A new logarithmic profile model and optimization design of cylindrical roller bearing

A new logarithmic profile model and optimization design of cylindrical roller bearing
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  Industrial Lubrication and Tribology A new logarithmic profile model and optimization design of cylindrical roller bearing Li Cui Yafei He  Article information: To cite this document:Li Cui Yafei He , (2015),"A new logarithmic profile model and optimization design of cylindrical roller bearing", IndustrialLubrication and Tribology, Vol. 67 Iss 5 pp. 498 - 508 Permanent link to this document: Downloaded on: 23 September 2015, At: 15:18 (PT)References: this document contains references to 16 other documents.To copy this document: permissions@emeraldinsight.comThe fulltext of this document has been downloaded 28 times since 2015* Users who downloaded this article also downloaded: Fanming Meng, Yuanpei Chen, (2015),"Analysis of elasto-hydrodynamic lubrication of journal bearing based on differentnumerical methods", Industrial Lubrication and Tribology, Vol. 67 Iss 5 pp. 486-497 Cheng, Xianghui Meng, Youbai Xie, Wenxiang Li, (2015),"On the running-in behavior of rough surface of piston ringsin mixed lubrication regime", Industrial Lubrication and Tribology, Vol. 67 Iss 5 pp. 468-485 Castro Lara, Henara Costa, José Daniel Biasoli de Mello, (2015),"Influence of layer thickness on sliding wear of multifunctional tribological coatings", Industrial Lubrication and Tribology, Vol. 67 Iss 5 pp. 460-467 Access to this document was granted through an Emerald subscription provided by emerald-srm:238453 [] For Authors If you would like to write for this, or any other Emerald publication, then please use our Emerald for Authors serviceinformation about how to choose which publication to write for and submission guidelines are available for all. Please for more information.  About Emerald Emerald is a global publisher linking research and practice to the benefit of society. The company manages a portfolio of more than 290 journals and over 2,350 books and book series volumes, as well as providing an extensive range of onlineproducts and additional customer resources and services. Emerald is both COUNTER 4 and TRANSFER compliant. The organization is a partner of the Committee on Publication Ethics(COPE) and also works with Portico and the LOCKSS initiative for digital archive preservation. *Related content and download information correct at time of download.    D  o  w  n   l  o  a   d  e   d   b  y   T  e  c   h  n   i  s  c   h  e   U  n   i  v  e  r  s   i   t  a  e   t   B  e  r   l   i  n   A   t   1   5  :   1   8   2   3   S  e  p   t  e  m   b  e  r   2   0   1   5   (   P   T   )  A new logarithmic profile model andoptimization design of cylindricalroller bearing Li Cui and Yafei He Faculty of Engineering, Shanghai Second Polytechnic University, Shanghai, China Abstract Purpose  – The purpose of this paper is to find a new logarithmic profile model of cylindrical roller bearing, which is expected to avoid edge effectand allow a straight portion on the roller considering uniform pressure distribution and easier manufacturing. Design/methodology/approach  – A new logarithmic cylindrical roller profile model using three parameters is proposed. Contact model betweenroller and rings and quasi-static model of roller bearing are given to obtain contact pressure distribution and solved by multi-grid andNewton–Raphson method. Optimization of modified reference rating life model of the roller bearing is proposed by using genetic algorithms. Findings  – Under heavy load or tilting moment conditions, modified reference rating life of cylindrical roller bearing may increase greatly byoptimization of three design parameters using the new logarithmic profile model. Originality/value  – The results of the present paper could aid in the design of logarithmic profile of cylindrical roller bearing and increase fatiguelife of cylindrical roller bearing. Keywords  Design, Optimization, Bearings, Fatigue life, Cylindrical roller bearing, Logarithmic profile, Optimization design Paper type  Research paper 1. Introduction Cylindrical roller bearing is used widely for its compactstructure and large carrying capacity. Concentrated stressoccurs at the end of contact area when a cylindrical rollercontacts with raceways, which results in early fatigue failure of the bearing, especially when the bearing is working undermisaligned loads condition. Therefore, the crowning design of roller to improve the contact condition between roller andrings has attracted substantial attention to increase the fatiguelife of cylindrical roller bearing.The early work by Lundberg (1939) introduced alogarithmic function to crowning of roller, which can ensureuniform contact pressure distribution between roller and ringsby using Lundberg’s profile. The function used very widelybecause of uniform contact pressure distribution and itsconvenience and simplicity. However, this curve isdiscontinuous at the end of roller because of an infinite dropat the end of the effective contact length and a finite value atthe end of roller, which results in difficult manufacturing. Johns-Gohar ( Johns and Gohar, 1981) improved Lundberg’s function by avoiding the infinite drop to obtain continuouslogarithmic curves; however, it was difficult to obtain uniformcontact pressure distribution, especially under heavy load andtilting moment conditions. Horng  et al.  (2000) provided adeformation formula for a circular crowned roller. Acomparison between crowned and non-crowned parts wasanalyzed. Zhu  et al.  (2012) gave a mixed lubrication model toinvestigate roller contact elastohydrodynamic lubrication(EHL) problems. Effect of roller length, crown radius andround corner radius were studied. Roller profile was optimizedto obtain uniform pressure distribution and film thicknessdistribution. Chen  et al.  (2001) studied the EHL characteristics in logarithmic profiled roller contact and theeffect of crown value. Urata (2000) proposed a crowningshape formed by combining two or more circular arcs. Edgestress disappeared in the crowning shape, but misalignmentcould not be considered in the model. Hiroki  et al.  (2009)proposed a new logarithm profile function. Three designparameters were introduced into the profile, and this profilecould prevent edge stress due to misalignment. However,these parameters are not easy to choose and also theparameters can only be optimized in a small range, thus theprofile is not used widely in the roller bearing design.Optimization of fatigue life of rolling bearing has beenstudied for its importance. Chakraborty  et al.  (2003)optimized ball bearing and roller bearings using geneticalgorithms. The results showed that the fatigue life of thebearing improved marginally compared with the fatigue lifegiven in standard catalogs. Rao and Tiwari (2007) improved The current issue and full text archive of this journal is available onEmerald Insight at: Industrial Lubrication and Tribology67/5 (2015) 498–508© Emerald Group Publishing Limited [ISSN 0036-8792][DOI 10.1108/ILT-01-2015-0006] This work was financially supported by Shanghai Natural ScienceFoundation (No. 14ZR1416800), Shanghai Alliance Plan (No.LM201408) and Foundation of Shanghai Second Polytechnic University(EGD14XQD12).Received 1 February 2015Revised 22 May 201523 June 2015Accepted 25 June 2015 498    D  o  w  n   l  o  a   d  e   d   b  y   T  e  c   h  n   i  s  c   h  e   U  n   i  v  e  r  s   i   t  a  e   t   B  e  r   l   i  n   A   t   1   5  :   1   8   2   3   S  e  p   t  e  m   b  e  r   2   0   1   5   (   P   T   )  the design optimization of deep-groove ball bearings usinggenetic algorithms, and the major improvement was informulating realistic constraints. However, these attempts foroptimum design of bearings were concentrated on the basicdesign geometric parameters of the bearings, such as diameterand number of rolling elements. Few works of literature areavailable that focus on the profile of rollers for theoptimization. Kamamoto  et al  . (2001) optimized the crowningprofile of cylindrical roller bearing. Results showed that thepressure became axially uniform and fatigue life wasimproved, but the effect of misalignment was not taken intoaccount. Kumar  et al.  (2009) optimized fatigue life of rollerbearing by genetic algorithm; geometric parameter and twologarithmic profile parameters of roller were chosen as designvariables. Results showed that the logarithmic profileparameters had a significant effect on the fatigue life.In the present paper, a new logarithmic roller profile modelon the basis of Johns-Gohar’s function is proposed. Themodel can be used in heavy load and tilting momentconditions, as well as is convenient in engineering application.Three design parameters of the profile are introduced, and theeffect of these parameters on the contact pressure and fatiguelife is analyzed. A procedure for optimizing fatigue life of the roller bearing is given by using genetic algorithm, theoptimized bearing life show improvement as compared with Johns-Gohar’s result. 2. Calculating of contact pressure of crownedroller 2.1 Three design parameters logarithmic profile Crowned roller in a cylindrical roller bearing is shown inFigure 1.  s  x   is crowning profile of roller.Lundberg gave a shape of crowning expressed in alogarithmic function; however, this curve had an infinite dropat the end of the effective contact length. Johns-Goharproposed an improved function based on Lundberg’s formula(Kabus  et al. , 2012): s  x     2 Q   l  we E  = ln   11    1    0.3033 ba   2 xl  we  2   (1)where  Q  is load on roller,  E  = is equivalent Young’s modulus,  x is position in the axial direction,  l  we  is effective roller length,  a is half length of the effective contact and  b  is half width of contact.However, edge stress cannot be avoided completely by Johns-Gohar model, especially under heavy load conditionand tilting moment conditions; also in addition, it is difficultto obtain uniform contact pressure distribution of roller. Tosolve these problems, a new logarithmic profile modelconsidering three design parameters is proposed: s  x    2 k 1 Q   l  we E  = ln   11     1    0.3033 k 2 ba   2 xl  we  2 k 3   (2)where  k 1  stands for multiple factor of the applied load,  k 2 stands for roller radius reduction at rolling contact surfaceends and  k 3  stands for ratio of straight portion to the profile. 2.2 Contact calculation model The contact between roller and ring is shown in Figure 2.EHL and dry contact are considered in contact calculationmodel.For EHL contact condition, Reynolds equation for aNewtonian lubricant can be expressed as follows: Figure 1  Crowned roller in a cylindrical roller bearing  j ZY O X Roller   s (  x ) Raceway (a)(b)Notes:  (a) Cylindrical roller bearing view; (b) crowned roller and raceway Figure 2  Line contact between roller and ring  C o  n  t a c  t  a  r e a Roller Ring A new logarithmic profile model Li Cui and Yafei He Industrial Lubrication and Tribology Volume 67 · Number 5 · 2015 · 498–508  499    D  o  w  n   l  o  a   d  e   d   b  y   T  e  c   h  n   i  s  c   h  e   U  n   i  v  e  r  s   i   t  a  e   t   B  e  r   l   i  n   A   t   1   5  :   1   8   2   3   S  e  p   t  e  m   b  e  r   2   0   1   5   (   P   T   )   x     h 3 12    p  x     y     h 3 12    p   y     u m     h    y    0 (3)where  p  is oil film pressure,  h  is film thickness,  u m  is the meansurface velocity of roller and ring in the flow direction,    isviscosity and     is density of the lubricant.Film thickness can be calculated by the following: h ( x ,  y )    h 00    s  x     x 2 2 R x   y 2 2 R  y   2   E  = · A  p  x = ,  y =  dx = dy =    x    x =  2     y    y =  2 (4)where  h 00  is minimum film thickness;  R x  and  R  y  are theeffective radius of curvature between roller and ring in the xand y direction; and elastic deformation due to the pressuredistribution is considered in the last item.Viscosity is varied as contact pressure (Harris, 2001):    p      0 exp   ln   0     9.67    1    1    p 1.96 · 10 8  0.6  (5)Density is given by Dowson–Higginson relation:     p       0 5.9 · 10 8   1.34  p 5.9 · 10 8   p (6)For dry contact condition, the distance between roller andring can also be calculated by the formula given in equation(4) because pressure occurs at the area where two surfacescome in contact with each other and no pressure occurs wherethere is no contact; the following relationships are obtained: h ( x ,  y )    0,  p ( x ,  y )    0 Non-contact area h ( x ,  y )    0,  p ( x ,  y )    0 Contact area (7)Pressure in dry contact condition is given by a semi-ellipticalpressure distribution based on the Hertz solution (Harris,2001):  p ( x ,  y )    p 0   1    xa  2    yb  2 (8)  p 0   3 A  p  x ,  y  dxdy 2   ab  (9)Multi-grid algorithm is used to calculate the equations(Venner and Lubrecht, 2000). Considering calculationaccuracy and speed, five-level grid is used and the finest gridhas 256   128 points. 2.3 Quasi-static model of cylindrical roller bearing Figure 3 shows geometry of roller–inner ring interaction of roller bearing, where inner ring is moved and rotated relativeto the outer ring. OXYZ is coordinate of the bearing.Considering tilting moment acting on the bearing, assumingload on the bearing is    F   y ,  F  z ,  M   y  , displacement of inner ringis     y ,  z ,     y  , tilting angle of   j  th roller is    j   and angularlocation of the  j  th roller is     j  .    j     2    Z   (  j     1)      o    t   (10)where  Z   is the number of rollers and    o  is rotating speed of roller.Assuming roller is divided into  n s  slices along the axialdirection, the minimum film thickness between inner ring and k th slice of   j  th roller can be obtained:  h 00  ijk   u r  4    12    y cos    j     z sin    j     x k    12 D i  tan  12     y cos    j   tan      y cos    j       j   (11)The minimum film thickness between outer ring and  k th sliceof   j  th roller can be given as follows:  h 00  ejk   u r  4    12    y cos    j     z sin    j     x k    12 D i  tan  12     y cos    j   tan     j    (12)where  u r   is radial clearance of roller bearing,  D i   is diameter of the inner raceway, cos    j   0 for upper and cos    j   0 for lowersign,  x k   k  0.5  l  we /n s  0.5 l  we .By solving the contact calculation model,  p ijk  and  p ejk , whichare contact pressure between inner rings, outer ring and  k thslice of   j  th roller can be obtained. Then, contact force can beobtained as follows: q ijk     A ijk  p ijk  x ,  y  dA ijk  (13) q ejk     A ejk  p ejk  x ,  y  dA ejk  (14)The centers of the inner ring must satisfy force equationconditions as follows: Figure 3  The geometry of roller and inner ring interaction X   Y ZO  D  w Δ θ  y α  j  k  t h  s l i c e A new logarithmic profile model Li Cui and Yafei He Industrial Lubrication and Tribology Volume 67 · Number 5 · 2015 · 498–508  500    D  o  w  n   l  o  a   d  e   d   b  y   T  e  c   h  n   i  s  c   h  e   U  n   i  v  e  r  s   i   t  a  e   t   B  e  r   l   i  n   A   t   1   5  :   1   8   2   3   S  e  p   t  e  m   b  e  r   2   0   1   5   (   P   T   )   F   y    1 n s   j   1  Z  ·  k  1 n s q ijk sin    j     0 (15)  F  z    1 n s   j   1  Z  ·  k  1 n s q ijk cos    j     0 (16)  M   y     j   1  Z   1 n s ·  k  1 n s q ijk  ·  k  1 n s q ijk  ·    k    0.5  l  we n s   0.5 l  we   k  1 n s q ijk · cos    j     0 (17)In radial direction, the  j  th roller equilibrium is as follows:1 n s  k  1 n s q ijk    0.5 m r    o 2 D  pw    1 n s  k  1 n s q ejk  (18)where  m r   is mass of roller and  D  pw  is pitch diameter of rollerbearing. Newton–Raphson method is used to solve theequations. 3. Fatigue life model of and optimization designof cylindrical roller bearing 3.1 Modified rating life model The basic rating life of rolling bearing  L 10  is used widely.However, for many applications, it has become desirable tocalculate the life more accurate under specified lubricationand contamination conditions. Accordingly, the modifiedrating life model of rolling bearing is given as follows: L nm    a 1 a iso L 10  (19)where  a 1  is reliability factor,  a iso  is life modification factor and n  means probability of failure. 3.2 Modified reference rating life model In equation (19), the influence of tilted or misaligned bearings and the influence of bearing clearance during operation cannot be considered in modified rating life calculation. To makethese improvements, modified reference rating life model isgiven according to contact load distribution results of rollingbearing.According to ISO/TS 16281 (2008), the basic dynamiccapacity between inner ring, outer ring and  k th slice of rollerare as follows: q kci     10.83 · C  r  0.378  Z   1    1.038  1      1       143/108  9/2  2/9 ·  1 n s  7/9 (20) q kce    10.83 · C  r  0.364  Z   1    1.038  1      1       143/108   9/2  2/9 ·  1 n s  7/9 (21)where       D we / D  pw ,  D we  is roller diameter and  C  r   is dynamicload rating of bearing.Assuming outer ring is stationary and inner ring is rotatingrelative to the load, the dynamic equivalent load betweeninner ring, outer ring and  k th slice roller are as follows: q kei     1  Z    j   1  Z     p ijk 271  2 D we  1       l  we n s  4  14 (22) q kee    1  Z    j   1  Z     p ejk 271  2 D we  1       l  we n s  4.5  14.5 (23)The modified reference rating life is described as follows: L nmr     a 1   k  1 n s  a iso  –9/8  q kci  q kei    9/2    q kce q kee   9/2     8/9 (24)In equations (19) and (24), reliability of bearing is defined as 90 per cent, reliability factor is 1.0 and  L 10 m  and  L 10 mr   arestudied. 3.3 Optimization design of crowned cylindrical roller bearing The single optimization objective function is maximizedmodified reference rating life:max   f    X      max  L 10 mr    (25)where  f    X    represents the objective function and  X   is thedesign variable vector.Assuming the basic geometric parameters of roller bearinghave been chosen, the three design parameters of roller profileare chosen as design variables for the optimization:  X      k 1 , k 2 , k 3   (26)To reduce the solution range, the strict upper and lowerbounds of the parameters are given as constraint:1    k 1    50    k 2    11    k 3    5 (27)Genetic algorithm is used as optimization design algorithms of bearing fatigue life (Deb and Tiwari, 2008). Initial populationsize is set as 500 and number of generations as 100, crossoverprobability is 0.90 and mutation probability is 0.01.Figure 4 gives flow chart for calculation and optimization of cylindrical roller bearing. First, the profile parameters areinputted. Then, the contact pressure and contact force arecalculated by multi-grid algorithm. The quasi-static equationsare solved by Newton–Raphson method. Finally, the bearingfatigue life is calculated and optimized by genetic algorithm. 4. Results and discussions The specifications of the roller bearing and parameter valuesused in the analysis are listed in Table I. Rotating speed is 2,000 rpm; it is assumed that only rollers are profiled, boththree design parameters roller profile model proposed in thispaper and Johns-Gohar’s profile model are calculated and A new logarithmic profile model Li Cui and Yafei He Industrial Lubrication and Tribology Volume 67 · Number 5 · 2015 · 498–508  501    D  o  w  n   l  o  a   d  e   d   b  y   T  e  c   h  n   i  s  c   h  e   U  n   i  v  e  r  s   i   t  a  e   t   B  e  r   l   i  n   A   t   1   5  :   1   8   2   3   S  e  p   t  e  m   b  e  r   2   0   1   5   (   P   T   )
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