DYNAMICS OF A FLEXIBLE ROTATING BEAM INTERACTING WITH A FLAT RIGID SURFACE, PART I: MODEL DEVELOPMENT

Abstract The dynamics of a flexible Euler-Bernoulli beam attached to a uniformly rotating hub and interacting at the tip with a plane rigid surface parallel to the hub axis is analyzed. Lagrange's equation and a generalized co-ordinates-eigenfunctions representation for the displacement fields are used to formulate the unconstrained motion equations. It is shown that one must account for the longitudinal displacements if dynamical stiffening is to be observed. A quadratic approximation for these displacements is used, and a computationally effective way of by-passing the calculation of the volume integrals at every integration step, by matrix-vector multiplications is implemented. The energy dissipating constraint, which requires the tip of the filament to move on the surface, is imposed with a Lagrange multiplier and a generalized force column vector corresponding to dry friction. A procedure of evaluating the generalized friction force in terms of the generalized co-ordinates and velocities is presented. The presence of dynamical stiffening is tested and the calculated values of the fundamental frequency are found to be in excellent agreement with data from literature.