Parametric Study of Nonlinear Beam Vibration Resting on Linear Elastic Foundation

Nonlinear beam resting on linear elastic foundation and subjected to harmonic excitation is investigated. The beam is simply supported at both ends. Both linear and nonlinear analyses are carried out. Hamilton’s principle is utilized in deriving the governing equations. Well known forced duffing oscillator equation is obtained. The equation is analyzed numerically using Runk-Kutta technique. Three main parameters are investigated: the damping coefficient, the natural frequency, and the coefficient of the nonlinearity. Stability regions for first mode analyses are unveiled. Comparison between the linear and the nonlinear model is presented. It is shown that first mode shape the natural frequency could be approximated as square root of the sum of squares of both natural frequency of the beam and the foundation. The stretching potential energy is proved to be responsible for generating the cubic nonlinearity in the system.