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Technical Briefs

# Modeling of Cross-Coupling Responses on Hingeless Helicopters via Gyroscopic Effect

[+] Author and Article Information
Tak Kit Lau

Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, N. T. Hong Kong, P. R. C.tklau@mae.cuhk.edu.hk

Yun-hui Liu

Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, N. T. Hong Kong, P. R. C.yhliu@mae.cuhk.edu.hk

J. Comput. Nonlinear Dynam 7(2), 024501 (Jan 06, 2012) (9 pages) doi:10.1115/1.4005473 History: Received June 27, 2011; Accepted November 22, 2011; Revised November 22, 2011; Published January 06, 2012; Online January 06, 2012

## Abstract

The perplexing cross-coupled responses between the control axes on hingeless helicopters have puzzled researchers for years. Unlike previous studies, which introduced a physically meaningless phase-lag to account for the cross-couplings, this paper proposes a new method to relate both on-axis and off-axis responses by the gyroscopic moments through the actuation mechanism of hingeless helicopters. This new method allows investigators to directly and analytically quantify the debatable cross-coupling due to the complex actuation of the rotor. This method is based on the fact that when the angular momentum of the spinning rotor is disturbed by an incremental lift along the main blades due to the varying cyclic pitch angle controlled by the servo mechanisms, off-axis moments are induced to counteract the changes in angular momentum according to the principles of the gyroscope, and hence these gyroscopic moments directly exhibit on-axis responses. This new method yields a parametric framework for examining the previously unexplained cross-coupled responses of hingeless helicopters and demonstrates that, in addition to aerodynamics, intricate dynamic nonlinearities also occur due to non-intuitive actuation mechanisms on the rotor of hingeless helicopters. Finally, simulations and experiments were performed to validate the proposed modeling method.

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## Figures

Figure 1

Left: A 120°-spacing swashplate of a hingeless helicopter. Right: Each joint on the swashplate connects with a servomechanism, which are placed below the plate. A naming convention is employed, such that (a) joint elevator, (b) joint aileron, and (c) joint pitch.

Figure 2

(a) A three-bar-linkage from the inner ring of the swashplate to the flybar. It varies the angle of attack of the paddles and, hence, alters the inclination of the hub plane of paddles. It serves as an indirect input to the cyclic pitch angle of the main blades. (b) A single-linkage that connects the inner ring to the leverage set. It directly alters the cyclic pitch angle, and therefore it serves as a direct input. The main blade and the flybar rod are not shown.

Figure 3

The side-view of the schematics of the flybar-rotor structure. When the swashplate tilts, it exhibits a displacement δ cyc . Then the flybar flaps for an angle β .

Figure 4

The direct measurements of the cyclic pitch angle. (a) The actual variation of the cyclic pitch angle on the main blades in a revolution. It can be measured directly from the blades. The rotation angle ψR is defined positively around the z-axis of the body frame. This variation cannot be intuitively understood because, for example, to roll left, more lift should be generated on the right hand side of the hub plane instead of the front side. (b) The compensated variation of the cyclic pitch angle after going through a conventional phase-lag treatment, which can be found in the previous work. Such that the cyclic pitch angles are added by a 90° as a phase-lag. It then becomes intuitively understandable because, for example, to roll left (blue solid line), more lift is generated on the right side of the hub plane.

Figure 5

The schematics of the indirect input. The indirect input directly influences the flapping angle of the flybar; hence, the cyclic pitch angle is eventually altered. δ E′′ is the input from the joint elevator.

Figure 6

Using the analytical forms (4), (6), and (3), this figure plots the variation of the cyclic pitch angles of a main blade in a function of the azimuth angle. The control commands are recorded from a commercially available R/C (remote control) transmitter (JR PCM9XII). Then, the collected input data goes into the derived analytical forms so as to obtain the resultant cyclic pitch angle at an azimuth angle.

Figure 7

When a moment (MA ) is applied on an axis perpendicular to the spinning axis of an object, the moment will change the direction of the object as well as the spinning direction. Therefore, by conservation, the change of angular momentum (ΔH = H′−H) will induce an opposite counterpart (−Δ H), and this angular momentum has a spinning velocity WP , which is known as the velocity of precession.

Figure 8

(a) At first, the helicopter is hovering horizontally. The spinning rotor possesses an angular momentum HR . (b) Under a command to pitch forward, more lift is generated on the right-hand-side of the hub plane. (c) Both rotor and fuselage experience the rotation about the body x-axis. The angular momentum is altered to H′R and induces a gyroscopic moment MG . (d) By conservation, an opposite gyroscopic moment is induced on the body y-axis, which is perpendicular to the axis of actuation, i.e., the x-axis of the body frame in the case of a forward pitching. Then, the helicopter pitches forward with a velocity of precession θ· < 0.

Figure 9

The responses on the attitude angles of a hingeless helicopter using the proposed modeling method. The helicopter is commanded to pitch forward then backward sinusoidally. The responses in the pitch channel are coherent with the commands and that it correctly exhibits a motion of pitching forward then a backward motion. Also, it demonstrates the off-axis responses on the roll and yaw channels. Extensive simulations are performed on other axis-pairs, and the on-axis and off-axis relations are summarized in Table 1. The off-axis responses in the yaw channel are described, however, as those results can open another discussion on the tail hunting, which is often attributed to the change of motor torque and insufficiency of feedback from the autogyro. We do not further discuss this aspect in this paper.

Figure 10

These figures plot the responses in the attitude angles due to different pitching and rolling commands. (a) Rolling left command: the coupled auxiliary response takes place positively in the pitch channel. (b) Rolling right command: the coupled auxiliary response takes place negatively in the pitch channel. (c) Pitching forward command: the coupled auxiliary response takes place negatively in the roll channel. (d) Pitching backward command: the coupled auxiliary response takes place positively in the roll channel. These results demonstrate that the predicted auxiliary responses do nonintuitively exist and influence the overall dynamics. Moreover, the signs of both on- and off-axis responses are coherent with the predictions by our method.

Figure 11

The schematics of the hingeless rotor design. Only a quarter of the structure is shown. The structures on the left and right planes are further explained in Fig. 3 and Fig. 5.

Figure 12

The definition of the coordinate frames on the helicopter

Figure 13

In the experiments, a fully instrumented hingeless helicopter (JR Voyager GSR) is commanded to pitch and roll on the ground. The onboard computer records the inputs to the rotor from servomechanisms, and an IMU measures the orientation.

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