《结构动力学》课程教学课件（讲稿）11 Multi-degree-of-freedom systems（MDOF）Undamped free vibrations

Wuhan University of TechnologyPartIIMulti-degree-of-freedom systems(MDOF)Chapter 11Undamped free vibrations4-1

4-1 Wuhan University of TechnologyChapter 11 Undamped free vibrations Part II Multi-degree-of-freedom systems (MDOF)

Wuhan University of TechnologyContents11.1Analysisof vibrationfrequencies11.2 Analysis of vibration mode shapes11.3 Flexibility formulation of vibration analysis11.4 Influence of axial forces11.5 Orthogonality conditions4-2

4-2 Wuhan University of Technology 11.1 Analysis of vibration frequencies 11.2 Analysis of vibration mode shapes 11.3 Flexibility formulation of vibration analysis 11.4 Influence of axial forces 11.5 Orthogonality conditions Contents

Wuhan University of Technology?11.1 Analysis of vibration frequenciesmv+kv=0v(t) = sin(wt +0)= -w2 v sin(wt +0) = -w? -w?mv sin(wt+)+kv sin(wt+)=0[k-w?m] v= 04-3

4-3 Wuhan University of Technology 11.1 Analysis of vibration frequencies

WuhanUniversityof Technology11.1Analysisof vibrationfrequenciesEquation(114)isonewayofexpressingwhatiscalledaneigenvalueorcharacteristicvalueproblem.Thequantitiesw2aretheeigenvaluesorcharacteristicvaluesindicatingthe squareof thefreevibrationfrequencies,whilethecorrespondingdisplacementvectorsyexpressthecorrespondingshapesofthevibrating system.knownastheeigenvectorsormodeshapes.Nowit canbeshownby Cramer's rulethatthe solutionof this set of simultaneous equationsisoftheform0k-w2mlk-w2m=00 =[01,02, 0..0, ]4-4

4-4 Wuhan University of Technology Equation (114) is one way of expressing what is called an eigenvalue or characteristic value problem. The quantities are the eigenvalues or characteristic values indicating the square of the freevibration frequencies, while the corresponding displacement vectors express the corresponding shapes of the vibrating system . known as the eigenvectors or mode shapes. Now it can be shown by Cramer's rule that the solution of this set of simultaneous equations is of the form 11.1 Analysis of vibration frequencies 123 [ , , . ] T ω    n

Wuhan University of Technology11.1 Analysis of vibration frequenciesExampleE111.Theanalysisofvibrationfrequenciesbythesolutionofthedeterminantalequation(116)willbedemonstratedwithreferencetothestructureofFig.E111,thesameframeforwhichanapproximationofthefundamental frequencywasobtainedbytheRayleighmethod inExampleE86.The stiffness matrix for this frame can be determined by applying a unitdisplacementtoeachstoryinsuccessionand evaluatingtheresultingstoryforces,asshown inthefigure.Becausethegirdersareassumedto be rigid,thestoryforces caneasilybedetermined herebymerelyaddingthesideswaystiffnessesoftheappropriatestories.4-5

4-5 Wuhan University of Technology Example E111. The analysis of vibration frequencies by the solution of the determinantal equation (116) will be demonstrated with reference to the structure of Fig. E111, the same frame for which an approximation of the fundamental frequency was obtained by the Rayleigh method in Example E86. The stiffness matrix for this frame can be determined by applying a unit displacement to each story in succession and evaluating the resulting story forces, as shown in the figure. Because the girders are assumed to be rigid, the story forces can easily be determined here by merely adding the sidesway stiffnesses of the appropriate stories. 11.1 Analysis of vibration frequencies

Wuhan Universityof Technology11.1 Analysis of vibration freguenciesU, = 1U=1U,=11.0kip.sec/inUk= 600k12=-600k13=0600kips/in1.5U2k21=-600k22=1,800k23=1,2001,2002.0U3k32=—1,200k33=3,000k31=01,800(a)(b)FIGUREE11-1Frameusedinexampleofvibrationanalysis:(a)structuralsystem;(b)stiffnessinfluencecoefficients4-6

4-6 Wuhan University of Technology 11.1 Analysis of vibration frequencies FIGURE E11-1 Frame used in example of vibration analysis: (a) structural system; (b) stiffness influence coefficients

Wuhan University of Technology11.1 Analysis of vibration freguencies0[1.00001.5m=(1kip·sec2/in)002.0]-103-2-1k=(600kips/in)0-25]1-B-10k-w2m=(600kips/in)3-1.5B-2-10-25-2B2B-6004-7

4-7 Wuhan University of Technology 11.1 Analysis of vibration frequencies

Wuhan University of Technology11.1 Analysis of vibration freguenciesB3-5.5B2+7.5B-2=0B1 = 0.3515B2= 1.6066.B3=3.5420210.8814.522W1we963.9631.048rad/secw2w2,125.2046.100w34-8

4-8 Wuhan University of Technology 11.1 Analysis of vibration frequencies

Wuhan UniversityofTechnology11.2 Analysis of vibration mode shapesEE(n)=飞-wmForthispurposeitwillbeassumedthatthefirstelementofthedisplacementvector has a unit amplitude; that is,0in02n02n03n03n.:UNN4-9

4-9 Wuhan University of Technology 11.2 Analysis of vibration mode shapes For this purpose it will be assumed that the first element of the displacement vector has a unit amplitude; that is

Wuhan University of Technology11.2 Analysis of vibration mode shapeseete(n)-01eiNelteee(n)em020e2Ne(n)es(n)(n)03n0e31e33e3Ne(n)emem(n)LeNIUNeNNEf)enEonEo(n)(n)Eo1+E00Von=0+E(n)=0Voneii4-10

4-10 Wuhan University of Technology 11.2 Analysis of vibration mode shapes