THE MODELING OF AXIALLY TRANSLATING FLEXIBLE BEAMS

The axially translating flexible beam with a prismatic joint can be mo delled by using the Euler-Bernoulli beam equation together with the co nvective terms. In general, the method of separation of variables cann ot be applied to solve this partial differential equation. In this pap er, a non-dimensional form of the Euler-Bernoulli beam equation is pre sented, obtained by using the concept of group velocity, and also the conditions under which separation of variables and assumed modes metho d can be used. The use of clamped-mass boundary conditions leads to a time-dependent frequency equation for the translating flexible beam. A novel method is presented for solving this time-dependent frequency e quation by using a differential form of the frequency equation. The as sumed mode/Lagrangian formulation of dynamics is employed to derive cl osed form equations of motion. It is shown by using Lyapunov's first m ethod that the dynamic responses of flexural modal variables become un stable during retraction of the flexible beam, while the dynamic respo nse during extention of the beam is stable. Numerical simulation resul ts are presented for the uniform axial motion induced transverse vibra tion for a typical flexible beam. (C) 1996 Academic Press Limited