Handbook of Turbomachinery Second Edition Revised - Ventech!

(11) for a simple case]:
Where
WA ¼ p XNc
i¼1
½AfAx sinðf fxÞŠ i
ð23Þ
Af is the amplitude of the induced aerodynamic force component on
mesh cell i in the direction of the vibratory motion.
Ax is the amplitude of the vibrating displacement of the cell.
f fx is the phase lead of the induced aerodynamic force relative to the
displacement for the cell.
Aerodynamic damping can be defined as
DA ¼ WA ð24Þ
The total damping of the blade aeroelastic system is
D ¼ DM þ DA
ð25Þ
A conservative prediction can be made by neglecting the mechanical
damping, DM, which is always positive. Aeroelastic stability of a blade is
directly indicated by a damping coefficient taking a form of
CA ¼ WA
A 2 m
ð26Þ
where Am is the maximum vibration amplitude taken in the unsteady
aerodynamic calculation. So, the somehow arbitrary vibrating amplitude
taken in the damping calculation does not matter provided the aerodynamic
response is of a linear nature.
Apparently, given natural vibration mode shapes and frequencies, a
blade flutter analysis using the energy method will be solely determined by
unsteady aerodynamic calculations. This is why unsteady aerodynamics
around oscillating blades in a specified mode has attracted so much
attention and effort in flutter analysis and predictions for turbomachinery
blades.
Forced Response Prediction
Forced response is more complex, because it involves two separate
aerodynamic elements, aerodynamic damping and aerodynamic forcing.
The analysis may be most effectively carried out in a Modal space, where
the original structural dynamic equations with N degrees-of-freedom are
decoupled into N single-degree-of-freedom equations. Each of these
Copyright © 2003 Marcel Dekker, Inc.