Explanation Given information : The coefficient of friction is μ s . The axial force is P . Explanation : Show the free body diagram of the conical bearing subjected to a constant pressure distribution as in Figure (1). Using Figure (1), Express the differential area ( dA ) (shaded) of the shaft as follows: d A = 2 π r d s (I) Here, the radius of the conical is r and the differential spiral area is ds . Express the pressure ( P ) distributed at the conical surface of contact as follows: P = ∫ 0 R p cos θ d A (II Here, the normal pressure is p . Express the differential normal force ( dN ) as follows: d N = p d A (III) Express the torque ( M ) required for overcoming the friction is μ s when the shaft supports an axial force P as follows: M = ∫ 0 R r d F = ∫ 0 R μ s r d N (IV) Conclusion : Substitute ( d r cos θ ) for ds in Equation (I). d A = 2 π r ( d r cos θ ) = 2 π r d r cos θ Substitute 2 π r d r cos θ for dA in Equation (II). P = ∫ 0 R p cos θ ( 2 π r d r cos θ ) = ( 2 π p cos θ cos θ ) ∫ 0 R r d r = 2 π p ∫ 0 R r d r (V) Differentiate Equation (V) with respect to r . P = 2 π p ( r 2 2 ) 0 R Apply the limits as follows: P = 2 π p ( R 2 2 − 0 2 2 ) = 2 π p R 2 2 = π p R 2 (VI) Simplify Equation (VI) and it can be written as follows: p = P π R 2 Substitute P π R 2 for p and 2 π r d r cos θ for dA in Equation (III)

Engineering Mechanics: Statics Plus Mastering Engineering with Pearson eText -- Access Card Package (14th Edition) (Hibbeler, The Engineering Mechanics: Statics & Dynamics Series, 14th Edition)

14th Edition

ISBN: 9780134160689

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 8.8, Problem 113P

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The torque required to overcome the friction is μs when the shaft supports an axial force P.

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Chapter 8.8, Problem 111P

Chapter 8.8, Problem 114P

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Engineering Mechanics: Statics Plus Mastering Engineering with Pearson eText -- Access Card Package (14th Edition) (Hibbeler, The Engineering Mechanics: Statics & Dynamics Series, 14th Edition)

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The torque required to overcome the friction is μ s when the shaft supports an axial force P .

Solution Summary: The author explains the torque required to overcome the friction in a conical bearing subjected to constant pressure distribution.

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