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Localization in Nonlinear Mistuned Systems with Cyclic Symmetry

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Normal Modes and Localization in Nonlinear Systems

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Abstract

In forced systems with cyclic symmetry localization can occur due to parameter uncertainties. Often, Monte-Carlo simulations are used to find regions, where the system response is sensitive to parameter uncertainties. These simulations require a large computation time. Therefore, an approximate method to calculate the envelopes of the frequency response functions is developed in this paper. An example of a nonlinear system with cyclic symmetry is a bladed disk assembly with friction dampers. Friction dampers can be installed underneath the blade platforms of turbine blades. Due to dry friction and the relative motion between blades and dampers, energy is dissipated, which results in a reduction of blade vibration amplitudes. By optimizing the mass of the friction dampers, the best damping effects are obtained, which lead to an increase in the reliability of the turbine. In this paper, the calculated response of a mistuned bladed disk assembly with friction dampers is discussed. An approximate method is developed to calculate the envelopes of the corresponding frequency response function for statistically varying eigenfrequencies of the blades. Regions where localization can occur with a high probability, are calculated by this method.

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Abbreviations

a Z :

= half width of contact area (m)

c R :

= global tangential contact stiffness (N/m)

E[ ]:

= mean value of [ ]

F :

= force (N)

F H :

= centrifugal force (N)

F N :

= average normal force (N)

i :

= complex unit

K :

= stiffness coefficient (N/m)

k :

= exponent of the Weibull-distribution

l K :

= length of contact area (m)

m R :

= mass of friction damper (Kg)

n M :

= number of modes for one blade

n S :

= number of blades

p :

= parameter

P H :

= Hertzian contact Pressure (N/m2)

q N :

= distributed normal load (N/m)

r C :

= distance form rotor axis to center of gravity of the friction damper (m)

R :

= radius (m)

s :

= measure for sensitivity or strength for localization (−)

S:

= sensitivity

u :

= relative displacement (m)

w :

= amplitude (m)

w 0 :

= parameter of the Weibull-distribution (m)

x, y, z :

= coordinates (m)

A :

= system matrix

B :

= matrix

F :

= generalized force vector

K :

= stiffness matrix

q :

= vector of modal coordinates

T :

= modal matrix of the bladed disk

u :

= vector of generalized relative displacements

w :

= vector of generalized displacements

β :

= blade root angle (rad)

η ζ :

= coordinates fixed to the platform (m)

Γ( ):

= gamma function

µ :

= friction coefficient (−)

γ :

= platform angle (rad)

σ :

= standard deviation

Ω:

= angular velocity (rad/s)

ω :

= excitation angular frequency (rad/s)

ω 0 :

= angular eigenfrequency (rad/s)

(ˆ):

= complex

( )i :

= imaginary part

( )r :

= real part

( )T :

= transpose of ()

( )*:

= dimensionless

( )E :

= related to points of excitation

( ) m :

= mean value

( )res :

= resultant

( ) S :

= related to the S-coordinate system

( ) w :

= related to the amplitude w

( ) Z :

= related to the Z-coordinate system

( ) η ζ :

= in η ζ-direction

( )0 :

= related to the point 0

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Author information

Authors and Affiliations

  1. Institute of Mechanics, University of Hannover, Appelstr. 11, D-30167, Hannover, Germany

    Walter Sextro & Karl Popp

  2. Faculty of Mechnical Engineering, Koszalin University of Technology, Raclawicka 15-17, PL-75620, Koszalin, Poland

    Tomasz Krzyzynski

Authors
  1. Walter Sextro
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  2. Karl Popp
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  3. Tomasz Krzyzynski
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Editor information

Editors and Affiliations

  1. University or Illinois at Urbana/Champaign, Urbana, Illinois, USA

    Alexander F. Vakakis

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Sextro, W., Popp, K., Krzyzynski, T. (2001). Localization in Nonlinear Mistuned Systems with Cyclic Symmetry. In: Vakakis, A.F. (eds) Normal Modes and Localization in Nonlinear Systems. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2452-4_11

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  • DOI: https://doi.org/10.1007/978-94-017-2452-4_11

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-5715-0

  • Online ISBN: 978-94-017-2452-4

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