Effects of approximations on the static and dynamic response of a cantilever with a tip mass

Abstract

The problem of determining the equilibrium deflections of a beam with a tip mass, and the frequency of infinitesimally small oscillations about the beam's equilibrium state, E, is analyzed. The beam is able to experience flexure along two normal directions in space (thus, flexure in any direction) and torsion. Numerical solutions of the full, nonlinear beam static equilibrium equations are obtained by direct integration of a two-point boundary value problem. The results obtained are compared with an approximate perturbation expansion solution, E*, for the equilibrium state E. The frequencies associated with the small oscillations about the equilibrium state arc determind by linearizing the equations of motion about the equilibrium state, and by using a transfer matrix technique on the resulting equations. The effect on the calculated frequencies of using the approximate solution E*, obtained by a perturbation method (instead of the more exact numerical solution E mentioned above) in the linearized equations, is assessed. The use of an alternative way to determine an approximation for the natural frequencies of the system is also assessed. For this, small motions are assumed and the equations of motion are first expanded about the undeformed state of the beam. The undeformed state is not a static equilibrium state if gravity is considered. The resulting equations, which contain only polynomial nonlinearities, are then used to analyze the motion. After the equilibrium solution to these equations, E* is determined by a perturbation expansion ; these same equations are then linearized about E* and the natural frequencies are also determined by a transfer matrix technique. It is shown that the approximation so obtained for the natural frequencies can be unsatisfactory for large values of the tip mass. All the results obtained in this paper are compared with published unite elements und with experimental results.