The present dynamic investigation of the stability of motion of airplanes is based on the well-known theory of small oscillations of rigid dynamics as first applied to airplanes by Bryan^ and extended by Bairstow.' The necessary coefficients for use in the equations of motion were determined by model tests in the wind tunnel of the Massachusetts Institute of Technology. After analyzing Clark's plane, the work was repeated for a model of a military plane known as Curtiss JN2.' The.
NO. 5 STABILITY OF AEROPLANES HUNSAKER AND OTHERS 7
3. MODEL
4. WING COEFFICIENTS
SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62
NO. 5 STABILITY OF AEROPLANES HUNSAKER AND OTHERS
10 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62 The lift and drift coefificients Ky and Kx were computed from the
NO. 5 STABILITY OF AEROPLANES HUNSAKER AND OTHERS I I
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5. LONGITUDINAL BALANCE
NO. 5 STABILITY OF AEROPLANES HUNSAKER AND OTHERS 13
For small values of L and angles of incidence between —2° and +2°, corresponding in practice to high flight speed, the curves are practically identical.
NO. STABILITY OF AEROPLANES — IIUXSAKER AND OTHERS
Then the 'pilot will have to carry his elevator up when flying at low speed. Case III, with stabilizer at -5°, appears to balance longitudinally at +2° of incidence, and at +12° of incidence to have (full size) a natural pitching moment reduced by a negative light on the elevator of only about 44 pounds can be held, which corresponds to approximately 4 ° elevator angle.
6. VECTOR REPRESENTATION
NO. 5 STABILITY OF AEROPLANES HUNSAKER AND OTHERS 1/
7. PERFORMANCE CURVES
8. AXES AND NOTATION
In the normal position, the axis of X is tangent to the trajectory of the center of mass. In normal calm flight, the apparent wind blows in the positive direction of the x-axis.
9. EQUILIBRIUM CONDITIONS AND DYNAMICAL EQUATIONS OF MOTION
Suppose this speed is produced by the aircraft's forward speed U during normal flight, f/ is a negative number of feet per second. In normal flight it is assumed that the available power maintains the aircraft at a speed sufficient to maintain weight and also that the normal attitude is appropriate for the speed.
22 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62 But the products of inertia (relative to moving axes fixed in the
In this investigation, the normal flight path is assumed to be horizontal or ^„=c>- The inertial product E is small for the normal. In view of the likely insignificance of E and the fact that E cannot be easily determined for an aircraft by simple experiments, it is neglected here.
24 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62 The right-hand members of these equations are no longer zero if
NO. 5 STABILITY OF AEROPLANES HUNSAKER AND OTHERS 25 In a similar manner the equations (ib) defining the asymmetric
10. CONVERSION TO MOVING AXES, LONGITUDINAL DATA
5 STABILITY OF AIRWAYS HUNSAKER ETC. 25 In a similar way are the equations (ib) that define the asymmetric. The axes are fixed by the equilibrium conditions and . different for each speed, as each speed requires a different attitude. Here L and Dare}) are Pounds on model, 6* is angle of pitch, and Z' and X' are Pounds force along the moving axes.
28 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL, 62 For this aeroplane we have, for example,
NO. 5 STABILITY OF AEROPLANES HUNSAKER AND OTHERS 29
30 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62 through the experimental points to show the consistency of the
11. RESISTANCE DERIVATIVES, LONGITUDINAL The longitudinal oscillations of the aeroplane are given by three
NO. 5 .STABILITY OF AEROPLANES MUNSAKER AND OTHERS 3I
12. DAMPING
32 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62 The equation of motion then is
NO. 5 STABILITY OF AEROPLANES HUNSAKER AND OTHERS 33 Likewise fjiiu was obtained by oscillating the apparatus without
13. OSCILLATIONS IN PITCH
34 SMITHSONIAN MISCELLANEOUS COLLECTIONS Apparatus and Model, Incidence of Wing, 6°
5 AIRCRAFT STABILITY HUNSAKER AND OTHERS 35 Apparatus and model, wing sweep, 12" Apparatus and model, wing sweep, 12".
36 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62
For most aircraft, this first factor corresponds to a short oscillation that is SO heavily damped that it doesn't matter. A period of 34.7 seconds for movement is large, and at high speed this aircraft, if left to its own devices after random longitudinal disturbances, should follow an undulating path with the center of gravity rising and falling, together with ])itching and occasional changes in forward speed. It is unlikely that this movement would be uncomfortable to the pilot, even if the initial disturbance from a jolt or other cause were serious.
At high speed, this airplane is very stable compared to other machines that have been tested. A Bleriot monoplane model tested by Bairstow had a takeoff period of 25 seconds, damped 50 percent to 15 seconds. The tested design has appeared in small periods as the Curtissand Bleriot, both considered very satisfactory in flight, along with greater damping.
38 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62 demanded at one time a neutral aeroplane with no stability whatever
NO. 5 STABILITY OF AEROPLANES HUNSAKER AND OTHERS 39
40 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62 of the pilot's losing control, yet it is clear that he cannot fly at this
5 STABILITY OF AIRCRAFT HUNSAKER AND OTHERS 4The second term is only ^ of its high-speed box. The third quarter is second quarter is only ^ of his high-speed box. we see by inspection that the main reduction in C\ at low speed. is due to smaller values U,Mw, Z^, and Mo, which greatly reduce the terms ZwMci and UMic- These two terms are the main numerical ones in the expression for C\. In general, £1= —gZuMiv will increase in value due to increase in Zu andMio, but the effect on the movement is not great.
It is seen that the quantities U, Zuu and Mq predominate in the numerical values of the coefficients D^C-^ and E^B-^. When sinking, M^, if large, will send the machine downward, speed will be gained with the dive and the resulting increase in lift will cause the plane to rise again. If too large, the machine is dynamically unstable in that the issue of safe flight at a standstill is complicated by the fact that the lateral controls become ineffective, but by manipulation of the power delivered by the motor combined with skillful use of the rudder, an expert can land an airplane at surprisingly low speeds.
15. CONCLUSIONS (LONGITUDINAL DYNAMICAL STABILITY)
44 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62 The following table facilitates comparison
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LATERAL MOTION
1. LATERAL OR ASYMMETRICAL TESTS
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This adjustment minus the observed M for each yaw angle is necessary to give the airplane longitudinal equilibrium when in normal attitude. If the airplane yawto rit;ht, it is practically slartini^' off on a turn to rii^ht. As is well known, to make such a turn an aircraft should be "banked" to such a roll angle that the centrifugal force acting to the left is approximately balanced by the horizontal component of the normal force acting to the right.
As in the case of the yawing moments, an excessive amount of natural banking can be uncomfortable, especially ingustyair.. if the wind shifts to the left, the relative yaw angle is positive, the aircraft tends to turn to the left due to its "directional" stability.. and to bank for a turn to the right due to the natural banking of rollingmomentL. The result might cause the plane to toss around in a somewhat violent manner, or it might overturn. Still air, if the apparent wind shifts io° to the left, the lateral force pushes the airplane to the right. This equates to a 55-pound unload on the right rudder and a 55-pound load on the left rudder.
52 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62 that the aileron control shall be very powerful compared with the
2. RESISTANCE DERIVATIVES
5 STABILITY OF AIRPLANE HUNSAKER AND OTHERS 53Fraction^^is the slope of the Nplottedonangleofyaw curve Fraction^^is the slope of the Nplottedonangleofyaw curve. The magnitude of this moment was in dispute in the recent Curtiss-Wright patent litigation.
54 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62 The derivative Lr is the rate of change of rolhng moment with an
5 STABILITY OF AIRPLANE IIUNSAKER AND OTHERS 55Hence the "rotating moment on an airplane is —2UKIr, and substituting- Hence the rolling"moment on an airplane is —2UKIr, and substituting the expression above,. When an airplane rotates at an angular velocity p radians per second (positive when the right wing is down), an element area of the right wing has its angle of incidence increased and a corresponding element of the left wing has its angle of incidence decreases by the same amount. Due to the greater angle of incidence, the resistance of the head of the element increases. Fig. 3, Part I) we can draw a tangent line at the point of the curve corresponding to the angle of incidence for normal flight.
For small changes in incidence from normal incidence, this tangent can be substituted for the actual curve without significant error.
56 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62 of the drift coefificient in pounds per square foot per foot-second is
57-3UKJ
NO. 5 STABILITY OF AEROPLANES HUNSAKER AND OTHERS 5/
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NO. 5 STABILITY OF AEROPLANES HUNSAKER AND OTHERS 59 INCIDENCE OF WINGS 12°
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62 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62
7. NEGLECTED COEFFICIENTS
8. INDEPENDENCE OF THE LONGITUDINAL AND LATERAL MOTION
5 STABILITY OF AIRWAYS HUNSAKER ETC. current state of our knowledge, the calculation of the stability of.
NO. 5 STABILITY OF AEROPLANES HUNSAKER AND OTHERS present state of our knowledge, the calculation of the stability of
64 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62
9. LATERAL STABILITY, DYNAMICAL
The condition that the motion will be stable is that A^, B^, Co, D., and each will be positive, as will Routh's discriminant. It turns out that for the particular aircraft in question, Routh's discriminant and the bi-squared coefficients are all positive at high and medium speeds. However, at low speed we see that £o becomes negative, indicating that the lateral motion is unstable.
In this case, Routh's discriminant remains positive, but it is small compared to its value at high speed. It is unfortunate that this lateral instability is related to the longitudinal instability found in Part I at low speed.
10. CHARACTER OF LATERAL MOTION
NO. 5 STABILITY OF AEROPLANES HUNSAKER AND OTHERS 67 This is a pair of imaginary roots indicating an oscillation of natural
68 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62 tion from normal attitude and to correct it by use of his controls
The drop of Lvo, observed experimentally, is therefore expected for an aircraft with raised wingtips. Until further experiments are made it is not profitable to speculate on this question, but one would see no a priori reason to expect that the coefficient Lr given by vertical fins depends in any way on the angle of incidence of the normal. flight attitude. This coefficient is a measure of the damping of the angular velocity during yaw, and can be made large by a vertical surface in front of and behind the center of gravity.
The derivative Lr is characteristic of the rolling moment due to the rate of yaw or spin, and it was found to be caused by the greater air velocity on the outer wing during turning. At the same time, due to spininyaw, the machine tends to oversteer due to the greater lift on the left wing. Considering the usual magnitudes of the derivatives entering Bo, Co, Do, E2, we can write very approximately.
2 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62 that of the longitudinal motion and unless strongly damped, the
Small yawing moment due to rolling A',^
5 STABILITY OF AIRCRAFT — IIUXSAKEK AND OTHERS 73 Similarly, the rolling moment, due to sideslip or restoring moment, as imparted by tall fins or raised wingtips, should be large to avoid spiral instability. Similarly, we want the natural bank due to directional rotation to be low for "spiral" stability, but now we want this coefficient to be large. The conflicting nature of the stability requirements is shown here using rather drastic simplifications in a more precise formula.
Indeed, in practice, airplanes with a large dihedral angle for the wings have been found so violent in their movement under certain circumstances that average pilots have a strong prejudice against using such a wing arrangement. this bias has some physical basis is shown here. The dihedral angle machine is not likely to encounter a "spiral dip" but. from the analogy of a well-known figure skating. "If left to itself because the nooscillation of the fast period is involved. Consideration of the magnitude of the derivatives leads us to the conclusion that in any plane, if A''^ becomes very small, then.
14. COMPARISON WITH OTHER AEROPLANES
5 AIRCRAFT STABILITY — HUNSAKER AND OTHERS 75 a motion called "Dutch roll" is likely to be unstable at low speeds, a motion called "Dutch roll" is likely to be unstable at low speeds where Np becomes large. At low speed, the longitudinal movement becomes unstable, as well as one or the other type of lateral movement. We don't have data for the Bleriot at low speed, but it looks like it will become a Clark model.
The reduction in Nv (or weather rudder) at high angles of attack cannot be attributed to straight wings. An increase in Lv and a decrease in A'^i for a Curtiss airplane, favorable for spiral stability, is not favorable for "Dutch roll" stability. )litu(l(*doubles in 7.66 seconds.
NO. 5 STARILITY OF AEROPLANES HUNSAKER AND OTHERS 77 The motion is a swaying of the aeroplane of increasing amphtnde and
78 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62 Curtiss has been given greater wing area in order to reduce the
