Fundamentals of
M e a s u r e m e n t Te c h n o l o g y
Problem Book
Wang Boxiong
Luo Xiuzhi
Dept. of Precision Instruments and Mechanology
Tsinghua University
November, 20
Preface
This problem book is compiled and used for the course
“ Fundamentals of Measurement Technology”. All
problems contained in this book are designed for the
students
with
the
aim
of
strengthening
their
understanding of theories of measurement technology
and enhancing their capabilities in analyzing and solving
problems relating measurement technology.
The authors
Chapter 2 Signal Representations
2-1 Two time sequences:
(1) xn   e
n 
j   
8

(2) xn  e cos
2
10n
Are they periodic functions? If so, what are the periods?
2-2 A periodic signal x(t ) has its Fourier coefficients a n ， bn ， cn . Prove that a
time-delayed signal x(t  t 0 ) has the following expression of Fourier series:
1



xt  t   a    a cos n t  b sin n t 


0
0
n 1


c
n  
where ： an


n
n
0
n
0
e jn 0t
 an cos n 0 t 0  bn sin n 0 t 0

bn  an sin n 0 t 0  bn cos n 0 t 0

cn  cn e  jn0t0
2-3 Determine the Fourier series of the following periodic signals and plot their
spectra.
(1)
x(t)
sawtooth wave
+1
…
…
T
2
T

2
0
t x(t )  2 / T  t
 T / 2  t  T / 2)
-1
Fig 2-1
(2)
x(t)
square wave
A
T
2
…

T
2
…
0
-A
Fig 2-2
(3)
2
 A
x(t )  
 A
t
 T/ 2  t  0
0  t  T/ 2
x(t)
full-rectified sinusoidal
A
x(t )  A sin  0 t
…
…
0
t
T
Fig 2-3
(4)
x(t)
exponential function
…
…
1

0
T
2
t
T
2
x(t )  e at
a  0
 T / 2  t  T/ 2
Fig 2-4
(5)
rectangular impulse
for T  3 and T  5
x(t)
1
…
…


2
0 
2
t
T
Fig 2-5
2-4 Determine the Fourier transforms of the following nonperiodic signals , and
plot their spectra.
x(t )  Ae  at (a  0, t  0) ；
(1) exponentially-decaying function
(2) signum function(Fig (a)), and unit-step function (Fig (b)).
3
x(t)
x(t)
1
1
t
0
t
0
-1
(a)
(b)
Fig 2-6
Hint ：a signum function is symbolized as Sgn(t ) ，Make first the Fourier
transform of the function
 e  t
x(t )   t
 e
t 0
t 0
 0
and let λ→0 to obtain the Fourier transform of Sgn(t ) . A unit-step function
u (t ) can be obtained by shifting a signum function along the ordinate.
(3)
x(t)
1
t
0
T
T’
Fig 2-7
truncated cosine wave
cos  0 t
x(t )  
 0
t  T'
t  T'
Using graphic method to discuss the effect of different T0 s on the spectrum for
4
T0  T ' , T0  T ' , T0  T ' .
(4) exponentially-decaying x(t )  e  at cos  0 t
(5) triangular impulse with time shift 
(Calculate with one method and present two other ideas for solutions)
x(t)
1
t
τ
0
W
.
Fig 2-8
(6) Two rectangular impulses, one centered at origin and the other centered at
x(t)
1
0
t
τ
T
Fig 2-9
(7) Plot the waveform of time signal x(t )  A cos(2 f 0t   )  k ，and its
Fourier spectrum.
(8) Find the spectrum of cosine impulse.
x(t)
cost τ
x(t )  
0

1
-τ/2
τ/2
0
Fig 2-10
5
 t
 τ / 2
 t  τ / 2
t
(9) Find the spectrum of two cosine impulses.
x(t)
。
0
τ0
t
Fig 2-11
(10) Find the spectrum of the periodic modulated signal by use of graphic
method:
x(t )  e  at cos  0 t
(a  0, t  0)
x(t)
…
t
0
Fig 2-12
2-5 Signal x(t ) and its frequency spectrum are shown in Fig 2-13. An oscillatory
signal cos 2f 0 t ( f 0  f m ) is multiplied with x(t ) .
(In this case, signal x(t ) is
called the modulating signal , and the oscillatory signal cos 2f 0 t is called
carrier .)
Find the Fourier transform of the modulated signal x(t ) cos 2f 0 t , and
plot its time waveform and frequency spectrum. What will be the situation if
f0  fm .
6
x(t)
x(f)
I
0
-fm
fm
t
I
t
fm
0
0
Fig 2-130
2-6 Two signals, x1 (t ) and x2 (t ) are shown in Fig 2-14 x1 (t ) has the frequency
spectrum X 1 ( f ) . Determine the frequency spectrum of x2 (t ) .
x2 (t)
x1 (t)
t0
t
t0
t
Fig 2-14
2-7 Find the convolution of x1 (t ) and x2 (t ) using graphic method.
X1(t)
X2(t)
*
t
t
T1
-T2
Fig 2-15
2-8 Find the Fourier transform of the signal
spectrum (a  0, b  0, t  0) .
7
e  e  ,
 at
 bt
and plot its
Chapter 3 Analysis of Measuring system
3-1 A piezoelectric transducer with a sensitivity S cp  9.00 PC / N is connected
with a charge amplifier of a sensitivity S vc  0.005V / PC , which are then
connected to a light trace oscilloscope of a sensitivity S xv  20mm / V . Draw the
whole system using block diagram, and determine the resultant sensitivity.
3-2 A micro-ammeter with a measurement range of 60 A is calibrated. The
following are the obtained data:
number of measurements
1
2
3
4
5
readings of calibrating meter 10
20
30
40
50
readings of calibrated meter 10 20.5 29.5 39 50.5
Calculate the linearity of the meter (using the least-square regression).
3-3 Find the total sensitivity S for the following systems formed by elements
connected in series, in parallel , and with negative feedback loop. Assuming that
all elements in the figure are linear ones.
x

S2
S1
y
Sn
S1
S2
y
y
x
S2
S1

x
S3
Sn
Fig 3-1
H s  
1
and a time
1  s
constant   0.35s is used to measure sinusoidal signals with period of 1 s ,
3-4 A first-order device with a transfer function
8
2 s and 5 s . Determine the relative amplitude error for each case.
3-5 A gear box with a reduction ratio of 5 : 1 has unbalances on its two shafts. A
signal y (t ) of the vibration caused by the unbalances is picked up by a sensor on
the box case, whose frequency spectrum is Yf  .
y(t)
t
|Y(f)|
f( Hz)
10
20
30
40
50
Fig 3-2
Question：
(1) What are the input and output of the whole system？
(2) Is the system a linear one?
(3) What are the two frequency components in the spectrum that are caused
by the unbalances of the two gear shafts? Assume that the meter has a rotation
speed of 3000 rpm .
3-6 A periodic signal x(t )  2 cos100t  cos(300t   / 4) ) passes through a
first-order system whose frequency response H ( j )  1 /( 0.05 j  1) .
Find its steady-state frequency response.
(1) Design a procedure for evaluating the output response;
(2) Use graphic method to determine:
①the synthesized wave forms of the input and the output; and
②the amplitude spectra and the phase spectra of the input and output
respectively.
3-7 Second-order measuring devices usually have a damping ratio
9
  0.6 ~ 0.7 . Explain the reason.
3-8 A second-order device has the transfer function

 2n
 H s  

 j 2  2 n  j    2 n


 . Its damping ratio   0.7 and the natural


frequency f n  50Hz . What is its steady-state output y (t ) under the excitation
x(t ) shown in Fig 3-3.
x(t)
1
·
·
·
t
0.025 0.05
Fig 3-3
3-9 A measuring device has its amplitude spectrum shown in Fig 3-4. The phase
spectrum shows a phase-shift of 75  for   125.5rad / s , a phase-shift of 90
for   150.6rad / s , and a phase-shift of 180 for   626rad / s . The device is
used to measure the following two signals:
x1 (t )  A1 sin 125.5t  A2 sin 150.6t
x2 (t )  A3 sin 626t  A4 sin 700t
Is it possible to obtain a distortionless measurement of x1 (t ) and x2 (t ) using
the device? Indicate the reason.
A(f)
14
19
f(Hz)
Fig 3-4
10
3-10 Find the steady-state response of the device with a transform function
1
H s  
when its input signal x(t )  0.5 cos10t  0.2 cos(10t  45 ) .
0.005s  1
3-11 Insert suddenly a thermometer from the air of 20℃ into the water of
80℃. If the thermometer has a time constant   3.5s ，what is the reading
indicated by the thermometer after 2 seconds？
3-12 A meteorological balloon with a thermometer having a time constant 15
(t he t he rm om et er c an be consi de red a s a fi rst -o rde r s yst em ) is
passing through the air at a speed of 5m/s. The temperature decreases 0.15℃ for
every 30m increase in altitude. The balloon transmits the altitude and the
temperature data to the ground .By calculation , the temperature at the altitude of
3000m is -1℃.What is the altitude when a temperature of -0.1℃ is practically
reached.？
Chapter 4 Transducers
4-1 A steel plate with a length of L  1m and an
strain
gage
elastic modulus E  2.1  10 6 kg  f / cm 2 is pulled
by a force P . The recorded tensile strain by an
HP-3 foil strain gage （ R  120 ,sensitivity
coefficient K  2 ） is 300  .Calculate
the
elongation L of the plate, its stress  and R / R
of the strain gage. If a strain of 1 must
L
steel
plate
P
Fig 4-1
be
measured what is the related R / R ？
4-2 A resistance strain gage has a sensitivity
S  2 and a resistance R  120 .If its strain
is 1000
when it operates. What is the
mA
resistance change R ?
If the strain gage is connected in a circuit
ma
11
(see Fig 4-2), determine:
(1) the current indicated when there is no strain;
Fig 4-2
(2) the current indicated when there is a strain；
(3) the relative variation of the current indication；
(4) whether it is possible to read out this variation from the ammeter.
4-3 A capacitance micro-displacement measuring
workpiece
instrument has an initial gap d 0  0.3mm between
its two plates in an air medium. Each plate has a
diameter r  4mm . Determine :
(1) the capacitance variation when the gap has
d
Fig 4-3
a displacement d  1m from its initial position；and
(2) the variation in graduations of the instrument .
Assuming that the amplification factor of the measuring circuit K 2  1mV / PF ，
the sensitivity K 0  5graduations / mV , and the displacement d  1m .
x
b
4-4 A planar plate-capacitor displacement sensor
with a width b  4cm and a plate gap
  0.2mm is used to measure displacement .What is its sensitivity?
Fig 4-4
4-5 A platinum-resistance thermometer is used to measure temperatures from
0 ~ 200  C .The resistance-temperature relationship is RT  R0 (1  T  T 2 ) .
For R0  100 ， R100  138.5
and R200=175.83,
determine:
(1) values of  and  ；and
(2) the nonlinearity the thermometer displays at 100  C .
12
4-6 A capacitance liquid level gage has two metal cylinders located in a liquid
tank. The outer cylinder has an inner diameter of 2 R and the inner cylinder has
an outer diameter of 2 r . If the liquid
ε1
L
constant is  0 ，the total length of the plate is
ε0
L1
dielectric constant is  1 ，the air dielectric
L and the level height is L1 , determine:
2r
2R
(1) the relationship between the level
height L1 and the capacitance C ；and
Fig 4-5
(2) the sensitivity S . Are L1 and C linearly related ?
4-7 Use the following two self-inductance transducers to show the sensibilities for
δ1
Δδ
δ0
δ0
Δδ
L1
L
L2
Fig 4-6
differential and nondifferential transducers.
4-8 An inertial velocity pickup of a natural frequency f n  15Hz ，and a damping
ratioζ=0.7 is used to measure the following two vibrations:
x1 (t )  5a sin( 2f1t ), x2 (t )  a sin( 2f 2t )
，where f1  10Hz ，f 2  100Hz
Calculate the pickup’s output. If a piezoelectric accelerometer is used for the
measurement, what is its output?
4-9 A velocity transducer is designed for measuring vibrations of frequencies
higher than 30Hz . The maximum amplitude error should not be larger than ±5%.
For a damping ratio ζ=0.6，find the natural frequency of the transducer .
13
4-10 What are the working frequency ranges of velocity transducers and
accelerometers ? Describe the influence of m , k and ζ on frequency range.
4-11
A pressure transducer (considered as a second-order vibration system)
has a natural frequency f 0  800 Hz and a damping ζ =0.14. It is used to
measure a sinusoidal force of f  400 Hz . Find its amplitude ratio A( ) and
the phase difference  ( ) . If ζis changed to 0.7，determine A( ) and  ( )
again.
4-12 A Hall-element，whose sensitivity is K H  1.2mV / mA  KGS ，is placed in
a magnetic field with a gradient of 5KGS / mm . If the rated control current is
20mA and the element vibrates about its equilibrium position with vibration
amplitude of  0.01mm ，calculate its output voltage.
4-13 Fig 4-7 shows the equivalent circuit of the combination of a piezoelectric
transducer and a change amplifier, where C is the sum of the inherent
capacitance of the transducer, the stray capacitance of the cable ,and the input
capacitance of the amplifier.
The transducer has a sensitivity S q  100 PC / g ，
and a feedback capacitance C f  1000 pF . Find the amplitude voltage of the
amplifier for a measured acceleration of 0.5 g .
Cf
q
C
v
14
Fig 4-7
Chapter 5 Signal Conditioning
5-1 A bridge consists of a resistance strain gage having a resistance R  120 ，a
sensitivity S  2 and a fixed resistor of 120 .The bridge is powered with an
excitation of 3V . Assume that the load resistance is infinite . For a strain of
2  and a strain of 2000  ，determine the output voltage of a single-arm
bridge and a double-arm bridge respectively, and compare their sensitivities.
5-2 Someone found in using a strain gage that the sensitivity of the gage is
insufficient. To raise the sensitivity，he increased the number of strain gages. Can
the sensitivity be raised under the following conditions when
（1）a strain gage is connected in series with each of the two arms of the
half bridge；and when
（2）a strain gage is connected in parallel with each of the two arms of the
half bridge?
Z2
Z1
5-3 An a. c. bridge is shown in Fig. 5-1.
Z3 Z4
For Z1  R1  500 ,
ey
Z 2  R2  1000 ,
e0
Z 3   j  1 / 0.2 , and
the power supply e0  10V of f  1000 Hz ,
Fig 5-1
calculate:
(1) Z4 when the bridge is balanced, and tell whether the reactance Z4 is a
capacitive one or an inductive one; and
(2) Z4 when Z 2 and Z 3 are interchanged.
5-4 A Wheatstone bridge composed of resistive strain gages is used to measure a
15
structure’s strain . The strain  (t )  A cos10t  B cos100t . The bridge has an
excitation voltage e0 (t )  E sin 100000t . Find the spectrum of the output.
5-5 An amplitude-modulated signal
xa (t )  (100  30 cos t  20 cos 3t )(cos  c t ) ,
where f c  10KHz and f   500Hz . Determine :
(1) the frequencies and amplitudes of the frequency components contained in
xa (t ) ；
(2) plot the frequency spectra of the modulating signal, the carrier and the
modulated signal respectively.
5-6 Two strain gages are connected in the opposite arms of a bridge (see Fig 5-2).
The excitation voltage e0  cos 2f 0 t ，where f 0  1000 Hz . Strain gage sensitivity
K  2 and R1  R when the bridge is initially balanced .
Find （1）the bridge output and plot its time waveform;
（2）plot the amplitude spectrum of the output.
ε(t)
ε(t)
R
R1
u(t)
ε(t)
R
0
R1
t
e0
Fig 5-2
5-7 A signal x(t )  e  at (a  0, t  0) is used to modulate a carrier cos  0 t ，The
modulated carrier is again used as a reference signal for synchronous
demodulotion and phase-sensitive detection. Determine the time waveform and
the amplitude spectrum of the signal after synchronous demodulation and
phase-sensitive detection. What is the cut-off frequency of a filter if it is used to
recover the original waveform.
16
5-8 Determine the output voltage e L when the load RL is connected to the
following bridge, and the relation between e L and the output for an open circuit
ey .
R1
R2
RL
eL
R3
R4
e0
Fig 5-3
5-9 Determine the amplitude spectrum of the signal
f (t )  A(1  m cos 2ft) sin 2f 0 t .
5-10 The following graph shows the strain curve measured by an a.c. bridge,
determine:
(1) the modulated time waveform；
(2) the waveform after a phase-sensitive detection；
(3) the amplitude spectrum of the signal after being modulated and phase
-sensitive detected.
ε(t)
1
t
0
Fig 5-4
5-11 A cosine wave signal of frequency f is used to modulate a sine wave carrier
of frequency f 0 . The modulated signal x(t )  A(1  m cos 2ft)  sin 2f 0 t
where m is the modulating factor, and f 0  f
17
Determine (1) the spectrum of x(t ) ; and
 m2
(2) prove that the average power of x(t ) is P  P0 1 
2




where P0 is the average power of the unmodulated carrier .
5-12 Two band-pass filters, one is an octave filter and the other is a 1/3-octave
filter, have the same lower cut-off frequency. How many times is the center
frequency of the former filter larger than that of the latter one?
5-13 A 1/3-octave band-pass filter has a center frequency f 0  80 Hz . Calculate
its lower and upper cut-off frequencies f c1 and f c 2 .
5-14 A band-pass filter has a lower cut-off frequency f c1 and upper cut-off
frequency f c 2 ，and a center frequency f 0 . Are the following statements correct ?
(1) for an octave filter, f c2  2 f c1
(2) f 0  1 2 f c1  f c2  ;
(3) f 0 
f c1  f c2 ;
(4) the cut-off frequency of a filter is the frequency at which the -3dB
magnitude of the passband is located; and
(5) the center frequency of an octave filter is
3
2 times that of a
1/3-octave filter, when the two filters have the same lower cut-off
frequency .
5-15 Two RC filters are shown in Fig 5-5. If the upper cut-off frequency of the
low-pass filter is to be f c 2  440 Hz ，and the lower cut-off frequency of the
high-pass filter f c1  360 Hz , select the resistor values, when the following
18
capacitors are available ：4700 PF ，0.15 F , 0.01 F ，0.022 F ，0.033 F ,
0.039 F ，0.047 F ，0.082 F .
（calculate only one pair of data for each filter）
C
R
R
C
low-pass filter
high-pass filter
Fig 5-5
5-16 For a filter with multi-channel negative feedback(shown in Fig 5-6)，
(1)determine its frequency response; and (2) identify what kind of filter it is ,
and calculate its cut-off frequency.
C1
R3
R1
Vin
Vout
C2
R2
Fig 5-6
5-17 For a filter with limited voltage amplification , determine
(1) its frequency response; and
(2) identify what kind of filter it is , and calculate its cut-off frequency.
19
C1
R1
R2
Vout
Vin
Rf
C2
R3
Fig 5-7
5-18 A low-pass filter x(t ) has its amplitude shown in the following figure and
its phase angle characteristic  ( )  0 . The input signal to the filter is a periodic
square wave of magnitude 1 and period T  1ms .
Determine its output y (t )
and the related frequency spectrum Y ( f ) , sketch the waveform of y (t ) and its
spectrum.
x(t)
H(f)
1
t
T

2
0
T
2
0
T
1
2
3
f(KHz)
Fig 5-8
5-19 A measurement system is shown in block diagram. The first stage has a form
of an exponential signal e ax and the second one is an amplifier. Determine the
characteristic of the third linear correction stage f. If f is now placed in a feedback
20
loop, what is its characteristic ?
x
x1
ax
e
x2
K
y
f
Fig 5-9
5-20 If a high-pass filter and a low-pass filter are connected in series (shown in
Fig 5-10), is it possible to form a band-pass filter? Derive the frequency response
function of the total network. Analyze its amplitude phase spectra.
C1
R2
R1
Vin
Vout
C2
Fig 5-10
5-21 How to distinguish between the two signals x(t ) and y (t ) (see Fig 5-11)
by both their time and frequency waveforms after they have been first amplitudemodulated and then demodulated.
x(t)
y(t)
T
t
t
T
2
T
2
T
Fig 5-11
5-22 Calculate the output of the network (see Fig 5-12) for the input of sin 10t .
100k
Sin10t
1 f
21
ey
Fig 5-12
5-23 The difference between two neighboring quantized levels of an A/D
converter is equal to the quantization unit q . Prove that the quantization unit q
for an N-bit A/D converter is
q
Vm
2N
where Vm is the maximum converted analog voltage of the converter.
5-24 Fig 5-13 shows a cantilever beam made of steel with a Young’s modulus
E  20  1010 Pa . Four resistive strain gages are cemented on the beam to form a
full bridge. Each strain gage has a rated resistance of 200 and a sensitivity
factor of 3.5. The bridge is powered by a dc power supply of 5.6V . Find the
output of the bridge when the load is 100N .
15
R1(R2)
R3(R4)
R1
R2
e0
F
R3
R4
R1，R3
6
R2，R4
60
70
Fig 5-13
22
ei
Chapter 6 Recording Instrument
6-1 The vibrator of an SC16 Light Trace Oscilloscope has a unit step input i (t )
and gives an output  (t ) (see Fig 6-1). What are the damping ratio  and the
natural frequency  n of the vibrator? Deduce its transfer function.
(t)
i(t)
mi =0.15
3
1
0
t
t2
t1
0
t
6.28ms
a
Fig 6-1
6-2 Several vibrators are used to record sine wave signals of different frequencies
and analyze the relationship between these different signals. How should one
choose the vibrators?
6-3 An FC6-1200 Vibrator is used to record signals from a velocity transducer.
Determine its working frequency range. If signals from piezoelectric
accelerometer are to be recorded, what is the working frequency range of the
accelerometer.
6-4 An FC6-30 vibrator is used to record a 100Hz sine wave signal of a
magnitude 2V . The desired deflection on the recording sheet should be  50mm .
Determine the values of the resistors to be connected in series and in parallel.
(Assume the inner resistance of the power supply is 200 )
6-5 A vibrator with a natural frequency of 1200Hz is used to record a squarewave signal with a fundamental frequency of 600Hz . Calculate and analyze the
recorded results , and sketch the recorded waveform.
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Chapter 7 Signal Processing and Analysis
7-1 Design a spectrum analyzer with multi-channel filters for a signal frequency
range of 0 ~ 8KHZ . Ten octave band-pass filters are employed to cover the
whole frequency range.
Determine ：(1) the center frequency of each filter, f 0 ；
(2) the band width of each filter.
7-2 A tracking filter spectrum analyzer is used to analyze a square wave signal of
a period of 0.1s . Calculate the band width of the band-pass filter.
7-3 Ideal sampling with a sampling rate f s  4 Hz is performed to three sinusoidal
signal
x1 (t )  cos 2t , x2 (t )  cos 6t
and x3 (t )  cos10t
respectively.
Determine and compare the sampled output sequences of the three signals, sketch
the time waveforms and the sample positions, and explain the frequency aliasing
phenomenon by use of frequency spectra.
7-4 Calculate the DFTs of the following discrete sequence .
（ 0,1
2 ,1,1
2 ,0,1
2 ,1,1
2）
7-5 Calculate the frequency spectrum of the sequence in Problem 4 using FFT
algorithm. Compare and explain the two results.
7-6 Calculate the probability density function of signal x(t )  A sin t ，sketch the
result graph.
7-7 Calculate the autocorrelation and the power spectrum of a periodic cosine
wave x(t )  A cos  0 t . Sketch their graphs.
7-8 Calculate the cross-correlation of a square wave signal and a sine wave signal
(see Fig 7-1). Compare the result with that of the autocorrelation of a sine wave
signal and explain the reason.
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x(t)
Sinωt
1
t
0
-1
y(t)
1
t
0
-1
Fig 7-1
7-9 For y  x 2 and the sequence x  0, 1, 2, 3, 4, 5 , calculate the
cross-correlation coefficient  xy of x and y . Explain the result.
7-10 Deduce the DFT of the exponential function
x(t )  e t
（t≥0）
The waveform of x(t ) and its amplitude spectrum X  f  are shown in Fig 7-2.
x(t)
|X(f )|
1
1
0
t
0
f
Fig 7-2
7-11 Digital Fourier transform is to be made with a stationary random signal
which has been low-pass filtered before hand. If the components with frequencies
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lower than 500Hz are to be analyzed, and the spectrum resolution must be
0.5Hz , determine
(1) the sampling frequency f s ;
(2) the number of samples N ; and
(3) the bandwidth of the window function , T .
Comprehensive Practical Problems
Use your knowledge to solve the following practical problems. You are
required to (1) put forward plans for measuring systems, their fundamental
arrangements and the necessary explanations ; (2) put forward solutions to the
problems these systems might encounter with in practice, such as temperature,
variation, and vibration , etc.
P
1．Truck deadweight gage
The truck body is fixed on the chasis
through springs. Indications of the gage
must be seen by the driver in driver’s cab.
Work out two possible solutions.
2．Rail-breadth measuring instrument ：
The inner width of rails is defined to be within A where A is the width , and  is
the tolerance. Width larger than the value will cause abnormal operation. Put
forward two possible schemes for measuring width using patrol train.
A
3．Automatic thickness measurement of rolled steel：
The thickness of rolled steel-plate must be ensured within . Propose a
solution for automatic measurement of plate thickness during rolling process.
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δ
4．Steel cable is made by twisting several strands of wire. How to detect the wire
breakage on-line.
5. It is found that a workpiece has vibration trace on its machined surface after
being processed by a grinding machine. To find the possible source for the
vibration, transducers and spectrum analysis are employed. Design a scheme for
the vibration source detection. (Assume that the grinding machine has 3 motors:
one for the spindle with a rotation speed of 1500rpm , one for the grinding wheel
with a rotation speed of 3000 rpm , and one for the cooling oil pump with a
rotation speed of 750rpm . No other vibration source is present.)
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