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Example 1 cont
1a
The following table shows the summations needed for the calculations of the constants in the regression model
θ 2θ θTRadians N-m Radians2 N-m-Radians
0698132 0188224 0487388 0131405
0959931 0209138 0921468 0200758
1134464 0230052 12870 0260986
1570796 0250965 24674 0394215
1919862 0313707 36859 0602274
62831 11921 88491 15896
Table Tabulation of data for calculation of important
sum=
=5
1i
5=nUsing equations described for
25
1
5
1
2
5
1
5
1
5
12
minus
minus=
sumsum
sumsumsum
==
===
ii
ii
ii
ii
iii
n
TTnk
θθ
θθ
( ) ( )( )( ) ( )228316849185
1921128316589615
minus
minus=
21060919 minustimes= N-mrad
summations
0aTand with
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Example 1 cont
n
TT i
isum==
5
1_
Use the average torque and average angle to calculate 1k
_
2
_
1 θkTk minus=
ni
isum==
5
1_
θθ
519211
=
11038422 minustimes=
528316
=
25661=
Using
)25661)(1060919(1038422 21 minusminus timesminustimes=11017671 minustimes= N-m
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Example 1 Results
Figure Linear regression of Torque versus Angle data
Using linear regression a trend line is found from the data
Can you find the energy in the spring if it is twisted from 0 to 180 degrees
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Linear Regression (special case)
Given
best fit
to the data
)( )()( 2211 nn yxyx yx
xay 1=
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Linear Regression (special case cont)
Is this correct
2
11
2
1111
minus
minus=
sumsum
sumsumsum
==
===
n
ii
n
ii
n
ii
n
ii
n
iii
xxn
yxyxna
xay 1=
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x
11 yx
ii xax 1
nnyx
iiyx
iii xay 1minus=ε
y
Linear Regression (special case cont)
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Linear Regression (special case cont)
iii xay 1minus=ε
sum=
=n
iirS
1
2ε
( )2
11sum
=
minus=n
iii xay
Residual at each data point
Sum of square of residuals
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Linear Regression (special case cont)
Differentiate with respect to
gives
( )( )sum=
minusminus=n
iiii
r xxaydadS
11
1
2
( )sum=
+minus=n
iiii xaxy
1
2122
01
=dadSr
sum
sum
=
== n
ii
n
iii
x
yxa
1
2
11
1a
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Linear Regression (special case cont)
sum
sum
=
== n
ii
n
iii
x
yxa
1
2
11
( )sum=
+minus=n
iiii
r xaxydadS
1
21
1
22
021
22
1
2
gt= sum=
n
ii
r xda
Sd
sum
sum
=
== n
ii
n
iii
x
yxa
1
2
11
Does this value of a1 correspond to a local minima or local maxima
Yes it corresponds to a local minima
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Linear Regression (special case cont)
Is this local minima of an absolute minimum of rS rS
1a
rS
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Example 2
Strain Stress
() (MPa)
0 0
0183 306
036 612
05324 917
0702 1223
0867 1529
10244 1835
11774 2140
1329 2446
1479 2752
15 2767
156 2896
To find the longitudinal modulus of composite the following data is collected Find the longitudinal modulus Table Stress vs Strain data
E using the regression modelεσ E= and the sum of the square of the
00E+00
10E+09
20E+09
30E+09
0 0005 001 0015 002
Strain ε (mm)
Stre
ss σ
(Pa)
residuals
Figure Data points for Stress vs Strain data