Static Aeroelasticity Solution For Divergence Dynamic Pressure Of Wing

ae qcae

Note that X2 and uv are independent of y since the wing is assumed to be uniform. The static aeroelastic equilibrium equation can now be written as d20

The general solution to this linear ordinary differential equation is

0 — A sin(Ày) + B cos(Xy) - (ar + âr). (3.49) Applying the boundary conditions, one finds that 0(0) — 0: B — ar + dr.

0'(i) — 0: A = B tan(Àf), where ( )' — d( )/dy. Thus, the elastic twist distribution becomes e = (ar + âr) [tan(Xl) sin(Ày) + cos(Ày) - 1]. (3.51)

Since 0 is now known, the spanwise lift distribution can be found using the relation

It is important to note from the above expression for elastic twist that 6 becomes infinite as XI approaches tz/2. This phenomenon is called "torsional divergence" and depends on the numerical value of ww-

Thus, it is apparent that there exists a value of the dynamic pressure q — qD, a t which X£ equals tt/2, where the elastic twist theoretically becomes infinite. In practice, this static aeroelastic instability causes catastrophic failure of the wing structure. The value qD is called the "divergence dynamic pressure" and is given by

eca \2I

Noting now that one can write with

the twist angle of the wing at the tip can be written as

where Eq. (3.47) can now be written as

_ ccmac U2Nmgd ae GJTt2q

Letting d be zero, so that ur becomes independent of q, one can examine the behavior of 9(1) versus q. Such a function is plotted in Fig. 3.13, where one sees that the tip twist angle goes to infinity as q approaches unity. Note that the character of the plot in Fig. 3.13 is quite similar to the prebuckling behavior of columns that have imperfections. It is of practical interest to note that the tip twist angle becomes sufficiently large to warrant concern about the structural integrity for dynamic pressures well below qD.

Because this instability occurs at a dynamic pressure that is independent of the right-hand side of Eq. (3.48), as long as the right-hand side is nonzero, it seems possible that the divergence condition could be obtained from the homogeneous equilibrium equation d20

The general solution to this eigenvalue problem of the Sturm-Liouville type is

Applying the boundary conditions, one obtains 0(0) = 0: fi = 0, 6\t) = 0: AAcos(Af) = 0. Figure 3.13 Plot of tip twist angle for wing versus q for ar + cir = 1,:.

If A — 0 in the last condition, there is no deflection; this is a so-called trivial solution. However, if X — 0 the given general solution is not valid. Thus, the desired result is obtained when cos(Af) = 0. This is the "characteristic equation" with solutions given by

These values are called "eigenvalues." Note that this set of As corresponds to a set of dynamic pressures t / 7T \2 GJ

The lowest of these values, q i, is equal to the divergence dynamic pressure, q/>, previously obtained from the nonhomogeneous equilibrium equation. This result implies that there exist nontrivial solutions of the homogeneous equation for the elastic twist. In other words, even for cases in which the right-hand side ofEq. (3.48) is zero (when ar + ur — 0), there is a nontrivial solution

for each of these discrete values of dynamic pressure. Since A„ is undetermined, the amplitude of 91, is arbitrary, which means that the effective torsional stiffness is zero whenever the dynamic pressure q — qn. The mode shape 9\ is the divergence mode shape, which must not be confused with the twist distribution obtained from the nonhomogeneous equation.

It may be noted that if the elastic axis is upstream of the aerodynamic center then e < 0 and X is imaginary in the preceding analysis. The characteristic equation for the divergence condition becomes cosh(|A|£) = 0. Because there is no real value of X that satisfies this equation, the divergence phenomenon will not occur in this case.

It has been observed that the spanwise lift distribution can be determined as

where we recall from Eq. (3.51) that

9 = (ar + ar) [tan(/.!:) sin(/.v) + cos(Ay) - 1] (3.66)

and where ur is given in Eq. (3.47). If the lifting surface is a wind tunnel model of a wing and is fastened to the wind tunnel wall, then the load factor, N, is equal to unity and ar can be specified. The resulting computation of L' is straightforward.

If, however, the lifting surface represents half the wing surface of a flying vehicle, the computation of L' is not as direct. It may be noted that the constant ur is a function of N. Thus, for given value of orr there will be a corresponding distribution of elastic twist and a particular airload distribution. This airload can be integrated over the vehicle to obtain the total lift, L. Recall that N — L/W, where W is the vehicle weight. It is thus apparent that the load factor, N, is related to the rigid angle of attack, aT, through the elastic twist angle, 9. For this reason either of the two variables ar and N can be specified; the other can then be obtained from the total lift L. Assuming a two-winged vehicle with all the lift being generated from the wings, one finds

Substituting for L' and ur as given above yields f

L — 2qca / [ar + (ar + ar) [tan(Af) sin(Xy) + cos(Ay) — 1]} dy

0 0