Shafts

1
Shafts Definition

• Generally shafts are members which rotate in
  order to transmit power or motion. They are
  usually circular in cross section, and thats the
  type we will analyze.
• Shafts do not always rotate themselves, as in the
  case of an axle but axles support rotating
  members.

2
Common Shaft Types



3
Elements Attached to a Shaft
Shoulders provide axial positioning location,
allow for larger center shaft diameter where
bending stress is highest.
4
Common Shaft Materials

• Typically shafts are machined or cold-drawn from
  plain hot-rolled carbon steel. Applications
  requiring greater strength often specify alloy
  steels (e.g., 4140).
• Some corrosion applications call for brass,
  stainless, Ti, or others.
• Aluminum is not commonly used (low modulus, low
  surface hardness).

5
Shafts for Steady Torsion

• Often the rotating mass static load on a
  shaft are neglected, and the shaft is sized
  simply to accommodate the transmitted power. In
  such cases, the engineer typically seeks to limit
  the maximum shear stress ?max to some value under
  the yield stress in shear (Sys), or to limit the
  twist angle ? .

6
Shafts in Steady Torsion

• Chapter 1 review equations
• kW FV/1000 Tn/9600
• hp FV/745.7 Tn/7121
• kW kilowatts of power
• F tangential force (N)
• V tangential velocity (m/s)
• T torque (N x m)
• n shaft speed (rpm)

7
U.S. Power Units

• Review equation
• hp FV/33,000 Tn/63,000
  where,
• hp horsepower
• F tangential force (lb.)
• V tangential velocity (ft/min)
• T torque (lb - in.)
• n shaft speed (rpm)

8
Steady State Shaft Design

• Because shafts are in torsion, the shear stress
  is generally the limiting factor. Recall that
• ?max Tc/J
• where c radius, and, for a circular shaft,
• J ?d4/32
• As always, use a safety factor of n to arrive at
• ?all ?max /n

9
Limiting the Twist Angle

• In some cases, it is desired to limit the twist
  angle to a certain value. Recall
• ? TL/GJ
• L length
• G shear modulus
• ? is always in radians (deg. x ?/180)

10
(No Transcript)
11
Combined Static Loads

• The axial stress is given by
• ?x Mc/I P/A 32M/ ?D3 4P/ ?D2
• (M bending moment, P axial load, D
  diameter)
• The torsional stress is given by
• ?xy Tc/J 16T/ ?D3
• (T Torque, J polar moment of inertia, c
  radius)
• (For circular cross sections.)

12
Maximum Shear Stress Theory

• Typically the axial load P is small compared
  to the bending moment M and the torque T, and so
  it is neglected. (Notice how direct shear is
  completely omitted.)
• Recall the maximum shear stress criterion
• Sy/n (?x2 4 ?xy 2)1/2

13
Maximum Shear Stress Theory

• Substitute the previous values for ?x and ?xy
  into MSST to obtain

This equation, or the related eq. for the maximum
energy of distortion theory (MDET), is useful for
finding either D or n. Note that this would be
for steady loads.
14
Fluctuating Loads

• In their support of rotating members, most
  shafts are subject to fluctuating loads, possibly
  including a shock component as well. Weve
  covered fatigue impact in previous lectures,
  and that material is directly applied to the
  design of shafts.

15
Shock Factors

• In shaft design, shock loading is typically
  accounted for by yet more fudge factors, Ksb
  (bending shock) and Kst (torsional shock). The
  values of these factors range from 1.0 to 2.0.
  The shock factors are applied to their respective
  stress components.

16
Shaft Design Formulas

• There are a number of shaft design formulas
  that incorporate failure theories (MSST or MDET)
  with fatigue theories (Goodman or Soderberg).
• In practice, using MDET with the Soderberg
  criterion is probably the most accurate.

17
Shaft Design Formulas
MDET with the Goodman criterion and shock
factors. For Soderberg, recall that you use Sy
instead of Su.
18
Fully-Reversed Bending

• In analyzing a rotating shaft for fatigue
  life, you will need to compute Mm and Ma. The
  moment might be due to a rotating imbalance or
  due to the tension from a belt, or radial loading
  from gears. No matter the case, because the
  shaft is rotating, it experiences both tension
  and compression from the bending loads
  therefore, typically, Mm 0, and Ma Mmax. (A
  sinusoidal variation about zero.)

19
Example 9.2
Find required dia. of shaft using MDET
Soderberg fatigue relation. Surface is ground.
Su 810 MPa, and Sy 605 MPa. Torque varies by
/- 10. The fatigue stress factor Kf 1.4.
Temp 500 oC, and n 2. Survival rate 50.
20
Critical Speeds of Shafts

• All structures exhibit one or more natural, or
  resonant frequencies. When a shaft rotates at
  speeds equal or close to the natural frequencies,
  resonance may occur. This is usually to be
  avoided, although some designs feature resonance.
• Generally the designer tries to keep the speed
  at least 25 lower than ?o. But in some cases,
  the operating speed is higher.

21
The Rayleigh Equation
ncr (1/2?) (g?W?)/(?W?2)1/2
ncr critical speed (rev/sec) g gravitational
acceleration (9.81 m2/s) W concentrated weight
including load (kg) ? respective static
deflection of the weight.
22
Shaft Attachments

• Many different methods, each with pros and cons
  of both function, ease of use, and cost the
  designer must balance between these factors.
• Some methods are very weak compared to the shaft
  (e.g., a set screw), others are stronger than the
  shaft itself.

23
ShaftAttachments Keys
Square (w D/4) Flat
Round (or tapered)
Gib head
Woodruff key
24
Shaft Attachments Pins
Straight Tapered
Roll
25
Shaft Attachments Tapered Clamps
www.ringfeder.com
26
Stresses in Keys
Distribution of force is quite complicated. The
common assumption is that the torque T is carried
by a tangential force F acting on radius r
T Fr
27
Stresses in Keys

• From T Fr, both shear and compressive bearing
  stresses may be calculated from the width and
  length of the key.
• The safety factor ranges from n 2 (ordinary
  service) to n 4.5 (shock).
• The stress concentration factor in the keyway
  ranges from 2 to 4.

28
Splines
Splines permit axial motion between matching
parts, but transmit torque. Common use is
automotive driveshafts check your car.
29
Couplings

• In many designs involving shafts, two shafts must
  be connected co-axially. Couplings are used to
  make these connections.
• Couplings are either rigid or flexible. Rigid
  couplings require very close alignment of the
  shafts, generally better than .001 per inch of
  separation.

30
Rigid Couplings Sleeves

• The simplest type of coupling is the simple
  sleeve coupling. But this also has the lowest
  torque capacity.

http//www.grainger.com/Grainger/wwg/start.shtml
31
Rigid Couplings - Flanged
Taper locked
Keyed to shaft
Great web resource http//www.powertransmission.c
om/pages/couplings.htm
32
Flexible Couplings

• There are many types of flexible couplings as
  well. Generally a flexible element is sandwiched
  in between, or connected to, rigid flanges
  attached to each shaft.
• Alignment is still important! Reaction forces
  increase with misalignment, and often bearings
  are not sized properly for reaction forces.
  Mechanics often assume that because the
  coupling is flexible, alignment is unimportant.

33
Two-piece Donut (or toroidal) flexible coupling
http//viva.rexnord.com/content/features.html
34
Universal Joints

• U-joints are considered linkages rather than
  couplings, but serve the same purpose of
  transmitting rotation.
• Very large angular displacements may be
  accommodated.
• Single joints are not constant-velocity. Almost
  always, two joints are used. The angles must be
  equal for uniform velocity.

35
Shafts parallel but offset
Shafts not parallel but intersecting
36
Its Not Nanotechnology, But You Could Get Rich!

• Despite decades of research and 1000s of Ph.D.
  theses, highly engineered shafts and components
  fail all too frequently. Even NASA cant always
  get it right.
• Often the connections are to blame keys,
  splines, couplings, and so on. Fatigue wear
  failure is the culprit.

37
Bearing Definition

• A device that supports, guides, and reduces the
  friction of motion between fixed and moving
  machine parts.

38
Bearing Types

• Three major types hydrodynamic or journal
  bearings, rolling-element bearings, and sleeve
  bearings.

39
Design of Journal Bearings

• Nomenclature
• r journal radius
• c radial clearance
• L length of bearing
• viscosity
• n speed (rps)
• W radial load
• P load per projected
• area (W/2rL)

In this figure, U tangential velocity and F
frictional force
40
Journal Bearing Design Charts

• Procedure generally, you first calculate the
  dimensionless Sommerfeld Number, from,
• S (r/c)2(?n/P)
• This characteristic number is used along with
  the L/D ratio of the bearing to enter the Design
  Charts. In some cases, you find the Sommerfeld
  Number from given data.

41
(No Transcript)
42
(No Transcript)
43
(No Transcript)
44
Journal Design Examples

• Problem 10.6
• A 4-in. diameter ? 2-in. long bearing turns at
  1800 rpm c/r 0.001 h0 0.001 in. SAE 30 oil
  is used at 200?F. Through the use of the design
  charts, find the load W.

45
Journal Design Example I

• Looking at Figure 10.7, find the viscosity for
  SAE 30 wt. Oil at 200oF, 1.2 x 10-6 reyns
• In this problem, we dont have enough data to
  calculate S, but we can look it up on the charts

46
Oil Viscosity
Fig. 10.7, p. 385
47
Journal Design Example I

• We are given r 2, and c/r .001. Therefore,
  the clearance c .002.
• We are also given the minimum film thickness, ho
  .001.
• This enables us to enter Design Chart with L/D
  0.5, and ho/c 0.5.
• Then, you can find S 0.5 on the chart.

48
(No Transcript)
49
Journal Design Example

• With S 0.5, we can go back to the definition
  of the Sommerfeld , from Equation
• S (r/c)2(?n/P)
• Rearranging this to solve for P, we have
• P (r/c)2 ?n/S

50
Journal Design Example I

• P (r/c)2 ?n/S
• S 0.5
• (c/r) .001, so (r/c) 1000
• 1.2 x 10-6 psi-sec
• N 1800 rpm 30 rps
• Therefore P 72 psi, and,
• W PLD 576 lbs.

51
Journal Design Example II

• A 25mm diameter by 25mm long bearing carries a
  radial load of 1.5 kN at 1000 rpm c/r 0.0008,
  ? 50 mPa-sec. Use charts to find
• A) The minimum oil film thickness ho
• B) The friction power loss

52
Journal Design Example II

• In this case, we have enough information to
  calculate the Sommerfeld ,
• S (r/c)2(?n/P)
• P W/DL 1500/(.025.025) 2.4 MPa
• n 1000 rpm 16.67 rps
• c/r .0008, so r/c 1250
• ? 50 mPa-sec
• S 0.543

53
Journal Design Example II

• With S 0.543, and L/D 1.0, we can once
  again chart to find
• ho/c 0.75
• We are given c/r .0008, and r 12.5mm, so c
  .01mm.
• Therefore, ho .75.01 .008mm (part A)

54
(No Transcript)
55
Journal Design Example II

• Next, to find the friction power loss, we can use
  chart. We have S .543, and L/D 1.0. From
  that we can look up the coefficient of friction
  variable
• (r/c)f 11
• Since c/r is given as .0008, f .0088

56
(No Transcript)
57
Journal Design Example II

• Knowing the coefficient of friction f, we can
  then use equation to calculate the friction
  torque, Tf
• Tf fWr .00881500.0125 .165 N-m
• Then the friction power loss is found from
  equation
• Power Tfn/159
• (.16516.67)/159 .017 kW

58
Rolling Element Bearings

• Rolling element, or anti-friction bearings,
  make use of spherical or cylindrical rolling
  elements captured between inner and outer rings.
  The rolling elements support the load, and
  transmit rotation by rolling, rather than
  sliding.

59
Rolling Element Bearings

• A major benefit of rolling versus sliding is
  that the coefficient of friction is much lower.
  Recall that for journal bearings operating
  hydrodynamically,
• 0.002 lt f lt 0.010
• For rolling element bearings,
• 0.001 lt f lt 0.002

60
Rolling Element Benefits

• Observe that f is much more uniform. In
  addition, f is much less a function of rotational
  speed. This means that friction power loss is
  more predictable, and remains constant over a
  range of speeds. Rolling element bearings also
  experience much less wear at slower speeds than
  do journal bearings.

61
Ball Bearings Roller Bearings

• There are two types of rolling element
  bearings, ball bearings and roller bearings.
• In general, ball bearings can operate at
  higher speeds (but with less load), and roller
  bearings operate at lower speeds but with heavier
  loads. The difference is due to point contact
  versus line contact.

62
Ball Bearings

• There are many types of ball bearings
  deep-groove, double or triple row, angular
  contact, thrust, cam followers, etc. Each is
  best suited for a particular application.
• For different types, there are series numbers,
  usually in increasing order of cross section
  (i.e., thicker rings, larger spheres, etc.)

63
(No Transcript)
64
Ball Bearing Dimensions
65
Roller Bearings

• The same situation exists with roller
  bearings there are single and double row,
  removable inner or outer race, tapered or
  straight rollers, thrust bearings, and spherical
  bearings. Again, each is best suited for a
  particular application.

66
Bearing Examples

• Double row spherical bearing from the axis of the
  earth high load rating with angular misalignment
  capability.
• NU bearing, straight cylindrical rollers, for
  radial loads only note translational ability
• Light-weight single row ball bearing
• Tapered roller bearing common type of automotive
  wheel bearing. Car example, 1.79 x 108
  revolutions with no maintenance.

67
Bearing Load Life

• There is a basic load rating associated with
  each bearing. It is nominally the radial load
  that a bearing can support for 106 revolutions.
  These numbers, however, are for comparison
  purposes only. In practice, the design load for
  most bearings is only a few of the basic load
  rating.

68
Equivalent Radial Load

• The basic load rating is given for purely
  radial loads only. However, most bearings need
  to support both radial and axial loads.
• Equations are used to calculate an equivalent
  radial load given actual radial and axial loads,
  and the geometry of the design

69
Equivalent Radial Load, P

• P XVFr YFa
• P VFr (cyl. rollers, gen.)
• Fr applied radial load
• Fa applied axial load (thrust)
• V rotation factor, 1.0 for inner-ring rotation,
  1.2 for outer-ring rotation
• X a radial factor
• Y a thrust factor
• NOTE that straight cyl. roller bearings cannot
  support much thrust.

70
Equivalent Load with Shock

• P Ks(XVFr YFa)
• P KsVFr
• Ks is a shock or service factor, find in
  table. Ks ranges from 1.0 to 3.0 depending on
  the type of bearing and the service.

71
The L10 Life

• Bearing life is an important consideration in
  many designs. The desired lifetime could range
  from a few million to a few billion revolutions.
  It doesnt take long for 1000 rpm running
  24/7/365 to add up. (0.5x109)
• The L10 life refers to the expected life (hours
  or revs) under a given load at which 90 of the
  bearings will survive.

72
L10 Life in Revolutions
L10 life rating in 106 revolutions C basic
load rating from manufacturer or Tables 10.3 and
10.4 NOTE difference between C and Cs. P
equivalent radial load a 3 for ball bearings or
10/3 for roller bearings.
73
L10 Life in Hours
L10 rating life in hours n rotational speed,
rpm
74
L5 and Beyond

• The L10 life is based on a 90 survival rate.
  If the application requires higher reliability,
  then a life adjustment fudge factor, Kr, is
  applied. Kr is found in chart, and ranges from
  1.0 (90 reliability) to 0.2 (99 reliability).
  L5 is the name given to any reliability gt 90.