Nonlinear Modal Analysis of Mechanical Systems with Frictionless Contact Interfaces

Transcription

1 Nonlinear Modal Analysis of Mechanical Systems with Frictionless Contact Interfaces Denis Laxalde, Mathias Legrand and Christophe Pierre Structural Dynamics and Vibration Laboratory Department of Mechanical Engineering, McGill University Montreal, Quebec, Canada ASME IDETC/CIE 2009 VIB 14

2 Outline Background and overview Nonlinear modes for contact systems Contact analysis for compressor blades Conclusions Laxalde, Legrand & Pierre (McGill) Nonlinear Modes for Contact Problems DETC IDETC/CIE 09 2 / 14

3 Rotor-stator interaction in turbomachinery Why does it happen? optimized designs, new materials, reduced operating tip clearances performance improvements and increased vibratory stresses Laxalde, Legrand & Pierre (McGill) Nonlinear Modes for Contact Problems DETC IDETC/CIE 09 3 / 14

4 Rotor-stator interaction in turbomachinery Why does it happen? optimized designs, new materials, reduced operating tip clearances performance improvements and increased vibratory stresses Contact occurrences normal operating conditions not only transient but also stationary phenomena instabilities Laxalde, Legrand & Pierre (McGill) Nonlinear Modes for Contact Problems DETC IDETC/CIE 09 3 / 14

5 Rotor-stator interaction in turbomachinery Why does it happen? optimized designs, new materials, reduced operating tip clearances performance improvements and increased vibratory stresses Contact occurrences normal operating conditions not only transient but also stationary phenomena instabilities Analysis of stationary phenomena time integration frequency-domain approaches Laxalde, Legrand & Pierre (McGill) Nonlinear Modes for Contact Problems DETC IDETC/CIE 09 3 / 14

6 Rotor-stator interaction in turbomachinery Why does it happen? optimized designs, new materials, reduced operating tip clearances performance improvements and increased vibratory stresses Contact occurrences normal operating conditions not only transient but also stationary phenomena instabilities Analysis of stationary phenomena time integration frequency-domain approaches Nonlinear modes non-smooth nonlinearities large-scale systems Laxalde, Legrand & Pierre (McGill) Nonlinear Modes for Contact Problems DETC IDETC/CIE 09 3 / 14

7 Contact modeling and constitutive equations Solid mechanics, small perturbations assumption Γ d Ω u (x) Γ c contact interface: Γ c no external forcing no friction or dissipation Laxalde, Legrand & Pierre (McGill) Nonlinear Modes for Contact Problems DETC IDETC/CIE 09 4 / 14

8 Contact modeling and constitutive equations Solid mechanics, small perturbations assumption Γ d Gap function defined on the contact interface: g(u) = u(x) n g 0 (x) Ω u (x) n, outward normal vector g 0, initial gap Contact conditions Γ c contact interface: Γ c no external forcing no friction or dissipation τ N 0, g (u) 0, g (u) τ N = 0 τ N =σ n, contact pressure Laxalde, Legrand & Pierre (McGill) Nonlinear Modes for Contact Problems DETC IDETC/CIE 09 4 / 14

9 Strong formulation of the eigenvalue problem Boundary value problem (Dirichlet and Signorini conditions): ρü divσ(u) = 0 u = 0 on Ω R + on Γ d R + g (u) 0, τ N 0 et g (u) τ N = 0 on Γ c R + Displacement field in the frequency domain: u(t) = û n e jnωt with û n = 1 T n Z Eigenvalue problem: Find{ω, û}, with û ={û n, n Z} such as: T divσ(û n ) = (nω) 2 û n û n = 0 u(t)e jnωt dt on Ω Z on Γ d Z g (û) 0, τ N 0 et g (û) τ N = 0 on Γ c [0, T] Laxalde, Legrand & Pierre (McGill) Nonlinear Modes for Contact Problems DETC IDETC/CIE 09 5 / 14

10 Rayleigh quotient Generalized Rayleigh quotient: r (û) = n Z k (û n, û n ) m (û n, û n ) with: m(u, v) = Ω ρuv dx et k (u, v) = σ (u) :ǫ (v) dx Ω Critical points û of this Rayleigh quotient are eigenvectors and eigenvalues are:ω 2 = r (û) Constrained minimization (Signorini conditions): min û n V r (û) with g (û) 0 on Γ c t [0, T] Laxalde, Legrand & Pierre (McGill) Nonlinear Modes for Contact Problems DETC IDETC/CIE 09 6 / 14

11 Optimality conditions and variational formulation Optimality conditions: λ(t) > 0 such as: ûn r (û) + λ(t), ûn g (û) dt = 0 n Z T λ(t)>0 et λ(t), g (û) = 0 t [0, T] Variation formulation Find{û n, n Z} such as v V 2 m(û n, û n ) 1( k (û n, v) n 2 ω 2 (û) m (û n, v) ) + λ, v e jnωt dt = 0 n Z T λ>0 and λ, g (û) = 0 t [0, T] Laxalde, Legrand & Pierre (McGill) Nonlinear Modes for Contact Problems DETC IDETC/CIE 09 7 / 14

12 Augmented Lagrangian Augmented Lagrangian: L κ (û,θ) = r (û) Optimality conditions: T κ (g (û) +θ) 2 + dt ûn L κ (û,θ) = 0 n Z Using the variational formulation: Find{û n, n Z} such as v V Multiplier updates: 2 m(û n, û n ) 1( k (û n, v) n 2 ω 2 (û) m (û n, v) ) + κ (g (û) +θ) +, v e jnωt dt = 0 T θ θ+ max (g (û), θ) x,t Laxalde, Legrand & Pierre (McGill) Nonlinear Modes for Contact Problems DETC IDETC/CIE 09 8 / 14

13 Numerical implementation Discretization Time discretization: t ={t k = kt/m, k = 1,..., m} Fourier series and discrete Fourier transform: û n = 1 T m k=1 2πkn j ū k e m and ū k = N n= N û n e j 2πkn m Gap function: g (U) = AU g 0 Discretized eigenvalue problem: 2M 1( K (nω) 2 M ) Û n +A T m k=1 ( ) ) 2πkn κ g(û +θ k ej m = 0 ( ) + General algorithm 1 continuation with respect to the modal amplitude q 2 calculateω (q) with û n (q) fixed 3 update û n(q) forω (q) Laxalde, Legrand & Pierre (McGill) Nonlinear Modes for Contact Problems DETC IDETC/CIE 09 9 / 14

14 Augmented Lagrangian algorithm Initialization i = 0, i max, G (0) = θ =θ (1),κ =κ (1) α>1,β> 1,ǫ>0 Iterations while G (i) >ǫ and i< i max do Find{û n } n= N,...,N solution of ( ); Evaluate{g(ū k )} k=1,...,m ; Define: l ={l : max (g(ū k ), θ k ) G (i) /α} Ḡ = max max (g (ū k ), θ k ) if Ḡ G(i) then l l,κ k,l βκ k,l andθ k,l θ k,l /β; else i i + 1θ (i) k θ k,κ (i) k κ k and G (i) Ḡ; θ k θ (i) k + max (g (ū k ), θ k ); if G (i) G (i 1) /α then l l,κ k,l βκ k,l ; Laxalde, Legrand & Pierre (McGill) Nonlinear Modes for Contact Problems DETC IDETC/CIE / 14

15 Blade and contact interface modeling casing r r z t Craig & Bampton reduced-order model 24 interface nodes and 40 normal modes contact in radial direction r uniform initial gap Laxalde, Legrand & Pierre (McGill) Nonlinear Modes for Contact Problems DETC IDETC/CIE / 14

16 Nonlinear modes: modal parameters Normalized frequency displ. nonlinear DOF Normalized modal amplitude vel. master coord. displ. master coord. Laxalde, Legrand & Pierre (McGill) Nonlinear Modes for Contact Problems DETC IDETC/CIE / 14

17 Effects of contact on mode shapes Normalized frequency Normalized modal amplitude displ. nonlinear DOF vel. master coord. displ. master coord. Laxalde, Legrand & Pierre (McGill) Nonlinear Modes for Contact Problems DETC IDETC/CIE / 14

18 Effects of contact on mode shapes Normalized frequency Normalized modal amplitude displ. nonlinear DOF vel. master coord. displ. master coord. Laxalde, Legrand & Pierre (McGill) Nonlinear Modes for Contact Problems DETC IDETC/CIE / 14

19 Effects of contact on mode shapes Normalized frequency Normalized modal amplitude displ. nonlinear DOF vel. master coord. displ. master coord. Laxalde, Legrand & Pierre (McGill) Nonlinear Modes for Contact Problems DETC IDETC/CIE / 14

20 Concluding remarks Summary: A method for nonlinear modal analysis of mechanical systems with contact interfaces has been proposed. Based on a mixed time-frequency formulation of the eigenvalue problem, a constrained minimization of a Rayleigh quotient is defined, which is then solved using an augmented Lagrangian approach. An application to a large-scale structure has been proposed. Parametric studies have shown the effects of contact on modal parameters. Laxalde, Legrand & Pierre (McGill) Nonlinear Modes for Contact Problems DETC IDETC/CIE / 14

21 Concluding remarks Summary: A method for nonlinear modal analysis of mechanical systems with contact interfaces has been proposed. Based on a mixed time-frequency formulation of the eigenvalue problem, a constrained minimization of a Rayleigh quotient is defined, which is then solved using an augmented Lagrangian approach. An application to a large-scale structure has been proposed. Parametric studies have shown the effects of contact on modal parameters. Future works: Introduction of friction and use of complex modes formalism Stability and bifurcation analysis Application to bladed disks with flexible casing Laxalde, Legrand & Pierre (McGill) Nonlinear Modes for Contact Problems DETC IDETC/CIE / 14