Question

A solid shaft of mild steel 200 mm in diameter is to be replaced by hollow shaft of alloy steel for which the allowable shear stress is 22% greater. If the power to be transmitted is to be increased by 20% and the speed of rotation increased by 6%, determine the maximum internal diameter of the hollow shaft. The external diameter of the hollow shaft is to be 200 mm.

Answer 1

Solid shaft dia d = 200 mm 

Outer dia of Hollow shaft = d_o = 200 mm 

Inner dia of Hollow shaft = d_i = ? 

Allowable shear stress ι_H = 1.22 ι_S 

Power transmitted P_H = 1.20 P_S 

Shaft speed N_H = 1.06 N_S 

Now use P_H = 1.20 P_S 

i.e.; 2π.N_HT_H/60 = 1.2 [2π.N_ST_S/60] 

N_HT_H = 1.2 N_ST_S

Since N_H = 1.06N_S, putting this values 

1.06 N_ST_H = 1.2 N_ST_S 

T_S = (1.06/1.2)T_H ...(i) 

Since T = (σ/16)D³.ι_max 

(σ/16)d³.ι_S = (1.06/1.2) [(σ/16)(d_O⁴ – d_i⁴)/d_O].ι_H 

putting ι_H = 1.22 ι_S 

(σ/16)d³.ι_S = (1.06/1.2) [(σ/16)(d_O⁴ – d_i⁴)/d_O].1.22ι_S 

200³ = (1.06/1.2)(1.22) [(200⁴ – d_i⁴)/200] 

d_i = 104 mm