Fundamentals of Machine Elements, 2nd Ed

Hamrock • Fundamentals of Machine Elements

Chapter 1: Introduction

The invention all admir’d, and each, how he

To be th’ inventor missed; so easy it seem’d

Once found, which yet unfound most would have thought

Impossible

John Milton


Figure 1.1 Approaches to product development. (a) Classic approach, with large design iterations typical of the over-the-wall engineering approach. (b) A more modern approach, showing a main design flow with minor iterations representing concurrent engineering inputs.

Product Development Approaches


Characteristica

a vg = very good, g = good, f = fair, and p = poor.A = quality of materials, workmanship, maintenance, and inspection.B = control over load applied to part.C = accuracy of stress analysis, experimental data, or experience with similar parts.

vg g f p

B =

C =

vggfp

C =

vggfp

C =

vggfp

A = vg

A = g

A = f

A = p C =

vggfp

1.11.21.31.4

1.51.71.92.1

1.31.451.61.75

1.71.952.22.45

1.31.451.61.75

1.82.052.32.55

1.551.751.952.15

2.152.352.652.95

1.51.71.92.1

2.12.42.73.0

1.82.052.32.55

2.42.753.13.45

1.71.952.22.45

2.42.753.13.45

2.052.352.652.95

2.753.153.553.95

Characteristica

ns s vs

D =

E =

nssvs

1.01.01.2

1.21.31.4

1.41.51.6

a vs = very serious, s = serious, and ns = not seriousD = danger to personnel.E = economic impact.

Figure 1.2 Safety factor characteristics A, B, and C.

Figure 1.3 Safety factor characteristics D and E.

Pugsley Method for Safety Factor

ns = nsxnsy

ns =!all

!dSafety Factor:

Pugsley Equation:


Screw

Screw

Screw

Screw

Screw

Cover

Bearing

Connecting rodPin

Pin

Bush

Seating

Piston

Washer

Blade clamp

Gear

Housing

Set screw

Guard

PlugScrew

Screw

Screw

Screw

Screw

Screw

Cover

Bearing

BearingConnecting rod

Pin

Pin

BushSeal

Seating

Seating

Piston

Washer

Washer

Needle bearing

Roll pinRivet

Blade clamp

Counterweight

Gear

Housing

Set screwGuard

(b)(a)

Figure 1.2 Effect of manufacturing and assembly considerations on design of a reciprocating power saw. (a) Original design, with 41 parts and 6.37-min assembly time; (b) modified design, with 29 parts and 2.58-min assembly time.

Design for Assembly


(a) SI unitsQuantity Unit SI symbol FormulaSI base unitsLength meter m -Mass kilogram kg -Time second s -Temperature kelvin K -

SI supplementary unitPlane angle radian rad -

SI derived unitsEnergy joule J N-mForce newton N kg-m/s2

Power watt W J/sPressure pascal Pa N/m2

Work joule J N-m

(b) SI prefixesSI symbol

Multiplication factor Prefix for prefix1,000,000,000,000 = 1012 tera T1,000,000,000 = 109 giga G1,000,000 = 106 mega M1000 = 103 kilo k100 = 102 hecto h10 = 101 deka da0.1 = 10−1 deci d0.01 = 10−2 centi c0.001 = 10−3 milli m0.000 001 = 10−6 micro µ0.000 000 001 = 10−9 nano n0.000 000 000 001 = 10−12 pico p Table 1.3 SI units and prefixes.

SI Units


(a) Fundamental conversion factorsEnglish Exact SI Approximate

unit value SI valueLength 1 in. 0.0254 m -Mass 1 lbm 0.453 592 37 kg 0.4536 kgTemperature 1 deg R 5/9 K -

(b) DefinitionsAcceleration of gravity 1 g = 9.8066 m/s2 (32.174 ft/s2)Energy Btu (British thermal unit) = amount of energy required to

raise 1 lbm of water 1 deg F (1 Btu = 778.2 ft-lb)kilocalorie = amount of energy required to raise 1 kg ofwater 1K (1 kcal = 4187 J)

Length 1 mile = 5280 ft; 1 nautical mile =6076.1 ftPower 1 horsepower = 550 ft-lb/sPressure 1 bar = 105 Pa

Temperature degree Fahrenheit tF =9

5tC + 32 (where tC is degrees

Celsius)degree Rankine tR = tF + 459.67Kelvin tK = tC + 273.15 (exact)

Kinematic viscosity 1 poise = 0.1 kg/m-s1 stoke = 0.0001 m2/s

Volume 1 cubic foot = 7.48 gal

(c) Useful conversion factors1 ft = 0.3048 m1 lb = 4.448 N1 lb = 386.1 lbm-in./s2

1 kgf = 9.807 N1 lb/in.2 = 6895 Pa1 ksi = 6.895 MPa1 Btu = 1055 J1 ft-lb = 1.356 J1 hp = 746 W = 2545 Btu/hra

1 kW = 3413 Btu/hr1 quart = 0.000946 m3 = 0.946 liter1 kcal =3.968 Btu

Table 1.4 Conversion factors and definitions.

Conversion Factors


Invisalign Product

Figure 1.3 The Invisalign® product. (a) An example of an Aligner; (b) a comparison of conventional orthodontic braces and a transparent Aligner.


Invisalign Part 1

Figure 1.4 The process used in application of Invisalign orthodontic treatment. (a) Impressions are made of the patient's teeth by the orthontist, and shipped to Align technology, Inc. These are used to make plaster models of the patient's teeth. (b) High-resolution three-dimensional representations of the teeth are produced from the plaster models. The correction plan is then developed using computer tools.


Invisalign Part 2

Figure 1.4 (cont.) (c) Rapid-prototyped molds of the teeth at incremental positions are produced through stereolithography. (d) An Aligner, produced by molding a transparent plastic over the stereolithography part. Each Aligner is work approximately two weeks. The patient is left with a healthy bite and a beautiful smile.


Chapter 2: Load, Stress and Strain

The careful text-books measure(Let all who build beware!)

The load, the shock, the pressureMaterial can bear.

So when the buckled girderLets down the grinding spanThe blame of loss, or murder

is laid upon the man.Not on the stuff - The Man!

Rudyard Kipling, “Hymn of Breaking Strain”


Establishing Critical Section

To establish the critical section and the critical loading, the designer:

1. Considers the external loads applied to a machine (e.g. an automobile).

2. Considers the external loads applied to an element within the machine (e.g. a cylindrical rolling-element bearing.

3. Located the critical section within the machine element (e.g., the inner race).

4. Determines the loading at the critical section (e.g., contact stress).


Figure 2.1 A simple crane and forces acting on it. (a) Assembly drawing; (b) free-body diagram of forces acting on the beam.

Example 2.1

P = 10,000 N0.25 m0.75 m

(b)

y

x

Wp

WrP = 10,000 N

0.25 m0.75 m

Roller

Pin

(a)


(a)

(b)

(d)

(e)

(f)

T

T

x

z

y

M

M

T

x

x

yM

P

PP

P V

V

y

x

P

P

P

P

V

(c)

y

Figure 2.2 Load classified as to location and method of application. (a) Normal, tensile; (b) normal, compressive; (c) shear; (d) bending; (e) torsion; (f) combined.

Types of Loads


y

(a)

x

y'' < 0 y'' > 0

M > 0 M < 0

y

(b)

x

y'' < 0 y'' > 0

M < 0 M > 0

Sign Convention in Bending

Figure 2.3 Sign convention used in bending. (a) Positive moment leads to tensile stress in the positive y direction; (b) positive moment acts in a positive direction on a positive face. The sign convention shown in (b) will be used in this book.


B

(1)

(2)

(3)

(4)

Normal, tensile

Shear

Bending

Torsion

Px = 0

T = –aP

B

B

B

(b)

Px

Py

Mz

My

T

Py = –P

Mz = - bP, My = 0

B

b

a

P

y

z

x

(a)

Figure 2.4 Lever assembly and results. (a) Lever assembly; (b) results showing (1) normal, tensile, (2) shear, (3) bending, and (40 torsion on section B of lever assembly.

Example 2.2


Table 2.1 Four types of support with their corresponding reactions.

Support Types

Cable

θ θ

Type of support Reaction

Roller

Pin

Fixed support

P

P

Px

Py

Px

Py

M


x0

20°

y

µP2

µP1

P2

P1

(ml + mp)g

µdP

µdPµdP

µdP

dP

dP

dP

4W4W 4W

4W

4W

4W

W

W

W

dP

(Rotation)

W

30∞ 30∞

20∞20∞130∞

130∞10 R.

A B

3 3

4

12

12

5 516

(a)

(b)

(Rotation)

Figure 2.5 Ladder in contact with the house and the ground while a painter is on the ladder.

Figure 2.6 External rim brake and forces acting on it. (a) External rim brake; (b) external rim brake with forces acting on each part.

Examples 2.4 and 2.5


Figure 2.7 Sphere and forces acting on it. (a) Sphere supported with wires from top and spring at bottom; (b) free-body diagram of forces acting on sphere.

Example 2.6

60° 60°

60° 60°

(a)

(b)ma g

P P

150 N


(a)

(b)

(c)

Figure 2.8 Three types of beam support. (a) Simply supported; (b) cantilevered; (c) overhanging.

Beam Support Types


Shear and Moment Diagrams

Procedure for Drawing Shear and Moment Diagrams:

1. Draw a free-body diagram, and determine all the support reactions. Resolve the forces into components acting perpendicular and parallel to the beam’s axis, which is assumed to be the x axis.

2. Choose a position x between the origin and the length of the beam l, thus dividing the beam into two segments. The origin is chosen at the beam’s left end to ensure that any x chosen will be positive.

3. Draw a free-body diagram of the two segments, and use the equilibrium equations to determine the transverse shear force V and the moment M

4. Plot the shear and moment functions versus x. Note the location of the maximum moment. Generally, it is convenient to show the shear and moment diagrams directly below the free-body diagram of the beam.


P

A C

B

M

P__2

P__2

P__2

(a)

(b)

l__2

l__2

P

A

x

P__2

(c)

l__2

x

y

x

x

V

A

M

V

y

P__2

x

V V =

M

P

P__2

P__2

M =P__2

V =P__2

(d)

x

–

x M =P__2

(l – x)

Mmax =Pl__4

x

y

Figure 2.9 Simply supported bar. (a) Midlength load and reactions; (b) free-body diagram for 0 < x < l/2; (c) free-body diagram l/2 ≤ x < l; (d) shear and moment diagrams.

Example 2.7


Concentrated moment

Singularity Graph of q(x) Expression for q(x)

q(x) = M ⟨x – a –2M

a

y

x

P

a

y

x

w0

a

y

x

w0

a

y

xb

ba

y

x

a

y

x

w0

Concentrated force

Unit step

Ramp

Inverse ramp

Parabolic shape

q(x) = P ⟨x – a –1

q(x) = w0 ⟨x – a 0

q(x) = ⟨x – a 1

q(x) = w0 ⟨x – a 0– ⟨x – a 1

q(x) = ⟨x – a 2

w0__b

w0__b

⟨

⟨

⟨

⟨

⟨ ⟨

⟨

Table 2.2 Singularity and load intensity functions with corresponding graphs and expressions.

Singularity Functions


Using Singularity Functions

Procedure for Drawing the Shear and Moment Diagrams by Making Use of Singularity Functions:

1. Draw a free-body diagram with all the singularities acting on the beam, and determine all support reactions. Resolve the forces into components acting perpendicular and parallel to the beam’s axis.

2. Referring to Table 2.2, write an expression for the load intensity function q(x) that describes all the singularities acting on the beam.

3. Integrate the negative load intensity function over the beam to get the shear force. Integrate the negative shear force over the beam length to get the moment (see Section 5.2).

4. Draw shear and moment diagrams from the expressions developed.


x

M

(b)

__2

x

M

M1

M2

M3

l l

__2

l l

V3

V1

V2

x

x

V

(a)

__2

l

__2

l l

l

P__2

P__2

–

V

P

0

P__2

–

Figure 2.10 (a) Shear and (b) moment diagrams for Example 2.8.

Example 2.8


x

l__6

w0l__8

w0 l__2

l__4

l__4

l__2

R1 R2P2P1

(b)

x, m

(d)

40.0

30.0

33.0

19.5

25.5

10.012.0

30 6 9 12

33.27 34.53 =

Parabolic

Mom

ent,

kN

-m

Cubic

Maximum

moment

(at x 6.9 m)

y w0

x

l__4

l__4

l__4

l__4

P1 P2

(a)

x, m

(c)

3

1.5

–2.5

5.0

7.58.5

10.5

–5.0

–8.5

6 9 12

ParabolicS

hear,

kN

Figure 2.11 Simply supported beam. (a) Forces acting on beam when P1=8 kN, P2=5 kN; w0=4 kN/m, l=12 m; (b) free-body diagram showing resulting forces; (c) shear and (d) moment diagrams for Example 2.9.

Example 2.9


1 m

A

B

C

3 m

1.5 m

ma1

ma2

(a)

1 m

RB

RC

0.5

m1.5 m

ma1g

ma2g

(b)

Figure 2.12 Figures used in Example 2.10. (a) Load assembly drawing; (b) free-body diagram.

Example 2.10


0

x

z

y

σy

τyxτyzτzy

τzx τxz

τxy

σxσz

y

x

(a)(b)

0

0x

z

y

xyτ yxτ

yσ yσ

xσ xσ

yxτ

xyτ

Figure 2.13 Stress element showing general state of three-dimensional stress with origin placed in center of element.

Figure 2.14 Stress element showing two-dimensional state of stress. (a) Three-dimensional view; (b) plane view.

Stress Elements


(a) (b)

x

y

yσ

xσ

yxτ

xyτ

φφ

σφτφ

A sin φ

A

A cos φ

Figure 2.15 Illustration of equivalent stress states. (a) Stress element oriented in the direction of applied stress. (b) stress element oriented in different (arbitrary) direction.

Figure 2.16 Stresses in an oblique plane at an angle φ.

Stresses in Arbitrary Directions


yσ xσ 1σ 2σ

y , xyσ τ

cσ σ

xyτ

x + yσ σ______

2

2 φτ

2 φσ

xyτ

x ,– xyσ τ

0

1τ

2τ

c =σ

yσ

xσ

τ

Figure 2.17 Mohr’s circle diagram of Equations (2.13) and (2.14).

Mohr’s Circle


The steps in constructing and using Mohr’s circle in two dimensions are as follows:

1. Calculate the plane stress state for any x-y coordinate system so that σx,σy, and τxy are known.

2. The center of the Mohr’s circle can be placed at(σx + σy

2, 0

)3. Two points diametrically opposite to each other on the circle correspond to the points (σx,−τxy) and

(σy, τxy). Using the center and either point allows one to draw the circle.

4. The radius of the circle can be calculated from stress transformation equations or through trigonometryby using the center and one point on the circle. For example, the radius is the distance between points(σx,−τxy) and the center, which directly leads to

r =

√(σx − σy

2

)2

+ τ2x

5. The principal stresses have the values σ1,2 = center ± radius.

6. The maximum shear stress is equal to the radius.

7. The principal axes can be found by calculating the angle between the x axis in the Mohr’s circle planeand the point (σx,−τxy). The principal axes in the real plane are rotated one-half this angle in thesame direction relative to the x axis in the real plane.

8. The stresses in an orientation rotated φ from the x axis in the real plane can be read by traversing anarc of 2φ in the same direction on the Mohr’s circle from the reference points(σx,−τxy) and (σy, τxy).The new points on the circle correspond to the new stresses (σx′ ,−τx′y′) and (σy′ , τx′y′), respectively.

Constructing Mohr’s Circle


y

x

D

B

c=14 ksiσ

1τ 2τ cσ

cσ

2σ

1=23.43 ksiσ

1σ

1=23.43 ksiσ yσ 2=4.57 ksiσ

x

y

A

(a)

(b) (c)

C

2 φσ=58°

2 φτ=32°

B

AC0

max = 9.43 ksiττ (cw)

–τ (ccw) (σx, -τxy)

= (9 ksi, -8 ksi)

5

8 9.43

(a)

29°σ2

=4.57 ksi

τ = τmax = 9.43 ksi

σc = 14 ksi

σc = 14 ksi

16°

Figure 2.18 Results from Example 2.12. (a) Mohr’s circle diagram; (b) stress element for proncipal normal stress shown in xy coordinates; (c) stress element for principal shear stresses shown in xy coordinates.

Example 2.12


1σ 1σ

3σ 2σ 1/2τ

1/2τ

1/3τ

2/3τ

τ

2σ

σσ

(a) (b)

Figure 2.19 Mohr’s circle for triaxial stress state. (a) Mohr’s circle representation; (b) principal stresses on two planes.

Three Dimensional Mohr’s Circle


1/2τ 1/3τ 2/3τ

1σ 2σ

3σ 10

10

y

x

20 300

Normal stress, σ

Normal stress, σ

Shear stress, τ

Shear stress, τ

20 30

1σ cσ

1τ

2τ

2σ

2 φτ

2 φσ

(σy, τxy)

(σx, –τxy)

1/2τ 1/3τ

2/3τ

3σ 1σ

100 20 30

Normal stress, σ

Shear stress, τ

(c)

(b)

(a)

Figure 2.20 Mohr’s circle diagrams for Example 2.13. (a) Triaxial stress state when σ1=23.43 ksi, σ2 = 4.57 ksi, and σ3 = 0; (b) biaxial stress state when σ1=30.76 ksi, σ2 = -2.76 ksi; (c) triaxial stress state when σ1=30.76 ksi, σ2 = 0, and σ3 = -2.76 ksi.

Example 2.13


3σ 1σ

2σ

octσ

octσ octσ

octσ

octτ

octτ

octτ

octτ

(a) (b) (c)

+∫

Figure 2.21 Stresses acting on octahedral planes. (a) General state of stress; (b) normal stress; (c) octahedral stress.

!oct =1

3

[("x−"y)2+("y−"z)2+("z−"x)2+6

(!2xy+ !2yz+ !2xz

)]1/2!oct =

!x+!y+!z

3

Octahedral Stresses


xz

z

y y

x

y

0

xδ y

z

– δ

– δx

(a) (b)

y

y

0

xδ

yxθ

x

xδ

(a) (b)

yxθ

x

z

y

Figure 2.22 Normal strain of a cubic element subjected to uniform tension in the x direction. (a) Three-dimensional view; (b) two-dimensional (or plane) view.

Figure 2.23 Shear strain of cubic element subjected to shear stress. (a) Three-dimensional view; (b) two-dimensional (or plane) view.

Strain in Cubic Elements


y

dx

xdxε

x

(a) (b) (c)

y

dy

ydyε

x

y

x

xy___

2

γ

xy___

2

γ

0 B

A

Figure 2.24 Graphical depiction of plane strain element. (a) Normal strain εx; (b) normal strain εy; (c) shear strain γxy

Plain Strain


90°

0°

45°

x

Strain Gage Rosette in Example 2.16

Figure 2.25 Strain gage rosette used in Example 2.16.


Roll

Sheet

Adhesive

Block

Slice

Expanded panel

75 70 70 70 75

(a)

(b)

(c)

(d)

400 N 400 N 400 N 400 N

RA RB

x

x

x

y

800

400

0

0

–400

–40

–800

–80

Mom

ent,

N-m

Shear,

N

Glue Spreader Shaft Case Study

Figure 2.26 Expansion process used in honeycomb materials. Figure 2.27 Glue spreader case study. (a)

Machine; (b) free-body diagram; (c) shear diagram; (d) moment diagram.

Hamrock • Fundamentals of machine Elements

Chapter 3

Give me matter, and I will construct a world out of it.

Immanuel Kant


Centroid of Area

Figure 3.1 Ductile material from a standard tensile test apparatus. (a) Necking; (b) failure.

(a) (b)

Figure 3.2 Failure of a brittle material from a standard tensile test apparatus.


0 100 200 300 400 500 600 × 104

Lead

Epoxy

Pure aluminum

Mat

eria

l

Wood

Steel

Nylon

Graphite

Ratio of strength to density, N-m/kg

12τ 1σ

2σ

1

2

Fiber

Fiber Reinforced Composites

Figure 3.3 Strength/density ratio for various materials.

Figure 3.4 Cross section of fiber-reinforced composite material.


P

0 0 ′

Y

U

R

0.002

__Slope, E =σε

__l

Strain, = ε δ

P __ AS

tres

s,

=

σ Sy

Su

0.002

PlasticElastic

00 ′

Y

E

P

__l

Strain, = ε δP __ A

Str

ess

,

=

σ

Sy

Figure 3.5 Typical stress-strain curve for a ductile material.

Stress-Strain Diagram for Ductile Material

Figure 3.6 Typical stress-strain behavior for ductile metal showing elastic and plastic deformations and yield strength Sy.


0C

Ductile

Brittle

C ′

B

B′

__l

Strain, = ε δ

P __ AS

tress

,

=

σ

Compression

Tensionx

__l

Strain, = ε δP __ A

Str

ess,

=

σ

Sfc

Sft

Comparison of Brittle and Ductile Materials

Figure 3.7 Typical tensile stress-strain diagrams for brittle and ductile metals loaded to fracture.

Figure 3.8 Stress-strain diagram for ceramic in tension and in compression.


Magnesia

Steel

Example 3.6

Figure 3.9 Bending strength of bar used in Example 3.6.


__l

Strain, = ε δ

P __ AS

tres

s,

=

σ

Sy

0.01

Brittle (T << Tg)

Viscous flow (T >> Tg)

Limited plasticity(T ≈ 0.8 Tg)

Extensive cold drawing(T ≈ Tg)

Behavior of Polymers

Figure 3.10 Stress-strain diagram for polymer below, at, and above its glass transition behavior.


Lead

Copper

Aluminum tin

Aluminum

Magnesium

SteelsCast iron

Zinc alloysSintered iron

Alumina

Silicon nitride

Silicon carbide

Graphite

CeramicsMetals Polymers

Silicone rubber

Natural rubberPolyethylene

AcetalPhenol formal- dehydeNylon

8 × 102

103

10 4D

ensi

ty, ρ,

kg

/m3

Density of Materials

Figure 3.11 Density for various metals, polymers, and ceramics at room temperature (20°C, 68°F).


Density, ρMaterial kg/m3 lbm/in.3

MetalsAluminum and its alloysa 2.7 × 103 0.097Aluminum tin 3.1× 103 0.11Babbitt, lead-based white metal 10.1 × 103 0.36Babbitt, tin-based white metal 7.4 × 103 0.27Brasses 8.6 × 103 0.31Bronze, aluminum 7.5 × 103 0.27Bronze, leaded 8.9 × 103 0.32Bronze, phosphor (cast)b 8.7 × 103 0.31Bronze, porous 6.4 × 103 0.23Copper 8.9 × 103 0.32Copper lead 9.5 × 103 0.34Iron, cast 7.4 × 103 0.27Iron, porous 6.1 × 103 0.22Iron, wrought 7.8 × 103 0.28Magnesium alloys 1.8 × 103 0.065Steelsc 7.8 × 103 0.28Zinc alloys 6.7 × 103 0.24

PolymersAcetal (polyformaldehyde) 1.4× 103 .051Nylons (polyamides) 1.14× 103 .041Polyethylene, high density .95× 103 .034Phenol formaldehyde 1.3× 103 .047Rubber, naturald 1.0× 103 .036Rubber, silicone 1.8× 103 .065

CeramicsAlumina (Al2O3) 3.9× 103 0.14Graphite, high strength 1.7× 103 0.061Silicon carbide (SiC) 2.9× 103 0.10Silicon nitride (Si2N4) 3.2× 103 0.12

aStructural alloysbBar stock, typically 8.8 ×103 kg/m3 (0.03 lbm/in.3).cExcluding “refractory” steels.d“Mechanical” rubber.

Table 3.1 Density for various metals, polymers, and ceramics at room temperature (20°C, 68°F).

Density of Materials


Phenol formaldehyde

SteelsCast ironBrass, bronzeAluminumZinc alloysMagnesium alloysBabbitts

CarbidesAlumina

Graphite


Acetal

Nylon

Polyethylene

Natural rubber

Mo

du

lus

of

elas

tici

ty, E

, P

a

106

107

108

109

1010

1011

1012

Figure 3.12 Modulus of elasticity for various metals, polymers, and ceramics at room temperature (20°C, 68°F).

Modulus of Elasticity


Modulus of Elasticity, EMaterial GPa MpsiMetals

Aluminum 62 9.0Aluminum alloysa 70 10.2Aluminum tin 63 9.1Babbitt, lead-based white metal 29 4.2Babbitt, tin-based white metal 52 7.5Brasses 100 14.5Bronze, aluminum 117 17.0Bronze, leaded 97 14.1Bronze, phosphor (cast)b 110 16.0Bronze, porous 60 8.7Copper 124 18.0Iron, gray cast 109 15.8Iron, malleable cast 170 24.7Iron, spheroidal graphiteb 159 23.1Iron, porous 80 11.6Iron, wrought 170 24.7Magnesium alloys 41 5.9Steel, low alloys 196 28.4Steel, medium and high alloys 200 29.0Steel, stainlessc 193 28.0Steel, high speed 212 30.7Zinc alloys d 50 7.3

PolymersAcetal (polyformaldehyde) 2.7 0.39Nylons (polyamides) 1.9 0.28Polyethylene, high density .9 0.13Phenol formaldehyde e 7.0 1.02Rubber, naturalf .004 0.0006

CeramicsAlumina (Al2O3) 390 56.6Graphite 27 3.9Cemented carbides 450 65.3Silicon carbide (SiC) 450 65.3Silicon nitride (Si2N4) 314 45.5

aStructural alloys.bFor bearings.cPrecipitation-hardened alloys up to 211 GPa (30 Mpsi).dSome alloys up to 96 GPa (14 Mpsi).eFilled.f25%-carbon-black “mechanical” rubber.

Table 3.2 Modulus of elasticity for various metals, polymers, and ceramics at room temperature (20°C, 68°F).

Modulus of Elasticity


Material Poisson’s ratio, νMetals

Aluminum and its alloysa 0.33Aluminum tin -Babbitt, lead-based white metal -Babbitt, tin-based white metal -Brasses 0.33Bronze 0.33Bronze, porous 0.22Copper 0.33Copper lead -Iron, cast 0.26Iron, porous 0.20Iron, wrought 0.30Magnesium alloys 0.33Steels, 0.30Zinc alloys d 0.27

PolymersAcetal (polyformaldehyde) -Nylons (polyamides) 0.40Polyethylene, high density 0.35Phenol formaldehyde -Rubber 0.50

CeramicsAlumina (Al2O3) 0.28Graphite, high strength -Cemented carbides 0.19Silicon carbide (SiC) 0.19Silicon nitride (Si2N4) 0.26

aStructural alloys.

Table 3.3 Poisson’s ratio for various metals, polymers, and ceramics at room temperature (20°C, 68°F).

Poisson’s Ratio


AluminumCopperBrass

Magnesium alloys

Stainless steel

Cast iron

BronzeSteel

Graphite

Alumina

Silicon carbide


Natural rubber

Polyethylene

Acetal, nylon

10–1

1

10

102

2

3 × 102T

her

mal

con

du

ctiv

ity

, Kt, W

/m-°

C

Figure 3.13 Thermal conductivity for various metals, polymers, and ceramics at room temperature (20°C, 68°F).

Thermal Conductivity


Thermal conductivity, Kt

Material W/m·◦C Btu/ft·hr·◦FMetals

Aluminum 209 120Aluminum alloys, casta 146 84Aluminum alloys, siliconb 170 98Aluminum alloys, wroughtc 151 87Aluminum tin 180 100Babbitt, lead-based white metal 24 14Babbitt, tin-based white metal 56 32Brassesa 120 69Bronze, aluminuma 50 29Bronze, leaded 47 27Bronze, phosphor (cast)d 50 29Bronze, porous 30 17Coppera 170 98Copper lead 30 17Iron, gray cast 50 29Iron, spheroidal graphite 30 17Iron, porous 28 16Iron, wrought 70 40Magnesium alloys 110 64Steel, low alloyse 35 20Steel, medium alloys 30 17Steel, stainlessf 15 8.7Zinc alloys 110 64

PolymersAcetal (polyformaldehyde) 0.24 0.14Nylons (polyamides) 0.25 0.14Polyethylene, high density 0.5 0.29Rubber, natural 1.6 0.92

CeramicsAlumina (Al2O3)g 25 14Graphite, high strength 125 72Silicon carbide (SiC) 15 8.6

aAt 100◦C.bAt 100◦C (∼ 150 W/m·◦C at 25◦C).c20 to 100◦C.dBar stock, typically 69 W/m·◦C.e20 to 200◦C.fTypically 22 W/m·◦C at 200◦C.gTypically 12 W/m·◦C at 400◦C.

Table 3.4 Thermal conductivity for various metals, polymers, and ceramics at room temperature (20°C, 68°F).

Thermal Conductivity


ZincMagnesiumAluminum

Brass, copperMost bronzes

BabbittsSteel

Leaded bronze

Cast irons

Sintered iron

AluminaSilicon carbide

Silicon nitride

Graphite


PolyethyleneSilicone rubberNatural rubberAcetal, nylon

Nitrile rubber

Phenol formaldehyde

10– 6

10– 5

10– 4

2 × 10– 4

Lin

ear

ther

mal

expan

sion c

oef

fici

ent,

a_, (°

C)–

1

Figure 3.14 Linear thermal expansion coefficient for various metals, polymers, and ceramics at room temperature (20°C, 68°F).

Linear Thermal Expansion Coefficient


Linear thermal expansion coefficient, aMaterial (◦C)−1 (◦F)−1

MetalsAluminum 23 ×10−6 12.8×10−6

Aluminum alloysa 24 ×10−6 13.3×10−6

Aluminum tin 24 ×10−6 13.3×10−6

Babbitt, lead-based white metal 20 ×10−6 11×10−6

Babbitt, tin-based white metal 23 ×10−6 13×10−6

Brasses 19×10−6 10.6×10−6

Bronzes 18 ×10−6 10.0×10−6

Copper 18×10−6 10.0×10−6

Copper lead 18×10−6 10.0×10−6

Iron, cast 11 ×10−6 6.1×10−6

Iron, porous 12×10−6 6.7×10−6

Iron, wrought 12 ×10−6 6.7×10−6

Magnesium alloys 27×10−6 15×10−6

Steel, alloysb 11 ×10−6 6.1×10−6

Steel, stainless 17 ×10−6 9.5×10−6

Steel, high speed 11 ×10−6 6.1×10−6

Zinc alloys 27 ×10−6 15×10−6

PolymersThermoplasticsc (60-100)×10−6 (33-56)×10−6

Thermosetsd (10-80)×10−6 (6-44)×10−6

Acetal (polyformaldehyde) 90×10−6 50×10−6

Nylons (polyamides) 100×10−6 56×10−6

Polyethylene, high density 126×10−6 70×10−6

Phenol formaldehydee (25-40)×10−6 (14-22) ×10−6

Rubber, naturalf (80-120)×10−6 (44-67)×10−6

Rubber, nitrileg 34×10−6 62×10−6

Rubber, silicone 57×10−6 103×10−6

CeramicsAlumina (Al2O3)h 5.0×10−6 2.8×10−6

Graphite, high strength 1.4-4.0×10−6 0.8-2.2×10−6

Silicon carbide (SiC) 4.3×10−6 2.4×10−6

Silicon nitride (Si3N4) 3.2×10−6 1.8×10−6

aStructural alloys.bCast alloys can be up to 15 ×10−6(◦C)−1.cTypical bearing materials.d25× 10−6(◦C)−1 to 80× 10−6(◦C)−1 when reinforced.eMineral filled.fFillers can reduce coefficients.gVaries with composition.h0 to 200◦C.

Table 3.5 Linear thermal expansion coefficient for various metals, polymers, and ceramics at room temperature (20°C, 68°F).

Thermal Expansion Coefficient


MagnesiumAluminum

SteelCast ironCopper

Graphite

Carbides, alumina


Thermoplastics

Natural rubber

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Spec

ific

hea

t ca

pac

ity, Cp, k

J/kg-°

C

Figure 3.15 Specific heat capacity for various metals, polymers, and ceramics at room temperature (20°C, 68°F).

Specific Heat Capacity


Specific heat capacity, Cp

Material kJ/kg·◦C Btu/lbm·◦FMetals

Aluminum and its alloys 0.9 0.22Aluminum tin 0.96 0.23Babbitt, lead-based white metal 0.15 0.036Babbitt, tin-based white metal 0.21 0.05Brasses 0.39 0.093Bronzes 0.38 0.091Coppera 0.38 0.091Copper lead 0.32 0.076Iron, cast 0.42 0.10Iron, porous 0.46 0.11Iron, wrought 0.46 0.11Magnesium alloys 1.0 0.24Steelsb 0.45 0.11Zinc alloys 0.4 0.096

PolymersThermoplastics 1.4 0.33Rubber, natural 2.0 0.48

CeramicsGraphite 0.8 0.2Cemented Carbides 0.7 0.17

aAluminum bronze up to 0.48 kJ/kg·◦C (0.12 Btu/lbm·◦F.bRising up to 0.55 kJ/kg·◦C (0.13 Btu/lbm·◦F) at 200 ◦C (392◦F).

Table 3.6 Specific heat capacity for various metals, polymers, and ceramics at room temperature (20°C, 68°F).

Specific Heat Capacity


P1 = 3 kN

l2 = 0.3 m

d = 0.1 m

l1 = 1 m

l3 = 0.5 m

Figure 3.16 Rigid beam assembly used in Example 3.12.

Example 3.12


WC-Co

Moalloys

W alloys

Ni alloys

Cu alloys

Zn alloysTi alloysAlalloys

Rock, stoneCement, concrete

Porousceramics

Engineeringalloys

Engineeringpolymers

Mgalloys

Tinalloys

Leadalloys

Engineeringceramics

Diamond

Aluminas

1000

100

10

1.0

0.1

0.01100 300 1000 3000 10 000 30 000

Mo

du

lus

of

elas

tici

ty,

E,

GP

a

BBe

SialonsSi

GlassesPottery

ZrO2

SiC

BeO

Si3N4

CFRPUniply

KFRPGFRP

GFRPKFRP

Laminates

Engineeringcomposites

Ash

MEL

Epoxies

PMMA

PVC

PP

HDPE

Density, ρ, kg/m3

PTFE

LDPE

PlasticizedPVC

Hardbutyl

Softbutyl

Silicone

Elastomers

PU

PS

PC

Nylon

Polyesters

OakPine

Fir

AshOak

PineFir

Spruce

Balsa

Cork

BalsaWood

products

Parallelto grain

Perpendicularto grain

Polymers,foams

Polymers,foams

Lower E limit for true solids

Woods

Guidelines for minimum- weight design

CFRP

103

104

3 × 103

3 × 102= C

E__ρ

1 /

2

= CE__ρ

= CE__ρ

1 /

3

E__ρ( )

1 /

2

(m/s)

(G ≈ 3E/8; K ≈ E)

Specific Elastic Modulus

Figure 3.17 Modulus of elasticity plotted against density.


Engineeringceramics


AshPP

PS

PU

LDPE

SiliconeSoftbutyl

PTFE

Elastomers

Engineeringpolymers

HDPEPolyesters

EpoxiesPVC

MEL

Nylons

PMMA

Woodproducts

OakPine

Fir

Parallelto grain

Balsa

Ash

OakPine

FirPerpendicular

to grain

Polymersfoams

Balsa

Cork

Pottery

Be

Engineeringalloys

W alloys

Mo alloys

Ni alloys

Cu alloys

Leadalloys

Znalloys

Cermets

Glasses

CFRPGFRPUniply

KFRPCFRP

LaminatesGFRPKFRP

SiC

B

Si

Si3N4

Diamond

Sialons

Al2O3

ZrO2

GeMgO

Castirons

Steels

Tialloys

Mgalloys

Al alloys

Stone,rock

Porousceramics

Woods

Engineeringalloys

Cement,concrete

Guidelinesfor minimum-weight design

= Cσf__ρ

= Cσf__ρ

2 /

3= C

σf__ρ

1 /

2

10 000

1000

100

10

1

0.1100 300 1000 3000 10 000 30 000

Str

eng

th, S

, M

Pa

Density, ρ, kg/m3

Specific Strength

Figure 3.18 Strength plotted against density.


= 10–2S__

E= 10–3S__

E

= 10–4S__

E

1000

100

10

1.0

0.1

0.010.1 1 10 100 1000 10 000

Modulu

s of

elas

tici

ty, E

, G

Pa

Strength, S, MPa

Diamond

WC

SiCBoron

Cermets

WMo alloys

Beryllium

Cast irons

Cu alloys

Al alloys

Common rocksZn alloys Ti alloys

Mg alloysSn

LeadAsh

OakPine

Mel

EpoxiesPMMA

Designguidelines

Nylons

Wood products

Polyester

PVCBalsa

Woods

PPAshOak

Pine

Balsa

HDPE

PTFE

LDPE

PU

Silicone

Cork

Hardbutyl

Softbutyl

PS

ll tograin

Cement +

Concrete

CFRPUniply

GFRP

GlassesBrick, etc.

LaminatesCFRPGFRP

Porousceramics


Engineeringpolymers

Engineeringceramics

Engineeringalloys

SteelsNi alloys

MgO

Ge

ZrO2

BeOSilicon

Si3N4Al2O3

Polymers,foams

⊥ tograin

Maximum energystorage per unit volume

Bucklingbefore yield

Minimum energystorage per unit volume

Yield beforebuckling

= CS__E

3 /

2

= CS__E

= 0.1S__E

= C S

2

__E

Elastomers

Modulus of Elasticity vs.

Strength

Figure 3.19 Modulus of elasticity plotted against strength.


Range ofKA for p << pmax

Rising KA as p

nears pmax

Maximum bearingpressure, Pmax

Engineeringalloys

Engineeringpolymers


Engineeringceramics

Al Alloys

Copper

PTFE

LDPE

HDPE

FilledPTFE

Nylons

Filled thermosets

Filled thermoplastics

Filled polymides

Unfilledthermoplastics

Mildsteel

Stainlesssteels

Medium-carbonsteels

High-carbonsteels

Toolssteels

Nitridedsteels

Cementedcarbides

Al2O3

Si3N4

SiC

Sailons

Diamond

Castirons

Bronzes

10–11

10–12

10–13

10–14

10–15

10–16

10–17

10–18

1 10 100 1000 10 000

10–10

= 10 –3W__A

10 –4

10 –5

10 –7

10 –8

10 –6

Arc

har

d w

ear

const

ant,

KA, m

2/N

Limiting pressure, pl , MPa

= 10 –9W__A

Wear Constant

Figure 3.20 Archard wear constant plotted against limiting pressure.


1000

100

10

1.0

0.1

0.010.1 1 10 100 1000 10 000

Mo

du

lus

of

elas

tici

ty, E

, G

Pa

Relative cost times density, Mg/m3

Designguidelines

Engineeringpolymers

Engineeringalloys


Cermets

Cualloys

KFRP

Si C

Designguidelines

PinesPF

PS

Softbutyl

Silicones

Elastomers

E–––CRρ

= C

E1/2

–––CRρ

= C

E1/3

–––CRρ

= C

Ash,oak

Ash,oak

PC PMMA

PTFE

NylonsEpoxies

Polyesters

Polymides

PVC

Balsawood

products

Parallelto grain

Porousceramics

Stone,brick,and

pottery Cement,concrete

Brick,stone,

concrete

Woods

Polymerfoams

Engineeringceramics

Perpendicularto grain

Pines

Balsa

Zn

alloysAL alloys

Mgalloys

Pballoys

GFRP

Nialloys

AL7 O3

Walloys

S alloys

Ti alloys

CFRP

Si3 N4

Zr O3

PP

HDPE

LDPE

PVC(plasticised)

Hardbutyl

Glasses

Mild steel

Cast irons Modulus of Elasticity vs. Cost

Figure 3.21 Modulus of elasticity plotted against cost times density.


Sand Casting

Figure 3.22 Schematic illustration of the sand casting process.


(a)

Container liner Container

Billet

Pressing stem

Dummy blockExtrusion

Die

Die backer

(b)

Figure 3.23 An example of the steps in forging a connecting rod for an internal combustion engine, and the die used.

Figure 3.24 The extrusion process. (a) Schematic illustration of the forward or direct extrusion process; (b) Examples of cross-sections commonly extruded.

Forging and Extrusion


Thrustbearing

HopperThroat

Screw

Barrelliner

Barrel

Barrelheater/cooler

Thermocouples

Filterscreen

Breakerplate

AdapterDie

Melt-pumping zoneMelting zone

Feed zoneThroat-cooling channel

Gearreducer

boxMotor

Meltthermocouple

Polymer Extruder

Figure 3.25 Schematic illustration of a typical extruder.


Material Available formsa

Aluminum B, F, I, P, S, T, WCeramics B, p, s, TCopper and brass B, f, I, P, s, T, WElastomers b, P, TGlass B, P, s, T, WGraphite B, P, s, T, WMagnesium B, I, P, S, T, wPlastics B, f, P, T, wPrecious metals B, F, I P, t, WSteels and stainless steels B, I, P, S, T, WZinc F, I, P, Wa B=bar and rod; F = foil; I = ingot; P = plateand sheet; S = structural shapes; T = tubing;W=wire. Lowercase letters indicate limited avail-ability. Most of the metals are also available inpowder form, including prealloyed powders.

Table 3.8 Commercially available forms of materials.

Available Forms of Materials


Tolerance vs. Surface

Roughness

Figure 3.26 A plot of achievable tolerance versus surface roughness for assorted manufacturing operations.


Chapter 4: Stresses and Strains

I am never content until I have constructed a mechanical model of the subject I am studying. If I succeed in making one, I understand; otherwise I do not.

William Thomson (Lord Kelvin)


Centroid of Area

Figure 4.1 Centroid of area.

A

C

x

x

y

y

y_

x_

dA

y=RA ydA

A=A1y1+A2y2+ · · ·A1+A2+ · · ·

x=RAxdA

A=A1x1+A2x2+ · · ·A1+A2+ · · ·


y

x

c

a

b

f

d

e

A

x

y

dA

x0

ry

Figure 4.2 Rectangular hole within a rectangular section used in Example 4.1.

Figure 4.3 Area with coordinates used in describing area moment of inertia.

Example 4.1


y

r

xφ

dy

Figure 4.4 Centroid of area.

Example 4.2


A

C

x ′

x

y

dA

0

0′

y

dy

Parallel-Axis Theorem

Figure 4.5 Coordinates and distance used in describing parallel-axis theorem.

Ix′ = Ix+Ad2y

Iy′ = Iy+Ad2x


Figure 4.6 Triangular cross section with circular hole in it, used in Example 4.3.

y′

x′dx = 3r

dy = 4r

x′c = 3r

r

y′c = 4r x

y


Figure 4.7 Circular cross-sectional area relative to x’ - y’ coordinates, used in Example 4.4.

x

b

a

y

1 cm

6 cm

60°60°

2 cm


Properties of Cross Sections

Table 4.1 Centroid, area moment of inertia, and area for seven cross sections

Circular area

Cross section Centroid Area moment of inertia Area

x_

= 0

y_

= 0x

y Ix = Ix_ = r4π__

4

Iy = Iy_ = r4π__

4

J = r4π__2

Ix = Iy =πr4____16

J =πr4____

8

x_

= 0

y_

= 0

A = πr2

r

Hollow circular area

x

y

ri

r

Ix = , bh3____12

bh3____36

Ix_ =Triangular area

Rectangular area

a + b_____3

x_

=

2___3

r sin α______αx

_ =

4r____3πx

_ = y

_ =

h__3

y_

=

J_

= bh___36

Ix = , bh3____3

hb3____3

Iy = ,

Ix_ =

bh3____12

hb3____12

Iy_ =

J_

= b2 + h2bh___12

( )

J = a2 + b2πab____16

( )

A = π r 2 – ri2( )

x_

C

b

b

Cr

x

x

x

x

y_

y

a

h

h

Area of circular sector

Quarter-circular area

y

y

y

C

A =bh____2

A =πr2___

4

A = bh

A = r 2α

y_

=

x_

= b__2

h__2

y_

=

x_

= 4a___3π4b___3π

αα

x_

x_

y_Cr

x

Area of elliptical quadrant

y x_

y_Cb

a

Ix = ,πab3_____

16Ix_ = ab3π___

164___

9π( )–

Iy = ,πa3b_____16

Iy_ = a3b

π___16

4___9π( )– A =

πab____4

Ix_ = Iy

_ = r4π___16

4___9π( )–

Ix = (α – sin 2α)r4___4

1___2

Iy = (α + sin 2α)r4___4

1___2

J = r4α1__2

Iy =bh(b2 + ab + a2)_______________

12

bh(b2 – ab + a2)_______________36

Iy_ =

Ix = Ix_ = r4 – ri

4( )π__4

J = r4 – ri4π__

2( )

Iy = Iy_ = r4 – ri

4π__4

( )

(b2 + h2 + a2 + ab)


y

x

xz

y

z

z2 + y2

x2 + y2

x2 + z2

dma

Figure 4.8 Mass element in three-dimensional coordinates and distance from the three axes.

Mass Element

x0

y

x

y

dma

r

Figure 4.9 Mass element in two-dimensional coordinates and distance from the two axes.


Table 4.2 Mass and mass moment of inertia of six solids.

Mass and Mass Moment of Inertia

Shape Equations

ma =πd2lρ_____

4

Imy = Imz =mal2_____

12

ma =

ma = abcρ

πd2thρ ______4

Imx =ma(a2 + b2)_____________

12

Imy =ma(a2 + c2)_____________

12

Imz =ma(b2 + c2)_____________

12

Imy = Imz =mad2_____

16

ma = πd3ρ _____

6

Imx = Imy = Imz =mad2____10

Imx =mad2_____

8

ma = πd2lρ ______

4

Imy = Imz =ma(3d2 + 4l2)_________________

48

Imx =mad2_____

8

ma = πlρ(do

2 – di2) ___________

4

Imy = Imz =ma(3do

2 + 3di2 + 4l2)____________________________

48

Imx =ma(do

2 – di2)___________

8

d

x

z

y

l

z

th

y

x

z

y

x

b

y

x

zc a

y

x

d

l

z

y

x

do

l

z

di

Rectangular prism

Cylinder

Hollow cylinder

Sphere

Disk

Rod

d


P

P

Internal load

External load

PP

Circular Bar with Tensile Load

Figure 4.10 Circular bar with tensile load applied.


θ

r

ro

z

T

l

γθz

Twist Due to Torque

Figure 4.11 Twisting of member due to applied torque.

!=TL

GJ

!=Tc

J


(a) (b)

Longitudinal linesbecome curved

Transverse lines remainstraight, yet rotate

y

M

M

xz

Figure 4.12 Bar made of elastomeric material to illustrate effect of bending. (a) Undeformed bar; (b) deformed bar.

Deformation in Bending


M

y

xz

l

Neutralsurface

Bendingaxis

Axis ofsymmetry

Neutralaxis

Figure 4.13 Bending occurring in cantilevered bar, showing neutral surface

y

(a) (b)

y

0

Neutral

axis∆x

∆x

∆s = ∆x

∆θ

rr

∆x

∆s ′

Figure 4.14 Undeformed and deformed elements in bending.

Neutral Surface and Deformation in Bending


σmax

σc M

y

x

Figure 4.15 Profile view of bending stress variation.

Stress in Bending

!=−McI


80 mm

120 mm

8 mm

8 mm

C

A

B

y

x

Neutral axis

Figure 4.16 U-shaped cross section experiencing bending moment, used in Example 4.10

Example 4.10


(a) (b)dφb Centroidal

surface

Center of initial curvature

Neutral

surface

d ′ d

e

c ′c

ci

a

φ

ro

co

y

c_

r_

rn

ri

Neutral

axis

Centroidal

axis

c_

r

r_

rn

y

CG

M M

e

Figure 4.17 Curved member in bending. (a) Circumferential view; (b) cross-sectional view

Deformation of Member in Bending

Neutral

axis

Centroidal

axis

r

dr

r_

rn

ro

ri

b

Figure 4.18 Rectangular cross section of curved member


P

P

(a)

(b)

Figure 4.19 How transverse shear is developed. (a) Boards not bonded together; (b) boards bonded together.

Transverse Shear


(a) (b)

V

Figure 4.20 Cantilevered bar made of highly deformable material and marked with horizontal and vertical grid lines to show deformation due to transverse shear. (a) Undeformed; (b) deformed.

Deformation Due to Transverse Shear


M

A′

σ′σ

σ′σ

τ

τ

M + dM

M + dMMy′wt

dx

(a) (b)

Figure 4.21 Three-dimensional and profile views of moments and stresses associated with shaded top segment of element that has been sectioned at y’ about neutral axis. Shear stresses have been omitted for clarity. (a) three-dimensional view; (b) profile view.

Moments and Stresses on Element


Rectangular

Cross section Maximum shear stress

Circular

Round tube

I-beam

τmax =3V___2A

τmax =4V___3A

τmax =2V___A

τmax =V___

Aweb

Table 4.3 Maximum shear stress for different beam cross sections.

Maximum Shear Stress for Different

Cross Sections


300 lb 150 lb

250 lb 200 lb

T = 1000 in-lbT = 1000 in-lb

50 lb75 lb25 lb

5 in.5 in. 5 in.

Example 4.13

Figure 4.22 Shaft with loading considered in Example 4.13.

200 lb 50 lb

–250 lb

V

x

M

x

1250 in-lb

1000 in-lb

P

x

-25 lb

75 lb

(a) (b) (c)

Figure 4.23 (a) Shear force; (b) normal force and (c) bending moment diagrams for the shaft in Fig. 4.22.


A

B

C

D

Figure 4.24 Cross section of shaft at x=5 in., with identification of stress elements considered in Example 4.13.

Stress Elements in Example 4.13.


Figure 4.25 Shear stress distributions. (a) Shear stress due to a vertical shear force; (b) Shear stress due to torsion.

Shear Stress Distributions


10 in. 5 in.6 in.

10 in.

RA

RB

500 lbf 180 lbf 540 lbf

Recoiler

Two-roll

tension stand

Operator's console

Slitter Entry pinch

rolls and guide table Peeler

Pay-off

reel (uncoiler)

Coil-loading car

Slitter bladeSet screw

Collar

Rubber roller

Steel spacers

Key

Driveshaft

Lower driveshaft

Rubber rollers

Lower slitter blade

Collar

Adjustable

height

(a)

(b) (c)

Steel spacers (behind slitter blade)

Figure 4.26 Design of shaft for coil slitting line. (a) Illustration of coil slitting line; (b) knife and shaft detail; (c) free-body diagram of simplified shaft for case study.

Coil Slitter


720 lb290 lbf

–430 lb

V

x

M

x

4300 in.-lb

Center at (21,900/d3, 0)

(σx, – τxy)= (43,800/d3, 11,000/d3)

Radius = –11,000_______

d3

221,900______

d3

24,510______

d3

2

+ =( ) ( )

Figure 4.27 Shear diagram (a) and moment diagram (b) for idealized coil slitter shaft.

Figure 4.28 Mohr's circle for location of maximum bending stress.

Coil Slitter Results


Chapter 5: Deformation

Knowing is not enough; we must apply.

Willing is not enough; we must do.

Johann Wolfgang von Goethe


x

y

P

P

l

V M

Example 5.1

Figure 5.1 Cantilevered beam with concentrated force applied at free end. Used in Example 5.1.


Deflection by Singularity Functions

1. Draw a free-body diagram showing the forces acting on the system.

2. Use force and moment equilibrium to establish reaction forces acting on the system.

3. Write an expression for the load-intensity function for all the loads acting on the system while makinguse of Table 2.2.

4. Integrate the negative load-intensity function to give the shear force and then integrate the negativeshear force to obtain the moment.

5. Make use of Eq. (5.9) to describe the deflection at any value.

6. Plot the following as a function of x:

(a) Shear

(b) Moment

(c) Slope

(d) Deflection


x

a b

y

l

Pb___l

PRA = RB =Pa___l

x

x

a x – a

y

P

V M

RA =Pb___l

(a)

(b)

Figure 5.2 Free-body diagram of force anywhere between simply-supported ends. (a) Complete bar; (b) portion of bar.

Force on Simply Supported Beam


y

a

A B

C

b

l

w0

x–yl

RA = w0b

(a)

(b)

(c)

a b

w0

MA = w0b a +b_2( )

a

x

V M

x – a

w0

w0b

w0b a +b_2( )

Figure 5.3 Cantilevered bar with unit step distribution over part of bar. (a) Loads and deflection acting on cantilevered bar; (b) free-body diagram of forces and moments acting on entire bar; (c) free-body diagram of forces and moments acting on portion of bar.

Cantilever with Step Load


y

a

A C

b

l

RC

(a)

(b)

MC

B

P

x

RA

a b

P

V

(c)

M

RA

a

x

x – a

P

Figure 5.4 Cantilevered bar with other end simply-supported and with concentrated force acting anywhere along bar. (a) Sketch of assembly; (b) free-body diagram of entire bar; (c) free-body diagram of part of bar.

Cantilever with Concentrated Force


Deflection for any x

y = – ( x – a 3 – x3 + 3x2a)P

–—6EI

w0–—EI

bx3–—6

bx2–—2

b–2

1—24

x

bP

y

Type of loading

Concentrated load at any x

Unit step distribution overpart or all of bar

Moment applied to free end

(a)

ly

l

y =

Mx2–——

2EIy = –

– x – a 4

a

x

b

M

y

ly

l

a

x

y

ly

l

w0 +– a

P–—6EI

b–l

y = x 3 – x – a 3 + 3a2x – 2alx –

y

a b

l

P

P

x

w0b–—––24lEI

b–2

b–2

y = –

+ x b3 + 6bc2 + 4b2c + 4c3 – 4l2 c +

4 c + x 3 – x – a 4 –

l–b

(b)

b–l

P a–l

y

a c

l

b

c +

x

b–2

a3x–––

l

x – a – b 4

w0

w0b–––

la +

b–2

w0b–––

l

Beam Deflections

Table 5.1 Deflection for three different situations when one end is fixed and one end is free and two different situations of simply supported ends.


y

a

AB

P

l

C

xyl

(a)

Mo

y

a

P

x

x

yl, 1

yl, 2

(b)

(c)

y

Mo

l

Figure 5.5 Bar fixed at one end and free at other with moment applied to free end and concentrated force at any distance from free end. (a) Complete assembly; (b) free-body diagram showing effect of concentrated force; (c) free-body diagram showing effect of moment.

Cantilever with Moment


dx

σz

dz

dy

dx

dz

dy

τγγdz

Figure 5.6 Element subjected to normal stress.

Figure 5.7 Element subjected to shear stress.

Stress Elements


Strain energy forspecial case whereall three factors are General expression

Loading type Factors involved constant with x for strain energy

Axial P,E,A U =P 2l

2EAU =

l∫0

P 2

2EAdx

Bending M,E, I U =M2l

2EIU =

l∫0

M2

2EIdx

Torson T,G, J U =T 2l

2GJU =

l∫0

T 2

2GJdx

Transverse shear V,G,A U =3V 2l

5GAU =

l∫0

3V 2

5GAdx

(rectangular section)

Strain Energy

Table 5.2 Strain energy for four types of loading.


Castigliano’s Theorem

yi =!U

!Qi

∂Qi(5.30)

The load Qi is applied to a particular point of deformation and therefore is not a function of x. Thus, it ispermissible to take the derivative with respect to Qi before integrating for the general expressions for the

The following procedure is to be employed in using Castigliano’s theorem:

1. Obtain an expression for the total strain energy including

(a) Loads (P , M , T , V ) acting on the object (use Table 5.2)(b) A fictitious force Q acting at the point and in the direction of the desired deflection

2. Obtain deflection from y = ∂U/∂Q.

3. If Q is fictitious, set Q = 0 and solve the resulting equation.


y

l

l

b

B CA

P

(a)

x

Q

bP

(b)

x

Figure 5.8 Cantilevered bar with concentrated force acting distance b from free end. (a) Coordinate system and important points shown; (b) fictitious force shown along with concentrated force.

Cantilever with Concentrated Force


P

A

(a)

θθ

l, A1, E1

l, A2, E2

P

(b)

P2

P1

A Qθθ

Figure 5.9 Linkage system arrangement. (a) Entire assembly; (b) free-body diagram of forces acting at point A.

Linkage System


PA

BC

l

y

x

Q

h

Example 5.10

Figure 5.10 Cantilevered bar with 90° bend acted upon by horizontal force at free end.


Chapter 6: Failure Prediction for Static Loading

The concept of failure is central to the design process, and it is by thinking in terms of obviating failure that successful designs are achieved.

Henry Petroski Design Paradigms


P

(a)

a

a

P

(b)

P

σmax b

σ σa

(c)

Figure 6.1 Rectangular plate with hole subjected to axial load. (a) Plate with cross-sectional plane; (b) one-half of plate with stress distribution; (c) plate with elliptical hole subjected to axial load.

Rectangular Plate with Hole

Kc = 1+2(a

b

)

Kc =actual maximum stress

average stress


01.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

Diameter-to-width ratio, d/b

0.1 0.2 0.3 0.4 0.5 0.6

P

P––A

P–––––––(b – d)h

Pb

nom= =

dh

Str

ess

concentr

ati

on f

acto

r, K

c

Figure 6.1 Stress concentration factors for rectangular plate with central hole. (a) Uniform tension.

Rectangular Plate with Hole


0


0.1 0.2 0.3 0.4 0.5 0.6

c/b = 0.35

c/b = 0.50

c/b ≥ 1.0

P/2

P––A

P–––––––(b – d)h

Pb

σnom= =

d

h

P/2

c

Str

ess

co

ncen

trati

on

facto

r, K

c

1.0

3.0

5.0

7.0

9.0

11.0

2.0

4.0

6.0

8.0

10.0

Figure 6.2 Stress concentration factors for rectangular plate with central hole. (b) pin-loaded hole.

Rectangular Plate with Pin-Loaded Hole


01.0

1.2

1.4

1.6

1.8

Str

ess

concentr

ati

on f

acto

r, K

c

2.0

2.2

2.4

2.6

2.8

3.0


0.1 0.2 0.3 0.4 0.5 0.6

Mc––––

I

6M–––––––(b – d)h2

b

σnom= =

dh

d/h = 0

0.25

0.5

1.0

2.0

∞

M M

Figure 6.2 Stress concentration factors for rectangular plate with central hole. (c) Bending.

Rectangular Plate with Hole in Bending


1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

H/h = 3

P––A

P––bh

nom= =

P P

H h

br

21.5

1.15

1.05

1.01

0

Radius-to-height ratio, r/h

0.05 0.10 0.15 0.20 0.25 0.30

Str

ess

con

cen

trat

ion

fac

tor,

Kc

Figure 6.3 Stress concentration factors for rectangular plate with fillet. (a) Axial load.

Axially Loaded Rectangular Plate with Fillet


1.0

1.2

1.4

1.6

1.8

2.0

Str

ess

conce

ntr

atio

n f

acto

r, K

c

2.2

2.4

2.6

2.8

3.0

Mc–––I

6M–––bh

2σnom= =

H h

br

M M

1.011.051.22H/h = 6

0


0.05 0.10 0.15 0.20 0.25 0.30

Figure 6.3 Stress concentration factors for rectangular plate with fillet. (b) Bending.

Rectangular Plate with Fillet in Bending


P

01.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0


0.05 0.10 0.15 0.20 0.25 0.30

P––A

P––bh

nom = =

b

r

P

h H

H/h = !1.51.15

1.05

1.01

Str

ess

con

cen

trat

ion

fac

tor,

Kc

Figure 6.4 Stress concentration factors for rectangular plate with groove. (a) Axial load.

Axially Loaded Rectangular Plate with Groove


0


0.05 0.10 0.15 0.20 0.25 0.301.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

(b)

br

h HM M

Mc–––I

6M–––bh

2σnom= =

H/h = ∞1.51.151.051.01

Str

ess

con

cen

trat

ion

fac

tor,

Kc

Figure 6.4 Stress concentration factors for rectangular plate with groove. (b) Bending.

Rectangular Plate with Groove in Bending


0

Radius-to-diameter ratio, r/d

0.1 0.2 0.3

(a)

P––A

4P–––

d2nom = =

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

r

D dP P

D/d = 2

1.5

1.2

1.05

1.01

Str

ess

co

ncen

trati

on

facto

r, K

c

Figure 6.5 Stress concentration factors for round bar with fillet. (a) Axial load.

Axially Loaded Round Bar with Fillet


0


0.1 0.2 0.3

3.0

M

Mc–––I

32M––––

d3nom = =

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8r

D dM

D/d = 6

3

1.5

1.1

1.03

1.01

Str

ess

co

ncen

trati

on

facto

r, K

c

Figure 6.5 Stress concentration factors for round bar with fillet. (b) Bending.

Round Bar with Fillet in Bending


0


0.1 0.2 0.3

(c)

Tc––J

16T–––πd

3τnom = =

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

r

D dT T

D/d = 2

1.2

1.09

Str

ess

co

ncen

trati

on

facto

r, K

c

Round Bar with Fillet in Torsion

Figure 6.5 Stress concentration factors for round bar with fillet. (c) Torsion.


0


0.1 0.2 0.3

3.0

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

r

D d

1.01

P P

P––A

4P–––

d2nom = =

D/d 2

1.1

1.03

Str

ess

concentr

ati

on f

acto

r, K

c

Figure 6.6 Stress concentration factors for round bar with groove. (a) Axial load.

Axially Loaded Round Bar with Groove


0


0.1 0.2 0.3

3.0

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

r

D d MM

Mc–––I

32M––––πd

3σnom = =

1.01

D/d > 2

1.1

1.03

Str

ess

co

ncen

trati

on

facto

r, K

c

Figure 6.6 Stress concentration factors for round bar with groove. (b) Bending.

Round Bar with Groove in Bending


r

D d

0


0.1 0.2 0.3

Tc––J

16T–––πd

3τnom = =

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

T T

D/d ≥ 2

1.1

1.01

Str

ess

co

ncen

trati

on

facto

r, K

c

Figure 6.6 Stress concentration factors for round bar with groove. (c) Torsion.

Round Bar with Groove in Torsion


01.0

1.2

1.4

1.6

1.8

Str

ess

con

cen

trat

ion

fac

tor,

Kc

2.0

2.2

2.4

2.6

2.8

3.0

0.1 0.2 0.3

Hole diameter-to-bar diameter ratio, d/D

T TMMD

d

P P

Mc–––I

σnom= = (πD3/32) - (dD2/6)

M

P––A

σnom = = P

(πD2/4) - Dd

Tc––J

τnom = = T

(πD3/16) - (dD2/6)

Axial load:

Bending (plane shown is critical):

Torsion:

Nominal stresses:

Axial

Bending

Torsion

Figure 6.7 Stress concentration factors for round bar with hole.

Round Bar with Hole


PP

(a)

PP

(b)

PP

(c)

PP

(d)

Figure 6.8 Flat plate with fillet axially loaded showing stress contours for (a) square corners; (b) rounded corners; (c) small grooves; and (d) small holes.

Stress Contours


(a)

A

(b) (c)

Modes of Fracture

Figure 6.9 Three modes of crack displacement. (a) Mode I, opening; (b) mode II, sliding; (c) Mode III, tearing.


Yield stress, Sy Fracture toughness, Kci

Material ksi MPa ksi√

in. MPa√

mMetals

Aluminum alloy 47 325 33 362024-T351

Aluminum alloy 73 505 26 297075-T651

Alloy steel 4340 238 1640 45.8 50.0tempered at 260◦C

Alloy steel 4340 206 1420 80.0 87.4tempered at 425◦C

Titanium alloy 130 910 40-60 44-66Ti-6Al-4V

CeramicsAluminum oxide — — 2.7-4.8 3.0-5.3Soda-lime glass — — 0.64-0.73 0.7-0.8Concrete — — 0.18-1.27 0.2-1.4

PolymersPolymethyl methacrylate — — 0.9 1.0Polystyrene — — 0.73-1.0 0.8-1.1

Table 6.1 Yield stress and fracture toughness data for selected engineering materials at room temperature.

Fracture Toughness

Kci = Y!nom√"a


Failure Prediction for Multiaxial StressesI. Ductile Materials

Maximum Shear Stress Theory (MSST):

Distortion-Energy Theory (DET)

!1−!2 =Sy

ns

1√2

[(!2−!1)2+(!3−!1)2+(!3−!2)2

]1/2 =Sy

ns


Maximum Normal Stress Theory (MNST)

Failure Prediction for Multiaxial StressesII. Brittle Materials

Internal Friction Theory (IFT)

Modified Mohr Theory

!1 ≥ Sut

nsor !3 ≤ Suc

ns

If !1 > 0 and !3 < 0,!1

Sut− !3

Suc=1

ns

If !3 > 0, !1 =Sut

ns

If !1 < 0, !3 =Suc

ns

If !1 > 0 and !3 <−Sut, !1− Sut!3

Suc−Sut =SucSut

nsSuc−SutIf !3 >−Sut !1 =

Sut

ns

If !1 < 0, !3 =Suc

ns


Centerline of cylinderand hexagon

View along axis of cylinder

σ1

σ1

σ2

σ3

σ3

σ2

MSST

DET

Figure 6.10 Three-dimensional yield locus for MSST and DET.

Three-Dimensional Yield Locus


+ 1

+ 2

– 2

– 1

A G

B

Shear diagonal

D

E

45

H C

F

Sy

Sy

Sy

–Sy

+ 1

+ 2

– 2

– 1

Syt

Syt

Syt

Syt45

–Syt

–Syt

–0.577 Syt

–Syt

–Syt

0.577 Syt

–0.577 Syt

0.577 Syt

Shear diagonal

Figure 6.11 Graphical representation of maximum-shear-stress theory for biaxial stress state.

Figure 6.12 Graphical representation of distortion energy theory for biaxial stress state.

MSST and DET for Biaxial Stress State


100 mm

Torsion barBearing

Arm

2500 N

300 mm

Figure 6.13 Rear wheel suspension used in Example 6.6

Example 6.6


x

y

T

z

(a)

(b)

1

(c)

1

13 0

2

2

x

y

Td

P

z

Element

(a)

l

(b)

xz

dz

dx

xxx

(c)

1 = max

13 2

–4 4 8 12 16 20 24 28

2

Figure 6.14 Cantilevered round bar with torsion applied to free end used in Example 6.7.

Figure 6.15 Cantilevered round bar with torsion and transverse force applied to free end used in Example 6.8.



Suc

Suc

0

Sut

Sut+ σ1

+ σ2

Maximum Normal Stress Theory

Figure 6.16 Graphical representation of maximum-normal-stress theory (MNST) for biaxial stress state.

Most suitable for fibrous brittle materials, glasses, and brittle materials in general.

!1 ≥ Sut

ns

!3 ≤ Sut

ns


σ2

σ1

Sut

Sut

Sut

Suc

Pure shear: σ1 = –σ2

IFTMMT

0

Figure 6.17 Internal friction theory and modified Mohr theory for failure prediction of brittle materials.

Internal Friction and Modified Mohr Theories


(a) (b)

1.0–1.0

–1.0

1.0

Maximumshearing stress

Maximumnormal stress Maximum distortion energy

0

σ2 σult

σ1 σult

Cast ironSteelCopperAluminum

SteelCast iron

σ2

σ1

Figure 6.18 Experimental verification of yield and fracture criteria for several materials. (a) Brittle fracture; (b) ductile yielding.

Experimental Verification


1.452

0.462

0.90

0.59 r

0.38 r

0.500 diam

0.25 r

0.06 r

4.54 r2.858

0.115

0.931

1.255

1.06

5∞

3∞ taper

36.5∞

5.39

A

A

B

B

C

C

Stress Analysis of Artificial Hip

Figure 6.19 Inserted total hip replacement. Figure 6.20 Dimension of

femoral implant (in inches).

Figure 6.21 Sections of femoral stem analyzed for static failure.


Chapter 7: Failure Prediction for Cyclic and Impact Loading

All machines and structural designs are problems in fatigue because the forces of Nature are always at work and each object must respond in some fashion.

Carl Osgood, Fatigue Design


Figure 7.1 “On the Bridge”, an illustration from Punch magazine in 1891 warning the populace that death was waiting for them on the next bridge. Note the cracks in the iron bridge.

On the Bridge!


Methods to Maximize Design Life

1. By minimizing initial flaws, especially surface flaws. Great care is taken to produce fatigue-insusceptible surfaces through processes, such as grinding or polishing, that leave exceptionally smooth surfaces. These surfaces are then carefully protected before being placed into service.

2. By maximizing crack initiation time. Surface residual stresses are imparted (or at least tensile residual stresses are relieved) through manufacturing processes, such as shot peening or burnishing, or by a number of surface treatments.

3. By maximizing crack propagation time. Substrate properties, especially those that retard crack growth, are also important. For example, fatigue cracks propagate more quickly along grain boundaries than through grains (because grains have much more efficient atomic packing). Thus, using a material that does not present elongated grains in the direction of fatigue crack growth can extend fatigue life (e.g., by using cold-worked components instead of castings).

4. By maximizing the critical crack length. Fracture toughness is an essential ingredient. (The material properties that allow for larger internal flaws are discussed in Chapter 6.)


Time

Ten

sion

+C

om

pre

ssio

n–

Str

ess

1 cycle

0

σmax

σa σ

r

σm

σmin 0.30

37__

16

9 R7–8

Stress Cycle and Test Specimen

Figure 7.2 Variation in nonzero cyclic mean stress.

Figure 7.3 R.R. Moore machine fatigue test specimen. Dimensions in inches.


Yield Fracture Fatigue Fatigue Fatiguestrength strength ductility strength ductility

Sy , σ′f coefficient, exponent, exponent,

Material Condition MPa MPa ε′f a α

Steel1015 Normalized 228 827 0.95 -0.110 -0.644340 Tempered 1172 1655 0.73 -0.076 -0.621045 Q&Ta 80◦F — 2140 — -0.065 -1.001045 Q&T 306◦F 1720 2720 0.07 -0.055 -0.601045 Q&T 500◦F 1275 2275 0.25 -0.080 -0.681045 Q&T 600◦F 965 1790 0.35 -0.070 -0.694142 Q&T 80◦F 2070 2585 — -0.075 -1.004142 Q&T 400◦F 1720 2650 0.07 -0.076 -0.764142 Q&T 600◦F 1340 2170 0.09 -0.081 -0.664142 Q&T 700◦F 1070 2000 0.40 -0.080 -0.734142 Q&T 840◦F 900 1550 0.45 -0.080 -0.75

Aluminum1100 Annealed 97 193 1.80 -0.106 -0.692014 T6 462 848 0.42 -0.106 -0.652024 T351 379 1103 0.22 -0.124 -0.595456 H311 234 724 0.46 -0.110 -0.677075 T6 469 1317 0.19 -0.126 -0.52

Table 7.1 Cyclic properties of some metals.

Cyclic Properties of Metals


10-2

10-4

10-6

10-8C

rack

gro

wth

rat

e

da/d

N (

mm

/cycl

e)log ∆K

1 mm/min

1 mm/hour

1 mm/day

1 mm/week

Cra

ck g

row

th r

ate

at 5

0 H

z

= C(∆K)mda

dN

Regime A Regime B

Regime C

one lattice

spacing

per cycle

m

1

Kc

Cra

ck l

ength

, a

Number of cylces, N

∆σ2 > ∆σ1

∆σ2

∆σ1

da

dN

(a) (b)

Fatigue Crack Growth

Figure 7.4 Illustration of fatigue crack growth. (a) Size of a fatigue crack for two different stress ratios as a function of the number of cycles; (b) rate of crack growth, illustrating three regimes.


Fatigued Part Cross-Section

Figure 7.5 Cross-section of a fatigued section, showing fatigue striations or beachmarks originating from a fatigue crack at B.


Figure 7.6 Typical fatigue fracture surfaces of smooth and notched cross-sections under different loading conditions and stress levels.

Fatigue Fracture Surfaces

High Nominal Stress Low Nominal Stress

No stressconcentration

Mild stressconcentration

Severe stressconcentration

No stressconcentration

Mild stressconcentration

Severe stressconcentration

Ten

sion-t

ensi

on

or

tensi

on-c

om

pre

ssio

nU

nid

irec

tional

ben

din

gR

ever

sed

ben

din

gR

ota

tional

ben

din

g

Beachmarks Fracture surface


Not broken

1.0

0.9

0.8

0.7

0.6

0.5

0.4

103 104 105 106 107

Number of cycles to failure, N′

Fat

igue

stre

ss r

atio

, S

f /S

ut

Figure 7.7 Fatigue strengths as a function of number of loading cycles. (a) Ferrous alloys, showing clear endurance limit.

Fatigue Strength of Ferrous Metals


103 104 105 106 107 108 109


80

70

60

50

40

35

30

25

20181614

12

10

87

6

5

Alt

ernat

ing s

tres

s, sa,

ksi

Wrought

Sand cast

Permanent mold cast

Figure 7.7 Fatigue strengths as a function of number of loading cycles. (b) Aluminum alloys, with less pronounced knee and no endurance limit.

Fatigue Strength of Aluminum Alloys


104103 105 106 107


Alt

ern

atin

g s

tres

s, sa,

MP

a

Alt

ern

atin

g s

tres

s, sa,

psi

0 0

2

4

6

83103

10

20

30

40

50

60

PTFE

Phenolic

Epoxy

Diallyl-phthalate

Alkyd

Nylon (dry)

Polycarbonate

Polysulfone

Figure 7.7 Fatigue strengths as a function of number of loading cycles. (c) Selected properties of assorted polymer classes.

Fatigue Strengths of Polymers


1603103

140

120

100

80

60

40

20

00 20 40 60 80 100 140 180 220 260 3003103120 160 200 240 280

S′e___

S u

= 0.6 0.5

0.4

Carbon steelsAlloy steelsWrought irons

Tensile strength, Sut, psi

Endura

nce

lim

it, S′ e,

psi 100 x103 psi

Figure 7.8 Endurance limit as function of ultimate strength for wrought steels.

Endurance Limit vs. Ultimate Strength


Material Number of cycles RelationMagnesium alloys 108 S′

e = 0.35Su

Copper alloys 108 0.25Su < S′e < 0.5Su

Nickel alloys 108 0.35Su < S′e < 0.5Su

Titanium 107 0.45Su < S′e < 0.65Su

Aluminum alloys 5× 108 S′e = 0.40Su (Su < 48 ksi

S′e = 19 ksi (Su ≥ 48 ksi

Table 7.2 Approximate endurance limit for various materials.

Endurance Limit


0 0.5 1.0 1.5 2.0

Notch radius, r, in.

Notch radius, r, mm

Notc

h s

ensi

tivit

y, q

n

1.0

Use these values with bending and axial loads Steel,

Su, ksi (MPa)as marked

Aluminum alloy (based on 2024–T6 data)

Use these values with torsion

0.8

0.6

0.4

0.2

02.5 3.0 3.5 4.0

0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

200 (1379) 180 (1241)

120 (827)

80 (552)14

0 (965)

100 (689)

60 (414)

80 (552)

60 (414)

50 (345)

Notch Sensitivity

Figure 7.9 Notch sensitivity as function of notch radius for several materials and types of loading.

Kf = 1+(Kc−1)qn

Usage:


60

0.6 0.80.4 1.0 1.2 1.4 1.6

80 100 120 140 160 180 200 220 240

0

0.2

0.4

0.6

0.8

Su

rfac

e fi

nis

h f

acto

r, kf

Ultimate strength in tension, Sut , GPa

Ultimate strength in tension, Sut , ksi

1.0

As forged

Hot rolled

Machined or cold drawn

PolishedGround

Manufacturing Factor eprocess MPa ksi Exponent fGrinding 1.58 1.34 -0.085Machining or 4.51 2.70 -0.265

cold drawingHot rolling 57.7 14.4 -0.718As forged 272.0 39.9 -0.995

k f = eSfut

Figure 7.10 Surface finish factors for steel. (a) As function of ultimate strength in tension for different manufacturing processes.

Table 7.3 Surface finish factor.

Surface Finish Factor


1.0

0.9

0.8

0.7

0.6

0.5

0.440 60 80

(b)

100 120 140 160 180 200 220 240x103

2000 1000500

250125

6332

168 4

12

Surface finishRa, µin.

Ultimate strength in tension, Sut, psi

Surf

ace

finis

h f

acto

r, kf

Roughness Effect on Surface Finish Factor

Figure 7.10 Surface finish factors for steel. (b) As function of ultimate strength and surface roughness as measured with a stylus profilometer.


Probability of Relaibility factor,survival, percent kr

50 1.0090 0.9095 0.8799 0.82

99.9 0.7599.99 0.70

Reliability Factor

Table 7.4 Reliability factor for six probabilities of survival.


Fat

igue

stre

ngth

at

two m

illi

on c

ycl

es (

MP

a)

1380

1035

690

345

0 0

50

100

150

200

ksi

Ultimate tensile strength, Sut, (MPa)

690 1380 2170

100 200 300

Peened - smooth

or notched

Not peened - notched (typical machined surface)

Not peened - smooth

MPa

483

414

345

276

207

138

70

60

50

40

30

20

Alt

ernat

ing s

tres

s, σ

a, M

Pa

ksi

104 105 106 107 108

Number of cycles to failure, N'

Al 7050-T7651

Ti-6Al-4V

Shot peened

Machined Polished

(a) (b)

Figure 7.11 The use of shot peening to improve fatigue properties. (a) Fatigue strength at two million cycles for high strength steel as a function of ultimate strength; (b) typical S-N curves for nonferrous metals.

Shot Peening


25 mm

25 mm

(a) (b)

P

P

P

P

25 mm

30 mm25 mm

r = 2.5 mm

r = 2.5 mm

Example 7.4

Figure 7.12 Tensile loaded bar. (a) Unnotched; (b) notched.


Goodman line

Gerber line

Yield line

Alt

ern

atin

g s

tres

s, σa

Mean stress, σm

Soderberg line

0Syt

Syt

Sut

Se

Influence of Non-Zero Mean Stress

Figure 7.13 Influence of nonzero mean stress on fatigue life for tensile loading as estimated by four empirical relationships.

Kfns!a

Se+

(ns!m

Sut

)2

= 1

Gerber Line

Goodman Line

Soderberg Line

Kf!a

Se+

(!m

Sut

)2

=1

ns

Kf!a

Se+

(!m

Syt

)2

=1

ns


a b

E

FG

H

A

Su

Sy

D

L

N

M

0

B Cσmax

σmaxσmin

σmin

σm

+σ

–σm σm

–σ

c d

45∞

45∞

Su

Sy

Su

–Se /Kf

Sy

–Sy

Se /Kf

Modified Goodman Diagram

Figure 7.14 Complete modified Goodman diagram, plotting stress as ordinate and mean stress as abscissa.


Line Equation Range

AB σmax =Se

Kf+ σm

(1− Se

SuKf

)0 ≤ σm ≤ Sy − Se/Kf

1− Se

Kf Su

BC σmax = Sy

Sy − Se

Kf

1− Se

Kf Su

≤ σm ≤ Sy

CD σmin = 2σm − Sy

Sy − Se

Kf

1− Se

Kf Su

≤ σm ≤ Sy

DE σmin =

(1 +

Se

Kf Su

)σm − Se

Kf0 ≤ σm ≤

Sy − Se

Kf

1− Se

Kf Su

EF σmin = σm − Se

Kf

Se

Kf− Sy ≤ σm ≤ 0

FG σmin = −Sy −Sy ≤ σm ≤ Se

Kf− Sy

GH σmax = 2σm + Sy −Sy ≤ σm ≤ Se

Kf− Sy

HA σmax = σm +Se

Kf

Se


Modified Goodman Criterion

Table 7.5 Equations and range of applicability for construction of complete modified Goodman diagram.


Region in Failure Validity limitsFig. 7.14 equation of equation

a σmax − 2σm = Sy/ns −Sy ≤ σm ≤ Se

Kf− Sy

b σmax − σm =Se

nsKf

Se


c σmax + σm

(Se

Kf Su− 1

)= Se

nsKf0 ≤ σm ≤

Sy − Se

Kf

1− Se

Kf Su

d σmax =Sy

ns

Sy − Se

Kf

1− Se

Kf Su

≤ σm ≤ Sy

Table 7.6 Failure equations and validity limits of equations for four regions of complete modified Goodman diagram.

Modified Goodman Criterion


90

60B C

F

D

A30 120

45°

Str

ess

, ksi

0

–30E 45

Mean stress, σm

, ksi

90

Example 7.7

Figure 7.15 Modified Goodman diagram for Example 7.7.


–4.0

Mean stress ratio, σm/Su

Alt

ern

atin

g s

tres

s ra

tio

, σ a

/Su

0

0.5

1.0

1.5

–3.5 –3.0 –2.5 –2.0 –1.5 –1.0 –0.5 0 0.5 1.0

≈ (0.4)(0.9) = 0.36Se—Su

Figure 7.16 Alternating stress ratio as function of mean stress ratio for axially loaded cast iron.

Alternating Stress Ratio for Cast Iron


100

80

Ult

imat

e an

d y

ield

str

esse

s, Su a

nd Sy, ksi

Elo

ngat

ion, per

cent

Sy/Su, per

cent

60

40

20

Average strain rate, s–1

0

100

80

60

40

20

010–6 10–5 10–4 10–3 10–2 10–1 1 10 102 103

Ratio Sy/Su

Ultimate strength Su

Yield strength Sy

Total elongation

Properties of Mild Steel

Figure 7.17 Mechanical properties of mild steel at room temperature as a function of strain rate.


Example 7.10

Figure 7.18 Diver impacting diving board, used in Example 7.10. (a) Side view; (b) front view; (c) side view showing forces and coordinates.

5 ft

(a) (c)(b)

2 ft

18 in.

1.5 in.

y

xP

V

M


2.25 in.1 in.1.375 in.3.0625 in.

R = 0.375 in.

Machine frame

A–A

P

x

Shoulder

P

V

Sh

eer

forc

e,

N

M

Mo

men

t, N

–m

Px

(a)

(b)

Time

Str

ess

, P

a σa

σm

Figure 7.19 Dimensions of existing brake drum design.

Figure 7.20 Press brake stud loading. (a) Shear and bending-moment diagrams for applied load; (b) stress cycle.

Brake Stud Design Analysis


Chapter 8: Lubrication, Friction and Wear

“...among all those who have written on the subject of moving forces, probably not a single one has given sufficient attention to the effect of friction in machines...”

Guillaume Amontons (1699)


Conformal and Nonconformal Surfaces

Figure 8.1 Conformal surfaces.

Sleeve

Journal

Fluid

film

Outerring

Innerring

Rollingelement

Figure 8.2 Nonconformal surfaces.


W

W

y

x

y

x

Solid a

Solid b

rby

ray

rbx

rax

Contacting Solids

Figure 8.3 Geometry of contacting solids.


rax = r

raxr

ray = r ray

rbx

rbx

–rbyRi

Ri–rby

ray ray

raxrax rax

Barrel shapeSphere Cylinder Conic frustum Concave shape

–ray

Thrust

(a)

(b)

(c)

Radial inner Radial outer

Thrust Cylindrical inner Cylindrical outer

rbx

rbx–rbx

–rbyrby

α−β

β

–rby–rby

–rbx

Figure 8.4 Sign designations for radii of curvature of various machine elements. (a) Rolling elements; (b) ball bearing races; (c) rolling bearing races.

Radii of Curvature


Dx––2

pmax

x

y

p

Dy––2

Figure 8.5 Pressure distribution in an ellipsoidal contact.

pH = pmax

[1−

(2x

Dx

)2

−(2y

Dy

)2]1/2

Ellipsoidal Contact

pmax =6W

!DxDy


0 4 8 12 16

Radius ratio, αr20 24 3228

Ell

ipti

c in

teg

rals

, E a

nd

F Elliptic integral E

Elliptic integral FEllipticity parameter ke

Ell

ipti

city

par

amet

er, ke

1

2

3

4

5

2

4

6

8

10Dy––2

Dx––2

x

y

Dy––2

Dx––2

x

y

1 ≤ αr ≤ 100 0.01 ≤ αr ≤ 1.0ke = α2/π

r ke = α2/πr

F = π2 + qa lnαr F = π

2 − qa lnαr

where qa = π2 − 1 where qa = π

2 − 1E = 1 + qa

αrE = 1 + qaαr

Figure 8.6 Variation of ellipticity parameter and elliptic integrals of first and second kinds as function of radius ratio.

Table 8.1 Simplified equations.

Elliptic Integrals


W

Conformal surfaces

pmax ≈ 5 MPa

hmin = f (W, ub, η0, Rx, Ry) > 1 µm

No elastic effect

ub

hmin

Figure 8.7 Characteristics of hydrodynamic lubrication.

Hydrodynamic Lubrication

wa

WW

ub

W

ps

(c)(a) (b)

pa

Figure 8.8 Mechanism of pressure development for hydrodynamic lubrication. (a) Slider bearing; (b) squeeze film bearing; (c) externally pressurized bearing.


W

Nonconformal surfacesHigh-elastic-modulus material (e.g., steel)pmax ≈ 0.5 to 4 GPahmin = f (W, ub, η0, Rx , Ry, E′, ξ) > 0.1 µmElastic and viscous effects both important

ub

hmin

W

Nonconformal surfaces (e.g., nitrite rubber)pmax ≈ 0.5 to 4 MPahmin = f (W, ub, η0, Rx , Ry, E ′) ≈ 1 µmElastic effects predominate

ub

hmin

Figure 8.9 Characteristics of hard elastohydrodynamic lubrication.

Figure 8.10 Characteristics of soft elastohydrodynamic lubrication.

Elastohydrodynamic Lubrication


Boundary film

Bulk lubricant

(a) (b) (c)

Figure 8.11 Regimes of lubrication. (a) Fluid film lubrication - surfaces separated by bulk lubricant film. This regime is sometimes further classified as thick or thin film lubrication; (b) partial lubrication - both bulk lubricant and boundary film play a role; (c) boundary lubrication - performance depends essentially on boundary film.

Regimes of Lubrication


Elasto-hydrodynamic

Hydrodynamic

Co

effi

cien

t o

f fr

icti

on

, µ

10–2

10–3

10–4

10–1

1

10

Boundary

Unlubricated

Friction and Lubrication Condition

Figure 8.12 Bar diagram showing coefficient of friction for various lubrication conditions.


Boundary

Relative load

UnlubricatedHydrodynamic

Wea

r ra

te

Elastohydro-dynamic

Severe wear

Seizure

Figure 8.13 Wear rate for various lubrication regimes.

Wear Rate and Lubrication


Mean reference

line

z

Figure 8.14 Surface profile showing surface height variation relative to mean reference line.

Surface Roughness

Root-mean-square roughness

Centerline average roughness

Ra =1

N!i=1N|zi|

Rq =

(1

N!i=1Nz2i

)1/2


Arithmetic average surface roughness, Ra

µm µin.Processes

Sand casting; hot rolling 12.5-25 500-1000Sawing 3.2-25 128-1000Planing and shaping 0.8-25 32-1000Forging 3.2-12.5 128-500Drilling 1.6-6.3 64-250Milling 0.8-6.3 32-250Boring; turning 0.4-6.3 16-250Broaching; reaming; cold rolling; drawing 0.8-3.2 32-128Die casting 0.8-1.6 32-64Grinding, coarse 0.4-1.6 16-64Grinding, fine 0.1-0.4 4-16Honing 0.03-0.4 1.2-16Polishing 0.02-0.2 0.8-8Lapping 0.005-0.1 0.2-4

ComponentsGears 0.25-10 10-400Plain bearings – journal (runner) 0.12-0.5 5-20Plain bearings – bearing (pad) 0.25-1.2 10-50Rolling bearings - rolling elements 0.015-.12 0.6-5Rolling bearings - tracks 0.1-0.3 4-12

Table 8.2 Typical arithmetic average surface roughness for various manufacturing processes and machine components.

Typical Surface Roughness


F

u

ub

ub

Friction force

h

z

Area A

Figure 8.15 Slider bearing illustrating absolute viscosity.

Viscosity

!=F/A

ub/h=

shear stress

shear strain rate


To convert Tofrom cP kgf·s/m2 N·s/m2 lb·s/in.2

Multiply bycP 1 1.02× 10−4 10−3 1.45× 10−7

kgf·s/m2 9.807× 103 1 9.807 1.422× 10−3

N·s/m2 103 1.02× 10−1 1 1.45× 10−4

reyn, or lb·s/in.2 6.9× 106 7.03× 102 6.9× 103 1

Table 8.3 Absolute viscosity conversion factors.

Viscosity Conversion Factors


Figure 8.16 Absolute viscosities of a number of fluids for a wide range of temperatures.

Viscosity Data

–5010–9

10–8

10–7

10–6

10–5

10–4

10–2

10–1

100

101

102

103

0

Saturated steam

Air

Superheated steam(14.7 psig)

Gasoline(specific gravity, 0.680)

Gasoline(specific gravity,

0.748)

Octane

Kerosene

Mercury

Water

Waterplus 23%

NaCl

20.7 MPa(3000 psi)

Hydrogen

100

Temperature, tm

, °C

Ab

solu

te v

isco

sity

, η,

lb

-s/i

n.2

Ab

solu

te v

isco

sity

, η,

cP

Temperature, tm

, °F

500400200 300

–50 0 100 600 800 1000200 400

Crude oil(specific gravity,

0.855)

Di(n-butyl)sebacate

DC 500 A

LB 100 X

SAE 10

NavySymbol

2135

FluorolubeFCD–331

Halocarbons

DC 200 E

Fluorolube light grease

Residuum (specificgravity, 0.968)

Polypropylene glycolderivatives

6.9 MPa(1000 psi)

SAE 70

LB 550 X

Polymethyl siloxanes(silicones)


Fluid Temperature, tm, ◦C38 99 149 38 99 149

Absolute viscosity Kinematic viscosity at p = 0at p = 0, ηk,

η0, cP m2/sAdvanced ester 25.3 4.75 2.06 2.58× 10−5 0.51× 10−5 0.23× 10−5

Formulated advanced ester 27.6 4.96 2.15 2.82× 10−5 0.53× 10−5 0.24× 10−5

Polyalkyl aromatic 25.5 4.08 1.80 3.0× 10−5 0.50× 10−5 0.23× 10−5

Synthetic paraffinic oil (lot 3) 414 34.3 10.9 49.3× 10−5 4.26× 10−5 1.4× 10−5



plus antiwear additiveSynthetic paraffinic oil (lot 4) 375 34.7 10.1 44.7× 10−5 4.04× 10−5 1.3× 10−5

plus antiwear additiveC-ether 29.5 4.67 2.20 2.5× 10−5 0.41× 10−5 0.20× 10−5

Superrefined napthenic mineral oil 68.1 6.86 2.74 7.8× 10−5 0.82× 10−5 0.33× 10−5

Synthetic hydrocarbon (traction fluid) 34.3 3.53 1.62 3.72× 10−5 0.40× 10−5 0.19× 10−5

Fluorinated polyether 181 20.2 6.68 9.66× 10−5 1.15× 10−5 0.4× 10−5

Viscosity of Selected Fluids

Table 8.4 Absolute and kinematic viscosities of various fluids at atmospheric pressure and different temperatures.


Fluid Temperature, tm, ◦C38 99 149Pressure-viscosity coefficient,

ξ, m2/NAdvanced ester 1.28× 10−8 0.987× 10−8 0.851× 10−8

Formulated advanced ester 1.37× 10−8 1.00× 10−8 0.874× 10−8

Polyalkyl aromatic 1.58× 10−8 1.25× 10−8 1.01× 10−8

Synthetic paraffinic oil (lot 3) 1.77× 10−8 1.51× 10−8 1.09× 10−8



plus antiwear additiveSynthetic paraffinic oil (lot 4) 1.96× 10−8 1.55× 10−8 1.25× 10−8

plus antiwear additiveC-ether 1.80× 10−8 0.980× 10−8 0.795× 10−8

Superrefined napthenic mineral oil 2.51× 10−8 1.54× 10−8 1.27× 10−8

Synthetic hydrocarbon (traction fluid) 3.12× 10−8 1.71× 10−8 0.939× 10−8

Fluorinated polyether 4.17× 10−8 3.24× 10−8 3.02× 10−8

Table 8.5 Pressure-viscosity coefficients of various fluids at different temperatures.

Piezoviscous Properties of Fluids


Temperature, °C

0 20 30 40 50 60 80 100 120 14070 90 110 130.002

.003

.004

.005

.01

.02

.03

.04

.06

.2

.1

.4

1

235 4

10

.5

.3

SAE 70

Abso

lute

vis

cosi

ty, η,

N-s

/m2

10

20

30

40

50

60

Abso

lute

vis

cosi

ty, η,

rey

n

Temperature, °F

32 50 75 100 125 150 200 250225175 275

3×10-7

10-3

10-6

10-5

10-4

Figure 8.17 Absolute viscosities of SAE lubricating oils at atmospheric pressure. (a) Single grade oils.

Viscosity of Single Grade SAE Oils


Temperature, °C

0 20 30 40 50 60 80 100 120 14070 90 110 130.002

.003

.004

.005

.01

.02

.03

.04

.06

.2

.1

.4

1

235 4

10

.5

.3

Ab

solu

te v

isco

sity

, η,

N-s

/m2

Ab

solu

te v

isco

sity

, η,

rey

n

Temperature, °F

32 50 75 100 125 150 200 250225175 275

3×10-7

10-3

10-6

10-5

10-4

10W

20W

5W-30

10W-30

20W-40

20W-50

Figure 8.17 Absolute viscosities of SAE lubricating oils at atmospheric pressure. (b) Multigrade oils.

Viscosity of Multigrade SAE Oils


SAE Grade Constant C1 Constant C2

reyn N-s/m2

10 1.58× 10−8 1.09× 10−4 1157.520 1.36× 10−8 9.38× 10−5 1271.630 1.41× 10−8 9.73× 10−5 1360.040 1.21× 10−8 8.35× 10−5 1474.450 1.70× 10−8 1.17× 10−4 1509.660 1.87× 10−8 1.29× 10−4 1564.0

Table 8.6 Curve fit data for SAE single-grade oils.

Viscosity of SAE Single-Grade Oils

!=C1 expC2

tF +95(English units)

!=C1 expC2

1.8tC+127(SI units)


W

F

(b)(a)

W

F

Figure 8.18 Friction force in (a) sliding and (b) rolling.

µ=F

W

Coulomb Friction Law:

Friction


Coefficientof friction, µ

Self-mated metals in airGold 2.5

Silver 0.8-1Tin 1Aluminum 0.8-1.2Copper 0.7-1.4Indium 2Magnesium 0.5Lead 1.5Cadmium 0.5Chromium 0.4

Pure metals and alloys sliding on steel (0.13% carbon) in airSilver 0.5Aluminum 0.5Cadmium 0.4Copper 0.8Chromium 0.5Indium 2Lead 1.2Copper - 20% lead 0.2Whitemetal (tin based) 0.8Whitemetal (lead based) 0.5α-brass (copper - 30% zinc) 0.5Leaded α/β brass (copper - 40% zinc) 0.2Gray cast iron 0.4Mid steel (0.13% carbon) 0.8

Table 8.7 Typical coefficients of friction for combinations of unlubricated metals in air.

Coefficient of Friction Data


W

F

L

θTransferred surface of softer metal

Asperities

Small spot welds

Intimate contact between metalsof two opposing surfaces

Abrasive and Adhesive Wear

Figure 8.19 Conical asperity having mean angle θ plowing through a softer material. This action simulates abrasive wear. Figure 8.20 Adhesive wear model.

Archard Wear Law: v=k1WL

3H


Coefficient of Adhesive wearRubbing materials friction, µ coefficient, k1

Gold on gold 2.5 0.1-1Copper on copper 1.2 0.01-0.1Mild steel on mild steel 0.6 10−2

Brass on hard steel 0.3 10−3

Lead on steel 0.2 2× 10−5

Polytetrafluoroethylene (teflon) on steel 0.2 2× 10−5

Stainless steel on hard steel 0.5 2× 10−5

Tungsten carbide on tungsten carbide 0.35 10−6

Polyethylene on steel 0.5 10−8 − 10−7

Table 8.8 Coefficients of rubbing friction and adhesive wear constant for selected rubbing materials.

Wear Coefficient Data


(a) (b)

(c) (d)

Periodicapplied load

Subsurface defects

Coalescence of defects

Wear particle

Fatiguespall

Figure 8.21 Fatigue wear simulation. (a) Machine element surface is subjected to cyclic loading; (b) defects and cracks develop near the surface; (c) the cracks grow and coalesce, eventually extending to the surface until (d) a wear particle is produced, leaving a fatigue spall in the material

Fatigue Wear


Chapter 9: Columns

And as imagination bodies forth the forms of things unknown, The poet’s pen turns them to shapesAnd gives to airy nothingness a local habitation and a name.

William ShakespeareA Midsummer Night’s Dream


(c)

0

P

We

(a)

0

P

We

Re

0

We

P

(b)

Re

Figure 9.1 Depiction of equilibrium regimes. (a) Stable; (b) neutral; (c) unstable.

Equilibrium Regimes


θl

θ

P

mag

Example 9.1

Figure 9.2 Pendulum used in Example 9.1.


P

P

(a)

y

x

y

dy

P

P

(b)

dxds

l

y

P

P′

V

M

(c)

Figure 9.3 Column with pinned ends. (a) Assembly; (b) deformation shape; (c) load acting.

Column with Pinned Ends


P

x

y

P

Figure 9.4 Buckling of rectangular section.

Buckling of Columns

Euler Equation: Pcr =n2!2EI

l2


l

le = 0.7l

P

P

Both ends pinnedEnd condition description

Theoretical effectivecolumn length

Illustration of end condition

le = l l

e = 0.7l l

e = 0.5l l

e = 2l

AISC (1989)– recommendedeffective column length

le = l l

e = 0.8l l

e = 0.65l l

e = 2.1l

One end pinned andone end fixed

Both ends fixed One end fixed and one end free

le = 0.5ll

P

P

le = 2l

l

P

P

l = le

P

P

Table 9.1 Effective length for four end conditions.

Column Effective Lengths


No

rmal

str

ess,

σ

(le /rg)T

Slenderness ratio, le /rg

Safe

Johnson equation

AISC Equations

Yield strength

Yielding

Buckling

Euler equationT

A

Figure 9.5 Normal stress as a function of slenderness ratio.

Buckling for Different Slenderness Ratio

Critical Slenderness Ratio

Euler Equation

Johnson Parabola

Cc =(le

rg

)E

=

√2E!2

Sy

(!cr)E =(Pcr)EA

="2E

(le/rg)2

(!cr)J =(Pcr)JA

= Sy−S2y

4"2E

(le

rg

)2


AISC Buckling Criterion

!all =12"2E

23(le/rg)2Elastic Buckling

Inelastic Buckling

Allowable Stress Reduction

!all =

{1−

[(le/rg)2

2C2c

]}Sy

n!

n! =5

3+3(le/rg)8Cc

− (le/rg)3

8C3c


27.6 mm

50 mm

(a)

Pcr = 738.6 N

(b)

(c)

Pcr = 4094 N

Pcr = 5672 N

(d)

Pcr = 773.5 N

50 mm

41.7 mm

43.6 mm

24.5 mm

Example 9.3

Figure 9.6 Cross-sectional areas, drawn to scale, from results of Exampe 9.3, aas well as critical buckling load for each cross-sectional area.


x

x

y

A

B

l

P

P

e

e

(a)

M′ = Pe

M′ = Pe

x

l

y

P

P

(b)

x

y

x

M

e

y

P

(c)

P

Figure 9.7 Eccentrically loaded column. (a) Eccentricity; (b) statically equivalent bending moment; (c) free-body diagram through arbitrary section.

Eccentrically Loaded Column

Secant Equation:

ymax = e

[sec

(le

2

√P

EI

)−1

]


Chapter 10: Stresses and Deformations in Cylinders

In all things, success depends on previous preparation. And without such preparation there is sure to be failure.

Confucius, Analects


Class Description Type Applications1 Loose Clearance Where accuracy is not essential, such as in

building and mining equipment2 Free Clearance In rotating journals with speeds of 600 rpm or greater,

such as in engines and some automotive parts3 Medium Clearance In rotating journals with speeds under 600 rpm, such as

in accurate machine tools and precise automotive parts4 Snug Clearance Where small clearance is permissible and where

moving parts are not intended to move freely under load5 Wringing Interference Where light tapping with a hammer is necessary to

assemble the parts6 Tight Interference In semipermanent assemblies suitable for drive or

shring fits on light sections7 Medium Interference Where considerable pressure is needed to assemble

and for shrink fits of medium sections; suitable forpress fits on generator and motor armatures and for carwheels

8 Heavy force Interference Where considerable bonding between surfaces isor shrink required, such as locomotive wheels and heavy

crankshaft disks of large engines

Table 10.1 Classes of fit.

Classes of Fit


Hub ShaftClass Allowance, a Interference, δ tolerance, tlh tolerance, tls1 0.0025d2/3 — 0.0025d1/3 0.0025d1/3

2 0.0014d2/3 — 0.0013d1/3 0.0013d1/3

3 0.0009d2/3 — 0.0008d1/3 0.0008d1/3

4 0.000 — 0.0006d1/3 0.0004d1/3

5 — 0.000 0.0006d1/3 0.0004d1/3

6 — 0.00025d 0.0006d1/3 0.0006d1/3

7 — 0.0005d 0.0006d1/3 0.0006d1/3

8 — 0.0010d 0.0006d1/3 0.0006d1/3

Table 10.2 Recommended tolerances in inches for classes of fit.

Recommended Tolerences for Classes of Fit


Hub ShaftClass Allowance, a Interference, δ tolerance, tlh tolerance, tls1 0.0073d2/3 — 0.0216d1/3 0.0216d1/3

2 0.0041d2/3 — 0.0112d1/3 0.0112d1/3

3 0.0026d2/3 — 0.0069d1/3 0.0069d1/3

4 0.000 — 0.0052d1/3 0.0035d1/3

5 — 0.000 0.0052d1/3 0.0035d1/3

6 — 0.00025d 0.0052d1/3 0.0052d1/3

7 — 0.0005d 0.0052d1/3 0.0052d1/3

8 — 0.0010d 0.0052d1/3 0.0052d1/3

Table 10.3 Recommended tolerances in millimeters for classes of fit.

Recommended Tolerances for Classes of Fit


Hub diameter Shaft diameterType Maximum, Minimum Maximum Minimumof fit dh.max dh,min ds,max ds,min

Clearance d + tlh d d− a d− a− tlsInterference d + tlh d d + δ + tls d + δ

Table 10.4 Maximum and minimum diameters of shaft and hub for two types of fit.

Maximum and Minimum Shaft and Hub Diameters


l

th

rz

pi

di

1

2

r

z

(a) (b)

Figure 10.1 Internally pressurized thin-walled cylinder. (a) Stress element on cylinder; (b) stresses acting on element.

Internally Pressurized Thin-Walled Cylinder

Stresses in Thin-Walled Cylinders:

!r = 0

!" =pir

th

!z =pir

2th


pi th

ri

d

ro

piri d dld /2

thdl

Figure 10.2 Front view of internally pressurized, thin-walled cylinder.

Thin-Walled Cylinder - Front View


po

ri

rdr

d

ro

(a)

(b)

pi

d /2

d /2

d /2 d /2

d /2

r + d r

r

sin d /2

Figure 10.3 Complete front view of thick-walled cylinder internally and externally pressurized. (a) With stresses acting on cylinder; (b) with stresses acting on element.

Thick-Walled Cylinder - Front View


d

d

rd

Initial element

Deformed element

r

dr

1–r

r–––

d–––dr

+ d–––

r + drr–––r

r

Figure 10.4 Cylindrical polar element before and after deformation.

Cylindrical Element


ro

ri

Normalstress,

Tension

Radius, rCompression

r

pi ro

ri

Normalstress,

Tension Radius, r

Compression

r

po

po

Figure 10.5 Internally pressurized, thick-walled cylinder showing circumferential (hoop) and radial stresses for various radii.

Figure 10.6 Externally pressurized, thick-walled cylinder showing circumferential (hoop) and radial stresses for various radii.

Pressurized Cylinders


ro

ri

Normalstress,

Radius, rr

ro

Normalstress,

Tension

Radius, r

Compression

r

Figure 10.7 Stresses in rotating cylinder with central hole and no pressurization.

Figure 10.8 Stresses in rotating solid cylinder and no pressurization.

Rotating Cylinders


δrh δrh

δrs

l

δrs

ri

rf

ro

Figure 10.9 Side view showing interference of press fit of hollow shaft to hub.

Press Fit - Side View


ri

rf ro

ri

rfrf

ro

pf

pf

(a)

(b)

Figure 10.10 Front view showing (a) cylinder assembled with an interference fit and (b) hub and hollow shaft disassembled (also showing interference pressure).

Press Fit - Front View


Chapter 11: Shafts

When a man has a vision, he cannot obtain the power from that vision until he has performed it on the Earth for the people to see.

Black Elk


1. Develop a free-body diagram by replacing the various machine elements mounted on the shaft by their statically equivalent load or torque components. To illustrate this, Fig. 11.1(a) shows two gears exerting forces on a shaft, and Fig. 11.1(b) then shows a free-body diagram of the gears acting on the shaft.

2. Draw a bending moment diagram in the x-y and x-z planes as shown in Fig. 11.1(c). The resultant internal moment at any section along the shaft may be expressed as

3. Develop a torque diagram as shown in Fig. 11.1(d). Torque developed from one power-transmitting element must balance the torque from other power-transmitting elements.

4. Establish the location of the critical cross-section, or the x location where the torque and moment are the largest.

5. For ductile materials use the maximum-shear-stress theory (MSST) or the distortion-energy theory (DET) covered in Sec. 6.7.1.

6. For brittle materials use the maximum-normal-stress theory (MNST), the internal friction theory (IFT), or the modified Mohr theory (MMT), covered in Sec. 6.7.2.

Shaft Design Procedure

Mx =√M2xy+M2

xz


P1

P2

A

B

P1

P2

y

z

x

Az

T

T

Ay

Bz

By

(a) (b)

My

x

Moment diagram caused byloads in x-z plane

Mz

x

Moment diagram caused byloads in x-y plane

(c)

(d)

Tx

T

x

Figure 11.1 Shaft assembly. (a) Shaft with two bearings at A and B and two gears with resulting forces P1 and P2; (b) free-body diagram of torque and forces resulting from assembly drawing; (c) moment diagram in xz and xy planes; (d) torque diagram.

Shaft Assembly


160 mm40 mm

160 mm40 mm

Chain

(a)

(b)

(c)

y

RA

RB

P

x

xM

–80 N-m

M

Example 11.1

Figure 11.2 Illustration for Example 11.1. (a) Chain drive assembly; (b) free-body diagram; (c) bending moment diagram.


(a)

(c)

(b)

My(N-m)

x(m)

150 N

475 N

7.5 N-m

7.5 N-m

950 Nx

y

z

0.250 m0.250 m

0.150 m

475 N

650 N

500 N

550 N

475 N 950 N 475 N

400 N

300 N

200 N

A

BC

D

DBCA

x

y

z

0.250 m0.250 m

0.050 m

0.075 m

0.150 m

0.250 m 0.250 m

My = 118.75 N-m

0.150 m

(d)

Mz(N-m)

x(m)

150 N 650 N

DBCA

500 N

0.250 m 0.250 m

Mz = 37.5 N-mMy = 75 N-m

0.150 m

(e)

Tx(N-m)

x(m)

DBCA

0.250 m 0.250 m

–7.5

7.5 N-m

7.5 N-m

0.150 m

Figure 11.3 Illustration for Example 11.2. (a) Assembly drawing; (b) free-body diagram; (c) moment diagram in xz plane; (d) moment diagram in xy plane; (e) torque diagram.

Example 11.2


m Kf a

m Kf a

m Kfs a

m Kfs a

(a)

(b)

m + Kfs a

m + Kfs a

m + Kf a

y

x

AA sin

A cos

m

a

Se/2ns

Sy/2ns

D

G

H0

F

Figure 11.4 Fluctuating normal and shear stresses acting on shaft. (a) Stresses acting on rectangular element; (b) stresses acting on oblique plane at angle ϕ.

Fluctuating Stresses on Shafts

Figure 11.5 Soderberg line for shear stresses.


2φ

2B~

A~

(A~ )

2 + 4

(B~ )

2

sin2!

cos2!= tan2!=

A

2B

Derivation in Eq. (11.29)

Figure 11.6 Illustration of relationship given in Eq. (11.29).


Shaft Design Equations

Using DET and Soderberg Criteria

ns =!d3Sy

32

√(Mm+

Sy

SeKfMa

)2

+3

4

(Tm+

Sy

SeKf sTa

)2

d =

32ns!Sy

√(Mm+

Sy

SeKfMa

)2

+3

4

(Tm+

Sy

SeKf sTa

)2

1/3


d3 d2

d1

Figure 11.7 Section of shaft in Example 11.4.

Example 11.4


ma

W(t)

y

k

Figure 11.8 Simple single-mass system.

Critical Frequency of Shafts

Single Mass

Multiple Mass - Dunkerly Equation

!=√

k

ma

=

√W/y

W/g=

√g

"

Multiple Mass - Raleigh Equation

!cr =

√g"i=1,...,nWi#i,max

"i=1,...,nWi#2

i,max

1

!2cr

=1

!21

+1

!22

+ · · ·+ 1

!2n


x1

R1

PA PB

A B

x2 x3

R2

y

x

Example 11.5

Figure 11.9 Simply supported shaft arrangement for Example 11.5.


Keys and Pins

Figure 11.10 Illustration of keys and pins. (a) Dimensions of shaft with keyway in shaft and hub; (b) square parallel key; (c) flat parallel key; (d) tapered key; (e) tapered key with Gib head, or Gib-head key. The Gib head assists in removal of the key; (f) round key; (g) Woodruff key with illustration of mountingl (h) pin, which is often grooved. The pin is slightly larger than the hole so that friction holds the pin in place; (i) roll pin. Elastic deformation of the pin in the smaller hole leads to friction forces that keep the pin in place.


Shaft Key Distance from Shaft Key Distance fromdiameter width keyseat to diameter width keyseat to

in. in. opposite side in. in. opposite sideof shaft, in. of shaft, in.

0.500 0.125 0.430 2.000 0.50 1.7180.625 0.1875 0.517 2.250 0.50 1.9720.750 0.1875 0.644 2.500 0.625 2.1480.875 0.1875 0.771 2.750 0.625 2.4021.000 0.25 0.859 3.000 0.75 2.5771.125 0.25 0.956 3.250 0.75 2.8311.250 0.25 1.112 3.500 0.875 3.0071.375 0.3125 1.201 3.750 0.875 3.2611.500 0.375 1.289 4.000 1.00 3.4371.675 0.375 1.416 4.500 1.00 3.9441.750 0.375 1.542 5.000 1.25 4.2961.875 0.50 1.591 6.000 1.50 5.155

w

w w–2

d

w

w

l

Table 11.1 Dimensions of selected square plain parallel stock keys.

Plain Parallel Keys


Shaft diameter Square type Flat type Available lengths, lin. Width Heighta Width Heighta Minimum Maximum Available

w h w h in. in. incrementsin. in. in. in. in.

0.5-0.5625 0.125 0.125 0.125 0.09375 0.50 2.00 0250.625-0.875 0.1875 0.1875 0.1875 0.125 0.75 3.00 0.3750.9375-1.25 0.25 0.25 0.25 0.1875 1.00 4.00 0.501.3125-1.375 0.3125 0.3125 0.3125 0.25 1.25 5.25 0.6251.4375-1.75 0375 0.375 0.375 0.25 1.50 6.00 0.751.8125-2.25 0.5 0.5 0.5 0.375 2.00 8.00 1.002.3125-2.75 0.625 0.625 0.625 0.4375 2.50 10.00 1.252.875-3.25 0.75 0.75 0.75 0.50 3.00 12.00 1.503.375-3.75 0.875 0.875 0.875 0.625 3.50 14.00 1.753.875-4.5 1.00 1.00 1.00 0.75 4.00 16.00 2.004.75-5.5 1.25 1.25 1.25 0.875 5.00 20.00 2.505.75-6 1.50 1.50 1.50 1.00 6.00 24.00 3.00

Table 11.2 Dimensions of square and flat taper stock keys.

Tapered Keys


Key No. Suggested Nominal Height of Shearingshaft sizes, in. key sizea, in. key, in. Area, in.2

w × l h204 0.3125-0.375 0.062× 0.500 0.203 0.030305 0.4375-0.50 0.094× 0.625 0.250 0.052405 0.6875-0.75 0.125× 0.625 0.250 0.072506 0.8125-0.9375 0.156× 0.750 0.313 0.109507 0.875-0.9375 0.156× 0.875 0.375 0.129608 1.00-1.1875 0.188× 1.000 0.438 0.178807 1.25-1.3125 0.250× 0.875 0.375 0.198809 1.25-1.75 0.250× 1.125 0.484 0.262810 1.25-1.75 0.250× 1.250 0.547 0.296812 1.25-1.75 0.250× 1.500 0.641 0.3561012 1.8125-2.5 0.312× 1.500 0.641 0.4381212 1.875-2.5 0.375× 1.500 0.641 0.517

Woodruff Keys

Table 11.3 Dimensions of selected Woodruff keys.


Screw Diameter (in.) Holding Force (lb)0.25 100

0.375 2500.50 5000.75 13001.0 2500

Set Screws

Table 11.4 Holding force generated by setscrews.


TlTm

Flywheel

Figure 11.11 Flywheel with driving (mean) torque Tm and load torque Tl.


Type of equipment Coefficient of fluctuation, Cf

Crushing machinery 0.200Electrical machinery 0.003Electrical machinery, direct driven 0.002Engines with belt transmissions 0.030Flour milling machinery 0.020Gear wheel transmission 0.020Hammering machinery 0.200Machine tools 0.030Paper-making machinery 0.025Pumping machinery 0.030-0.050Shearing machinery 0.030-0.050Spinning machinery 0.010-0.020Textile machinery 0.025

Table 11.5 Coefficient of fluctuation for various types of equipment.

Cf =!max−!min

!avg

=2(!max−!min)!max+!min

Coefficient of Fluctuation


Design Procedure for Sizing a Flywheel

1. Plot the load torque Tl versus θ for one cycle.2. Determine Tl,avg over one cycle.3. Find the locations θωmax and θωmin.4. Determine kinetic energy by integrating the torque curve.5. Determine ωavg.6. Determine Im from Eq. (11.72).

7. Find the dimensions of the flywheel.

Im =Ke

Cf!2ave


θ2ππ0

0

40

80

120

160

3π––2

π–2

Torq

ue,

T, N

-m

12 N-m

144 N-m

12 N-m

Example 11.7

Figure 11.12 Load or output torque variation for one cycle used in Example 11.7.


Performanceindex, Mf

Material kJ/kg CommentCeramics 200-2000 Brittle and weak in tension. Use is

(compression only) usually discouragedComposites:

Ceramic-fiber-reinforced polymer 200-500 The best performance; a good choice.Graphite-fiber-reinforced polymer 100-400 Almost as good as CFRP and cheaper;

an excellent choice.Berylium 300 Good but expensive, difficult to work,

and toxicHigh-strength steel 100-200 All about equal in performance;High-strength aluminum (Al) alloys 100-200 steel and Al-alloys less expensiveHigh-strength magnesium (Mg) alloys 100-200 than Mg and Ti alloysTitanium alloys 100-200Lead alloys 3 High density makes these a good (andCast iron 8-10 traditional) selection when

performance is velocity limited, notstrength limited.

Materials for Flywheels

Table 11.6 Materials for flywheels.


Chapter 12: Hydrodynamic and Hydrostatic Bearings

A Kingsbury Bearing.

A cup of tea, standing in a dry saucer, is apt to slip about in an awkward manner, for which a remedy is found in introduction of a few drops of water, or tea, wetting the parts in contact.

Lord Rayleigh (1918)


ρp

ub

p

ub(x)

Figure 12.1 Density wedge mechanism.

Figure 12.2 Stretch mechanism.

Density Wedge and Stretch


p

ub

p

wa

wb

Physical Wedge and Normal Squeeze

Figure 12.3 Physical wedge mechanism.

Figure 12.4 Normal squeexe mechanism.


ua

ub

p

Heat

p

Figure 12.5 Translation squeeze mechanism.

Figure 12.6 Local expansion mechanism.

Translation Squeeze and Local Expansion


z

ua

Stationary

BA

B′A′

h

Wz

z

x

B

A

(c)

L M

N

BC J KD

Aua ua ua

H

J

I

B

Moving

StationaryA

E

(a)

B

C DA

ua ua

B

A

(b)

BA

Figure 12.7 Velocity profiles in a parallel-surface slider bearing. Figure 12.8 Flow within a fixed-incline

slider bearing (a) Couette flow; (b) Poiseuille flow; (c) resulting velocity profile.

Velocity Profiles inSlider Bearings


A A′Wt

ω

ro

ri

Lubricant

Thrust bearing

Bearing pad

Bearing pad

Figure 12.9 Thrust slider bearing geometry.

Thrust Slider Bearing


Wz

ho

qs

sh

q

qs (both sides)

Film pressuredistribution

q

ub

l

ho

ub

z

x

sh

Fb

Fa

Wxa

Wa

Wza

Wzb

Figure 12.10 Force components and oil film geometry in a hydrodynamically lubricated thrust slider bearing.

Figure 12.11 Side view of fixed-incline slider bearing.

Force and Pressure Profiles for Slider Bearing


Design Procedure for Fixed-Incline Thrust Bearing1. Choose a pad length-to-width ratio. A square pad (λ = 1) is generally thought to give good

performance. If it is known whether maximum load or minimum power loss is more important in a particular application, the outlet film thickness ratio Ho can be determined

from Fig. 12.13.2. Once λ and Ho are known, Fig. 12.14 can be used to obtain the bearing number Bt.

3. From Fig. 12.15 determine the temperature rise due to shear heating for a given λ and Bt.

The volumetric specific heat Cs = ρCp, which is the dimensionless temperature rise

parameter, is relatively constant for mineral oils and is equivalent to 1.36 × 106 N/(m2°C). 4. Determine lubricant temperature. Mean temperature can be expressed as

where tmi = inlet temperature, °C. The inlet temperature is usually known beforehand.

Once the mean temperature tm is known, it can be used in Fig. 8.17 to determine the

viscosity of SAE oils, or Fig. 8.16 or Table 8.4 can be used. In using Table 8.4 if the temperature is different from the three temperatures given, a linear interpolation can be used.

(continued)

tm = tmi+!tm

2


Design Procedure for Fixed-Incline Thrust Bearing5. Make use of Eqs. (12.34) and (12.68) to get the outlet (minimum) film thickness h0 as

Once the outlet film thickness is known, the shoulder height sh can be directly obtained from sh

= ho/Ho. If in some applications the outlet film thickness is specified and either the velocity ub

or the normal applied load Wz is not known, Eq. (12.72) can be rewritten to establish ub or Wz.

6. Check Table 12.1 to see if the outlet (minimum) film thickness is sufficient for the pressurized surface finish. If ho from Eq. (12.72) ≥ ho from Table 12.1, go to step 7. If ho from Eq. (12.72)

< ho from Table 12.1, consider one or both of the following steps:

a. Increase the bearing speed.b. Decrease the load, the surface finish, or the inlet temperature. Upon making this change

return to step 3.7. Evaluate the other performance parameters. Once an adequate minimum film thickness and a

proper lubricant temperature have been determined, the performance parameters can be evaluated. Specifically, from Fig. 12.16 the power loss, the coefficient of friction, and the total and side flows can be determined.

h0 = Hol

√!0ubwt

WzBt


Sliding surfaceor runner

ri Na

ro

l

Pads

0

.2

.4

.6

.8

1.0

1 2

Length-to-width ratio, λ = l /wt

Maximumnormal

load

Dim

ensi

onle

ss m

inim

um

fil

m t

hic

knes

s, H

o =

ho/sh

Minimum powerconsumed

3 4

Figure 12.12 Configuration of multiple fixed-incline thrust slider bearing

Figure 12.13 Chart for determining minimum film thickness corresponding to maximum load or minimum power loss for various pad proportions in fixed-incline bearings.

Slider Bearings: Configuration and Film

Thickness


Surface finish Examples of Approximate Allowable outlet (minimum)(centerline average), Ra manufacturing relative film thicknessa, ho

µm µin. Description of surface methods costs µm µin.0.1-0.2 4-8 Mirror-like surface Grind, lap, 17-20 2.5 100

without toolmarks; and super-close tolerances finish

.2-.4 8-16 Smooth surface with- Grind and lap 17-20 6.2 250out scratches; closetolerances

.4-.8 16-32 Smooth surfaces; close Grind, file, 10 12.5 500tolderances and lap

.8-1.6 32-63 Accurate bearing sur- Grind, precision 7 25 1000face without toolmarks mill, and file

1.6-3.2 63-125 Smooth surface with- Shape, mill, 5 50 2000out objectionable tool- grind andmarks; moderate turntolerances

aThe values of film thickness are given only for guidance. They indicate the film thickness required to avoid metal-to-metalcontact under clean oil conditions with no misalignment. It may be necessary to take a larger film thickness than thatindicated (e.g., to obtain an acceptable temperature rise). It has been assumed that the average surface finish of the padsis the same as that of the runner.

Film Thickness for Hydrodynamic Bearings

Table 12.1 Allowable outlet (minimum) film thickness for a given surface finish.


Figure 12.14 Chart for determining minimum film thickness for fixed-incline thrust bearings.

Thrust Bearing - Film Thickness

Dim

ensi

onle

ss m

inim

um

fil

m t

hic

knes

s, H

o =

ho/s

h

Bearing number, Bt

10

.1

2

.2

.4

.6

1

2

4

6

10

4 6 10 20 40 60 100 200 400 1000

Length-to-width ratio,

λ

2

1

1/2

0

3

4


Figure 12.14 Chart for determining dimensionless temperature rise due to viscous shear heating of lubricant for fixed-incline thrust bearings.

Thrust Bearings - Temperature Rise

Dim

ensi

onle

ss t

emp

erat

ure

ris

e, 0

.9 C

s w

t l

∆tm

/Wz

Bearing number, Bt

10

4

2

10

6

20

40

60

200

100

400

600

1000

4 6 10 20 40 60 100 200 400 1000


λ

4

3

2

1

1/2

0


Figure 12.16a Chart for determining friction coefficient for fixed-incline thrust bearings.

Thrust Bearings - Friction Coefficient

Bearing number, Bt

Dim

ensi

onle

ss c

oef

fici

ent

of

fric

tion, µl

/ s

h

10

1

2

2

4

6

10

20

40

60

100

200

400

600

1000

4 6 10 20 40 60 100 200 400 1000

21

1/2

0

34


λ


Po

wer

lo

ss v

aria

ble

, 1

.5 h

pl /

Wz u

b s

h,

W-s

/N-m

Bearing number, Bt

10

1

2

2

4

6

10

20

40

60

100

200

400

600

1000

4 6 10 20

(b)

40 60 100 200 400 1000

21

1/2

0

34


Figure 12.16b Chart for determining power loss for fixed-incline thrust bearings.

Thrust Bearings - Power Loss


Dim

ensi

onle

ss f

low

, q/w

t u

b s

h

Bearing number, Bt

10 2

1

2

3

4

4 6 10 20 40 60 100 200 400 1000

1

0

4

(c)

qsq

qs (both sides)

q –


1/2

32

Figure 12.16c Chart for determining lubricant flow for fixed-incline thrust bearings.

Thrust Bearings - Lubricant Flow


Figure 12.16d Chart for determining lubricant side flow for fixed-incline thrust bearings.

Thrust Bearings - Side Flow

Vo

lum

etri

c fl

ow

rat

io, qs/q

Bearing number, Bt

10 2

.2

.4

.6

.8

1.0

4 6 10 20 40 60 100 200 400 1000

2

1

1/2

0

3

4


λ


Wr

Film pressure, p

pmax

e

0

max

hmin

b

00

Journal Bearing - Pressure Distribution

Figure 12.17 Pressure distribution around a journal bearing.


h = c

2πr

u u

Bearing

Journal

wt

Na

u

r

c

Figure 12.18 Concentric Journal Bearing

Figure 12.19 Developed journal bearing surfaces for a concentric journal bearing.

Concentric Journal Bearing


Average radial load per area, W ∗r

Application psi MPaAutomotive engines:

Main bearings 600-750 4-5Connecting rod bearing 1700-2300 10-15

Diesel engines:Main bearings 900-1700 6-12

Connecting rod bearing 1150-2300 8-15Electric motors 120-250 0.8-1.5Steam turbines 150-300 1.0-2.0Gear reducers 120-250 0.8-1.5Centrifugal pumps 100-180 0.6-1.2Air compressors:

Main bearings 140-280 1-2Crankpin 280-500 2-4

Centrifugal pumps 100-180 0.6-1.2

Table 12.2 Typical radial load per area Wr* in use for journal bearings.

Typical Radial Load for Journal Bearings


Figure 12.20 Effect of bearing number on minimum film thickness for four diameter-to-width ratios.

Journal Bearing - Film Thickness


40

60

80

100

Att

itu

de

ang

le,

Φ,

deg

Bearing number, Bj

10110010–110–2

20

0

Diameter-to-width ratio,

λ j

0

1

2

4

Figure 12.21 Effect of bearing number on attitude angle for four different diameter-to-width ratios.

Journal Bearings - Attitude Angle


Dim

ensi

on

less

coef

fici

ent

of

fric

tio

n v

aria

ble

, r b

µ/c

Bearing number, Bj

101100

100

101

102

2 × 102

10–110–20


λ j

2

10

4

Figure 12.21 Effect of bearing number on coefficient of friction for four different diameter-to-width ratios.

Journal Bearing - Friction Coefficient


Dim

ensi

on

less

vo

lum

etri

cfl

ow

rat

e, Q

= 2

πq/rbcw

tωb

Bearing number, Bj

101100

2

4

6

8

10–110–20


λ j

4

2

1

0

Figure 12.23 Effect of bearing number on dimensionless volume flow rate for four different diameter-to-width ratios.

Journal Bearing - Flow Rate


Sid

e-le

akag

e fl

ow

rat

io, qs/q

Bearing number, Bj

101100

.4

.6

.8

1.0

10–110–20

.2


λ j

4

2

1

Figure 12.21 Effect of bearing number on side-flow leakage for four different diameter-to-width ratios.

Journal Bearing - Side Flow


Dim

ensi

on

less

max

imu

m f

ilm

pre

ssure

, P

max

= W

r/2

r b w

t p

max

Bearing number, Bj

101100

.4

.6

.8

1.0

10–110–20

.2


λ j0

1

2

4

Figure 12.25 Effect of bearing number on dimensionless maximum film pressure for four different diameter-to-width ratios.

Journal Bearing - Maximum Pressure


Loca

tion o

f te

rmin

atin

g p

ress

ure

, φ 0

, deg

Lo

cati

on o

f m

axim

um

pre

ssure

, φ m

ax, deg

Bearing number, Bj

101100

40

60

80

100

10–110–2

20

0

10

15

20

25

5

0


λ j

φ0

φmax

0 1 24

4

2

1

0

Figure 12.21 Effect of bearing number on location of terminating and maximum pressure for four different diameter-to-width ratios.

Journal Bearing - Maximum and Terminating Pressure Location


Min

imu

m f

ilm

th

ick

nes

s, h

min

; p

ow

er l

oss

, hp;

ou

tlet

tem

per

atu

re, t mo;

vo

lum

etri

c fl

ow

rat

e, q

Radial clearance, c, in.

q

tmo

hp

hmin

0 .5 1.0 1.5 2.0 2.5 3.0 × 10–3

Journal Bearing - Effect of Radial Clearance

Figure 12.27 Effect of radial clearance on some performance parameters for a particular case.


Figure 12.21 Parallel-surface squeeze film bearing

Squeeze Film Bearing

x

Surface b

w = –

ho

∂h∂t

Surface a

z

l


Wz

Bearing runner

Bearing pad

Recess pressure,pr = 0 Flow,

q = 0Supply pressure,ps = 0

Bearingrecess

Manifold

Restrictor

(a)

Wz

pr > 0

ps > 0

(b)

p = pl

p = pl

Wz

q = 0

(c)

Wz

p = pr

ho

q

p = ps

(d)

Wz + ∆Wz

p = pr + ∆ pr

ho – ∆ho

q

p = ps

(e)

Wz – ∆Wz

p = pr – ∆ pr

ho + ∆ho

q

p = ps

(f)

Fluid Film in Hydrostatic

Bearing

Figure 12.29 Formation of fluid film in hydrostatic bearing system. (a) Pump off; (b) pressure buildup; (c) pressure times recess area equals normal applied load; (d) bearing operation; (e) increased load; (f) decreased load.


p = 0

hosh

Wz

ro

ri

pr

q

Radial Flow Hydrostatic Bearing

Figure 12.30 Radial flow hydrostratic thrust bearing with circular step pad.

Hamrock • Fundamentlas of Machine Elements

Chapter 13: Rolling Element Bearings

Since there is no model in nature for guiding wheels on axles or axle journals, man faced a great task in designing bearings - a task which has not lost its importance and attraction to this day.


Type Approximate Relative capacity Limiting Tolerancerange of bore Radial Thrust speed to mis-

sizes, mm factor alignmentMini- Maxi-mum mum

Conrad ordeep groove

3 1060 1.00 a0.7 1.0 ±0◦15′

Maximum capacityor filling notch

10 130 1.2-1.4 a0.2 1.0 ±0◦3′

Self-aligning,internal

5 120 0.7 b0.2 1.0 ±2◦30′

Self-aligning,external

— — 1.0 a0.7 1.0 High

Double row,maximum

6 110 1.5 a0.2 1.0 ±0◦3′

Double row,deep groove

6 110 1.5 a1.4 1.0 0◦

Table 13.1 Characteristics of representative radial ball bearings.

Radial Ball Bearings


Type Approximate Relative capacity Limiting Tolerancemaximum Radial Thrust speed to mis-bore size, factor alignment

mmOne-directionalthrust

320 b1.00-1.15 a,b1.5-2.3 b1.1-3.0 ±0◦2′

Duplex, backto back

320 1.85 c1.5 3.0 0◦

Duplex, faceto face

320 1.85 c1.5 3.0 0◦

Duplex, tandem 320 1.85 a2.4 3.0 0◦

Two directionalor split ring

110 1.15 c 1.5 3.0 ±0◦2′

Double row 140 1.5 c 1.85 0.8 0◦

Table 13.2 Characteristics of representative angular-contact ball bearings.

Angular-Contact Ball Bearings


Type Approximate range Relative Limiting Toleranceof bore sizes, mm thrust speed to mis-

Minimum MaximumOne directional,flat race

6.45 88.9 a0.7 0.10 b0◦

One directional,grooved race

6.45 1180 a1.5 0.30 0◦

Two directional,grooved race

15 220 c1.5 0.30 0◦

Table 13.1 Characteristics of representative thrust ball bearings.

Thrust Ball Bearings


Table 13.1 Characteristics of representative cylindrical roller bearings.

Type Approximate Relative capacity Limiting Tolerancerange of bore Radial thrust speed to mis-


Separable outerring, nonlocating (N)

10 320 1.55 0 1.20 ±0◦5′

Separable innerring, nonlocating (NU)

12 500 1.55 0 1.20 ±0◦5′

Separable inner ring,one-direction locating(NJ)

12 320 1.55 aLocating 1.15 ±0◦5′

Separable inner ring,two-direction locating

20 320 1.55 bLocating 1.15 ±0◦5′

Cylindrical Roller Bearings


Type Approximate Relative capacity Limiting Tolerancerange of bore Radial Thrust speed to mis-


Single row, barrelor convex

20 320 2.10 0.20 0.50 ±2◦

Double row, barrelor convex

25 1250 2.40 0.70 0.50 ±1◦30′

Thrust 85 360 a0.10 a1.80 0.35-0.50 ±3◦b0.10 b2.40

Double row,concave

50 130 2.40 0.70 0.50 ±1◦30′

aSymmetric rollers.bAsymmetric rollers.

Table 13.5 Characteristics of representative spherical roller bearings.

Spherical Roller Bearings


do

cd

bw

db

d

dideda

Figure 13.1 (a) Cross section through radial single-row ball bearings; (b) examples of radial single-row ball bearings.

Radial Single-Row Ball Bearings


ro

(a) (b)

x

Axis of rotation Axis of rotation

ri

d–2

cd–4

cd–4

ro

ri

crcr –

βf

cd—2

ce—2

r

d

Figure 13.2 Cross section of ball and outer race, showing race conformity.

Figure 13.1 Cross section of radial bearing, showing ball-race contact due to axial shift of inner and outer races. (a) Initial position; (b) shifted position.

Race Conformity and Axial Shift


0.001 .002 .003 .004 .005 .006 .007 .008 .009

4

8

12

16

20

0

.008

.016

.024

.032

.040

24

28

32

36

40

Fre

e co

nta

ct a

ng

le,

β f

Dim

ensi

on

less

en

dp

lay

, c e

/2d

Dimensionless diametral clearance, cd/2d

44

48

.06

.04

.06

.08

.08

.04

.02

0.02

B

Total

conformity ratio,

Free contact angle

Dimensionless

endplay

Figure 13.4 (a) Free contact angle and endplay as function of cd/2d for four values of total conformity ratio.

Free Contact Angle and Endplay


r

dθs

sh

d

de/2

de + d cos β—————

2

de – d cos β—————

2

β

ωi ωo

L C

Figure 13.5 Shoulder height in a ball bearing.

Figure 13.6 Cross section of ball bearing.

Ball Bearing


d

(a)

ll

lt

rr

d

(b)

le

rr

ll

Figure 13.7 (a) Spherical roller (fully crowned) and (b) cylindrical roller; (c) section of toroidal roller bearing.

Spherical, Cylindrical and Toroidal Bearings


dero

d

β

ri

rrrr

ro

Outer ring

Axis of rotation

cd—2

ro – cd/2

ce—2

γd

β

Figure 13.8 Geometry of spherical roller bearing.

Figure 13.9 Schematic diagram of spherical roller bearing, showing diametral play and endplay.

Spherical Roller Bearing


Pzi

Pzo

Centrifugal

force

βi

βo

φs

βo

ωb

ωro

ωso

ωoωi

ωsiωri

βi

Figure 13.10 Contact angles in ball bearing at appreciable speeds.

Figure 13.11 Angular velocities of ball

Contact Angle and Angular Velocity of Ball


(a) (b)

ωb

ωb

Figure 13.12 Ball-spin orientations for (a) outer-race and (b) inner race control.

Ball Spin Orientations


de/2

β

d

Figure 13.13 Tapered-roller bearing. (a) Tapered-roller bearing with outer race removed to show rolling elements; (b) simplified geometry for tapered-roller bearing.

Tapered Roller Bearing


di di + 2d

cd /2

c

(a) (b)

cd/2

do

cd /2

δmax

δψ

δmax

δm

(c)

ψl

ψ

ψ

Figure 13.14 Radial loaded rolling-element bearing. (a) Concentric arrangement; (b) initial contact; (c) interference.

Radially Loaded Rolling Element

Bearing


ββf

θs

rP

sh

DxDy

Outer race

Inner race

P

ri

cr

cr

cr – cd /2

cr – cd /2

ce /2

cr + δm

δt

δt

δt

δmβf

β

βf

β

ro

CL

Figure 13.15 Contact ellipse in bearing race under load.

Figure 13.16 Angular-contact ball bearing under thrust load.

Contact Ellipse and Angular Contact Ball Bearing


StickoutStickout

Pa

Pa

(a)

(b)

Bearing a Bearing b

Figure 13.17 Angular-contact bearings in back-to-back arrangement, shown (a) individually as manufactured and (b) as mounted with preload.

Back-to-Back Arrangement


Bearing a

load

Bearing b

load

Appli

ed t

hru

st l

oad

, Pa

Axial deflection, δa

∆a ∆b

Stickout

Preload

System thrust

load, Pt

Figure 13.18 Thrust load-axial deflection curve for typical ball bearing.

Load-Deflection Curve


Principal Basic load ratings Allowable Speed ratings Mass Designationdimensions Dynamic Static load Grease Oil

limitdb da bw C C0 wall

mm N N kgin. lb lb rpm lbm —15 32 8 5590 2850 120 22,000 28,000 0.025 160020.5906 1.2598 0.3510 1260 641 27.0 0.055

32 8 5590 2850 120 22,000 28,000 0.030 60021.2598 0.3543 1260 641 27.0 0.06635 11 7800 3750 160 19,000 24,000 0.045 62021.3780 0.4331 1750 843 36.0 0.09935 13 11,400 5400 228 17,000 20,000 0.082 63021.3780 0.5118 2560 1210 51.3 0.18

20 42 8 6890 4050 173 17,000 20,000 0.050 160040.7874 16535 0.3150 1550 910 38.9 0.11

42 12 9360 5000 212 17,000 20,000 0.090 60041.6535 0.4724 2100 1120 47.7 0.1547 14 12,700 6550 280 15,000 18,000 0.11 62041.8504 0.5512 2860 1470 62.9 0.1552 15 15,900 7800 335 13,000 16,000 0.14 63042.0472 0.5906 3570 1750 75.3 0.3172 19 30,700 15,000 640 10,000 13,000 0.40 64062.8346 0.7480 6900 3370 144 0.88

25 47 12 11,200 6550 275 15,000 18,000 0.080 60050.9843 1.8504 0.4724 5520 1470 61.8 0.18

52 15 14,000 7800 335 12,000 15,000 0.13 62052.0472 0.5906 3150 1750 75.3 0.2962 17 22,500 11,600 490 11,000 14,000 0.23 63052.4409 0.6693 5060 2610 110 0.5180 21 35,800 19,300 815 9000 11,000 0.53 64053.1496 0.8268 8050 4340 183 0.51

30 55 15 13,300 8300 355 12,000 15,000 0.12 60061.1811 2.1654 0.5118 2990 1870 79.8 0.26

62 16 19,500 11,200 475 10,000 13,000 0.20 62062.4409 0.6299 4380 2520 107 0.4472 19 28,100 16,000 670 9000 11,000 0.35 63062.8346 0.7480 6320 3600 151 0.7790 23 43,600 23,600 1000 8500 10,000 0.74 64063.5433 0.9055 9800 5310 225 1653

35 62 14 15,900 10,200 440 10,000 13,000 0.16 60071.3780 2.4409 0.5512 3570 2290 98.9 0.35

72 17 25,500 15,300 655 9000 11,000 0.29 62072.8346 0.6693 5370 3440 147 0.6480 21 33,200 19,000 815 8500 10,000 0.46 63073.1496 0.8268 7460 4270 183 1.00100 25 55,300 31,000 1290 7000 8500 0.95 64073.9370 0.9843 12,400 6970 290 2.10

40 68 15 16,800 11,600 490 9500 12,000 0.19 60081.5748 2.6672 0.5906 3780 2610 110 0.42

80 18 30,700 19,000 800 8500 10,000 0.37 62083.1496 0.7087 6900 4270 147 0.8290 23 41,000 24,000 1020 7500 9000 0.63 63083.5433 0.9055 9220 5400 229 1.40110 27 63,700 36,500 1530 6700 8000 1.25 64084.3307 1.0630 14,300 8210 344 2.75

Bearing Ratings

Table 13.1 Single-row, deep-groove ball bearings.


Principal Basic load ratings Allowable Speed ratings Mass Designationdimensions Dynamic Static load Grease Oil

limitdb da bw C C0 wall

mm N N kgin. lb lb rpm lbm —15 35 11 12,500 10,200 1200 18,000 22,000 0.047 NU 202 EC0.5906 1.3780 0.4331 2810 2290 274 0.10

42 13 19,400 15,300 1860 16,000 19,000 0.086 NU 302 EC1.6535 0.5118 4360 3440 418 0.19

20 47 14 25,100 22,000 2750 13,000 16,000 0.11 NU 204 EC0.7874 1,8504 0.5512 5640 4950 618 0.24

52 15 30,800 26,000 3250 12,000 15,000 0.15 NU 304 EC2.0472 0.5906 6920 5850 731 0.33

25 52 15 28,600 27,000 3350 11,000 14,000 0.13 NU 205 EC0.9843 2.0472 0.5906 6430 6070 753 0.29

62 17 40,200 36,500 4550 9500 12,000 0.24 NU 305 EC2.4409 0.6693 9040 8210 1020 0.53

30 62 16 38,000 36,500 4450 9500 12000 0.20 NU 206 EC1.811 2.4409 0.6299 8540 8210 1020 0.44

72 19 51,200 48,000 6200 9000 11,000 0.36 NU 306 EC2.8346 0.7480 11,500 10,800 1390 0.79

35 72 17 48,400 48,000 6100 8500 10,000 0.30 NU 207 EC1.3780 2.8346 0.693 10,900 10,800 1390 0.66

80 21 64,400 63,000 8150 8000 9500 0.48 NU 307 EC3.1496 0.8268 14,500 14,200 1830 1.05

40 80 18 53,900 53,000 6700 7500 9000 0.37 NU 208 EC1.5748 3.1496 0.7087 12,100 11,900 1510 0.82

90 23 80,900 78,000 10,200 6700 8000 0.65 NU 308 EC3.5433 0.9055 18,200 17,500 2290 1.05

45 85 19 60,500 64,000 8150 6700 8000 0.43 NU 209 EC1.7717 3.3465 0.7480 13,600 14,400 1830 0.95

100 25 99,000 100,000 12,900 6300 7500 0.90 NU 309 EC3.9370 0.9843 22,300 22,500 2900 2.00

50 90 20 64,400 69,500 8800 6300 7500 0.48 NU 210 EC1.9685 3.5433 0.7874 14,500 15,600 1980 1.05

110 27 110,000 112,000 15,000 5000 6000 1.15 NU 310 EC4.3307 1.0630 24,700 25,200 3370 2.55

Table 13.7 Single-row cylindrical roller bearings.

Bearing Ratings


Pa = Psinαp

Pr = Pcosαp

αp

P

Figure 13.19 Combined load acting on radial deep-groove ball bearing.

Combined Load

P0 = X0Pr+Y0Pa

Bearing type Single row Double rowX0 Y0 X0 Y0

Radial deep-groove ball 0.6 0.5 0.6 0.5Radial angular-contact ball β = 20◦ 0.5 0.42 1 0.84

β = 25◦ 0.5 0.38 1 0.76β = 30◦ 0.5 0.33 1 0.66β = 35◦ 0.5 0.29 1 0.58β = 40◦ 0.5 0.26 1 0.52

Radial self-aligning ball 0.5 0.22 cot β 1 0.44 cot βRadial spherical roller 0.5 0.22 cot β 1 0.44 cot βRadial tapered roller 0.5 0.22 cot β 1 0.44 cot β

Table 13.8 Radial factor X0 and thrust factor Y0 for statically stressed radial bearings.


(a) (b)

Figure 13.20 Typical fatigue spall (a) Spall on tapered roller bearing; (b) detail of fatigue spall.

Fatigue Spall


4

3

2

1

10080604020

Lif

e

Bearings failed, percent

0

L = 0.2~

L~

S1 = 1 – M1 S2 = 1 – M2 S3 = 1 – M3 Sm = 1 – Mm…

Figure 13.21 Distribution of bearing fatigue failures.

Figure 13.22 Representation of m similar stressed volumes.

Fatigue Failure


9590

80

70

60

50

40

30

20

10

5

21000100604020106421

Spec

imen

s te

sted

, per

cent

Specimen life, millions of stress cycles

Figure 13.23 Typical Weibull plot of bearing fatigue failures.

Bearing Fatigue Failures


Bearing type Single-row bearings Double-row bearings

ePa

Pr≤ e

Pa

Pr> e

Pa

Pr≤ e

Pa

Pr> e

X Y X Y X Y X YDeep-groove Pa/C0 = 0.025 0.22 1 0 0.56 2.0ball bearings Pa/C0 = 0.04 0.24 1 0 0.56 1.8

Pa/C0 = 0.07 0.27 1 0 0.56 1.6Pa/C0 = 0.13 0.31 1 0 0.56 1.4Pa/C0 = 0.25 0.37 1 0 0.56 1.2Pa/C0 = 0.50 0.44 1 0 0.56 1

Angular- β = 20◦ 0.57 1 0 0.43 1 1 1.09 0.70 1.63contact ball β = 25◦ 0.68 1 0 0.41 0.87 1 0.92 0.67 1.41bearings β = 30◦ 0.80 1 0 0.39 0.76 1 0.78 0.63 1.24

β = 35◦ 0.95 1 0 0.37 0.66 1 0.66 0.60 1.07β = 40◦ 1.14 1 0 0.35 0.57 1 0.55 0.57 0.93β = 45◦ 1.33 1 0 0.33 0.50 1 0.47 0.54 0.81

Self-aligning ball bearings 1.5 tan β 1 0.42 cot β 0.65 0.65 cot βSpherical roller bearings 1.5 tan β 1 0.45 cot β 0.67 0.67 cot βTapered-roller bearings 1.5 tan β 1 0 0.40 0.4 cot β 1 0.42 cot β 0.67 0.67 cot β

Table 13.9 Capacity formulas for rectangular and elliptical conjunctions for radial and angular bearings.

Capacity Formulas


Bearing type Single acting Double acting

ePa

Pr> e

Pa

Pr≤ e

Pa

Pr> e

X Y X Y X YThrust ball β = 45◦ 1.25 0.66 1 1.18 0.59 0.66 1

β = 60◦ 2.17 0.92 1 1.90 0.55 0.92 1β = 75◦ 4.67 1.66 1 3.89 0.52 1.66 1

Spherical roller thrust 1.5 tan β tan β 1 1.5 tan β 0.67 tan β 1Tapered roller 1.5 tan β tan β 1 1.5 tan β 0.67 tan β 1

Table 13.10 Radial factor X and thrust factor Y for thrust bearings.

Radial and Thrust Factor for Thrust Bearings


Material Material factor, D52100 2.0M-1 0.6M-2 0.6M-10 2.0M-50 2.0T-1 0.6Halmo 2.0M-42 0.2WB 49 0.6440C 0.6-0.8

Table 13.11 Material factors for through-hardened bearing materials.

Material Factors


300

250

200

150

100

50

02 4 6 8 101.8.6

Dimensionless film parameter, Λ

Region of

lubrication-related

surface distress

Region of

possible

surface

distress

L10 l

ife,

per

cen

t~

L10 life~

3.5

3.0

2.5

2.0

1.5

1.0

.5

02 4 6 8 101.8.6

Dimensionless film parameter, Λ

Mean curve

recommended

for use

Lu

bri

cati

on

fac

tor,

Fl

From Tallian (1967)

Calculated from AFBMA

From Skurka (1970)

Figure 13.24 Group fatigue life L10 as a function of dimensionless film parameter.

Figure 13.25 Lubrication factor as a function of dimensionless film parameter.

Lubrication and Bearing Life


Chapter 14

Just stare at the machine. There is nothing wrong with that. Just live with it for a while. Watch it the way you watch a line when fishing and before long, as sure as you live, you’ll get a little nibble, a little fact asking in a timid, humble way if you’re interested in it. That’s the way the world keeps on happening. Be interested in it.

Robert Pirsig, Zen and the Art of Motorcycle Maintenance


Spur Gears

Figure 14.1 Spur gear drive. (a) Schematic illustration of meshing spur gears; (b) a collection of spur gears.


Helical Gears

Figure 14.2 Helical gear drive. (a) Schematic illustration of meshing helical gears; (b) a collection of helical gears.


Bevel Gears

Figure 14.3 Bevel gear drive. (a) Schematic illustration of meshing bevel gears; (b) a collection of bevel gears.


(a) (b)

Worm Gears

Figure 14.4 Worm gear drive. (a) Cylindrical teeth; (b) double enveloping; (c) a collection of worm gears.


Tooth profile

Pitch circle

Whole depth, ht

Addendum, a

Dedendum, b

Root (tooth)Fillet

Topland

Pitch point

Pitch circleBase circle

Pressureangle,

Line ofaction

Circular pitch, pc

Base diameter, dbg

Workingdepth, hk

Clearance, cr

Circular tooththickness

Chordal tooththickness

Gear

Pinion

Centerdistance, cd

Pitch

dia

met

er, dg

Pitch (pd )

Root diameter

r bp

r p

ropOutside diameter, d

op

rbg

rg

rog

Figure 14.5 Basic spur gear geometry.

Spur Gear Geometry


Outside circle

Flank

Face

Tooththickness

Pitchcircle

Widthof space

Circular pitch

Bot

tom

land

Top la

ndFace w

idth

Clearance Filletradius Clearance

circle

Dedendumcircle

Addendum

Dedendum

Figure 14.6 Nomenclature of gear teeth.

Gear Teeth


2 3 4 5

6 7 8 9 10 12 14 16

22

1

Figure 14.7 Standard diametral pitches compared with tooth size.

Standard Tooth Size

Diametral pitch,Class pd, in−1

Coarse 1/2, 1, 2, 4, 6, 8, 10Medium coarse 12, 14, 16, 18Fine 20, 24, 32, 48, 64

72, 80, 96, 120, 128Ultrafine 150, 180, 200

Table 14.1 Preferred diametral pitches for four tooth classes


200

100

70

50

30

20

10.0

7.0

5.0

3.0

2.0

1.0

0.7

0.5

Pow

er t

ransm

itte

d, h

p

0 600 1200 1800 2400 3000 3600

Pinion speed, rpm

100

70

50

30

20

10.0

7.0

5.0

3.0

2.0

1.0

0.7

0.5

Pow

er t

ransm

itte

d, k

W

150

60

15

40

6.0

4.0

1.5

Data for all curves:

gr = 4, NP=24

Ka = 1.0

20° full depth teeth

24 pitch; d=1.00 in (m=1; d=25 mm)

12 pitch; d = 2.00 in (m=2; d=50 mm)

6 pitch; d=4.00 in (m=4; d=100 mm)

3 p

itch

; d =

8.00 in (m = 8; d = 200

Figure 14.8 Transmitted power as a function of pinion speed for a number of diametral pitches.

Power vs. Pinion Speed


MetricCoarse pitch Fine pitch module

Parameter Symbol (pd < 20 in.−1) (pd ≥ 20 in.−1) systemAddendum a 1/pd 1/pd 1.00 mDedendum b 1.25/pd 1.200/pd + 0.002 1.25 mClearance c 0.25/pd 0.200/pd + 0.002 0.25 m

Table 14.2 Formulas for addendum, dedendum, and clearance (pressure angle, 20°; full-depth involute).

Gear Geometry Formulas


Base circle

Pitch circle

Pitch circle

Pitch point, pp

Gear

Base circle

Pinion

a

b

φ

φ

rp

rg

rbg

rbp

ωg

ωp

Figure 14.9 Pitch and base circles for pinion and gear as well as line of action and pressure angle.

Pitch and Base Circles


A4

A3

A2

A1C1

C2

B1

B2

B3

B4

C4

C3

A0

Base circle

0

Involute

Figure 14.10 Construction of the involute curve.

Involute Curve


Construction of the Involute Curve

1. Divide the base circle into a number of equal distances, thus constructing A0, A1, A2,...

2. Beginning at A1, construct the straight line A1B1, perpendicular with 0A1, and likewise beginning at A2 and A3.

3. Along A1B1, lay off the distance A1A0, thus establishing C1. Along A2B2, lay off twice A1A0, thus establishing C2, etc.

4. Establish the involute curve by using points A0, C1, C2, C3,... Gears made from the involute curve have at least one pair of teeth in contact with each other.


Arc of approach qa Arc of recess qr

PBA

aOutside circle

Pitch circle Outside circle

Motion

b

Lab

Line of action

Figure 14.11 Illustration of parameters important in defining contact.

Contact Parameters


Figure 14.12 Details of line of action, showing angles of approach and recess for both pinion and gear.

Line of Action

Lab =√r2op− r2bp+

√r2og− r2bg− cd sin!

Cr =1

pc cos!

[√r2op− r2bp+

√r2og− r2bg

]− cd tan!

pc

Length of line of action:

Contact ratio:


Base circle

Base circle

Pitch circle

Pitch circle

0

Backlash

Pressure line

Backlash

φ

Figure 14.13 Illustration of backlash in gears.

Backlash

Diametralpitch Center distance, cd, in.pd, in.−1 2 4 8 16 3218 0.005 0.006 — — —12 0.006 0.007 0.009 — —8 0.007 0.008 0.010 0.014 —5 — 0.010 0.012 0.016 —3 — 0.014 0.016 0.020 0.0282 — — 0.021 0.025 0.0331.25 — — — 0.034 0.042

Table 14.3 Recommended minimum backlash for coarse-pitched gears.


Gear 1

(N1 teeth)

Gear 2

(N2 teeth)

(+) (–)1

2

r1

r2

1

2

Gear 1

(N1)

Gear 2

(N2)

r1

r2

Figure 14.14 Externally meshing gears.

Meshing Gears

Figure 14.15 Internally meshing gears.


N2

N1

N1

N2 N5

N6

N7 N8N3 N4

Figure 14.16 Simple gear train.

Gear Trains

Figure 14.17 Compound gear train.


Only pitchcircles of gears shown

Shaft 1

Input

Shaft 2

Shaft 3

Shaft 4

Output

A

B

NA = 20

NB = 70

NC = 18

ND = 22

NE = 54

A

B

C

D

E

C

D

E

Example 14.7

Figure 14.18 Gear train used in Example 14.7.


RingR R

P

P

S

P

SA

P

Planet

Arm

Sun

(a) (b)

Planetary Gear Trains

Figure 14.19 Illustration of planetary gear train. (a) With three planets; (b) with one planet (for analysis only).

!ring−!arm

!sun−!arm

=−NsunNring

!planet−!arm

!sun−!arm

=− Nsun

Nplanet

Nring = Nsun+2Nplanet

Zp =!L−!A

!F−!A

Important planet gear equations:


Rel

ativ

e co

st

10

100

1

0.5

1 2 3 4 5 6 7 8 9 1 0 1 1 1 2

DIN quality number

(4)(6) (5)(7)891011121315(17)

14(16)

AGMA quality index

Special methods

Production grinding

Shaving

Gear shaper-hobbing

0.0

0005 i

n.

0.0

0010

0.0

005

0.0

010

0.0

10

0.0

15

Gear Quality

Figure 14.20 Gear cost as a function of gear quality. The numbers along the vertical lines indicate tolerances.

Application Quality index, Qv

Cement mixer drum driver 3-5Cement kiln 5-6Steel mill drives 5-6Corn pickers 5-7Punch press 5-7Mining conveyor 5-7Clothes washing machine 8-10Printing press 9-11Automotive transmission 10-11Marine propulsion drive 10-12Aircraft engine drive 10-13Gyroscope 12-14

Pitch velocity Quality index, Qv

ft/min m/s0-800 0-4 6-8800-2000 4-10 8-102000-4000 10-20 10-12> 4000 > 20 12-14

Table 14.4 Quality index Qv for various applications.


Form Cutting

Figure 14.21 Form cutting of teeth. (a) A form cutter. Notice that the tooth profile is defined by the cutter profile. (b) Schematic illustration of the form cutting process. (c) Form cutting of teeth on a bevel gear.

Gear

blank

Form

cutter

(a) (b) (c)


Pinion-Shaped Cutter

Figure 14.22 Production of gear teeth with a pinion-shaped cutter. (a) Schematic illustration of the process; (b) photograph of the process with gear and cutter motions indicated.


Gear Hobbing

Figure 14.23 Production of gears through the hobbing process. (a) A hob, along with a schematic illustration of the process; (b) production of a worm gear through hobbing.

Gearblank

Gearblank

Hob

Top view

(a)

Hob

Helical gear

Hob rotation

(b)

(b)


10150120 200 250 300 350 400 450

20

100

150

200

250

300

350

400

30

40

50

60

All

ow

able

ben

din

g s

tres

s n

um

ber

, S

b,

MP

a

ksi

Brinell hardness, HB

Grade 1

Through-hardenedThrough-h

ardened

Grade 2

Nitrided

Nitrided

Material Grade Allowable bending stress number

Through-hardened steels 12

Nitriding through- hardened steels

12

MPa

0.703 HB + 1130.533 HB + 88.3

0.0823 HB + 12.150.1086 HB + 15.89

ksi

0.0773 HB + 12.80.102 HB + 16.4

0.568 HB + 83.80.749 HB + 110

Allowable Bending Stress

Figure 14.24 Effect of Brinell hardness on allowable bending stress number for steel gears. (a) Through-hardened steels. Note that the Brinell hardness refers to the case hardness for these gears.


Material designation Grade Typical Allowable bending stress, σall,b Allowable contact stress, σall,b

Hardnessa lb/in.2 MPa lb/in.2 MPaSteel

Through-hardened 1 — See Fig. 14.24a See Fig. 14.252 — See Fig. 14.24a See Fig. 14.25

Carburized & hardened 1 55-64 HRC 55,000 380 180,000 12402 58-64 HRC 65,000b 450b 225,000 15503 58-64 HRC 75,000 515 275,000 1895

Nitrided and through- 1 83.5 HR15N See Fig. 14.24b 150,000 1035hardened 2 — See Fig. 14.24b 163,000 1125

Nitralloy 135M and 1 87.5 HR15N See Fig. 14.24b 170,000 1170Nitralloy N, nitrided 2 87.5 HR15N See Fig. 14.24b 183,000 1260

2.5% Chrome, nitrided 1 87.5 HR15N See Fig. 14.24b 155,000 10702 87.5 HR15N See Fig. 14.24b 172,000 11853 87.5 HR15N See Fig. 14.24b 189,000 1305

Cast IronASTM A48 gray cast Class 20 — 5000 34.5 50,000-60,000 345-415

iron, as-cast Class 30 174 HB 8500 59 65,000-75,000 450-520Class 40 201 HB 13,000 90 75,000-85,000 520-585

ASTM A536 ductile 60-40-18 140 HB 22,000-33,000 150-230 77,000-92,000 530-635(nodular) iron 80-55-06 179 HB 22,000-33,000 150-230 77,000-92,000 530-635

100-70-03 229 HB 27,000-40,000 185-275 92,000-112,000 635-770120-90-02 269 HB 31,000-44,000 215-305 103,000-126,000 710-870

BronzeSut > 40, 000 psi 5700 39.5 30,000 205

(Sut > 275GPa)Sut > 90, 000 psi 23,600 165 65,000 450

(Sut > 620GPa)

Allowable Bending and Contact Stress

Table 14.5 Allowable bending and contact stresses for selected gear materials.


Material Grade Allowable bending stress number

MPa ksi

Niralloy 12

0.594 HB + 87.760.784 HB + 114.81

0.0862 HB + 12.730.1138 HB + 16.65

2.5% Chrome 123

0.7255 HB + 63.890.7255 HB + 153.630.7255 HB + 201.91

0.1052 HB + 9.280.1052 HB + 22.280.1052 HB + 29.28Grade 1 - Nitralloy

250 300 350275 325

100

200

300

400

500

All

ow

able

ben

din

g s

tres

s n

um

ber

, S

b,

MP

a


Grade 3 - 2.5% Chrome


Grade 2 - Nitralloy


70

20

30

40

50

60

ksi

Allowable Bending Stress

Figure 14.24 Effect of Brinell hardness on allowable bending stress number for steel gears. (b) Flame or induction-hardened nitriding steels. Note that the Brinell hardness refers to the case hardness for these gears.


75150 200 250 300 350 400 450

100

600

700

800

900

1000

1100

1200

1300

125

150

175A

llow

able

conta

ct

stre

ss n

um

ber,

Sc, M

Pa

ksi


Grade 1

Grade 2

1400

2.22 HB +200 (MPa)

0.322 HB + 29.1 (ksi)Sc = {

Grade 2:

2.41 HB +237 (MPa)

0.349 HB + 34.3 (ksi)Sc = {

Grade 1:

Allowable Contact Stress

Figure 14.25 Effect of Brinell hardness on allowable contact stress number for two grades of through-hardened steel.


Number of load cycles, N

102 103 104 105 106 107 108 109 1010

Str

ess

cycl

e fa

ctor,

Yn

2.0

1.0

0.9

0.8

0.7

0.6

0.5

3.0

4.0400 HB: Y

N = 9.4518 N-0.148

Case Carb.: YN = 6.1514N

-0.1192

250 HB: YN = 4.9404 N-0.1045

Nitrided: YN = 3.517 N-0.0817

160 HB: YN = 2.3194 N-0.0538

YN = 1.3558 N-0.0178

YN = 1.6831 N-0.0323

Stress Cycle Factor

Figure 14.26 Stress cycle factor. (a) Bending stress cycle factor YN.


Str

ess

cycl

e fa

ctor,

Zn

2.0

1.1

1.0

0.9

0.8

0.7

0.6

0.5

Number of load cycles, N

102 103 104 105 106 107 108 109 1010

Nitrided

Zn = 1.249 N-0.0138

Zn = 2.466 N-0.056

Zn = 1.4488 N-0.023

1.5

Stress Cycle Factor

Figure 14.26 Stress cycle factor. (a) pitting resistance cycle factor ZN.


Reliability Factor

Table 14.6 Reliability factor, KR.

Probability of Reliability factora,survival, percent KR

50 0.70b

90 0.85b

99 1.0099.9 1.2599.99 1.50

a Based on surface pitting. If tooth breakageis considered a greater hazard, a largervalue may be required.

b At this value plastic flow may occur ratherthan pitting.


Har

dnes

s ra

tio f

acto

r,C

H

1.16

1.14

1.12

1.10

1.08

1.06

1.04

1.02

1.00180 200 250 300 350 400

Brinell hardness of gear, HB

Ra,p = 0.4 µm

(16 µin.)

Ra,p = 0.8 µm

(32 µin.)Rap = 1.6 µm (64 µin.)

For Rap > 1.6,

use CH = 1.0

Hardness Ratio Factor

Figure 14.27 Hardness ratio factor CH for surface hardened pinions and through-hardened gears.


Loads on Gear Tooth

Figure 14.24 Loads acting on an individual gear tooth.

Pitch circle

φWr

Wt

W


rf

t

x

(a) (b)

Wt

Wt

Wr

W

a

φ

bw

t

l

l

Loads and Dimensions of Gear Tooth

Figure 14.29 Loads and length dimensions used in determining tooth bending stress. (a) Tooth; (b) cantilevered beam.


Bending and Contact Stress Equations

Lewis Equation

AGMA Bending Stress Equation

!t =Wtpd

bwY

!t =WtpdKaKsKmKvKiKb

bwYj

Hertz Stress

AGMA Contact Stress Equation

pH = E ′(W ′

2!

)1/2

!c = pH(KaKsKmKv)1/2


Lewis Form Factor

Table 14.7 Lewis form factor for various numbers of teeth (pressure angle, 20°; full-depth involute).

Lewis LewisNumber form Number formof teeth factor of teeth factor

10 0.176 34 0.32511 0.192 36 0.32912 0.210 38 0.33213 0.223 40 0.33614 0.236 45 0.34015 0.245 50 0.34616 0.256 55 0.35217 0.264 60 0.35518 0.270 65 0.35819 0.277 70 0.36020 0.283 75 0.36122 0.292 80 0.36324 0.302 90 0.36626 0.308 100 0.36828 0.314 150 0.37530 0.318 200 0.37832 0.322 300 0.382


Spur Gear Geometry Factors

01512 20 25 30 40 60 80 275

.10

.20

.30

.40

.50G

eom

etry

fac

tor,

Yj

Number of teeth, N

Number of

teeth in

mating gear.

Load considered

applied at

highest point

of single-tooth

contact.

10001708550352517

Load applied at

tip of tooth

∞125

Figure 14.30 Spur gear geometry factors for pressure angle of 20° and full-depth involute profile.


Application and Size Factors

Table 14.8 Application factor as function of driving power source and driven machine.

Driven MachinesPower source Uniform Light shock Moderate shock heavy shock

Application factor, Ka

Uniform 1.00 1.25 1.50 1.75Light shock 1.20 1.40 1.75 2.25Moderate shock 1.30 1.70 2.00 2.75

Diametral pitch, pd, Module, m,in.−1 mm Size factor, Ks

≥ 5 ≤ 5 1.004 6 1.053 8 1.153 12 1.25

1.25 20 1.40

Table 14.9 Size factor as a function of diametral pitch or module.


Load Distribution Factor

where

Cmc ={1.0 for uncrowned teeth0.8 for crowned teeth

Km = 1.0+Cmc(CpfCpm+CmaCe)

Ce =

0.80 when gearing is adjusted at assembly0.80 when compatability between gear teeth is improved by lapping1.0 for all other conditions


0.60

0.50

0.40

0.30

0.20

0.10

0.00

b w/d p

= 2.00

1.50

1.00

0.50

For bw/dp < 0.5 use

curve for bw/dp = 0.5

Pin

ion p

roport

ion f

acto

r, C

pf

0.70

0 200 400 600 800 1000

Face width, bw (mm)

400 10 20 30

Face width, bw (in.)

Pinion Proportion Factor

Figure 14.31 Pinion proportion factor Cpf.

Cpf =

bw

10dp−0.025 bw ≤ 25 mm

bw

10dp−0.0375+0.000492bw 25 mm< bw ≤ 432 mm

bw

10dp−0.1109+0.000815bw− (3.53×10−7)b2w432 mm< bw ≤ 1020 mm


S

S/2S1

Pinion Proportion Modifier

Figure 14.32 Evaluation of S and S1.

Cpm ={1.0,(S1/S) < 0.1751.1,(S1/S)≥ 0.175


0 200 400 600 800 1000

Face width, bw (mm)

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00

Mes

h a

lig

nm

ent

fact

or,

Cma

Face width, bw (in)

403020100

Open gea

ring

Commercial enclosed gear u

nits

Precision enclosed gear units

Extra-precision enclosed gear units

Cma = A + Bbw + Cbw

2

If bw is in inches:

Open gearing

Commercial enclosed gears

Precision enclosed gears

Extraprecision enclosed gears

0.247

0.127

0.0675

0.000380

0.0167

0.0158

0.0128

0.0102

-0.765 10-4

-1.093 10-4

-0.926 10-4

-0.822 10-4

Condition A B C

Open gearing

Commercial enclosed gears

Precision enclosed gears

Extraprecision enclosed gears

0.247

0.127

0.0675

0.000360

6.57 10-4

6.22 10-4

5.04 10-4

4.02 10-4

-1.186 10-7

-1.69 10-7

-1.44 10-7

-1.27 10-7

Condition A B C

If bw is in mm:

Mesh Alignment Factor

Figure 14.33 Mesh alignment factor.


Dynam

ic f

acto

r, K

v

1.8

1.7

1.6

1.5

1.4

1.3

1.2

1.1

1.0

Pitch line velocity, m/s

0 10 20 30 40 50

Qv = 5 Qv = 6

Qv = 7

Qv = 8

Qv = 9

Qv = 10

Qv = 11

"Very accurate" gearing

0

Pitch line velocity, ft/min

750050002500

Dynamic Factor

Figure 14.34 Dynamic factor as a function of pitch-line velocity and transmission accuracy level number.


Chapter 15: Helical, Bevel and Worm Gears

The main object of science is the freedom and happiness of man.

Thomas Jefferson


Gear type Advantages DisadvantagesSpur Inexpensive, simple to design, no thrust load is

developed by the gearing, wide variety of manu-facturing options.

Can generate significant noise, especially at highspeeds, and are usually restricted to pitch-linespeeds below 20 m/s (4000 ft/min).

Helical Useful for high speed and high power applica-tions, quiet at high speeds. Often used in lieu ofspur gears for high speed applications.

Generate a thrust load on a single face, more ex-pensive than spur gears.

Bevel High efficiency (can be 98% or higher), can trans-fer power across nonintersecting shafts. Spiralbevel gears transmit loads evenly and are quieterthan straight bevel.

Shaft alignment is critical, rolling element bear-ings are therefore often used with bevel gears.This limits power transfer for high speed applica-tions (where a journal bearing is preferable). Canbe expensive.

Worm Compact designs for large gear ratios. Efficiencycan be 90% or higher.

Wear by abrasion is of higher concern than othergear types, can be expensive. Generate very highthrust loads. Worm cannot be driven by gear;worm must drive gear.

Table 15.1 Design considerations for gears.

Summary of Gear Design


Tangent to helical tooth

(a) (b)

Helixangle, ψ

Element of pitchcylinder (or gear's axis)

Pitch cylinder

Figure 13.1 Helical gear. (a) Front view; (b) side view.

Helical Gear


pc p

a

(a)

p cn

ψ

(b)

ψ

Figure 15.2 Pitches of helical gears. (a) Circular; (b) axial.

Helical Gear Pitches


AGMA Equations for Helical Gears

!t =WtpdKaKsKmKvKiKb

bwYh

!c = pH

(KaKsKmKv

Ih

)1

2

Bending Stress:

Pitting Resistance:


Pinion TeethGear 12 14 17 21 26 35 55 135teeth P G P G P G P G P G P G P G P G

12 Ua

14 Ua Ua

17 Ua Ua Ua

21 Ih Ua Ua Ua 0.127Yh Ua Ua Ua 0.46 0.46

26 Ih Ua Ua Ua 0.143 0.131Yh Ua Ua Ua 0.47 0.49 0.49 0.49

35 Ih Ua Ua Ua 0.164 0.153 0.136Yh Ua Ua Ua 0.48 0.52 0.50 0.53 0.54 0.54

55 Ih Ua Ua Ua 0.195 0.186 0.170 0.143Yh Ua Ua Ua 0.49 0.55 0.52 0.56 0.55 0.57 0.59 0.59

135 Ih Ua Ua Ua 0.241 0.237 0.228 0.209 0.151Yh Ua Ua Ua 0.50 0.60 0.53 0.61 0.57 0.62 0.60 0.63 0.65 0.65

Geometry Factors for Helical Gears

Table 15.2 Geometry factors Yh and Ih for helical gears loaded at tooth tip.

!= 20◦, "= 10

◦



12 Ua

14 Ua Ua

17 Ih Ua Ua 0.125Yh Ua Ua 0.44 0.44

21 Ih Ua Ua 0.140 0.129Yh Ua Ua 0.45 0.46 0.47 0.47

26 Ih Ua Ua 0.156 0.145 0.133Yh Ua Ua 0.45 0.49 0.48 0.49 0.50 0.50

35 Ih Ua Ua 0.177 0.167 0.155 0.138Yh Ua Ua 0.46 0.51 0.49 0.52 0.51 0.53 0.54 0.54

55 Ih Ua Ua 0.205 0.197 0.188 0.172 0.144Yh Ua Ua 0.47 0.54 0.50 0.55 0.52 0.56 0.55 0.57 0.58 0.58

135 Ih Ua Ua 0.245 0.242 0.238 0.229 0.209 0.151Yh Ua Ua 0.48 0.58 0.51 0.59 0.54 0.60 0.57 0.61 0.60 0.62 0.64 0.64



!= 20◦, "= 20

◦



12 Ua

14 Ih Ua 0.125Yh Ua 0.39 0.39

17 Ih Ua 0.139 0.128Yh Ua 0.39 0.41 0.41 0.41

21 Ih Ua 0.154 0.144 0.132Yh Ua 0.40 0.42 0.42 0.43 0.44 0.44

26 Ih Ua 0.169 0.159 0.148 0.135Yh Ua 0.41 0.44 0.43 0.45 0.45 0.46 0.46 0.46

35 Ih Ua 0.189 0.180 0.170 0.158 0.139Yh Ua 0.41 0.46 0.43 0.47 0.45 0.48 0.47 0.48 0.49 0.49

55 Ih Ua 0.215 0.208 0.200 0.190 0.174 0.145Yh Ua 0.42 0.49 0.44 0.49 0.46 0.50 0.48 0.50 0.50 0.51 0.52 0.52

135 Ih Ua 0.250 0.248 0.245 0.240 0.231 0.210 0.151Yh Ua 0.43 0.51 0.45 0.52 0.47 0.53 0.49 0.53 0.51 0.54 0.53 0.55 0.56 0.56

a A ‘U’ indicates that this geometry would produce an undercut tooth form and should be avoided.



!= 20◦, "= 30

◦



12 Ua

14 Ih Ua 0.129Yh Ua 0.47 0.47

17 Ih Ua 0.144 0.133Yh Ua 0.48 0.51 0.52 0.52

21 Ih Ua 0.159 0.149 0.136Yh Ua 0.48 0.55 0.52 0.55 0.56 0.56

26 Ih Ua 0.175 0.165 0.152 0.139Yh Ua 0.49 0.58 0.53 0.58 0.57 0.59 0.60 0.60

35 Ih Ua Ua 0.195 0.186 0.175 0.162Yh Ua 0.50 0.61 0.54 0.62 0.57 0.63 0.61 0.64 0.64 0.64

55 Ih Ua 0.221 0.215 0.206 0.195 0.178 0.148Yh Ua Ua 0.51 0.65 0.55 0.66 0.58 0.67 0.62 0.68 0.65 0.69 0.70 0.70

135 Ih Ua 0.257 0.255 0.251 0.246 0.236 0.215 0.154Yh Ua 0.52 0.70 0.56 0.71 0.60 0.72 0.63 0.73 0.67 0.74 0.71 0.75 0.76 0.76



!= 25◦, "= 10◦



12 Ih 0.128Yh 0.47 0.47

14 Ih 0.140 0.131Yh 0.47 0.50 0.50 0.50

17 Ih 0.154 0.145 0.134Yh 0.48 0.53 0.51 0.54 0.54 0.54

21 Ih 0.169 0.161 0.150 0.137Yh 0.48 0.56 0.51 0.57 0.55 0.58 0.58 0.58

26 Ih 0.184 0.176 0.166 0.153 0.140Yh 0.49 0.59 0.52 0.60 0.55 0.60 0.59 0.61 0.62 0.62

35 Ih 0.202 0.196 0.187 0.176 0.163 0.144Yh 0.49 0.62 0.53 0.63 0.56 0.64 0.60 0.64 0.62 0.65 0.66 0.66

55 Ih 0.227 0.222 0.215 0.206 0.196 0.178 0.148Yh 0.50 0.66 0.53 0.67 0.57 0.67 0.60 0.68 0.63 0.69 0.67 0.70 0.71 0.71

135 Ih 0.258 0.257 0.255 0.251 0.246 0.236 0.214 0.153Yh 0.51 0.70 0.54 0.71 0.58 0.72 0.62 0.72 0.65 0.73 0.68 0.74 0.72 0.75 0.76 0.76



!= 25◦, "= 20◦



12 Ih 0.130Yh 0.46 0.46

14 Ih 0.142 0.133Yh 0.47 0.49 0.49 0.49

17 Ih 0.156 0.147 0.136Yh 0.47 0.51 0.50 0.52 0.52 0.52

21 Ih 0.171 0.163 0.151 0.138Yh 0.48 0.54 0.50 0.54 0.53 0.55 0.55 0.55

26 Ih 0.186 0.178 0.167 0.154 0.141Yh 0.48 0.56 0.51 0.56 0.53 0.57 0.56 0.57 0.58 0.58

35 Ih 0.204 0.198 0.188 0.176 0.163 0.144Yh 0.49 0.58 0.51 0.59 0.54 0.59 0.56 0.60 0.58 0.60 0.61 0.61

55 Ih 0.228 0.223 0.216 0.207 0.196 0.178 0.147Yh 0.49 0.61 0.52 0.61 0.54 0.62 0.57 0.62 0.59 0.63 0.62 0.64 0.64 0.64

135 Ih 0.259 0.257 0.255 0.251 0.245 0.234 0.212 0.151Yh 0.50 0.64 0.53 0.64 0.55 0.65 0.58 0.66 0.60 0.66 0.62 0.67 0.65 0.68 0.68 0.68



!= 25◦, "= 30◦


Gear

Pinion

Pitch apex to backCrown to back

Pitch apex to crown

Crown

Pitchapex

Shaft angle

Uniform clearance

Root anglePitchangle

Frontangle

Dedendumangle

Facewidth

Faceangle

Backangle

Pitch diameter

Outside diameter

Bac

k co

ne d

ista

nce

Backcone

Figure 15.3 Terminology for bevel gears.

Bevel Gears


Figure 15.4 Bevel gears with curved teeth. (a) Spiral bevel gears; (b) Zerol®.

Bevel Gears


Figure 15.5 Forces acting on a bevel gear

Bevel Gear Forces

W =Wt

cos!=

T

rcos!

Wa =Wt tan!sin"

Wr =Wt tan!cos"


Design of Bevel Gears

!t =

2Tp

bwdp

pdKaKvKsKm

KxYbEnglish units

2Tp

bwdp

KaKvKsKm

mpKxYbSI units

!c =

√TpE

′

"bwd2pIbKaKvKmKsKx English units

Cp

√2TpE

′

"bwd2pIbKaKvKmKsKx SI units

Bending Stress:

Pitting Resistance:


Load Distribution Factor for Bevel Gears

Km ={Kmb+0.0036b2w English unitsKmb+5.6b2w SI units

where Kmb = 1.00 for both gear and pinion straddle mounted (bearings on both

sides of gear)= 1.10 for only one member straddle mounted= 1.25 for neither member straddle mounted


Outer transverse module, met

1.6 5 10 20 30 40 50

16 5 2.5 1.25 0.8 0.6 0.5

Outer transverse pitch, pd, in.-1

Siz

e fa

ctor,

Ks

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Face width, bw, mm

25 50 75 100 125

0 1.0 2.0 3.0 4.0 5.0

Face width, bw, in.

Siz

e fa

ctor,

Ks

1.00

0.75

0.50

0.25

(a) (b)

Ks = 0.5 for met < 1.6 (pd < 16 in.-1)

Ks = 0.4867 + 0.2133/pd

= 0.4867 + 0.008399 met

Ks = 0.5 for bw < 12.7 mm (0.5 in.)

Ks = 1.0 for bw > 114.3 mm (4.5 in.)

Ks = 0.00492 bw + 0.4375 (bw in mm)

= 0.125 bw + 0.4375 (bw in in.)

Figure 15.6 Size factor for bevel gears. (a) Size factor for bending stress; (b) size factor for contact stress or pitting resistance.

Size Factor for Bevel Gears


Figure 15.7 Geometry factor for straight bevel gears with pressure angle = 20° and shaft angle 90°. (a) Geometry factor for contact stress, Ib.

Straight Bevel Gear Geometry Factor for Contact Stess


13 15 20 25 30 35 40 45 50

60

70

80

90

100100

90

80

70

60

50

40

30

20

10

Num

ber

of

teet

h o

n g

ear

for

whic

h g

eom

etry

fac

tor

is d

esir

ed

Number of teeth in mate

0.16 0.20 0.24 0.28 0.32 0.36 0.40Geometry factor, Yb

Figure 15.7 Geometry factor for straight bevel gears with pressure angle = 20° and shaft angle 90°. (b) Geometry factor for bending, Yb.

Straight Bevel Gear Geometry Factor for Bending


Number of teeth in gear

0.04 0.06 0.08 0.10 0.12 0.14 0.16

50

40

30

20

10

Nu

mb

er o

f p

inio

n t

eeth

15

25

30

35

40

45

Geometry factor, Ib

50 60 70 80 90 100

20

Figure 15.8 Geometry factor for spiral bevel gears with pressure angle = 20°, spiral angle = 25°, and shaft angle 90°. (a) Geometry factor for contact stress, Ib.

Spiral Bevel Gears Geometry Factor for Contact Stress


100

90

80

70

60

50

40

30

20

10

Num

ber

of

teet

h o

n g

ear

for

whic

h

geo

met

ry f

acto

r is

des

ired

0.16 0.20 0.24 0.28 0.32 0.36

Geometry factor, Yb

0.12

Number of teeth in mate12 20 30 40 50

60708090

100

Figure 15.8 Geometry factor for spiral bevel gears with pressure angle = 20°, spiral angle = 25°, and shaft angle 90°. (a) Geometry factor for bending, Yb.

Spiral Bevel Gears Geometry Factor for Bending


Num

ber

of

pin

ion t

eeth

10

20

30

40

50

0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11Geometry factor, Ib

Number of gear teeth

15

20

35

45

25

30

40

50 60 70 80 90 100

Figure 15.9 Geometry factor for Zerol bevel gears with pressure angle = 20°, spiral angle = 25°, and shaft angle 90°. (a) Geometry factor for contact stress, Ib.

Zerol Bevel Gears Geometry Factor for Contact Stress


Nu

mb

er o

f te

eth

in

gea

r fo

r w

hic

h

geo

met

ry f

acto

r is

des

ired

100

90

80

70

60

50

40

30

20

100.16 0.20 0.24 0.28 0.32 0.36 0.40

Geometry factor, Yb

Number of teeth in mate13 15 20 25 30 35 40 45 50

60 70 80 90 100

Figure 15.9 Geometry factor for Zerol bevel gears with pressure angle = 20°, spiral angle = 25°, and shaft angle 90°. (a) Geometry factor for bending, Yb.

Zerol Bevel Gears Geometry Factor for Bending


Figure 15.10 Illustration of worm contact with a worm gear, showing multiple teeth in contact.

Worm Gear Contact


Pressure angle, φ Minimum number ofdeg wormgear teeth14.5 4017.5 2720 2122.5 1725 1427.5 1230 10

Table 15.3 Suggested minimum number of worm gear teeth for customary designs.

Minimum Number of Worm Gear Teeth


Figure 15.11 Forces acting on a worm. (a) Side view, showing forces acting on worm and worm gear. (b) Three-dimensional view of worm, showing worm forces. The worm gear has been removed for clarity.

Forces on a Worm Gear


AGMA Equations for Worm Gears

hpi =

NWtdg

(126,000)Z+

vtWf

33,000English units

!Wtdg

2Z+ vtWf SI units

vt =

!Ndwm

12cos"English units

#dwm

2cos"SI units

The rated input power is

where


AGMA Equations for Worm Gears (cont.)Tangential Force

Wt =

Csd0.8gmbwCmCv English units

Csd0.8gmbwCmCv

75.948SI units

Cm =

0.0200

(−Z2+40Z−76)0.5+0.463≤ Z < 20

0.0107(−Z2+56Z+5145

)0.520≤ Z < 76

1.1483−0.00658Z 76≤ Z

Cv =

0.659exp(−0.0011vt)0< vt ≤ 700 ft/min13.31v(−0.571)t 700 ft/min< vt ≤ 3000 ft/min65.52v(−0.774)t 3000 ft/min< vt

where


AGMA Equations for Worm Gears (cont.)Friction Force

where

Wf =µWt

cos!cos"n

µ=

0.150 vt = 0 ft/min

0.124exp(−0.074v0.645t

)0< vt ≤ 10 ft/min

0.103exp(−0.110v0.450t

)+0.01210 ft/min< vt


(b)

1 IO00

I i i i i 1 i i i i i i i i i i i i 3 i in

I I I I I I I I CALlbN I I I

A. .-..a. r*al m-r A ..IlT

800 1-i I i i i i i i Y

C;HtC;K I-ItiUHt 1 HIWJ

USE THE LOWER OF

/ THE TWO VALUES 3

0.5 1.0 C::TER DISTANCET:NCHES

2.5 3.0

12 20 30 40 50 60 70 75

Center distance, cd (in.)0.5 1.0 1.5 2.0 2.5 3.0

20 30 40 50 60 70 75

Center distance, cd (mm)

1000

900

800

700

Mat

eria

ls f

acto

r, C

s

(a)

800

600

í FACTOR FOR CENTER

t DISTANCES c 3.00 IN (76 mm) I I ! I Y I

500 1 í I I I I III I I I I I I I I

2.5 3 4 5 6 7 8910 15 20 25 30 40 50 60 70 1 MEAN GEAR DIAMETER, Dm -INCHES

60 90

800

600

í

t 500 1

2.5 5 20 30 40 60 MEAN GEAR DIAMETER, Dm -INCHES

90

Sand cast

Static chill cast or forged

Centrifugally cast

2.5 5 10 20 30 40 60 90Mean gear pitch diameter, d (in.)

500

600

700

800

900

1000M

ater

ials

fac

tor,

Cs

II 111 I I I I I I I Ill I I I 70 100 200 500 1000 2500

Mean gear pitch diameter, d (mm)

Figure 15.12 Materials parameter Cs for bronze worm gears and worm minimum surface hardness of 58 Rc. (a) Materials factor for center distances cd greater than 76 mm (3 in); (b) Materials factor for center distances cd less than 76 mm (3 in). When using the figure in (b), the value from part (a) should be checked and the lower value used. See also Table 15.4.

Materials Parameter for Worm Gears


Manufacturing Pitch Units for pitch diameterProcess diameter in. mm

Sand casting d ≤ 64 mm (2.5 in.) 1000 1000d ≥ 64 mm 1189.6365− 476.5454 log d 1859.104− 476.5454 log d

Static chill cast d ≤ 200 mm (8 in.) 1000 1000or forged d > 200 mm 1411.6518− 455.8259 log d 2052.012− 455.8259 log dCentrifugally cast d ≤ 625 mm (25 in.) 1000 1000

d > 625 mm 1251.2913− 179.7503 log d 1503.811− 179.7503 log d

Table 15.4 Materials factor Cs for bronze worm gears with worm having surface hardness of 58 Rc.

Materials Factor for Worm Gear


Tmax=20 ft-lbf

Maximumcurrent

Motor current, A

Speed Torque

Food Mixer Gear Train Case Study

Table 15.13 The gears used to transmit power from an electric motor to the agitators of a commercial mixer.

Table 15.14 Torque and speed of motor as a function of current for the industrial mixer used in the Case Study.


Chapter 16: Fasteners and Power Screws

Engineers need to be continually reminded that nearly all

engineering failures result from faulty judgments rather than

faulty calculations.

Eugene S. Ferguson, Engineering and the Mind’s Eye


Figure 16.1 Parameters used in defining terminology of thread profile.

Thread Geometry

Crest

ht

p

d

dc

dpdr

Root

β

(a) (b) (c)

p pβ

lα

l

p

l

Figure 16.2 (a) Single-, (b) double-, and (c) triple-threaded screws.


β

Pitch

diameter0.25 ht

0.125 ht

0.625 ht0.375 ht

0.25 p

0.5 p

0.5 p

0.125 p

29∞

(b)(a)

60∞

Acme, UN, and M Threads

Figure 16.3 Thread profiles. (a) Acme; (b) UN.

Figure 16.4 Details of M and UN thread profiles.


p

β = 29∞

dr

dpdc

p

2.7– 0.052

p

2.7

0.5p + 0.01

Inch series Metric seriesBolts Nuts Bolts Nuts

1A 1B 8g 7H2A 2B 6g 6H3A 3B 8h 5H

Equivalent Threads and Acme Profile

Table 16.1 Inch and metric equivalent thread classifications.

Figure 16.5 Details of Acme thread profile. (All dimensions are in inches.)


Crest Number ofdiameter, dc, threads Tensile stress Shear stress

in. per inch, n area, At, in.2 area, As, in.2

1/4 16 0.02632 0.33555/16 14 0.04438 0.43443/8 12 0.06589 0.52767/16 12 0.09720 0.63961/2 10 0.1225 0.72785/8 8 0.1955 0.91803/4 6 0.2732 1.0847/8 6 0.4003 1.3131 5 0.5175 1.493

1 1/8 5 0.6881 1.7221 1/4 5 0.8831 1.9521 3/8 4 1.030 2.1101 1/2 4 1.266 2.3411 3/4 4 1.811 2.803

2 4 2.454 3.2622 1/4 3 2.982 3.6102 1/2 3 3.802 4.0752 3/4 3 4.711 4.538

3 2 5.181 4.7573 1/2 2 7.338 5.700

4 2 9.985 6.6404 1/2 2 12.972 7.577

5 2 16.351 8.511

Table 16.2 Crest diameters, threads per inch, and stress areas for Acme threads.

Acme Threads


dp/2

rc

Equal

Thrustcollar

β/2

α

β/2

Load, W(Screw is threaded into W)

Pitch, p

Power Screws

Tr =W

[(dp/2)(cos!n tan"+µ)

cos!n−µtan" + rcµc

]

Tl =−W[(dp/2)(µ− cos!n tan")

cos!n+µtan"+ rcµc

]


D

B

H

0

E

C

H

W

Pα 0

A

(a)

(c)

(b)

αβ/2

AA

Pn cosθn cosα

Pn cosθn cosα tan(β/2)

Pn cosθn cosα

µPn cosα

µPn sinα µPn

Pn cosθn sinα

Pn cosθn

B

E 0A

xis

of

scre

w

C

dp/2

θnPn

α

µcW

β/2

Forces on Power Screw

Figure 16.7 Forces acting in raising load of power screw. (a) Forces acting on parallelepiped; (b) forces acting on axial section; (c) forces acting on tangential plane.


(a) (b) (c)

Figure 16.8 Three types of threaded fastener. (a) Bolt and nut; (b) cap screw; (c) stud.

Types of Threaded Fasteners


Illustration Type Description Application notesHex-head bolt An externally threaded fastener with a

trimmed hex head, often with a washerface on the bearing side.

Used in a variety of general purpose ap-plications in different grades dependingon the required loads and material be-ing joined.

Carriage bolt A round head bolt with a square neckunder the head and a standard thread.

Used in slots where the square neckkeeps the bolt from turning when be-ing tightened.

Elevator bolt A bolt with a wide, countersunk flathead, a shallow conical bearing surface,an integrally-formed square neck underthe head and a standard thread.

Used in belting and elevator applica-tions where head clearances must beminimal.

or belt bolt

Serrated flange bolt A hex bolt with integrated washer, butwider than standard washers and incor-porating serrations on the bearing sur-face side.

Used in applications where looseninghazard exists, such as vibration appli-cations. The serrations grip the surfaceso that more torque is needed to loosenthan tighten the bolt.

Flat cap screw A flat, countersunk screw with a flattop surface and conical bearing surface.

A common fastener for assemblingjoints where head clearance is critical.(slotted head shown)

Buttunhead cap screw Dome shaped head that is wider andhas a lower profile than a flat cap screw.

Designed for light fastening applica-tions where their appearance is desired.Not recommended for high-strength ap-plications.

(socket head shown)

Lag screw A screw with spaced threads, a hexhead, and a gimlet point. (Can alsobe made with a square head.)

Used to fasten metal to wood or withexpansion fittings in masonry.

Step bolt A plain, circular, oval head bolt witha square neck. The head diameter isabout three times the bolt diameter.

Used to join resilient materials or sheetmetal to supporting structures, or forjoining wood since the large head willnot pull through.

Table 16.3 Common types of bolts and screws.

Types of Bolts and Screws


Illustration Type Description Application notesNuts

Hex nut A six-sided internally threaded fas-tener. Specific dimensions are pre-scribed in industry standards.

The most commonly used general-purpose nut.

Nylon insert stop A nut with a hex profile and an integralnylon insert.

The nylon insert exerts friction on thethreads and prevents loosening due tovibration or corrosion.

Cap nut Similar to a hex nut with a dome top. Used to cover exposed, dangerous boltthreads or for aesthetic reasons.

Castle nut A type of slotted nut. Used for general purpose fastening andlocking. A cotter pin or wire can be in-serted through the slots and the drilledshank of the fastener.

Coupling nut A six-sided double chamfered nut. Used to join two externally threadedparts of equal thread diameter andpitch.

Hex jam nut A six-sided internally threaded fas-tener, thinner than a normal hex nut.

Used in combination with a hex nut tokeep the nut from loosening.

K-lock or keplock nut A hex nut preassembled with a freespinning external tool lock washer.When tightened, the teeth bite into themember to achieve locking.

A popular lock nut because of ease ofuse and low cost.

Wing nut An internally threaded nut with inte-gral pronounced flat tabs.

Used for applications where repetitivehand tightening is required.

Serrated nut A hex nut with integrated washer, butwider than standard washers and incor-porating serrations on the bearing sur-face side.

Used in applications where looseninghazard exists, such as vibration appli-cations. The serrations grip the surfaceso that more torque is needed to loosenthan tighten the bolt.

Table 16.4 Common nuts and washers for use with threaded fasteners.

Types of Nuts


WashersFlat washer A circular disk with circular hole,

produced in accordance with indus-try standards. Fender washers havelarger surface area than conventionalflat washer.

Designed for general-purpose mechani-cal and structural use.

Belleville washer A conical disk spring. Used to maintain load in bolted con-nections.

Split lock washer A coiled, hardened, split circularwasher with a slightly trapezoidalcross-section

Preferred for use with hardened bear-ing surfaces. Applies high bolt tensionper torque, resists loosening caused byvibration and corrosion.

Tooth lock washer A hardened circular washer withtwisted teeth or prongs.

Internal teeth are preferred fro aesth-stics since the teeth are hidden un-der the bolt head. External teeth givegreater locking efficiency. Combinationteeth are used for oversized or out-of-round holes or for electrical connec-tions.

Table 16.4 (cont.) Common nuts and washers for use with threaded fasteners.

Types of Washers


kb

kj

Pb (

ten

sio

n)

Pj (c

om

pre

ssio

n)

0b

0b

Pi

Pb Pj

0j

0j

b (extension)δ

Extension

(b)

(a)

j (contraction)δ

Figure 16.9 Bolt-and-nut assembly modeled as bolt-and-joint spring.

Figure 16.10 Force versus deflection of bolt and member. (a) Separated bolt and joint; (b) assembled bolt and joint.

Bolt and Nut Forces


0b 0j

Pj

ek

Pb

Pi

Lo

ad

Deflection

(extension of bolt = reduction in contraction of joint)

Pi + kbek

Pi – kjek

P = increase in Pb plus decrease in Pj

Figure 16.11 Forces versus deflection of bolt and joint when external load is applied.

Force vs. Deflection


Ls

Lt

dc

dr

Lse = Ls + 0.4dc

Lte = Lt + 0.4dr

Bolt Stiffness

Figure 16.12 Bolt and nut. (a) Assembled; (b) stepped-shaft representation of shank and threaded section.

1

kb=4

!E

(Ls+0.4dc

d2c

+Lt +0.4dr

d2r

)


αf

dc

dw

L

Figure 16.13 Bolt and nut assembly with conical frustum stress representation of joint.

Member Stiffness as Conical Frustum


Decription Member stiffness, km

Single member, general case kji =πEjdc tan αf

ln

[(2Li tan αf + di − dc)(di + dc)

(2Li tan αf + di + dc)(di − dc)

]Single member, αf = 30◦ kji =

1.813Ejdc

ln

[(1.15Li + di − dc)(di + dc)

(1.15Li + di + dc)(di − dc)

]Two members, same Young’s modulus,E, back-to-back frustaa

kj =πEjdc tan αf

2 ln

[(2Li tan αf + di − dc)(di + dc)

(2Li tan αf + di + dc)(di − dc)

]Two members, same Young’s modulus,back-to-back frusta, α = 30◦, di =dw = 1.5dc

a

kj =1.813Ejdc

2 ln

[(2.885Li + 2.5dc)

(0.577Li + 2.5dc)

]Two members, same material, Wilemanmethoda,b

kj = EidcAieBidc/Li

a Note that this is stiffness for the complete joint, not a member in the joint.b See Table 16.6 for values of Ai and Bi for various materials.

Table 16.5 Member stiffness equations for common bolted joint configurations.

Member Stiffness Equations


Poisson’s ratio, Modulus of elasticity, Numerical constants,Material ν E, GPa Ai Bi

Steel 0.291 206.8 0.78715 0.62873Aluminum 0.334 71.0 0.79670 0.63816Copper 0.326 118.6 0.79568 0.63553Gray cast iron 0.211 100.0 0.77871 0.61616

Wileman Member Stiffness

Table 16.6 Constants used in joint stiffness formula.

k j = EidcAieBjdc/L


8 10 10

(a) (b)

12.5

25

15

3/2 dc

dc

d2

30∞1

23

Figure 16.14 Hexagonal bolt-and-nut assembly used in Example 16.6. (a) Assembly and dimensions; (b) dimensions of frustrun cone.

Example 16.6


SAE Head Range of crest Ultimate tensile strength Yield strength, Proof strength,grade marking diameters, in. Su, ksi Sy , ksi Sp, ksi

1 1/4 – 1 1/2 60 36 33

2 1/4 – 3/4 74 57 55> 3/4 – 1 1/2 60 36 33

4 1/4 – 1 1/2 115 100 65

5 1/4 – 1 120 92 85>1 - 1 1/2 105 81 74

5.2 1/4-1 120 92 85

7 1/4 – 1 1/2 133 115 105

8 1/4 – 1 1/2 150 130 120

8.2 1/4 -1 150 130 120

Strength of Steel Bolts

Table 16.7 Strength of steel bolts for various sizes in inches.


Metric Head Crest diameter, Ultimate tensile Yield strength, Proof strength,grade marking dc, mm strength, Su, MPa Sy , MPa Sp, MPa

4.6 M5 – M36 400 240 2254.6

4.8 M1.6 – M16 420 340a 3104.8

5.8 M5 – M24 520 415a 3805.8

8.8 M17 – M36 830 660 6008.8

9.8 M1.6 – M16 900 720a 6509.8

10.9 M6 – M36 1040 940 83010.9

12.9 M1.6 – M36 1220 1100 97012.9

Strength of Steel Bolts

Table 16.8 Strength of steel bolts for various sizes in millimeters.


Coarse threads (UNC) Fine threads (UNF)Number of Root Tensile stress Number of Root Tensile stress

Crest diameter, threads per, diameter, area, At, threads per, diameter, area, At,dc, in. inch, n dr, in. in.2 inch, n dr, in. in.2

0.0600 — — — 80 0.04647 0.001800.0730 64 0.05609 0.00263 72 0.05796 0.002780.0860 56 0.06667 0.00370 64 0.06909 0.003940.0990 48 0.07645 0.00487 56 0.07967 0.005230.1120 40 0.08494 0.00604 48 0.08945 0.006610.1250 40 0.09794 0.00796 44 0.1004 0.008300.1380 32 0.1042 0.00909 40 0.1109 0.010150.1640 32 0.1302 0.0140 36 0.1339 0.014740.1900 24 0.1449 0.0175 32 0.1562 0.02000.2160 24 0.1709 0.0242 28 0.1773 0.02580.2500 20 0.1959 0.0318 28 0.2113 0.03640.3125 18 0.2523 0.0524 24 0.2674 0.05800.3750 16 0.3073 0.0775 24 0.3299 0.08780.4750 14 0.3962 0.1063 20 0.4194 0.11870.5000 13 0.4167 0.1419 20 0.4459 0.15990.5625 12 0.4723 0.182 18 0.5023 0.2030.6250 11 0.5266 0.226 18 0.5648 0.2560.7500 10 0.6417 0.334 16 0.6823 0.3730.8750 9 0.7547 0.462 14 0.7977 0.5091.000 8 0.8647 0.606 12 0.9098 0.6631.125 7 0.9703 0.763 12 1.035 0.8561.250 7 1.095 0.969 12 1.160 1.0731.375 6 1.195 1.155 12 1.285 1.3150.500 6 1.320 1.405 12 1.140 1.5811.750 5 1.533 1.90 — — —2.000 4.5 1.759 2.5 — — —

Table 16.9 Dimensions and tensile stress areas for UN coarse and fine threads. Root diameter is calculated from Eq. (16.2) and Fig. 16.4.

UN Coarse and Fine Threads


Coarse threads (MC) Fine threads (MF)Crest Root Tensile Root Tensile

diameter, Pitch, diameter, stress area, Pitch, diameter, stress area,dc, mm p, mm dr, mm At, mm2 p, mm dr, mm At, mm2

1 0.25 0.7294 0.460 — — —1.6 0.35 1.221 1.27 0.20 1.383 1.572 0.4 1.567 2.07 0.25 1.729 2.45

2.5 0.45 2.013 3.39 0.35 2.121 3.703 0.5 2.459 5.03 0.35 2.621 5.614 0.7 3.242 8.78 0.5 3.459 9.795 0.8 4.134 14.2 0.5 4.459 16.16 1.0 4.917 20.1 0.75 5.188 228 1.25 6.647 36.6 1.0 6.917 39.210 1.5 8.376 58.0 1.25 8.647 61.212 1.75 10.11 84.3 1.25 10.65 92.116 2.0 13.83 157 1.5 14.38 16720 2.5 17.29 245 1.5 18.38 27224 3.0 20.75 353 2.0 21.83 38430 3.5 26.21 561 2.0 27.83 62136 4.0 31.67 817 3.0 32.75 86542 4.5 37.13 1121 — — —48 5.0 42.59 1473 — — —

Table 16.10 Dimensions and tensile stress areas for M coarse and fine threads. Root diameter is calculated from Eq. (16.2) and Fig. 16.4.

M Coarse and Fine Threads


Pj

PjFigure 16.15 Separation of joint.

Separation of Joint


Pb

0b 0j

Pj, min

Deflection ∆δ

PiPi

Pj

Lo

ad o

n b

olt

Lo

ad o

n j

oin

t

Pb, max

SAE grade Metric grade Rolled threads Cut threads Fillet0-2 3.6-5.8 2.2 2.8 2.14-8 6.6-10.9 3.0 3.8 2.3

Figure 16.16 Forces versus deflection of bolt and joint as function of time.

Table 16.11 Fatigue stress concentration factors for threaded elements.

Threads in Fatigue Loading


Gasket

Gaskets

Figure 16.17 Threaded fastener with unconfined gasket and two other members.


(a) (b) (c) (d)

Failure Modes for Fasteners in Shear

Figure 16.18 Failure modes due to shear loading of riveted fasteners. (a) Bending of member; (b) shear of rivet; (c) tensile failure of member; (d) bearing of member on rivet.


A

2

2

7 8

P = 1000 lb

B

C

2.365

4.635

α

β3

3

Dx

y

A

rA

rC rD

rC

rD

rB

B

C

(a) (b)

(c) (d) (e)

D

3

3

5

5

3

3

A

τtC

τtD

τtA

τtB

τd τd

τd τd

B

C D

7–8

1–2

7–8

Figure 16.19 Group of riveted fasteners used in Example 16.9. (a) Assembly of rivet group; (b) radii from centroid to center of rivets; (c) resulting triangles; (d) direct and torsional shear acting on each rivet; (e) side view of member. (All dimensions in inches.)

Example 16.9


Bead FilletPlug

orslot Square V Bevel U J

Groove

Basic arc and gas weld symbols

Spot Projection SeamFlash

orupset

Basic resistance weld symbols

Finish symbol

Contour symbol

Root opening, depthof filling for plugand slot welds

Effective throat

Depth of preparationor size in inches

Reference line

Specification, processor other reference

Tail (omitted whenreference is not used)

Basic weld symbolor detail reference

T

S(E)

(Both

s

ides

)

(Arr

ow

si

de)

(Oth

er si

de)

L @ P

Arrow connects reference line to arrow sideof joint. Use break as at A or B to signifythat arrow is pointing to the grooved memberin bevel or J-grooved joints.

A B

Weld-all-around symbol

Field weld symbol

Pitch (center-to-center spacing)of welds in inches

Length of weld in inches

Groove angle or includedangle of countersinkfor plug welds

F

A

R

Figure 16.20 Basic weld symbols.

Weld Symbols


Shear planes

te te

Shear stress

Shear stress

Load

in.

Actual weld configuration

Assumed weld configuration

Shear plane of weld at throat

(a)

(b)

in. clear for plates in. thick

LL

1—16

1—16

1–4

te

he

he

Figure 16.21 Fillet weld. (a) Cross-section of weld showing throat and legs; (b) shear planes.

Fillet Weld


Weld

BendingDimensions

of weld

Torsion

Iu = d2/6

M = Pa

A = d

P

x

xxd

x

a

Weld

Ju = d3/12

c = d/2

T = Pa

P

a

Weld

Iu = d2/3

A = 2d

P

x

xxd

x

a

b

Weld

Ju =

P

P a d(3b2 + d2)

–––––––––6

Weld

Iu = bdA = 2b

P

x

xxd

x

a

b

Weld

Ju =

P

P a b3 + 3bd

2

––––––––6

Table 16.12 Geometry of welds and parameters used when considering various types of loading.

Geometry of Welds


Weld

At top

At bottom

Ju =

Pa

4bd + d2

–––––––6

(b + d)4 – 6b2d2

–––––––––––––12(b + d)

b2(b + d)2

–––––––––(2 b + d)

Iu =d2(4b + d)––––––––6(2b + d)

b2

––––––2(b + d)

A = b + d

x

y–

y =–

x =–

x–

xd

b

Weld

Iu =d2

––––––2(b + d)

P

x

xa

Weld

Weld

Ju = –

Pa

(2b + d)3

––––––12

Iu = bd + d2/6b2

––––––2(b + d)

A = d + 2b

x

x =–

x–

xd

b

Weld

Weld

P

x

x

x

x

a

x

x

Table 16.12 (cont.) Geometry of welds and parameters used when considering various types of loading.

Geometry of Welds


At top

At bottom

2bd + d2

–––––––3

Iu =d2(2b + d)––––––––3(b + d)

A = b + 2d

A = 2b + 2d

x

y–

y =–

xd

b

Weld

Weld WeldIu =

d2

––––––(b + 2d)

P

x

x

aWeld

x

x

P

P a

d2(b + d)2

–––––––––(b + 2d)

Ju = –(b + 2d)3

––––––12

Weld allaround

Iu = bd + d2/3

P

x

xxd

xab

Ju =

P

Pa

(b + d)3

––––––––6

Weld allaround

BendingDimensions

of weld

Torsion


Geometry of Welds


Weld

Weld all

around

Weld

x

x

A = 2b + 2d

A = πb

Weld

Weld

Iu = bd + d2/3

Iu = π(d2/4)

Ju = π(d3/4)

P

xx

xx

d

d

ab

Ju =

P

P a b3 + 3bd

2+ d3

––––––––——6

x

x

x

x P

a

x

x P

P a

Weld all

around


Geometry of Welds


Ultimate tensile Yield strength, Elongation, ek,Electrode number strength, Su, ksi Sy, ksi percent

E60XX 62 50 17-25E70XX 70 57 22E80XX 80 67 19E90XX 90 77 14-17E100XX 100 87 13-16E120XX 120 107 14

Electrode Properties

Table 16.13 Minimum strength properties of electrode classes.

Geometry of Welds


(b)

150

45

τtxτty

τty

τtx

A

B

80

100

y

x

(a)

300

20 kN

l1

x

y

B

Al2

Figure 16.22 Welded bracket used in Example 16.10. (a) Dimensions, load and coordinates; (b) torsional shear stress components at points A and B. (All dimensions in millemeters.)

Example 16.10


Fatigue stress concentrationType of weld factor, Kf

Reinforced butt weld 1.2Tow of transverse fillet weld 1.5End of parallel fillet weld 2.7T-butt joint with sharp corners 2.0

Table 16.14 Fatigue strength reduction factors for welds.

Welds in Fatigue


b

L

(b)

(c)

(d)

(a)

Figure 16.23 Four methods of applying adhesive bonding. (a) Lap; (b) butt; (c) scarf; (d) double lap.

Adhesive Joints


P

M M

T

x

l

rori

y

T

O

A

A

θP

(a)

(b)

(c)

tm

tm

θτ

σx

σn

θ

Figure 16.24 Scarf joint. (a) Axial loading; (b) bending; (c) torsion.

Scarf Joint


DeflectedRigid

(b)

(c)

h

(a)

h

2

Integrated Snap Fasteners

Figure 16.25 Common examples of integrated fasteners. (a) Module with four cantilever lugs; (b) cover with two cantilever and two rigid lugs; (c) separable snap joint for chassis cover.


l

h

h_2

yP

α

l

h

P

y

y

z

h_2

b_4

ch

b

c

c2h

b

a

c1

Shape of cross section

(Per

mis

sib

le)

def

lect

ion

Type of design

y = 0.67εl2___h

εl2___h

y = 0.86εl2___h

y = 1.09εl2___h

Cross section constantover length

All dimensions indirection y (e.g., h)decrease to one-half

All dimensions indirection z (e.g., b and a) decrease to one-quarter

a + b(1)_______2a + b

y =

εl2___h

a + b(1)_______2a + b

y = 1.64

εl2___h

a + b(1)_______2a + b

y = 1.28

A

Rectangle

B

Trapezoid

1

2

3

h

b

Snap Joint Design

Figure 16.26 Cantilever snap joint.

Figure 16.27 Permissible deflection of different snap fastener cantilever shapes.


Coefficient of frictionOn self-mated

Material On steel polymerPolytetrafluoroethylene PTFE (teflon) 0.12-0.22 —Polyethylene (rigid) 0.20-0.25 0.40-0.50Polyethylene (flexible) 0.55-0.60 0.66-0.72Polypropylene 0.25-0.30 0.38-0.45Polymethylmethacrylate (PMMA) 0.50-0.60 0.60-0.72Acrylonitrile-butadiene-styrene (ABS) 0.50-0.65 0.60-0.78Polyvinylchloride (PVC) 0.55-0.60 0.55-0.60Polystyrene 0.40-0.50 0.48-0.60Polycarbonate 0.45-0.55 0.54-0.66

Friction for Integrated Snap Fastener Mateirals

Table 16.15 Coefficients of friction for common snap fastener polymers


Cylinderflange

Seal

End cap

Hydraulic Baler Case Study

Figure 16.28 End cap and cylinder flange.


Chapter 17: Springs

Entia non multiplicantor sunt prater necessitatum.

(Do not complicate matters more than necessary.)

Galileo Gallilei


Figure 17.1 Stress-strain curve for one loading cycle.

Stress Cycle

Strain

∆UUS

tress


Shear MaximumModulus of modulus of serviceelasticity, E, elasticity, G Density, ρ, temperature, Principal

Common name Specification psi psi lb/in.3 ◦F characteristicsHigh-carbon steelsMusic wire ASTM A228 30× 106 11.5× 106 0.283 250 High strength; excellent

fatigue lifeHard drawn ASTM A227 20× 106 11.5× 106 0.283 250 General purpose use; poor

fatigue lifeStainless steelsMartensitic AISI 410, 420 29× 106 11× 106 0.280 500 Unsatisfactory for subzero

applicationsAustenitic AISI 301, 302 28× 106 10× 106 0.282 600 Good strength at moderate

temperatures; low stressrelaxation

Copper-based alloysSpring brass ASTM B134 16× 106 6× 106 0.308 200 Low cost; high conductivity;

poor mechanical propertiesPhosphor bronze ASTM B159 15× 106 6.3× 106 0.320 200 Ability to withstand repeated

flexures; popular alloyBeryllium copper ASTM B197 19× 106 6.5× 106 0.297 400 High elastic and fatigue

strength; hardenableNickel-based alloysInconel 600 — 31× 106 11× 106 0.307 600 Good strength; high

corrosion resistanceInconel X-750 — 31× 106 11× 106 0.298 1100 Precipitation hardening; for

high temperaturesNi-Span C — 27× 106 9.6× 106 0.294 200 Constant modulus over a

wide temperature range

Spring Materials

Table 17.1 Typical properties of common spring materials.


Size range Exponent, Constant, Ap

Material in. mm m ksi MPaMusic wirea 0.004-0.250 0.10-6.5 0.146 196 2170Oil-tempered wireb 0.020-0.500 0.50-12 0.186 149 1880Hard-drawn wirec 0.028-0.500 0.70-12 0.192 136 1750Chromium vanadiumd 0.032-0.437 0.80-12 0.167 169 2000Chromium siliconee 0.063-0.375 1.6-10 0.112 202 2000

Strength of Spring Materials

Sut =Ap

dm

Table 17.2 Coefficients used in Equation (17.2) for five spring materials.


P

P

P

P

PP

P

P

R

R

D

d

l = 2πRN

(b)

(a)

(c)

R

T = PR

R

Helical Coil

Figure 17.2 Helical coil. (a) Straight wire before coiling; (b) coiled wire showing transverse (or direct) shear; (c) coiled wire showing torsional shear.


Spring axis

(c)

(d)

(a)

(b)

Spring axis

d d

d

D/2

D/2

d

Figure 17.3 Shear stresses acting on wire and coil. (a) Pure torsional loading; (b) transverse loading; (c) torsional and transverse loading with no curvature effects; (d) torsional and transverse loading with curvature effects.

Shear Stresses on Wire and Coil


(a) (b) (c) (d)

Figure 17.4 Four end types commonly used in compression springs. (a) Plain; (b) plain and ground; (c) squared; (d) squared and ground.

Compression Spring End Types


Type of spring endTerm Plain Plain and ground Squared or closed Squared and groundNumber of end coils, Ne 0 1 2 2Total number of coils, Nt Na Na + 1 Na + 2 Na + 2Free length, lf pNa + d p(Na + 1) pNa + 3d pNa + 2dSolid length, ls d(Nt + 1) dNt d(Nt + 1) dNt

Pitch, p (lf − d)/Na lf /(Na + 1) (lf − 3d)/Na (lf − 2d)/Na

Table 17.3 Useful formulas for compression springs with four end conditions.

Compression Spring Formulas


(a) (b) (c) (d)

(P = 0)

Pr

Po

ga

Ps

lf

li

lo ls

Figure 17.5 Various lengths and forces applicable to helical compression springs. (a) Unloaded; (b) under initial load; (c) under solid load.

Lengths and Forces in Helical Springs


Ps 0

Po

Pi

δi δo δs

lsLength, l

00

Deflection, δ

Spri

ng f

orc

e, P

lo

li

lf

Figure 17.6 Graphical representation of deflection, force and length for four spring positions.

Force vs. Deflection


Rat

io o

f def

lect

ion t

o f

ree

length

, δ/l f

Ratio of free length to mean coil diameter, lf /D

3 4 5 6

Nonparallel ends

Parallel ends

Stable Unstable

7 8 9 10

0.20

0

0.40

0.60

0.80

Stable Unstable

Figure 17.7 Critical buckling conditions for parallel and nonparallel ends of compression springs.

Buckling Conditions


(a) (b)

d A

P

r3

r1

d

B

P

r2r4

(d)(c)

d

B

P

r2r4

P

Ar3

r1

d

Figure 17.8 Ends for extension springs. (a) Conventional design; (b) Side view of Fig. 17.8(a); (c) improved design; (d) side view of Fig. 17.8(c).

Extension Spring Ends


do

di

ll

lb

lf

lh

ga

Pre

load

str

ess,

MP

a

Pre

load

str

ess,

ksi

Spring index

Preferred range

4 6 8 10 12 14 16

30

28

26

24

22

20

18

16

14

12

10

8

6

4

200

175

150

125

100

75

50

25

Helical Extension Springs

Figure 17.9 Dimensions of helical extension spring.

Figure 17.10 Preferred range of preload stress for various spring indexes.


D

P

a

d

P

Figure 17.11 Helical torsion spring.

Helical Torsion Spring


nb

P

P

b

b

2b

2

b

2b

2

(a)

(b)

lx

x

tl

t

b

P

P

Figure 17.12 Leaf spring. (a) Triangular plate, cantilever spring; (b) equivalent multiple-leaf spring.

Leaf Spring


Di

h

t

Do

Per

cent

forc

e to

fla

t

Percent deflection to flat

0 20 40 60 80 100

Height-to-thickness ratio, 2.828

2.275

1.414

0.400

1.000

120 140 160 180 2000

40

80

120

160

200

Figure 17.13 Typical Belleville Spring.

Figure 17.14 Force-deflection response of Belleville spring.

Belleville Springs


(b)(a)

Stacking of Belleville Springs

Figure 17.15 Stacking of Belleville springs. (a) in parallel; (b) in series.


CamGripping unit(sliding)

S-ringFixed rear guide

Gripping unit(fixed)

Max

imum

forc

e, P

max

, lb

f

Wire diameter, d, in.

0.04 0.08 0.12 0.16 0.20

Wire diameter, d, in.

0.04

0

0 0.08 0.12 0.16 0.20

600

Saf

ety f

acto

r, ns

500

400

300

200

100

1.4

1.2

1.0

0.8

0.6

0.4

0.2

Dickerman Feed Case Study

Figure 17.16 Dickerman feed unit.

Figure 17.17 Performance of the spring in the case study.


Chapter 18: Brakes and Clutches

Nothing has such power to broaden the mind as the ability to

investigate systematically and truly all that comes under thy

observation in life.

Marcus Aurelius, Roman Emperor


Figure 18.1 Five types of brake and clutch. (a) internal, expanding rim type; (b) external contracting rim type; (c) band brake; (d) thrust disk; (e) cone disk.

Brake and Clutch Types

P

P

(a)

(b)

(c)

(d)

P

PP

P

P

P

P

(e)

P

P


ro

ri

r

drθ

0.6

0.5

0.4

0.3

0.20 0.2

Tp––

––Tw

0.4

Radius ratio, β = ri /ro

0.6 0.8 1.0D

imen

sionle

ss torq

ue,

T =

T/2

µPr o

––

Figure 18.2 Thrust disk clutch surface with various radii.

Figure 18.3 Effect of radius ratio on dimensionless torque for uniform pressure and uniform wear models.

Thrust Disk


Maximum contact Maximum bulkCoefficient of pressure,b pmax temperature, tm,max

Friction materiala friction, µ psi kPa ◦F ◦CMolded 0.25-0.45 150-300 1030-2070 400-500 204-260Woven 0.25-0.45 50-100 345-690 400-500 204-260Sintered metal 0.15-0.45 150-300 1030-2070 400-1250 204-677Cork 0.30-0.50 8-14 55-95 180 82Wood 0.20-0.30 50-90 345-620 200 93Cast iron; hard steel 0.15-0.25 100-250 690-1720 500 260

Properties of Common Friction Materials

Table 18.1 Representative properties of contacting materials operating dry.


Friction materiala Coefficient of friction, µMolded 0.06-0.09Woven 0.08-0.10Sintered metal 0.05-0.08Paper 0.10-0.14Graphitic 0.12 (avg.)Polymeric 0.11 (avg.)Cork 0.15-0.25Wood 0.12-0.16Cast iron; hard steel 0.03-0.16a When rubbing against smooth cas iron or seel.

Table 18.2 Coefficient of friction for contacting materials operating in oil.

Friction for Wet Clutch Materials


b

dD

r

drα

dA

dw

dP

rdθ

dr——sin α

dθ θ

Figure 18.4 Forces acting on elements of cone clutch.

Cone Clutch


d3

µP

ω

Pd2

d1D

C

B

W

d4

r

Short-Shoe Brake

Figure 18.5 Block, or short-shoe brake, with two configurations.


1.5 in.

14 in.

14 in.

36 in.

WP

Example 18.3

Figure 18.6 Short-shoe brake used in Example 18.3.


Rotation

Drum

W W

d5

Lining

d5

d6

θ1

θ2

θd7

rA

Figure 18.7 Long-shoe, internal, expanding rim brake with two shoes.

Long-Shoe, Internal Expanding Rim Brake


θ1

θ

θ2

d6

Rx

Wx

y

x

Wy

W

A

Ryr

Rotation

dP cos θ

µdPµdP cos θ

r – d7 cos θ

d7 sin θ

µdP sin θ

dP dP sin θ

θθ

d7

Figure 18.8 Forces and dimensions of long-shoe, internal, expanding rim brake.

Forces and Dimensions


Example 18.4

Figure 18.9 Four-long-shoe, internal expanding rim brakes used in Example 18.4.

15° 15°

W W

W W

15°

ω

15°

10°

10°

10°

10°

y

x

a a

r

A B

b

bd d

d d


θ1

θ

θ2

Rx

Wx

y

x

WyW

A

Ry

r

Rotation

µdP

dPdP sin θ

θ

d7

d6

θ

dP cos θ

µdP sin θ

µdP cos θ

Long-Shoe, External, Contracting Rim Brake

Figure 18.10 Forces and dimensions of long-shoe, external, contracting rim brake.


x

y

θ

θ1

θ2

Rotation

Rx

Ry

dP

dP

sin

θdP cos θ

r cos θ

d7 cos θ – r

µdP

µdP sin θ

µdP cos θ

r

d7

Pivot-Shoe Brake

Figure 18.11 Symmetrically loaded pivot-shoe brake.


θφ

dθ

Drum rotation

F1 F2

(b)(a)

0

µdP

dPF + dF

F

0

r

dθ

dθ—2

F sindθ—2

(F + dF) sindθ—2

(F + dF) cosdθ—2

F cosdθ—2

r dθ

dθ—2

Figure 18.12 Band brake. (a) Forces acting on band; (b) forces acting on element.

Band Brake


Rotation

φ

d9

F1

F2

d10

W

Cutting plane forfree-body diagram

d8

Example 18.7

Figure 18.13 Band brake used in Example 18.7.


puOperating condition (kPa)(m/s) (psi)(ft/min)Continuous: poor heat dissipation 1050 30,000Occasional: poor heat dissipation 2100 60,000Continuous: good heat dissipation as in oil bath 3000 85,000

Capacities of Brake and Clutch Materials

Table 18.3 Product of contact pressure and sliding velocity for brakes and clutches.


30°

30°17.5 in.

18 in.

W

Hydraulic Crane Brake Case Study

Figure 18.14 Hoist line brake for mobile hydraulic crane. The cross-section of brake with relevant dimensions is given.


Chapter 19: Flexible Machine Elements

Scientists study the world as it is; engineers create the world that

has never been.

Theodore von Karmen


Figure 19.1 Dimensions, angles of contact, and center distance of open flat belt.

Flat Belt

D1

D2

01 02

α

α

α

α

α

φ1φ2

A

B

D

cd


Pivot

Idler

Weight

Tightside

Figure 19.2 Weighted idler used to maintain desired belt tension.

Weighted Idler


Circularpitch

Tooth includedangle Backing

Facing

Pulley face radius

Pulley pitch radius

Neoprene-encasedtension member

Neoprenetooth cord

A

A

Section A–A

Figure 19.3 Synchronous, or timing, belt.

Synchronous Belt


ht

wt

dN/2––––sin β

dN––––

2

2β≈36∞

V-Belt in Groove

Figure 19.4 V-belt in sheave groove.


Driven unit Overload factorAgitatorsLiquid 1.2Semisolid 1.4CompressorCentrifugal 1.2Reciprocating 1.4Conveyors and elevatorsPackage and oven 1.2Belt 1.4Fans and blowersCentrifugal, calculating 1.2Exhausters 1.4Food machinerySlicers 1.2Grinders and mixers 1.4GeneratorsFarm lighting and exciters 1.2Heating and VentilatingFans and oil burners 1.2Stokers 1.4Laundry machineryDryers and ironers 1.2Washers 1.4Machine toolsHome workshop and woodworking 1.4PumpsCentrifugal 1.2Reciprocating 1.4RefrigerationCentrifugal 1.2Reciprocating 1.4Worm gear speed reducers, input side 1.0

Table 19.1 Overload service factors f1 for various types of driven unit.

Overload Service Factors


Belt Size of Minimum pitch diameter, in.type belt, in. Recommended Absolute2L 1

4 × 18 1.0 1.0

3L 38 × 7

32 1.5 1.5

4L 12 × 5

16 2.5 1.8

Motor speed, rpmMotor 575 695 870 1160 1750

horsepower, hp Recommended pulley diameter, in.0.50 2.50 2.50 2.50 — —0.75 3.00 2.50 2.50 2.50 —1.00 3.00 3.00 2.50 2.50 2.25

Table 19.2 Recommended minimum pitch diameters of pulley for three belt sizes.

Table 19.3 Recommended pulley dimensions in inches for three electric motor sizes.

Pulley Design Considerations


Loss in arc of Correctioncontact, deg. factor

0 1.005 0.9910 0.9815 0.9620 0.9525 0.9330 0.9235 0.8940 0.8945 0.8750 0.8655 0.8460 0.8365 0.8170 0.7975 0.7680 0.7485 0.7190 0.69

Table 19.4 Arc correction factor for various angle of loss in arc of contact.

Arc Correction Factor


Speed of Pulley effective outside diameter, in.faster shaft 1.00 2.00 3.00 4.00 5.00 6.00

rpm Rated horsepower, hp500 0.04 0.05 0.06 0.08 0.10 0.121000 0.05 0.08 0.12 0.16 0.19 0.211500 0.06 0.12 0.17 0.22 0.25 0.302000 0.08 0.15 0.23 0.28 0.32 0.392500 0.10 0.18 0.27 0.34 0.39 0.44

3000 0.12 0.21 0.31 0.40 0.31 0.473500 0.14 0.24 0.35 0.44 0.444000 0.16 0.28 0.38 0.464500 0.18 0.31 0.425000 0.20 0.35

(a)

Table 19.5 Power ratings for light-duty V-belts. (a) 2L section with wt = 1/4 in and ht=1/8 in.

Power Ratings for V-Belts


Speed of Pulley effective outside diameter, in.faster shaft 1.50 1.75 2.00 2.25 2.50 2.7 3.00

rpm Rated horsepower, hp500 0.04 0.07 0.09 0.12 0.14 0.17 0.191000 0.07 0.12 0.16 0.21 0.25 0.30 0.341500 0.09 0.15 0.22 0.29 0.35 0.41 0.472000 0.10 0.19 0.27 0.35 0.43 0.51 0.592500 0.11 0.21 0.31 0.41 0.51 0.60 0.69

3000 0.11 0.23 0.35 0.45 0.57 0.68 0.783500 0.11 0.25 0.38 0.50 0.62 0.74 0.844000 0.11 0.26 0.40 0.54 0.66 0.78 0.884500 0.10 0.25 0.42 0.56 0.68 0.80 0.905000 0.09 0.26 0.42 0.57 0.69 0.80 0.89

(b)

Table 19.5 Power ratings for light-duty V-belts. (b) 3L section with wt = 3/8 in and ht=1/4 in.

Power Ratings for V-Belts (cont.)


Speed of Pulley effective outside diameter, in.faster shaft 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00

rpm Rated horsepower, hp500 0.08 0.14 0.19 0.24 0.29 0.34 0.39 0.44 0.491000 0.11 0.21 0.31 0.41 0.50 0.60 0.69 0.78 0.871500 0.12 0.26 0.40 0.54 0.67 0.81 0.94 1.07 1.202000 0.11 0.30 0.47 0.65 0.82 0.99 1.15 1.31 1.472500 0.09 0.31 0.53 0.73 0.94 1.13 1.32 1.51 1.69

3000 0.06 0.31 0.56 0.79 1.02 1.24 1.45 1.65 1.843500 0.02 0.30 0.57 0.83 1.07 1.31 1.53 1.73 1.924000 — 0.27 0.56 0.83 1.09 1.33 1.55 1.75 1.924500 — 0.22 0.54 0.81 1.07 1.30 1.51 1.69 1.845000 — 0.15 0.47 0.75 1.01 1.23 1.41 1.66 1.65

(c)

Table 19.5 Power ratings for light-duty V-belts. (c) 4L section with wt = 1/2 in and ht=9/32 in.

Power Ratings for V-Belts (cont.)


Pulley combination Nominal center distanceDriver Driven Short center Medium centerpitch pitch Center Center

diameter, diameter, Belt distance, Belt distance,in. in. number in. number in.2.0 2.0 3L200 6.4 3L250 9.43.0 3.0 3L250 7.4 3L310 10.42.0 2.5 3L210 6.6 3L270 9.62.0 3.0 3L220 6.7 3L280 9.73.0 4.5 3L290 8.2 3L350 11.2

2.0 3.5 3L240 7.3 3L300 10.32.0 4.0 3L250 7.2 3L310 10.32.25 4.5 3L270 7.7 3L330 10.72.5 5.0 3L290 8.1 3L350 11.13.0 6.0 3L330 8.9 3L390 11.9

2.0 5.0 3L250 8.0 3L340 11.02.0 6.0 3L310 8.6 3L370 11.63.0 9.0 3L410 10.3 3L470 13.42.0 7.0 3L340 9.2 3L400 12.22.0 9.0 3L390 9.9 3L450 13.0

2.0 10.0 3L420 10.4 3L480 13.61.5 9.0 3L390 10.1 3L450 13.3

(a)

Center Distance for V-Belts

Table 19.6 Center distances for various pitch diameters of driver and driven pulleys. (a) 3L type of V-Belt.


Pulley combination Nominal center distanceDriver Driven Minimum center Short center Medium centerpitch pitch Center Center Center

diameter, diameter, Belt distance, Belt distance, Belt distance,in. in. number in. number in. number in.2.5 2.5 4L170 4.0 4L150 8.0 4L330 12.03.0 3.0 4L200 4.8 4L280 8.8 4L360 12.83.0 4.5 4L240 5.5 4L320 9.6 4L400 13.64.0 6.0 4L300 6.5 4L380 10.6 4L460 14.53.0 6.0 4L280 6.2 4L360 10.3 4L440 14.33.5 7.0 4L320 7.0 4L400 11.1 4L480 15.13.0 7.5 4L320 6.8 4L400 11.0 4L480 15.04.0 10.0 4L410 8.5 4L490 12.5 4L570 16.73.0 9.0 4L360 7.5 4L440 11.7 4L520 15.83.5 10.5 4L420 8.8 4L500 13.0 4L580 17.14.0 12.0 4L470 9.6 4L550 13.8 4L630 18.03.0 10.5 4L410 8.6 4L490 12.9 4L570 17.04.0 14.0 4L530 10.5 4L610 15.0 4L690 19.13.0 12.0 4L450 9.1 4L530 13.4 4L610 17.63.5 14.0 4L520 10.4 4L600 14.8 4L680 19.02.0 9.0 4L350 7.5 4L430 11.8 4L510 15.94.0 18.0 4L650 12.8 4L730 17.3 4L810 21.62.4 12.0 4L440 8.9 4L520 13.3 4L600 17.52.5 18.0 4L480 9.3 4L560 14.3 4L640 18.52.8 14.0 4L510 10.3 4L590 14.7 4L670 19.02.0 11.0 4L400 8.0 4L480 12.5 4L560 16.72.0 12.0 4L430 8.5 4L510 13.0 4L590 17.32.5 15.0 4L530 10.3 4L610 14.9 4L690 19.23.0 13.0 4L630 12.2 4L710 16.8 4L790 21.22.0 14.0 4L490 9.5 4L570 14.1 4L650 18.4

(b)

Center Distance for V-Belts (cont.)

Table 19.6 Center distances for various pitch diameters of driver and driven pulleys. (b) 4L type of V-Belt.


(b)(a)

Figure 19.6 Two lays of wire rope. (a) Lang; (b) regular.

Figure 19.5 Cross-section of wire rope.

Wire Rope


Minimum ModulusWeight sheave Rope Size of of

per height, diameter diameter, outer elasticity,a Strength,b

Rope lb/ft in. d, in. Material wires psi psi6× 7 Haulage 1.50d2 42d 1

4 − 1 12 Monior steel d/9 14× 106 100× 103

Plow steel d/9 14× 106 88× 103

Mild plow steel d/9 14× 106 76× 103

6× 19 Standard 1.60d2 26d− 34d 14 − 2 3

4 Monior steel d/13− d/16 12× 106 106× 103

hoisting Plow steel d/13− d/16 12× 106 93× 103

Mild plow steel d/13− d/16 12× 106 80× 103

6× 37 Special 1.55d2 18d 14 − 3 1

2 Monior steel d/22 11× 106 100× 103

flexible Plow steel d/22 11× 106 88× 103

8× 19 Extra 1.45d2 21d− 26d 14 − 1 1

2 Monior steel d/15− d/19 10× 106 92× 103

flexible Plow steel d/15− d/19 10× 106 80× 103

7× 7 Aircraft 1.70d2 — 116 − 3

8 Corrosion-resistant — — 124× 103

steelCarbon steel — — 124× 103

7× 9 Aircraft 1.75d2 — 18 − 1 3



19-Wire aircraft 2.15d2 — 132 − 5



a The modulus of elasticity is only approximate; it is affected by the loads on the rope and, in general, increases with the lifeof the rope.b The strength is based on the nominal area of the rope. The figures given are only approximate and are based on 1-in. ropesizes and 1/4-in. aircraft cable sizes.

Table 19.7 Wire rope data.

Wire Rope Data


Application Safety factor,a ns

Track cables 3.2Guys 3.5Mine shafts, ft

Up tp 500 8.01000-2000 7.02000-3000 6.0Over 3000 5.0

Hoisting 5.0Haulage 6.0Cranes and derricks 6.0Electric hoists 7.0Hand elevators 5.0Private elevators 7.5Hand dumbwaiters 4.5Grain elevators 7.5Passenger elevators, ft/min

50 7.60300 9.20800 11.251200 11.801500 11.90

Freight elevators, ft/min50 6.65300 8.20800 10.001200 10.501500 10.55

Powered dumbwaiters, ft/min50 4.8300 6.6800 8.0

a Use of these factors does not preclude a fatigue failure

Table 19.8 Minimum safety factors for a variety of wire rope applications.

Minimum Safety Factors


50

40

30

20

10

0 10 20 30 40

D/d ratio

Str

eng

th l

oss

, p

erce

nt

100

80

60

40

20

0 10 20 30 40 50 60

D/d ratio

Rel

ativ

e se

rvic

e li

fe, per

cent

Figure 19.7 Percent strength loss in wire rope for different D/d ratios.

Figure 19.8 Service life for different D/d ratios.

Effect of D/d Ratio


MaterialCast Cast Chilled Manganese

Wooda iron b steelc cast irond steele

Rope Allowable bearing pressure, pall, psiRegular lay6× 7 150 300 550 650 14706× 19 250 480 900 1100 24006× 37 300 585 1075 1325 30008× 19 350 680 1260 1550 3500Lang lay6× 7 165 350 600 715 16506× 19 275 550 1000 1210 27506× 37 330 660 1180 1450 3300

Table 19.9 Maximum allowable bearing pressures for various sleeve materials and types of rope.

Allowable Bearing Pressures


Roller link

Pitch Pitch

Linkplate

Pinlink

RollerPin

Bushing

pt

rc

θr

pt/2

A

Pitch

circle

B C

0

Figure 19.9 Various parts of rolling chain.

Figure 19.10 Chordal rise in rolling chains.

Rolling Chains


Link AverageRoller Pin plate ultimate Weight

Chain Pitch, Diameter, Width, diameter, thickness, strength, per foot,number pt, in. in. in. d, in. a, in. Su, lb lb

a25 1/4 0.130 1/8 0.0905 0.030 875 0.084a35 3/8 0.200 3/16 0.141 0.050 2100 0.21a41 1/2 0.306 1/4 0.141 0.050 2000 0.2840 1/2 5/16 5/16 0.156 0.060 3700 0.4150 5/8 2/5 3/8 0.200 0.080 6100 0.6860 3/4 15/32 1/2 0.234 0.094 8500 1.0080 1 5/8 5/8 0.312 0.125 14,500 1.69100 1 1

4 3/4 3/4 0.375 0.156 24,000 2.49120 1 1

2 7/8 1 0.437 0.187 34,000 3.67140 1 3

4 1 1 0.500 0.219 46,000 4.93160 2 1 1

8 1 14 0.562 0.250 58,000 6.43

180 2 14 1 13

32 1 1332 0.687 0.281 76,000 8.70

200 2 12 1 9

16 1 12 0.781 0.312 95,000 10.51

240 3 1 78 1 7

8 0.937 0.375 130,000 16.90

Table 19.10 Standard sizes and strengths of rolling chains.

Strength of Rolling Chains


Table 19.11: Transmitted power of single-strand, No. 25 rolling chain.

Number of Small sprocket speed, rpmteeth in 100 500 900 1200 1800 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000 7500 8000 8500 9000 10000

small sprocket Type I Type II lubrication11 0.054 0.23 0.39 0.50 0.73 0.98 1.15 1.32 1.42 1.19 1.01 0.88 0.77 0.68 0.61 0.55 0.50 0.46 0.42 0.3612 0.059 0.25 0.43 0.55 0.80 1.07 1.26 1.45 1.62 1.36 1.16 1.00 0.88 0.78 0.70 0.63 0.57 0.52 0.48 0.4113 0.064 0.27 0.47 0.60 0.87 1.17 1.38 1.58 1.78 1.53 1.30 1.13 0.99 0.88 0.79 0.71 0.64 0.59 1.54 0.4614 0.070 0.30 0.50 0.65 0.94 1.27 1.49 1.71 1.93 1.71 1.46 1.26 1.11 0.94 0.88 0.79 0.72 0.66 0.60 0.5115 0.075 0.32 0.54 0.70 1.01 1.36 1.61 1.85 2.06 1.89 1.62 1.40 1.23 1.09 0.96 0.88 0.80 0.73 0.67 0.5716 0.081 0.34 0.58 0.75 1.09 1.46 1.72 1.98 2.23 2.08 1.78 1.54 1.35 1.20 1.07 0.97 0.88 0.80 0.74 0.6317 0.086 0.37 0.62 0.81 1.16 1.56 1.84 2.11 2.38 2.28 1.95 1.69 1.48 1.31 1.18 1.06 0.96 0.88 0.81 0.6918 0.097 0.39 0.66 0.86 1.23 1.66 1.95 2.25 2.53 2.49 2.12 1.84 1.52 1.43 1.28 1.16 1.05 0.96 0.88 0.7519 0.097 0.41 0.70 0.91 1.31 1.76 2.07 2.38 2.69 2.70 2.30 2.00 1.75 1.55 1.39 1.25 1.14 1.04 0.96 0.8120 0.103 0.44 0.74 0.96 1.38 1.86 2.19 2.52 2.84 2.91 2.49 2.16 1.89 1.68 1.50 1.35 1.23 1.12 1.03 0.8821 0.108 0.46 0.78 1.01 1.46 1.96 2.31 2.65 2.99 3.13 2.68 2.32 2.04 1.80 1.61 1.46 1.32 1.21 1.11 0.9522 0.114 0.48 0.82 1.06 1.53 2.06 2.43 2.79 3.15 3.36 2.87 2.49 2.18 1.93 1.73 1.56 1.42 1.29 1.19 1.0123 0.119 0.51 0.86 1.12 1.61 2.16 2.55 2.93 3.30 3.59 3.07 2.66 2.33 2.07 1.85 1.67 1.51 1.38 1.27 1.0624 0.125 0.53 0.90 1.17 1.69 2.26 2.67 3.07 3.46 3.83 3.27 2.83 2.48 2.20 1.97 1.78 1.61 1.47 1.35 1.1625 0.131 0.56 0.94 1.22 1.76 2.37 2.79 3.20 3.61 4.07 3.48 3.01 2.64 2.34 2.10 1.89 1.72 1.52 1.44 1.2328 0.148 0.63 1.07 1.38 1.99 2.67 3.15 3.62 4.28 4.54 4.12 3.57 3.13 2.78 2.49 2.24 2.04 1.86 1.71 1.4630 0.159 0.68 1.15 1.49 2.14 2.88 3.39 3.90 4.40 4.89 4.57 3.96 3.47 3.08 2.76 2.49 2.26 2.06 1.89 1.6232 0.170 0.73 1.23 1.60 2.30 3.09 3.64 4.18 4.71 5.24 5.03 4.36 3.83 3.39 3.04 2.74 2.49 2.27 2.06 1.7835 0.188 0.80 1.36 2.76 2.53 3.40 4.31 4.62 5.19 5.78 5.76 4.99 4.38 3.88 3.48 3.13 2.85 2.60 2.38 2.0440 0.217 0.92 1.57 2.03 2.93 3.93 4.63 5.32 6.00 6.67 7.04 6.10 5.35 4.75 4.25 3.83 3.48 3.17 2.91 2.4945 0.246 1.05 1.78 2.31 3.32 4.46 5.26 6.04 6.81 7.58 8.33 7.28 6.39 5.66 5.07 4.57 4.15 3.79 3.48 2.9750 0.276 1.18 1.99 2.58 3.72 5.00 5.89 6.77 7.64 8.49 9.33 8.52 7.48 6.63 5.93 5.35 4.96 4.44 4.07 3.4855 0.306 1.30 2.21 2.96 4.12 5.54 6.53 7.51 8.46 9.41 10.3 9.83 8.63 7.65 6.85 6.17 5.60 5.12 4.76 4.0160 0.336 1.43 2.43 3.15 4.53 6.09 7.18 8.25 9.30 10.3 11.3 11.2 9.83 8.72 7.80 7.03 6.38 5.83 5.35 4.57

Type II III Type IV lubrication

Chain Power Ratings


Type of input powerInternal combustion Electric motor Internal combustion

engine with or engine withType of driven load hydraulic drive turbine mechanical driveSmooth 1.0 1.0 1.2Moderate shock 1.2 1.3 1.4Heavy shock 1.4 1.5 1.7

Table 19.13: Multiple-strand factors for rolling chains.

Number of strands Multiple-strand factor, a2

2 1.73 2.54 3.3

Rolling Chain Data

Table 19.12 Service factors for rolling chains.


φ

Drag rope

Gantry line system

θ

15 ft

5 ft

Mast (20 ft)

Multi-part line

Hoist rope

Dragline

Figure 19.11 Typical dragline.


Chapter 20: Elements of Microelectromechanical Systems (MEMS)

There is plenty of room at the bottom.

Richard Feynman


Figure 20.1 The Texas Instruments digital pixel technology (DPT) device.

DPT Device

Mirror

Mirror support post

Landing tips Torsion hinge

Addresselectrode

Yoke Electrodesupport postHinge

support post

To SRAM

Metal 3 address padsLandingsites

Bias/ResetBus

Mirror -10°Mirror +10°

HingeYoke

Landing tip CMOSsubstrate

(a) (b)

(c) (d)


Figure 20.2 Pattern transfer by lithography. Note that the mask in step 3 can be a positive or negative image of the pattern.

Lithography


Etch front

Mask layer

Etch material(e.g. silicon)

{111} face

Etch front

Final shape

54.7°

Etch front

(a) (b) (c)undercut

Figure 20.3 Etching directionality. (a) Isotropic etching: etch proceeds vertically and horizontally at approximately the same rate, with significant mask undercut. (b) Orientation-dependant etching (ODE): etch proceeds vertically, terminating on {111} crystal planes with little mask undercut. (c) Vertical etching: etch proceeds vertically with little mask undercut.

Etching Directionality


Diffused layer(e.g. p-type Si)

Substrate(e.g. n-type Si)

Non-etching mask(e.g. silicon nitride)

Freestandingcantilever

(111) planes

(a)

(b)

(c) Figure 20.4 Schematic illustration of bulk micromachining. (a) Diffuse dopant in desired pattern. (b) Deposit and pattern masking film. (c) Orientation-dependant etch (ODE), leaving behind a freestanding structure.

Bulk Micromachining


Figure 20.5 Schematic illustration of the steps in surface micromachining. (a) deposition of a phosphosilicate glass (PSG) spacer layer; (b) etching of spacer layer; (c) deposition of polysilicon; (d) etching of polysilicon; (e) selective wet etching of PSG, leaving the silicon substrate and deposited polysilicon unaffected.

Surface Micromachining


(a) (b)

Figure 20.6 (a) SEM image of a deployed micromirror. (b) Detail of the micromirror hinge. (Source: Sandia National Laboratories.)

Micromirror


Spacer layer 1

Silicon

Poly1

Spacer Layer 2

(a) (b) (c)

(d)(e)

Poly2

Figure 20.7 Schematic illustration of the steps required to manufacture a hinge. (a) Deposition of a phosphosilicate glass (PSG) spacer layer and polysilicon layer. (b) deposition of a second spacer layer; (c) Selective etching of the PSG; (d) depostion of polysilicon to form a staple for the hinge; (e) After selective wet etching of the PSG, the hinge can rotate.

Micro-Hinge Manufacture


LIGA

Figure 20.8 The LIGA (lithography, electrodeposition and molding) technique. (a) Primary production of a metal final product or mold insert. (b) Use of the primary part for secondary operations, or replication


Table 20.1 Summary of important beam situations for MEMS devices.

Beams in MEMS


(a) (b)

Atomic Force Microscope Probe

Figure 20.9 Scanning electron microscope images of a diamond-tipped cantilever probe used in atomic force microscopy. (a) Side view with detail of diamond; (b) bottom view of entire cantilever.


a/b 1.0 1.2 1.4 1.6 1.8 2.0 ∞α 0.0138 0.0188 0.0226 0.0251 0.0267 0.0277 0.0284β 0.3078 0.3834 0.4356 0.4680 0.4872 0.4974 0.5000

Table 20.2 Coefficients α and β for analysis of rectangular plate pressure sensor.

Rectangular Plate

!max =−"pb4

Et3

!max = "pb2

t2


V (volts)

yx

h

a

V (volts)y

x h

a

Fy

Fy

Stationary comb

Moving comb

Beam spring suspension

Anchors

(a)

(b)

(c)

Figure 20.10 Illustration of electrostatic actuatuation. (a) Attractive forces between charged plates; (b) forces resulting from eccentric charged plate between two other plates; (c) schematic illustration of a comb drive.

Electrostatic Actuation


Figure 20.11 A comb drive. Note the springs in the center provide a restoring force to return the electrostatic comb teeth to their original position. From Sandia National Laboratories.

Comb Drive


Rotor

Stator

(a) (b)

18°

12°

27°

Figure 20.12 (a) Schematic illustration of a rotary electrostatic motor, sometimes called a slide motor; (b) scanning electron microscope image of a rotary micromotor.

Rotary Electrostatic Motor


Flow

Piezoelectric coatingwith transducer

Flexible tube wall

Traveling wave direction

Figure 20.13 Capilary tube for microflow. (a) Schamitic illustration of tube construction; (b) induced traveing wave and fluid flow.

Capilary Tube for Microflow


a) Actuation

Heating element

Ink

Bubble

b) Droplet formation

c) Droplet ejection

Satellite dropletsd) Liquid refills

Thermal Inkjet Printer

Figure 20.14 (a) Sequence of operation of a thermal inkjet printer. (a) Resistive heating element is turned on, rapidly vaporizing ink and forming a bubble. (b) Within five microseconds, the bubble has expanded and displaced liquid ink from the nozzle. (c) Surface tension breaks the ink stream into a bubble, which is discharged at high velocity. The heating element is turned off at this time, so that the bubble collapses as heat is transferred to the surrounding ink. (d) Within 24 microseconds, an ink droplet (and undesirable satellite droplets) are ejected, and surface tension of the ink draws more liquid from the reservoir.


Ink reservoir

Teflon coating

PZT actuator

Nozzle

Paper Ink dot

Ink droplet

Piezoelectric Inkjet Mechanism

Figure 20.15 Schematic illustration of a piezoelectric driven inkjet printer head.


Metal Coating Catalyst Detected gasBaTiO3/CuO La2O3, CaCO3 CO2

SNO2 Pt + Sb COSNO2 Pt AlcoholsSNO2 Sb2O3 H2, O2, H2SSNO2 CuO H2SWO3 Pt NH3

Fe2O3 Ti-doped Au COGa2O3 Au COMoO3 — NO2, COIn2O3 — O3

Metal Oxide Sensors

Table 20.4 Common metal oxide sensors.


Stationarypolysilicon fingers

Spring(beam)

Suspendedinertial mass

Anchor to substrate

C1

C2

Direction of acceleration

(a) (b)

Accellerometer

Figure 20.16 (a) Schematic illustration of accellerometer; (b) photograph of Analog Devices’ ADXL-50 accelerometer with a surface micromachined capacitive sensor (center), on-chip excitation, self-test and signal conditioning circuitry. The entire chip measures 0.500 by 0.625 mm.

Documents

Fundamentals of Machine Elements, 2nd Ed