ME- 495
Mechanical and Thermal Systems Lab
Fall 2011
Chapter 4 - THE ANALOG MEASURAND:
TIME-DEPENDANT CHARACTERISTICS
Professor: Sam Kassegne
THE ANALOG MEASURAND:
TIME-DEPENDANT CHARACTERISTICS
Two types of measurement
Static – constant over time
– Dynamic – varies over time
–
state – periodic over time
 Non-repetitive or Transient
 Steady
–
–
Single Pulse – aperiodic
Continuing or random
SIMPLE HARMONIC RELATIONS

S = Sosin(t)
–
–
–
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Velocity = ds/dt = So  cos(t)
–

S = instantaneous displacement from equilibrium
So = amplitude (maximum displacement)
 = circular frequency (rad/s)
Vo = So 
Acceleration = dV/dt = - So 2 sin(t)
–
ao = - So 2
Exercise: Give some practical
examples of such a motion.
CIRCULAR & CYCLIC FREQUENCY
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1Hz (hertz) = 1 cycle/second
f = cyclic frequency (Hz)
 = circular frequency
 = 2f
S = Sosin(2ft)
Scotch-yoke mechanism – Harmonic motion
COMPLEX RELATIONS
Most complex dynamic mechanical signals, steady-state or
transient (such as pressure, displacement, or strain) may be
expressed as a combination of simple harmonic components.
Ao 
y(t ) 
  An cosnt   Bn sin nt 
2 n1
–
–
Ao, An, Bn = amplitude determining constants called
harmonic constants.
n = integers from 1 to  called “harmonic orders”
The above equation is also called Fourier Series.
Example:
A 2-harmonic term pressure-time function.
P = 100 sin 80t + 50 cos (160t-/4)
The circular frequency of the fundamental harmonic
= 80 rad/sec or 80/2 = 12.7 Hz.
The period = 1/12.7 = 0.0788 sec
The circular frequency of the second harmonic =
160rad/sec or 160/2p = 25.4 Hz.
The period = 1/25.4 = 0.039 sec
It also lags the fundamental by 1/8th cycle (/4)
FREQUENCY SPECTRUM
Type of plot where frequency (instead of time) is the
independent variable and amplitude is the ordinate.
Special Wave-forms.
(infinite series)
FREQUENCY SPECTRUM (ctd.)
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Frequency spectrum is useful because it allows identifying
– at a glance – the frequencies present in a signal (say
natural frequencies).
Interest in FS plots increased because of spectrum
analyzers.
Two ways to perform a spectrum analysis


Spectrum Analyzers  electronic device which displays frequency
spectrum on CRT using circuitry
Fast Fourier Transform  computer algorithm that computes
spectrum
Harmonic or Fourier Analysis
The process of determining the frequency spectrum
of a known waveform is called harmonic analysis, or
Fourier analysis.
The case where y(t) is known only at discrete points
in time is important is practice because of wide use
of computers for recording signals.
DISCREET FOURIER TRANSFORM
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N = total points recorded
T = time period
t = time interval
n = harmonic order
r = sample count = 1,2,3,…N
DISCREET FOURIER TRANSFORM (ctd)
If one assumes that one’s experimental data can be
expressed as cosine/sine function
(say y = cos(-x+phi); y = sin (x+phi1), etc),
the harmonic analysis then becomes a problem of
determining appropriate values for harmonic orders (n),
coefficients and phase angles.
• The process starts with an analog time dependent signal which
is then digitized by selecting discrete values at predetermined
time intervals.
• These values are then processed by harmonic analysis through
which frequency contribution and relative amplitudes are
determined.
• This provides a composite functional relationship that defines the
original relationship (Equation in previous slide).
EXAMPLE OF DFT ANALYSIS
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3 signals: 500, 1000, 1500
Same amplitude: 50 mV
Out of phase with each other
Sample rate fs = 9000Hz
N = 18
flowest = 500 Hz therefore T = 1/fL=0.002sec
and t= 1/fs
t=0.11ms
Harmonic orders evaluated: n=N/2=9
Cn  A n  B n
2
2
Method for calculating the Harmonic constants using
Excel Spreadsheet
EXAMPLE OF DFT ANALYSIS (ctd.)