It has been revealed in the proposed work that a pair of delay positive position feedback control can lessen the vibration response of double Van der Pol oscillators with external forces. We also studied the effects of both the control and the delayed feedback signal gains to illustrate the low vibration amplitudes. The averaging perturbation process has been used to consider the frequency-response equations of amplitudes and modulation phases at the primary resonance and one-to-one internal resonances. According to the perturbation solutions for the four-degrees-of-freedom system, we presented the frequency response curves that were periodic in the time delays. The stability analysis presented in this study has shown optimum stable ranges. If the time delays increase, the steady-state amplitudes of the oscillator’s system will periodically result in few stable regions and more unstable ones. The numerical simulation has been introduced to check the analytical approximation. It was also found to be almost identical after presenting the comparison of the results.

在提出的工作中已经揭示，一对延迟正位置反馈控制可以减轻双Van der Pol振荡器在外力作用下的振动响应。我们还研究了控制增益和延迟反馈信号增益的影响，以说明低振动幅度。平均摄动过程已被用于考虑在初级共振和一对一内部共振时的振幅和调制相位的频率响应方程。根据四自由度系统的摄动解，我们给出了时滞中周期性的频率响应曲线。这项研究中提出的稳定性分析显示了最佳的稳定范围。如果时间延迟增加，振荡器系统的稳态振幅会周期性地产生很少的稳定区域，而产生更不稳定的区域。引入了数值模拟以检查解析近似。提出结果的比较后，发现几乎相同。