Environmental Engineering Reference
In-Depth Information
Direct numerical integration of the forced equations of motion in the time domain (
i.e.
,
output is load
vs.
time) is the most straightforward and informative method of solution.
This procedure, which is often referred to as
system simulation
,
handles both steady-state
and transient response. Since the governing equations of motion contain time-varying
coefficients and are generally nonlinear, frequency domain techniques have limited value.
Furthermore, seeking closed-form analytical solutions to these complex equations is usually
unnecessary with the capabilities of today's digital computers. The engineer's time is best
spent in understanding the results of a system simulation and then exploring methods for
reducing dynamic loads.
Aeroelastic Stability Models
Aeroelastic stability analysis is often separate and distinct from the modal and loads
analyses for two reasons: First, it is preferable to linearize the equations of motion and
examine the resulting
eigenvalues
for stability, rather than to examine a myriad of time
histories to determine if the response is stable. Of course, if the mechanism of instability
is highly nonlinear, simulation in the time domain may be the only recourse. Most potential
wind turbine instabilities can be adequately assessed using linearized models.
The second reason for a separate model is that the modes which are important for
aeroelastic stability often differ from those needed for loads analysis. In a successful
system, instabilities will occur outside the limits of planned operation. Hence, a blade
torsional mode which incites instability at a high rotor speed may well be dormant during nor-
mal operation and, hence, be omitted in the loads analysis. It should be noted that aeroelastic
stability analysis, like modal analysis, can be carried out with a finite-element model.
Dynamic Load Model of a HAWT Blade
A beam model of a wind turbine blade is generally suitable for structural-dynamic
analysis. It will differ from the small-deflection theory beam models used in conventional
analyses of non-rotating structures, however, because the axial loads on the blade
significantly effect the lateral and torsional deflections. In this respect it is more like the
beam-column representations used in elastic stability analysis. General three-dimensional
theory is quite complicated and has provided fodder for more than one doctoral thesis. It
is still subject to controversy as to which terms are important and which are not. Rather
than beginning with the general case, we will develop the equations for uncoupled
flapwise
(
i.e.
out-of-plane or flatwise) motions to acquaint the reader with the physics involved. In
many cases, these will be suitable for the analysis of HAWT rotor loads.
Elastic blade flapping equations appear in many sources and are in a sense, classical.
A relatively brief derivation is given here to highlight the approximations and assumptions
that are implicit in the equations which are commonly used. We begin by assuming that
the strains everywhere in the blade are small (less than 10 percent) and below the elastic limit.
Other assumptions will be introduced at appropriate points in the derivation.
1
Equations of
motion for wind turbine components are generally derived using a “strength of materials”
rather than a “theory of elasticity” approach.
1
Note here that the assumptions of small strain and elasticity do not mean that the
deflections must be small. Everyone is aware that a steel ruler can be bent into an arc with
deflections much greater than its cross-sectional dimensions, and yet it will spring back
elastically to its original shape.
Search WWH ::
Custom Search