Improved modal convergence using the assumed modes method for rods carrying various lumped elements

In this paper, the assumed modes method is used to determine the modes of vibration of an arbitrarily supported uniform and nonuniform rods carrying various lumped elements, including a lumped mass, a grounded spring, a grounded viscous damper, and an undamped and damped oscillator. In applying the assumed modes method, the set of trial functions used in the expansion can be arbitrary as long as they satisfy the geometric boundary conditions of the system. In practice, the trial functions are often selected to correspond to the eigenfunctions of the bare uniform rod. Numerical experiments show that while this set of trial functions converges to the exact results, the rate of convergence can be exceedingly slow. In order to expedite modal convergence, the eigenfunctions are augmented with piecewise linear functions that capture the slope discontinuities of the mode shapes at the attachment locations due to the presence of the lumped elements. The results obtained using the two sets of trial functions are compared with those obtained exactly. It is shown that including the piecewise linear functions significantly improves the accuracy of the modes of vibration of the system while drastically reducing the computational time and effort