## 1 Introduction  
In modern aviation aeroengines, fatigue fracture of the blades caused by vibration is one of the key problems that affect engine reliability. Dry friction damping structures have been widely applied in aeroengines due to their obvious vibration reduction effect, simple structure, and high reliability. Common dry friction damping structures consist of intermediate shrouded blades and under-platform dampers (UPDs) [1]. Generally speaking, UPDs are applied to the vibration reduction design of fan blades and turbine blades because of their advantages of a small additional centrifugal load and no adverse effect on the aerodynamic performance. Examples include the fan blades of the CFM56-3 and Taihang engines, and the high-pressure turbine blades of the JF9D, CF6, RB211, BR715, and AJI-31 engines. Studying the contact mechanical properties of asperities on rough surfaces can not only play a necessary guiding role in the vibration reduction characteristics of aeroengine blades with an under-platform damper structure, but it can also help improve the service life of aeroengines and enhance the reliability and safety of aeroengines during operation. Great efforts have been made in the academic community to accurately describe the mechanical vibration characteristics of the real contact surface of a blade with an under-platform damper. For the contact surfaces of different under-platform dampers, the initial models used were finite element models [2,3] or half-space models [4,5], both of which have their advantages and disadvantages. The advantage of the spatial model is that the calculation efficiency is greatly improved, but the description of the elastic-plastic deformation is inaccurate. Therefore, a characterization method of complex contact surface roughness on account of the FGT is established in this paper. The primary task for studying under-platform dampers is to quantitatively characterize the mechanical properties of the contact surface. Most early studies have used the macro-slip model, which assumes that the deformation and pressure on the contact surface are evenly distributed and that the contact surfaces enter the stick or slip state simultaneously. The representative work consists of the coupled double contact surface model proposed by [PERSON] [6], the small degree of freedom theoretical model established by [PERSON] [7], and the B-G simplified model adopted by [PERSON]. It is worth noting that the fretting sliding model can better describe the local slip phenomenon between the contact surfaces and predict the mechanical properties of the contact surface more accurately [8]. For example, [PERSON] et al. used multiple contact pairs in parallel form to establish a micro-slip model, and [PERSON] [9] performed a micro-slip model comparative analysis of the damping effect of a wedge-shaped under-platform damper and a cylindrical under-platform damper. Additionally, [PERSON] [10] analyzed the vibration reduction effect of different damping structures using a self-developed dry friction nonlinear vibration response solving program. In the early studies of dry friction structures, the contact interface was usually described by the macroscopic slip model. The smooth and flat surface of the part at the macroscopic angle shows different degrees of roughness and unevenness under the microscope, making the real area of contact during the vibration process smaller than the theoretical area [11]. With this model, a large load is distributed on the small contact area of the UPD, which aggravates its wear failure. First, the deformation of the interface and the uniform arrangement of the interface pressure are assumed. Then, the friction characteristics on the contact surface are described by the single-point Coulomb friction model. The advantages of this model include simplicity, intuition, and less computation, making the macroscopic slip model widely applied in early studies.