DYNAMIC STABILITY OF FLEXIBLE S PINNING MISSILES Part 11: Vibration and stability analysis of a structurally damped controlled free-free Bernoulli-Euler beam, as a model for flexible missiles.

controlled free-free Timoshenko beams under a constant follower load, as a model for flexible missiles, and will In this paper, part 11, a mathematical tool is developed to present an extensive comparison for Bernoulli-Euler and solve the general system of equations, describing the lateral Timoshenko beam theories. vibration of a Timoshenko beam and related boundary conditions, in order to obtain a virtually exact solution. 11. LITERATURE REVIEW Through the literature review, it becomes evident that the problem of a structurally damped controlled Bernoulli-Euler Study of the buckling load for beam columns under a 2 beam under the influence an thrust has been tangential load goes back to Beck who, as a pioneer, used discussed. Therefore, here, as an application of the the kinematic method instead of static method to compute the the problem dynamic stability of critical load. As we may recall, due to the nonconservative undamped and damped controlled free-free Bernoulli-Euler nature of the follower forces the static methods, i.e., the beam under a constant follower load is addressed. In follow geometric and energy methods based on the Euler static UP Paper, Part Irl, stability boundaries, using the developed approach, are not applicable. In real life, missiles are too1, for and damped classified as examples of Beck's model. In fact, there are a controlled free-free Timoshenko beams under a constant few papers that have the stability of flexible follower load are established and an extensive comparison missiles directly, but there is a range of papers between these two dominant beam theories is presented. discussing the related areas, i.e., vibration of pinned-pinned, clamped-free, clamped-clamped beams under axial loads. missiles, suitable for low drag, in the presence of an axial missiles, Wu and Celep studied the stability of a free-free thrust is responsible for structural instability in the form of Bernoulli-Euler beam under a direction-controlled thrust. divergence, or flutter. In addition, slender missiles are Using finite element methods, park and Mote computed the suspected to be sensitive to dynamic coupling between the optimal position of concentrated mass and the gain of the longitudinal and transverse axes. The latter may cause control system to maximize the follower force for stable considerable flight abnormalities due to possible twist of the transverse motion of a uniform Bernoulli-Euler beam. missile structure in the launch phase and, later on during the flight, transfer of the twisting energy from the longitudinal Feodosev examined the dynamic stability of a slender axis to the lateral axes, ~~d~lli~~ the flexible missile as a Bernoulli-Euler beam subjected to an uncontrolled follower free-free beam, most of the researchers have treated this force. Matsumoto and Mote studied the stability of a problem through the Bernoulli-Euler beam theory. As we Bernoulli-Euler beam under a controlled follower force. It know, this theory ignores the rotary inertia and shear has been shown 93 lo that in systems with combinations of deformation effects, which is not always justified. Some conservative and nonconservative forces and without researchers have investigated this problem by the directional control systems, the of instability can be effect the TimOshenkO beam theory using changed from fluner to divergence and vice versa depending approximate methods, i.e., finite element approaches. In fact, Timoshenko beam theory is the only available on the ratio of these forces. Also it has been shown that approximate theory which leads to more accurate solutions damping can change the mode of instability in above than Bernoulli-Euler beam theory does. To study the mentioned systems depending on the combined influence of the load distribution pattern, internal damping, and ratio of stability, following the development in part I l, we consider a system of coupled partial differential equations with the two types of forces. Beal 12, l3 used the Galerkin variable coefficients resulting from the case of a lateral method to investigate the stability of a flexible missile, vibration of a uniform damped free-free Timoshenko beam modelled as a Bernoulli-Euler beam, acted upon by an end under the influence of a controlled end thrust. Separating thrust with and without directional control. Dailey l4 time and space variables, we end up with a system of considered the same problem and applied a series solution to coupled ordinary differential equations with variable the single equation, describing the lateral vibration of a coefficients for which the stability can be investigated. It is damped uncontrolled Bernoulli-Euler beam, and related realized that the derived general system of equations and boundary conditions. In fact, he presents only one graph for related boundary conditions can not always be manipulated c=0.001, the nondimensional damping Parameter, with in order to obtain a single equation with required boundary thrust in the stable region. Peters and Wu l5 and Wu ''3 l7 conditions in terms of just one variable as can the Bernoulli- used finite element approaches to study the stability of a Euler beam. Therefore, a mathematical tool, which may be flexible missile modeled as a free-free Bernoulli-Euler beam called an extension of Frobenius method, is developed for canying a concentrated mass and subjected to a controlled "lving this 'ystem of equations and constant thrust. Sundararamaiah and Johns used finite conditions in order to obtain a virtually exact solution. As an element approaches with a larger stiffness and pointed