Analytical solutions for fixed-free beam dynamics in thin rib machining

Abstract

Two different analytical approaches for predicting thin rib, fixed-free beam dynamics with varying geometries are presented. The first approach uses the Rayleigh method to determine the effective mass for the fundamental bending mode of the stepped thickness beams and Castigliano’s theorem to calculate the stiffness both at the beam’s free end and at the change in thickness. The second method uses receptance coupling substructure analysis (RCSA) to predict the beam receptances (or frequency response functions) at the same two locations by rigidly connecting receptances that describe the individual stepped beam sections, where the receptances are derived from the Timoshenko beam model. Comparisons with finite element calculations are completed to verify the two techniques. It is observed that the RCSA predictions agree more closely with finite element results. Experiments are also performed, where the stepped beam thickness is changed by multiple machining passes, and receptance measurements are carried out between passes. The RCSA predictions are compared to experimental results for natural frequency and stiffness. Agreement in natural frequency to within a few percent is reported.