Abstract
The response of a dynamical system of two-degree-of-freedom with parametrically excited pendulum is solved and studied. The delayed feedback control is applied to suppress or stabilize the vibration of the system. The case of 1:2 sub-harmonic resonances between pendulum and primary system is studied; the method of multiple scales is applied to obtain second-order approximations of the response of the system. It is shown that the delayed feedback control can be used to suppress the vibration or stabilize the system when the saturation control is invalid, the vibration of the system can be suppressed by the delayed feedback control. The effect of delay on the suppression is discussed; the vibration of the system can be suppressed at some values of the delay. Key words: Frequency response, delayed feedback control, multiple times scale, vibration suppression.  
Highlights
In the last years, many papers have been devoted to the control of resonantly forced systems in various engineering fields
It is shown that the delayed feedback control can be used to suppress the vibration or stabilize the system when the saturation control is invalid, the vibration of the system can be suppressed by the delayed feedback control
Forced nonlinear systems under delay control have been analyzed by Plaut and Hsieh (1987) in the case of nonlinear structural vibrations with a time delay in damping
Summary
Many papers have been devoted to the control of resonantly forced systems in various engineering fields. The delayed feedback control is applied to suppress or stabilize the vibration of the system. The case of 1:2 sub-harmonic resonances between pendulum and primary system is studied; the method of multiple scales is applied to obtain second-order approximations of the response of the system.
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