Stability and Routh’s Discriminant

As the minutes of their first meeting show, G. H. Bryan had been in touch with the Advisory Committee and, though not a member, was considered central to their effort to understand stability.88 Bryan (see fig. 1.6) was a versa­tile applied mathematician who wrote on thermodynamics and fluid dynam­ics but had become interested in aeronautics through contact with Sir Hiram

figure i.6. George Hartley Bryan (1864-1928), a British pioneer in the analysis of aircraft stability. Bryan applied the mathematical techniques that had been developed by his Cambridge coach Edward Routh. (By permission of the Royal Society of London)

Maxim and the pioneer glider flyer Percy Pilcher.89 Bryan had been publishing calculations on stability since 1904 and was, without doubt, the leading Brit­ish authority in the field.90 It was failure to understand stability, he argued, that led to so many fatal accidents.91 Lanchester had written on stability, but in Bryan’s eyes, and judged by Tripos standards, this work “certainly appears wanting in rigour.”92 Lanchester’s approach was original, conceded Bryan, and he avoided the errors that had vitiated many other attempts, but he did not deduce his conclusions from clearly stated assumptions. Describing how he had arrived at his own, highly mathematical, analysis Bryan recalled that “about the year 1903 I noticed that if a glider or other body is moving in a resisting medium, such as air, in a vertical plane with respect to which it is symmetrical, the small oscillations about steady motion in that plane are determined by a biquadratic equation; and Prof. Love directed my attention to the condition of stability given by Routh.”93 A quadratic equation has the form ax 2 + bx + c = 0, whereas a biquadratic of the kind referred to by Bryan has a term in x 4 and takes the form ax4 + bx3 + cx2 + dx + e = 0. When Bryan said that he “noticed” that the oscillations of a glider were determined by a biquadratic equation, he did not mean that he drew this conclusion simply by looking at a model glider in flight. He meant that he noticed this math­ematical fact in the course of using Newton’s laws to write down the general equations of motion of a body, such as an airplane, moving with a specified velocity and subject to specified forces such as gravity, lift, and drag.

Bryan posed the following question: If an airplane was in steady flight and hence in dynamic equilibrium, and was then subject to small, disturbing forces, such as a gust of wind or a sudden alteration of the control surfaces, what would happen? Would the disturbance die away or would it get bigger and bigger? If the disturbance died away, the machine would count as stable; if the result was that the disturbance became amplified and disruptive, then the machine would count as unstable. He treated the airplane as a rigid body subject to forces of acceleration and rotation. Given the force of gravity and the aerodynamic forces to which it was subject, how did this mechanical sys­tem respond? What sort of longitudinal or lateral oscillations would follow from the disturbance? At this point Prof. Love stepped in. A. E. H. Love, a student at St. John’s, was second wrangler in 1885 and first Smith’s prizeman in 1887. He later became professor of mathematics at Oxford.94 Love appar­ently reminded Bryan that the techniques and concepts he needed to answer his question about stability had already been worked out by Routh, who had been Bryan’s old Cambridge coach. E. J. Routh’s Adams Prize essay of 1877 and his textbook, Dynamics of a System of Rigid Bodies, contained a general analysis of stability for mechanical systems. Both of these works had shown the importance of a mathematical device that came to be known as Routh’s discriminant, an expression whose negative or positive value indicated the stability or instability of the system under analysis.95

Following Routh’s methods, and citing Routh’s results, Bryan was able to reduce the problem of the stability of an aircraft subject to small disturbances to the behavior of an equation of the general form

AX4 + BX3 + CX2 + DX + E = o,

where X was the symbol for the modulus of decay or the strength of the damp­ing tendency on the oscillations that were being investigated. (This equation in X was the “biquadratic” that Bryan “noticed.”) The coefficients A, B, C, etc., in Bryan’s equation were complicated mathematical expressions involv­ing terms that were called “resistance derivatives” and “rotary derivatives.” These described the rate of change of the various forces, and their leverage on the aircraft, relative to its varying conditions of speed and orientation. The values of the derivatives, and hence the values of the coefficients A, B, etc., depend on the details of the particular machine. They could not be calculated from first principles but could be given numerical values on the basis of ap­propriate measurements made on models in a wind channel.

An examination of the four roots, that is, the values of X that satisfy the equation, would determine whether the machine were stable. As Bryan put it, “the small oscillations. . . are determined by an equation of the fourth degree, so the conditions for stable steady motion are those obtained by Routh.”96 Routh had discovered the general result that the stability of an oscillating system required that the coefficients A, B, C, D, and E should all be positive and that the quantity BCD – AD2 – EB2 should also be positive. This latter expression was called Routh’s discriminant. Abstract though it was, it cast light on design features that unwittingly rendered many aircraft dangerous to fly and prone to accidents. The proper mathematical understanding of an aircraft in terms of this equation, argued Bryan, could diminish the risks. In his 1904 paper he had recommended that mathematical investigations should be carried out on any “aerial machines that may be designed or constructed” (115) before they take to the air. Like Haldane, he had no reservations about asserting the priority of theory over practice.97 Bryan’s studies culminated in 1911 in a treatise titled Stability in Aviation

Rayleigh used to say that when he hit a hard mathematical problem he would pick up pen and paper, call to mind his old coach, and “write it out for Routh.”99 This may also have been Bryan’s procedure. That he too was writing it out for Routh is suggested by the way he echoed the title of Routh’s book when he projected a second volume to follow from his own 1911 book.

Bryan intended to call the combined, two-volume work The Rigid Dynam­ics of Aeroplane Motions. The aim was to carry the analysis into much more difficult problems, such as that of circling and helical flight, which would generate an equation with terms involving X to the power eight.100 Whatever the underlying psychological processes, however, there can be no doubt that the skills honed in the Tripos classes and coaching rooms of Cambridge were about to be given a new application, and one whose potential importance would be inestimable.

If he were given the right empirical data about an aircraft, Bryan was in a position to make predictions about its stability. Now the question became: Were those predictions correct? It was not evident, a priori, that even Bryan’s sophisticated mathematics would capture the complex reality of the behav­ior of a real aircraft. At a discussion at the Aeronautical Society, Greenhill, with considerable experience in ballistics to back up his words, expressed his concern that gyroscopic effects such as those from the engine and propeller had been neglected. “I must confess it alarms me,” he said in response to an exposition of the theory by E. H. Harper, a co-worker of Bryan’s, “that w, p, q, have no influence on u, v, r, especially with gyroscopic influence,” where the first three letters referred to rotations around the axes of the aircraft and the latter three to velocities of translation along those axes. Greenhill could not resist a further dig at Bryan by adding that of course the pioneers of flight “could not wait for the solution of a differential equation or its determinantel quartic.” Greenhill’s reservations could only have emboldened the “practical men” in the audience, who also suspected that all manner of simplifications must have been introduced into the calculations. Bryan’s colleague and rep­resentative was questioned closely by Handley Page and others. What about the tangential forces on the wings? Would this approach be of help designing a new machine rather than comparing two given machines?101

Such suspicions were shared by the reviewers of Stability in Aviation in the scientific press. The review in Nature was signed W. H.W.102 The writer was clearly impressed by the book but drew attention to the problematic relation between mathematics and reality, and to Bryan’s uncompromising attitude. The reviewer quoted the following passage, observing dryly, “it strikes the keynote of the book itself.” In this book, said Bryan,

attention is concentrated on the mathematical aspect of the problem for sev­eral reasons. In the first place, there is no obvious alternative between de­veloping the mathematical theory fairly thoroughly and leaving it altogether alone; any attempt at a via media would probably lead to erroneous conclu­sions. In the second place, the formulae arrived at, even in the simplest cases, are such that it is difficult to see how they could be established without a mathematical theory. In the third place, there is probably no lack of com­petent workers in the practical and experimental side of aviation, and under these conditions it is evident that the balance between theory and practice can be improved by throwing as much weight as possible on the mathematical side of the scale.103

Bryan’s position, first stated in his 1904 paper, was that even if the analysis was wrong, provided it was not too wrong, it would provide a “basis of com­parison” and the means for interpreting experimental results “in their true light” (100). As for the problem created for his theory by gyroscopic effects, of the kind that worried Greenhill, Bryan took the view that the fault was with reality not with his theory: “surely it may be left to practical men to get rid of these objectionable influences by proper balancing.”104 This attitude was precisely what worried W. H.W.

W. H.W. was probably Sir William H. White, FRS, an expert in naval ar­chitecture. If so, then the reviewer and Bryan had crossed swords before. At a heated meeting of the British Association in 1910, White had taken Bryan to task for insisting that mathematicians and practical men should stick to their own, separate spheres of activity. The report of this confrontation, as given in the Aero, is worth quoting:

The advocacy of watertight compartments, so to speak, drew from Sir William White a strong protest against drawing any such sharp demarcation, for he conceived the existence of an engineer who was a mathematician and a math­ematician who was an engineer. Sir William White was also somewhat severe on a suggestion made by Dr. Bryan that had the mathematical problems been sufficiently studied many, if not all, of the unfortunate fatal accidents to flying men would have been avoided, and that the practical man’s refusal to work on these lines rendered the accidents the results of foolhardiness rather than bravery.105

Another reviewer of Bryan’s book, this time in the Mathematical Gazette, went into the presuppositions behind the analysis of stability in some detail and remarked:

the author is obliged to make a series of assumptions—that the air resistance on the planes are linear functions of the small changes in linear and angu­lar velocities; that in steady motion they are proportional to the square of the velocity; that they are normal to the planes; that they are proportional to sin a; that the angle of attack a is small; that the pressure on an element of a narrow plane is independent of the motion of neighbouring elements, etc. Methods of approximation are also at times employed to simplify the alge­bra. The cumulative effect of small inaccuracies in each assumption may be considerable.106

Only experiment would reveal if the approximations were cumulative and failed to cancel out. If this were so, then the predictions would fail, however elegant the mathematics and however pure its Tripos pedigree.

A young scientist called Leonard Bairstow—a product of London Uni­versity rather than Cambridge—led the wind-channel work at the NPL that gave empirical content to Bryan’s equations. Working with Nayler and Ben­nett Melvill Jones, a Cambridge engineering graduate, Bairstow provided the data needed to attach values to the coefficients in the equations and hence to check on the viability of the assumptions behind the calculations.107 The mea­surements were delicate, involving the timing of oscillations on models of complete aircraft supported on a spindle, and damped by a spring, when they were exposed, respectively, to still and moving streams of air. As might be ex­pected with difficult experiments, there were problems behind the scenes that were not always apparent in the published reports. As the aircraft designer J. D. North pointed out, “torsional oscillations in the spindle connecting the model with the indicating or recording apparatus” was a disturbing fac­tor and gave rise to “varying results with different moments of inertia of the apparatus.”108 Despite these complications, Bairstow’s experiments seemed to show the models in the wind channel behaving in the manner predicted from Bryan’s equations. There was a gratifying coordination between experiment and theory.

Because both the experiments and the theory concerned small distur­bances, the results necessarily had their limitations, and the scope of the agreement between fact and theory was still open to discussion. Bairstow vigorously defended the work on stability by insisting that some, at least, of the limitations were “more apparent than real.” Consider, for example, “the necessity for assuming infinitesimally small disturbances from the path of flight.” A similar assumption had to be made, said Bairstow, invoking one of the classic achievements of mechanics, when setting up the differential equation for the motion of a simple pendulum. But the solution can then be “applied to oscillations of finite magnitude, without sacrificing any great proportion of accuracy.”109 The appeal to infinitesimal motions does not viti­ate the empirical significance of the inquiry. Rhetorically this was a powerful comparison, and the move from infinitesimal to small, finite disturbances can be justified by the analogy. Cautious persons, however, would note that this argument still left the move from small finite disturbances to large finite disturbances unaccounted for. The inference from the stability of an aircraft under small disturbing forces to its stability when confronted with larger forces therefore remained problematic. Bairstow’s colleague Melvill Jones, who worked on control during slow flying and stalling, and who was strongly supportive of the stability research program, nevertheless acknowledged that Bryan’s equations became inapplicable under these circumstances.110 Some experts also remained troubled by the points raised by the reviewers—that the forces and couples were assumed to depend on linear and angular veloci­ties but not on accelerations.111

The most visible symbol of the British preoccupation with the problem of stability was the excellent BE2, the machine subject to so much hatred in the aviation press. Even here it could not be asserted that de Havilland’s original machine had been stable because it had been designed according to Bryan’s equations. It had not. The aircraft had been the result of good judgment and had then been further improved and, in the form of the BE2C, rendered in­herently stable by subsequent trial and error. This result had been achieved not just by mathematics and the wind tunnel, but also through the danger­ous flying experiments of Busk and his colleagues. Furthermore, whatever may be true of the relation between Bryan’s equations and small-scale mod­els, J. D. North continued to argue that the relation between these equations and full-size aircraft remained problematic. Speaking specifically of the lon­gitudinal damping of the BE2, North said that it was “the only rotary de­rivative deduced from quantitative results” and insisted that even then it had “not shown good agreement with the estimated figures.”112 Added to reser­vations of this kind was a more general issue. Stability was only one of the competing virtues that might be desirable in a design. Maneuverability was another, and often incompatible, demand. Contrary to the critics, Bairstow always maintained that, scientifically and technically, the BE2C was “one of the most interesting aeroplanes ever built.”113 Despite the confidence of men such as Bairstow, however, the politics of stability would not be resolved in the laboratory.

The undoubted achievement of a greater understanding of stability was sufficient to impress at least one practical man, Archibald Reith Low, of Vick­ers (see fig. 1.7). Low was himself a pilot and had designed the Vickers “Gun Bus” of 1913, a machine that earned the accolade of being the first purely military aircraft.114 Low had been to an evening lecture given by Bairstow to the Aeronautical Society on January 21, 1914. Bairstow reported on the NPL’s stability work and illustrated the findings with model gliders.115 In the discus­sion after the demonstration Low, who had previously expressed reservations about Bryan, affirmed his enthusiastic conversion. The NPL work, he said, “constituted a triumphant vindication of Professor Bryan and the Advisory Committee.” He promised (generously if not perhaps entirely seriously) to spend the next couple of years digesting the theory of small oscillations and learning about Routh’s discriminant. Low recalled that, despite the advances

figure 1.7. Grahame-White type 10 aero-charabanc, 1912. A. R. Low is seated third from the left; J. D. North is fourth from the right. (By permission of the Royal Aeronautical Society Library)

he had made, Bryan had been laughed out of a British Association meeting by “so called ‘practical’ engineers.” Low also expressed the hope that the “igno­rant agitation” in the press would be stopped by the dawning realization on the part of those responsible that “there were problems in aviation that they had not begun to be able to understand.”116

Certainly the scientists directly responsible for mastering the problem of stability were in no doubt about the value of their achievements—even if there was more work yet to be done. It was clear to them that mathemati­cians could now contribute to the design of inherently stable aircraft (and they were beginning to convince at least some practical men). In 1915 the Aeronautical Society awarded Bryan their Gold Medal and, if the subsequent history of aeronautics is to be the judge, the honor was well deserved: Bryan’s equations are still used.117 Understandably, Bryan was deeply grateful to those who had rendered his theory applicable. In a letter of February 21, 1916, he said it was an “extraordinary feat” that Bairstow and E. T. Busk and their colleagues had got inherent stability “into a sufficiently practical form to be incorporated into military aeroplanes.” But, he went on, in the present war­time conditions it was necessary for everyone to keep working in both pure and applied research. In the prewar days, “Reissner and Bader were running us pretty hard on the mathematical side,” so no one could be complacent. He ended with a warning: “the Germans are probably putting their best brains into improving their aeroplanes.”118 Bryan was right, and his sentiments did not fall on deaf ears. As Greenhill had said in 1914 in the pages of Nature, this was a “Mathematical War.119 Despite the scoffing directed at mathematicians, the exponents of scientific aerodynamics were proud of their contribution to the understanding of stability and the progress that had been made. “It can­not be regarded otherwise,” said W. L. Cowley and H. Levy, two of the lead­ing experts at the National Physical Laboratory, “than in the light of a signal triumph for mathematical science.”120