Time-delay feedback control of a cantilever beam with concentrated mass based on the homotopy analysis method.

• Frequency response of a strongly nonlinear time delay system is obtained. • Optimal time delays of control system are given. • First statement about the feedback control relationship between displacement and velocity. • Appropriate time delay parameters can make the system softening or hardening. In this paper, the strongly nonlinear vibration of a cantilever beam system with concentrated mass at an intermediate position controlled by displacement and velocity time delay is investigated. The nonlinear governing equation is studied using the homotopy analysis method. The effects of the time delay, displacement and velocity feedback gain coefficients, as well as frequency on the amplitude of the system were studied in detail. The results indicate that the velocity feedback control is not necessarily superior to displacement feedback control. Reasonable selection of the displacement and velocity time delay parameters can avoid the Hopf bifurcation, adjust the number, amplitude and stability of periodic solutions, and exhibit the softening behavior at low frequency and hardening spring characteristic at high frequency. The theoretical research in this paper will promote the application of homotopy analysis method in the field of time delay control, and serve as a theoretical reference for the design and optimization of cantilever beam control systems with concentrated mass. [ABSTRACT FROM AUTHOR]