## 1 Introduction  
Multi-rope friction hoists are widely used in deep and ultra-deep coal mine hoisting for their remarkable advantages, such as their large capacity, high lifting height, high safety factor, low power consumption, small machine size and cost-effectiveness. The friction hoist (Figure 1) relies on the friction force between the rope and the pulley to transport goods or staff. With the development of modern large-scale friction hoisting, the problems of dynamic characteristics, which are not obvious in low-speed and light-load conditions, are gradually reflected, for example, in rope vibration and slippage [1, 2]. During dynamic operations, especially in severe conditions such as acceleration and emergency braking, the long-distance suspended rope will produce large elastic vibrations and tension impacts under the influence of the terminal heavy container [3, 4, 5]. The vibration impact produces not only longitudinal tension, but also horizontal and torsional vibration due to the transmission of the shock wave [6, 7]. Especially when the depth and speed of the mine are continuously increasing, the time-varying characteristics of the rope length and the swirl of the wellbore gas flow [8] at the bottom of the container caused by the rapid speed change will intensify the vibration [3, 9]. When the vibration is transmitted upward to the friction pulley along the rope, it leads to abnormal contact, dynamic slip and even disengagement [10] between the rope and pulley, which will seriously reduce the friction stability. Therefore, it is of great importance to combine the suspended ropes on both sides of the pulley with the wrapped rope on the pulley in order to accurately obtain their dynamic behaviors. Especially for the rope on the pulley, its effective friction contact with the pulley is an important guarantee for the working efficiency and anti-skid safety of the hoist. Many scholars have conducted abundant research on the dynamics of suspension rope hoisting systems. [PERSON] [11, 12] established the classical distributed parameter dynamic model of the kilometre hoisting rope based on [PERSON]'s principle and described the transverse-longitudinal coupling dynamic response for the first time. On this basis, [PERSON] [13] calculated the dynamic responses of a one-dimensional system with a spatial discretization method. The method was applied to study the longitudinal, transverse, and coupled vibrations in moving elevator cable-car systems [14]. Meanwhile, a rope vibration parameter analysis based on [PERSON]'s principle has also been widely conducted [15; 16; 17]. The above studies are mainly focused on the dynamic characteristics of the rope itself. However, in the present numerical analysis method based on [PERSON]'s principle, the initial mechanical boundary conditions at the point where the rope meets or separates from the pulley are either set to zero or applied with a constant-amplitude sinusoidal excitation [3; 18], which causes the \"mechanical link\" between the suspended rope and the wrapped rope to be truncated. Therefore, these numerical methods do not establish the relationship between the suspended rope dynamics and the friction contact in the wrap angle, and are thus unable of determining the rope vibration entering the wrap angle during operation. Currently, either the classical Euler formula or the deformation formula are used in the design or calculation of friction transmission. The theoretical calculation theory of friction transmission of modern large-scale equipment seems inaccurate to some extent. At the beginning, the formula was used to calculate the belt transmission, while the calculation method of wire rope friction transmission is a direct reference to the Euler formula. It is considered that the contact process between the belt and pulley can be divided into two stages. First, the viscous stage is when the belt and pulley are relatively static, which is also called the static angle.