# A Fractal Prediction Model for the Friction Coefficient of Wet Clutch Friction Plates
## 1 Introduction  
With the development of technology and the rapid advancement in the field of helicopters, there is an increasing demand for variable-speed helicopter transmission systems. The confirmation of the friction factor is crucial for predicting the engagement state of clutches. The M-B fractal contact model is an effective model considering the morphology characteristics of the contact surface at the microscopic level [1]. In the model, the contact is regarded as the contact between a rough surface and a smooth plane. It is assumed that the rough surface is defined by the W-M function, and the deformation state of the micro-convex body is related to its contact area. [PERSON] [2] proposed a size-independent M-B contact model for his research on the \"island area distribution theory\". The contact model based on the M-B fractal model can objectively reflect the properties of the friction plate's engagement state to some extent. Therefore, establishing a friction factor calculation model based on the M-B fractal model allows for the development of a calculation method for the dynamic rough surface engagement state, considering the rough surface condition. Fractal theory was first proposed by [PERSON]. The concept of fractal theory was introduced from the problem of coastline measurement: the length of coastline is not a fixed value but has a self-similarity that changes with the change of measurement scale. When a curve has self-similarity, the dimension used to describe its characteristics is called fractal dimension [3]. [PERSON] [4] correctly wrote that fractals are irregular and complex shapes that, however, have important properties in the form of self-similarity to each other and scale invariance. Therefore, rough surfaces can be described by multifractals. [PERSON] [5] studied the coefficient of friction of clutch friction discs under various material formulations, considering the effects of temperature and sliding velocity, using Taguchi methods and pin-on-disk experiments for testing. [PERSON] [6] integrated a friction model suitable for wet friction clutches into a four-degree-of-freedom lumped mass-spring-damper system representing a typical SAE#2 test rig. A degradation model expressing the evolution of friction model parameters was proposed to simulate the engagement dynamics of the clutch system during its operational period over its lifetime. [PERSON] [7] adopted a hybrid friction model, based on the asperity model and [PERSON]'s viscosity law, to establish a friction coefficient model for wet clutches based on the variation of engagement rotational speed, solving for the prediction of friction coefficients under different sliding velocities and different load pressures. This can be used to predict the boundary friction of wet friction elements. [PERSON] [8] proposed a model that combines macroscopic and microscopic phenomena; the macroscopic model obtains the distribution of fluid pressure and fiber contact pressure over the entire engagement surface, while the microscopic model obtains the friction coefficient for each fiber contact through a detailed model of single asperity fiber contact. [PERSON] [9] proposed a finite element analysis method to simulate the engaging phase of a wet clutch to capture the pressure overshoot in the engaging hydrodynamic state and combined the Reynolds equations with the force and torque balance equations to simulate the engaging of a two-piece clutch system. Based on the dynamic lubrication model of oil film and the elastic contact model of a convex and convex body, [PERSON] [10] established the kinematic coupling model of the rotating axial engagement process of friction elements. Considering the viscous torque and contact torque, the influence of piston pressure and the number of friction pairs on the speed and transmission torque during clutch engagement was analyzed. Finally, the effect of engaging pressure on clutch torque and engaging time was verified.