Nonlinear Vibrations of Aerospace Structures

Transcription

1 Noliear Vibratios of Aerospace Structures Uiversity of Liège, Belgium L Itroductio Course objectives Review of liear theory

2 Istructors: G. Kersche, J.P. Noel, T. Detroux Cotact details: Space Structures ad Systems Lab (S3L) Aerospace ad Mechaical Egieerig Departmet Room: +/4 (B5 buildig) g.kersche, jp.oel, Course details:

3 What Is a Noliearity? LINEAR NONLINEARITY Force Force Displacemet Velocity Displacemet Velocity Force Displacemet Velocity 3

4 What Is a Noliear Vibratio? Airbus satellite 4

5 Course Motivatio THEORETICAL STANDPOINT PRACTICAL STANDPOINT Studied extesively! 5

6 It Is Noliear, So What? 6

7 It Is Noliear, So What? 6 x -4 x (m) 3 REGIME : weakly oliear effects REGIME : strogly oliear effects Noliear Liear F (N) 7

8 It Is Noliear, So What? 6 x -4 x (m) 3 REGIME : weakly oliear effects REGIME : strogly oliear effects Noliear Liear F (N) 8

9 Course Objectives At the ed of this course, you will Uderstad the impact of oliearity o system dyamics. Master the cocepts of mode shape, resoace freuecy ad freuecy respose fuctio of oliear systems. Be familiar with ew oliear cocepts icudig stability ad bifurcatios. Recogize oliearity i real-world (aerospace) structures. Kow how to use the NID software. You will be exposed to ew theoretical cocepts, advaced computatioal methods ad practical experimetal techiues 9

10 The Noliear Idetificatio to Desig Software

11 Course Outlie. Brief review of liear theory. Impact of oliearity, oliear FRFs ad 4 ew cocepts 3. Mathematical modelig ad umerical computatio 4. Noliear modes 5. Itroductio to system idetificatio ad oliearity detectio 6. Noliearity characterizatio 7. Noliear parameter estimatio 8. Advaced cocepts: bifurcatios, modal iteractios, isolas. 9. Idustrial case study

12 Noliear Vibratios of Aerospace Structures Uiversity of Liège, Belgium L Itroductio Course objectives Review of liear theory

13 How To Write the Goverig Euatios? d dt T s + T s V s D s + Q s t =, s =,. Lagrage euatios for geeralized coordiates T = ml θ, V = mgl( cosθ) θ + g si θ = L 3

14 Let s Calculate the Period of the Motio T + V = E ml θ + mgl cosθ = mgl( cosθ ) θ = dθ dt = ± g L (cosθ cosθ ) dt = dθ θ = L g(cosθ cosθ ) dθ Period = 4 L g θ dθ cosθ cosθ = π L g + θ 6 + 4

15 The Liearizatio Aroud a Euilibrium V = V + s= V s = + s= r= V s r = r s + V = k rs r s = T K s= r= T = m rs r s = T M s= r= M + K = 5

16 The Liearizatio of the Pedulum θ + g L si θ = θ + g L θ3 θ 6 + = θ θ + g L θ = Period = π L g 6

17 3 Mai Assumptios i Liear Structural Dyamics M + C + K = Liear elasticity oliear materials Small displ. ad rotatios geometrical oliearity oliear boudary coditios Viscous dampig oliear dampig mechaisms 7

18 Assumptio : Noliear Materials Stress Hyperelastic material (e.g., rubber) Strai Stress Shape memory alloy Strai 8

19 Assumptio : Ligamet i Your Kee Joit Load Huma cadaveric aterior cruciate ligamet i kee joit (Dr. Ziv, MAE, Buffalo) Toe regio: ormal rage Liear Yield Extesio 9

20 Assumptio : Geometrical Noliearities Gree s strai tesor l l x F = k l l l x + l x l = kx l x + l = x l + 3x4 8l 4 + O(x6 )

21 Assumptio : A Geometrical Beam i Our Lab

22 Assumptio 3: Noliear Dampig Viscous dampig but also Coulomb frictio Aerodyamic dampig

23 The Cocept of Mode Shape M + K = + = x φ(t) Sychroous vibratio of the structure det K ω M = K ω (r) M x (r) = atural freuecies ormal modes φ r t = α r cos ω r t + β r si ω r t ormal coordiates 3

24 Normal Modes: Importat Properties Clear physical meaig: Structural deformatio at resoace Sychroous vibratio of the structure Importat mathematical properties: Orthogoality Decouplig of the euatios of motio (modal superpositio) Ivariace 4

25 The Cocept of FRF M + K = s cos ωt + = x cos ωt x = FRF K ω M s = H ω s H ω = x (s) x (s) T ω s ω μ s Clear lik betwee the FRF ad the modal parameter 5

26 FRF: Importat Properties The FRF is a costat system properties for a liear system FRF ca be easily estimated from measured data Very coveiet way of locatig resoace freuecies 6

27 Time Respose: Mode Displacemet Method M + K = p t = g φ(t) t = s= x s x s T g μ s ω s t siω s (t τ) φ(t)dτ Exact t = k< s= x s x s T g μ s ω s t siω s (t τ) φ(t)dτ Approximate 7

28 Time Respose: Numerical Itegratio M, C, K, Newmark itegratio Compute scheme for liear systems Time icremetatio t t h * * Predictio h h.5 h Computatio of acceleratios S M h C h K S p C * K * Correctio h * * h 8

29 9 Time Respose: Numerical Itegratio Compute Time icremetatio h t t Predictio.5 h h h Residual vector evaluatio g f M r Calculatio of the correctio,,,, S p f M Covergece? f r ), ( f g M ) ( r S Correctio h h Yes No Newmark itegratio scheme for oliear systems

30 3 Very Importat: Samplig Freuecy h h h h O h h h T T h acceleratio (modified) Average costat acceleratio Average costat acceleratio Liear & Goodwi Fox differece Cetral explicit Purely Algorithm Amplitude error Periodicity error Stability limit Accuracy

31 I Summary: Commo Sources of Noliearity Bolts, joits ad gaps Elastomers ad composites Frictio ad cotact Large amplitudes 3

32 I Summary: Importat Liear Cocepts/Methods Mode shapes, resoace freuecies, dampig ratios Freuecy respose fuctios (FRFs) Modal superpositio/umerical itegratio OPEN QUESTION: Will they remai valid/useful for oliear systems? 3