A time-delayed absorber is utilized to suppress the vibration of a primary system excited by a simple harmonic force. The inherent and intentional time delays in the feedback control loop are taken into consideration. The value of the former is fixed, while the value of the latter is tunable in the controller. To begin with, the mechanical model of the system is established and the acceleration transfer functions of the system are derived. Consequently, the stability analysis of the coupled system is carried out. Finally, the experimental studies on the performance of the time-delayed absorber are conducted. Both experimental and theoretical results show that the time-delayed absorber with proper values of feedback gain coefficient and intentional time delay greatly suppresses the vibration of the primary system. The numerical results validate the correctness of the experimental and theoretical ones.

Time delay is inherent in the active control loop, which is derived from the signal acquisition and processing, filtering, the action of the actuator, etc [

Originally, the time delay was taken as a negative factor in the active control. It may result in the degradation of the control performance and the instability of the controlled system. Therefore, several methods were employed to compensate the adverse influence of time delay, such as phase-shift method [

A time-delayed absorber is a new technique in the field of active vibration control. The key idea of the time-delayed absorber is the introduction of an actuator controlled via time-delayed feedback control. In 1994, Olgac and Holm-Hansen [

In our previous work [

The present paper is organized as follows. The mechanical model and stability analysis are shown in Section

Figure

Mechanical model of the coupled system.

Assuming that simply harmonic excitation is applied to the primary mass, the governing equations of the coupled system are given by

Substituting equation (

The external excitation can be rewritten as

The solutions of equations (

Substituting equations (

To facilitate the following analysis, new variables

It is seen from equation (

It is known that the values of feedback gain coefficients and time delays determine the stability of the system when the physical parameters of the coupled system are fixed. Hence, it is necessary to analyze the stability of the system before the experiments are carried out.

In Laplace domain, equations (

The characteristic equation of the coupled system is det (

The coupled system is stable if and only if all characteristic roots of equation (

Substituting equation (

Using

Since equation (

Through the above computation, the stable and unstable ranges of

The photo of the experimental setup is shown in Figure

Photo of experiment setup: (a) front view; (b) right view (1: absorber mass, 2: primary mass, 3: servo motor, 4: controlled steel sheet, 5: base, and 6: shaker).

Figure

Step 1: the acceleration signals of the absorber mass (1) and the primary mass enter into signal conditioning instrument, in which the functions of signal amplification and low-pass filtering are achieved to improve the signal-to-noise ratio.

Step 2: the processed signals go into the voltage lifting device, where the voltage of the input signal is raised 5 volts.

Step 3: the raised voltage signal enters into Trio motion controller, where control commands are written. The values of feedback control feedback gains and intentional time delay can be adjusted in the control commands.

Step 4: the control commands are transferred into the servo controller, which guides the shaft rotation of the servo motor (3).

Step 5: driven by the rotation of the servo motor shaft, the lower end of the controlled steel sheet (4) realizes the horizontal reciprocating motion and applies the time-delayed feedback control force.

Schematic of time-delayed feedback control (1: absorber mass, 2: primary mass, 3: servo motor, 4: controlled steel sheet, 5: shaker, 6: force sensor, and 7 and 8: acceleration sensors).

As a preliminary, the values of the physical parameters of the 2-dof coupled system need to be identified when the time-delayed feedback control is absent (i.e.,

Physical parameters of the coupled system.

0.667 | 6.8 | 2431.72 | 13502.4 | 198.96 | 0.17 | 1 |

Figure

Comparison of the experimental and theoretical results of

In this subsection, the effects of

Figure

Stability charts of the coupled system. (a)

Figure

Figure

Table

Hereinafter,

To verify the experimental and theoretical results, numerical results are obtained by numerical integration of equations (

Measured time histories of excitation force and system accelerations when

Effect of

9.75 | — | 0 | 0 | 0.037 | 0 |

9.75 | 50 | −0.4 | −0.33 | 0.020 | −45.95 |

9.75 | 50 | −0.4 | −0.055 | 0.096 | 159.46 |

Numerical simulations of system responses for

The variations of

Figure

Table

Figure

Measured time histories of excitation force and system acceleration when

Influence of

10 | — | 0 | 0 | 0.166 | 0 |

10 | 30 | 0.25 | −0.455 | 0.035 | −78.92 |

10 | 30 | 0.25 | −0.065 | 0.126 | −24.10 |

Numerical simulations of the time histories of system responses for

The variations of

Time history of excitation force and system responses when

Table

Influence of

10.25 | — | 0 | 0 | 0.538 | 0 |

10.25 | 36 | 0.5 | 0.5 | 3.272 | 508.18 |

10.25 | 84 | 0.5 | 0.5 | 0.165 | −69.33 |

Figure

Numerical simulations of the time histories of system responses for

An active vibration suppression via a time-delayed absorber is presented. Case studies are provided to demonstrate the effects of feedback gain coefficient and intentional time delay on the vibration suppression performance of the time-delayed absorber. The following points are concluded:

From the viewpoint of vibration suppression, the time-delayed absorber has advantages and disadvantages over the passive one, which depend on the values of the feedback gain coefficient and intentional time delay. The time-delayed absorber with proper choices of the two parameters decreases the values of acceleration transfer function of the primary system by 45.95%, 78.92%, and 69.33% for

When the values of feedback gain coefficients are fixed, the value of intentional time delay determines the vibration suppression effect of the time-delayed absorber. It acts as a double-edged sword. Reasonable values of intentional time delay effectively improve the vibration suppression effect without changing the mass or stiffness of the absorber. However, unreasonable values of intentional time delay greatly intensify the vibration of the primary system. This situation should be avoided in practical engineering application.

One has

The data used to support the findings of this study are available from the corresponding author upon request.

This work was based on the manuscript presented in the 9th European Nonlinear Dynamics Conference.

The author declares that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China under Grant no. 11602135.