Power Transmission: Belts, Chains, Clutches, and Brakes

1
Power Transmission Belts, Chains, Clutches, and
Brakes
2
Power Transmission

• In many machine designs, power must be
  transmitted from a driving source to the rest of
  the machine. The driving source may be an
  electric motor, engine, windmill, whatever
  often, the energy must be transmitted to the rest
  of the machine to accomplish useful work.
• We will talk about flexible elements used to
  transmit power, and the control of that power.

3
Belts and Chains

• Flexible elements, which are economical
  alternatives to gears for transmitting rotary
  motion between shafts. Can be lighter, smaller,
  and as efficient as gears.
• However, usually the life is more limited.
• Bicycle gears (chain drive) versus automotive
  gears efficient, lightweight, compact, vastly
  greater range of ratios.

4
Types of Belts

• Common belts include flat, round, V, and
  timing belts. Flat belts can be very efficient,
  but require high tension levels, which require
  beefier parts. Round belts are similar,
  outstanding for non-parallel shafts. V belts are
  most common, as we will discuss. These belts all
  depend on friction between the belt and the
  pulley (or sheave). Most slip a few .

5
Timing Belts

• Toothed or synchronous belts dont slip, and
  therefore transmit torque at a constant ratio
  great for applications requiring precise timing,
  such as driving an automotive camshaft from the
  crankshaft.
• Very efficient. More than other types of belts.

6
Timing Belt Nomenclature
7
Flat, Round, and V Belts

• Flat and round belts work very well. Flat belts
  must work under higher tension than V belts to
  transmit the same torque as V belts. Therefore
  they require more rigid shafts, larger bearings,
  and so on.
• V belts create greater friction by wedging into
  the groove on the pulley or sheave. This greater
  friction great torque capacity.

8
V Belt Sheave Cross Section
Wedging action along sides high torque cap.
The included angle 2? ranges between 34o and 40o.
9
V Belt Cross Sections (U.S.)
10
Flat Belts vs. V Belts

• Flat belt drives can have an efficiency close to
  98, about the same as a gear drive.
• V belt drive efficiency varies between 70 and
  96, but they can transmit more power for a
  similar size. (Think of the wedged belt having
  to come un-wedged.)
• In low power applications (most industrial uses),
  the cheaper installed cost wins vs. their greater
  efficiency V belts are very common.

11
Flat Belt Drive Nomenclature
12
Force Profile in Belt Drive
13
Flat Belt Drive Equations

• T (F1 F2)r
  
• hp (F1 F2)V/33,000 Tn/63,000
• speed ratio n1/n2 r2/r1
  
• sin? (r2 r1)/c
  
• contact angle (small) ? ? 2 ?
• cent. force in belt Fc (w/g)V2
• F1 Fc ? /(?-1)(T1/r1)
  
• ? ef?/sin?
  
• (Where ? 90o for a
  flat belt)

14
Chains

• Compared to belts, chains can transmit more power
  for a given size, and can maintain more precise
  speed ratios.
• Like belts, chains may suffer from a shorter life
  than a gear drive. Flexibility is limited by the
  link-length, which can cause a non-uniform output
  at high speeds.

15
Chains

• Chain drives can be very efficient. Bicycle
  example there are very few belt-drive bicycles.
• The fact that the user controls the length (with
  master links) is a plus. However, the sprockets
  wear out much more frequently than does a belt
  sheave. Take your pick!

16
Chain Nomenclature
17
View with side plates omitted
18
Chain Sizes (U.S.)

• The chain number is nominally the
  roller-to-roller pitch in 1/80 inch increments.
• Size 40 chain ½ pitch bike chain.
• Size 80 chain 1 pitch.
• Size 120 chain 1 ½ pitch.

19
Clutches Brakes

• Both use friction to control rotational power.
• Clutches are used to couple decouple rotating
  members typically a power source from the rest
  of a machine. Auto example.
• Brakes are used to dissipate rotational energy.

20
Friction Materials

• Clutches and brakes depend on friction to
  operate. Typically one surface is metal, either
  steel or cast iron. The other surface is usually
  of a composite nature, for example soft metal
  particles embedded with reinforcing fibers in a
  bonding matrix.
• Conflicting requirements of minimal wear, but
  acceptable f.

21
Single Plate Disc Clutch
(for yoke shifter)
22
Hydraulically Actuated Multiple Plate Clutch,
Wet or Dry
Whats wrong with this picture?
23
Wet Clutches

• Why on earth would an engineer design a clutch
  where the plates operate in an oil bath? Isnt
  friction the idea?
• Cooling, smooth operation (no grabbing, and
  reduced wear, thats why.
• True that f is reduced and so sizes must be
  increased but a worthwhile tradeoff.

24
Cone Clutches Brakes
25
Simple Band Brake
Very similar to a belt drive torque capacity is
T (F1 F2)r
26
Differential Band Brake
The friction force helps to apply the band
therefore it is self-energizing. Can become
self-locking Fa (1/a)(cF2 sF1)
27
Short-Shoe Drum Brakes
If the shoe is short (less than 45o contact
angle), a uniform pressure distribution may be
assumed which simplifies the analysis in
comparison to long-shoe brakes.
28
Self-Energizing Self-Locking Brakes
If the rotation is as shown, then Fa (Fn/a)(b
fc). If b lt fc, then the brake is
self-locking. Think of a door stop, that is a
self-locking short shoe brake.
29
Long-Shoe Drum Brakes
Cannot assume uniform pressure distribution, so
the analysis is more involved.
30
Internal Long-Shoe Drum Brakes
Formerly in wide automotive use being replaced
by caliper disc brakes, which offer better
cooling capacity (and many other
advantages). Brakes can dissipate enormous
amounts of power.
31
  The band brake shown has a power capacity of
40 kW at 600 rpm. Determine the belt
tensions.   Given ? 250?, r 250 mm, a 500
mm, and f 0.4.
32

• Torque T (9549 x kW)/n
• (9549 x 40)/600 636.6 N-m
• F1 F2e f?,
• where ? is in radians, so
• F1 F2e (.4)(4.363) 5.727F2
• T (F1 F2)r
• Or, T (.25)(5.727F2 F2) 1.182F2

33

• Therefore, F2 538.6 N, and,
• F1 3,085 N

34
Power Screws, Fasteners, and Connections
35
Threads and Connections

• We will start off discussing the mechanics of
  screw threads. Next, power screws threaded
  fasteners will be examined. Since threaded
  fasteners are often used to make connections, we
  will end with that topic.

36
The Inclined Plane

• Wrapped into a helix, this becomes one of the
  worlds great inventions.
• By inspection, a steeper angle gains you
  elevation more quickly, but the applied force
  must increase.

W
fN
Q
?
N
37
Helically-Inclined Planes
Differential element of one thread transferring
force to the mating thread. The helix or lead
angle ? the slope of the ramp, and is a
critical design parameter. ? is the thread angle,
and is another important parameter.
38
?, ?, and f

• On a screw thread, the helix angle ? controls the
  distance traveled per revolution and the force
  exerted.
• ?, the thread angle, effects the friction force
  resisting motion. Sometimes friction is
  desirable (e.g., so that threads wont loosen),
  and sometimes it is not.
• f is the coefficient of friction, and plays an
  important role in all threads.

39
? and ?

• ?, the helix angle, is given by
• tan ? L/(?dm)
• where,
• L the lead or pitch (threads per unit length)
• dm the mean dia. of the thread contact surface.
  As dm increases, ? decreases.
• ?, the thread angle, is determined by the design
  of the threads not a function of L or dm.

40
Thread Friction Examples

• Acme Threads
• Bolt Threads
• Pipe Threads

41
Power Screws
Force F acts on moment arm a to produce a torque
T. Tables show standard sizes of power screw
threads. In this drawing, only the nut rotates.
42
Power Screw Thread Types
Acme in wide use, but less efficient.
Square most efficient, but hard to make.
Modified Square compromise.
43
Self-Locking of Power Screws

• Self locking is an important design feature
  for jacks. Occurs when the coefficient of
  thread friction is gt the tangent of the helix
  angle the cosine of the thread angle
• f gt cos?ntan ?

44
Power Screw Efficiency
Note the wide range as a function of both f and ?.
45
Threaded Fasteners Nomenclature
46
Threaded Fasteners Thread Forms
Note that the crests roots may be either flat
or rounded
47
Threaded Fasteners UNS ISO

• UNS Unified National Standard. Threads are
  specified by the bolt or screw diameter (also
  called the major diameter)in inches, and the
  number of threads per inch.
• ISO International Standards Organization.
  Threads are specified by the major diameter in
  mm, and the pitch, or, number of mm per thread.
• Generally UNS and ISO threads are NOT
  interchangeable. (3mm is close to 1/8.)

48
Threaded Fasteners UNS

• The specification is written in the format
• Dia threads/in UNC or UNF class and
  internal or external RH or LH.
• UNC Unified National Coarse
• UNF Unified National Fine
• Class ranges from 1 (cheap inaccurate) to 3
  (expensive precise). Class 2 is common.
• A external, B internal

49
Threaded Fasteners UNS

• RH right hand threads, LH left hand
• Example thus would be
• ½ 13 UNC 2A RH
• Notes
• UNF and UNC are redundant information.
• For diameters less than ¼, a numeric size is
  specified instead of the diameter. (000 14)

50
Threaded Fasteners ISO

• Metric designations are a little simpler.
  Preceded by an M, then the diameter in mm, then
  the pitch (mm per thread, not threads per mm).
  There are also coarse and fine threads in the ISO
  system.
• Examples M10 x 1.5
• M10 x 1.25

51
Coarse Versus Fine Threads

• Coarse threads are fine ? for normal
  applications. They are easier to assemble, a
  little more forgiving of dings, possibly cheaper
  to make, and for a given size of bolt, they exert
  less force than do fine threads good for softer
  materials bolted together.
• Fine threads develop greater force per applied
  torque, and are more effective at resisting
  vibration-induced loosening.

52
Bolts, Screws, and Studs
The same fastener could be a bolt or a screw,
depending on if a nut is used. Studs are
threaded at both ends.
53
Bolt Grades

• Bolts (and nuts) are made from a variety of
  materials. The SAE Grade is an indication of the
  strength of the material, based on the proof
  stress, Sp (slightly less than the yield stress).
  Sp ranges from 33 ksi for a grade1 bolt, up to
  120 ksi for a grade 8 bolt. The proof load of a
  bolt is the load at which permanent deformation
  commences.

54
SAE Bolt Head Markings
Hexagonal bolt heads are stamped with radial
lines to indicate the grade. The grade the
number of lines 2.
http//raskcycle.com/techtip/webdoc14.html
55
Thread Manufacture

• Threads are generally produced by either
  rolling (forming with a specialized die) or by
  cutting, as on a lathe. Rolled threads are
  stronger and have better fatigue properties due
  to the cold work put into the material.
• Power screw threads may be ground to achieve a
  very smooth surface to reduce f. Threads may
  also be cast into a part.

56
Stresses in Threaded Fasteners

• Due to imperfect thread spacing, most of the
  load between a bolt and a nut is taken by the
  first pair of threads. This is partially
  relieved by bending and localized yielding,
  however most thread failures occur in that
  region. The stress concentration ranges from 2
  to 4.

57
Major-Diameter Stresses

• Axial stress is given by the familiar
• ? P/A
• For A, use either the root diameter for power
  screws, or tabulated values for fasteners.
• Torsional stress is given by the familiar
• ? T/J 16T/ ?d3
• for interpretation of T and d. T is the
  applied torque for power screws, or ½ the wrench
  torque, for fasteners.

58
Bearing Stress

• Bearing stress, the compressive stress between
  the surfaces of the threads, is given by ?b
  P/(?dmhne)
• P load,
• dm pitch or mean screw thread diameter,
• h depth of thread, and
• ne number of threads in engagement.
• ?b is usually not a limiting design factor.

59
Nomenclature for Thread Stress Analysis
60
Direct Shear Stress on Threads

• In addition to the torsional shear stress we
  just discussed, the threads also experience
  direct shear stress. The threads are considered
  to be loaded as a cantilever beam (wrapped around
  a cylinder), with the load evenly distributed
  over the mean screw diameter. Because the nut
  threads are wrapped inside of a larger cylinder
  than the bolt threads, they experience less
  stress.

61
Direct Shear Stress on Threads

• Then we have,
• ? 3P/(2 ?dbne), where,
• d root dia. for the screw or major dia. for the
  nut,
• b the thread thickness at the root, and
• ne the number of threads in engagement.
• Note that ? can be a limiting factor.

62
Bolt Tightening Preload

• Bolted joints commonly hold parts together in
  opposition to both normal and shear forces.
• In certain applications it is desirable to
  tighten a bolted joint to a specified preload Fi,
  which is some fraction of the bolts proof load,
  Fp.

63
Bolt Tightening Preload

• An engineer would specify a preload in the
  case of fatigue applications, in order to
  minimize the relative magnitude of the
  alternating load Pa compared to the average load
  Pmean.
• Preloading is also important in sealing
  applications, as in a gasketed joint. Both
  reasons are important for auto cylinder heads.

64
Preload Values

• The optimum preload is often given by eq. 15.20
• Fi 0.75 Fp for connections to be reused, or
• Fi 0.90 Fp for permanent connections.
• The proof load Fp is, Fp SpAt, where the proof
  stress Sp is an SAE
• specification, and tension area At is found in
  Tables.

65
Tightening Torque

• To develop the specified preload, the tightening
  torque is given by
• T KdFi, where
• T the tightening torque,
• d the nominal bolt diameter (e.g., ½),
• Fi the desired preload, and
• K a torque coefficient

66
Tightening Torque

• For dry, unlubricated, or average threads,
• K 0.2. For lubricated threads, K 0.15.
• Rewrite eq. as,
• Fi T/(Kd) to see that, for a given torque, Fi
  increases with lubricated threads.

67
Relaxation and Exactness

• Most joints lose on the order of 5 of the
  original preload over time, due to relaxation
  effects (usually over the course of 100s or 1000s
  of hours).
• By now it should be clear that threaded fasteners
  are extremely complex. Often extensive testing
  is done for critical applications.

68
Tension Joints

• Bolted joints are frequently used to clamp
  together parts that themselves carry additional
  loads these additional loads increase the bolt
  tension. The engineer often must determine
  acceptable loads for such joints.
• We consider both the joints and the parts as
  springs, with spring constants kb and kp.

69
Tension Joints
After assembly with preload Fi, applied load P
will change the force in the bolt and the parts.
70
Tension Joints

• P ?Fb ?Fp, where
• ?Fb the increased tension in the bolt, and
• ?Fp the decreased compression force in the
  parts. The deformations are given by
• ?b ?Fb/kb, and ?p ?Fp/kp
• Then compatibility requires that
• ?Fb/kb ?Fp/kp

71
Joint Constant C

• The joint stiffness factor, or joint constant, is
  defined in eq. 15.22 as C kb/(kb kp).
  Then the preceding equations yield
• ?Fb CP and ?Fp (1 C)P
• kb is usually small compared to kp, and so C is a
  small fraction.

72
Forces in Bolted Joints

• When a load P is applied to a bolted joint, the
  tensile force Fb in the bolt increases, and the
  compressive force Fp in the parts decreases. As
  long as Fp gt 0, the forces are
• Fb CP Fi
• and,
• Fp (1 C)P Fi

73
Determination of C

• Deflection ? is given by ? PL/AE, and the
  spring rate k is given by k P/ ?. Combining
  these we obtain
• kb AbEb/L
• and
• kp ApEp/L

74
Determination of kb
In determining kb, the threaded and the
unthreaded parts of the bolt are considered as
separate springs in series. 1/kb Lt/AtEb
Ls/AbEb
75
Determination of kp
kp is more complex the stress distribution in
the parts is clearly non-uniform, and depends on
factors like washers, etc. It is approximated by
the double-cone illustrated.
76
Determination of kp

• Estimate of kp for standard hex-head bolts and
  washers is given by
• kp (.5 ?Epd)/2 ln 5(.58L.5d)/(.58L2.5d)
• d bolt diameter and
• L grip (thickness of bolted assembly).
• Alternatively, just use kp 3kb !

77
Some Rules of Thumb for Threaded Fasteners

• Threaded depth for a bolt diameter d, the length
  of full thread engagement should be 1.0d in
  steel, 1.5d in cast iron, and 2.0d in aluminum.
• In gasketed joints, bolts are arrayed in a bolt
  circle or other pattern. The bolt-to-bolt
  spacing should not exceed about 6d to maintain
  uniform pressure.

78
Rivets
Rivets often find application in larger
structures such as bridges and towers. They are
also used extensively in aircraft construction.
A rivet starts off as a cylinder with one head
(usually rounded). The protruding cylinder is
deformed to create a second head, which locks the
joint in compression.
79
Joints Primarily in Shear

• Both bolts and rivets are used in connections
  that primarily experience shear loading (separate
  from the case of axial or normal loading which we
  just examined).
• Such connections may experience any of several
  failure modes, and the engineer must analyze for
  each mode.

80
Shear Joint Failure Modes
Shearing Failure of Fastener ? 4P/ ?d2 d
diameter of fastener
81
Shear Joint Failure Modes
Tensile Failure of Plate ?t P/(w de)t,
where de effective hole dia., w width, and t
thickness of thinnest plate
82
Effective Hole Diameter

• In analyzing potential tensile failure of the
  plate, the effective hole diameter is used rather
  than the diameter of the fastener.
• de the fastener diameter 1/16 for drilled
  holes, or,
• de the fastener diameter 1/8 for punched
  holes (this is usually used).

83
Shear Joint Failure Modes
Bearing Failure of Plate or Fastener ?b P/dt,
where d diameter of fastener and t thickness
of the thinnest plate.
84
Shear Joint Failure Modes
a gt 1.5d
Shearing Failure of Plate ? t P/2at, where t
thickness of thinnest plate and a closest
distance from fastener to edge.
85
Joint Efficiency

• The efficiency of a joint is defined as
• e Pall/Pt,
  
• where
• Pall is the smallest of the allowable loads in
  the preceding failure mode examples, and
• Pt is the static tensile strength of the plate
  with no holes. e is always less than 100.

86
Welded Joints

• Welded joints are produced by localized melting
  of the parts to be joined, in the region of the
  joint. Often a filler metal (or plastic, in the
  case of plastics) is added, creating a chemical
  bond in the parts that may be stronger than the
  base material.
• There are many, many welding processes an
  entire engineering major.

87
Strength of Butt Welds
The height h does not include the crowned region
generally it is just the plate thickness.
88
Strength of Fillet Welds
Specified size is based on h, but stress is
calculated with t, the region of minimum cross
sectional area.
89
Factor of Safety for Welds in Shear

• Just as many riveted or bolted joints are in
  shear, so too are many welded joints. The factor
  of safety for a welded joint is given by
• n Sys/ ? 0.5Sy/ ? (eq.
  15.44)

90
Purchased vs. Designed Components

• We went into shafts in some detail because shafts
  tend to be custom-designed for each application.
• Often the components that the engineer puts onto
  the shaft, however, are purchased. These
  components can be analyzed as much as one likes,
  but its usually best to work with the
  manufacturers application data.

91
Polar Moment of Inertia

• The polar moment of inertia is the sum of Ix
  and Iy for each weld about the centroid of the
  weld group. Knowing J, apply
• ?t Tr/J
  
• to find ?t at a given point, and then use
  
• ? (?t2 ?d2)½
• to find the max ? , which is used to find the
  required weld size.

92
Eccentric Loading of Welded Joints (P.E.
Question!)
Determination of the exact stress distribution is
very complicated. With some simplifying
assumptions, the following procedure gives
reasonably accurate results. Direct shear stress
is given by ?d P/A, where A the throat area
of all the welds.
93
Eccentric Loading of Welded Joints

• ?d is taken to be uniformly distributed over
  the length of all the welds.
• Due to the eccentricity e, a torque T is
  developed about the centroid C of the weld group
  T Pe. The torque causes an additional shear
  stress in the welds
• ?t Tr/J
  
• J polar moment of inertia of the weld group
  about C, based on the throat area.

94
Eccentric Loading of Welded Joints

• ?t Tr/J
• In this equation, r is the distance from C to
  the point in the weld of interest. ?t is not
  uniform across the weld group, and one point will
  experience the greatest stress resultant
• ? (?t2 ?d2)½

95
Location of the Centroid (Review)
C is located at coordinates x-bar and y-bar,
where x-bar (?Aixi)/ ?Ai, and y-bar (?Aiyi)/
?Ai, where i denotes a given weld segment, and
the coordinate origin is conveniently chosen. A
key is that the weld throat t is assumed to be
very small, sometimes 0.
96
Moment of Inertia of a Weld and the Parallel Axis
Theorem.
Use the familiar bh3/12, substituting t and L for
b and h as appropriate. However, assume t3 0
to simplify. Remember the parallel axis theorem,
Ix Ix Ay12, to find the moment of inertia
about the centroid of the weld group. (So even
if Ix 0, you still have A.)