Download MECH373-Final-2010-F-QES

CONCORDIA UNIVERSITY
FACULTY OF ENGINEERING AND COMPUTER SCIENCE
Department of Mechanical and Industrial Engineering
Final Exam
MECH 373 Instrumentation and Measurements: MECH 373 -T and X - FALL 2010
Date: 8-Dec-2010
Time: 14:00 to 17:00
NAME ________________________________________________________________
(Please Print) SURNAME
GIVEN NAME
STUDENT ID ___________________ SECTION: _____
COURSE GIVEN BY: Dr A Kaushal and Dr J Cheung
Name and student I/D must be written in INK.
You must show all the steps in the calculation in order to gain the full mark assigned
to the question.
INDIVIDUAL WORK –CLOSED BOOK TEST
Material allowed: Approved calculator.
Answer the questions in the space provided.
Return the paper at the end of the exam.
Q1
Q2
Q3
Q4
Q5
1
Q6
Total
(100%)
Question 1 (10 marks)
Notes: Select only one appropriate answer for the multiple choice questions. Record your
answers in the given table.
1. Linearity of an instrument defines:
A, The maximum deviation of the output from true value.
B, The minimum change in input required to reflect a change in the output.
C, The deviation of the output from the ‘best-fit’ straight line.
D, None of the A, B and C.
2. Precision of an instrument expresses:
A, The closeness of the data to the mean value.
B, The closeness of the data to the true value.
C, The closeness of the data to the median value.
D, None of the A, B and C.
3. What is the total error for the instrument system shown schematically below?
A,  12.25%
B,  6.5%
C,  3.5%
D, None of the A, B and C
4. The output-input relationship of an instrument is given by:
qo  0.1  2qi
where qo is an output in volts and qi
is an input in Pa.
The static sensitivity and zero-drift of the instrument are:
A, 2 V/Pa, 50 mV
B, 2 V/Pa, 100 mV
C, 0.1 V/Pa, 2 V
D, None of the A, B and C
5. A single 120 strain gauge with a gauge factor S = 2.1 is used on an aluminum (Elastic
Modulus E = 69 x 103 MPa) tensile strength specimen. Upon loading the resistance change,
as found using a ohmmeter is 21.9 m, The applied axial stress is approximately:
A, 5 MPa
6.
B, 6 MPa
C, 7 MPa
D, None of the A, B and C
What type of error would you expect to see in the results if the measurement system is
unable to exactly follow changes in the measurand?
A, Resolution error
B, Loading error
C, Spatial error
D, Linearity error
7. Which of the following pressure transducers is the most appropriate in terms of accuracy to
measure pressure as low as 0.1 Pa?
A, Strain gauge
B, Linear variable differential transformer
C, Capacitive
D, Piezoelectric
8. Which of the following measuring devices is most suitable to measure structural vibrations
having high frequencies and small displacement amplitudes?
A, Linear
potentiometer
B, Accelerometer
C, Vibrometer
2
D, Linear variable
differential transformer.
9. A piezoelectric accelerometer mounted on a bearing pedestal casing has a resonance
frequency of 50 kHz. What is the maximum useful frequency range of the accelerometer in
which it is capable of measuring accurate vibration response?
A, 5 to 2500Hz
B, 5 to 45 kHz
C, 5 to 50 KHz
D, 5 to 25kHz
10. Which of the following statement(s) is (are) correct?
The random errors can be minimized using:
1. Shielding the measuring system.
2. Grounding the measuring system.
3. Statistical analysis of measured data
4. Calibration.
A, 1 and 2
1
B, 1 and 3.
i
3
4
C, 1, 2 and 3.
5
6
7
D, 1, 2, 3, and 4.
8
9
10
Question 2: (15 Marks) - Ashok
A data acquisition system is to be used to measure the value of a maximum of 120 V line voltage.
The maximum input voltage to the data acquisition system is 8 V and its input impedance is 1 M
(Ro). The output impedance of the power line circuit is 0.5  (Rs).
A. Determine the value for resistor R2 in Figure below if R1 has a value of 100 k (neglect
loading effects resulting from Rs and Ro).
B. Determine the power dissipated by the two resistors R1 and R2 (neglect loading effects
resulting from Rs and Ro).
C. Determine the loading error of the line voltage at the attenuator input – include all
resistors in the calculation.
D. If the loading effects are to be considered, what is the value for resistor R2, assuming the
input voltage to the DAS is 8 V.
A, R2 (No loading effects)
B, Power dissipation (No loading effects)
C, Loading error
D, R2 (Loading effects)
3
Question 2 continued:
4
Question 3: (15 Marks) - JOHN
A vibration survey has been carried out on a bearing structure of a variable speed compressor in
a gas pipe-line. The compressor has an operating speed range from 0 to 10000 rpm.
At 6000 rpm, it is found that there are major resonance frequency components after the recorded
data have been analyzed using an optimum sampling frequency. These are 1 st harmonic with
vibration amplitude of ± 150 mV and third harmonic’s amplitude of ± 50 mV.
A. Calculate the first and third harmonic frequencies.
B. What would be the aliasing frequency or frequencies if the above vibration data is
sampled at 350 samples per second?
C. Comment on the effect of the aliasing frequency or frequencies on the response of the
most dominant frequency component.
Answer:
A, First and Third harmonics
B, Aliasing frequencies or frequencies
C, Aliasing frequency effect
5
Question 3 continued:
6
Question 4: (15 Marks) - JOHN
A sample of 15 high tension bolts has been tested from a batch. The average yield strength and
standard deviation are 850N/mm² and 16N/mm², respectively.
A. Find the 95% confidence interval on the mean yield strength. (MPa)
B. Find the upper and lower limits of the mean yield strength. (MPa)
C. If we want to reduce the 95% confidence interval found in Part 1 by 20%, how many bolts
should we need to test? (Assuming the same standard deviation as used in Part 1.)
Answer:
A, 95% Confidence level
B, Upper and lower limits
C, Number of bolts
7
Question 4 continued:
8
Question 5: (20 Marks) - Ashok
Part A:
The crashworthiness tests of automobile body structure require to determine the kinetic energy of
the vehicle before impact:
Ek 
1
mv2
2
As shown in the sketch of the test stand, the vehicle mass is measured by weighing of the car : m
=W/g, the vehicle velocity is measured by an automatic clocking of the time t, required by the car,
to travel a known distance 𝐿 = 𝑣⁄𝑡
For the values and measurement uncertainties of the individual variables given above, calculate
the total absolute uncertainty  Ek in the kinetic energy measurement.
Part B:
The capitulating mechanism of the test stand is designed to select and control the car impact
velocity. The table below shows the vehicle velocity measured in five tests with the catapult
velocity selector set at 7.0 m/s
Test #
m/s
1
5.97
2
7.07
3
7.10
4
7.02
5
6.95
Determine the mean deviation, standard deviation, % precision, % accuracy and bias of the
catapult velocity control system. The velocity measurement is considered to be a part of the
system.
Part A: % Uncertainty of the kinetic
energy measurement
Part B: Mean deviation,
Standard deviation,
% Precision,
% Accuracy,
Bias.
9
Question 5 Continued:
10
Question 6: (25 Marks) - JOHN
Part A:
A load cell is to be built using four uniaxial strain gauges (R1, R2, R3 and R4), instrumented onto a
cantilevered beam. It is used to measure the applied end load A, as shown in the sketch.
i.
Find the best axial location for the four strain gauges along the beam, which will yield the
optimum output. Illustrate your results in the sketch shown below.
ii.
Show the orientation and configuration of the four strain gauges; such that the output of
the bridge is four times that of a single gauge. [See Wheatstone bridge equation from the
formula sheet). Label R1 to R4 in the sketch shown below.
iii.
Sketch the Wheatstone bridge circuit showing the positions of the four strain gauges (R1,
R2, R3 and R4), supply voltage and output voltage.
Answer:
11
Question 6 –Continued:
Part B:
The cantilevered beam load cell, as shown in Part A, is used to measure strain in the beam due
to an applied load.
The nominal resistance and the gauge factor of each gauge are 120Ω and 2.0 respectively. The
supply voltage is 5.0 V.
A. What are the strains in the gauges on the upper and lower surfaces of the beam when
the bridge output voltage is 50 mV due to a bending load?
B. If the temperature of the four gauges changes by the same amount, what effect will it
have in the results?
C. Can this configuration of the strain gauges be used to measure strain due to an axial
force on the beam? Why?
Answer:
A, Upper and lower
surfaces’ strains
B, Temperature effect
C, Wheatstone bridge for
axial loading applications
12
Question 6 continued:
13
Formula Sheet

GdB  20 log 10 G  20 log 10

Amplifier (input loop),

Amplifier (output loop),

Butterworth filter, G 
Vo
Vi
RiVs
Rs  Ri
R GV
VL  L i
Ro  RL
1
Vi 
 f 
1   
 fc 
2n

Butterworth filter Roll-off = 6n dB/octave

Probability calculation – Normal distribution , z 






1
x
N
sx 
x 

N
x
i 1
i
1 n
2
 xi  x 

n  1 i 1
Sx
n
x
z
x
x
t
S/ n
  x  z / 2
Sx
n
n

x
Correlation coefficient, rxy 
 x
i 1
i
n
 x  yi  y 
n
n
2


 yi  y 2 
x

x

 i
i 1
 i 1

14
1/ 2
, x
n
 xi
i 1
n
,
y
y
i 1
n
i
Formula Sheet (continued)



n
 n  n 
n xi y i    xi   y i 
i 1
 i 1  i 1  b 
Least-squares fit, a 
,
2
n
 n 
2
n xi    xi 
i 1
 i 1 
n
wR max   wxi
i 1
 n 
n x    xi 
i 1
 i 1 
n
2
i
n
n
n
i 1
i 1
i 1
 yi2  b yi  a xi yi
S yx 
Standard error of estimate,
n
 n  n

2
x
y

  xi   xi y i 

i  i
i 1
i 1
 i 1  i 1

n
n2
R
xi
1/ 2

2
 n 



R

wR    wxi

 i 1 
xi  



2
2
2
 wn  
wR 
 w1   w2 

 
  a    b    N
R  x1   x2 
 xn  



k

Bx   Bi2 
 i 1 

Random uncertainty for a single measurement: 𝑃𝑥 = 𝑡𝑆𝑥

Random uncertainty in the mean: 𝑃𝑥̅ =
1/ 2
1/ 2
1/ 2
m 
, S x   Si2 
 i1 
,
𝑡𝑆𝑥
⁄
√𝑀
2 2
[∑𝑚
𝑖=0 𝑆𝑖 ]

Combined degree of freedom: 𝜗𝑥

Total uncertainty with random uncertainty of a single measurement:
o

=
𝑆𝑖4
⁄ )
∑𝑚
(
𝑖=0
𝜗𝑖
1/2
𝜔𝑥 = (𝐵𝑥 2 + 𝑃𝑥 2 )
Total uncertainty with random uncertainty in the mean:
o
1/2
𝜔𝑥 = (𝐵𝑥 2 + 𝑃𝑥̅ 2 )
15
2
Formula Sheet (continued)

Fourier-series analysis:

Periodic function f(t) = 𝑎0 + 𝑎1 𝑐𝑜𝑠 𝜔0 𝑡 + 𝑎2 𝑐𝑜𝑠 2𝜔0 𝑡 + ∙ ∙ ∙ ∙ + 𝑎𝑛 𝑐𝑜𝑠 𝑛𝜔0 𝑡
+ 𝑏1 𝑠𝑖𝑛 𝜔0 𝑡 + 𝑏2 𝑠𝑖𝑛 2𝜔0 𝑡 + ∙ ∙ ∙ ∙ + 𝑏𝑛 𝑠𝑖𝑛 𝑛𝜔0 𝑡
1
𝑇
2
𝑇
1
𝑇
 𝑎0 = ( ) ∫0 𝑓(𝑡) 𝑑𝑡
𝑇
 𝑎𝑛 = ( ) ∫0 𝑓(𝑡) 𝑐𝑜𝑠 𝑛𝜔0 𝑡 𝑑𝑡
𝑇
 𝑏𝑛 = ( ) ∫0 𝑓(𝑡) 𝑠𝑖𝑛 𝑛𝜔0 𝑡 𝑑𝑡
𝑇

where T = Period, 𝜔0 = Fundamental angular frequency (rad/s)

Strain gauge factor:

Wheatstone bridge output:
o
𝑆=
𝜕𝑅⁄
𝑅
𝜖𝑎
𝑉 𝑅2 𝑅3
𝜕𝑅3
(
2
𝑅3
2 + 𝑅3 )
𝑉𝑜 = [(𝑅 𝑠
+
𝜕𝑅1
𝑅1
16
−
𝜕𝑅2
𝑅2
−
𝜕𝑅4
𝑅4
)]
Folding diagram:
17
Formula Sheet (continued)

Input resolution error =  0.5
Vru  Vrl
volts
2N
18
Formula Sheet (continued)

Normal z distribution:
19
Formula Sheet (continued)

Student’s T distribution:
20
Formula Sheet (continued)

Common thermocouples:
21