Vibration frequencies of a constrained flexible arm carrying an end mass

The natural vibration frequencies of a constrained arm carrying an end mass are studied in this paper. A clamped-free Euler-Bernoulli beam is used to model the arm. An axial compressive force which is derived from the contact force between the tip of the arm and the constrained curve is applied at the free end. Hamilton's principle is used to derive the governing equation and the boundary conditions of the beam. By defining a new variable, the non-homogeneous boundary condition is transformed into a homogeneous one. An exact characteristic equation is derived giving the relationship between the non-dimensional frequency and the two non-dimensional parameters, i.e., the axially compressed force and the end mass. Frequencies are obtained for the first three modes by solving numerically the transcendental equation. Results are presented for the frequencies of this beam under different force and mass conditions. The natural frequency characteristics of the constrained damped-free beam carrying an end mass are extremely important in predicting and understanding the dynamic behavior of the flexible arm. (C) 1997 Academic Press Limited.