# Dynamic Characteristics Analysis of a Rod Fastening Rotor System Considering Contact Roughness
## 2 Dynamics Modeling of Rod Fastening Rotor  
The rotor system considering the disc contact roughness of rods is established in this section. As shown in Figure 1, the global coordinate system is _o-xyz_. Through several rods, discs 1 and 2 are connected, which are installed in the middle of the flexible shaft. Since the mass of the flexible shaft is much smaller than that of the two discs, it is ignored in the modeling process of the system. Meanwhile, the boundary conditions of the flexible shaft are the fixed supports.  
Assuming that the effects reduced by the contact roughness of the joint's interface are temporarily put aside, the two discs are integrated into a whole. Under this circumstance, the rod fastening rotor system shown in Figure 1 can be seen as a classical Jeffcott rotor system, as shown in Figure 2.  
Figure 1: A rod fastening rotor with disc contact.  
When disc 3 is installed in the middle of the shaft, the bending stiffness of the left and right parts of the massless shaft can be obtained according to the compliance coefficient method, namely  
\[k_{1}=k_{2}=\frac{48E_{rs}I}{L^{3}} \tag{1}\]  
where \(E_{rs}\) denotes the elastic modulus of the massless shaft, \(I\) denotes the second moment of area of the cross-section of the shaft and \(L\) denotes the length of the shaft. It is noted that the rotor system is subjected to centrifugal action, and the whirling motion happens synchronously. If the integrated disc is installed in the middle of the shaft, the normal of the disc is parallel to axis \(oz\). This means that the gyroscopic effect of the disc does not exist in the rotor system.  
According to [PERSON]'s second law, the governing equation of motion of the dynamic model in the coordinate plane of \(xoz\) can be obtained as follows:  
\[(m_{1}+m_{2})\ddot{x}+(k_{1}+k_{2})x=(m_{1}+m_{2})e\omega^{2}\cos\omega t \tag{2}\]  
where \(m_{1}\) and \(m_{2}\) denote the mass of discs 1 and 2, respectively. \(e\) denotes the eccentricity of the integrated disc, and \(\omega\) denotes the rotational speed of the system.  
The above expression suggests that according to the theory of linear vibration, when discs 1 and 2 are rigidly connected, the natural frequency of the Jeffcott rotor system (see Figure 2) is mainly determined by the integrated disc mass and the bending stiffness of the massless flexible shaft.  
For the rod fastening rotor system, the contact roughness of the joint's interface is a crucial factor that directly affects the dynamic characteristics of the system. Therefore, a schematic diagram of an equivalent rotor system is depicted in Figure 3, in which the equivalent contact spring is used to describe the contact mechanism of the joint's interface. Therefore, the governing equation of motion of the equivalent rotor system in the coordinate plane of \(xoz\) can be expressed as  
\[\begin{split}&\left(\begin{array}{cc}m_{1}&0\\ 0&m_{2}\end{array}\right)\left(\begin{array}{c}\ddot{x}_{1}\\ \ddot{x}_{2}\end{array}\right)+\left(\begin{array}{cc}k_{1}+k_{2}&-k_{3}\\ -k_{3}&k_{2}+k_{3}\end{array}\right)\left(\begin{array}{c}x_{1}\\ x_{2}\end{array}\right)\\ &=\left(\begin{array}{c}m_{1}e_{1}\omega^{2}\cos\omega t\\ m_{2}e_{2}\omega^{2}\cos\omega t\end{array}\right)\end{split} \tag{3}\]  
where \(k_{3}\) denotes the equivalent bending stiffness of the joint's interface. \(e_{1}\) and \(e_{2}\) denotes the eccentricities of discs 1 and 2.  
This illustrates that the first and second natural frequencies of the rotor system are closely related to the mass of the two discs, the bending stiffness of the massless flexible shaft and also the bending stiffness of the equivalent contact spring. Therefore, in order to predict the dynamic characteristics of the rod fastening rotor system more accurately, the contact mechanism of the joint's interface should be investigated in advance.  
Figure 2: Schematic diagram of an integrated rotor system.