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Two ball bearings from different manufacturers are being considered for a certain application. Bearing A has a catalog rating of 2.12 kN based on a catalog rating system of 3000 hours at 500 rev/min. Bearing B has a catalog rating of 7.5 kN based on a catalog that rates at 106 cycles. For a given application, determine which bearing can carry the larger load.

Sagot :

nuhulawal20

Answer:

F[tex]_D[/tex] for A > F[tex]_D[/tex] for B

Hence, Bearing A can carry the larger load

Explanation:

Given the data in the question,

First lets consider an application which requires desired speed of n₀ and a desired life of L₀.

Lets start with Bearing A

so we write the relation between desired load and life catalog load and life;

[tex]F_R(L_Rn_R60)^{1/a}[/tex] = [tex]F_D(L_Dn_D60)^{1/a}[/tex]

where F[tex]_R[/tex] is the catalog rating( 2.12 kN)

L[tex]_R[/tex] is the rating life ( 3000 hours )

n[tex]_R[/tex] is the rating speed ( 500 rev/min )

F[tex]_D[/tex] is the desired load

L[tex]_D[/tex] is the desired life ( L₀ )

n[tex]_D[/tex]  is the the desired speed ( n₀ )

Now as we know, a = 3 for ball bearings

so we substitute

[tex]2.12( 3000 * 500 * 60 )^{1/3[/tex]  =  [tex]F_D( L_0n_060)^{1/3[/tex]    

950.0578 = [tex]F_D( L_0n_0)^{1/3} 3.914867[/tex]    

950.0578 / 3.914867 = [tex]F_D( L_0n_0)^{1/3}[/tex]

242.6794 =   [tex]F_D( L_0n_0)^{1/3}[/tex]

F[tex]_D[/tex] for A =  (242.6794 / [tex]( L_0n_0)^{1/3}[/tex] ) kN

Therefore the load that bearing A can carry is  (242.6794 / [tex]( L_0n_0)^{1/3}[/tex] ) kN

Next is Bearing B

[tex]F_R(L_Rn_R60)^{1/a}[/tex] = [tex]F_D(L_Dn_D60)^{1/a}[/tex]

F[tex]_R[/tex] = 7.5 kN, [tex](L_Rn_R60) = 10^6[/tex]

Also, for ball bearings, a = 3

so we substitute

[tex]7.5(10^6)^{1/3[/tex] = [tex]F_D(L_0n_060)^{1/3}[/tex]

750 =  [tex]F_D(L_0n_0)^{1/3} 3.914867[/tex]

750 / 3.914867  =  [tex]F_D(L_0n_0)^{1/3}[/tex]

191.5773 = [tex]F_D(L_0n_0)^{1/3}[/tex]

F[tex]_D[/tex] for B = ( 191.5773 / [tex](L_0n_0)^{1/3}[/tex] ) kN

Therefore, the load that bearing B can carry is  ( 191.5773 / [tex](L_0n_0)^{1/3}[/tex] ) kN

Now, comparing the Two results above,

we can say;

F[tex]_D[/tex] for A > F[tex]_D[/tex] for B

Hence, Bearing A can carry the larger load

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