MECHANICAL MEASUREMENTS

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MECHANICAL MEASUREMENTS
Prof. Dr. Ing. Andrei Szuder Tel.
40.2.1.4112604 Fax. [phone] www.labsmn.pub.
ro szuder_at_labsmn.pub.ro
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SIGNALS
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Characteristics of Signals
By signal, we generally refer to an amplitude of
something that varied in time. Signals can be
classified as analog, discrete time, or
digital. Analog Both amplitude and time vary
continuously. Real processes are generally
analog. Discrete time While the amplitude may
vary continuously, it is only known at several
points in time. Not very common Digital Amplit
ude and time have discrete values.
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ANALOG SIGNALS
Analog signal signal with continuous variation
Signal
(50 mA) 10 V
t1
time
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Captori numerici semnal frecvential (numar de
impulsuri pe unitatea de timp functie de marimea
masurata). Ex Tahimetrul optic Captori digitali
semnal direct codificat binar. Ex. Codificatori
optici
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Modurile de operare LOGIC
Marime logica semnal cu doua stari (totul sau
nimic) Captor logic detector ( key sensors)
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The Nature of Electronic Signals

• Signals are generally classified in terms of
  their time or frequency domain behaviour
• Time domain classifications include
• Static Unchanging (DC)
• Quasi static Slowly changing (amplifier drift)
• Periodic Sine wave
• Repetitive Periodic but varying
  (Electrocardiograph)
• Transient Occur once only (impulse)
• Quasi transient Repetitive but seldom (radar
  pulses)

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Classification of Waveforms
The shape of the signal on an x-y plot is called
a Waveform. Static Does not change with
time. y(t) Ao Dynamic Does vary with
time Deterministic Varies in time in predictable
way Periodic Repeats at regular intervals
y(t) Ao sinwt Complex Periodic More than
a single frequency Nonperiodic Does not
repeat y(t) Ao for t gt 0 Nondeterministic
Varies in time in a chaotic way. Usually
described statistically
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Signals
Static
Dynamic
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Signals
Static
Stationary
Dynamic
Transient
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Signals
Static
Aperiodic
Stationary
Dynamic
Periodic
Transient
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Signals
Static
Time
Aperiodic
Stationary
Time
Dynamic
Periodic
Time
Transient
Time
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Static Signal Characteristics
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Static Signal Characteristics

• Error - The difference between a measured and a
  true value

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Static Signal Characteristics

• Error - The difference between a measured and a
  true value
• Uncertainty -The estimated error limit around the
  measured value for given odds. For symmetric
  case, the total error range is twice the
  uncertainty

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Static Signal Characteristics

• Error - The difference between a measured and a
  true value
• Uncertainty -The estimated error limit around the
  measured value for given odds. For symmetric
  case, the total error range is twice the
  uncertainty
• Accuracy - The opposite of uncertainty, but
  sometimes used interchangeably. Thus 99
  accurate and accurate to within 1 mean the
  same.

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Static Signal Characteristics

• Error - The difference between a measured and a
  true value
• Uncertainty -The estimated error limit around the
  measured value for given odds. For symmetric
  case, the total error range is twice the
  uncertainty
• Accuracy - The opposite of uncertainty, but
  sometimes used interchangeably. Thus 99
  accurate and accurate to within 1 mean the
  same.
• Bias - Systematic departure from the true value.
• Precision - Repeatability of measurement results.

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Relationship between accuracy and precision
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More Static Characteristics

• Resolution - The smallest increment that can be
  resolved at the instrument output.
• Sensitivity - The change of output per unit
  change in measured quantity.
• Zero-Offset - Constant portion of bias.
• Hysteresis - An error component dependant upon
  the direction in which the measurements are taken
  (a slack).
• Nonlinearity - The departure of input/output
  curve from the straight line.

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More Static Characteristics

• Resolution - The smallest increment that can be
  resolved at the instrument output.
• Sensitivity - The change of output per unit
  change in measured quantity.
• Zero-Offset - Constant portion of bias.
• Hysteresis - An error component dependant upon
  the direction in which the measurements are taken
  (a slack).
• Nonlinearity - The departure of input/output
  curve from the straight line.

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More Static Characteristics

• Resolution - The smallest increment that can be
  resolved at the instrument output.
• Sensitivity - The change of output per unit
  change in measured quantity.
• Zero-Offset - Constant portion of bias.
• Hysteresis - An error component dependant upon
  the direction in which the measurements are taken
  (a slack).
• Nonlinearity - The departure of input/output
  curve from the straight line.

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More Static Characteristics

• Resolution - The smallest increment that can be
  resolved at the instrument output.
• Sensitivity - The change of output per unit
  change in measured quantity.
• Zero-Offset - Constant portion of bias.
• Hysteresis - An error component dependant upon
  the direction in which the measurements are taken
  (a slack).
• Nonlinearity - The departure of input/output
  curve from the straight line.

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More Static Characteristics

• Resolution - The smallest increment that can be
  resolved at the instrument output.
• Sensitivity - The change of output per unit
  change in measured quantity.
• Zero-Offset - Constant portion of bias.
• Hysteresis - An error component dependant upon
  the direction in which the measurements are taken
  (a slack).
• Nonlinearity - The departure of input/output
  curve from the straight line.

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More Static Characteristics

• Resolution - The smallest increment that can be
  resolved at the instrument output.
• Sensitivity - The change of output per unit
  change in measured quantity.
• Zero-Offset - Constant portion of bias.
• Hysteresis - An error component dependant upon
  the direction in which the measurements are taken
  (a slack).
• Nonlinearity - The departure of input/output
  curve from the straight line.

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Signal Analysis
It is often useful to think of a signal as having
two parts, one that is constant in time (the DC
component) and one that varies in time with zero
mean (the AC component). The average value, or
the mean of a signal y(t).
The AC component is often characterized by its
rms value, defined as
If the mean of the signal is zero, or if it has
been removed, then the rms is a statistical
measure of the amplitude of the signal.
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Mean and RMS
A signal y(t) A1 Ao sin(2pft). Clearly, the
mean of this signal over any integer number of
cycles is A1 (the DC part of the signal). If we
subtract of this DC value and take the rms over 1
period T 1/f, then we find that
y
A0
A1
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Discrete form Mean and RMS
More times than not, modern measurements are made
using digital data acquisition equipment (DAQ).
Most real processes are analog in nature. As
such, we need to be able to move freely between
these two ways of thinking. Notation to convert
from a continuous function y(t) to a discrete set
of N numbers between time t1 and t2
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Discrete form Mean and RMS

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Discrete form Mean and RMS
Remember that an integral is the limit of an
infinite sum as the space between each increment
goes to zero. Each time we sample our analog
waveform y(t), we obtain the value of y at that
instant, yi. If we want an estimate of the mean
of y,
which is probably much closer to what most of you
think of as an average. This definition assumes
that the time interval between the samples is
fixed. We say estimate because we dont
actually know what happens to the signal between
samples.
Similarly,
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Averaging Period

• If the signal is periodic, we get the same mean
  and rms result if we average over any integer
  number of periods.
• If we average over a non-integer number of
  periods, we will get a different result, and the
  difference will be larger for smaller time
  intervals.
• If we are not sure of the period, we are safe if
  we take a very long average, many periods.
• The averaging period should be much longer than
  the longest period of any waveform.
• If the period is sufficient, then extending it
  further will have no effect.

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AC and DC signals
In general, a signal has both an AC and a DC
part. Often, we are interested in only one or
the other. If we want only the DC part, we can
time-average the wave form to obtain it. If we
want only the AC part, we can subtract the DC
value from the waveform to obtain the AC signal.
Doing so will make the features that are
important to us much easier to see. This is the
reason for the AC coupling button on the
oscilloscope. This conveniently removes the DC
from the signal. Additionally, all volt meters
allow us to easily measure only the DC part of
the signal, or the rms of the AC part.
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Signal Amplitude and Frequency
You have (hopefully) already learned that any
signal can be described exactly as an infinite
sum of sine waves of arbitrary amplitude (Fourier
series). A fine example of this is white light,
which we know is made up of all the colors of the
rainbow. Remembering that in light, we perceive
different frequencies as color, we see that white
light is the sum of several different colors. A
prism can be used to separate the different
colors. Fourier analysis of signals is
essentially the same thing--taking a wave that is
made up of many frequencies and separating them
for analysis.
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Sinusoidal Signals

• Most acoustic and electromagnetic sensors exploit
  the properties of sinusoidal signals
• In the time domain, such signals are constructed
  of sinusoidally varying voltages or currents
  constrained within wires
• Where vc(t) Signal
• Ac Signal amplitude (V)
• ?c Frequency (rad/s)
• fc Frequency (hz)
• t Time (s)

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Generating Sinusoidal Signals
Feedback
Frequency selective network
Gain
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Sinusoidal Signals in the Frequency Domain
No uncertainty in the measurement of the frequency
2/? Measurement Uncertainty
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Fourier Series
For a periodic function y(t) with period T
2p/w, y is equivalent to the infinite fourier
series
The various n values correspond to higher and
higher frequencies, and the As and Bs are the
amplitudes of each of these frequencies.
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Fourier Transform and the Frequency Spectrum

• Most signals that we want to look at are not
  periodic (or we dont know the period exactly)
  and we dont know their functional form.
• We may have a measured signal. If we let the
  period of our function get very long (or if we
  look at many periods), the fourier series will
  approach an integral as the spacing between the
  frequency components becomes infinitesimal.
• The As and Bs therefore go from discrete values
  to continuous functions of frequency

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Fourier Transform
We define the FT
and introduce the identity eiq cosq isinq
Reverse Transform
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The Fourier Series

• All continuous periodic signals can be
  represented by a fundamental frequency sine wave
  and a collection of sine and/or cosine harmonics
  of that fundamental sine wave
• The Fourier series for any waveform can be
  expressed in the following form
• where an bn Amplitudes of the harmonics (can be
  zero)
• n Integer

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Calculating the Fourier Coefficients

• The amplitude coefficients can be calculated as
  follows
• Because only certain frequencies, determined by
  integer n, are allowed, the spectrum is discrete
• The term a0/2 is the average value of v(t) over a
  complete cycle
• Though the harmonic series is infinite, the
  coefficients become so small that their
  contribution is considered to be negligible.

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Number of Harmonics Required

• An ECG trace, with a fundamental frequency of
  about 1.2Hz can be reproduced with 70 to 80
  harmonics (a bandwidth of about 100Hz)
• A square wave on the other hand may require up to
  1000 harmonics, because extremely high frequency
  terms are contained within the transitions

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Amplitude Modulation (AM)

• A modulation technique in which the amplitude of
  the carrier is varied in accordance with some
  characteristic of the baseband modulating signal.
• It is the most common form of modulation because
  of the ease with which the baseband signal can be
  recovered from the transmitted signal

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Percentage Modulation

• In general Aamlt1 otherwise a phase reversal
  occurs and demodulation becomes more difficult.
• The extent to which the carrier has been
  amplitude modulated is expressed in terms of a
  percentage modulation which is just calculated by
  multiplying Aam by 100.

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AM in the Frequency Domain

• To determine the characteristics of the signal in
  the frequency domain, it can be rewritten in the
  following form (using a trig identity)
• It can be seen that this is made up from three
  independent frequencies
• The original carrier
• A frequency at the difference between the carrier
  and the baseband signal
• A frequency at the sum of the carrier and the
  baseband signal

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AM Demodulation
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Frequency Modulation (FM)

• A modulation technique in which the frequency of
  the carrier is varied in accordance with some
  characteristic of the baseband signal.
• The modulating signal is integrated because
  variations in the modulating term equate to
  variations in the carrier phase.
• The instantaneous angular frequency can be
  obtained by differentiating the instantaneous
  phase as shown

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Sinusoidal FM Modulation

• For sinusoidal modulation, the formula for FM is
  as follows
• In this case ? which is the maximum phase
  deviation is usually referred to as the
  modulation index
• The instantaneous frequency in this case is
• So the maximum frequency deviation defined as ?f

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FM Spectrum

• Even though the instantaneous frequency lies
  within the range fc/-?f, the spectral components
  of the signal dont lie within this range
• The spectrum comprises a carrier with amplitude
  Jo(?) with sidebands on either side of the
  carrier at offsets of ?a, 2?a, 3?a, .
• The bandwidth is infinite, however, for any ?,
  most of the power is confined within finite
  limits
• Carsons Rule states that the bandwidth is about
  twice the sum of the maximum frequency deviation
  plus the modulating frequency

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Linear Frequency Modulation

• In most active sensors, the frequency is
  modulated in a linear manner with time
• Substituting into the standard equation for FM,
  we obtain the following result

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Frequency Modulated Continuous Wave Processing

• In Frequency Modulated Continuous Wave (FMCW)
  systems, a portion of the transmitted signal is
  mixed with (multiplied by) the returned echo.
• The transmit signal will be shifted from that of
  the received signal because of the round trip
  time ?
• Calculating the product

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FMCW continued..

• Equating using the trig identity for the product
  of two sines CosAcosB0.5cos(AB)cos(A-B)
• The first cos term describes a linearly
  increasing FM signal (chirp) at about twice the
  carrier frequency. This term is generally
  filtered out.
• The second cos term describes a beat signal at a
  fixed frequency
• The signal frequency is directly proportional to
  the delay time ?, and hence is directly
  proportional to the round trip time to the target

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Pulse Amplitude Modulation

• Modulation in which the amplitude of individual,
  regularly spaced pulses in a pulse train is
  varied in accordance with some characteristic of
  the modulating signal
• For time-of-flight sensors, the amplitude is
  generally constant
• The ability of a pulsed sensor to resolve two
  closely spaced targets is determined by the pulse
  width
• The 3dB bandwidth of a pulse is
• where ? - Spectral Width (Hz)
• ? - Pulsewidth (sec)

?
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Repeated Pulses
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Frequency Shift Keying (FSK)

• This modulation technique is the digital
  equivalent of linear FM where only two different
  frequencies are utilised
• A single bit can be represented by a single cycle
  of the carrier, but if the data rate is not
  critical, then multiple cycles can be used
• Demodulation can be achieved by detecting the
  outputs of a pair of filters centred at the two
  modulation frequencies

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Effect of the Number of Cycles per Bit on the
Signal Spectrum
5 Cycles per Bit
One Cycle per Bit
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Phase Shift Keying (PSK)

• Usually binary phase coding. The carrier is
  switched between /-180? according to a digital
  baseband sequence.
• This modulation technique can be implemented
  quite easily using a balanced mixer, or with a
  dedicated BPSK modulator
• Demodulation is achieved by multiplying the
  modulated signal by a coherent carrier (a carrier
  that is identical in frequency and phase to the
  carrier that originally modulated the BPSK
  signal).
• This produces the original BPSK signal plus a
  signal at twice the carrier which can be filtered
  out.

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Spectrum of a BPSK SignalOne Cycle per Bit
Random Modulation
Unmodulated Carrier
Modulated Carrier
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Stepped Frequency Modulation

• A sequence of pulses are transmitted each at a
  slightly different frequency
• The pulse width (in the radar case) is made
  sufficiently wide to span the region of interest,
  but because it is so wide, it cannot resolve
  individual targets within that region
• If all of the pulses are processed together, the
  effective resolution is improved because the
  total bandwidth is widened by the total frequency
  deviation of the sequence of pulses.

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Binary Numbering System

• Before we can undertake how the encoder works, we
  must first understand binary code. Specifically,
  we need to know
• How to convert a decimal number to a binary
  number
• How to convert a binary number to a decimal
  number
• What is meant by 2s complement binary numbers
  and their use

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Digital Data and Binary Numbering System
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The Decimal System
Decimal System means we can use up to 10
possibilities per digit 0, 1, 2, 3, 4, 5, 6, 7,
8, 9
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2 tens 4 ones
The decimal numbering system is called a
base-10 system.
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The Decimal System
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Binary System
2 digits 2 possibilities 0, 1
So this is a Base-2 system.
1011
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Binary Numbers
1 0 1 0 1 0 1 0 128 64 32 16 8
4 2 1
170
0000 1111 b 15 d
1111 1111 b 255 d 0000
1111 1111 1111 b 4095 d 1111 1111 1111 1111 b
65,535d
8 bits 1 byte
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Converting Binary to Decimal
0000 0101 b ? d 0000 1111 b ? d 0010
1010 b ? d 1101 0101 b ? d
5
15
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213
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Adding in Binary
0 0 1 1 0 1 0 1 0 1
1 10
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Subtracting in Binary
1 1 11 10 -0 -1 -1 -1 1 0 10
1
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More Adding in Binary
3 3 6
11 11 110
compare to decimal
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Converting Decimal to Binary
Assume we are in an 8-Bit System
Least Significant Bit (LSB)
Most Significant Bit (MSB)
92 d 0101 1100 b
zero padding
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Converting Decimal to Binary
Example
12 d ? b 75 d ? b 1215 d ? b
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Representing Negative Numbers
0000 0001 0010 0011 0100 0101 0110 0111 1000 1001
1010 1011 1100 1101 1110 1111
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 1 2 3 4 5 6 7 -8 -7 -6 -5 -4 -3 -2 -1
3 bits of numbers, 1 bit of sign
4 bits of numbers
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Positive Negative Numbers
To get positive numbers
1) Convert the magnitude to binary - but you must
have at least one leading zero
To get negative numbers
1) Convert the magnitude of the number to binary
- have at least one leading zero 2) Invert all
of the bits - 0s become 1s, 0s become 1s 3)
Add 1 to the result
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To get negative numbers
Example - 92
1) Convert the magnitude of the number to binary
- have at least one leading zero 2) Invert
all of the bits - 0s become 1s, 0s become
1s 3) Add 1 to the result
01011100
10100011
10100011 1 10100100
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Exemple
Use one byte 8 bits for both
19 d ? b
-19 d ? b
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A One-bit A/D Converter
Input
Output
Vin
Vin gt 5 V Þ output on 1 Vin lt 5 V Þ output
off 0
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A Two-bit (Unipolar) A/D Converter

• Output has 2N possible values

Input
Output

• Range?
• Span?

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A Two-bit (Bipolar) A/D Converter
Input
Output (2s complement)
5.0 V
1d
Range is 5 to 5 volts. Span is 5 (-5) or
10 volts.
2.5 V
0d
0.0 V
-1d
- 2.5 V
-2d
- 5.0 V
N 2
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A Two-bit (Unipolar) A/D Converter

• How big is each input bin ?

Input
Output
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Input Resolution Error Quantization Error
10.0 V
3d
7.5 V
2d
5.0 V
1d
2.5 V
0d
0 V
N 2
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Saturation
Input
Output
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A Two-bit (Bipolar) A/D Converter
Input
Output (2s complement)
5.0 V
1d
2.5 V
0.5 (Vru - Vrl) / 2N
0d
0.0 V
-1d
- 2.5 V
-2d
- 5.0 V
N 2
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Calculating the Digital Output

• To estimate the digital output of an A/D
  converter, see page 83.
• Example for a simple binary A/D converter

Do digital output (bin number as a decimal
number) Vin analog input voltage Vru upper
value of input range Vrl lower value of input
range N number of bits
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A Two-bit (Unipolar) A/D Converter

• How big is each input bin?

Input
Output
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Calculating the Digital Output
Example 8-bit, simple binary A/D
converter Range is 0 to 5V. Input is 3.15V.
Find output.
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Choosing an A/D Converter Resolution
National Instruments model 16E-4 16?10-4
12 bits
National Instruments model 16E-50 16?10-50
16 bits
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Choosing an A/D Converter Speed
National Instruments model 16E-4 16?10-4
(kiloSamples/second)
500 kS/s
National Instruments model 16E-50 16?10-50
20 kS/s
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Choosing an A/D Converter Input Range
National Instruments model 16E-4 16?10-4
? 10 V
National Instruments model 16E-50 16?10-50
? 10 V
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Measurement of Electrical Signals

• Sampling

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Sampled Signals and Digitisation

• To process signals within a computer requires
  that they be sampled periodically and then
  converted to a digital representation
• To ensure accurate reconstruction
• the signal must be sampled at a rate which is at
  least double the highest significant frequency
  component of the signal. This is known as the
  Nyquist rate.
• The number of discrete levels to which the signal
  is quantised must also be sufficient.
• Signal reconstruction involves holding the
  signal constant (zero order hold) during the
  period between samples then passing the signal
  through low-pass filter to remove high frequency
  components generated by the sampling process

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SAMPLING
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Sampling and A/D Conversion

• a) An analog signal has been sampled and then
  converted to digital (2s complement).
• b) This quantization results in error.

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Sampling
When sampling a waveform, it is important to
sample frequently enough to accurately represent
the wave.
The sampling theorem, which we will prove later,
tells us that we can get accurate frequency
information if we sample at least twice as fast
as the highest frequency in our signal, fs gt 2fm,
or dt lt 1/2fm
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Selecting the sampling rateand number of samples
The sampling rate

• This number needs to be sufficiently small to
  provide the desired frequency resolution.
  Decreasing it decreases leakage effects.
• fs must be large enough to be at least (and
  preferably much more than) twice the highest
  frequency in your signal. Once fs is chosen, it
  is a good idea to apply a low-pass filter to your
  signal fs /2 to remove any aliasing.

96
Sampling of Time-Varying Signals (Measurands)

• When using a computerized data acquisition
  system, measurements are only made at a discrete
  set of times, not continuously.
• For example, a temperature or voltage reading
  (called a sample) may be taken every 0.1 s or
  every 2 min, and no information is taken for the
  time periods in between the samples.

97
Sampling of Time-Varying Signals (Measurands)

• The rate at which measurements are made is known
  as the sampling rate, expressed in samples/sec or
  Hz.
• Incorrect selection of the sampling rate can lead
  to misleading results.

98
Sampling of Time-Varying Signals (Measurands)

• a) time-varying signal
• (e.g., voltage)
• b) signal being sampled
• c) the sampled points
• (dots)
• d) signal can be reconstructed by connecting the
  dots

99
The Problem of Aliasing
10 Hz input sampled at 11 Hz
Output looks like 1 Hz !
100
Aliasing

• In the frequency domain, an analog signal may be
  represented in terms of its amplitude and total
  bandwidth as shown in the figure.
• A sampled version of the same signal can be
  represented by a repeated sequence spaced at the
  sample frequency (generally denoted fs)
• If the sample rate is not sufficiently high, then
  the sequences will overlap, and high frequency
  components will appear at a lower frequency
  (albeit with reduced amplitude)

Chirp to 500Hz sampled at 1kHz
Sampled at 500Hz
Sampled at 250Hz
101
The Problem of Aliasing (continued)
10 Hz input sampled at 12 Hz
Output looks like 2 Hz
102
Another pathology
10 Hz input sampled at 5 Hz
Beware if
Output looks like 0 Hz (dc)
103
Avoiding aliasing
To avoid aliasing, sample your signal at greater
than twice the maximum frequency of interest.
This is a minimum 10 X the maximum frequency
of interest would be better.
Another way to state this rule is the Nyquist
criterion
104
Summary of Sampling and Aliasing Frequencies
Forbidden Region!!
105
A-D Conversion
Say we have a voltage range from 0-4V and a 2 bit
A-D converter. Then our resolution, Q 1V.
106
A-D CONVERSION
107
A-D CONVERSION
FM QUALITY SAMPLINQ FREQUENCE 32 Khz 12 BITS
CD QUALITY SAMPLINQ FREQUENCE 44,1 Khz 16 BITS
108
Digital to Analog Converters
Currently, a lot of analog data is stored
digitally. A very common example is the compact
disk. The heart of a CD player is a D-A
converter that takes the digital information on
the disk and creates the analog signal that
drives the speakers.
109
Analog to Digital Converters

• The flip side of this is an analog to digital
  converter. These are also very common today.
• More important than how it works is knowing what
  its limitations are. The two most important
  numbers about an AD converter are its voltage
  range Efsr and the number of bits M.
• Some A-D converters are 12 bits and have several
  different possible voltage ranges including
  5Volts. This makes Efsr 10V. For this
  configuration, the voltage resolution of the
  system is
• Q Efsr/2M 0.00244 Volts.
• Current automobiles need to take data on their
  performance in order to keep emissions low.
  Measurements such as temperature are digitized so
  they can be analyzed by the cars computer.

110
Computer Number System ExamplesFrom Computer
Science I
10110101110001011001110011110110 binary number
11 5 12 5 9 12 15 6

equivalent base 10 value for each group of 4
consecutive binary digits (bits)
B 5 C 5 9 C F
6
corresponding hexadecimal (base 16) digit
B5C59CF6
equivalent hexadecimal number
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Counting and Converting Binary Numbers
8 1x23 0x22 0x21 0x20 01000 in Binary
Calculator Applet
112
Communicating With Pulses

• PCM Pulse Code Modulation

113
PCM Pulse Code Modulation
114
PCM Pulse Code Modulation
115
Binary Counter

• Animations showing counter operation
• http//www.play-hookey.com/digital/synchronous_c
  ounter.html
• Counter from IEE

116
Typical Output for Binary Counter

• Note how the Q outputs form 4 bit numbers