Vibration control of a coupled nonlinear string system in transverse and longitudinal directions

In this paper, vibration control is considered in both transverse and longitudinal directions for a coupled nonlinear flexible string system. The vibrating string is nonlinear due to the consideration of the coupling between transverse and longitudinal displacements. By using Hamilton´s principle, the dynamics of the nonlinear string are presented by two partial differential equations (PDEs) and four ordinary differential equations (ODEs). Based on Lyapunov´s direct method, boundary control is constructed to regulate the string´s vibration and stabilize the system. Numerical simulations are provided to demonstrate the effectiveness of the proposed control.