Multi-objective optimization in geometric design of tapered roller bearings based on fatigue, wear and thermal considerations through genetic algorithms

Abstract

To improve the fatigue, wear and thermal based failures of Tapered Roller Bearings (TRBs) a multi-objective optimization technique has been proposed. Objective functions considered are: the dynamic capacity (C_d) that is related to fatigue life, the elasto-hydrodynamic minimum film thickness (h_min) that is associated to the wear life, and the maximum bearing temperature (T_max) that is related to the lubricant life. This paper presents a non-linear constrained optimization problem of three objectives with eleven design variables and twenty-eight constraints. The said objectives have been optimized individually (i.e., the single-objective optimization) and concurrently (i.e., the multi-objective optimization) through a multi-objective evolutionary procedure, titled as the Elitist Non-dominated Sorting Genetic Algorithm. A set of standard TRBs have been selected for the optimization. Pareto-optimal fronts (POFs) and Pareto-optimal surfaces (POSs) are obtained for one representative standard TRB. Out of many solutions on the POFs/POSs only the knee-point solution has been shown in a tabular form. Life comparison factors have been calculated based on both the optimized and standard TRBs, and results indicate that the optimized TRBs got enhanced lives than standard bearings. To get the graphical impression of optimized TRBs, a skeleton of radial dimensions of all seven optimized bearings based on various combinations of objectives has been shown for one of the representative standard TRB. In few cases the multi-objective optimization has better convergence as compared to single objective optimization due to its inherent diversity by the principle of dominance. The sensitivity investigation has also been conducted to observe the sensitivity of three objectives with design variables. From the sensitivity analysis data, tolerances have been provided for design variables. These tolerances could be used by the manufacturing industry while producing TRBs.

Appendix A: Geometrical Parameters and Their Relationships for TRBs

Following equations are used in the present study on the MOO of TRBs [13]

The minimum thickness of the front face of the cup, (refer figure 2)

$$ S_{{2_{min} }}^{o} = \frac{1}{2}D - \left\{ {\frac{1}{2}D_{m} {\text{cosec}}\left( {\alpha_{o} - \beta^{o} } \right) + \frac{1}{2}l} \right\}\sec \beta^{o} \sin \alpha_{o} $$

(A.1)

The minimum thickness of the front face of the cone, (refer figure 2)

$$ S_{{1_{min} }}^{i} = \left\{ {\frac{1}{2}D_{m} {\text{cosec}}\left( {\alpha_{o} - \beta^{o} } \right) - \frac{1}{2}l} \right\}\sec \beta^{o} \sin \left( {\alpha_{o} - 2\beta^{o} } \right) - \frac{1}{2}d $$

(A.2)

The minimum width of the front face of the cup, (refer figure 2)

$$ C_{{2_{min} }} = \left[ {C + \frac{1}{2}D_{{o_{i} }} \cot \alpha_{o} } \right] - \left[ {\left\{ {\frac{1}{2}D_{m} {\text{cosec}}\left( {\alpha_{o} - \beta^{o} } \right) + \frac{1}{2}l} \right\}\sec \beta^{o} \cos \alpha_{o} } \right] $$

(A.3)

The minimum thickness of the back face of the cup, (refer figure 2)

$$ C_{{1_{min} }} = C - C_{{2_{min} }} - \frac{{l\cos \alpha_{o} }}{{\cos \beta^{o} }} $$

(A.4)

The minimum width of the back face of the cup, (refer figure 2)

$$ B_{{2_{min} }} = \left[ {T + \frac{1}{2}D_{{o_{i} }} \cot \alpha_{o} } \right] - \left[ {\left\{ {\frac{1}{2}D_{m} {\text{cosec}}\left( {\alpha_{o} - \beta^{o} } \right) + \frac{1}{2}l} \right\}\sec \beta^{o} \cos \left( {\alpha_{o} - 2\beta^{o} } \right)} \right] $$

(A.5)

The minimum width of the front face of the cone, (refer figure 2)

$$ B_{{1_{min} }} = B - B_{{2_{min} }} - \frac{l}{{\cos \beta^{o} }}\cos \left( {\alpha_{o} - 2\beta^{o} } \right) $$

(A.6)

The minimum thickness of the back face of the cup, (refer figure 2)

$$ S_{{1_{min} }}^{o} = \frac{1}{2}D - \left\{ {\frac{1}{2}D_{m} - \frac{1}{2}l\sin \left( {\alpha_{o} - \beta^{o} } \right)} \right\} + \left\{ {\left( {\frac{1}{2}D_{r} - \frac{1}{2}l\sin \beta^{o} } \right)\cos \left( {\alpha_{o} - \beta^{o} } \right)} \right\} $$

(A.7)

The minimum thickness of the back face of the cone, (refer figure 2)

$$ S_{{2_{min} }}^{i} = \left\{ {\frac{1}{2}D_{m} - \frac{{D_{r} }}{2}\cos \left( {\alpha_{o} - \beta^{o} } \right) + \frac{l}{2}\sin \beta^{o} } \right\} - \frac{d}{2} $$

(A.8)

Force on the spherical face of the roller

$$ Q_{f} = Q_{i} \cos \alpha_{i} \frac{{\left( {\sin \alpha_{o} - \tan \alpha_{i} \cos \alpha_{o} } \right)}}{{\sin \left( {\alpha_{o} + \alpha_{f} } \right)}} $$

(A.9)

Normal force on the cup

$$ Q_{o} = Q_{i} \cos \alpha_{i} \frac{{\left( {\sin \alpha_{f} + \tan \alpha_{i} \cos \alpha_{f} } \right)}}{{\sin \left( {\alpha_{o} + \alpha_{f} } \right)}} $$

(A.10)

Tensile stress in the flange

$$ \sigma_{{t_{f} }} = \frac{{Q_{f} \sin \alpha_{f} }}{{A_{f} }} $$

(A.11)

Area of the flange

$$ A_{f} = \frac{{\pi \left( {\frac{1}{2}d + S_{{2_{min} }}^{i} } \right)B_{{2_{min} }} }}{Z} $$

(A.12)

Maximum shear stress in the flange

$$ \tau_{f} = 1.5\frac{{Q_{f} \cos \alpha_{f} }}{{A_{f} }} $$

(A.13)

Bending stress in the flange

$$ \sigma_{{b_{f} }} = \frac{{Q_{f} h_{f} \cos^{2} \alpha_{f} }}{EI} $$

(A.14)

Position of the rib-roller contact on the flange face

$$ h_{f} = \frac{{\frac{1}{2}d_{1} - \left( {\frac{1}{2}d + S_{{2_{min} }}^{i} } \right)}}{{2\cos \alpha_{f} }} $$

(A.15)

Maximum principal stress in the flange

$$ \sigma_{{f_{\hbox{max} } }} = \frac{{\sigma_{{t_{f} }} + \sigma_{{b_{f} }} }}{2} + \sqrt {\left( {\frac{{\sigma_{{t_{f} }} + \sigma_{{b_{f} }} }}{2}} \right)^{2} + \tau_{f}^{2} } $$

(A.16)