Finite element method for structural dynamic and stability analyses

Transcription

1 Finie elemen mehod for srucural dynamic and sabiliy analyses Module-3 Analysis of equaions of moion Lecure-8: FRF-s and Damping models Prof C S Manohar Deparmen of Civil Engineering IISc, Bangalore 56 India

2 Dynamic response analysis MU CU KU G U, U, F U() U ; U U Frequency domain mehods Time domain mehods Response specrum based mehods Linear ime invarian sysems Time varying sysems Nonlinear sysems Quaniaive mehods Direc mehods Mode superposiion mehods Qualiaive mehods Bifurcaions and sabiliy

3 Recall Linear Damping models Viscous Srucural Classical Non-Classical Classical Non-Classical Classificaion ino viscous and srucural depends upon behavior of energy dissipaed under harmonic seady sae as a funcion of frequency. Classificaion ino classical and non-classical depends upon orhogonaliy (or lack of orhogonaliy) of damping marix wih respec o undamped normal modal marix. 3

4 Inpu-oupu relaions for linear ime invarian sysems f () LTI x ) h( ( ) f ( ) d f ( ) F( ) x( ) X ( ) h( ) H ( ) F() LTI () LTI h() X ( ) H( ) F( ) exp( i) LTI H( )exp( i) 4

5 Damped forced response analysis ; Z M Z C Z K Z F() IZ C Z Z F() MX CX KX F X X X X X () M Z CZ KZ F() If C is no a diagonal marix, he equaions of moion would sill remain coupled. 5

6 Classical damping models If he damping marix C is such ha C is a diagonal marix, hen equaions would ge uncoupled. Such C marices are called classical damping marices. Example Rayleigh's proporional damping marix C M K C M K I 6

7 C M K T T C [ M K] T T I K [ I] Diag[ i ] c c n n n n n n n How o find and? We need o know damping raios a leas for wo modes. For example, n Knowing and, solve for and 7

8 ea - mass and siffness proporional siffness proporional mass proporional frequency rad/s 8

9 IZ CZ Z F() Z MX () & Z MX () z z z f r n r rr r r r r ;,,, wih z & z specified. r r exp cos sin z a b r r r r dr r dr exp rr fr d dr 9

10 z exp a cos b sin exp f d r r r r dr r dr r r r dr X Z k kr r r n x z n kr exp rrar cosdr br sindr exp rr fr d r dr k,,, n

11 Remarks The success of C M K in making C diagonal depends upon he wo orhogonaliy relaions: and K. M I The fac ha he model C M K admis wo open parameers, and, is relaed o he fac ha we have wo orhogonaliy relaions. Having only wo model parameers permis us o o capure damping in only wo modes of oscillaions. This is resricive, since, once and are chosen, he variaion of w.r.. frequency is also arifically fixed. How o make he model less inflexible? Inroduce more model parameers. Do we have newer orhogonaliy relaions?

12 The wo infinie families of orhogonaliy relaions s famil Consider K M wih M I & K K M M K y KM K K is orhogonal w.r.. KM M K 3 KM KM K KM K is orhogonal w.r.. K KM KM K a diagonal marix a diagonal marix 4 KM KM KM K KM KM K is orhogonal w.r.. KM KM KM K a diagonal marix

13 nd fa Consider K M wih M I & K K MK mily M M M is orhogonal w.r.. MK K M MK MK M MK M M a diagonal marix is orhogonal w. r.. M is orhogonal w.r.. K MK M MK MK MK M a diagonal marix 3 MK MK MK M MK MK M a diagonal marix 3

14 Generalized classical damping model C M MK M MK MK M MK MK MK M 3 4 K KM K KM KM K KM KM KM K C 3 4 C is diagonal. is orhogonal o C. 4 n n 4 n n n n 3 n n n 3 5 n n n 4

15 Generalized classical damping model can also be wrien as C M a n M K n Example C M a n M K n Remarks Clearly n n M a K MK M a K M a a M K a M KM K a MK MK M a MK M a M a K a KM K C a MK MK M a MK M a M is diagonal. a K a KM K Wih a, a, a, we ge C a M a K, which is he Rayleigh's proporional damping model. 5

16 n 3 n n 3 5 n n n Remarks We can inroduce as many model parameers as he number of modes for which we have damping raios known. Damping raios for modes can be arbirarily fixed (for example, he damping raio can be he same for all he paricpaing modes) wih an undersanding ha we can arrive a a classical damping model based on he above model. Noice ha C C n n n n This approach would no be pracicable since we seldom deermine for all possible modes of he sysem. Cauion: for hose modes for which damping raios are no explicily specified, bu are infered from model for C, care mus be aken o ensure ha ha raios are physically meaningful. 6

17 FRF nomenclaure and modal represenaions SDOF SYSTEMS Recepance, mobiliy, and accelerance Dynamic siffness, Mechanical impedance, apparen mass FRFs for viscously and srucurally damped sysems Asympoes MDOF SYSTEMS Modal represenaion of FRFs for proporional and non-proporional damping models Direc and cross FRFs Resonance and ani resonance 7

18 Viscously damped sdof sysem mx cx kx F exp( i) F x( ) X ( )exp( i) X ( ) Recepance k m ic X ( ) ( ) F k m ic Dynamic siffness x( ) i X ( )exp( i) V ( )exp( i) Mobiliy V( ) i Y ( ) F k m ic Mechanical impedance x( ) X ( )exp( i ) a i Accelerance a( ( )exp( ) ) A( ) F k m ic Apparen mass 8

19 Srucurally damped sdof sysem c mx x kx F exp( i) F x( ) X ( )exp( i) X ( ) Recepance k m ic X ( ) ( ) F k m ic Dynamic siffness x( ) i X ( )exp( i) V ( )exp( i) Mobiliy V( ) i Y ( ) F k m ic Mechanical impedance x X i a i ( ) ( )exp( ) ( )exp( ) Accelerance a ( ) A( ) F k m ic Apparen mass 9

20 Nomenclaure for FRF Response Quaniy R F R F R Recepance Displacemen Admiance Dynamic siffness Dynamic compliance Dynamic flexibiliy Velociy Mobiliy Mechanical impedance Acceleraion Accelerance Apparen mass

21 Nomenclaure for graphical display of FRF-s No. Deails of graphical display Nomenclaure Remarks Modulus (FRF) vs frequency Phase (FRF) vs frequency Real (FRF) vs frequency Imag (FRF) vs frequency 3 Real (FRF) vs Imag (FRF) Bode s plo A pair of plos - A pair of plos Nyquis s plo Single plo; frequency does no appear explicily.

22 real(alpha) imag(alpha) alpha m/n phase alpha rad Viscously damped sysem.5 x Bode s Plo frequency rad/s frequency rad/s x -3 x frequency rad/s frequency rad/s Recepance

23 imag(alpha) Viscously damped sysem x Nyquis s Plo real(alpha) x -3 Recepance 3

24 real(y) imag(y) Y m/s/n phase Y rad Viscously damped sysem Bode s plo frequency rad/s - 3 frequency rad/s frequency rad/s Mobiliy -. 3 frequency rad/s 4

25 imag(alpha) m/s/n Viscously damped sysem..8 Nyquis s plo real(alpha) m/s/n 5 Mobiliy

26 real(a) imag(a) A m/s/s/n phase A rad Viscously damped sysem Bode s plo frequency rad/s frequency rad/s frequency rad/s Accelerance frequency rad/s 6

27 imag(alpha) m/s/n Viscously damped sysem.8.6 Nyquis s plo real(alpha) m/s/n Accelerance 7

28 real(alpha) imag(alpha) alpha m/n phase alpha rad Srucurally damped sysem 4 x -3 Bode s plo x -3 frequency rad/s -4 3 frequency rad/s x frequency rad/s -4 3 frequency rad/s Recepance 8

29 imag(alpha) Srucurally damped sysem x -3 Nyquis s plo real(alpha) x -3 Recepance 9

30 real(y) imag(y) Y m/s/n phase Y rad Srucurally damped sysem.4.3 Bode s plo frequency rad/s - 3. frequency rad/s frequency rad/s Mobiliy -. 3 frequency rad/s 3

31 imag(y) m/s/n Srucurally damped sysem..5 Nyquis s plo real(y) m/s/n 3 Mobiliy

32 real(a) imag(a) A m/s/s/n phase A rad Srucurally damped sysem Bode s plo.. 3 frequency rad/s 3 frequency rad/s frequency rad/s Accelerance 3 frequency rad/s 3

33 imag(a) m/s/n Srucurally damped sysem.35 Nyquis s plo real(a) m/s/n Accelerance 33

34 Asympoic properies of FRF - s for viscously damped sysem lim lim m ic k m k i i i Y lim Y limy m ic k m k A lim A lim A m ic k m k 34

35 FRF of discree mass and siffness elemens FRF parameer m, k c m, k, c m k log log m log log k i i Y m k log Y log m log log log k A m k log A log m log log k 35

36 recepance m/n Ampliude of recepance for a Viscously damped sysem recepance mass siffness w rad/s 36

37 mobiliy m//s/n Ampliude of Mobiliy for a Viscously damped sysem - mobiliy mass siffness w rad/s 37

38 accelerance m//s/s/n Ampliude of Accelerance for a Viscously damped sysem accelerance mass siffness w rad/s 38

39 Nyquis s plo for mobiliy of a viscously damped sysem is a circle wih radius /c and cenre a (/c,) i Y ( ) k m ic c Re Y ( ) ( k m ) ( c) Im Y ( ) Define U V ( k m ) ( k m ) ( c) U Re Y ( ) & V Im Y ( ) c c 39

40 Nyquis s plo for recepance of a srucurally damped sysem is a circle wih radius /h and cenre a (,-/h) ( ) Re ( ) Im ( ) Define U k m ih ( ) k m ( k m ) h h ( k m ) h V Re ( ) & Im ( ) U V h h 4

41 FRFs for MDOF sysems r exp i x x rr rs s x x rr rs ( ) ( ) X rs X X rr rs ( )exp( i) ( )exp( i) ( ) X ( ) sr r x x sr ss s exp i Recepance marix X [ ( )] X rr sr ( ) ( ) Symmeric No Hermiian X X rs ss ( ) ( ) X X rr rs ( ) : Poin recepance/ Poin Mobiliy/ Poin Accelerance ( ) : Transfer recepance/ Transfer Mobiliy/ Transfer Accelerance

42 exp i 4 3 ( ) ( ) ( ) 3 ( ) ( )exp( i) ( )exp( i) ( )exp( i) ( )exp( i) 3 4 ( ) Direc poin recepance (mobiliy/accelerance) ( ) Cross poin recepance (mobiliy/accelerance) ( ) Direc ransfer recepance (mobiliy/accelerance) ( ) Cross ransfer recepance (mobiliy/accelerance) 4

43 Y A Displ [ ( )] Force ; [ ( )] Recepance Vel [ ( )] Force ; [ Y( )] Mobiliy Accl [ ( )] Force ; [ A( )] Accelrance Y A ( ) Displ = Force ; ( ) ( ) Dynamic siffness marix ( ) Vel = Force ; ( ) Y ( ) Mechanical impedance marix ( ) Accl = Force ; ( ) A ( ) Apparen mass marix Remarks We ofen measure accelerance Obviously recepance, mobiliy and accelerance are easier o measure han Dynamic siffness, mechanical impedance and apparen mass. Cauion does no imply ij ( ) ij ( ) 43

44 Viscously damped MDOF sysem wih s-h dof driven by an uni harmonic force MX CX KX F exp i F sr s h enry response of he -h coordinae due o X r uni harmonic driving a s-h coordinae. lim X? sr 44

45 MX CX KX F exp i F lim exp X X i X X iexpi exp X X i exp exp MX exp i CX i exp i KX exp i F exp i M ic K X i F i M i C K X F 45

46 M i C K X F exp exp X X i Z i M I & K C is classical C (Diagonal) wih nn n n M i C K Z F M i C K Z F M i C K Z F I i Z F Diagonal 46

47 Z N N nk Fk knfk k k n Recall F n i nn n i nn ( s-h enry=; res=) sn n n i n n lim X Z exp i X Z exp i Z I i Z F N rn sn n n i n n rs r rn n n exp X H i H rs rs N rn sn n n i n n exp i N 47

48 X exp rs i H rs rs rs N rn sn n n i n n N rn sn n n i n n sr X X H Remarks H sr H H rs H is symmeric bu no Hermiian N rn H M ic k 48 sn n n i n n

49 H M ic k Concepually simple Compuaionally difficul o implemen H * N N rn sn n n i nn Compuaionally easier o implemen No all modes need o be included (Nor i is advisable o include all modes) 49

50 mobiliy m/s/n H jk m r jk R M jk rm r i rr A K R jk Red: n=:5 Black: n=5: Green: n=:5 Magena: n=: frequency rad/s x 4 5

51 mobiliy m/s/n Red: n=:5 Black: n=5: Green: n=:5 Magena: n=: frequency rad/s 5

52 Beam on an inclined roller wih an inermediae hinge BCS: u u u Consrains u u u u u u EI, AE, m, l co 9 EI, AE, m, l 8 Hinge 5 6 A 5 7 B 7 u sin u 7 5 cos E GPa; =78 kg/m ; B=. m; D=.3 m; l m; l 3 m; =4 3

53 G I = - ; M Na freq HZ K

54 Damping: =. for all modes C C

55 accelerance(,) m/s/s/n frequency rad/s 55

56 phase accelerance(,) rad frequency rad/s 56

57 accelerance(,6) m/s/s/n frequency rad/s 57

58 phase accelerance(,6) rad frequency rad/s 58

59 imag(accelerance(,)) m/s/s/n real(accelerance(,)) m/s/s/n 59

60 imag(accelerance(,6)) m/s/s/n real(accelerance(,6)) m/s/s/n 6

61 Recepance m/n frequency rad/s 6

62 Incidence of resonan peaks, aniresonances, and minima Le damping be low so ha we can ignore he damping erms in he expression for he FRF-s. Consider he case of poin recepance rr n rn n Noe The numeraor for all erms is nonnegaive. h h Le n n and consider conribuions from n and ( n) modes n h N mode: ( ) mode : rn h rn n n n Negaive Posiive * There exiss a poin n n a which he conribuions from he wo erms ge cancelled. This poin is he poin of aniresonance. The locaion of hese poins depend upon naural frequency and mode shapes. 6

63 Now consider he case of cross recepance rs n rn sn n h h Le n n and consider conribuions from n and ( n) modes n h N sn rn sn h r n mode: ( n) mode : n n Num- Num- Term- Term- Remark Aniresonance Minimum Minimum Aniresonance 63

64 Remarks All FRF-s peak a he same frequencies (naural frequencies) This is no rue for aniresonace and poins of minima. For a poin FRF, beween every wo resonances, an aniresonance occurs wihou excepion. Trasnfer FRF-s show a mixure of aniresonance and minima. Resonan peaks are accompanied by large responses and rapid changes in phase angle. Aniresonance poins are accompanied by low responses and rapid changes in phase angle. Presence of damping could make idenificaion of resonance, aniresonance, and minima difficul. 64

65 Remarks (coninued) In sysems wih closely space modes he behavior of FRF-s a any frequency would be affeced by more han he wo neares modes. The inerpreaion of resonance, aniresonance, and minima would no be sraighforward. If poin of driving/measuremen coincides wih he zero of a mode shape, he corresponding resonan peak would no show up in he FRF-s. A B C I is reasonable o expec ha alpha(ac) will have more minima han ani-resonance as compared wih alpha(a,b). 65

66 MDOF sysem wih s-h dof driven by an uni impulse force MX CX KX F X ; X F s h enry rs response of he -h coordinae due o X r uni impulse driving a s-h coordinae. 66

67 MX CX KX F F X Z M I & K C is classical C (Diagonal) wih M Z CZ KZ F M Z CZ KZ F IZ Z Z F nn n n 67

68 IZ Z Z F N n n n n n n jn j sn j z z z F z n ; z sn zn snhn exp nn sindn X Z N X z r rn n n n N h exp sin dn rs rn sn n n dn n dn 68

69 N X h exp sin rs rs rn sn n n dn n dn Remarks rs h h sr h h rs Marix of impulse response funcions h h No all modes need o be included in he summaion If an arbirary load f uni impulse exciaion) X h f d rs rs s s is applied a he s-h dof (insead of N fs rnsn exp nn sin dn d n dn 69