Rotor Dynamics and Helicopter Stability: Development of a Unified Mathematical Theory of the Steady and Disturbed Motion of a Helicopter with Particular Emphasis on the Dynamical Aspects of the Problem

Abstract

The component of velocity, μ0ΩR, along the x‐axis is specified, and also X, the angle of inclination of the flight path. The concept of the ‘internal’ and ‘external’ parameters of the motion is introduced, the ‘internal’ parameters consisting of certain combinations of the control and flapping angles θ0, θ1, θ2, β0, β1, β2 and the ‘external’ parameters consisting of μ0, Λ0, Θ, X, ∈ and z*, where Θ is the inclination of the x‐axis to the horizontal, and e the inclination to the vertical of the rotor resultant force‐vector. For a given centre of gravity position of the helicopter the ‘external’ parameters are shown to be determinable independently of the ‘internal’ parameters, subject to certain assumptions regarding the fuselage drag coefficient. The fundamental ‘internal’ parameter then emerges as (β1—Θ), which physically is the fore or aft inclination to the vertical of the rotor cone axis. The value of this parameter is found as the root of a quadratic equation, and not a linear equation as hitherto. It is shown that at high values of μ0 the rotor force‐vector docs not coincide with the rotor cone axis (as it is commonly supposed to do), the difference between (β1—Θ) and ∈ amounting possibly to 5 deg., of which no more than 1½ deg. can be accounted for by the rotor mean drag force component. Particular attention is paid to the case of horizontal flight, in which the true speed is μ0ΩR see Θ, the factor sec Θ being important. It is shown that at a given true speed the following quantities are independent of the centre of gravity position of the helicopter: (i) ∈, (ii) the quantity (λ1—Λ0+λ0β1), i.e. the component of the air velocity along the rotor cone axis, (iii) the horse‐power of the engine. In addition the following ‘internal’ parameters, (iv) (β1—Θ), (v) (θ2—β1), (vi) (θ1—β2), are very nearly independent of C.G. position, their variations with C.G. position only just becoming noticeable at large values of μ0. This feature enables a much simplified solution to the trim problem to be obtained by working in terms of a fictitious C.G. position chosen to make Θ zero. The genuinely C.G.‐sensitive parameters θ1, θ2, β1, β2 and Θ are subsequently converted to their true values corresponding to the actual C.G. position. The effect of variation in blade moment of inertia is examined through variations in the associated parameter γ, and it is shown that both the ‘external’ and ‘internal’ conditions are substantially unaltered, save for (θ1+ β2) and the mean coning angle β0, which do vary considerably with γ. The analysis is accurate at least so far as μ05 and the results are shown to be consistent with an energy equation. The only sources of error in the dynamics of the problem lie in the rejection of higher powers of μ0 and of flapping harmonics of β beyond the first. In order to assess the effects of this rejection, an analysis is made of a rigid rotor system free only to feather without flapping, and an exact dynamical solution for horizontal flight is obtained, subject to the same aerodynamical assumptions as before. At a given value of μ0 see Θ the results differ slightly from the case of the flapping rotor because now the purely feathering rotor applies a pitching couple mechanically to the fuselage via the shaft, which the flapping rotor is not able to do. However, by calculating the value of this couple, and supposing that the helicopter with the flapping rotor experiences either an aerodynamic fuselage pitching moment of the same amount or alternatively an appropriate C.G. shift, the two cases are reduced to essentially equivalent physical systems, particularly if the mean coning angle, β0, is deliberately arranged to be zero by proper choice of γ. A recalculation of the flapping case is then made; if its theory were exact, its solution would coincide with the known exact solution of the purely feathering rotor case. The discrepancies are shown to be very small, and thus the validity of rejecting μ06 is established for the earlier flapping rotor analysis. Part III is concerned with the theoretical solution of the feathering‐cumflapping rotor case, and also with a geometrical account of the trim configuration. Part IV contains the solution for the purely feathering rotor together with a discussion of the large number of numerical results and related tables, obtained on a Pegasus Computer.

Citation

Price, H.L. (1963), "Rotor Dynamics and Helicopter Stability: Development of a Unified Mathematical Theory of the Steady and Disturbed Motion of a Helicopter with Particular Emphasis on the Dynamical Aspects of the Problem", Aircraft Engineering and Aerospace Technology, Vol. 35 No. 11, pp. 332-338. https://doi.org/10.1108/eb033802