Duffing Oscillator. Mike Brennan (UNESP) Bin Tang (Dalian University of Technology, China)

Transcription

1 On th Impuls Rspons of th Duffing Oscillator Mik Brnnan (UNESP) Gianluca Gatti (Univrsity of Calabria, Italy) Bin Tang (Dalian Univrsity of Tchnology, China) 1

2 Outlin Objctiv: (1) Undrstanding four basic analysis mthod. () ( ) Transint rspons of Duffing oscillator.

3 Outlin 1 Rviw Ky Faturs of th Impuls Rspons 3 Conclusion 3

4 Rviw-Introduction Fr vibration of a Duffing oscillator Initial displacmnt / Impuls rspons (Initial vlocity)? Th Straightforward xpansion Th Lindstdt-Poincaré Mthod Th Mthod of Multipl Scals Th Mthod of Harmonic Balanc Can ths rsults captur th ky faturs for th impuls rspons of a dampd systm? 4

5 Rviw-Equation of Motion Impuls xcitation Aδ(t) m c k 1,k 3 O x Non-dimnsional i () mx cx k x k x A t y y y y ( ) 3 5

6 Th Mthod of Multipl Scals (MMS) Damping is small 3 y 0 sin 1 16 γ 0 y i 0 sin Not corrct for dampd systm! y ( ) sin 1 1 Only valid for lightly dampd systm! 6

7 Th Modifid KBM mthod (MKBMM) / y( ) sin 1 ln (1 ) sin 3 1 ln 1 3/ (1 ) 3/ cos 3 1 ln 1 3/ (1 )

8 Th Modifid KBM mthod (MKBMM) 1 1/ 3 1 y( ) sin 1 ln 1 8(1 ) y( ) sin γ 0 y( ) sin 1 1 8

9 Lindstdt-Poincaré mthod (LPM) 1 3 y ( ) sin sin cos sin3 1 cos

10 Lindstdt-Poincaré mthod (LPM) y 0 3 sin Dampd frquncy is indpndnt with tim whn 1, 1 d d 3 y ( ) sin MKBMM MKBMM

11 Comparison of th Rsults Tim Domain Non n-dimns sional di isplacm mnt y( )1 )0.6 γ = 0.; γ = 0.; ζ =0.5 ζ = Non n-dimns sional di isplacm mnt y( Non-dimnsional tim Non-dimnsional / i tim /, Rung-Kutta mthod ;, MMS;, MKBMM;, LPM. 11

12 Comparison of th Rsults Frq. Domain γ =0.;ζ =0.05 Non-dimn nsional amp plitud y FEM; Envlop DSM; of th rspons Thory Solution Non-dimnsional frquncy s angl (D Dg ) Pha , Rung-Kutta mthod ;, MKBMM;, LPM Non-dimnsional frquncy 1

13 Comparison of th Rsults Frq. Domain γ =0.;ζ =0.5 Non-dimn nsional amp plitud y FEM; Envlop DSM; of th rspons Thory Solution Non-dimnsional frquncy Phas s angl (D g ) , Rung-Kutta mthod ;, MKBMM;, LPM Non-dimnsional frquncy 13

14 Ky Faturs Envlop of th rspons Envlop 1 1 1/ Whn Envlop 1 14

15 Ky Faturs Dampd natural frquncy which changs with tim ( ) y y Whn and nonlinarity is vry small Whn ( ) ( ) 1 15

16 nvlop A() A Non-di imnsional Ky Faturs γ =0.;ζ =0.05 Envlop IF (Instan. Frq.) Non-dimnsional tim /, Envlop 1 +, MKBMM Numrical rsults + HT, Non-dim. instantano ous frquncy () Non-dimnsional tim / 3 ( ) 1 8 1, +, MKBMM, Numrical rsults + HT 16

17 Concluding Rmarks Analytical mthod Mthod of multipl scals (MMS) Good Modifid Krylov-Bogoliubov-Mitropolskiy mthod (MKBMM) Good Lindstdt-Poincaré mthod (LPM) cannot captur tim dpndnt dampd natural frq. Two ky faturs of th impuls rspons Th nvlop of th dcay of fr vibration Lightly dampd cas: approximatly xponntial dcay Th tim dpndnt dampd natural frquncy 17

18 Rfrncs [1] A.H. Nayfh, D.T. Mook, Nonlinar Oscillations. Wily, Nw York, [] J.J. J Thomsn, Vibrations and Stability, Advancd d Thory, Analysis, and Tools, nd d., Springr, Brlin, 003. [3] S. Martin. Th Voltrra and Winr Thoris of Nonlinar Systms, John Wily & Sons, Nw York, [4] I. Kovacic, M.J. Brnnan, Th Duffing Equation: Nonlinar Oscillators and thir Bhaviour, Wily, Chichstr, 011. [5] K.S. Mndlson, Prturbation thory for dampd nonlinar oscillations. Journal of Mathmatical Physics, 11, , [6] R.G. Whit, Effcts of non-linarity du to larg dflctions in th drivation of frquncy rspons data from th impuls rspons of structurs. Journal of Sound and Vibration, 9, ,

19 Rfrncs [7] M. Fldman, Non-linar systm vibration analysis using Hilbrt transform-i. Fr vibration analysis mthod FREEVIB. Mchanical Systms and Signal Procssing, 8, , [8] Bin Tang, M.J. Brnnan, On th impuls rspons of th Duffing oscillator, Intrnational Confrnc on Vibration and Vibro-acoustics (ICVV014), January 13-15, 014, Harbin, China. 19

20 Thank You for Your Attntion!!! Any Qustions ar wlcom! 谢谢 (Xièxiè)!( ) Bin Tang Institut of Intrnal Combustion Engin, Dalian Univrsity of Tchnology, China. 0