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Abstract
Bearing standards impose restrictions on the internal geometry of spherical roller bearings. Geometrical and strength constraints conditions have been formulated for the optimization of bearing design. The long fatigue life is one of the most important criteria in the optimum design of bearing. The life is directly proportional to the dynamic capacity; hence, the objective function has been chosen as the maximization of dynamic capacity. The effect of speed and static loads acting on the bearing are also taken into account. Design variables for the bearing include five geometrical parameters: the roller diameter, the roller length, the bearing pitch diameter, the number of rollers, and the contact angle. There are a few design constraint parameters which are also included in the optimization, the bounds of which are obtained by initial runs of the optimization. The optimization program is made to run for different values of these design constraint parameters and a range of the parameters is obtained for which the objective function has a higher value. The artificial bee colony algorithm (ABCA) has been used to solve the constrained optimized problem and the optimum design is compared with the one obtained from the grid search method (GSM), both operating independently. Both the ABCA and the GSM have been finally combined together to reach the global optimum point. A constraint violation study has also been carried out to give priority to the constraint having greater possibility of violations. Optimized bearing designs show a better performance parameter with those specified in bearing catalogs. The sensitivity analysis of bearing parameters has also been carried out to see the effect of manufacturing tolerance on the objective function.
NOMENCLATURE
| Symbol | = | Description Units |
| bm | = | Rating factor for contemporary material |
| B | = | Bearing width mm |
| Cd | = | Basic dynamic load rating kN |
| d | = | Bearing bore diameter mm |
| di/2 | = | Inner raceway radius of curvature mm |
| do/2 | = | Outer raceway radius of curvature mm |
| D | = | Bearing outside diameter mm |
| Dm | = | Bearing pitch diameter mm |
| Dr | = | Roller nominal diameter mm |
| i | = | Number of rows of rollers |
| KDmin | = | Minimum roller diameter limiter |
| KDmax | = | Maximum roller diameter limiter |
| le | = | Roller effective length mm |
| L10 | = | Bearing fatigue life Cycles |
| P | = | Equivalent radial load N |
| Pe | = | Bearing free endplay mm |
| Q | = | Roller normal load N |
| rc | = | Bearing corner radius mm |
| R | = | Roller contour radius mm |
| Sd | = | Bearing diametral play mm |
| X | = | Design variable vector |
| Z | = | Number of rolling elements |
| α | = | Contact angle degree |
| γ | = | Dr cos α /Dm |
| λ | = | Reduction factor to account for edge loading and non-uniform stress |
| λl | = | Bearing life comparison parameter |
| ν | = | Reduction factor used in conjunction with a load-life exponent n = 10.3 |
| ϵ | = | Parameter for outer ring strength consideration |
| χ | = | Ratio (Dr /le) |
| θ | = | Roller semi angle degree |
| σ | = | Contact stresses N/mm2 |
| ρ | = | Curvature mm−1 |
| F(ρ) | = | Curvature difference mm−1 |
| ∑ρ | = | Curvature sum mm−2 |
| ϕ | = | Osculation |
| lt | = | Roller total length mm |
| β | = | Initial contact angle at the limiting axial position degree |
Subscripts
| Hybrid | = | |
| i | = | Inner ring or raceway |
| o | = | Outer ring or raceway |
| s | = | standard bearing |