Advertisement

A Comparison of Two Reduction Techniques for Forced Response of Shrouded Blades with Contact Interfaces

  • Fahimeh Mashayekhi
  • Stefano Zucca
  • Ali Salehzadeh Nobari
Conference paper
Part of the Conference Proceedings of the Society for Experimental Mechanics Series book series (CPSEMS)

Abstract

Two reduction methods for dynamic analysis of structures with local nonlinearity are compared. Dual and primal formulation have the same projection basis including flexibility residual attachment modes and free interface modes, but there are significant differences in their implementation. Both methods can be applied to nonlinear forced response analysis of turbine blades with contact interfaces in shroud. In this study, the shroud contact elements are employed using the adequate description of friction and 3D tangential coupled contact forces considering the effect of normal load variation. In order to examine and compare the accuracy of the two formulations, a rod and a simplified shrouded turbine blade was considered as case studies.

Keywords

Reduced order models Primal formulation Dual formulation Component mode synthesis Shroud contact Forced response 

Nomenclature

[M],[C], and [K]

Mass, damping, and stiffness matrix

X, F

Displacement and Force vector

[Φ]

Matrix of free interface dynamic modes

[Ψ], [Ψ]ar

Matrix of attachment modes and residual attachment modes

ωp and Φp

pth eigenvalue and eigenvectors of free interface body

N and ∗L

Pertaining to nonlinear(interface) and linear(inner) dofs of substructure

R

Reduction matrix

h

Harmonic Component Number

α

Mass proportional damping

C and ∗E

Pertaining to Contact and excitation

\( {{\bar{\mathrm{X}}}}^{\mathrm{h}} \)

Vector of Fourier coefficient of hth harmonic of X

ω

Circular frequency

n

Pertaining to nth contact nodes

[\( \tilde{\mathrm{A}} \)]

Pertaining to characteristic matrix [A] of system with interfaces added stiffness in fully suck condition

[kc, n]

nth sub matrix regarding the added stiffness to nth contact node

kt, kn

Tangential and normal contact stiffness parameters

α

Proportional damping constant

N0

Static normal preload on contact area

References

  1. Petrov, E.: A method for use of cyclic symmetry properties in analysis of nonlinear multiharmonic vibrations of bladed discs. In ASME Turbo Expo 2003, collocated with the 2003 International Joint Power Generation Conference, pp. 235–245. (2003)Google Scholar
  2. 2.
    Craig Jr., R.: A brief tutorial on substructure analysis and testing,# 325. Proc. IMAC-XVIII Conf. Struct. Dyn. 4062, 899 (2000)Google Scholar
  3. Charleux, D., Gibert, C., Thouverez, F., Dupeux, J.: Numerical and experimental study of friction damping blade attachments of rotating bladed disks. Int. J. Rotating Mach. 2006, 1–13 (2006)CrossRefGoogle Scholar
  4. 4.
    Guyan, R.J.: Reduction of stiffness and mass matrices. AIAA J. 3(2), 380 (1965)CrossRefGoogle Scholar
  5. 5.
    Tran, D.-M.: Component mode synthesis methods using partial interface modes: application to tuned and mistuned structures with cyclic symmetry. Comput. Struct. 87(17), 1141–1153 (2009)CrossRefGoogle Scholar
  6. 6.
    Rubin, S.: Improved component-mode representation for structural dynamic analysis. AIAA J. 13(8), 995–1006 (1975)CrossRefGoogle Scholar
  7. 7.
    Zucca, S.: On the dual Craig–Bampton method for the forced response of structures with contact interfaces. Nonlinear Dynamics. 87(4), 2445–2455 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Rixen, D.J.: A dual Craig–Bampton method for dynamic substructuring. J. Comput. Appl. Math. 168(1), 383–391 (2004)MathSciNetCrossRefGoogle Scholar
  9. Van der Valk, P.: Model reduction & interface modeling in dynamic substructuring: application to a multi-megawatt wind turbine, Master’s thesis. TU Delft, The Netherlands, pp. 40–43 (2010)Google Scholar
  10. 10.
    Martinez, D.R., Carrie, T.G., Gregory, D.L., Miller, A.K.: Combined experimental/analytical modeling using component mode synthesis. Comput. Struct. 78, 583–590 (1984)Google Scholar
  11. 11.
    Craig, R.R.: Coupling of substructures for dynamic analyses: an overview, structures, structural dynamics and material conference, (2000)Google Scholar
  12. 12.
    Afzal, M., Arteaga, I.L., Kari, L.: An analytical calculation of the Jacobian matrix for 3D friction contact model applied to turbine blade shroud contact. Comput. Struct. 177, 204–217 (2016)CrossRefGoogle Scholar

Copyright information

© The Society for Experimental Mechanics, Inc. 2018

Authors and Affiliations

  • Fahimeh Mashayekhi
    • 1
  • Stefano Zucca
    • 2
  • Ali Salehzadeh Nobari
    • 1
  1. 1.Department of Aerospace EngineeringAmirkabir University of TechnologyTehranIran
  2. 2.Department of Mechanical and Aerospace EngineeringPolitecnico di TorinoTorinoItaly

Personalised recommendations