Phugoid Motion for Grazing-Entry Trajectories at Near-Circular Speeds

We analyze the trajectory perturbations that result in phugoid oscillations during the grazing planetary entry of space vehicles starting with near-circular speed, in ballistic and equilibrium glide modes. The equations for planar entry are transformed into a dimensionless system appropriate for an analytical integration that provides accurate results in comparison with those obtained from a numerical integration of the original system. For ballistic entry, the small perturbation in the flight-path angle is a lightly damped oscillation with long period, while the perturbation in the density (altitude) increases as the speed decreases. Nevertheless, the relative change in the density is decreasing, and the phugoid in ballistic entry is termed stable in both the flight-path angle and the altitude. For glide entry, in the reference solution where there is equilibrium of forces along the normal to the flight path, both the flight-path angle and the altitude have a steady decrease along the flight path. With a slight perturbatipn in either the initial speed or the initial flight-path angle, or both, the trajectory variables undergo a damped oscillation about the reference trajectory. Both the damping and the frequency are obtained explicitly, and they correctly predict the phugoid motion as seen in the numerical solution.