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)


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.


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


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

Mass, damping, and stiffness matrix

X, F

Displacement and Force vector


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


Reduction matrix


Harmonic Component Number


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


Circular frequency


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


Proportional damping constant


Static normal preload on contact area


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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

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