Dynamic inflow modelling for autorotating rotors

Abstract

A linearised model system matrix B linearised model control matrix C L aerodynamic perturbation in roll moment C M aerodynamic perturbation in pitch moment C T aerodynamic perturbation in thrust p,q,r perturbed angular velocity components along airframe axes, rad/s r radial position on rotor disc, m R rotor radius, m U perturbation velocity vector, U = (u,v,w), dimensionless on ΩR t non-dimensional time u,v,w perturbed velocity components along airframe axes, dimensionless on ΩR u control vector v 0 component of induced velocity, mean, ms -1 v c1 component of induced velocity, longitudinal, ms -1 v s1 component of induced velocity, lateral, ms -1 v m wake mass flow velocity, ms -1 V general mass-flow parameter, dimensionless on ΩR V m mass-flow parameter, V m /ΩR V m+ mass-flow parameter for normal working state, dimensionless on ΩR V m-mass-flow parameter for windmill-brake state, dimensionless on ΩR V T non-dimensional total flow at rotor disc, dimensionless on ΩR ABSTRACTThe dynamic inflow model is a powerful tool for predicting the induced velocity distribution over a rotor disc. On account of its closed form and simplicity, the model is especially practical for studying flight mechanics or for designing control systems for helicopters. Scant attention has, however, been paid so far in utilising the dynamic inflow model to analyse an autorotating rotor, which is different from a powered rotor in the geometric relation between the direction of the inflow and the rotor disc. Autorotation is an abnormal condition for helicopters, but for gyroplanes it is the normal mode of operation. Therefore the theoretical discussion on an autorotating rotor is of importance not only to improve the understanding of present gyroplanes, but also in the development of new gyroplanes and to analyse the windmill-brake state of helicopters. Dynamic inflow modelling is reviewed from first principles, and this identifies a modification to the mass flow parameter. A qualitative assessment of this change indicates that it is likely to have a negligible impact on the trim state of rotorcraft in autorotation, but a significant effect on the dynamic inflow modes in certain flight conditions. This is confirmed by numerical simulation, although considerable differences only become apparent for steep descents with low forward speed. It is concluded that while modification of the mass flow parameter is perhaps mathematically accurate, for practical purposes it is required only in a limited area of the flight envelope of autorotating rotorcraft.

Correction of the definition of mass flow parameter in dynamic inflow modelling

“…In fact, Glauert [14] pointed out that the mass flow should be defined with modulus sign so as to be positive either in the normal working or windmill-brake state as early as in 1927. The modulus sign in equation (12) can be also explained in the same manner as in reference [14], and Glauert's argument underpins the discussion in this article. Unification of these two approaches is therefore considered a timely and significant result.…”

“…2. The relationship between the original definition of the mass flow parameter and its modified version for the windmill-brake state [12] is both mathematically and consistently enunciated and hence the necessity of the modification for the windmillbrake state is clearly confirmed than by the physical argument in reference [12]. …”

“…Reference [12] is arguably the first study in which a necessary change to equation (2) for the windmillbrake state has been discussed. Reference [12] elucidated the modification of equation (2) to…”

“…where the minus sign in the subscript means the windmill-brake state, following the denomination used in reference [12]. In reference [12], the necessity of the sign change is discussed from the purely physical point of view, considering the fact that while in the normal working state the inflow is given energy and accelerated, in the wind-mill brake state the inflow is taken energy and decelerated. Also, regarding the definition of wake skew angle, χ , reference [12] confirmed that χ should be defined in Peters' manner as…”

“…In response, questions were raised by readers about how the modified and original inflow parameters should consistently be explained, why the original model of Peters is insufficient for autorotation, and how the validity of the modified mass flow parameter can mathematically be enunciated. Indeed, the discussion in the article by Murakami and Houston (2008) was held in a manner that readers might have received an impression that the modified model was derived independently of the original model, and the connection to the original model was not clearly presented. This technical note is written with an aim to elucidate the mathematically consistent relationship between the original and modified mass flow parameters, and it is confirmed that the original definition of the mass flow parameter contains an error in its definition.…”