Explanation Write the equation of the magnitude of the couple as follows; M = 1 2 μ k P ( R 1 + R 2 ) Here, the coefficient of kinetic friction is μ k , the magnitude of the total axial force is P , the outer radii of the collar is R 1 , and the inner radii of the collar is R 2 . Show the free-body diagram of the thrust bearing as in Figure 1. It has been assumed that the normal force per unit area is inversely proportional to r . Δ A = r Δ θ Δ r Here, the distance between the point to the axis of the shaft is r , the change is angle is Δ θ , and the change in distance is Δ r . Find the normal force exerted Δ N due to the element of area Δ A using the relation; Δ N = k r Δ A Substitute r Δ θ Δ r for Δ A . Δ N = k r r Δ θ Δ r = k Δ θ Δ r (1) Find the value of k using the relation. P = ∑ Δ N = ∫ d N = ∫ 0 2 π ∫ R 1 R 2 Δ N Substitute k Δ θ Δ r for Δ N . P = ∫ 0 2 π ∫ R 1 R 2 k Δ θ Δ r = k [ θ ] 0 2 π [ r ] R 1 R 2 = k [ 2 π − 0 ] [ R 2 − R 1 ] P = 2 k π ( R 2 − R 1 ) k = P 2 π ( R 2 − R 1 ) Substitute P 2 π ( R 2 − R 1 ) for k in Equation (1). Δ N = P 2 π ( R 2 − R 1 ) Δ θ Δ r Find the change in friction force Δ F using the relation. Δ F = μ k Δ N Substitute P 2 π ( R 2 − R 1 ) Δ θ Δ r for Δ N

Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9780073398242

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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Chapter 8.3, Problem 8.95P

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Proof that the magnitude M of the couple required to overcome the frictional resistance of a worn-out collar bearing is M=12μkP(R1+R2).

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Chapter 8.3, Problem 8.94P

Chapter 8.3, Problem 8.96P

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Proof that the magnitude M of the couple required to overcome the frictional resistance of a worn-out collar bearing is M = 1 2 μ k P ( R 1 + R 2 ) .

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Proof that the magnitude M of the couple required to overcome the frictional resistance of a worn-out collar bearing is M=12μkP(R1+R2).

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Chapter 8 Solutions