# Dynamic and Stability Analysis of Multibolt Plane Joints under Normal Forces
## 3 Dynamic Model of the Bolt Joint Interface  
In this section, a model of the spindle box in contact with the machine bed is built to analyze the dynamic characteristic of the fixed joint parts. For simplicity, contact objects are considered to be a series of asperities according to the equivalent surface on base B, two elastic smooth bases A (with a rigid bottom plane), B connected by bolts and two rigid bodies A, B, which is performed in Figure 2a. It has been assumed that pressures on bases are evenly distributed. The stiffness of serial bulks is defined as \(K_{buss}=K_{1}+K_{2}\), in which \(K_{1}=E_{1}A_{a}/l_{1}\) and \(K_{2}=E_{2}A_{a}/l_{2}\), where \(l_{1}\) and \(l_{2}\) are the thickness of bulks respectively. Deformation occurs at those asperities leading to a contact stiffness \(K_{s}\). The total stiffness of bolts \(K_{bolts}=mk_{bolt}\) is determined by the type and number \(m\). As demonstrated in Figure 2a, \(K_{s}\) and \(K_{buss}\) are in series, and in parallel with \(K_{bolts}\). On the basis of the structure, a single-degree-of-freedom model of the system is established in Figure 2b. The parameter \(c\) is the damping coefficient of the system. The distance variation \(\Delta x\) between the most remote two surfaces belong to bases and the variation \(\Delta x_{s}\) are caused by a harmonic excitation \(F\)sin(\(\omega t\)) in this model. Initial distances, \(x_{pre}=x+\Delta x\) and \(x_{spre}=x_{s}+\Delta x_{s}\), corresponding to these two variables are both due to the mass of the spindle box \(M\) and the pretension force of bolts \(F_{pre}=mf_{pre}\), where the subscript pre denotes the system is static equilibrium and preloaded by bolts. In order to obtain the force between the interfaceundervibration condition, aphenomenon is introducedthat even though the large plastic deformation occurs during the load process, the unloading process is completely elastic [26]. Therefore, the stressbetweencontact surfaces of a mechanical joint under repeated loading can reconsidered as an elastic force only, which is acquired by Equation (7). The kinetic equation is given by  
\[\left\{ \begin{array}{ll}M\Delta\ddot{x}+c\Delta\dot{x}+K_{bolts}\Delta x+P_{ e}\Big{(}x_{spre}-\Delta x_{s}\Big{)}=Mg+F_{pre}+F\sin(\omega t),&\Delta x\leq \Delta x_{spre}\\ M\Delta\ddot{x}+c\Delta\dot{x}+P_{e}\Big{(}x_{spre}-\Delta x_{s}\Big{)}=Mg+F\sin (\omega t),&\Delta x>\Delta x_{spre}\end{array} \right. \tag{13}\]  
in which \(x_{spre}\) satisfied the function \(P_{e}\Big{(}x_{spre}\Big{)}=Mg+F_{pre}\), and \(\Delta x_{spre}=f_{pre}/k_{bolt}\) is the elongation of preloaded bolts. Moreover, \(P_{e}\Big{(}x_{spre}-\Delta x_{s}\Big{)}-P_{e}\Big{(}x_{spre}\Big{)}=K_{buss} (\Delta x-\Delta x_{s})\) is known according to Figure 2b to deduce the function \(\Delta x=f(\Delta x_{s})=\Big{[}P_{e}\Big{(}x_{spre}-\Delta x_{s}\Big{)}-P_{e }\Big{(}x_{spre}\Big{)}\Big{]}/K_{buss}+\Delta x_{s}\). Substituting this relationship into Equation (13) yields  
\[\left\{ \begin{array}{ll}M\Delta\ddot{x}+c\Delta\dot{x}+(K_{buss}+K_{bolts}) \Delta x-K_{buss}f^{-1}(\Delta x)=F\sin(\omega t),&\Delta x\leq\Delta x_{spre} \\ M\Delta\ddot{x}+c\Delta\dot{x}+K_{buss}\Delta x-K_{buss}f^{-1}(\Delta x)=F\sin (\omega t)-F_{pre},&\Delta x>\Delta x_{spre}\end{array} \right.. \tag{14}\]  
Figure 2: Simplified model of the spindle box in contact with the machine bed: (**a**) the schematic diagram of the system, (**b**) the single-degree-of-freedom model of the system.