11 - Nonconservative systems

Abstract

Static analysis of elastic systems subjected to follower forces may erroneously show that the system is free of instability. To ascertain whether such a system is stable requires a kinetic analysis. For problems that lose their stability by buckling, the kinetic method predicts that one of the system–natural frequencies tends to zero as the critical load is approached. For nonconservative systems, flutter instabilities may occur in addition to possible buckling instabilities. This means that small perturbations about the static equilibrium state oscillate with increasing amplitude. This chapter presents several examples and presents an alternative solution method based on the fully intrinsic equations of beam vibration. The first of these examples is a mechanical analog to the Beck column, which consists of a cantilevered beam undergoing a compressive concentrated follower force at its free end. Both exact and approximate analyses of the Beck column itself are presented. Then, a column undergoing a compressive and uniformly distributed follower force is treated. The next example is a shaft subject to a tangential follower torque. The final example presented in the chapter is a deep cantilevered beam with a lateral follower force applied at the tip and in the plane of greatest flexural rigidity.