1
A component mode synthesis approach for dynamic analysis of viscoelastically damped structures Lucie Rouleau 1,2 , Jean-Fran¸ cois De¨ u 1 , Antoine Legay 1 , Jean-Fran¸ cois Sigrist 2 1 LMSSC - Cnam Paris, Paris, France 2 DCNS Research, La Montagne, France Structural Mechanics and Coupled Systems Laboratory, Cnam, Paris, France e-mail: [email protected] Website: www.lmssc.cnam.fr/ 1. Context and Motivation The noise radiated by the vibrations of some parts of the submarine, such as the propulsion system, may be detected by enemy radars. Viscoelastic material are used to damp the vibrations of the structure and enhance submarine stealthiness performance. I How to calculate efficiently the dynamic response of viscoelastically damped structures? Figure 1: Detection of a submarine by a passive sonar (http://marport.com). 2. Finite element model of the sandwich structure Figure 2: Turbine nozzle (http://www.propellerpages.com). Viscoelastic layers Figure 3: Finite element model of the turbine nozzle with viscoelastic material. - Modeling of the viscoelastic material by a fractional derivative model: G * (ω )= G 0 + G ∞ (iωτ ) α 1 + (iωτ ) α - Identification of the parameters by a least square method from DMA measurements: Environmental Thermometer Samples Grips chamber Figure 4: Master curves of Deltane 350 (Paulstrar) at T ref = 12 o C. 10 0 10 5 10 7 10 8 Reduced frequency [Hz] Storage modulus [Pa] 10 0 10 5 0 5 10 15 20 25 30 35 40 45 50 55 Reduced frequency [Hz] Loss angle [ o ] Figure 5: Master curves of Deltane 350 (Paulstrar) at T ref = 12 o C. I After finite element discretisation, the matrix system to be solved is: K e + G * (ω )K v - ω 2 M U = F K e Stiffness matrix of the elastic part of the structure K v Stiffness matrix of the viscoelastic part of the structure calculated with unitary shear modulus M Mass matrix of the overall structure 3. Component mode synthesis - Substructuring: decomposition of the original structure into several components Component 1 Component 2 U 2 = U b U 2 i U 1 = U b U 1 i Figure 6: Decomposition of the turbine nozzle into 2 components. For each component, the degrees of freedom (dofs) U j are divided into interface dofs U b and internal dofs U j i . - Calculation of the Craig-Bampton basis for each component. For component 1: T 1 = [Ψ stat , Φ 0 , Φ v ] ψ stat Static responses of the structure to U b (k )=1, U b (j )=0 ∀j 6= k φ 0 Fixed interface vibration modes calculated with G * = <(G * (ω = 0)) φ v Fixed interface vibration modes calculated with G * = <(G * (ω = ω max )) u b = 1 0 0 0 ... 0 u b = 0 1 0 0 ... 0 u b = 0 0 1 0 ... 0 ψ 2 stat ψ 3 stat ψ 1 stat φ 32 0 φ 2 0 φ 32 v φ 3 v φ 3 0 φ 2 v Figure 7: Deformed shape of static modes (column 1) and fixed-interface vibration modes (columns 2 and 3) - Projection of the matrix system on the reduction basis for each component i T i T K e + G * (ω )K v - ω 2 M T i Y i = T i T F U i = T i Y i I The Craig-Bampton basis of component 1 is enriched with modes computed for a high frequency modulus in order to take into account the changes induced by the frequency- dependent viscoelastic properties on mode shapes. 5. Application and validation of the reduction method on the 3D turbine nozzle Modal truncation criteria: modes with f < 1.5f max I 32 modes ψ 0 I 12 modes ψ v (computed for ω max = 400 Hz) Reduction of the number of dofs: I 109161 dofs (unreduced) I 137 dofs (reduced) Reduction of the computational time of the frequency response on [0, 400] Hz with a frequency step of 1 Hz Direct CMS 0 200 400 600 800 15000 Computational time (s) Calculation of Ke Kv and M Calculation of reduction basis Projection on reduction basis Calculation of FRF Figure 8: Distribution of computational time of the FRF for the direct method and the component mode synthesis (CMS) method. I The presented component mode synthesis method allows to take into account the frequency-dependency of the viscoelastic properties, and reduces significantly the computational time of the frequency reponse. 0 50 100 150 200 250 300 350 400 -240 -220 -200 -180 -160 -140 -120 Frequency [Hz] FRF [dB] Undamped structure Damped structure (CMS method) Damped structure (Direct method) Figure 9: Frequency response of the undamped and damped structure. Validation of the component mode synthesis method by comparison to the direct method. WORK IN PROGRESS Characterisation of viscoelastic materials Introduction of the causality Kramers-Kronig relationships G 0 (ω )= G ∞ + 2 π Z ∞ 0 uG 00 (u ) ω 2 - u 2 u G 00 (ω )= 2ω π Z ∞ 0 G 0 (u ) u 2 - ω 2 u in the procedure to obtain the master curves of a viscoelastic material from DMA measurements. Validation of the finite element model Comparison of the numerical frequency response of a sandwich beam to the experimental one obtained by laser vibrometer measurements. Modeling of the external fluid Methods investigated : I Semi-analytical added mass operator [3] I Infinite elements with absorbing boundary conditions Modeling of viscoelastic interfaces Development of interfaces elements for constrained thin rubber layers. u + 1 u + 2 u + 3 u - 1 u - 2 u - 3 Figure 10: Interface element. REFERENCES [1] Rouleau, L., De¨ u, J.-F., Legay, A., Sigrist, J.-F., Marin-Curtoud, P., Reduced order model for noise and vibration attenuation of water immersed viscoelastic sandwich structures, Acoustics 2012 [2] Rouleau, L., De¨ u, J.-F., Legay, A., Sigrist, Le Lay, F., J.-F., Marin-Curtoud, P., A component mode synthesis approach for dynamic analysis of viscoelastically damped structures, WCCM 2012 [3] Rouleau, L., De¨ u, J.-F., Legay, A., Sigrist, J.-F., ”Vibro-acoustic study of a viscoelastic sandwich ring immersed in water”, J. of Sound and Vibration, 331, 522-539, 2012 This research is funded by :
