Effects of Stiffness and the Variation of Center of Mass on Rocket Motion

Abstract

Newton’s law, Hamilton’s principle, or Lagrange’s equations cannot be applied directly for variable mass system. This paper presents a method to derive and solve the Lagrange equation applied to a system of a rocket and its exhaust while considering the flexibility of the rocket body. This method may help avoid the application of the Lagrange equation to an open system, in which thrust is regarded as a follower force. The derived system of equations of motion can express the effects of the stiffness and the variation of the center of mass on the dynamic parameters of the rocket. In the present system of equations of motion, some normal force components due to the loss of mass, and the bending vibration and rotation of the rocket body are added. In addition, the equations of motion indicate the mutual interaction between the movement of the center of mass and the bending vibration of the rocket body. A finite element method is employed together with the Newmark and Newton-Raphson methods in order to solve the system of equations.