IE 305
Dev Rajani, Abhishek Patel, Eduardo Dominguez, Gaunte shoter
LABORATORY M_5
MEASUREMENT UNCERTAINTY
Objective:
This metrology Laboratory exercise will give you an opportunity to carry
out the metrology and the measurement uncertainty associated with that measurement.
Your results, analysis and conclusions should be reported in well-structured and easy to
read quality report.
Deliverable: One group report of activities due one week from today, in Lab.
Laboratory activities:
Your group will be provided with a ring work piece manufactured using the machining
processes of the FAME Lab. The objective of the lab is to use a set of metrology
instruments to get an indirect measurement of the inside diameter of the ring and then carry
out the analysis to determine the uncertainty associated with your measurement The steps
needed for the above tasks are listed below:
1. Each group will be given a ring to be measured and a set of 4 high precision spheres
(the small spheres are .75 in. nominal diameter and the larger sphere is 1.0 in.
diameter). Set the ring on top of your granite or cast iron plate and place the spheres
inside the ring as per the drawing below. Use your Vernier height gage to take three
(10) repeated measurements of the distance H from the top of the ring to the
reference granite or cast iron plate as per the drawing. The 10 measurements should
be taken under conditions of “Repeatability” (What does this mean?). Repeat the
procedure given above using the Mitutoyo digital height gage (this is a more
expensive and better quality instrument than the one you have on your table) by
taking 3 repeated measurements.. Compute the averages for each set of
measurements and set them aside.
D2
D1
H
Measurements:
Vernier Height Gage Height
Measurement Hv (in.)
Measurement # (i)
Reading (Hvi)
in
1.575
1
1.576 in
2
3
1.575 in
4
1.575 in
1.579 in
5
6
1.575 in
1.575in
7
8
1.575 in
1.579 in
9
1.575 in
10
1.5763 in
∑10 𝐻𝐻
�𝐻𝐻���𝑣𝑣 = 1 𝑣𝑣𝑣𝑣
𝑛𝑛
𝑠𝑠����
𝐻𝐻𝑣𝑣
���� 2
∑10
1 (𝐻𝐻𝑣𝑣𝑣𝑣 − 𝐻𝐻𝑣𝑣 ) 1/2
=[
]
𝑛𝑛(𝑛𝑛 − 1)
INSTRUMENT
Mitutoyo
Digital Height
Gage
5.96
10-4
x
Meas. #1
Meas. #1
Meas. #1
1.577 in
1.572 in
1.577in
in
Instrument
Average
����
𝐻𝐻𝑚𝑚 = 1.5753
in
2. Determine the functional relationship between the inner ring diameter D and the
variables D1, D2 and H.
D1
D2
B
H
A
C
R
Inner ring diameter = D = 2R
Large sphere diameter = D1 = 2R1
Small sphere diameter = D2 = 2R2
From the figure above, we have the following relations:
𝑅𝑅 = 𝐴𝐴𝐴𝐴 + 𝑅𝑅2
𝐴𝐴𝐴𝐴 = �𝐴𝐴𝐴𝐴2 − 𝐴𝐴𝐴𝐴 2
𝐴𝐴𝐴𝐴 = 𝐻𝐻 − 𝑅𝑅1 − 𝑅𝑅2 𝑎𝑎𝑛𝑛𝑎𝑎 𝐴𝐴𝐴𝐴 = (𝑅𝑅1 + 𝑅𝑅2 )
𝐴𝐴𝐴𝐴 = �(𝑅𝑅1 + 𝑅𝑅2 )2 − (𝐻𝐻 − 𝑅𝑅1 − 𝑅𝑅2 )2
𝐴𝐴𝐴𝐴 = �𝐻𝐻(2𝑅𝑅2 + 2𝑅𝑅1 − 𝐻𝐻)
1
𝑎𝑎𝑛𝑛𝑎𝑎 𝐷𝐷 = 2(𝐴𝐴𝐴𝐴 + 𝑅𝑅2 ) = 2[𝐻𝐻(𝐷𝐷1 + 𝐷𝐷2 − 𝐻𝐻)]2 + 𝐷𝐷2
3. Use the relationship given above to compute the probable value for the inner ring
diameter using the measurements retuned by your table Vernier height gage (Dv)
and the probable value for the inner ring diameter using the measurements returned
by the Mitutoyo Digital height gage (Dm).
1.2733 in
Dv = _______________________________
1.2746 in
Dm = _______________________________
4. Use the Law of Propagation of Uncertainties to derive an expression for the
combined uncertainty of the inner ring diameter as a function of the standard
uncertainties of the independent variables D1, D2 and H.
𝑢𝑢𝑐𝑐2 (𝐷𝐷)
𝜕𝜕𝜕𝜕 2 2
𝜕𝜕𝜕𝜕 2 2
𝜕𝜕𝜕𝜕 2 2
=�
� . 𝑢𝑢 (𝐷𝐷1 ) + �
� . 𝑢𝑢 (𝐷𝐷2 ) + � � . 𝑢𝑢 (𝐻𝐻)
𝜕𝜕𝐷𝐷1
𝜕𝜕𝐷𝐷2
𝜕𝜕𝐻𝐻
S
5. We now need to determine expressions for the coefficients (the partial derivative
terms in the equation above) of each standard uncertainty above, as well as
estimates for the standard uncertainties u2(D1), u2(D2) and u2(H).
5.1. Compute each partial derivative and enter the expression and its evaluation in
the table below:
Coefficient
Algebraic Expression
Numerical Evaluation
𝜕𝜕𝜕𝜕 2
)
𝜕𝜕𝐷𝐷1
(e)
HV=14-3046 in
(
𝜕𝜕𝜕𝜕 2
(
)
𝜕𝜕𝐷𝐷2
𝜕𝜕𝜕𝜕
( )2
𝜕𝜕𝐻𝐻
-nt)
(
I-e
Hm--14.2047s in
2
Mr 22.8689
=
Hm
Mr
in
22-74259 in
8.6973 in
=
"m 8.60235 in
=
5.2 We will now determine estimates for the standard uncertainties for each
variable as follows:
a. We will use a type B standard uncertainty evaluation for D1 and D2 as
follows. We know that the spheres D1 and D2 are Grade 25 steel calibration
spheres. Conduct an internet search to find out the standard tolerance associated
with the diameter of a Grade 25 calibration sphere. Assume that this tolerance
spans the bounds of a uniform distribution and calculate the standard deviation
of this distribution (you will find an expression for the standard deviation of the
uniform distribution in your Probability and Statistics book). Use the estimate
of the standard deviation as your estimate for the standard uncertainty
associated with spheres D1 and D2.
To
pool
0
X
↳*
9.7733418
=
u (D1) = u (D2) = ________________________
b. We will also use a Type B evaluation for the standard uncertainty of the
height Hm returned by the Mitutoyo Digital height gage as follows: The
Mitutoyo Digital height gage specifications show that this type of instrument
has a tolerance of ±.0002 in. on its readings. Assume again that this tolerance
spans the bounds of a uniform distribution and compute the standard deviation
of this distribution. Use the estimate of the standard deviation as your estimate
for the standard uncertainty associated with measured height Hm.
1.1547 X104
u (Hm) = ______________________________
c. Finally, we will use a Type A evaluation for the standard uncertainty associated
with the height Hv returned by the Vernier height gage. You have already
performed 10 measurements for Hv and computed the average value for Hv and
the sample estimate of the standard deviation of this average in the last two
rows of the Vernier measurement table. The average value is your estimate for
H and the sample estimate of the standard deviation of the average becomes
your estimate for u(H).
in
3.96x10-4
u (Hv) = ______________________________
6. You now have all the components required to calculate the combined uncertainty
of D using the Vernier height gage measurements and the combined uncertainty
of D using the Mitutoyo height gage measurements. Compute and compare these
two combined uncertainties:
14.3046(5.7753x105)22.8683(5.7753x10-5):8.697343.96x104,
𝑢𝑢2 (𝐷𝐷𝑣𝑣 ) = ______________________________________________________
=0.00733
𝑢𝑢2 (𝐷𝐷𝑚𝑚 ) = ______________________________________________________
14.2075(5.7753x10-5) 22.74259(5.7753x1-) 8.60235(1.1547x8
+
+
=0.003 1
7. Draw your conclusions if you know that the Mitutoyo Digital Gage costs about
$3,720, the Vernier height gage about $274 and Grade 25 calibration steel spheres
about $25 each.
It is better to get Mitutoyo Digital Gage because there is a difference between Dm
and Dv.
Digital gives us a better and clear measurement.