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??座右銘：行百里者，半于九十。

??????本文目錄如下：??????

目錄

??1 概述

??2 運行結(jié)果

??3?參考文獻

??4 Matlab代碼及詳細文章

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??1 概述

文獻來源：

這項工作提出了一類檢測和表征暫態(tài)響應測量非線性的時頻方法。這些方法適用于響應隨著響應幅度的衰減而變得越來越線性的系統(tǒng)。響應數(shù)據(jù)的離散傅里葉變換是在初始響應的各個部分都為零的情況下得到的。這些頻率響應，被稱為零早期快速傅立葉變換(ZEFFTs)，獲得線性頻率響應函數(shù)(frf)的通常形狀，因為更多的初始非線性響應被取消。因此，非線性是由一個質(zhì)變的形狀的ZEFFT作為初始無效部分的長度變化證明。這些光譜顯示出對非線性的敏感性，即使它僅在響應的前幾個周期中活躍，也能顯示出它的存在，就像機械關節(jié)的宏觀滑移一樣。它們還提供了對非線性特性的洞察，潛在地揭示了模態(tài)之間的非線性能量傳遞或系統(tǒng)線性行為的模態(tài)振幅。在某些情況下，人們可以從后期線性響應中識別線性模型，并使用它來重建系統(tǒng)在以前的時間執(zhí)行的響應，如果它是線性的。這表明了非線性的嚴重程度及其對測量響應的影響。用滑移或沖擊非線性系統(tǒng)的分析和實驗數(shù)據(jù)驗證了這些方法。

原文摘要：

This work presents a class of time-frequency methods for detecting and characterizing?
nonlinearity in transient response measurements. ?The methods are intended for systems whose? response becomes increasingly linear as the response amplitude decays. The discrete Fourier Transform?of the response data is found with various sections of the initial response set to zero. ?These frequency?responses, dubbed Zeroed Early-time Fast Fourier Transforms (ZEFFTs), acquire the usual shape of?linear Frequency Response Functions (FRFs) as more of the initial nonlinear response is nullified. ?Hence, nonlinearity is evidenced by a qualitative change in the shape of the ZEFFT as the length of the?initial nullified section is varied. ?These spectra are shown to be sensitive to nonlinearity, revealing its?presence even if it is active in only the first few cycles of a response, as may be the case with macro-slip?in mechanical joints. ?They also give insight into the character of the nonlinearity, potentially revealing?nonlinear energy transfer between modes or the modal amplitudes below which a system behaves?linearly. ?In some cases one can identify a linear model from the late time, linear response, and use it to?reconstruct the response that the system would have executed at previous times if it had been linear. ?
This gives an indication of the severity of the nonlinearity and its effect on the measured response. ?The?methods are demonstrated on both analytical and experimental data from systems with slip or impact?nonlinearities.?

??2 運行結(jié)果

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部分代碼：

% Specify how systems are connected
? ? at_ns = [2,6]; % attachment happens between this pair of nodes.
? ? fext_ns = 5; % Node at which external force is applied
? ? fnl_vec = zeros(Ntot,1); fnl_vec(at_ns(1)) = -1; fnl_vec(at_ns(2)) = 1;
? ? fext_vec = zeros(Ntot,1); fext_vec(fext_ns(1)) = 1;
? ? dnodes = [1:Ntot];
? ? vnodes = [(Ntot+1):(Ntot*2)];
? ??
? ? % Damping Matrix ? ?
? ? cfactk = 0.00003; % multiplied by K to get damping.
? ? cfactm = 8; % Multiplied by M to get damping
? ? % for k = 1:5; eval(['c',num2str(k),' = cfact*k',num2str(k),';']); end
? ? C1 = cfactk*K1 + cfactm*M1;
? ? C2 = cfactk*K2 + cfactm*M2; % proportional damping
? ? Ctot = [C1, zeros(size(C1,1),size(C2,2));
? ? ? ? zeros(size(C2,1),size(C1,2)), C2];
? ??
? ? % Linearized System Analysis %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
? ? ? ? % add a linear spring between attachment nodes
? ? ? ? Mlin = Mtot;
? ? ? ? Klin = Ktot;
? ? ? ? Klin(at_ns,at_ns) = Klin(at_ns,at_ns) + kat*[1, -1; -1, 1];
? ? ? ? % Linear Damping matrix:
? ? ? ? Clin = Ctot;
? ? ? ? Clin(at_ns,at_ns) = Clin(at_ns,at_ns) + cfactk*kat*[1, -1; -1, 1];

? ? % State Space Eigenanalysis
? ? Slin = [Clin, Mlin; Mlin, zeros(size(Mlin))];
? ? Rlin = [-Klin, zeros(size(Mlin));
? ? ? ? zeros(size(Mlin)), Mlin];
? ? Alin = (Slin\Rlin);
? ? [Philin,lamlin] = eig(Alin);
? ? lamlin = diag(lamlin);
? ? [junk,sind] = sort(abs(lamlin) - 0.001*min(abs(lamlin))*(imag(lamlin) > 0));
? ? lamlin = lamlin(sind);
? ? Philin = Philin(:,sind);

% ? ? ? ? % Plot Mode Shapes
% ? ? ? ? figure(1)
% ? ? ? ? hls = plot([1:Ntot], imag(Philin(1:Ntot,1:2:end ))); grid on;
% ? ? ? ? legend(hls, num2str([1:Ntot].'));
% ? ? ? ? xlabel('X-coordinate');
% ? ? ? ? ylabel('Im\{Mode Shape\}');

? ? wns = abs(lamlin);
? ? fns = wns/2/pi;
? ? zts = -real(lamlin)./abs(lamlin);
? ? disp('Natural Frequencies:, Damping Ratios:');
? ? [fns, zts]
% ? ? DispFnZt(lamlin) - replace 2 lines abouve with this if you have the EMA Functions toolbox
? ??
? ? % Nonlinear Parameters
? ? NLType = 'bang'
? ? if strcmp(NLType,'bang');
? ? ? ? % Bang (Contact) Nonlinearity
? ? ? ? delcont = 1e-3;
? ? ? ? k4mult = 20; % Factor by which k4 increases: k4_contact = k4*k4mult
? ? ? ? c4mult = 1; % Factor by which c4 increases: c4_contact = c4*c4mult
? ? elseif strcmp(NLType,'cubic');
? ? ? ? % Cubic Spring
? ? ? ? katnl = 1e8;
? ? else
? ? ? ? error('NLType not recognized');
? ? end
? ? % Force paramters
? ? ? ? % length of half-sine force pulse
? ? ? ? % This is normalized in the EOM to unit area and multiplied by Afnl
? ? ? ? tfp = 1e-4;
? ? ? ? Afnl = 4e9;
? ? ? ??
? ? % Which Response to Use in evaluating Nonlinearity (i.e. x1, x2, x3..?)
? ? respind = 6;
? ??
? ? % Number of numerical derivatives to evaluate. ?This M-file simulates
? ? % the displacement response of the 5-DOF system. ?To simulate the
? ? % measurement of the velocity or acceleration response, the
? ? % displacement response is differentiated using finite differences
? ? % (i.e. Matlab's 'diff' command.) ?This parameter sets the number of
? ? % derivatives to perform:
? ? nders = 2;
? ? ? ? % nders = 0; => use displacement

??3?參考文獻

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??4 Matlab代碼及詳細文章

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