Capítulo 17 elementos mecânicos flexíveis

Chapter 17
17-1 Preliminaries to give some checkpoints:
VR =
1
Angular velocity ratio
=
1
0.5
= 2
Hnom = 2 hp, 1750 rev/min, C = 9(12) = 108 in, Ks = 1.25, nd = 1
dmin = 1 in, Fa = 35 lbf/in, γ = 0.035 lbf/in3
, f = 0.50, b = 6 in, d = 2 in,
from Table 17-2 for F-1 Polyamide: t = 0.05 in; from Table 17-4, Cp = 0.70.
w = 12γ bt = 12(0.035)(6)(0.05) = 0.126 lbf/ft
θd = 3.123 rad, exp( f θ) = 4.766 (perhaps)
V =
πdn
12
=
π(2)(1750)
12
= 916.3 ft/min
(a) Eq. (e): Fc =
w
32.174
V
60
2
=
0.126
32.174
916.3
60
2
= 0.913 lbf Ans.
T =
63 025(2)(1.25)(1)
1750
= 90.0 lbf · in
F =
2T
d
=
2(90)
2
= 90 lbf
Eq. (17-12): (F1)a = bFaCpCv = 6(35)(0.70)(1) = 147 lbf Ans.
F2 = F1a − F = 147 − 90 = 57 lbf Ans.
Do not use Eq. (17-9) because we do not yet know f .
Eq. (i) Fi =
F1a + F2
2
− Fc =
147 + 57
2
− 0.913 = 101.1 lbf Ans.
f =
1
θd
ln
(F1)a − Fc
F2 − Fc
=
1
3.123
ln
147 − 0.913
57 − 0.913
= 0.307
The friction is thus undeveloped.
(b) The transmitted horsepower is,
H =
( F)V
33 000
=
90(916.3)
33 000
= 2.5 hp Ans.
nf s =
H
HnomKs
=
2.5
2(1.25)
= 1
From Eq. (17-2), L = 225.3 in Ans.
(c) From Eq. (17-13), dip =
3C2
w
2Fi
where C is the center-to-center distance in feet.
dip =
3(108/12)2
(0.126)
2(101.1)
= 0.151 in Ans.
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432 Solutions Manual • Instructor’s Solution Manual to Accompany Mechanical Engineering Design
Comment: The friction is under-developed. Narrowing the belt width to 5 in (if size is
available) will increase f . The limit of narrowing is bmin = 4.680 in, whence
w = 0.0983 lbf/ft (F1)a = 114.7 lbf
Fc = 0.712 lbf F2 = 24.6 lbf
T = 90 lbf · in (same) f = f = 0.50
F = (F1)a − F2 = 90 lbf dip = 0.173 in
Fi = 68.9 lbf
Longer life can be obtained with a 6-inch wide belt by reducing Fi to attain f = 0.50.
Prob. 17-8 develops an equation we can use here
Fi =
( F + Fc) exp( f θ) − Fc
exp( f θ) − 1
F2 = F1 − F
Fi =
F1 + F2
2
− Fc
f =
1
θd
ln
F1 − Fc
F2 − Fc
dip =
3(C D/12)2
w
2Fi
which in this case gives
F1 = 114.9 lbf Fc = 0.913 lbf
F2 = 24.8 lbf f = 0.50
Fi = 68.9 lbf dip = 0.222 in
So, reducing Fi from 101.1 lbf to 68.9 lbf will bring the undeveloped friction up to
0.50, with a corresponding dip of 0.222 in. Having reduced F1 and F2 , the endurance
of the belt is improved. Power, service factor and design factor have remained in tack.
17-2 There are practical limitations on doubling the iconic scale. We can double pulley diame-
ters and the center-to-center distance. With the belt we could:
• Use the same A-3 belt and double its width;
• Change the belt to A-5 which has a thickness 0.25 in rather than 2(0.13) = 0.26 in, and
an increased Fa ;
• Double the thickness and double tabulated Fa which is based on table thickness.
The object of the problem is to reveal where the non-proportionalities occur and the nature
of scaling a ﬂat belt drive.
We will utilize the third alternative, choosing anA-3 polyamide belt of double thickness,
assuming it is available. We will also remember to double the tabulated Fa from 100 lbf/in to
200 lbf/in.
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Chapter 17 433
Ex. 17-2: b = 10 in, d = 16 in, D = 32 in,
Polyamide A-3, t = 0.13 in, γ = 0.042, Fa =
100 lbf/in, Cp = 0.94, Cv = 1, f = 0.8
T =
63 025(60)(1.15)(1.05)
860
= 5313 lbf · in
w = 12 γ bt = 12(0.042)(10)(0.13)
= 0.655 lbf/ft
V = πdn/12 = π(16)(860/12) = 3602 ft/min
θd = 3.037 rad
For fully-developed friction:
exp( f θd) = [0.8(3.037)] = 11.35
Fc =
wV2
g
=
0.655(3602/60)2
32.174
= 73.4 lbf
(F1)a = F1 = bFaCpCv
= 10(100)(0.94)(1) = 940 lbf
F = 2T/D = 2(5313)/(16) = 664 lbf
F2 = F1 − F = 940 − 664 = 276 lbf
Fi =
F1 + F2
2
− Fc
=
940 + 276
2
− 73.4 = 535 lbf
Transmitted power H (or Ha):
H =
F(V)
33 000
=
664(3602)
33 000
= 72.5 hp
f =
1
θd
ln
F1 − Fc
F2 − Fc
=
1
3.037
ln
940 − 73.4
276 − 73.4
= 0.479 undeveloped
Note, in this as well as in the double-size case,
exp( f θd) is not used. It will show up if we
relax Fi (and change other parameters to trans-
mit the required power), in order to bring f up
to f = 0.80, and increase belt life.
You may wish to suggest to your students
that solving comparison problems in this man-
ner assists in the design process.
Doubled: b = 20 in, d = 32 in, D = 72 in,
Polyamide A-3, t = 0.26 in, γ = 0.042,
Fa = 2(100) = 200 lbf/in, Cp = 1, Cv = 1,
f = 0.8
T = 4(5313) = 21 252 lbf · in
w = 12(0.042)(20)(0.26) = 2.62 lbf/ft
V = π(32)(860/12) = 7205 ft/min
θ = 3.037 rad
For fully-developed friction:
exp( f θd) = exp[0.8(3.037)] = 11.35
Fc =
wV2
g
=
0.262(7205/60)2
32.174
= 1174.3 lbf
(F1)a = 20(200)(1)(1)
= 4000 lbf = F1
F = 2T/D = 2(21 252)/(32) = 1328.3 lbf
F2 = F1 − F = 4000 − 1328.3 = 2671.7 lbf
Fi =
F1 + F2
2
− Fc
=
4000 + 2671.7
2
− 1174.3 = 2161.6 lbf
Transmitted power H:
H =
F(V)
33 000
=
1328.3(7205)
33 000
= 290 hp
f =
1
θd
ln
F1 − Fc
F2 − Fc
=
1
3.037
ln
4000 − 1174.3
2671.7 − 1174.3
= 0.209 undeveloped
There was a small change in Cp .
Parameter Change Parameter Change
V 2-fold F 2-fold
Fc 16-fold Fi 4-fold
F1 4.26-fold Ht 4-fold
F2 9.7-fold f 0.48-fold
Note the change in Fc !
In assigning this problem, you could outline (or solicit) the three alternatives just mentioned
and assign the one of your choice–alternative 3:
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17-3
As a design task, the decision set on p. 881 is useful.
A priori decisions:
• Function: Hnom = 60 hp, n = 380 rev/min, VR = 1, C = 192 in, Ks = 1.1
• Design factor: nd = 1
• Initial tension: Catenary
• Belt material: Polyamide A-3
• Drive geometry: d = D = 48 in
• Belt thickness: t = 0.13 in
Design variable: Belt width of 6 in
Use a method of trials. Initially choose b = 6 in
V =
πdn
12
=
π(48)(380)
12
= 4775 ft/min
w = 12γ bt = 12(0.042)(6)(0.13) = 0.393 lbf/ft
Fc =
wV2
g
=
0.393(4775/60)2
32.174
= 77.4 lbf
T =
63 025(1.1)(1)(60)
380
= 10 946 lbf · in
F =
2T
d
=
2(10 946)
48
= 456.1 lbf
F1 = (F1)a = bFaCpCv = 6(100)(1)(1) = 600 lbf
F2 = F1 − F = 600 − 456.1 = 143.9 lbf
Transmitted power H
H =
F(V)
33 000
=
456.1(4775)
33 000
= 66 hp
Fi =
F1 + F2
2
− Fc =
600 + 143.9
2
− 77.4 = 294.6 lbf
f =
1
θd
ln
F1 − Fc
F2 − Fc
=
1
π
ln
600 − 77.4
143.9 − 77.4
= 0.656
L = 534.8 in, from Eq. (17-2)
Friction is not fully developed, so bmin is just a little smaller than 6 in (5.7 in). Not having
a ﬁgure of merit, we choose the most narrow belt available (6 in). We can improve the
48"
192"
shi20396_ch17.qxd 8/28/03 3:58 PM Page 434

Chapter 17 435
design by reducing the initial tension, which reduces F1 and F2, thereby increasing belt life.
This will bring f to 0.80
F1 =
( F + Fc) exp( f θ) − Fc
exp( f θ) − 1
exp( f θ) = exp(0.80π) = 12.345
Therefore
F1 =
(456.1 + 77.4)(12.345) − 77.4
12.345 − 1
= 573.7 lbf
F2 = F1 − F = 573.7 − 456.1 = 117.6 lbf
Fi =
F1 + F2
2
− Fc =
573.7 + 117.6
2
− 77.4 = 268.3 lbf
These are small reductions since f is close to f , but improvements nevertheless.
dip =
3C2
w
2Fi
=
3(192/12)2
(0.393)
2(268.3)
= 0.562 in
17-4 From the last equation given in the Problem Statement,
exp( f φ) =
1
1 − {2T/[d(a0 − a2)b]}
1 −
2T
d(a0 − a2)b
exp( f φ) = 1
2T
d(a0 − a2)b
exp( f φ) = exp( f φ) − 1
b =
1
a0 − a2
2T
d
exp( f φ)
exp( f φ) − 1
But 2T/d = 33 000Hd/V
Thus,
b =
1
a0 − a2
33 000Hd
V
exp( f φ)
exp( f φ) − 1
Q.E.D.
17-5 Refer to Ex. 17-1 on p. 878 for the values used below.
(a) The maximum torque prior to slip is,
T =
63 025HnomKsnd
n
=
63 025(15)(1.25)(1.1)
1750
= 742.8 lbf · in Ans.
The corresponding initial tension is,
Fi =
T
D
exp( f θ) + 1
exp( f θ) − 1
=
742.8
6
11.17 + 1
11.17 − 1
= 148.1 lbf Ans.
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(b) See Prob. 17-4 statement. The ﬁnal relation can be written
bmin =
1
FaCpCv − (12γ t/32.174)(V/60)2
33 000Ha exp( f θ)
V[exp( f θ) − 1]
=
1
100(0.7)(1) −{[12(0.042)(0.13)]/32.174}(2749/60)2
33 000(20.6)(11.17)
2749(11.17 − 1)
= 4.13 in Ans.
This is the minimum belt width since the belt is at the point of slip. The design must
round up to an available width.
Eq. (17-1):
θd = π − 2 sin−1 D − d
2C
= π − 2 sin−1 18 − 6
2(96)
= 3.016 511 rad
θD = π + 2 sin−1 D − d
2C
= π + 2 sin−1 18 − 6
2(96)
= 3.266 674
Eq. (17-2):
L = [4(96)2
− (18 − 6)2
]1/2
+
1
2
[18(3.266 674) + 6(3.016 511)]
= 230.074 in Ans.
(c) F =
2T
d
=
2(742.8)
6
= 247.6 lbf
(F1)a = bFaCpCv = F1 = 4.13(100)(0.70)(1) = 289.1 lbf
F2 = F1 − F = 289.1 − 247.6 = 41.5 lbf
Fc = 25.6
0.271
0.393
= 17.7 lbf
Fi =
F1 + F2
2
− Fc =
289.1 + 41.5
2
− 17.7 = 147.6 lbf
Transmitted belt power H
H =
F(V)
33 000
=
247.6(2749)
33 000
= 20.6 hp
nf s =
H
HnomKs
=
20.6
15(1.25)
= 1.1
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Chapter 17 437
If you only change the belt width, the parameters in the following table change as shown.
Ex. 17-1 This Problem
b 6.00 4.13
w 0.393 0.271
Fc 25.6 17.6
(F1)a 420 289
F2 172.4 42
Fi 270.6 147.7
f 0.33* 0.80**
dip 0.139 0.176
*Friction underdeveloped
**Friction fully developed
17-6 The transmitted power is the same.
n-Fold
b = 6 in b = 12 in Change
Fc 25.65 51.3 2
Fi 270.35 664.9 2.46
(F1)a 420 840 2
F2 172.4 592.4 3.44
Ha 20.62 20.62 1
nfs 1.1 1.1 1
f 0.139 0.125 0.90
dip 0.328 0.114 0.34
If we relax Fi to develop full friction ( f = 0.80) and obtain longer life, then
n-Fold
b = 6 in b = 12 in Change
Fc 25.6 51.3 2
Fi 148.1 148.1 1
F1 297.6 323.2 1.09
F2 50 75.6 1.51
f 0.80 0.80 1
dip 0.255 0.503 2
shi20396_ch17.qxd 8/28/03 3:58 PM Page 437

17-7
Find the resultant of F1 and F2 :
α = sin−1 D − d
2C
sin α =
D − d
2C
cos α ˙= 1 −
1
2
D − d
2C
2
Rx
= F1 cos α + F2 cos α = (F1 + F2) 1 −
1
2
D − d
2C
2
Ans.
Ry
= F1 sin α − F2 sin α = (F1 − F2)
D − d
2C
Ans.
From Ex. 17-2, d = 16 in, D = 36 in, C = 16(12) = 192 in, F1 = 940 lbf, F2 = 276 lbf
α = sin−1 36 − 16
2(192)
= 2.9855◦
Rx
= (940 + 276) 1 −
1
2
36 − 16
2(192)
2
= 1214.4 lbf
Ry
= (940 − 276)
36 − 16
2(192)
= 34.6 lbf
T = (F1 − F2)
d
2
= (940 − 276)
16
2
= 5312 lbf · in
17-8 Begin with Eq. (17-10),
F1 = Fc + Fi
2 exp( f θ)
exp( f θ) − 1
Introduce Eq. (17-9):
F1 = Fc +
T
D
exp( f θ) + 1
exp( f θ) − 1
2 exp( f θ)
exp( f θ) + 1
= Fc +
2T
D
exp( f θ)
exp( f θ) − 1
F1 = Fc + F
exp( f θ)
exp( f θ) − 1
d D
C
F1
Rx
Ry
x
y
F2
␣
␣
shi20396_ch17.qxd 8/28/03 3:58 PM Page 438

Chapter 17 439
Now add and subtract Fc
exp( f θ)
exp( f θ) − 1
F1 = Fc + Fc
exp( f θ)
exp( f θ) − 1
+ F
exp( f θ)
exp( f θ) − 1
− Fc
exp( f θ)
exp( f θ) − 1
F1 = (Fc + F)
exp( f θ)
exp( f θ) − 1
+ Fc − Fc
exp( f θ)
exp( f θ) − 1
F1 = (Fc + F)
exp( f θ)
exp( f θ) − 1
−
Fc
exp( f θ) − 1
F1 =
(Fc + F) exp( f θ) − Fc
exp( f θ) − 1
Q.E.D.
From Ex. 17-2: θd = 3.037 rad, F = 664 lbf, exp( f θ) = exp[0.80(3.037)] = 11.35,
and Fc = 73.4 lbf.
F1 =
(73.4 + 664)(11.35 − 73.4)
(11.35 − 1)
= 802 lbf
F2 = F1 − F = 802 − 664 = 138 lbf
Fi =
802 + 138
2
− 73.4 = 396.6 lbf
f =
1
θd
ln
F1 − Fc
F2 − Fc
=
1
3.037
ln
802 − 73.4
138 − 73.4
= 0.80 Ans.
17-9 This is a good class project. Form four groups, each with a belt to design. Once each group
agrees internally, all four should report their designs including the forces and torques on the
line shaft. If you give them the pulley locations, they could design the line shaft when they
get to Chap. 18. For now you could have the groups exchange group reports to determine
if they agree or have counter suggestions.
17-10 If you have the students implement a computer program, the design problem selections
may differ, and the students will be able to explore them. For Ks = 1.25, nd = 1.1,
d = 14 in and D = 28 in, a polyamideA-5 belt, 8 inches wide, will do(bmin = 6.58 in)
17-11 An efﬁciency of less than unity lowers the output for a given input. Since the object of the
drive is the output, the efﬁciency must be incorporated such that the belt’s capacity is in-
creased. The design power would thus be expressed as
Hd =
HnomKsnd
eff
Ans.
17-12 Some perspective on the size of Fc can be obtained from
Fc =
w
g
V
60
2
=
12γ bt
g
V
60
2
shi20396_ch17.qxd 8/28/03 3:58 PM Page 439

An approximate comparison of non-metal and metal belts is presented in the table
below.
Non-metal Metal
γ, lbf/in3
0.04 0.280
b, in 5.00 1.000
t, in 0.20 0.005
The ratio w/wm is
w
wm
=
12(0.04)(5)(0.2)
12(0.28)(1)(0.005)
˙= 29
The second contribution to Fc is the belt peripheral velocity which tends to be low in
metal belts used in instrument, printer, plotter and similar drives. The velocity ratio
squared inﬂuences any Fc/(Fc)m ratio.
It is common for engineers to treat Fc as negligible compared to other tensions in the
belting problem. However, when developing a computer code, one should include Fc .
17-13 Eq. (17-8):
F = F1 − F2 = (F1 − Fc)
exp( f θ) − 1
exp( f θ)
Assuming negligible centrifugal force and setting F1 = ab from step 3,
bmin =
F
a
exp( f θ)
exp( f θ) − 1
(1)
Also, Hd = HnomKsnd =
( F)V
33 000
F =
33 000HnomKsnd
V
Substituting into (1), bmin =
1
a
33 000Hd
V
exp( f θ)
exp( f θ) − 1
Ans.
17-14 The decision set for the friction metal ﬂat-belt drive is:
A priori decisions
• Function: Hnom = 1 hp, n = 1750 rev/min, V R = 2, C ˙= 15 in, Ks = 1.2,
Np = 106
belt passes.
• Design factor: nd = 1.05
• Belt material and properties: 301/302 stainless steel
Table 17-8: Sy = 175 000 psi, E = 28 Mpsi, ν = 0.285
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Chapter 17 441
• Drive geometry: d = 2 in, D = 4 in
Design variables:
• Belt width b
• Belt loop periphery
Preliminaries
Hd = HnomKsnd = 1(1.2)(1.05) = 1.26 hp
T =
63 025(1.26)
1750
= 45.38 lbf · in
A 15 in center-to-center distance corresponds to a belt loop periphery of 39.5 in. The
40 in loop available corresponds to a 15.254 in center distance.
θd = π − 2 sin−1 4 − 2
2(15.254)
= 3.010 rad
θD = π + 2 sin−1 4 − 2
2(15.274)
= 3.273 rad
For full friction development
exp( f θd) = exp[0.35(3.010)] = 2.868
V =
πdn
12
=
π(2)(1750)
12
= 916.3 ft/s
Sy = 175 000 psi
Eq. (17-15):
Sf = 14.17(106
)(106
)−0.407
= 51 212 psi
From selection step 3
a = Sf −
Et
(1 − ν2)d
t = 51 212 −
28(106
)(0.003)
(1 − 0.2852)(2)
(0.003)
= 16.50 lbf/in of belt width
(F1)a = ab = 16.50b
For full friction development, from Prob. 17-13,
bmin =
F
a
exp( f θd)
exp( f θd) − 1
F =
2T
d
=
2(45.38)
2
= 45.38 lbf
So
bmin =
45.38
16.50
2.868
2.868 − 1
= 4.23 in
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Decision #1: b = 4.5 in
F1 = (F1)a = ab = 16.5(4.5) = 74.25 lbf
F2 = F1 − F = 74.25 − 45.38 = 28.87 lbf
Fi =
F1 + F2
2
=
74.25 + 28.87
2
= 51.56 lbf
Existing friction
f =
1
θd
ln
F1
F2
=
1
3.010
ln
74.25
28.87
= 0.314
Ht =
( F)V
33 000
=
45.38(916.3)
33 000
= 1.26 hp
nf s =
Ht
HnomKs
=
1.26
1(1.2)
= 1.05
This is a non-trivial point. The methodology preserved the factor of safety corresponding
to nd = 1.1 even as we rounded bmin up to b.
Decision #2 was taken care of with the adjustment of the center-to-center distance to
accommodate the belt loop. Use Eq. (17-2) as is and solve for C to assist in this. Remem-
ber to subsequently recalculate θd and θD .
17-15 Decision set:
A priori decisions
• Function: Hnom = 5 hp, N = 1125 rev/min, V R = 3, C ˙= 20 in, Ks = 1.25,
Np = 106
belt passes
• Design factor: nd = 1.1
• Belt material: BeCu, Sy = 170 000 psi, E = 17(106
) psi, ν = 0.220
• Belt geometry: d = 3 in, D = 9 in
Design decisions
• Belt loop periphery
• Belt width b
Preliminaries:
Hd = HnomKsnd = 5(1.25)(1.1) = 6.875 hp
T =
63 025(6.875)
1125
= 385.2 lbf · in
Decision #1: Choose a 60-in belt loop with a center-to-center distance of 20.3 in.
θd = π − 2 sin−1 9 − 3
2(20.3)
= 2.845 rad
θD = π + 2 sin−1 9 − 3
2(20.3)
= 3.438 rad
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Chapter 17 443
For full friction development:
exp( f θd) = exp[0.32(2.845)] = 2.485
V =
πdn
12
=
π(3)(1125)
12
= 883.6 ft/min
Sf = 56 670 psi
From selection step 3
a = Sf −
Et
(1 − ν2)d
t = 56 670 −
17(106
)(0.003)
(1 − 0.222)(3)
(0.003) = 116.4 lbf/in
F =
2T
d
=
2(385.2)
3
= 256.8 lbf
bmin =
F
a
exp( f θd)
exp( f θd) − 1
=
256.8
116.4
2.485
2.485 − 1
= 3.69 in
Decision #2: b = 4 in
F1 = (F1)a = ab = 116.4(4) = 465.6 lbf
F2 = F1 − F = 465.6 − 256.8 = 208.8 lbf
Fi =
F1 + F2
2
=
465.6 + 208.8
2
= 337.3 lbf
Existing friction
f =
1
θd
ln
F1
F2
=
1
2.845
ln
465.6
208.8
= 0.282
H =
( F)V
33 000
=
256.8(883.6)
33 000
= 6.88 hp
nf s =
H
5(1.25)
=
6.88
5(1.25)
= 1.1
Fi can be reduced only to the point at which f = f = 0.32. From Eq. (17-9)
Fi =
T
d
exp( f θd) + 1
exp( f θd) − 1
=
385.2
3
2.485 + 1
2.485 − 1
= 301.3 lbf
Eq. (17-10):
F1 = Fi
2 exp( f θd)
exp( f θd) + 1
= 301.3
2(2.485)
2.485 + 1
= 429.7 lbf
F2 = F1 − F = 429.7 − 256.8 = 172.9 lbf
and f = f = 0.32
17-16 This solution is the result of a series of ﬁve design tasks involving different belt thick-
nesses. The results are to be compared as a matter of perspective. These design tasks are
accomplished in the same manner as in Probs. 17-14 and 17-15 solutions.
shi20396_ch17.qxd 8/28/03 3:58 PM Page 443

The details will not be presented here, but the table is provided as a means of learning.
Five groups of students could each be assigned a belt thickness. You can form a table from
their results or use the table below
t, in
0.002 0.003 0.005 0.008 0.010
b 4.000 3.500 4.000 1.500 1.500
CD 20.300 20.300 20.300 18.700 20.200
a 109.700 131.900 110.900 194.900 221.800
d 3.000 3.000 3.000 5.000 6.000
D 9.000 9.000 9.000 15.000 18.000
Fi 310.600 333.300 315.200 215.300 268.500
F1 439.000 461.700 443.600 292.300 332.700
F2 182.200 209.000 186.800 138.200 204.300
nf s 1.100 1.100 1.100 1.100 1.100
L 60.000 60.000 60.000 70.000 80.000
f 0.309 0.285 0.304 0.288 0.192
Fi 301.200 301.200 301.200 195.700 166.600
F1 429.600 429.600 429.600 272.700 230.800
F2 172.800 172.800 172.800 118.700 102.400
f 0.320 0.320 0.320 0.320 0.320
The first three thicknesses result in the same adjusted Fi , F1 and F2 (why?). We have no
figure of merit, but the costs of the belt and the pulleys is about the same for these three
thicknesses. Since the same power is transmitted and the belts are widening, belt forces
are lessening.
17-17 This is a design task. The decision variables would be belt length and belt section, which
could be combined into one, such as B90. The number of belts is not an issue.
We have no figure of merit, which is not practical in a text for this application. I sug-
gest you gather sheave dimensions and costs and V-belt costs from a principal vendor and
construct a figure of merit based on the costs. Here is one trial.
Preliminaries: For a single V-belt drive with Hnom = 3 hp, n = 3100 rev/min,
D = 12 in, and d = 6.2 in, choose a B90 belt, Ks = 1.3 and nd = 1.
Lp = 90 + 1.8 = 91.8 in
Eq. (17-16b):
C = 0.25



91.8 −
π
2
(12 + 6.2) + 91.8 −
π
2
(12 + 6.2)
2
− 2(12 − 6.2)2



= 31.47 in
θd = π − 2 sin−1 12 − 6.2
2(31.47)
= 2.9570 rad
exp( f θd) = exp[0.5123(2.9570)] = 4.5489
V =
πdn
12
=
π(6.2)(3100)
12
= 5031.8 ft/min
shi20396_ch17.qxd 8/28/03 3:58 PM Page 444

Chapter 17 445
Table 17-13:
Angle θ = θd
180°
π
= (2.957 rad)
180°
π
= 169.42°
The footnote regression equation gives K1 without interpolation:
K1 = 0.143 543 + 0.007 468(169.42°) − 0.000 015 052(169.42°)2
= 0.9767
The design power is
Hd = HnomKsnd = 3(1.3)(1) = 3.9 hp
From Table 17-14 for B90, K2 = 1. From Table 17-12 take a marginal entry of Htab = 4,
although extrapolation would give a slightly lower Htab .
Eq. (17-17): Ha = K1K2 Htab
= 0.9767(1)(4) = 3.91 hp
The allowable Fa is given by
Fa =
63 025Ha
n(d/2)
=
63 025(3.91)
3100(6.2/2)
= 25.6 lbf
The allowable torque Ta is
Ta =
Fad
2
=
25.6(6.2)
2
= 79.4 lbf · in
From Table 17-16, Kc = 0.965. Thus, Eq. (17-21) gives,
Fc = 0.965
5031.8
1000
2
= 24.4 lbf
At incipient slip, Eq. (17-9) provides:
Fi =
T
d
exp( f θ) + 1
exp( f θ) − 1
=
79.4
6.2
4.5489 + 1
4.5489 − 1
= 20.0 lbf
Eq. (17-10):
F1 = Fc + Fi
2 exp( f θ)
exp( f θ) + 1
= 24.4 + 20
2(4.5489)
4.5489 + 1
= 57.2 lbf
Thus, F2 = F1 − Fa = 57.2 − 25.6 = 31.6 lbf
Eq. (17-26): nf s =
Ha Nb
Hd
=
(3.91)(1)
3.9
= 1.003 Ans.
If we had extrapolated for Htab, the factor of safety would have been slightly less
than one.
Life Use Table 17-16 to ﬁnd equivalent tensions T1 and T2 .
T1 = F1 + (Fb)1 = F1 +
Kb
d
= 57.2 +
576
6.2
= 150.1 lbf
T2 = F1 + (Fb)2 = F1 +
Kb
D
= 57.2 +
576
12
= 105.2 lbf
shi20396_ch17.qxd 8/28/03 3:58 PM Page 445

From Eq. (17-27), the number of belt passes is:
NP =
1193
150.1
−10.929
+
1193
105.2
−10.929 −1
= 6.76(109
)
From Eq. (17-28) for NP > 109
,
t =
NP Lp
720V
>
109
(91.8)
720(5031.8)
t > 25 340 h Ans.
Suppose nf s was too small. Compare these results with a 2-belt solution.
Htab = 4 hp/belt, Ta = 39.6 lbf · in/belt,
Fa = 12.8 lbf/belt, Ha = 3.91 hp/belt
nf s =
Nb Ha
Hd
=
Nb Ha
HnomKs
=
2(3.91)
3(1.3)
= 2.0
Also, F1 = 40.8 lbf/belt, F2 = 28.0 lbf/belt,
Fi = 9.99 lbf/belt, Fc = 24.4 lbf/belt
(Fb)1 = 92.9 lbf/belt, (Fb)2 = 48 lbf/belt
T1 = 133.7 lbf/belt, T2 = 88.8 lbf/belt
NP = 2.39(1010
) passes, t > 605 600 h
Initial tension of the drive:
(Fi )drive = Nb Fi = 2(9.99) = 20 lbf
17-18 Given: two B85 V-belts with d = 5.4 in, D = 16 in, n = 1200 rev/min, and Ks = 1.25
Table 17-11: Lp = 85 + 1.8 = 86.8 in
Eq. (17-17b):
C = 0.25



86.8 −
π
2
(16 + 5.4) + 86.8 −
π
2
(16 + 5.4)
2
− 2(16 − 5.4)2



= 26.05 in Ans.
Eq. (17-1):
θd = 180° − 2 sin−1 16 − 5.4
2(26.05)
= 156.5°
From table 17-13 footnote:
K1 = 0.143 543 + 0.007 468(156.5°) − 0.000 015 052(156.5°)2
= 0.944
Table 17-14: K2 = 1
Belt speed: V =
π(5.4)(1200)
12
= 1696 ft/min
shi20396_ch17.qxd 8/28/03 3:58 PM Page 446

Chapter 17 447
Use Table 17-12 to interpolate for Htab.
Htab = 1.59 +
2.62 − 1.59
2000 − 1000
(1696 − 1000) = 2.31 hp/belt
Ha = K1K2 Nb Htab = 1(0.944)(2)(2.31) = 4.36 hp
Assuming nd = 1
Hd = Ks Hnomnd = 1.25(1)Hnom
For a factor of safety of one,
Ha = Hd
4.36 = 1.25Hnom
Hnom =
4.36
1.25
= 3.49 hp Ans.
17-19 Given: Hnom = 60 hp, n = 400 rev/min, Ks = 1.4, d = D = 26 in on 12 ft centers.
Designtask:specifyV-beltandnumberofstrands(belts).Tentativedecision:UseD360belts.
Table 17-11: Lp = 360 + 3.3 = 363.3 in
Eq. (17-16b):
C = 0.25



363.3 −
π
2
(26 + 26) + 363.3 −
π
2
(26 + 26)
2
− 2(26 − 26)2



= 140.8 in (nearly 144 in)
θd = π, θD = π, exp[0.5123π] = 5.0,
V =
πdn
12
=
π(26)(400)
12
= 2722.7 ft/min
Table 17-13: For θ = 180°, K1 = 1
Table 17-14: For D360, K2 = 1.10
Table 17-12: Htab = 16.94 hp by interpolation
Thus, Ha = K1K2 Htab = 1(1.1)(16.94) = 18.63 hp
Hd = Ks Hnom = 1.4(60) = 84 hp
Number of belts, Nb
Nb =
Ks Hnom
K1K2 Htab
=
Hd
Ha
=
84
18.63
= 4.51
Round up to ﬁve belts. It is left to the reader to repeat the above for belts such as C360
and E360.
Fa =
63 025Ha
n(d/2)
=
63 025(18.63)
400(26/2)
= 225.8 lbf/belt
Ta =
( Fa)d
2
=
225.8(26)
2
= 2935 lbf · in/belt
shi20396_ch17.qxd 8/28/03 3:58 PM Page 447

Eq. (17-21):
Fc = 3.498
V
1000
2
= 3.498
2722.7
1000
2
= 25.9 lbf/belt
At fully developed friction, Eq. (17-9) gives
Fi =
T
d
exp( f θ) + 1
exp( f θ) − 1
=
2935
26
5 + 1
5 − 1
= 169.3 lbf/belt
Eq. (17-10): F1 = Fc + Fi
2 exp( f θ)
exp( f θ) + 1
= 25.9 + 169.3
2(5)
5 + 1
= 308.1 lbf/belt
F2 = F1 − Fa = 308.1 − 225.8 = 82.3 lbf/belt
nf s =
Ha Nb
Hd
=
(185.63)
84
= 1.109 Ans.
Reminder: Initial tension is for the drive
(Fi )drive = Nb Fi = 5(169.3) = 846.5 lbf
A 360 belt is at the right-hand edge of the range of center-to-center pulley distances.
D ≤ C ≤ 3(D + d)
26 ≤ C ≤ 3(26 + 26)
17-20 Preliminaries: D ˙= 60 in, 14-in wide rim, Hnom = 50 hp, n = 875 rev/min, Ks = 1.2,
nd = 1.1, mG = 875/170 = 5.147, d ˙= 60/5.147 = 11.65 in
(a) From Table 17-9, an 11-in sheave exceeds C-section minimum diameter and pre-
cludes D- and E-section V-belts.
Decision: Use d = 11 in, C270 belts
Table 17-11: Lp = 270 + 2.9 = 272.9 in
C = 0.25



272.9 −
π
2
(60 + 11) + 272.9 −
π
2
(60 + 11)
2
− 2(60 − 11)2



= 76.78 in
This ﬁts in the range
D < C < 3(D + d)
60 < C < 3(60 + 11)
60 in < C < 213 in
θd = π − 2 sin−1 60 − 11
2(76.78)
= 2.492 rad
θD = π + 2 sin−1 60 − 11
2(76.78)
= 3.791 rad
exp[0.5123(2.492)] = 3.5846
shi20396_ch17.qxd 8/28/03 3:58 PM Page 448

Chapter 17 449
For the ﬂat on ﬂywheel
exp[0.13(3.791)] = 1.637
V =
πdn
12
=
π(11)(875)
12
= 2519.8 ft/min
Table 17-13: Regression equation gives K1 = 0.90
Table 17-14: K2 = 1.15
Table 17-12: Htab = 7.83 hp/belt by interpolation
Eq. (17-17): Ha = K1K2 Htab = 0.905(1.15)(7.83) = 8.15 hp
Eq. (17-19): Hd = HnomKsnd = 50(1.2)(1.1) = 66 hp
Eq. (17-20): Nb =
Hd
Ha
=
66
8.15
= 8.1 belts
Decision: Use 9 belts. On a per belt basis,
Fa =
63 025Ha
n(d/2)
=
63 025(8.15)
875(11/2)
= 106.7 lbf/belt
Ta =
Fad
2
=
106.7(11)
2
= 586.9 lbf per belt
Fc = 1.716
V
1000
2
= 1.716
2519.8
1000
2
= 10.9 lbf/belt
At fully developed friction, Eq. (17-9) gives
Fi =
T
d
exp( f θd) + 1
exp( f θd) − 1
=
586.9
11
3.5846 + 1
3.5846 − 1
= 94.6 lbf/belt
Eq. (17-10):
F1 = Fc + Fi
2 exp( f θd)
exp( f θd) + 1
= 10.9 + 94.6
2(3.5846)
3.5846 + 1
= 158.8 lbf/belt
F2 = F1 − Fa = 158.8 − 106.7 = 52.1 lbf/belt
nf s =
Nb Ha
Hd
=
9(8.15)
66
= 1.11 O.K. Ans.
Durability:
(Fb)1 = 145.45 lbf/belt, (Fb)2 = 76.7 lbf/belt
T1 = 304.4 lbf/belt, T2 = 185.6 lbf/belt
and t > 150 000 h
Remember: (Fi )drive = 9(94.6) = 851.4 lbf
Table 17-9: C-section belts are 7/8" wide. Check sheave groove spacing to see if
14"-width is accommodating.
shi20396_ch17.qxd 8/28/03 3:58 PM Page 449

(b) The fully developed friction torque on the flywheel using the flats of the V-belts is
Tflat = Fi
exp( f θ) − 1
exp( f θ) + 1
= 60(94.6)
1.637 − 1
1.637 + 1
= 1371 lbf · in per belt
The flywheel torque should be
Tfly = mGTa = 5.147(586.9) = 3021 lbf · in per belt
but it is not. There are applications, however, in which it will work. For example,
make the flywheel controlling. Yes. Ans.
17-21
(a) S is the spliced-in string segment length
De is the equatorial diameter
D is the spliced string diameter
δ is the radial clearance
S + π De = π D = π(De + 2δ) = π De + 2πδ
From which
δ =
S
2π
The radial clearance is thus independent of De .
δ =
12(6)
2π
= 11.5 in Ans.
This is true whether the sphere is the earth, the moon or a marble. Thinking in terms
of a radial or diametral increment removes the basic size from the problem. Viewpoint
again!
(b) and (c)
Table 17-9: For an E210 belt, the thickness is 1 in.
dP − di =
210 + 4.5
π
−
210
π
=
4.5
π
2δ =
4.5
π
δ =
4.5
2π
= 0.716 in
0.716"
1"
dp
Dp
␦
␦
Pitch surface
60"
DЈ
S
De ␦
shi20396_ch17.qxd 8/28/03 3:58 PM Page 450

Chapter 17 451
The pitch diameter of the flywheel is
DP − 2δ = D
DP = D + 2δ = 60 + 2(0.716) = 61.43 in
We could make a table:
Diametral Section
Growth A B C D E
2δ
1.3
π
1.8
π
2.9
π
3.3
π
4.5
π
The velocity ratio for the D-section belt of Prob. 17-20 is
mG =
D + 2δ
d
=
60 + 3.3/π
11
= 5.55 Ans.
for the V-flat drive as compared to ma = 60/11 = 5.455 for the VV drive.
The pitch diameter of the pulley is still d = 11 in, so the new angle of wrap, θd, is
θd = π − 2 sin−1 D + 2δ − d
2C
Ans.
θD = π + 2 sin−1 D + 2δ − d
2C
Ans.
Equations (17-16a) and (17-16b) are modified as follows
Lp = 2C +
π
2
(D + 2δ + d) +
(D + δ − d)2
4C
Ans.
Cp = 0.25



Lp −
π
2
(D + 2δ + d)
+ Lp −
π
2
(D + 2δ + d)
2
− 2(D + 2δ − d)2



Ans.
The changes are small, but if you are writing a computer code for a V-flat drive,
remember that θd and θD changes are exponential.
17-22 This design task involves specifying a drive to couple an electric motor running at
1720 rev/min to a blower running at 240 rev/min, transmitting two horsepower with a
center distance of at least 22 inches. Instead of focusing on the steps, we will display two
different designs side-by-side for study. Parameters are in a “per belt” basis with per drive
quantities shown along side, where helpful.
shi20396_ch17.qxd 8/28/03 3:58 PM Page 451

Parameter Four A-90 Belts Two A-120 Belts
mG 7.33 7.142
Ks 1.1 1.1
nd 1.1 1.1
K1 0.877 0.869
K2 1.05 1.15
d, in 3.0 4.2
D, in 22 30
θd, rad 2.333 2.287
V, ft/min 1350.9 1891
exp( f θd) 3.304 3.2266
Lp, in 91.3 101.3
C, in 24.1 31
Htab, uncorr. 0.783 1.662
Nb Htab, uncorr. 3.13 3.326
Ta, lbf · in 26.45(105.8) 60.87(121.7)
Fa, lbf 17.6(70.4) 29.0(58)
Ha, hp 0.721(2.88) 1.667(3.33)
nf s 1.192 1.372
F1, lbf 26.28(105.2) 44(88)
F2, lbf 8.67(34.7) 15(30)
(Fb)1, lbf 73.3(293.2) 52.4(109.8)
(Fb)2, lbf 10(40) 7.33(14.7)
Fc, lbf 1.024 2.0
Fi , lbf 16.45(65.8) 27.5(55)
T1, lbf · in 99.2 96.4
T2, lbf · in 36.3 57.4
N , passes 1.61(109
) 2.3(109
)
t > h 93 869 89 080
Conclusions:
• Smaller sheaves lead to more belts.
• Larger sheaves lead to larger D and larger V.
• Larger sheaves lead to larger tabulated power.
• The discrete numbers of belts obscures some of the variation. The factors of safety
exceed the design factor by differing amounts.
17-23 In Ex. 17-5 the selected chain was 140-3, making the pitch of this 140 chain14/8 = 1.75 in.
Table 17-19 conﬁrms.
17-24
(a) Eq. (17-32): H1 = 0.004N1.08
1 n0.9
1 p(3−0.07p)
Eq. (17-33): H2 =
1000Kr N1.5
1 p0.8
n1.5
1
shi20396_ch17.qxd 8/28/03 3:58 PM Page 452

Chapter 17 453
Equating and solving for n1 gives
n1 =
0.25(106
)Kr N0.42
1
p(2.2−0.07p)
1/2.4
Ans.
(b) For a No. 60 chain, p = 6/8 = 0.75 in, N1 = 17, Kr = 17
n1 =
0.25(106
)(17)(17)0.42
0.75[2.2−0.07(0.75)]
1/2.4
= 1227 rev/min Ans.
Table 17-20 conﬁrms that this point occurs at 1200 ± 200 rev/min.
(c) Life predictions using Eq. (17-40) are possible at speeds greater than 1227 rev/min.
Ans.
17-25 Given: a double strand No. 60 roller chain with p = 0.75 in, N1 = 13 teeth at 300 rev/min,
N2 = 52 teeth.
(a) Table 17-20: Htab = 6.20 hp
Table 17-22: K1 = 0.75
Table 17-23: K2 = 1.7
Use Ks = 1
Eq. (17-37):
Ha = K1K2 Htab = 0.75(1.7)(6.20) = 7.91 hp Ans.
(b) Eqs. (17-35) and (17-36) with L/p = 82
A =
13 + 52
2
− 82 = −49.5[
C =
p
4

49.5 + 49.52 − 8
52 − 13
2π
2

 = 23.95p
C = 23.95(0.75) = 17.96 in, round up to 18 in Ans.
(c) For 30 percent less power transmission,
H = 0.7(7.91) = 5.54 hp
T =
63 025(5.54)
300
= 1164 lbf · in Ans.
Eq. (17-29):
D =
0.75
sin(180◦/13)
= 3.13 in
F =
T
r
=
1164
3.13/2
= 744 lbf Ans.
shi20396_ch17.qxd 8/28/03 3:58 PM Page 453

17-26 Given: No. 40-4 chain, N1 = 21 teeth for n = 2000 rev/min, N2 = 84 teeth,
h = 20 000 hours.
(a) Chain pitch is p = 4/8 = 0.500 in and C ˙= 20 in.
Eq. (17-34):
L
p
=
2(20)
0.5
+
21 + 84
2
+
(84 − 21)2
4π2(20/0.5)
= 135 pitches (or links)
L = 135(0.500) = 67.5 in Ans.
(b) Table 17-20: Htab = 7.72 hp (post-extreme power)
Eq. (17-40): Since K1 is required, the N3.75
1 term is omitted.
const =
7.722.5
(15 000)
135
= 18 399
Htab =
18 399(135)
20 000
1/2.5
= 6.88 hp Ans.
(c) Table 17-22:
K1 =
21
17
1.5
= 1.37
Table 17-23: K2 = 3.3
Ha = K1K2 Htab = 1.37(3.3)(6.88) = 31.1 hp Ans.
(d) V =
N1 pn
12
=
21(0.5)(2000)
12
= 1750 ft/min
F1 =
33 000(31.1)
1750
= 586 lbf Ans.
17-27 This is our ﬁrst design/selection task for chain drives. A possible decision set:
A priori decisions
• Function: Hnom, n1, space, life, Ks
• Design factor: nd
• Sprockets: Tooth counts N1 and N2, factors K1 and K2
Decision variables
• Chain number
• Strand count
• Lubrication type
• Chain length in pitches
Function: Motor with Hnom = 25 hp at n = 700 rev/min; pump at n = 140 rev/min;
mG = 700/140 = 5
Design Factor: nd = 1.1
Sprockets: Tooth count N2 = mG N1 = 5(17) = 85 teeth–odd and unavailable. Choose
84 teeth. Decision: N1 = 17, N2 = 84
shi20396_ch17.qxd 8/28/03 3:58 PM Page 454

Chapter 17 455
Evaluate K1 and K2
Eq. (17-38): Hd = HnomKsnd
Eq. (17-37): Ha = K1K2 Htab
Equate Hd to Ha and solve for Htab :
Htab =
Ksnd Hnom
K1K2
Table 17-22: K1 = 1
Table 17-23: K2 = 1, 1.7, 2.5, 3.3 for 1 through 4 strands
Htab =
1.5(1.1)(25)
(1)K2
=
41.25
K2
Prepare a table to help with the design decisions:
Chain Lub.
Strands K2 Htab No. Htab nf s Type
1 1.0 41.3 100 59.4 1.58 B
2 1.7 24.3 80 31.0 1.40 B
3 2.5 16.5 80 31.0 2.07 B
4 3.3 12.5 60 13.3 1.17 B
Design Decisions
We need a ﬁgure of merit to help with the choice. If the best was 4 strands of No. 60
chain, then
Decision #1 and #2: Choose four strand No. 60 roller chain with nf s = 1.17.
nf s =
K1K2 Htab
Ks Hnom
=
1(3.3)(13.3)
1.5(25)
= 1.17
Decision #3: Choose Type B lubrication
Analysis:
Table 17-20: Htab = 13.3 hp
Table 17-19: p = 0.75 in
Try C = 30 in in Eq. (17-34):
L
p
=
2C
p
+
N1 + N2
2
+
(N2 − N1)2
4π2C/p
= 2(30/0.75) +
17 + 84
2
+
(84 − 17)2
4π2(30/0.75)
= 133.3 → 134
From Eq. (17-35) with p = 0.75 in,C = 30.26 in.
Decision #4: Choose C = 30.26 in.
shi20396_ch17.qxd 8/28/03 3:58 PM Page 455

17-28 Follow the decision set outlined in Prob. 17-27 solution. We will form two tables, the ﬁrst
for a 15 000 h life goal, and a second for a 50 000 h life goal. The comparison is useful.
Function: Hnom = 50 hp at n = 1800 rev/min, npump = 900 rev/min
mG = 1800/900 = 2, Ks = 1.2
life = 15 000 h, then repeat with life = 50 000 h
Design factor: nd = 1.1
Sprockets: N1 = 19 teeth, N2 = 38 teeth
Table 17-22 (post extreme):
K1 =
N1
17
1.5
=
19
17
1.5
= 1.18
Table 17-23:
K2 = 1, 1.7, 2.5, 3.3, 3.9, 4.6, 6.0
Decision variables for 15 000 h life goal:
Htab =
Ksnd Hnom
K1K2
=
1.2(1.1)(50)
1.18K2
=
55.9
K2
(1)
nf s =
K1K2 Htab
Ks Hnom
=
1.18K2 Htab
1.2(50)
= 0.0197K2 Htab
Form a table for a 15 000 h life goal using these equations.
K2 Htab Chain # Htab nf s Lub
1 1.0 55.90 120 21.6 0.423 C
2 1.7 32.90 120 21.6 0.923 C
3 2.5 22.40 120 21.6 1.064 C
4 3.3 16.90 120 21.6 1.404 C
5 3.9 14.30 80 15.6 1.106 C
6 4.6 12.20 60 12.4 1.126 C
8 6.0 9.32 60 12.4 1.416 C
There are 4 possibilities where nf s ≥ 1.1
Decision variables for 50 000h life goal
From Eq. (17-40), the power-life tradeoff is:
(Htab)2.5
15 000 = (Htab)2.5
50 000
Htab =
15 000
50 000
(Htab)2.5
1/2.5
= 0.618 Htab
Substituting from (1),
Htab = 0.618
55.9
K2
=
34.5
K2
shi20396_ch17.qxd 8/28/03 3:58 PM Page 456

Chapter 17 457
The H notation is only necessary because we constructed the first table, which we nor-
mally would not do.
nf s =
K1K2 Htab
Ks Hnom
=
K1K2(0.618Htab)
Ks Hnom
= 0.618[(0.0197)K2 Htab] = 0.0122K2 Htab
Form a table for a 50 000 h life goal.
K2 Htab Chain # Htab nf s Lub
1 1.0 34.50 120 21.6 0.264 C
2 1.7 20.30 120 21.6 0.448 C
3 2.5 13.80 120 21.6 0.656 C
4 3.3 10.50 120 21.6 0.870 C
5 3.9 8.85 120 21.6 1.028 C
6 4.6 7.60 120 21.6 1.210 C
8 6.0 5.80 80 15.6 1.140 C
There are two possibilities in the second table with nf s ≥ 1.1. (The tables allow for the
identification of a longer life one of the outcomes.) We need a figure of merit to help with
the choice; costs of sprockets and chains are thus needed, but is more information than
we have.
Decision #1: #80 Chain (smaller installation) Ans.
nf s = 0.0122K2 Htab = 0.0122(8.0)(15.6) = 1.14 O.K.
Decision #2: 8-Strand, No. 80 Ans.
Decision #3: Type C Lubrication Ans.
Decision #4: p = 1.0 in, C is in midrange of 40 pitches
L
p
=
2C
p
+
N1 + N2
2
+
(N2 − N1)2
4π2C/p
= 2(40) +
19 + 38
2
+
(38 − 19)2
4π2(40)
= 108.7 ⇒ 110 even integer Ans.
Eq. (17-36):
A =
N1 + N2
2
−
L
p
=
19 + 38
2
− 110 = −81.5
Eq. (17-35):
C
p
=
1
4

81.5 + 81.52 − 8
38 − 19
2π
2

 = 40.64
C = p(C/p) = 1.0(40.64) = 40.64 in (for reference) Ans.
shi20396_ch17.qxd 8/28/03 3:58 PM Page 457

17-29 The objective of the problem is to explore factors of safety in wire rope. We will express
strengths as tensions.
(a) Monitor steel 2-in 6 × 19 rope, 480 ft long
Table 17-2: Minimum diameter of a sheave is 30d = 30(2) = 60 in, preferably
45(2) = 90 in. The hoist abuses the wire when it is bent around a sheave. Table 17-24
gives the nominal tensile strength as 106 kpsi. The ultimate load is
Fu = (Su)nom Anom = 106
π(2)2
4
= 333 kip Ans.
The tensile loading of the wire is given by Eq. (17-46)
Ft =
W
m
+ wl 1 +
a
g
W = 4(2) = 8 kip, m = 1
Table (17-24):
wl = 1.60d2
l = 1.60(22
)(480) = 3072 lbf or 3.072 kip
Therefore,
Ft = (8 + 3.072) 1 +
2
32.2
= 11.76 kip Ans.
Eq. (17-48):
Fb =
Er dw Am
D
and for the 72-in drum
Fb =
12(106
)(2/13)(0.38)(22
)(10−3
)
72
= 39 kip Ans.
For use in Eq. (17-44), from Fig. 17-21
( p/Su) = 0.0014
Su = 240 kpsi, p. 908
Ff =
0.0014(240)(2)(72)
2
= 24.2 kip Ans.
(b) Factors of safety
Static, no bending:
n =
Fu
Ft
=
333
11.76
= 28.3 Ans.
Static, with bending:
Eq. (17-49): ns =
Fu − Fb
Ft
=
333 − 39
11.76
= 25.0 Ans.
shi20396_ch17.qxd 8/28/03 3:58 PM Page 458

Chapter 17 459
Fatigue without bending:
nf =
Ff
Ft
=
24.2
11.76
= 2.06 Ans.
Fatigue, with bending: For a life of 0.1(106
) cycles, from Fig. 17-21
( p/Su) = 4/1000 = 0.004
Ff =
0.004(240)(2)(72)
2
= 69.1 kip
Eq. (17-50): nf =
69.1 − 39
11.76
= 2.56 Ans.
If we were to use the endurance strength at 106
cycles (Ff = 24.2 kip) the factor of
safety would be less than 1 indicating 106
cycle life impossible.
Comments:
• There are a number of factors of safety used in wire rope analysis. They are differ-
ent, with different meanings. There is no substitute for knowing exactly which fac-
tor of safety is written or spoken.
• Static performance of a rope in tension is impressive.
• In this problem, at the drum, we have a ﬁnite life.
• The remedy for fatigue is the use of smaller diameter ropes, with multiple ropes
supporting the load. See Ex. 17-6 for the effectiveness of this approach. It will also
be used in Prob. 17-30.
• Remind students that wire ropes do not fail suddenly due to fatigue. The outer
wires gradually show wear and breaks; such ropes should be retired. Periodic in-
spections prevent fatigue failures by parting of the rope.
17-30 Since this is a design task, a decision set is useful.
A priori decisions
• Function: load, height, acceleration, velocity, life goal
• Design Factor: nd
• Material: IPS, PS, MPS or other
• Rope: Lay, number of strands, number of wires per strand
Decision variables:
• Nominal wire size: d
• Number of load-supporting wires: m
From experience with Prob. 17-29, a 1-in diameter rope is not likely to have much of a
life, so approach the problem with the d and m decisions open.
Function: 5000 lbf load, 90 foot lift,acceleration = 4 ft/s2
,velocity = 2 ft/s, life
goal = 105
cycles
Design Factor: nd = 2
Material: IPS
Rope: Regular lay, 1-in plow-steel 6 × 19 hoisting
shi20396_ch17.qxd 8/28/03 3:58 PM Page 459

Design variables
Choose 30-in Dmin. Table 17-27: w = 1.60d2
lbf/ft
wl = 1.60d2
l = 1.60d2
(90) = 144d2
lbf, ea.
Eq. (17-46):
Ft =
W
m
+ wl 1 +
a
g
=
5000
m
+ 144d2
1 +
4
32.2
=
5620
m
+ 162d2
lbf, each wire
Eq. (17-47):
Ff =
( p/Su)Su Dd
2
From Fig. 17-21 for 105
cycles, p/Su = 0.004; from p. 908, Su = 240 000 psi, based on
metal area.
Ff =
0.004(240 000)(30d)
2
= 14 400d lbf each wire
Eq. (17-48) and Table 17-27:
Fb =
Ewdw Am
D
=
12(106
)(0.067d)(0.4d2
)
30
= 10 720d3
lbf, each wire
Eq. (17-45):
nf =
Ff − Fb
Ft
=
14 400d − 10 720d3
(5620/m) + 162d2
We could use a computer program to build a table similar to that of Ex. 17-6. Alterna-
tively, we could recognize that 162d2
is small compared to 5620/m, and therefore elimi-
nate the 162d2
term.
nf ˙=
14 400d − 10 720d3
5620/m
=
m
5620
(14 400d − 10 720d3
)
Maximize nf ,
∂nf
∂d
= 0 =
m
5620
[14 400 − 3(10 720)d2
]
From which
d* =
14 400
32 160
= 0.669 in
Back-substituting
nf =
m
5620
[14 400(0.669) − 10 720(0.6693
)] = 1.14 m
Thus nf =1.14, 2.28, 3.42, 4.56 for m =1, 2, 3, 4 respectively. If we choose d =0.50 in,
then m = 2.
nf =
14 400(0.5) − 10 720(0.53
)
(5620/2) + 162(0.5)2
= 2.06
shi20396_ch17.qxd 8/28/03 3:58 PM Page 460

Chapter 17 461
This exceeds nd = 2
Decision #1: d = 1/2 in
Decision #2: m = 2 ropes supporting load. Rope should be inspected weekly for any
signs of fatigue (broken outer wires).
Comment: Table 17-25 gives n for freight elevators in terms of velocity.
Fu = (Su)nom Anom = 106 000
πd2
4
= 83 252d2
lbf, each wire
n =
Fu
Ft
=
83 452(0.5)2
(5620/2) + 162(0.5)2
= 7.32
By comparison, interpolation for 120 ft/min gives 7.08-close. The category of construc-
tion hoists is not addressed in Table 17-25. We should investigate this before proceeding
further.
17-31 2000 ft lift, 72 in drum, 6 × 19 MS rope. Cage and load 8000 lbf, acceleration = 2 ft/s2
.
(a) Table 17-24: (Su)nom = 106 kpsi; Su = 240 kpsi (p. 1093, metal area); Fig. 17-22:
( p/Su)106 = 0.0014
Ff =
0.0014(240)(72)d
2
= 12.1d kip
Table 17-24: wl = 1.6d2
2000(10−3
) = 3.2d2
kip
Eq. (17-46): Ft = (W + wl) 1 +
a
g
= (8 + 3.2d2
) 1 +
2
32.2
= 8.5 + 3.4d2
kip
Note that bending is not included.
n =
Ff
Ft
=
12.1d
8.5 + 3.4d2
← maximum n Ans.
d, in n
0.500 0.650
1.000 1.020
1.500 1.124
1.625 1.125
1.750 1.120
2.000 1.095
shi20396_ch17.qxd 8/28/03 3:58 PM Page 461

(b) Try m = 4 strands
Ft =
8
4
+ 3.2d2
1 +
2
32.2
= 2.12 + 3.4d2
kip
Ff = 12.1d kip
n =
12.1d
2.12 + 3.4d2
Comparing tables, multiple ropes supporting the load increases the factor of safety,
and reduces the corresponding wire rope diameter, a useful perspective.
17-32
n =
ad
b/m + cd2
dn
dd
=
(b/m + cd2
)a − ad(2cd)
(b/m + cd2)2
= 0
From which
d* =
b
mc
Ans.
n* =
a
√
b/(mc)
(b/m) + c[b/(mc)]
=
a
2
m
bc
Ans.
These results agree closely with Prob. 17-31 solution. The small differences are due to
rounding in Prob. 17-31.
17-33 From Prob. 17-32 solution:
n1 =
ad
b/m + cd2
Solve the above equation for m
m =
b
ad/n1 − cd2
(1)
dm
ad
= 0 =
[(ad/n1) − ad2
](0) − b[(a/n1) − 2cd]
[(ad/n1) − cd2]2
← maximum n Ans.
d, in n
0.5000 2.037
0.5625 2.130
0.6250 2.193
0.7500 2.250
0.8750 2.242
1.0000 2.192
shi20396_ch17.qxd 8/28/03 3:58 PM Page 462

Chapter 17 463
From which d* =
a
2cn1
Ans.
Substituting this result for d in Eq. (1) gives
m* =
4bcn1
a2
Ans.
17-34
Am = 0.40d2
= 0.40(22
) = 1.6 in2
Er = 12 Mpsi, w = 1.6d2
= 1.6(22
) = 6.4 lbf/ft
wl = 6.4(480) = 3072 lbf
Treat the rest of the system as rigid, so that all of the stretch is due to the cage weighing
1000 lbf and the wire’s weight. From Prob. 5-6
δ1 =
Pl
AE
+
(wl)l
2AE
=
1000(480)(12)
1.6(12)(106)
+
3072(480)(12)
2(1.6)(12)(106)
= 0.3 + 0.461 = 0.761 in
due to cage and wire. The stretch due to the wire, the cart and the cage is
δ2 =
9000(480)(12)
1.6(12)(106)
+ 0.761 = 3.461 in Ans.
17-35 to 17-38 Computer programs will vary.
shi20396_ch17.qxd 8/28/03 3:58 PM Page 463

Capítulo 17 elementos mecânicos flexíveis

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