F - Ma = 0 The unique certitude in Aerospace?

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1 3rd European LS-DYNA Conference F - Ma = 0 The unique certitude in Aerospace? Dr Yves GOURINAT ENSICA Professor Mechanics & Space Techniques, Toulouse

2 Acceleration Landscape Dynamics in Aerospace

3 Acceleration Landscape General Formalism of Dynamics Stability and Linearity Technique of Model

4 Acceleration The origin : free fall flight Galilee Kinematics F - Mg = 0

5 Kepler Orbital Kinematics Newton General Dynamics F - Ma = 0

6 General Dynamic Formalism Constitutive equations Newton : F - Ma = 0 Derivative & vectorial application General Resulting Theorems : R ext - p = 0 sum (& moment) applied to external system forces General 1st degree natural kinetik conservations

7 Newton (cont d) : Real Power Formulation : P abs - E kin = 0 for working forces General 1st degree natural energy conservation (E pot ) Lagrange - Hamilton : (F - Ma)q* = 0 Virtual Power Formulation : [L k ](E kin ) - Q k = 0 applied to canonical virtual motions on q k geometrical parameters

8 Lagrange (cont d) : [L k ] d and Q k = Q (load)k + Q (diss)k + Q (n.-hol.)k dt q k q k Lagrange-Routh Kinetik Prime Integral Routh E lag / q k - G valid if k [L k ](E lag ) = 0 and G(q j ; t ) E lag / q k = G Lagrange-Hamilton Energy Prime Integral E Hamilton E lag2(qj ) - E lag0(qj ) + F valid if k, Q (load)k = [L k ](E pot ) and Q (diss)k = 0, and if F(q j ; t ) t : E lag / t= F, and if in addition all non-holonomic relations are homogeneous in the q j. That was explicitated by Painlevé : E Painlevé E kin2 - E kin0 + E pot - j [q k ( E pot / q k )]

9 Acceleration Landscape General Formalism of Dynamics Stability and Linearity Technique of Model

10 Linear Dynamics Rational Dynamics : [L k ](E kin ) - Q k = 0 ; [M]{q} + [K]{q} - {q} e = 0 Continuous Beams & Shells : ρh W - EI W - λ Z = 0 ρs U - ES U - λ X = 0

11 Linear Harmonic Analysis (Modes & Waves) Harmonic Diagonalization / free solution Effective Modes ω rigid 0 M µ 1 ω 1 ω 2 µ 2... ωn µ N ω resid µ resid Modes : ω i = k i m i Waves : c j = E j ρ j

12 Stability & Linearity y ± y - a = 0 (Newton/Lagrange) s 2 ± s - α = 0 (Laplace/Fourier) exp complex - evolution : real - vibration : imag Static E pot E pot Geometrical and/or topological non-linearities

13 Dynamic instability Coupling & Resonance t ω

14 Solid Damping Viscous Structural Slide Real κ mx + rx + kx - f = 0 mx + κx - f = 0 mx + f(x / x ) + kx - f = 0 mx + f damp + f elast - f = 0 2 P visc = r (x ) 2 κ C Heaviside Polynomial

15 Acceleration Landscape General Formalism of Dynamics Stability and Linearity Technique of Model

16 Numerical Modelling Lagrange-Ritz Deflection FEM Mesh Autom triang/tetra Quad/hexa Local element matrix Assembly Inversion

17 Natural FEM Developp ts - Substructure Superelement Condensation Frontier elements Acc t - Explicitation Evolutive geometry topology materials t t+dt V t+dt = V t + acc t. dt F t+dt = F t + K t+dt.v t+dt. dt Acc t+dt = F t+dt / M t+dt

18 SPH evolution Newton s Solid Percussion Fluid Mechanics 2p = ρ v 2 cos 2 i Valid in rarefied atm and/or hypersonic Not valid for classical fluid Smooth Particles Natural Lagrangian modern extension

19 The Engineer s Panoply Model Test all by hand numerical solving analogical solving numerical model physical analogy scale specimen prototype Context : Goal/facilities Performance / Cost / Time

20 Gen Energy [M]{q} + [K]{q} - {q} e = 0 Lin Dyn Structures [L k ](E kin ) - Q k = 0 Elast Beams Shells Theoretical foundations (F-Ma)δ*=0 Mech Energy F-Ma=0

21 HAMILTON PAINLEVÉ APELL KANE Gen Energy GAUSS KRAMER LAPLACE FOURIER RAYLEIGH [M]{q} + [K]{q} - {q} e = 0 [L k ](E kin ) - Q k = 0 LAGRANGE Lin Dyn Structures Elast Beams Shells GALILEE HOOKE KIRCHOFF BRESSE MINDLIN LOVE REISSNER Theoretical foundations (F-Ma)δ*=0 d ALEMBERT Mech Energy F-Ma=0 NEWTON / LEIBNIZ

22 Implicit Explicit Fin Diff Depl FEM SPH NUMERICAL METHODS DAILY PARAPHERNALIA

23 Dynamics in Aerospace Airships Aircrafts Spacecrafts

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29 Structure : rods, beams, threads & sails Non-développability : - Dynamic stability - Command

30 Dynamics in Aerospace Airships Aircrafts Spacecrafts

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35 Quasistatic Certification Loads - Dynamic Behavior - Vibrational Stability Flutter Dynamic Stability Inertial solution Hybrid solution Engine Dynamic Sizing Vertical & longi LL Lateral Dyn

36 Shock & Crashworthiness - Global crash test / model - Depressurization - Global shock analysis (impact & ingestion, shock protection) - Local material Damage Tolerance.