Topic: Rolling Contact Bearings
Subject: Machine Design
Tool: Scilab & QCAD
Let us use an illustrated problem shown in Figure 1 to demonstrate the basic process of selecting an appropriate deep groove ball bearing that satisfy the minimum requirements for an application. The design on Figure 1 requires a deep groove ball bearing with a fixed bore diameter, \(d\), subjected to a pure constant and steady radial load, \(F_r\). The assembly requires the outer ring of the bearing to rotate at 1200 \(rpm\).
Figure 1. Deep Groove Ball Bearing Sample Design Requirements.
For the application with a simple pure radial load, a deep groove ball bearing will be used. For this example, a Timken brand will be chosen to select an appropriate bearing number that will satisfy requirements for static, dynamic load rating and bearing rating life.
- Static Load Rating
The basic static load rating, \(C_{0r}\), is based on a maximum contact stress within a non-rotating bearing at the center of the most heavily loaded rolling element and raceway contact. Equation (1) is the requirement for the static load rating.
$$C_{0r} > P_r \tag{1}$$
The equivalent radial load is defined in Equation (2)
$$P_r = C_1 V_1 F_r \tag{2}$$
Calculating the equivalent radial load of the given design problem will arrived at
$$P_r = (1) (1.2) (1\;\text{kN}) = 1.2 \; \text{kN}$$
From the given \(P_r\), a bearing with static load rating, \(C_{0r}\), is chosen together with the information of bore diameter, \(d = 10 \; mm\). In reference to Timken’s catalog for deep groove ball bearing, there are three possible bearing numbers that satisfy the two requirements and they are Bearing no. 6000, 6200, and 6300 as shown in Table 1.
Table 1. Basic Information of Timken Bearing no. 6000, 6200, and 6300.
- Dynamic Load Rating and Rating Life
Dynamic load rating, designated as \(C\), is defined as the radial load under which a population of bearings will achieve an \(L_{10}\) life of one million revolutions. Bearing life is defined as the length of time, or number of revolutions, until a fatigue spall of \(6 \; {mm}^2\) develops. The rating life, \(L_{10}\), is the life that 90 percent of a group of identical bearings will complete or exceed before a fatigue spall develops. The \(L_{10}\) life is associated with 90 percent reliability for a single bearing under a certain load. It has been calculated using the dynamic equivalent radial load, \(P_r\), and the dynamic load rating, \(C\), based on one million cycles.
$$L_{10} = \left(\frac{C}{P_r}\right)^e \; \left(\frac{1\, \times \,{10}^6}{60 \, n} \right) \tag{3}$$
Using the rating life defined in Equation (3), the appropriate bearing number can be selected based on the number of hours it can be run until it completes or exceeds the fatigue limit. Below is the calculation for the illustrated problem with the use of Scilab script.
- Scilab Script
// Copyright (C) 2025 - Gani Comia // Date of creation: 5 Mar 2025 // Deep Groove Ball Bearing Selection clear;clc; // (1) static load rating calculation // (1-1) equivalent radial load function function Pr=eqRadLoad(Fr) C1 = 1.0; // constant and steady load factor // V1 = 1.0; // inner ring rotation factor V1 = 1.2; // outer ring rotation factor Pr = C1.*V1.*Fr; // kN, equivalent radial load endfunction Fr = 1; // kN, pure radial load Pr = eqRadLoad(Fr); // kN, equivalent radial load mprintf("Equivalent Radial Load, Pr = %3.2f kN\n", Pr); mprintf("Required Static Load Rating, Cor > %3.2f kN\n", Pr); // (1-2) design decision // for the given d = 10 and equivalent radial load for // deep groove ball bearing, 6000, 6200 and 6300 can be used since // their static load rating, Cor > Pr. // (2) dynamic load rating and bearing life calculation // bearing life, L10 (hrs) as a function of dynamic load ratings function L10=bLife(Cr, Pr, n) e = 3.0; // for ball bearings L10 = ((Cr./Pr).^e).*(1e6/(60*n)); endfunction // (2-1) dynamic load ratings for Bearing No. 6000, 6200, and 6300 Cr = [4.60 5.10 8.10]; // kN, dynamic load ratings nCr = length(Cr); // number of bearing samples Pr = 0.1:0.005:2.0; // domain of equivalent radial load n = 1200; // rpm, design rotation // (2-2) calculation and plot clf; f = gcf(); f.figure_size = [700,700]; // (2-3) calculation of bearing life, L10 for i = 1:nCr L10(i,:) = bLife(Cr(i),Pr,n); end // (2-4) plot of L10 and Pr plot("nl",Pr,L10(1,:),"b-","linewidth",3); plot("nl",Pr,L10(2,:),"r-","linewidth",3); plot("nl",Pr,L10(3,:),"m-","linewidth",3); title("Timken Deep Groove Ball Bearing","fontsize",4.0); xlabel(["$\mathbf{P_r}$",",Equiv Radial Load, kN"],"fontsize",3.5); ylabel(["$\mathbf{L_{10}}$",",Bearing Life, hr"],"fontsize",3.5); note1 = "$\Large\text{Bearing no. 6000}$"; note2 = "$\Large\text{Bearing no. 6200}$"; note3 = "$\Large\text{Bearing no. 6300}$"; legend([note1,note2,note3],with_box=%F); xgrid(3,0); // (2-5) plot of L10 for Prad = 1.2 kN for 3 bearing nos. Prad = 1.2; // kN, equivalent radial load for j = 1:nCr L10(j) = bLife(Cr(j),Prad,n); end // (2-5-1) vertical line or Prad vX = [Prad Prad]; vY = [1e2 L10(3)]; plot(vX,vY,"k--"); // (2-5-2) horizontal line for L10 for 3 bearing nos. for k = 1:nCr hX = [0 Prad]; hY = [L10(k) L10(k)]; plot(hX,hY,"k--"); plot(Prad,L10(k),"marker","o","markerFaceColor","black"); end // (2-5-3) L10 labels format(7); xstring(0.1,L10(1)-375,["$\large \mathbf {L_{10}=}$",string(L10(1))]); xstring(0.1,L10(2),["$\large \mathbf {L_{10}=}$",string(L10(2))]); xstring(0.1,L10(3),["$\large \mathbf {L_{10}=}$",string(L10(3))]); xstring(1.4,1e6,["@ n = ", string(n), "rpm"]); xstring(0.4,5e6,"https://gani-mech-toolbox.blogspot.com"); mprintf("Bearing no. 6300 has L10 = %6.1f hrs\n", L10(3));
- Scilab Output (Figure 2)
Figure 2. Bearing’s Rating Life for an Equivalent Radial Load.
Though the three bearing numbers satisfy the bore dimensions and static load rating, the suitable bearing can be further trimmed down on the requirements of applicable rating life as shown in Figure 2.
- Bearing Survival
If the machine is assembled with a total number
of \(N\) bearings and each has the same reliability, \(R\), then the reliability of the
group of bearings or assembly is
$$R_N = R^N \tag{4}$$
Let us assume that there are two similar
bearings in the machine assembly, then its reliability is
The presented factors as basis for bearing selection are some of the important considerations. Actual design might need to take additional considerations into account and it is encouraged the reader to refer to the rolling contact bearing manufacturers information for more details.
Feel free to comment for inquiry,
clarification, or suggestion for improvement. Drop your email to request the
soft copy of the code.
References
- Timken Engineering Manual, Catalog No. 10424.
- Timken Deep Groove Ball Bearing, Catalog No. 10857.
- M.F. Spotts. Design of Machine Elements. 6th Ed. Englewood Cliffs, N.J. 07632. Prentice-Hall, Inc. 1985.
- J.E. Shigley and C.R. Mischke. Mechanical Engineering Design. 5th Ed. New York, New York 10020. McGraw-Hill Publishing Company. 1989.
