## 1 Introduction
An analytical contact stiffness model under tangential loading was constructed by [PERSON] [4], and with this model, it had been revealed that the contact stiffness was proportional to the normal load and was independent of [PERSON]'s Modulus, which was in good agreement at low loads with numerical analysis. [PERSON] [5] proposed a normal contact stiffness fractal prediction model of dry-friction on a rough surface considering the friction factor, and with the work by [PERSON] [6] as the foundation, he treated the tangential contact stiffness as proportional to the normal one with a fixed coefficient of 0.25-0.35. All these works mainly focus on the deformation and the force along the normal and tangential directions but barely reveal the specific form of the relationship between the frictional coefficient, the normal and tangential force and the tangential stiffness. [PERSON] [7] investigated the interface character of Ti-6 Al-4V (an important alloy in the aerospace industry), and with the foundation of the interface test and theoretical contact model, he found the linear decrease in the tangential contact stiffness with further application of a tangential load, and this had significant importance for the prediction of the coefficient of friction in contact surfaces, while whether this rule was suitable for other materials is to be verified. Meanwhile, throughout other spheres, except manufacturing industry, tangential contact stiffness is also essential. [PERSON] [8] analyzed the differences between the initial structural stiffness Rayleigh damping model and the updated tangent stiffness-based Rayleigh damping model in seismic motion. [PERSON] [9] revealed the tangential stiffness matrix in analytical manipulations and computations to investigate the stiffness and the stability properties of the prestress-able structures. [PERSON] [10] studied the characteristic of the distributed stiffness degradation of fabric-cement composites. [PERSON] [11] examined the effect on the track stiffness when the ballast is reinforced with a cross-linked urethane polymer. [PERSON] [12] contended that at least one negative eigenvalue in the second-order tangent stiffness plays a key role in static instability during the analysis by severe seismic excitation. [PERSON] [13] measured the tangential stiffness of each specimen during the stiffness testing of laser stake-welded T-joints in web-core sandwich structures. To study the fresh concrete with different grades and motion states, [PERSON] [14] proposed a new dynamic coupled discrete-element contact model, and the calculation of tangential stiffness in this model was indispensable. [PERSON] [15] studied the friction coefficient between the wheel and rail, and the calculation of contact stiffness was indispensable considering the relationship between the contact stiffness and the friction coefficient. [PERSON] [16] constructed a tangential stiffness model of the plate-springed parallel elastic actuator in a snake robot to help its design for energy efficiency improvement. To better the design of a robotic gripper, [PERSON] [17] discussed the role of stiffness among the contact area (contact stiffness), gripper deflection, contact pressure and load capacity, which was based on an analytical model constructed with elastic beam theory, the JKR adhesive model and the Hertz contact theory. [PERSON] [18] proposed an adaptive fuzzy global coupled nonsingular fast terminal sliding mode control (NFTSMC) method for the n-rigid-link elastic-joint robot, and the elastic-joint robot manipulator was modeled with joint interactions, which could be classified as a stiffness problem. While for the manufacturing industry, the scale of tangential stiffness research becomes relatively small.