## 1 Introduction  
The elastic beams have wide application in many engineering fields. Therefore, it is important to investigate the vibration problem of the elastic beams. As a very important topic in structural dynamics, the dynamics problem of the elastic beam is of practical importance in civil engineering [1, 2, 3]. At the same time, the vibration control problem of flexible structures has also received extensive attention. Different vibration control strategies are used to study vortex-induced vibrations of a bridge deck [4, 5, 6], building structure [7, 8, 9, 10, 11, 12], and other structures [13, 14, 15]. It is worth mentioning that scholars have carried out much research on piezoelectric-based vibration control [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. As a control strategy, the time-delayed feedback control technology, along with the rapid development of control theory [27, 28, 29], sensor testing technology, and computer technology, has attracted widespread attention and practical application in the field of aerospace engineering [30], vehicles engineering [31], mechanical engineering [32], civil engineering [33], etc. Delay feedback control can improve the stability of the controlled system. Based on the time-delayed displacement feedback, the time-delayed velocity feedback and the time-delayed acceleration feedback control strategy, the vibration absorber has excellent effectiveness to suppress the vibration of the system. In the past few years, the time-delayed feedback control technology has received much attention. An optimal control method for seismic-excited building structures with multiple time delays is investigated [34]. The time delay may be used to improve the system stability [35; 36]. The delayed position-feedback technique is used to reduce the payload pendulations [37]. [PERSON] et al. [38] presented a comprehensive investigation of the effect of feedback delays on the non-linear vibrations of a piezoelectrically-actuated cantilever beam. [PERSON] and [PERSON] [39] studied the time delay control and presented that it can achieve good control performance of a dynamic beam structure system. [PERSON] and [PERSON] [40] investigated the bifurcations due to time delay in the feedback control system with excitation. [PERSON], [PERSON], and [PERSON] [41] studied the existence of a subcritical Hopf bifurcation in the delay-differential equation model of the so-called regenerative machine tool vibration. [PERSON] et al. [42] investigated the stability and bifurcation of an SDOFSystem with time-delayed feedback. [PERSON] et al. [43] investigated the nonlinear dynamics of a Duffing-van der Pol oscillator under linear-plus-nonlinear state feedback control with a time delay. [PERSON] and [PERSON] [29] conceived of a concept study that explores new directions to enhance the performance of such energy-harvesting devices from base excitation. [PERSON] and [PERSON] [44] investigated the stabilization of traffic flow in an optimal velocity model via delayed-feedback control. [PERSON] and [PERSON] [45; 46] investigated nonlinear vibration suppression of flexible structures using the nonlinear modified positive position feedback approach. [PERSON] et al. [47] investigated the positive position feedback (PPF) controller for suppression of nonlinear system vibration. [PERSON] et al. [48] selected control algorithms for active suppression of nonlinear composite beam vibrations. [PERSON] et al. [49; 50] investigated energy harvesting in a Mathieu-van der Pol-Duffing MEMS device using time delay and quasi-periodic vibrations in a delayed van der Pol oscillator with time-periodic delay amplitude. [PERSON] et al. [51; 52; 53] presented modeling and tuning for a time-delayed vibration absorber with friction and investigated sub-harmonic resonances and periodic and chaotic motion of a time-delayed nonlinear system. These results show that the new control method (time-delayed feedback control) makes the system more stable and improves the control performance.