Post on 09-Dec-2020
1 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
y = 11.947x - 0.0861R2 = 0.9987
0
2
4
6
8
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
Interweaving Numerical Methods Techniques in Multi-Semester Projects
2 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
• 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
3 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
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
4 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
Regression analysis
y = 0.0043x3 - 0.15x2 + 1.8202x - 0.1147R2 = 0.9981
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5 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
Regression analysis
y = 11.947x - 0.0861R2 = 0.9987
0
2
4
6
8
10
12
0 0.2 0.4 0.6 0.8 1
Displacement (in)
Mag
nitu
de (V
)• Vital tool used widely in engineering analysis• Traditional presentation:
Students are given a simple data setPerform regressionReport R2 value & equation of line
6 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
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
7 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
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
0 0.5 1 1.5 2 2.5
Displacement (in)
Mag
nitu
de (V
)
8 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
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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9 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
Regression analysis
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0 0.5 1 1.5 2 2.5
Data set #1
• Bi-Linear• Calibration data from an LVDT
10 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
Regression analysis
y = 5.8456x + 2.9486R2 = 0.7926
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0.0 0.5 1.0 1.5 2.0 2.5
Fit regression line through all data points
11 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
Regression analysis
LVDT Calibration Data
y = 12.085x - 0.1201R2 = 0.9988
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0.0 0.5 1.0 1.5 2.0 2.5
Displacement (in)
Out
put (
V)
But an LVDT is linear only over a certain range
12 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
Regression analysis
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Data set #2
• Appears cubic• Data can be fit in multiple ways
13 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
Regression analysis
y = 0.1667x + 5.6667R2 = 1
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Central portion of data is linear
14 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
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
15 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
Regression analysis
y = 0.0043x3 - 0.15x2 + 1.8202x - 0.1147R2 = 0.9981
0
2
4
6
8
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0 5 10 15 20 25
Over entire range, data is cubic
16 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
Students can then use Matlab GUI to further explore regression
Regression analysis
17 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
Regression analysis
Multiple data sets availableOr user can input data set
18 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
Regression analysis
Different order regressions may be performed
19 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
Regression analysis
The equation of the line and the R2
value are reported
20 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
Regression analysis
Points may be deselected, and they will not be included in the regression calculation
21 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
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
22 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
Regression analysis
Therefore, if a data point lies outside the 3 σrange, either the data point or the regression line should probably be questioned
23 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
Regression analysis
In this case, ignoring one data point greatly improves the regression
24 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
Numerical differentiation and integration
25 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
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
26 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
Numerical differentiation and integration
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
27 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
Numerical differentiation and integration
Acceleration of falling object
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Tim e (sec)A
ccel
erat
ion
(ft/s
ec^2
)
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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Tim e (sec)
Dis
plac
emen
t (ft
)
28 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
Numerical differentiation and integration
Students given data with too-large ∆T spacing
1st Integration of Sine Wave, 30 deg increments
-1.500
-1.000
-0.500
0.000
0.500
1.000
1.500
0 30 60 90 120
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Angle (degrees)
Am
plitu
de
Numerical SolutionAnalytical Solution
2nd Integration of Sine Wave, 30 deg increments
-1.500
-1.000
-0.500
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0.500
1.000
1.500
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150
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360
Angle (degrees)
Am
plitu
de
Numerical SolutionAnalytical Solution
29 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
Numerical differentiation and integration
Students given data with small bias added
1st Integration
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Time (sec)
Am
plitu
de
2nd Integration
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Time (sec)
Am
plitu
de
-40
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First integration Second integration
30 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
Numerical differentiation and integration
Students can then use the GUI to further explore numerical integration & differentiation
31 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
Numerical differentiation and integration
Add different types of errors to original signalThese replicate problems seen with real lab data
32 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
Numerical differentiation and integration
Differentiate and integrate signal
33 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
Numerical differentiation and integration
34 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
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
35 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
Webpage http://dynsys.uml.edu
Project OverviewTechnical PapersTutorialsOnline AcquisitionDownloadsAcknowledgementsPeopleFeedback
Tutorials cover a wide assortment of integrated material Matlab GUIs are available for download
36 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
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
37 Tracy Van Zandt, UMass LowellMechanical Engineering Department
Interweaving Numerical Methods Techniques in Multi-Semester Projects
y = 11.947x - 0.0861R2 = 0.9987
0
2
4
6
8
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
Interweaving Numerical Methods Techniques in Multi-Semester Projects