Structural Dynamic Behavior of Wind Turbines - Wind Turbine Technology: Fundamental Concepts of Wind Turbine Engineering - page 623

blade. The time-history of s k ( t ) is determined by numerical integration of the equation of
motion during consecutive revolutions of the rotor, until each s k and its first time derivative
at y = 2p are equal to the same parameters at y = 0, within a set tolerance. This is the
so-called trim solution.
The flap mode shapes of importance to the structural-dynamic analysis of HAWT rotors
can be divided into the following two classes: (1) dependent blade modes, in which the
motions of a given blade are constrained or influenced by the motions of another blade in
the rotor, and (2) independent blade modes, in which the motions of a given blade can be
analyzed separately from the other blades. An example of a dependent mode is the rigid-
body teetering motion of two-bladed HAWT rotor, in which the mode shapes of opposite
blades must always be antisymmetric and without curvature at the hub. Dependent blade
motions will not be discussed further here because of the additional complexity introduced
by the constraints at the hub. The reader is referred to Wright et al. [1988] and [Wright
and Butterfield 1991] for the equations required to perform a dynamic analysis of a teetered
HAWT rotor and sample comparisons with test data.
The two classic examples of independent blade modes are the cantilevered shape, in
which both v and v ¢ are zero at the hub, but v ¢¢ may be non-zero; and the flapping hinge
shape, in which both v and v ¢¢ are zero at the hub, but v ¢ may be non-zero [Spera 1975].
In the remainder of this analysis, we will limit ourselves to one independent flapwise mode,
for simplicity ( i.e. , k = 1). Fortunately, experience has shown that one cantilever flapwise
mode is usually sufficient for the calculation of blade loads in a rigid-hub rotor. Moreover,
blade loads are much more sensitive to the scale of the wind input models (for wind shear,
turbulence, and tower-shadow) than they are to higher blade modes. The exception to this
general observation is the case in which the natural frequency of the next-higher mode is
equal to an integer multiple of the rotor speed, and a resonance condition may be present.

Equation of Motion for One Independent Blade Mode

Substitution of the assumed form of the blade displacement from Equation (11-28) into
Equation (11-6a) with k set equal to 1 gives

(11-29)

[- ( s g ) EI x ] + ( Ts g ) + q x + p z = 0

Now, use of a Galerkin approach [ e.g. Dym and Shames 1973] gives us

R

ò

(11-30)

[- ( s g) EI x ] + ( Ts g ) + q x + p z g d x = 0

0

Integrating this equation by parts and using the boundary conditions required of g (x) lead
to an ordinary differential equation in s(t) , which can then be integrated numerically in the
time domain.
The numerical integration approach is best explained by rearranging Equation (11-29)
so that the acceleration term remains on the left-hand side but all other terms are moved to
the right-hand side. If we perform our time-domain integration in time steps of duration
dt , we can now calculate the acceleration at time t in terms of the right-hand side of the
equation evaluated with calculations made previously at time t - dt. We continue this
forward-integration process in time until convergence to a trim solution is obtained. As
discussed previously, trim is achieved when the flap displacement and velocity at y = 2p
are equal to the same parameters at y = 0, within a specified tolerance.