Flight Mechanics of an Elastic Symmetric Missile

Abstract

The free-flight motion of an elastic missile is approximated with three bodies connected by two massless elastic cantilever beams. If the mass distribution of the three bodies is 1-2-1, the frequency of the symmetric oscillation of the outer bodies is within 5% of the classical frequency of the oscillation of a free-free beam. A second combined pitching antisymmetric flexing motion can occur with a frequency that is almost twice that of the symmetric flexing motion. As the beam stiffness is reduced, the symmetric flexing motion frequency approaches the rigid body aerodynamic zero-spin frequency, and the flight zero-spin aerodynamic frequency is considerably reduced. Moderate beam damping can cause dynamic instability for spins greater than the zero-spin aerodynamic frequency. Resonance mode amplification can occur when the spin is equal to the zero-spin aerodynamic frequency, but more importantly it can occur when the spin is equal to the two elastic flexing frequencies. Spin-yaw lock-in occurs at the lower elastic frequency.

A New Approach to Aeroelastic Response, Stability and Loads of Missiles and Projectiles

“…Mikhail showed examples of spin lock-in when fin damage produces a roll inducing moment sufficient to cause a steady state spin greater than the lowest elastic frequency and the initial spin rate was zero. However, Murphy and Mermagen (2000) insisted that results obtained by the former should be dismissed due to incorrect expressions for the angular momentum. Murphy and Mermagen (2000) approximated the elastic missile by three rigid bodies connected by two massless elastic beams and showed that it is impossible to cause spin lock-in by roll inducing moment and zero initial spin alone.…”

“…However, Murphy and Mermagen (2000) insisted that results obtained by the former should be dismissed due to incorrect expressions for the angular momentum. Murphy and Mermagen (2000) approximated the elastic missile by three rigid bodies connected by two massless elastic beams and showed that it is impossible to cause spin lock-in by roll inducing moment and zero initial spin alone. It should be noted that the use of the three-body model is a major simplification of the actual physical problem.…”

“…Special solutions showing spin lock-in at the lowest elastic frequency were developed by Mikhail (1996) and Murphy and Mermagen (2000). Mikhail showed examples of spin lock-in when fin damage produces a roll inducing moment sufficient to cause a steady state spin greater than the lowest elastic frequency and the initial spin rate was zero.…”

“…It should be noted that the use of the three-body model is a major simplification of the actual physical problem. Later they replaced the three-body model with a continuous elastic model using differential equations in Murphy and Mermagen (2000) and obtained numerical results for the natural frequencies, flexing waveforms and equilibrium spins for a specific missile. Reis and Sundberg (1967) investigated the causes of large coning angle that a Nike-Tomahawk sounding rocket experienced during flight.…”

Flight motion of a continuously elastic finned missile

“…Because these projectiles have to resist very large aerodynamic loads due to their high-velocity flight at sea level, designers have been concerned about the possibility of aeroelastic deformations [1][2][3][4][5][6][7]. There is some evidence that a small number of these projectiles have been forced to spin at rates close to their lowest elastic frequency and have been subject to large inelastic deformations.…”