Size Tolerances - Concordia Universityusers.encs.concordia.ca/~nrskumar/Index_files/Mech6491/Lecture 4.pdf · Tolerance Stack-up Tolerances taken from the same direction from one

1

Size Tolerances

2

Form Tolerances

3

Orientation Tolerances

4

Location Tolerances

5

Positional tolerancing

6

Coordinate Tolerancing

Coordinate dimensions and tolerances may be applied to the location of a

single hole, as shown in Fig. 16-9-2. It should be noted that the tolerance

zone extends for the full depth of the hole, that is, the whole length of the

axis 16-9-3.


7


• For the same tolerance values,

the positional tolerance gives

more room compared to

coordinate tolerancing

• This is due the fact that in

coordinate method the diagonal

becomes responsible for limits.

• If we use the same limits in

positional tolerancing we will get

more parts accepted without any

effect on assembly

Comparison between positional and coordinate tolerancing

8

Tolerance Buildup

During assembly, small variations in the part dimensions can multiply until the

final assembled result is unacceptable from the original design.

Though each part variation is small by themselves; however, with each added

part, the errors can compound to a defected final part.

Size tolerances is fairly straightforward. However, when added with form,

location and/or orientation variances, then it becomes more difficult to predict

dimensional fit.

9

Tolerance Stack-up

Tolerances taken from the same direction from one

reference are additive

Tolerances taken to the same point in different directions

are additive in both directions

10

Tolerance Stack-up

Case 1 - .020 between centers

11

Tolerance Stack-up


12

Tolerance Stack-up


13

Tolerance Analysis Methods

There are many different approaches that

are utilized in industry for tolerance analysis.

The more tradition methods include:

Worst-Case analysis

Root Sum of Squares

Taguchi tolerance design

14

Worst-Case Methodology

This is not a statistical procedure but is used often for tolerance

analysis and allocation

Provides a basis to establish the dimensions and tolerances such

that any combination will produce a functioning assembly

Extreme or most liberal condition of tolerance buildup

“…tolerances must be assigned to the component parts of the

mechanism in such a manner that the probability that a mechanism

will not function is zero…”

- Evans (1974)

15

Worst Case Scenario Example

Source: Tolerance Design, pp 109-111

In this example, we see a mating hole and pin assembly. The nominal

dimensions are given in the second figure.

16


Here, we can see the two worst

case situations where the pins are in

the extreme outer edges or inner

edges.

The analysis begins on the right

edge of the right pin.

You should always try to pick a

logical starting point for stack

analysis.

Note that the stack up dimensions

are summed according to their sign

(the arrows are like displacement

vectors).

17


• Largest => 0.05 + 0.093 = 0.143

• Smallest => 0.05 - 0.093 = -0.043

From the stack up, we can determine the tolerance calculations shown in table.

Analyzing the results, we find that there is a +0.05 nominal gap and +0.093 tolerance buildup

for the worst case in the positive direction.

This gives us a total worst-case largest gap of +0.143. It gives us a worst case smallest gap

of -0.043 which is an interference fit.

Thus, in this worst-case scenario, the parts will not fit and one needs to reconsider the

dimension or the tolerance.

18

Calculating Stackups

19


A (-)B (-)

C (-)

D (+)

E (-)

All dimensions have tolerance of ±0.2, unless

specified

20


Part Name Dim Tol (+ve) Tol (-ve) Direction

Cylinder1 A 54 +0.2 -0.2 -ve

Cylinder2 B 54 +0.2 -0.2 -ve

Holder C 20 +0.2 -0.2 -ve

Holder D 150 +0.2 -0.2 +ve

Holder E 20 +0.2 -0.2 -ve

Worst Case 2 1 -1

Worst case (Max) = 2 + 1 = 3 Worst case (Min) = 2 - 1 = 2

21


2 ± 1

A (-)B (-)

C (-)

D (+)

E (-)

All dimensions have tolerance of ±0.2, unless

specified

22

Statistical Principles

23


24

Excerpted from The Complete Guide to the

CQE by Thomas Pyzdek. 1996. Tucson: Quality

Publishing Inc.


± 6 translates to 2 parts

per billion

25

MECH 6491 Engineering Metrology

and Measurement Systems

Lecture 4

Instructor: N R Sivakumar

26

Basic optical principles

Wave optics

Geometric optics

Reflection, refraction and dispersion

Electromagnetism and polarization

Monochromatic and chromatic aberrations

Outline

27

Optical Metrology History

Vision-Based Metrology

refers to the technology

using optical sensors and

digital image processing

hardware and software to:

Identify

Guide

Inspect

Measure objects

28

Vision-Based Metrology

inspection systems evolved

from the combination of

microscopes, cameras and

optical comparators


29

Vision-Based Metrology is extensively used in general

industrial applications such as the manufacturing of:

Electronics

Automotive

Aerospace

Pharmaceutical

Consumer products

Vision-Based Metrology is being utilized in the automatic

identification and data collection market as a complementary

or alternative technology to traditional laser scanning devices

for reading bar codes


30

Early systems were integrated into

packaging lines for optical character

recognition to check the accuracy of

product codes and label information.

Today, high-resolution cameras,

advances in software and imaging

processors, and the availability of

powerful, inexpensive compact

computers have made vision

systems faster and more reliable

than ever.


31

Who needs a vision system?

Vision system may be needed for high

production product inspection CD and

pharma industries

They provide a means of increasing

yield-that is, the ratio of good parts to

bad parts.

When a serial defect is spotted, the

system not only recognizes it but can

stop the conveyor and inform the

operator of the defect and its

magnitude.

32

Vision Based Metrology is

now being used to focus on

the movement of objects

along with their deformation

This is being used in many

car wreck investigations

Vision system - Automobiles

33

Two consecutive images were grabbed from a high speed video

sequence

A displacement field of a car at a certain moment is presented

The deformation pattern was obtained from the principle vector

analysis

This analysis allows the representation of the deformation pattern

Vision system - Automobiles

34

Golf Ball Specifications

Weight: Less than or equal to 1.620

Ounces

Size: Greater than or equal to 1.680

Inches

Shape: Must be symmetrical

A ball passes USGA size inspection if it falls, under its own weight,

through a 1.680 inch diameter ring gauge fewer than 25/100 times

in randomly selected positions.

– Temperature is constant at 23° C (73.4 ° F).

– Humidity is held constant.

35

Procedure and Test Outline

Place the ball on the test stand

Take a picture from a standard height for each golf ball being

tested

Analyze the image using National Instruments® vision analysis

software

Compare the image to the standard size for the USGA ball

specification

Compile and analyze the data from the testing

Present information in graphical form

36

Test Device

Designed using Solidworks® CAD

program

Made of extruded aluminum

Center positioned ball holder that

provides consistent images for each

ball tested

Camera is secured using the tripod

mount

37

Images from Machine Vision

38

Lots of facts to know about light

Newton did the early work of

understanding light

Confusion whether light is made of

Waves

particles?

Optics around 1700

39

Waves or particles?

Light travels in straight lines

Waves travel in circles (throwing stone in water)

But particles in crossed beams would collide?

Light reflects off mirrors and leaves at the

same angle as it came in

Makes sense for particles (conservation of

momentum)

40

Waves or particles?

Light bends (refracts) when moving

between different media

n1sin(1) = n2sin(2)

Newton had a semi-plausible explanation

for particles

Easy to explain for waves if they travel in

straight lines!

Diffraction effects not really understood

Newton’s rings provide excellent evidence for wave behavior, but Newton was

unhappy with the wave model

Underlying basis of color hardly understood at all

Polarization only recently discovered

41

Particles

have mass (inertia)

respond to forces (acceleration)

have momentum (mass x velocity)

Waves

transfer energy from one place to another

mechanical (require a medium) and non-mechanical

Waves or particles?

42

Corpuscular theory of Newton (1670)

light corpuscles have mass and travel at extremely high speeds

in straight lines

rectilinear propagation - blocked by large objects (well-defined

shadows)

obey the law of reflection when bounced off a surface

speed up when they enter denser media (gravitational force of

attraction, net F = ma)

paths in denser media "bend towards the normal"

prism dispersion - contradicted corpuscular theory

Waves or particles?

43

Wave Particle Duality

Does light act as a wave or particle?

Huygens, Young – Wave nature of light

Newton – Particle nature of light. - photon

Experiments such as the photoelectric effect could not

be explained until light was treated as a particle

Geometric Optics - Situations where light can be treated as a

particle

Wave Optics - situations where light acts as waves (diffraction,

interference)

44

Huygens Principle (1680)

wavelet envelop model (each point on a wavefront acts as a

source for the next wavefront)

plane waves generate plane waves, circular waves generate

circular waves

light was composed of longitudinal waves like sound

obey the law of reflection when bounced off a surface

waves slowed down when they entered a denser medium

causing their paths to "bend towards the normal"

light SHOULD produce interference patterns and diffraction

patterns

Waves or particles?

45

Huygens Principle Light is made up of a series of pulsations in the ether, an

otherwise undectable substance filling all space

Each pulsation causes a chain of secondary pulsations to

spread out ahead

In certain directions these pulsations reinforce one another,

creating an intense pulsation that appears as visible light

Every point on wavefront may be regarded as a source of

secondary wavelets which spread out with wave velocity

The new wavefront is the envelope of these secondary

wavelets.

46

Huygen’s Wavelet Concept

Collimated Plane Wave

Spherical Wave

At time 0, wavefront is defined

by line (or curve) AB. Each

point on the original wavefront

emits a spherical wavelet which

propagates at speed c away

from the origin.

At time t, the new wavefront is

defined such that it is tangent to

the wavelets from each of the

time 0 source points. A ray of

light in geometric optics is

found by drawing a line from

the source point to the tangent

point for each wavelet.

47

Straight lines

Straight wavefronts stay straight

Points on the wavefront all move forward at the

same speed in a direction normal to the

wavefront. All points on a wavefront correspond

to the same point in time

Light rays travel along these normals

48

Speed of light in vacuum is constant ~ 186,000

mi/hr 2.997 x 108 m/s

The speed of light is different in different media,

however

Light gets absorbed, so

It takes longer to pass through

A material.

n = c/v n = index of refraction

Light

49

Optics deals with the propagation of light and its

interaction with mirrors, lenses, slits, etc.

Optical effects can be divided into two classes:

without considering light as a wave

only on the basis that light is a wave

wave effects are only important when the wave-length

is either comparable to, or larger than, the size of the

interacting objects

When the wave-length of the wave becomes much

smaller than the size of the interacting objects then the

interactions can be accounted in geometric manner

Geometric Optics

50

Geometric Optics – when?

51

Wave-length of visible light is only of order a micron,

it is easy to find situations where light wave-length is

smaller than the size of the interacting objects.

Thus, ``wave-less'' (geometric) optics,has a very

wide range of applications.

In geometric optics, light is treated as a set of rays,

emanating from a source, propagating through

transparent media based on 3 simple laws.

law of rectilinear propagation

law of reflection

law of refraction

Geometric Optics

52

Law of rectilinear propagation

states that light rays propagating through a homogeneous

transparent medium propagate in straight lines

Law of reflection

governs the interaction of light rays with conducting

surfaces (e.g., metallic mirrors)

Law of refraction

governs the behavior of light rays as they traverse a sharp

boundary between two different media (e.g., air and glass)

Geometric Optics - Laws

53

Law of rectilinear propagation

According to geometric optics, an opaque object illuminated

by a point source of light casts a sharp shadow whose

dimensions can be calculated using geometry.

The light-rays propagate

away from the source until

they encounter an opaque

object, at which point

they stop.


54

Law of reflection

Consider a light-ray incident on a plane mirror

The law of reflection states that the incident ray, the reflected

ray, and the normal to the surface of the mirror all lie in the

same plane

the angle of reflection =

the angle of incidence

Both angles are measured

WRT the normal to the mirror


551 = 2

Light - Reflection

56

Law of refraction (Snell’s Law)

Consider a light-ray incident on a interface between two transparent media.

The law of refraction states that the incident ray, the refracted ray, and the

normal to the interface, all lie in the same plane


n1sin(1) = n2sin(2), where 1 is the between

the incident ray and the normal, and 2 is the

between the refracted ray and the normal

The quantities n1 and n2 are termed the

refractive indices of media 1 and 2

Thus, the law predicts that a light deviates

towards the normal in the optically denser

medium (higher refractive index)

57

Law of refraction (Snell’s Law)

The law follows directly from the fact that the speed of light in medium is

inversely proportional to the refractive index n of the medium, i.e. = c/n

(where c is the speed of light in vacuum)


Consider two parallel light-rays, a, b incident at an

angle 1 wrt the normal. refractive indices of the two

media be n1 and n2 (n1 < n2). It is clear that ray b must

move from B to Q, in medium 1, in the same time

interval, t, in which ray a moves between points A and

P, in medium 2.

Now, the speed of light in medium 1 is 1=c/n1 ,

whereas the speed of light in medium 2 is 2=c/n2 .

By trigonometry, BQ = t 1 and AP = t 2

n1sin(1) = n2sin(2),

58

2211 sinsin nn

Snell’s Law

Light - Refraction

59

But it is really here!!

He sees the

fish here….

Light – Refraction

60

Suppose that light crosses an interface from medium 1 to medium 2,

where (n2 < n1). According to Snell's law,

In an interface, part of light will be refracted and part reflected. if at

any angle, the refracted ray follows the interface, that angle is called

critical angle c

Suppose if 1 > c, there will be no

refraction, and all the light will be

reflected This is Total Internal

Reflection

Example ???????

Where is it useful ??????

Light – Total Internal Dispersion

61

1

2

1

2 90sinsinn

n

n

nc

Fiber Optics

Light – Total Internal Dispersion

62

When a wave is refracted into a medium whose

refractive index varies with wave-length then the

angle of refraction also varies with wave-length.

If the incident wave is composed of a mixture of

wave-lengths, then each component is refracted

through a different angle - called dispersion

In the particular case of light, the refractive

indices of some common materials which vary

with wave-length are shown in the figure above.

It can be seen that the refractive index always

decreases with increasing wave-length in the

visible range. In other words, violet light is

always refracted more strongly than red light.

Light - Dispersion

63

The index of refraction is not

constant for all materials

Some n’s are wavelength

dependent.

Light - Dispersion

64

In 1669, Huygens studied light through a calcite crystal – observed two

rays (birefringence) .

Here, we are talking about separating out different parts of light when we

discuss polarization. Specifically, we are interested in the electric field of

the electromagnetic wave.

Light –Polarization

65

Electric field only going

up and down – say it is

linearly polarized.

Light can have other types of polarizations such as circularly polarized

or elliptically polarized. We will only look at linearly polarized light.

Net electric field is zero – Unpolarized light!


66

Vertical

Horizontal

)/sin( txAEy

Plane-polarized light

)/sin( txAEz

67

Right circular

Left circular

Circularly polarized light

)90/sin( txAEy

)/sin( txAEz

)90/sin( txAEy

)/sin( txAEz

68

I. Polarizers-

Polarizers are made of long

chained molecules which absorb

light with electric fields

perpendicular to the axis.

2

2

cosoII

EI

Because

Malus’s

Law


69

II. Scattering -


70

II. Reflection -

1

2

21

2

221

tan

cossin

90

sinsin

n

n

nn

nn

p

pp

p

o

p

Brewster’s

Law


71

Spherical Mirrors

A spherical mirror is a mirror which has the shape of

a piece cut out of a spherical surface.

There are two types concave, and convex mirrors.

Concave mirrors magnify objects placed close to

them

shaving mirrors and makeup mirrors.

Convex mirrors have wider fields of view

passenger-side wing mirrors of cars

but objects which appear in them generally look

smaller (and, therefore, farther away) than they

actually are.

72

The normal to the mirror centre is called principal axis.

V, at which the principal axis touches the mirror surface is called the

vertex.

The point C, on the principal axis, equidistant from all points on the

reflecting surface of the mirror is called the centre of curvature.

C to V is called the radius of curvature of the mirror R

Spherical Mirrors - Concave

Rays parallel principal axis striking a

concave mirror, are reflected by the mirror

at F (between C and V) is focal point.

The distance along the principal axis from

the focus to the vertex is called the focal

length of the mirror, and is denoted f.

73

The definitions of the principal axis, C, R, V of a convex mirror are

same as that of concave mirror.

When parallel light-rays strike a convex mirror they are reflected

such that they appear to emanate from a single point F located

behind the mirror.

Spherical Mirrors - Convex

This point is called the virtual focus

of the mirror.

The focal length f of the mirror is

simply the distance between V and

F.

f is half of R

74

Graphical method - just four simple rules:

An incident ray // to principal axis is reflected through the focus F of the mirror

An incident ray which passes through the F of the mirror is reflected // to the

principal axis

An incident ray which passes through the C of the mirror is reflected back along

its own path (since it is normally incident on the mirror)

An incident ray which strikes the mirror at V is reflected such that its angle of

incidence wrt the principal axis is equal to its angle of reflection.

Image Formation - Concave

ST is the object at distance p from the mirror (P > f).

Consider 4 light rays from tip T to strike the mirror

1 is // to axis so reflects through point F. 2 is through point F

so reflects // to the axis. 3 is through C so it traces its path

back. 4 is towards V so it has same angles of incidence and

reflectance.

The point at which all these meet is where the object will be

S’T’ at distace Q (if seen further from q, the image is seen)

This is a real image

75

We only need 2 rays (minimum) to get the image position:

In this case, the object ST is located within the focal length of the mirror f

Consider 2 lines from tip T of the object

Line 2, passes through F and reflects // to the principal axis

Line 3 passes through C and is reflected along its path

If these two lines are connected, the image is formed on the other side of the

lens

Here the image is magnified, but not inverted like the previous case


There are no real light-rays behind the mirror

Image cannot be viewed by projecting onto a screen

This type of image is termed a virtual image

The difference between a real and virtual image is,

immediately after reflection from the mirror, light-rays

from object converge on a real image, but diverge

from a virtual image.

76

Graphical method - just four simple rules:

Incident ray // to principal axis is reflected as if it is from virtual focus F of mirror.

Incident ray directed towards the virtual focus F of the mirror is reflected // to

principal axis.

Incident ray directed towards C of the mirror is reflected back along its own path

(since it is normally incident on the mirror).

Incident ray which strikes the mirror at V is reflected such that its angle of

incidence wrt the principal axis is equal to its angle of reflection. .

Image Formation - Convex

ST is the object. Consider 2 light rays from tip T to

strike the mirror

1 is // to axis so appears to reflect through point F.

2 is through point C so so it traces its path back.

The point of intersection of these two lines is

where the image will be

Vitural or Real ????

Inverted and Magnified or otherwise ?????

77

If we do this by analytical method, we will get the following formula

Magnification M is given by the object and image distances

It is negative if the image is inverted and positive if it is not

For expression that relates the object and image distances to radius of

curvature


If object is far away (p = ) all the lines are //

and focus on the focal point F

The focal length is R/2 which can be combine

to give

78

For plane mirrors the radius of curvature R =

If R = , then f = ±R/2 =

because 1/f = 0

1/p + 1/q = 1/f = 0

q = -p (which means that it is a virtual image far behind

the mirror as much the object is in front)

Magnification is –q/p = 1

So plane mirror does not magnify or invert the image

Image Formation – plane mirror

79

Image Formation – plane mirror

Sign conventions may vary based

on different text books. So follow

consistently which ever method is

used

80

Image Formation – LensesFor thin lenses this

distance is taken as 0

81

Image Formation – Lenses A lens is a transparent medium bounded by two

curved surfaces (spherical or cylindrical)

Line passing normally through both bounding

surfaces of a lens is called the optic axis.

The point O on the optic axis midway between the

two bounding surfaces is called the optic centre.

There are 2 basic kinds: converging, diverging

Converging lens - brings all incident light-rays

parallel to its optic axis together at a point F, behind

the lens, called the focal point, or focus.

Diverging lens spreads out all incident light-rays parallel to its optic

axis so that they appear to diverge from a virtual focal point F in front

of the lens.

Front side is conventionally to be the side from which the light is

incident.

82

Image Formation – Lenses Relationship between object and image distances to

focal length is given by

Magnification of the lens is given by

Example (Object outside Focal Point)

Object distance S = 200mm Object height h = 1mm

Focal length of the lens f = 50mm

Find image distance S’ and Magnification m

83




Example (Object inside Focal Point)


Focal length of the lens f = 50mm


84




Example (Object at Focal Point)


Focal length of the lens f = -50mm (diverging lens)


85

F-Number and NA The calculations used to determine

lens dia are based on the concepts

of focal ratio (f-number) and

numerical aperture (NA).

The f-number is the ratio of the

lens focal length of the to its clear

aperture (effective diameter ).

The f-number defines the angle of the cone of

light leaving the lens which ultimately forms

the image.

The other term used commonly in defining

this cone angle is numerical aperture NA.

NA is the sine of the angle made by the

marginal ray with the optical axis. By using

simple trigonometry, it can be seen that

86

Different Lenses

87

Spherical aberration comes from the spherical

surface of a lens

The further away the rays from the lens center, the

bigger the error is

Common in single lenses.

The distance along the optical axis between the

closest and farthest focal points is called (LSA)

The height at which these rays is called (TSA)

TSA = LSA X tan u″

Spherical aberration is dependent on lens shape,

orientation and index of refraction of the lens

Aspherical lenses offer best solution, but difficult

to manufacture

So cemented doublets (+ve and –ve) are used to

eliminate this aberration

Spherical Aberration

88

When an off-axis object is focused by a spherical lens, the natural

asymmetry leads to astigmatism.

The system appears to have two different focal lengths. Saggital and

tangential planes

Between these conjugates, the image is either an elliptical or a circular

blur. Astigmatism is defined as the separation of these conjugates.

Astigmatism

The amount of astigmatism depends on lens

shape

89

Optics, E. Hecht, p. 224.

Astigmatism

90

Modern Optics, R. Guenther, p. 207.

Least Astig.

Most Astig.

Astigmatism

9191

When an off-axis object is focused by a spherical lens, the natural

asymmetry leads to astigmatism.

The system appears to have two different focal lengths. Saggital and

tangential planes

Between these conjugates, the image is either an elliptical or a circular blur.

Astigmatism is defined as the separation of these conjugates.

Astigmatism

The amount of astigmatism depends on lens

shape

http://upload.wikimedia.org/wikipedia/commons/b/b3/Astigmatism.svg

http://upload.wikimedia.org/wikipedia/commons/b/b3/Astigmatism.svg

9292

Astigmatism

http://upload.wikimedia.org/wikipedia/en/4/42/Astigmatism_text_blur.png

http://upload.wikimedia.org/wikipedia/en/4/42/Astigmatism_text_blur.png

93

Chromatic Aberration

Material usually have

different refractive indices

for different wavelengths

nblue>nred

This is dispersion

blue reflects more than the

red, blue has a closer

focus

94

As in the case of spherical aberration, positive

and negative elements have opposite signs of

chromatic aberration.

By combining elements of nearly opposite

aberration to form a doublet, chromatic

aberration can be partially corrected

It is necessary to use two glasses with

different dispersion characteristics, so that the

weaker negative element can balance the

aberration of the stronger, positive element.

Achromatic Doublets

95

n1

n2

R1R2

R3

Achromatic doublet (achormat) is often used to compensate

for the chromatic aberration

the focuses for red and blue is the same if

0)11

)(()11

)((32

22

21

11 RR

nnRR

nn rbrb

Achromatic Doublets

Lens maker’s formula

96

97

1 enters from O' through C, follows same path

2 enters from O' through F, leaves // to axis

3 enters through O' // to axis, leaves F

CFO I

1

2

3

I'

O'

V

Image Formation - Exercise

S = 7cm R = +8cm

S’ = ? m = ? sfs

11

'

1

s

sm

'

2

Rf

98

CFO I

1

2

3

I'

O'

V

Image Formation - Exercise

1 enters from O' through C, follows same path

2 enters from O' through F, leaves // to axis

3 enters through O' // to axis, leaves F

S = 17cm R = -8cm

S’ = ? m = ? sfs

11

'

1

s

sm

'

2

Rf

Documents

Size Tolerances - Concordia Universityusers.encs.concordia.ca/~nrskumar/Index_files/Mech6491/Lecture 4.pdf · Tolerance Stack-up Tolerances taken from the same direction from one