A description of a vector method approach to the analysis of dynamic stability problems is presented. In addition to supplying information on the damping of the dutch-roll oscillation, the phase relation between roll to yaw, and the amplitude ratio of roll to yaw, the method makes possible an excellent physical visualization of the contribution of the various stability derivatives and mass characteristics to the overall motion of the airplane. The application of the vector method to several problems is briefly discussed.
DISCUSSION
I N THE SPRING OF 1951, two British scientists visitedthe Langley Laboratory of the NACA, and in the course of reporting on several phases of research related to dynamic stability of aircraft, they mentioned that Dr. K. H. Doetsch, a German scientist now working in Britain, had developed a vector method analysis of dynamic stability problems. Unfortunately, the British representatives were not familiar enough with the method to describe it in detail. Our curiosity about a so-called "vector method" stimulated thought on the problem, which resulted in the development of a method and the design of an analog computer, to be described in this paper. Recently, W. O. Breuhaus, of Cornell Aeronautical Laboratory, visited with Dr. Doetsch and published a paper 1 describing his method. It appears that, in principle, the vector method developed at Langley is similar to Dr. Doetsch's method.The fundamental principle involved in the method is well known from the solution of linear differential equations with constant coefficients and will be illustrated by considering first a simple spring-mass system and then introducing a dashpot into the system. In Fig. 1 we have a sketch of the spring-mass system and the differential equation describing the system, mx + kx = 0, where the two dots signify the second derivative of x with respect to time. One complimentary solution of the differential equation is x = C\ sin co/, where co is the natural frequency of the system -and d is an arbitrary constant. Thus, each term of the differential equation could be plotted as a vector, with the vector moment or force due to displacement along the positive x axis of magnitude k and the vector moment or force Presented at the Aerodynamics Session, Annual Summer Meeting, IAS, Los Angeles, July 15-17, 1953.* Head, Stability Analysis Section.due to acceleration 180° out of phase with the displacement of magnitude mco 2 . As indicated in the equation, the sum of the two terms is zero, or the resultant of the two vectors is zero. As shown on the plot, the velocity is 90° out of phase with the displacement. Fig. 2 shows the case of a dashpot added to the system. The differential equation becomes mx + dx + kx = 0. One complimentary solution is x = C\e at sin co/, a damped sinusoidal oscillation, and the expressions for the velocity and acceleration are given on Fig. 2. It should be noted in the analytical expression for the velocity that a phase angle is introduced which is a function of the damping and frequency of the system...
