At Jtan

Since N — L/W, the above expression can be divided by the vehicle weight to yield a relation for N in terms of ur and aT. This relation can then be solved simultaneously with the preceding expression for ar, Eq. (3.47), in terms of uv and N. In this manner uv can be eliminated, providing either a relation that expresses ar in terms of N, given by

or a relation that expresses uv in terms of N,

NWle

2G/Aitan(Ai)

These relations permit one to specify a constant ar and find N{q ) or, alternatively, to specify a constant N and find ar(q). One finds that N{q ) starts out at zero for <7=0. On the other hand, ar(q) starts out at infinity for <7 = 0. The limiting values as q —»• qo depend on the other parameters. These equations can be used to find the torsional deformation and the resulting airload distribution for a specified flight condition.

The calculation of the spanwise aeroelastic airload distribution is immensely practical and is used in industry in two separate ways. One way is to satisfy a requirement of the aerodynamicist or performance engineer who needs to know the total forces and moments on the flight vehicle as a function of its altitude and flight condition. In this instance the dynamic pressure q (and altitude or Mach number) and ar are specified, and the load factor N or total lift L is computed using Eq. (3.69). A second requirement is that of the structural engineer, who must ensure the structural integrity of the lifting surface for a specified load factor N and flight condition. Such a specification is normally described by what is called a V-N diagram. For the conditions of given load factor and flight condition it is necessary for the structural engineer to know the airload distribution to conduct a subsequent loads and stress analysis. When q (and altitude or Mach number) and N are specified, ar is then determined from Eq. (3.70). Knowing q, ar, and N, one then uses Eq. (3.47) to find ar. The torsional deformation, 0, then follows from Eq. (3.66), and the spanwise airload distribution follows from Eq. (3.65). From this, the bending moment distribution along the wing can be found, leading directly to the maximum stress in the wing, generally somewhere in the root cross section.

It may be observed that the overall effect of torsional flexibility on the unswept lifting surface is to significantly change the spanwise airload distribution. This effect can be seen as the presence of the elastic part of the lift coefficient, which is of course proportional to 9(y). Since this elastic torsional rotation will generally increase as the distance from the root (i.e., out along the span), so also will the resultant airload distribution. The net effect will depend on whether ar or N is specified. If ar is specified as in the case of a wall-mounted elastic wind tunnel model (N — 1) or as in performance computations, then the total lift will increase with the additional load appearing in the outboard region as shown in Fig. 3.14.

Lift distribution for elastic wing

Lift distribution for elastic wing

Figure 3.14 Rigid and elastic wing lift distributions holding ar constant.

Lift distribution for rigid wing

Figure 3.14 Rigid and elastic wing lift distributions holding ar constant.

In the other case, when N is specified by the structural engineer, the total lift (area under L' versus v ) is unchanged, as shown in Fig. 3.15. The addition of lift in the outboard region must be balanced by a decrease inboard. This is accomplished by decreasing ar as the surface is made more flexible.

All the preceding equations for torsional divergence and airload distribution have been based on a strip-theory aerodynamic representation. It may be noted that a slight numerical improvement in their predictive capability can be obtained if the two-dimensional lift-curve slope, a, is replaced everywhere by the total (three-dimensional) lift-curve slope. Although there is little theoretical justification for this modification, it does alter the numerical results in the direction of the exact answer. Also, it is important to note that the lift distributions depicted in Figs. 3.14 and 3.15 cannot be generated with strip-theory aerodynamics, because strip theory fails to pick up the dropoff of the airload to zero at the wing tip caused by three-dimensional effects. A theory at least as sophisticated as lifting-line theory would have to be used to capture that effect.

3.2.4 Sweep Effects

To observe the effect of sweeping a wing aft or forward on the aeroelastic characteristics, it will be presumed that the swept geometry is obtained by rotating the surface about the root of the elastic axis as illustrated in Fig. 3.16. The aerodynamic reactions will

Lift distribution for elastic wing

Lift distribution for elastic wing

Figure 3.15 Rigid and elastic wing lift distributions holding total lift constant.

Figure 3.16 Schematic of swept wing (positive A).

depend on the angle of attack as measured in the streamwise direction as a=ar + 0, (3.71)

where 0 is the change in the streamwise angle of attack caused by elastic deformation. To develop a kinematical relation for 0, we introduce the unit vectors ai and tb, aligned with the v axis and the freestream, respectively. Another set of unit vectors, bi and bj, are obtained by rotating ai and ao by the sweep angle A as shown in Fig. 3.16, so that bi is aligned with the elastic axis (i.e., the y axis). From Fig. 3.16 one sees that bi = cos(A)ai + sin(A)ai,

It should be observed that the total rotation of the local wing cross-sectional frame caused by elastic deformation can be written as the combination of rotations caused by wing torsion, 0 about bi, and wing bending, dw/dy about b2, where w is the wing bending deflection (positive up). Now, 0 is the component of this total rotation about a i, that is,

From this relation, it can be noted that, as the result of sweep, the effective angle of attack is altered by bending. This coupling between bending and torsion will affect both the static aeroelastic response of the wing in flight as well as the conditions under which divergence occurs. Also, it can be observed that, for combined bending and torsion of a swept elastic wing, the section in the direction of the streamwise airflow exhibits a change in camber, a higher-order effect that is here neglected.

To facilitate direct comparison with the previous unswept results, to the extent possible the same structural and aerodynamic notation will be retained as was used for the unswept planform. To determine the total elastic deflection two equilibrium equations are required,

one for torsional moment equilibrium as in the unswept case and one for transverse force equilibrium (associated with bending). These equations can be written as

 d / de
+1 0