**Development of Graphical Solution to Determine Optimum Hollowness of Hollow Cylindrical Roller Bearing Using Elastic Finite Element Analysis**

## P.H. Darji and D.P. Vakharia

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/46160

## **1. Introduction**

Technological progress creates increasingly arduous conditions for rolling mechanisms. Advances in many fields including gas turbine design, aeronautics, space and atomic power, involve extreme operating speeds, load, temperatures, environments which increases power and load on machinery and demand high strength to weight ratio of the rolling element bearings. Also bearing stiffness is an important parameter in the designing. Bearing design calculations require a good understanding of the Hertzian contact stress due to which high stress concentration is produced which greatly influence the fatigue life and dominate the upper speed limits as in the case of solid rolling elements. Since being originally introduced, cylindrical rolling element bearings have been significantly improved, in terms of their performance and working life. A major objective has been to decrease the Hertz contact stresses at the roller–raceway interfaces, because these are the most heavily stressed areas in a bearing. It has been shown that bearing life is inversely proportional to the stress raised to the ninth power (even higher). Whereas making the rollers hollow which are flexible enough reduces stress concentration and finally increase the fatigue life of bearing.

Investigators have proposed that under large normal loads a hollow element with a sufficiently thin wall thickness will deflect appreciably more than a solid element of the same size. An improvement in load distribution and thus load capacity may be realized, as well as contact stress is also reduced considerably by using a bearing with hollow rollers. Since for hollow roller bearing no method is available for the calculation of hollowness, contact stresses and deformation. The contact stresses in hollow members are often calculated by using the same equations and procedures as for solid specimens. This approach seems to be incorrect.

© 2012 Darji and Vakharia, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Darji and Vakharia, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Initially in the present work author has carried out sufficient literature review (Somasundar & Krishnamurthy, 1984; Harris & Aaronson, 1967; Bamberger, Parker & Dietrich, 1976; Bhateja & Hahn, 1980; Murthy & Rao, 1983; Hong & Jianjun, 1998; Zhao, 1998; Yangang, Raj & Qingyu, 2004; Darji & Vakharia, 2008) and market survey to understand the practical application of hollow cylindrical roller bearing and its advantages in comparison with solid roller bearing. It is concluded that bearing manufacturer are production these type of bearing as per the requirement, but no standard formula or catalogue is available through which user can directly select hollow cylindrical roller bearing. Thus no standard formula (method) is available to find the optimum hollowness for the given loading condition and dimensions of bearing. Hollowness of the roller bearing is mainly dependent of applied load, dimensions of roller and endurance limit of the material used. Calculation of exact contact pressure for the hollow roller requires a finite element approach, and this has not been carried out yet. Present work is aimed to identify optimum hollowness irrespective of the geometry of the bearing and applied load.

Development of Graphical Solution to Determine Optimum

Hollowness of Hollow Cylindrical Roller Bearing Using Elastic Finite Element Analysis 239

present work 2206, 2210, 2215, 2220 and 2224 bearings are selected for analytical analysis.

Before executing the FE analysis for cylindrical roller bearing to understand the contact behavior, it is possible to execute the same contact behavior for roller-flat interaction in place of roller-race as shown in Fig. 2. It is very easy to check the contact behavior of roller and flat. To check the contact behavior of roller and flat, if we will take the same diameter of roller which is used in the corresponding roller bearing then contact width will be changed. So comparison of contact behavior of roller and flat with roller-race as in case of bearing is not possible. Using the equation for equivalent diameter in the present work roller diameter is identified in such a way that for the same loading condition contact width of roller and flat interaction will remain same as the interaction of roller with race in bearing. These equivalent diameter of roller is designated by 'Roller 1' for NU 2206 bearing, 'Roller 2' for NU 2210 bearing, 'Roller 3' for NU 2215 bearing, 'Roller 4' for NU 2220 bearing and 'Roller 5' for NU 2224 bearing. Contact interaction between roller and flat plate is shown in Fig. 1, which is a part of the contact between inner-race and roller in the cylindrical roller bearing as described. This contact interaction is studied in detail by analytically using Hertz theory in the present work.

Contact behavior of all these bearing is studied using Hertz theory.

**Figure 1.** Load distribution in roller bearing

**Figure 2.** Schematic of contact profile of roller on flat race

To meet the requirement, in the first part of the present work contact analysis has been carried out for contact between roller and flat. Dimensions of the rollers are calculated using equation of equivalent diameter corresponding to the five different cylindrical roller bearing i. e. 2206, 2210, 2215, 2220 and 2224 to get the large data range. Value of applied load is taken from minimum to maximum. Finite element analysis is carried out for the same rollerflat contact and results are compared with analytical solution given by Hertz. This step is required to check the feasibility of FE procedure. In the second part of the work FE analysis has been carried out for the same applied load and material for all five cases, only the change is rollers are taken hollow. Roller hollowness is ranging from 10% to the hollowness for which bending stress at the inner bore should not exceed endurance limit of the material is taken for the consideration. Flexural fatigue failures occurred in hollow roller when the maximum bending stress at the bore cross the limit of endurance limit of the material. The fatigue cracks always began in the bore of the hollow roller. Those that propagated to the roller surface resulted in surface cracks and spallig and finally it fails the bearing. Around seventy FE analysis are done to generate the large data range. Finally graphical solution has been proposed to identify optimum hollowness irrespective of geometry of bearing and material properties.

## **2. Analytical study of solid cylindrical roller bearings**

In the present work five different cylindrical roller bearing of NU 22 series are selected. First of all load distribution (Fig. 1) is calculated by applying load equal to static load carrying capacity of the bearing and the values of contact pressure, deflection, contact width and von Mises stress induced in the roller-race interface are determined (SKF General Catalogue, 1989; Design Data, 1994; Harris, 2001; Shigley, 1983; Nortron, 2010; I. S. 9202, 2001; Harris & Kotzalas, 2007; Horng, Ju & Cha, 2000; Demirhan & Kanber, 2008; Kania, 2006). Taking modulus of elasticity E = 201330 N/mm2 and Poission ratio v = 0.277 for the bearing material AISI 52100 steel (Guo & Liu, 2002). These five cylindrical roller bearings are selected in such a way that the size of roller should be in step of 5 mm approximately. So we can get the wide data range of load distribution for further analysis. Considering this point in the present work 2206, 2210, 2215, 2220 and 2224 bearings are selected for analytical analysis. Contact behavior of all these bearing is studied using Hertz theory.

**Figure 1.** Load distribution in roller bearing

238 Finite Element Analysis – Applications in Mechanical Engineering

the geometry of the bearing and applied load.

**2. Analytical study of solid cylindrical roller bearings** 

material properties.

Initially in the present work author has carried out sufficient literature review (Somasundar & Krishnamurthy, 1984; Harris & Aaronson, 1967; Bamberger, Parker & Dietrich, 1976; Bhateja & Hahn, 1980; Murthy & Rao, 1983; Hong & Jianjun, 1998; Zhao, 1998; Yangang, Raj & Qingyu, 2004; Darji & Vakharia, 2008) and market survey to understand the practical application of hollow cylindrical roller bearing and its advantages in comparison with solid roller bearing. It is concluded that bearing manufacturer are production these type of bearing as per the requirement, but no standard formula or catalogue is available through which user can directly select hollow cylindrical roller bearing. Thus no standard formula (method) is available to find the optimum hollowness for the given loading condition and dimensions of bearing. Hollowness of the roller bearing is mainly dependent of applied load, dimensions of roller and endurance limit of the material used. Calculation of exact contact pressure for the hollow roller requires a finite element approach, and this has not been carried out yet. Present work is aimed to identify optimum hollowness irrespective of

To meet the requirement, in the first part of the present work contact analysis has been carried out for contact between roller and flat. Dimensions of the rollers are calculated using equation of equivalent diameter corresponding to the five different cylindrical roller bearing i. e. 2206, 2210, 2215, 2220 and 2224 to get the large data range. Value of applied load is taken from minimum to maximum. Finite element analysis is carried out for the same rollerflat contact and results are compared with analytical solution given by Hertz. This step is required to check the feasibility of FE procedure. In the second part of the work FE analysis has been carried out for the same applied load and material for all five cases, only the change is rollers are taken hollow. Roller hollowness is ranging from 10% to the hollowness for which bending stress at the inner bore should not exceed endurance limit of the material is taken for the consideration. Flexural fatigue failures occurred in hollow roller when the maximum bending stress at the bore cross the limit of endurance limit of the material. The fatigue cracks always began in the bore of the hollow roller. Those that propagated to the roller surface resulted in surface cracks and spallig and finally it fails the bearing. Around seventy FE analysis are done to generate the large data range. Finally graphical solution has been proposed to identify optimum hollowness irrespective of geometry of bearing and

In the present work five different cylindrical roller bearing of NU 22 series are selected. First of all load distribution (Fig. 1) is calculated by applying load equal to static load carrying capacity of the bearing and the values of contact pressure, deflection, contact width and von Mises stress induced in the roller-race interface are determined (SKF General Catalogue, 1989; Design Data, 1994; Harris, 2001; Shigley, 1983; Nortron, 2010; I. S. 9202, 2001; Harris & Kotzalas, 2007; Horng, Ju & Cha, 2000; Demirhan & Kanber, 2008; Kania, 2006). Taking modulus of elasticity E = 201330 N/mm2 and Poission ratio v = 0.277 for the bearing material AISI 52100 steel (Guo & Liu, 2002). These five cylindrical roller bearings are selected in such a way that the size of roller should be in step of 5 mm approximately. So we can get the wide data range of load distribution for further analysis. Considering this point in the Before executing the FE analysis for cylindrical roller bearing to understand the contact behavior, it is possible to execute the same contact behavior for roller-flat interaction in place of roller-race as shown in Fig. 2. It is very easy to check the contact behavior of roller and flat. To check the contact behavior of roller and flat, if we will take the same diameter of roller which is used in the corresponding roller bearing then contact width will be changed. So comparison of contact behavior of roller and flat with roller-race as in case of bearing is not possible. Using the equation for equivalent diameter in the present work roller diameter is identified in such a way that for the same loading condition contact width of roller and flat interaction will remain same as the interaction of roller with race in bearing. These equivalent diameter of roller is designated by 'Roller 1' for NU 2206 bearing, 'Roller 2' for NU 2210 bearing, 'Roller 3' for NU 2215 bearing, 'Roller 4' for NU 2220 bearing and 'Roller 5' for NU 2224 bearing. Contact interaction between roller and flat plate is shown in Fig. 1, which is a part of the contact between inner-race and roller in the cylindrical roller bearing as described. This contact interaction is studied in detail by analytically using Hertz theory in the present work.

**Figure 2.** Schematic of contact profile of roller on flat race


Table 1 shows the value of all these analytical results corresponding to roller-flat contact.

Development of Graphical Solution to Determine Optimum

Deformation (δ) mm

Hollowness of Hollow Cylindrical Roller Bearing Using Elastic Finite Element Analysis 241

von Mises stress (σVM) N/mm2

Roller position Load (Q) N Contact

width (b) mm

N/mm2 of roller material AISI 52100 steel (Guo and Liu, 2002).

**3. FE analysis of solid cylinder and flat contact** 

element analysis for the present case is carried out.

**3.2. Finite element analysis details** 

**Figure 3.** Sketch of roller-plat contact model

*3.2.1. Model description* 

presented in Fig. 3.

**3.1. Existing FE models** 

Contact pressure (pmax)

N/mm2

0 73318.39 0.4946 2622.82 1169.78 0.05224 1 & 2 66316.76 0.4704 2494.44 1112.52 0.04773 3 & 4 46938.59 0.3957 2098.59 935.97 0.03497 5 & 6 19911.01 0.2577 1366.81 609.6 0.01616

The induced von Mises stress in the cylinder/roller is less then the yield strength 1410.17

Since the first mathematical treatment of the contact problem of ideally smooth elastic solids, presented by Hertz in 1882, significant progress has been made in the field of contact mechanics. In particular, the deformation characteristics of semi-infinite elastic media subjected to concentrated and distributed surface traction have been elucidated, and analytical solutions for the contact pressure distributions and subsurface stress fields have been obtained for elastic bodies of different shapes and various interfacial friction conditions (Timoshenko & Godier, 1970). The results of these studies have been invaluable in the design of durable mechanical components, such as rolling element bearings (Komvopoulos & Choi, 1992). Existing FE Models like GW Model (Greenwood & Williamson, 1966), KE Model (Kogut & Etsion, 2002) and JG Model (Jackson & Green, 2005) are studied and finite

In order to validate the relationship of load vs deflection, load vs contact width etc., an FE model of an un-profiled roller contacting a flat plate was set up. A sketch of the problem is

**Table 5.** Analytical results for 2224 bearing : Equivalent diameter – 20.56 mm (Roller 5)

**Table 1.** Analytical results for 2206 bearing : Equivalent diameter – 6.62 mm (Roller 1)


**Table 2.** Analytical results for 2210 bearing : Equivalent diameter – 8.58 mm (Roller 2)


**Table 3.** Analytical results for 2215 bearing : Equivalent diameter – 12.82 mm (Roller 3)


**Table 4.** Analytical results for 2220 bearing : Equivalent diameter – 17.14 mm (Roller 4)

Development of Graphical Solution to Determine Optimum Hollowness of Hollow Cylindrical Roller Bearing Using Elastic Finite Element Analysis 241


**Table 5.** Analytical results for 2224 bearing : Equivalent diameter – 20.56 mm (Roller 5)

The induced von Mises stress in the cylinder/roller is less then the yield strength 1410.17 N/mm2 of roller material AISI 52100 steel (Guo and Liu, 2002).

## **3. FE analysis of solid cylinder and flat contact**

## **3.1. Existing FE models**

240 Finite Element Analysis – Applications in Mechanical Engineering

Contact width (b)mm

Load (Q)

N

Load (Q) N

Roller position

Roller Position

Roller position

Roller position

Table 1 shows the value of all these analytical results corresponding to roller-flat contact.

Contact

N/mm2

0 5414.28 0.1321 2175.58 970.31 0.01206 1 & 2 4730.5 0.1235 2033.57 906.97 0.01068 3 & 4 2890.5 0.0965 1589.61 708.97 0.00685 5 & 6 517.68 0.0408 672.72 300.03 0.00146

Contact

N/mm2

0 10397.88 0.1929 2451.83 1093.51 0.01917 1 & 2 9404.92 0.1835 2331.82 1039.99 0.01752 3 & 4 6656.75 0.1544 1961.77 874.95 0.01283 5 & 6 2823.74 0.1005 1277.7 569.86 0.00593

0 21620.59 0.2713 2307.3 1029.05 0.02581 1 & 2 19801.61 0.2596 2208.11 984.81 0.02385 3 & 4 14716.21 0.2238 1903.56 848.99 0.01826 5 & 6 7444.23 0.1592 1353.88 603.83 0.00989

Contact

N/mm2

0 50656.34 0.4112 2615.63 1166.57 0.04333 1 & 2 45818.86 0.3911 2487.6 1109.47 0.03959 3 & 4 32430.3 0.329 2092.83 933.4 0.02901 5 & 6 13756.7 0.2143 1363.06 607.93 0.01341

pressure (pmax)

**Table 3.** Analytical results for 2215 bearing : Equivalent diameter – 12.82 mm (Roller 3)

**Table 4.** Analytical results for 2220 bearing : Equivalent diameter – 17.14 mm (Roller 4)

pressure (pmax)

Contact pressure (pmax) N/mm2

**Table 1.** Analytical results for 2206 bearing : Equivalent diameter – 6.62 mm (Roller 1)

**Table 2.** Analytical results for 2210 bearing : Equivalent diameter – 8.58 mm (Roller 2)

width (b) mm

Contact width (b) mm

Contact width (b) mm

Load (Q) N Contact

Load (Q)

N

pressure (pmax)

von Mises stress (σVM) N/mm2

von Mises stress (σVM) N/mm2

> von Mises stress (σVM) N/mm2

von Mises stress (σVM) N/mm2

Deformation (δ) mm

Deformation (δ) mm

Deformation (δ) mm

> Deformation (δ) mm

Since the first mathematical treatment of the contact problem of ideally smooth elastic solids, presented by Hertz in 1882, significant progress has been made in the field of contact mechanics. In particular, the deformation characteristics of semi-infinite elastic media subjected to concentrated and distributed surface traction have been elucidated, and analytical solutions for the contact pressure distributions and subsurface stress fields have been obtained for elastic bodies of different shapes and various interfacial friction conditions (Timoshenko & Godier, 1970). The results of these studies have been invaluable in the design of durable mechanical components, such as rolling element bearings (Komvopoulos & Choi, 1992). Existing FE Models like GW Model (Greenwood & Williamson, 1966), KE Model (Kogut & Etsion, 2002) and JG Model (Jackson & Green, 2005) are studied and finite element analysis for the present case is carried out.

## **3.2. Finite element analysis details**

## *3.2.1. Model description*

In order to validate the relationship of load vs deflection, load vs contact width etc., an FE model of an un-profiled roller contacting a flat plate was set up. A sketch of the problem is presented in Fig. 3.

**Figure 3.** Sketch of roller-plat contact model

A commercial package ANSYS 9.0 was used to solve the non linear contact problem. Initially for Roller 1, first of all three dimensional axis symmetric model was developed to form the single asperity contact between half cylinder and flat plate as shown in Fig. 6. For bearing 2206 taking equivalent diameter of roller as 6.62 mm and length 12mm. Dimensions of flat plat are taken as 12 x 8 x 2 mm3. The circular surface of cylinder and contact flat surface of plate was discretized by SOLID 185 elements. SOLID185 is used for the 3-D modeling of solid structures. It is defined by eight nodes having three degrees of freedom at each node: translations in the nodal x, y, and z directions see Fig. 4. The element has plasticity, hyperelasticity, stress stiffening, creep, large deflection, and large strain capabilities. It also has mixed formulation capability for simulating deformations of nearly incompressible elastoplastic materials, and fully incompressible hyperelastic materials.

Development of Graphical Solution to Determine Optimum

Hollowness of Hollow Cylindrical Roller Bearing Using Elastic Finite Element Analysis 243

converged with coarse to fine meshes as shown in Fig. 5. The number of elements and nodes

It is very clear from the Fig. 5 that for the last three points value of von Mises stress is approximately same and its value are 242.09 N/mm2, 259.94 N/mm2 and 263.52 N/mm2 for

0 2000 4000 6000 8000 10000 12000 14000 16000 **Number of Nodes**

the corresponding values of 0.08, 0.05 and 0.03 element edge length.

of models will increase as size of model will increase.

**Figure 5.** Von Mises stress vs number of nodes

**von Mises Stress (N/mm^2)**

**Figure 6.** Finite Element Model

**Figure 7.** Densely meshed regions of the contact model

**Figure 4.** SOLID 185 geometry

#### *3.2.2. Mesh convergence*

A converged solution is one that is nearly independent of meshing errors. An extremely coarse mesh would give a very approximate solution, which is far from reality. As the mesh is refined by reducing the size of the elements, the solution slowly approaches an exact solution. It should be noted that, in theory, the solution will not be exact until the mesh size is zero, which is obviously impossible. However, it is possible to fix a tolerance to the solution error and this can be achieved by solving the problem on several meshes. In order to ensure that the solution obtained is as close as possible to reality, solutions should be obtained from several meshes starting with a very coarse mesh and finishing with a very fine mesh. Once these solutions are available, many key quantities can be compared and plotted against mesh densities (or number of points) as shown in Fig. 5.

In order to investigate the convergence of the solutions, all models have been solved with increasing numbers of elements. The elements around the roller contact region are subdivided into number of elements as shown in Fig. 7. Although the stresses and displacements at different regions are investigated in this work the convergence check has been made for only point A as shown in Fig. 7 It is a common point of contact of roller and flat plate where induced von Mises stress should be investigated. The von Mises stress (σVM) converged with coarse to fine meshes as shown in Fig. 5. The number of elements and nodes of models will increase as size of model will increase.

**Figure 5.** Von Mises stress vs number of nodes

242 Finite Element Analysis – Applications in Mechanical Engineering

**Figure 4.** SOLID 185 geometry

*3.2.2. Mesh convergence* 

A commercial package ANSYS 9.0 was used to solve the non linear contact problem. Initially for Roller 1, first of all three dimensional axis symmetric model was developed to form the single asperity contact between half cylinder and flat plate as shown in Fig. 6. For bearing 2206 taking equivalent diameter of roller as 6.62 mm and length 12mm. Dimensions of flat plat are taken as 12 x 8 x 2 mm3. The circular surface of cylinder and contact flat surface of plate was discretized by SOLID 185 elements. SOLID185 is used for the 3-D modeling of solid structures. It is defined by eight nodes having three degrees of freedom at each node: translations in the nodal x, y, and z directions see Fig. 4. The element has plasticity, hyperelasticity, stress stiffening, creep, large deflection, and large strain capabilities. It also has mixed formulation capability for simulating deformations of nearly incompressible elastoplastic materials, and fully incompressible hyperelastic materials.

A converged solution is one that is nearly independent of meshing errors. An extremely coarse mesh would give a very approximate solution, which is far from reality. As the mesh is refined by reducing the size of the elements, the solution slowly approaches an exact solution. It should be noted that, in theory, the solution will not be exact until the mesh size is zero, which is obviously impossible. However, it is possible to fix a tolerance to the solution error and this can be achieved by solving the problem on several meshes. In order to ensure that the solution obtained is as close as possible to reality, solutions should be obtained from several meshes starting with a very coarse mesh and finishing with a very fine mesh. Once these solutions are available, many key quantities can be compared and

In order to investigate the convergence of the solutions, all models have been solved with increasing numbers of elements. The elements around the roller contact region are subdivided into number of elements as shown in Fig. 7. Although the stresses and displacements at different regions are investigated in this work the convergence check has been made for only point A as shown in Fig. 7 It is a common point of contact of roller and flat plate where induced von Mises stress should be investigated. The von Mises stress (σVM)

plotted against mesh densities (or number of points) as shown in Fig. 5.

It is very clear from the Fig. 5 that for the last three points value of von Mises stress is approximately same and its value are 242.09 N/mm2, 259.94 N/mm2 and 263.52 N/mm2 for the corresponding values of 0.08, 0.05 and 0.03 element edge length.

**Figure 7.** Densely meshed regions of the contact model

The region of most interest is adjacent to the contact interface and has the greatest concentration of elements for lower interferences. Away from the contact region, the mesh becomes coarser to minimize the computational efforts. In the present all FE models in contact region element edge length is taken as 0.08 and in other area it is taken as 0.5, the details of which is shown in Fig 8. The total numbers of elements generated are 14120 and nodes generated are 16977 for this model.

Development of Graphical Solution to Determine Optimum

Hollowness of Hollow Cylindrical Roller Bearing Using Elastic Finite Element Analysis 245

**Figure 9.** Applied pressure and boundary condition

*3.2.6. Evaluation of the finite element model* 

contact width and deformation.

with the percentage error of 5.6 % only.

To examine the appropriateness of the finite element mesh and modeling assumptions, such as the dimensions and fineness of the mesh and the imposed boundary conditions, finite element results for an elastic half-space indented by a rigid cylindrical asperity were compared with analytical results for line contacts. The FE model was first verified by comparing its output with the analytical results of the Hertz solution in the elastic regime. The verification included the contact pressure, contact stresses, deformation and contact width. For the evaluation of Finite Element Model initially von Mises stress criteria is taken for the consideration because it is the final output of analytical study as discussed in section 2. Also it is an important stress which should remain within limit with respect to yield stress of the material. Figure 10 shows the contour plot of von Mises stress for the applied load of 5414.28 N. As it is clear from the figure that at contact zone induced stress is higher which is marked by red colour. Figure 11 shows the detail view of contact zone with node numbers. Value of von Mises stress is to be identifying for the node no 5258 which is on contact surfaces and it is 1031.1 N/mm2. Whereas analytical result gives 970.31 N/mm2 (Table 1). Thus the von Mises stress of FE model differs from the Hertz solution by 5.8% which is acceptable for the present analysis. The small differences between analytical and FEA solutions near the contact edge may be attributed to the fineness of the mesh. The favorable comparison of the results illustrates the suitability of the finite element model for the present analysis involving only global variables, such as von Mises stress, contact pressure,

Figure 12 shows the contour plot for shear stress distribution. It is clear from the FE analysis that value of induced shear stress is 457.88 N/mm2 and analytical result gives 485.16 N/mm2

## *3.2.3. Contact model*

In order to create contact models in ANSYS, a contact pair of elements must be created a contact element and a target element. ANSYS has general guidelines as to what line, surface, or volume these elements should be applied. Perhaps the most critical feature is the mesh size. For example, a large target element size and very fine contact element will not work. The sizes of the contact and target elements should be fairly close to one another. It is possible to get a solution to converge, but the results will most likely be incorrect. That is why there is a densely meshed region in both the bottom part of the half-cylinder and on the surface of the block shown in Fig. 8.

**Figure 8.** Contact and target elements

## *3.2.4. Boundary condition and application of pressure*

The boundary conditions are presented in Fig. 9. The nodes on the bottom surface of the flat plat all degree of freedoms are restricted and rigidly constrained from translating in the x, y and z direction. Where as on top surface of half cylinder uniform pressure of 68.16 N/mm2 is applied which is related with Qmax (5414.28 N) for bearing no. 2206

### *3.2.5. Solutions*

The solutions have been carried out by means of a PC. The hardware configuration consists or an Intel Pentium IV 2.53 GHz CPU with 1 GB of RAM. The models were solved in round 25 minutes to 7 hours. All models have been solved as 3D static with Newton Raphson option.

**Figure 9.** Applied pressure and boundary condition

nodes generated are 16977 for this model.

surface of the block shown in Fig. 8.

**Figure 8.** Contact and target elements

*3.2.5. Solutions* 

*3.2.4. Boundary condition and application of pressure* 

applied which is related with Qmax (5414.28 N) for bearing no. 2206

*3.2.3. Contact model* 

The region of most interest is adjacent to the contact interface and has the greatest concentration of elements for lower interferences. Away from the contact region, the mesh becomes coarser to minimize the computational efforts. In the present all FE models in contact region element edge length is taken as 0.08 and in other area it is taken as 0.5, the details of which is shown in Fig 8. The total numbers of elements generated are 14120 and

In order to create contact models in ANSYS, a contact pair of elements must be created a contact element and a target element. ANSYS has general guidelines as to what line, surface, or volume these elements should be applied. Perhaps the most critical feature is the mesh size. For example, a large target element size and very fine contact element will not work. The sizes of the contact and target elements should be fairly close to one another. It is possible to get a solution to converge, but the results will most likely be incorrect. That is why there is a densely meshed region in both the bottom part of the half-cylinder and on the

The boundary conditions are presented in Fig. 9. The nodes on the bottom surface of the flat plat all degree of freedoms are restricted and rigidly constrained from translating in the x, y and z direction. Where as on top surface of half cylinder uniform pressure of 68.16 N/mm2 is

The solutions have been carried out by means of a PC. The hardware configuration consists or an Intel Pentium IV 2.53 GHz CPU with 1 GB of RAM. The models were solved in round 25 minutes to 7 hours. All models have been solved as 3D static with Newton Raphson option.

## *3.2.6. Evaluation of the finite element model*

To examine the appropriateness of the finite element mesh and modeling assumptions, such as the dimensions and fineness of the mesh and the imposed boundary conditions, finite element results for an elastic half-space indented by a rigid cylindrical asperity were compared with analytical results for line contacts. The FE model was first verified by comparing its output with the analytical results of the Hertz solution in the elastic regime. The verification included the contact pressure, contact stresses, deformation and contact width. For the evaluation of Finite Element Model initially von Mises stress criteria is taken for the consideration because it is the final output of analytical study as discussed in section 2. Also it is an important stress which should remain within limit with respect to yield stress of the material. Figure 10 shows the contour plot of von Mises stress for the applied load of 5414.28 N. As it is clear from the figure that at contact zone induced stress is higher which is marked by red colour. Figure 11 shows the detail view of contact zone with node numbers. Value of von Mises stress is to be identifying for the node no 5258 which is on contact surfaces and it is 1031.1 N/mm2. Whereas analytical result gives 970.31 N/mm2 (Table 1). Thus the von Mises stress of FE model differs from the Hertz solution by 5.8% which is acceptable for the present analysis. The small differences between analytical and FEA solutions near the contact edge may be attributed to the fineness of the mesh. The favorable comparison of the results illustrates the suitability of the finite element model for the present analysis involving only global variables, such as von Mises stress, contact pressure, contact width and deformation.

Figure 12 shows the contour plot for shear stress distribution. It is clear from the FE analysis that value of induced shear stress is 457.88 N/mm2 and analytical result gives 485.16 N/mm2 with the percentage error of 5.6 % only.

Development of Graphical Solution to Determine Optimum

Deformation, δ

(mm)

Hollowness of Hollow Cylindrical Roller Bearing Using Elastic Finite Element Analysis 247

Contact pressure, p (N/mm2)

Now using similar contact model and boundary conditions FE analysis has been carried out for all five rollers. Applied loads are taken as per the calculated load distribution among the rollers. Table 6 shows the validation of meshing scheme employed by comparing the

Figure 13 to 16 shows the graphical comparison of Table 6. Four important parameters von Mises stress, Contact Pressure, Deformation and Contact Width are plotted for different

Analytical FEA Analytical FEA Analytical FEA Analytical FEA

5414.28 970.31 1031.1 0.1321 0.1412 2175.58 2034.5 0.01206 0.01288 4730.5 906.97 958.92 0.1235 0.1322 2033.57 1898.2 0.01068 0.01148 2890.5 708.97 719.39 0.0965 0.1066 1589.61 1437.9 0.00685 0.00742 517.68 300.03 242.09 0.0408 0.0467 672.72 587.2 0.00146 0.00184

10397.88 1093.51 1083.7 0.1929 0.2097 2451.83 2255.4 0.01917 0.02028 9404.92 1039.99 1018.9 0.1835 0.2003 2331.82 2136 0.01752 0.0186 6656.75 874.95 840.21 0.1544 0.1726 1961.77 1754.5 0.01283 0.01376 2823.74 569.86 519.65 0.1005 0.1218 1277.7 1053.9 0.00593 0.006469

21620.59 1029.05 1110 0.2713 0.2991 2307.3 2092.5 0.02581 0.02643 19801.61 984.81 1050.2 0.2596 0.2877 2208.11 1992.4 0.02385 0.02451 14716.21 848.99 914.4 0.2238 0.2529 1903.56 1684.6 0.01826 0.01899 7444.23 603.83 616.54 0.1592 0.1879 1353.88 1146.5 0.00989 0.01057

50656.34 1166.57 1245.6 0.4112 0.4659 2615.63 2308.2 0.04333 0.04357 45818.86 1109.47 1176.8 0.3911 0.4466 2487.6 2177.8 0.03959 0.04 32430.3 933.4 957.3 0.329 0.3860 2092.83 1783.4 0.02901 0.02989 13756.7 607.93 592.31 0.2143 0.2723 1363.06 1072.4 0.01341 0.01442

73318.39 1169.78 1237.3 0.4946 0.5677 2622.82 2285 0.05224 0.05235 66316.76 1112.52 1174.7 0.4704 0.5442 2494.44 2155.9 0.04773 0.04811 46938.59 935.97 963.4 0.3957 0.4706 2098.59 1764.6 0.03497 0.03604 19911.01 609.6 582.93 0.2577 0.3306 1366.81 1065.4 0.01616 0.01758

Contact width, b

(mm)

analytical results with FEA.

Load (N) Von Mises stress, σVM (N/mm2)

applied load.

Roller 1

Roller 2

Roller 3

Roller 4

Roller 5

**Table 6.** Validation of meshing scheme employed

**Figure 10.** von Mises stress distribution over solid cylinder-flat

**Figure 11.** Detail of contact zone for von Mises stress distribution

**Figure 12.** Contour plots for shear stress

Now using similar contact model and boundary conditions FE analysis has been carried out for all five rollers. Applied loads are taken as per the calculated load distribution among the rollers. Table 6 shows the validation of meshing scheme employed by comparing the analytical results with FEA.

246 Finite Element Analysis – Applications in Mechanical Engineering

**Figure 10.** von Mises stress distribution over solid cylinder-flat

**Figure 11.** Detail of contact zone for von Mises stress distribution

**Figure 12.** Contour plots for shear stress

Figure 13 to 16 shows the graphical comparison of Table 6. Four important parameters von Mises stress, Contact Pressure, Deformation and Contact Width are plotted for different applied load.


**Table 6.** Validation of meshing scheme employed

Development of Graphical Solution to Determine Optimum

Hollowness of Hollow Cylindrical Roller Bearing Using Elastic Finite Element Analysis 249

The finite element method was used to analyze the contact mechanics aspects of nominally flat single-asperity surfaces and to identify the effect of important parameters like von Mises stress, contact pressure, contact width and deformation for the given load. On the basis of

The smaller error in the FE model is attributed to overall balance (static equilibrium) enforced by the FEM package. The smaller differences between analytical and FEA solutions

On the basis of the results discussed, it may be concluded that the finite element configuration shown in Fig. 10 and the invoked modeling approximations are acceptable for

Figure 14 to 17 shows that the agreement between analytical and finite element results from different rollers and various load is appreciably good. The maximum disagreement between the FEA value and analytical values occurs at the lowest applied load. The accord between the FEA and analytical results gets progressively better as higher applied load. Thus smaller the interference the smaller number of contact elements are in effect, leading to a large error

Different hollowness percentage ranging from 10% to 80% (in step of 10%) has been investigated for Roller 1 of diameter 6.62 mm which is equivalent of 8 mm diameter roller co-relate with bearing 2206. Figure 17 shows the finite element model for 40 % hollowness. Same surfaces as taken in the contact model of solid roller and flat i.e. outer surface of roller and top surface of flat plate are selected for contact element and target element respectively.

the presented results, the following major conclusions can be drawn.

near the contact edge may be attributed to the fineness of the mesh.

**4. Finite element analysis for hollow roller and flat contact** 

*3.2.7. Summary of FE analysis* 

the purpose of the present analysis.

and visa versa.

**4.1. Finite element model** 

**Figure 17.** Finite element model for 40% Hollowness

**Figure 13.** von Mises stress vs Applied load

**Figure 14.** Contact Pressure vs Applied load

**Figure 15.** Deformation vs Applied load

**Figure 16.** Contact Width vs Applied load

## *3.2.7. Summary of FE analysis*

248 Finite Element Analysis – Applications in Mechanical Engineering

**Figure 13.** von Mises stress vs Applied load

**von Mises stress (N/mm^2)**

0 10000 20000 30000 40000 50000 60000 70000 80000 **Applied Load (N)**

0 20000 40000 60000 80000 **Applied Load (N)**

0 10000 20000 30000 40000 50000 60000 70000 80000 **Applied Load (N)**

0 20000 40000 60000 80000 **Applied Load (N)**

Roller 1 Analytical Roller 2 Analytical Roller 3 Analytical Roller 4 Analytical Roller 5 Analytical Roller 1 FEM Roller 2 FEM Roller 3 FEM Roller 4 FEM Roller 5 FEM

> Roller 1 Analytical Roller 2 Analytical Roller 3 Analytical Roller 4 Analytical Roller 5 Analytical Roller 1 FEM Roller 2 FEM Roller 3 FEM Roller 4 FEM Roller 5 FEM

Roller 1 Analytical Roller 2 Analytical Roller 3 Analytical Roller 4 Analytical Roller 5 Analytical Roller 1 FEM Roller 2 FEM Roller 3 FEM Roller 4 FEM Roller 5 FEM

Roller 1 Analytical Roller 2 Analytical Roller 3 Analytical Roller 4 Analytical Roller 5 Analytical Roller 1 FEM Roller 2 FEM Roller 3 FEM Roller 4 FEM Roller 5 FEM

**Figure 14.** Contact Pressure vs Applied load

**Contact Pressure (N/mm^2)**

**Figure 15.** Deformation vs Applied load

**Contact Width (mm)**

**Deformation (mm)**

0 0.01 0.02 0.03 0.04 0.05 0.06

**Figure 16.** Contact Width vs Applied load

0 0.1 0.2 0.3 0.4 0.5 0.6 The finite element method was used to analyze the contact mechanics aspects of nominally flat single-asperity surfaces and to identify the effect of important parameters like von Mises stress, contact pressure, contact width and deformation for the given load. On the basis of the presented results, the following major conclusions can be drawn.

The smaller error in the FE model is attributed to overall balance (static equilibrium) enforced by the FEM package. The smaller differences between analytical and FEA solutions near the contact edge may be attributed to the fineness of the mesh.

On the basis of the results discussed, it may be concluded that the finite element configuration shown in Fig. 10 and the invoked modeling approximations are acceptable for the purpose of the present analysis.

Figure 14 to 17 shows that the agreement between analytical and finite element results from different rollers and various load is appreciably good. The maximum disagreement between the FEA value and analytical values occurs at the lowest applied load. The accord between the FEA and analytical results gets progressively better as higher applied load. Thus smaller the interference the smaller number of contact elements are in effect, leading to a large error and visa versa.

## **4. Finite element analysis for hollow roller and flat contact**

## **4.1. Finite element model**

Different hollowness percentage ranging from 10% to 80% (in step of 10%) has been investigated for Roller 1 of diameter 6.62 mm which is equivalent of 8 mm diameter roller co-relate with bearing 2206. Figure 17 shows the finite element model for 40 % hollowness. Same surfaces as taken in the contact model of solid roller and flat i.e. outer surface of roller and top surface of flat plate are selected for contact element and target element respectively.

**Figure 17.** Finite element model for 40% Hollowness

## **4.2. Meshing**

Now in this case also for the contact surface element edge length is taken as 0.08. Also as discussed by Murthy & Rao (1983) that, in addition to the contact stresses at the outer contact zone, the hollow specimens are subjected to tangential stresses (bending stress) at inner surface. Thus for the inner surface element edge length should be high and is taken as 0.08 as shown in Fig. 18. For other remaining area it is taken as 0.5.

Development of Graphical Solution to Determine Optimum

Hollowness of Hollow Cylindrical Roller Bearing Using Elastic Finite Element Analysis 251

Bending stress (N/mm2) Deformation

(mm)

**Figure 19.** von Mises stress plots for hollow roller

Contact pressure (N/mm2)

**Table 7.** Values of parameters for different hollowness for Roller 1

Roller 1 Endurance Limit

**Figure 20.** Bending stress vs Hollowness for Roller 1

**Bending Stress (N/mm^2)**

von Mises stress (N/mm2)

(a) 10% hollowness (b) 80% hollowness

0 10 20 30 40 50 60 70 80 90 **% Hollowness**

10 1901 952.92 377.07 0.01253 20 1812.3 895.99 422.55 0.01264 30 1679.6 843.84 463.1 0.01345 40 1536.3 785.46 539.91 0.01517 50 1391.9 741.49 650.03 0.01885 60 1247.1 704.71 822.71 0.02663 70 1093.3 832.79 1123.8 0.04516 80 897.31 1379.1 1721.4 0.10102

Hollowness

%

**Figure 18.** Densely meshed regions of the contact model of hollow roller and flat

Same boundary condition and pressure is applied as discussed in section 3.2.4. For each hollowness 10% to 80%, FE model is developed as shown in Fig. 19, maximum applied load 5414.28 N is taken and results are observed.

## **5. Results and discussion**

Due to thin section very less material is available to resist the force so von Mises stress is increase after 60% hollowness. Also at this stage plastic deformation will take place and failure will occur due to permanent deformation, which is not desirable and should be avoided.

An added criterion for evaluation in a bearing with hollow rollers is the roller bending stress. To evaluate the life integrals, the value of the fatigue limit stress must be known for the bearing component material. This can be determined by endurance testing of bearings or selected components. Performance analyses were conducted, using the von Mises stress as the fatigue failure-initiating criterion. Based on this subsequent study fatigue limit stress for bearing material AISI 52100 is 684 N/mm2 (Harris and Kotzalas, 2007). From Table 7 it is very clear that bending stress is continuously increase from 377.07 N/mm2 to 1721.4 N/mm2 as hollowness increase from 10% to 80% respectively. But the practical limit of this stress is 684 N/mm2. So the hollowness should be restricted upto 52% which is clear from Fig. 20.

#### Development of Graphical Solution to Determine Optimum

Hollowness of Hollow Cylindrical Roller Bearing Using Elastic Finite Element Analysis 251

**Figure 19.** von Mises stress plots for hollow roller

250 Finite Element Analysis – Applications in Mechanical Engineering

0.08 as shown in Fig. 18. For other remaining area it is taken as 0.5.

**Figure 18.** Densely meshed regions of the contact model of hollow roller and flat

5414.28 N is taken and results are observed.

**5. Results and discussion** 

avoided.

from Fig. 20.

Same boundary condition and pressure is applied as discussed in section 3.2.4. For each hollowness 10% to 80%, FE model is developed as shown in Fig. 19, maximum applied load

Due to thin section very less material is available to resist the force so von Mises stress is increase after 60% hollowness. Also at this stage plastic deformation will take place and failure will occur due to permanent deformation, which is not desirable and should be

An added criterion for evaluation in a bearing with hollow rollers is the roller bending stress. To evaluate the life integrals, the value of the fatigue limit stress must be known for the bearing component material. This can be determined by endurance testing of bearings or selected components. Performance analyses were conducted, using the von Mises stress as the fatigue failure-initiating criterion. Based on this subsequent study fatigue limit stress for bearing material AISI 52100 is 684 N/mm2 (Harris and Kotzalas, 2007). From Table 7 it is very clear that bending stress is continuously increase from 377.07 N/mm2 to 1721.4 N/mm2 as hollowness increase from 10% to 80% respectively. But the practical limit of this stress is 684 N/mm2. So the hollowness should be restricted upto 52% which is clear

Now in this case also for the contact surface element edge length is taken as 0.08. Also as discussed by Murthy & Rao (1983) that, in addition to the contact stresses at the outer contact zone, the hollow specimens are subjected to tangential stresses (bending stress) at inner surface. Thus for the inner surface element edge length should be high and is taken as

**4.2. Meshing** 


**Table 7.** Values of parameters for different hollowness for Roller 1

**Figure 20.** Bending stress vs Hollowness for Roller 1

Thus for the present case of Roller 1, for the applied load of 5414.28 N the % hollowness of the hollow roller should not exceed 52%, otherwise induced stress at the bore of the roller will increase beyond the endurance limit and cause fatigue failure of roller.

Development of Graphical Solution to Determine Optimum

Hollowness of Hollow Cylindrical Roller Bearing Using Elastic Finite Element Analysis 253

0 10 20 30 40 50 60 70 80 **% Hollowness**

0 10 20 30 40 50 60 70 80 **% Hollowness**

**Figure 23.** Deformation vs Hollowness

**von Mises stress (N/mm^2)**

**Deformation (mm)**

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

**Figure 24.** Mises stress vs Hollowness

**Contact Pressure (N/mm^2)**

**Figure 25.** Contact pressure vs Hollowness

0

500

1000

1500

2000

2500

The term "hollowness" as referred to here for the rollers in the ratio of the inner diameter to the outer diameter expressed as a percentage and is an important parameter in the design of the hollow roller bearing. It determines not only the overall bearing stiffness but also the amount of preload most desirable, the bearing's load capacity and its life. In fact, hollowness

0 10 20 30 40 50 60 70 80 **% Hollowness**

Great care must be given to the smooth finishing of the inside surface of a hollow roller during manufacturing as the stress raisers that offer due to poorly finished inside surface will reduce the allowable roller hollowness ratios still further than indicated by Fig. 20.

**Figure 21.** Contour plot for maximum shear stress for hollow roller

Figure 21 shows the contour plot of maximum shear stress for 52% hollowness. The induced shear stress is 273.68 N/mm2 which is approximately half than the shear stress induced in solid roller for the same load of 5414.28 N. Thus reduction in shear stress gives improvement in fatigue life of bearing.

Figure 22 to 25 shows the effect of hollowness on different parameters.

**Figure 22.** Effect of hollowness on the deformation for same applied load

**Figure 23.** Deformation vs Hollowness

Thus for the present case of Roller 1, for the applied load of 5414.28 N the % hollowness of the hollow roller should not exceed 52%, otherwise induced stress at the bore of the roller

Great care must be given to the smooth finishing of the inside surface of a hollow roller during manufacturing as the stress raisers that offer due to poorly finished inside surface will reduce the allowable roller hollowness ratios still further than indicated by Fig. 20.

Figure 21 shows the contour plot of maximum shear stress for 52% hollowness. The induced shear stress is 273.68 N/mm2 which is approximately half than the shear stress induced in solid roller for the same load of 5414.28 N. Thus reduction in shear stress gives

> 0 1000 2000 3000 4000 5000 6000 **Applied Load (N)**

will increase beyond the endurance limit and cause fatigue failure of roller.

**Figure 21.** Contour plot for maximum shear stress for hollow roller

Figure 22 to 25 shows the effect of hollowness on different parameters.

10% 20% 30% 40% 50% 60% 70%

**Figure 22.** Effect of hollowness on the deformation for same applied load

improvement in fatigue life of bearing.

**Deformation (mm)**

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

**Figure 24.** Mises stress vs Hollowness

**Figure 25.** Contact pressure vs Hollowness

The term "hollowness" as referred to here for the rollers in the ratio of the inner diameter to the outer diameter expressed as a percentage and is an important parameter in the design of the hollow roller bearing. It determines not only the overall bearing stiffness but also the amount of preload most desirable, the bearing's load capacity and its life. In fact, hollowness

is a control parameter used to optimize the bearing design. In the present case of Roller 1, load is applied in such a way that induced bending stress at inner bore should cross endurance limit of the material i.e. 684 N/mm2 for each hollowness. Result of FE analysis is shown in Fig. 26. The roller hollowness values from 10% to 80% have been analyze by Finite Element as discuss above for Roller 1 and the roller load, deflection and stress curves of Fig. 27 have been developed. The dotted line across these curves show the points of constant maximum roller bore stress for values of 684 N/mm2.

Development of Graphical Solution to Determine Optimum

Hollowness of Hollow Cylindrical Roller Bearing Using Elastic Finite Element Analysis 255

Load (N)

identified. Thus similar analysis can also be carried out for remaining four rollers to find the optimum hollowness and results are given in Table 9. For each roller these analyses have been carried out upto the hollowness where induced bending stress should just cross the

% Hollowness Max. Applied

10 9850 20 8800 30 8000 40 6900 50 5700 60 4500 70 3300 80 2520

**Table 8.** Maximum applied load for different hollowness for Roller 1

% Hollowness

**Table 9.** Values of parameters for different hollowness for Roller 2, 3, 4 & 5

Contact pressure (N/mm2) von Mises stress (N/mm2)

10 2153.5 1067.8 500.59 0.0195 20 1984.2 1045.7 548.37 0.02 30 1812.5 969.11 599.86 0.02149 40 1652.5 900.87 693.55 0.02461

10 1917.9 975.09 468.32 0.026 20 1769.9 909.63 491.53 0.02671 30 1629.6 855.49 538.21 0.02871 40 1484 818.1 619.64 0.03284 50 1324.6 804.27 746.49 0.04103

10 2126.9 1094.9 620.35 0.0428 20 1951.7 1021.8 634.68 0.04448 30 1802.2 984.04 689.2 0.04793

10 2105.2 1084.6 633.57 0.05157 20 1922 1009.9 640.66 0.05374 30 1820.2 993.23 692.28 0.05704

Bending stress (N/mm2)

Deformation

(mm)

endurance limit.

Roller no Maximum

Roller 2 10397.88

Roller 3 21620.59

Roller 4 50656.34

Roller 5 73318.39

Load (Qmax) N

**Figure 26.** Bending stress crosses the endurance limit for different hollowness

**Figure 27.** Relationship between the roller hollowness, Deflection and bore stress for 6.62 mm diameter roller

Corresponding to each roller hollowness value, there is a specific optimum load for each bearing design, which is indicated in Table 8.

Figure 27 gives the best solution to find the optimum hollowness for verities of load. But this is not the final solution, because solution given in Fig. 27 is only applicable for Roller 1 i.e. equivalent roller of bearing 2206. As bearing/roller geometry will change again same procedure of FE analysis should be carried out and again optimum hollowness is to be identified. Thus similar analysis can also be carried out for remaining four rollers to find the optimum hollowness and results are given in Table 9. For each roller these analyses have been carried out upto the hollowness where induced bending stress should just cross the endurance limit.


**Table 8.** Maximum applied load for different hollowness for Roller 1

254 Finite Element Analysis – Applications in Mechanical Engineering

maximum roller bore stress for values of 684 N/mm2.

**Figure 26.** Bending stress crosses the endurance limit for different hollowness

roller

**Bending Stress (N/m**

 **m ^2)**

bearing design, which is indicated in Table 8.

**Figure 27.** Relationship between the roller hollowness, Deflection and bore stress for 6.62 mm diameter

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000 7500 8000 8500 9000 9500 10000 10500 11000 11500 12000 12500 13000 **Applied Load (N)**

10% 20% 30% 40% 50% 60% 70% 80% Endurance limit

Corresponding to each roller hollowness value, there is a specific optimum load for each

Figure 27 gives the best solution to find the optimum hollowness for verities of load. But this is not the final solution, because solution given in Fig. 27 is only applicable for Roller 1 i.e. equivalent roller of bearing 2206. As bearing/roller geometry will change again same procedure of FE analysis should be carried out and again optimum hollowness is to be

is a control parameter used to optimize the bearing design. In the present case of Roller 1, load is applied in such a way that induced bending stress at inner bore should cross endurance limit of the material i.e. 684 N/mm2 for each hollowness. Result of FE analysis is shown in Fig. 26. The roller hollowness values from 10% to 80% have been analyze by Finite Element as discuss above for Roller 1 and the roller load, deflection and stress curves of Fig. 27 have been developed. The dotted line across these curves show the points of constant


**Table 9.** Values of parameters for different hollowness for Roller 2, 3, 4 & 5

Development of Graphical Solution to Determine Optimum

Hollowness of Hollow Cylindrical Roller Bearing Using Elastic Finite Element Analysis 257

1% 5% 10% 20% 30% 40% 50% 60% 70% 80% 90% 95%

68.16 330.68 347.21 377.07 422.55 463.1 539.91 650.03 822.71 1123.8 1721.4 3365 4653.6

Applied pressure (N/mm2) Bending Stress (N/mm2)

99.06 515.11 526 633.57 640.66 692.28 98.52 497.6 512.3 620.35 634.68 689.2

86.56 441.46 450.112 500.59 548.37 599.86 693.55 78.3 419.68 418.9 452.84 496.14 542.76 627.65 76.66 371.8 401.3 468.32 491.53 538.21 619.64 746.49 70.21 344.029 359.13 429 450.31 493.08 567.54 683.78

55.42 277.1 282.64 321.12 351.66 384.46 444.33 52.18 263.5 273.95 319.16 334.9 366.59 422.04 508.42 36.39 189.23 192.867 201.72 225.7 247.51 288.52 347.05 439.33 26.39 129.311 138.475 161.84 169.76 185.62 213.61 257.35 23.51 112.848 124.6 136.5 149.44 163.28 188.71

59.55 303.71 309.66 329.92 369.25 404.64 471.74 567.96 718.81

6.52 33.9 34.88 36.36 40.636 44.509 51.777 62.158 78.697 **Table 10.** Value of bending stress corresponding to the applied pressure

**Figure 29.** Bending stress vs applied pressure for different hollowness

In case of solid roller bearing induced sub-surface stresses are the limiting criteria for the fatigue life of bearing whereas for hollow roller bearing bending stress is the limiting criteria. The bending stresses on the internal diameter of the roller in the plane of the loading forces are the most critical for destructions. In the present work graphical solution was developed to determine optimum hollowness of cylindrical roller bearing for which

**6. Conclusion** 

**Figure 28.** Comparison of hollowness for different rollers

From Fig. 28 it is clear that for

Roller 1 optimum hollowness should be 52% for the applied load of 5414.28 N, Roller 2 optimum hollowness should be 39% for the applied load of 10397.88 N, Roller 3 optimum hollowness should be 45% for the applied load of 21620.59 N, Roller 4 optimum hollowness should be 29% for the applied load of 50656.34 N, Roller 5 optimum hollowness should be 28% for the applied load of 73318.39 N.

It is very clear from the results and discussion of all five rollers that optimum value of hollowness is dependent on magnitude of applied load, bearing geometry i.e. diameter and length of roller and mechanical properties of material used. If the value of applied load will increase than hollowness should be reduced to maintain the bending stress within endurance limit of the material. Change in bearing geometry will change the applied pressure and resulted into change in hollowness. Thus the solution given in Fig. 26 and 27 is not a generalized solution and it can not be applicable to any bearing geometry for any load. It is applicable to specific type of bearing and for specific load only respectively. If applied load will change one can't use the results shown in Fig. 28 and lengthy FE procedure should be again carried out to get the result in the form of hollowness.

## **5.1. Generalized graphical solution**

To find the optimum hollowness for any material and for any applied load irrespective of bearing geometry, in the present work large data are generated by FE analysis. To get the generalized solution FE analysis for the hollowness percentage ranging from 1% to 95% (after 95% ANSYS solution was not supported) is carried out and following Table 10 has been developed. This table shows the values of bending stress corresponding to the applied pressure.

Table 10 is presented in graphical form in Fig. 29. This diagram shows the value of bending stress for different applied pressure with respect to hollowness ranging from 1% to 95%.

#### Development of Graphical Solution to Determine Optimum Hollowness of Hollow Cylindrical Roller Bearing Using Elastic Finite Element Analysis 257


**Table 10.** Value of bending stress corresponding to the applied pressure

**Figure 29.** Bending stress vs applied pressure for different hollowness

## **6. Conclusion**

256 Finite Element Analysis – Applications in Mechanical Engineering

**Figure 28.** Comparison of hollowness for different rollers

be again carried out to get the result in the form of hollowness.

**5.1. Generalized graphical solution** 

applied pressure.

Roller 1 optimum hollowness should be 52% for the applied load of 5414.28 N, Roller 2 optimum hollowness should be 39% for the applied load of 10397.88 N, Roller 3 optimum hollowness should be 45% for the applied load of 21620.59 N, Roller 4 optimum hollowness should be 29% for the applied load of 50656.34 N, Roller 5 optimum hollowness should be 28% for the applied load of 73318.39 N.

It is very clear from the results and discussion of all five rollers that optimum value of hollowness is dependent on magnitude of applied load, bearing geometry i.e. diameter and length of roller and mechanical properties of material used. If the value of applied load will increase than hollowness should be reduced to maintain the bending stress within endurance limit of the material. Change in bearing geometry will change the applied pressure and resulted into change in hollowness. Thus the solution given in Fig. 26 and 27 is not a generalized solution and it can not be applicable to any bearing geometry for any load. It is applicable to specific type of bearing and for specific load only respectively. If applied load will change one can't use the results shown in Fig. 28 and lengthy FE procedure should

0 10 20 30 40 50 60 70 80 90 **% Hollowness**

Roller 1 Roller 2 Roller 3 Roller 4 Roller 5 Endurance limit

To find the optimum hollowness for any material and for any applied load irrespective of bearing geometry, in the present work large data are generated by FE analysis. To get the generalized solution FE analysis for the hollowness percentage ranging from 1% to 95% (after 95% ANSYS solution was not supported) is carried out and following Table 10 has been developed. This table shows the values of bending stress corresponding to the

Table 10 is presented in graphical form in Fig. 29. This diagram shows the value of bending stress for different applied pressure with respect to hollowness ranging from 1% to 95%.

From Fig. 28 it is clear that for

**Bending stress (N/mm^2)**

> In case of solid roller bearing induced sub-surface stresses are the limiting criteria for the fatigue life of bearing whereas for hollow roller bearing bending stress is the limiting criteria. The bending stresses on the internal diameter of the roller in the plane of the loading forces are the most critical for destructions. In the present work graphical solution was developed to determine optimum hollowness of cylindrical roller bearing for which

induced bending stress should be within the endurance limit of the material. Figure 29 shows the generalized diagram for bending stress vs applied pressure. Following are the major outcomes from this diagram.

Development of Graphical Solution to Determine Optimum

Hollowness of Hollow Cylindrical Roller Bearing Using Elastic Finite Element Analysis 259

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Komvopoulos, K. and Choi, D. –H. (1992) Elastic Finite Element Analysis of Multi-Asperity

Murthy, C.S.C. and Rao, A. R. (1983) Mechanics and Behaviour of Hollow Cylindrical

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Timoshenko S. P. and Godier J. N. (1970) Theory of Elasticity. 3rd ed., McGraw-Hall

Yangang, W., Yi, Q., Raj, B. and Qingyu, J., 2004, "FE analysis of a novel roller form : A deep end cavity roller for roller type bearings," Elsevier, Journal of Materials Processing

Soc. London, Ser A. 295, 300 – 319. ISSN 1471-2946.

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I. S. 9202 (2001) Specification for Cylindrical Rollers. Edition 1.1.

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0-471-35457-0

For the same value of applied pressure, Fig 29 shows that there is very small variation in the value of bending stress by increase the hollowness from 10% to 30%.

If the hollowness increases from 1% to 95% the slop of line will also increase accordingly.

The durability of the bearings with hollow rollers operating on cycles not exceeding the maximum permitted level of bending stresses can be substantially greater than the durability of similar bearings with solid rollers.

For the applied load on equivalent size of roller initially applied pressure is to be calculated. As per the endurance limit of the material used and calculated applied pressure optimum hollowness can be identified from the diagram.

For the particular hollowness diagram gives the maximum limit of applied pressure and hence applied load. The developed graphical solution can be applicable for any material of bearing.

## **Author details**

P.H. Darji *Department of Mechanical Engineering, C. U. Shah College of Engineering & Technology, Surendranagar, India* 

D.P. Vakharia *Department of Mechanical Engineering, S.V. National Institute of Technology, Surat, India* 

## **7. References**


Design Data (1994). PSG College of Technology, Coimbatore.


durability of similar bearings with solid rollers.

hollowness can be identified from the diagram.

bearing.

P.H. Darji

**Author details** 

*Surendranagar, India* 

D.P. Vakharia

**7. References** 

major outcomes from this diagram.

induced bending stress should be within the endurance limit of the material. Figure 29 shows the generalized diagram for bending stress vs applied pressure. Following are the

For the same value of applied pressure, Fig 29 shows that there is very small variation in the

The durability of the bearings with hollow rollers operating on cycles not exceeding the maximum permitted level of bending stresses can be substantially greater than the

For the applied load on equivalent size of roller initially applied pressure is to be calculated. As per the endurance limit of the material used and calculated applied pressure optimum

For the particular hollowness diagram gives the maximum limit of applied pressure and hence applied load. The developed graphical solution can be applicable for any material of

*Department of Mechanical Engineering, C. U. Shah College of Engineering & Technology,* 

*Department of Mechanical Engineering, S.V. National Institute of Technology, Surat, India* 

Rolling Elements," NASA TN D – 8313, Washington.

Nov 6, Paper No. IMECE2008-67294. ISBN: 978-0-7918-4873-9

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Bamberger, E. N., Parker, R. J. and Dietrich, M. W., 1976, "Flexural Fatigue of Hollow

Bhateja, C. P. and Hahn, R. S., 1980, "A Hollow Roller Bearing for Use in Precision Machine

Darji, P. H. and Vakharia, D. P., (2008), "Determination of Optimum hollowness for hollow cylindrical rolling element bearing", Proceedings of ASME 2008 International Mechanical Engineering congress and exposition, Boston, Massachusetts, USA, Oct 31-

Demirhan, N. and Kanber, B. (2008) Stress and Displacement Distributions of Cylindrical Roller Bearing Rings Using FEM. Taylor & Francis, Mechanics Based Design of

If the hollowness increases from 1% to 95% the slop of line will also increase accordingly.

value of bending stress by increase the hollowness from 10% to 30%.


Zhao, H., 1998, "Analysis of Load Distributions within Solid and Hollow Roller Bearings", ASME J. Tribol., 120, pp. 134 – 139. ISSN 0742-4787.

**Chapter 12** 

© 2012 Saito, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Saito, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Finite Element Analysis** 

Additional information is available at the end of the chapter

Toshihiro Saito

**1. Introduction** 

http://dx.doi.org/10.5772/46194

engine rpm while shifting the gear ratio.

**Coupled with Feedback Control for** 

overall efficiency of a vehicle is the efficiency of its transmission.

**Dynamics of Metal Pushing V-Belt CVT** 

Nowadays automobiles are required to meet environmental requirements, such as lower exhaust emissions and higher fuel economy. One of the key factors for improving the

A CVT has a greater potential for improving fuel economy than a step-type automatic transmission (AT), because of its integrated control with the engine [1]. That is, CVTs are capable of continuously tracing engine operating ranges with high fuel efficiency. Another advantage is that CVTs allow vehicles to drive without lowering the driving torque or the

However, when the transmission efficiency of a CVT by itself is compared with a step-type AT, CVT is known to have lower efficiency because its driving torque is transferred by means of contact and friction [2]. The transmission efficiency of a CVT is determined by friction loss at its oil pump and metal pushing V-belt. The oil pump must produce enough pulley pressure so that the metal V-belt mounted between two pulleys does not slip. A

As for the metal V-belt, gradually lowering pulley pressure while maintaining a constant transmission torque increases the transmission efficiency of the belt by itself, as long as it does not slip on the pulleys. However, the transmission efficiency begins to drop under a certain operating condition. This implies the existence of an optimum operating condition that maximizes the transmission efficiency of the belt [4]. To find this condition, it is important to predict friction loss at each portion of the metal V-belt during CVT operation. A considerable amount of research has so far been made on methods for calculating friction loss that occurs at each part of the V-belt, but many of them use simple equations that are

higher pulley pressure, however, means a greater friction loss at the oil pump [3].
