Analytical method of the modal damping ratio of the beam with distributed dissipative oscillators and application in broadband vibration mitigation

•The analytical formulas for calculating the modal damping of the beam distributed with dissipative oscillators.•The modal damping formulas in case of single-frequency or dual-frequency oscillators.•Application of modal damping formula in Minimax optimization for broadband vibration reduction.•Formula used for multi-modal vibration reduction design in a wide frequency band.•Frequency shift based on fixed point and bandgap.