The present work discusses the strongly nonlinear dynamical characteristics of the secondary resonance in a cantilever beam system with concentrated mass regulated through displacement and velocity time delay. Through the homotopy analysis method, the potential superharmonic and subharmonic resonance issues are resolved. A detailed analysis is conducted to investigate the influences of the time delay, feedback gain coefficient, excitation amplitude, and damping coefficient on the nonlinear response. The results show that the system exhibits hardening behavior for superharmonic resonance with one or two peaks and softening behavior for subharmonic resonance. In time delay control systems, increasing the damping coefficient occasionally causes an increase in amplitude. Due to the complexity of the system, it is challenging to definitively determine whether displacement or velocity time delay control is more suitable. Only when specific working conditions are given can it be compared which way is better. Prudently choosing the velocity and displacement time delay factors can restrain the amplitude of the system, adjust the response region, jump frequency and the number of solutions. The findings of this paper are thought to be useful in the design of time delay feedback controllers.