The problem of a cruise missile aeroelastic vibrations based on Ritz method
The article presents the study of aeroelastic stability of the elastic cruise missile based on Ritz method employing a different number of approximating functions and comparative analysis with the similar computations, performed on the basis of the finite elements method (FEM). A cruise missile performs a longitudinal short-periodical motion in an incompressible airflow. The missile is being modeled employing a beam scheme, including a fuselage, two wing straight consoles and deflectable controls, i.e. elevation rudders. The wing is considered as an elastic beam, working in bending with transverse shear and torsion, while the missile hull and stabilizer are assumed sheer rigid. It is assumed, that the wing consoles elastic vibrations occur in line with the flat sections hypothesis. Aerodynamic loads are being determined based on the quasi-stationary theory of plane-parallel flow of the wing cross-sections flow-around. The longitudinal compression of the missile body under the engine thrust impact is accounted for as well. The unknown functions of the transversal fuselage axis displacements, the wing axis transverse displacements and torsion angle are presented in the form of expansions in generalized coordinates, representing the movement along the natural forms of oscillations of a free structure with fixed controls. Computation results convergence to a certain value close to the exact analytical solution, which is achieved by increasing the number of approximating functions, is demonstrated on an example. The graphs of the missile natural frequencies dependence from flight speed variation and engine thrust were plotted. Critical velocities and stability regions of the missile flight were respectively determined for each method. All calculations were performed in “Wolfram Mathematica 8”.