Analytical research on a single degree-of-freedom semi-active oscillator with time delay

In this paper a single degree-of-freedom semi-active oscillator with time delay is researched. By averaging method, the first-order approximately analytical solution is obtained, and the stability condition is also established based on the Lyapunov theory. The analytical results show that the amplitude and the stability condition of the steady-state solution are all periodic functions of time delay, with the same period as the excitation one. Moreover, another simple case, namely the semi-active oscillator without time delay, is also investigated based on the first-order approximately analytical solution, and the result shows that the steady-state solution in this case is unconditionally stable. The comparisons of the analytical solution and the numerical one are fulfilled, and the results verify the correctness and satisfactory precision of the first-order approximately analytical solution. At last, the selection or design of an appropriate time delay to improve control performance through the first-order approximately analytical solution is studied.