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NONLINEAR VIBRATIONS ANALYSIS OF VISCOELASTICALLY DAMPED STRUCTURES

NONLINEAR VIBRATIONS ANALYSIS OF VISCOELASTICALLY DAMPED STRUCTURES. CONTENTS 1- Introduction 2- Formulation 3- An approximated harmonic balance method 4- An amplitude equation 5- Applications 6- Conclusions and perspectives.

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NONLINEAR VIBRATIONS ANALYSIS OF VISCOELASTICALLY DAMPED STRUCTURES

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  1. NONLINEAR VIBRATIONS ANALYSIS OF VISCOELASTICALLY DAMPED STRUCTURES

  2. CONTENTS1- Introduction2- Formulation3- An approximated harmonic balance method4- An amplitude equation5- Applications6- Conclusions and perspectives

  3. Considering the non linear geometrical effectDampingproperties depend on amplitude vibrationHow to get these dependencies ? Viscoelastic material Elastic layers 1- Introduction

  4. Viscoelastic Behaviorlaw : 2- Formulation VWP L(.): Linear operator, Q(.,.): Bilinear operator, M: Masse operator, U=(u, S): Mixed vector, u: Displacement, S: Stress ( Formulation in displacement: Cubic non linearities ) Y: Relaxation function g : Strain Harmonic motion Complex Young’s Modulus

  5. 3-An Approximated harmonic balance method (Exact B.H. Induces coupling) H 1 :  a : Complex amplitude w : Complex frequency Um : linear real eigenmode wm : linear frequency H 2 : The effect the secondary frequencies is small than the principal one

  6. H 3 :Vibration near linear frequency Uinduces three harmonics

  7. 4-An amplitude equation Galerkin’s procedure with one mode K1: Modal complex stiffness, M: Modal mass, Knl: Non linear stiffness ( coupling of viscoelastic and non linear geometrical effects)

  8. Nonlinear frequency and loss factor

  9. 5- Applications Simply supported viscoelastic sandwich beam Linear loss factors

  10. Non linear loss factors

  11. Fsin( wt) Present Method h h h f f f l l l 48 2.48 10-2 48 2.25 10-2 49 5,62 10-2 213 8.80 10-2 211 8.45 10-2 218 1,18 10-1 450 1.25 10-1 459 1.37 10-1 455 1.36 10-1 5- Applications Viscoelastic sandwich plate Viscoelastic behaviour Generalized Maxwell Model Experimental results ANM (2001)

  12. . ANM (2001) Present Method h h f f l l 47 2.40 10-2 47 2,74 10-2 199 9,54 10-2 199 1,05 10-1 423 9.22 10-1 423 9,75 10-2 Non linear frequency-amplitude curves corresponding to the first mode at various excitations

  13. 5- Applications Circular viscoelastic ring Moderate rotations Shear deformation is neglected Square section Complex Young’s Modulus E = E0(1+ihl) E0: Real modul of elasticity hl : Linear loss factor of the viscoelastic material h: Thickness

  14. Backbone curve of the Frequency-amplitude for three modes (m=2,6,10) , (R=100, h=1)

  15. Loss factor-amplitude curves for three modes (m=2,6,10) (R=100, h=1)

  16. 6-Conclusion and perspectives • The presented approach permits to reduce the Non linear vibrations problem of viscoelastic structures to an amplitude equation whose coefficients are obtained by solving • - 1 Linear eigenvalue problem • - 2 Classical linear problems • By this way, one obtains Frequency-amplitude and loss factor-amplitude relationships. • - Under development : Sandwich structures(Ring, Arch, Cylinder… ) • - Future investigation: non linear vibration of piezoelectric sandwich structures

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