u4e00\u4e8b\u5b9e\u4e86\uff09\u3002\u5728\u5e94\u53d8\u63a7\u5236\u7684\u5b9e\u9a8c\u4e2d\uff0c\u6211\u4eec\u65bd\u52a0\u4e00\u4e2a\u6b63\u5f26\u7684\u5e94\u53d8\"\gamma\left(t\right)\"\uff0c\u6d4b\u91cf\u5e94\u529b\u7684\u54cd\u5e94\"\sigma\left(t\right)\"\uff0c\u5373
\n$$\\begin{cases}\\gamma\\left(t\\right)=\\sin\\omega t\\\\ \\sigma\\left(t\\right)=\\sum_{n=0:\\text{odd}}^\\infty\\sigma_n\\sin\\left(\\omega t+\\delta_n\\right)\\end{cases}$$\u7531\u6b64\u5f0f\u53ef\u5b9a\u4e49LAOS\u5b9e\u9a8c\u7ed3\u679c\u7684\u8868\u5f81\u53c2\u6570\"I_{n\/1}\equiv\frac{\sigma_n}{\sigma_1}\"\uff0c\u4ee5\u53ca\"\mathit{\Phi}_n=\delta_n-n\delta_1\"\uff0c\u6765\u8868\u5f81\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7684\u7a0b\u5ea6\uff08\u5373\"\sigma\left(t\right)\"\u504f\u79bb\u6b63\u5f26\u51fd\u6570\u7684\u7a0b\u5ea6\u3002\u6b64\u5916\uff0c\u6211\u4eec\u8fd8\u4f1a\u4ee5\"\sigma\left(t\right)\"\u5bf9\"\gamma\left(t\right)\"\u4f5c\u56fe\uff08\u53c2\u6570\u65b9\u7a0b\u4f5c\u56fe\uff09\uff0c\u5f97\u5230Lissajous\u66f2\u7ebf\u3002\u7136\u540e\u7528\u66f2\u7ebf\u7684\u5f62\u72b6\u7279\u5f81\u6765\u8868\u5f81\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7684\u7a0b\u5ea6\uff08\u5373\u66f2\u7ebf\u504f\u79bb\u692d\u5706\u7684\u7a0b\u5ea6\uff09\u3002\u63cf\u8ff0\u66f2\u7ebf\u5f62\u72b6\u7279\u5f81\uff0c\u5f80\u5f80\u5c31\u662f\u5404\u79cd\u5404\u6837\u7684\u5207\u7ebf\u659c\u7387\uff08\u53ef\u81ea\u884c\u4e0a\u7f51\u641c\u7d22\u590d\u4e60\u53c2\u6570\u65b9\u7a0b\u6c42\u5bfc<\/a>\uff09\u3002\u5728LAOS\u4e2d\u5e38\u7528\uff1a$$G_\\text{M}=\\left.\\frac{d\\sigma}{d\\gamma}\\right|_{\\gamma=0}$$$$G_\\text{L}=\\frac{\\sigma{\\gamma_0}}{\\gamma_0}$$$$G_\\text{k}=\\left.\\frac{d\\sigma}{d\\gamma}\\right|_{\\sigma=0}$$<\/p>\n

MATLAB\u7f16\u7a0b\u6b65\u9aa4<\/h2>\n

\u5728\u8be6\u7ec6\u4ecb\u7ecdMATLAB\u4ee3\u7801\u4e4b\u524d\uff0c\u5148\u603b\u7ed3\u4e00\u4e0b\u6211\u4eec\u8981\u5b8c\u6210\u4ec0\u4e48\u4efb\u52a1\uff0c\u8ddf\u4ece\u54ea\u4e9b\u6b65\u9aa4\u3002\u6211\u4eec\u7684\u4efb\u52a1\u5c31\u662f\u8981\u4ece\u5e94\u529b\u548c\u5e94\u53d8\u7684\u4fe1\u53f7\u8ba1\u7b97\u51fa\u4e0a\u8ff0\u7684\u8fd9\u4e48\u591a\u53c2\u6570\u3002\u5b9e\u9a8c\u65f6\uff0c\u6211\u4eec\u4f1a\u89c4\u5b9a\u5e94\u53d8\u7684\u5e45\u5ea6\"\gamma_0\"\u548c\u9891\u7387\"\omega\"\uff0c\u4f46\u662f\uff0c\u7531\u4e8e\u4eea\u5668\u8bef\u5dee\uff0c\u5b9e\u9645\u65bd\u52a0\u5728\u6837\u54c1\u4e0a\u7684\u5e94\u53d8\u5e45\u5ea6\u548c\u9891\u7387\uff0c\u672a\u5fc5\u5b8c\u5168\u7b49\u4e8e\u7684\u547d\u4ee4\u503c\u3002\u5c3d\u7ba1\u8fd9\u4e00\u8bef\u5dee\u975e\u5e38\u5c0f\uff0c\u4f46\u4ece\u505a\u6cd5\u4e0a\u6211\u4eec\u8fd8\u662f\u8981\u4ee5\u5b9e\u9645\u5e94\u53d8\u4fe1\u53f7\u4e3a\u51c6\u3002\u6240\u4ee5\uff0c\u6211\u4eec\u9996\u5148\u8981\u5bf9\u5b9e\u9645\u5e94\u53d8\u4fe1\u53f7\u8fdb\u884c\u5224\u65ad\uff0c\u83b7\u5f97\u5b9e\u9645\u7684\u5e94\u53d8\u5e45\u5ea6\u548c\u9891\u7387\u3002\u5982\u4f55\u5224\u65ad\u4e00\u4e2a\u672a\u77e5\u6b63\u5f26\u4fe1\u53f7\u7684\u632f\u5e45\u3001\u9891\u7387\u548c\u76f8\u4f4d\u7684\u65b9\u6cd5\u5df2\u7ecf\u5728\u4e0a\u4e00\u7bc7\u6587\u7ae0\u4e2d\u4ecb\u7ecd\u4e86\u3002<\/p>\n

\u7136\u540e\uff0c\u5c31\u8981\u5904\u7406\u5e94\u529b\u7684\u4fe1\u53f7\u3002\u4e3a\u4e86\u83b7\u5f97\"\sigma_n\"\u548c\"\delta_n\"\u540c\u6837\u9700\u8981\u5bf9\"\sigma\left(t\right)\"\u8fdb\u884cfft\u5904\u7406\uff0c\u65b9\u6cd5\u8ddf\u4e0a\u4e00\u7bc7\u6587\u7ae0\u4ecb\u7ecd\u7684\u7c7b\u4f3c\uff0c\u533a\u522b\u53ea\u5728\u4e8e\u6211\u4eec\u9664\u4e86\u8981\u83b7\u5f97\u57fa\u9891\u7684\u5e45\u503c\u548c\u76f8\u4f4d\u4e4b\u5916\uff0c\u8fd8\u8981\u83b7\u5f97\u9ad8\u6b21\u8c10\u6ce2\u7684\u5e45\u503c\u548c\u76f8\u4f4d\u3002\u7406\u8bba\u4e0a\u6765\u8bf4\"n=\infty\"\uff0c\u4f46\u662f\u5b9e\u9645\u4e0a\u6211\u4eec\u6839\u636e\u9700\u8981\u53d6\u6709\u9650\u7684n<\/i>\u4e2a\u9ad8\u6b21\u8c10\u6ce2\uff0c\u5c31\u53ef\u4ee5\u6bd4\u8f83\u51c6\u786e\u5730\u91cd\u6784\u51fa\"\sigma\left(t\right)\"\u7684\u6ce2\u5f62\u4e86\u3002\u63d0\u9192\u4e00\u4e0b\uff0c\u5b9a\u4e49\u5f0f\u91cc\u5e94\u53d8\u662f\u4e00\u4e2a\u96f6\u76f8\u4f4d\u7684\u6807\u51c6\u6b63\u5f26\u51fd\u6570\uff0c\u4f46\u5b9e\u9645\u91cd\u6784\u7684\u5e94\u53d8\u6ce2\u5f62\u53ef\u80fd\u4f1a\u6709\u975e\u96f6\u7684\u76f8\u4f4d\uff0c\u8fd9\u65f6\uff0c\u5e94\u529b\u7684\u5404\u4e2a\u8c10\u6ce2\u7684\u76f8\u4f4d\u8981\u90fd\u51cf\u53bb\u5e94\u53d8\u7684\u76f8\u4f4d\uff0c\u624d\u662f\u6211\u4eec\u8981\u6c42\u7684\"\delta_n\"\u3002<\/p>\n

Lissajous\u66f2\u7ebf\u7684\u5404\u79cd\u5bfc\u6570\uff0c\u8981\u540c\u65f6\u5728\"\gamma\left(t\right)\"\u548c\"\sigma\left(t\right)\"\u90fd\u91cd\u6784\u4e86\u4e4b\u540e\uff0c\u518d\u8fdb\u884c\u6c42\u5bfc\u8ba1\u7b97\u5f97\u5230\u3002\u603b\u800c\u8a00\u4e4b\uff0c\u8981\u5b8c\u6210\u4ee5\u4e0b\u4e09\u4e2a\u6b65\u9aa4\uff1a<\/p>\n

    \n
  1. \u91cd\u6784\u5e94\u53d8\"\gamma\left(t\right)\"\uff0c\u83b7\u5f97\"\gamma_0\"\u3001\"\omega\"\u548c\u76f8\u4f4d<\/li>\n
  2. \u91cd\u6784\"\sigma\left(t\right)\"\uff0c\u83b7\u5f97\"\sigma_n\"\u548c\"\delta_n\"\uff0c\u8ba1\u7b97\"I_{n\/1}\"<\/li>\n
  3. \u7528\u91cd\u6784\u7684\"\sigma\left(t\right)\"\u548c\"\gamma\left(t\right)\"\u4f5cLissajous\u66f2\u7ebf\uff0c\u4ee5\u53ca\u8ba1\u7b97\"G_\text{M}\"\u3001\"G_\text{L}\"\u548c\"G_\text{k}\"<\/li>\n<\/ol>\n

    \u91cd\u6784\"\gamma\left(t\right)\"\u548c\"\sigma\left(t\right)\"<\/h3>\n

    MATLAB\u5e2e\u52a9\u6587\u4ef6\u91ccfft\u547d\u4ee4\u7684\u6761\u76ee\u6709\u4e00\u4e2a\u8303\u4f8b\uff0c\u5e94\u8be5\u80fd\u89e3\u51b3\u5927\u90e8\u5206\u95ee\u9898\u4e86\u3002\u5176\u4e2d\u5728coherent sampling\uff08\u89c1\u4e0a\u7bc7\u6587\u7ae0\uff09\u5904\uff0c\u8303\u4f8b\u7528next power of 2\uff0c\u5176\u5b9e\u6ca1\u6709\u5fc5\u8981\u3002\u4e0b\u9762\u7684\u4ee3\u7801\uff0c\u6211\u5c31\u4e0d\u591a\u4f5c\u89e3\u91ca\u4e86\u3002<\/p>\n


    \nclear all
    \nclc<\/code><\/p>\n

    % the strain signal
    \nfs=1000; % sampling frequency
    \nt=0:1\/fs:30.2; % the time series of the strain signal
    \nt=t(:); % convert to colume vector<\/p>\n

    w=5; % (rad\/s) the angular frequency of the strain signal
    \nf=w\/(2*pi); % (Hz)<\/p>\n

    x=250sin(2<\/em>pif<\/em>t+1)+25randn(size(t)); % strain signal with noise
    \ny=1.001+8.1532.<\/em>sin(2pi<\/em>f.t+2.5936)+2.4952.<\/em>sin(6pi<\/em>f.t+0.5199)+...
    \n0.8856.<\/em>sin(10pi<\/em>f.t+4.3353)+0.2932.<\/em>sin(14pi<\/em>f.t+1.8641)+...
    \n0.0992.<\/em>sin(18pi<\/em>f.t-0.6417)+0.0282<\/em>sin(22pi<\/em>f.+3.3818)+...
    \n0.2<\/em>randn(size(t)); % stress signal with noise<\/p>\n

    <\/code>[AX,H1,H2]=plotyy(t,x,t,y,'plot'); % Figure 1
    \nset(AX(1),'XLim',[0 8])
    \nset(AX(2),'XLim',[0 8]) % set the time axes
    \n<\/code><\/p>\n

    \"Figure

    Figure 1 Parts of the strain and the stress signals<\/p><\/div>\n

    \u4e24\u4e2a\u4fe1\u53f7\u90fd\u52a0\u5165\u4e86\u566a\u97f3\uff0c\u800c\u4e14\u5e94\u529b\u4fe1\u53f7\u52a0\u5165\u4e86\u76f4\u6d41\u5206\u91cf\uff0c\u66f4\u63a5\u8fd1\u771f\u5b9e\u7684\u60c5\u51b5\u3002\u53e6\u4e00\u4e2a\u771f\u5b9e\u7684\u60c5\u51b5\u662f\uff0c\u5e94\u529b\u54cd\u5e94\u7684\u6ce2\u5f62\u5f80\u5f80\u9700\u8981\u7ecf\u8fc7\u51e0\u4e2a\u5468\u671f\u4e4b\u540e\u624d\u8fbe\u5230\u7a33\u5b9a\u7684\u5f62\u72b6\u3002\u6211\u4eec\u5f53\u7136\u662f\u53d6\u7a33\u5b9a\u4e4b\u540e\u7684\u6ce2\u5f62\u6765\u8fdb\u884cFourier\u53d8\u6362\uff0c\u56e0\u6b64\u5c31\u8981\u622a\u53d6\u540e\u51e0\u4e2a\u5468\u671f\u3002<\/p>\n


    \nN=length(x);
    \nN_per_cycle = fs\/f; % average points per cycle
    \nN_cycles = floor(N \/ N_per_cycle); % maximum number of complete cycles<\/code><\/p>\n


    \n<\/code>N_cut=round(15*N_per_cycle); % use the last 15 cycles
    \nt=t(length(t)-N_cut:end);
    \nx=x(length(x)-N_cut:end);
    \ny=y(length(y)-N_cut:end);
    \n% crop the last N_cut data.
    \n<\/code><\/p>\n

    \u63a5\u4e0b\u6765\uff0c\u5148\u5bf9\u5e94\u53d8\u4fe1\u53f7\u8fdb\u884c\u91cd\u6784\uff0c\u65b9\u6cd5\u8ddf\u4e0a\u7bc7\u6587\u7ae0\u4e00\u6837\u3002<\/p>\n


    \nXraw=2fft(x)\/N_cut;
    \nXraw=Xraw(1:ceil(N_cut\/2));
    \nfreq=0:fs\/N_cut:fs\/2-fs\/N_cut;
    \nA = abs(Xraw(round(N_cut<\/em>f\/fs)+1))
    \ndelta = wrapTo2Pi(angle(Xraw(round(N_cutf\/fs)+1))+0.5<\/em>pi)
    \nfrq = freq(round(N_cutf\/fs)+1)
    \nx_rec=A.<\/em>sin(2pi<\/em>frq.*t+delta);
    \nplot(t,x,'b-',t,x_rec,'-r')
    \n<\/code><\/p>\n

    \"Figure

    Figure 2 Reconstruct the strain signal<\/p><\/div>\n

    \u7136\u540e\u8f6e\u5230\u5e94\u529b\u7684\u6ce2\u5f62\u3002\u5148\u770b\u770b\u5b83\u7684Fourier\u53d8\u6362\u7ed3\u679c\uff1a<\/p>\n


    \nYraw=2*fft(y)\/N_cut;
    \nYraw=Yraw(1:ceil(N_cut\/2));<\/code><\/p>\n


    \n<\/code>plot(freq, abs(Yraw)) % Figure 3
    \n<\/code><\/p>\n

    \"Figure

    Figure 3 Amplitude spectrum of the stress signal<\/p><\/div>\n

    \u653e\u5927\u6700\u5de6\u8fb9\u7684\u90e8\u5206\uff0c\u53ef\u4ee5\u770b\u5230\u597d\u591a\u9ad8\u6b21\u8c10\u6ce2\u3002\u4e3a\u4ec0\u4e48\u53f3\u8fb9\u6709\u8fd9\u4e48\u957f\u7684\u591a\u4f59\u90e8\u5206\uff1f\u56de\u5fc6\u4e0a\u7bc7\u6587\u7ae0\u3002Fourier\u53d8\u6362\u540e\u7684\u9891\u7387\u8303\u56f4\u7b49\u4e8ef<\/i>s<\/sub>\/2\u3002\u6211\u4eecf<\/i>s<\/sub>\u8fbe\u52301000 Hz\uff0c\u800c\u4fe1\u53f7\u4e3b\u9891\u53ea\u6709\u7ea60.8 Hz\uff0c\u51e0\u4e2a\u9ad8\u6b21\u8c10\u6ce2\u90fd\u96c6\u4e2d\u5728\u6700\u5de6\u4fa7\u4e86\u3002\u6709\u65f6\u95f4\u7684\u8bdd\uff0c\u6700\u597d\u591a\u70b9\u5c1d\u8bd5\u4e0d\u540c\u4fe1\u53f7\u7684FFT\uff0c\u719f\u6089\u8fd9\u4e00\u8f6c\u6362\u662f\u8ba4\u8bc6\u5927\u81ea\u7136\u7684\u4e00\u4e2a\u65b0\u7684\u601d\u60f3\u9ad8\u5ea6\u3002\u56e0\u4e3a\u5927\u81ea\u7136\u5f88\u591a\u81ea\u7136\u8fc7\u7a0b\u5c31\u662f\u5728\u505aFourier\u53d8\u6362\u3002\u4f8b\u5982X\u7ebf\u7a7f\u8fc7\u6676\u4f53\u53d1\u751f\u884d\u5c04\uff0c\u5149\u5c31\u88abFourier\u53d8\u6362\u4e86\uff0c\u5f62\u6210\u7684\u4e8c\u7ef4\u6591\u70b9\u56fe\u53ef\u901a\u8fc7\u8fdb\u884c\u4e8c\u7ef4\u7684\u53cdFourier\u53d8\u6362\uff0c\u8fd8\u539f\u6210\u6837\u54c1\u4e2d\u7684\u6676\u683c\u6392\u5e03\u3002\u6240\u4ee5\u6211\u4eec\u603b\u662f\u8bf4\uff0cX\u5c04\u7ebf\u7684\u56fe\u8c31\u662f\u201c\u5012\u6613\u7a7a\u95f4\u201d\uff08Fourire\u53d8\u6362\u524d\u540e\u5750\u6807\u53d8\u5012\u6570\u4e86\uff09\u3002<\/p>\n

    Figure 3\u8fd8\u53ef\u770b\u5230\uff0c\u5728\u9891\u7387\u4e3a0\u5904\u6709\u4e00\u4e2a\u5cf0\u3002\u8fd9\u4e2a\u5cf0\u662f\u539f\u6ce2\u5f62\u7684\u76f4\u6d41\u5206\u91cf\uff08\u89c1\u6587\u7ae0\u5f00\u5934\u6784\u5efa\u4fe1\u53f7\u7684\u5f0f\u5b50\uff09\uff0c\u6211\u4eec\u9700\u8981\u77e5\u9053\u5404\u4e2a\u8c10\u6ce2\u5728\u4fe1\u53f7\u5e8f\u5217\u4e2d\u7684\u5177\u4f53\u4f4d\u7f6e\u3002\u4e0a\u7bc7\u6587\u7ae0\u5df2\u7ecf\u89e3\u51b3\u4e86\u6b63\u5f26\u4fe1\u53f7\u7684\u60c5\u51b5\uff0c\u5bf9\u4e8e\u9ad8\u6b21\u8c10\u6ce2\uff08\u90fd\u662f\u5947\u6570\u500d\u9891\uff09\uff0c\u53ea\u8981\u76f8\u5e94\u5730\u4e58\u4ee53\u30015\u30017\u5373\u53ef\u3002\u4f46\u662f\uff0c\u6839\u636e\u5b9a\u4e49\u5f0f\uff0c\u5e94\u529b\u7684Fourier\u5c55\u5f00\u7406\u8bba\u4e0a\u662f\u65e0\u7a77\u591a\u9879\u7684\uff0c\u800c\u5b9e\u9645\u4e0a\u4e5f\u9700\u8981\u53d6\u5c3d\u91cf\u591a\u9879\u4ee5\u786e\u4fdd\u6700\u51c6\u786e\u5730\u91cd\u6784\u539f\u6ce2\u5f62\u3002\u5230\u5e95Fourier\u53d8\u6362\u4e4b\u540e\uff0c\u6700\u591a\u5141\u8bb8\u770b\u5230\u591a\u5c11\u4e2a\u9ad8\u6b21\u8c10\u6ce2\u5462\uff1f\u8fd9\u5c31\u76f8\u5f53\u4e8e\u8981\u627e\u4f7fnf<\/i><f<\/i>s<\/sub>\/2\u7684\u6700\u5927\u5947\u6570n<\/i>\u3002\u4e0d\u8fc7\uff0c\u5982\u679c\u4fe1\u53f7\u7684\u53d6\u6837\u9891\u7387f<\/i>s<\/sub>\u76f8\u5f53\u5927\uff0c\u6700\u5927\u8c10\u6ce2\u6b21\u6570n<\/i>\u4e5f\u4f1a\u975e\u5e38\u5927\uff0c\u6211\u4eec\u5b9e\u5728\u6ca1\u5fc5\u8981\u53d6\u8fd9\u4e48\u5927\u7684n<\/i>\uff08\u5982Figure 3\u7684\u60c5\u51b5\u53f3\u8fb9\u5927\u90e8\u5206\u662f\u591a\u4f59\u7684\u3002\u56e0\u6b64\u6211\u4eec\u53c8\u4eba\u4e3a\u89c4\u5b9a\u6700\u5927\u8c10\u6ce2\u6b21\u6570\u4e0d\u8d85\u8fc725\u4e2a\uff0c\u5c31\u662f\u8bf4\u6211\u4eec\u5047\u5b9a25\u4e2a\u9ad8\u6b21\u9879\u5bf9\u4e8e\u518d\u600e\u4e48\u975e\u7ebf\u6027\u7684\u4fe1\u53f7\u90fd\u80af\u5b9a\u8db3\u591f\u4e86\u3002\u4ee5\u4e0b\u662f\u76f8\u5173\u7684\u4ee3\u7801\uff1a<\/p>\n


    \nmax_nr_harm=floor(fs\/(2*f)); % Maximum number of harmonics
    \nif max_nr_harm\/2==floor(max_nr_harm\/2)
    \nmax_nr_harm=max_nr_harm-1;
    \nend
    \n% maxmum odd number of harmonic<\/code><\/p>\n

    max_nr_harm=max_nr_harm-2; % fall back one more odd harmonic for safety<\/p>\n

    <\/code>if max_nr_harm>25
    \nmax_nr_harm=25;
    \nend
    \n<\/code><\/p>\n

    \u63a5\u4e0b\u6765\u5c31\u662f\u8981\u786e\u5b9a\u8fd9\u4e9b\u8c10\u6ce2\u7684\u5e45\u503c\u548c\u8f90\u89d2\uff1a<\/p>\n


    \n% create the variables for harmonic amplitudes and phase angles
    \nI_n=zeros((max_nr_harm+1)\/2,1);
    \ndelta_n=zeros((max_nr_harm+1)\/2,1);<\/code><\/p>\n


    \n<\/code>for i=1:(max_nr_harm+1)\/2
    \nI_n(i)=abs(Yraw(round((2i-1)<\/em>N_cutf\/fs)+1));
    \ndelta_n(i)=angle(Yraw(round((2<\/em>i-1)N_cut<\/em>f\/fs)+1))+0.5pi;
    \nend
    \nDC=abs(Yraw(1))<\/em>cos(angle(Yraw(1)))\/2;
    \n<\/code><\/p>\n

    \u5176\u4e2d\uff0c\u76f4\u6d41\u5206\u91cf\u7684\u5927\u5c0f\u8ba1\u7b97\u65b9\u6cd5\u4e3a\u4ec0\u4e48\u662f\u8fd9\u4e2a\uff0c\u6211\u4e0d\u592a\u8bb0\u5f97\u4e86\u3002\u8fd9\u65f6\uff0c\u4f60\u5df2\u7ecf\u53ef\u4ee5\u901a\u8fc7I_n(i)\u548cdelta_n(i)\u6765\u8ba1\u7b97\u76f8\u5e94\u7684\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u53c2\u6570\"I_{n\/1}\equiv\frac{\sigma_n}{\sigma_1}\"\u548c\"\mathit{\Phi}_n=\delta_n-n\delta_1\"\u4e86\u3002\u518d\u6b21\u63d0\u9192\uff0c\u5e94\u529b\u6ce2\u5f62\u7684\u5404\u8c10\u6d41\u76f8\u4f4d\u89d2\u8981\u51cf\u53bb\u5e94\u53d8\u6ce2\u5f62\u7684\u76f8\u4f4d\u89d2\uff0c\u624d\u662f\u5b9a\u4e49\u5f0f\u4e2d\u7684\u76f8\u4f4d\u89d2\u3002\u6211\u4eec\u4e5f\u53ef\u4ee5\u68c0\u9a8c\u4e00\u4e0b\u91cd\u6784\u7684\u5e94\u529b\u6ce2\u5f62\uff1a<\/p>\n


    \ny_rec=zeros(length(y),1); % create the variables for the reconstructed signals
    \nfor i=1:2:max_nr_harm
    \ny_rec=y_rec+I_n((i+1)\/2).sin(i<\/em>2pi<\/em>f.*t+delta_n((i+1)\/2));
    \nend
    \ny_rec=y_rec+DC;
    \nplot(t,y,'-b',t,y_rec,'-r') % Figure 4
    \n<\/code><\/p>\n

    \"Figure

    Figure 4 Reconstructed stress signal<\/p><\/div>\n

    \u4f5c\u6210Lissajous\u66f2\u7ebf\u6765\u770b\uff1a<\/p>\n


    \nplot(x,y,'b-',x_rec,y_rec,'r-')
    \n<\/code><\/p>\n

    \"Figure

    Figure 5 Reconstructed Lissajous curve<\/p><\/div>\n

    \u53ef\u89c1\uff0c\u901a\u8fc7Fourier\u53d8\u6362\u91cd\u6784\u6ce2\u5f62\uff0c\u53ef\u4ee5\u5f88\u597d\u5730\u628a\u566a\u97f3\u6392\u9664\u6389\u3002Lissajous\u66f2\u7ebf\u7684\u5404\u4e2a\u5f62\u72b6\u53c2\u6570\uff0c\u9700\u8981\u5728\u66f2\u7ebf\u4e0a\u627e\u5404\u79cd\u4ea4\u70b9\u6c42\u5207\u7ebf\uff08\u89c1\u5b9a\u4e49\u5f0f\uff09\u3002\u5982\u679c\u5728\u539f\u4fe1\u53f7\u7684\u57fa\u7840\u4e0a\u76f4\u63a5\u505a\uff0c\u53ef\u60f3\u800c\u77e5\u8bef\u5dee\u4f1a\u6709\u591a\u5927\u3002\u8fd9\u5c31\u662f\u4e3a\u4ec0\u4e48\u6211\u4eec\u660e\u660e\u5df2\u7ecf\u6709\u4fe1\u53f7\u4e0d\u76f4\u63a5\u7528\u8981\u5148\u91cd\u6784\u4e00\u756a\u7684\u539f\u56e0\u3002\u5177\u4f53\u5982\u4f55\u5728Lissajous\u66f2\u7ebf\u4e0a\u627e\u70b9\u3001\u6c42\u5207\u7ebf\u7684\u65b9\u6cd5\uff0c\u5728\u4e0b\u4e00\u7bc7\u6587\u7ae0\u4e2d\u4ecb\u7ecd\u3002<\/p>\n

    References<\/h2>\n
      \n
    1. <\/a>\nK. Hyun, M. Wilhelm, C.O. Klein, K.S. Cho, J.G. Nam, K.H. Ahn, S.J. Lee, R.H. Ewoldt, and G.H. McKinley, \"A review of nonlinear oscillatory shear tests: Analysis and application of large amplitude oscillatory shear (LAOS)\", Progress in Polymer Science<\/i>, vol. 36, pp. 1697-1753, 2011. http:\/\/dx.doi.org\/10.1016\/j.progpolymsci.2011.02.002<\/a>\n\n\n<\/li>\n<\/ol>\n\n<\/div> ","protected":false},"excerpt":{"rendered":"

      [mathjax] \u4e0a\u4e00\u7bc7\u4ecb\u7ecd\u4e86\u7528Fourier\u53d8\u6362\u6765\u91cd\u6784\u4e00\u4e2a\u672a\u77e5\u6b63\u5f26\u4fe1\u53f7\u7684\u65b9\u6cd5\u3002\u672c\u7bc7\u4ecb\u7ecd\u5bf9\u4e8e\u672a\u77e5\u975e\u6b63\u5f26\u4fe1\u53f7\u7684\u60c5\u51b5\u3002 \u5927\u5e45\u632f\u8361\u526a\u5207\u7684\u53c2\u6570 \u5927\u5e45\u632f\u8361\u526a\u5207\u5b9e\u9a8c\u4ea7\u751f\u7684\u6ce2\u5f62\u662f\u975e\u6b63\u5f26\u4fe1\u53f7\uff0c\u975e\u6b63\u5f26\u662f\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7684\u7279\u70b9\u3002\u6240\u4ee5\u5bf9\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7a0b\u5ea6\u7684\u8868\u5f81\uff0c\u5c31\u662f\u5bf9\u6ce2\u5f62\u6b63\u5f26\u5f62\u72b6\u504f\u79bb\u7a0b\u5ea6\u7684\u8868\u5f81\u3002\u5173\u4e8e\u5927\u5e45\u632f\u8361\u526a\u5207\u548c\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7684\u539f\u7406\u4e0e\u5b9e\u8df5\uff0c\u53ef\u89c1Hyhn\u7b49\u7684\u7efc\u8ff0\uff0c\u6216\u8005\u53c2\u9605\u6211\u7684\u535a\u58eb\u5b66\u4f4d\u8bba\u6587\u7b2c1.2.1\u8282\u3001\u7b2c1.3\u8282\u548c\u7b2c2\u7ae0\u3002\u672c\u6587\u53ea\u89e3\u51b3MATLAB\u7f16\u7a0b\u5b9e\u73b0\u95ee\u9898\u3002 \u5927\u5e45\u632f\u8361\u526a\u5207\uff08LAOS\uff09\u5b9e\u9a8c\u4e2d\uff0c\u6211\u4eec\u4ece\u6d41\u53d8\u4eea\u540c\u65f6\u83b7\u53d6\u4e24\u4e2a\u4fe1\u53f7\u3002\u4e00\u4e2a\u662f\u5e94\u53d8\u7684\u4fe1\u53f7\uff0c\u4e00\u4e2a\u662f\u5e94\u529b\u7684\u4fe1\u53f7\uff08\u4e8b\u5b9e\u4e0a\uff0c\u4e00\u4e2a\u662f\u9a6c\u8fbe\u4f4d\u79fb\u4fe1\u53f7\uff0c\u53e6\u4e00\u4e2a\u662f\u529b\u77e9\u4fe1\u53f7\uff0c\u5b83\u4eec\u90fd\u8981\u4e58\u4ee5\u4e00\u4e2a\u7cfb\u6570\uff0c\u624d\u662f\u5e94\u53d8\u548c\u5e94\u529b\u3002\u6b64\u6587\u5148\u7565\u53bb\u8fd9\u4e00\u4e8b\u5b9e\u4e86\uff09\u3002\u5728\u5e94\u53d8\u63a7\u5236\u7684\u5b9e\u9a8c\u4e2d\uff0c\u6211\u4eec\u65bd\u52a0\u4e00\u4e2a\u6b63\u5f26\u7684\u5e94\u53d8\uff0c\u6d4b\u91cf\u5e94\u529b\u7684\u54cd\u5e94\uff0c\u5373 $$\\begin{cases}\\gamma\\left(t\\right)=\\sin\\omega t\\\\ \\sigma\\left(t\\right)=\\sum_{n=0:\\text{odd}}^\\infty\\sigma_n\\sin\\left(\\omega t+\\delta_n\\right)\\end{cases}$$\u7531\u6b64\u5f0f\u53ef\u5b9a\u4e49LAOS\u5b9e\u9a8c\u7ed3\u679c\u7684\u8868\u5f81\u53c2\u6570\uff0c\u4ee5\u53ca\uff0c\u6765\u8868\u5f81\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7684\u7a0b\u5ea6\uff08\u5373\u504f\u79bb\u6b63\u5f26\u51fd\u6570\u7684\u7a0b\u5ea6\u3002\u6b64\u5916\uff0c\u6211\u4eec\u8fd8\u4f1a\u4ee5\u5bf9\u4f5c\u56fe\uff08\u53c2\u6570\u65b9\u7a0b\u4f5c\u56fe\uff09\uff0c\u5f97\u5230Lissajous\u66f2\u7ebf\u3002\u7136\u540e\u7528\u66f2\u7ebf\u7684\u5f62\u72b6\u7279\u5f81\u6765\u8868\u5f81\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7684\u7a0b\u5ea6\uff08\u5373\u66f2\u7ebf\u504f\u79bb\u692d\u5706\u7684\u7a0b\u5ea6\uff09\u3002\u63cf\u8ff0\u66f2\u7ebf\u5f62\u72b6\u7279\u5f81\uff0c\u5f80\u5f80\u5c31\u662f\u5404\u79cd\u5404\u6837\u7684\u5207\u7ebf\u659c\u7387\uff08\u53ef\u81ea\u884c\u4e0a\u7f51\u641c\u7d22\u590d\u4e60\u53c2\u6570\u65b9\u7a0b\u6c42\u5bfc\uff09\u3002\u5728LAOS\u4e2d\u5e38\u7528\uff1a$$G_\\text{M}=\\left.\\frac{d\\sigma}{d\\gamma}\\right|_{\\gamma=0}$$$$G_\\text{L}=\\frac{\\sigma{\\gamma_0}}{\\gamma_0}$$$$G_\\text{k}=\\left.\\frac{d\\sigma}{d\\gamma}\\right|_{\\sigma=0}$$ MATLAB\u7f16\u7a0b\u6b65\u9aa4 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fs=1000; % sampling frequency t=0:1\/fs:30.2; % the time series of the strain signal t=t(:); % convert to colume vector w=5; % (rad\/s) the angular frequency of 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2005, 438, 997-1100\uff09\u8003\u5bdf\u4e86\u60ac\u6d6e\u6db2\u5728\u632f\u8361\u526a\u5207\u4e0b\u60ac\u6d6e\u7c92\u5b50\u7684\u884c\u8e2a\u95ee\u9898\u3002\u53d1\u73b0\u5982\u679c\u632f\u8361\u526a\u5207\u7684\u5e45\u5ea6\u8db3\u591f\u5c0f\uff0c\u60ac\u6d6e\u7c92\u5b50\u57fa\u672c\u4e0a\u539f\u5730\u4e0d\u52a8\uff1b\u5f53\u5e45\u5ea6\u5927\u4e8e\u4e00\u5b9a\u7a0b\u5ea6\uff0c\u7c92\u5b50\u5c31\u6df7\u8d77\u6765\u4e86\uff0c\u5177\u6709\u6269\u6563\u7684\u6027\u8d28\u3002\u8be5\u5c0f\u7ec4\u8fdb\u4e00\u6b65\uff08Nat. Phys. 2008, 4, 420-424\uff09\u901a\u8fc7\u5efa\u6a21\u5728\u4e8c\u7ef4\u7a7a\u95f4\u91cd\u73b0\u4e86\u8fd9\u4e00\u73b0\u8c61\uff0c\u8868\u660e\u8fd9\u662f\u4e00\u79cd\u76f8\u53d8\uff0c\u6709\u6e1d\u6e17\u9608\u503c\u3002\u8fd9\u51e0\u5929\u53c8\u770b\u5230Menon\u5bf9\u8fd9\u4e00\u73b0\u8c61\u8fdb\u884c\u540e\u7eed\u7814\u7a76\uff08Phys. Rev. E 2009, 79, 061108\uff09\uff0c\u8c8c\u4f3c\u60f3\u8ba8\u8bbauniversality\u7684\u95ee\u9898\uff0c\u4f46\u5177\u4f53\u6211\u8fd8\u6ca1\u770b\u61c2\u3002\u8fd9\u4e00\u7cfb\u5217\u7814\u7a76\u5df2\u7ecf\u662f\u5728\u8ba8\u8bba\u632f\u8361\u526a\u5207\u7684\u5e45\u5ea6\u95ee\u9898\u4e86\uff0c\u53ea\u662f\u6ca1\u6709\u8ddf\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u8054\u7cfb\u8d77\u6765\u3002\u4f5c\u8005\u662f\u5177\u4f53\u5230\u60ac\u6d6e\u6db2\u8fd9\u4e00\u4f53\u7cfb\uff0c\u4ececreep flow\u7684\u65b9\u7a0b\u51fa\u53d1\u8003\u8651\u95ee\u9898\uff0c\u76f8\u53d8\u539f\u56e0\u89e3\u91ca\u4e3a\u7c92\u5b50\u4e4b\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\u5bfc\u81f4\u7684\u975e\u5e73\u8861\u6001\u3002\u800c\u5728\u6d41\u53d8\u5b66\u4e2d\uff0c\u5927\u5e45\u5f62\u53d8\u4e0b\u8bd5\u6837\u51fa\u73b0\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7684\u89e3\u91ca\u4e5f\u662f\u67d0\u79cd\u5e73\u8861\u7684\u7834\u574f\u3002\u6211\u731c\u6d4b\u8fd9\u4e24\u8005\u662f\u540c\u4e00\u4ef6\u4e8b\uff0c\u51fa\u73b0\u6df7\u6c8c\u7684\u90a3\u4e2a\u76f8\u53d8\u9608\u632f\u5e45\u5c31\u5bf9\u5e94\u7740\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u4e0d\u53ef\u5ffd\u7565\u7684\u90a3\u4e2a\u632f\u5e45\u3002Nat. Phys.\u90a3\u7bc7\u6587\u7ae0\u8fd8\u8bf4\u76f8\u53d8\u4e4b\u540e\u6709\u81ea\u7ec4\u7ec7\u884c\u4e3a\uff0c\u800c\u8bb8\u591a\u8bd5\u6837\u5728\u5927\u5e45\u526a\u5207\u4e0b\u4e5f\u7684\u786e\u51fa\u73b0\u8bf8\u5982shear banding\u3001shear-induced ordering\u7684\u73b0\u8c61\uff0c\u6211\u731c\u6d4b\u4e8c\u8005\u4e5f\u662f\u5bf9\u5e94\u7684\u3002\u4e0d\u8fc7\uff0c\u5176\u4ed6\u8bd5\u6837\u60ac\u6d6e\u4f53\u7cfb\uff0c\u5982\u679c\u8981\u8bc1\u660e\u8fd9\u4e9b\u90fd\u662f\u540c\u4e00\u7c7b\u76f8\u53d8\u7684universality\uff0c\u4f3c\u4e4e\u8981\u5728creep flow\u65b9\u7a0b\u7684\u57fa\u7840\u4e0a\u8fdb\u4e00\u6b65\u62bd\u8c61\u7684\u7406\u8bba\u624d\u884c\u3002\u6211\u662f\u505a\u5b9e\u9a8c\u7684\u4e0d\u662f\u505a\u7406\u8bba\u7684\uff0c\u6211\u7422\u78e8\u7740\u8fd9\u4e00\u76f8\u53d8\u5728\u6d41\u53d8\u5b66\u51fd\u6570\u4e2d\u80fd\u4e0d\u80fd\u4f53\u73b0\u51fa\u6765\uff0c\u80fd\u4e0d\u80fd\u628a\u5b83\u4ece\u5b9e\u9a8c\u4e0a\u505a\u51fa\u6765\u3002Menon, G., & Ramaswamy, S. (2009). Universality class of the reversible-irreversible transition in sheared suspensions Physical Review E, 79 (6) DOI: 10.1103\/PhysRevE.79.061108","rel":"","context":"In "\u4ee5tag\u5206\u7c7b\u7684\u6587\u7ae0"","block_context":{"text":"\u4ee5tag\u5206\u7c7b\u7684\u6587\u7ae0","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/category\/tagged"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":5377,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/5377","url_meta":{"origin":5297,"position":2},"title":"Fourier\u53d8\u6362\uff08\u4e09\uff09","author":"Andrew","date":"2013\u5e745\u67082\u65e5","format":false,"excerpt":"[mathjax] 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sweep\u662f\u56fa\u5b9a\u632f\u8361\u7684\u5e94\u53d8\u5e45\u5ea6\u03b30\u548c\u9891\u7387\u03c9\uff0c\u6d4b\u91cfG'\u548cG''\u968f\u65f6\u95f4\u53d8\u5316\u7684\u6d4b\u8bd5\uff0c\u6240\u4ee5\u6b63\u9002\u5408\u6211\u4eec\u3002\u5728Orchestrator\u8f6f\u4ef6\u4e2d\u8bbe\u5b9a\u597ddynamic time sweep\u7684\u5e94\u53d8\u548c\u9891\u7387\uff0c\u4eea\u5668\u5c31\u4f1a\u4ee5\u76f8\u5e94\u7684\u6b63\u5f26\u51fd\u6570\u53bb\u63a7\u5236\u9a6c\u8fbe\u4e86\u3002\u6b64\u65f6\uff0c\u6211\u4eec\u9700\u8981\u9a6c\u4e0a\u4ece\u4eea\u5668\u80cc\u540e\u7684\u6a21\u62df\u8f93\u51fa\u7aef\u53e3\u540c\u65f6\u8bfb\u51fa\u4eea\u5668\u7684\u6d4b\u91cf\u4fe1\u53f7\uff0c\u5e76\u7528\u4e00\u4e2a\u6570\u6a21\u8f6c\u6362\u5668\uff08analog-to-digital converter, ADC\uff09\u8f6c\u6362\u6210\u6570\u5b57\u4fe1\u53f7\uff0c\u8f93\u5165\u5230\u8ba1\u7b97\u673a\u3002\u6211\u4eec\u7528\u7684ADC\u662fNI 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Meassurement\u63a7\u5236ADC\u3002\u8fd9\u4e00\u6d41\u7a0b\u56fe\u53ef\u53c2\u8003\u548c\u4e2d\u7684\u7528\u56fe\u3002\u5177\u4f53\u4fe1\u53f7\u7ebf\u63a5\u6cd5\u548c\u5904\u7406\u65b9\u5f0f\u89c1\u4e0b\u9762\u7684PPT\uff08\u53ef\u4e0b\u8f7d\uff09\u3002 ARES\u6a21\u62df\u8f93\u51fa View more presentations or Upload your own. \u4e0a\u7bc7\u6587\u7ae0\u4e5f\u8bf4\u8fc7\uff0c\u4ece\u6d41\u53d8\u4eea\u91c7\u96c6\u5230\u7684\u4e00\u4e2a\u662f\u9a6c\u8fbe\u4f4d\u79fb\u7684\u7535\u538b\u4fe1\u53f7\uff0c\u53e6\u4e00\u4e2a\u662f\u529b\u77e9\u7684\u7535\u538b\u4fe1\u53f7\uff0c\u5b83\u4eec\u90fd\u8981\u4e58\u4ee5\u4e00\u4e2a\u7cfb\u6570\uff0c\u624d\u662f\u5e94\u53d8\u548c\u5e94\u529b\u3002\u4eceARES\u6d41\u53d8\u4eea\u7684\u8bf4\u4e66\u4e2d\u53ef\u4ee5\u67e5\u5230\u80cc\u677f\u8f93\u51fa\u4fe1\u53f7\u7684\u5177\u4f53\u60c5\u51b5\uff1a TORQUE OUT\uff1a\u4ece0\u5230\u6ee1\u91cf\u7a0b\uff0c\u5bf9\u5e94\u7535\u538b\u662f0 ~ 5 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