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 (Cd) that is related to fatigue life, the elasto-hydrodynamic minimum film thickness (hmin) that is associated to the wear life, and the maximum bearing temperature (Tmax) 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.

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Abbreviations

\( A_{{}} \) :

Area normal to the heat flow, m2

\( A_{f} \) :

Area of the flange, mm2

b m :

Rating factor for the contemporary material

B :

Width of the cone, mm

\( B_{{1_{ \hbox{min} } }} \) :

Width of the narrow front face of cone, mm

\( B_{{2_{ \hbox{min} } }} \) :

Width of the flange, mm

C :

Width of the cup, mm

\( C_{d} \) :

Dynamic load capacity of the bearing, N

\( C_{d\_new} \) :

Dynamic load capacity of the bearing obtained using NSGA-II, N

\( C_{d\_std} \) :

Dynamic load capacity of the bearing given in bearing catalogues, N

C p :

Specific heat of a lubricant at a conSOOnt pressure, J/Kg-K

\( C_{{1_{ \hbox{min} } }} \) :

Minimum width of the back-face of the cup, mm

\( C_{{2_{ \hbox{min} } }} \) :

Minimum width of the front-face of the cup, mm

\( d \) :

Inner (or bore) diameter of the bearing, mm

d i :

inner raceway mean diameter, mm

d o :

outer raceway mean diameter, mm

\( D \) :

Outer diameter of the bearing, mm

\( D_{m} \) :

Bearing pitch diameter, mm

\( D_{r} \) :

Roller mean diameter, mm

\( D_{{r_{LL} }} \) :

Lower limit of the roller mean diameter, mm

\( D_{{r_{UL} }} \) :

Upper limit of the roller mean diameter, mm

\( D_{{o_{i} }} \) :

Minimum diameter of the cup, mm

\( e \) :

Parameter for the mobility condition

E :

Modulus of elasticity, Pa

\( E^{\prime} \) :

Equivalent modulus of elasticity, Pa

\( EI \) :

Section modulus of the flange section subjected to bending, N-mm2

\( f(X) \) :

Objective vector

Gr :

Grashof number

\( h_{f} \) :

Position of the rib-roller contact on the flange face, mm

h min :

Elasto-hydrodynamic minimum film thickness, m

h r :

Radiation heat transfer coefficient

H w :

Housing width, mm

i :

Number of rows of the roller

k :

Thermal conductivity of rings and the rolling element

\( K_{D\hbox{min} } \) :

Minimum roller diameter limiter

\( K_{D\hbox{max} } \) :

Maximum roller diameter limiter

ko :

Thermal conductivity of the lubricant, W/m°C

l :

Total length of the roller, mm

\( l_{e} \) :

Effective length of the roller, mm

\( \ell \) :

DiSOOnce between two points (i, j) of the heat transfer, m

\( L_{10} \) :

Rating life (Bearing fatigue life cycles)

no :

Outer raceway speed, rpm

\( P \) :

Equivalent radial load, N

Pr :

Prandtl number of the lubrication oil

\( Q_{i} \) :

Load on the inner ring at the most heavily loaded roller, N

Qmax :

Contact force on the interior raceway at the heaviest loaded roller, N

\( Q_{o} \) :

Load on the outer ring at the most heavily loaded roller, N

\( Q_{f} \) :

Load on the flange at the most heavily loaded roller, N

r:

Corner radius of the roller, mm

\( r_{1} \) :

Cone back-face chamfer height, mm

\( r_{2} \) :

Cone back-face chamfer width, mm

\( r_{3} \) :

Cup back-face chamfer height, mm

\( r_{4} \) :

Cup back-face chamfer width, mm

\( r_{5} \) :

Chamfer height and width of the front-face the cone and the cup, mm

Re:

Reynolds number

\( R_{{e_{i,o} }} \) :

Equivalent radius (m)

\( S_{{1_{ \hbox{min} } }}^{i} \) :

Minimum thickness of the front-face of the cone, mm

\( S_{{2_{ \hbox{min} } }}^{i} \) :

Minimum thickness of the back face of the cone, mm

\( S_{{1_{ \hbox{min} } }}^{o} \) :

Minimum thickness of the back-face of the cup, mm

\( S_{{2_{ \hbox{min} } }}^{o} \) :

Minimum thickness of the front-face of the cup, mm

th:

Seal thickness, mm

\( T \) :

Total width of the bearing, mm

T i and T j :

Temperatures of the two points (i, j) between which the heat transfer is taking place

T l :

Lubricant temperature, °C

T max :

Maximum bearing temperature, °C

u i,o :

Entrainment velocity, m/s

u s :

1/3rd of the surface velocity of the housing, m/s

X :

Design variable vector

\( Z \) :

Number of rollers

\( \alpha_{f} \) :

Flange angle

\( \alpha_{i} \) :

Contact angle of the inner raceway (i.e., cone)

\( \alpha_{o} \) :

Outer raceway contact angle

αp :

Pressure viscosity coefficient of lubricant, m2/N

\( \beta \) :

Parameter for the effective length of the roller

\( \beta^{o} \) :

Semi taper angle of the roller

\( \gamma \) :

Ratio, \( D_{{r_{mean} }} \cos \alpha_{o} /D_{m} \)

\( \varepsilon \) :

Parameter for outer ring strength consideration

\( \varepsilon_{h} \) :

Thermal emissivity of the housing

ηo :

Dynamic viscosity of lubricant, N-s/m2

\( \lambda \) :

Reduction factor to account for the edge loading and the non-uniform stress

\( \lambda_{l} \) :

Life comparison factor

\( \nu \) :

Factor to account for the edge loading

νo :

Kinematic viscosity of lubricant, \( {\text{m}}^{ 2} / {\text{s}} \)

\( \sigma_{{b_{f} }} \) :

Bending stress in the flange, \( {\text{N/mm}}^{ 2} \)

\( \sigma_{{f_{ \hbox{max} } }} \) :

Maximum stress in the flange, \( {\text{N/mm}}^{ 2} \)

\( \sigma_{ \hbox{max} }^{l} \) :

Maximum contact stress, \( {\text{N/mm}}^{ 2} \)

\( \sigma_{safe} \) :

Safe contact stress, \( {\text{N/mm}}^{ 2} \)

\( \sigma_{tf} \) :

Direct tensile stress in the flange, \( {\text{N/mm}}^{ 2} \)

\( \tau_{f} \) :

Shear stress in the flange, \( {\text{N/mm}}^{ 2} \)

\( \upsilon \) :

Poisson’s ratio

ωm :

Angular velocity of the cage, rad/s

ωr :

Angular velocity of the roller, rad/s

ωo :

Angular velocity of the outer raceway, rad/s

\( \varpi \) :

Width of the structure, m

f:

Flange

\( i \) :

Represents the inner raceway or cone

\( o \) :

Represents the outer raceway or cup

DOO:

Dual Objective Optimization

MOO:

Multi Objective Optimization

POF:

Pareto Optimal Front

POS:

Pareto Optimal Surface

SOO:

Single Objective Optimization

TOO:

Triple Objective Optimization

TRB:

Tapered Roller Bearing

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Correspondence to Rajiv Tiwari.

Appendix A: Geometrical Parameters and Their Relationships for 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)

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Kalyan, M., Tiwari, R. & Ahmad, M.S. Multi-objective optimization in geometric design of tapered roller bearings based on fatigue, wear and thermal considerations through genetic algorithms. Sādhanā 45, 142 (2020). https://doi.org/10.1007/s12046-020-01385-3

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Keywords

  • Tapered roller bearings
  • dynamic capacity
  • elasto-hydrodynamic lubrication
  • maximum temperature
  • multiple objectives
  • NSGA-II
  • sensitivity analysis