2. The tapered cylindrical bar in Fig. 2 is subjected to opposing torques T on its free ends. For the tapered section only, the angle of twist 0 can be expressed as: 32 TI(d? + d,d2 + d?) Gd d? (a) Derive an expression for the effective spring constant keff for the entire bar, treating each segment as a torsional spring in series. Fig. 2 (b) For the bar shown, let; I, = 12 = 12 in., 1= 24 in., d= 0.25 in., d2= 1 in., T= 1000 lbf-in., and G = 13.5 x 103 kpsi. Calculate the angle twist 0 for the entire bar using the expression for kef determined in (a). Verify that this value is the sum the angles of twist for each individual segment. %3D

2. The tapered cylindrical bar in Fig. 2 is subjected to opposing torques T on its free ends. For the tapered section only, the angle of twist 0 can be expressed as: 32 TI(d? + d,d2 + d?) Gd d? (a) Derive an expression for the effective spring constant keff for the entire bar, treating each segment as a torsional spring in series. Fig. 2 (b) For the bar shown, let; I, = 12 = 12 in., 1= 24 in., d= 0.25 in., d2= 1 in., T= 1000 lbf-in., and G = 13.5 x 103 kpsi. Calculate the angle twist 0 for the entire bar using the expression for kef determined in (a). Verify that this value is the sum the angles of twist for each individual segment. %3D

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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Concept explainers

Combined Loading

Any functioning component, such as machine parts, structural members, pressure vessels, and so on, are all under the influence of more than one force. When anybody is under the influence of more than one force such as shear forces, bending moments, axial forces, torsion, and so on, then such body is said to be under combined loading. A very practical example of combined loading is the crankshaft of an internal combustion engine (IC engine). The crankshaft of the IC engine is under high rpm, which is caused due to the torsional forces. Due to the presence of piston, counterbalances, and internal imbalances, the crankshaft experiences lateral forces and due to the rise in temperature, the crankshaft undergoes thermal expansion, which pulls the crankshaft along the axial direction and lateral direction. The presence of unbalanced loads along the periphery of the crankshaft also induces a bending moment.

Question

2. The tapered cylindrical bar in Fig. 2 is subjected to opposing torques T on its free ends. For the
tapered section only, the angle of twist 0 can be expressed as:
32 TI(d? + d,d2 + dž)
Gd d?
37n
(a) Derive an expression for the effective spring constant keff
for the entire bar, treating each segment as a torsional spring in series.
Fig. 2
(b) For the bar shown, let; I, = 12 = 12 in., 1= 24 in., d1= 0.25 in., d2= 1 in., T = 1000 lbf-in., and G
= 13.5 x 103 kpsi. Calculate the angle twist 0 for the entire bar using the expression for kef
determined in (a). Verify that this value is the sum the angles of twist for each individual
%3D
%3D
segment.

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