General representations
Rolling bearings are the most important elements of the of rotor system models that determine the total support stiffness in the absence of flexible elements. Various of theoretical descriptions describe the bearing stiffness. The stiffness coefficients values verify in a large range, which in turn produces a scatter in the rotor system dynamic characteristics.
In the simplest cases the radial stiffness may be calculated with finite equations. The papers by Palmgren and Gargiulo include empirical equations for the bearing radial stiffness coefficient assuming rigid rings.
For a thrust radial bearing, the Gargiulo equation is:
For a low to medium speed radial roller bearing with contact along the entire roller length in both raceways, the radial stiffness
Kxx = Kyy can be obtained using the Palmgren equation:
where: z is the number of rolling elements, Z; L is the roller effective length, for example, without chamfers, m; D is the rolling element diameter, m; P is the radial load, N; α is the contact angle, deg.
More details in the articles below and the site calculators:
Read more:
Palmgren Calculator,
>> Roller bearing stiffness calculator
Read more:
Gargiulo Calculator,
>> Ball bearing stiffness calculator
- Gargiulo E. P. A simple way to estimate bearing stiffness // Machine Design. – 1980. – Т. 52. – №. 17. – С. 107-110.
- Palmgren A. Ball and roller bearing engineering // Philadelphia: SKF Industries Inc. – 1959. – Т. 1.
Rolling bearing models in DYNAMICS R4
DYNAMICS R4 allows application of 2-degree or 5-degree models for the both roller bearing and angular contact ball bearing. The high speed of models calculation allows their use in non-stationary problems. Also it is possible to take into account the variety of load factors at different modes.
The models take into account clearances along the inner and outer races, the number and inertia of rolling elements and the loading zone contact stiffness. For ball bearings, an axial force is taken into account together with the radial load. Additionally are input the contact angle and the curvatures of outer and inner races. The Hertz theory is used to determine the contact stiffness.
The bearing also operates in the environment of elements. The bearing rings surfaces contact flexibility is usually not taken into account, but it can change the bearing reaction forces.
The models built in DYNAMICS R4 take into account interferences and clearances in the bearing rings. Local flexibilities of the ring surrounding elements, shaft and housing are also taken into account. This requires large computational resources so the analysis is performed in static state for calculation of the bearing elastic performance. The resulting stiffness can be applied to the dynamic model. The result of dynamic system non-stationary calculation with non-linear roller bearings are the amplitude-time characteristics, that can be presented in various forms, for example, RMS or amplitude vibration displacements, vibration velocities or vibration accelerations. The rotor orbit motion in the gap can be shown. The resulting map can be converted into the frequency domain and displayed as a spectrum or waterfall plot.
Rolling bearing life in DYNAMICS R4
The company experts have developed a method for calculation of the equivalent radial load on ball or roller bearing. The method is based on calculations confirmed by operating experience. The proposed approach is based on the advanced standard for life of rolling bearings calculation ISO 16281, which is supplemented by algorithms for the bearing mounting and operating clearances. The approach involves the action of external radial and axial loads, including taking into account the deformation of rings and thin-wall shaft and housing.
As a result, DYNAMICS R4 uses a complete approach to solving the problem of the life, or endurance of a bearing unit, which takes into account:
- The operational clearance;
- Rings and support assembly flexibility;
- Introduction of rings non-circularity of;
- Rotor rotation speed;
- Operation temperature;
- The relative rings misalignment.
For roller bearings, the calculation takes into account the profile of rolling element generatrix and the possible cone shape of the rings. In a ball bearing, the possibility of individual balls being in three-point contact, while the transition from 2-point operation to the 3 point one is fully automatic. The both methods include 3 main steps:
- Determination of the bearing mounting and operating clearances;
- Determination of the radial loads distribution;
- Life calculation according to ISO 16281.
Thanks to this unification, it is quite convenient to move from mathematical expressions to implementation in the form of numerical algorithms, and due to modularity, to verify each step separately.
The key innovative component of the updated methods is the misalignment consideration, which required a significant modernization of the basic balance equations, and had an impact upon the distribution of internal contact forces.
The curves of internal forces obtained in the calculations are in good agreement with the published results, as well as those obtained in FEM.
It worth mentioning that there are few available works that cover the issue internal contact forces distribution. Not a single work has yet considered the loads cumulative effect on the bearing durability.
In this regard, the approaches used in DYNAMICS R4 are unique.
