## 1 Introduction
Compared to other exciting manners (such as the axial vibration tool Agitator, the lateral vibration tool Xcite, and the torsional vibration system Slider), the technology of drill-string axial vibration at the surface proposed in this paper has the following advantages: Firstly, no drilling fluid energy is needed for vibrating the drill-string, which make it feasible to use and suited for a deep well. Secondly, exciting parameters are adjustable at any time according to the real-time drilling situation through simple surface tools. Thirdly, key seating and pressure differential sticking can also be overcome by applying a higher amplitude and lower frequency excitation. Scholars have developed many torque and drag calculation models in the last decades, which can be divided into soft-string models [10; 11; 12; 13; 14] and stiff-string models [15; 16; 17; 18]. However, these models can only be used for calculating the drag while tripping in or tripping out, and the torque while rotating drilling. When the drill-string is vibrated axially at the surface, the drill-string elements are frequently in the states of forward sliding, resting, and backward sliding, which makes the above models not applicable. Meanwhile, dynamic models [19] can be used to study the motion and force characteristics along the whole drill-string, and the load transfer efficiency can be obtained based on the deformation of the drill-string. However, these dynamic models usually focus on the rotating drilling mode, and neglect the axial friction or assume it as linear viscous damping. The study of friction reduction and load transfer in drill-strings under vibrational conditions is significantly lagging behind the advancements made in friction reduction technologies through the applications of vibrating drill-strings. Several scholars have devised models for determining the load transfer of a drill-string subjected to vibration [20; 21; 22; 23]. However, the friction model in these models is the Coulomb model that does not consider the viscous effect of friction. The research on the mechanism of friction reduction by applying vibration can provide a reference for choosing a friction model [24]. The friction reduction mechanisms of vibration mainly include: (a) the static friction transforms to dynamic friction due to axial and torsional vibrations [6]; (b) during the cycle of vibration, the direction of dynamic friction is altered by both axial and torsional vibrations, leading to a reduction in the mean friction force [25; 26]; and (c) the lateral vibration decreases the normal contact force periodically [27]. Based on these mechanisms, under vibration conditions, the dynamic friction models that take pre-slip displacement into account are better choices to simulate drill-string dynamics. In this paper, the load transfer characteristics of the drill-string while being vibrated axially at the surface is studied. Firstly, friction is incorporated based on the pre-slip displacement of the drill-string to accurately calculate the friction distribution of the whole drill-string during its axial vibration. The dynamic friction model is used to capture the change in coefficient and a basic model is derived according to the equilibrium of forces. The established numerical model is solved using a finite difference method. Subsequently, evaluations of the load transfer characteristics of the drill-string under both displacement and force excitations are conducted and compared with other types of vibration patterns. The main objective of this study is to demonstrate the viability and benefits of a new technique, which involves axial vibration at the surface to reduce friction. By comparing its load transfer properties with other types of vibration, we aim to establish a strong theoretical basis for the application of this technology in actual drilling.