Comment on "A Direct Matrix Method for the Divergence Problem"

Abstract

T is curiously refreshing to be reminded from time to time of the struggle that accompanies the advancing state-of-theart of the aeronautical sciences. Lest we take our present position too much for granted, Ref. 1 has just appeared from out of the past to recall the period of the 1950's during which the field of static aeroelasticity made the transition from the slide-rule to the high-speed digital computer. The following historical recollections are offered as a more complete review of that dynamic period of progress. Equation (4) of Ref. (1) was almost implicit in Eq. (8-55) of Bisplinghoff, Ashley, and Halfman.2 The definition of static and oscillatory aerodynamic influence coefficients (AIC's) in Ref. 3 [Eq. (6) of Ref. 1 appears as Eq. (8) in Ref. 3] provided for a general formulation of the various aeroelastic problems which was outlined in the introduction to Ref. 4. A general purpose computer program was developed in Ref. 5 for the static aeroelastic analysis of the problems of rigid and flexible load distributions, divergence, estimation of rigid and flexible static and dynamic stability derivatives, and the correction of wind-tunnel data measured on flexible models. Both structural influence coefficients and AIC's were used and Eq. (4) of Ref. 1 is found as Eq. (23) in Ref. 5. The example problem used to demonstrate the computer program of Ref. 5 was also the jet transport wing of Bisplinghoff, Ashley, and Halfman.2 However, the AIC's were based on the subsonic lifting surface theory (at a Mach number of zero) of Runyan and Woolston.6'7 The first two divergence dynamic pressures were calculated to illustrate the divergence option, and were#i = 3786 and#2 = 26,951 psf. The fundamental divergence dynamic pressure corresponds to the velocity Vt = 1784 fps at sea level, with which the value in Ref. 1 using lifting line theory agrees well. Later calculations of the divergence characteristics of the jet transport wing were performed using incompressible strip theory in connection with a method for transient flutter analysis.8 The five divergence dynamic pressures were found to be q, = 2397, q2 = 9510, q3 = 23,555, q* = 38,208, and q$ = 69,914 psf. The fundamental divergence velocity at sea level is Ki = 1420 fps, as was mentioned in Ref. 8. If the aspect ratio correction used in Refs. 1 and 2 is made, the divergence velocity at sea level becomes Vi = 1950 fps and agrees with the result in Ref. 2, p. 440. The concern in Ref. 1 over the null eigenvalues from bending is unwarranted. The five nonzero eigenvalues of the example wing (using strip theory) were found easily by the power method because of their wide separation. The sixth through tenth eigenvalues were not zero (infinite divergence speeds) because of round-off errors in the eigen-matrix deflation required in the power method; however, the sixth eigenvalue was four orders of magnitude lower than the fifth and thereby gave some clue to what should have been expected.