Vibration measurements2

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Vibration of
Structures

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Vibration Measurements

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Course Contents
SDOF
M-DOF
Cables/String
 Bars
 Shafts
 Vibration Attenuation
 Beams
 Membranes & Thin Plates
o Vibration Testing and Measurements

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Objectives
• Vibration Measurements Hardware
• Introduction to signal Processing
• Vibration Measurements Terminology

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Measurements Hardware

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Active health monitoring of an aircraft wing with embedded piezoelectric sensor/actuator network, Smart Mater. Struct. 16 (2007) 1208–1217

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• Transducers
• Exciters
• Signal Conditioning
• Analyzer

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Measurements HW: Transducers
• Load cell: a piezoceramic device
configured to produce a voltage
proportional to force
• Accelerometer: (piezoceramic) produce
a signal proportional to acceleration at
the point of attachment
• Proximity probes: magnetic device
gives a signal proportional to local
displacement

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• Laser vibrometers: give a signal proportional
to local velocity (or scanned to give velocity
field)
• Strain gauges: (local strain or strain rate)
Measurements HW: Transducers

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• Electromagnetic shakers: which may
apply a force through a range of
frequencies (harmonic or random inputs)
• Instrumented hammers: which
simulate an impact
Measurements HW: Exciters

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• The direct out put of a transducer not
usually well suited for input into an
analyzer: Impedance miss matched,
voltage or current levels too low
• SC is a charge or voltage amp designed
to take an accelerometer signal and
match it to the input requirements of the
analyzer
Measurements HW: SC

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• Electronic boxes (dedicated computers)
gathers signals and manipulate them
mathematically
• Modern analyzers have evolved almost
in to chip sized devices
• Main source of manipulation is Digital
Fourier Transforms for manipulating
the vibration data in the frequency
domain
Measurements HW: Analyzer

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Objectives
• Vibration Measurements Hardware
• Introduction to Signal Processing
• Vibration Measurements Terminology

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Fourier Series: Physical Meaning

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Fourier Series of F(t) of period T
 
,....3,2,1
)sin()(
2
,)cos()(
2
)(
2
2
)sin()cos(
2
)(
00
0
0
1
0










n
dttntF
T
bdttntF
T
a
dttF
T
a
T
tnbtna
a
tF
T
Tn
T
Tn
T
T
n
TnTn





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Digital Signal Processing
The analyzer takes signals form the
transducer and puts the signal into a form
that can be mathematically `manipulated

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Analog-to-Digital Converter
Analog signal x(t) is sampled at many
equally spaced time intervals to produce
the digital record:
{x(t1), x(t2), ….x(tN)}
where x(tk) is the discrete value of x(t) at
time tk and N is the number of samples
taken

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What happens if Sampling rate is too
large?
sec1
)
4
7
sin(
)
4
sin(
2
1



T
tx
tx



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What happens if Sampling rate is too
large?

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• Aliasing: When sampling rate is too
large, the digital record does not catch the
details of the analog signal
• What is the optimum Sampling rate?
• Min. 2 Samples / Cycle
• Experience: 2.5 Samples / Cycle
Shannon’s sampling theorem

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• Antialiasing Filter: low-pass Filter(only
allows low frequencies through).
• The filter cuts off frequencies higher than
about half the maximum frequency of
interest: (ωmax , Nyquist frequency).
• Most modern digital analyzers provide
built-in antialiasing filters.

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Digital Fourier Transform
,....3,2,1
2
sin
1
,
2
cos
1
1
2
sin
2
cos
2
)(
11
1
0
2/
1
0








































n
t
T
n
x
N
bt
T
n
x
N
a
x
N
a
t
T
n
bt
T
n
a
a
txx
N
k
kkn
N
k
kkn
N
k
k
N
n
knknkk


N is fixed by hardware (a power of 2)

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Above becomes the matrix equation
x=Ca
Where:
C: contains the sin and cos terms
x: is the vector of samples and
a: is the vector of Fourier Coefficients
• Analyzer computes the coefficients
in the DFT formula:
a=C-1x

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What happens if Samples do not match
the periods?

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Leakage: If the signal is not periodic in N
samples (signal cut off mid period) the DFT will
produce extra frequencies

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Windowing: multiply signal by a function
which is zero at the end points

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Radom Signals
• Autocorrelation: Tells how fast x(t) is changing
 

T
T
kxx dtxtx
T
tR
0
)()(
1
lim)(


 




 dexRS j
xxxx )(
2
1
)(
• Power Spectral Density (PSD): Frequency
domain of Autocorrelation [ FT(Rxx) ]

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Radom Signals
• Crosscorrelation: Tells how fast x(t) is
changing w.r.t. f(x)
 

T
T
kxf dtftx
T
tR
0
)()(
1
lim)(


 




 dexRS j
xfxf )(
2
1
)(
• Cross Spectral Density (PSD): Frequency
domain of Autocorrelation FT(Rxf)

Vibration measurements2

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