Coefficient Method for Solving the Flutter Frequency Equation

The flutter frequency equation for a three-degree-of-freedom system is a cubic equation with complex coefficients. The usual methods of solving this equation involve the solution of a cubic equation for every assumed value of the "reduced velocity/ ' The coefficient method makes it possible to determine the stability of the motion from the relations of the coefficients of the frequency equation and eliminates the necessity of solving any cubic equation to determine the critical conditions. The method also makes it possible to obtain a "bracket" of the critical conditions, thereby reducing the number of trial values of the reduced velocity. The intersection of the curves indicating the critical conditions is, in general, more definitely established by the coefficient method than by other methods. The method may be easily extended to systems with more than three degrees of freedom. INTRODUCTION THERE HAVE BEEN many methods devised for solving the stability determinant or the frequency equation that arises in determining the flutter speed of a portion of the aircraft structure. The equation representing the critical conditions for a three-degreeof-freedom system—for example, wing-bending, wingtorsion, and aileron-rotation—is a cubic equation with complex coefficients. The coefficients of the equation are functions of a large number of structural and frequency parameters. The nature of the problem is such that the unknowns cannot be separated, thus making it impossible to solve the cubic frequency equation for the critical conditions directly. A trial-and-error solution is therefore indicated. The straightforward procedure of solution consists of assuming one of the unknown parameters, usually the "reduced velocity," and determining the corresponding cubic frequency equation. The equation is then divided into its real and imaginary parts and the roots of these two equations are determined in the usual way. The roots from the real and the imaginary equations are plotted and the intersection of their graphs indicate the values of the parameters for the critical conditions. For each assumed value of the reduced velocity there must be solved at least one cubic equation in order to determine the roots for plotting. The method proposed here would obviate the necessity for solving any cubic Received June 12, 1944. * The author has recently learned tha t Dr. S. Sherman, of Curtiss-Wright Corporation, has had considerable success in solving this problem by means of a generalization of the Routh Discriminant. t Professor of Aeronautical Engineering. equations. I t will be shown that an examination of the coefficients of the frequency equation is sufficient for determining the critical stability conditions. FREQUENCY EQUATION FOR Two DEGREES OF FREEDOM Although the coefficient method is more valuable when applied to a three-degree-of-freedom system, a binary system will first be discussed in order to indicate the nature of the problem. Consider a vibrating system involving only two flexibilities, such as wing-bending and wing-torsion. A mechanical model of such a system might be represented as shown in Fig. 1, in which the restraint to the bending and the twisting of the wing section is provided by springs. The other quantities to be considered