# Vibration of Rotating and Revolving Planet Rings with Discrete and Partially Distributed Stiffness
## 3 Vibration Analysis
### Influences of Parameters  
The influences of several parameters, such as the number of partially distributed stiffnesses (NPDS), area of partially distributed stiffnesses, and discrete and partially distributed stiffnesses, on the vibration properties of planet rings are investigated in the following. Different types of bearings with the same outer diameter may have different numbers of rollers or balls. Therefore, the influence of numbers of balls or rollers, that is, the number of partially distributed stiffnesses, is analyzed. Figure 6 shows the natural frequencies of a planet ring with discrete stiffnesses and changing the number of partially distributed stiffnesses. With the increase of the number of partially distributed stiffnesses, the natural frequencies increase, as it increases the stiffness of the whole ring partially. Moreover, from Figure 6, it can be observed that the lower frequencies are affected slightly, while the higher frequencies are increasing sharply with NPDS. Therefore, it could be a method for designers to avoid some natural frequencies by changing the number of partially distributed stiffnesses. Figure 4: Frequency veering with vibration mode exchange between (a) \(\omega=3.4\) and (b) \(\omega=3.6\). The dash and solid lines denote the original position and vibration modes, respectively. In applications of planetary gear systems, the loads on the gears may change at different working status. The load changing on the planet gear rings may change the contact area with the bearings, that is, the number of partially distributed stiffnesses. To analyze the influence, the number of rolling elements is set to 36, and the change range of the area is set as 90\({}^{\circ}\)-180\({}^{\circ}\). Figure 7 shows the natural frequency variation versus partially distributed area at a rotating speed of \(\omega=1\). Figure 7 reveals that the support area will influence the natural frequencies heavily. The natural frequencies are all increase with the increase of support area, and it especially has a great influence on the higher-order natural frequencies. This means, in heavy load conditions, the natural frequencies will get lower, and in light load conditions, the natural frequencies will get higher. Figure 5: Natural frequencies and vibration modes of a planet ring with discrete and partial distributed stiffnesses (\(N_{f}=20\)): (**a**) natural frequencies vs. rotation speed \(\omega\); (**b**) typical vibration modes of planet rings at \(\omega=1\) (dash line = original position; solid line = vibration mode). Figure 6: Natural frequencies of a planet ring with the different number of partially distributed stiffnesses at \(\omega=1\). Figure 8a shows the influence of partially distributed stiffnesses to natural frequencies. It can be seen that only higher-order frequencies are affected largely by partially distributed stiffnesses. Moreover, in the analysis range of partially distributed stiffnesses, there is no obvious frequency veering phenomenon in Figure 8a. To analyze the mode veering properties, a veering parameter, \(\xi_{mn}\), is chosen, as done in Reference [25], and can be calculated as follows. Equation (24) can be rewritten in the state space form, as follows:  
\[\lambda\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{x}=0 \tag{27}\]  
where  
\[\mathbf{x}=[\lambda\mathbf{x},\mathbf{x}]^{\mathrm{T}},\;\mathbf{A}=\left[ \begin{array}{cc}\mathbf{M}&0\\ 0&\mathbf{K}\end{array} \right],\mathbf{B}=\left[ \begin{array}{cc}(\Omega+ \omega)\mathbf{G}&\mathbf{K}\\ -\mathbf{K}&0\end{array} \right],\]  
then  
\[\xi_{mn}=\frac{\langle\mathbf{x}_{m},\lambda_{m}\mathbf{A}_{1}\mathbf{x}_{n}+ \mathbf{B}_{1}\mathbf{x}_{n}\rangle\langle\mathbf{x}_{n},\lambda_{n}\mathbf{ A}_{1}\mathbf{x}_{n}+\mathbf{B}_{1}\mathbf{x}_{m}\rangle}{\lambda_{m}-\lambda_{n}} \tag{28}\]  
where \(m\), \(n\) denotes the order of frequencies; \(<\bullet,\bullet>\) denotes the inner production; \(\mathbf{A}_{1}\) and \(\mathbf{B}_{1}\) are the first derivative of A and B, with respect parameters such as stiffness here.