Page 421 - Marks Calculation for Machine Design

P1: Naresh
January 4, 2005
15:28
Brown.cls
Brown˙C09
MACHINE ENERGY
For each spoke, the mass moment of inertia is given by Eq. (9.79)
403
1 2 2
I spoke = m spoke L spoke + L cg (9.79)
12
where the mass of each spoke (m spoke ) is given by Eq. (9.80).
1 2
2
m spoke = ρπ L spoke r spoke = ρπ L spoke d spoke (9.80)
4
Therefore, the total mass moment of inertia of the flywheel (I flywheel ) is
I flywheel = I hub + I rim + N spokes I spoke (9.81)
where (N spokes ) is the number of spokes.
Consider the following design option. In Example 1, the mass moment of inertia was
calculated for a thin solid circular disk. To improve the efficiency of this flywheel, a com-
posite design is considered. If the overall outside diameter remains the same, and using four
circular rods as spokes, compare the weight of a composite flywheel similar to Fig. 9.14
with the weight of a solid disk flywheel of equal mass moment of inertia.
U.S. Customary SI/Metric
Example 5. Compare the weight of a com- Example 5. Compare the weight of a com-
posite flywheel design as shown in Fig. 9.14 posite flywheel design like shown in Fig. 9.14
with the thin, solid disk flywheel of Example 1, with the thin solid disk flywheel of Example 1,
where where
2
2
I flywheel = 29.80 slug · ft (Example 1) I flywheel = 39.85 kg · m (Example 1)
3
3
ρ = 15.2 slug/ft (steel) ρ = 7,850 kg/m (steel)
D o = 3 ft (Example 1) D o = 90 cm = 0.9 m (Example 1)
d o = 6in = 0.5 ft (hub) d o = 16 cm = 0.16 m (hub)
d i = 3in = 0.25 ft (hub) d i = 8cm = 0.08 m (hub)
w hub = 3in = 0.25 ft w hub = 8cm = 0.08 m
t = 3in = 0.25 ft (rim) t = 7cm = 0.07 m (rim)
L spoke = 12 in = 1ft L spoke = 30 cm = 0.3 m
d spoke = 1.5 in = 0.125 ft d spoke = 4cm = 0.04 m
L cg = 9in = 0.75 ft L cg = 23 cm = 0.23 m
w rim is the only unknown dimension w rim is the only unknown dimension
solution solution
Step 1. Calculate the weight of the thin solid Step 1. Calculate the weight of the thin solid
disk flywheel in Example 1. disk flywheel in Example 1.
W solid disk = m solid disk g W solid disk = m solid disk g
where the mass of the solid disk is found by where the mass of the solid disk is found by
multiplying density times its volume. multiplying density times its volume.
1 1
2
2
m solid disk = ρ πt d − d 2 i m solid disk = ρ πt d − d i 2
o
o
4 4

volume volume

slug 1 kg 1
= 15.2 π(0.25 ft) = 7,850 π(0.08 m)
ft 3 4 m 3 4
2
2
2
2
×((3ft) − (0.25 ft) ) ×((0.9m) − (0.08 m) )