Carnegie Mellon UniversityResearch Showcase @ CMURobotics Institute School of Computer Science1981Te use of matrix displacement method forvibrational analysis of structuresWilliam O. HughesCarnegie Mellon UniversityFollow this and additional works at: htp://repository.cmu.edu/roboticsTis Technical Report is brought to you for free and open access by the School of Computer Science at Research Showcase @ CMU. It has beenaccepted for inclusion in Robotics Institute by an authorized administrator of Research Showcase @ CMU. For more information, please contactresearch-showcase@andrew.cmu.edu.Recommended CitationHughes, William O., "Te use of matrix displacement method for vibrational analysis of structures" (1981). Robotics Institute. Paper479.htp://repository.cmu.edu/robotics/479NOTICE WARNINGCONCERNINGCOPYRIGHTRESTRICTIONS:The copyrightlaw of the UnitedStates (title17, U.S. Code) governs the makingof photocopies or other reproductionsof copyrighted material.Any copying of thisdocument without permissionof its author may be prohibited by law.The Use of Matrix Displacement Method forVibrational Analysis of StructuresbyWilliam 0. Hughes1Department of Mechanical Engineering andRoboticsInstituteCarnegie-MellonUniversityPittsburgh, PA15213May, 1981ABSTRACTA studyof the matrixdisplacementmethodformodelingthe vibrations ofstructures is presented in this report.The model can analyze both the free andforced vibrations of a structure.Staticloading on a structure is treated as aspecial case of the forced vibration analysis.The workwas supported in part by Carnegie-MellonUniversityInternal funding, through a FordFoundation Research Grant, and by the Robotics Institute.1fjwi@r the Supanrisioni of Prof. W.L.. Wh&aJcar md Prof. A.A HolierTableofContents1Introduction12 The Finite Element Method Fundamental Concepts and Applications13 Explanation of the Model 23.1 Equations of Motion23.2 The Matrix DisplacementMethod43.3Specific Aspects of Model-54 The Model:Examples and Accuracy74.1 Example1: Free Vibrati.on of a Fixed-Free Uniform Beam94.2 Example 2:Free Vibration of a Fixed-Fixed Uniform Beam104.3Example 3:Forced Vibration of a Fixed-Free UniformBeam104.4 Example 4:StaticDeflection of a Fixed-FreeUniform Beam134.5Example 5:Static Deflection of a Fixed-Free Non-UniformBeam155 The Extension of the Model to Model A Turbine Blade*186 Conclusion*21I.AppendixI InfluenceCoefficient Method24II. Appendix II Variational Method26III. Appendix 111 Computer Code of Model3111ListofFigures.Figure 1:The beamelement and its forces, afterPrzemieniecki [7]6Figure 2:StiffnessMatrixofBeamElementofFigure1[AfterPrzmieniecki].[Thesheer8deformationparameters 4>and 4> can be considered to be zero.]Figure 3:Consistent Mass Matrixfor a Beam Element (After Przemieniecki[7]) 99Figure 4:Example1: Fixed-Free UniformBeam.9Figure 5:First five bending mode shapes of Example 1. 12Figure 6:First four axial mode shapes of Example1.14Figure 7:Example 2:Fixed-FixedUniformBeam.15Figure 8:Example3:Fixed-Free UniformBeam With Dynamic Load,15Figure 9:Magnitude versus ForcingFrequencyforExainple 3.16Figure10:Example 4: Fixed-Free UniformBeam With Static Load .17Figure11:Static Deflectionof a UniformBeam, Example 4#17Figure 12:Example 5: Static Deflectionof aFixed-Free Non-UniformBeam -18Figure 13:Static Deflectionof a Non-UniformBeam, Example 5 ,19Figure 14:Element StiffiiessInfluenceCoefficients(AfterWhite, et al [10])424Figure 15:Stiffnessmatrix of prismatic elementsof Figure14 _25Figure 16:Axial element, cross-sectional area A, modulus E.-.29IllListofTablesTable 1:UniformBeam Properties %". 8Table 2:Natural frequencies (radians/sec)andPercentageError(%)as afunctionofnumberof11elements for Example 1.Table 3:CalculatedandExactNaturalFrequenciesinAxialMode.Calculatedvalueusedfive13element model, for Example 2.Table 4:Calculated and Exact Values of Deflections for Example 4 181IntroductionAstudyof thematrixdisplacementmethodformodelingthevibrationsof structuresispresentedinthisreport.The model can analyze boththe free and forced vibrations of a structure.Static loading on a structureis treated as a special caseof theforcedvibrationanalysis.Abrief reviewof theFiniteElementMethodanditspresentuseisfirstgiven.Thisisfollowedbyadiscussionof themethodologyof thematrixdisplacementapproachandadescriptionof thespecificmodelused.Examplesof theuseof themodeltoanalyzethefrequenciesandmodeshapesof thefreeandforcedresponseof a beamstructureandthestaticdeflections of a beamstructureare shownandcompared withtheclosed form solutions.Finally,ways of extending the model toa more complicated structure,a turbine blade,are discussed-Conclusions are then drawn.2 TheFinite Element Method--FundamentalConceptsandApplicationsTherearemanymethodsavailabletodaywhichperformtheanalysisof structures.Forexample,inonemethodthestructureisdescribedbydifferentialequations.Thedifferentialequationsarethensolvedbyanalyticalor numerical methods.Another method of analysis is the finiteelement method (FEM).Inthismethod,thestructureisidealizedintoanassemblyof discretestructuralelements,eachhavinganassumedformof displacement orstressdistribution.Thecompletesolutionisthenobtainedbyassemblingtheseindividual,approximate,displacementorstressdistributionsinawaysatisfyingtheforceequilibriumequations,theconstitutiverelationshipsof thematerial,thedisplacementcompatibility betweenandwithintheelementsand theboundary conditions of the structure.Methods based on discrete element idealizationhavebeen usedextensivelyinstructuralanalysis/The earlypioneering works of Turner, et al.,in1956[1],and Argyris in1960 [2] led to theapplicationof this method tostaticanddynamicanalysisof aircraftstructures.Otherfieldsof structuralengineering,suchasnuclearreactor design and ship construction have since employed this method.Nor istheidea of discreteelementslimitedinuse tostructuralanalysis only.Thefundamentalconcept ofthefiniteelement methodisthatanycontinuousquantity,suchasdisplacements,temperature,or pressure,canbeapproximated by a finite number of elements.Thus,this, approachcanbeusedtosolveproblems inheatflow,fluiddynamics,electro-magnetics*fracturemechanicsandseepageflowtoname justafew "otherareas of usage.The representation of a continuous structure by structuralelementsof finitesize results in large systems ofalgebraicequations.Aconvenientway of handlingthesesetsofequationsis bytheuseofmatrixalgebra,which also has the advantage of being ideally suitedforcomputationsonhigh-speeddigital computers.Forthis reason, expressionssuchas "matrixmethodsofstructuralanalysis"aresometimesusedto describethemethod.More commonthough is the term"finiteelement method",whichemphasizesthediscretisationofthe structure.The finite element method actually encompasses three classes of matrix methods of structural analysis.Thefirstisthedisplacement(orstiffnessmethod),wherethedisplacementsofthenodesareconsideredtheunknowns.The correct set of displacements resultsfromsatisfyingtheequationsof forceequilibrium.Thesecond method is the force (or flexibility) method.Herethe nodal forces are the unknowns and are foundbysatisfyingthe conditions of compatible of deformationsof the members.The thirdclass of matrix method isthe mixed method, which is a combined force-displacementmethod.One last commentonthe finiteelement methodingeneralis necessary.Anerroris introducedinto thesolutionof theoriginalproblemas soon asthecontinuousstructureisreplacedbydiscreteelements.Thiserror remains, even when the discrete element analysisis performedexactly.Ingeneralthis error is reducedby increasingthe numberof discreteelements, therebydecreasingtheelementsize andthus giving a betteridealizationofthecontinuousstructure.Zienkiewicz,BrottonandMorton[3]suggestthattheusermaydetermine thelimits of his error by:"(a) comparisonoffiniteelementcalculationswithexact solutionsforcases similar to his specificproblem;(b) a'convergencestudy* inwhichtwoormoresolutions areobtainedusing progessively finer subdivisions and the results plottedto establishtheir trend or(c) using experience ofprevious calculationsasa guideto thetreatmentofthespecificproblem."Furtherinformationonmatrixstructural analysis and the finite element method may be found in many sources.[4-11]3 Explanation of the ModelThefollowingdiscussionisdividedintothreesections.Firstlydieequationsofmotionwillbestated.Secondly, the matrix displacement methodfor solving such equations will be described.Fmally some specificaspects of the particular model being used will be discussed3.1 Equations of MotionThemotionofavibratingsystem,consistingofmassandstiffness,ofndegreesoffreedomcanberepresented by n differentia! equations of motion.These equations of motionmaybeobtained byNewton'ssecond law of motion, byLag3range*s equationor by theInfluenceCoefficientsmethod.Since theequationsofmo t i o n ,i ng e n e r a l ,a r e n o ti n d e p e n d e n to fe a c ho t h e r ,as i mu l t a n e o u ss ol ut i onoft h e s ee q u a t i o n sisr equi r edt o c a l c ul a t et h e f r equenci esof t h es y s t e m.T h ema t r i xe q u a t i o nfor t h e free v i b r a t i o ncas e i s :[ K - < J2M ] [ X ]=[0](1)where[K]r e p r e s e n t st h e stiffnessma t r i xo f t h es t r u c t u r e ,[M]r e p r e s e n t st h e i ne r t i a l(mass)ma t r i xo f t hes t r u c t u r e ,idr e p r e s e n t st h e s e t o f e i ge nva l ue sof t h ee q u a t i o n sc o r r e s p o n d i n gt o t h e s e t o f n a t u r a lf r e que nc i e s ,PC]r e p r e s e n t st h e s e t o f ei genf unct i onso f t hee q u a t i o n sc o r r e s p o n d i n gt o t h e s e t o f d i s p l a c e me n t sFo rt hefreev i b r a t i o ncaset he s et of forcesis j u s tz e r o .Th ema t r i x[K-and $zcan be taken as zero.This matrix may be obtained in various ways, two of which are the influencecoefficientsmethod and the variational method, which arc outlined in Appendices I andII.The inertia! matrix for the beam element is shown in Fig.3.This matrixis obtained by the same methods asthe stiffiiessmatrix, as described in Appendices I andII.Liepe^s [13] gives a third way of calculating the stiffnessand inertial matrices.The structuralmatrixfor both stiffnessand inertiais obtainedby superpositionof the individual elementalmatrices-Actual superposition occurs only when degrees of freedomare common to more than one clement1Computer CodingThe computer code itself contains ten subroutines, calledby the mainprogram,entitledVIBRAT.A briefexplana&on of the subroutines will now be given.INPUT-Thissubroutineaskstheuserforthenecessaryinformationwhichisneededtoassemblethestructure.Informationsuchas:freeorforcedcase,numberofelements,coordinatesofnodes,physicalparameters,structuralloading,andconstraineddegreesoffreedomareinputted in this section;CONECT-Thissubroutineestablishesthegeometryofthemodel.Itdeterminesthedistancesbetweenadjacentnodes of thestructure.KMAT-Thissubroutinecalculatestheelementalstiffnessmatrixforeachelementandthenassemblesthestructural stiffnessmatrixfromthem.MM AT - This is similar to KMAT only here the mass or inertial matrices are calculated.EIGEN - This subroutineis calledforthe free vibrationcase.The purposeof it is to calculate theeigenvalues(naturalfrequencies)and eigenvectors (mode- shapes) of equation (1).This subroutinecallstwoothersubroutines;EIGZF,anIMSLroutinewhichactuallydoesthesolving,andCLAMPR,which determines which degrees of freedomare constrained.SOLVE-Thissubroutineis calledfortheforcedvibrationcase.Thisroutinesolvesequation(2)forthedisplacementThis subroutine also calls two other subroutines: LEQT1F, an IMSLroutinewhich does the solving, and CLAMPR,which determines theproper degrees offreedomtobeconstrained.REMARK-isasubroutinewhosepurposeistoexplaintheuseofthemainprogramVIBRATanditssubroutines.Information.onthenomenclatureandfilestructureusedcanbefoundinREMARK.Theuserofthemodelis recommendedtorefertoREMARKifhehasanyquestions on the computer code used in this model.The code for all of these routines may be foundin AppendixIII.4 The Model: ExamplesandAccuracyThis sectionpresentsvarious examples ofuseofthemodel.Theexampleschosenrepresentfivetypesofpossible problems.They are:1. free vibration of a fixed-freeuniformbeam2. free vibration of a fixed-fixeduniformbeam3. forcedvibrationof a fixed-freeuniformbeam4static deflectionof a fixed-freeuniformbeam5. static deflectionof a fixed-free non-uniformbeam.The accuracy of each example is discussed.s ,s