This document discusses using wavelet transforms to identify modal parameters like natural frequencies and damping ratios from vibration response data of structures. It introduces wavelet transforms as a time-frequency analysis method that can handle non-stationary signals, unlike Fourier transforms. The document then describes how wavelet transforms can be applied to extract natural frequencies and damping ratios from the free vibration responses of linear systems by decomposing the governing equations of motion into single degree of freedom systems. It provides an example of using this process on a simulated 3 degree of freedom system and a real test on a clamped-clamped beam.

1. 1 INTRODUCTION
Modelling of structural systems involves an inverse procedure to identify the structural parame-
ters from the recorded response of structures. The goal is to estimate the dynamic properties
such as natural frequencies and damping ratios of vibrating systems from the responses of
structures under different excitation forces such as impact or random forces (Torrésan 1995 and
Chui 1992). The identified parameters provide a design criterion for the measured structures.
Analysis of response signals of structures may be performed in two different ways: (i) time-
domain analysis and (ii) frequency-domain analysis. Several approaches to time-domain system
identification have been developed. The classical method of frequency-domain analysis is by
means of Fourier Transform (Peeters and De Roeck 1999) and its numerical implementation
Fast Fourier Transform (FFT). Though FFT has been widely used, it has several limitations.
FFT is useful for stationary signals. An approach in signal analysis which deals with non sta-
tionary signals is the time-frequency analysis (Kulla and Jarvenin 1999). The alternative solu-
tion is wavelet transform which has been successfully used for signal analysis. Wavelets pro-
duce presentation of a signal using time-limited local functions having variable scales. There
have been several articles in the literature on the use of wavelets to determine the modal pa-
rameters (Feng et all 1998). In some of these studies natural frequencies and damping ratios
were identified. In this paper wavelet transform is used for identification of the natural frequen-
cies and damping ratios of a 3 DOFs viscously damped system. The time signals of free vibra-
tion of the system are generated in the simulated test and used in the wavelet algorithm to iden-
tify the modal parameters. Then the method is applied to a clamped-clamped beam in a real test
and the results are compared with the exact values.
Modal Identification Process Using Wavelet Transform
M. Jafari, M.R. Ashory, M.M. Khatibi
Semnan University, Department of Mechanical Engineering, Semnan, Iran
ABSTRACT: There are different methods to extract the modal parameters of vibrating systems
in terms of natural frequencies, damping coefficients and mode shapes. The wavelet transform
is a time-frequency identification method which can be used for determination of the vibrating
characteristics of structures under their non stationary natural loading conditions. In this paper,
the wavelet transform is applied to obtain the natural frequencies and damping ratios of a nu-
merical model of viscously damped three degrees of freedom system. Also the method is used
to extract the natural frequencies and damping ratios of a clamped-clamped beam by measuring
the responses of beam in a real test.

2. 2 IOMAC'09 – 3rd
International Operational Modal Analysis Conference
2 CONTINIOUS WAVELETS
2.1 Wavelet transform
The wavelet transform has been used by research workers for several important applications
such as: damage detection, transient analysis, data compression, system identification (Newland
1996 and Meyer and Ryan 1993). The capability of wavelet transform in time-frequency analy-
sis has been used extensively. In wavelet analysis, a time signal x(t) is decomposed to several
time localized shifted and scaled basis functions )/)(( st τψ − , where "τ " is the shifting pa-
rameter and "s" is the dilation or scaling parameter. "τ " centers the wavelet function so that the
information about the signal can be obtained around time τt . The dilation parameter "s"
controls the range of frequencies about which information can be obtained in the vicinity of
time τt . The mother wavelet is defined as:
ℜ∈
−
 τ
τ
ψψ τ ,0)(
1
)(, s
s
t
s
ts
(1)
A signal x(t) can be decomposed into wavelet coefficients ),( τψ s using the son wavelet
)(, ts τψ . x(t) is required to decay to zero as ∞→t , mathematically:
∫
∞
∞−
∞dttx
2
)( (2)
The wavelet transform of x(t) is expressed by the inner product (Soman and Ramachandran
2005 and Tan et all 2008 and Yu et all 2003) in the Hilbert space as:
∫
∞
∞−
dtttxttxs ss )()()(),(),( *
,, ττψ ψψτω (3)
The required condition for the function )(tψ to be qualified as an analyzing wavelet (Soman
and Ramachandran 2005) is:
∫ ∞
∞
∞−
ω
ω
ωψ
ψ dC
2
)(
0 (4)
where )(ωψ is the Fourier transform of )(tψ . Also for practical purposes, the possibility of
time-frequency localization arises if the wavelet acts as a window function, this means )(tψ
decays as ∞→t , namely:
∫
∞
∞−
∞dtt)(ψ (5)
2.2 Modal parameter identification
Wavelet transform can be applied to extract the natural frequencies and damping ratios from the
free vibration responses of a linear vibrating system. The governing equation of the free vibra-
tion of a m Degrees Of Freedom (mDOFs) system (De Silva 2007 and Staszewski 1997) is:
0 KxxCxM &&& (6)
where M, C and K are the mass, damping and stiffness matrices respectively. If the damping
matrix C is proportional, namely:
KMC βα  (7)
Eq. (6) can be decoupled (Yu et all 2003) in to m Single Degree Of Freedom (SDOF) system
as:
mkqqq knkknkkk ,,2,102 2
K&&&  ωωξ (8)
where kξ is the kth
damping ratio, nkω is the kth
natural frequency , kq is the kth
coordinate and
can be represented by a linear combination of mode shapes and modal responses as: