1. It is important to mention that the flap-lag stability problem, without elastic coupling, exhibits exaggerated sensitivity to small effects which influence the damping in the lag degree of freedom. Thus for example dynamic inflowcan affectflap-lag stability, when elastic coupling is set equal to zero, however with elastic couplingpresent the effect of dynamic inflow, on this instability in hover, is small213·215. Additional results on this instability for an elastic hingeless blade in hover, using an analysis based on a number of elastic modes are available in Ref. 166. An interesting result182based ona finiteelement model fora hingeless blade, shown in Fig. 8, is presented in Fig. 10. The blade has only flap and lag degrees of freedom and each degree of freedom is represented by a varying number of elastic global modes (between 1-3, per degree of freedom). The unstable region of the stability boundaries is denoted by the letter U on the boundary. The figure illustrates the combined effect of the number of modes used in theanalysis and the elastic coupling parameter Re.When R.=1.0elastic coupling ispresent, when Re= 0, elastic coupling is neglected, and values of 0.0 < Re< 1.0 represent varying amounts of elastic coupling. FromFig. 10,for Re= 0.0, itisalways thefundamental lag mode which yields the lowest stability boundary, and for Re= 1.0theflap-laginstabilityisvirtuallyeliminated. The interesting results shown in the figure are the unstable regions associated with the second lag mode, which for intermediate values of the elastic coupling parameter R.= 0.60, becomes unstable at lower values of the critical collectivepitch angle than the firstlag mode. The implication of this result is that tl:e second elastic lag mode should be retained in the stability analysis of hingeless and bearinglessrotor blades.
2. Consider next the coupled flap-lag-torsional aeroelastic problem in hover. Numerous results on the coupled flap-lag-torsional behavior of hingeless rotor blades in hover can be found in Refs. 166and 180. Various physical aspects of flap-lag-torsional coupling are also described in Refs. 222-224. The research carried out has shown that soft-in-plane hingeless rotor blades in hover are usually stable. A typical flap-lag-torsional stability boundary taken from Ref. 180 is shown in Fig. 11. The main item of interest in this figure is the bubble-like region of instability present at low values of collective pitch 0. This instability occurs only in the presence of precone and is a flap-lag type of instability. Sometimes it is called the precone induced flap-lag instability. It was also obtained in Refs. 166,224 and 225.The unstable region decreases as the torsional stiffness ro,1is increased from 4.5 to 6.0. Very small amounts of torsional damping (Tlsu = 0.0025, 0.25%of critical in lag) reduce this unstable region for the torsionally soft.!_>ladeand completely eliminate it for the stiffer blade (ro,1= 6.0). Other results not shown here222·225indicate thatdroopandsweepcanhaveastrong beneficial as well as negative effect on the hingeless blade stability. In addition offsets between cross sectional elastic axis, aerodynamic center and center of mass can also influence blade stability. A similar study clarifying the effects of modeling assumptions on the coupled flap-lagtorsional stability of a stiff-in-plane hingeless blade, including comparisons with experimental data, was conducted on Ref. 298.
3. Among the data sets available for hover, the results obtained by Sharpe230have proven themselves useful for correlation studies on hingeless rotors. Full scale hover tests on a four bladed B0-105 soft-inz:Rlanehingeless rotor were also conducted at NASA Ames1•Reasonably good agreement with predictions using the CAMRAD code were obtained. 4.2 Aeroelastic Behavior in Forward Flight
4. Figure 14, from Ref. 107, illustrates a number of important effects. The label CFLT on the curves denotes the results form coupled flap-lag-torsional analysis. The label flap-lag denotes results from a flap-lag analysis. The full lines are results from a converged nonlinear analysis, the dashed lines are results for the case when geometrical nonlinearities are neglected. The results shown, depicting the real part of the characteristic exponent for lag, as a function ofµ, are for propulsive trim. From the figureit is evident that blade stability increases with forward flight for soft-in-plane configurations. The importance of the geometrically nonlinear terms is also evient from the figure. Comparing the stability margin (as represented by LAG) from a flap-lag analysis with that obtained from a flap-lag-torsional analysis it is clear that damping from a flap-lag analysis is 250-300% lower than that obtained from the accurate flap-lag-torsional analysis. Therefore flap-lag analyses in forward flight can be misleading and should be avoied in trend type studies.
5. Subuent research on hingeless rotor stability in forward flight34•235as well as more recent research, Refs. 49, 77, 167, 168, 170, 209-211, 215, 216 have confirmed the conclusions presented in Ref. 107. In Refs. 234 and 167 the effect of dynamic inflow was also included and was found to be relatively small. All the studies mentioned indicated that stiff-in-plane confugurations are destabilized by forward flight, while the stability of soft-inplane configurations increases with forward flight This is clearly evident from a recent study by Dull and Chopra299where the analytical predictions are inexcellent agreement with experimental results. A considerable amount of experimental data showing this behavior is also presented inRef. 10.