## 1 Introduction
In the literature [8], the authors considered that the contact area and the micro-convex had scale correlation, which solved the contradiction between the elastic deformation and the plastic deformation transformation of the MB model so that it conformed to the prediction of classical contact mechanics. In the literature [9,10], the tangential contact properties between the two rough planes and between the rough plane and the sphere are studied, respectively. It is found that the normal load, material properties, surface morphology, tangential load, and friction coefficient all have influences on the tangential contact stiffness. [PERSON]. [11] studied the contact properties of chamfered cuboids under tangential force in the condition of elastic contact conditions. In the literature [12, 13, 14], the authors studied the contact properties of simple geometrical bodies such as spheres and ellipsoids and extended them with the GW model to obtain the displacement expression under the tangential load. It had become a widely used method for studying the tangential contact performance of rough surfaces. [PERSON]. [15] investigated the effects of surface morphology, elastic layer thickness, and matrix material on the maximum Von [PERSON] stress and shear stress distribution by sliding contact analysis between the smooth sphere and a flat surface with a rough surface contact layer. In the literature [16, 17, 18, 19, 20, 21], the finite element method is used to study the contact problem between rough surfaces. The above reports were the research of contact interface, which is essential to the study of the RFR. What follows are some studies directly on the dynamic characteristics of the RFR. [PERSON]. [3] analyzed comprehensively the structural characteristics of the combined rotor, and proposed a mechanical model of RFR which is proved by experiment and calculation. [PERSON] et al. [22] investigated the vibration characteristics of a gas turbine rotor, which considered the contact interface and preload. In their study, an improved 2-D finite element method (FEM) considered the contact interface was presented, which is useful to evaluate the critical speed of the rotor. [PERSON] et al. [23] provided a calculation method of contact stiffness by modal test and FEM analysis. [PERSON] et al. [24] advanced a FEM dynamic analysis model of the RFR, which is proved to be effective. [PERSON] et al. [25] analyzed the vibration characteristic of the RFR by the FEM method. [PERSON] et al. [26] investigated the dynamic characteristics of the RFR based on elastic-plastic contact. [PERSON] et al. [27] studied the influence of the moment and preload on the bending stiffness of contact interfaces. [PERSON] et al. [28] advanced a calculation method of vibration characteristics of the rotor with initial bending, and it has been proved correct. [PERSON] et al. [PERSON] et al. (2018) investigated the influence of the thermal effects on the tensile forces of rods in the combined rotor of heavy-duty gas turbine. [PERSON] et al. [PERSON] et al. (2018) studied the effect of detuning of clamping force of tie rods on dynamic performance of the combined rotor. [PERSON] and [PERSON] (2018); [PERSON] and [PERSON] (2018), [PERSON] (2018), [PERSON] and [PERSON] (2018) investigated the non-linear dynamic characteristics of a circumferential RFR. The contact interface and tie rod are the most obvious features in the rod-fastened rotor. These reports mostly focus on the dynamic model of RFR and the calculation of contact stiffness of the rough contact interfaces. However, the characteristics of tie rods are also essential. It will also influence the dynamic performance of the rotor. Thus the dynamic analysis of RFR considering the characteristics of circumferential tie rods needs to be studied. In this study, for a RFR system, the vibration equation of RFR is deduced in detail using the lumped parameter method. The equation fully considers all factors brought by the tie rods.