Nonlinear Dyn https://doi.org/10.1007/s11071-021-06271-z (0123456789().,-volV)(0123456789().,-volV) ORIGINAL PAPER Stability analysis of whirl flutter in rotor-nacelle systems with freeplay nonlinearity Christopher Mair . Branislav Titurus . Djamel Rezgui Received: 16 July 2020 / Accepted: 29 January 2021 Ó The Author(s) 2021 Abstract Tiltrotor aircraft are growing in preva- consequently the change in the stability boundary is lence due to the usefulness of their unique flight quantified. Furthermore, the effects on freeplay envelope. However, aeroelastic stability—particularly behaviour of (a) model complexity and (b) deadband whirl flutter stability—is a major design influence that edge sharpness are studied. Ultimately, the freeplay demands accurate prediction. Several nonlinearities nonlinearity is shown to have a complex effect on the that may be present in tiltrotor systems, such as dynamics of both systems, even creating the possibil- freeplay, are often neglected for simplicity, either in ity of whirl flutter in parameter ranges that linear the modelling or the stability analysis. However, the analysis methods predict to be stable. While the size of effects of such nonlinearities can be significant, this additional whirl flutter region is finite and sometimes even invalidating the stability predictions bounded for the basic model, it is unbounded for the from linear analysis methods. Freeplay is a nonlinear- higher complexity model. ity that may arise in tiltrotor nacelle rotation actuators due to the tension–compression loading cycles they Keywords Whirl flutter Á Continuation and undergo. This paper investigates the effect of a bifurcation methods Á Nonlinear aeroelasticity freeplay structural nonlinearity in the nacelle pitch degree of freedom. Two rotor-nacelle models of Abbreviations contrasting complexity are studied: one represents VTOL Vertical Take-Off and Landing classical whirl flutter (propellers) and the other V/STOL Vertical and/or Short Take-Off and captures the main effects of tiltrotor aeroelasticity Landing (proprotors). The manifestation of the freeplay in the CBM Continuation and Bifurcation Methods systems’ dynamical behaviour is mapped out using LCO Limit Cycle Oscillation Continuation and Bifurcation Methods, and DoF Degree of Freedom C. Mair (&) Á B. Titurus Á D. Rezgui Department of Aerospace Engineering, University of Bristol, Queens Building, Bristol, UK 1 Introduction e-mail: [email protected] B. Titurus Tiltrotor aircraft such as the XV-15 shown in Fig. 1 e-mail: [email protected] aim to combine the speed and range of turboprop D. Rezgui aircraft with the VTOL capabilities of helicopters. e-mail: [email protected] 123 C. Mair et al. early as 1963 [6]. However, with slender, highly twisted and flexible blades and heavy engine nacelles mounted upon wingtips to provide clearance of the long blades from the fuselage, tiltrotors are promi- nently vulnerable to whirl flutter. The dynamics of a tiltrotor system—with flapping blades and a flexible wing—are more complex than the classical model. For instance, the manner in which the precession-gener- ated aerodynamic loads act on the pylon/wing is significantly different [7] to the classical case. The phenomenon limits the performance of tiltrotor air- Fig. 1 XV-15 tiltrotor aircraft [1] craft; it either imposes a direct limit on the maximum safe cruise speed, or the increased stiffness (and This relatively large flight envelope makes them therefore thickness) of the wings necessary to guar- highly versatile and consequently they are attractive antee aeroelastic stability up to a certain design speed to both civil and military operators. It is in increasing results in reduced aerodynamic efficiency [8]. their maximum cruising speed that the aeroelastic Several lines of research have been well explored in instability known as whirl flutter is encountered. efforts to delay the onset airspeed of whirl flutter in Whirl flutter affects propellers or rotors mounted on tiltrotors. Active control has been applied to the flexible structures. It is caused by the interaction of the proprotor swashplate and to various aerodynamic wing’s elasticity, gyroscopic moments acting on the surfaces, while passive measures such as aeroelastic rotor as a whole and aerodynamic forces and moments tailoring or installing winglets constitute design acting on the rotor disc. Motion-dependent in-plane refinements that act against physical drivers of the forces are the most significant contributor to the whirl flutter instability. Furthermore, at the heart of instability [2]. The physical origin is typically the any whirl flutter research lies the matter of accurate coupling between the wing torsional motion and rotor prediction, and to this end, dedicated research has been in-plane forces [3]. Additionally, these in-plane forces conducted, broadly focusing on closing gaps between may destabilise the whole aircraft’s short period flight the predictions of various tools and their experiment modes [4]. From the designer’s perspective, an equivalents. aircraft’s whirl flutter stability is a function of its The vast majority of the available literature is various physical properties, such as the damping and limited in its treatment of nonlinearities present in real stiffness of various structural components or the tiltrotor aircraft. In many cases, available studies placement of wing modal frequencies relative to one restricted the modelling of the structural stiffness to another. However, from a pilot’s perspective, it is linear approximations, which is contingent on the encountered at or beyond a certain onset speed, when assumption of small deformations. Where nonlinear the damping ratio of one or more whirl flutter modes structural stiffnesses were used, linear stability anal- becomes negative. ysis methods were ultimately employed once lin- In its canonical form, the instability is usually earization about a nonlinear trim point had been referred to as ‘‘propeller whirl flutter’’ or ‘‘classical obtained. Park et al. investigated whirl flutter with a whirl flutter’’. This usage strictly refers to a model of nonlinear structural model [9], though the focus of the the form first derived in [5]: a basic system with rigid paper was an overall design optimization framework blades and a rigid shaft, all of which is able to pitch as opposed to impacts on the whirl flutter predictions and yaw elastically about an effective pivot point. made by using nonlinear elements in the model. Although turboprop aircraft such as the Electra were Additionally, whirl flutter stability analysis in Park’s the context for the first (classical) whirl flutter studies, work was conducted using time-domain methods: a the search for feasible high-speed V/STOL concepts simulation was conducted for each case under exam- had by that time recognised the promise of the tiltrotor ination to see if whirl flutter was encountered. configuration, and the aeroelastic properties of tiltro- Similarly, an investigation by Janetzke et al. [10] tor-like V/STOL configurations were considered as used nonlinear aerodynamic models adapted from 123 Stability analysis of whirl flutter in rotor-nacelle systems with freeplay nonlinearity aerofoil data, though the structural aspects of the Other aerospace contexts with similarly reduced model did not appear to have benefitted from the same instability boundaries include landing gear shimmy approach. [14], flutter of wing-mounted stores [18] and freeplay- However, nonlinearities of various types can have a induced oscillations in helicopter pitch-links [24]. significant impact on system behaviour, as has been Freeplay is well known to arise gradually in mechan- evident in a number of works. Masarati et al. [11] ical systems due to ordinary wear, though structural showed within a tiltrotor context that deformability damage may cause freeplay deadbands to appear nonlinearities within the blades can have effects that instantly. Additionally, freeplay oscillations them- manifest in the overall system stability. Krueger [12] selves may directly cause the deadband to grow [19]. showed that differences in the dynamic behaviour of a Exemplified by Breitbach [20], a significant body rotor between windmilling and thrust mode can arise exists of nonlinear aeroelastic research by a number of from nonlinearities introduced by the influence of the authors into basic systems such as pitch-plunge drivetrain such as freeplay and backlash. In general, models. Nonlinearity types are categorised and their limit cycle oscillations (LCOs) are necessarily non- respective detrimental impacts on stability character- linear phenomena and are commonly observed in istics are shown in isolation. However, tiltrotor practical aeroelastic contexts. Their study in compu- aeroelastic systems have not received such treatment. tational works is not possible without consideration of In order to fulfil their flight envelope, tiltrotor the nonlinearities that give rise to their existence, such aircraft employ nacelle tilting actuators. These actu- as Gandhi and Chopra’s work on an elastomeric lag ators are able to rotate each nacelle to any point damper that combats air and ground resonance in a between horizontal and vertical, and hold the nacelle bearingless helicopter main rotor [13], which models still. The two-stage telescopic ballscrew design that is the damper as a nonlinear spring in series with a typically employed [21] undergoes a range of com- Kelvin chain to permit the prediction of LCOs. The pressive and tensile loads within