2005 American Society of Engineering Education Annual ...€¦ · Analytical Solution 2nd...

1 Tracy Van Zandt, UMass LowellMechanical Engineering Department

Interweaving Numerical Methods Techniques in Multi-Semester Projects

y = 11.947x - 0.0861R2 = 0.9987

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Dr. Peter Avitabile, Dr. John McKelliget, Tracy Van ZandtMechanical Engineering DepartmentUniversity of Massachusetts Lowell

2005 American Society of Engineering Education Annual Conference and ExpositionJune 12-15, 2005, Portland Oregon




• Do not understandneed for basicSTEM material

• Course materialappears disjointed

• Modular course environment reinforces this disjointed appearance

• Students hit the RESET button after each course

Students have difficulties because:

Student views materialin a disjointed fashion

Professor clearly seeshow pieces fit together

Problems teaching mechanical engineering



Problems teaching mechanical engineering

• Students take Math Methods in Junior year• Learn numerical integration and differentiation and regression in preparation for lab

• Two projects have been added to this course to go beyond the typical textbook presentations of these topics

• Students can then use MATLAB GUIs to further their understanding



Regression analysis

y = 0.0043x3 - 0.15x2 + 1.8202x - 0.1147R2 = 0.9981

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Regression analysis

y = 11.947x - 0.0861R2 = 0.9987

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)• Vital tool used widely in engineering analysis• Traditional presentation:

Students are given a simple data setPerform regressionReport R2 value & equation of line



Regression analysis

Problems with traditional approach:• Students simply apply regression over all data points

• Students think that increasing the order of regression always improves the result, because it improves the R2 value

We want to make them THINK about how to best perform regression on their data



Regression analysis

Students perform regression using hand calculations, MATLAB, and/or ExcelTutorial material is given Part A: LVDT Calibration Data

y = 5.8456x + 2.9486R2 = 0.7926

02468

1012141618

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Displacement (in)

Mag

nitu

de (V

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Two major points we’d like them to consider:

1. Should the regression be performed over the whole range of the data?

2. What order of regression is most appropriate?

Regression analysis

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Students are given two data sets to study

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Regression analysis

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Data set #1

• Bi-Linear• Calibration data from an LVDT



Regression analysis

y = 5.8456x + 2.9486R2 = 0.7926

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Fit regression line through all data points



Regression analysis

LVDT Calibration Data

y = 12.085x - 0.1201R2 = 0.9988

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But an LVDT is linear only over a certain range



Regression analysis

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Data set #2

• Appears cubic• Data can be fit in multiple ways



Regression analysis

y = 0.1667x + 5.6667R2 = 1

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Central portion of data is linear



Regression analysis

y = -0.0606x2 + 1.2776x + 0.7459R2 = 0.998

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y = 0.0355x2 - 0.6844x + 10.417R2 = 0.9596

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Upper and lower portions of data are somewhat parabolic



Regression analysis

y = 0.0043x3 - 0.15x2 + 1.8202x - 0.1147R2 = 0.9981

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Over entire range, data is cubic



Students can then use Matlab GUI to further explore regression

Regression analysis



Regression analysis

Multiple data sets availableOr user can input data set



Regression analysis

Different order regressions may be performed



Regression analysis

The equation of the line and the R2

value are reported



Regression analysis

Points may be deselected, and they will not be included in the regression calculation



Regression analysis

The GUI can also plot the y-variance of the data

Lines are drawn above and below the regression line at a distance of 3 σ



Regression analysis

Therefore, if a data point lies outside the 3 σrange, either the data point or the regression line should probably be questioned



Regression analysis

In this case, ignoring one data point greatly improves the regression



Numerical differentiation and integration




Traditional teaching method:Students use different differentiation and integration tools on simple, well-behaved analytical data setsProblem with traditional method:Students get to the lab course, and don’t understand how to handle real data




Concepts that students have trouble with:• Understanding the use of initial conditions in integration

• Processing data that has noise, bias, drift• Processing data with large time steps

This project forces the students to consider these issues when they first see the material, so they will be better prepared when they see these problems in lab




Acceleration of falling object

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Students are given the displacement and acceleration of a falling objectObject has initial velocity – students must find this I.C. by differentiating the displacement

Displacement of falling object

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Students given data with too-large ∆T spacing

1st Integration of Sine Wave, 30 deg increments

-1.500

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Numerical SolutionAnalytical Solution

2nd Integration of Sine Wave, 30 deg increments

-1.500

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Numerical SolutionAnalytical Solution




Students given data with small bias added

1st Integration

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2nd Integration

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First integration Second integration




Students can then use the GUI to further explore numerical integration & differentiation




Add different types of errors to original signalThese replicate problems seen with real lab data




Differentiate and integrate signal



Conclusions

•Students have trouble retaining basic theory•Projects can help them better understand the need for the material

•Two projects were developed to help them better understand numerical integration and differentiation and regression

•MATLAB GUIs are available to help them further explore these topics as they relate to real measurements



Webpage http://dynsys.uml.edu

Project OverviewTechnical PapersTutorialsOnline AcquisitionDownloadsAcknowledgementsPeopleFeedback

Tutorials cover a wide assortment of integrated material Matlab GUIs are available for download



This project is partially supported by NSF Engineering Education Division Grant EEC-0314875

Multi-Semester Interwoven Project for Teaching Basic Core STEM Material Critical for Solving Dynamic Systems Problems

Dr. Peter Avitabile, Dr. John McKelliget

Acknowledgements

TIME

FREQUENCY

ACCELEROMETER

IMPACT HAMMER

FORCE GAGEHAMMER TIP

FOURIERTRANSFORM

LVDT

DISPLACEMENT

ACCELERATION

DIGITALANALOG TO

DIGITIAL DATA ACQUISITION

NUMERICAL PROCESSINGINTEGRATION / DIFFERENTIATION

( )i1ii1i

1ii xx2yyII −++= +

+−

QUANTIZATIONSAMPLINGALIASINGLEAKAGEWINDOWS

DYNAMIC TESTINGPULLS ALL THE

PIECES TOGETHER !!!

TIME

FREQUENCY

X

Y

TRANSDUCERCALIBRATION

REGRESSION ANALYSIS

HAMMER TIP CHARACTERIZATION

FOURIER SERIES & FFT

m

k c

x(t) f(t)

SDOF DYNAMIC

MODEL APPROXIMATION

SYSTEM MODEL

100

10

1

ω/ωn

ζ=0.1%

ζ=1%

ζ=2%

ζ=5%

ζ=10%

ζ=20%

ζ=0.1%

ζ=1%

ζ=2%

ζ=5%

ζ=10%

ζ=20%

0

-90

-180ω/ω n

)t(fxkdtdxc

dtxdm 2

2

=++

DIFFERENT PULSE SHAPES

)ps(a

)ps(a)s(h *

1

*1

1

1

−+

−=

FREE BODY DIAGRAM& EQUATION OF MOTION

LAPLACE & TRANSFER FUNCTION

tsinem1)t(h d

t

d

ωω

= ζω−

FIRST & SECOND ORDER SYSTEMS

SIGNAL CONDITIONER RISE & SETTLING TIME



y = 11.947x - 0.0861R2 = 0.9987

0

2

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6

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10

12

0 0.2 0.4 0.6 0.8 1

Displacement (in)

Mag

nitu

de (V

)

Dr. Peter Avitabile, Dr. John McKelliget, Tracy Van ZandtMechanical Engineering DepartmentUniversity of Massachusetts Lowell

2005 American Society of Engineering Education Annual Conference and ExpositionJune 12-15, 2005, Portland Oregon


2005 American Society of Engineering Education Annual ...€¦ · Analytical Solution 2nd...

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