Chapter 19: Flexible Machine
Elements
Scientists study the world as it is,
engineers create the world that never
has been.
Theodore von Karman
A rolling chain on a sprocket. Source:
Shutterstock.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Comparison of Power Transmission
Devices
Constraint
Flat Belt
Synchronization
i
1
Efficenc y
1
Anti-shock
4
N oise/ vibration
4
Com pactness
High speed / low load
2
Low speed / hig h load
1
Lubrication
None
Bearing load s
2
Long evity
1
1, Poor; 2, Fair; 3, G ood ; 4, Excellent.
V-Belt
1
1
4
4
3
1
None
1
2
Power Transmission Device
Synchronous Belt Roller Chain
1
4
2
4
3
2
3
2
3
2
None
2
2
1
4
R equired
4
3
Silent Chain
4
4
3
3
Spur Gear
4
4
1
1
4
3
R equired
3
3
3
2
R equired
4
4
Table 19.1: Comparison of selected power transmission devices.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Belt Dimensions
Figure 19.1: Dimensions, angles of contact, and center distance of open flat belt.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Weighted Idler
Figure 19.2: Weighted idler used to maintain desired belt tension.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Synchronous Belt
Figure 19.3: Synchronous, or timing, belt.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
V-Belts
Figure 19.4: Design and construction of V-belts. (a) Standard V-belts cross sections with
dimensions; (b) Typical single-belt, showing reinforcing cords and wear resistant
exterior; (c) Double V-belt, used for higher power transmission than single belts. Up to
five belts can be combined in this fashion.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
V-Belt in Groove
Figure 19.5: V-belt in sheave groove.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Power by Belt Class
Figure 19.6: Guide to selection of belt cross section as a function of power transmitted
and shaft speed.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Overload Factors
Driven machine
Power source
Normala
Demanding b
Service (hrs/day)
Service (hrs/day)
3-5
8-10
16-24
3-5 8-10
16-24
1.0
1.1
1.2
1.1
1.2
1.3
Dispensing, display equipment, measuring equipment, office and projection equipment
Liquid agitators, appliances, sewing m achines, sweepers, lig ht duty
1.1
1.2
1.3
1.2
1.3
conveyors, fans, lig ht duty m achine tools (drill presses, lathes, saws),
woodworking equipment.
Semi-liquid agitators, centrifug es, centrifug al com pressors, heavy-d uty 1.1
1.2
1.4
1.2
1.3
conveyors, dough mixers, generators, laundry equipment, heavy duty
m achine tools (boring mills, grinders, mills, shapers), presses, shears,
printing m achinery, centrifug al and gear pumps.
Brick m achinery, piston com pressors, screw conveyors, bucket ele1.2
1.3
1.5
1.4
1.5
vators, extractors, ham m er mills, paper pulpers, pulverisers, piston
pumps, extruders, rubber calend ar mills, textile m achinery.
Jaw crushers, hoists, ball mills, rod and tube mills, sawmill m achinery.
1.3
1.4
1.6
1.5
1.6
a Includ es normal torque, squirrel cag e, synchronous and split phase AC motors; shunt wound DC motors;
multiple cylind er internal com bustion engines.
b Includ es hig h torque, hig h slip, repulsion-ind uction, single phase, series wound AC motors; series wound,
com pound wound DC motors; single piston internal com bustion engines.
1.4
1.5
1.6
1.8
Table 19.2: Typical overload service factors, f1. Source: Courtesy of Gates Corp.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Minimum Datum Diameters
Motor Power,
hp
0.5
0.75
1
1.5
2
5
10
15
20
30
50
75
100
200
575
2.5
3.0
3.0
3.0
3.8
4.5
6.0
6.8
8.2
10.0
11.0
14.0
18.0
22.0
690
2.5
2.5
3.0
3.0
3.0
4.5
5.2
6.0
6.8
9.0
10.0
13.0
15.0
22.0
Motor rpm
870
1160
2.2
–
2.4
2.2
2.4
2.4
2.4
2.4
3.0
2.4
3.8
3.0
4.6
4.4
5.4
4.6
6.0
5.4
6.8
6.8
9.0
8.2
10.5
10.0
12.5
11.0
22.0
–
1750
–
–
2.2
2.4
2.4
3.0
3.8
4.4
4.6
5.4
6.8
9.0
10.0
–
3450
–
–
–
2.2
2.4
2.6
3.0
3.8
4.4
–
–
–
–
–
Table 19.3: Recommended minimum datum diameters, in inches, of sheaves for general
purpose 60-cycle electric motors. Source: Courtesy of Gates Corp.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Design Procedure 19.1: V-Belt Drives
It will be assumed that a belt drive will be designed for power transmission
where the shaft speeds (and hence speed ratio) and desired center distance are
known. The power available can be obtained from the rating of the motor, or
else it can be obtained from design requirements. Based on these quantities,
this design procedure provides a methodology for selecting a cross-section of
a belt, choosing sheaves and number of belts required.
1. Estimate the overload service factor from Table 19.2 and use it to obtain the
required belt power rating using Eq. (19.11).
2. Select a cross section of the belt from the required belt power rating and
the shaft speed using Fig. 19.6.
3. Obtain the minimum allowable sheave datum diameter from Table 19.3.
4. Locate the sheave diameter combinations in Table 19.4 that are suitable for
a desired speed ratio. Disregard from consideration any candidates that
are smaller than the minimum values obtained in Step 3. From the
remaining candidates, select a sheave size that is consistent with space
requirements.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Design Procedure 19.1 (concluded)
5. From Table 19.4, locate the center distance that most closely matches design
constraints, and obtain the power correction factor, f2. Note that the belt
length can be calculated from Eq. (19.5) or read directly from Table 19.4.
6. From Table 19.5, locate the proper belt cross section and center distance, and
obtain the basic power rating per belt, h1. Note that for very high speeds or
small sheaves, an additional power may be required. This is usually a small
amount and is neglected in this design procedure.
7. The rated power per belt is given by
8. The number of belts required can be obtained from the required power from
Step 1:
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Center
Distance
Table 19.4: Center distance and power correction factor, f2, for standard sheaves. Source:
Courtesy of Gates Corp.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Rated Horsepower per Belt
Table 19.5: Rated power in horsepower per belt for selected 3V and 5V cross sections.
Source: Courtesy of Gates Corp.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Wire Rope
Figure 19.7: Cross sections of selected wire
rope. (a) 6 x 19 fiber core; (b) 1 x 19; (c) 6 x
36 wire core; (d) 18 x 7 fiber core.
Figure 19.8: Two lays of wire rope. (a)
Lang; (b) regular.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Wire Rope Data
Weight
per height,
lb/ft
1.50d 2
Minimum
sheave
diameter
in.
42d
Rope
diameter,
d, in.
1
− 1 12
4
Size of
outer
Stiffness,a
Strength, b
Rope
Material
wires
psi
psi
6 × 7 Haulage
Monitor steel
d/ 9
14 × 10 6
100 × 10 3
Plow steel
d/ 9
14 × 10 6
88 × 10 3
6
Mild plow steel
d/ 9
14 × 10
76 × 10 3
1
3
2
6
6 × 19 Standard
1.60d
26d − 34d
− 24
Monitor steel
d/ 13 − d/ 16
12 × 10
106 × 10 3
4
6
hoisting
Plow steel
d/ 13 − d/ 16
12 × 10
93 × 10 3
6
Mild plow steel
d/ 13 − d/ 16
12 × 10
80 × 10 3
1
1
2
6
6 × 37 Special
1.55d
18d
− 32
Monitor steel
d/ 22
11 × 10
100 × 10 3
4
6
flexible
Plow steel
d/ 22
11 × 10
88 × 10 3
1
1
2
6
8 × 19 Extra
1.45d
21d − 26d
− 12
Monitor steel
d/ 15 − d/ 19
10 × 10
92 × 10 3
4
6
flexible
Plow steel
d/ 15 − d/ 19
10 × 10
80 × 10 3
1
3
7 × 7 Aircraft
1.70d 2
—
−
Corrosion-resistant
steel
—
—
124
× 10 3
16
8
Carbon steel
—
—
124 × 10 3
1
7 × 9 Aircraft
1.75d 2
—
− 1 38
Corrosion-resistant steel
—
—
135 × 10 3
8
Carbon steel
—
—
143 × 10 3
1
5
19-Wire aircraft
2.15d 2
—
−
Corrosion-resistant
steel
—
—
165 × 10 3
32
16
Carbon steel
—
—
165 × 10 3
a The stiffness is only approximate; it is affected by the loads on the rope and, in general, increases with the life of the rope.
b The strength is based on the nominal area of the rope.uThe fig res given are only approximate and are based on 1-in. rope sizes and
1/ 4-in. aircraft cable sizes.
Table 19.6: Wire rope data. Source: From Shigley and Mitchell [1983].
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Sheave to Rope Diameter Effects
Figure 19.9: Percent strength loss in
wire rope for different D/d ratios.
Figure 19.10: Service life for different
D/d ratios.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Application
Track cables
G uy s
Mine shafts, ft
Up tp 500
1000-2000
2000-3000
Over 3000
Hoisting
Haulage
Cranes and derricks
Electric hoists
Hand elevators
Private elevators
Hand dumbwaiters
Grain elevators
Passenger elevators
up to 50 ft/ min
50-300
300-800
800-1200
1200-1500
Freight elevators
up to 50 ft/ min
50-300
300-800
800-1200
1200-1500
Powered dumbwaiters
up to 50 ft/ min
50-300
300-800
Safety
factor,
ns
3.2
3.5
Safety Factors
8.0
7.0
6.0
5.0
5.0
6.0
6.0
7.0
5.0
7.5
4.5
7.5
7.60
9.20
11.25
11.80
11.90
6.65
8.20
10.00
10.50
10.55
4.8
6.6
8.0
Table 19.7: Minimum safety factors for a variety of wire
rope applications. Note that the use of these safety
factors does not preclude a fatigue failure. Source: From
Shigley and Mitchell [1983].
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Maximum Bearing Pressure
Table 19.8: Maximum allowable bearing pressures for various sheave materials and
types of rope Source: From Shigley and Mitchell [1983]
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Rolling Chain
Figure 19.11: Various parts of
rolling chain.
Figure 19.12: Typical rolling chain.
(a) One-strand rolling chain; (b)
three-strand chain.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Standard Rolling Chain Sizes
Table 19.9: Standard sizes and strengths of rolling chains.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Chordal Rise
Figure 19.13: Chordal rise in rolling chains. Note that the chain link travels upwards as
well as horizontally when moving from position A to position B.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Service and Multiple Strand Factors
Number
of
strands
1
2
3
4
5
6
Table 19.10: Service factor, a1, for
rolling chains.
Multiple-strand
factor,
a2
1.0
1.7
2.5
3.3
3.9
4.6
Table 19.11: Multiple-strand factor for
rolling chains.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Power Rating
for Rolling
Chains
Table 19.12: Power rating of selected
standard roller chains, in horsepower.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Chain Power Ratings
Figure 19.14: Design guideline for standard roller chains.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Silent Chain
Figure 19.15: A silent chain drive. (a)
Silent chain with sprockets; (b) detail of
silent chain links. Source: Courtesy of
Ramsey Products Corp.
Figure 19.16: The use of guide links in
silent chains. (a) One guide link in center
of chain; (b) two center guide links; (c)
Two side guide links. Source: Courtesy of
Ramsey Products Corp.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Design Procedure 19.2: Design of
Chain Drives
For the purposes of this Design Procedure, the power transmitted (or chain force and
speed), power source, speed ratio, and loading environment need to be known, or at
least be somewhat constrained.
1. Obtain the service factor from Table 19.10. Calculate the chain's required power
rating from Eq. (19.31), taking a2=1.0.
2. Select a chain size from Fig. 19.14 using the required power rating and the small
sprocket speed. Note that using the fewest number of chain strands while satisfying
power requirements usually results in the most economical design.
3. Obtain the strand factor, a2, from Table 19.11.
4. The required power rating, given by Eq. (19.31), needs to be recalculated if a
multiple strand chain is to be used.
5. Referring to Table 19.12, identify the column of the table that corresponds to the
small sprocket's speed. Reading down from the top, find the number of teeth in the
smaller sprocket that produces the required modified power rating, h’pr. This is the
minimum number of teeth that are required for the application. Larger sprockets
can be used if desired.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Design Procedure 19.2 (concluded)
6. If a multiple strand chain is being considered, record the modified power
rating from Table 19.12, and multiply by the strand factor to obtain the
chain's power rating.
7. Note the required lubrication method in Table 19.12. Variation from the
lubrication approach may compromise chain longevity.
8. The number of teeth on the larger sprocket can be calculated from the
desired velocity ratio by using Eq.~(19.22).
9. If the center distance has not been prescribed, it can be estimated by
recognizing that cd/pt should be between 30 and 50, although larger lengths
can be allowed if chain guides are incorporated into the design. If the center
distance exceeds space limitations, increase the number of strands or select
the next largest pitch chain and return to Step 4.
10. The number of links in the chain can be calculated from Eq. (19.26), rounded
up to the next highest even integer.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Case Study: Drag Line Gantry
Figure 19.17: Typical dragline.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press