## Aeroelasticity Efficiency Change As Q Approaches

Figure 3.26 Sweep angle for which divergence dynamic pressure is infinite for a wing with e/( = 0.02; solid line is for GJ/EI = 1.0; dashed line is for GJ/EI = 0.25.

### 3.3 Epilogue

In this chapter we have considered divergence and aileron reversal of simple wind tunnel models, torsional divergence and load redistribution in flexible beam representations of lifting surfaces, the effects of sweep on coupled bending-torsion divergence, and the role of aeroelastic tailoring. In all these cases, the inertial loads are inconsequential and have thus been neglected. In Chapter 4, inertial loads are introduced into the aeroelastic analysis of flight vehicles, and the flutter problem is explored.

### Problems

1. Consider a rigid, wind tunnel model of a uniform wing, which is pivoted in pitch about the mid-chord and elastically restrained in pitch by a linear spring with spring constant of 225 lb/in mounted at the trailing edge. The model has a symmetric airfoil, a span of 3 ft, and a chord of 6 in. The total lift-curve slope is 6 per rad. The aerodynamic center is located at the quarter-chord, and the mass centroid is at the mid-chord.

(a) Calculate the divergence dynamic pressure at sea level.

(b) Calculate the divergence airspeed at sea level.

2. For the model of Problem 1, for a dynamic pressure of 30 lb/ft2, compute the percentage change in lift caused by the aeroelastic effect.

Answer. 25%

3. For the model of Problem 1, propose design changes in the support system that would double the divergence dynamic pressure by

(a) changing the stiffness of the restraining spring

(b) relocating the pivot point

4. For the model of Problem 1 as altered bythe design changes of Problem 3, calculate the percentage change in lift caused by the aeroelastic effect for a dynamic pressure of 30 lb/ft2, a weight of 3 lb, ar = 0.5°, and for

(a) the design change of Problem 3a (b ) the design change of Problem 3b

5. Consider a strut-mounted wing similar to the one discussed in Section 3.1.3, except that the two springs may have different stiffnesses. Denoting the leading-edge spring constant by k\ and the trailing-edge spring constant by k2, show that divergence can be eliminated if k\/ k2 > 3.

6. Using Excel or a similar tool, plot a family of curves that depicts the relationship of the aileron elastic efficiency, )], versus normalized dynamic pressure, q — q/qD, for various values of R — qR IqD and 0 < q < 1. You should make two plots on the following scales to reduce confusion:

Hint: Do not compute values for the cases where 1 < R < 1.1; Excel does not handle these well and you may get confused. For some cases you may want to plot symbols only and nicely sketch the lines that form the curves.

Answer the following questions: Where does aileron reversal occur? If you had to design a wing, what R would you try to match (or approach) and why? What happens when qR — q^l How does the efficiency change as q approaches qR? Why do you think this happens? What other pertinent features can you extract from these plots? Explain how you came to these conclusions.

7. Consider a torsionally elastic (GJ — 8000 lb in2) wind tunnel model of a uniform wing, the ends of which are rigidly fastened to the wind tunnel walls. The model has a symmetric airfoil, a span of 3 ft, and a chord of 6 in. The sectional lift-curve slope is 6 per rad. The aerodynamic center is located at the quarter-chord, and both the mass centroid and the elastic axis are at the mid-chord.

(a) Calculate the divergence dynamic pressure at sea level.

(b) Calculate the divergence airspeed at sea level.

Answers: (a) qD = 162.46 lb/ft2; (b) UD = 369.65 ft/s

8. For the model of Problem 7, propose design changes in the model that would double the divergence dynamic pressure by

(a ) changing the torsional stiffness of the wing (b) relocating the elastic axis

Answers: (a) GJ — 16,000 lb in2; (b) xea — 2.25 in

9. For the model of Problem 7, for a dynamic pressure of 30 lb/ft2, compute the percentage increase in the sectional lift at mid-span caused by the aeroelastic effect.

Answer: 28.094%

10. For the model of Problem 7, for a dynamic pressure of 30 lb/ft2, compute the percentage increase in the total lift caused by the aeroelastic effect.

Answer: 18.59%

11. Consider a swept clamped-free wing, as described in Section 3.2.4. The governing partial differential equations are given in Eqs. (3.76) and the boundary conditions in Eqs. (3.77). An approximate solution is sought for a wing with a symmetric airfoil, using a truncated set of assumed modes and the generalized equations of equilibrium - a specialized version of the generalized equations of motion for which all time-dependent terms are zero. Note that what is being asked for here is equivalent to the application of the method of Ritz to the principle of virtual work; see Section 2.4. With the wing weight ignored, only structural and aerodynamic terms are involved. The structural terms of the generalized equations of equilibrium are based on the potential energy (here the strain energy ) given by rt

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