Dynamic Modeling and Control of Elastic Beam Fixed on A Moving Cart and Carrying Lumped Tip

In this paper, a Bernoulli – Euler beam fixed on a moving cart and carrying a lumped tip mass is considered. The equations of motion which describe the global motion as well as the vibrational motion were derived using the extended Hamilton’s principle. The obtained equations of motion are analyzed by means of the unconstrained modal analysis. In order to eliminate the need of sensor placement at the tip of the cantilever beam, as most of the practical control implementations require that sensors and actuators to be placed in accessible locations, the Linear Quadratic Estimator (LQE) approach was utilized to estimate the vibrations of any point on the span of the elastic cantilever beam in the presence of process and measurement noises. For the purpose of suppressing the vibrations at the tip of the beam, an active optimal controller was designed based on the Linear Quadratic Gaussian (LQG) method. Numerical simulation results demonstrated the capability of the developed optimal controller in eliminating the vibrations at the tip of the beam as well as to improve the positioning accuracy.