UNIVERSITY OF CINCINNATI
May 31 2005
Date:___________________
SIDDHARTH SINHA
I, _________________________________________________________,
hereby submit this work as part of the requirements for the degree of:
(MS) MASTER OF SCIENCE
in:
MECHANICAL ENGINEERING
It is entitled:
Techniques for Real Normalization of Complex Modal
Parameters for Updating and Correlation with FEM Models
This work and its defense approved by:
Dr R.J. Allemang
Chair: _______________________________
Dr D.L. Brown
_______________________________
Dr. T.C. Lim
_______________________________
_______________________________
_______________________________
Techniques for Real Normalization of Complex Modal Parameters for Updating and
Correlation with FEM Models
A Dissertation submitted to the
Division of Research and Advanced Studies
Of the University of Cincinnati
In Partial Fulfillment of the Degree of
MASTER OF SCIENCE (MS)
In the Department of Mechanical Industrial and Nuclear Engineering
Of the College of Engineering
2005
by
Siddharth Sinha
B.E., Veermata Jijabai Technological Institute, India 2001
Committee Chair: Dr R.J. Allemang
Abstract
The research done during the course of this thesis leads to the development of three real
normalization techniques for complex modal parameters. The real modal parameter estimates
obtained would enable the correlation/correction with real modal parameters obtained from a
conservative finite element or modal model.
Finite element methodologies (FEM) generally result in real valued modal vectors and
frequencies due to lack of damping information provided or due to the assumption of
proportional damping. Modal models developed for structural dynamics modifications also work
better with real modal parameters. Experimental modal analysis on the other hand generates
complex modal parameters by default. Hence in order to correlate/compare experimental and
analytical results, the real normalization techniques developed would be useful in converting the
complex domain modal parameters into the real domain. This would lead to better and quicker
correlation and updating of FEM and Modal models with experimental results. Overall this
would aid in shortening the design cycle of a structure or a system and reduce the number of
prototypes required in the design cycle.
The research identifies possible reasons for generation of complex modal parameters and also
reviews past work done on their real normalization. It also develops techniques to gauge the level
of complexity of the modal vectors and develops two simple yet effective methods based on a
least squares solution to obtain preliminary real modal vector estimates for the complex modal
vectors.
Finally the research leads to the development of three real normalization techniques
•
Rescaling of Complex Modal Vectors Technique (RCMVT)
•
Information Matrix Subspace Technique (IMST)
•
Effective Independence Subspace Technique (EIST)
Experimental validation was provided by performing an experimental modal analysis on a
structure and then normalizing the complex modal parameters obtained. The normalized results
generated by the techniques fitted the complex modal parameters extremely well. A detailed
validation study with different test cases should be done to observe how the techniques work in
normalizing different data sets. The research also leads to the development of a stand alone
Matlab GUI based software which implements all the techniques and formulations developed
during the course of this research.
Preface
I would like to express my profound gratitude to my advisor Prof. Randall J. Allemang for his
tireless guidance, patience and encouragement during the course of my Master’s Thesis. His
assistance and encouragement has played a major role in the success of this thesis. I would also
like to thank Prof. David Brown and Prof. Teik Lim for serving on the committee and for their
guidance. I would also like to acknowledge the staff of the Structural Dynamics Research
Laboratory and QRDC, Inc. for the use of instruments and facilities in their laboratories. I would
like to thank the staff and professors of the MINE department for their assistance and sincerely
thank them for the knowledge that they have provided me during the course of my Masters
Degree. I greatly appreciate the efforts put in by the University of Cincinnati and the College of
Engineering in making my stay at the university a memorable and cherished one.
Lastly, but not the least I wish to express my gratitude for my parents Dr. R. P. Sinha, Madhu
Sinha and sister Menaka Deshpande without whose support and encouragement it would have
been impossible to achieve this accomplishment and today I am because they are.
Table of Contents
Section
Page
1.0 Introduction
1.1 Research Objectives and Applications
1.2 Research Outline
1
1
2
2.0 Overview of Modal Analysis
2.1 Analytical Modal Analysis
2.1.1 Laplace Domain Solution
2.1.2 Real Normal Modes Solution
2.1.3 Complex Modes Solution
2.2 Experimental Modal Analysis
2.2.1 Partial Fraction Model
2.2.2 Unified Matrix Polynomial Approach
2.2.3 Experimentally Determined Modal Scaling Factors
2.2.3.1 Real Normal Modes
2.2.3.2 Complex Modes
2.3 Conclusion
4
5
5
6
8
11
11
13
16
16
17
18
3.0 Origin and Identification of Complex Modal Vectors
3.1 Possible source for complex modal vectors
3.2 Complexity of Modal Vectors
3.2.1 Plotting the Complex Modal Vector in the Complex Plane
3.2.2 Modal Phase Collinearity
3.2.3 Mean Phase Deviation
19
19
22
22
25
26
4.0 Historical Methods for Real Normalization of Complex Modal Vectors
4.1 Computation of Normal Modes from Identified Complex Modes- S.R Ibrahim[5, 6]
4.1.1 Using an Oversized Mathematical Model
4.1.2 Assumed Modes Solution
4.2 Time Domain Subspace Iteration Technique
27
27
27
30
32
5.0 Least Squares Solution
5.1 The Least Squares Method
5.2 The Weighted Least Squares Method
5.3 The Correlation Coefficient
5.4 Least Squares Line Fit for Complex Modal Vectors
5.5 Weighted Least Squares Line Fit for Complex Modal Vectors
5.6 Conclusion
36
36
38
39
40
41
46
6.0 Techniques for Real Normalization of Complex Modal Parameters
6.1 Rescaling of Complex Modal Vectors Technique
47
47
-i-
6.1.1 Theory
6.1.2 Proof
6.2 Theory for other Normalization Techniques
6.3 Information Matrix Subspace Technique
6.3.1 Theory
6.3.2 Normalization Procedure
6.4 Effective Independence Subspace Technique
6.4.1 Theory
6.4.2 Normalization Technique
6.5 Conclusion
48
48
52
54
54
56
59
60
63
65
7.0 Experimental Analysis and Validation
7.1 Experimental Modal Analysis
7.2 Test Analysis and Results
7.3 The Real Normalization Process
7.4 Results Validation
7.4.1 Consistency of Modal Assurance Criterion (MAC) Values
7.4.2 Comparison of Natural Frequencies
7.4.3 Preservation of underlying motion characteristics
66
67
73
75
75
76
81
82
8.0 Conclusion
8.1 Future Work
85
86
9.0 References
88
Appendix
90
- ii -
List of Tables
Table 7.1 Equipment for Vibration Measurements………………………………………………70
Table 7.2 Calibration of Accelerometers………………………………………………………...70
Table 7.3 Vibration Acquisition Setup for CV395 Analyzer……………………………………71
Table 7.4 Vibration Analysis Setup for CV395 Analyzer……………………………………….71
Table 7.5 The extracted complex modal parameters for the glass plate…………………………75
Table 7.6 MAC Values between the complex modal vectors obtained …………………………76
from Experimental Modal
Table 7.7 MAC Values between real modal vectors obtained by………………………………..77
Rescaling of Complex Modal Vectors Technique
Table 7.8 MAC Values between real modal vectors obtained by………………………………..77
Information Matrix Subspace Iteration Technique
Table 7.9 MAC Values between real modal vectors obtained by………………………………..78
Effective Independence Subspace Iteration Technique
Table 7.10 Cross MAC Values between complex modal vectors …………………………........78
and real modal vectors obtained by Rescaling of Complex
Modal Vectors Technique (RCMVT)
Table 7.11 Cross MAC Values between complex modal vectors ………………………………79
and real modal vectors obtained by Information Matrix
Subspace Technique (IMST)
- iii -
Table 7.12 Cross MAC Values between complex modal vectors ………………………………79
and real modal vectors obtained by Effective Independence
Subspace Technique (EIST)
Table 7.13 Comparison of Undamped and Complex Natural Frequencies……………………...81
Table 7.14 Plots of Complex and Real Normalized Modal Vectors……………………………..83
- iv -
List of Figures
Fig 3.1: An almost purely imaginary complex modal vector……………………………………23
Fig 3.2: A complex modal vector with a constant phase shift…………………………………...23
Fig 3.3: A slightly complex modal vector……………………………………………………….24
Fig 3.4: A highly complex modal vector………………………………………………………...24
Fig 5.1: An almost purely complex modal vector………………………………………………..43
Fig 5.2: LS and WLS Estimates for the almost purely complex modal vector……….…………43
Fig 5.3: A highly complex modal vector………………………….……………………………..44
Fig 5.4: LS and WLS Estimates for the highly complex modal vector……………………..…...44
Fig 5.5: Real Modal Vector Estimates for a highly complex modal vector……………………..45
Fig 5.6: Imaginary Modal Vector Estimates for a highly complex modal vector……………….45
Fig 7.1 Solid Model of Test Fixture……………………………………………………………...69
Fig 7.2 Experimental Test Fixture……………………………………………………………….69
Fig 7.3 Modal Extraction Grid for Glass Plate…………………………………………………..72
Fig 7.4 Experimental glass plate vibration measurement points grid……………………………72
Fig A.1 Matlab GUI software main screen………………………………………………………91
Fig A.2 Matlab GUI software post normalization screen………………………………………..92
-v-
Nomenclature
N : Number of modes of a structure obtained by modal analysis
[M ] : Mass matrix of the structure in the physical coordinates
[C ] : Damping matrix of the structure in the physical coordinates
[K ] : Stiffness matrix of the structure in the physical coordinates
{x} : Displacement Vector of the structure in the physical coordinates
*
{x} : Velocity Vector of the structure in the physical coordinates
**
{x} : Acceleration Vector of the structure in the physical coordinates
{ f (t )} : Forcing Function Vector of the structure in the physical coordinates
[ B ( s )] : System Impedance Matrix of the structure
H (s ) : Transfer Function Matrix of the structure
[ H ( w)] : Frequency Response Function Matrix of the structure
{X } : Response Vector of the structure in the frequency domain
{F } : Forcing Vector of the structure in the frequency domain
λr : Natural frequency of a non-proportionally damped system
Ω r : Undamped Natural frequency of the mode r
{φ }r : Real Normal Modal Vector for mode r
{ψ }r : Complex Modal Vector for mode r
[φ ]
: Real Normal Modal Vector Matrix
- vi -
[ψ ] : Complex Modal Vector Matrix
[ M r ] : Modal Mass Matrix
[C r ] : Modal Damping Matrix
[ K r ] : Modal Stiffness Matrix
{ y} : State space displacement vector of the structure
*
{ y} : State space velocity vector of the structure
__
{ f } : State space forcing function vector of the structure
{ψ ss }r
: State space complex eigen vector of mode r
Ma r : Modal A of mode r
Mbr : Modal B of mode r
[ Ar ] : Residue Matrix of mode r
No
: Number of Output measurement degrees of freedom
Ni
: Number of Input measurement degrees of freedom
[ L] : Modal Participation Matrix of the structure
H pq (w)
: Frequency Response Function between measurement point p and input point q
[E ] : Companion Matrix developed in a state space solution
[T ] : Fisher Information Matrix of the parameter estimation process
p (t ) : Time domain displacement vector in the modal coordinates
*
p (t ) : Time domain velocity vector in the modal coordinates
- vii -
**
p (t ) : Time domain acceleration vector in the modal coordinates
[ E N ] : Companion Matrix developed in Modal coordinates
[ FE ] : Fractional Eigen Value Distribution Matrix
{E D } : Effective Independence Contribution Vector
- viii -
1.0 Introduction
1.1 Research Objectives and Applications
Modal parameters obtained from an experimental or analytical modal analysis are often used in
many industrial applications for
•
Updating a finite element model
•
Sensitivity analysis
•
Modal model updating and structural dynamics modifications
•
Control theory for vibrations and acoustics
•
Mechanical diagnosis
It is often seen that these obtained modal parameters are complex rather than real. There are
many reasons for developing normalization techniques of these complex modal parameters in
order to obtain real estimates of modal vectors and natural frequencies.
•
First, most of today’s advanced modal parameter estimation algorithms use a general case
non proportional damping approach which results in complex modal parameters even in
cases where real modal parameters are expected. Thus normalization of these complex
modal parameters to get real modal parameter estimates would be needed to obtain the
correct modal parameters of the structure
•
The finite element model (FEM) developed for a structure generally assumes a case of
proportional damping due to lack of information about the damping distribution in the
structure. Thus in order to update, correlate and optimize such conservative finite element
models of the structure with experimentally obtained complex modal parameters it would
-1-
be extremely helpful to have normalization techniques which would give accurate real
modal parameter estimates for the obtained complex parameters.
•
Modal model techniques to carry out structural dynamics modifications work much better
and are easier to develop for a set of real modal parameters as opposed to complex modal
parameters.
•
In many industries like the automobile or aircraft industry the thrust of the research is to
come out with new and improved products with shorter design cycles and reduced
development costs. These industries use modal parameters for design, model updating,
correlation and optimization and would benefit from these real normalization techniques
which would give accurate real modal parameter estimates for complex modal
parameters. The real estimates would be easier to comprehend, lead to quicker FEM and
modal model updating, correlation and optimization as opposed to the complex modal
parameters. This would lead to shorter design cycles and lesser number of prototypes
Thus recognizing the potential benefits of having accurate real modal parameter estimates
of complex modal parameters the primary goal of this research is to develop efficient
algorithms which can perform real normalization of complex modal parameters. The real
modal parameters provided by these techniques should retain almost all the information
content of the complex modal parameters.
1.2 Research Outline
During the course of this research the first task was to get a thorough understanding of modal
analysis from both an analytical as well as an experimental approach. A detailed study of these
modal analysis techniques is presented which would help identify the reasons for and the scope
-2-
of this research. A study of the possible reasons for the origin of complex modal vectors is also
done along with development and review of techniques to gauge the complexity of modal
vectors.
Many real normalization techniques and approaches have been tried in the past and thus a
literature review of those techniques is presented. Two of those techniques which have been used
extensively during the course of this research are explained in detail.
Two simple real normalization techniques using a least squares approach and a weighted least
squares approach were also developed during the course of this research and have been used to
get preliminary real modal vector estimates of the complex modal vectors. The development and
theory behind these simple yet effective techniques is presented.
With the ground work for development of the real normalization algorithms laid, a detailed
presentation of the three real normalization techniques developed for accurately normalizing the
complex modal parameters is presented. This study provides the theoretical justification and
proof for the three techniques developed which are
•
Rescaling of Complex Modal Vectors Technique (RCMVT)
•
Information Matrix Subspace Technique (IMST)
•
Effective Independence Subspace Technique (EIST)
An experimental validation of the three developed techniques is provided by performing an
experimental modal extraction on a laminated rectangular glass plate and obtaining the complex
modal parameters. The complex parameters were normalized by the three techniques and the
validity of the results was gauged against several stringent parameters all of which were satisfied
by the normalized modal parameters. During the course of this research a stand alone Matlab
GUI based software (Appendix) to do the above normalizations was created.
-3-
2.0 Overview of Modal Analysis
The research done during the course of this thesis aims to correlate the results (complex modal
parameters) obtained from experimental modal analysis with those obtained (generally real
modal parameters) from an analytical modal analysis (mathematical models). The modal
parameters of a system which are the natural frequencies, mode shapes and modal scaling factors
(Modal Mass, Stiffness, Damping or Modal A and B) are characteristic properties of the system
which govern the dynamic response of the structure. A thorough understanding of modal analysis
forms the basis of this research and a review of analytical and experimental modal analysis
techniques is essential to appreciate the reasons for and the scope of this research.
Analytical modal analysis[1,2] begins with discretizing a structure into mass, stiffness and
damping components. This discretization is done either in a lumped mass approach[1] or using
finite element models to divide the structure into elements and assigning these elements material
properties. This leads to the generation of a mathematical model based on the equation of motion
where in the system mass, stiffness and damping matrices are obtained and an eigen value
problem is solved to obtain the modal parameters.
Experimental modal analysis[2,3] on the other hand uses measured input-output data generally in
the form of transfer functions (e.g. the frequency response function) to derive the modal
parameters without any knowledge of the mass, stiffness and damping distribution in the system.
The key difference between the two approaches is the spatial basis of the resulting modes. The
results obtained from the two approaches to modal analysis are similar and the reasons for this
can be understood by studying the theoretical background to the two techniques.
-4-
2.1 Analytical Modal Analysis[1]
The equation of motion of an N degree of freedom system with viscous damping can be
expressed as
**
*
[ M ]{x} + [C ]{x} + [ K ]{x} = { f (t )}
(2.1)
Where [M], [C], [K] are the mass, damping and stiffness matrices of the system each of order
**
*
N*N. {x} , {x} and {x} are the system’s acceleration, velocity and displacement vectors in the
physical coordinates (each vector of length N*1).
The above equation of motion can be solved and the modal parameters be obtained in a number
of ways. Some of the solutions are valid for undamped/proportionally damped systems and
others for non-proportionally damped systems. The solution techniques for obtaining the modal
parameters are discussed below.
2.1.1 Laplace Domain Solution[1]
The above equation of motion (2.1) can be solved using a Laplace domain solution by assuming
a solution of the form {x} = { X }e st or taking Laplace transforms and assuming zero initial
conditions to arrive at a set of algebraic equations.
Thus
[ [ M ]s 2 + [C ]s + [ K ] ]{ X } = F ( s )
(2.2)
Or
[ B ( s )]{ X } = F ( s )
(2.3)
Where [ B ( s )] is called the System Impedance Matrix
-5-
From linear algebra, (2.2) has a solution if the determinant of the system impedance matrix is
zero [ B( s )] = 0 . Setting the determinant to zero would give the characteristic equation of the
system the solution of which would give the eigen values of the system. The System Impedance
matrix has quadratic (s2) functions of s thus it has 2N solution and since the system matrices [M],
[C], [K] are real matrices hence the poles (eigen values) λr occur in complex conjugate pairs.
Eigen vectors are obtained by substituting eigen values in (2.2) and obtaining {X}.
The inverse of the System Impedance Matrix is called the Transfer Function Matrix
[ H ( s )] = [ B( s )]−1
(2.4)
The frequency response function matrix H (ω ) is the transfer function matrix H (s ) analyzed
along the jω axis of the complex plane.
2.1.2 Real Normal Modes Solution[1]
If the system has no damping or is proportionally damped i.e. the damping matrix can be
expressed as [C ] = α [ M ] + β [ K ] where α , β are arbitrary constants, the eigen value analysis
will lead to real normal modes. The equation of motion (2.1) can be solved by assuming a
harmonic solution of the form {x} = { X }e jωt and the forcing function is also harmonic of the
form { f } = {F }e jωt .
The displacement, velocity and acceleration vectors become
{x} = { X }e jωt
(2.5)
*
{x} = { X } jωe jωt
(2.6)
**
{x} = − { X }ω 2 e jωt
(2.7)
-6-
Thus the equation of motion becomes
[ − [ M ]ω 2 + [C ] jω + [ K ] ]{ X } = {F }
(2.8)
As the damping matrix is a linear combination of the mass and stiffness matrix it can be
combined into the mass and stiffness matrices and an eigen value problem can be set up.
Since the [M], [C] and [K] are real matrices and the equation is a quadratic function in ω the
solution consists of 2N eigen values λr which occur in N complex conjugate pairs. The solution
also gives N real normal modes {φ}r which satisfy the following weighted orthogonality relations
{φ}Tr [ M ]{φ}s = 0
r≠s
(2.9)
= Mr r = s
{φ}Tr [ K ]{φ}s = 0
r≠ s
(2.10)
= Kr r = s
{φ}Tr [C ]{φ}s = 0
r≠s
(2.11)
= Cr r = s
Where M r , C r , K r are the modal mass, modal damping and modal stiffness of the mode r . The
above weighted orthogonality relations show that the N eigen vectors represents N linearly
independent vectors in an N dimensional space and any vector can be represented as a linear
combination of these N eigen vectors. Hence the displacement response can be represented as
N
{X } =
∑γ
r
{φ ) r
(2.12)
r =1
Substituting (2.12) in (2.1) and pre-multiplying by {φ }Ts
N
∑ γ [{φ}
r
T
s
]
[ M ]{φ }r (−ω 2 ) + {φ}Ts [C ]{φ}r jω + {φ }Ts [ K ]{φ }r = {φ}Ts {F }
(2.13)
r =1
Using the weighted orthogonality relations (2.9), (2.10), (2.11) the equation (2.13) becomes
-7-
γ r [− M r ω 2 + C r jω + K r ] = {φ}Tr {F }
Or
{φ}Tr {F }
γr =
M r [−ω 2 + 2 jωξ r Ω r + Ω 2r ]
(2.14)
Kr
is the undamped natural frequency of mode r.
Mr
Where Ω r =
Thus the displacement response in (2.12) can be expressed using (2.14) as
{φ}Tr {F }{φ}r
∑
2
2
r =1 M r [ −ω + 2 jωξ r Ω r + Ω r ]
N
{X } =
(2.15)
Equation (2.15) represents the response of an N degree of freedom system in terms of the
undamped natural modes. Rewriting (2.15) for response at a point p due to excitation at a point q
the frequency response function H pq can be written as
H pq =
Xp
Fq
N
=
∑M
r =1
φ pr φ qr
+ 2 jωξ r Ω r + Ω 2r ]
r [ −ω
2
(2.16)
Thus using the real normal modes solution the modal parameters of a discretized system can be
obtained and the frequency response function between any two measured degrees of freedom can
be synthesized.
2.1.3 Complex Modes Solution[1]
For non-proportionally damped system the damping matrix cannot be expressed as a linear
combination of the mass and stiffness matrix. Thus there are three independent matrices in the
equation of motion and hence an eigen value problem cannot be set up. However following a
state space solution the problem can be reduced to a standard eigen value problem using
-8-
*
*
[ M ]{x} − [ M ]{x} = 0
(2.17)
Combining equations (1) and (17)
*
**
 0 [ M ]  x  − [ M ] 0   x   0 
 = 
[ M ] [C ]   *  +  0
[ K ]  x  { f }

  x  
 
(2.18)
In (2.18) the state space displacement, velocity and force vectors can be substituted as
** *
x
 *  = { y}
 x 
 * 
x
  = { y}
 x 
  __
0
  ={ f }
{ f }
(2.19)
Equation (2.18) can be represented as
__
*
[ A] 2 N *2 N { y} + [ B ] 2 N *2 N { y} = { f }
(2.20)
The equation of motion (2.20) can be solved by assuming a harmonic solution of the form
__
__
{ y} = {Y }e jωt and the forcing function is also harmonic of the form { f } = {F }e jωt .A standard
eigen value is solved to obtain 2N eigen values λr , r = 1, 2…2N (which for a resonant system
occur in complex conjugate pairs) and 2N eigen vectors {ψ ss }r , r = 1, 2…2N.
Where
λ {ψ }r 
{ψ ss }r =  r

 {ψ }r  2 N *1
For complex eigen values the eigen vectors are also complex.
For a non-proportionally damped system the following weighted orthogonality relations hold
{ψ ss }Tr [ A]{ψ ss }s = 0 r ≠ s
(2.21)
= Ma r r = s
{ψ ss }Tr [ B]{ψ ss }s = 0 r ≠ s
(2.22)
= Mbr = − Ma r λ r r = s
-9-
Where Ma r , Mbr are the Modal A and Modal B for the mode r.
The above weighted orthogonality relations show that the 2N eigen vectors form a set of 2N
linearly independent vectors in a 2N dimensional vector space. Any vector in that 2N
dimensional space can be represented as a linear combination of these 2N eigen vectors.
Hence the state space displacement vector can be represented as
2N
{Y } =
∑γ
r
{ψ ss }r
(2.23)
r =1
Substituting (2.23) in (2.20) and pre multiplying by {ψ }Ts
2N
∑ γ [ jω{ψ
r
ss
]
__
}Ts [ A]{ψ ss }r + {ψ ss }Ts [ B ]{ψ ss }r = {ψ ss }Ts {F }
(2.24)
r =1
Using the weighted orthogonality relations (2.21), (2.22) in (2.24)
__
γ r [ jωMa r + Mbr ] = {ψ ss }Tr {F }
Or
__
{ψ ss }Tr {F }
γr =
Ma r ( jω − λ r )
(2.25)
Substituting (2.25) in (2.23)
__
{ψ }T {F }{ψ ss }r
{Y } = ∑ ss r
Ma r ( jω − λ r )
r =1
2N
(2.26)
Hence
 {0} 
{ψ ss }Tr  {ψ }r
{ jωX }
{F }

=∑
 { X }  r =1 Ma r ( jω − λ r )
2N
(2.27)
- 10 -
Equation (2.27) represents the response of an N degree of freedom system in terms of the
damped natural modes. Rewriting (2.27) for response at a point p due to excitation at a point q
the frequency response function H pq can be written as
H pq =
Xp
Fq
2N
=
ψ prψ qr
r ( jω − λ r )
∑ Ma
r =1
(2.28)
Thus using the complex modes solution the modal parameters of a discretized system can be
obtained and the frequency response function between any two measured degrees of freedom can
be synthesized.
2.2 Experimental Modal Analysis[3]
Experimental modal analysis[3] uses measured input-output data usually in the form of frequency
response functions to obtain the modal parameters of the structure. Modal parameter estimation
in experimental modal analysis uses statistical tools to obtain the poles, vectors and scaling
factors of the structure under test without any knowledge of the mass, stiffness and damping
distribution of the structure. The theoretical background and steps involved in the different
modal parameter estimation algorithms can be explained by the Unified Matrix Polynomial
Approach (UMPA) [3].
2.2.1 Partial Fraction Model
The frequency response function can be represented in the partial fraction form as
[ Ar ] No*Ni
[ Ar* ] No*Ni
+
∑
( jω − λ*r )
r =1 ( jω − λ r )
N
[ H (ω )] No*Ni =
(2.29)
- 11 -
Where λ r are the poles of the system and [ Ar ] is the residue matrix which is a complex No*Ni
matrix, No being the number of output measurement degrees of freedom and Ni being the force
input degrees of freedom. The above equation represents the multiple degrees of freedom
measurement as the contribution of the individual degrees of freedom.
Alternatively Equation (2.29) can be reformulated by splitting the residue matrices into two
matrices each
 1 
[ H (ω )] No*Ni = [ψ ] No*2 N 
[ L]T2 N *Ni

 jω − λ r  2 N *2 N
(2.30)
Where [ψ ] 2 N *No is the modal vector matrix of the structure and [ L] 2 N *Ni is the modal participation
vector matrix. The advantage of the above representation of the residue matrices is that the
summation is eliminated and the spatial information is represented as two distinct components,
the input and output measurement degrees of freedom.
In the time domain the impulse response function (generally obtained from the frequency
response function by the inverse FFT algorithm) can also be formulated as
[h(t )] = [ψ ] No*2 N [e λrt ] 2 N *2 N [ L]T2 N *Ni
(2.31)
The Equations (2.29), (2.30) and (2.31) indicate the measured frequency response function
matrix at any frequency (or impulse response function at any time step) can be represented as a
summation of the residue matrices and the modal frequencies. They also shows that the residue
matrices can be further split to represent the frequency response function matrix at any frequency
as a matrix multiplication of the modal vector matrix, the modal frequency matrix and the modal
participation vector matrix.
The different multiple degree of freedom modal parameter estimation algorithms essentially
calculate the above three matrices from the measured data using a general matrix polynomial
- 12 -
approach. The difference between the various parameter estimation algorithms lies in the fact
that they could operate in different domains (time/frequency), have different model order (low
order/high order) and have different complex matrix polynomial coefficients (scalar/matrix
polynomial coefficients). However in spite of these differences the working of the different multi
degree of freedom modal parameter algorithms is alike and can be explained broadly by the
Unified Matrix Polynomial Approach[3].
2.2.2 Unified Matrix Polynomial Approach (UMPA)
The common characteristics of different modal parameter estimation algorithms can be easily
identified by using a general matrix polynomial model which can be represented as
β n ( jω ) n + β n−1 ( jω ) n−1 + ... + β1 ( jω )1 + β 0 ( jω ) 0
[ H pq (ω )] =
=
Fq
α m ( jω ) m + α m−1 ( jω ) m −1 + ... + α 1 ( jω )1 + α 0 ( jω ) 0
Xp
(2.32)
Which for a generalized case of multiple input, multiple output can be represented as
m
∑ [[α
k =o
]
k
k ]( jω ) { X (ω )} =
n
∑ [[β
k
]
]( jω ) k {F (ω )}
(2.33)
k =0
The unknowns in the above equations are the polynomial coefficient matrices [α k ] and [ β k ] .
The above Equation (2.33) is valid at each frequency and as there are a total of m+n+2
unknowns hence the coefficient matrices can theoretically be determined if the frequency
response function has m+n+2 or more discrete frequencies (which is usually the case).
Multiplying (2.33) by {F (ω )}H and using the fact that { X (ω )}{F (ω )}H = ([G xf (ω )]) is the
cross power spectrum and {F (ω )}{F (ω )}H = ([G ff (ω )]) is the auto power spectrum for one
ensemble, Equation (2.33) can be reformulated as
- 13 -
m
∑ [[α
k
]( jω )
k
][G
n
xf
(ω )] =
∑ [[ β
k =o
k
]
]( jω ) k [G ff (ω )]
(2.34)
k =0
Multiplying both sides of equation (2.34) by [G ff (ω )] −1 the multiple-input multiple-output FRF
model can be represented as
m
∑ [[α
]
k
k ]( jω ) [ H (ω )] =
k =0
n
∑ [[β
k
]
]( jω ) k [ I ]
(2.35)
k =0
The size of the unknown coefficient matrix [α k ] will generally be No*No and the coefficient
matrix [ β k ] will generally be No*Ni when the equations are developed from experimental data.
Similarly a time domain model for the impulse response function data (which assumes the
forcing function to be zero for all time greater than zero) can be developed with unknown
coefficients matrices [α k ] .
m
∑ [α
k
]{h pq (t i + k )} = 0
(2.36)
k =0
The characteristic matrix polynomial equation for both time and frequency domain data are
essentially the same and are given below.
For frequency domain data case, this yields:
[α m ]s m + [α m −1 ]s m−1 + ... + [α 1 ]s 1 + [α 0 ]s 0 = 0
(2.37)
For time domain data case, this yields:
[α m ]z m + [α m−1 ]z m−1 + ... + [α 1 ]z 1 + [α 0 ]z 0 = 0
(2.38)
The time domain equation (2.38) is formulated in the z-domain ( z r = e λr∆t ) . The number m is the
model order of the characteristic polynomial equation and its choice is an extremely important
- 14 -
decision. The number of modal frequencies that will be found from the characteristic matrix
polynomial equation will be m times the size of the coefficient matrices [α ] .
Thus from (2.35) and (2.36) it is evident that most of the modal parameter estimation algorithms
can be developed by starting from a general matrix polynomial formulation that is justifiably
based upon the underlying matrix differential equation. From (2.37) and (2.38) it is evident that
the general matrix polynomial formulation yields essentially the same characteristic matrix
polynomial equation for both time and frequency domain data. Hence according to the Unified
Matrix Polynomial Approach, the solution process of any multiple degree of freedom modal
parameter estimation algorithms can be explained as a two stage linear process. The modal
frequencies and modal participation vectors are usually estimated in the first stage while the
modal vectors and modal scaling coefficients are estimated in the next stage.
Thus according to UMPA the modal parameter estimation process of any algorithm can be
outlined as
First Stage of Modal Parameter Estimation
•
Load the measured data into linear equation form i.e. Equations (2.35) or (2.36)
depending on the domain
•
•
Find the scalar or matrix coefficients ([α k ])
•
Normalize the frequency range (frequency domain only)
•
Utilize orthogonal polynomials (frequency domain only)
Solve matrix polynomials for modal frequencies
•
Formulate companion matrix
•
Obtain eigen values of the companion matrix (λr or z r )
- 15 -
•
Convert eigen values from z r to λr (time domain only)
•
Obtain modal participation vectors Lqr or modal vectors {ψ }r from the eigen vectors
of the companion matrix
Second Stage of Modal Parameter Estimation
•
Load data into linear equation form i.e. Equations (2.30) or (2.31)
•
Find the modal vectors {ψ }r and modal scaling from above
2.2.3 Experimentally Determined Modal Scaling Factors
The modal scaling parameters (modal mass for real normal modes, modal A for complex modes
solution) determined experimentally are dependant on the scaling chosen for the modal vectors.
From (2.29) a single-input single-output frequency response function can be represented as
N
H pq (ω ) =
A pqr
∑ ( jω − λ )
r =1
r
+
A*pqr
(2.39)
( jω − λ*r )
Rearranging
H pq (ω ) =
N
jω ( A pqr + A*pqr ) − A*pqr λr − A pqr λ*r
r =1
( jω − λr )( jω − λ*r )
∑
(2.40)
2.2.3.1 Real Normal Modes
For real normal modes the residue is purely imaginary
Hence A = − j A & A* = j A
And A pqr + A*pqr = 0
(2.41)
Therefore
- 16 -
H pq (ω ) =
H pq (ω ) =
j A λ*r − j A λ r
( jω − λ r )( jω − λ*r )
j A ( −2 jω r )
(2.42)
( jω − λ r )( jω − λ*r )
Also
λ r = − ξ r Ω r ± jΩ r (1 − ξ r2 )
(2.43)
Using equations (2.43) in (2.42)
2 A pqr Ω r (1 − ξ 2 )
N
H pq (ω ) =
∑ [−ω
2
r =1
(2.44)
+ 2ξ r ωΩ r + Ω r2 ]
From (2.16) and (2.43) the experimentally measured modal mass for the mode r
Mr =
φ pr φ qr
(2.45)
2
r
2 A pqr Ω r (1 − ξ )
Where φ pr , φ qr are the real modal vector coefficients for the measurement degrees of freedom
p and q for the mode r
As the residue A pqr is determined experimentally from the measured frequency response function
thus from (2.45) it is seen that the experimentally determined modal mass M r for a real normal
mode r is dependant on how the modal vector is scaled and on the modal frequency and
damping.
2.2.3.2 Complex Modes
Equation (2.27) can be rewritten as
ψ prψ qr
H pq (ω ) = ∑
+
r =1 Ma r ( jω − λ r )
N
N
ψ *prψ qr*
∑ Ma ( jω − λ )
r =1
r
*
r
- 17 -
(2.46)
From equations (2.40) and (2.46)
A pqr =
ψ prψ qr
Ma r
Thus the modal A for a mode r
Mar =
ψ prψ qr
(2.47)
Apqr
As the residue A pqr is determined experimentally from the measured frequency response function
thus, from (2.47), it is seen that the experimentally determined modal A Ma r for a complex mode
r is dependant purely on how the modal vector is scaled.
2.3 Conclusion
Thus it is seen that modal parameters for a structure can be obtained by both numerical as well
as experimental analysis and the modal parameters extracted accurately for a structure from
either of the two approaches are very close. The goal of this research is to obtain real modal
parameter estimates from complex modal parameters (modal vectors and modal frequencies)
obtained from either analysis techniques (either due to non proportional damping or other
reasons discussed in subsequent sections).
- 18 -
3.0 Origin and Identification of Complex Modal Vectors
Undamped and proportionally damped systems (systems in which the damping distribution
throughout the system is proportional to the mass and stiffness distribution) show real normal
modal vectors. Real modal vectors have modal vector coefficients that are either in phase or 180o
out of phase with each other and the measurement locations on the structure reach their maxima
and minima at the same time. Nonproportionally damped systems on the other hand have been
shown to have complex modal vectors wherein the modal vector coefficients have random phase
angles. Structures showing complex modes have their measurement degrees of freedom reaching
their maxima or minima at different time instances.
3.1 Possible sources for complex modal vectors
This chapter deals with the possible sources of origin of complex modal vectors and also
identifies techniques to gauge the complexity of the complex modal vector. It is often observed
that even in structures assumed to be proportionally damped the modal vectors obtained are often
complex. The possible reasons for this could be
•
Invalid assumption of proportional damping in the structures. It is understood that
systems with nonproportional damping will show complex modal vectors.
Measurement errors can also give rise to complex modes even in proportionally damped systems
during experimental modal analysis. Typical errors during the measurement stage which can give
rise to complex modal vectors are
- 19 -
•
Aliasing[2,3,4] (due to under sampling or insufficient number of measurement degrees of
freedom on the system) can cause highly complex modes. This is because although
aliasing doesn’t influence the damping and frequency estimates by much but it will cause
large errors in the residue or modal vector calculations.
•
Leakage[2,3,4] which is another common measurement error can cause errors in frequency
and damping estimates. A result of this invalid pole estimation is the estimation of
complex modal vectors. For lightly damped systems if significant leakage errors are
present in the measurement then the modal vector drawn in the complex (real v/s
imaginary) plane (which is generally a straight line) will form an angle with the
imaginary axis. The magnitude of this angle is dependant on the extent of leakage in the
measurement.
•
Insufficient frequency resolution[5] can cause poor estimates of frequency and damping
resulting in complex modal vectors[5]
•
Mass loading[3,4] is another measurement error which can result highly complex modal
vectors especially when using modal parameter estimation techniques like the
Polyreference Time Domain (PTD). Mass loading could occur when a transducer is
moved over a structure during measurement. This is analogous to moving a small mass
over the structure for each measurement. This mass addition results in the small mass
term being added to the diagonal term of the corresponding point in the modal mass
matrix. Thus the modal mass matrix changes during each measurement, changing the
system properties each time the transducer is moved. The consequence of this is that the
frequency for every pole changes from point to point due to data set inconsistency. The
modal parameter estimation algorithm (e.g. PTD) interprets this shift as another system
- 20 -
pole and would try to estimate as many poles as there are frequency shifts. Thus the result
is that it is difficult to obtain a true estimate of the number of poles as well as determine
which pole estimate is valid. Thus due to mass loading the pole frequencies are
underestimated and damping error estimates are unreliable. Thus the modal vectors
estimated by the modal parameter estimation algorithm are highly complex with respect
to their phase scatter although the error on the magnitude as compared to the no mass
loading case is slight
Errors during the parameter estimation phase can also give rise to complex modal vectors. The
errors during the parameter estimation phase cause unreliable estimates of the poles and damping
which result in complex modal vectors. The reasons for these errors are
•
Taking too many or too few poles in the frequency range of interest[4] could cause
improper damping estimates.
•
Taking poles too close to edges of the frequency range of interest[3,4] (especially the lower
frequency) can cause improper damping estimates.
•
If a pole is omitted in the frequency range of interest[4], the modal vectors of the pole
found will be highly complex.
•
Even if all the poles are found but the relative error in the damping estimate is more than
5% the modal vectors could still be very complex.
•
Insufficient number of references in the data set[3,4] can cause poor pole calculations
which can result in complex modal vectors especially for closely spaced modes.
- 21 -
3.2 Complexity of Modal Vectors
Complex modal vectors can have varying degrees of complexity. Some complex modal vectors
could essentially be an almost purely real or imaginary modal vector. Alternatively the complex
modal vector could be a real modal vector but with a constant phase shift i.e. all the modal vector
coefficients of the modal vector have almost the same phase shift angle. A highly complex modal
vector has modal vector coefficients which have considerable phase scatter. If an indication of
the complexity of the mode is known, then it can be estimated whether the mode is a pure
complex mode or just essentially a real mode with a phase shift. It can also be inferred whether
the complex mode is due to errors in the modal vector estimation (highly complex modal vectors
for systems thought to be proportionally damped).
The degree of complexity of modal vectors can be determined by a number of techniques some
of which are described below.
3.2.1 Plotting the Complex Modal Vector in the Complex Plane
A simple yet effective technique would be to plot the complex modal vector in a complex plane
(real vs. imaginary plane). A plot of the real modal vector in the complex plane would be a
straight line and would lie along the real axis. A highly complex modal vector’s plot would be
scattered all over the plane while a complex modal vector with an almost constant phase shift
would be a straight line at an angle from the real axis. This angle would be equal to the phase
angle of the modal vector coefficients.
Thus by plotting the modal vector in a real vs. imaginary plot the complexity of the mode can be
gauged.
- 22 -
Fig 3.1: An almost purely imaginary complex modal vector
Fig 3.2: A complex modal vector with a constant phase shift
- 23 -
Fig 3.3: A slightly complex modal vector
Fig 3.4: A highly complex modal vector
- 24 -
3.2.2 Modal Phase Collinearity
The modal phase collinearity[2] (MPC) is an indicator that checks the degree of complexity of the
modal vector coefficients. It evaluates the functional linear relation between the real and
imaginary parts of the modal vector coefficients. For each mode shape r a vector is calculated by
subtracting from each element the mean complex value of all elements.
No
∑ψ
_
ψ ir = ψ ir −
or
o =1
NO
for i = 1,2,3,..., N O
(3.1)
Using the quantities
_
ε=
_
|| Im{ψ }r || 2 − || Re{ψ }r || 2
_
_
2(Re{ψ }tr Im{ψ }r )
(3.2)
And
θ = arctan(| ε | + sign(ε ) 1 + ε 2 )
(3.3)
The modal phase collinearity is given by
_
MPC r =
_
_
|| Re{ψ }r || 2 + (Re{ψ }tr Im{ψ }r )(2(ε 2 + 1) sin 2 θ − 1) / ε
_
_
|| Re{ψ }r || 2 + || Re{ψ }r || 2
(3.4)
For real modes this approaches unity. A mode with a low index is rather complex, indicating a
computational or noisy mode in the cases where nearly normal modes are expected
(proportionally or lightly damped systems). In order to make the method applicable for modes
with arbitrary phase, the modal vector coefficients are first rotated over the mean phase of the
vector.
- 25 -
3.2.3 Mean Phase Deviation
The mean phase deviation[2] (MPD) is another statistical indicator of the complexity of the mode
shape.
The mean phase of the mode r is
No
∑w ϕ
o
MPH r =
or
o =1
No
∑w
o
(3.6)
o =1
Where
wo
: A weighting factor (e.g. = 1, or = ψ or )
ϕ or = arctan(Re{ψ or } / Im{ψ or }) if arctan (Re{ψ or } / Im{ψ or }) ≥ 0
ϕ or = arctan(Re{ψ or } / Im{ψ or }) + Π if arctan (Re{ψ or } / Im{ψ or }) < 0
The corresponding mean phase deviation of the mode is
2
No
∑ w (ϕ
o
MPDr =
or
− MPr )
o =1
No
∑w
o
o =1
(3.7)
The modal phase deviation expresses the phase scatter (in degrees) of each mode shape. For a
normal mode shape its value should be zero.
- 26 -
4.0 Historical Methods for Real Normalization of Complex Modal Vectors
The need for real estimates for the complex modal vectors obtained from experimental analysis
was felt early on when analytical models were developed and needed real modal vectors for the
weighted orthogonality relations
[φ ]T [ M ][φ ] = [ M r ]
(4.1)
[φ ]T [ K ][φ ] = [ K r ]
(4.2)
[φ ]T [C ][φ ] = [C r ]
(4.3)
In all these equations [φ ] are real normal modes whereas the modal parameters obtained from
experimental modal parameter estimation algorithms (both time and frequency domain
algorithms) are complex modal frequencies and modal vector estimates.
A number of real normalization techniques were developed to get real vector estimates. Two of
the algorithms referenced during the course of this research are explained in detail below
4.1 Computation of Normal Modes from Identified Complex Modes - S.R Ibrahim[5,6]
S.R Ibrahim[5,6] developed two approaches to obtain real modal vectors and normal modes from
complex modal vectors and frequencies obtained by experimental modal extraction
4.1.1 Using an Oversized Mathematical Model
This technique uses a set of complex modal vectors each having Nm measurement degrees of
freedom [ψ i ] , i = 1 to N (where Nm > N) and complex natural frequencies λi, i= 1 to N (and their
- 27 -
complex conjugates) obtained experimentally to generate free response time domain
displacement, velocity and acceleration responses as
2N
x(t ) =∑ψ r e λr t +[n1 (t )]
(4.4)
r =1
2N
*
x(t ) = ∑ψ r λ r e λr t + [n2 (t )]
(4.5)
r =1
2N
**
x (t ) = ∑ψ r λ2r e λr t + [n3 (t )]
(4.6)
r =1
Where n1 (t ), n2 (t ), n31 (t ) are the added random noises of random distribution
The time domain equations of motion can be represented in a state space solution as
 * 
0
 x (t )  
 **  = 
−1
 x ( t )   − [M ] [K ]
[I ]
  x ( t ) 
 * 
−1
− [M ] [C ]  x ( t ) 


(4.7)
Or
*
{ y (t )} = [E ]{y (t )}
(4.8)

0
[I ]


−1
−1
− [ M ] [ K ] − [ M ] [C ]
Where matrix [E ] = 
*
And { y (t )} and { y (t )} are the state space velocity and displacement responses.
Using Equations (4.4), (4.5), (4.6) and (4.7) the Equation (4.8) is repeated 2*Nm time instants such that
.
*
*
[ y (t ),......., y (t 2 Nm )] = [ E ][ y (t1 ),........, y (t 2 Nm )]
(4.9)
Or
*
[Y ] = [ E ][Y ]
(4.10)
*
Here all matrices [ E ], [Y ], [Y ] are of size 2Nm *2Nm and [E ] is obtained as
- 28 -
*
[ E ] = [Y ][Y ] −1
(4.11)
From (4.11) and definition of [E ] the − [ M ]−1 [ K ] portion of [E ] is retained.
Using the − [ M ]−1 [ K ] matrix an eigenvalue problem is set up as
[ [ M ] −1 [ K ] − Ω 2 * [ I ] ] * {φ} = {0}
(4.12)
An eigenvalue decomposition of this − [ M ]−1 [ K ] ] would give the undamped natural frequencies
(Ω r ) and real modal vectors {φ r } of the system.
It is obvious that without any noise, the state displacement matrix [Y ] would be singular since
the number of measurement degrees of freedom is greater than the number of modes present.
However a small amount of noise makes the inversion possible. Even signal to noise ratios of
0.000001 could be used to invert a 600*600 matrix of rank 4 without signs of ill-conditioning on
a 60 bit word computer.
The mechanism can be explained as follows
The state space vector’s 2Nm free time responses contain modal information from N structural
modes and can be represented as
2N
y (t ) = ∑ p r e λrt
(4.13)
r =1
Where [ p ] represents the N complex pairs of the state space independent eigen vectors.
If noise free measurements were used then the mathematical model must have exactly N
measurement degrees of freedom for unique identification. If more than N degrees of freedom
are present, then the state displacement matrix [Y ] is singular.
However in experiments the measured responses always contain a small amount of noise (or
small amounts of noise could be added on purpose). These noisy responses can be expressed as
- 29 -
2N
y (t ) = ∑ p r e λrt + [n(t )]
(4.14)
r =1
It was previously concluded that using noisy responses in the identification process with
measurement degrees of freedom greater than N, yielded good results without encountering
singularity. The explanation for this could be that the extra degrees of freedom (Nm-N) could act
as outlets for the noise. Thus the noise could be modeled into the extra degrees of freedom as
2N
y (t ) = ∑ p r e λrt +
r =1
2 Nm
∑N
k
e λrt
(4.15)
r = 2 N +1
Where the noise is modeled as (2Nm-2N) complex exponential functions
Since the number of modes N is a characteristic of the structure and not the data analysis process,
the additional exponential functions are allowed to represent the noise in the mathematical model
as Nm is increased. This results in a higher order fit for the noise portion of the responses,
reducing the residuals that would otherwise be included in the signal portion of the responses. It
should be noted that for the above to hold true the noise should be truly random and not biased
else it will affect p r and λ r
4.1.2 Assumed Mode Shapes Solution
This method uses a set of assumed modes (Nm – N) which are linearly independent of the N
complex modes present. The justification for this technique is as follows
The complex modal parameters obtained from an experimental modal analysis satisfy the
following equation
ψ 
[[ M ]−1 [ K ] [ M ]−1 [C ]] i  = − λi2ψ i
λiψ i 
{
}
where i =1to N
(4.16)
- 30 -
But since there are Nm measurement degrees of freedom and only N modes (Nm >N) the above
equation cannot be solved for [[ M ]−1 [ K ] [ M ]−1 [C ]] .
Now if it is assumed that there are a set of vectors [ R j ] and a set of characteristic roots s j ,
j=N+1 to Nm such that this set of assumed parameters satisfy the relation
λi ≠ s i
and
[ Ri ] ≠ [ψ 1 ,ψ 2 ,ψ 3 ,......ψ N ]{a}
Here {a} is any vector of coefficients.
The above implies that [R ] and [ψ ] together form a set of Nm linearly independent vectors and
similarly λ and s form a set of Nm different eigen values.
These Nm set of linearly independent modal vectors and eigen frequencies can be used to form a
hypothetical system having Nm measurement degrees of freedom and Nm modes. The free
displacement time domain response of such a system can be represented as
2N
x(t ) = ∑ψ r e λrt +
r =1
2 Nm − 2 N
∑R e
sjt
j
j =1
(4.17)
Also the Equation (4.16) can now be formulated for i = 1 to Nm thus obtaining [ M ] −1 [ K ] which
can be solved for real modes by setting up an eigenvalue solution.
An appropriate set of assumed modes would be from the structure’s finite element model. Higher
analytical modes other than the experimentally measured modes are highly recommended. A
quick orthogonality check could be done between the assumed modes and measured ones to
establish that they are linearly independent. An extremely important note is that
[ M ] −1 [ K ] obtained from either of the two explained approaches is not unique but is a function of
- 31 -
the introduced noise or assumed modes. However the normal modes obtained from the complex
modes was found to be independent of the introduced small levels of noise or the assumed
modes.
4.2 Time Domain Subspace Iteration Technique
The technique was derived by Wei, M[7] for obtaining real modal vector estimates and undamped
natural frequencies from complex modal parameters. The method uses a transformation matrix to
convert the normalization process from the physical coordinates vector space to a modal vector
subspace. It computes the matrix [E ] (Equation (4.11)) in the modal subspace. It then derives the
real modal parameters in the same modal subspace and then expands the solution to the physical
space.
As shown previously, that in the absence of noise, the derivation of the matrix [E ] (Equation
(4.11)), hence the − [ M ]−1 [ K ] system matrix, is difficult because of the singularity of the state
space displacement matrix [Y ] when the number of measurement degrees of freedom Nm are
greater than the number of modes N. This technique thus converts the problem to a modal
^
subspace using a transformation matrix [φ ] which is related the complex modal matrix [ψ ] as
follows
^
[ψ ] = [φ ][ X ]
(4.18)
Here [ X ] is a N*N complex matrix which can be derived through a pseudo inverse technique as
^
[ X ] = [φ ] + [ψ ]
(4.19)
- 32 -
^
Since the transformation matrix [φ ] represents the N-dimensional subspace of the physical Nm
dimensional vector space from which the matrix [ E N ] and normal modes are derived, it cannot
be arbitrarily chosen.
^
The transformation matrix [φ ] must satisfy the following requisites
^
1. [φ ] must not be orthogonal to the normal modal matrix [ φ ] of the undamped system but
in fact should be as close as possible to [ φ ]
^
2. [φ ] must have a rank of N
Fillod[10] normalized the complex modal vectors using the following equation
2λ r {ψ }Tr [ M ]{ψ ] r + {ψ }Tr [C ]{ψ ] r = 2 jω r
(4.20)
This normalization minimizes the imaginary part of {ψ }r and hence maximizes the real part of
{ψ }r which contains the maximum useful information of the identified complex modes.
^
The technique derives the transformation matrix [φ ] by scaling the complex modal matrix [ψ ]
using the above equation and then retaining the real part of the complex modal vectors. Thus the
^
derived transformation matrix [φ ] has a rank N because the identified complex modes are
^
independent of each other and the norms of the columns of [φ ] have the same order of
magnitude. This transformation matrix can be considered as a kind of preliminary estimate of the
^
real modal vector matrix [ φ ]. The transformation matrix [φ ] is now used to transform the free
decay displacement, velocity and accelerations in the physical coordinates into the modal space
by the following relations
- 33 -
^
{x(t )} = [φ ]{ p(t )}
(4.21)
Or
^
{ p(t )} = [φ ] + {x(t )}
(4.22)
Substituting Equations (4.4) (assuming no noise) and equation (4.19) into the above Equation
(4.22), yields:
^
{ p(t )} = [φ ] + 2 Re{[ψ ]{e λrt }}
^
= 2[φ ]+ [φ ] Re{[ X ]{e λrt }}
= 2 Re{[ X ]{e λrt }}
(4.23)
Where “Re” represents the real part of a matrix
Similarly the velocity and acceleration responses in the modal coordinates can be derived as:
*
{ p(t )} = 2 Re{[ X ]{λe λrt }}
(4.24)
**
{ p(t )} = 2 Re{[ X ]{λ2 e λrt }}
(4.25)
Using the above modal space responses the state space solution in the modal vector subspace can
be set up as:
*
{q (t )} = [ E N ]{q (t )}
(4.26)
The matrix [ E N ] in the modal subspace is defined as:
[0]
[I ]


EN = 

−1
− [ M N ] [ K N ] − [ M N ][C N ]
(4.27)
[MN], [KN], [CN] are the reduced system mass, stiffness and damping matrices respectively
which can be represented as:
^
^
[MN]N*N = [ φ ]T[M] [φ ]
- 34 -
^
^
^
^
[KN]N*N = [ φ ]T[K] [φ ]
[CN]N*N = [ φ ]T[C] [φ ]
*
Once again { q (t)} and { q (t)} are computed at 2*N time instants to obtain the matrix [ E N ] as:
*
[ E N ] = [Q ] 2 N *2 N [Q ] 2−1N *2 N
(4.28)
Inversion of [ Q ] is possible because it has a full rank 2*N.
From the matrix [ E N ] the N*N system matrices − [ M N ]−1 [ K N ] are obtained and are used to set
up an eigenvalue problem in the modal subspace as before the obtain the undamped natural
frequencies Ω r and the real modal vectors [φ ] where r = 1,2….N
The real modal vectors [φ1 ] in the physical coordinates are then derived from the subspace real
^
modal vector matrix [φ ] using the transformation matrix [φ ] as follows
^
[φ1 ] = [φ ] [φ ]
^
In order to improve (if possible) and check the accuracy of [φ1 ] , [φ1 ] is compared with [φ ]
column by column for convergence. If not convergent, then the entire normalization process
from Equation (4.18) onwards is repeated using [φ1 ] as the new transformation matrix to obtain
the new normal modal matrix [φ 2 ] . This process continues till [φ m ] converges to [ φ m −1 ] where m
^
is the number of iterations. Generally m is 2 or 3 i.e. the transformation matrix [φ ] converges
very fast as all iterations are done in the same vector subspace which is defined by the initial
^
estimated transformation matrix [φ ] .
- 35 -
5.0 LEAST SQUARES SOLUTION
Purely real or imaginary estimates for complex modal vectors can be obtained using a simple
least squares solution or a weighted least squares solution. These methods are simple to
implement and provide good preliminary estimates for the complex modal vectors. Additionally
viewing the line fitted modal vectors along with the complex vectors on a real vs. imaginary plot
gives a good indication of the complexity (phase scatter of modal coefficients) of the vectors.
The effectiveness of the least squares line fit process can be ascertained by calculating the
correlation coefficient between the modal vector coefficients and the estimated line fit. The
Modal Assurance Criterion (MAC) values between the least squares line fitted modal vectors can
be calculated to check if the linear independence of the vectors has been preserved. In order to
ascertain the preservation of the underlying motion characteristics, the line fitted vectors and the
complex modal vectors could be scaled alike (largest modal vector coefficient (in terms of
magnitude) for each modal vector be scaled to 1). Once scaled (and phased) alike the modal
vectors can be subtracted to view the difference.
5.1 The Least Squares Method
The Least squares method[3] is a popular method for parameter estimation. It is used to define the
relationship between two groups of variables x and y which can be represented by a linear
relationship
(5.1)
y = a1 x + a0
- 36 -
The variables a0 and a1 could be solved by arbitrarily choosing two sets of x and y quantities. Yet
this line constructed may not pass through all sets of x and y as the information associated with
those sets was not used to calculate a0 and a1.
Alternatively the least squares criterion uses all these x and y sets to obtain a linear fit (best
estimates of a0 and a1) which minimize the deviation of the sum of squares of the data points
from the estimated curve. These deviations are simply the difference between the estimated
~
values of y (or y r ) and the actual value of y (or y r )
Thus
∂E N
= ∑ 2 [ y r − (a1 x r + a o )][−1] = 0
∂a o r =1
N
N
r =1
r =1
~
N
(5.2)
E = ∑ er2 = ∑ [ y r − y r ] 2 = ∑ [ y r − (a1 x r + a o )]
2
(5.3)
r =1
Minimization of the error above with respect to a0 and a1 results in the following equations which
are called normal equations for the least squared problem
∂E N
= ∑ 2 [ y r − (a1 x r + a o )][− x r ]
∂a1 r =1
(5.4)
The above equations can be solved for the unknown parameters a0 and a1. These variables best
describe a line which minimizes the sum of the square of the errors in the y direction (ey)
The least squares problem can be formulated in matrix notation as
{Y} = [X] {a}
(5.5)
- 37 -
Where
 y1 
 x1
y 
x
 
{Y } =  2  [ X ] =  2
 ..
 .. 

 y n 
 xn
1
1
a 
{a} =  1 
..
a o 

1
These represent a set of inconsistent (no single {a} can satisfy all equations) and over determined
(more equations than unknowns) equations. These are solved as follows to get {a}
[X]T{Y} = [X]T[X] {a}
(5.6)
{a} = ([X]T[X])-1[X]T{Y}
(5.7)
Using {a} the least squares line can be plotted using the x and y values of the data points
5.2 The Weighted Least Squares Method
The individual equations could be multiplied by a weighting factor to give them more or less
importance in the least squares parameter estimation process. This weighting factor could be a
(N*N) diagonal matrix, W  . The diagonal term in row (i) represents the weighting of equation
(i) and off diagonal terms are all zero. Thus the weighted least squares line fit could be
represented as
W  {Y} = W  [X] {a}
(5.8)
Where
 w1
0

W
=
  0

.
 0
0
w2
0
.
0
0
0
w3
.
0
. 0 
. 0 
. 0 

. . 
. wN 
- 38 -
Thus we get {a} as
([X]T W  T) W  {Y} = ([X]T W  T) W  [X] {a}
(5.9)
{a} = (([X]T W  T) W  [X] )-1([X]T W  T) W  {Y}
(5.10)
Using this {a} and the x and y values of the data points the weighted least squares line can be
plotted.
5.3 The Correlation Coefficient
The quality of the least squares line fit is defined by the correlation coefficient of the process.
The correlation coefficient is defined as the ratio of the explained variation to the total variation.
N
The total variation of y is defined as
∑(y
__
r
− y ) 2 which is the sum of squares of the deviations yr
r =1
__
from the mean value y . The total variation has two parts (1) the explained variation,
N
_
~
∑(y
r
N
2
− y ) which follows a definite pattern and (2) the unexplained variation
∑(y
~
r
2
− yr )
r =1
r =1
which is unpredictable and random.
Thus the correlation coefficient γ2 is given by
_
 N ~
 ∑ ( y r − y) 2
γ 2 =  rN=1
_

2
 ∑ ( yr − y)
 r =1






(5.11)
The magnitude of the correlation coefficient is between 0 and 1 with 0 indicating that the fitted
least squared line does not have any correlation to the data set while a value of 1 indicates perfect
correlation.
- 39 -
5.4 Least Squares Normalization Technique (LSNT)
A simple least squares linear fit[3] can be used to fit a line through the modal vector coefficients.
The least squared method determines the coefficients of a line ( y = a1 x + a0 ) which minimizes
the square of the error between the estimated line and the modal vector coefficients.
A set of simultaneous equations can be expressed as:
{Y} = [X]{a}
(5.12)
Where
X = real part of each of the Nm modal vector coefficients
Y = imaginary part of each of the Nm modal vector coefficients
 imag (ψ 1r ) 
 real (ψ 1r )
 imag (ψ ) 
 real (ψ )
2r 
2r

{Y }= 
[
X
]
=


.
.



imag (ψ Nmr )
real (ψ Nmr )
1
1
a 
{a} =  1 
.
a o 

1
Hence the line coefficients are obtained by least squares method as:
[X]T{Y} = [X]T[X] {a}
(5.13)
{a} = ([X]T[X])-1[X]T{Y}
(5.14)
After getting a1 and a0 the least squares line can be plotted as Y = X* a1 + a0 .
After the line best describing the modal coefficients in a least squares sense is estimated, it can
be rotated by a certain phase angle (depending upon the existing phase angle of the estimated
line) to align either along the real or imaginary axis. This basically amounts to a phase shift of
the modal vector. Thus, the resultant vector is either purely real or purely imaginary. The
correlation coefficient between the estimated line and the modal vector coefficients is calculated
to check how well the line describes the points.
- 40 -
Using appropriate formulas for the explained and unexplained variations, the correlation
coefficient between the line and modal coefficients can be calculated.
A high correlation coefficient (close to 1) indicates that the line fits the modal vector coefficients
very well and all modal vector coefficients lie very close to the line. A low value of the
correlation coefficient (close to 0) indicates that the modal vector coefficients are scattered, also
the mode is more complex and the line does not describe the modal vector coefficients well.
5.5 Weighted Least Squares Normalization Technique (WLSNT)
This method
[3]
is slight variant of the least squares method. Here a weighting matrix is used to
estimate a weighted least squares line estimate. In this method, first the normal least squares
estimate line for the modal vector coefficients is found from the previous Section (5.4). Then a
weighting matrix which is a square diagonal matrix of size Nm* Nm is obtained by taking the
inverse of the square of the difference between a modal vector coefficient and its least squares
estimate (i.e. the unexplained variation of the modal vector coefficient) for each of the Nm modal
vector coefficients. This inverse term is then placed at its appropriate Nm diagonal position to
obtain the Nm* Nm weighting matrix.
 w1
0

W
=
  0

.
 0
0
w2
0
.
0
0
0
w3
.
0
0
0 
0
0 
0
0 

.
. 
0 w Nm 
Where wn = 1/ (unexplained variation between the modal vector coefficients and the initially
estimated line)
- 41 -
Thus the closer a point is to the least squared estimate line the greater will be its weighting and
the further a point is from the line the line the smaller will be its weighting. Using this weighting
matrix in the calculations, a new weighted least squares estimate line is found by the following
method.
W  {Y} = W  [X] {a}
(5.15)
Thus {W} is obtained as
([X]T W  T) W  {Y} = ([X]T W  T) W  [X] {a}
(5.16)
{a} = ( ([X]T W  T) W  [X] )-1([X]T W  T) W  {Y}
(5.17)
As in the previous case, the weighted least squares line can be calculated using the real and
imaginary coefficients of the modal vector and {a}.
The correlation coefficient is found between the weighted least squares estimate line and the
weighted modal vector coefficients. In this case, a weighted mean is used in the calculations.
It is seen that for modal vectors that fall almost along a straight line both methods (Sections 5.4
& 5.5) give almost identical pure real/imaginary modal vector estimates but for considerably
complex modes, the line estimates from the two methods show marked difference.
The figures below show the LSNT and the WLSNT estimates of an almost purely imaginary
modal vector and of a highly complex modal vector. Note that in the first case both the
normalization lines by the two techniques almost coincide while in the second case there is
marked difference between the two.
- 42 -
Fig 5.1: An almost purely complex modal vector
Fig 5.2: LSNT and WLS NT estimates for the almost purely complex modal vector
- 43 -
Fig 5.3: A highly complex modal vector
Fig 5.4: LSNT and WLSNT estimates for the highly complex modal vector
The LSNT/ WLSNT estimated line can be rotated by a certain phase angle (depending on the
existing phase angle of the estimated line) to align either with the real or imaginary axis to get
- 44 -
purely real or imaginary modal vector estimates. The purely real and imaginary estimates for a
highly complex modal vector are shown below
Fig 5.5: Real modal vector estimates for a highly complex modal vector
Fig 5.6: Imaginary modal vector estimates for a highly complex modal vector
- 45 -
5.6 Conclusion
The LSNT technique essentially line fits the modal vector coefficients into a straight line by
minimizing the square of the errors between the estimated line and the coefficients.
The WLSNT technique is a modification over the previous technique. Here the results of the
previous technique are used to generate a weighting matrix. The weighting matrix weights the
modal vector coefficients which are calculated as the inverse of the square of the difference
between the modal vector coefficients and their simple least square estimates (i.e. the
unexplained variation). The weighting matrix is used to generate another line estimate in a least
squared sense.
The vectors obtained by the above methods can be rotated by a certain phase angle (depending
upon the existing phase angle of the estimated line) to make them as purely real or purely
imaginary whichever is required.
The success of the least squares line fit in both cases can be evaluated calculating the correlation
coefficients. For complex modal vectors with little phase scatter the normalized estimates from
either of the two methods are good with correlation coefficients close to 1 (+0.99). But for highly
complex modal vectors with considerable phase scatter the two methods give different line fits
Both the methods explained are purely statistical techniques. They do not involve any modal
analysis principles during normalization, but the vectors obtained (purely real or imaginary) have
MAC values very close to that shown by their complex counterparts. Also these simple
normalization techniques preserve the underlying motion characteristics of the vectors very well.
- 46 -
6.0 Techniques for Real Normalization of the Complex Modal Parameters
The following section presents the three real normalization techniques[13] developed and/or
utilized during the course of this research. The theory, derivations, development and procedures
for the three techniques are explained below. The three techniques require, from an experimental
or an analytical modal analysis, a set of complex modal parameters namely the complex modal
frequencies (in terms of real and imaginary parts) and the associated complex modal vectors. The
Rescaling of Complex Modal Vectors Technique (RCMVT) gives the real normalized modal
vector matrix while the Information Matrix Subspace Technique (IMST) and the Effective
Independence Subspace Technique (EIST) give the undamped natural frequencies and the real
normalized modal vector matrix after the normalization process.
6.1 Rescaling of Complex Modal Vectors Technique (RCMVT)
The complex modal vectors obtained are rescaled in a way such that the largest modal coefficient
(i.e. the complex number with the largest complex modulus (magnitude)) for each vector is 1.0.
Scaling of complex modal vectors in this way ensures that the modal masses, computed from the
associated real modal vectors, of the system are between 0 and the physical mass of the system.
The real modal vector estimates are obtained by retaining the real part of these rescaled complex
modal vectors. The real modal vectors obtained retain the underlying motion characteristics and
linear independence of the complex modal vectors.
- 47 -
6.1.1 Theory
The complex modal vectors are rescaled such that the largest modal vector component (in terms
of magnitude) for each vector is 1. The real part of the modal vectors scaled in the above way
contains maximum information of the complex modal vectors as the imaginary part is
minimized. Thus the real part of thus scaled complex vectors is retained to get an estimate of the
real modal vectors.
It is proved in the following section that for an undamped system perturbed to the first order by a
nonproportional damping matrix, if the complex modal vectors obtained are scaled in the above
way then the real part of the complex modes is a very close approximation of the real normal
modes of the undamped system. The following proof was originally developed by Fillod R.[7, 10].
6.1.2 Proof
The system equations for an undamped and a damped system are
[ K ][φ ] − [ M ][φ ] Ω 2  = 0
(6.1)
[ M ][ψ ] λ2  + [C ][ψ ] λ  + [ K ] [ψ ] = 0
Where [φ ] : Real Modal Vector Matrix, [ψ ] : Complex Modal Vector Matrix
Ω 2  : Diagonal Real Modal Frequency Matrix
λ2  : Diagonal Complex Modal Frequency Matrix
If nonproportional damping is introduced into an undamped or proportionally damped system
then this nonproportionality it can be represented as[7, 10]
- 48 -
[C ] = ε [C ]
(6.2)
Here [C ] is the damping matrix for a proportionately damped system, ε is a small perturbation
(1st order) and [C ] is the damping matrix for the corresponding nonproportionally damped
system. The complex modal vectors are now scaled such that the largest modal coefficient (in
terms of magnitude) for each modal vector is 1.0.
It is assumed that the introduced nonproportionality in damping affects the natural frequencies
and mode shapes in the way indicated below[7, 10] where in the real modal vectors and natural
frequencies are perturbed to complex due to the introduced non proportionality.
λ  = j Ω + ε L
(6.3)
[ψ ] = [φ ][ [I ] + jε [P]]
The matrices L  and [P ] are assumed to be complex initially but are shown to be real matrices if
orthogonality relations between the modal vectors is to hold true.
Substituting Equations (6.3) and (6.2) in Equation (6.1) for the damped system
[M ][φ ][ [I ]+ jε [P] ][ j Ω + ε L  ]2 + ε [C ][φ ][ [I ]+ jε [P] ][ j Ω + ε L  ]
+ [K ][φ ][ [I ]+ jε [P ] ]= 0
Collecting terms containing powers of ε ( ε 0 , ε 1 , ε 2 , ε 3 ) in the above expression and equating to
0
− [M ][φ ] Ω 2  + [K ][φ ]= 0
2[M ][φ ] L Ω − [M ][φ ][P ] Ω 2  + [C ][φ ] Ω  + [K ][φ ][P ] = 0
[M ][φ ] L2  − 2 [M ][φ ][P] L Ω + [C ][φ ] L − [C ][φ ][P] Ω = 0
- 49 -
(6.4)
[M ][φ ][P] L2  + [C ][φ ][P] L = 0
The real modal vectors satisfy the orthogonality conditions and hence following relationships
exist
[φ ]T [M ][φ ] = M r 
[φ ]T [K ][φ ] = K r 
And [φ ]T [C ][φ ] = C r 
(6.5)
Pre-multiplying Equations (6.4) by [φ ]T and applying (6.5)
− M r  Ω 2  + K r  = 0
(6.6)
2 M r  L Ω  − M r  [P ] Ω 2  + C r Ω  + K r  [P ] = 0
(6.7)
2
M r  L  − 2 M r  [P ] L Ω + C r L  − C r  [P ] Ω = 0
(6.8)
2
M r  [P ] L  + C r  [P ] L  = 0
(6.9)
Substituting for K r  from Equation (6.6) into Equation (6.7) and rearranging
2 M r  LΩ + Cr Ω + M r  [ Ω2  [P] − [P] Ω2 ] = 0
(6.10)
From Equation (6.10) it can be inferred that the diagonal terms of the last term
( M
r
2
2
 [ Ω  [P ]− [P ] Ω ] ) will be zeros. Also as M r  , Ω and C r  are all real diagonal
matrices (due to the type of scaling of the real modal vectors) hence for Equation (6.10) to hold
true the diagonal matrix L  must also be a real.
Since L  is now a real diagonal matrix hence, for Equation (6.10) to hold true, the non diagonal
terms of [P] cannot be complex. On further examining Equations (6.8) and (6.9) it can be
inferred that the diagonal terms of [P] also cannot be complex for the equations to hold true.
Hence [P] must be a real matrix
- 50 -
Hence it is concluded that
•
L  is a real diagonal matrix. Thus the effect of 1sr order perturbation damping matrix on the
modal frequencies is purely real.
•
[P] is a real matrix. Thus the effect of the 1st order perturbation damping matrix on the
modal vectors is purely imaginary.
Thus from above it is proved that if an undamped system (or proportionally damped system) has
slight nonproportional damping (1st order) introduced in it and the complex modal vectors
obtained are scaled such that the largest modal vector coefficients (in terms of magnitude) are 1.0
then the real part of the complex modal vectors and the damped natural frequencies can be used
as an accurate approximate of the real modal vectors and undamped natural frequencies of the
associated undamped system.
The same conclusions regarding the matrices L  and [P ] can be reached if the damping model
is considered of the form
[C ]= j [C ] ± ε [C ]
The above proof depends upon the initial assumption of Equation 6.2 or the above assumption.
These assumptions are quite limiting and may yield a very limited solution application. A more
reasonable assumption would be:
[C ] = [C ] ± ε [C ]
This however does not easily yield the same conclusion. This proof needs to be reevaluated in
the light of these concerns.
- 51 -
6.2 Theory for other Normalization Techniques
The free decay time domain displacement, velocity and acceleration responses for a general
nonproportionally damped system having complex natural frequencies and modal vectors are
given by the equations
{ }
x(t ) Nm*1 = [ψ r ]Nm*2 N e λr t
*
{
x(t ) Nm*1 = [ψ r ]Nm*2 N λ r e λr t
{
**
}
x(t ) Nm*1 = [ψ r ]Nm*2 N λ r e λr t
2
(6.11)
2 N *1
(6.12)
2 N *1
}
(6.13)
2 N *1
The time domain equations of motion can be represented in a state space solution as
 * 
0
[I]   x(t )
 x(t ) 
 **  = 
 * 
-1
-1
 x(t ) - [M] [K] - [M] [C]  x(t )
(6.14)
Or
*
{ y (t )} = [ E ]{ y (t )}
(6.15)
0
[I ]


Where the matrix [ E ] = 

−1
−1
− [ M ] [ K ] − [ M ] [C ]
*
{ y (t )} and { y (t )} are the state space velocity and displacement responses
Using Equations (6.11), (6.12), (6.13) and (6.14) the Equation (6.15) is repeated 2*Nm time
instants such that
.
*
*
[ y (t ),......., y (t 2 Nm )] = [ E ][ y (t1 ),........, y (t 2 Nm )]
Or
- 52 -
(6.16)
*
[Y ] = [ E ][Y ]
(6.17)
*
Here all matrices [E], [Y ] and [Y] are of size 2Nm *2Nm and [E] is obtained as
*
[ E ] = [Y ][Y ] −1
(6.18)
From Equation (6.18) and the definition of [E] the - [M]-1[K] portion of [E] is retained.
Using the - [M]-1[K] matrix an eigenvalue problem is set up as
[[M ]
−1
]
[ K ] − Ω 2 [ I ] * {φ} = {0}
(6.19)
An eigenvalue decomposition of this − [ M ]−1 [ K ] would give the undamped natural frequencies
(Ω) r and real modal vector matrix [φ ] of the corresponding undamped system.
This is theoretically possible but in actuality it is seen that the physical degrees of freedom Nm
are usually much greater than the number of modes N resulting in lack of linear independence in
the column space of [Y ] . This results in the state space displacement matrix [Y ] (6.17) being
singular and hence its inversion (6.18) is not possible.
Ibrahim[5, 6] suggested that by the addition of small amounts of noise or if noise is already present
in the experimental data obtained, the inversion of this state space displacement matrix [Y ] is
possible.
To make the inversion of the state space displacement matrix [Y ] possible, the methods derived
reduce the size of the problem from physical coordinates Nm vector space to N modal vector
space as done in the Time Domain Subspace Iteration Technique[7]. Two of the normalization
techniques presented in the following sections differ in the method and type of coordinate
transformation matrix (size N*N) generated used to reduce the system from physical space to a
- 53 -
modal subspace. In this subspace, the inversion of the state displacement matrix is possible as it
has a full rank of 2N.
6.3 Information Matrix Subspace Technique (IMST)
In this approach the Fisher Information Matrix[8] is used for coordinate transformation from the
Nm physical vector space to the N modal vector space. This Fisher Information Matrix is an
indicator of the information content of the system.
6.3.1 Theory
As per Udwadia and Garba[9] the output from the sensors is given by
x (t ) = V ( p ) + N
(6.20)
Where
V ( p ) : The process measurement.
N : Stationary Gaussian white noise with variance as γ 02
γ 02 : Variance of the stationary Gaussian white noise N
For spatial independence of the modal vectors to be true, at any instant of time the sensor output
equation is given by (with absence of noise)
x(t ) = [φ ] * p (t )
(6.21)
Where
x(t ) : Output from the sensors.
[φ ] : Real modal vector matrix of size Nm*N.
p : Vector of target modal coordinates.
- 54 -
Thus taking the noise into consideration
x(t ) = [φ ]* p + N
(6.22)
To get the best estimate of the output from the sensors (i.e. best estimates of the modal vectors),
an efficient unbiased estimator needs to be selected which will minimize the covariance matrix
of the estimate errors. For such an unbiased estimator the covariance matrix of the estimate error
is given by
 ∂V
P = E [( p − p)( p − p) ] = 
 ∂p
^
^
T
T
 2 −1  ∂V
 [γ o ] 

 ∂p



(6.23)
^
Where E is the expected value and p is the target state to be achieved
In this formulation, it will be assumed that the sensors measure displacement, but similar results
are obtained for velocity and acceleration measurements. Therefore, given V ( p) = [φ ] * p , the
covariance matrix is given by
P = [ [φ ] (γ 02 ) −1 [φ ]] = J −1
T
(6.24)
In which J is the Fisher Information Matrix. As seen from above maximizing J would lead to
^
minimization of the covariance matrix and the best state estimate q .
J
φ ]T [φ ]
[
=
=
2
0
(γ )
[T ]
(γ 0 )
(6.25)
To simplify the analysis it is assumed that the noise is uncorrelated and possesses identical
statistical properties of each sensor ( (γ 0 ) can be ignored). Hence [T ] could now be referred to as
the Fisher information matrix. Thus the Fisher Information Matrix for the system is given by:
- 55 -
[T ] = [φ ]T * [φ ]
(6.26)
Where [φ ] is the Nm*N modal vector matrix and [T ] is N*N Fisher Information Matrix. In order
to minimize P , a suitable norm of [T ] must be maximized. The trace norm which is the most
useful and physically meaningful matrix norm is maximized. It is noted that the determinant of
Fisher Information Matrix for the best linear estimate is largest for all linear unbiased estimators.
In terms of contribution of each degree of freedom [T ] can be expressed as:
S
S
[T ] = ∑ φ S φ i S = ∑ Ti
iT
i =1
i =1
i
Where φ S i is the ith row of the modal partition corresponding to the ith degree of freedom or
sensor location. Thus the Fisher Information Matrix gains or loses information as measurement
degrees of freedom are added or subtracted. It is shown in subsequent sections that the
determinant of the Fisher Information Matrix [T ] gives an indication of the linear independence
of the modal vectors. A zero determinant indicates that the modal vectors are not linearly
independent.
6.3.2 Normalization Procedure
The real modal vector matrix obtained in Section 6.1 is used as a first estimate for this technique.
Thus the complex modal vector matrix [ψ ] is scaled by keeping the largest modal vector
coefficient (in terms of magnitude) for each modal vector as 1.0. The real valued modal vector
^
matrix [φ ] is obtained by taking the real part of these scaled complex modal vectors
^
This real valued modal vector matrix [φ ] is used to obtain the Fisher Information Matrix
[T ] (6.26). This N*N matrix [T ] is considered as the modal vector matrix in modal subspace or
- 56 -
coordinate transformation matrix. The use of [T ] as the coordinate transformation matrix as well
as the modal vector matrix in the modal subspace can be justified by the fact that [T ] is the
product of transpose of the linearly independent physical space modal vector matrix and the
physical space modal vector matrix itself. This ensures that [T ] is positive definite and
symmetric and also ensures the linear independence in the column space of [T ] giving it a rank
of N. It. The use of [T ] as the coordinate transformation matrix converting the problem from the
physical Nm vector space to the modal N vector space is derived as follows.
Let p (t ) be the time domain free displacement vector in the modal coordinates and x(t ) the time
domain free displacement vector in the physical coordinates.
Then (ignoring the complex conjugate roots)
^
{p(t )}N *1 = [φ ]TN *Nm * {x(t )}Nm*1
^
(6.27)
^
{p(t )}N *1 = [φ ]TN *Nm * [φ ] Nm*N {e λ t }N *1.
(6.28)
{ p(t ) }N *1 = [T ] N *N {e λ t }. N *1
(6.29)
r
r
Where [T ] is the Fisher Information Matrix of the system
Or assuming complex conjugate pairs of λ r
{p(t )} = 2 * real {[T ]{e λ t }}
r
r = 1 to N
(6.30)
Thus, using this matrix, the free decay displacement, velocity and acceleration response of the
structure in the modal coordinates can be calculated as follows
{p(t )} = 2 * real {[T ]{e λ t }}
r
*
{ {
{ p(t )} = 2 * real [T ] λ * e λr t
(6.31)
}}
(6.32)
- 57 -
**
{ {
{ p(t )} = 2 * real [T ] λ2 * e λr t
}}
(6.33)
Using (6.31), (6.32), (6. 33) and a state space approach in the modal coordinates, the matrix
[ E N ] in the modal coordinates is obtained.
*
{q (t )} = [ E N ]{q (t )}
(6.34)
Equation (6.34) can be repeated for 2*N time instants to get
*
[Q] = [ E N ][Q]
*
[ E N ] = [Q][Q] −1
Inversion of state displacement matrix (in modal coordinates) is now possible as it has a full rank
2N.
Where:
[0]
[I ]


[E N ] = 

−1
−1
− [ M N ] [ K N ] − [ M N ] [C N ]
Using (size (N*N) an eigenvalue solution can be obtained to get the undamped natural
[
]
Using the − [ M N ] −1 [ K N ] N *N matrix an eigenvalue solution can be obtained to get the undamped
natural frequencies (Ω r ) and the real modal vectors [φ ] in the selected vector subspace. The
__
subspace modal vectors are then extracted back into the physical coordinate space using the
following technique
__
[φ1 ]= [φ ] * [φ ]
(6.35)
__
- 58 -
Where:
__
^
[φ ] = [φ ] * [T ] −1
[φ1 ] is the required real modal vector matrix of size Nm *N in the required physical coordinates.
6.4 Effective Independence Subspace Technique (EIST)
This technique uses the Effective Independence Technique [11, 12] developed for on orbit sensor
placement in large space structures. This technique ranks the output measurement degrees of
freedom (Nm) according to their contribution to the linear independence of the N target modes on
a 0 to 1 scale. In an iterative manner, locations that do not contribute significantly to the linear
independence of the target mode set can be removed, reducing the number of sensors required
for on orbit modal identification and correlation. The final sensor configuration tends to
maximize the trace and determinant and minimize the condition number of the Fisher
Information Matrix. This feature is used to obtain the effective independence values (ranging
between 0 and 1) of the Nm degrees of freedom. These Nm degrees of freedom are ranked
according to their effective independence values (largest to smallest) and only the first N degrees
of freedom are retained while the rest of the degrees of freedom are discarded to reduce the size
of the modal vector matrix from Nm * N to N*N. In this method the reduction in the vector
subspace is obtained by removing the less significant degrees of freedom from the modal vector
matrix and then proceeding as though the system was an N degree of freedom system.
- 59 -
6.4.1 Theory
The Effective Independence Technique uses the Fisher Information Matrix[8, 9, 10] developed in
the previous section.
The Fisher Information Matrix is solved for an eigenvalue solution as
[[T ] − η * [ I ]]*τ = 0
(6.36)
Where
[T ] N *N = [φ ]TN *Nm * [φ ] Nm*N
Thus obtaining τ and η
As [T ] is obtained from the modal vector matrix [φ ] which is linearly independent, it is positive
definite and symmetric. Thus eigenvalues of [T ] , η are then real and positive and the eigen
vectors [τ ] , are orthonormal resulting in the relationships.
[τ ]T [T ][τ ] =η and [τ ]T * [τ ] = [ I ]
(6.37)
Because [τ ] are orthonormal vectors, they represent N orthogonal directions in N dimensional
space which is called absolute identification space.
Next
[G ] = [[φ ] * [τ ]]@[[φ ] * [τ ]]
(6.38)
Where @ represents term by term multiplication.
Now each row of [G ] contains the square of the components of the rows of [φ ] in terms of the
coordinate system defined by the columns of [τ ] , which span the absolute identification space.
Each column of [G ] sums to the corresponding eigenvalue of [T ] . Thus the ith term of a column
represents the contribution of the ith sensor location to the associated eigenvalue.
If [G ] is post multiplied by the inverse of the matrix of eigenvalues η such that
- 60 -
[ FE ] = [[φ ] * [τ ]]@[[φ ] * [τ ]]*η −1
(6.39)
Each direction within the absolute identification space will now have equal importance. The ith
term in the jth column of [ FE ] represents the fractional contribution of the ith sensor location to
the jth eigen value.
Thus summing up all the terms in each row of the matrix FE which is called as the Fractional
Eigen Value Distribution Matrix
k
k
k
j =1
j =1
j =1
{E D } = [∑ FE1 j ;∑ FE 2 j ;........∑ FEsj ;]T
(6.40)
Column vector {E D } referred to as Effective Independence Contribution Vector of the output
sensor set indicates the importance of each sensor location to the linear independence of the
modes present. It is hypothesized that the ith term within {E D } is the fractional contribution of
the ith sensor location to the linear independence of the modal vector set.
The physical significance of the column vector {E D } can be obtained by considering a Ndimensional ellipsoid surface defined by
{x}T η −1 {x} = 1
This can be referred to as the Absolute Identification Ellipsoid.
The principal directions of this ellipsoid are given by the eigen vectors [τ ] and the length of the
ith principal axis isη i1 / 2 . The magnitude of the ith eigen value give a measure of how identifiable
or how independent the modal vectors are if they are viewed along the corresponding principal
axis. If one of the eigenvalues becomes zero as sensor locations are deleted from the candidate
set, the ellipsoid collapses, its volume vanishes and the modal vectors [φ ] are no longer linearly
independent. The volume of the Absolute Identification Ellipsoid is proportional to the square
- 61 -
root of the determinant of the Fisher Information Matrix [T ] . Hence the determinant of the Fisher
Information Matrix can be considered as formally analogous to the definition of information
because as sensor locations are added or removed from the candidate set, information is added or
subtracted from the Fisher Information Matrix and the determinant changes, if it becomes zero
then this is an indication that the modal vectors are no longer linearly independent. Therefore as
sensor locations are removed from the candidate set it is desirable to maintain the determinant of
[T ] and thus the volume of the absolute identification ellipsoid
The effective independence contribution vector {E D } can be alternatively formulated as the
diagonal matrix
−1
T
T
−1
T
E  = [φ ] * [τ ] *η * [τ ] [φ ] = [φ ][T ] [φ ]
Using (6.37) E  can be written as
T
−1
T
E  = [φ ][[φ ] [φ ]] [φ ]
The matrix in the above equation can be identified as an orthogonal projector onto the column
space of the modal vector matrix [φ ] with rank equal to the number of target modes. The
projector E  is also an idempotent matrix which means that E  = E  . A well known
2
characteristic of idempotent matrices is that their trace is equal to their rank. Therefore the terms
on the diagonal of E  and also within {E D } represent the contributions of the corresponding
sensor locations to the rank of [φ ] or the linear independence of its columns as hypothesized.
It is further shown
[11]
that the value of terms in {E D } lies only between 0 and 1 with 0
indicating that the sensor has no contribution to the linear independence of the modes and a value
of 1 indicates that the particular sensor location is essential for the modes to be obtained.
- 62 -
Thus the sensor outputs are ranked on a 0 to 1 scale depending on their contribution to the linear
independence to the modes. The sensor locations with smaller effective independence values are
removed in an iterative manner to get the desired candidate sensor set. The accuracy of the
estimation or goodness of the sensor can be monitored during the iteration process by tracking
firstly the trace of the Fisher Information Matrix [T ] , secondly the determinant of [T ] which is
an indication of the amount of information contained in the measurements and lastly the
condition number of [T ] which yields a measure of the estimation’s robustness to modeling
errors in the measurement matrix [φ ] .
6.4.2 Normalization Technique
As in the previous case the complex modal vector matrix [ψ ] is scaled using the technique
described in Section 6.1 by keeping the largest vector component (in terms of magnitude) for
^
each vector as 1. The real valued modal vector matrix [φ ] is obtained by taking the real part of
these scaled complex modal vectors.
^
The effective independence technique is applied on the real modal vector matrix [φ ] to rank the
Nm output sensor locations. Then the N sensors having largest effective independence values are
retained and the remaining output measurement degrees of freedom discarded in an iterative
^
manner so that the real modal vector matrix [φ ] reduces from size Nm * N to the reduced real
modal vector matrix [φ Mod ] of size N*N. This reduced real modal vector matrix is used as the
coordinate transformation matrix.
- 63 -
This reduced real modal vector matrix is again used to calculate the state space companion
matrix as
{p(t )} = 2 * real {[φ Mod ]{e λ t }}
r
*
(6.41)
{
}
{
}
{ p (t )} = 2 * real [φ Mod ]{λ * e λr t }
**
(6.42)
{ p (t )} = 2 * real [φ Mod ]{λ2 * e λr t }
(6.43)
Again a state space approach in the modal coordinates can be adopted to get the matrix [ E N ] in
the modal coordinates
*
{q (t )} = [ E N ]{q (t )}
(6.44)
Equation (6.44) can be repeated for 2*N time instants to get
*
[Q] = [ E N ][Q]
*
[ E N ] = [Q][Q] −1
Inversion of state displacement matrix (in modal coordinates) is now possible as it has a full rank
2N. Where
[0]
[I ]


[E N ] = 

−1
−1
− [ M N ] [ K N ] − [ M N ] [C N ]
[
]
Using the − [ M N ] −1 [ K N ] N *N matrix an eigenvalue solution can be set up to get the undamped
natural frequencies (Ω r ) and the real modal vectors [φ ] in the selected vector subspace. The
__
subspace modal vectors are then extracted back into the physical coordinate space using the
following technique
- 64 -
__
[φ1 ]= [φ ] * [φ ]
(6.45)
__
Where
__
^
[φ ] = [φ ] * [φ Mod ] −1
[φ1 ] is the required real modal vector matrix of size Nm *N in the required physical coordinates.
6.5 Conclusion
The theory of the three real normalization techniques developed has been explained in detail in
the above sections. These methods utilize complex modal parameters as data and give real modal
estimates (undamped natural frequencies and real mode shapes) as results. The validation of
these developed techniques is done in the subsequent chapters where in complex modal
parameters obtained by experimental modal analysis on a laminated glass plate are real
normalized using the above techniques.
- 65 -
7.0 Experimental Analysis & Validation
The validation of the real normalization techniques was done by conducting an experimental
modal analysis on a test specimen resulting in a set of complex modal parameters. The real
normalization techniques developed in Section 6.0 were used to normalize the complex modal
parameters and obtain real modal parameter estimates. The validity of the results obtained was
checked in a number of ways[13].
First, the Modal Assurance Criterion (MAC) values between the complex modal vectors and
those between the real modal vectors obtained using the techniques were checked to verify that
the linear independence of the modal vectors was maintained. The Information Matrix Subspace
Technique (IMST) and the Effective Independence Subspace Technique (EIST) gave the
undamped natural frequencies along with the real modal vectors by solving a normal modes
eigenvalue problem. These natural frequencies were compared with the complex modal
frequencies obtained through experimental modal analysis to check for closeness. It is also
necessary that the underlying motion characteristics of the complex modal vectors be preserved
as accurately as possible during the normalization process. Hence, if in a complex modal vector a
measurement degree of freedom has a large displacement when compared to another
measurement degree of freedom then after normalization, this should be maintained. The
preservation of underlying motion characteristics was checked by visual verification. The
complex modal vectors were plotted along with the real normalized modal vectors and it was
visually verified that the plots were similar in shape.
- 66 -
7.1 Experimental Modal Analysis
Experimental modal analysis was performed on a laminated rectangular glass plate at QRDC,
Inc. Vibration and Acoustic Laboratory. A test fixture was fabricated which was capable of
clamping the exterior perimeter of the glass plate, inducing a fixed boundary condition. Figure
7.1 and Figure 7.2 show the solid model and experimental implementation of the test fixture
respectively, without the plate in place. In total, there were 64 threaded rods composed of ¼-28
B7 Alloy steel. The fixture constrained 1” of the perimeter of the glass plate. Glazing tape was
used for two purposes, (1) secure the plate in the test fixture, and (2) provide a uniform resistive
boundary to prevent breaking of glass. Exposed surface of the glass plate is 18” (H) x 30” (W).
Bolt torque was limited to 12 ft-lbf to prevent damage to the test fixture or glass plate and
applied with a criss-crossing tightening pattern. The procedure is similar to tightening lug nuts
on a vehicle, increasing repeatability of the boundary conditions.
Impact testing was conducted on the laminated glass plate for the extraction of modal data with
the Star Modal analysis package. Complex natural frequencies and mode shapes were extracted.
The impact tests were conducted on a 20” (H) x 32” (W) x 3/16” (T) glass plate under fixed
boundary conditions.
Calibrations of the accelerometers were recorded prior to making the vibration measurements to
verify sensitivity. Output of the calibrator is 10 m/s2 at 159.3 Hz. The tables below contain
information on the test equipment and analyzer configurations. Test equipment for vibration
measurements is provided in Table 7.1.
Results of calibration are shown in Table 7.2.
Configuration settings of the CV395 analyzer used to collect the modal analysis data are shown
in Table 7.3 for Acquisition Setup and Table 7.4 for Analysis Setup. Measurements were limited
to 2000 Hz.
- 67 -
Full-system excitation was measured at one location while the response point was changed for
each measurement. Only excitation normal to the glass panes (z direction) was applied and used
for the modal extraction. The impact excitation point and measurement grid was laid out as
described in Figure 7.3. Experimental grid layout of the glass plate can be seen in Figure 7.4.
The exposed surface of the glass plate after installation is 18” (H) x 30” (W) and used for
creation of the measurement grid. Measurement locations were determined using 1” spacing
along both dimensions, totaling 589 points in the Star Modal model. There were 19 rows and 31
columns in the measurement grid totaling 589 measurement degrees of freedom at which
measurements were made. The impact location was held fixed at point 65 of the exterior surface.
Accelerometers were calibrated prior to measurement.
- 68 -
Fig 7.1 Solid Model of Test Fixture
Fig 7.2 Experimental Test Fixture
- 69 -
Equipment
Manufacturer
Type
Four-Channel Data
Analyzer
Uniaxial Accelerometer
Cognitive Vision
CV395
PCB Piezotronics
353B16
Bruel & Kjaer
8202
71503
71505
71506
1122758
PCB Piezotronics
208B02
13574
Medium Impact
Hammer
Force Transducer
Serial
Number
-
Table 7.1 Equipment for Vibration Measurements
71503
Sensitivity
[mV/EU]
9.14
Output
[m/s2]
10.03
08:28
71505
9.91
9.98
08:35
71506
9.98
9.97
Condition
Time
Serial #
Prior to
baseline
Prior to
baseline
Prior to
baseline
08:25
Table 7.2 Calibration of Accelerometers
- 70 -
Channel
1
2
3
FS rms
200
mV
200
mV
200
mV
MV/EU
9.14
EU Name
G
dB Ref
0
V Ref
0
9.91
G
0
0
9.98
G
0
0
Trigger
Averages
Free
Run
Linear
20
seconds
Table 7.3 Vibration Acquisition Setup for CV395 Analyzer
Narrow
Band
Block Size
Window
4096 Samples, 1600
Lines
Hanning
Table 7.4 Vibration Analysis Setup for CV395 Analyzer
- 71 -
Fig 7.3 Modal Extraction Grid for Glass Plate
Fig 7.4 Experimental glass plate vibration measurement points grid
- 72 -
7.2 Test Analysis and Results
As a basic component, the laminated glass plate consists of two equal plates of glass with a
polyvinyl butyral (PVB) with a boundary condition that is clamped on the perimeter. Thus, the
classic plate modes occur within the glass plate structure. The extraction of natural frequencies
and mode shapes for the glass plate was successful.
The first ten extracted modes were used as the data set of complex modal parameters. These
mode shapes along with their measure of complexity (real vs. imaginary plane plot) and their
least squares and weighted least squares estimates are given in the table 7.5 below
Extracted
Modal Extracted Complex Modal Vector
Frequency
Complexity of the
Modal
Vector (Real vs. Imaginary
Plane Plot)
Mode 1: 67.29 Hz
Mode 2 : 180.00 Hz
Mode 3 : 273.20 Hz
- 73 -
Mode 4 : 355.58
Mode 5 : 433.86Hz
Mode 6 : 574.80
Mode 7 : 603.10
Mode 8 : 703.46
Mode 9 : 751.18
- 74 -
Mode 10 : 832.36
Table 7.5 The extracted complex modal parameters for the glass plate
7.3 The Real Normalization Process
The complex modal frequencies and modal vectors obtained from the experimental modal
analysis was the data provided to the three real normalization techniques developed in Section
6.0 Each of the three techniques gave the real normalized modal vectors matrix. The Information
Matrix Subspace Technique and the Effective Independence Technique also gave the undamped
natural frequencies during the normalization process.
7.4 Results Validation
The Validation of the results obtained from the three normalization processes was done by the
following three criteria.
- 75 -
7.4.1 Consistency of the Modal Assurance Criterion (MAC) Values
The Modal Assurance Criterion (MAC) values between the complex modal vectors were
calculated and are presented in Table 7.6. The MAC values between the real normalized modal
vectors obtained by the Rescaling of Complex Modal Vectors Technique, the Information Matrix
Subspace Technique and the Effective Independence Subspace Technique are presented in
Tables 7.7, 7.8 and 7.9 respectively. The cross MAC values between the complex modal vectors
and the real modal vector estimates obtained by the three different techniques are presented in
Tables 7.10, 7.11 and 7.12 respectively.
Mode
1
2
3
4
5
6
7
8
9
10
1
1.0000
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
2
0.00
1.0000
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
3
0.00
0.00
1.0000
0.00
0.00
0.00
0.00
0.00
0.00
0.00
4
0.00
0.00
0.00
1.0000
0.00
0.00
0.00
0.00
0.00
0.00
5
0.00
0.00
0.00
0.00
1.0000
0.00
0.00
0.00
0.00
0.00
6
0.00
0.00
0.00
0.00
0.00
1.0000
0.3044
0.00
0.00
0.00
7
0.00
0.00
0.00
0.00
0.00
0.3044
1.0000
0.00
0.00
0.00
8
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.0000
0.0976
0.00
9
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.0976
1.0000
0.00
10
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.0000
Table 7.6 MAC Values between the complex modal vectors obtained from experimental
modal analysis
- 76 -
Mode
1
2
3
4
5
6
7
8
9
10
1
1.0000
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
2
0.00
1.0000
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
3
0.00
0.00
1.0000
0.00
0.00
0.00
0.00
0.00
0.00
0.00
4
0.00
0.00
0.00
1.0000
0.00
0.00
0.00
0.00
0.00
0.00
5
0.00
0.00
0.00
0.00
1.0000
0.00
0.00
0.00
0.00
0.00
6
0.00
0.00
0.00
0.00
0.00
1.0000
0.3592
0.00
0.00
0.0122
7
0.00
0.00
0.00
0.00
0.00
0.3592
1.0000
0.00
0.00
0.00
8
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.0000
0.0921
0.00
9
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.0921
1.0000
0.00
10
0.00
0.00
0.00
0.00
0.00
0.0122
0.00
0.00
0.00
1.0000
Table 7.7 MAC Values between real modal vectors obtained by Rescaling of
Complex Modal Vectors Technique (RCMVT)
Mode
1
2
3
4
5
6
7
8
9
10
1
1.0000
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
2
0.00
1.0000
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
3
0.00
0.00
1.0000
0.00
0.00
0.00
0.00
0.00
0.00
0.00
4
0.00
0.00
0.00
1.0000
0.00
0.00
0.00
0.00
0.00
0.00
5
0.00
0.00
0.00
0.00
1.0000
0.00
0.00
0.00
0.00
0.00
6
0.00
0.00
0.00
0.00
0.00
1.0000
0.3592
0.00
0.00
0.0122
7
0.00
0.00
0.00
0.00
0.00
0.3592
1.0000
0.00
0.00
0.00
8
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.0000
0.0921
0.00
9
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.0921
1.0000
0.00
10
0.00
0.00
0.00
0.00
0.00
0.0122
0.00
0.00
0.00
1.0000
Table 7.8 MAC Values between real modal vectors obtained by Information
Matrix Subspace Technique (IMST)
- 77 -
Mode
1
2
3
4
5
6
7
8
9
10
1
1.0000
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
2
0.00
1.0000
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
3
0.00
0.00
1.0000
0.00
0.00
0.00
0.00
0.00
0.00
0.00
4
0.00
0.00
0.00
1.0000
0.00
0.00
0.00
0.00
0.00
0.00
5
0.00
0.00
0.00
0.00
1.0000
0.00
0.00
0.00
0.00
0.00
6
0.00
0.00
0.00
0.00
0.00
1.0000
0.3592
0.00
0.00
0.0122
7
0
0.00
0.00
0.00
0.00
0.3592
1.0000
0.00
0.00
0.00
8
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.0000
0.0921
0.00
9
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.0921
1.0000
0.00
10
0.00
0.00
0.00
0.00
0.00
0.0122
0.00
0.00
0.00
1.0000
Table 7.9 MAC Values between real modal vectors obtained by Effective
Independence Subspace Technique (EIST)
Mode
1
2
3
4
5
6
7
8
9
10
1
0.9631
0
0
0
0
0
0
0
0
0
0
0.9750
0
0.0158
0
0
0
0
0
0
0
0
0.9798
0
0
0
0
0
0
0
0
0.0158
0
0.9389
0
0
0
0
0
0
0
0
0
0
0.9224
0
0
0.0217
0
0
0
0
0
0
0
0.8765
0.3016
0
0
0.0158
0
0
0
0
0
0.3016
0.8397
0
0
0
0
0
0
0
0.0217
0
0
0.9467
0.0901
0
0
0
0
0
0
0
0
0.0901
0.9786
0
0
0
0
0
0
0.0158
0
0
0
0.8289
2
3
4
5
6
7
8
9
10
Table 7.10 Cross MAC Values between complex modal vectors and real modal vectors
obtained by Rescaling of Complex Modal Vectors Technique (RCMVT)
- 78 -
Mode
1
2
3
4
5
6
7
8
9
10
1
0.9631
0
0
0
0
0
0
0
0
0
0
0.9750
0
0.0138
0
0
0
0
0
0
0
0
0.9798
0
0
0
0
0
0
0
0
0.0138
0
0.9389
0
0
0
0
0
0
0
0
0
0
0.9224
0
0
0.0135
0
0
0
0
0
0
0
0.8765
0.2149
0
0
0.0330
0
0
0
0
0
0.2149
0.8397
0.0107
0
0
0
0
0
0
0.0135
0
0.0107
0.9467
0.07996
0
0
0
0
0
0
0
0
0.0799
0.9786
0.0167
0
0
0
0
0
0.0330
0
0
0.01676
0.8289
2
3
4
5
6
7
8
9
10
Table 7.11 Cross MAC Values between complex modal vectors and real modal vectors
obtained by Information Matrix Subspace Technique (IMST)
Mode
1
2
3
4
5
6
7
8
9
10
1
0.9631
0
0
0
0
0
0
0
0
0
0
0.9750
0
0
0
0
0
0
0
0
0
0
0.9798
0
0
0
0
0
0
0
0
0
0
0.9389
0
0
0
0
0
0
0
0
0
0
0.9224
0
0
0.0142
0
0
0
0
0
0
0
0.8765
0.2812
0
0
0.0241
0
0
0
0
0
0.2812
0.8397
0
0
0
0
0
0
0
0.0142
0
0
0.9467
0.0973
0
0
0
0
0
0
0
0
0.0973
0.9786
0.0101
0
0
0
0
0
0.0241
0
0
0.0101
0.8289
2
3
4
5
6
7
8
9
10
Table 7.12 Cross MAC Values between complex modal vectors and real modal vectors
obtained by Effective Independence Subspace Technique (EIST)
- 79 -
From Tables 7.6, 7.7, 7.8 and 7.9 it is seen that the MAC values for the different real as well as
complex modal vector estimates are very close. The cross MAC values between the complex
modal vectors and the different real modal vector estimates given in Tables 7.10, 7.11 and 7.12
also similar to the MAC values between the real modal vectors. Thus the real normalization
techniques developed preserve the linear independence of the complex modal vectors
- 80 -
7.4.2 Comparison of Natural Frequencies
The Information Matrix Subspace Technique and the Effective Independence Subspace
Technique give the associated undamped natural frequencies of the real normal modes while
solving a real normal modes eigenvalue problem as shown in Section 6.0. These undamped
natural frequencies are compared with the complex modal frequencies obtained experimentally
to check for closeness. A close match between the undamped eigen values and their
corresponding complex modal frequencies would indicate that the real normal modes obtained
from these two normalization techniques are essentially the same as the complex modes
Mode Number
Extracted Complex Modal
Undamped Frequency
Undamped Frequency
Frequency (Star Modal)
(Hz) – IMST
(Hz) – EIST
(Hz)
1
67.2967
67.2967
67.2968
2
180.009
180.009
180.0089
3
273.2075
273.2075
273.2075
4
355.5801
355.5801
355.5802
5
433.8691
433.8691
433.8691
6
574.802
574.802
574.802
7
603.1085
603.1085
603.1085
8
703.4672
703.4672
703.4672
9
751.1799
751.1799
751.1799
10
832.3683
832.3683
832.3683
Table 7.13 Comparison of Undamped and Complex Natural Frequencies
- 81 -
From Table 7.13 it can be concluded that the complex modal frequencies and the undamped
natural frequencies obtained from the two techniques are very close and represent the same
modes
7.4.3 Preservation of underlying motion characteristics
The real normalization techniques developed should maintain the underlying motion
characteristics of the mode shapes. The simplest way to verify this is to view the plot of the
normalized modal parameters. The displacement plots of the complex modal vectors and the real
normalized modal vectors obtained from the three methods are shown in the Table 7.11 below
Mode
Experimental
(Hz)
(StarModal)
Modal
Vector
Modal Vector- RCMVT
Modal VectorIMST
67.29
180.00
273.20
- 82 -
Modal Vector- EIST
355.58
433.861
574.80
603.10
703.46
751.18
832.37
Table 7.14 Plots of Complex and Real Normalized Modal Vectors
- 83 -
The plots of Table 7.14 indicate that the complex modal vectors and their real normalized
estimates obtained using the three normalization techniques have very similar shapes. Thus the
normalization techniques successfully preserve the underlying motion characteristics of the
complex modal vectors.
- 84 -
8.0 Conclusions
The experimental modal analysis conducted on a rectangular laminated glass plate gave a set of
complex modal parameters. The real normalization of these complex modal parameters was
estimated using the three normalization techniques developed during the course of this research.
From results and validations presented, it can be concluded that the techniques preserve the
linear independence of the complex modal vectors, the techniques also give the undamped
natural frequencies which are extremely close to the complex modal frequencies and the
associated real modal vectors can be considered as reasonable estimates of the complex modal
frequencies. The plots of the modal vectors verify that the real modal vector estimates have the
same profiles as the complex modal vectors. The techniques developed have previously been
applied to complex modal parameters obtained through a simulated modal analysis performed on
a rotor[13] and the results were presented as part of a conference proceeding. The real normal
modal parameters obtained on that data set also verify the normalization techniques developed.
Thus the Real Normalization Techniques developed, successfully provided real modal parameter
estimates for complex modal parameters for two test cases. These real modal parameter estimates
can now be used for FEM and Modal Model correlation and updating of these two test cases. A
rigorous validation of the developed techniques needs to be done using complex modal
parameters obtained under diverse testing conditions. Such a comprehensive validation would
further verify these techniques as well as demonstrate any potential shortcomings.
- 85 -
8.1 Future Work
The methodologies developed could be further enhanced by introducing techniques to calculate
the modal scaling based on these real normalized modal parameters. A suitable damping model
(e.g. proportional damping) could be incorporated to get the modal scaling. Such an
enhancement could make it possible to re-synthesize the frequency response functions which can
then be compared to the frequency response functions of the original complex system. A
comparison of the frequency response functions (especially those not used in the parameter
estimation process) could serve as yet another validation tool for the algorithms developed. A
potential shortcoming of these methods is that any errors in the data acquisition as well as the
parameter estimation process are carried beyond the real normalization process. Thus no attempt
is made to check for the errors present in the complex modal parameters used in the
normalization process. Thus if the techniques developed could in the future give frequency
response functions for the normalized system it would be easier to identify if this is the expected
data for the idealized system.
The developed tools can be used during the design phase of any structural system. A robust
normalization technique would enable quick and accurate updating of conservative FEM and
modal models. The primary focus of research in the industry is to develop technologies which
shorten the design cycle and decrease the cost by decreasing the number of prototypes required
in the design process. The developed algorithms fit in well within this research objective.
Further evaluation of the Rescaling of Complex Modal Vectors techniques (RCMVT) regarding
the complex modal vector truncation approach suggested by Fillod[7, 10] needs to be performed.
The current presentation appears to greatly limit the set of potential applications cases by the
initial assumption concerning the form of the damping matrix.
- 86 -
During the course of this research a complete stand alone Matlab GUI based software was
created. This software requires data in form of complex modal frequency and complex modal
vector matrices. The software can visually indicate the complexity of the vectors (real vs.
imaginary plots) as well calculate the Mean Phase Deviation and Mean Phase Collinearity of the
complex modal vectors. It can also provide the Least Squares and Weighted Least Squares
estimates for the complex modal vectors.
It can provide the real modal parameter estimates for any of the developed techniques and can
plot the real modal parameter estimates and calculate the MAC values. The code of this software
being very long is not presented in this thesis. Screen shots of the developed software and a brief
explanation of its capabilities is provided in the Appendix A1.
- 87 -
9.0 References
[1]:
Allemang, R.J., Analytical and Experimental Modal Analysis, UC-SDRL-CN-20-263662, February 1999
[2]:
K. U. L., Leuven, Modal Analysis Theory and Testing
[3]:
Allemang, R.J., Experimental Modal Analysis, UC-SDRL-CN-20-263-663/664, March
1999
[4]:
Deblauwe, F., Allemang, R.J., A Possible Origin of Complex Modal Vectors, Proceedings
of 11th International Seminar on Modal Analysis, K. U. L., Leuven, September 1986
[5]
Ibrahim, S. R., Computation of Normal Modes from Identified Complex Modes, AIAA
Journal, Vol. 21, No 3 (March 1983), pp 446-451.
[6]
Ibrahim, S. R., Dynamic Modeling of Structures from Measured Complex Modes, AIAA
Journal, Vol. 21, No 6 (1985), pp 898-901.
[7]
Wei, M.L., Allemang R. J., Brown D. L., A Time Domain Subspace Iteration Method for
the Normalization of Measured Complex Modes, Proceedings of 12th International
Seminar on Modal Analysis, K. U. L., Leuven, September 1986, ppS2-3.
- 88 -
[8]
Goodwin, G. C., Payne, R. L., Dynamic System Identification: Experimental Design and
Data Analysis, Academic Press (1977).
[9]
Qureshi, Z. H., Ng, T. S. and Goodwin, G. C., Optimum Experimental Design for
Identification of Distributed Parameter Systems, International Journal of Control, Vol.
31, No 1 (1980), pp 21-29.
[10]
Fillod R., Contribution on Identification of Linear Mechanical Structures, Doctor of
Sciences Physics Dissertation, University of Besançon, France (1980).
[11]
Kammer, Daniel C., Sensor Placement for On-Orbit Modal Identification and
Correlation of Large Space Structures, Journal of Guidance, Control and Dynamics, Vol.
14, No 2 (1991), pp 251-259.
[12]
Kammer, Daniel C., Optimal Sensor Placement for Modal Identification Using System
Realization Methods, Journal of Guidance, Control and Dynamics, Vol. 19, No 3 (1995),
pp 729-731
[13]
Sinha, S., Allemang R.J., Techniques for Real Normalization of Complex Modal Vectors
for Updating and Correlation with FEM Models,, Proceedings of 2004 International
Seminar on Modal Analysis, K. U. L., Leuven, September 2004
- 89 -
Appendix
Matlab GUI Based software
A stand-alone Matlab GUI based software was developed as part of this research. This software
implements the techniques reviewed and developed during the course of this research. The
software can be operated by executing the SiddGUI.m file in Matlab. The software loads the
modal information in the *.uf file format (universal file format) or in a Matlab matrix format.
The software sorts and arranges the modal parameter information and populates a drop down
menu with the modal frequencies. On selecting a modal frequency from the drop down menu, the
corresponding modal vector can be plotted in a real vs. imaginary plane along with its least
squares and weighted least squares line estimate. The correlation coefficient of the least squares
line fit is also displayed. The mean phase deviation and modal phase collinearity of the vector is
calculated and displayed giving an idea of the complexity of the modal vector. The MAC values
between the complex modal vectors can also be calculated.
The normalization technique to be used can be selected from a drop down menu. The available
normalization techniques are real least squares line estimate, imaginary least squares line
estimate, information matrix subspace technique (IMST) and effective independence subspace
technique (EIST). On selecting a normalization technique, the complex modal parameters are
normalized accordingly and a new window opens up displaying the results. The MAC values
between the normalized modal vectors can now be viewed.
Screen snapshots of the developed software are displayed below.
- 90 -
Fig A.1 Matlab GUI software main screen
- 91 -
Fig A.2 Matlab GUI software post normalization screen
A
=
[
φ
]
T
r
*
[
φ
]
r
- 92 -