Dynamic stability of a flexible missile under constant and pulsating thrusts

The stability of a flexible missile, idealized as a uniform free beam under an end thrust, is investigated. A simplified control system is incorporated to obtain directional stability. It is shown that, in the absence of the control system, the critical thrust magnitude is associated with coalescence of the two lowest bending frequencies. With the control system included, it is found that the critical thrust magnitude corresponds to a reduction of the lowest frequency to zero. The results are related to typical thrust levels found in large, present-day missiles. Also considered is the effect of periodically varying thrust perturbations on the stability of the vehicle. In this case, parametric instabilities are found to exist. It is found that, when the beam is very stiff longitudinally, instabilities are most likely to occur when the frequency of the thrust variation is in the vicinity of 1) twice any of the bending frequencies, 2) the sum of any two of the bending frequencies, or 3) the difference of any two of the bending frequencies. With low longitudinal compliance, significant instabilities are also found to occur when the frequency of the thrust variation is in the vicinity of one of the longitudinal natural frequencies.