## 1 Introduction  
Beams are important structural components used in many practical engineering structures, such as gantry crane, etc. Under the dynamic loads of vehicles, wind, and earthquakes, the dynamic behaviors become more complicated [1; 2; 3]. As a very important topic in structural dynamics, the moving load problem of the beam is of practical importance in engineering. [PERSON] et al. [4] proposed a simple and compact formula to determine the free vibration responses of a uniform beam, which was applied to both lightly and heavily damped beams. [PERSON] and [PERSON] [5] described a procedure to estimate the nonlinear ultimate load-carrying capacity of continuous beams by use of a combination of linear elastic influence lines for segments of the beam in appropriately selected damaged configurations. Up to now, many studies have reported investigating vibration control of structures [6]. [PERSON] et al. [7] studied the PTMD control on a benchmark tv tower under earthquake and wind load excitations. [PERSON] et al. [8] investigated the exact H2 optimal solutions to inerter-based isolation systems for building structures. Besides, as a kind of active control, the time-delay feedback control technology has attracted wide attention because of its easy adjustment, wide application range, and high control efficiency. The robust control and the time delay control that could achieve good control performance of a dynamic beam structure system were presented [9; 10]. [PERSON] and [PERSON] [11] studied the primary, superharmonic, and subharmonic resonances of a harmonically excited non-linear s.d.o.f. system with two distinct time-delays in the linear state feedback. [PERSON] et al. [12] studied feedback control with time-delay, which was employed to minimize the maximum deflection of a beam subjected to impulsive and/or short-time loads. [PERSON] and [PERSON] [13] investigated the dynamics of a parametrically excited quarter-car model with time delayed feedback position and linear velocity terms. [PERSON] et al. [14] studied the dynamics of a system of two Van der Pol oscillators with delayed position and velocity coupling by the method of averaging together with truncation of Taylor expansions. [PERSON] et al. [15] presented analytical and numerical studies of the primary resonance and the \(1/3\) subharmonic resonance of a harmonically forced Duffing oscillator under state feedback control with a time delay. [PERSON] and [PERSON] [16] studied the stabilization of vibration systems via delayed state difference feedback. [PERSON] [17] investigated the primary resonance of a cantilever beam under state feedback control with a time delay. [PERSON] and [PERSON] [18] applied the delayed feedback control to suppress the vibration of vertical displacement in a two-degree-of-freedom nonlinear system with external excitation. [PERSON] et al. [19] investigated the non-linear vibrations of parametrically excited cantilever beams subjected to non-linear delayed-feedback control. More specifically, three non-linear cubic delayed-feedback control methodologies: position, velocity, and acceleration delayed feedback were examined. [PERSON] et al. [20; 21; 22] studied the time delayed feedback control for beam structures. [PERSON] et al. [23] investigated the local dynamics of an axially moving string under aerodynamic forces with a time-delayed velocity feedback controller. In all the studies cited above, the linear and nonlinear dynamics were investigated. The primary resonance, super/subharmonic resonance and multi-frequency excitation resonance responses are all involved. These resonance forms are particularly important in flexible structures [24; 25]. So the primary and \(1/3\) subharmonic resonance responses of a dynamics beam with moving load and time-delayed feedback is investigated in this study. Moreover, the vibration control equations we derived belong to duffing equations [26], which are delay differential equations with cubic nonlinearity.