Application of Transmissibility Measurement in Estimation of Modal Parameters for a Structure

59 Pages·2013·3.71 MB·English

Master's Degree Thesis
ISRN: BTH-AMT-EX--2013/D06--SE

Application of Transmissibility

Measurement in Estimation of

Modal Parameter s for a Structure

AmirHossein Aghdasi

Department of Mechanical Engineering
Blekinge Institute of Technology
Karlskrona, Sweden
2013

Supervisors: Andreas Josefsson, BTH
Ansel Berghuvud, BTH

Application of transmissibility
measurement in estimation of
modal parameters for a structure

AmirHossein Aghdasi

Department of Mechanical Engineering
Blekinge Institute of Technology
Karlskrona, Sweden
2012
Thesis submitted for completion of Master of Science in Mechanical
Engineering with emphasis on Structural Mechanics at the Department of
Mechanical Engineering, Blekinge Institute of Technology, Karlskrona,
Sweden.
Abstract:
Identification of modal parameters from output-only data is studied in this
work. The proposed methodology uses transmissibility functions obtained
under different loading conditions to identify resonances and mode shapes.
The technique is demonstrated with numerical simulations on a 2-DOF
system and a cantilever beam. To underpin the simulations, a real test was
done on a beam to show the efficiency of the method. A practical
application of the method can be to identify structures subjected to moving
loads. This is demonstrated by using FEM-model of a cantilever beam with
a moving load.
Keywords:
Operational Modal Analysis, Transmissibility measurement, mode shape,
MATLAB.
Acknowledgements
This work was carried out at the Department of Mechanical Engineering,
Blekinge Institute of Technology, Karlskrona, Sweden, under the
supervision of Andreas Josefsson and Ansel Berghuvud.
First of all, I do appreciate my parents that they always support me in any
aspect of my life.
I am so grateful to Dr.Andreas Josefsson for all that he has done for me
throughout this paper. I really thank Dr.Ansel Berghuvud for his support all
the time.
Special thanks should be dedicated to Mohammad Mahdi Khatibi and also
my professor Mohammad Reza Ashory that he kindly and patiently helped
me out in this regard.

Karlskrona, March 2013
AmirHossein Aghdasi

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Contents

1 Notation 4
2 Introduction 6
2.1 Aim and scope 7
2.2 Modal Analysis 7
2.2.1 Theoretical track of a vibrational analysis 8
2.2.2 Experimental track of a vibrational analysis 9
2.3 Research methods 12
3 Operational Modal Analysis 12
4 Transmissibility concept 14
5 Simulation study 15
5.1 2-DOF system 16
5.1.1 Conclusion 23
5.2 Beam system 23
5.2.1 Conclusion 37
6 Experimental validation 38
6.1 Conclusion 45
7 Harmonic in Transmissibility 46
8 Conclusion 52
9 Appendices 53
10 References 55

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1 Notation

A Residue
B Band width
c Damping
E Young’s modulus
F Force
f Frequency
H Frequency Response Function
I Moment of inertia
k Stiffness
m Mass
S Second decomposed matrix
s Pole
T Transmissibility
U First decomposed matrix
V Third decomposed matrix
X(s) Displacement in frequency domain
X Displacement
x Length

λ Pole
ζ Damping ratio
ω Imaginary part of the system pole or damped natural frequency
σ2 Variance

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Abbreviations

DOF Degree Of Freedom
EMA Experimental Modal Analysis
EFFD Enhanced Frequency Domain Decomposition
FDD Frequency Domain Decomposition
FRF Frequency Response Function
MDOF Multiple Degree of Freedom
OMA Operational Modal Analysis
SVD Singular Value Decomposition
TR Transmissibility matrix

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2 Introduction

Operational Modal Analysis (OMA) is being increasingly used as an
experimental technique to identify the dynamic properties of many types of
large-scale constructed systems. For industrial structures in real operation
condition, it is sometimes hard or impossible to measure excitation forces
so some identification techniques have been developed to work on response
data only.
In Operational Modal Analysis (OMA), there is no need to make any
assumption about the nature of force and consequently resonance
frequencies are recognised. In this thesis, one application of Operational
Modal Analysis which is called transmissibility measurement is supposed
to be engaged to find resonance frequencies.
Operational Modal Analysis has become a valid alternative for structures
where a classic forced-vibration test would be difficult to conduct. The use
of in-operation modal analysis is particularly interesting because the
structure remains in its normal in-operating condition during the test. This
is an important advantage because the conditions during a laboratory
forced-vibration test are often considerably different from the real working
conditions regardless of constraints and limitations. Therefore, this method
enables us to find system parameters without any assumption about the
nature of the forces.

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2.1 Aim and scope
In this thesis, first of all, a concise explanation of Modal is presented. We
will be dealing with Operational Modal Analysis and its differences with
Experimental Modal Analysis in chapter 3.
We will then discuss about the concept of transmissibility in chapter 4.
Afterwards, we will turn into the simulation to explain the method. It has
been used a 2-DOF system and a multiple degree of freedom system such
as a cantilever beam in chapter 5.
To verify the simulation study, an experiment has been done on a both end
free beam to find modal parameters by means of transmissibility
measurement. This validation has been presented in chapter 6.
We will be dealing with a challenging matter in Operational Modal
Analysis in chapter 7. To do so, an investigation of moving load has been
taken into account.
Finally, we will be dealing with conclusion and discussion about the results
in chapter 8.

2.2 Modal Analysis
Modal analysis is the process of determining the dynamic behaviour of a
system in forms of natural frequencies, damping factors and mode shapes,
and using them to formulate a mathematical model for its dynamic
characteristic. The formulated mathematical model is referred to as the
modal of the system and the information for the characteristics is known as
its modal data [1]. For further information see Ref [1].
Design, repair and protection of various structures such as buildings,
bridges, dams, trains are important and necessary. One of the design and
repair tools for protecting a structure is dynamic analysis. There are a few
problems and errors with respect to applying improper assumptions and
theories, the lack of information about the material properties and system
modelling, numerical methods such as Finite Element Method have posed
inaccurate analytical solution for complex structures. Therefore, Modal test
has been known a useful tool to achieve dynamic properties of a structure.
Modal test has basically two problems in big structures:
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1. Excitation is difficult for big structure such as bridges and
buildings.
2. The presence of noise in environment.
Once the structure is excited for Modal test, the level of excitation should
be great as much as the structure equilibrium is unbalanced. Therefore, a
big excitation force is demanded for doing so. On the other hand, the
excitation level could not have been very big because it poses non-linear
behaviour and distraction in the structure. Moreover, some factors such as
wind, noise contamination, traffic harden the process of test measurement.
Therefore, the Modal Analysis theory has a great effect on a successful test
because this theory allots dominant part of concept using various steps in an
experiment.
In general, vibrational analysis of a structure is possible to consider in two
approaches: 1) theoretical method 2) experimental method.

2.2.1 Theoretical track of a vibrational analysis
Different steps of Modal analysis have been shown in figure 2.2.1.1. The
figure shows three different steps of a sample of vibrational analysis.
Spatial Model Modal Model Response Model

Description of Vibration Response
Structure Modes Levels

Mass damping Natural frequencies Frequency
Stiffness & Mode shapes Responses &
Impulse
Responses
Figure 2.2.1.1- Theoretical track in vibration analysis

In general, in a spatial model, physical properties of a structure is divided in
a form of mass, stiffness and damping. By means of applying Modal theory
on spatial model, the behaviour of a structure is defined in a form of
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