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Early Numerical Work
Useful solutions to Bryan’s equations of airplane motion for scientific or engineering uses are either roots or eigenvalues or actual time histories, which give airplane responses to specific control or disturbance inputs. Either type of solution was essentially out of the question with the means available in 1911. However, by 1920 Bairstow had found useful approximations that served as starting points for developing eigenvalues from the Bryan equations.
When, later on, research engineers in both the United States and in Britain generated time history solutions to the linearized Bryan equations, it was only with great labor. Early step-by-step numerical solutions were published for the S. E.-5 airplane of World War I fame by F. Workman in 1924. A year later, B. Melvill Jones and A. Trevelyan (1925) published step-by-step solutions for the lateral or asymmetrical motions.
As an advance over step-by-step methods, B. Melvill Jones (1934) applied the formal mathematical theory of differential equations to the linearized Bryan equations, producing a marvelously complete set of time histories for the B. F.2b Bristol Fighter at an altitude of 6,000 feet (Figure 18.5). A generation of pre-electronic-computer engineers struggled through those formal solutions. The complementary function is found first. In addition to using a considerable amount of algebra, one has to find the real and complex roots of a fourth-degree polynomial. The complementary function gives the time histories of the variables of motion under no applied forces and moments, but with arbitrary initial conditions.
The last step in the formal solution is finding a particular integral of the equations. This adds to the complementary function the effects of constant applied moments, such as are produced by deflections of the airplane’s control surfaces. In Jones’ own words, “The numerical computations involved… are heavy, they involve amongst other things, the solution of four simultaneous equations with four variables.” It is little wonder that numerical time history calculations languished for years, until electronic analog computers were commercially available, about the year 1950.
