OFthe many dangers encountered in aeroplaning one of the most clearly defined, as also one of the most seductive, results from fancy flying-from wheeling round sharp horizontal curves, from conic spiraling, from cascading, swooping and undulating in vertical plane curves. The danger is particularly alluring to reckless young aviators engaged in public exhibitions, partly because they do not adequately appreciate the risk, partly because of the intrinsic charm and delight of such varied motion, and largely no doubt because of the liberal applause received from the admiring throng and the reciprocal desire to please that throng, too captivated by the fantastic grace of curvilinear flight, and by the bravery and abandon of the impetuous airman, to realize the hazardous nature of his 'fascinating maneuvers. Singularly enough the exact magnitude of such hazard, or say more accurately, the increased stress in the machine, though beyond the approximate guess of the practical aviator, is capable of nice computation in terms of the speed and curvature of flight. It seems important therefore to place the computations in ready form before aU persons concerned in such maneuvers-the reckless pilot, perhaps with a wife or family dependent upon him, the applauding spectators who encourage his daring, and the ea utious manager or company. that, for reasons manifold and obvious, wishes to avoid disaster both to the man and to the machine. During his recent exhibition in Washington, Mr. Glenn H. Curtiss found difficulty in restraining one of his intrepid young aviators from executing various hair-raising maneuvers. The boy loved to frolic in the air like a swallow or bird of prey, to the joy of a light-hearted throng, except possibly his wife and friends. He would . plunge from a great elevation to acquire the utmost speed, then suddenly rebound and shoot far aloft. He would undulate about the feld like a galloping horse, or an impetuous torrent vaulting over rocks or a rugged bed. He would bank his machine till its wings stood apparently vertical. Mr. Curtiss solemnly warned the aviator and earnestly restrained him, pointing out the danger of sweeping sharp curves at high speed. He then turned to me and expressed the wish that Bore one would determine exactly the amount of the added stress in curvilinear flight. The following paragraphs embody my response to his wish. With the death of st. Croix Johnstone fresh in the public mind, let us hope that they may be taken to heart by pilots who strive to please the multitude. Whe n a body_ pursues a curvilinear path in space the centripetal force urging it at any instant may be expressed by the equation: Fn = m V/R (absolute units) m = - V'/R (gravitational units) g in which Fn is the centripetal foree, ' the mass of the body, V its velocity and R the instantaneous radius of c1ryature 'of the.path pursued' by its center of mass: Since the mass may he regarded as constant for any short period, the equation may be verbally expressed by this simple law: the centripetal force varies directly as the square of the velocity of flight and inveysely as the instantaneous radius of curvature of the path. In applying the above equation to compute the stress in an aeroplane of given mass m., we may assume a series o f values for V and R, compute the corresponding values for Fn , and tabulate the results for reference. In such manner the following table was obtained. It may be remarked that, on substituting in the equation, V has been taken as so many miles an hour, R as feet, ind g as 22 miles an hour, for convenience, this being 32.1 feet per second. This table shows at a glance the centripetal force Centripetal Force on an Aeroplane for Various 8pcn and Curvatures Of J?light. Hadius of Curvature, R. Velocity, V", of the Aeroplane. ,,1«,. *.«* R” *-,«*, k,«». Miles per Hour. Wei!ht. Weight. Weight. Weight. Weight. 30 0.41 0.20 0.14 0.10 0.08 40 0.73 0,36 0.24 0.18 0.15 50 1.14 0.57 0.38 0.28 0.23 60 1.64 0.82 0,55 0.41 0.33 70 2,23 1.11 0.74 0.56 0.45 80 2.91 1.45 \ 0.97 0.73 0.58 90 3 68 1.81 1.23 0.92 0.74 100 4.55 2.27 - 1.52 1.14 0.!1 acting on an aeroplane, as a fractional part of the gravitational force, or weight, of the machine and load. If, for example, the flier is rounding a curve of 300 feet radius, at 60 miles per hour, the centripetal force is 0.55 of the total weight. At the excessively high speed of 100 miles an hour and the extremely short radius of 100 feet the centripetal force would be 4.55 times the weight of the moving mass. The pilot would then feel heavier on his seat than he would sitting still with a man of his own weight on either shoulder. Again for speeds below 60 miles an hour, and radii of curvature above 500 'feet, the centripetal force is below one-third '0'£ the. weight. The entire straining force on the aeroplane in horizontal flight being substantially the resultant of the total weight and the centripetal force, can be found by compounding them. Thus in horizontal wheeling, the resultant force, as seen from Fig. 4 (page 196) is approximately: F = VFn 2 + W* In swooping, Or undulating in a vertical plane, the resultant force at the bottom of the curve has its maximum value: F = F", + W and at any other part of the vertical path it has a more complex, though smaller, value which need not engage us here. It is obvious from the foregoing discussion that the greatest stress in the machine occurs at the bottom of a swoop, if the machine be made to rebound on a sharp curve. The total force En + W sustained at this point may be found from the table, if V and R be known, by adding unity to the figures given, then multiplying by the weight of the machine. For example, if the speed should be 90 miles an 'hour and the radius of curvature 200 feet, the total force on the sustaining surfaces would be 2.84 times the total weight of the machine. In this case the stress in all parts of the framing would be 2.84 times its value in level flight, when only the weight has to be sustained. The pilot would feel nearly three times his usual weight. We may conclude from the foregoing investigation that in ordinary banking at moderate speeds 'on mod erat e curves, the additional stress due to centripetal force is usually well below that due to the weight of the machine, and th a t in violent flying the added stress may eonsiderably exceed that due to the weight of the maehine, and may be dangerous, unless the aeroplane be constructed with a specially high factor of safety. But there is nothing in the results here obtained that seems to make sharp curving and swooping prohibitive; for if the framing of the machine be given an extra factor of safety, at the expense may be of endurance and speed, it may be made practically unbreakable by such maneuvers, and still afford to the operator and spectators a li k e all the pleasures of (Continued on page 196.) Stress in Aeroplanes in Curvilinear and Fancy Flight (Continued from paae 189.) frolic and fantastic flying. In order to obtain actual data for the fluctuations of stress in an aeroplane in varied flight, I would suggest that a record be taken of the stress or strain of some tension or compression member of the machine in action; or simpler still, perhaps, that a record of the aeroplane's acceleration be taken, and more partiClIlarly its transverse acceleration. A very simple device to reveal the transverse acceleration of the aeroplane in flight would be a massive index elastically supported. A lath or flat bar stretching lengthwise of the machine, one end fixed, the other free to vibrate, and carrying a pencil along a vertical chronograph drum would serve such purpose. The whole could be incased from the wind as shown in the sketch. (Fig. 2.) An adjustable sliding weight could be set to increase or diminish the amplitude of the tracing, and an aerial or liquid damper could be added to smooth the tracing. The zero line would be midway between the tracings made on the drum by the stationary instrument when resting alternately in its normal position and upside down; the distance from this zero line to the actual tracing of the stationary instrument would be proportional to the aeroplane stresses in level rectilinear fight; while in level flight on a curve,Fig. i.- Resultant force due to horizontal wheehng. either horizontal or vertic: the deviation of the mean tracing from the zero line would indicate the actual stress in the machine during, such accelerated fight. Of course t h e drum could be omitted and a simple scale put in its place, so that the pilot could observe the mean excursion of the pencil or pointer from instant to instant. Also, the damping of such excursion could be adjusted to any amount in the proposed instrument, if the vibrating lath fitted its incasing box closely, with an adjustable passage for the air as it moved to and fro, or if light damping wings were added to the lath, or flat pencil bar. Another method would be to obtain by instantaneous photography the position of the centroid of the aeroplane at a number of successive instants, from which could be determined its speed and path, or V and R of the first equation, by which data therefore the stress could be read from the first table. Perhaps the simplest plan would be to add an acceleration penholder, with its spring and damper, to any recording drum the aeroplane may carry for recording the air pressure, temperature, speed, etc. Indeed, all such records could be taken on a single drum. A score of devices,more or less siming stress in an aerop lane, will occur to lactic acid, whose formula is I any engineer who may give the subject CH. his attention, and it is desirable that I some one obtain roughly accurate data HO - C - H for the stresses developed in actual I flight. This is desirable in the interest COOH both of aeroplane design and of.prudent manipulation. It is commonly supposed by aviators that the increment of speed due to diving is very prodigious. An easy form ula will determine t he major limit of such flPeed increment. If the i nitial and n atural speed of the aeroplane be v, and the change of level in diving be h, while the speed at the end of the dive is V, the minimum change of level necessary to acquire any increment of s peed V-v, may be found from the equation h = (V-v) /2g And if, as before, g be taken as 22 miles per hour, the equation reduces to the conven ient formula h = (V-v) /30 in which V and v are taken in miles pel' hou r . AssUIl1ing v:lrious values for V and v the following table has been found for th e corresponding values of h in feet. Mi nimum Change 01 Level Necessary to Produce Various Speed Increments. For example, if the natural speed o f the aeroplane in level flight be 50 miles hou ::, and th e aviator wishes to increase the speed by 20 miles per hour, he must dive at I east 80 feet, assuming t hat the aeroplane fa lls freely, like a body in vacu o, or that its propeller o vercomes the air resistance completely ; o therwise the fall must be rather more than 80 feet. A very instructive and spectacular contest could be arranged to determine which aviator could dive most swiftly and rebound most sudden ly, the prize going to the one who should stress his machine most, as indicated by th e ace celerograph above proposed ; but to avoid danger the contest would would have to be supervi sed by competent experimentalists and would best be conducted over water. It is safe to say that more than one weil known aeroplane would be denied entry in such a contest, for lack of a sufficient factor of safety.
This article was originally published with the title "Stress in Aeroplanes in Curvilinear and Fancy Flight" in Scientific American 105, 9, 189 (August 1911)