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G. H. Bryan and the Equations of Motion
The mathematical theory of the motion of an airplane in flight, considered as a rigid body with 6 degrees of freedom, was put into essentially its present form by Professor George Hartley Bryan (frontispiece) in England in 1911. In an earlier (1903) collaboration with W. E. Williams, Bryan had developed the longitudinal equations of airplane motion only. Bryan’s important contribution rested on fundamental theories of Sir Isaac Newton (1642-1727) and Leonhard Euler (1707-1783). Today’s stability and control engineers are generally
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Figure 1.7 The perturbation form of Bryan’s equations of airplane motion. The longitudinal equations are above, the lateral equations below. Note the absence of control derivatives. (From Bryan, Stability in Aviation, 1911) |
astonished when they first see these equations (Bryan, 1911). As his book’s (Bryan, 1911) title indicated, he focused on airplane stability, not control. Aside from minor notational differences, Bryan’s equations are identical to those used in analysis and simulation for the most advanced of today’s aircraft (Figures 1.6 and 1.7).
Not surprisingly, at this early date he does not cover in detail control force and moments, nor does he treat the airplane as an object of control. The perturbation equations in Fig. 1.7 include stability but not control derivatives. The influence of external disturbances such as gusts is also not addressed, although he recognizes this and other problems by presenting a summary of questions not covered in his book that set an agenda for years of research.
Bryan calculated stability derivatives based on the assumption that the force on an airfoil is perpendicular to the airfoil chord. W. Hewitt Phillips points out that while this theory is not the most accurate for subsonic aircraft, it is quite accurate for supersonic aircraft, particularly those with nearly unswept wings, such as the Lockheed F-104. Thus, Bryan might be considered even more ahead of his time than is usually acknowledged.
Bryan obtained solutions for his equations and arrived at correct modes of airplane longitudinal and lateral motion. At the end of Stability in Aviation, Bryan reviews earlier stability and control theories by Captain Ferber, Professor Marcel Brillouin, and MM. Soreau and Lecornu of France; Dr. Hans Reissner of Germany; and Lieutenant Luigi Crocco of Italy.
Little progress was made at first in the application of Bryan’s equations because of the difficultiesof performing the calculationsand the uncertaintiesin estimating the airloadscor – responding to airplane motions. The airloads associated with rolling, pitching, and yawing motions, the so-called rotary loads, were a particular problem. Early efforts were made at the National Physical Laboratory in England to measure these rotary airloads in a wind tunnel.
The evolution of Bryan’s equations of airplane motion into an indispensable tool for stability and control researchers and designers is traced in Chapter 18 of this book.
