This document proposes a new method for determining mode shapes of dynamic systems using wavelet transform of free vibration data. The method requires time signals of responses and known natural frequencies as input. A Morlet wavelet function with an adjusting parameter is used to identify natural frequencies and damping ratios. The extracted mode shapes are then scaled using a mass change method. The validity of the method is demonstrated through numerical and experimental case studies.

1. Arch Appl Mech (2013) 83:907–921
DOI 10.1007/s00419-012-0726-1
ORIGINAL
Determination of mode shapes using wavelet transform
of free vibration data
Received: 3 February 2012 / Accepted: 21 December 2012 / Published online: 5 January 2013
© Springer-Verlag 2013
Abstract In this article, a new method is proposed to determine the mode shapes of linear dynamic systems
with proportional viscous damping excited by an impact force. The time signals of responses and a priori
knowledge of the natural frequencies are required in this method. The method is particularly suitable for the
wavelet techniques which can precisely estimate the natural frequencies. 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, and the mode shapes are estimated using the proposed method in this work. It
is shown that the extracted mode shapes are not scaled. Therefore, mass change method is used for scaling the
mode shapes. Moreover, the effect of noise on the extracted modal parameters is investigated. The validity of
method is demonstrated using numerical and experimental case studies.
Keywords Mode shape · Wavelet transform · Scaling · Mass change · Free vibration data · Morlet wavelet
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 methods used for identification
of modal parameters can be categorized into single degree of freedom (SDOF) methods and multi-degrees
of freedom (MDOFs) methods. Pick peaking method, circle fit method, and line fit method are the classical
methods for the identification of the modal parameters [1]. The recent method of three point finite difference
method [2] gives more accurate results compared to the traditional methods. Least square complex exponential
method [3], poly-reference time domain method [4], Ibrahim time domain method [5], automated parameter
identification, and order reduction for discrete time series [6] are among the MDOFs methods for the determi-
nation of the modal parameters [7]. The basis of most of these methods is Fourier analysis which transforms the
time data to the frequency data for the estimation of frequency response functions (FRFs). However, Fourier
analysis cannot determine the modal parameters accurately in the noisy environments. Some methods consist
of pre-filtering of the signals can improve the results. Moreover, close modes may hardly be identified using
the techniques based on the Fourier analysis [1].
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 time and frequency domain [8]. By using this property,
M. R. Ashory · M. Jafari · A. Malekjafarian
School of Mechanical Engineering, Semnan University P.O. Box 35195-363, Semnan, Iran
M. M. Khatibi (B)
Department of Mechanical Engineering, Semnan Branch, Islamic Azad University, P.O. Box 35145-179, Semnan, Iran
E-mail: m.m.khatybi@gmail.com
Tel.: +98-231-3354122
Fax.: +98-231-3354122
M. R. Ashory · M. Jafari · M. Khatibi · A. Malekjafarian