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264 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES
It is seen that the damping ratio for the second mode is under half that for the
first.
5.8.5 Response to deterministic loads—step-by-step dynamic
analysis
As set out in Section 5.8.1, blade dynamic response to time varying loading is best
analysed in terms of the separate excitation of each blade mode of vibration, for
which, under the assumptions of unstalled flow and mass-proportional aerody-
namic damping , the governing equation is
ð R
_
€
2
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 ¼ Q i (t) (5:70)
f
f
i
0
where f i (t) and ì i (r) are the tip displacement and mode shape for the ith mode
respectively. Starting with the initial tip displacement, velocity and acceleration
arbitrarily set at zero, this equation can be used to derive values for these quantities
at successive time steps over a complete blade revolution by numerical integration.
The procedure is then repeated for several more revolutions until the cyclic blade
response to the periodic loading becomes sensibly invariant from one revolution to
the next.
Linear acceleration method
The precise form of the equations linking the tip displacement, velocity and
acceleration at the end of a time step to those at the beginning depends on how the
acceleration is assumed to vary over the time step. Newmark has classified
alternative assumptions in terms of a parameter â which measures the relative
weightings placed on the initial and final accelerations in deriving the final
displacement The simplest assumption is that the acceleration takes a constant
value equal to the average of the initial and final values (â ¼ 1=4). Clough and
Penzien (1993), however, recommend that the acceleration is assumed to vary
linearly between the initial and final values, as this will be a closer approximation
to the actual variation. Step-by-step integration with this assumption is known as
either the linear acceleration method or the Newmark â ¼ 1=6 method.
Expressions for the tip displacement, velocity and acceleration at the end of the
€
_
first time step – f i1 , f i1 and f i1 respectively – are derived in terms of the initial
f
f
_
€
f
f
values – f i0 , f i0 and f i0 – as follows. The acceleration at time t during the time step
of total duration h is
€ €
f
€ € f f i1 f i0 t (5:81)
f f i (t) ¼ f i0 þ
f
h
This can be integrated to give the velocity at the end of the time step as