Input shaping control of a flexible cantilever beam excited by a moving hub

Abstract

Introduction: A cantilever beam is a well-known structural element in engineering, which is only fixed at one end. This structure can be used to describe a manipulator, whose stiffness is large to ensure rigidity and stability of the system. A flexible cantilever beam provides a light-weight structure and high cost efficiency but generates vibration under high-speed positioning. In this paper, we aim to control the vibratory behavior of a flexible cantilever beam attached to a moving hub. Method: The mathematical model of the flexible beam is described by partial differential equations (PDEs) using Euler-Bernoulli beam theory. Then, The PDE model is approximated by using the Galerkin method, which is resulted in a set of ordinary differential equations (ODEs). Experiment is used to determine unknown parameters of the system to complete the model. The ODE model enables the control design of three input shapers: (i) Zero-Vibration (ZV), (ii) Zero-Vibration-Derivative (ZVD), and (iii) Zero-Vibration-Derivative-Derivative (ZVDD), which are employed to drive the flexible beam to the desired position and to reduce vibrations of the beam. Results and conclusion: The dynamic model is obtained in term of ordinary differential equations. Unknown parameters of the system are determined by experimental process. Various controllers are designed to eliminate the vibration of the beam. The simulation is applied to predict the dynamic response of the beam to verify the designed controllers numerically. Experiment shows the validity of the mathematical model through the consistency between the simulation and experimental data and the effectiveness of the controllers for the real system. These controllers show several advantages such as no need of extra equipment; the positioning controller is intact, which means it may be applied to many existing systems.