Chaotic Dynamics of Cage Behavior in a High-Speed Cylindrical Roller Bearing

Hindawi Publishing Corporation Shock and Vibration Volume 2016, Article ID 9120505, 12 pages http://dx.doi.org/10.1155/2016/9120505 Research Article Chaotic Dynamics of Cage Behavior in a High-Speed Cylindrical Roller Bearing Long Chen, Xintao Xia, Haotian Zheng, and Ming Qiu School of Mechatronics Engineering, Henan University of Science and Technology, Luoyang 471039, China Correspondence should be addressed to Long Chen; Received 7 August 2015; Revised 14 December 2015; Accepted 15 December 2015 Academic Editor: Jussi Sopanen Copyright © 2016 Long Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper presents a mathematical model to investigate the nonlinear dynamic behavior of cage in high-speed cylindrical bearing. Variations of cage behavior due to varying cage eccentricity and cage guidance gap are observed. Hydrodynamic behavior in cage contacts is taken into consideration for a more realistic calculation of acting forces owing to high working speed. Analysis of realtime cage dynamic behavior on radial plane is carried out using chaos theory based on the theoretical and mathematical model established in the paper. The analytical results of this paper provide a solid foundation for designing and manufacturing of highspeed cylindrical roller bearing. 1. Introductions Radial cylindrical roller bearings are designed to carry radial loads and be applied under high-speed conditions. Usually, cylindrical roller bearings may be obtained as a unit, which includes two steel rings each of which having a hardened raceway on which hardened cylindrical rollers roll. The rollers are usually held in an angularly spaced relationship by a cage. The cage is made from machined brass or pressed steel. Brass cage is widely used in high-speed application. There are normally two methods to distinguish the type of cylindrical roller bearings. One is classified by arrangement of the ribs. Depending on the type of bearing, either the inner or the outer ring has two roller guiding ribs. The other is classified by the types of cage guidance. There are three types of cage guidance, as briefly demonstrated in Figure 1. They are outer-ring-rib guidance (Figure 1(a)), inner-ringrib guidance (Figure 1(b)), and roller guidance (Figure 1(c)), respectively. The weight of the cage acts on rollers directly of roller guidance bearing and it acts on inner/outer ring, respectively, of inner/outer guidance bearing when it is mounted horizontally in most common application. As shown in Figure 1, four kinds of gaps between components can be found in the section. They are radial clearance (𝐺𝑟 ), roller gap (Δ), cage guidance gap (𝐶𝑟 ), and cage axial gap (𝐶𝑎 ), respectively. These gaps are defined as the maximum possible displacements between relative components in radial/axial direction. There is another important variable named pocket clearance (𝑇) in rolling bearing design, and it can be observed from Figure 2. Obviously, pocket clearance can be defined by the difference between cage pocket diameter and roller diameter. The cage pocket diameter has to be optimized to avoid faster wear of cage in terms of better lubrication film forming and decreased roller-cage bridge impact forces. According to the general design guidelines of rolling bearings, the value of cage axial gap (𝐶𝑎 ) is larger than roller axial gap (Δ), the value of pocket clearance (𝑇) is larger than cage guidance gap (𝐶𝑟 ), and the value of cage guidance gap (𝐶𝑟 ) is larger than bearing radial clearance (𝐺𝑟 ) usually. These rules ensure that the rollers contact with raceways directly. The rollers’ skewing can be adjusted by ring ribs according to the rules. Then the cage is in a certain state of being “free” in the bearing. When cylindrical bearings operate at a high speed, they generate vibrations and noise. The principal forces, which drive these vibrations, are time varying nonlinear contact forces, which exist among the various components of the bearings: rings, rollers, and cage. In the last decades, a lot of efforts have been devoted to studying the stability 2 Shock and Vibration Ca /2 Ca /2 Ca /2 Ca /2 Δ/2 Δ/2 Ca /2 Δ/2 Δ/2 Gr Outer ring Ca /2 Gr Δ/2 Gr Δ/2 Cr Cage Rollers (a) Outer-ring-rib guidance (b) Inner-ring-rib guidance Cr Cr Inner ring (c) Roller guidance Figure 1: Cage guidance in cylindrical roller bearings. and nonlinear dynamic behavior of flexible rotor bearings. Obviously, one of the most important mechanical elements to be taken into account is bearings due to their large influence on the dynamic behavior of rotating machinery (Tiwari et al. [1], Adam Jr. [2]). Many researchers developed special technique on dynamic behavior considering details of geometric parameters and applicable conditions of rolling bearings. Aktürk et al. [3] and Upadhyay et al. [4] presented theoretical investigations of varying preload, the influence of the number of the balls, and ball diameters on vibration characteristics of a rotor bearing system. Aktürk [5] and Harsha et al. [6] researched the effect of surface waviness on vibrations of ball bearings. Sopanen and Mikkola [7] and Upadhyay et al. [8] investigated dynamic behaviors of high-speed rotation with localized and distributed defects. Harsha [9] and Villa et al. [10] presented a nonlinear dynamic analysis of a flexible unbalanced rotor supported by ball bearings. The importance of energy efficiency has been increasing and has become a quality criterion for bearing producers and users in recent years. Hence, more and more researchers drew their attention on dynamic behaviors of the cage in rolling bearing. Houpert [11] developed a simulation software to simulate cage behavior and carried out relative experimental validation. Harsha [12] analyzed the nonlinear dynamics of ball bearings due to cage run-out and varying number of balls. He presented the results in the form of fast Fourier transformations (FFT) and phase trajectories. It is implied from the obtained FFT that due to the nonuniform spacing the ball passage frequency is modulated with the cage frequency. Bercea et al. [13] and Sakaguchi and Harada [14] investigated cage behavior of tapered roller bearings and made a comparison between numerical and experimental results. In this paper, a theoretical investigation considering all possible contacts of cage is conducted to observe its dynamic behavior. Owing to the investigation object applied in highspeed working condition, hydrodynamics in the contacts is taken into consideration for a more realistic calculation of acting forces by simulation model with the hypothesis of enough lubrication to supply all the contacts. Brass is chosen as the raw material of the cage in the paper considering the working speed of the bearing. Brass cages are machined by lathe an (...truncated)