Modal Testing (Lecture 1)

Transcription

1 Modal Tesing Lecue D. Hamid Ahmadian School of Mechanical Engineeing Ian Univesiy of Science and Technology

2 Oveview Inoducion o Modal Tesing Applicaions of Modal Tesing Philosophy of Modal Tesing Summay of Theoy Summay of Measuemen Mehods Summay of Modal Analysis Pocesses Review of Tes Pocedues and Levels Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

3 Inoducion o Modal Tesing Expeimenal Sucual Dynamics To undesand and o conol he many vibaion phenomenon in pacice Sucual inegiy Tubine blades- Suspension Bidges Pefomance malfuncion, disubance, discomfo Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

4 Inoducion o Modal Tesing coninued Necessiies fo expeimenal obsevaions Naue and exend of vibaion in opeaion Veifying heoeical models Maeial popeies unde dynamic loading damping capaciy, ficion, Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

5 Inoducion o Modal Tesing coninued Tes ypes coesponding o objecives: Opeaional Foce/Response measuemens Response measuemen of PZL Mielec Skyuck Mode Shapes 3.7 Hz,.6 %, 8.39 Hz,.93 % Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

6 Inoducion o Modal Tesing coninued Modal Tesing in a conolled envionmen/ Resonance Tesing/ Mechanical Impedance Mehod Tesing a componen o a sucue wih he objecive of obaining mahemaical model of dynamical/vibaion behavio Sucual Analysis of ULTRA Mio Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

7 Inoducion o Modal Tesing coninued Milesones in he developmen: Kennedy and Pancu 947 Naual fequencies and damping of aicafs Bishop and Gladwell 96 Theoy of esonance esing ISMA bi-annual since 975 IMAC annual since 98 Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

8 Applicaions of Modal Tesing Model Validaion/Coelaion: Poducing majo es modes validaes he model Naual fequencies Mode shapes Damping infomaion ae no available in FE models Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

9 Applicaions of Modal Tesing coninued Model Updaing Coelaion of expeimenal/analyical model Adjus/coec he analyical model Opimizaion pocedues ae used fo updaing. Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

10 Applicaions of Modal Tesing coninued Componen Model Idenificaion Subsucue pocess The componen model is incopoaed ino he sucual assembly Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

11 Applicaions of Modal Tesing coninued Foce Deeminaion Knowledge of dynamic foce is equied Diec foce measuemen is no possible Measuemen of esponse Analyical Model esuls he exenal foce [ K ] [ M ]{ x} { f } Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

12 Philosophy of Modal Tesing Inegaion of hee componens: Theoy of vibaion Accuae vibaion measuemen Realisic and deailed daa analysis Examples: Qualiy and suiabiliy of daa fo pocess Exciaion ype Undesanding of foms and ends of plos Choice of cuve fiing Aveaging Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

13 Summay of Theoy SDOF Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

14 Summay of Theoy MDOF Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

15 Summay of Theoy Definiion of FRF: H h jk [ K] [ M ] i[ D] x j f k N Cuve-fiing he measued FRF: φ jφk. Modal Model is obained Spaial Model is obained Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

16 Summay of Measuemen Mehods Basic measuemen sysem: Single poin exciaion Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

17 Summay of Modal Analysis Pocesses Analysis of measued FRF daa Appopiae ype of model SDOF,MDOF, Appopiae paamees fo chosen model Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

18 Review of Tes Pocedues and Levels The pocedue consiss of: FRF measuemen Cuve-Fiing Consuc he equied model Diffeen level of deails and accuacy in above pocedue is equied depending on he applicaion. Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

19 Review of Tes Pocedues and Levels Levels accoding o Dynamic Tesing Agency: Level Naual Feq Damping aio Mode Shapes Usable fo validaion Ou of ange esidues Updaing Only in few poins 3 4 Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

20 Tex Books Ewins, D.J.,, Modal Tesing; heoy, pacice and applicaion, nd ediion, Reseach sudies pess Ld. McConnell, K.G., 995, Vibaion esing; heoy and pacice, John Wiley & Sons. Maia, e. al., 997, Theoeical and Expeimenal Modal Analysis, Reseach sudies pess Ld. Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

21 Evaluaion Scheme Home Woks % Mid-em Exam % Couse Pojec 3% Final Exam 3% Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

22 Modal Tesing Lecue D. Hamid Ahmadian School of Mechanical Engineeing Ian Univesiy of Science and Technology

23 Theoeical Basis Analysis of weakly nonlinea sucues Appoximae analysis of nonlinea sucues Cubic siffness nonlineaiy Coulomb ficion nonlineaiy Ohe nonlineaiies and ohe descipions Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

24 Analysis of weakly nonlinea sucues The whole bases of modal esing assumes lineaiy: Response linealy elaed o he exciaion Response o simulaneous applicaion of seveal foces can be obained by supeposiion of esponses o individual foces An inoducion o chaaceisics of weakly nonlinea sysems is given o deec if any nonlineaiy is involved duing modal es. Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

25 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Cubic siffness nonlineaiy sin sin3 4 sin 4 3 sin cos sin sin sin sin cos sin sin sin φ φ φ F X k kx X c X m F X k kx X c X m X x F x k kx cx x m & &&

26 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Cubic siffness nonlineaiy sin cos 4 3 sin cos cos sin sin3 4 sin 4 3 sin cos sin φ φ φ φ F X c F X k kx X m F F X k kx X c X m

27 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Cubic siffness nonlineaiy X k k k c X k k m F X eq

28 Cubic siffness nonlineaiy Theoeical Basis Sofening effec Hadening effec IUST,Modal Tesing Lab,D H Ahmadian

29 Sofening-siffness effec Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

30 Sofening-siffness effec Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

31 Sofening-siffness effec Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

32 Sofening-siffness effec Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

33 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Coulomb ficion nonlineaiy Δ Δ X c c i m k F X X c d x E c X c E x x c cx f F F eq F F d π π π / & & & &

34 Coulomb ficion nonlineaiy X F k m i c 4cF πx X inceasing Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

35 Ohe nonlineaiies and ohe descipions Backlash Bilinea Siffness Micoslip ficion damping Quadaic and ohe powe law damping.. Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

36 Modal Tesing Lecue D. Hamid Ahmadian School of Mechanical Engineeing Ian Univesiy of Science and Technology

37 MODAL ANALYSIS THEORY Undesanding of how he sucual paamees of mass, damping, and siffness elae o he impulse esponse funcion ime domain, he fequency esponse funcion Fouie, o fequency domain, and he ansfe funcion Laplace domain fo single and muliple degee of feedom sysems. Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

38 Theoeical Basis SDOF sysem Time Domain: Impulse Response Funcion Pesenaion of FRF Popeies of FRF Undamped MDOF sysem MDOF sysem wih popoional damping Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

39 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis SDOF Sysem Thee classes of sysem: Undamped Viscously-damped Sucually Damped Response Models: id m k ic m k m k F X H

40 Time Domain: Impulse Response Funcion Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

41 Fequency Domain: Fequency Response Funcion Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

42 Alenaive Foms of FRF Recepance Invese is Dynamic Siffness Mobiliy Invese is Dynamic Impedance Ineance Invese is Appaen mass X F V F A F X i F X F Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

43 Gaphical Display of FRF Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

44 Gaphical Display of FRF The magniude of he hee mobiliy funcions acceleance, mobiliy and compliance Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

45 Siffness and Mass Lines Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

46 Recipocal Plos The invese o F Re k m ecipocal plos X Real pa Imaginay pa F Im c X Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

47 Nyquis Plo Fo viscous damping he Mobiliy plo is a cicle. Fo sucual damping he Recepance and Ineance plos ae cicles. Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

48 3D FRF Plo SDOF Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

49 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Popeies of SDOF FRF Plos Nyquis Mobiliy fo viscose damping 4 Im, Re Im Re c c m k c c m k V U Y V c Y U c m k m k Y c m k c Y ic m k i Y

50 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Popeies of SDOF FRF Plos Nyquis Recepance fo sucual damping, d d V U d m k d V d m k m k U d m k id m k m id k H

51 A Demo Basic Assumpions The sucue is assumed o be linea The sucue is ime invaian The sucue obeys Maxwell s ecipociy The sucue is obsevable loose componens, o degees-offeedom of moion ha ae no measued, ae no compleely obsevable. Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

52 Modal Tesing Lecue 3 D. Hamid Ahmadian School of Mechanical Engineeing Ian Univesiy of Science and Technology ahmadian@ius.ac.i

53 Theoeical Basis Undamped MDOF Sysems MDOF Sysems wih Popoional Damping MDOF Sysems wih Geneal Sucual Damping Geneal Foce Veco Undamped Nomal Mode Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

54 Undamped MDOF Sysems The equaion of moion: [ M ]{ && x } [ K]{ x } { f } The modal model: The ohogonaliy: [ Φ], Γ diag,, K, [ ] T [ ][ ] [ ] [ ][ T ][ ] [ ]. Φ M Φ I, Φ K Φ Γ Foced esponse soluion: i K M X e F e i [ ] [ ]{ } { } { } [ ] X K [ M ] N { F} { X } [ α ]{ F} Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

55 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Undamped MDOF Sysems coninued Response Model [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] T T T T T I I I M K M K Φ Γ Φ Φ Γ Φ Φ Φ Γ Φ Φ Φ Φ α α α α α

56 Undamped MDOF Sysems coninued The ecepance maix is symmeic. α α jk jk X F k j N φ φ N A j k jk α kj X F k j, Single Inpu Modal Consan/ Modal Residue Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

57 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Example: [ ] [ ] [ ] [ ] / α Φ e e e e e e e m MN K kg M

58 Example coninued: Zeo.5 5.e6 α 4 4e5 e6 8e.4e6. Theoeical Basis Poles IUST,Modal Tesing Lab,D H Ahmadian

59 Example coninued: α.5 5 8e4 4 4e5 e6 8e.4e6. Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

60 MDOF Sysems wih Popoional Damping A popoionally damped maix is diagonalized by nomal modes of he coesponding undamped sysem [ ] T Φ [ D][ Φ] diag d, d, L, d N Special cases: [ D ] β [ K ], [ D] δ [ M ], [ D] β [ K] δ [ M ]. Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

61 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis MDOF Sysems wih Sucually Popoional Damping Response Model [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [] [ ] [ ] [ ] [ ] [ ] [ ] [] [ ] [ ] [ ][ ] [] [ ] Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ N k j jk T T T T T i I i I i I i M D i K M D i K η φ φ α η α η α α η α α Real Residue Complex Pole

62 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis MDOF Sysems wih Viscously Popoional Damping Response Model [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ N k j jk T T T T T I i I i I i M C i K M C i K ζ φ φ α ζ α ζ α α ζ α α

63 MDOF Sysems wih Geneal Sucual Damping The equaion of moion: M && x K i D x f [ ]{ } [ ] [ ]{ } { } The ohogonaliy: [ ] T [ ][ ] [ ] [ ] T Φ M Φ I, Φ [ K id][ Φ] [ Γ]. Complex Mode Shapes Foced esponse soluion: K i D M X e i [ ] [ ] [ ]{ } { F} e i { F} { X } [ α ]{ F} { } [ ] [ ] X K i D [ M ] Complex Eigen-values Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

64 Example: Model m k j.5kg, m e3n / m, Undamped 95 Γ 335 kg, m 3 j,...,6, kg [ Φ] Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

65 Example: P opoional [ D].5[ K] 95 Γ i.5 Non P opoional d 957 i.67 Γ [ Φ].3k, d o o.537. o.636. Theoeical Basis j., j 335,...,6, 6698 [ Φ] i.4, 669 i.78 Almos eal modes o o o o o o IUST,Modal Tesing Lab,D H Ahmadian

66 Example: Model m k j kg, m e3n / m, Undamped 999 Γ 389 P opoional [ D].5[ K].95kg, m j,...,6, Γ i.5 389, 3.5kg [ Φ] , 44 [ Φ] Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

67 Example: Non P opoional d 6 i. Γ [ Φ].3k, d o o.569 o.588 j., j,...,6 394 i o o o, 467 i.9 o.6854 o.976 o.56 5 Heavily complex modes Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

68 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis MDOF Sysems wih Geneal Sucual Damping [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [] [ ] [ ] [ ] [ ] [ ] [ ] [] [ ] [ ] [ ][ ] [] [ ] Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ N k j jk T T T T T i I i I i I i M D i K M D i K η φ φ α η α η α α η α α Complex Residues Complex Poles

69 Geneal Foce Veco In many siuaions he sysem is excied a seveal poins. Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

70 Geneal Foce Veco coninued The esponse is govened by: [ ] [ ]{ } i M X e { F} e i K id The soluion: { X } N { } T φ { F}{ φ} i All foces have he same fequency bu may vay in magniude and phase. η Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

71 Geneal Foce Veco coninued The esponse veco is efeed o: Foced Vibaion Mode o Opeaing Deflecion Shape ODS When he exciaion fequency is close o he naual fequency: ODS eflecs he shape of neaby mode Bu no idenical due o conibuions of ohe modes. Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

72 Geneal Foce Veco coninued Damped sysem nomal mode: By caefully uning he foce veco he esponse can be conolled by a single mode. The is aained if { }{ T φ } F s δ s Depending upon damping condiion he foce veco enies may well be complex hey have diffeen phases Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

73 Undamped Nomal Mode Special Case of inees: Hamonic exciaion of mono-phased foces Same fequency Same phase Magniudes may vay Is i possible o obain mono-phased esponse? Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

74 Undamped Nomal Mode coninued The eal foce esponse ampliudes: { } { ˆ} i f F e { } { ˆ } [ ] K id [ M ] Xˆ i θ x X e Real and imaginay pas: { } i { } e Fˆ e i [ ] [ ] [ ] { } { } K M cosθ D sinθ Xˆ Fˆ [ ] [ ] sin [ ] cos { ˆ } K M θ D θ X {} The nd equaion is an eigen-value poblem; is soluions leads o eal { Fˆ } Theoeical Basis N soluions IUST,Modal Tesing Lab,D H Ahmadian

75 Undamped Nomal Mode coninued A a fequency ha he phase lag beween all foces and all esponses is 9 degee hen [ ] [ ] [ ] { } K M sinθ D cosθ Xˆ { } Resuls Undamped nomal modes Naual fequencies of undamped sysem Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

76 Undamped Nomal Mode coninued The base fo mulishake es pocedues. Modal Analysis of Lage Sucues: Muliple Excie Sysems By: M. Phil. K. Zavei Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

77 Modal Tesing Lecue 4 D. Hamid Ahmadian School of Mechanical Engineeing Ian Univesiy of Science and Technology ahmadian@ius.ac.i

78 Theoeical Basis Geneal Foce Veco Undamped Nomal Mode MDOF Sysem wih Geneal Viscous Damping Foce Response Soluion/ Geneal Viscous Damping Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

79 Geneal Foce Veco In many siuaions he sysem is excied a seveal poins. Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

80 Geneal Foce Veco Ohewise you end up damaging he sucue!!!! Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

81 Geneal Foce Veco coninued The esponse is govened by: [ ] [ ]{ } i K id M X e { F} e i All foces have he same fequency bu may vay in magniude and phase. The soluion: { X } N {}{ T φ F}{ φ} i η Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

82 Geneal Foce Veco coninued The esponse veco is efeed o: Foced Vibaion Mode o Opeaing Deflecion Shape ODS When he exciaion fequency is close o he naual fequency: ODS eflecs he shape of neaby mode Bu no idenical due o conibuions of ohe modes. Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

83 Geneal Foce Veco coninued Damped sysem nomal mode: By caefully uning he foce veco he esponse can be conolled by a single mode. This is aained if { }{ T φ } F s δ s Depending upon damping condiion he foce veco enies may well be complex hey have diffeen phases Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

84 Undamped Nomal Mode Special Case of inees: Hamonic exciaion of mono-phased foces Same fequency Same phase Magniudes may vay Is i possible o obain mono-phased esponse? Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

85 Undamped Nomal Mode 5 channel 37 Shakes Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

86 Undamped Nomal Mode coninued The eal foce esponse ampliudes: ˆ i f F e { } { } [ K id] [ M ] { } { ˆ } i θ x X e Real and imaginay pas: { Xˆ } i { } θ e Fˆ e i [ ] [ ] [ ] { } { } K M cosθ D sinθ Xˆ Fˆ [ ] [ ] sin [ ] cos { ˆ } K M θ D θ X {} The nd equaion is an eigen-value poblem; is soluions leads o eal { Fˆ } Theoeical Basis N soluions IUST,Modal Tesing Lab,D H Ahmadian

87 Undamped Nomal Mode coninued A a fequency ha he phase lag beween all foces and all esponses is 9 degee hen [ ] [ ] [ ] { } K M sinθ D cosθ Xˆ { } Resuls [ ] [ ]{ ˆ } K M X {} Undamped nomal modes Naual fequencies of undamped sysem Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

88 Undamped Nomal Mode coninued The base fo mulishake es pocedues. Modal Analysis of Lage Sucues: Muliple Excie Sysems By: M. Phil. K. Zavei Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

89 MDOF Sysem wih Geneal Viscous Damping E. O. M. [ M ]{ && x} [ C]{ x& } [ K ]{ x} { f } { f } { F} e i { x } { X} K M i C { F} { X } [ ] [ ] [ ] Nex he ohogonaliy popeies of he sysem in N space is used fo foce esponse soluion. e i Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

90 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Foce Response Soluion {} {} {} { } { }.,,,,. N T T s s s diag U M K U I U M M C U u M K M M C s soluion Eigen u M K u M M C Vib Fee f x x M K x x M M C EOM L & & && &

91 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Foce Response Soluion * H N T N T s i u F u s i u F u s i u F u X i X The above simplificaion is due o he fac ha eigen-values and eigen-vecos occu in complex conjugae pais.

92 Foce Response Soluion Single poin exciaion: α jk N j u u u u j k i s i k s Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

93 Modal Tesing Lecue 5 D. Hamid Ahmadian School of Mechanical Engineeing Ian Univesiy of Science and Technology ahmadian@ius.ac.i

94 Modal Analysis of Roaing Sucues Non-symmey in sysem maices Modes of undamped oaing sysem Symmeic Sao Non-Symmeic Sao FRF s of oaing sysem Ou-of-balance exciaion Synchonous exciaion Non-Synchonous exciaion Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

95 Non-symmey in Sysem Maices The oaing sucues ae subjec o addiional foces: Gyoscopic foces Roo-sao ub foces Elecodynamic foces Unseady aeodynamic foces Time vaying fluid foces These foces can desoy he symmey of he sysem maices. Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

96 Non-oaing sysem popeies A igid disc mouned on he fee end of a igid shaf of lengh L, The ohe end of is effecively pin-joined. Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

97 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Modes of Undamped Roaing Sysem. / / / / Ω Ω y x L k L k y x L J L J y x L I L I y x z z & & && &&

98 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Symmeic sao., / / / /,,, 4 Ω Ω Ω I kl I J I kl Y X L I k L J i L J i L I k Ye y Xe x k k k z z z i i y x Suppo is symmeic Simple hamonic moion

99 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Naual Fequencies Ω Ω ± Ω Ω, 4, 4. I J I kl I kl I J I kl z γ γ γ γ

100 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Mode Shapes. / / / /, / / / / Ω Ω Ω Ω i L I k L J i L J i L I k i L I k L J i L J i L I k z z z z

101 Non-symmeic Sao k k x y Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

102 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis FRF of he Roaing Sucue, / / / / Ω Ω y x z z f f y x k k y x c L J L J c y x L I L I & & && && [ ] Ω Ω / / / / α α α α α yx xy yy xx z z ic L I k L J i L J i ic L I k Loss of Recipociy Exenal Damping Coupling Effec

103 FRF of he Roaing Sucue wih Exenal Damping Complex Mode Shapes due o significan imaginay pa Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

104 Ou-of-balance exciaion Response analysis fo he paicula case of exciaion povided by ou-ofbalance foces is invesigaed: When he foce esuls fom an ou-ofbalance mass on he oo, i is of a synchonous naue When he foce esuls fom an ou-ofbalance mass on a co/coune oaing shaf, i is of a non-synchonous naue Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

105 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Synchonous OOB Exciaion { } : sin cos γ Ω Ω Ω Ω Ω Ω Ω I L A e ia A F e Y X Sao Symmeic e i F m F i OOB i i OOB

106 Synchonous OOB Exciaion Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

107 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Non-Synchonous OOB Exciaion Foce is geneaed by anohe oo a diffeen speed γ β β β β β Ω Ω Ω Ω I L A e ia A F e Y X Exciaion i OOB i The essenial esuls ae he same as fo synchonous case.

108 Modal Tesing Lecue 6 D. Hamid Ahmadian School of Mechanical Engineeing Ian Univesiy of Science and Technology ahmadian@ius.ac.i

109 Theoeical Basis Analysis using oaing fame Damping in oaing and saionay fames Dynamic analysis of geneal oo-sao sysems Linea Time Invaian Roo-Sao Sysems LTI Roo-Sao Viscous Damp Sysem LTI Sysems Eigen-Popeies Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

110 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Analysis using oaing fame Ω Ω Ω Ω y x y x cos sin sin cos Ω Ω Ω Ω y x y x cos sin sin cos Y X Y X Ω

111 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Analysis using oaing fame [ ] [ ] [ ] [ ] [ ] [ ] Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω x x T y x T y x T y x y x y x y x y x T y x T y x y x y x y x T y x y x cos sin sin cos sin cos cos sin cos sin sin cos, sin cos cos sin cos sin sin cos, cos sin sin cos & & && && & & && && && && & & & & & & Tansfomaion Maices

112 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Analysis using oaing fame z z z z z z Ω Ω Ω Ω Ω Ω / / / / γ γ γ γ. / / / / / / / / / / Ω Ω Ω Ω Ω Ω Ω Ω y x z z x y x y y x z z z z z z y x s k c k L J L I k k cs k k cs s k c k L J L I y x L J L I L J L I y x L I L I & & && && Equaion of Moion in Saionay Coodinaes Equaion of Moion in Roaing Coodinaes / / / /, / / / / z z z z y x z z y x k k y x L J L J y x L I L I Ω Ω Ω Ω Ω Ω γ γ γ γ & & && && Noe: Eigenvecos emain unchanged

113 Analysis using oaing fame F F x y cos Ω sin Ω sin Ω F cos Ω F x y. Fo Example: F F x y cos Ω sin Ω F cos Ω sin Ω sin Ω F cos Ω cos cos Ω sin Ω Response hamonies no pesen in he exciaion Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

114 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Inenal Damping in oaing and saionay fames. / / / / / / / / / / Ω Ω Ω Ω Ω Ω Ω Ω z z z z I z z z z I y x k L J L I k L J L I y x c L J L I L J L I c y x L I L I & & && &&, / / / / Ω Ω Ω Ω y x k c c k y x c L J L J c y x L I L I y I z I z x I z z I & & && && Equaion of Moion in Roaing Coodinaes Equaion of Moion in Saionay Coodinaes

115 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Inenal/Exenal Damping in DOF Sysem, / / / / Ω Ω Ω Ω y x k c c k y x c c L J L J c c y x L I L I y I z I z x I E z z I E & & && && A supe ciical speeds he eal pas of eigen-values may become posiive, i.e. unsable sysem

116 Dynamic Analysis of Geneal Roo-Sao Sysems The oaing machines and hei modal esing is much moe complex Non-symmeic beaing suppo Fixed/Roaing obsevaion fame Non-axisymmeic oos Inenal/Exenal damping These lead o: Time-vaying modal popeies Response hamonies no pesen in he exciaion Insabilies negaive modal damping Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

117 Dynamic Analysis of Geneal Roo-Sao Sysems Equaion of moion of oaing sysems ae pone: o lose he symmey o geneae complex eigen-values/vecos fom velociy/displacemen elaed nonsymmey o include ime vaying coefficiens as appose o convenional Linea Time Invaian LTI sysems Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

118 Dynamic Analysis of Geneal Roo-Sao Sysems Sysem Type R-symm;S-symm Saionay Cood. LTI Roaing Cood. LTI R-symm;S-nonsymm LTI L R-nonsymm;S-symm L LTI R-nonsymm;S-nonsymm L L LTI: Linea Time Invaian L: Linea Time Dependen Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

119 Linea Time Invaian Roo- Sao Sysems [ M ]{} && x [ C] [ G Ω ]{ x& } [ K] i[ D] [ E Ω ]{ x} { f } [ M ][, C][, K][, D] Symm. [ G Ω ][, E Ω ] Skew symm. Soluion of equaions will follow diffeen ous depending upon he specific feaues. Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

120 LTI Roo-Sao Sysems Viscous Damping Only [ A]{} u& [ B]{ u} { } [ A] [ B] {} u C G Ω M K E Ω x x& Theoeical Basis, M M The sysem maices ae non-symmeic Complex eigenvals Two eigenvec ses: RH; mode shapes LH; nomal exciaion shapes IUST,Modal Tesing Lab,D H Ahmadian

121 LTI Roo-Sao Sysems Viscous Damping Only Symmeic Roo/ Non-symmeic Suppo Fowads Whil Backwads Whil Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

122 FRF of LTI Roo-Sao Sysems [ ] [ ][ ] α V λ i [ V ] H RH LH Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

123 LTI Sysems Eigen-Popeies Skew-symmey in damping Maix [ M ],[ K].5.5 3, 3.5 [ C] ΔC ΔC.5 Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

124 LTI Sysems Eigen-Popeies λ ΔC i i i i -.3.4i -.7.8i.i X X i -.8.8i i i i i Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

125 LTI Sysems Eigen-Popeies Skew-symmey in siffness Maix [ M ],[ C] 3 [ K] ΔK ΔK, 3 Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

126 LTI Sysems Eigen-Popeies λ ΔK.i.9i.65i.3i.3.i.57.79i.7.7i X X Infiniy.58i.i i Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

127 Modal Tesing Lecue 7 D. Hamid Ahmadian School of Mechanical Engineeing Ian Univesiy of Science and Technology ahmadian@ius.ac.i

128 Complex Measued Modes Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

129 Complex Measued Modes Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

130 Display of Mode Complexiy Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

131 Analyical Real Modes Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

132 Exacing eal modes fom complex measued modes H Ahmadian, GML Gladwell - Poceedings of he 3h Inenaional Modal Analysis 995: The opimum eal mode is he one wih maximum coelaion wih he complex measued one: Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

133 Exacing eal modes Nomalizing he complex measued mode shape: The poblem is ewien as: Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

134 Exacing eal modes Symmeic Skew-symmeic Rank maices Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

135 Exacing eal modes Since V is skew symmeic, Theefoe he poblem is equivalen o: Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

136 Exacing eal modes Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

137 Exacing eal modes Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

138 Exacing eal modes Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

139 Exacing eal modes Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

140 Exacing eal modes Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

141 Follow-ups: E. Folee, J. Pianda, Tansfoming Complex Eigenmodes ino Real Ones Based on an Appopiaion Technique, Jounal of Vibaion and Acousics, JANUARY, Vol. 3 S.D. GARVEY, J.E.T. PENNY, THE RELATIONSHIP BETWEEN THE REAL AND IMAGINARY PARTS OF COMPLEX MODES, Jounal of Sound and Vibaion 998,,75-83 Oveview of Modal Tesing IUST,Modal Tesing Lab,D H Ahmadian

142 Modal Tesing Lecue 8 D. Hamid Ahmadian School of Mechanical Engineeing Ian Univesiy of Science and Technology ahmadian@ius.ac.i

143 Theoeical Basis Non-sinusoidal Vibaion and FRF Popeies: Peiodic Vibaion Tansien Vibaion Random Vibaion Theoeical Basis Violaion of Diichle s condiions Auocoelaion and PSD funcions H and H Incomplee Response Models IUST,Modal Tesing Lab,D H Ahmadian

144 Non-sinusoidal Vibaion and FRF Popeies Wih he FRF daa, esponse of a MDOF sysem o a se of hamonic loads: { X } e i [ α ]{ F} e i Diffeen ampliudes and phases The same fequency We shall now un ou aenion o a ange of ohe exciaion/esponse siuvaions. Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

145 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Peiodic Vibaion Exciaion is no simply sinusoidal bu eain peiodiciy. The easies way of compuing he esponse is by means of Fouie Seies, i n nk n jk j n i n nk k n n e F x T e F f α π

146 Peiodic Vibaion To deive FRF fom peiodic vibaion signals: Deemine he Fouie Seies componens of he inpu foce and he elevan esponse Boh seies conain componens a he same se of discee fequencies The FRF can be defined a he same se of fequency poins by compuing he aio of esponse o inpu componens. Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

147 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Tansien Vibaion Analysis via Fouie Tansfom π d e F H x F H X d e f F i i

148 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Tansien Vibaion Response via ime domain supeposiion h d e H x Then F f Le d f h x i π π δ τ τ τ

149 Tansien Vibaion To deive FRF fom ansien vibaion signals: Calculaion of he Fouie Tansfoms of boh exciaion and esponse signals Compuing he aio of boh signals a he same fequency In pacice i is common o compue a DFT of he signals. Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

150 Random Vibaion Neihe exciaion no esponse signal can be subjec o a valid Fouie Tansfom: Violaion of Diichle Condiions Finie numbe of isolaed min and max Finie numbe of poins of finie disconinuiy Hee we assume he andom signals o be egodic Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

151 Random Vibaion f Time Signal Auocoelaion Funcion R S ff ff τ f f π R ff τ τ e d Powe Specal Densiy i τ d τ Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

152 Random Vibaion Time Signal Auocoelaion Powe Specal Densiy Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

153 Random Vibaion Sinusoidal Signal Auocoelaion Powe Specal Densiy Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

154 Random Vibaion Random Signal Auocoelaion Powe Specal Densiy Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

155 Random Vibaion Noisy Signal Auocoelaion Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

156 Random Vibaion The auocoelaion funcion is eal and even: R ff τ Theoeical Basis f u f u τ f f τ d τ u du R ff τ The Auo/Powe Specal Densiy funcion is eal and even. IUST,Modal Tesing Lab,D H Ahmadian

157 Random Vibaion Coss Coelaion / Specal Densiies R xf τ i x f τ d S R τ e τ xf xf dτ π Coss Coelaion funcions ae eal bu no always even. Coss Specal Densiies ae complex funcions. Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

158 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Random Vibaion * * * τ τ τ τ τ τ X X S d x x R F X S d f x R F F S d f f R xx xx xf xf ff ff Time Domain Fequency Domain

159 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Random Vibaion To deive FRF fom andom vibaion signals: * * * * ff fx xf xx S S F F X F H S S F X X X H F X H

160 Complee/ Incomplee Models I is no possible o measue he esponse a all DOF o all modes of sucue N by N Diffeen incomplee models: Reduced size fom N o n by deleing some DOFs Numbe of modes ae a educed as well fom N o m, usually m<n Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

161 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Incomplee Response Models [ ] [ ] Φ < m n m m N m jk jk i A η α

162 Incomplee Response Models Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

163 Modal Tesing Lecue 9 D. Hamid Ahmadian School of Mechanical Engineeing Ian Univesiy of Science and Technology ahmadian@ius.ac.i

164 Theoeical Basis Sensiiviy of Models Modal Sensiiviy SDOF eigen sensiiviy MDOF sysem naual fequency sensiiviy MDOF sysem mode shape sensiiviy FRF Sensiiviy SDOF FRF sensiiviy MDOF FRF sensiiviy Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

165 Sensiiviy of Models The sensiiviy analysis ae equied: o help locae eos in models in updaing o guide design opimizaion pocedues hey ae used in he couse of cuve fiing A sho summey on deducing sensiiviies fom expeimenal and analyical models is given. Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

166 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Modal Sensiiviies SDOF., 3 k mk k m m k m m k

167 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Modal Sensiiviies MDOF [ ] [ ] { } { } [ ] [ ] { } { } [ ] [ ] { } [ ] [ ] [ ] { } { },,, p M M p p K p M K M K p M K φ φ φ φ

168 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Eigenvalue Sensiiviy MDOF { } { } [ ] [ ] { } { } [ ] [ ] [ ] { } { } { } [ ] [ ] { } { } [ ]{ } T T T T T M p M p K p esuls p M M p p K p M K Muliply by φ φ φ φ φ φ φ φ φ,

169 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Eigenveco Sensiiviy MDOF [ ] [ ] { } [ ] [ ] [ ] { } { } { } { } [ ] [ ] { } [ ] [ ] [ ] { } { }, : N j j j j N j j j j p M M p p K M K p and aking p M M p p K p M K fom Saing φ φ γ φ γ φ φ φ

170 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Eigenveco Sensiiviy MDOF [ ] [ ] { } [ ] [ ] [ ] { } { } { } [ ] [ ] { } { } [ ] [ ] [ ] { } { } { } [ ] [ ] { } { } T s s s T s N j j j j T s N j j j j p M p K p M M p p K M K p M M p p K M K φ φ γ φ φ φ γ φ φ φ γ

171 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Eigenveco Sensiiviy MDOF { } [ ] [ ] { } { } [ ] [ ] { } { } { } [ ] [ ] { } { } N s s s s T s s T s s T s s s p M p K p p M p K p M p K φ φ φ φ φ φ γ φ φ γ

172 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis Updaing, Redesign, Reanalysis { } { } { } { } { } { } { } { } Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ M O M M M K K K M M M K K M M p p p p p p p p p p p p p p p φ φ φ φ φ φ φ φ

173 Updaing, Redesign, Reanalysis The change in paamees mus be vey small fo accuae analysis When he change in paamees is no small: Highe ode sensiiviy analysis Ieaive linea sensiiviy analysis Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

174 IUST,Modal Tesing Lab,D H Ahmadian Theoeical Basis FRF Sensiiviies SDOF m c i k m m c i k i c m c i k k m c i k α α α α

175 FRF Sensiiviies MDOF [ Z ] [ K ] i[ C] [ M ], [ A] [ B] [ A] [ A] [ B] [ B][ A] ake [ A] [ Z ], [ A B] [ Z ] A x hen [ Z ] [ Z ] [ Z ] [ Z ] [ Z ] [ Z ] x A [ α ] x [ α ] A [ α ] x [ ΔZ ][ α ] A x x A A Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

176 FRF Sensiiviies MDOF [ α ] [ α ] [ α ] [ ΔZ ][ α ] x A x A, { } T { } T α [ ][ ] x α A α x ΔZ α A j j Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

177 FRF Sensiiviies MDOF Theoeical Basis IUST,Modal Tesing Lab,D H Ahmadian

178 FRF Sensiiviies MDOF [ α ] [ Z ] p Z p [ ] [ Z ] [ Z ] p [ α ] p [ α ] [ Z ] [ α ] p [ α ] p Theoeical Basis [ α ] [ K ] [ C] [ M ] p i p p [ α ] IUST,Modal Tesing Lab,D H Ahmadian