Porpoising
1. Introduction 2. Problem layout 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).


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 b_{0} 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 closedform 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 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 ft^{2}. 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 68 seconds. This is a fairly short porpoising period, and could therefore be potentially hazardous if not selfdamped, 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 bladeprop 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 ft^{2}, 30 ft^{2}, and 40 ft^{2}. 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 ft^{2}, 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 