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Please look at the Project guidelines and the Description,,,

Please look at the Project guidelines and the Description,,,

Thank you,,,

Math 250A

Spring 2018

How to Work on Your Project; What is Expected

Oyez! Oyez! Come hither and hearken to these words of advice, encouragement (I

hope!) and Rules of the Game!

This is an important, sizable, lengthy assignment. To do well you should start

immediately, and work on it every day. You may need all of the time you have been given to

complete your project by the due date.

1.

Start today. Let your subconscious work for you; it can do amazing things. If you

immerse yourself in the project, solutions will come to you at the most unexpected

times.

2.

Read the entire project to see what it encompasses. Do not worry about the details

the first time through. Do this today.

3.

Next, read the project very carefully and make a list of any unfamiliar words or

concepts you encounter. Should concepts arise about which you are unsure, you

must learn about and understand them before you can do the project. Even if you

understand all of the terms and concepts, do not assume that the project is easy; it is

not. Should you wait till the last few days to start, it is doubtful that you will be able

to finish on time.

4.

You may need to do outside reading. In addition to your textbook there are lots of

books in the library that contain information you might find helpful. You may also

find help on the internet.

5.

While I expect all groups to work independently, I do not expect that you can finish

the project without assistance. Your first source of assistance should be the other

members of your group. I also encourage you to come to talk to me about your

project. I will not solve any problem you have, but I may be able to suggest a helpful

direction to take. Bring specific questions, and be prepared to show clear, written

work that you have prepared. You may not consult with anyone other than your

group members and me.

6.

When you have finished the work on your project, you need to prepare it in written

form. The paper you turn in should have a mix of equations, formulas, and prose to

support your conclusions; it should resemble a mathematics textbook presenting a

complete idea. In particular, it should not sound as if you are merely answering a

series of questions. Use complete sentences, proper grammar, and correct

punctuation. Correct spelling is important. The prose should be written to convey to

the reader an explanation of what you have done and why you have done it. It should

be written in such a way that anyone who knows the material in this course can read

and understand it. Your grade will depend on the written presentation as well as the

mathematical content.

Math 250A

Spring 2018

Here are some suggestions for group work:

1.

You should plan your first group meeting as soon as possible, but before the first

meeting, you should have read the project carefully and given it some thought. At

your first meeting, you should plan a method of attack and you may wish to divide

the labor amongst the group members. Different members of the group may perform

different tasks, but I expect all members of the group to understand all parts of the

solution. Regular group meetings to discuss the progress of the solution are

important.

2.

Group projects often work well when each group member first spends some time

working individually on the problem followed by a group meeting in which the ideas

and progress of the members are discussed. I encourage you to try this. Again, this

discussion of results and details should be confined to the other members of your

group and me.

3.

Your group will turn in one project report with each group memberâ€™s name clearly

shown on the cover. Also, each group member will turn in separately an assessment

of the other group membersâ€™ contributions to the solution and report. This

assessment will address the following questions:

a. Did each group member attend and participate in each meeting?

b. Did each group member complete his/her assigned task(s)?

c. Did each group member read the final report?

All group members may not necessarily receive the same grade on the project.

Your score out of 100 points will be based upon your written presentation as well as the

mathematical content according to the following distribution:

Mathematical Content (50 points): Your approach, procedure, derivations, and

calculations should be clear, complete, and correct.

Presentation and Style (30 points): Your paper should include the general background

of the project along with your work and conclusions. The report should be a cohesive

presentation that a reasonably literate individual who knows calculus can follow and

appreciate. Your report should not be written as though you were merely solving a

series of separate homework problems.

Grammar (10 points): You should use correct English.

Mathematical Notation (10 points): You should use correct and appropriate

mathematical notation.

Extra Credit (up to 10 points): I will reward any imaginative ideas, mathematical or

artistic additions, historical background, and so on, which go beyond normal

expectations.

Math 250A

Spring 2018

Project Group Assessments

Name

List your fellow group members and answer the following questions for each member:

Group Member

1. Did this member attend and participate in each meeting?

2. Did this group member complete his/her assigned task(s)?

3. Did this group member read the final report?

Group Member

1. Did this member attend and participate in each meeting?

2. Did this group member complete his/her assigned task(s)?

3. Did this group member read the final report?

Group Member

1. Did this member attend and participate in each meeting?

2. Did this group member complete his/her assigned task(s)?

3. Did this group member read the final report?

Group Member

1. Did this member attend and participate in each meeting?

2. Did this group member complete his/her assigned task(s)?

3. Did this group member read the final report?

Group Member

1. Did this member attend and participate in each meeting?

2. Did this group member complete his/her assigned task(s)?

3. Did this group member read the final report?

Math 250A

Spring 2018

PROJECT

INSTRUCTIONS: Refer to the project guidelines for complete instructions and advice.

This project is due by the beginning of class on Wednesday, April 25, 2018. A first draft

is due by the beginning of class on Monday, April 2, 2018. Your (entire) team will meet

with your instructor by Wednesday, March 21, 2018 to discuss the project and your

teamâ€™s progress. Team members who fail to show up for their scheduled meeting will be

docked 5 points on their project grade.

INSPECTING COMPLEX HARDWARE

Complex machined hardware components are designed to fit together with very tight

tolerances, sometimes as small as 0.001â€. These parts are manufactured using

computerized numerically controlled (CNC) milling machines that are programmed to cut

elaborate contours. In a typical CNC mill, the contour consists of segments of straight

lines and arcs of circles. After the parts are manufactured, they must be inspected to

ensure that the required tolerances are maintained. The inspection process consists of

two steps: collecting the inspection data and analyzing the data.

The data are collected using a coÃ¶rdinate measuring machine (CMM). The CMM has a

probe with a small ruby sphere at the end. The probe follows the contour of the part and

periodically records the x- and y-coÃ¶rdinates of the center of the sphere. For a typical

inspection the coÃ¶rdinates might be recorded every 0.020â€ around the contour, or fifty

points per inch.

The data are analyzed by comparing the recorded coÃ¶rdinates with a computer design file

giving the nominal, or ideal, contour for the part. Should the coÃ¶rdinate system for the

data points and the coÃ¶rdinate system for the design file be identical, the analyst can

compute the distance from each data point to the corresponding point in the design. If the

part were manufactured perfectly, all of these distances should equal the radius of the

probe sphere. Distances computed to be too large indicate that the part is oversized;

distances computed to be too small indicate that the part is undersized. The objective of

analyzing the inspection data is to determine whether the part is in tolerance everywhere,

or to note where it is out of tolerance. It may be possible to remachine an out-oftolerance part to bring it into tolerance. Failing this, a variance may be granted for the

part, or it may be scrapped.

The typical situation is that the coÃ¶rdinate systems for the design file and for the

inspection data are not identical. In this case, the first thing the analyst must do is to

transform the inspection coÃ¶rdinates so that they â€œfitâ€ the design file coÃ¶rdinates; only

then can the analyst determine whether the part is in tolerance or not. The purpose of this

project is to investigate how this transformation is accomplished, and how the â€œfitâ€ of the

inspection data to the design file is determined.

The fit of the inspection data to the design file is done using the Least Squares method:

for each data point, the analyst computes the square of the distance from that point to the

Math 250A

Spring 2018

nearest point on the design file. The degree of fit is the sum of these squared distances

for all of the data points. The â€œbestâ€ fit is obtained when this sum is minimized, hence

the name â€œLeastâ€ Squares. The first step in the process is to develop a method for

determining the distance from a data point to the nearest point in the design file.

Step 1: Distance from a Data Point to a Design File Contour

The contour consists of line segments and arcs of circles. We assume that the analyst has

determined which segment or arc is closest to a given data point.

A. Given a data point with coÃ¶rdinates ï€¨ x, y ï€© , and a line segment with start point

and end point ï€¨ xe , ye ï€© , compute the (shortest) distance from the data

point to the line segment. There are three distinct cases you have to consider,

depending on the relative location of the data point and the segment.

B. Given a data point with coÃ¶rdinates ï€¨ x, y ï€© , and an arc of a circle with center

ï€¨ xs , ys ï€©

ï€¨ xc , yc ï€© ,

radius r , starting and ending angles ï¡ and ï¢ , respectively,

(0 ï‚£ ï¡ ï€¼ 2ï° , ï¢ ï€¾ ï¡ , ï¢ ï‚£ ï¡ ï€« 2ï° ) , compute the (shortest) distance from the data

point to the arc. There are several cases you have to consider, depending on the

relative location of the data point and the arc â€“ in particular, whether the data

point is inside the circle or outside the circle.

Step 2: Transformation of a Data Point

The data points will undergo a transformation consisting of a rotation about

ï€¨ 0, 0 ï€©

through an angle ï± , followed by a translation of ï€¨ ï„x, ï„y ï€© .

A. Given a data point with coÃ¶rdinates ï€¨ x, y ï€© , compute the coÃ¶rdinates of the data

point after undergoing the given transformation.

B. Explain why the order of the operations in the transformation â€“ rotation followed

by translation â€“ is important. Be specific in your explanation.

We will refer to the data point before the transformation as the original data point, and

after the transformation as the transformed data point.

Step 3: Distance from a Data Point to a Contour as a Function of Three Variables

The distance from a given transformed data point to a design file contour is a function of

three variables.

A. What are the three independent variables for this function?

B. Using the results from Steps 1 and 2, write the formula(e) for the distance from a

transformed data point to a line segment.

Math 250A

Spring 2018

C. Using the results from Steps 1 and 2, write the formula(e) for the distance from a

transformed data point to an arc of a circle.

D. List and explain any difficulties with using the formulae from parts B and C. In

particular, what assumptions might be violated by a transformed data point?

(Figure 1 might help.)

Step 4: The Function to be Minimized

The Least Squares function to be minimized is the sum of the squares of the distances

from the data points to the contour. To make the problem tractable, assume that there are

only two line segments in the contour â€“ xs1 , ys1 â€“ xe1 , ye1 and xs2 , ys2 â€“ xe2 , ye2

ï»ï€¨

ï€© ï€¨

ï€©ï½

ï»ï€¨

ï€©ï½

ï€© ï€¨

â€“ and that all transformed data points have distances measured normal to the appropriate

segment (see Figure 2). The n1 data points for segment 1 have (initial, untransformed)

coÃ¶rdinates

ï€¨x

1,1

, y1,1 ï€© , ï€¨ x1,2 , y1,2 ï€© ,

ï€¨

ï€©

, x1, n 1 , y1, n 1 ; the n2 data points for segment 2 have

(initial, untransformed) coÃ¶rdinates ï€¨ x2,1 , y2,1 ï€© , ï€¨ x2,2 , y2,2 ï€© ,

ï€¨

ï€©

, x2,n 2 , y2,n 2 .

A. Write the formula for the squared distance from the transformed data point having

initial coÃ¶rdinates ï€¨ x1,i , y1,i ï€© to the first segment with end points xs1 , ys1 and

ï€¨x

e1

ï€©

ï€¨

ï€©

, ye1 . Do the same for the points for the second segment.

B. Write the formula for the sum of the squared distances for all of the transformed

data points. Use sigma notation. This is the function to be minimized; for

convenience, this function will be known as F ï€¨ v1 , v2 , v3 ï€© . (You didnâ€™t think I was

going to name the real variables and give away the answer to 3A, did you?)

Step 5: Minimizing F, Part I

A. Explain in general how you would minimize a function of three independent

variables.

B. Write the three equations that must be solved to minimize F .

C. Explain why a solution to these three equations might not give you a minimum

value for F . What would such a solution represent physically in the inspection

problem? Draw a picture of this sort of situation.

Math 250A

Spring 2018

Step 6: Minimizing F, Part II

The difficulty in finding a solution to the set of equations in Step 5B is that the equations

are nonlinear. However, numerical techniques can be used to approximate a solution.

These techniques can be repeated to get closer and closer approximations to the actual

solution.

One such technique is to substitute a linear function for each nonlinear function in F .

The linear function should be chosen so that it gives the best approximation possible to

the nonlinear function it is replacing.

A. Identify the nonlinear functions that you intend to replace with linear

approximations.

B. For each of the nonlinear functions, explain how you will select an appropriate

linear function with which to replace it. (Hint: think about Taylor polynomials.)

C. List and explain any assumptions needed to justify the use of the linear functions

identified in part B. In particular, explain why these assumptions are reasonable.

D. Write equations for the linear approximations to the three equations you derived

in 5B.

Step 7: Minimizing F, Part III

When you solve the set of equations in 6D, the solution you get is an approximation to

the solution for the set of real equations in 5B.

A. Explain how you will use the solution to the set of equations in 6D; i.e., what will

you do with the values of the variables obtained in the solution?

B. Because the solution is only approximate, you will repeat some of the steps above

to get a better approximation. Explain which steps you will repeat, and how the

physical situation changes with each repetition.

C. Explain how you will decide when your approximation is close enough to stop the

repetitions. (Hint: you havenâ€™t yet used the tolerance on the part.)

Step 8: Finishing Up

You now have a solution that minimizes F .

A. Explain what minimizing F means physically.

B. Reiterate the goal of the inspection process; i.e., what were you really trying to

accomplish here?

C. Explain how you will use the solution found in Step 7 to accomplish this goal.

Math 250A

Spring 2018

Step 9: Extra Credit

Because F is nonlinear, the actual solution to minimizing F cannot be obtained by

simply adding the incremental values from each iteration. However, for a final

inspection report, the values used to minimize F should be included.

A. Explain how you would determine the actual solution for minimizing F using the

incremental values. (Hint: look at Step 2B.)

Many machined parts have contours and tolerances in three dimensions, rather than two.

B. What transformations would be necessary for a three-dimensional inspection data

analysis? How many variables would be involved?

Math 250A

Spring 2018

Figure 1

Math 250A

Spring 2018

Figure 2

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