Tvr

^Mac

Figure 3.12 Cross section of spanwise uniform lifting surface.

Now, a static equilibrium equation for the elastic torsional rotation, 0, about the elastic axis can be obtained from the fundamental torsional relation d0

dy where G J is the effective torsional stiffness and T is the twisting moment about the elastic axis. One can obtain an equilibrium equation by equating the rate of change of twisting moment to the negative of the applied torque distribution so that dT d ( d0\

Recognizing that uniformity implies GJ is constant over the length, substituting Eqs. (3.36) into Eq. (3.34) to obtain the applied torque, and finally substituting the applied torque and Eq. (3.38) for the internal torque into the equilibrium equation, Eq. (3.39), one obtains d20 2

The sectional lift coefficient can be related to the angle of attack by an appropriate aerodynamic theory as some function ce(a), where the functional relation generally involves integration over the planform. To simplify the calculation, the wing can be broken up into spanwise segments of infinitesimal length, where the local lift can be estimated from two-dimensional theory. This theory, commonly known as strip theory, frequently makes use of table look-up for efficient calculation. Here we will use an even simpler, linear form in which the lift-curve slope is assumed to be a constant along the span. Thus, dc(

da where the constant sectional lift-curve slope is denoted by a.

The angle of attack will be represented by two components. The first is a rigid contribution, «r, from a "rigid" rotation of the surface (plus any built-in twist, although none is assumed to exist here). The second component is the elastic torsional rotation 0. Hence, a(y) = av + 0(y). (3.42)

Associated with each angle of attack contribution is a component of sectional lift coefficient given by strip theory as

This aerodynamic representation can be substituted into the equilibrium equation to yield its final form as d20 qcae 1

—T + ~-\$ —--(<7c~c»iac + qcaear — Nmgd). (3.44)

Finally, a complete description of this equilibrium condition requires specification of the boundary conditions. Since the surface is built-in at the root and free at the tip, these conditions can be written as y — 0: 0 — 0 (zero deflection), d0 (3.45)

Obviously, these boundary conditions are only valid for the clamped-free condition. The boundary conditions for other end conditions for beams in torsion are given in Chapter 2, Section 2.2.2.

3.2.2 Torsional Divergence

If it is presumed that the configuration parameters of the above uniform wing are known, then it should be possible to solve Eq. (3.44) to determine the resulting twist distribution and associated airload. To simplify the notation let

0 0