Datum Systems (1)

7/28/2019 Datum Systems (1)

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DATUM SYSTEMS

UNIT II

The term "Datum" was first adopted by the International Standards Organization (ISO

1081-1980) and most recently by the Rubber Manufacturers Association Engineering Standard

for Classical V-Belts and Sheaves (IP- 20-1988. This datum system defines specific sheave and

belt dimensions previously known as the pitch system for classical belts and sheaves. What were

previously identified as pitch diameter or pitch length are now known as datum diameter or

datum length. These are the new catalog identifiers which allow all belts and sheave

manufacturers and users to use the same nomenclature.

The change to the datum system addresses some fundamental problems with the pitch

system. The previous PD to OD [Pitch Diameter to Outside Diameter] values were based on the

old multiple unit constructions, which have long since been replaced by the newer single unit

constructions. The pitch line of these single unit construction belts was determined to ride closer

to the outside diameter of the sheaves. The true pitch diameter and pitch length are not

predetermined values, however, but change according to the belt dimensions and construction. It

is very difficult to come up with PD to OD values which cover single and joined, wrapped and

raw edge, new and old, and Gates and its competitors belts. The datum system dimensions, as

outlined in the RMA standard, are a compromise of all belt types and belt manufacturers.The new pitch diameter values are equal to the outside diameter of a standard (not deep

groove or combination groove) sheave, since this is judged to be the best approximation of the

pitch line location of most belt products manufactured. This will yield a sufficiently accurate

speed ratio calculation for virtually all industrial drives. It is important to note that this change is

for classical belts and sheaves only and does not affect the Super HC line [Narrow Belts]

which is based on the effective system. Most V-belt users and drive designers need only to

understand the concept of this new system. All of the calculations and necessary steps for proper

drive analysis have been incorporated in Design Flexibility and the upcoming edition of the

Heavy Duty Drive Design Manual. Although these two design tools will assist in most common

drive analyses, there are those who deal with more unique drives and require a more

comprehensive understanding of the datum system. The question then will arise, "When is the

datum diameter used for calculations, and when the pitch is or outside diameter used?" Although


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all of the formulas have remained the same, different values must be used for some of the

calculations as shown below.


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Degree of freedom:-

All parts have six degrees of freedom. There are three translational degrees of freedom

and three rotational degrees of freedom. The term degrees of freedom mean the number of

coordinates it takes to exclusively control the position of a part. The term translational refers to

uniform movement without rotation, and the term rotational refers to movement around an axis.

Considering the datum reference frame in Example 3-11, the part can move without rotation in

each of the three directions from the mutually perpendicular planes. This is called the three

degrees of translation. The part can also rotate about each of the axes, which is referred to as the

three degrees of rotation. Refer again to Example 3-11 and notice that the three translational

degrees of freedom are labeled x, y, and z. The three rotational degrees of freedom are labeled u,

v, and w. The following demonstrates the degrees of freedom related to the primary, secondary,

and tertiary datums:

The primary datum plane constrains three degrees of freedom:

One translational in z.

One rotational in u.

One rotational in v

Example 3-11. All parts have six degrees of freedom. The three translational degrees of freedom

for this part are labeled x, y, and z. The three rotational degrees of freedom for this part are

labeled u, v, and w.

The secondary datum plane constrains two degrees of freedom:


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One translational iny.

One rotational in w.

The tertiary datum plane constrains one degree of freedom:

One translational inx.

Different types:-

Datum Feature Symbol

The datum f eature symbolis placed on the drawing to identify the features of the object that are

specified as datums and referred to as datum features. The datum feature symbol identifies

physical features and shall not be applied to centerlines, center planes, or axes. This symbol is

placed in the following locations on a drawing:

On the outline of a feature surface in the view where the surface appears as an edge.

On a leader line directed to the surface. The leader line can be shown as a dashed line if the

datum feature is not on the visible surface.

On an extension line projecting from the edge view of a surface, clearly separated from the

dimension line.

On a chain line next to a partial datum surface.

On the dimension line or an extension of the dimension line of a feature of size when the datum

is an axis or center plane.

On the outline of a cylindrical feature or an extension line of the feature outline, separated from

the size dimension, when the datum is an axis.

Above or below and attached to a feature control frame.

Datum feature symbols are commonly drawn using thin lines with the symbol size related to the

drawing lettering height. The triangular base on the datum feature symbol can be filled or

unfilled, depending on the company or school preference. The filled base helps easily locate

these symbols on the drawing. Each datum feature on a part requiring identification must be

assigned a different datum identification letter. Uppercase letters of the alphabet, except theletters I, O, and Q, are used for datum feature symbol letters. These letters are not used because

they can be mistaken for numbers. Example 3-1 shows the specifications for drawing a datum

feature symbol.


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Datum Feature

The datum feature is the actual feature of the part that is used to establish the datum. When the

datum feature is a surface, it is the actual surface of the object that is identified as the datum.Look at the magnified view of a datum feature placed on the simulated datum in Example 3-2.

Study the following terms: Actual mating envelope: The smallest size that can be contracted

about an external feature or the largest size that can be expanded within an internal feature.

Datum feature: The actual feature of the part (such as a surface). Datum feature simulator: The

opposite shape of the datum feature. The datum feature simulator is one of two types:

1. The theoretical datum feature simulator is a perfect boundary used to establish a datum from a

specified datum feature.

2. The physical datum feature simulator is the physical boundary used to establish a simulated

datum from a specified datum feature. The manufacturing inspection equipment associated with

the datum feature or features is used as the physical object to establish the simulated datum or

datums. Physical datum feature simulators represent the theoretical datum feature simulators

during manufacturing and inspection.

A datum feature simulator can be one of the following:

Maximum material boundary (MMB).

Least material boundary (LMB).

The actual mating envelope.

A tangent plane.

A mathematically defined contour.

Datum feature simulators shall have the following requirements:


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Perfect form.

Basic orientation to each other for all the datum references in the feature control frame.

Basic location relative to other datum feature simulators for all the datum references in the

feature control frame, unless a translation modifier or movable datum target symbol is specified.

Movable location when a translation modifier or movable datum target symbol is specified.

In actual practice, measurements cannot be made from theoretical datum features or datum

feature simulators. This is why manufacturing inspection equipment is of the highest quality for

making measurements and verifying dimensions even though they are not perfect.

Datum plane: The theoretically exact plane established by the simulated datum of the

datum feature.

Simulated datum: A point, axis, line, or plane consistent with or resulting from processing or

inspection equipment, such as a surface plate, inspection table, gage surface, or a mandrel. The

simulated datum plane in Example 3-2 is the plane derived from the physical datum feature

simulator and coincides with the datum plane when the datum plane is in contact with the

simulated datum plane.

When a surface is used to establish a datum plane on a part, the datum feature symbol is placed

on the edge view of the surface or on an extension line in the view where the surface appears as a

line. Refer to Example 3-3. A leader line can also be used to connect the datum feature symbol

to the view in some applications.


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Geometric Control of Datum Surface

The datum surface can be controlled by a geometric tolerance such as flatness,

straightness, angularity, perpendicularity, or parallelism. Measurements taken from a datum

plane do not take into account any variations of the datum surface from the datum plane. Any

geometric tolerance applied to a datum should only be specified if the design requires the

control. Example 3-4 shows the feature control frame and datum feature symbol together.

Example 3-5 is a magnified representation that shows the meaning of the drawing in Example

3-4. The geometric tolerance of 0.1 is specified in the feature control frame in Example 3-4. The

maximum size that the part can be produced is the upper limit of the dimensional tolerance, or

MMC. The MMC is 12.5 + 0.3 = 12.8. The minimum size that the part can be produced is the

lower limit of the dimensional tolerance, or LMC. The LMC is 12.50.3 = 12.2.


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The DatumReference F Datum Reference Frame Concept

Datum features are selected based on their importance to the design of the part. Generally

three datum features are selected that are perpendicular to each other. These three datums are

called the datum reference frame (DRF). The datums that make up the datum reference frame

are referred to as the primary datum, secondary datum, and tertiary datum. As their names imply,

the primary datum is the most important, followed by the other two in order of importance. Refer

to Example 3-6 and notice how the direction of measurement is projected to various features on

the object from the three common perpendicular planes of the datum reference frame. Theprimary datum must be inspected first, the secondary datum inspected second, and the tertiary

datum inspected third regardless of the letters. For example, the letters in the feature control

frame might be C, B, A, where C is the primary datum, B is the secondary datum, and A is the

tertiary datum. The datum feature symbols on the drawing relate to the datum features on the

part. Notice datum feature symbols A, B, and C as you look at Example 3-7. Also, notice the

datum reference order A, B, C in the feature control frame. The datum reference in the feature

control frame tells you that Datum A is primary, Datum B is


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Two and three perpendicular

Primary, Secondary, and Tertiary Features & Datums:

The primary is the first feature contacted (minimum contact at 3 points), the secondary

feature is the second feature contacted (minimum contact at 2 points), and the tertiary is the third

feature contacted (minimum contact at 1 point). Contacting the three (3) datum features

simultaneously establishes the three (3) mutually perpendicular datum planes or the datum

reference frame. If the part has a circular feature that is identified as the primary datum feature

then as discussed later a datum axis is obtained which allows two (2) mutually perpendicular

planes to intersect the axis which will be the primary and secondary datum planes. Another

feature is needed (tertiary) to be contacted in order orientate (fix the two planes that intersect the

datum axis) and to establish the datum reference frame. Datum features have to be specified in

an order of precedence to properly position a part on the Datum Reference Frame. The desired

order of precedence is obtained by entering the appropriate datum feature letter from left to right

in the Feature Control Frame (FCF) (see Section 5 for explanation for FCF). The first letter is the

primary datum, the second letter is the secondary datum, and the third letter is the tertiary datum.

The letter identifies the datum feature that is to be contacted however the letter in the FCF is the

datum plane or axis of the datum simulators. See Fig. 5-3 for Datum Features & Planes.


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Datum Feature vs Datum Plane:

The datum features are the features (surfaces) on the part that will be contacted by the

datum simulators. The symbol is a capital letter (except I,O, and Q) in a box such as A used in

the 1994 ASME Y14.5 or -A- used on drawings made to the Y14.5 before 1994. The features are

selected for datums based on their relationship to toleranced features, i.e., function, however they

must be accessible, discernible, and of sufficient size to be useful. A datum plane is a datum

simulator such as a surface plate. See Fig. 5-4 for a Datum Feature vs a Datum Plane.

Datum Plane vs Datum Axis:

A datum plane is the datum simulator such as a surface plate. A datum axis is also the

axis of a datum simulator such as a three (3) jaw chuck or an expandable collet (adjustable gage).

It is important to note that two (2) mutually perpendicular planes can intersect a datum axis

however there are an infinite number of planes that can intersect this axis (straight line). Only

one (1) set of mutually perpendicular planes have to be established in order to stabilize the part

(everyone has to get the same answerdoes the part meet the drawing requirements?) therefore

a feature that will orientate or clock or stabilize has to be contacted. The datum planes and

datum axis establish the datum reference frame and are where measurements are made from. See

Fig. 5-5 for Datum Feature vs Datum Axis.


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GEOMETRIC ANALYSIS AND APPLICATION:-

The subject of Geometric Analysis is motivated by viewing analytical problems via an

understanding the Geometric properties of the functionals associated with these problems. As an

illustration of this type of analysis consider the solution for the following linear system

a11x1 + a12x2 + + a1nxn =b1

a21x1 + a22x2 + + a2nxn =b2

.

.

.

am1x1 + am2x2 + + amnxn =bm

(0.1)

For instance, each individual equation in (0.1) can be viewed geometrically as an affine hyper

plane, thus the analytical problem of finding solutions is now shifted to a geometric problem of

intersections of hyper surfaces. As an illustration, consider the following:

(0.2) 2x1 + 3x2 + 4x3 =0

(0.3) 4x1 + 6x2 + 8x3 =1

Then, subtracting (0.3) from (0.2) we have

2x1 + 3x2 + 4x3 = 1


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Which contradicts the initial set of equations; hence there cannot be a solution. Alternatively,

each equation (0.2) and (0.3) gives an affine hyper plane with normal h2, 3, 4i, therefore they are

parallel. Since the distance to the origin (given by bi) is different, they cannot intersect, hence a

solution does not exist. This simple idea of viewing the interplay between Analysis and

Geometry is a very influential and classical area of Mathematics, some famous problems are as

follows.

1. Variational Calculus

Find a solution of f(x) = 0 by finding Critical Points of the antiderivative

F(x)

2. Yamabe Problem - Uniformization Theorem for Surfaces (M2, g)

Can we find a metric conformal to g with constant scalar curvature? That

is, solve

_u K(x) e2u = 0.

3. Minimal Surface Theory and Bubbles Given a simple wire frame in R3, can you find a

smooth surface with the given frame as its boundary with least area? That is, minimize the

energy associated to the frame and show that the minimize corresponds to a (classical) smooth

surface. In these notes we will focus on ideas from Variational Calculus, in particular, Critical

Point Theory and discuss the celebrated Mountain Pass Theorem of Ambrosetti and Rabinowitz

[1]. The approach here follows the presentation in the book of Jabri [3].

1. Geometric AnalysisVariational and topological methods have proved to be powerful tools in the solution of concrete

nonlinear boundary value problems appearing in many disciplines where classical methods have

failed. The ideas to be presented here use the analysis inspired in the geometry of a mountain

pass. The abstract process in modern critical point theory has its roots in the Dirichlet principle

which he postulated at Gottingen. Given an open bounded set in the plane and a continuous

function h : @ R, the boundary value problem

(1.1) __u = 0 , in

u = h , in @

admits a smooth solution u that minimizes the functional

(1.2) _(u) = Z

2


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Xi=1

(Dih(x))2 dx

in the set of smooth functions defined on that are equal to h on @. The Euler equation

corresponding to (1.2) is the equation (1.1). Euler established the principle above by showing

that any smooth minimizer of (1.2), such that u = h on @, is a solution of (1.1). Weierstrass

pointed out in 1870 that the existence of the minimum is not assured in spite of the fact that the

functional _ may be bounded from below. The subtle difference between minimum and infimum,

not yet perceived in those these early times, was made. He proved that the functional

(1.3) (u) = Z 1

1

(xu(x))2dx

possesses an infimum but does not admit any minimum in the set

C = _u C1[1, 1]; u(1) = 0, u(1) = 1 .

Indeed, if we consider the sequence

un =1

2

+arctan( x

n)

2 arctan( 1

n),

then, un C and (un) 0. If some u was a minimum, then xu(x) = 0 on [1, 1].

Therefore, u = constant, in contradiction with u(1) = 0 and u(1) = 1. Major contributions

looking for absolute minimizers of functionals bounded from below were also made by pioneers

like Lagrange, Legendre, Jacobi, Hamilton, Poincare, etc. This was revisited by Birkhoff in

1917 who succeeded to obtain a minimax principle were critical points u are such that _(u) =

infAAsupxA _(x) and A is a family of particular sets. The remaining ingredient for the

modern theory of minimax theorems was a notion of compactness introduced in the 1960s by

Palais, Smale, and Rothe as we will see in the sequel.

Geometric Analysis

The research area of Geometric Analysis historically has grown out of the study of

calculus and differential equations involving curves and surfaces or domains with curved


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boundaries. This has been the origin of many mathematical disciplines such as Differential

Geometry, Lie Group Theory and the Calculus of Variations, which are subareas of Geometric

Analysis. The development of techniques to handle the mathematical intricacies of doing

analysis on manifolds (the higher dimensional and intrinsic analogues of curves, surfaces and

domains) is fundamental to much of Engineering and Physics. It is therefore not surprising that

much of the impetus for current research in Geometric Analysis comes from these disciplines.

Physics in particular, through theoretical developments in relativity theory, quantum mechanics

and string theory, has spurred many exciting new research directions in geometric analysis. For

instance, as a consequence of these motivations, tremendous progress has been made on

topological classification problems for manifolds.

Problems in Applied Mathematics or Engineering can also give rise to deep and

fascinating geometric questions. An example is the following: by carefully listening to a drum

being played can you reconstruct the drum? More specifically do the characteristic frequencies

found in a Fourier decomposition of the sounds coming from the drum allow you to

mathematically recover the particulars of the drum. Mark Kac put it succinctly, "Can you hear

the shape of a drum"? One can pose a similar inverse problem on a general Riemannian

manifold, which is a manifold endowed with a local metric (distance function); namely, from a

knowledge of the frequencies of the standing solutions of the wave equation on a Riemannian

manifold, how much of the manifold and its metrical properties can one recover?

This kind of broad interplay between physical motivations, geometric reasoning and

various analytical methods is characteristic of the activities of our faculty and graduate students

doing research in Geometric Analysis.

Applications to the field of vision and pattern recognition will follow. More generally,

topics of study are motivated by nonlinear physics and will include areas such as integrable

systems, optimal control, Virasoro and Krichever-Novikov algebras, Loewner equations, and

sub-Riemannian/Lorentzian geometric analysis. In other words, we shall work on a theory of

constrained motion in difficult, mostly infinite dimensional, spaces

Datum Systems (1)

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