Page 38 - Marks Calculation for Machine Design

P1: Shibu
January 4, 2005
Brown˙C01
Brown.cls
20
U.S. Customary 12:26 STRENGTH OF MACHINES SI/Metric
Example 3. Calculate the angle of twist (φ) Example 3. Calculate the angle of twist (φ)
for a solid circular shaft, where for a solid circular shaft, where
T = 6,000 ft · lb = 72,000 in · lb T = 9,000 N · m
D = 3in = 2R D = 8cm = 0.08 m = 2R
L = 6ft = 72 in L = 2m
9
2
6
2
G = 11.7 × 10 lb/in (steel) G = 80.8 × 10 N/m (steel)
solution solution
Step 1. Calculate the polar moment of inertia Step 1. Calculate the polar moment of inertia
(J) of the shaft using Eq. (1.22). (J) of the shaft using Eq. (1.22).
1 4 1 4 1 4 1 4
J = πR = π(1.5in) J = πR = π(0.04 m)
2 2 2 2
= 7.95 in 4 = 0.00000402 m 4
Step 2. Substitute this value of the polar Step 2. Substitute this value of the polar
moment of inertia(J), the torque (T ), the length moment of inertia(J), the torque (T ), the length
(L), and the shear modulus of elasticity (G) into (L), and the shear modulus of elasticity (G) into
Eq. (1.29) to give the angle of twist (φ) as Eq. (1.29) to give the angle of twist (φ) as
TL TL
φ = φ =
GJ GJ
(72,000 in · lb)(72 in) (9,000 N · m)(2m)
= 2 4 =
9
6
2
4
(11.7 × 10 lb/in )(7.95 in ) (80.8 × 10 N/m )(0.00000402 m )
5,184,000 lb · in 2 18,000 N · m 2
= =
93,015,000 lb · in 2 323,200 N · m 2
= 0.056 rad = 0.056 rad
Example 4. Determine the maximum torque Example 4. Determine the maximum torque
(T max ) that can be applied to a solid circular (T max ) that can be applied to a solid circular
shaft if there is a maximum allowable shear shaft if there is a maximum allowable shear
stress (τ max ) and a maximum allowable angle stress (τ max ) and a maximum allowable angle
of twist (φ), where of twist (φ), where
τ max = 60 kpsi = 60,000 lb/in 2 τ max = 410 MPa = 410,000,000 N/m 2
◦
φ max = 1.5 = 0.026 rad φ max = 1.5 = 0.026 rad
◦
D = 6in = 2R D = 15 cm = 0.15 m = 2R
L = 3ft = 36 in L = 1m
6
9
2
2
G = 4.1 × 10 lb/in (aluminum) G = 26.7 × 10 N/m (aluminum)
solution solution
Step 1. For a maximum shear stress (τ max ), Step 1. For a maximum shear stress (τ max ),
solve for the maximum torque (T max ) in solve for the maximum torque (T max ) in
Eq. (1.21) to give Eq. (1.21) to give
τ max J τ max J
T max = T max =
R R
Step 2. Calculate the polar moment of inertia Step 2. Calculate the polar moment of inertia
(J) of the shaft using Eq. (1.22). (J) of the shaft using Eq. (1.22).
1 1 1 1
4
4
J = πR = π(3in) 4 J = πR = π(0.075 m) 4
2 2 2 2
= 127.2in 4 = 0.0000497 m 4