Non-smooth and time-varying systems of geometrically nonlinear beams

Bending vibrations of geometrically nonlinear beams, which are connected with some clearance in their contact areas, are analyzed during dynamic extending and retracting motion of the different segments. For the physical model of a fork lifter, as an example of application, the governing system equations are derived by applying Hamiltonʹs principle. Using a discretization procedure, based on admissible shape functions, a system of coupled, nonlinear, time-varying, ordinary differential equations is generated. Linearization and model reduction leads to a sequence of simple models. On the basis of these models, an adaptive state regulator and an adaptive full-state observer (Luenberger Observer) are designed for vibration suppression using the optimal linear quadratic regulator (LQR). The adaptive controller and observer are applied to the original, significantly more complicated, geometrically nonlinear and time-varying system with clearance so that the robustness of the controlled system can be studied during dynamic extending and retracting motions.