Pitch-damping coefficients at hypersonic speeds.

Abstract

Equation (A6) may be recognized as the Mathieu equation; two of the zones of parametric resonance determined by this equation are shown in Fig. 6 (zones A and B in the figure). Such zones define regions where the solutions of Eq. (A6) exhibit an exponential growth of the form e^. The iso-ju lines shown in the figure connect points of equal growth rate, the boundaries of the zone corresponding to ju = 0. Considering these lines as a measure of the strength of the parametric excitation, we see that the excitation is stronger in the interior regions of the zones. The figure also shows straight line loci of the values of the load ratio qi/qi for fixed values of Tr/ri. These loci intersect the zones of parametric resonance. As qi/qi is increased from zero, for fixed rr/ri, the loci first intersect zone A. If the excitation in the region of the zone traversed by the loci is sufficiently strong, snapping will occur at some critical value of the load ratio. However, as TV/TI is raised, the regions of the zone traversed by the loci become increasingly weaker in excitation. Eventually the excitation in this zone becomes too weak to precipitate snapping. As a result the load ratios must be increased above unity to satisfy the conditions of parametric resonance in zone B, the snapping phenomenon now being precipitated by the excitation in this zone. Thus, it is seen that the change in the zone of parametric resonance accounts for both the jump in critical load ratio and the appearance of load ratios greater than unity (see Fig. 2) .