## 3 Results and Discussion
### Discussion
#### 3.2.1 Support without Rubber O-rings and Support with Rubber O-rings  
In the analysis model, the support without rubber O-rings only considers the action of the bearing, and ignores the rubber O-ring. So, the supporting stiffness coefficient of the rolling bearing is that of the support without rubber O-rings, and its damping effect is not considered.  
Figure 13: Amplitude-frequency characteristics at the measuring point \(C\) under speed-up time of 50 s.  
To understand the importance of support with rubber O-rings, especially in high-power and high-speed applications, the dynamic response of the winding rotor at point A was investigated under support without rubber O-rings and support with rubber O-rings, as shown in Figure 14.  
Research on the dynamic characteristics of the winding rotor with support without rubber O-rings and support with rubber O-rings through the simulation, the results are as follows:  
* The multiple rubber O-rings are installed in groups to form flexible support, and its equivalent overall supporting stiffness coefficient could reach the order of \(10^{7}\) N/m, which is equal in the order of magnitude with the equivalent stiffness coefficient of the rolling bearings, as shown in Figure 8a. When the winding rotor adopts support with rubber O-rings, its first-order critical speed point is near 986 rpm. When the winding rotor adopts support without rubber O-rings, its first-order critical speed point is near 1063 rpm. The main reason for this is high stiffness through the interference connection, which leads to an increase in the critical speed of the rotor. In addition, when the rotational speed of the winding rotor is much lower than the first-order critical speed, the winding rotor with support without rubber O-rings and support with rubber O-rings both have relatively small vibration amplitude, as shown in Figure 14;
* As the rotational speed increases, the residual unbalance force acting on the rotor, including centrifugal force, increases correspondingly, which leads to the heavy vibration of the winding rotor when the rotational speed passes through the critical  
Figure 14: Dynamic response of point A (node 51) of support without rubber O-rings and support with rubber O-rings under the speed-up time of 60 s. (**a**) Displacement amplitude curve; (**b**) velocity amplitude curve; (**c**) acceleration amplitude curve.  
speed range. Especially after passing through the second-order critical speed, the winding rotor with support without rubber O-rings has constant heavy vibration. Due to the heavy vibration after passing through the second-order critical speed point, the winding rotor cannot work anymore in the textile process;
* When the winding rotor rotates within 7730 rpm, according to two different supports, the dynamic characteristics of the winding rotor are divided into two different parts by the first-order critical speed point. For the winding rotor with the support without rubber O-rings, as the residual unbalance force acting on the rotor increases, the displacement amplitude, velocity and acceleration amplitude of point A gradually increase. For the winding rotor with the support with rubber O-rings, the vibration amplitude increases slowly and the acceleration amplitude was stable after the rotational speed passing through the second-order critical speed point of the winding rotor;
* By contrast, the winding rotor with support with rubber O-rings has a very attractive dynamic character to control the vibration of the rotor system when rotational speed passes through the critical speed range.