For the clamped-free boundary conditions C (0) = ('((') — ("(-(') = 0, this equation has a known analytical solution that yields a divergence dynamic pressure of qD = -6.32970---—-. (3.79)

The minus sign implies that this bending divergence instability only takes place for forward-swept wings, that is, where A < 0.

Examination of Eqs. (3.76) illustrates that there are two ways in which the sweep influences the aeroelastic behavior. One is the loss of aerodynamic effectiveness as exhibited by the change in the second term of the torsion equations from qeca— qeca— ,

Note that this effect is independent of the direction of sweep. The second effect is the influence of bending slope on the effective angle of attack (see Eq. 3.73), which leads to bending-torsion coupling. This coupling has a strong influence on both divergence and load distribution. The total effect of sweep depends strongly on whether the surface is swept back or forward. This can be illustrated by its influence on the divergence dynamic pressure, qo, as shown in Fig. 3.17. It is apparent that forward sweep causes the surface to be more susceptible to divergence whereas backward sweep increases the divergence dynamic pressure. Indeed, a small amount of backward sweep (for the idealized case under consideration, depending on e/l and GJ/EI, only 5° or 10°) can cause the divergence dynamic pressure to become infinite, thus eliminating the instability altogether. Some specific cases are discussed later in this section in conjunction with an approximate solution of the governing equations.

The overall effect of sweep on the aeroelastic load distribution also strongly depends on whether the surface is swept forward or aft. This is illustrated in Fig. 3.18, which shows spanwise load distributions for an elastic surface for which the total lift (or N) is held constant by adjusting «,.. From a structural loads standpoint it is apparent that the root bending moment is significantly greater for forward sweep than for backward sweep at a given value of total lift.

Figure 3.17 Divergence dynamic pressure versus A.
Lifting Surface Drag Distribution
Figure 3.18 Lift distribution for positive, zero, and negative A.

The primary motivation for sweeping a lifting surface is to improve the vehicle performance through drag reduction, although some loss in lifting capability may be experienced. However, the above aeroelastic considerations can have a significant impact on design decisions. From an aeroelastic standpoint, forward sweep exacerbates divergence instability and increases structural loads whereas backward sweep can alleviate these concerns. The advent of composite lifting surfaces has enabled the use of bending-twist elastic coupling to passively stabilize forward sweep, making it possible to use forward-swept wings. Indeed, the X-29 could not have been flown without some means to stabilize the wings against divergence. We will discuss this further below under aeroelastic tailoring.

Exact Solution for Bending—Torsion Divergence

Extraction of the analytical solution of the above set of coupled ordinary differential equations, Eqs. (3.76), is very complicated. The exact analytical solution is most easily obtained by first converting the coupled set of equations into a single equation governing the elastic component of the angle of attack. For calculation of only the divergence dynamic pressure, one can consider just the homogeneous parts of Eqs. (3.76):

0 + --0 cos"(A) - --w sin(A) cos(A) = 0, ca ca- (381)

w"" + yjw' sin(A) cos(A) - ^j-0 cos2(A) = 0.

To get a single equation one differentiates the first equation with respect to y and multiplies it by cos( A ). From this modified first equation, one subtracts sin( A ) times the second equation, replacing 0 cos(A) — w' sin(A) with 0, to obtain

Introducing a dimensionless axial coordinate r] — y/l, the above equation can be written as

0"' + qe™j cos2(A)0' + ^- sin(A) cos(A)0 = 0, (3.83)

where ()' now denotes d( )/drj. The boundary conditions can be derived from Eqs. (3.77) as

0(0) = 0'(1) = 0"(1) + cos2(A)0(l) = 0. (3.84)

0 0

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