Study on the Strong Nonlinear Dynamics of Nonlocal Nanobeam Under Time-Delayed Feedback Using Homotopy Analysis Method

Time-delay effects are ubiquitous in active control systems and can significantly influence the dynamic characteristics. This paper examines an active control system for a nanobeam and analyzes the strong nonlinear vibrations influenced by displacement and velocity time-delayed feedback.

The periodic solutions of the system are first derived using the homotopy analysis method. Then, the Hopf bifurcation of the system is discussed with time delay as a parameter. Finally, the accuracy of the above study is verified using the Runge-Kutta method.

The study reveals that the system can exhibit multiple periodic solutions, and the time delays and feedback gain coefficients can alter the quantity and amplitude of the periodic solutions. Furthermore, positive feedback gain coefficients provide a larger stability region for the system compared to their negative counterparts. Nonlocal parameters not only increase the response region and peak values of the system but also affect its hardening behavior. A comparison with numerical solutions demonstrates that the homotopy analysis method not only provides accurate approximations but also predicts periodic solutions that numerical methods fail to capture.