The aeroelastic
analysis of laminated composite wings is vital to the prevention of failures
induced by oscillatory motion. The aeroelastic instabilities, however, will
change, when a crack has initiated in a wing structure and must be accounted for
by adjustment to the structural and dynamic model. An aeroelastic normal mode
analysis greatly depends on the free vibration modes of the wing. To achieve
accurate results a new Dynamic Finite Cracked Element (DFCE) (refer to Chapter
2) is implemented. From the previous chapters both the DFE and DFCE both show
excellent accuracy for preliminary coarse meshes. Composite wings consist of two
types of bending-torsion couplings, Geometric and Material. Geometric coupling
originates from an offset of the centre of gravity (CG) axis from the elastic
axis (EA), where material coupling arises from material anisotropy. Geometric or
material couplings can cause flutter instabilities in a wing. Wings modeled as
beam assemblies can produce various couplings. Since the incentive of this
chapter is to study the aeroelastic flutter and divergence of a defective wing,
only bending-torsion couplings are considered. To achieve purely bending-torsion
behaviour in a composite beam or wing structure, specific laminate stacking
sequence must be considered (i.e. Symmetric or unidirectional unbalanced
laminates). The beams used in this chapter, to approximate a wing, are assumed
to be based on classical laminate theory with solid rectangular cross section
and unidirectional plies.
Figure 1:
Cracked wing approximated using unidirectional composite solid rectangular cross
section beam elements.
A composite
unidirectional wing is considered, length, L=1 m, base b=0.25 m
and thickness, t=0.02 m. The laminated composite material properties and
geometric properties for this wing configuration can be found in Table 1 and
Table 2 respectively. The flow is assumed to be
quasi-steady, incompressible flow with a lift curve slope of
.
From Figure 2 the instabilities are
displayed for the current composite wing profile for a wing with a crack located
at the 0.2L with a crack ratio of a/b=0.3. In the range of ply
angles from 0 to 97 degrees, the type of instability is flutter. Then for ply
angles of 97 to 141 degrees the wing will diverge, then begin to flutter again
from 141 to 180 degrees.
The free vibration natural frequencies and
modes are extracted using Dynamic Finite Cracked Elements (DFCE) for an intact
wing and displayed in Figure 3 for various unidirectional plies. Usually the
frequencies only need to be plotted for the first 90 degrees, as the frequencies
are generally a mirror image across the 90 degree line. For a wing, the natural
frequencies must be plotted for all 180 degrees due to the geometric coupling,
which is developed by an offset of the mass axis from the elastic axis.
In Figure 4 and Figure 5 the normalized
divergence and flutter speeds are plotted for various angles ranging from 0 to
180 degrees. In Figure 4, the divergence speeds are observed to change
significantly across this range of angles, specifically the divergence speed
quickly rises at approximately 30 degrees until 90 degrees. The influence of a
static non-propagating crack is observed to have a consistent drop in the
normalized divergence speed with increasing crack ratio. The lowest divergence
speeds are found for composite wings with a unidirectional ply angles between 96
degrees 146 degrees. In Figure 5, much like the divergence speed plot, the
flutter speeds tend to be sensitive to ply angle and crack ratio. When the crack
ratio is gradually increased from no crack up a/b =0.6 a drop in flutter
speed is observed for most ply angles, except for the unidirectional plies set
in the region of 96 degrees to 146 degrees. In this region a flip flop in this
trend is seen, where the flutter speed increases with a larger crack.
From the frequencies plots in Figure 6
to Figure 13
large deviations are observed for both test cases (crack located at 20% and 50%
span). Although it’s interesting to note that the different mode numbers are
affected by the ply angle in much different ways. For example, for the first
mode the highest change in frequency cause by the crack is observed near the 90
degree ply. The data collected at 0, 90, 180 degrees are not valid results and
should be ignored, since for these special cross-ply laminates an alternative
formulation is required. However the results near 90 degrees are valid. For the
second natural frequency the largest difference is notice in the 80 and 110
degree plies, and for the third frequency the there is little to no noticeable
change in the frequency near the 90 degree plies. The largest difference from
the intact frequencies for the third mode is observed for plies at 70 and 125
degrees.
An aeroelastic study has been accomplished
and validated for a laminated composite cracked wing using Dynamic Finite
Cracked Elements. The influence of a static crack on a laminated composite wing
is significant in both the free vibration modes as shown previously in Chapter 2
and on the flutter and divergence speeds observed in this chapter.
A reduction in the
divergence speeds occurs when the crack size is increased for most
unidirectional ply angles considered. Whereas, the flutter speeds are observed
to increase in specific ranges of ply angles. The V-g method has proven to be an
excellent method for the extraction of the flutter speeds and readily extendible
to more complex formulations particularly ones that include unsteady flow.
