The equations of motion for a parametrically excited cantilever beam

Parametric vibration of a simple cantilevered structure has been shown to be capable of generating complex multi-modal instabilities within the structure, and while many such resonances are physically significant the level of mathematical modelling required can be prohibitive. In this paper a descriptive, yet manageable, set of governing equations of motion are derived for a cantilever beam beam with a lumped end mass by a straightforward treatment of the system kinematics, and the subsequent use of Lagrange's formulation in conjunction with the kinetic and potential energy functions. The intention is to illustrate the use of classical engineering theories in the accurate modelling of a very simple structure, and to highlight the conceptualization of such a three-dimensional problem.