Page 339 - Wind Energy Handbook

FREQUENCY RESPONSE FUNCTION 313

Appendix: Dynamic Response of Stationary Blade in
Turbulent Wind

A5.1 Introduction

As described in Chapter 2, the turbulent wind contains wind speed fluctuations
over a wide range of frequencies, as described by the power spectrum. Although
the bulk of the turbulent energy is normally at frequencies much lower than the
blade first mode out-of-plane frequency, which is typically over 1 Hz, the fraction
close to the first mode frequency will excite resonant blade oscillations. This
appendix describes the method by which the resonant response may be determined.
Working in the frequency domain, expressions for the standard deviations of both
the tip displacement and root bending moment responses are derived, and then the
method of deriving the peak value in a given period is described. Initially the wind
is assumed to be perfectly correlated along the blade, but subsequently the
treatment is extended to include the effect of spatial variation.

A5.2 Frequency Response Function

A5.2.1 Equation of motion

The dynamic response of a cantilever blade to the fluctuating aerodynamic loads
upon it is most conveniently investigated by means of modal analysis, in which the
the excitations of the various different natural modes of vibration are computed
separately and the results superposed. Thus the deflection x(r, t) at radius r is given
by:

X
1
x(r, t) ¼ f i (t)ì i (r)
i¼1

Normally, in the case of a stationary blade, the first mode dominates and higher
modes do not need to be considered. The equation of motion for the ith mode,
which is derived in Section 5.8.1, is as follows:

ð R
€
_
2
f
f
m i f i (t) þ c i f i (t) þ m i ø f i (t) ¼ ì i (r)q(r, t)dr (A5:1)
i
0
where q(r, t) is the applied loading, f i (t) is the tip displacement, ì i (r) is the non-
dimensional mode shape of the ith mode, normalized to give a tip displacement of
unity, ø i is the natural frequency in radians per second, m i is the generalized mass,
Ð Ð
R 2 R 2
i
i
0 m(r)ì (r)dr, and c i is the generalized damping, 0 ^ c c(r)ì (r)dr.