Page 401 - Shigley's Mechanical Engineering Design

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376 Mechanical Engineering Design

Assume generous fillet radius for gear at I.
From Table 7–1, estimate K t = 1.7, K ts = 1.5. For quick, conservative first
pass, assume K f = K t , K fs = K ts .
Choose inexpensive steel, 1020 CD, with S ut = 68 kpsi. For S e ,
b
Eq. (6–19) k a = aS = 2.7(68) −0.265 = 0.883
ut
Guess k b = 0.9. Check later when d is known.
k c = k d = k e = 1
Eq. (6–18) S e = (0.883)(0.9)(0.5)(68) = 27.0 kpsi
For first estimate of the small diameter at the shoulder at point I, use the
DE-Goodman criterion of Eq. (7–8). This criterion is good for the initial design,
since it is simple and conservative. With M m = T a = 0, Eq. (7–8) reduces to
⎧ ⎞⎫ 1/3
⎛ 1/2
2

⎪ ⎪
⎨ 3 K fs T m ⎬
2 K f M a
⎟
16n ⎜
d = +
⎪ π ⎝ S e S ut ⎠ ⎪
⎩ ⎭
1/3

2 1/2
16(1.5) 2 (1.7)(3651) 3 [(1.5)(3240)]
d = +
π 27 000 68 000
d = 1.65 in
All estimates have probably been conservative, so select the next standard size
below 1.65 in. and check, d 1.625 in.
A typical D/d ratio for support at a shoulder is D/d 1.2, thus, D 1.2(1.625)
1.95 in. Increase to D 2.0 in. A nominal 2 in. cold-drawn shaft diameter can be
used. Check if estimates were acceptable.
D/d = 2/1.625 = 1.23
∼
Assume fillet radius r = d/10 = 0.16 in. r/d = 0.1
K t = 1.6 (Fig. A–15–9), q = 0.82 (Fig. 6–20)
Eq. (6–32) K f = 1 + 0.82(1.6 − 1) = 1.49
K ts = 1.35 (Fig. A–15–8), q s = 0.85 (Fig. 6–21)
K fs = 1 + 0.85(1.35 − 1) = 1.30
k a = 0.883 (no change)
−0.107
1.625

Eq. (6–20) k b = = 0.835
0.3
S e = (0.883)(0.835)(0.5)(68) = 25.1 kpsi
32K f M a 32(1.49)(3651)

Eq. (7–5) σ = = = 12 910 psi
a
πd 3 π(1.625) 3
1/2 √
2

16K fs T m 3(16)(1.30)(3240)
Eq. (7–6) σ = 3 = = 8659 psi

m
πd 3 π(1.625) 3
Using Goodman criterion
1 σ a σ m 129 10 8659
= + = + = 0.642
n f S e S ut 25 100 68 000
n f = 1.56