Page 305 - Wind Energy Handbook

BLADE DYNAMIC RESPONSE 279

ð
X X R
€
M ¼ r cos ł J q(r, t)dr r cos ł J m 1 (r)ì(r)dr: f J (t)
f
N N 0
X ð R
_
c
r cos ł J ^ c(r)ì(r)dr:f J (t) (5:113)
f
N 0
The suffix J refers to the Jth blade, and N in the summations is the total number of
blades.
Hence

X X ð R
€
F þ M=L ¼ ì TJ q(r, t)dr m 1 (r)ì(r)ì TJ (r)dr: f J (t)
f
N N 0
ð
X R
_
^ c c(r)ì(r)ì TJ (r)dr: f J (t)
f
N 0
and Equation (5.109) becomes
ð
X X R
€
€
_
2
f
m T1 f T (t) þ c T1 f T (t) þ m T1 ø f T (t) ¼ ì TJ q(r, t)dr m 1 (r)ì(r)ì TJ (r)dr: f J (t)
f
f
T
N N 0
X ð R
_
f
^ c c(r)ì(r)ì TJ (r)dr:f J (t) (5:114)
N 0
omitting the term for loading on the tower itself.
Equations (5.108) and (5.114) provide (N þ 1) simultaneous equations of motion
with periodic coefficients ì TJ corresponding to the (N þ 1) degrees of freedom
assumed. The procedure for the step-by-step dynamic analysis which is based on
these equations may be summarized as follows:
(1) Substitute the displacements, velocities and aerodynamic loads at the beginning
of the first time step into Equations (5.108) and (5.114), and solve for the initial
accelerations.

(2) Formulate the incremental equations of motion for the time step, based on
Equations (5.108) and (5.114), retaining the coupled terms on the right-hand
side, i.e., as pseudo forces.

(3) Assume initially that the coupled terms are constant over the duration of the
time step, so that they disappear from the incremental equations of motion
altogether, rendering them uncoupled.

(4) Solve the uncoupled incremental equations of motion to obtain the increments
of displacement and velocity over the time step. Adopting the linear accelera-