Modal Testing - Iran University Of Science &
Modal Testing(Lecture 1)Dr. Hamid AhmadianSchool of Mechanical EngineeringIran University of Science and Technologyahmadian@iust.ac.ir
Overview Introduction to Modal TestingApplications of Modal TestingPhilosophy of Modal TestingSummary of TheorySummary of Measurement MethodsSummary of Modal Analysis ProcessesReview of Test Procedures and LevelsOverview of Modal TestingIUST ,Modal Testing Lab ,Dr H Ahmadian
Introduction to Modal Testing Experimental Structural Dynamics To understand and to control the many vibrationphenomenon in practice Structural integrity (Turbine blades- Suspension Bridges)Performance ( malfunction, disturbance, discomfort)Overview of Modal TestingIUST ,Modal Testing Lab ,Dr H Ahmadian
Introduction to Modal Testing (continued) Necessities for experimentalobservations Nature and extend of vibration in operationVerifying theoretical modelsMaterial properties under dynamic loading(damping capacity, friction, )Overview of Modal TestingIUST ,Modal Testing Lab ,Dr H Ahmadian
Introduction to Modal Testing (continued) Test types corresponding to objectives: Operational Force/Response measurements Response measurement of PZL Mielec Skytruck ModeShapes (3.17 Hz, 1.62 %), (8.39 Hz, 1.93 %)Overview of Modal TestingIUST ,Modal Testing Lab ,Dr H Ahmadian
Introduction to Modal Testing (continued) Modal Testing in acontrolled environment/Resonance Testing/Mechanical ImpedanceMethod Testing a component ora structure with theobjective of obtainingmathematical model ofdynamical/vibrationbehaviorStructural Analysis ofULTRA MirrorOverview of Modal TestingIUST ,Modal Testing Lab ,Dr H Ahmadian
Introduction to Modal Testing (continued) Milestones in the development: Kennedy and Pancu (1947) Bishop and Gladwell (1962) Natural frequencies and damping of aircraftsTheory of resonance testingISMA (bi-annual since 1975)IMAC (annual since 1982)Overview of Modal TestingIUST ,Modal Testing Lab ,Dr H Ahmadian
Applications of Modal Testing Model Validation/Correlation: Producing major test modes validates the model Natural frequenciesMode shapesDamping information are not available in FE modelsOverview of Modal TestingIUST ,Modal Testing Lab ,Dr H Ahmadian
Applications of Modal Testing (continued) Model Updating Correlation of experimental/analyticalmodelAdjust/correct the analytical modelOptimization procedures are used forupdating.Overview of Modal TestingIUST ,Modal Testing Lab ,Dr H Ahmadian
Applications of Modal Testing (continued) Component ModelIdentification Substructure processThe component modelis incorporated into thestructural assemblyOverview of Modal TestingIUST ,Modal Testing Lab ,Dr H Ahmadian
Applications of Modal Testing (continued) Force Determination Knowledge of dynamic force is requiredDirect force measurement is not possibleMeasurement of response Analytical Modelresults the external force([K ] ω [M ]){x} { f }2Overview of Modal TestingIUST ,Modal Testing Lab ,Dr H Ahmadian
Philosophy of Modal Testing Integration of three components: Theory of vibrationAccurate vibration measurementRealistic and detailed data analysisExamples: Quality and suitability of data for processExcitation typeUnderstanding of forms and trends of plotsChoice of curve fittingAveragingOverview of Modal TestingIUST ,Modal Testing Lab ,Dr H Ahmadian
Summary of Theory (SDOF)Overview of Modal TestingIUST ,Modal Testing Lab ,Dr H Ahmadian
Summary of Theory (MDOF)Overview of Modal TestingIUST ,Modal Testing Lab ,Dr H Ahmadian
Summary of Theory Definition of FRF:H (ω ) ([K ] ω 2 [M ] i[D ])x j (ω ) N φ jrφkr.h jk (ω ) 22f k (ω ) r 1 ω r ω 1 Curve-fitting themeasured FRF: Modal Model is obtainedSpatial Model is obtainedOverview of Modal TestingIUST ,Modal Testing Lab ,Dr H Ahmadian
Summary of MeasurementMethods Basic measurement system: Single point excitationOverview of Modal TestingIUST ,Modal Testing Lab ,Dr H Ahmadian
Summary of Modal AnalysisProcesses Analysis of measured FRF data Appropriate type of model (SDOF,MDOF, )Appropriate parameters for chosen modelOverview of Modal TestingIUST ,Modal Testing Lab ,Dr H Ahmadian
Review of Test Proceduresand Levels The procedure consists of: FRF measurementCurve-FittingConstruct the required modelDifferent level of details and accuracy inabove procedure is required dependingon the application.Overview of Modal TestingIUST ,Modal Testing Lab ,Dr H Ahmadian
Review of Test Proceduresand Levels Levels according to Dynamic Testing Agency:LevelNaturalFreqDampingratioMode ShapesUsable forvalidationOut of rangeresiduesUpdating01Only in fewpoints234Overview of Modal TestingIUST ,Modal Testing Lab ,Dr H Ahmadian
Text Books Ewins, D.J. , 2000, “Modal Testing; theory,practice and application”, 2nd edition,Research studies press Ltd.McConnell, K.G., 1995, “Vibration testing;theory and practice”, John Wiley & Sons.Maia, et. al. , 1997, “Theoretical andExperimental Modal Analysis”, Researchstudies press Ltd.Overview of Modal TestingIUST ,Modal Testing Lab ,Dr H Ahmadian
Evaluation Scheme Home Works (20%)Mid-term Exam (20%)Course Project (30%)Final Exam(30%)Overview of Modal TestingIUST ,Modal Testing Lab ,Dr H Ahmadian
Modal Testing(Lecture 10)Dr. Hamid AhmadianSchool of Mechanical EngineeringIran University of Science and Technologyahmadian@iust.ac.ir
Theoretical Basis Analysis of weakly nonlinear structuresApproximate analysis of nonlinearstructuresCubic stiffness nonlinearityCoulomb friction nonlinearityOther nonlinearities and otherdescriptionsTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Analysis of weakly nonlinearstructures The whole bases of modal testing assumeslinearity: Response linearly related to the excitationResponse to simultaneous application of severalforces can be obtained by superposition ofresponses to individual forcesAn introduction to characteristics of weaklynonlinear systems is given to detect if anynonlinearity is involved during modal test.Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Cubic stiffness nonlinearitym&x& cx& kx k3 x 3 F sin(ωt φ ) x(t ) X sin(ωt ) mω 2 X sin(ωt ) cωX cos(ωt ) kX sin(ωt ) k3 X 3 sin 3 (ωt ) F sin(ωt φ ) mω 2 X sin(ωt ) cωX cos(ωt ) kX sin(ωt ) 1 3 k3 X sin(ωt ) sin(3ωt ) F sin(ωt φ )4 4 3Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Cubic stiffness nonlinearity mω X sin(ωt ) cωX cos(ωt ) kX sin(ωt ) 21 3 k3 X sin(ωt ) sin(3ωt ) 4 4 F sin(ωt ) cos(φ ) F cos(ωt ) sin(φ )33 23 mω X kX k3 X F cos(φ ) 4 cωX F sin(φ )Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Cubic stiffness nonlinearityX1 2F3 222 mω k k3 X (cω )4 32keq k k3 X4Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Cubic stiffness nonlinearitySoftening effectTheoretical BasisHardening effectIUST ,Modal Testing Lab ,Dr H Ahmadian
Softening-stiffness effectTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Softening-stiffness effectTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Softening-stiffness effectTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Softening-stiffness effectTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Coulomb friction nonlinearityx& (t )f d (t ) cx& (t ) cFx& (t )ΔE4cFΔE 4cF X ceq 2π / ω 2πωX&()xtdt 0X FTheoretical Basis14c F 2k mω i cω πωX IUST ,Modal Testing Lab ,Dr H Ahmadian
Coulomb friction nonlinearityX F14c F 2k mω i cω πωX X increasingTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Other nonlinearities and otherdescriptions BacklashBilinear StiffnessMicroslip friction dampingQuadratic (and other power lawdamping) .Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Modal Testing(Lecture 2)Dr. Hamid AhmadianSchool of Mechanical EngineeringIran University of Science and Technologyahmadian@iust.ac.ir
MODAL ANALYSIS THEORY Understanding of how the structuralparameters of mass, damping, and stiffnessrelate to the impulse response function (time domain),the frequency response function (Fourier, orfrequency domain), andthe transfer function (Laplace domain)for single and multiple degree of freedomsystems.Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Theoretical Basis SDOF system Time Domain: Impulse Response FunctionPresentation of FRFProperties of FRFUndamped MDOF systemMDOF system with proportionaldampingTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
SDOF System Three classes of system: 1 2 k mωResponse Models:1X (ω ) H (ω ) 2F (ω ) k mω icω1 k mω 2 id UndampedViscously-dampedStructurally DampedTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Time Domain: ImpulseResponse FunctionTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Frequency Domain:Frequency Response FunctionTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Alternative Forms of FRF Receptance Mobility Inverse is “DynamicStiffness”Inverse is “DynamicImpedance”Inertance Inverse is “Apparentmass”Theoretical BasisX (ω )F (ω )V (ω )X (ω ) iωF (ω )F (ω )A(ω )2 X (ω ) ωF (ω )F (ω )IUST ,Modal Testing Lab ,Dr H Ahmadian
Graphical Display of FRFTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Graphical Display of FRFThe magnitude of the three mobility functions(accelerance, mobility and compliance)Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Stiffness and Mass LinesTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Reciprocal Plots F (ω ) 2The “inverse” orRe() kmω X (ω )“reciprocal” plots F (ω ) Real part Im() cω X (ω ) Imaginary partTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Nyquist Plot For viscous damping the Mobility plot is acircle. For structural damping the Receptance andInertance plots are circles.Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
3D FRF Plot (SDOF)Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Properties of SDOF FRF Plots Nyquist Mobility for viscose dampingY (ω ) iωk mω 2 icωcω 2Re(Y ) ( k mω 2 ) 2 ( c ω ) 2ω ( k mω 2 )Im(Y ) ( k mω 2 ) 2 ( c ω ) 21 U Re(Y ) , V Im(Y )2c U 2 V 2 Theoretical Basis((k mω ) (cω ) )4c (( k mω ) ( cω ) )2 222 22 22 2 1 2c 2IUST ,Modal Testing Lab ,Dr H Ahmadian
Properties of SDOF FRF Plots Nyquist Receptance for structural damping(1k mω 2 ) idH (ω ) 222k id mω(k mω ) d 2(k mω )U (k mω ) d22 22,V 21 1 U V 2d 2d d(k mω )2 2 d222Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
A Demo Basic Assumptions The structure is assumed tobe linearThe structure is time invariantThe structure obeys Maxwell’sreciprocityThe structure is observable loose components, or degrees-offreedom of motion that are notmeasured, are not completelyobservable.Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Modal Testing(Lecture 3)Dr. Hamid AhmadianSchool of Mechanical EngineeringIran University of Science and Technologyahmadian@iust.ac.ir
Theoretical Basis Undamped MDOF SystemsMDOF Systems with ProportionalDampingMDOF Systems with General StructuralDampingGeneral Force VectorUndamped Normal ModeTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Undamped MDOF Systems The equation of motion:[M ]{&x&(t )} [K ]{x(t )} { f (t )}The modal model:The orthogonality:[Φ ], Γ diag (ω12 , ω 22 ,K, ω N2 )[Φ ] [M ][Φ ] [I ], [Φ ] [K ][Φ ] [Γ].T TForced response solution:([K ] ω 2 [M ]){X }eiωt {F }eiωt{X } ([K ] ω [M ]) {F } {X } [α (ω )]{F }2Theoretical Basis 1IUST ,Modal Testing Lab ,Dr H Ahmadian
Undamped MDOF Systems(continued) Response Model([K ] ω [M ]) [α (ω )][Φ ] ([K ] ω [M ])[Φ ] [Φ ] [α (ω )] [Φ ]([Γ] ω [I ]) [Φ ] [α (ω )] [Φ ][α (ω )] [Φ ] ([Γ] ω [I ])[Φ ][α (ω )] [Φ ]([Γ] ω [I ]) [Φ ] 12T 1T2 1 T 122Theoretical Basis 1T2 1TIUST ,Modal Testing Lab ,Dr H Ahmadian
Undamped MDOF Systems(continued) The receptance matrix is symmetric.Single InputXjXk,α jk α kj FjFkModal Constant/Modal ResidueNφ jrφkrr A jkα jk (ω ) 2 222r 1 ω r ωr 1 ω r ωNTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Example: 1.2 0.8 [K ] MN / m 0.8 1.2 1 1 1 [Φ ] 2 e 6 2 1 1 0.5051.2e6 ω 2 α11 (ω ) .22244e5 ω2 e6 ω8e11 2.4e6ω ω 1[M ] 4e52ωr [ ]Theoretical Basis kg 1 IUST ,Modal Testing Lab ,Dr H Ahmadian
Example:(continued)Zero0.5051.2e6 ω 2α11 (ω ) .22244e5 ω2 e6 ω8e11 2.4e6ω ωTheoretical BasisPolesIUST ,Modal Testing Lab ,Dr H Ahmadian
Example:(continued)0.5058e4.α12 (ω ) 22244e5 ω2 e6 ω8e11 2.4e6ω ωTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
MDOF Systems withProportional Damping A proportionally damped matrix isdiagonalized by normal modes of thecorresponding undamped system[Φ ] [D ][Φ ] diag (d1, d 2 ,L, d N )T Special cases:Theoretical Basis[D ] β [K ],[D ] δ [M ],[D ] β [K ] δ [M ].IUST ,Modal Testing Lab ,Dr H Ahmadian
MDOF Systems with StructurallyProportional Damping Response Model([K ] i[D] ω 2 [M ]) [α (ω )] 1[Φ ]T ([K ] i[D ] ω 2 [M ])[Φ ] [Φ ]T [α (ω )] 1[Φ ]([ωr2 (1 iηr2 )] ω 2 [I ]) [Φ ]T [α (ω )] 1[Φ ][α (ω )] 1 [Φ ] T ([ω r2 (1 iηr2 )] ω 2 [I ])[Φ ] 1 1222[α (ω )] [Φ ]([ω r (1 iηr )] ω [I ]) [Φ ]Tφ jrφkrα jk (ω ) 222ω(1 iη) ωr 1rrNTheoretical BasisReal ResidueComplex PoleIUST ,Modal Testing Lab ,Dr H Ahmadian
MDOF Systems with ViscouslyProportional Damping Response Model([K ] iω [C ] ω [M ]) [α (ω )][Φ ] ([K ] iω [C ] ω [M ])[Φ ] [Φ ] [α (ω )] [Φ ]([ω ] iω [2ζ ω ] ω [I ]) [Φ ] [α (ω )] [Φ ][α (ω )] [Φ ] ([ω ] iω [2ζ ω ] ω [I ])[Φ ][α (ω )] [Φ ]([ω ] iω [2ζ ω ] ω [I ]) [Φ ] 12T22r2r 1 1T 1Tr T2r2r 12rr2r 1Trφ jrφkrα jk (ω ) 22ω ω 2ζ rω rωr 1rNTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
MDOF Systems with GeneralStructural Damping The equation of motion:[M ]{&x&(t )} ([K ] i[D ]){x(t )} { f (t )}The orthogonality:[Φ ] [M ][Φ ] [I ], [Φ ] [K iD ][Φ ] [Γ].TTComplex Mode Shapes Complex Eigen-valuesForced response solution:([K ] i[D] ω 2 [M ]){X }eiωt {F }eiωt{X } ([K ] i[D ] ω [M ]) {F } {X } [α (ω )]{F }2Theoretical Basis 1IUST ,Modal Testing Lab ,Dr H Ahmadian
Example:Model 1m1 0.5kg , m2 1kg , m3 1.5kgk j 1e3N / m, j 1,.,6Undamped 950 0.464 0.218 1.318 , [Φ ] 0.536 0.782 0.318 Γ 3352 6698 0.142 0.635 0.493Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Example:Pr oportional[D ] 0.05[K ] 950 0.464 0.218 1.318 , [Φ ] 0.536 0.782 0.318 3352Γ (1 i 0.05) 6698 0.142 0.635 0.493Non Pr oportionald1 0.3k1 , d j 0.0, j 2,.,6 957(1 i 0.067) ,3354(1 i 0.042)Γ 6690(1 i 0.078) 0.463( 5.5 ) 0.217(173 ) 1.318(181 ) [Φ ] 0.537(0.0o ) 0.784(181o ) 0.318( 6.7o ) 0.636(1.0o ) 0.492( 1.3o ) 0.142( 3.1o ) oTheoretical BasisooAlmost real modesIUST ,Modal Testing Lab ,Dr H Ahmadian
Example:Model 2m1 1kg , m2 0.95kg , m3 1.05kgk j 1e3N / m, j 1,.,6Undamped 999 0.577 0.602 0.552 , [Φ ] 0.567 0.215 0.827 Γ 3892 4124 0.207 0.587 0.752Pr oportional[D ] 0.05[K ], 999 0.577 0.602 0.552 , [Φ ] 0.567 0.215 0.827 Γ (1 i 0.05) 3892 4124 0.207 0.587 0.752Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Example:Non Pr oportionald1 0.3k1 , d j 0.0, j 2,.,6 1006(1 i 0.1) ,3942(1 i 0.031)Γ 4067(1 i 0.019) 0.578( 4o ) 0.851(162o ) 0.685( 40o ) [Φ ] 0.569(2o ) 0.570(101o ) 1.019(176o ) 0.588( 2o ) 0.848(12o ) 0.560( 50o ) Heavily complex modesTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
MDOF Systems with GeneralStructural Damping([K ] i[D] ω [M ]) [α (ω )][Φ ] ([K ] i[D ] ω [M ])[Φ ] [Φ ] [α (ω )] [Φ ]([ω (1 iη )] ω [I ]) [Φ ] [α (ω )] [Φ ][α (ω )] [Φ ] ([ω (1 iη )] ω [I ])[Φ ][α (ω )] [Φ ]([ω (1 iη )] ω [I ]) [Φ ] 12T2r2r 12r2r2r2r 122φ jrφkrα jk (ω ) 222ω(1 iη) ωr 1rrN 1T2 T 1T2 1TComplex ResiduesComplex PolesTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
General Force Vector In many situationsthe system isexcited at severalpoints.Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
General Force Vector The response is governed by:([K iD] ω [M ]){X }e2 The solution:N{X } r 1 (continued)iωtiωt{} Fe{φ } {F }{φ }rTrω (1 iη ) ω2r2r2All forces have the same frequency butmay vary in magnitude and phase.Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
General Force Vector The response vector is referred to: (continued)Forced Vibration Modeor Operating Deflection Shape (ODS)When the excitation frequency is closeto the natural frequency: ODS reflects the shape of nearby modeBut not identical due to contributions ofother modes.Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
General Force Vector (continued)Damped system normal mode: By carefully tuning the force vector theresponse can be controlled by a singlemode.T{}The is attained if φ r {F }s δ rsDepending upon damping condition theforce vector entries may well be complex(they have different phases)Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Undamped Normal Mode Special Case of interest: Harmonic excitation of mono-phased forces Same frequencySame phaseMagnitudes may varyIs it possible to obtain mono-phasedresponse?Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Undamped Normal Mode(continued) The real force response amplitudes:{}{}iωtˆ{ f (t )} F e2ˆ }e iωt {Fˆ }e iωt{([][])K iD ωMX{x(t )} Xˆ ei (ωt θ )Real and imaginary parts:(([K ] ω [M ])cosθ [D]sin θ ){Xˆ } {Fˆ }(([K ] ω [M ])sin θ [D]cosθ ){Xˆ } {0}22 The 2nd equation is an eigen-value problem;its solutions leads to real {F̂ }Theoretical BasisN solutionsIUST ,Modal Testing Lab ,Dr H Ahmadian
Undamped Normal Mode(continued) At a frequency that the phase lagbetween all forces and all responses is90 degree then(([K ] ω 2 [M ])sin θ [D]cosθ ){Xˆ } {0}Results Undamped normal modesNatural frequencies of undamped systemTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Undamped Normal Mode(continued) The base for multishaker test procedures.Modal Analysis of LargeStructures: MultipleExciter Systems By: M.Phil. K. ZaveriTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Modal Testing(Lecture 4)Dr. Hamid AhmadianSchool of Mechanical EngineeringIran University of Science and Technologyahmadian@iust.ac.ir
Theoretical Basis General Force VectorUndamped Normal ModeMDOF System with General ViscousDampingForce Response Solution/ GeneralViscous DampingTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
General Force Vector In many situationsthe system isexcited at severalpoints.Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
General Force Vector Otherwise you end updamaging the structure!!!!Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
General Force Vector The response is governed by:([K iD] ω [M ]){X }e {F }eiωtAll forces have the same frequency butmay vary in magnitude and phase.The solution:T2 (continued)N{X } r 1Theoretical Basisiωt{φ }r {F }{φ }rω (1 iη ) ω2r2r2IUST ,Modal Testing Lab ,Dr H Ahmadian
General Force Vector The response vector is referred to: (continued)Forced Vibration Modeor Operating Deflection Shape (ODS)When the excitation frequency is closeto the natural frequency: ODS reflects the shape of nearby modeBut not identical due to contributions ofother modes.Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
General Force Vector (continued)Damped system normal mode: By carefully tuning the force vector theresponse can be controlled by a singlemode.T{}This is attained if φ r {F }s δ rsDepending upon damping condition theforce vector entries may well be complex(they have different phases)Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Undamped Normal Mode Special Case of interest: Harmonic excitation of mono-phased forces Same frequencySame phaseMagnitudes may varyIs it possible to obtain mono-phasedresponse?Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Undamped Normal Mode512 channel37 ShakersTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Undamped Normal Mode(continued) The real force response amplitudes:{}{}iωtˆ{ f (t )} F e([K iD ] ω 2 [M ]){Xˆ }ei (ωt θ ) {Fˆ }eiωt{x(t )} Xˆ ei (ωt θ )Real and imaginary parts:(([K ] ω [M ])cosθ [D]sin θ ){Xˆ } {Fˆ }(([K ] ω [M ])sin θ [D]cosθ ){Xˆ } {0}22 The 2nd equation is an eigen-value problem;its solutions leads to real {F̂ }Theoretical BasisN solutionsIUST ,Modal Testing Lab ,Dr H Ahmadian
Undamped Normal Mode(continued) At a frequency that the phase lagbetween all forces and all responses is90 degree then(([K ] ω [M ])sin θ [D]cosθ ){Xˆ } {0}Results ([K ] ω [M ]){Xˆ } {0}2 2 Undamped normal modesNatural frequencies of undamped systemTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Undamped Normal Mode(continued) The base for multishaker test procedures.Modal Analysis of LargeStructures: MultipleExciter Systems By:M. Phil. K. ZaveriTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
MDOF System with GeneralViscous DampingE.O.M . [M ]{&x&} [C ]{x&} [K ]{x} { f }{ f (t )} {F }e {x(t )} {X }e 12{X } ([K ] ω [M ] iω[C ]) {F }i ωt i ωtNext the orthogonality properties of thesystem in 2N space is used for force responsesolution.Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Force Response Solution CEOM MM x& K 0 &x& 0 CFree Vib. MM K{u&} 0 0 CEigen solution sr M C U MTTheoretical Basis0 x f M x& 0 M T KU I ,U 0 00 {u} {0} M M K 0 00 {ur } {0} M 0 U diag ( s1 , s2 ,L , s2 N ). M IUST ,Modal Testing Lab ,Dr H Ahmadian
Force Response Solution F T F H F uuuuuu rrrrr2NN0 0 0 X *iωXiωsiωsiω s rr 1 r 1rrTr The above simplification is due to the factthat eigen-values and eigen-vectors occur incomplex conjugate pairs.Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Force Response Solution Single point excitation:Nα jk (ω ) r 1Theoretical Basisu jr ukriω sr jr kru uiω s rIUST ,Modal Testing Lab ,Dr H Ahmadian
Modal Testing(Lecture 5)Dr. Hamid AhmadianSchool of Mechanical EngineeringIran University of Science and Technologyahmadian@iust.ac.ir
Modal Analysis of RotatingStructures Non-symmetry in systemmatricesModes of undamped rotatingsystem Symmetric StatorNon-Symmetric StatorFRF’s of rotating systemOut-of-balance excitation Synchronous excitationNon-Synchronous excitationTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Non-symmetry in SystemMatrices The rotating structures are subject toadditional forces: Gyroscopic forcesRotor-stator rub forcesElectrodynamic forcesUnsteady aerodynamic forcesTime varying fluid forcesThese forces can destroy the symmetry of thesystem matrices.Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Non-rotating systemproperties A rigid disc mounted onthe free end of a rigidshaft of length L,The other end of iseffectively pin-jointed.Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Modes of Undamped RotatingSystem0 &x& 0JΩ z / L x& k x L 0 x 0 I0 / L . 0 I 0 / L &y& JΩ z / L0 y& 0 k y L y 0 Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Symmetric statorkx k y k,Support is symmetricx Xeiωt ,y Ye iωt , (k ω 2 I 0 / L2 ) ( iωJΩ z / L2 )Simple harmonic motion(iωJΩ2) X 0 ,/Lz 22 (k ω I 0 / L ) Y 0 22 kL2 JΩ 2 kL 42z 0. ω ω 2 I0 I0 I0 Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Natural Frequencies22 kL2 JΩ 2 kLz ω 2 0. ω4 2 I0 I0 I0 2 22 2 2ΩΩγγ22 Ω ωωγω 1, 20024 2kLJ ω02 ,γ I0I0Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Mode Shapes (k ω12 I 0 / L2 ) ( iω1JΩ z / L2 )(iω JΩ / L ) 1 0 ,(k ω I / L ) i 0 21z21 0Theoretical Basis2 (k ω 22 I 0 / L2 ) ( iω2 JΩ z / L2 )(iω JΩ / L ) i 0 .(k ω I / L ) 1 0 22z22 02IUST ,Modal Testing Lab ,Dr H Ahmadian
Non-symmetric Statorkx k yTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
FRF of the Rotating StructureExternal Damping I 0 / L2 00 &x& c 2 I 0 / L &y& JΩ z / L2JΩ z / L2 x& k 0 x f x , & c y 0 k y f y (k ω I 0 / L icω )(iωJΩ z / L ) [α (ω )] 222()() Ω ω/ω/ωiJLkILic0z Loss of Reciprocity α xx (ω ) α yy (ω ) α xy (ω ) α yx (ω )222 1Coupling EffectTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
FRF of the Rotating Structurewith External DampingComplex ModeShapesdue to significantimaginary partTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Out-of-balance excitation Response analysis for the particularcase of excitation provided by out-ofbalance forces is investigated: When the force results from an out-ofbalance mass on the rotor, it is of asynchronous natureWhen the force results from an out-ofbalance mass on a co/counter rotatingshaft, it is of a non-synchronous natureTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Synchronous OOB Excitation cos(Ωt ) 1 iΩt{F } mrΩ FOOB e sin(Ωt ) i 2Symmetric Stator : A iΩt X iΩt e e FOOB Y iA 2LA I 0 (ω02 Ω 2 (1 γ ) )Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Synchronous OOB ExcitationTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Non-Synchronous OOBExcitation Force is generated by another rotor atdifferent speedExcitation βΩ A iβΩt X iβΩt e e FOOB iA Y L2A I 0 (ω 02 βΩ 2 ( β γ ) )The essential results are the same as forsynchronous case.Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Modal Testing(Lecture 6)Dr. Hamid AhmadianSchool of Mechanical EngineeringIran University of Science and Technologyahmadian@iust.ac.ir
Theoretical Basis Analysis using rotating frameDamping in rotating and stationaryframesDynamic analysis of general rotor-statorsystems Linear Time Invariant Rotor-StatorSystemsLTI Rotor-Stator Viscous Damp SystemLTI Systems Eigen-PropertiesTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Analysis using rotating frameYrYXrΩt x cos(Ωt ) sin(Ωt ) xr y ΩΩysin(t)cos(t) r X xr cos(Ωt ) sin(Ωt ) x y y ΩΩttsin()cos() r Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Analysis using rotating frame xr cos(Ωt ) sin(Ωt ) x x [] T 1 , y ysin()cos() ΩΩtt y r Transformation Matrices x& r cos(Ωt ) sin(Ωt ) x& x& x sin(Ωt ) cos(Ωt ) x y& Ω cos(Ωt ) sin(Ωt ) y [T1 ] y& Ω[T2 ] y ,&ysin()cos() ΩΩtt r sin(Ωt ) x &x&r cos(Ωt ) sin(Ωt ) &x& sin(Ωt ) cos(Ωt ) x& 2 cos(Ωt )2 Ω Ω sin(Ωt ) cos(Ωt ) y cos(Ωt ) sin(Ωt ) y& &y& &&ysin()cos()tt ΩΩ r &x& x& x [T1 ] 2Ω[T2 ] Ω 2 [T1 ] &y& y& x Theoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Analysis using rotating frameEquation of Motion in Stationary Coordinates I 0 / L2 00 &x& 0 I 0 / L2 &y& JΩ z / L2ω1 ω02 (γΩ z / 2) 2 γΩ z / 2 I0 / L 02JΩ z / L2 x& k x & 00 y 0 x 0 , k y y 0 ω 2 ω02 (γΩ z / 2)2 γΩ z / 2Equation of Motion in Rotating Coordinates0 &x&r 0 I 0 / L2 &y&r 2Ω z I 0 / L2 JΩ z / L2 Ω 2z I 0 / L2 JΩ 2z / L2 k x c 2 k y s 2 cs (k y k x ) 2Ω z I 0 / L2 JΩ z / L2 x&r & 0 yr xr 0 .222222 Ω z I 0 / L JΩ z / L k x c k y s yr 0 ω1 ω02 (γΩ z / 2)2 γΩ z / 2 Ω zcs (k y k x )ω 2 ω02 (γΩ z / 2) 2 γΩ z / 2 Ω zNote: Eigenvectors remain unchangedTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Analysis using rotating frame Fxr cos(Ωt ) sin(Ωt ) Fx . Fyr sin(Ωt ) cos(Ωt ) Fy For Example: Fxr cos(Ωt ) sin(Ωt ) F0 cos(ωt ) Fyr sin(Ωt ) cos(Ωt ) 0 F0 cos(ω Ω)t cos(ω Ω)t 2 sin(ω Ω)t sin(ω Ω)t Theoretical BasisResponse harmonies notpresent in the excitationIUST ,Modal Testing Lab ,Dr H Ahmadian
Internal Damping in rotatingand stationary framesEquation of Motion in Rotating Coordinates I 0 / L2 00 &x&r cI 2 &&I 0 / L yr 2Ω z I 0 / L2 JΩ z / L2 Ω 2z I 0 / L2 JΩ 2z / L2 k 0 2Ω z I 0 / L2 JΩ z / L2 x& r & cI yr xr 0 0 .2222 Ω z I 0 / L JΩ z / L k yr 0 Equation of Motion in Stationary Coordinates I 0 / L2 00 &x& cI I 0 / L2 &y& JΩ z / L2Theoretical BasisJΩ z / L2 x& k x & Ω ccI y z IΩ z cI x 0 , k y y 0 IUST ,Modal Testing Lab ,Dr H Ahmadian
Internal/External Damping in2DOF System I0 / L 0 kx c ΩzI 20 &x& cE cI 2 2&&I 0 / L y JΩ z / LJΩ z / LcE c I2 x& & y Ω z cI x 0 , k y y 0 At super critical speeds the real parts ofeigen-values may become positive,i.e. unstable systemTheoretical BasisIUST ,Modal Testing Lab ,Dr H Ahmadian
Dynamic Analysis of GeneralRotor-S
Overview of Modal Testing IUST ,Modal Testing Lab ,Dr H Ahmadian Introduction to Modal Testing (continued) Milestones in the development: Kennedy and Pancu (1947) Natural frequencies and damping of aircrafts Bishop and Gladwell (1962) Theory of resonance testing ISMA (bi-annual since 1975) IMAC (annual since 1982)
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Experimental Modal Analysis (EMA) modal model, a Finite Element Analysis (FEA) modal model, or a Hybrid modal model consisting of both EMA and FEA modal parameters. EMA mode shapes are obtained from experimental data and FEA mode shapes are obtained from an analytical finite element computer model.
LANDASAN TEORI A. Pengertian Pasar Modal Pengertian Pasar Modal adalah menurut para ahli yang diharapkan dapat menjadi rujukan penulisan sahabat ekoonomi Pengertian Pasar Modal Pasar modal adalah sebuah lembaga keuangan negara yang kegiatannya dalam hal penawaran dan perdagangan efek (surat berharga). Pasar modal bisa diartikan sebuah lembaga .
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AutoCAD education, and apply them right away to your fi rst real drawing. Let us take a look at the Layers Properties Manager. This is AutoCAD’s Layers dialog box, where everything important related to layers happens. Open a new fi le and, if you decide to use toolbars, also bring up the Layer toolbar. We take a closer look at that toolbar in Section 3.3 , but for now you need only the .
