Doctoral Dissertations

Date of Award

8-1991

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mechanical Engineering

Major Professor

James A. Euler

Abstract

Our aim is to study the dynamic stability of flexible missiles. The bending characteristics of the high length-to-diameter missiles, suitable for low drag, in the presence of an axial thrust is responsible for structural instability in form of divergence, or flutter. In addition, the slender missiles are suspected to be sensitive to dynamic coupling between the longitudinal and transverse axes. The latter may cause considerable flight abnormalities due to possible twist of the missile structure in the launch phase and, later on during the flight, transfer of the twisting energy from the longitudinal axis to the lateral axes. In modelling the flexible missile as a free-free beam, most of the researchers have used the Bemoulli-Euler beam theory. As we may know, this theory ignores the rotary inertia and shear deformation effects, which is not always justified. In fact, the shear deformation effect adds more deflection to the beam which increases the flexibility of the beam, while the rotary inertia effect will introduce more dynamic loading on the beam. Some researchers have investigated the stability problem by ignoring the damping effect through the Timoshenko beam theory using approximate methods, i.e., finite element approaches. This is due to the fact that Timoshenko beam theory leads to more accurate solutions than Bemoulli-Euler beam theory. In this dissertation, the general equations of motion, which clearly reflect the coupling between the longitudinal axis and lateral axes are derived for a general path in space, which include the rotational and translational motions. These equations represent the motion of a nonuniform pretwisted structurally damped rotating free-free Timoshenko beam subjected to aerodynamic forces, gravity, and a directionally controlled pulsating end thrust. Nonuniformity is considered for both the cross sectional area and mass distribution, and the missile is pretwisted due to the launch phase. To study the stability, a system of coupled partial differential equations resulting from the case of lateral vibration of a uniform damped free-free Timoshenko beam under the influence of a controlled end thrust is considered. The above system of partial differential equations with variable coefficients and related boundary conditions can not be formulated in terms of one variable. In other words, for certain boundary conditions, namely the free-free Timoshenko Beam with a directionally controlled end thrust, the equation of motion in terms of one of the describing variables, i.e., moment, shear, displacement, or slope can be obtained but the boundary conditions can not be separated into one variable. To deal with this problem, a mathematical tool is developed which results in a virtually exact solution. Utilizing this tool in order to study the lateral vibration of the beam, the stability boundaries for more than 100 different combinations of the system parameters, namely thrust, rotary inertia, shear deformation, damping, position of the attitude sensor, and the gain of the control system are investigated. An extensive comparison between the Bemoulli-Euler and Timoshenko Beam theories is presented, and possible transition between divergence and flutter instabilities are discussed. In longitudinal direction, a partial differential equation representing the longitudinal vibration of the beam is derived and a closed form solution for the travelling waves in axial direction is presented. The axial load distribution along the beam is deduced from the above closed form solution and later on is used in the equations representing the lateral vibration of the beam. As far as damping is concerned, the only literature available for the damped Bemoulli-Euler beam is for the uncontrolled beam [35] which has a graph only for c=0.001, nondimensional damping coefficient, with thrust in the stable region.

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