03-02ChapGere.0005 - Solution 3.4-14 Wire inside a flexible...

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Problem 3.4-14 A magnesium-alloy wire of diameter d 5 4 mm and length L rotates inside a flexible tube in order to open or close a switch from a remote location (see figure). A torque T is applied manually (either clockwise or counterclockwise) at end B , thus twisting the wire inside the tube. At the other end A , the rotation of the wire operates a handle that opens or closes the switch. A torque T 0 5 0.2 N ? m is required to operate the switch. The torsional stiffness of the tube, combined with friction between the tube and the wire, induces a distributed torque of constant intensity t 5 0.04 N ? m/m (torque per unit distance) acting along the entire length of the wire. (a) If the allowable shear stress in the wire is t allow 5 30 MPa, what is the longest permissible length L max of the wire? (b) If the wire has length L 5 4.0 m and the shear modulus of elasticity for the wire is G 5 15 GPa, what is the angle of twist f (in degrees) between the ends of the wire?
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Unformatted text preview: Solution 3.4-14 Wire inside a flexible tube 208 CHAPTER 3 Torsion t d A B T T = torque Flexible tube d 5 4 mm T 5 0.2 N ? m t 5 0.04 N ? m/m (a) M AXIMUM LENGTH L max t allow 5 30 MPa Equilibrium: T 5 tL 1 T From Eq. (3-12): Substitute numerical values: L max 5 4.42 m L max 5 1 16 t ( p d 3 t allow 2 16 T ) L 5 1 16 t ( p d 3 t max 2 16 T ) tL 1 T 5 p d 3 t max 16 t max 5 16 T p d 3 T 5 p d 3 t max 16 (b) A NGLE OF TWIST f L 5 4 m G 5 15 GPa f 1 5 angle of twist due to distributed torque t (from problem 3.4-12) f 2 5 angle of twist due to torque T (from Eq. 3-15) f 5 total angle of twist 5 f 1 1 f 2 Substitute numerical values: f 5 2.971 rad 5 170 8 f 5 16 L p Gd 4 ( tL 1 2 T ) 5 T L GI P 5 32 T L p Gd 4 5 16 tL 2 p Gd 4 t T T L d A-PDF Split DEMO : Purchase from www.A-PDF.com to remove the watermark...
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This note was uploaded on 09/20/2009 for the course COE 3001 taught by Professor Armanios during the Spring '08 term at Georgia Institute of Technology.

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