spatial mode shape identification using continuous wavelet transform
DESCRIPTION
This paper presents an experimental modal analysis of damped multi-degree-of-freedommechanical system using continuous wavelet transform. An approximation of wavelet transformof impulse response function is deduced, which serves as a basis for extraction of naturalfrequencies, damping ratios and mode shapes. Due to approximation with finite Taylor series acomputational error of identified oscillatory amplitude occurs, which is observed on simulatedsystem response. Presented approach of modal identification is applied to real mechanicalsystems, such as steel beam and horizontal tail of ultralight motor craft. Using proposedmeasurement methodology, it is possible to reconstruct spatial mode shapes of any dynamiclinear system with arbitrary geometry.TRANSCRIPT
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Spatial mode shape identification using continuous wavelet transform
Martin Česnik 1, Janko Slavič1, Miha Boltežar 1 (
1Faculty of Mechanical Engineering, University of Ljubljana)
V članku je predstavljena eksperimentalno modalna analiza dušenega mehanskega sistema z več
prostostnimi stopnjami z uporabo zvezne valčne transformacije. Izpeljana je aproksimacija
zvezne valčne transformacije pričakovanega impulznega odziva sistema, na podlagi katere je
razvit postopek za določitev lastnih frekvenc, razmernikov dušenja in lastnih oblik . Prikazana je
računska napaka pri identifikaciji amplitude, ki se pojavi kot posledica aproksimacije.
Predstavljeni pristop je uporabljen na realnih primerih jeklenega nosilca in horizontalnega repa
ultra lahkega letala. S prikazano metodologijo je možno rekonstruirati trirazsežne lastne oblike
sistema s poljubno geometrijo.
This paper presents an experimental modal analysis of damped multi-degree-of-freedom
mechanical system using continuous wavelet transform. An approximation of wavelet transform
of impulse response function is deduced, which serves as a basis for extraction of natural
frequencies, damping ratios and mode shapes. Due to approximation with finite Taylor series a
computational error of identified oscillatory amplitude occurs, which is observed on simulated
system response. Presented approach of modal identification is applied to real mechanical
systems, such as steel beam and horizontal tail of ultralight motor craft. Using proposed
measurement methodology, it is possible to reconstruct spatial mode shapes of any dynamic
linear system with arbitrary geometry.
(c) 2009 Journal of Mechanical Engineering. All rights reserved.
(Keywords: modal parameters, continuous wavelet transform, spatial vibrations)
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0 INTRODUCTION
One of the key steps at development of
numerical model is its validation, where the
modal analysis of numerical model and the
experimental modal analysis (EMA) of real
system are compared. With the results of the
comparison it is possible to determine the
adequateness of the numerical model.
The most common method to perform
modal analysis is done by computing
frequency response functions (FRF) [1, 2].
This is done by signal transformation from
time domain to frequency domain using
Fourier transformation. With the use of fast
Fourier transform (FFT) algorithm the
calculation of FRF is a simple and useful
method to perform EMA [3, 4, 5]; however,
its feasibility is restricted due to the absence
of time component in transformed signal.
Described method is therefore suitable for
analyzing stationary signals. For analysis of
non-stationary signals it is convenient to use
transformations in time-frequency domain,
for instance short-time Fourier transform [6],
continuous wavelet transform (CWT), etc.
An impulse response of mechanical
system is a typical non-stationary process
where the oscillation amplitude decreases
exponentially with time. The idea of
applying CWT on dynamic system response
for damping identification was first
introduced by Staszewski [7]. While
Staszewski used Morlet wavelet function,
Slavič et. al [8, 9, 10] developed their idea
using Gabor wavelet function and focusing it
on the edge effect and relatively short
signals. The properties of the Gabor wavelet
have been studied in detail by Simonovski
and Boltežar [11], who used it also for fault
detection in DC electro motors [12].
Le and Argoul [13] extended the work
of previous researches and identified all
modal parameters by applying CWT on
theoretical impulse response of spatial 4
degree-of-freedom (DOF) model with
Morlet, Cauchy and harmonic wavelet
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function. They also explored the problem of
edge-effect and time-frequency localization.
Furthermore, Lardies and Gouttebroze
[14] applied CWT to analyze real signals
measured on tower, excited by wind and
determined tower’s eigen frequencies,
damping ratios and one-dimensional mode
shapes. They compared results, obtained by
CWT and by autocorrelation method. Their
work was enhanced by Huang and Su [15]
who applied CWT on discrete equations of
motion in order it identify modal parameters
of steel frame subjected to earthquake
excitation. As a result they obtained frame
mode shapes by determination of planar
displacements in horizontal plane.
To continue their work, Česnik et. al
[16] tried to extract spatial mode shapes of
mechanical system with arbitrary geometry.
This work represents a continuation and
enhancement of their research.
In this paper, a procedure for
identification of modal parameters is
introduced, based on CWT of impulse
response function. Due to the low order
approximation, the computation error of
oscillatory amplitude occurs when dealing
with systems with high damping, which is
presented on numerical example. The
method is also tested on real system of steel
beam and compared to theoretical mode
shapes. With identification of spatial mode
shapes of horizontal tail of ultralight aircraft
the feasibility of developed method is
shown.
1 CONTINUOUS WAVELETTRANSFORM
Introductory part of CWT theory will be
summarized according to Mallat [6] and
Tchamitchian et. al [17]. Continuous wavelet
transformation of time function )(t f , that
satisfies )()( 2 Lt f , is defined as
)()(),( *
, t t f suWf sud t , (1)
where )(t represents wavelet function,
superscript * denotes complex conjugate, u
is translation parameter related to time and
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s is scale parameter that serves as an
inverse of frequency. Figure 1 shows the
influence of parameters s and u on wavelet
function.
Function 2( ) ( ) f t L may be treated
as a mother wavelet function when it
satisfies oscillatory, energy preservation and
admissibility conditions [6]. A modified
wavelet function )(, t su, translated in time
domain and dilated in frequency domain,
also has to satisfy energy preservation
condition, we obtain
,
1,u s
t ut
s s (2)
i
,ˆ ˆ ,
u
u s s e s (3)
where,
ˆu s
denotes integral Fourier
transformation of wavelet function )(, t su
in frequency domain.
An impulse response of multi-degree-
of-freedom (MDOF) system is expected to
be a summation of N mutually independent
function pairs [7]
i
1
i
N t
i
i
f t A t e , (4)
wherei A t denotes amplitude part and
i t phase part (9). Therefore, a linearity
property of CWT [6] can be used
1 1
, , . N N
i i i i
i i
W f u s Wf u s
(5)
1.1 Gabor Wavelet Function
Gabor mother wavelet function is
defined as [6]
2
2 i2
1 42
1, , .
t
t
Gabor t e e
(6)
By introducing translation and dilation
parameters into (6), we obtain a family of
Gabor wavelet functions, defined in time and
frequency domain as [11]
2
2 2 i2
, 1 42 2
1, , ,
t u t u
s su s t e e
s (7)
22 2
1 42 2 i2
,
ˆ , , 4 .
s s
u
u s s e e
(8)
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Figure 1: (─) mother , 0, 1Gabor t u s and
(▬) ,, 4, 0.5
u sGabor t u s.
1.2 Approximation of CWT of
Asymptotic FunctionAn impulse response of viscously
damped single-degree-of-freedom (SDOF)
system can be written as [1]
0
0d 0cos ,
t f t A e t
(9)
where denotes damping ratio, 0 phase
shift,0 undamped and
2
0d 0 1
damped angular frequency. In order to
separate amplitude and phase information of
response function it can be written in more
general form as an analytic function [17]
i. f t
a f f t A t e (10)
Similarly, a wavelet function may also be
given in form of analytic function as
i t
a t A t e . With regards to a
relation , 1 2 ,aWf u s Wf u s [6] one can
deduce
i
1( , )
2
d . f
f
t ut
s
t uWf u s A t A
s
e t (11)
In order to solve equation (11) few
approximations have to be made:
function f A t is approximated with zero
order Taylor series (i.e. constant value A),
function f t is approximated with first
order Taylor series. By using this
approximations, a general solution of CWT
can be deduced [8]
i1ˆ( , ) ' , ,
2
'' , ' ,
f u
f f
f f
Wf u s A u e u
Er A u u (12)
where '' , ' f f Er A u u denotes error
function due to usage of finite Taylor series.
Due to linearity property of CWT (5) one
can apply an approximation (12) to multi –
component functions, such as impulse
response of MDOF system (4), in order to
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identify modal parameters of particular
natural frequency.
1.3 Modal Parameter Identification
Due to the nature of CWT it is possible
to transform measured signal in form of time
series to frequency domain without the loss
of time domain information. Furthermore,
CWT offers a very good resistance to noise
in the measured signal. Regarding the
CWT’s nature it is possible to observe the
development of natural frequencies with
time. In order to apply CWT to modal
analysis it is essential to analytically
describe distinctive properties of impulse
response’s wavelet transformation. The
deduction of impulse response CWT will be
presented according to Staszewski [7] and
Slavič [8], who analyzed Morlet and Gabor
wavelet, respectively.
From equation (11) one can see that the
essential contribution to the magnitude is
provided by stationary points of the
argument of the integrand [16], i.e. the
points st such that
1
' ' . s
f s
t u
t s s (13)
Canonical pairs ,u s , which satisfy
condition ( , ) , st u s u define ridge of the
wavelet transform [17]. Ridge can be
detected with different approaches,
Staszewski presented three: cross-section
method, amplitude method and phase
method [7]. In this work the phase method is
used, which is based on frequency match
(13). In order to determine ridge an initial
estimation of instantaneous frequency 1 has
to be made. It is then entered into recursive
equation
1 11
1 1
, ,,
i i i ii
i i
u u
u u (14)
which gradually converges to instantaneous
frequency in signal (Fig 2).
The values of the wavelet transform
restricted to its ridge are called the skeleton
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of the wavelet transform; they serve for
modal parameter identification (16, 17).
When considering an impulse response
function (9) an assumption of constant
frequency can be used. Due to linear relation
between frequency and scale for Gabor
wavelet ' ,u s s one can conclude, that
the scale variable on the ridge s u is
constant 0 s u s .
Figure 2: Ridge detection using phase
method with recursive equation (14).
By introducing Gabor wavelet function
into equation (12), regarding frequency
match and neglecting error function, it
follows [6]
0d1 4
2 20 0
1, 4 .
2
uWf u s Ae s
(15)
Finally, one can deduce equations for
direct identification of damping ratio and
initial amplitude . A
0
0d
ln ,
const.,
Wf u s
u
(16)
0d0
1 4220d
2 ,
const.4
ue Wf u s
A
(17)
1.4 A t Variation Influence on
Identified Amplitude
An approximation for theoretical CWT
of impulse response is deduced by
approximating an impulse response (9) as a
finite Taylor series. An approximation of
const. A t is exact only for asymptotic
signals, like the undamped response, where
the phase variations are considerably faster
than amplitude variations
'' .
f
f f
A t t
A t (18)
When considering signals, by which the
asymptotic condition is not entirely satisfied,
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e.g. highly damped impulse response, an
error occurs at amplitude A identification.
Presented error represents a deviation of
reconstructed amplitude reconstr A from real
amplitude in system response real A . It can be
noticed on numerical example of highly
damped impulse response and depends of
damping ratio and parameter (Fig. 3
and 4); however it is independent of real
amplitude value real A . Consequently, a ratio
between real amplitude real A and
reconstructed amplitude reconstr A , which serves
for mode shape reconstruction, is constant
1 real 2 real
1reconstr 2 reconstr
const. A A
A A (19)
This finding is important when a
reconstruction of mode shape is made only
by comparing the amplitudes of
displacements on different measurement
positions (Section 2.1).
Figure 3: Reconstructed initial amplitude at
variable parameter and real 0 1 A t .
Fig
Figure 4: Reconstructed initial amplitude at
1real 0 1 A t and 2 real 0 2 A t .
1.5 Spatial Mode Shapes – Geometry
In odrer to recunstruct spatial mode shapes
the vibrations in three spatial directions have
to be measured with response accelerometers
on pre-defined measurment positions. When
performing a modal analysis on simple
geometry, such as steel beam (Section 2.1),
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the response accelerometers can be oriented
in the same direction and no issues arise
from the geometry aspect. When analyzing a
system with complex geometry a placement
of reference accelerometers in the same
directions is not always possible. If this is
the case, a global coordinate system has to
be defined. Furthermore, all local coordinate
systems have to be defined and a mutual
connection between global and local
coordinate system has to be detremined. One
way to describe this connection is with
Euler angles (Eq. 20-22).
Figure 10: Problem of spatial vibration
measurements.
To reconstruct defined spatial mode
shape one has transform displacements,
measured in local coordinate systems, into
displacements in global coordinate system
with use of rotational matrices
,
1 0 0
0 cos sin
0 sin cos
x R , (20)
,
cos 0 sin
0 1 0
sin 0 cos
y R , (21)
,
cos sin 0
sin cos 0
0 0 1
z R , (22)
where , and denote Euler rotation
angles about x, y or z axis, respectively.
1.6 Identification of Spatial ModeShapes
The procedure to reconstruct defined spatial
mode shape is presented in the following
steps:
1. Definition of the excitation position.
2. Definition of global coordinate
system, n response positions and
belonging local coordinate systems.
3. Selection of the reference position
and reference direction (this should
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be the position/direction, where large
oscillations are expected).
4. Excitation of the system with
impulse blow, acceleration
measurement on reference
position/direction and on chosen
response position(s) in chosen
direction(s).
5. Identification of the ridge (14) and
oscillation amplitudes ref j A and
resp ,i j A (17) for reference and
response signal, respectively; j
denotes the consecutive number of
impulse blow and i is the serial
number of response position
(i=1,… ,n).
6. Calculation of the normed
displacements regarding oscillation
amplitude ref j A on reference position/
direction norm resp , ref i i j j A A A
7. Transformation of the normed
displacements norm i A from local to
global coordinate system (Eq. 20-
22).
8. Graphical presentation of the spatial
mode shape.
2 EXPERIMENT
For a relatively simple steel beam the
presented approach for modal identification
is experimentally compared to the theoretical
mode shapes. Furthermore, to demonstrate
an experimental identification of spatial
modal analysis a horizontal tail of ultralight
motor craft is presented.
2.1 Experimental Validation
In order to perform specific numerical
manipulation custom programs and
algorithms were developed by the authors.
Custom made programs offer higher
flexibility and enable the author ’s insight
into the computation process. To check the
correctness of developed programs a simple
experiment with well-defined theoretical
mode shapes was carried out. With
algorithms, which are verified on simple
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models, it is possible to perform a reliable
modal analysis on more complexed spatial
structures.
Simple validation experiment for
method verification was performed on free-
free supported steel beam with dimensions
5x40x1000 mm, excited by an impulse
hammer on pre-defined ‘’excitation’’
position. The experiment was made with
several impulse excitations; at each
excitation the response accelerometer was at
a different position, as described in the
following lines.
For the sake of experiment simplicity
only two accelerometers were used for
measurements. One accelerometer was
denoted as ‘reference’ sensor and was during
the experiment always on the ‘reference’
position. The second accelerometer, denoted
as ‘mobile’ accelerometer was placed on
different ‘selected’ position at each impulse
excitation.
Eleven measuring positions were set
along the beam, position 1 was defined as a
‘reference’ position, remaining positions
were defined as ‘selected’ positions, as
shown on Fig. 5.
Figure 5: Measuring positions placement on
the analyzed beam.
After the whole set of measurements
was carried out one could obtain impulse
response from any pre-defined position
along the beam.
Figure 6: The validation experiment
configuration set-up.
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With the use of CWT on measured
responses it is possible to identify modal
parameters according theoretical
background, described in Section 1.3.
Equation (14) is used to define the time
development of observed natural frequency.
Once the value of natural frequency is
known, the equations (15) and (16) are used
to define damping ratio of the system and
initial amplitude A at the location of
measurement. Based on the ratio between
displacements on reference and on selected
positions the first nine normalized mode
shapes were reconstructed and compared to
theoretical mode shape; some of them are
shown on Figure 7. In Table 1 belonging
natural frequencies i f and damping ratios i
are shown.
a) First mode shape
b) Fourth mode shape
c) Nineth mode shape
Figure 7: Reconstructed normalized beam’s
mode shapes: ― theoretical shape and ( )
measured values.
Table 1: Beam’s natural frequencies and
damping ratios.
i i f [Hz]410i
1. 26.08 3.567
4. 232.73 1.311
9. 1040.7 0.5266
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2.2 Horizontal Tail Experiment
An experiment, similar to one described
in Section 2.1, was carried out on horizontal
tail of ultra light motor craft (Fig 8).
Figure 8: Horizontal tail in isometric view.
In order to reduce the experiment
complexity and the amount of nonlinearities
in system the tail was supported free-free.
Twenty-eight measuring positions were
defined on the tail surface, as shown on Fig.
9.
Figure 9: Measuring positions placement and
a global coordinate system of horizontal tail.
At the simple validation experiment
(Section 2.1) all accelerometers were
orientated in the same direction as the
beam‘s geometry could be assumed as a
plane. In the case of horizontal tail the
accelerometers cannot be oriented in the
same direction due to its complex geometry.
Therefore a global coordinate system of the
horizontal tail (Fig. 9) and local coordinate
systems of each measuring position were
defined.
Figure 10: Problem of spatial vibration
measurements.
Responses, measured in local
coordinate systems are transformed into
global coordinate system with use of Euler
angles and rotational matrices
,
1 0 0
0 cos sin
0 sin cos
x R ,
(20)
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,
cos 0 sin
0 1 0
sin 0 cos
y R ,
(21)
,
cos sin 0
sin cos 0
0 0 1
z R ,
(22)
where denotes rotation angle about x, y or z
axis, respectively.
Altogether 28 local coordinate systems
were defined in the way that the z axis was
normal to the surface and x axis followed
obvious lines in tail geometry. As a
reference a measurement position 11 was
chosen (Fig. 9, 11), oriented in local
direction z .
Figure 11: A horizontal tail with
accelerometers placed on reference position
(left) and on selected position (right).
During each measurement two signals
were sampled:
- acceleration on reference position in
reference direction,
- acceleration on selected position in
one of the defined local axis.
Altogether 84 separate measurements were
done (28 measuring positions in 3
directions).
In order to transform identified
displacements from local to global
coordinate system special software was
developed by the authors, which an arbitrary
geometry can be imported to. Geometry
characteristics are imported into software in
form of finite element method (FEM) mesh,
which consists of volume elements. User
defines positions and orientations of local
coordinate systems and corresponding
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identified displacements of single mode
shape. The software generates a script,
which can be run in ANSYS and gives a
reconstructed mode shape in form of static
problem solution.
Identified natural frequencies and
damping ratios are shown in Table 2,
reconstructed normalized mode shapes are
shown on Fig. 12.
Table 2: Horizontal tail’s natural frequencies
and damping ratios.
i i f [Hz]4
10i
1. 22.8 63.17
2. 43.4 83.89
3. 75.5 141.8
Figure 12: Reconstructed normalized
horizontal tail’s mode shapes.
3 DISCUSSION
From validation experiment, which
served as an ideal real system, it is obvious
that identified mode shapes agree with
theoretical mode shapes to a high degree.
With simple beam experiment the
applicability of continuous wavelet
transform to determine modal parameters is
confirmed. A measurement approach with an
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introduction of a reference accelerometer
was proved to be suitable.
According to experiment realization and
results, performed and obtained in section
2.2, the presented approach provides a
simple and feasible method for experimental
determination of modal parameters.
Although the mechanical system, such as a
horizontal tail, is not an ideal system without
nonlinearities, the extraction of its modal
parameters with CWT showed to be a robust
and effective method.
4 CONCLUSIONS
In this paper a modal parameter
identification of dynamic system using
continuous wavelet transformation of system
impulse response was presented. A
computation error due to the finite Taylor
series was indicated and its influence on
mode shape reconstruction was shown. Steel
beam mode shapes were reconstructed and
compared to its theoretical mode shapes.
Spatial mode shapes of free - free supported
horizontal tail of ultra light motor craft were
reconstructed.
A comparison between beam’s
theoretical and experimental mode shapes
confirmed suitability of proposed method for
mode shape reconstruction. The main
contribution of this paper presents an
enhancement of known methods for EMA
using CWT for one-dimensional cases with a
new method for reconstruction of spatial
mode shapes for arbitrary geometry, in this
case a horizontal tail. Experimental results
can be used for validation of numerical
model. Method, described in the paper, can
be used to identify spatial mode shapes of
any dynamical system with low damping
ratios.
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