## 1 Introduction  
In the manufacturing process of aerospace blades, the workpiece fixed with the fixture and the damper, including the machine tool, composes a complex mechanical engineering system, with an assembly of its subunits transferring the motion and forces to their neighboring ones via the frictional interfaces. To better understand the dynamic behavior of the fixture-workpiece system during the manufacturing of the aerospace blade, a system dynamic model, or this joined-subunits dynamic model, is necessary, which is also the foundation of the vibration depression for meloirating the manufacturing quality of aerospace blades. In order to construct this model, to describe the motion and forces transferring phenomenon, the effect of the joined interface has to be considered, which makes the contact stiffness along normal and tangential direction, the factors belonging to the joined interface, be a first key step to establish the system dynamic model. Given all that, to get a more accurate dynamic model of the fixture-workpiece system for the vibration reduction of aerospace blade manufacturing, the study of the contact stiffness between the system's units is inevitable. The analysis of the contact stiffness has been attracting the attention of researchers for a long time, for this is not only a key factor of dynamic analysis but also has a great influence on the wear, heat transfer and the electric conductance among interfaces, with relatively complex slip process between two solid elastic surfaces, which is also a task for the stiffness model's construction. To be more specific on a theoretical level, during the collision between two elastic solid surfaces, there is a small region of deformation that surrounds the initial contact point [1]. In this contact region, there is not only the normal motion but also the tangential one named slip. In many cases, the initial slip is treated as a large one, which makes the tangential compliance negligible. While this is obviously not sufficient to study those tiny slips between two rough surfaces and, in the contact of the workpiece and the fixture, this kind of slip occupies in too many situations to be ignored. What is more, if the contact region's tangential compliance is non-negligible, there is almost no frictional dissipation if the angle of incidence during the collision (specific contact process happened in many cases) is small [1]. In other words, to dissipate the energy from the impact process when the incidence angle is not small enough, it is inevitable to study the tangential compliance's behavior, because, in most cases, the impact of the manufacturing processes system units is not central collision. And the tangential stiffness is the reciprocal of tangential compliance, and that takes equal significance and complexity of tangential stiffness modeling. In summary, it is of great significance to analyze the tangential contact stiffness in a machining system by considering the friction's squeezing and slipping interactions because this is not only a theoretical modeling task but also a key step in establishing the system's dynamic model with the effect of a joined interface, which is also an inevitable procedure of vibration suppression for aerospace blade manufacturing. In every walk of life, tangential contact stiffness has a wide range of use and even plays an irreplaceable role in some research areas. Focusing on theoretical contact mechanics itself, [PERSON] [2] made a systematic derivation about contact tangential strain and stress based on [PERSON]'s contact theory, but his main focus was on the force and the deformation itself, and its results need the geometry of the contact area, which was hard to measure. [PERSON] [3] derived a normal and tangential contact stiffness model from analyzing the characteristics between rough surfaces of machined joints considering fully elastic, elastic-plastic and fully plastic contact deformations.