1. A circular shaft with a keyway can be approximated by the section shown in Fig. r=b r 0 r = 2acose Figure 1: Circular shaft with a keyway. The keyway is represented by the boundary equation r = b and the shaft is represented by the boundary equation r = 2a cos 0. Show that using a Prandtl stress function of the form b) = K (6² – 7²) (1 - 7²) (1 - 2a cos) will solve the problem of torsion on this shaft and find the constant K (assume an applied torque T). Compute the shear stress components Taz and Tyz. (Hint: start by converting the stress function to Cartesian coordinates).

1. A circular shaft with a keyway can be approximated by the section shown in Fig. r=b r 0 r = 2acose Figure 1: Circular shaft with a keyway. The keyway is represented by the boundary equation r = b and the shaft is represented by the boundary equation r = 2a cos 0. Show that using a Prandtl stress function of the form b) = K (6² – 7²) (1 - 7²) (1 - 2a cos) will solve the problem of torsion on this shaft and find the constant K (assume an applied torque T). Compute the shear stress components Taz and Tyz. (Hint: start by converting the stress function to Cartesian coordinates).

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Question

1.
A circular shaft with a keyway can be approximated by the section shown in Fig.
r=b
Ay
r
r=2acose
Figure 1: Circular shaft with a keyway.
The keyway is represented by the boundary equation r = b and the shaft is represented by
the boundary equation r = 2a cos 0. Show that using a Prandtl stress function of the form
²) (₁
(1-2a cos)
X
v = K (b² − ²) (1 – :
will solve the problem of torsion on this shaft and find the constant K (assume an applied
torque T). Compute the shear stress components Txz and Tyz. (Hint: start by converting the
stress function to Cartesian coordinates).

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