Tools › Aerodynamics › Stability and Control › Flutter and Divergence › Example

Bending-torsion flutter of a typical section with Theodorsen's function: the worked case of NACA Report 685

Tool	Flutter and Divergence
Source	Theodorsen, T., and Garrick, I. E. (1940). Mechanism of Flutter: A Theoretical and Experimental Investigation of the Flutter Problem. NACA Report 685, National Advisory Committee for Aeronautics, Washington, DC. Pages 8-9 (Figure 1) and 17; recomputed in Perry, B., III (2020), NASA/TP-2020-220562, Table 3.
Level	Graduate

Problem

A wing section of 12 ft chord has its elastic axis at 30 percent chord (a = -0.4), its centre of gravity at 40 percent chord (x_alpha = 0.2), r_alpha^2 = 0.25 and a mass ratio of 4 (kappa = 1/4). The torsion frequency is 90 rad/s and the bending frequency a quarter of it. Find the flutter speed, the reduced frequency and the divergence speed in incompressible flow.

Inputs to Enter

Section: NACA Report 685, Figure 1 case; semichord b = 6 ft, mass ratio 4, a = -0.4, x_alpha = 0.2, r_alpha = 0.5
Torsion frequency 14.324 Hz (90 rad/s), bending frequency 3.581 Hz, no structural damping, sea level
Aileron chord ratio 0.25, dive speed 400 ft/s, factor 1.15, altitude table to 20,000 ft

Open the Flutter and Divergence tool (loaded with this example)

Answers

Reference speed b omega_alpha = 540 ft/s; Theodorsen's C(k_f) = 0.6229 - 0.1640 i
U-g method: V_f / (b omega_alpha) = 1.5448, 1/k_f = 2.460, flutter at 834.2 ft/s (report 1.542, 2.46 and 833 ft/s; recomputed by NASA 834.4 ft/s, k = 0.407), frequency 8.996 Hz
p-k method: the same flutter point to 1e-13; quasi-steady (C = 1) 555.7 ft/s and steady frequency merging 565.3 ft/s, both about two thirds of the true value
Divergence V_D / (b omega_alpha) = sqrt(mu r_alpha^2 / (1 + 2a)) = 2.2361, 1,207 ft/s; approximate flutter formula of the report 697.1 ft/s (outside its stated range of kappa below 1/10)
At constant mass the flutter equivalent airspeed falls from 834.2 ft/s at sea level to 721.6 ft/s at 20,000 ft; ratio to 1.15 V_D = 460 ft/s is 1.81 at sea level

Notes

The tool's default inputs are this case; press Submit. The flutter speed is a Mach number of 0.75 at sea level, where the incompressible theory of the report is only an estimate; the page says so. test_flutter_divergence.php also reproduces the standard case of NACA Report 496 (173.26 ft/s, k = 0.4355, NASA/TP-2015-218765 Table CI) and 20 seeded sections against an independent numpy solution.