《结构动力学》课程教学课件（讲稿）13 Vibration analysis by matrix iteration

Wuhan University of TechnologyChapter13Vibration analysis by matrixiteration13-1

13-1 Wuhan University of Technology Chapter 13 Vibration analysis by matrix iteration

Wuhan University of TechnologyContents13.1 Preliminary comments13.2 Fundamental mode analysis13.3 Proof of convergence13.4 Analysis of higher modes13.5 Buckling analysis by matrix iteration13.6Special eigenproblemtopics13-2

13-2 Wuhan University of Technology 13.1 Preliminary comments 13.2 Fundamental mode analysis 13.3 Proof of convergence 13.4 Analysis of higher modes 13.5 Buckling analysis by matrix iteration 13.6 Special eigenproblem topics Contents

WuhanUniversityof Technology13.1 Preliminary commentsThe mathematical models developed to solvepractical problems in structuraldynamics range from very simplified systems having only a few degrees offreedomtohighlysophisticatedfiniteelementmodelsincludinghundredsoreventhousands of degrees of freedom in which as many as 50 to 100 modes maycontributesignificantlytotheresponseTodealeffectivelywiththesepracticalproblems,muchmoreefficientmeansofvibrationanalysisareneededthanthedeterminantal solutionproceduredescribed earlier, andthis chapter describesthematrix iterationapproach whichisthebasisof manyofthevibrationor"eigenproblem"solutiontechniquesthatareusedinpractice13-3

13-3 Wuhan University of Technology 13.1 Preliminary comments  The mathematical models developed to solve practical problems in structural dynamics range from very simplified systems having only a few degrees of freedom to highly sophisticated finiteelement models including hundreds or even thousands of degrees of freedom in which as many as 50 to 100 modes may contribute significantly to the response. To deal effectively with these practical problems, much more efficient means of vibration analysis are needed than the determinantal solution procedure described earlier, and this chapter describes the matrix iteration approach which is the basis of many of the vibration or “eigenproblem” solution techniques that are used in practice

Wuhan University of Technology13.2 Fundamental mode analysiskVn=wmVnfin =wm VnVn =k-1 finVn=wzk-1mVnD=k-1 mn=wDVn13-4

13-4 Wuhan University of Technology 13.2 Fundamental mode analysis

Wuhan Universityof Technology13.2 Fundamental mode analysisTo initiatethe iterationprocedurefor evaluatingthefirstmode shape,atrialdisplacement vector v(0) is assumed that is a reasonable estimate of this shape.The zero superscript indicates that this is the initial shape used in the iterationsequence;forconveniencethevectorisnormalized sothat a selected referenceelement is unity.Introducing this on the right sideof Eq.(131)gives anexpressionfortheinertialforcesinducedbythesystemmassesmovingharmonically in this shape at the as yet unknown vibration frequencywimv(0)(0(1)=W?DvV(1) = D v(0)13-5

13-5 Wuhan University of Technology 13.2 Fundamental mode analysis To initiate the iteration procedure for evaluating the firstmode shape, a trial displacement vector is assumed that is a reasonable estimate of this shape. The zero superscript indicates that this is the initial shape used in the iteration sequence; for convenience the vector is normalized so that a selected reference element is unity. Introducing this on the right side of Eq. (131) gives an expression for the inertial forces induced by the system masses moving harmonically in this shape at the as yet unknown vibration frequency

Wuhan University of Technology13.2 Fundamental mode analysis(1)Vref(v()(1)= w v(1) = v(0),(0)Ukwi-.(0)(0)UklUkiuminmaxmv(owimv(1)13-6

13-6 Wuhan University of Technology 13.2 Fundamental mode analysis

Wuhan University of Technology13.2 Fundamental mode analysis11SDmax(v(s-1)1max(v(s))max(v(s))13-7

13-7 Wuhan University of Technology 13.2 Fundamental mode analysis

Wuhan University of Technology13.3 Proof of convergenceThattheStodolaiterationprocessmustconvergetothefirstmodeshape,ingeneral, canbedemonstrated byrecognizingthat itessentially involvescomputing the inertial forces corresponding to any assumed shape, thencomputingthedeflectionsresultingfromthoseforces,thencomputingtheinertialforcesduetothecomputeddeflections,etc.Theconceptisillustrated in Fig. 131 and explained mathematically in the followingparagraph.13-8

13-8 Wuhan University of Technology 13.3 Proof of convergence That the Stodola iteration process must converge to the firstmode shape, in general, can be demonstrated by recognizing that it essentially involves computing the inertial forces corresponding to any assumed shape, then computing the deflections resulting from those forces, then computing the inertial forces due to the computed deflections, etc. The concept is illustrated in Fig. 131 and explained mathematically in the following paragraph

Wuhan Universityof Technology13.3 Proof of convergenceComputed shape u,(l)Assumed shape u,(0)U1,(0)02;(0)U3;(0)D2,(1)U,,(1)Us(1)0?OResultinginertial forcesfi(t)Resulting inertial forcesf,(o)faroJiafi."fh.(0)fi.rofia)Etc.FIGURE13-1PhysicalinterpretationofStodolaiterationsequence.13-9

13-9 Wuhan University of Technology 13.3 Proof of convergence FIGURE 13-1 Physical interpretation of Stodola iteration sequence

Wuhan Universityof Technology-13.3 Proof of convergenceTheinitiallyassumedshapeisexpressedinnormalcoordinates[seeEg(122)) asv(0) = (0) = 1Y(0) + 2) + g0)Theinertialforcesassociatedwiththisshapevibratingatthefirstmodefrequency will be [see Eq. (1133)]f,(0) =wm v(0) =wim±y(0)=m011(0+020)+03gy(0)f.(o)w3DD:v(1) = k-1f,(0) = k-1mpiwry(0)+02wY0wo13-10

13-10 Wuhan University of Technology 13.3 Proof of convergence The initially assumed shape is expressed in normal coordinates [see Eq. (122)] as The inertial forces associated with this shape vibrating at the firstmode frequency will be [see Eq. (1133)]