Spatial mode shape identification using continuous wavelet transform

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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)



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



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



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)



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



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



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,



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),



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



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



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.



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



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)



,

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



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



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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Spatial mode shape identification using continuous wavelet transform

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