No abstract

In this article, a nonlinear control algorithm has been presented, in order to reduce vibrational movements of a flexible beam cantilevered in a cart supported by nonlinear stiffness, actuated by a hardly constrained second-order dynamic system. In the presented control algorithm, the interactive model of the solid body and a flexible beam has been extracted and utilized to design an algorithm, based on which the nonlinear vibrations of the tip point of the beam whose function is smooth and nonoscillatory tracking of the predefined desired trajectory are diminished. To design a nonlinear vibration control system of the beam, it has been assumed that the feedback system includes three modules: a vibrational beam cantilevered on a base, a base affected by nonlinear stiffness and actuator’s control force, and a force imposing system with constraints of bandwidth frequency limitation and actuation saturation boundary. In order to design this control system, the generalized tracking error function has been defined to nonlinearly amplify the vibrationally induced error and its rate, which by properly adjusting them, a new quantity can be extracted that satisfies the control system design constraints. To design a tracking control system for a nonlinear vibrating beam for the aforementioned three-modulus system, first, the generalized tracking error for the three modules is separately defined and parametrized, and then, by simultaneously adjusting all the defined parameters systematically, the predominant system presence in Lyapunov stability conditions is guaranteed. To illustrate the multimodule imitative (MMI) controller design algorithm, a moving support connected to a nonlinear spring and damper is considered, which carries a flexible cantilevered beam. The applied actuation system has second-order linear dynamics with the presence of command control saturation boundaries. For each of the abovementioned modules, a generalized tracking error is defined, and then, it is explained to how simultaneously adjust the parameters based on the stability and actuation constraints. The MMI controller is applied to the mentioned mechanical system modeled in the ANSYS® Mechanical APDL environment, and then, the necessary conclusions are discussed about the performance of the control system in eliminating the vibrations of the flexible arm, considering the actuation constraints while possessing the dominant Lyapunov stability.