Dynamical analysis of a beam with a tip mass

The nonlinear dynamics of a beam with a tip mass attached to the end of the beam subjected to an arbitrary harmonic excitation at the clamped end was investigated. The equation of motio~ in the form of an integro-differential equation governing the system dynamics, was obtained using D'Alembert's Principle. The resulting equation contained nonlinearities up to the cubic order denying a closed form solution. One of the weighted residual methods, Galerkin's method, was applied to the nonlinear partial differential equation to reduce it to an ordinary differential equation. Next, a perturbation method, method of multiple scales, was applied to the resulting ordinary differential equation to obtain an approximate solution. The primary resonance case, the subharmonic resonance case of order 1/2, the superharmonic resonance case of order 2, the supersubharmonic resonance cases of orders 2/3 and 3/2 were investigated. The resulting phase portraits and the bifurcation diagrams were plotted. In addition to the above, the problem was solved by means of numerical methods using different system parameters, and the results of the theoretical analysis and the numerical analysis were compared. The results indicated close agreement between theory and numerical simulations.