Vibration Measurement
1
Introduction
• Why we need to measure vibrations:
– To detect shifts in ωn which indicates
possible failure
– To select operational speeds to avoid
resonance
– Measured values may be different from
theoretical values
– To design active vibration isolation systems
– To identify mass, stiffness and damping of a
system
– To verify the approximated model
2
Introduction
• Type of vibration measuring instrument
used will depend on:
– Expected range of frequencies and
amplitudes
– Size of machine/structure involved
– Conditions of operation of the
machine/structure
– Type of data processing used
3
Equipmental Modal Testing
• General arrangement for experimental modal analysis:
&x&(t )
f(t)
v(t)
4
Vibration Measurement
• Dynamics testing is a valuable
complement to dynamics analysis
• Modal testing
– Natural frequencies
– Damping
– Mode shapes
• Operational testing
– In-service loads (often random)
5
Transducers
• Translates motion or forces into
electrical signals
• Size of transducer is important (micro)
– Ideally does not influence the structure’s
dynamics through added mass or stiffness
– Analytical models often include effects of
transducer mass
6
Variable resistance transducer
x (t )
Strain gage
m
Cantilever beam
ε = const x(t )
base
Sec. 10.2
7
Variable resistance transducer
http://www.blh.de
8
Piezoelectric transducer
F=A p
h
Qx=d F=d A p
F
Vout(t)=ν h p(t)
d = piezoelectric constant
v =voltage sensitivity coeff
h = thickness of transducer
http://www.bksv.com/
11
Electrodynamic transducer
Vout
Sensor: vel. →Volt
Vout(t)=B l v(t)
Exciter: I → F
B l =V/v=F/I
F=Bl I
13
Response Transducers
(acceleration)
• Accelerometers are very common
– Based on piezoelectric elements
piezoresistive, capacitive, etc.
– See Figures 10.12 in book www.bksv.com
• Accelerometers themselves are m/c/k
systems
– Must watch for internal resonances
14
Response Transducers
(velocity)
• MHD (magneto-hydrodynamics)
ATA Sensors Inc, USA
http://www.aptec.com/Sensors2/ars01_01s_mhd_angular_rate_sensor.htm
• Lasers vibrometer
– Doppler shifts
– Can scan large structures
– Line of sight
– Noncontact
15
Response Transducers
(displacement)
• LVDT
– linear variable differential transformers
– Magnets and coils
– Noisy; best for low-freq, highdisplacement application
• RVDT
– rotary variable differential transformers
• Optical sensors (noncontact)
6
http://www.transtekinc.com/
Linear variable differential transformer
transducer
•
•
17
Output voltage depends on the axial displacement of the core
Insensitive to temp and high output
http://www.daytronic.com/Products/trans/lvdt/default.htm
Modal Testing
The use of our analytical methods
to date to interpret vibration
measurements
A standard skill used in industry
18
Measurement Hardware
Load cell
Accelerometer
Laser, etc
Structure
Exciter
Power
supply
SC
SC
Display
Signal Analyzer
Modal Software
Signal
generator
SC = Singal Conditioning
19
Exciters
• Electromagnetic shakers which may apply a
force through a range of frequencies
(harmonic or random inputs)
• Instrumented hammers which simulate an
impact (recall the impulse response)
B&K 4808,
5~10 kHz, 112 Newtons
20
Mechanical Exciters
• Force applied as an
inertia force:
• Force applied as an
elastic spring force:
• Used for frequency <30
Hz and loads <700N
22
Scotch yoke
mechanism
Mechanical Exciters
23
• Makes use of unbalance created by 2
masses rotating at same speed in
opposite directions, load: 250N-25kN
• F(t) = 2mRω2cosωt
Signal Conditioning
• The direct output 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
B&K 2635 charge amplifier
http://www.bksv.com/3073.asp
24
Analyzer
• Electronic boxes (really dedicated computers)
which gather signals and manipulate them
mathematically
• Like all other computer based technologies
the analyzer “boxes” have evolved almost in
to chip sized devices
• Essentially their main source of manipulation
is digital Fourier Transforms for manipulating
the vibration data in the frequency domain
25
Fourier Series of F(t) of period T
a0 ∞
F(t ) =
+ ∑ (an cosnωT t +bn sin nωT t)
2 n=1
where ωT =
2π
T
2
a0 = ∫ F(t)dt
T 0
T
2
an = ∫ F(t)cosnω Ttdt n = 1,2,3....
T 0
T
2T
bn = ∫ F(t)sin nωT tdt n = 1,2,3....
T0
26
Basic idea of the Analyzer
• Analog voltage in from force f(t) and one
of x(t), v(t) or a(t) transducers
• Signals are filtered, “digitized” and
transformed to the frequency domian
• Manipulated to produce digital
frequency response functions from
which vibration data is extracted
27
Digital signal processing
• The analyzer takes signals form the
transducer and puts the signal into a
form that can be mathematically
manipulated
• This of course is best performed with
digital computers, hence we rely on
some basic principles of DSP (Matlab)
28
Forming a digital signal:
Analog to Digital Conversion
• The 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
• Constructed by A/D converter (eg gated)
29
Sampling rate
• Aliasing is caused by not sampling enough of
the signal so the digital record does not catch
the details of the analog signal
• Problem is solved by choosing a sample rate
of 2.5 times the largest frequency of interest
and by
• Anti-aliasing filters (cut off filter) allowing a
sane choice of sampling rate
30
Digital Fourier Transform
a0 N / 2 ⎛
2 πitk
2π itk ⎞
xk = x(t k ) =
+ ∑ ai cos
+ bi sin
,k = 1,2...N
⎝
⎠
2 i =1
T
T
where
1
a0 =
N
N
∑x
k =1
k
1 N
2 πik
ai = ∑ xk cos
N k =1
T
1
bi =
N
31
N
∑ x k sin
k =1
2 πik
T
FFT/DFT Analyzer
• 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
• The analyzer computes the coefficients
in the DFT formula by a=C-1x
• N is fixed by hardware (a power of 2)
32
Spectral Leakage
• If the signal is not periodic in N samples
(signal cut off mid period) the DFT will
produce extra frequencies, called leakage
• Fixed by windowing, multiply signal by a
function which is zero at the end points, or
tapers off (many kinds, depending on nature
of signal)
• Downside, it adds damping to signal
33
Random Signal Analysis
1
Autocorrelation : Rxx (τ ) = lim ∫ x(t)x(t + τ )dτ
T→∞ T
Tells how fast x(t) is changing
0
T
1 ∞
− jωτ
Power Spectral Density (PSD) : Sxx (ω) =
R
(
τ
)e
dτ
xx
∫
2π −∞
Fourier transform of R
1
Crosscorrelation : Rxf (τ ) = lim ∫ x(t) f (t + τ )dτ
T→∞ T
0
Tells how fast one signal changes
T
∞
relative to another
1
− jωτ
Cross Spectral Density : Sxf (ω ) =
R
(
τ
)e
dτ
xf
∫
2 π −∞
R and S are available from DFT and
connect response to system frequency
response function
S fx (ω) = H ( jω )S ff
Sxx (ω ) = H( jω )Sxf
2
Sxx (ω ) = H( j ω) S ff
35
1
G(s) =
2
ms + cs + k
G( jω ) = H(ω ) =
Transfer function
1
Frequency response function
k − mω + cjω
1 − ζωt
h(t) =
e sin ω d t
mωd
2
Impulse response function
1
LT[h(t)] = G(s) = 2
ms + cs + k
f (t) deterministic
f (t) random
X(s) = G(s)F(s)
t
x(t) = ∫ h(t − τ ) f ( τ)dτ
0
36
2
Sxx (ω ) = H(ω ) Sff ( ω)
E[x 2 ] =
∞
∫
−∞
2
H(ω ) S ff (ω )dω
The way it works
• PSD’s calculated in analyzer
• Used to form H(w) for force in, response
out (velocity, position or acceleration)
• H(w) used to extract modal data of
structure in a number of different ways,
forming the topic of Modal Testing
37
Coherence
γ =
2
Sxf ( ω)
2
Sxx (ω )S ff (ω )
0 ≤ γ2 ≤ 1
should be 1, especially
near resonance
38
• Compute H(w) a
number of different
ways
• Compare the
various measured
values of H(w)
• Indicates how good
the measurement is
Transfer function nomenclature
2
sX ( s )
s X (s)
X (s)
= G ( s ), or
= G ( s ), or
= G(s)
F ( s)
F (s)
F (s)
response
standard
reciprical
Displacement compliance dynamic stiffness
Velocity
mobility
impedance
Acceleration
39
inertance
apparent mass
Measured FRF
Natural frequency
taken as peak value
Illustrates peak picking method
of determining modal parameters
40
Damping ratio from peak picking
H (ωd )
H( ωa ) = H(ω b ) =
2
ωb − ω a
ζ=
2ω d
41
• Use the two
frequencies
determined from the
half power (0.707)
points
• Gives the modal
damping ratio
Example 1 : Response with two modes:
First mode
ζ1 =
=
42
ω b − ωa
2ω1
10.16 − 9.75
= 0.02
20
second mode
ζ2 =
ω b − ωa
2ω2
21.67 − 17.10
=
40
= 0.11
Mode shape measurement
j ωt
jωt
&
&
&
Mx + Cx + Kx = fe , x = ue
( K − ω M + jωC )u = f
2
u = ( K − ω M + j ωC ) f
2
−1
α (ω ) = ( K − ω M + jωC )
2
−1
= [ Sdiag(ω − ω + 2ζ iω i j ) S ]
2
i
43
2
T −1
P = [u1 : u2 :Lun ]
S = M1 / 2 P,
⎡
⎤ −1
1
S
α (ω ) = S diag ⎢ 2
2
⎣ ω − ω + 2ζ ω ωj ⎥⎦
−T
i
i=1
i
T
ui u i
n
α (ω ) = ∑
i
2
i
2
ω − ω + 2ζiωi ωj
in element form at resonance
T
u iu i
sr
= 2ζiω Hsr (ω i ), where Hsr = α sr
2
i
Measurement points
44
Need only one column to get
mode shape
Suppose ui = [a1 a2 a3 ]
T
⎡ a12 a1a2 a1 a3 ⎤
ui uTi = ⎢ a2 a1 a22 a2 a3 ⎥
2 ⎥
⎢a a a a
a
3 2
3 ⎦
⎣ 3 1
really only 3 unkowns a1 , a2 and a3 . So just three
elements of α (ω) need be measured per mode to get
the mode shape
45
Measure MDOF FRF
Freq Response Fun.
Force
Transd.
Accel. #1
Shaker
Accel. #2
Accel. #3
46
Impulse Response Testing
47
49
Inertial sensor model
case
m
x
m
. .
c(x-y)
k(x-y)
y
k/2
c
k/2
z = x− y
0
c
k
&z& + z& + z = − &y& = ω 2Y sin ω t
m
m
Frequency Response Function for Vibration Sensor
r2
Z
=Y
Y
(1 − r 2 ) 2 + (2ς r ) 2
2ζ r
φ = tan
1− r 2
−1
ω 2Y sin(ω t − φ ) Acceleration
z (t ) =
=
2
ωn
ω n2
z (t ) = Y sin(ω t − φ )
51
Frequency Response Function for Vibration Sensor
If stiff spring, big k → high ωn
r=ω /ωn → small
ω 2Y sin(ω t − φ ) Acceleration
=
z (t ) =
2
ωn
ω n2
a) If no damping (ζ = 0) → r < 0.1
r = .1 output value has 1% error
b) If ζ = 0.7, r < 0.25
If soft spring, small k → low ωn
r=ω /ωn → large
52
z(t) ~ y(t)
Example 10.2
• A vibrometer having a natural frequency
of 4 rad/s and ζ = 0.2 is attached to a
structure that performs a harmonic motion.
If the difference between the mximum
and the minimum recorded values is 8
mm, find the amplitude of motion of the
vibrating structure when its frequency is
40 rad/s.
54
Solution
• Amplitude of recorded motion:
Z=
Y (10 )
2
(1 − 10 ) + [2(0.2)(10)]
2 2
2
= 1.0093Y = 4 mm
• Amplitude of vibration of structure:
• Y = Z/1.0093 = 3.9631 mm
55
10.3.2 Accelerometer
• Measures acceleration
of a vibrating body.
− z (t )ωn2 =
If
− Yω 2 sin (ωt − φ )
(1 − r ) + (2ζr )
2 2
2
1
(1 − r ) + (2ζr )
2 2
2
≈ 1,
− z (t )ω ≈ −Yω sin (ωt − φ )
2
n
56
2
10.3.2 Accelerometer
• If 0.65< ζ < 0.7,
1
0.96 ≤
≤ 1.04 for 0 ≤ r ≤ 0.6
2
2 2
(1 − r ) + (2ζr )
57
• Accelerometers are preferred due their
small size.
Example 10.3
• An accelerometer has a suspended mass
of 0.01 kg with a damped natural
frequency of vibration of 150 Hz. When
mounted on an engine undergoing an
acceleration of 1 g at an operating speed
of 6000 rpm, the acceleration is recorded
as 9.5 m/s2 by the instrument. Find the
damping constant and the spring stiffness
of the accelerometer.
58
Solution
1
Measured value 9.5
=
=
= 0.9684
True value
9.81
(1 − r ) + (2ζr )
or (1 − r ) + (2ζr ) = (1 / 0.9684 )
2 2
2 2
2
2
2
= 1.0663
6000(2π )
Operating speed ω =
= 628.32 rad/s
60
(E.1)
ωd = 1 − ζ 2 ωn = 150(2π ) = 942.48 rad/s
ω
ω
628.32
r
Thus
=
=
=
= 0.6667
2
2
ωd
942.48
1 − ζ ωn
1− ζ
∴ r = 0.6667 1 − ζ
59
2
(
or r = 0.4444 1 − ζ
2
2
)
(E.2)
Solution
•
•
•
•
Substitute (E.2) into (E.1):
1.5801ζ4 – 2.2714ζ2 + 0.7576 = 0
Solution gives ζ2 = 0.7253, 0.9547
Choosing ζ= 0.7253 arbitrarily,
ωn =
ωd
1− ζ
2
=
942.48
1 − 0.7253
2
= 1368.8889 rad/s
k = mωn2 = (0.01)(1368.8889 ) = 18738.5628 N/m
2
Damping constant
c = 2mωnζ = 2(0.01)(1368.8889)(0.7253)
= 19.8571 N - s/m
60
Machine Maintenance Techniques
• Life of machine follows the bathtub curve:
61
Machine fatique, wear, deformation, etc, lead to increase in the
clearances between mating parts, misalignments in shaft, crack
initiation in parts and unbalance in rotors
Machine Maintenance Techniques
Breakdown maintenance:
• Allow the machine to fail and then replace
it with a new machine
• This strategy is used when machine is
inexpensive and no other damage is
caused by the breakdown.
6
Machine Maintenance Techniques
Preventive maintenance:
• Maintenance performed at fixed intervals
• Intervals determined statistically from past experience
• This method is uneconomical
Condition-based/Predictive maintenance:
• Replace fixed-interval overhaul with fixed-interval
measurements
• Can extrapolate measured vibration levels to predict
when they will reach unacceptable values
63
Machine Maintenance Techniques
• Maintenance costs are greatly reduced
64
Machine Condition Monitoring Technique
• Following methods are used to monitor
machine conditions:
• Aural and visual – a skilled technician will listen
and see the vibrations produced by the
machine
• Operational variables monitoring – performance
is monitored wrt intended duty. Deviation
denotes a malfunction.
65
Machine Condition Monitoring Technique
• Temperature monitoring – rapid increase in
temperature is an indication of malfunction
• Wear debris found in lubricating oils can be used to
assess extent of damage by observing
concentration, size, shape and colour of the
particles
• Available vibration monitoring techniques
66
Vibration Monitoring Techniques
Time domain analysis
• E.g. following is an acceleration waveform of a
faulty gearbox. Pinion is coupled to 2685 rpm
motor.
• Period of waveform same as period of pinion.
• This implies a broken gear tooth on the pinion.
67
Vibration Monitoring Techniques
• Peak level, RMS level and crest factor may be
used as indices to identify damage
• Changes in Lissajous figures can be used to
identify faults.
68
Relationship between machine components
and the vibration spectrum
Each rotating element generates identifiable frequency. Thus
changes in the spectrum at a given freq can be attributed to the
corresponding element.
69
Vibration Monitoring Techniques
• 2nd gear was at
fault although 1st
gear was engaged.
70