by Eric M. Hollmann, Ph.D. Copyright © January 2, 2007

1. Introduction
Porpoising has been suggested as a possible source of accidents in gyroplanes. Porpoising, or "phugoid" oscillations are well-known in helicopters and fixed-wing aircraft but generally have a long period (~ 20 sec) and can therefore be corrected for by the pilot [1]. In small gyroplanes, it is possible that these oscillations can be fast enough to be considered dangerous. This is especially true if the gyroplane body pitching motion falls sufficiently out of phase with the blade disk oscillations, in which case the porpoising could cause an impact between the rotor disk and the pusher propeller [2].
Here, the basic porpoising motion of a small gyroplane is calculated numerically. It is found that the porpoising motion has a fairly short period (~6 sec) and can diverge if not sufficiently damped by the horizontal tail. Porpoising can therefore be dangerous for small gyroplanes and should be taken into account during the aircraft design. The blade disk and gyroplane body are found to remain fairly well in-phase during the porpoising motion, so impact between the rotor and propeller should not normally occur unless the mode is allowed to grow to very large amplitudes.

2. Problem layout
The basic coordinate system and forces considered here are shown in Fig. 1. Only pitching motion will be considered; rolling and yawing are assumed to be decoupled from the pitching motion. The lab coordinate system is (x,y,z), while the gyroplane body is described by a coordinate system (x′,y′,z′) pitched down at an angle f from the lab coordinate system. The rotor blade disk is described by a coordinate system (x′′,y′′,z′′) which is assumed to be pitched up by an angle g from the lab coordinates.

The body drag force D is assumed to be antiparallel to the body velocity vector v and is assumed to act on the body center of mass (which is assumed to equal the aerodynamic center).

                                       (1)

where A is the fuselage flat plate drag area and r is the air density. The gravity force simply points down:

                                                                                         (2)

where M is the gyroplane mass and g is the acceleration of gravity. The blade lift force is assumed to be perpendicular to the blade disk:

                                                 (3)

The horizontal tail is assumed to provide a lift which is perpendicular to the air flow and proportional to the sin of the angle of attack:

                             (4)

where A_H is the tail lifting area (area times lift coefficient). The tail rotor thrust is assumed to be constant in magnitude in the x′ direction:

                                                                          (5)

Fig. 1. Coordinate axes and body forces

The pusher thrust is determined by the empirical form [3]:

                                                                (6)

where

,  

, and  

In Eq. (6), T is the thrust [lbs], v the velocity [ft/s], P the propeller power [hp], W_T the propeller angular velocity [rpm], and D_T the propeller diameter [ft]. The thrust is assumed to remain constant during the pitching motion of the gyroplane.

The total force on the gyroplane is given by the sum of Eqs. (1)-(5). This force produces a net acceleration on the body mass M. The effective body forces seen in the accelerating frame are obtained by subtracting the total force from the CM; the resulting forces are shown in Fig. 2. These forces produce a torque N_y about the rotor hub and cause changes in the gyroplane pitch angle f. The torque about the rotor hub due to the pusher prop is:

                                                                                  (7)

While the torque due to the tail lift L_H is:

                           (8)

and the lift force on the rotor hub results in a body torque:

(9)

The total torque about the rotor hub in the accelerating frame is the sum of Eqs. (7)-(9). This torque produces pendulum-like oscillations of the hanging gyroplane body. Approximating the gyroplane mass distribution as a point mass M at the CM, the moment of inertia about the rotor hub is: .

Fig. 2. Body forces in accelerating body frame

Fig. 3. Airflow components seen by blade section

To calculate the lift and torque on the rotor disk, the rotor section forces need to be considered. The airflow velocity components seen by a blade section are shown in Fig. 3. A teetering hub with fixed cone angle b₀ is assumed. The rotation angle of a blade is described by the angle y, while radial position along a rotor blade is given by r. The angular velocity of the rotor blades is given by W. The airflow components seen by a blade section are:

                                              (10)

 
                                                                                                       (11)

The angle q seen in the insert of Fig. 3 is the cyclic pitch angle set by the pitch of the rotor disk, which is in turn set by the position of the control stick. We define a cyclic pitch , where the flapping angle is , and f_c is the angle between the x′ axis and the rotor disk. f_c is held fixed by the pilot. The perpendicular section lift force F_p is:

                         (12)

where a is the section lift slope (per radian) and c is the section chord. The rotor lift force can now be found by integrating over the section lift forces. To do this, the induced velocity v_i is assumed to be constant over the rotor disk. Also, we assume a small flapping angle, i.e. . Also, the blade rotation is assumed to be rapid compared with the dynamics studied here. The rotor lift then becomes:

  (13)

Applying momentum and energy conservation to the induced velocity flow through a lifting wing gives the standard approximation for the lift in terms of induced velocity [4]:

                                                                                (14)

Combining Eqs. (12) and (13) then gives a closed-form estimate for the blade lift:

               (15)

Since we are assuming that the blade rotation is fast compared with other time scales, changes in the rotor disk orientation can be described as a slow precession of the spinning rotor. Changes in pitch will be caused by the torque about the x′′ axis, which can be estimated by integrating over the blade section forces:

(16)

The equations of motion of the gyroplane are now known:

                                             (17a) 

                                              (17b)

                                          (17c)

                                          (17d)

Eqs. (17a) and (17b) describe the translation of the gyroplane, with F_x and F_z given by summing Eqs. (1)-(5). Eq. (17c) desribes the pitching motion of the gyroplane body, with N_y determined by summing Eqs. (7)-(9). Finally, Eq. (17d) describes the precession of the rotor disk, with given by Eq. (16) and the rotor angular momentum given by:

                             (18)

where M_B is the total blade mass.

3. Results
The equations of motion (17) can be integrated forward in time numerically for a given initial flight condition (determined from the four equations of motion (17) with d/dt = 0 ).

Figure 5 shows the gyroplane velocity components v_x = dx/dt and v_z = dz/dt as a function of time for the sample gyroplane. The horizontal tail lift area is taken to be A_H = 30 ft². The other dimensions used are listed in the appendix. To initiate the porpoising motion, a vertical gust of 30 mph is applied at t = 1 and turned off at t = 3 seconds. It can be seen that, after an initial transient, the gyroplane velocity oscillates at a period of roughly 6-8 seconds. This is a fairly short porpoising period, and could therefore be potentially hazardous if not self-damped, since the pilot may not have time to correct for the oscillations.

Fig. 5. Airflow components seen by blade section

Perhaps more important than the gyroplane translation is the pitching motion of the gyroplane body and the rotor disk. This is shown in Fig. 6. It can be seen in Fig. 6(a) that the gyroplane body and rotor disk tend to oscillate in phase. However, a slight relative pitch motion can be seen in Fig. 6(b). This pitch difference is especially dangerous if it grows to the point where a blade-prop collision can result.

Fig. 6. Airflow components seen by blade section
The oscillations seen in Figs. 5,6 are slowly damped, dominantly by the horizontal tail. The effect of the horizontal tail area is illustrated in Fig. 7. Here, the relative pitch angle (g+f) is plotted as a function of time. The simulation is run with horizontal tail lift areas of 20 ft², 30 ft², and 40 ft². For the smallest tail, it can be seen that the oscillations diverge and are therefore dangerous if not corrected by the pilot. For the largest tail area, 40 ft², the oscillations are quickly damped. A large tail, while providing more drag, therefore provides added stability against porpoising, as expected.

Fig. 7. Airflow components seen by blade section
The Pitch Stability Computer Program is now available with the book "Modern Gyroplane Design" for $220 from ADI. See books.

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