Shigley’s Mechanical Engineering Design
9th Edition in SI units
Richard G. Budynas and J. Keith Nisbett
Chapter 17
Flexible Mechanical Elements
Prepared by
Kuei-Yuan Chan
Associate Professor of Mechanical Engineering
National Cheng Kung University
Copyright © 2011 by The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
17 Flexible Mechanical Elements
Chapter
Outline
17-1
Belts
17-2
Flat- and Round-Belt Drives
17-3
V Belts
17-4
Timing Belts
17-5
Roller Chain
17-6
Wire Rope
17-7
Flexible Shafts
Belts
• Most flexible elements do not have an infinite life.
• Characteristics of belts include
 They may be used for long center distances.
 Except for timing belts, there is some slip and creep, and so the angular-velocity
ratio between the driving and driven shafts is neither constant nor exactly equal
to the ratio of the pulley diameters.
 In some cases an idler or tension pulley can be used to avoid adjustments in
center distance that are ordinarily necessitated by age or the installation of new
belts.
• Flat-belt geometry.
3
Belts (Cont.)
• Belt drives are either reversing or nonreversing.
• The shafts need not be at right angles as in a flat-belt drive with outof-plane pulleys.
• In contrast with flat belts, V belts are used with similar sheaves and
at shorter center distances.
• For timing belts, no initial tension is necessary, so that fixed-center
drives may be used. The restriction on speeds has also been
eliminated.
4
Flat-Belt Drivers
•
A flat-belt drive has an efficiency of about 98 percent, which is about the
same as for a gear drive. On the other hand, the efficiency of a V-belt drive
ranges from about 70 to 96 percent.
•
When an open-belt drive is used, the contact angles are found to be
•
The length of the belt is found by summing the two arc lengths with twice
the distance between the beginning and end of contact.
•
For crossed belt, the angle of wrap is the same for both pulleys and is
•
The belt length for crossed belts is found to be
5
Mechanics of Flat-Belt Drives
• A change in belt tension due to friction forces between
the belt and pulley will cause the belt to elongate or
contract and move relative to the surface of the pulley.
• Assuming that the friction force on the belt is proportional to the
normal pressure along the arc of contact, a relationship between
the tight side tension and slack side tension, follows
• Fc is found as
• The tight side tension F1 and the loose side tension F2 on a pulley
have the following additive components:
6
Mechanics of Flat-Belt Drives (Cont.)
• Solving for the initial tension, we have
• The initial tension needs to be sufficient so
that the difference between the F1 and F2
curve is 2T/D.
• Initial tension is the key to the functioning
of the flat belt as intended.
• The weight of the belt itself can also
provide the initial tension resulting in a dip.
where d = dip, in
L = center-to-center distance, ft
w = weight per foot of the belt, lbf/ft
Fi = initial tension, lbf
7
Analysis of Flat-Belt Drives
•
•
The transmitted horsepower is
given by
geometry and friction
 From belt geometry and speed find
Fc
 From T = HnomKsnd /n find necessary
torque
 From torque T find the necessary
(F1)a − F2 = 2T /D
 Find F2 from (F1)a − [(F1)a − F2]
 From Eq. (i) find the necessary
initial tension Fi
 Check the friction development, f ′ <
f . Use Eq. (17–7) solved for f ′:
Corrections on allowable tension give
where (F1)a = allowable largest tension,
lbf
b = belt width, in
Fa = manufacturer’s allowed
tension, lbf/in
Cp = pulley correction factor
(Table 17–4)
Cv = velocity correction factor
•
The steps in analyzing a flat-belt drive
can include
 Find exp(f φ) from belt-drive
8
 Find the factor of safety from
nf s = Ha /(HnomKs)
Flat-Metal Belts
• Thin metal belts exhibit





High strength-to-weight ratio
Dimensional stability
Accurate timing
Usefulness to temperatures up to 700°F
Good electrical and thermal conduction properties
• The selection of a metal flat belt can consist of the following steps:
9
V Belts
• The cross-sectional dimensions of V belts have been standardized
by manufacturers, with each section designated by a letter of the
alphabet for sizes in inch dimensions.
• To specify a V belt, give the belt-section letter, followed by the inside
circumference in inches.
• The pitch length is obtained by adding a quantity to the inside
circumference.
• For best results, a V belt should be run quite fast: 20 m/s is a good
speed. Trouble may be encountered if the belt runs much faster than
25 m/s or much slower than 5 m/s .
10
Analysis of V Belts
• The analysis of a V-belt drive can consist of the following steps:




Find V, Lp, C, φ, and exp(0.5123φ)
Find Hd , Ha , and Nb from Hd /Ha and round up
Find Fc, F, F1, F2, and Fi , and nf s
Find belt life in number of passes, or hours, if possible
• Pitch length :
• Allowable Power :
where Ha = allowable power, per belt, Table 17–12
K1 = angle-of-wrap correction factor, Table 17–13
K2 = belt length correction factor, Table 17–14
• Design Power :
where Hnom is the nominal power, Ks is the service factor given in Table 17–15, and nd
is the design factor.
• Lifetime in hours :
11
Timing Belts
• A timing belt does not stretch appreciably or slip and consequently
transmits power at a constant angular-velocity ratio.
• Timing belts can operate over a very wide range of speeds, have
efficiencies in the range of 97 to 99 percent, require no lubrication,
and are quieter than chain drives.
• The five standard inch-series pitches available are listed in Table
17–18 with their letter designations.
• The design and selection process for timing belts is similar to that for
V belts.
12
Roller Chain
• Basic features of chain drives include a constant ratio,
since no slippage or creep is involved; long life; and
the ability to drive a number of shafts from a single
source of power.
• The pitch diameter of the sprocket by D can be written
• The chain velocity V is defined as the number of feet
coming off the sprocket per unit time.
where N = number of sprocket teeth, p = chain pitch, in ,n =
sprocket speed, rev/min
• The maximum exit velocity of the chain is
and the minimum exit velocity is
13
Analysis of Roller Chains
• The chordal speed variation is
• For smooth operation at moderate and high speeds it is considered
good practice to use a driving sprocket with at least 17 teeth and no
more than 12 teeth.
• The maximum speed (rev/min) for a chain drive is limited by galling
between the pin and the bushing.
where F is the chain tension in pounds.
• Lubrication of roller chains is essential in order to obtain a long and
trouble-free life.
14
Wire Rope
• Wire rope is made with two types of winding, the regular lay and the
lang-lay.
• A wire rope tension giving the same tensile stress as the sheave
bending is called the equivalent bending load Fb, given by
• A wire rope may fail because the static load exceeds the ultimate
strength of the rope. For an average operation, use a factor of safety
of 5. Factors of safety up to 8 or 9 are used if there is danger to
human life and for very critical situations.
15
Wire Rope (Cont.)
•
The fatigue tensile strength in pounds for a specified life Ff is
where (p /Su) = specified life, from Fig. 17–21
Su = ultimate tensile strength of the wires, psi
D = sheave or winch drum diameter, in
d = nominal wire rope size, in
•
The equivalent bending load Fb is
where Er = Young’s modulus for the wire rope, Table 17–24 or 17–27, psi
dw = diameter of the wires, in
Am = metal cross-sectional area, Table 17–24 or 17–28, in2
D = sheave or winch drum diameter, in
•
The static factor of safety ns is
and the fatigue factor of safety nf is
16