Roller Bearing equations

The rollers must roll without slipping on both the inner and outer race, otherwise the wear rate would be very high.

Inner race stationary

Holding the inner race stationery implies the outer race must move at speed

s o = ω o r o = ω CM r CM + ω r r r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadohadaWgaa WcbaGaam4BaaqabaGccqGH9aqpcqaHjpWDdaWgaaWcbaGaam4Baaqa baGccaWGYbWaaSbaaSqaaiaad+gaaeqaaOGaeyypa0JaeqyYdC3aaS baaSqaaiaadoeacaWGnbaabeaakiaadkhadaWgaaWcbaGaam4qaiaa d2eaaeqaaOGaey4kaSIaeqyYdC3aaSbaaSqaaiaadkhaaeqaaOGaam OCamaaBaaaleaacaWGYbaabeaaaaa@4B8B@  

(1.1)

 

where subscript o stands for outer, CM stands for the center of mass of the rollers, and r stands for the roller bearing.  The ω’s are angular rotation rates.  rCM is the radius from the center of the bearing to the center of the roller.  ro is the inside radius of the outer race.   

We also have the relations:

r CM = r o − r r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkhadaWgaa WcbaGaam4qaiaad2eaaeqaaOGaeyypa0JaamOCamaaBaaaleaacaWG VbaabeaakiabgkHiTiaadkhadaWgaaWcbaGaamOCaaqabaaaaa@3EE0@  

(1.2)

We can solve equations(1.1) and (1.2) for the spin rate of the rollers:

ω r = ω CM ( 1+ r CM − r r r r )= ω CM r CM r r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeM8a3naaBa aaleaacaWGYbaabeaakiabg2da9iabeM8a3naaBaaaleaacaWGdbGa amytaaqabaGcdaqadaqaaiaaigdacqGHRaWkdaWcaaqaaiaadkhada WgaaWcbaGaam4qaiaad2eaaeqaaOGaeyOeI0IaamOCamaaBaaaleaa caWGYbaabeaaaOqaaiaadkhadaWgaaWcbaGaamOCaaqabaaaaaGcca GLOaGaayzkaaGaeyypa0JaeqyYdC3aaSbaaSqaaiaadoeacaWGnbaa beaakmaalaaabaGaamOCamaaBaaaleaacaWGdbGaamytaaqabaaake aacaWGYbWaaSbaaSqaaiaadkhaaeqaaaaaaaa@524E@  

(1.3)

Substituting(1.3)  into (1.1) we have:

s o = ω o r o =2 ω CM ( r 0 − r r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadohadaWgaa WcbaGaam4BaaqabaGccqGH9aqpcqaHjpWDdaWgaaWcbaGaam4Baaqa baGccaWGYbWaaSbaaSqaaiaad+gaaeqaaOGaeyypa0JaaGOmaiabeM 8a3naaBaaaleaacaWGdbGaamytaaqabaGccaGGOaGaamOCamaaBaaa leaacaaIWaaabeaakiabgkHiTiaadkhadaWgaaWcbaGaamOCaaqaba GccaGGPaaaaa@49DB@  

(1.4)

Solving for equation (1.4) for ωo we have:

ω o = 2 ω CM ( r o − r r ) r o =2 ω CM ( 1− r r r o ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeM8a3naaBa aaleaacaWGVbaabeaakiabg2da9maalaaabaGaaGOmaiabeM8a3naa BaaaleaacaWGdbGaamytaaqabaGccaGGOaGaamOCamaaBaaaleaaca WGVbaabeaakiabgkHiTiaadkhadaWgaaWcbaGaamOCaaqabaGccaGG PaaabaGaamOCamaaBaaaleaacaWGVbaabeaaaaGccqGH9aqpcaaIYa GaeqyYdC3aaSbaaSqaaiaadoeacaWGnbaabeaakmaabmaabaGaaGym aiabgkHiTmaalaaabaGaamOCamaaBaaaleaacaWGYbaabeaaaOqaai aadkhadaWgaaWcbaGaam4BaaqabaaaaaGccaGLOaGaayzkaaaaaa@53E2@  

(1.5)

Outer race stationary

Outer race is stationery implies that the inner race (axle) must move at speed:

s I = ω I r I = ω CM r CM + ω r r r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadohadaWgaa WcbaGaamysaaqabaGccqGH9aqpcqaHjpWDdaWgaaWcbaGaamysaaqa baGccaWGYbWaaSbaaSqaaiaadMeaaeqaaOGaeyypa0JaeqyYdC3aaS baaSqaaiaadoeacaWGnbaabeaakiaadkhadaWgaaWcbaGaam4qaiaa d2eaaeqaaOGaey4kaSIaeqyYdC3aaSbaaSqaaiaadkhaaeqaaOGaam OCamaaBaaaleaacaWGYbaabeaaaaa@4B19@  

(1.6)

Where rI is the radius of the outside of the inner race (or axle).

We also have the relations:

r CM = r I + r r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkhadaWgaa WcbaGaam4qaiaad2eaaeqaaOGaeyypa0JaamOCamaaBaaaleaacaWG jbaabeaakiabgUcaRiaadkhadaWgaaWcbaGaamOCaaqabaaaaa@3EAF@  

(1.7)

and we can solve (1.6) for ω r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeM8a3naaBa aaleaacaWGYbaabeaaaaa@38DC@  

           

ω r = ω CM ( 1+ r CM − r r r r )= ω CM r CM r r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeM8a3naaBa aaleaacaWGYbaabeaakiabg2da9iabeM8a3naaBaaaleaacaWGdbGa amytaaqabaGcdaqadaqaaiaaigdacqGHRaWkdaWcaaqaaiaadkhada WgaaWcbaGaam4qaiaad2eaaeqaaOGaeyOeI0IaamOCamaaBaaaleaa caWGYbaabeaaaOqaaiaadkhadaWgaaWcbaGaamOCaaqabaaaaaGcca GLOaGaayzkaaGaeyypa0JaeqyYdC3aaSbaaSqaaiaadoeacaWGnbaa beaakmaalaaabaGaamOCamaaBaaaleaacaWGdbGaamytaaqabaaake aacaWGYbWaaSbaaSqaaiaadkhaaeqaaaaaaaa@524E@  

(1.8)

Substituting (1.8) into (1.6) we obtain:

s I = ω I r I = ω CM ( r I + r r )+ ω CM ( r I + r r )=2 ω CM ( r I + r r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadohadaWgaa WcbaGaamysaaqabaGccqGH9aqpcqaHjpWDdaWgaaWcbaGaamysaaqa baGccaWGYbWaaSbaaSqaaiaadMeaaeqaaOGaeyypa0JaeqyYdC3aaS baaSqaaiaadoeacaWGnbaabeaakiaacIcacaWGYbWaaSbaaSqaaiaa dMeaaeqaaOGaey4kaSIaamOCamaaBaaaleaacaWGYbaabeaakiaacM cacqGHRaWkcqaHjpWDdaWgaaWcbaGaam4qaiaad2eaaeqaaOGaaiik aiaadkhadaWgaaWcbaGaamysaaqabaGccqGHRaWkcaWGYbWaaSbaaS qaaiaadkhaaeqaaOGaaiykaiabg2da9iaaikdacqaHjpWDdaWgaaWc baGaam4qaiaad2eaaeqaaOGaaiikaiaadkhadaWgaaWcbaGaamysaa qabaGccqGHRaWkcaWGYbWaaSbaaSqaaiaadkhaaeqaaOGaaiykaaaa @5F48@  

(1.9)

Solving equation (1.9) for ωI we have:

                                                                       

ω I =2 ω CM ( r I + r r r I )=2 ω CM ( 1+ r r r I ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeM8a3naaBa aaleaacaWGjbaabeaakiabg2da9iaaikdacqaHjpWDdaWgaaWcbaGa am4qaiaad2eaaeqaaOWaaeWaaeaadaWcaaqaaiaadkhadaWgaaWcba GaamysaaqabaGccqGHRaWkcaWGYbWaaSbaaSqaaiaadkhaaeqaaaGc baGaamOCamaaBaaaleaacaWGjbaabeaaaaaakiaawIcacaGLPaaacq GH9aqpcaaIYaGaeqyYdC3aaSbaaSqaaiaadoeacaWGnbaabeaakmaa bmaabaGaaGymaiabgUcaRmaalaaabaGaamOCamaaBaaaleaacaWGYb aabeaaaOqaaiaadkhadaWgaaWcbaGaamysaaqabaaaaaGccaGLOaGa ayzkaaaaaa@5364@  

(1.10)

 

Conclusions:

 

In terms of the diameters, the equations are:

 

θ Roller = θ CM ( 1+ D Inner D Roller ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzajGaeqiUde xddaWgaaWcbaqcLbKacaWGsbGaam4BaiaadYgacaWGSbGaamyzaiaa dkhaaSqabaqcLbKacqGH9aqpcqaH4oqCnmaaBaaaleaajugqciaado eacaWGnbaaleqaa0WaaeWaaOqaaKqzajGaaGymaiabgUcaR0WaaSaa aOqaaKqzajGaamira0WaaSbaaSqaaKqzajGaamysaiaad6gacaWGUb GaamyzaiaadkhaaSqabaaakeaajugqciaadseanmaaBaaaleaajugq ciaadkfacaWGVbGaamiBaiaadYgacaWGLbGaamOCaaWcbeaaaaaaki aawIcacaGLPaaaaaa@57F2@

 

 

 

 

When the inner race rotates and the outer race is stationery:

θ Inner =2 θ CM ( 1+ D roller D Inner ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzajGaeqiUde xddaWgaaWcbaqcLbKacaWGjbGaamOBaiaad6gacaWGLbGaamOCaaWc beaajugqciabg2da9iaaikdacqaH4oqCnmaaBaaaleaajugqciaado eacaWGnbaaleqaa0WaaeWaaOqaaKqzajGaaGymaiabgUcaR0WaaSaa aOqaaKqzajGaamira0WaaSbaaSqaaKqzajGaamOCaiaad+gacaWGSb GaamiBaiaadwgacaWGYbaaleqaaaGcbaqcLbKacaWGebqddaWgaaWc baqcLbKacaWGjbGaamOBaiaad6gacaWGLbGaamOCaaWcbeaaaaaaki aawIcacaGLPaaaaaa@57D5@

When the outer race rotates and the inner race is stationery:

θ Outer =2 θ CM ( 1− D roller D Outer ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzajGaeqiUde xddaWgaaWcbaqcLbKacaWGpbGaamyDaiaadshacaWGLbGaamOCaaWc beaajugqciabg2da9iaaikdacqaH4oqCnmaaBaaaleaajugqciaado eacaWGnbaaleqaa0WaaeWaaOqaaKqzajGaaGymaiabgkHiT0WaaSaa aOqaaKqzajGaamira0WaaSbaaSqaaKqzajGaamOCaiaad+gacaWGSb GaamiBaiaadwgacaWGYbaaleqaaaGcbaqcLbKacaWGebqddaWgaaWc baqcLbKacaWGpbGaamyDaiaadshacaWGLbGaamOCaaWcbeaaaaaaki aawIcacaGLPaaaaaa@5806@

 

θ CM MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzajGaeqiUde xddaWgaaWcbaqcLbKacaWGdbGaamytaaWcbeaaaaa@3ADF@  is the angle moved by the center of mass of the bearing ball