Question

7. Figure 7 shows a truncated conical shaft of length L whose diameter changes linearly from d at one end to 2d at the other. It transmits a torque T. Find the relative rotation between the ends (the twist) as a function of d, L and the shear modulus G. Compare your result with the twist of a shaft of the same material and length, but with a uniform diameter 3d/2. diameter 2d diameter d Figure 7

Answer 1

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Sol:7 Twisting of truncated conical shaft Tapered Truncated conical shaft may be assumed to be h no of uniform diameter shaft. O = twist angle of fat tapered cone On=twist angle of nth tapered cone Ardu 1 Ostrip truncated conical shaft Tror du = 0 + 0₂ t... + On - - [ Txx) du Jo (GJ)xx Txx T Gxx=G, Jxx - T (dxx) dx = d = d , + (dz -d; ) 2 and in touncated conical = shaft ( 32T de do RG [d, + (d₂d, ) (2)] + Here, d = 2d diod & .32 TL K9 l d + 2d²+ 4d²7 3xd x 8 32 TL 7027 TG 24 d 8d3 J 48 7- KG = 28 TL 3G dt - stan [aka] febre 9333 (Rap) 9.333/TI KG dt

diameter GJ for the uniform shaft, d, = d₂ = 3d Also, 0= TL = TL - D G A {3} 0 = 32TL X 16 X G X 81 dt Benitong huge = 6321 () = 512 TL 81 a G dt Clearly, Ore o uniform dia. shaft