1. 1 INTRODUCTION
Estimation of the modal parameters in terms of natural frequencies, damping coefficients and
mode shapes from experimental data is a fundamental problem in structural dynamics. The mo-
dal parameter identification methods can be categorized in to the Single Degree Of Freedom
(SDOF) methods and the Multi Degrees Of Freedom (MDOF) methods. Pick peaking method,
circle fit method and line fit method are the classical methods for modal parameter identifica-
tion (Ewins 2000). The recent method of three point finite difference method (Yin and Du-
hamel 2000) gives more accurate results compared to the traditional methods. Least square
complex exponential method (Smith 1981), poly-reference time domain method (Zhang 1987),
Ibrahim time domain method (Ibrahim and Mikulcik 1977), automated parameter identification
and order reduction for discrete time series (Hollkamp and Batill 1991) are among the MDOF
methods for modal parameter determination (Feng et al. 1998). The basis of most of these
methods is Fourier analysis which transforms the time data to the frequency data. However,
Fourier analysis cannot determine the modal parameters accurately in the noisy environments.
Some methods consist of pre-filtering of the input signals can improve the results. Moreover,
close modes may hardly be identified using the techniques based on the Fourier analysis. In
contrast to the Fourier transform which has a uniform resolution in frequency domain, the
wavelet transform has the property of double resolution in both the time and frequency domain
(Miranda 2008). By using this property, the wavelet transform can be adjusted to analyze the
non-stationary signals. Also, the strongly coupled modes can be identified by tuning the wave-
lets. Moreover, the inherent ability of wavelet transform in filtering out the noise contaminating
a signal is an important advantage for identifying the modal parameters.
Three methods for estimating the damping ratios based on the Continuous Wavelet Trans-
form (CWT) were proposed in (Staszewski 1997). A procedure of identification of natural fre-
quencies and damping ratios of the system from its free decays using wavelet transform was
presented in (Ruzzene et al. 1997). In order to improve the accuracy of the modal parameters
identification a modified Morlet wavelet function with an adjusting parameter was proposed by
Mode shapes identification from the results of wavelet transform:
Theory and simulation
M.R. Ashory, M. Jafari
Modal Analysis Lab., School of Mechanical Engineering, Semnan University, Semnan, Iran
M.M. Khatibi
Department of Mechanical Engineering, Islamic Azad University - Semnan Branch, Semnan, Iran
A. Malekjafarian
Palande-Saf Center; University of Applied Science and Technology, Semnan, Iran
ABSTRACT: In this article a new method is presented to determine the mode shapes of linear
dynamic systems with proportional viscous damping excited by an impact force. The time sig-
nals of responses and a priori knowledge of the natural frequencies are required. The method is
particularly suitable for the wavelet techniques which can estimate the natural frequencies and
damping ratios. A previously proposed method based on a modified Morlet wavelet function
with an adjusting parameter is used to identify the natural frequencies and damping ratios of
system. Then the mode shapes are estimated using the new method. It is shown that the ex-
tracted mode shapes are not scaled. Therefore, mass change method is used for scaling the
mode shapes. Also the effect of noise on the extracted modal parameters is investigated. The
validity of method is demonstrated by a numerical case study.

2. 2 IOMAC'11 – 4th
International Operational Modal Analysis Conference
Lardies and Gouttebroze (2002). A modal parameter identification procedure using continuous
wavelet transform including the mode shape identification was presented by Le and Argoul
(2004). There is no complete procedure for mode shape identification in most of the wavelet
based identification methods.
The contribution of this paper consists essentially of introducing a mode shape identification
method from the wavelet analysis of response data for the systems with proportional viscous
damping excited by an impact force. A Morlet wavelet with an adjusting parameter is used for
identification of the natural frequencies and damping coefficients. A numerical case study con-
firms the accuracy of method.
2 THEORY
2.1 Mode shape identification method
For a MDOF system with proportional viscous damping, the mode shapes are identical to those
of the undamped system (Ewins 2000). The response for each degree of freedom for viscously
damped system can be calculated by (Ewins 2000):
∑
−
−
n
r
rrr
ti
ri tety rr
1
2
))1(cos(.)( θζωωζ
A (1)
when the system is undamped:
nrr ,...,2,10 ζ (2)
Eq. (1) can be formulated in the matrix form as:
  ∑
Υ
n
r
rrr tt
1
)cos(. θωA (3)
where rA is the unscaled mode shapes of system. If the system is excited by an impact force,
the initial conditions are:
0(0 Υ ) (4)
0V(0) Υ& (5)
Inserting Eq. (4) into Eq. (3); gives:
0cos.
1
∑
r
n
r
r θA (6)
As the mode shape vectors are linearly independent, it is concluded that:
nrr ,...,2,10cos θ (7)
which results in:
,...2,1,0
2
2  kkr
π
πθ (8)
By inserting Eq. (8) in to Eq. (3), the following equation is obtained:
  ∑
Υ
n
r
rr tt
1
sin. ωA (9)
In which the elements of rA can also be negative to compensate the negative sign of
)cos( rr t θω  when ,...2,1,0,
2
2  kkr
ππθ Eq. (9) can be rewritten as: