Effect of the Simultaneous Variation in Blade Root Chord Length and Blade Taper on Maneuvering Manned Helicopter Control Effort

In this article, the influence of simultaneous variation in blade root chord length and blade taper on the flight control effort of maneuvering helicopter is researched. For this intention, helicopter models which are complex, control-oriented, physics-based models and capturing the main-physics and essential-dynamics are benefited. The influence of simultaneous variation in the blade root chord length and blade taper (i.e., in both chordwise and lengthwise directions dependently) on the control effort of a maneuvering manned helicopter and also on the closed-loop responses are worked. Comparisons in terms of the control effort and peak-values with and without variations in the blade root chord and blade taper changes are followed. For helicopter control variance-constrained controllers (i.e. output variance-constrained controllers) are useful. Shortly, in also maneuvering flight conditions effects and benefits of simultaneous variation in blade root chord length and blade taper on the flight control effort is tried to be proved.


Introduction
Methods for minimizing the control effort of a flight control system (FCS) of a helicopter were frequently and commonly examined in the literature (Bluman & Gandhi, 2011;D Fusato & Celi, 2006;Dario Fusato, Guglieri, & Celi, 2001;Luo, Liu, Yang, & Chang, 2003;Tugrul Oktay & Sal, 2015;T Oktay & Şal, 2015;Tugrul Oktay & Sultan, 2013a, 2013b, 2013c, 2014. Limited number of studies studies on the helicopter blade root chord length and blade taper have been done recently for different reasons (Kambampati, Ganguli, & Mani, 2013;Tugrul Oktay & Sal, 2017;Özdemir & Kaya, 2006;Ozgumus & Kaya, 2007;Vu & Lee, 2015). For instance, in (Özdemir & Kaya, 2006), flapwise bending vibration analysis of a rotating, tapered cantilever Bernoulli-Euler beam (e.g., helicopter blade) was followed via benefiting the differential transform method. The effect of blade taper on natural frequencies was also worked. It was obtained that the natural frequencies of a rotating, tapered cantilever Bernoulli-Euler beam can be determined with high correctness via benefitting the differential transform method. In addition to the previous study, the effect of blade taper on the control effort of the flight control system of a helicopter was also studied in (Tugrul Oktay & Sal, 2017). In that article, blade root chord, blade taper and the gains of the output variance-constrained controller (OVC)-based helicopter FCS were the optimization parameters during straight level flight.It was obtained in this article that when blade taper is required to applied due to the performance reasons, blade root chord has to be used in order not to increase control effort.
Numerous control approaches for the helicopter FCS have been found recently (Dalamagkidis, Valavanis, & Piegl, 2011;Hsieh, Skelton, & Damra, 1989;Li, Liu, & Song, 2011;Skelton, 1987;Skelton, Iwasaki, & Grigoriadis, 1998). The use of variance-constrained controllers is one these approaches (Skelton, 1987). These kinds of controllers have many advantages with respect to the other existing common controllers. One of these advantages is that variance-constrained controllers are enhanced LQG controllers and they include Kalman filters for state estimators. In this article, a specific variance-constrained controller, the OVC, is used for helicopter FCS during maneuvering flight.
In this article, the simultaneous effect of the helicopter blade taper and blade root chord length on the control effort of the FCS of a helicopter during maneuvering flight is for the first time examined with OVCs. Furthermore, comparisons in terms of the peak values of closed-loop responses with and without variations in the blade root chord and blade taper changes are followed for the first time in the literature.

Our Helicopter models
In this article, the helicopter models summarized in (Tugrul Oktay & Sal, 2015;Tugrul Oktay & Sal, 2017;T Oktay & Şal, 2015) are benefited. These models are concisely summarized next. They captures the physics principles leading to dynamic models consisting of a finite set of ordinary differential equations. These models includes the fuselage, empennage, landing gear, fully articulated main rotor (i.e., with four blades), main-rotor downwash, and tail rotor. As a result, these models are accurately complex with a total of 29 equations:

Variance Constrained Control
For a clear continuous linear time invariant (LTI), stabilizable and detectable plant (Dalamagkidis et al., 2011;Hsieh et al., 1989;Li et al., 2011;Skelton, 1987;Skelton et al., 1998)  Here, X and K are solutions of solutions of two algebraic Riccati equations given next:

Results
In this article, the effect of combined change in blade root chord length and blade taper on the control effort of the FCS of a helicopter is examined. For this intention, complex, control-oriented, physics-based helicopter models (see (Tugrul Oktay & Sal, 2015;Tugrul Oktay & Sal, 2017;T Oktay & Şal, 2015) for further information) are trimmed and linearized around hover helical turn, 40-knots helical turn and 80-knots helical turn maneuvering conditions are applied. OVCs are designed with output variance constraints on helicopter Euler angles of 10^(-4), while all four helicopter controls are used as inputs. The noise intensities are W=10^(-7) x I_25 and V=10^(-7) x I_4.
In Figs. 3, 4 and 5 for hover helical turn, 40 kts helical turn and 80 kts helical turn, control effort (J) as functions of taper ratio (Ω) and root chord (Ԑ) is given. It can be seen from these figures that for any of maneuvering flight condition when taper ratio is applied, the control effort increases. On the other hand, if root chord is extended the control effort decreases. These results are similar with the results found for straight level flight as in (Tugrul Oktay & Sal, 2017).
In Figs. 6 and 7 effect of simultaneous variations in blade root chord and blade taper on some states and controls for 40 kts helical turn is given. Three scenarios are considered. The first one is original blade. The second one is for the situation that the taper ratio and root chord are applied in borders (version I). The third one is for the situation that taper ratio is chosen at the border and the root chord constant is chosen in order to keep blade area constant (version II). It is found that when it is required to use taper due to the performance reasons, the length of the blade root chord must also be increased in order to not rise the control effort of the FCS of the helicopter.

Conclusions and Recommendations
In this study, the effect of combined variation in blade root chord length and blade taper on the flight control effort of maneuvering helicopter was investigated. For this purpose, helicopter models which are complex, control-oriented, physics-based models and capturing the main-physics and essential-dynamics are applied. For the helical turn maneuvering flight conditions it is found that when taper ratio is applied, the control effort increases. On the other hand, if root chord is extended the control effort decreases. Moreover, when it is required to use taper due to the performance reasons, the length of the blade root chord must also be increased in order not to rise the control effort of the flight control system of the helicopter for also maneuvering flight. Consequently, in also maneuvering flight conditions effects and benefits of simultaneous variation in blade root chord length and blade taper on the flight control effort were proved in this study.