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MATLAB文涛,胡青春(,510640):多自由度振动系统固有频率及主振型计算分析是研究其振动特性的基础,矩阵迭代法是计算固有频率及主振型的基本方法之一根据矩阵迭代的方法,利用MATLAB编程并验证程序的正确性通过程序的运行,能快速获得多自由度振动系统的固有频率以及主振型,为设计人员提供了防止系统共振的理论依据,也为初步分析各构件的振动情况以及解耦分析系统响应奠定了基础:MATLAB;多自由度;振动系统;固有频率;主振型:TH113:A:1672-1616(2007)01-0078-04,,,(),,,2[1]:(1);(2)2,,,,,,,,,,,,1MATLABMATLAB,[2],,VisiualBasicFortran[3]VisiualBasicFortran,,,,,,MATLAB2MATALAB2.1矩阵迭代求解系统固有频率以及主振型分析[4,5],12n,j{A(j)}=[][M][A(j)](1)2n,jA(j)=[M]-1[K][A(j)](2),(1),:2006-09-25:(1982-),,,,7820071361,(1)(1)12n,j{A(j)}=[][M][A(j)]=[D]{A(j)}(3):n,jj;{A(j)}j;[];[M];[D],(j=1){A0}(A0A10,A20,,Aj-10,Aj0),(3){A}1=[D]{A}0=An,1{A}1,{A}2=[D]{A}1=An,2{A}2,{A}j-1=[D]{A}j-2=An,j-1{A}j-1{A}j=[D]{A}j=An,j{A}j(3)12n,j-1{A(j-1)}=An,j-1{A}j-112n,j{A(j)}=An,j{A}jAn,j{A}j(),2n,j=1An,j,2n,j-1(j=1,2n,0=0),,|2n,j-2n,j-1|2n,j!,{Aj}={Aj-1}j,fn,j=n,j2=21An,j(Hz)[D],,j=1,[D],[D]*():{A}0[D],{A}1{A}1=[D]{A}0=[D](c1{A(1)}+c2{A(2)}++cn{A(n)})=c1[D]{A(1)}+c2[D]{A(2)}++cn[D]{A(n)}=c12n,1{A(1)}+c22n,2{A(2)}++cn2n,n{A(n)}(4),{A}112n,1{A(1)}{A}1c12n,1{A(1)},({A}1-c12n,1{A(1)}),(1),{A(i)}T[M][A(j)]=Mi(i=j)0(i∀j){A}0=c1{A(1)}+c2{A(2)}++cn{A(n)}{A(1)}T[M],c1={A(1)}T[M]{A}0{A(1)}T[M]{A(1)}={A(1)}T[M]{A}0[M1](5)c1(4),{A}1-c12n,1{A(1)}=[D]{A}0-12n,1{A(1)}1M1{A(1)}T[M]{A}0=([D]-1M12n,1{A(1)}{A(1)}T[M]){A}0=[D]*{A}0,[D]*,{A}0,{A}1-c12n,1{A(1)},,,[D],()[D]*[D],,[D][D]*=[D]-1M12n,1{A(1)}{A(1)}T[M]-1M12n,2{A(2)}{A(2)}T[M]--1Mn-j2n,n-j{A(n-j)}{A(n-j)}T[M]=[D]-#n-1i=1(1Mi2n,i{A(i)}{A(i)}T[M])[D]*,{A}0,[D]*[D],,2.2程序流程179∃∃MATLAB32,:,,k(6),3图1迭代计算绘图流程图图2等直悬臂梁图3悬臂梁离散后简化模型m,m,m,m,m/2,m/2,,k,:M=m10000010000010000010000005K=k2-1000-12-1000-120000-12-1000-11MATLAB,{A}0=[5;4;3;2;1],:W=km[3.128689300440963e-001;9.079809993470356e-001;1.414213562644755e+000;1.782013047831442e+000;1.975376681190276e+000]A:3.090169944099627e-0015.877852523406680e-0018.090169944099636e-0019.510565163065312e-0011.000000000000000e+000;8020071361-8.090169944490573e-001-9.510565160832560e-001-3.090169939390841e-0015.877852525082009e-0011.000000000000000e+000;9.999999979223553e-0011.510136460905555e-010-9.999999992064176e-001-9.330403120376382e-0111.000000000000000e+000;-8.090169992430517e-0019.510565213984441e-001-3.090169966484762e-001-5.877852519307574e-0011.000000000000000e+000;3.090169907000778e-001-5.877852472344104e-0018.090169907000161e-001-9.510565151010871e-0011.000000000000000e+0004图4各阶主振型示意图4,,(FortranC),MATLAB,,,MATLAB%,,,,:[1],.[M].:,1998.[2]HanselmanD,LittlefieldBMatlab[M].().:,2002.[3],.[M].:,1993.[4],.[M]:,2004.[5].[M].:,2004.AnalysisoftheNaturalFrequencyandMainVibrationModeforMulti-FreedomSystemBasedonMATLABWENTao,HUQing-chun(SouthChinaUniversityofScienceandTechnology,GuangdongGuangzhou,510640,China)Abstract:Thenaturalfrequencyandmainvibrationmodeofmulti-freedomvibrationsystemarethebaseoftheresearchonthevibrationcharacteristics,matrixiterativealgorithmisoneofthebasicmethodstocaculatethesedynamicfeatures.ItprogrammesmatrixiterativealgorithmonMATLAB.Thisprogramcanquicklygetthenaturalfrequencyandthemainvibrationmodeofmulti-freedomvibrationsystems,providetheoryfoundationfordesignerstoavoidsystemresonance.Keywords:MATLAB;Multi-Freedom;VibrationSystems;NaturalFrequency;MainVibrationMode81∃∃MATLAB