UNIVERSITY OF CINCINNATI
August 12
02
_____________
, 20 _____
Jeffrey Edward Hylok
I,______________________________________________,
hereby submit this as part of the requirements for the
degree of:
Masters of Science
________________________________________________
in:
Mechanical Engineering
________________________________________________
It is entitled:
Experimental Identification of Distributed Damping Matrices
________________________________________________
Using the Dynamic Stiffness Matrix
________________________________________________
________________________________________________
________________________________________________
Approved by:
David Brown
________________________
Jay Kim
________________________
Randal Allemang
________________________
Allyn Phyllips
________________________
________________________
Experimental Identification of
Distributed Damping Matrices Using
the Dynamic Stiffness Matrix
A thesis submitted to the
Division of Research and Advanced Studies
of the University of Cincinnati
in partial fulfillment
of the requirements for the degree of
MASTERS OF SCIENCE
In the Department of Mechanical, Nuclear, and Industrial
Engineering of the College of Engineering
2002
by
Jeffrey E. Hylok
B.S.M.E. Michigan Technological University 1996
Committee Chair: Dr. David L. Brown
Abstract
Modeling of distributed damping characteristics are increasingly important for validating
analytical structural models and correlating experimental and analytical data. In addition,
damping mechanisms are a measure of structural conditions since they are sensitive to small
structural changes. Thus, identification of localized changes in damping has uses in the
field of damage identification [7].
A new method to experimentally identify spatial damping, which was developed by Lee and
Kim [1,2], has been studied in this research. The method fits matrix coefficient polynomials
to the real and imaginary parts of the dynamic stiffness matrix (DSM, the inverse of the
frequency response matrix). The real DSM coefficients represent dynamically conservative
features of the system, namely mass and stiffness. The imaginary DSM coefficients model
the dynamic energy removal mechanisms of the structure, namely damping. The greatest
strength of the method is its simplicity and computational efficiency. Once data is collected
experimentally, only two steps are required: a FRF matrix inversion (frequency line by line),
and a polynomial fit for the real and imaginary components of the DSM. Since both DSM
components are fit independently, the damping properties of a system may be identified
without any prior knowledge about the system’s mass and stiffness. Likewise, the damping
calculation is not subject to errors in any pre-determined structural values.
The work is broken up into three segments. First, the DSM algorithm is derived and
features of the algorithm are discussed. Next, the DSM algorithm is applied to several
analytical systems and several qualitative and quantitative validation tools are presented.
i
Finally, the DSM algorithm and validation tools are applied to three experimental case
studies. Practical issues are discussed and benefits and limitations of the algorithm are
observed.
ii
iii
Preface
No man is an island, and so I am deeply grateful to the many, many individuals who
have helped me achieve my academic, research, and personal goals. First, thanks go to Dr.
Jay Kim for approaching me with this research opportunity. I am grateful for his help
crunching through unexpected results and keeping me focused. My advisor, Dr. David
Brown, can always be counted on for new ideas and wisdom from experience. Dr. Randy
Allemang always pushed me further, uncovered things I missed, and asked the question I
couldn’t answer. Dr. Allyn Phillips was always the source of that one bit of knowledge, or
the right tip that made my task go a whole lot better. Thank-you and good luck to all of the
students of the SDRL (in no particular order): Bruce Fouts, Dan Lazor, Doug Coombs, Tom
Terrell, Amit Shukla, Gokhan Ozgen, Bill Fladung, Susan DeClerq, Jason Grundey, John
Kantura, and others, who have all been there at some time to answer a question, work on
homework, or discuss the politics of football stadiums and the Reds.
Then there are my family and friends. My parents exposed me to many wonderful
things in this world, and have always encouraged me to fulfill my potential. There is not
enough room here for me to articulate how grateful I am to have them in my life. Jeanine
has been there for those key words of encouragement, advice, and a greater understanding of
myself than I give anyone credit for having. You were right, sis, I am moving to New
Mexico. My girlfriend, Karen Mitchell shows me the world though eyes of wonder and
optimism. I can always count on her to find the best of any situation. Thanks to all of my
UC mountaineering club and Michigan friends for always being more than willing to go on
an adventure and keep me sane. Finally, I am indebted to John Schultze for giving me the
opportunity at Los Alamos and for all of his support throughout this last year of school.
iv
Table of Content
Abstract…………………………………………………………………………i
Preface………………………………………………………………………....ii
Table of Contents………………………………………………………………v
Table of Figures……………………………………………………………....vii
Table of Tables……………………………………………………………….xii
Nomenclature…………………………………………………………….......xiv
Abbreviations…………………………………………………………….......xvi
1
2
3
OVERVIEW OF WORK............................................................................. 1
1.1
MOTIVATION ......................................................................................................................................1
1.2
CHALLENGES IN SPATIAL DAMPING IDENTIFICATION ........................................................................1
1.3
STATE OF THE ART FOR SPATIAL DAMPING IDENTIFICATION ...............................................................4
1.4
GOAL OF THE NEW SPATIAL DAMPING IDENTIFICATION ALGORITHM ..................................................6
DSM DAMPING IDENTIFICATION METHOD...................................... 7
2.1
IDENTIFICATION METHOD DEVELOPED BY LEE AND KIM ..................................................................7
2.2
PRACTICAL ISSUES IN IMPLEMENTING DSM ALGORITHM ................................................................10
DSM ALGORITHM VALIDITY STUDY USING ANALYTICAL
DATA. .............................................................................................................. 12
3.1
THE ANALYTICAL MODEL ...............................................................................................................13
3.2
FRF MATRIX INVERSION ..................................................................................................................16
3.3
APPLYING THE DSM ALGORITHM ....................................................................................................22
v
3.3.1
Full system and model degrees of freedom.................................................................................22
3.3.2
Effect of Spatial Truncation........................................................................................................35
3.4
4
ANALYTICAL MODEL CONCLUSIONS ...............................................................................................54
EXPERIMENTAL SPATIAL DAMPING MATRIX IDENTIFICATION
USING DSM..................................................................................................... 56
4.1
HARDWARE CONSIDERATIONS AND SENSOR CALIBRATION .............................................................57
4.2
SPATIAL SENSITIVITY .......................................................................................................................62
4.3
DISCUSSION OF EXPERIMENTAL EVALUATION DEFINITIONS ............................................................63
4.4
EXPERIMENTAL CASE STUDY 1: ELASTOMERIC SOFT-SPRING BOUNDARY CONDITIONS .................64
4.4.1
Modal Analysis of Both Tests Arrangements ..............................................................................66
4.4.2
DSM Algorithm Identification of Spatial Damping: Case 1 .......................................................68
4.4.3
Summary of the Free-Free Boundary Condition Experiment .....................................................92
4.5
EXPERIMENTAL CASE STUDY 2: CABLE DAMPING TEST ..................................................................94
4.5.1
Modal Analysis of Cable Damping Tests....................................................................................95
4.5.2
DSM Algorithm Identification of Spatial Damping: Case 2 .......................................................96
4.5.3
Summary of Cable Damping Experiment ................................................................................. 112
4.6
CASE 3: FIXED-FIXED BOUNDARY CONDITION TEST ..................................................................... 114
4.6.1
Modal Analysis of Clamped Beam Tests................................................................................... 116
4.6.2
Graphical Examination of the Clamped Beam Test data ......................................................... 116
4.6.3
Summary of Clamped Beam Experiment .................................................................................. 122
5
CONCLUSION AND FUTURE WORK ................................................ 124
6
REFERENCES ........................................................................................ 129
vi
TABLE OF FIGURES
Figure 1. N degree of freedom analytical model ................................................................... 13
Figure 2. Analytical model matrices...................................................................................... 14
Figure 3. Driving Point FRF, dof 1. ...................................................................................... 16
Figure 4. Driving Point DSM dof 1. ...................................................................................... 16
Figure 5 Superposition of Single Degree of Freedom Modes. .............................................. 18
Figure 6 CMIF plot for FRF.................................................................................................. 19
Figure 7 CMIF plot for DSM................................................................................................. 21
Figure 8 Matrices from 9 dof model. ..................................................................................... 23
Figure 9 Driving Point FRFs (entry 1,1 on left; entry 9,9 on right) .....................................24
Figure 10 CMIF Plot of 9 dof system. .................................................................................. 25
Figure 11 Driving Point DSM Magnitude/Phase (entry 1,1 on the left, 9,9 on the right).....26
Figure 12 Driving Point DSM Real/Imaginary Plot.............................................................. 27
Figure 13 CMIF of 9 degree of freedom DSM matrix. .......................................................... 29
Figure 14 DSM Plot of Synthesized ( | ) and Actual Driving Point Values, Real-Im (B11 to
B99, Left to Right) ................................................................................................................... 33
Figure 15 DSM Plot of Synthesized ( | ) and Actual Driving Point Values ( | ), PhaseMagnitude. ............................................................................................................................. 34
Figure 16 FRF Plot of Synthesized ( | ) and Actual ( | ) Driving Point Values, Phase
Magnitude. ............................................................................................................................. 34
Figure 17 CMIF Plot of Synthesized (Dashed Line) Actual DSM Data Sets. ....................... 35
Figure 18 Driving Point FRF Matrix Plots for Spatialy Truncated Data Set. ...................... 37
vii
Figure 19 CMIF Plot of Truncated FRF Data Set (9 d.o.f.)..................................................38
Figure 20 Driving Point DSM Matrix Plots, Real/Imaginary, for Spatially Truncated Data
Set........................................................................................................................................... 38
Figure 21 Synthesized ( | ) and Orig. ( | ) Driving Point FRF Measurements Using
Truncated Datasets (9 d.o.f.). ................................................................................................ 39
Figure 22 CMIF Plot of Synthesized (dotted) and Original FRF Data (9 d.o.f. Dataset). ... 40
Figure 23 FRF Plot of Pseudo-synthesized ( | ) and Original ( | ) Data (9 d.o.f.). .............. 41
Figure 24 CMIF Plot of Pseudo-synthesized and Original Data (9 d.o.f.). .......................... 41
Figure 25. Second Row of DSM Matrix (B2,1 –B2,9) 9 D.O.F. Marginal Spatially Truncated
Data Set.................................................................................................................................. 43
Figure 26 CMIF Plot of Truncated FRF Data Set (8d.o.f.)...................................................46
Figure 27 Reflection Point in CMIF Plot of Truncated FRF data Set (8d.o.f.),.................... 47
Figure 28 CMIF Plot of Truncated DSM data Set (8d.o.f.)...................................................48
Figure 29. Driving Point DSM Matrix Plots, Real/Imaginary, for Spatially Truncated Data
Set, 8 D.O.F. .......................................................................................................................... 48
Figure 30 Imaginary DSM Component for Nodes 4 and 5.................................................... 49
Figure 31 Synthesized ( | ) and Original ( | ) FRF Data for 8 d.o.f. Truncated Data Set. .... 50
Figure 32 CMIF Plot of Original ( | ) and Synthesized (: ) Data sets, 8 d.o.f. Model...........50
Figure 33 Pseudo-synthesized ( | ) and Original ( | ) FRF Data for 8 d.o.f. Truncated Data
Set........................................................................................................................................... 51
Figure 34 Driving Point DSM Measurements of Bar using PCB UT333 Accels and Hot Glue
Adhesive ................................................................................................................................. 58
viii
Figure 35 Driving Point DSM Measurements of Bar using PCB UT352 Accels and Super
Glue Adhesive. ....................................................................................................................... 59
Figure 36 Ratio Calibration Set-up ....................................................................................... 60
Figure 37 Aluminum Bar Dimensions ...................................................................................64
Figure 38 Set-up and Sensor Location for Free-Free Boundary Condition ......................... 65
Figure 39 Mode Shapes for Bungee Cord Tests, B.C.’s Between Locations 1,2 and 5,6...... 67
Figure 40 CMIF Plots of FRF Data from Tests 1 and 2 ....................................................... 69
Figure 41 CMIF Plot of DSM Data from Bungee Cord Tests 1 and 2..................................69
Figure 42 Bode Plot of Test 1 FRF Data............................................................................... 70
Figure 43 Real, Imaginary Plot of Test 1 FRF Data............................................................. 71
Figure 44 DSM Driving Point Measurement for Test 1, Bungee Cord Boundary Conditions.
................................................................................................................................................ 73
Figure 45 DSM Driving Point Measurement for Test 2 Bungee Cord Boundary Conditions.
................................................................................................................................................ 73
Figure 46 Real-Imaginary DSM Plots of Synthesized ( | ) and Original ( | ) Data from Test
1.............................................................................................................................................. 75
Figure 47 Real-Imaginary Plot, Third Row of the Experimental ( | ) and Synthesized ( | )
DSM Matrices, Test 1 ............................................................................................................ 76
Figure 48 Synthesized ( | ) and Experimental ( | ) FRF Bode Plots for Test 1 and 2 Data... 77
Figure 49 Real-Imaginary Plots of Synthesized ( | ) and Experimental ( | ) FRF Data from
Test 1 and 2............................................................................................................................ 78
Figure 50 Bode Plot of Pseudo-Synthesized ( | ) and Experimental ( | ) FRFs from Test 1
and 2....................................................................................................................................... 79
ix
Figure 51 CMIF Plot of Pseudo-Synthesized and Original FRF Data from Test 1 and 2 .... 79
Figure 52 Pseudo-synthesized ( | ) and Experimental ( | ) FRF Plot for Driving Point 1, Test
1.............................................................................................................................................. 81
Figure 53 Cable Damping Set-up .......................................................................................... 95
Figure 54Mode Shapes and Location of Additional Mass, Test 2......................................... 96
Figure 55 CMIF Plots of FRF Data from Test 1 and Test 2 ................................................. 97
Figure 56 CMIF Plot of DSM Data from Cable Damping Tests 1 and 2.............................. 98
Figure 57 Driving Point FRF Plot of Plate Without Cable Damping.................................. 98
Figure 58 DSM Driving Point Measurement for Plate without Damping.............................99
Figure 59 DSM Driving Point Measurement for Plate with Damping..................................99
Figure 60 Synthesized ( | ) and Experimental ( | ) DSM Driving Points, Plate without
Damping............................................................................................................................... 101
Figure 61 Synthesized ( | ) DSM and Experimental ( | ) Driving Points, Plate with Damping
.............................................................................................................................................. 101
Figure 62 Synthesized ( | ) and Experimental ( | ) FRF Driving Point for Location 1........102
Figure 63 Pseudo-Synthesized ( | ) and Experimental ( | ) Driving Point FRF, Measurement
Location 1 ............................................................................................................................ 103
Figure 64 CMIF Plot of Pseudo-Synthesized FRF Data, Case 2 ........................................ 104
Figure 65 Synthesized ( | ) and Experimental ( | ) DSM Data, 190-1600 Hz Input to
Algorithm, without Cable Damping..................................................................................... 105
Figure 66 Pseudo-Synthesized ( | ) and Experimental ( | ) Data using Two Different Input
Frequency Ranges, without Damping.................................................................................. 106
Figure 67 Case 3 Experimental Structure ........................................................................... 114
x
Figure 68 Clamped Beam Dimensions ................................................................................ 114
Figure 69 CMIF Plot of Clamped Beam FRF, with and without Dashpot Damper............117
Figure 70 CMIF Plot of Clamped Beam DSM, with and without Dashpot Damper........... 117
Figure 71 FRF Plot of Clamped Bar, without Damper ....................................................... 118
Figure 72 FRF Plot of Clamped Bar, with Damper ............................................................ 118
Figure 73 DSM Plot of Clamped Bar, without Damper ...................................................... 119
Figure 74 DSM Plot of Clamped Bar, with Damper ........................................................... 120
Figure 75 Bar without Damping DSM, Entry B44, Synthesized ( | ) using Two Different
Frequency Ranges................................................................................................................ 121
xi
TABLE OF TABLES
Table 1. Structural Damping Matrices, Full D.O.F. Dataset................................................ 30
Table 2. Viscous Damping Matrices, Full D.O.F. Dataset.................................................... 31
Table 3. Mass Matrices, Full D.O.F. Dataset. ...................................................................... 31
Table 4. Stiffness Matrices, Full D.O.F. Data Set. ................................................................ 32
Table 5. Calculated Mass Matrix, 9 D.O.F. Marginally Spatially Truncated Data Set........42
Table 7 Calculated System Properties, 9 D.O.F. Truncated Data Set .................................. 45
Table 9 Modal Properties of a Component Calculated from Two Different Combinations of
Sensors. .................................................................................................................................. 58
Table 11 Modal Properties of Pseudo-Synthesized FRF Data from Tests 1 and 2 ............... 80
Table 12 Modal Damping for SVD Conditioned Pseudo-synthesized FRF Data..................82
Table 13 Calculated Real Mass Coefficients, Test 1 and 2, 400 Hz- 1600 Hz. .....................84
Table 14 Calculated K Coefficents, Test 1 and 2, 400 Hz-1600 Hz. .....................................85
Table 15 Calculated D (zero order) Coefficients, Test 1 and 2, 400-1600 Hz ...................... 86
Table 16 Calculated Visous Damping (First Order) Coefficients, Test 1 and 2, 400-1600 Hz.
................................................................................................................................................ 88
Table 17 Calculated Second Order Coefficients, Test 1 and 2, 400-1600 Hz. ...................... 90
Table 18.Mass Matrix Coefficient of SVD Conditioned Test 1 Data..................................... 91
Table 19 Viscous Damping Matrix Coefficient of SVD Conditioned Test 1 Data................. 91
Table 20 Modal Properties for Cable Damping Tests........................................................... 95
Table 21 Calculated Modal Properties of Pseudo-Synthesized FRF Data ......................... 104
xii
Table 26 Case 2 Second Order Damping Matrices, Plate with and without Cable Damping
.............................................................................................................................................. 111
Table 27 Modal Properties of Clamped Beam Tests ........................................................... 116
xiii
Nomenclature
[ ]-1
matrix operator denoting inverse
[ ]+
matrix operator denoting pseudo-inverse
[ ]T
matrix operator denoting transpose
m
mass
[M] or M
mass matrix
[K] or K
stiffness matrix
[D] or D
structural damping matrix, zero order coefficient
[C] or C
viscous damping matrix, first order coefficient
[E] or E
second order damping coefficient
[H]
frequency response function matrix
[H]C
frequency response function matrix, C denoting complex values
[B]
dynamic stiffness matrix
Hij
entry row i, column j of frequency response matrix
Bij
entry row i, column j of dynamic stiffness matrix
ω
rotational frequency, radians/second
[I] or I
identity matrix
X(ω)
displacement in the frequency domain
F(ω)
force in the frequency domain
Ar (ω)
frequency domain acceleration of accelerometer r
[U] or U
U matrix of the singular value decomposition
[S] or S
singular value matrix of singular value decomposition
[V] or V
V matrix of the singular value decomposition
xiv
imag([B])
imaginary component of dynamic stiffness matrix
real([B])
real component of dynamic stiffness matrix
σr
rth singular value entry in the S matrix
xv
Abbreviations
FRF
frequency response function
DSM
dynamic stiffness matrix
SVD
singular value decomposition
CMIF
complex mode indicator function
d.o.f.
degrees of freedom
xvi
1 Overview of work
1.1 Motivation
The desire for effective experimental identification of the spatial distribution of damping is
motivated by the need for model verification/correlation in many fields including
automotive, aerospace, and appliances. Damping matrices become more crucial in some
problems that require increased dynamic model fidelity (as in the case of stability in
rotational dynamics [1-2, 7, 8]). Spatial damping description represents energy dissipation
mechanisms, which are sensitive to changes in energy paths and interfaces (joint preloads or
corrosion), therefore spatial damping can be used very effectively for damage detection [7].
1.2 Challenges in Spatial Damping Identification
Linear dynamic responses of structures are made up of two general types of internal force
mechanisms: conservative and non-conservative [8]. Identifying the characteristics of both
mechanisms is required for a high-fidelity model of a structure. Conservative characteristics
trade dynamic energy between each other without loss. These are made up of inertial (mass)
and stiffness components. Both types of components can be experimentally identified in
several manners, and modeling of the properties is well developed and many verification
methods exist. Non-conservative (damping) characteristics on the other hand, remove
kinetic energy from the system in an irreversible manner. Due to their function, non-
conservative features can only be observed while there is kinetic energy to dissipate. This
requirement is one of several issues that make analytic or experimental spatial damping
identification a challenge.
The major issues for damping identification fall under two categories: observability and
mechanism modeling. As mentioned previously, damping effects can only be observed
when kinetic energy exists [7]. Therefore, any experimental technique must rely on
dynamic measurements, thus limiting the scope of possible measurement tools. Typical
vibration/resonance problems involve lightly damped structures. By definition, such
structures produce much smaller damping forces than the inertial or stiffness mechanisms.
The dynamic resolution of sensors will be dominated by the contributions of M and K,
therefore the contributions of damping features may be lost in the noise floor of the
instrumentation and data acquisition system. Spatial and temporal phasing characteristics of
a structure are functions of its damping mechanisms, which again tend to be weak actors
(creating small phase values). In this case, small errors in sensor phasing or other
unidentified noise sources can create substantial errors in the measured damping values.
Damping modeling can be daunting since the mechanisms are extremely complex, even in
simple structures, and are very case dependent. For this reason, many damping models
exist, however none are all encompassing [3-10]. Proportional damping attempts to model
the non-conservative forces using a spatial relationship, however, the mechanism is assumed
to be a function of the geometry (namely, M and K). The greatest strength of this model is
the benefit of simplifying the system’s eigenvalue solution. Modal damping assumes that
2
damping is a temporal phenomenon and follows the mode shape at a particular modal
frequency. Viscous damping approaches a true spatial method, but assumes the damping
force is a linear function of frequency. Structural damping models damping as a frequency
independent spatial feature which exhibits hysteric properties. The large drawback of
structural damping is a violation of the causality requirement when applied to an analytical
model [10]. Time based simulations will produce damping forces before an external
excitation force can be applied. Due to the usual existence of non-frequency dependent
damping, structural damping is typically required for modeling, and in turn, the causality
error is of a small magnitude and may be considered negligible. Other models exist, but
these are the most prevalent.
3
1.3 State of the art for spatial damping identification
Many spatial damping identification methods exist. Some are frequency domain-based,
while others operate within the time or Laplace domain.
Various methods have been proposed as reviewed by Pilkey and Inman [7] in a recent
survey paper. Practically all reported damping identification methods assume measured
system responses are available in the form of time or frequency domain frequency response
functions, and most cases assume the damping mechanism is viscous, structural or their
combinations, leaving out friction and viscoelastic damping, two commonly encountered
energy dissipation mechanisms. Often, a minimization process is performed and utilizes
various error norms defined by response functions, modal vectors, or the equation of motion
[3,7]. Also, damping systems are often approximated using a perturbation of the undamped
system to relate damping matrices to undamped modal parameters or orthogonality of modal
vectors. Some of theses methods are effective if applied to systems with dominant real
modes (lightly damped), as is the case with Rayleigh damping [3,8].
Many of the methods share one or several issues. First, all the methods begin with some
sort of a priori information. This data may range from analytical M and K matrices
(calculated from finite element models) to modal frequencies and vectors (curve fit from
experimental data) [7 ]. The quality of the distributed damping identification is a function
of the accuracy of the input information. Again, non-conservative internal structural forces
tend to have very low contributions when compared to forces caused by M and K.
4
Therefore, a small percentage of error in the M and K matrices (or modal frequencies and
vectors) may overshadow the true contribution of damping to the dynamic behavior of the
system. Ideally, a damping identification method should not rely on M and K dependent
information.
Second, many of the methods require several computational steps, iterations, and inverses.
Each computational step introduces some amount of numerical error (in addition to any
input data error). Again, the damping values have a small contribution to the overall
response of the system. As in the a priori data error, the computational error may
overshadow the true damping values of the system. A computationally simple identification
algorithm will be more robust to numerical error.
Many of the damping identification algorithms begin with an assumption about the
underlying damping mechanism. Ideally, an identification algorithm should allow for
flexibility in identifying particular mechanisms.
Finally, complicated methods inhibit the user from gaining an intuitive understanding of the
method and its output. All of the existing a priori information based methods are
susceptible to experimental error. Unfortunately, this is an issue that affects any damping
identification method that employs standard modal analysis techniques. Quantifying errors
can be difficult; however, an intuitive understanding will allow the user to observe
deviations from an expected model and identify experimental or numerical error.
5
1.4
Goal of the new spatial damping identification algorithm
The proposed damping algorithm seeks to fulfill the points described previously, in addition
to having additional features. The method should:
•
Minimize numerical error through computational simplicity.
•
Have enough flexibility to differentiate between different damping mechanisms.
•
Not rely on a priori information and be as independent of M and K properties as
possible.
•
Use several methods to identify data quality and to generate an intuitive feel for the
properties of the algorithm’s input data.
•
Be straightforward in order to generate an intuitive feel for the algorithm’s output.
•
Be sensitive to the spatial properties of the structure’s damping.
6
2 DSM Damping Identification Method
2.1 Identification Method Developed by Lee and Kim
The dynamic stiffness distributed damping algorithm was initially derived by J.-H. Lee and
J. Kim [1, 2]. For an N-d.o.f. linear dynamic system under harmonic excitation, the
equation of motion is given as:
M&x&(t ) + Cx& (t ) + ( jD + K ) x(t ) = f (t )
(2.1)
Assuming f(t)=Fejωt (and therefore x(t)=Xejωt) and performing the required substitutions, the
equation is rewritten in the frequency domain:
[( K − Mω 2 ) + j (ωC + D)] X (ω ) = F (ω )
(2.2)
Re-arranging terms produces the frequency response function (FRF):
[H
C
]
(ω ) k × k × n =
X (ω )
1
=
2
F (ω ) ( K − Mω ) + j (ωC + D)
(2.3)
The superscript c denotes the complex numerical make-up of the FRF. The dynamic
stiffness matrix is simply defined as:
[ H C (ω )]−1 = ( K − Mω 2 ) + j (ωC + D )
(2.4)
7
For derivation purposes, it would be simple to rearrange Equation 2 in order to achieve
(2.4). However, it is much easier to experimentally measure a system’s FRF than its DSM.
Therefore, an experimental DSM matrix is derived from inverting the frequency response
function matrix.
Next, the DSM is broken into its real and imaginary components:
real ([ H C (ω )]−1 ) = K − Mω 2
(2.5a)
imag ([ H C (ω )]−1 ) = ωC + D,
(2.5b)
Equations 2.5a-b are rearranged into matrix notation:
[I
[I
]
K
2
− ω I = real ( H C (ω ) −1 )
M
(2.6a)
D
ωI ] = imag ( H C (ω ) −1 ),
C
(2.6b)
where I is an kxk identity matrix.
K and M can be solved by using a pseudo-inverse procedure:
K
M
2 k ×k
[
[
I
I
=
I
[
]
]
+
2
real ( H C (ω1 ) −1 )
− ω1 I
2
C
−1
−ω2 I
real ( H (ω 2 ) )
M
M
M
M
2
− ω n I 2 k×k ∗n real ( H C (ω n ) −1 ) k ∗n×k
]
(2.7a)
Likewise for C and D:
D
C
2 k ×k
[
[
]
]
+
I ω1 I
imag ( H C (ω1 ) −1 )
C
−1
I ω2 I
imag ( H (ω2 ) )
=
M
M
M
M
I ω I
imag ( H C (ω ) −1 )
n
k ∗n×k
n
2 k×k*n
[
(2.7b)
]
8
The + sign on the middle matrices of Equations 2.7a-b signifies the pseudo-inverse of the
matrices. A minimum of two frequencies are required in order to satisfy the equations,
however, this procedure allows for the right hand side of Equations 2.7a-b to be overdetermined over a large range of ω values.
Intuitively, Equations 2.7a-b are simply second and first order polynomial fits (respectively)
of the DSM data. The conservative K and M forces should resemble a parabola, while the
non-conservative C and D forces should be linear. Note that while C and D (zero and first
order) where chosen as polynomial coefficients, any number of coefficients (or orders) may
be selected. Therefore, the method may be tailored to the qualitative nature of the
experimental data. The general form of the imaginary DSM equations becomes:
imag ([ H (ω )] ) =
C
−1
N −1
∑ [C ]ω
r =0
r
r
(2.8)
Cr is the rth order matrix coefficient; N is the number of polynomials used to fit the
imaginary DSM data. In a full matrix form the equations become:
[ I ω11 I
1
[ I ω2 I
M
M
1
[ I ωn I
imag ( H C (ω1 ) −1 )
L ω1N I ]
C1
imag ( H C (ω 2 ) −1 )
C
N
L ω2 I
2
=
M
M
O
M
M
N
C
L ω n I ] k ∗n×N ∗n N N ∗k×k
C
−1
(
(
)
)
imag
H
ω
n
k ∗n×k
(2.9)
The number of terms or dominance of certain terms may serve as clue to identify the
damping mechanism.
9
2.2 Practical Issues in Implementing DSM Algorithm
There are many issues pertaining to implementing the DSM damping identification
algorithm which require discussion. Some of these issues are included within the scope of
this thesis, while other areas will remain for future work.
First, a significant amount of measurement equipment is required. The algorithm’s data
inspection and conditioning techniques require the use of square FRF matrices for an input.
Therefore, driving point measurements must be made at every degree of freedom on the
structure. The minimum number of degrees of freedom is dictated by the algorithm’s
sensitivity to spatial truncation.
Establishing the minimum number of d.o.f.s requires significant knowledge about the
structure, including: number of modes in the system, number of significant residual modes
outside the frequency range of interest, the system’s boundary conditions, and an estimate of
the modal vectors. The algorithm should therefore be applied as a second or third step in an
experimental structural analysis.
The degrees of freedom of experimentally identified matrices must be expanded in order to
match the size of analytical models. Typically, the algorithm’s results will be used to refine
analytical models. Large experimental tests might contain several hundred degrees of
freedom (or less, considering the hardware requirements of the DSM algorithm), while
analytical models handle millions of degree of freedom easily. A method must be
10
developed in order to expand the experimental degrees of freedom to match the analytical
model.
Interactions between the test structure and instrumentation greatly affect the DSM data.
Any time an experiment is run, one runs the risk of affecting test results due to influences of
the instrumentation. Sensor mass loading can cause natural frequencies to shift in a modal
analysis test. The effect may be observable; however, the modal test is not effective
qualitatively. The DSM is sensitive to changes in residuals and noise. The addition of
instrumentation mass does not only change temporal based qualities of the data, but the
overall quality of the can be affected (creating discontinuities, etc…)
Finally, the algorithm operates on data consisting of inverted FRF singular values. These
values are dominated by residual modes and noise. Even if identified (and quantified),
system noise can be difficult to remove or minimize, and few signal conditioning options
currently exist.
11
3 DSM Algorithm Validity Study Using Analytical Data.
J. –H. Lee published an initial DSM algorithm analytical study using a 3-d.o.f. lumped mass
system [1,2]. His findings validated the algorithm using clean and random-noise tainted
data. He also discovered that by averaging off-diagonal terms in the DSM (enforcing the
assumption of reciprocity), the damping identification algorithm became more robust to
random noise. Lee’s work was only an initial study, and the following work builds upon his
findings and explores several other characteristics of the DSM damping identification
method.
In this section, three key concepts will be discussed. First, time will be spent discussing the
mechanics behind the FRF inversion. This will allow the reader to gain a better feel for the
system features being identified by the curve fit process. Next, a graphical representation of
the DSM data will be presented. The qualitative data analysis allows the user to identify
encouraging data features and noise levels. Finally, an optimal degree of freedom study will
be presented. In any experimental case, the number of modes in the system is infinite.
Granted, only a finite number of modal frequencies will exist within a given frequency
range, however, the residuals from out of band modes will still contribute to the data within
the selected frequency range.
12
3.1 The Analytical Model
All of the analytical data was generated from lumped mass analytical models containing
either 50 or 9 degrees of freedom. Both models have the same stiffness, damping and mass
values for each element or node. The only difference between the two models are the
number of degrees of freedom (structural elements, or masses) used to generate the raw
data. Both d.o.f. systems have two connections to ground, one at each end:
K1=1500000 N/m
C1=120 Ns/m
K2=1500000 N/m
M1=10 Kg
C2=120 Ns/m
D1=2000 N/m
KN=1500000 N/m
M2=10 Kg
CN=120 Ns/m
DN=2000 N/m
D2=2000 N/m
X(t)1
•••••••••
X(t)2
KN+1=1500000 N/m
CN+1=120 Ns/m
MN=10 Kg
DN+1=2000 N/m
X(t)N
Figure 1. N degree of freedom analytical model
As shown in Figure 1, four system features are modeled: mass, stiffness, viscous damping
and structural (hysteretic) damping. The subscript N denotes the number of degrees of
freedom (N=9 or 50). The resulting matrices have the following form:
13
10
0
0
M =
0
M
0
0
10
0
0
M
0
0
0
10
0
L
L
0
0
0
O
L
L
L
L
M
M
10
0
0
0
M
,
M
0
10 N×N
L
0
0
0
240 − 120
− 120 240 − 120
L
0
0
0
M
M
− 120 240 − 120
C=
,
O
M
M
−
0
0
120
M
M
L
L
240 − 120
L
L
− 120 240 N × N
0
0
0
L
0
30 − 15 0
− 15 30 − 15 0
L
0
0 − 15 30 − 15 M
M
5
K =
×10 ,
−
0
0
15
O
M
M
M
M
L L 30 − 15
0
L L − 15 30 N×N
0
0 L 0
4 −2 0
− 2 4 − 2 0 L 0
0 −2 4 −2 M
M
D=
× 10 3 ,
0 −2 O
M
M
0
M
M
L L 4 − 2
0 L L − 2 4 N ×N
0
Figure 2. Analytical model matrices.
The resulting M, K, C, D matrices are symmetric, thus displaying evenly distributed
structural properties. The form in Figure 2 is the same for the 50 d.o.f. and the 9 d.o.f.
model, except the overall matrix dimension N. The intent is to simulate a system with
similar characteristics as the beam structures tested in the experimental validation work.
The analytical data FRFs are generated using Equation 3.1 for a given frequency range.
[H
C
(ω1×n )
]
N × N ×n
[
]
= [K ] − [M ][I ]ω12×n + j (ω1×n [I ][C ] + [D ])
−1
(3.1)
For some analytical cases, the FRF matrix is spatially reduced in order to observe the effects
of spatial truncation on the DSM damping identification algorithm. The truncated data sets
are generated by using only a select number of rows and columns from the original 50 d.o.f.
FRF dataset. For example, if only 5 degrees of freedom were processed (say, nodes 5, 15,
25, 35, 45), the new FRF matrix will be composed of entries from only those rows and
columns of the original 50 d.o.f. FRF matrix:
14
H 50C x 50 xn
H 1,1
M
H 13,1
H 14,1
H 15,1
M
= H 34,1
H 35,1
H
36,1
M
H 45,1
M
H 50,1
L H 1,13
O
M
L H 13,13
H 1,14
M
H 13,14
H 1,15
M
H 13,15
L H 1,34
O
M
K H 13,34
H 1,35
M
H 13,35
H 1,36
M
H 13,36
L H 1, 45
O
M
L H 13, 45
L H 14,13
L H 15,13
O
M
H 14,14
H 15,14
M
H 14,15 L H 14,34
H 15,15 L H 15,34
M
O
M
H 14,35
H 15,35
M
H 14,36
H 15,36
M
L H 14, 45
L H 15, 45
O
M
L H 34,13
L H 35,13
H 34,14
H 35,14
H 34,15 L H 34,34
H 35,15 L H 35,34
H 34,35
H 35,35
H 34,36 L H 34, 45
H 35,36 L H 35, 45
L H 36,13
O
M
L H 45,13
H 36,14
M
H 45,14
H 36,15 L H 36,34
M
O
M
H 45,15 L H 45,34
H 36,35
M
H 45,35
H 36,36 L H 36, 45
M
O
M
H 45,36 L H 45, 45
O
M
L H 50,13
M
H 50,14
M
O
M
H 50,15 L H 50,34
M
H 50,35
M
O
M
H 50,36 L H 50, 45
L H 1,50
O
M
L H 13,50
L H 14,50
L H 15,50
O
M
L H 34,50
L H 35,50
L H 36,50
O
M
L H 45,50
O
M
L H 50,50
(3.2)
Resampled
H
C
5×5× N
H 5, 5
H
15 ,5
= H 25 , 5
H 35 , 5
H 45 , 5
H 5,15
H 5, 25
H 5,35
H 15 ,15
H 15 , 25
H 15 ,35
H 25 ,15
H 25 , 25
H 25 , 35
H 35 ,15
H 35 , 25
H 35 , 35
H 45 ,15
H 45 , 25
H 35 , 45
H 5, 45
H 15 , 45
H 25 , 45
H 35 , 45
H 45 , 45 5×5
(3.3)
The FRF data in each entry in Equation 3.3 is still identical to the original 50 d.o.f. FRF
matrix, only the overall dimension of the new matrix is different. In other words, the spatial
information of the matrix will be truncated; however, the temporal data will remain
unchanged.
15
3.2 FRF matrix inversion
Below is the H11 driving point entry from a 9 d.o.f. FRF matrix generated from the
analytical model. The system contains 9 degrees of freedom and the analytical DSM matrix
has a 9x9 spatial dimension.
H11 Driving Pont FRF
-5
0
1
0.5
-100
Real
Phase
-50
0
-150
-200
-0.5
0
10
20
30
Freq (Hz)
40
50
-1
60
0
20
30
Freq (Hz)
20
30
Freq (Hz)
40
50
60
10
20
30
Freq (Hz)
40
50
60
-1
-8
10
10
40
50
x 10
-0.5
Imaginary
-6
10
-1.5
60
0
Figure 3. Driving Point FRF, dof 1.
Figure 3 displays three well spaced natural frequencies. Now observe the DSM for the same
degree of freedom:
B11 Driving Point DSM
6
5
3
B11 Driving Point DSM
x 10
2.5
3
Real
P hase
4
2
0
2
1.5
1
0
10
20
30
Freq (Hz)
40
50
1
60
0
10
20
30
Freq (Hz)
40
50
60
10
20
30
Freq (Hz)
40
50
60
4
15
x 10
6.4
Im aginary
10
M agnitude
Magnitude
10
0
0
-5
-4
10
H11 Driving Point FRF
x 10
6.3
10
6.2
10
5
10
0
10
20
30
Freq (Hz)
40
50
60
0
0
Figure 4. Driving Point DSM dof 1.
16
Figure 4 shows no signs of the three modes, which may be different than expected. Keep in
mind that the DSM is the matrix inverse of the FRF. For each frequency line, the whole
FRF matrix is inverted to produce the corresponding DSM. If the system only had one
degree of freedom, its DSM plot would simply have poles looking like zeros, and zeros
looking like poles. The degree-of-freedom inverse dependence is due to how the FRF’s
singular values are modified during the inverse process. The most effective way to
demonstrate this affect is by using the singular value decomposition (SVD).
The SVD separates a given H(ωk)mxn matrix into three matrices [11]:
[H ] = [U ][S ][V ]H
(3.4)
where [U] is an mxm orthonormal matrix, whose columns are the dominant vectors
contained in the [H] column space. The [V] matrix is a nxn orthonormal matrix whose rows
are the dominant vectors contained in the [H] matrix row space. The [S] matrix is an mxn
diagonal matrix of s non-zero singular values of [H]. The values are ordered with the
highest value in the upper left entry, the lowest value (or zero) in the lower right entry. The
[S] matrix serves as the scale factor for the row and column vectors that comprise the [H]
matrix.
σ 1 0 0 0
0 σ
0 0
2
[S ] =
0
0 O M
0 L σs
0
(3.5)
17
The dominant vector in the [H] column space will be in the left column of [U] and will have
the highest singular value scale factor, σ1. At, or near a natural frequency, the dominant [U]
column represents an estimate of the mode shape at that natural frequency. The remaining
vectors are composed of residuals from neighboring modes. The weak vectors may also
represent noise in the system, especially if residual mode contributions are below the noise
floor. Figure 5 demonstrates the superposition assumption, derived from modal analysis
theory [14]:
1
2
3
4
5
6
Figure 5 Superposition of Single Degree of Freedom Modes.
The total FRF measurement is the solid line, while the dotted lines are the contributions
from all of the modes in the system. As displayed in Figure 5, each data point in the total
FRF is the summation of all the system’s modes contributing at that frequency. Observe the
1 Hz mode in Figure 5. The dominant signal (1) makes up the majority of the peak,
18
however Curves 2-6 still contribute to the data point. In terms of the SVD, the first singular
value of [S] will represent Curve 1; the last [S] entry will be for Curve 6. Likewise, the
dominant modal vector of Curve 1 will be estimated in the left column of the [U] matrix.
Keep in mind that the SVD is a spatial decomposition of the data at a particular frequency
value. Processing the SVD of [H] at 2 Hz may produce the same vectors in the [U] matrix;
however, the columns will not be in the same order as the 1 Hz calculation. Likewise, the
[S] will have different values, thus changing the scaling of each vector in [U].
Below is the complex mode indicator function (CMIF) plot for the same [H] data as used in
Figure 3 [12,15]. The CMIF is simply a plot of all the singular values (the diagonal entries
of [S]) at each frequency point.
CMIF Plot of FRF
-3
10
-4
10
-5
10
-6
10
-7
10
0
10
20
30
Freq (Hz)
40
50
60
Figure 6 CMIF plot for FRF.
19
Figure 6 displays all three modes and also the contribution of the neighboring modes’
residuals. The top (blue) curve is the dominant singular value at each frequency; the second
(green) line is the second highest value and so forth. The data in Figure 6 is purely
analytical and thus is free of noise. Therefore, the bottom curves in Figure 6 are only due to
the residuals of high frequency modes.
The inverse of H(ωk)mxn matrix is:
[B] = [H ]−1 = ([U ][S ][V ]H )
−1
(3.6)
[B] is the impedance matrix, or dynamic stiffness matrix (DSM). Since the [V] and [U]
matrices are both orthonormal, their inverses are simply their matrix transposes. The [S]
matrix is the only component of the decomposition which is inverted. Since [S] is diagonal,
the inverse is simply the inverse of each entry along the diagonal.
[B] = [H ]−1 = [V ][S ]−1 [U ]H
(3.7)
and:
1
σ
1
[S ]−1 = 0
0
0
0
1
σ2
0
0
0
0 0
O M
1
L
σ s
0
(3.8)
The largest singular value becomes the smallest and vice versa. Since the singular values
are vector scale factors, the weakly represented vectors of the [H] matrix dominate the [B]
20
matrix and the strong [H] vectors are attenuated. Therefore, the [B] matrix is dominated by
residual mode contributions, and potentially by noise. Residuals from high frequency, outof-band, modes appear as stiffness lines at lower frequencies. The residuals, therefore,
appear as straight lines in the data, as represented in the CMIF plot of Figure 7.
CMIF of DSM Matrix
7
10
6
10
5
10
4
10
3
10
0
10
20
30
Freq (Hz)
40
50
60
Figure 7 CMIF plot for DSM.
Unlike the stark difference between Figure 3 and Figure 4, the CMIF plots in Figure 6 and
Figure 7 appear to be simple inverses of each other. The residuals of higher frequency
modes make up the dominant curves in Figure 7. Since the weakest residual is at a much
higher frequency than 150 Hz, its residual values appear as a horizontal stiffness line. As
one progresses to the stronger residuals (lower curves in Figure 7), the residual’s natural
frequency approaches the observed frequency band. Residuals from just above 150 Hz
display more curvature at higher frequencies than residuals from much higher frequency.
21
In summary, care must be taken when performing matrix inverses. As shown in the above
FRF and CMIF plots, the matrix inverse is not an inverse of each matrix entry. Instead, the
operation merely inverses the singular values of the matrix, while keeping the values of the
column and row vector space the same. The result is an amplification of weak modes or
noise and an attenuation of strong modal contributions. Therefore, the data in a DSM matrix
is dominated by residuals and (potentially) by noise sources. The fit of the distributed
structural parameters (M, K, C, D, etc…) is really a fit of the higher mode residuals.
3.3 Applying the DSM algorithm
3.3.1 Full system and model degrees of freedom
In this section, the basic DSM damping identification algorithm will be demonstrated on an
ideal 9 d.o.f. dataset. The system model will be the same as shown in Figure 1, with the
N=9. Nine modes will be calculated, and the spatial dimensions of the DSM will be 9x9.
This set of analytical data was chosen since all of the system modes are observed within the
selected frequency range, and the algorithm will contain enough degrees of freedom to
properly identify the system matrices. System degrees of freedom are defined as the number
of modes in the system. Algorithm degrees of freedom are the number of available spatial
points in which to assign structural characteristics (M, K, C, D). This value is equal to the
smallest dimension of the B matrix. A later demonstration will investigate the case when
the system d.o.f.’s exceed the algorithm d.o.f.’s. This is the case with any real structure,
22
since the number of modes is infinite, and there is a practical limit to the number of sensors
and amount of sensing equipment that can be used.
The object of this analytical case will be threefold. First, some pre-processing graphical
diagnosis tools are introduced. Next, some comments will be made to better relate the
intuitive nature of the DSM algorithm matrix polynomial fit process. Finally, some postprocessing graphic methods will be demonstrated in order to verify the quality of the data
fit.
Figure 8 shows the system matrices obtained for this system:
10 0 0 0 L 0
0 10 0 0 L 0
0 0 10 0 M
M
M =
,
M
0 0 0 O M
M
M L L 10 0
0 0 L L 0 10 9×9
0
0
L
30 − 15 0
− 15 30 − 15 0
0
L
0 − 15 30 − 15 M
M
5
K=
× 10
0
0
15
O
M
M
−
M
M
L L 30 − 15
0
L L − 15 30 9 × 9
0
4
0
0
0
L
240 − 120
− 2
− 120 240 − 120
0
0
L
0
0
− 120 240 − 120
M
M
D=
C=
,
0
0
− 120 O
M
M
0
M
M
240 − 120
M
L
L
0
0
− 120 240 9×9
L
L
0
−2
4
−2
0
M
0
0
−2
4
−2
L
L
0 L 0
0 L 0
M
−2 M
3
×10
O M
M
L 4 − 2
L − 2 4 9×9
Figure 8 Matrices from 9 dof model.
The above matrices are of the same form as those in Figure 2, only with a different
dimension. 0.5% random noise has been added to the analytical FRF data in order to
23
simulate a real situation. Figure 9 displays all nine driving point FRFs in a phase/magnitude
plot:
50
50
50
50
50
50
50
50
0
0
0
0
0
0
0
0
0
-50
-50
-50
-50
-50
-50
-50
-50
-50
-100
-100
-100
-100
-100
-100
-100
-100
-100
-150
-150
-150
-150
-150
-150
-150
-150
-150
Phase
50
-200
0
50
100
Freq (Hz)
-200
150
0
-5
50
100
Freq (Hz)
-4
10
10
-200
150
0
50
100
Freq (Hz)
-200
0
150
50
100
Freq (Hz)
10
-5
10
50
100
Freq (Hz)
-2
-4
-4
10
-200
0
150
-200
0
150
50
100
Freq (Hz)
-4
10
10
-200
0
150
50
100
Freq (Hz)
-200
0
150
50
100
Freq (Hz)
10
-5
10
50
100
Freq (Hz)
150
50
100
Freq (Hz)
150
-5
10
-4
-4
10
-200
0
150
-3
10
-5
10
-5
-5
10
10
-6
-5
10
-6
10
-4
10
Magnitude
10
-6
-5
-6
-6
10
-6
10
10
10
-7
10
-6
-6
10
-7
10
10
-7
10
-6
10
10
-7
-7
-7
-7
10
10
10
10
-7
10
-7
10
-8
10
-8
0
50
100
Freq (Hz)
150
10
0
50
100
Freq (Hz)
10
150
0
-8
-8
-8
50
100
Freq (Hz)
10
150 0
50
100
Freq (Hz)
10
0
150
-8
50
100
Freq (Hz)
10
150
0
10
150
0
-8
-8
-8
50
100
Freq (Hz)
50
100
Freq (Hz)
10
0
150
50
100
Freq (Hz)
10
0
150
Figure 9 Driving Point FRFs (entry 1,1 on left; entry 9,9 on right)
All nine modes can be observed between 0-150 Hz. Typically, this is the first data that one
would obtain. Many of the same features required for good modal data are also required for
the damping matrix fit. Is the data set clean? Are the magnitude and phase (real and
imaginary) plots indicative of good driving point measurements? Was the coherence near 1
for all the measurements? Are there signs of leakage in the data sets?
24
Next, the FRF’s CMIF plot is plotted in Figure 10:
-3
10
-4
10
-5
10
-6
10
-7
10
0
50
100
150
Freq (Hz)
Figure 10 CMIF Plot of 9 dof system.
Again, note the nine peaks, each denoting a strong singular value at each of the modal
frequencies. Also note the absence of flat lines at the bottom of the plot, because all of the
modes are observed within the frequency range of interest, there are no out of frequency
range residuals in the dataset. The number of peaks lines up with the number of singular
values. Therefore, the system d.o.f.’s and the algorithm d.o.f.’s are equal (the FRF
dimensions, and therefore, the DSM dimensions, are 9x9). Observe how the singular value
decomposition assigned the random noise to the lower singular value curves. Small residual
mode contributions become less distinguishable from the random noise.
25
Now examine the DSM matrix driving point measurements:
200
200
200
200
200
200
200
200
150
150
150
150
150
150
150
150
150
100
100
100
100
100
100
100
100
100
50
50
50
50
50
50
50
50
50
0
0
0
0
0
0
0
0
0
Phase
200
-50
0
7
-50
50 100 150
0
Freq (Hz)
7
10
10
-50
50 100 150
0
Freq (Hz)
7
10
-50
50 100 150
0
Freq (Hz)
7
10
-50
50 100 150
0
Freq (Hz)
7
10
6
Magnitude
6
6
10
6
10
6
4
0
10
50 100 150
0
Freq (Hz)
6
10
5
5
7
10
-50
50 100 150
0
Freq (Hz)
-50
50 100 150
0
Freq (Hz)
7
50 100 150
Freq (Hz)
7
10
10
10
10
6
6
10
10
5
10
10
7
10
-50
50 100 150
0
Freq (Hz)
6
10
10
-50
50 100 150
0
Freq (Hz)
10
5
10
50 100 150
0
Freq (Hz)
5
10
50 100 150
0
Freq (Hz)
5
10
50 100 150
0
Freq (Hz)
4
10
50 100 150
0
Freq (Hz)
5
10
50 100 150
0
Freq (Hz)
5
10
50 100 150
0
Freq (Hz)
5
10
50 100 150
0
Freq (Hz)
50 100 150
Freq (Hz)
Figure 11 Driving Point DSM Magnitude/Phase (entry 1,1 on the left, 9,9 on the right)
Comparing Figure 9 and Figure 11, the DSM again appears significantly different that the
FRF plots. None of the DSM driving points display peaks, and only one zero is present in
the plots.
26
6
5
x 10
x 10
0
6
5
6
x 10
5
0
x 10
0
6
5
-5
-10
-5
0
-10
50 100 150
0
Freq (Hz)
5
Imaginary
6
-5
x 10
5
6
x 10
-5
-10
50 100 150
0
Freq (Hz)
5
6
-10
50 100 150
0
Freq (Hz)
5
x 10
6
4
x 10
50 100 150
Freq (Hz)
6
4
3
2
2
1
1
0
0
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
5
6
x 10
3
0
-5
-10
50 100 150
0
Freq (Hz)
6
x 10
Real
0
6
5
0
50 100 150
Freq (Hz)
-6
5
x 10
6
x 10
6
x 10
0
5
6
5
0
0
-5
-5
-10
50 100 150
0
Freq (Hz)
5
6
x 10
-10
50 100 150
0
Freq (Hz)
5
x 10
6
x 10
5
5
5
5
5
5
5
5
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
-1
-1
-1
-1
-1
-1
-1
-1
-1
0
50 100 150
Freq (Hz)
-2
0
50 100 150
Freq (Hz)
-2
0
50 100 150
Freq (Hz)
-2
0
50 100 150
Freq (Hz)
-2
0
50 100 150
Freq (Hz)
-2
0
50 100 150
Freq (Hz)
-2
0
50 100 150
Freq (Hz)
-2
0
50 100 150
Freq (Hz)
50 100 150
Freq (Hz)
5
6
5
-2
x 10
-2
x 10
0
50 100 150
Freq (Hz)
Figure 12 Driving Point DSM Real/Imaginary Plot.
Figure 12 displays the actual data used for the matrix coefficient fit. Note the magnitude
of the noise on the imaginary component of the B matrix.
The real data was fit using the following equation (the same as 2.5a):
real ([ B(ω )]) = K − Mω 2
(3.9a)
K represents the y-intercept for the real data, and M is the frequency squared polynomial
term. The zero point displayed in each magnitude plot of Figure 11 occurs when the K
and Mω2 terms are equal. The imaginary data in Figure 12 is fit in a similar manner
using Equation 3.9b, which is the same as 2.5b:
27
imag ([ B (ω )]) = ωC + D,
(3.9b)
The imaginary data appears to be a sloped line, even with all the noise. In this case the
curve is fit with two matrix coefficient terms. D is the frequency independent damping
matrix, and is therefore equivalent to the y-intercept of the imaginary data. Viscous
damping is a linear function of frequency; therefore, the C matrix represents the slope of
the imaginary data in Figure 12. If the imaginary data appeared to have curvature, or a
higher order form, more terms could be added to Equation 3.9b.
As discussed previously, only two frequency points are required in order to solve for
the M, K, C, and D coefficients. More points results in an over-determine equation,
which gives a least squares best-fit solution. Any range of the frequency points may be
included or excluded, depending on data quality. The CMIF plot of the B matrix can be
used, along with the DSM plots, to identify good frequency ranges for processing data:
28
7
10
6
10
5
10
4
10
3
10
0
50
100
150
Freq (Hz)
Figure 13 CMIF of 9 degree of freedom DSM matrix.
Examining Figure 12, the data above 100 Hz has a significant increase in noise. Therefore,
the data between 0 and 100 Hz was used to curve fit the data. Another tool to improve the
data fit is enforcing the reciprocity assumption for cross terms in the DSM. All of the off
diagonal terms in the DSM are averaged with its compliment term (entry B1,4 with B4,1, for
example). The data is then fit using Equations 3.9a-b.
29
The resulting matrix coefficients using the averaged DSM, and 0-100 Hz frequency range,
are show below:
[D]calculated
6847.5
-5616.3
1915.3
-1696.5
-1053.5
1642.6
1102.7
-2143.7
399.8
-5616.3
6629.7
-3300.3
-949.9
913.8
-1588.6
-1007.3
1016.6
388.1
1915.3
-3300.3
3087.6
-1146.8
323.6
383.9
-761.9
84.9
186.1
-1696.5
-949.9
-1146.8
6123.7
-4435.2
1849.7
243.8
-37.7
-751.1
-1053.5
913.8
323.6
-4435.2
7420.2
-4876.8
1575.2
-849.1
124.4
1642.6
-1588.6
383.9
1849.7
-4876.8
7592
-4472.3
335.4
1379.1
1102.7
-1007.3
-761.9
243.8
1575.2
-4472.3
2245.9
2463.2
-3089.9
-2143.7
1016.6
84.9
-37.7
-849.1
335.4
2463.2
-2674.4
-263.5
399.8
388.1
186.1
-751.1
124.4
1379.1
-3089.9
-263.5
3265.1
[D]original
4000
-2000
0
0
0
0
0
0
0
-2000
4000
-2000
0
0
0
0
0
0
0
-2000
4000
-2000
0
0
0
0
0
0
0
-2000
4000
-2000
0
0
0
0
0
0
0
-2000
4000
-2000
0
0
0
0
0
0
0
-2000
4000
-2000
0
0
0
0
0
0
0
-2000
4000
-2000
0
0
0
0
0
0
0
-2000
4000
-2000
0
0
0
0
0
0
0
-2000
4000
Table 1. Structural Damping Matrices, Full D.O.F. Dataset.
30
[C]calculated
235.11
-111.51
-4.10
5.14
2.73
-1.09
-2.00
4.26
0.28
-111.51
235.78
-115.04
2.06
-1.67
2.25
3.09
-3.46
-0.01
-4.10
-115.04
239.02
-119.92
-1.59
-1.19
2.45
0.25
-2.26
5.14
2.06
-119.92
232.12
-116.87
-3.39
-0.46
-2.04
5.30
2.73
-1.67
-1.59
-116.87
233.92
-116.17
-3.54
2.42
-2.35
-1.09
2.25
-1.19
-3.39
-116.17
233.83
-115.42
0.23
-2.63
-2.00
3.09
2.45
-0.46
-3.54
-115.42
243.71
-130.70
7.14
4.26
-3.46
0.25
-2.04
2.42
0.23
-130.70
258.36
-122.99
0.28
-0.01
-2.26
5.30
-2.35
-2.63
7.14
-122.99
242.17
0
0
0
0
-120
240
-120
0
0
0
0
0
0
0
-120
240
-120
0
0
0
0
0
0
0
-120
240
-120
0
0
0
0
0
0
0
-120
240
0.00
-0.01
0.00
-0.01
0.02
9.99
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
9.99
0.01
0.00
0.01
-0.01
0.00
0.01
0.00
0.00
0.01
10.00
0.01
-0.02
0.01
0.00
-0.01
0.00
0.00
0.00
0.01
9.99
0
0
0
0
0
0
10
0
0
0
0
0
0
0
0
0
10
0
0
0
0
0
0
0
0
0
10
[C]original
240
-120
0
0
0
0
0
0
0
-120
240
-120
0
0
0
0
0
0
0
-120
240
-120
0
0
0
0
0
0
0
-120
240
-120
0
0
0
0
0
0
0
-120
240
-120
0
0
0
Table 2. Viscous Damping Matrices, Full D.O.F. Dataset.
[M]calculated
10.00
0.01
0.00
0.00
0.00
0.00
0.00
0.01
-0.02
0.01
10.00
0.00
-0.01
0.01
-0.01
0.00
-0.01
0.01
0.00
0.00
10.00
0.00
-0.01
0.00
0.00
0.00
0.00
0.00
-0.01
0.00
9.98
0.01
-0.01
0.00
0.01
-0.01
0.00
0.01
-0.01
0.01
9.99
0.02
0.00
0.00
0.00
[M]original
10
0
0
0
0
0
0
0
0
0
10
0
0
0
0
0
0
0
0
0
10
0
0
0
0
0
0
0
0
0
10
0
0
0
0
0
0
0
0
0
10
0
0
0
0
0
0
0
0
0
10
0
0
0
Table 3. Mass Matrices, Full D.O.F. Dataset.
31
[K]calculated
x10^3
2999.6
-1499.2
-0.2
0.4
-0.8
0
-0.1
3.1
-4.8
x10^3
3000
-1500
0
0
0
0
0
0
0
-1499.2
2999.5
-1499.7
-1.9
2.1
-1.9
1.1
-4
3.8
-0.2
-1499.7
2999.7
-1498.8
-1.4
0.4
0.3
-0.9
0.8
0.4
-1.9
-1498.8
2997
-1497.9
-1
-0.8
3
-2.6
-0.8
2.1
-1.4
-1497.9
2997
-1496.9
-0.5
-1.3
1.2
0
-1.9
0.4
-1
-1496.9
2997.2
-1498
-0.9
0.4
-0.1
1.1
0.3
-0.8
-0.5
-1498
2997.7
-1497.7
-1
3.1
-4
-0.9
3
-1.3
-0.9
-1497.7
2998.1
-1498.4
-4.8
3.8
0.8
-2.6
1.2
0.4
-1
-1498.4
2998.9
[K]original
-1500
3000
-1500
0
0
0
0
0
0
0
-1500
3000
-1500
0
0
0
0
0
0
0
-1500
3000
-1500
0
0
0
0
0
0
0
-1500
3000
-1500
0
0
0
0
0
0
0
-1500
3000
-1500
0
0
0
0
0
0
0
-1500
3000
-1500
0
0
0
0
0
0
0
-1500
3000
-1500
0
0
0
0
0
0
0
-1500
3000
Table 4. Stiffness Matrices, Full D.O.F. Data Set.
The M, K, and C matrices are fit with a small amount of error (~7.5% for the C entries).
However, the strong noise content on the imaginary DSM data produces almost random
matrix entries in the D matrix. Entries off the diagonal which should be zero are given
values within the same magnitude as the diagonal entries. Also, the signs of each entry
are inconsistent with the underlying physics of the system. A qualitative way to evaluate
the significance of the error is using the calculated matrices to synthesize the original
data.
32
Below, the synthesized data is plotted on top of the original data:
6
5
6
x 10
5
0
6
6
x 10
5
0
x 10
5
5
0
4
0
Real
0
x 10
-5
-10
-5
0
50 100
Freq (Hz)
-10
150 0
5
Imaginary
6
50 100
Freq (Hz)
-10
150 0
5
x 10
6
-5
-5
50
100
Freq (Hz)
-10
150 0
6
50 100
Freq (Hz)
-10
150 0
5
5
x 10
-5
x 10
6
50
100
Freq (Hz)
6
x 10
4
3
3
2
2
1
1
0
0
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
50 100
Freq (Hz)
150
-6
6
x 10
0
50 100
Freq (Hz)
150
x 10
6
6
x 10
5
0
-5
-5
-10
0
x 10
5
0
50 100
Freq (Hz)
150
-10
0
5
5
5
5
x 10
x 10
-6
150 0
6
6
6
6
x 10
x 10
6
6
5
5
5
5
5
5
5
5
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
-1
-1
-1
-1
-1
-1
-1
-1
-2
-2
-2
-2
-2
0
50 100
Freq (Hz)
150
0
50 100
Freq (Hz)
150
0
50 100
Freq (Hz)
150
0
50 100
Freq (Hz)
150
0
50 100
Freq (Hz)
-2
150 0
50 100
Freq (Hz)
150
-2
0
50 100
Freq (Hz)
150
-2
0
50 100
Freq (Hz)
150
5
x 10
5
-1
50 100
Freq (Hz)
150
-2
x 10
0
50 100
Freq (Hz)
150
Figure 14 DSM Plot of Synthesized ( | ) and Actual Driving Point Values, Real-Im (B11 to B99, Left to
Right)
Figure 14 displays the difficulty of fitting the damping data. The algorithm is able to
identify a slope in the data set, but the Y-intercept of the line is sensitive to the heavy noise
content in the data. The high variability in the driving point data is also represented in a
similar magnitude in the cross-term measurements. The same variability is therefore
observed in all measurements at the lower frequencies, and results in D matrix entries with
similar magnitudes. How does the structural damping error affect the overall system model?
Figure 15 through Figure 17 plot the synthesized data (using the M, K, C, and D matrices) in
a phase/magnitude DSM and FRF plot and also a CMIF plot respectively.
33
200
200
200
200
200
200
200
200
150
150
150
150
150
150
150
150
150
100
100
100
100
100
100
100
100
100
50
50
50
50
50
50
50
50
50
0
0
0
0
0
0
0
0
0
Phase
200
-50
0
7
-50
50 100 150 0
Freq (Hz)
7
10
10
-50
50 100 150 0
Freq (Hz)
7
10
-50
50 100 150 0
Freq (Hz)
7
10
-50
50 100 150 0
Freq (Hz)
7
10
6
Magnitude
6
6
10
6
10
6
4
0
10
50 100 150 0
Freq (Hz)
7
10
-50
50 100 150 0
Freq (Hz)
50 100 150
Freq (Hz)
7
10
6
10
5
5
7
-50
50 100 150 0
Freq (Hz)
10
10
6
10
6
10
10
5
10
10
7
10
-50
50 100 150 0
Freq (Hz)
6
10
10
-50
50 100 150 0
Freq (Hz)
10
5
10
50 100 150 0
Freq (Hz)
5
10
50 100 150 0
Freq (Hz)
5
10
50 100 150 0
Freq (Hz)
4
10
50 100 150 0
Freq (Hz)
5
10
50 100 150 0
Freq (Hz)
5
10
50 100 150 0
Freq (Hz)
5
10
50 100 150 0
Freq (Hz)
50 100 150
Freq (Hz)
Figure 15 DSM Plot of Synthesized ( | ) and Actual Driving Point Values ( | ), Phase-Magnitude.
50
50
50
50
50
50
50
50
0
0
0
0
0
0
0
0
0
-50
-50
-50
-50
-50
-50
-50
-50
-50
-100
-100
-100
-100
-100
-100
-100
-100
-100
-150
-150
-150
-150
-150
-150
-150
-150
-150
Phase
50
-200
0
-4
-200
50 100 150
0
Freq (Hz)
-4
10
10
-200
50 100 150
0
Freq (Hz)
-4
10
-200
50 100 150
0
Freq (Hz)
-2
10
-3
10
-5
-5
10
10
Magnitude
-6
10
-6
10
-5
10
-8
0
10
50 100 150
0
Freq (Hz)
-5
10
-6
10
-8
10
50 100 150
0
Freq (Hz)
-8
-8
10
50 100 150
0
Freq (Hz)
-4
10
-5
10
-5
10
-4
-5
10
-6
10
-6
10
-6
10
-6
10
-7
10
50 100 150
0
Freq (Hz)
50 100 150
Freq (Hz)
10
10
-7
10
-4
10
-200
50 100 150
0
Freq (Hz)
-3
-7
-7
-8
-4
10
-200
50 100 150
0
Freq (Hz)
10
10
10
10
-4
10
-5
-6
-7
10
-200
50 100 150
0
Freq (Hz)
10
10
-7
10
-3
10
-2
10
-5
-4
-6
-2
10
-200
50 100 150
0
Freq (Hz)
10
10
10
-200
50 100 150
0
Freq (Hz)
-7
10
-7
10
-7
10
-8
10
50 100 150
0
Freq (Hz)
-8
10
50 100 150
0
Freq (Hz)
-8
10
50 100 150
0
Freq (Hz)
-8
10
50 100 150
0
Freq (Hz)
50 100 150
Freq (Hz)
Figure 16 FRF Plot of Synthesized ( | ) and Actual ( | ) Driving Point Values, Phase Magnitude.
34
CMIF Plot of Synthesized and Original DSM Data
7
10
6
10
5
10
4
10
3
10
0
50
100
150
Freq (Hz)
Figure 17 CMIF Plot of Synthesized (Dashed Line) Actual DSM Data Sets.
All three above plots show little to no error between the original and synthesized data sets.
This calls into question the significance of the D matrix data.
3.3.2 Effect of Spatial Truncation
In the previous example, the DSM algorithm was applied to a system containing 9 degrees
of freedom, while the algorithm model also used 9 degrees of freedom. Real structures
contain an infinite number of degrees of freedom, and only a few degrees of freedom can be
measured during an experiment. The difference in system and experimental degrees of
freedom results in a spatial truncation of the DSM data. The truncation produces frequency
correlated effects [13] and thus induces errors in the calculated matrix coefficients. In some
cases, the truncation error in the DSM is small, and the resulting matrix coefficients are
35
sufficient enough to model the given frequency range. In other cases, the truncation
produces temporal effects in the data, and therefore, erroneous data fits. This example will
examine some of the features that signify a severe truncation in the dataset.
The data will again be generated by the system shown in Figure 1, which contains 50
lumped masses and 51 springs and dampers. The result will be a 50x50x1024 system
frequency response matrix, with data from 0 to 20 Hz. Like the previous example, 0.5%
random noise is introduced. This matrix is down-sampled in order to create spatially
truncated datasets. The information contained in each entry of the matrix will be the same
as the system matrix, only certain rows and columns of data have been excluded.
36
3.3.2.1 Marginal Spatial Truncation
The first truncated data set will have a 9x9x1024 FRF matrix. The driving point
FRF’s are displayed below:
50
0
Phase
-50
-100
-150
-200
0
10
Freq (Hz)
-2
0
0
0
0
0
0
-20
-20
-20
-20
-20
-20
-40
-40
-40
-40
-40
-40
-40
-40
-60
-60
-60
-60
-60
-60
-60
-60
-80
-80
-80
-80
-80
-80
-80
-80
-100
-100
-100
-100
-100
-100
-100
-100
-120
-120
-120
-120
-120
-120
-120
-120
-140
-140
-140
-140
-140
-140
-140
-140
-160
-160
-160
-160
-160
-160
-160
-160
-180
-180
-180
-180
-180
-180
-180
-200
20
0
10
Freq (Hz)
-3
-3
-4
-5
-5
-5
10
10
-7
-8
-8
0
10
Freq (Hz)
10
20
0
-8
10
Freq (Hz)
10
20
0
-7
10
-8
10
Freq (Hz)
10
20
0
10
20
0
10
20
0
10
20
0
-7
10
20
0
20
-7
10
-8
10
Freq (Hz)
10
Freq (Hz)
10
10
-8
10
Freq (Hz)
-6
10
10
-8
10
Freq (Hz)
-5
10
-6
-7
10
-8
10
Freq (Hz)
-5
10
-7
10
-4
10
10
-6
10
-3
-4
-5
20
10
10
10
-6
10
-7
10
-5
10
Freq (Hz)
-2
-3
-4
-200
20
0
10
10
10
10
-6
10
-7
10
-5
-180
10
Freq (Hz)
-2
-3
-4
-200
20
0
10
10
10
10
-6
10
-7
10
-5
10
Freq (Hz)
-2
-3
-4
-200
20
0
10
10
10
10
-6
10
Freq (Hz)
-2
-3
-4
-200
20
0
10
10
10
10
-6
10
Freq (Hz)
-2
-3
-4
-200
20
0
10
10
10
10
-6
10
Freq (Hz)
-2
-3
-4
-200
20
0
10
10
10
10
10
Freq (Hz)
10
10
10
-200
20
0
-2
10
10
Magnitude
0
-20
-2
10
10
0
-20
-8
10
Freq (Hz)
10
20
0
Figure 18 Driving Point FRF Matrix Plots for Spatialy Truncated Data Set.
The truncated data sets contain five modes between 0 and 20 Hz. The CMIF plot of the FRF
data also shows the five modes, but also shows the existence of residuals in the data set from
high frequency modes. Some of the noise can also be observed in the smaller singular
values:
37
CMIF of Truncated FRF Data Set (9 d.o.f.)
-2
10
-3
10
-4
10
-5
10
-6
10
-7
10
0
2
4
6
8
10
Freq (Hz)
12
14
16
18
20
Figure 19 CMIF Plot of Truncated FRF Data Set (9 d.o.f.).
Like the previous case, the imaginary component of the DSM is influenced more by the
random noise.
5
Real
8
5
x 10
8
5
x 10
8
5
x 10
8
5
x 10
8
5
x 10
8
5
x 10
8
5
x 10
8
5
x 10
6
7
7
7
7
7
7
7
7
5
6
6
6
6
6
6
6
6
4
5
5
5
5
5
5
5
5
3
4
4
4
4
4
4
4
4
2
3
3
3
3
3
3
3
3
1
2
2
2
2
2
2
2
2
0
1
1
1
1
1
1
1
1
-1
0
0
0
0
0
0
0
0
-2
-1
-1
-1
-1
-1
-1
-1
-1
-3
-2
-2
-2
-2
-2
-2
-2
-2
0
10
Freq (Hz)
10000
20
0
10
Freq (Hz)
20
0
10
Freq (Hz)
20
0
10
Freq (Hz)
20
0
10
Freq (Hz)
20
0
10
Freq (Hz)
20
0
10
Freq (Hz)
20
0
10
Freq (Hz)
20
-4
15000
15000
15000
15000
15000
15000
15000
15000
10000
10000
10000
10000
10000
10000
10000
10000
5000
5000
5000
5000
5000
5000
5000
5000
0
0
0
0
0
0
0
0
x 10
0
10
Freq (Hz)
20
10
Freq (Hz)
20
Imaginary
5000
0
-5000
0
10
Freq (Hz)
-5000
20
0
10
Freq (Hz)
-5000
20
0
10
Freq (Hz)
-5000
20
0
10
Freq (Hz)
-5000
20
0
10
Freq (Hz)
-5000
20
0
10
Freq (Hz)
-5000
20
0
10
Freq (Hz)
-5000
20
0
10
Freq (Hz)
-5000
20
0
Figure 20 Driving Point DSM Matrix Plots, Real/Imaginary, for Spatially Truncated Data Set.
38
Figure 20 also has the same qualitative form as the previous example. The real DSM
component appears to have a second order polynomial shape. The imaginary data appears
to be a linear, first order polynomial.
The DSM data is fit using M, K, C, and D matrix coefficients. The equations are overdetermined by using a frequency range of 0-18 Hz. The synthesized FRF real/imaginary
and CMIF plots are shown below:
50
50
50
50
50
50
50
50
0
0
0
0
0
0
0
0
0
-50
-50
-50
-50
-50
-50
-50
-50
-50
-100
-100
-100
-100
-100
-100
-100
-100
-100
-150
-150
-150
-150
-150
-150
-150
-150
-150
Phase
50
-200
0
10
Freq (Hz)
-2
-3
-3
-4
-5
-5
-5
-6
-6
10
-6
10
-7
-8
-8
0
10
Freq (Hz)
10
20
0
-8
10
Freq (Hz)
10
20
0
-7
10
-8
10
Freq (Hz)
10
20
0
10
20
0
10
20
0
-6
10
20
0
10
20
0
20
10
-8
10
Freq (Hz)
10
Freq (Hz)
-7
10
-8
10
Freq (Hz)
-6
10
-7
10
-8
10
Freq (Hz)
-5
10
10
-7
10
-8
10
Freq (Hz)
-5
10
-7
10
-4
10
10
-6
10
-3
-4
-5
20
10
10
10
-6
10
-7
10
-5
10
Freq (Hz)
-2
-3
-4
-200
20
0
10
10
10
10
-6
10
-7
10
-5
10
Freq (Hz)
-2
-3
-4
-200
20
0
10
10
10
10
-6
10
-7
10
-5
10
Freq (Hz)
-2
-3
-4
-200
20
0
10
10
10
10
10
Freq (Hz)
-2
-3
-4
-200
20
0
10
10
10
10
10
Freq (Hz)
-2
-3
-4
-200
20
0
10
10
10
10
10
Freq (Hz)
-2
-3
-4
-200
20
0
10
10
10
10
10
Freq (Hz)
10
10
10
-200
20
0
-2
10
10
Magnitude
10
Freq (Hz)
-2
10
10
-200
20
0
-8
10
Freq (Hz)
10
20
0
Figure 21 Synthesized ( | ) and Orig. ( | ) Driving Point FRF Measurements Using Truncated Datasets (9
d.o.f.).
39
-2
10
-3
10
-4
10
-5
10
-6
10
-7
10
0
2
4
6
8
10
Freq (Hz)
12
14
16
18
20
Figure 22 CMIF Plot of Synthesized (dotted) and Original FRF Data (9 d.o.f. Dataset).
Both Figure 21 and Figure 22 display some inaccuracy in the synthesized data. Most of the
modal frequencies are not calculated at their actual value. Note that the frequency shift is a
function of mass and stiffness error. The magnitude of the synthesized peaks appears to be
in agreement with the original date, which is a function of the system damping properties.
There appears to be more error in the fit of the real DSM data than in the imaginary DSM
data fit. For most applications, the mass and stiffness matrices will already have be
calculated by using some other method. The true goal of the research is to identify damping
properties; therefore, error in the mass and stiffness matrices is not of great concern. In
order to evaluate the damping matrix fit alone, the data is quasi-synthesized. The original
real DSM data is retained, and only the imaginary DSM data is synthesized using calculated
matrix coefficients. Inverting the modified synthesized DSM data creates a pseudosynthesized FRF matrix:
40
50
50
50
50
50
50
50
50
0
0
0
0
0
0
0
0
0
-50
-50
-50
-50
-50
-50
-50
-50
-50
-100
-100
-100
-100
-100
-100
-100
-100
-100
-150
-150
-150
-150
-150
-150
-150
-150
-150
Phase
50
-200
0
10
Freq (Hz)
-2
-3
-3
-4
-6
-6
-6
10
-7
-8
-8
0
10
Freq (Hz)
10
20
0
-7
10
10
20
0
-8
10
Freq (Hz)
10
20
0
10
20
0
10
20
0
-6
10
20
0
10
20
0
20
10
-8
10
Freq (Hz)
10
Freq (Hz)
-7
10
-8
10
Freq (Hz)
-6
10
-7
10
-8
10
Freq (Hz)
-5
10
10
-7
10
-8
10
Freq (Hz)
-5
10
-7
10
-4
10
10
-6
10
-7
10
-8
10
Freq (Hz)
-6
10
-3
10
-4
-5
20
-2
10
10
10
Freq (Hz)
10
-3
-4
-5
-200
20
0
10
10
10
-6
10
-7
10
-5
10
Freq (Hz)
-2
-3
-4
-200
20
0
10
10
10
10
-6
10
-7
10
-5
10
Freq (Hz)
-2
-3
-4
-200
20
0
10
10
10
10
10
Freq (Hz)
-2
-3
-4
-200
20
0
10
10
10
-5
10
Freq (Hz)
-2
-3
10
-200
20
0
10
10
-4
-5
10
-2
10
10
10
Freq (Hz)
10
-3
-4
-5
-200
20
0
10
10
10
10
Freq (Hz)
10
10
10
-200
20
0
-2
10
10
Magnitude
10
Freq (Hz)
-2
10
10
-200
20
0
-8
10
Freq (Hz)
10
20
0
Figure 23 FRF Plot of Pseudo-synthesized ( | ) and Original ( | ) Data (9 d.o.f.).
The resulting pseudo-synthesized FRF plots lay on top of the original data with only a little
error in the magnitude. Most of the error appears in the phase plot of Figure 23.
-2
10
Small
Deviation
-3
10
-4
10
-5
10
-6
10
-7
10
0
2
4
6
8
10
Freq (Hz)
12
14
16
18
20
Figure 24 CMIF Plot of Pseudo-synthesized and Original Data (9 d.o.f.).
41
The CMIF plot in Figure 24 agrees with the results of Figure 23. Each resonant frequency
demonstrates only minimal deviation between the synthesized and the original data sets.
Now, let’s examine the calculated matrix coefficients. Table 5 shows the M matrix, which
was calculated using the averaged DSM matrix by forcing reciprocity:
[M]calculated
39
10
0
0
0
0
0
0
0
10
39
10
0
0
0
0
0
0
0
10
39
10
0
0
0
0
0
0
0
10
39
10
0
0
0
0
0
0
0
10
39
10
0
0
0
0
0
0
0
10
39
10
0
0
0
0
0
0
0
10
39
10
0
0
0
0
0
0
0
10
39
10
0
0
0
0
0
0
0
10
44
Table 5. Calculated Mass Matrix, 9 D.O.F. Marginally Spatially Truncated Data Set.
The calculated mass matrix contains non-zero cross terms, or in other words, is no longer
diagonal as it should be (see Table 3). The truncation of degrees of freedom results in a
coupling of mass matrix elements. Figure 25 displays the real and imaginary components of
the second row of the 9 degree of freedom DSM. The blue curves are the original, truncated
data. The red curves are the synthesized data created from the calculated matrix
coefficients.
42
5
8
x 10
6
Real
8
x 10
8
5
5
5
x 10
8
x 10
8
x 10
8
5
x 10
8
5
x 10
8
6
6
6
6
6
6
6
4
4
4
4
4
4
4
4
2
2
2
2
2
2
2
2
2
0
0
0
0
0
0
0
0
0
-2
-2
-2
-2
-2
-2
-2
-2
-2
-4
10
Freq (Hz)
20
-4
-4
-4
0
Imaginary
x 10
6
4
5
5
5
8
0
10
Freq (Hz)
0
20
10
Freq (Hz)
20
10
Freq (Hz)
20
-4
-4
-4
0
0
10
Freq (Hz)
0
20
10
Freq (Hz)
20
-4
0
10
Freq (Hz)
20
x 10
-4
0
10
Freq (Hz)
20
0
14000
14000
14000
14000
14000
14000
14000
14000
14000
12000
12000
12000
12000
12000
12000
12000
12000
12000
10000
10000
10000
10000
10000
10000
10000
10000
10000
8000
8000
8000
8000
8000
8000
8000
8000
8000
6000
6000
6000
6000
6000
6000
6000
6000
6000
4000
4000
4000
4000
4000
4000
4000
4000
4000
2000
2000
2000
2000
2000
2000
2000
2000
2000
0
0
0
0
0
0
0
0
0
-2000
-2000
-2000
-2000
-2000
-2000
-2000
-2000
-2000
-4000
-4000
-6000
0
10
Freq (Hz)
20
-6000
-4000
0
10
Freq (Hz)
20
-6000
-4000
-4000
0
10
Freq (Hz)
20
-6000
0
10
Freq (Hz)
20
-6000
-4000
-4000
0
10
Freq (Hz)
20
-6000
0
10
Freq (Hz)
20
-6000
-4000
0
10
Freq (Hz)
20
-6000
10
Freq (Hz)
20
-4000
0
10
Freq (Hz)
20
-6000
0
10
Freq (Hz)
20
Figure 25. Second Row of DSM Matrix (B2,1 –B2,9) 9 D.O.F. Marginal Spatially Truncated Data Set.
The off diagonal entries in the calculated mass matrices are not anomalies of the curve fit
algorithm. As observed in Figure 25 mass effects are observed in the diagonal element (the
second plot from the left, B2,2 ), in the form of an inverted parabola (from the -Mω2 term).
The same inverted parabola shape is in the original data in entries B2,1 and B2,3 . All of the
other plots display data around zero. Therefore, the second row of the calculated mass
matrix contains three entries which have non-zero positive values. The reason for the nonzero entries is not known. However, the entry values do make sense when they are analyzed
in the same manner as the coupled K matrix. The magnitude of each entry’s value will be
discussed later, along with the values of the K, C, and D matrices.
43
[K]calculated
x10^2
6068
-2969
0
0
0
0
0
0
0
-2969
6068
-2969
0
0
0
0
0
0
0
-2969
6068
-2969
0
-1
0
0
0
0
0
-2969
6069
-2969
1
-1
0
1
52
-19
-1
1
-1
1
0
1
0
-19
52
-21
-1
1
-1
0
0
0
-1
-21
54
-20
-2
1
0
0
0
1
-1
-20
54
-20
-1
0
0
0
0
0
0
-2969
6069
-2970
1
0
0
0
0
-1
1
-2970
6069
-2969
0
0
0
0
0
-1
1
-2969
6069
-2969
0
0
0
0
0
0
0
-2969
6068
-2970
0
0
0
1
0
0
0
-2970
5603
1
-1
1
-1
-19
52
-21
-1
-2
0
0
0
0
-2
-21
52
-21
1
1
0
0
0
0
-1
-21
53
-23
0
0
0
0
0
-2
1
-23
54
-75
90
-78
61
-610
721
-543
16
98
41
7
-13
-32
112
-543
706
-491
-58
-64
11
13
26
-17
16
-491
588
-382
30
17
-15
14
-36
98
-58
-382
332
[C]calculated
-1
1
-2
-20
52
-19
-2
0
0
[D]calculated
715
-627
94
-76
63
-75
41
-64
30
-627
737
-535
69
-106
90
7
11
17
94
-535
598
-557
116
-78
-13
13
-15
-76
69
-557
641
-569
61
-32
26
14
63
-106
116
-569
734
-610
112
-17
-36
Table 6 Calculated Stiffness and Damping Matrices, 9 D.O.F. Marginally Spatially Truncated Data Set.
The remaining three matrix coefficients display strong diagonal and strong, negative, offdiagonal entries, same as the original system’s full d.o.f. matrices. The structural damping
matrix also contains non-zero entries further off the diagonal, whose values are within an
44
order of magnitude of the diagonal entry values. The error is expected due to the high
amount of noise on the imaginary component of the DSM, as shown in Figure 25.
Observe the magnitude of the diagonal and just off-diagonal entries of all the matrix
coefficients. Table 7 shows the values of each component of a 9 degree freedom system that
would also generate the matrices in Table 5 and Table 6 (assuming the same boundary
conditions as the 50 d.o.f. model):
Table 7 Calculated System Properties, 9 D.O.F. Truncated Data Set
Element
Number
1
2
3
4
5
6
7
8
9
10
M
K
C
D
49
49
49
49
49
49
49
49
54
N/A
309930
30993
30989
30992
30991
30994
30996
30987
29695
26334
33
31
34
34
32
31
31
31
23
32
88
201
41
71
124
178
214
205
382
-51
Referencing Figure 1, the original system properties for the 50 degree of freedom system
were: m=10 kg, k=1500000 N/m, c=120 N/m/s, D=2000 N/m. The values in Table 7 shows
the nominal effect of spatial truncation, which may be summed up as:
•
Mass is lumped to the measured degrees of freedom
•
Stiffness is decreased, in order to maintain the system’s modal properties
•
Damping is decreased in order to maintain the modal damping ratios
This study presents some preliminary ideas about how experimental models should be
expanded to the number of d.o.f.s of the analytical mode.
45
3.3.2.2 Excessive Spatial Truncation
The second truncated data set will have a 8x8x1024 FRF matrix. The FRF data set will be
identical to the previous 9x9 example, minus the fifth row and column of the FRF matrix.
The driving point FRF’s will appear the same as Figure 18, but without the middle
magnitude and phase plots. The resulting CMIF plot of the 8 d.o.f. FRF is shown in Figure
26:
-2
10
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
2
4
6
8
10
12
Freq (Hz)
14
16
18
20
Figure 26 CMIF Plot of Truncated FRF Data Set (8d.o.f.).
Comparing Figure 19 and Figure 26, both data sets contain the same number of resonant
frequencies, due to the required temporal properties. Figure 26 now displays an antiresonance around 19.5 Hz, instead of the mass line shown in Figure 19. The truncation of
spatial information resulted in an insufficient number of degrees of freedom for describing
the system within this frequency range. The anti-resonance at 19.5 Hz represents the change
46
in modal vectors associated with the singular value. Below 19.5 Hz, the lowest singular
value can be traced to the highest singular value at 3 Hz. The continuous curve between the
two features (peak and valley) represents the scale factor associated with the modal vector
of the first flexible mode. Above 19.5 Hz, the same singular value scales a new vector. The
spatial truncation can only allow for a maximum of eight orthogonal vectors. The ninth
flexible mode’s lower (flexible) residual begins to have a higher contribution than the first
flexible mode’s inertance residual. This is shown in Figure 27, which is the same data as
Figure 26, but with a maximum frequency of 35 Hz:
-2
10
Initial Modal
Vector
-3
10
New Modal
Vector
-4
10
-5
10
-6
10
-7
10
-8
10
Transition
Frequency
-9
10
-10
10
0
5
10
15
20
Freq (Hz)
25
30
35
Figure 27 Reflection Point in CMIF Plot of Truncated FRF data Set (8d.o.f.),
The CMIF of the 8 degree of freedom DSM is shown in Figure 28:
47
8
10
Cut-off
Frequency
7
10
6
10
5
10
4
10
3
10
2
10
0
2
4
6
8
10
Freq (Hz)
12
14
16
18
20
Figure 28 CMIF Plot of Truncated DSM data Set (8d.o.f.).
As shown in Section 3.2, the CMIF plot of the DSM is simply the FRF CMIF plot inverted.
The top curves of Figure 28 are the dominant singular values in the DSM. Above 14.5 Hz,
the DSM becomes dominated by spatially trucated data, and therefore cannot be used.
6
Real
5
6
x 10
5
5
6
x 10
5
6
x 10
5
6
x 10
5
6
x 10
5
6
x 10
5
0
0
0
0
0
0
0
0
-5
-5
-5
-5
-5
-5
-5
-5
-10
-10
-10
-10
-10
-10
-10
-10
-15
0
10
Freq (Hz)
20
-15
4
3
0
10
Freq (Hz)
20
-15
4
x 10
3
2.5
Imaginary
6
x 10
0
10
Freq (Hz)
20
-15
4
x 10
3
2.5
0
10
Freq (Hz)
20
-15
7
x 10
3
2.5
0
10
Freq (Hz)
20
-15
7
x 10
3
2.5
0
10
Freq (Hz)
20
-15
4
x 10
3
2.5
0
10
Freq (Hz)
20
-15
4
x 10
3
2.5
3
2.5
2
2
2
2
2
2
2
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1
1
1
1
1
1
1
1
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
0
0
-0.5
-0.5
-0.5
-0.5
-0.5
-0.5
-0.5
-0.5
10
Freq (Hz)
20
-1
0
10
Freq (Hz)
20
-1
0
10
Freq (Hz)
20
-1
0
10
Freq (Hz)
20
-1
0
10
Freq (Hz)
20
-1
0
10
Freq (Hz)
20
-1
10
Freq (Hz)
20
x 10
2.5
2
0
0
4
x 10
1.5
-1
x 10
0
10
Freq (Hz)
20
-1
0
10
Freq (Hz)
Figure 29. Driving Point DSM Matrix Plots, Real/Imaginary, for Spatially Truncated Data Set, 8 D.O.F.
48
20
Observing the DSM driving point measurements is the second graphical tool used to identify
the data’s quality. Figure 29 displays the absence of a critical degree of freedom between
Nodes four and five. Data at both nodes deviates from the expected polynomial form as
observed at the other nodes. In fact, the data now closely resembles the real and imaginary
components of an FRF. Note that the plot scales of Nodes 4 and 5 imaginary components
are different than the other nodes. Figure 30 zooms into the imaginary data at Nodes 4 and
5:
4
3
4
x 10
3
2.5
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
0
5
10
15
Freq (Hz)
20
x 10
0
5
10
15
Freq (Hz)
20
Figure 30 Imaginary DSM Component for Nodes 4 and 5.
The lower frequency data still retains the linear shape observed at the other nodes. The
curves begin to deviate from linear above 14 Hz, which agrees with the cut-off frequency
observed in the CMIF plot of the DSM data. Therefore, the matrix coefficients will be fit
using frequency data between 0 and 14 Hz. The results are shown below:
49
50
50
50
50
50
50
50
0
0
0
0
0
0
0
0
-50
-50
-50
-50
-50
-50
-50
-50
-100
-100
-100
-100
-100
-100
-100
-100
-150
-150
-150
-150
-150
-150
-150
-150
Phase
50
-200
0
10
Freq (Hz)
-200
20
0
-2
-2
10
-3
-3
-4
Magnitude
-5
-5
-6
-6
10
-7
-7
10
10
Freq (Hz)
10
20
0
-8
10
Freq (Hz)
10
20
0
10
Freq (Hz)
10
20
0
-6
20
10
10
Freq (Hz)
10
20
0
20
10
-8
0
10
Freq (Hz)
-7
10
-8
10
Freq (Hz)
-6
10
-7
10
-8
10
20
0
-5
10
10
-7
10
-8
10
Freq (Hz)
-5
10
-7
10
-4
10
10
-6
10
-7
10
-8
0
-6
10
-7
10
-8
-6
10
-3
-4
-5
20
10
10
10
10
Freq (Hz)
-2
-3
-4
-200
20
0
10
10
10
-5
10
Freq (Hz)
-2
-3
10
-200
20
0
10
10
-4
-5
-6
10
-2
10
10
10
Freq (Hz)
10
-3
-4
-5
-200
20
0
10
10
10
10
Freq (Hz)
-2
-3
-4
-200
20
0
10
10
10
10
10
Freq (Hz)
-2
-3
-4
-200
20
0
10
10
10
10
10
Freq (Hz)
10
10
10
-200
20
0
-2
10
10
10
10
Freq (Hz)
-8
10
Freq (Hz)
10
20
0
Figure 31 Synthesized ( | ) and Original ( | ) FRF Data for 8 d.o.f. Truncated Data Set.
The amplitudes of the synthesized FRF curves appear to agree with the original data set.
Error exists in the mass and stiffness terms, so the natural frequencies do not line up
between the synthesized and original FRFs. The same observation is made by looking at the
original and synthesized CMIF plots:
-2
10
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
0
2
4
6
8
10
Freq (Hz)
12
14
16
18
20
Figure 32 CMIF Plot of Original ( | ) and Synthesized (: ) Data sets, 8 d.o.f. Model.
50
The pseudo-synthesized FRFs, using calculated matrices for only the imaginary part of the
DSM, are calculated and plotted in Figure 33:
50
50
50
0
0
0
-50
-50
-50
100
50
50
50
50
0
0
0
0
-50
-50
-50
-50
-100
-100
-100
-100
-150
-150
-150
-150
50
Phase (degrees)
0
-50
-100
-100
-100
-150
-200
-150
0
10
Freq (Hz)
-2
10
Freq (Hz)
-3
-3
-4
-5
-5
-6
-6
10
-7
-7
10
10
Freq (Hz)
10
20
0
-8
10
Freq (Hz)
10
20
0
10
20
0
10
20
0
-6
20
10
10
Freq (Hz)
10
20
0
20
10
-8
0
10
Freq (Hz)
-7
10
-8
10
Freq (Hz)
-6
10
-7
10
-8
10
Freq (Hz)
-5
10
10
-7
10
-8
10
Freq (Hz)
-5
10
-7
10
-4
10
10
-6
10
-7
10
-8
0
-6
10
-7
10
-8
-6
10
-3
-4
-5
20
10
10
10
10
Freq (Hz)
-2
-3
-4
-200
20
0
10
10
10
-5
10
Freq (Hz)
-2
-3
10
-200
20
0
10
10
-4
-5
-6
10
-2
10
10
10
Freq (Hz)
10
-3
-4
-5
-200
20
0
10
10
10
10
Freq (Hz)
-2
-3
-4
-200
20
0
10
10
10
10
10
Freq (Hz)
-2
-3
-4
-200
20
0
10
10
10
10
10
Freq (Hz)
10
10
10
-200
20
0
-150
-2
10
10
10
-200
20
0
-150
-2
10
Magnitude (m/N)
-100
-8
10
Freq (Hz)
10
20
0
Figure 33 Pseudo-synthesized ( | ) and Original ( | ) FRF Data for 8 d.o.f. Truncated Data Set.
The modified synthesized data plots on top of the original data, with little observable error.
Therefore, the [C] and [D] matrices appear to have fit the data reasonably well, even with
the error in the DSM data that was induced by spatial truncation. Next, let’s examine the
damping matrices themselves:
51
[D]calculated
856
-581
118
-55
-10
32
-45
41
-581
917
-493
26
-13
23
-39
11
118
-493
740
-423
41
-50
51
-12
-55
26
-423
53
-732
1
15
-41
-10
-13
41
-732
42
-443
-60
71
32
23
-50
1
-443
803
-400
-109
-45
-39
51
15
-60
-400
675
-309
41
11
-12
-41
71
-109
-309
518
0
-1
1
0
-23
49
-24
2
0
1
-1
0
2
-24
51
-25
-1
0
0
1
-1
2
-25
49
0
0
0
0
9
37
9
0
0
0
0
0
0
9
37
9
0
0
0
0
0
0
9
41
-1
1
0
-1
-2990
6023
-2989
0
1
-1
1
0
0
-2989
6023
-2991
0
0
-1
0
0
0
-2991
5534
[C]calculated
48
-20
-2
1
0
0
0
-1
-20
47
-22
-1
0
-1
1
0
-2
-22
50
-23
-1
1
-1
0
1
-1
-23
56
7
0
0
1
0
0
-1
7
57
-23
2
-1
[M]calculated
37
9
0
0
0
0
0
0
9
37
9
0
0
0
0
0
0
9
37
9
0
0
0
0
0
0
0
32
69
9
0
0
[K]calculated
x10^2
6023
-2990
0
0
0
-1
1
0
0
0
9
69
32
0
0
0
-2990
6023
-2990
0
0
1
-1
0
0
-2990
6023
-2990
-1
0
1
-1
0
0
-2990
4680
-1342
-1
0
0
0
0
-1
-1342
4680
-2990
0
0
Table 8 Damping Matrices Calculated From 8 d.o.f. Spatially Over-truncated Data Set.
52
The above matrices show a localized affect of spatial truncation. The previous 9 d.o.f.
spatially truncated example displayed the scaling phenomenon when going from describing
a 50 d.o.f. system with only 9 d.o.f. (Table 7). However, the observed values were
confounded with the fact that each nodal point in the algorithm model was equally
distributed across the model. Each node should have similar mass properties, and similar
stiffness and damping values should exist between nodes. The results in Table 7 display this
property well. The lost degree of freedom in the 8 d.o.f. model is located between Nodes
four and six; the system characteristics are no longer evenly distributed.
3.3.2.3 Complex Structures and Spatial Truncation
The analytical models in section 4 and the experimental tests of Section 5 all focused on
simple continuous systems. In each case (except for experiment Case 3), the systems
contained evenly distributed mass, stiffness, and internal damping properties. Any results
and conclusions about spatial truncation are confounded with these systems. Some initial
analytical work examined the effects of analyzing complex structure containing abrupt
changes in mass, stiffness, or damping. The resulting DSM data sets were extremely
sensitive to spatial truncation, and introduced behaviors that were not observed in the
experimental data. Due to the project’s scope, efforts focused on identifying DSM
algorithm issues resulting from simple systems. Future experimental and analytical work
should focus on systems with much more complex properties.
53
3.4 Analytical Model Conclusions
Three main topics were discussed in this section:
•
Matrix inverses and the effects on DSM data content
•
Graphical tools for observing and evaluating DSM characteristics
•
System degrees of freedom and the effect of spatial truncation on the DSM
calculation and curve fit.
The matrix inverse of frequency response matrices is not the same as inverting a single entry
of the matrix. Through the singular value decomposition, it was shown that the matrix
inverts only inverses the singular values of the FRF matrix. The row and column vectors do
not change form. The resulting DSM contains an amplification of small FRF singular
values and an attenuation of dominant values. Residual modes and noise comprise the small
contributions of the FRF and therefore dominate the DSM data. The CMIF plot is a good
tool for observing these small and large singular values, and offers a method for identifying
noise levels in the data. Plotting the real and imaginary components of the DSM allow the
user to identify the data quality and the effectiveness of applying a polynomial model to the
system. The same graphical techniques are also effective at evaluating the resulting curve
fit. Finally, the DSM calculation was shown to be sensitive to the number of measurement
degrees of freedom used versus the number of significant degrees of freedom in the system
under analysis. The amount of allowable spatial truncation is dependent upon the frequency
range of interest and the dynamic characteristics of the system. Over-truncation of spatial
degrees of freedom can cause error in the temporal data, and therefore error in the DSM
54
calculation. The error may be significant enough to invalidate the polynomial model
assumption for damping identification.
55
4 Experimental Spatial Damping Matrix Identification Using
DSM
The DSM damping matrix identification algorithm utilizes FRFs as an input. From a modal
analysis perspective, measuring good FRFs is a mature and well developed experimental
technique. All of the classical experimental techniques required for good FRF
measurements apply to the input data for the DSM algorithm. However, there are a few
more considerations that must be applied for the DSM algorithm. This section will discuss
many of those additional requirements.
Three experimental examples are included. Each case study utilized simple structures tested
under two different conditions (the set of conditions are different for each example). The
different test arrangement must produce a measurable change in damping properties which
is verified by an established analysis technique. For all three case studies, a modal
parameter estimation algorithm was able to observe significant changes between the two test
arrangements. The modal parameters are tabulated at the beginning of each case study.
Each case studied different damping mechanisms and/or boundary conditions in order to
identify strengths and weaknesses of the DSM calculation and matrix coefficient polynomial
fit. One structure was suspended in a free-free condition with soft springs. The locations of
the spring supports were moved between tests in order to change the boundary conditions,
while keeping the structure’s properties constant. The second system was suspended in a
56
free-free condition by a thin wire. Localized damping was added to the structure between
tests. The third structure was mounted in a fixed-fixed boundary condition. Localized
damping was added to the structure between tests. The test combinations asked three
questions: can good DSM data be collected from a real system, will the algorithm be
sensitive enough to discriminate between the two different test conditions, and will the
algorithm identify spatially localized properties of each test conditions.
4.1 Hardware Considerations and Sensor Calibration
Data collection during an experiment seeks to observe the behavior of a system without
influencing the system, or misrepresenting the system. During any experiment, there exists
the possibility of interaction between the test component and the sensors measuring the
component’s response. The interaction can either be physical, or can occur within (or
between) the transducers. Several steps were taken to minimize both of these effects during
the experimental examples.
Physical interaction between the test component and transducer can result in temporal and
spatial error within the dataset. The mass of each sensor can load the system, and shift the
natural frequencies of the system under examination. The sensor and its adhesive can
influence the system’s damping. The experiments described in Section 4.4 were initially
performed with PCB UT333M07 accelerometers (4 gm mass, ~127 mm2 contact area) with
hot glue adhesive. The tests were repeated using PCB UT352B22 accelerometers (0.5 gms
mass, ~32 mm2 contact area) with superglue. Table 9 contains the modal properties of the
component/sensor combination:
57
Table 9 Modal Properties of a Component Calculated from Two Different Combinations of Sensors.
Modal Frequency (Hz)
UT333
UT352
233
247
645
684
1245
1339
Mode
Number
1
2
3
% Difference
5
6
7
Modal Damping (%)
UT333
UT352
1.028
0.432
0.865
0.064
1.253
0.072
% Difference
58
92
94
The different sensor mass did have an observable affect on the temporal properties of the
component, as noted by the 5-7% shift in natural frequencies. However, the sensor/adhesive
properties had a greater affect on the modal damping properties. The physical sensor effects
also resulted in a qualitative change in the measurements. Figure 34 and Figure 35 display
the driving point DSM real and imaginary measurements of the same component with
UT333 and UT352 sensors, respectively.
6
Real
1.5
6
x 10
1.5
1.5
6
x 10
1.5
6
x 10
1.5
6
x 10
1.5
1
1
1
1
1
1
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
-0.5
-0.5
-0.5
-0.5
-0.5
-0.5
-1
-1
-1
-1
-1
-1
-1.5
-1.5
-1.5
-1.5
-1.5
-1.5
-2
-2
-2
-2
-2
-2
-2.5
-2.5
-2.5
-2.5
-2.5
-2.5
0
500
1000
Freq (Hz)
1500
5
1.5
Imaginary
6
x 10
0
500
1000
Freq (Hz)
1500
5
x 10
1.5
0
500
1000
Freq (Hz)
1500
5
x 10
1.5
0
500
1000
Freq (Hz)
1500
5
x 10
1.5
0
500
1000
Freq (Hz)
1500
5
x 10
1.5
1.5
1
1
1
1
1
1
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
-0.5
-0.5
-0.5
-0.5
-0.5
-0.5
-1
-1
-1
-1
-1
-1
-1.5
-1.5
-1.5
-1.5
-1.5
-1.5
-2
-2
-2
-2
-2
-2
-2.5
-2.5
-2.5
-2.5
-2.5
-2.5
-3
-3
-3
-3
-3
-3
-3.5
-3.5
-3.5
-3.5
-3.5
-3.5
0
500
1000
Freq (Hz)
1500
-4
0
500
1000
Freq (Hz)
1500
-4
0
500
1000
Freq (Hz)
1500
-4
0
500
1000
Freq (Hz)
1500
-4
0
0
500
1000
Freq (Hz)
1500
500
1000
Freq (Hz)
1500
5
x 10
0.5
-4
x 10
500
1000
Freq (Hz)
1500
-4
x 10
0
Figure 34 Driving Point DSM Measurements of Bar using PCB UT333 Accels and Hot Glue Adhesive
58
9
9
Real
x 10
9
x 10
9
x 10
9
x 10
x 10
6
6
6
6
6
4
4
4
4
4
4
2
2
2
2
2
2
0
0
0
0
0
0
-2
-2
-2
-2
-2
-2
-4
-4
-4
-4
-4
-4
-6
-6
-6
-6
-6
-6
-8
-8
-8
-8
-8
-8
-10
0
500 1000
Freq (Hz)
1500
-10
9
1
Imaginary
9
x 10
6
0
500 1000
Freq (Hz)
1500
-10
9
x 10
1
0
500 1000
Freq (Hz)
1500
-10
9
x 10
1
0
500 1000
Freq (Hz)
1500
-10
9
x 10
1
0
500 1000
Freq (Hz)
1500
-10
9
x 10
1
1
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
-0.5
-0.5
-0.5
-0.5
-0.5
-0.5
-1
-1
-1
-1
-1
-1
-1.5
-1.5
-1.5
-1.5
-1.5
-1.5
0
500 1000
Freq (Hz)
1500
-2
0
500 1000
Freq (Hz)
1500
-2
0
500 1000
Freq (Hz)
1500
-2
0
500 1000
Freq (Hz)
1500
-2
0
500 1000
Freq (Hz)
1500
500 1000
Freq (Hz)
1500
9
x 10
0.5
-2
0
500 1000
Freq (Hz)
1500
-2
x 10
0
Figure 35 Driving Point DSM Measurements of Bar using PCB UT352 Accels and Super Glue Adhesive.
Both the real and imaginary data components in Figure 35 display similar characteristics to
the analytical data in Figure 20. Both are continuous at almost all frequencies, and an
identifiable shape, as opposed to random noise, dominates the data over the entire frequency
range. The imaginary components of Figure 34 contain discontinuous data and do not
display an identifiable shape across the frequency range. Continuous data and a dominant
shape across the frequency range are necessary for having confidence in the results
generated by the polynomial curve fit of the DSM algorithm.
The above observations represent a sensor’s affect on the structure under study. The sensors
electromechanical properties can also induce error in the spatial data generated in the
experiment. Lee and Kim discuss the effects of phase shift as a function of frequency and
59
the effects on the DSM algorithm [1, 2]. The experimental examples inSsections 4.4
through 4.6 take the phase calibration vs. frequency a step further by using a complex
calibration function. All of the experimental examples used a modal hammer as a force
input. The hammer and accelerometer were ratio calibrated using a swinging mass.
Ar
F
Mass
Modal
Hammer
Accel
Load
Cell
Figure 36 Ratio Calibration Set-up
From Newton’s Law:
Fr (ω ) = mAr (ω )
(4.1)
The subscript r denotes the particular accelerometer being calibrated to the given impact
hammer. In FRF form, Equation 4.1 becomes:
Ar
(ω ) = 1
Fr
m
(4.2)
Experimental FRF data is a ratio of the load cell and accelerometer voltages:
60
Cal r (ω ) ∗
V Ar
(ω ) = 1
VF
m
(4.3)
Engineering units of m/s2/N are achieved by the frequency dependent calibration value. For
a given load cell/accelerometer pair, the calibration array becomes:
Cal r (ω ) =
1 VF
(ω )
∗
m V Ar
(4.4)
The values of the voltage FRF will be complex, and therefore the Calr array will also made
up of complex numbers. Applying the calibration array will scale the experimental FRFs for
variations in phase and in magnitude as a function of frequency. Each of the Calr arrays are
stored in a diagonal matrix:
0
0
0
Cal1 (ω )
0
0
Cal 2 (ω ) 0
0
0
O
M
0
L Cal k (ω ) k ×k × N
0
(4.5)
N is the number of frequency points. The experimental FRFs are calibrated to engineering
units via Equation 5.6:
[Cal (ω )]k ×k × N [H exp ]k ×k × N = [H calibrated ]k×k × N
(4.6)
61
4.2 Spatial Sensitivity
The DSM damping identification algorithm was shown to be sensitive to spatial truncation
(Section 3.3). Real structures will always have an infinite number of modes, therefore
spatial truncation is inevitable. Graphical tools were introduced in Section 3.3 in order to
help identify when the truncation is excessive.
Unfortunately, many of the data validation tools require a minimum number of experimental
spatial degrees of freedom and a complete (square) FRF matrix. A significant amount of
spatial and temporal data points are required in order to validate an experiment set-up.
Therefore, experimental damping matrix identification testing should be initiated after other
temporal/spatial tools have been applied. A quality experimental modal model of the
system will give insight to the system’s modal density and comparison of the resulting
modal vectors (MAC) will bring intuition about the instrumentation’s ability to distinguish
modes within the system. Modal analysis will not validate the magnitude or spatial
accuracy of residuals in the data set. Other methods may be required (column by column
orthogonality check of the SVD vectors, for example). If a finite element model exists, it
can give insight in the minimum number of measurement degrees of freedom by performing
reductions on the analytical data [16]. The primary point is that experimentally applying the
DSM algorithm should not be the primary step when analyzing a structure. The method’s
data requirements are significant, and other simpler methods can help identify potential
pitfalls before investing time, effort, and equipment.
62
4.3 Discussion of Experimental Evaluation Definitions
The experimental examples in this work seek to identify the benefits, shortcomings, and
general features of the DSM algorithm when used on real-world structures. To make this
task more straightforward, some terminology must be defined or clarified. Any evaluation
requires some type of benchmark, or expectation, and an appropriate measurement in which
to compare items to that standard. In this investigation, the DSM algorithm results will be
analyzed qualitatively and quantitatively by how they compare to several reference points:
the analytical examples in Section 3, physical properties dictated by the test set-up (structure
and sensors), and results from well-established experimental modal analysis techniques.
The work in this thesis is still in the preliminary stages of validating the DSM algorithm.
Therefore, the experimental work below does not attempt to provide a pass/fail judgment
upon the DSM algorithm. Instead, the benchmarks and correlating measurements attempt to
identify encouraging features of the algorithm and areas of concern. Further study will be
required in order to verify if the algorithm’s properties are significant, or useless.
63
4.4 Experimental Case Study 1: Elastomeric Soft-Spring
Boundary Conditions
The first experimental test used a monolithic aluminum bar with dimensions given in Figure
37
408mm
73mm
1
2
3
4
5
6
7.6mm
Figure 37 Aluminum Bar Dimensions
The bar was divided up into six measurement locations. The points were about 76 mm
apart and symmetric across the middle of the beam. All of the measurement points were
located along the center of the beam in order to only excite and measure transverse bending
modes. The hammer input went into the measurement locations on the top surface. The
accelerometers were located on the underside of the beam, directly below the measurement
locations (Figure 38):
64
Figure 38 Set-up and Sensor Location for Free-Free Boundary Condition
The bar was suspended by bungee cords wrapped in latex tubing. The cords acted as soft
springs, and as energy removal mechanisms. For the first test, the bungee cords were
symmetrically located between the outer pairs of measurement points. The second test relocated
the springs inward one measurement point, as displayed in Figure 38. Driving point FRFs were
measured at every location, thus producing a 6x6xN FRF matrix.
This experiment was initiated with several objectives:
•
Create a measurable change in a structure’s damping properties through a change in
boundary conditions.
•
Evaluate the quality and spatial truncation of the experimental data using several
pre-processing graphical methods.
•
Fit the data and use several post-processing graphical methods to validate the quality
of the polynomial fit.
•
Differentiate between the two test set-ups by identify the location and magnitude of
localized structural properties.
65
The changing boundary conditions are intended to modify the dynamic behavior of the
structure without changing mass or stiffness properties. This objective is unique to Case
Study 1; the remaining objectives are consistent for all of the experimental cases.
4.4.1 Modal Analysis of Both Tests Arrangements
Modal frequencies, mode shapes, and percent damping were calculated for both test set-ups
in order to compare with the DSM identification results. The DSM algorithm should be able
to identify a similar magnitude of difference between the tests, and display the localized
effects of the bungee cords.
Three modes were observed between 0 and 1600 Hz for both boundary conditions. Table 10
contains the modal properties from a polyreference time domain (PTD) fit of the impact data.
PTD Modal Fit of Bungee Cord Tests
Location of
Bungee Cords
Mode Number
Frequency
(Hz)
Damping (%)
1,2 / 5,6
Test 1
1
247.4
0.4316
2
683.9
0.0637
3
1339.3
0.0715
1
247.7
0.1700
2
685.6
0.2170
3
1339.2
0.0701
2,3 / 4,5
Test 2
Table 10 Modal Properties of Bungee Cord Tests
66
Table 10 displays a negligible difference in modal frequencies between the two sets of boundary
conditions. Damping values for two modes do change significantly with the change in B.C.’s.
The changing damping values are most likely due to the interaction of the cords and the
participation of the nodes at the contact points. Figure 39 displays the mode shapes for the first
set of boundary conditions:
BC at 1,2-5,6
BC at 2,3-4,5
Figure 39 Mode Shapes for Bungee Cord Tests, B.C.’s Between Locations 1,2 and 5,6.
The solid (blue) lines in the wire frames of Figure 39 denote the locations of the bungee
cords for the first test, and the dashed (red) lines are the support locations for the second
test. For Mode 1, the blue and red support locations display a similar amount of
displacement. If anything, the first set of support points, between 1,2 and 5,6, display
slightly more displacement than the second set of support points. This would explain the
67
different damping values for Mode 1, 0.4% and 0.2%, for Test 1 and Test 2 respectively.
Mode 2 displayed the greatest difference in damping for the two tests. For Test 1, the
supports were located at a node, and therefore experience little to no displacement and a
damping ratio of 0.0637%. The supports in Test 2 were located at a location of maximum
displacement, and a damping ratio of 0.217 %, a 240% increase over the damping ratio of
Test 1. Mode 3 displayed a relatively small change in damping between the two tests.
Figure 39 shows both sets of supports sitting on node points for Mode 3. Therefore, the
supports were not allowed to participate in Mode 3 due to the minimal displacement at their
contact points.
4.4.2 DSM Algorithm Identification of Spatial Damping: Case 1
The DSM algorithm should be able to discriminate between the two tests and also should be
able to identify the locations of the supports.
4.4.2.1 Graphical Evaluation of Experimental Data and Calculation of
DSM
The same graphical tools introduced in Section 3 will be used to identify the quality of the
experimental data. The CMIF plot for the FRF data in Tests 1 and 2 are displayed in Figure
40:
68
CMIF Plot of Test 1 FRF Data
-4
10
-5
-5
10
10
-6
-6
10
10
-7
-7
10
10
-8
-8
10
10
-9
-9
10
10
-10
-10
10
10
-11
-11
10
10
-12
10
CMIF Plot of Test 2 FRF Data
-4
10
-12
0
200
400
600
800
Freq (Hz)
1000
1200
1400
10
1600
0
200
400
600
800
Freq (Hz)
1000
1200
1400
1600
Figure 40 CMIF Plots of FRF Data from Tests 1 and 2
Both curves are qualitatively similar. The three modes are obvious in both plots and all the
residuals appear to be well defined (greater than the noise floor). Two of the residuals
appear to identify the bar’s rigid body modes. The test set-up attempted to restrict the
observed rigid body modes to the vertical translation and rotation around the y axis (Figure
39). The CMIF plots of the DSM data are displayed in Figure 41. As expected, they are
simply the inverse of the data shown in Figure 40.
CMIF Plot of DSM Data from Test 1
11
10
10
10
9
10
8
10
7
10
6
10
5
10
4
10
10
9
10
8
10
7
10
6
10
5
10
4
10
3
3
10
CMIF Plot of DSM Data from Test 2
11
10
10
0
200
400
600
800
Freq (Hz)
1000
1200
1400
1600
10
0
200
400
600
800
Freq (Hz)
1000
1200
1400
1600
Figure 41 CMIF Plot of DSM Data from Bungee Cord Tests 1 and 2
The majority of the DSM’s noise appears to exist at lower frequencies (below 400 Hz), and
is confounded with the highest DSM singular value. Therefore, the data should be curve fit
with data above 400 Hz. One caveat, the selected minimum frequency will exclude data
from the first resonant frequency.
69
800
500
500
700
400
600
300
500
200
400
100
300
500
600
0
500
0
Phase
400
300
-500
200
-500
100
-1000
0
200
-100
100
-200
0
0
-500
0
-300
-100
-200
0
500
1000
Freq (Hz)
1500
-4
0
500
1000
Freq (Hz)
1500
-4
10
10
-8
10
0
500
1000
Freq (Hz)
1500
-400
-100
0
500
1000
Freq (Hz)
1500
-4
10
-6
-6
-1500
-4
10
10
Magnitude
-1000
-6
0
500
1000
Freq (Hz)
1500
-4
10
-1000
0
500
1000
Freq (Hz)
1500
0
500
1000
Freq (Hz)
1500
-4
10
10
-6
10
-6
10
-200
10
-6
10
-8
-8
10
-8
10
-8
10
10
-8
10
-10
10
-10
10
-10
-10
-10
10
10
10
-10
10
-12
10
0
500
1000
Freq (Hz)
1500
0
500
1000
Freq (Hz)
1500
0
500
1000
Freq (Hz)
1500
0
500
1000
Freq (Hz)
1500
0
500
1000
Freq (Hz)
1500
Figure 42 Bode Plot of Test 1 FRF Data
The driving point magnitude plots appear clean with lightly damped peaks paired with antiresonances. The phase plot displays an interesting pattern. The driving point phase does not
cycle between 0 and 180 degrees. Instead, the phase at the anti-resonances appears to fall.
This can be an artifact of several causes including small leakage error (can independently
occur on any signal) and error in Matlab’s phase unwrap command. The FRF’s real and
imaginary plots appear normal in Figure 43:
70
-7
-8
x 10
-8
-8
x 10
3
0
1.5
2
1
-2
-0.5
0.5
0
0
0
-1
-1
-0.5
-2
0
-7
500
1000
Freq (Hz)
1500
-8
x 10
1500
-8
-8
-2
-3
-7
0
0
-1
-0.5
-2
-1
-4
-4
-3
-5
-5
-4
-3
-1.5
-6
-1.4
-1.6
1000 1500
Freq (Hz)
x 10
x 10
-3
-1
-1.2
500
-1
-2
-2
500 1000 1500 2000
Freq (Hz)
x 10
-1
-1
-0.6
500
1000
Freq (Hz)
0
0
-0.2
-0.8
0
-8
0
-0.4
1500
x 10
x 10
0
Imaginary
500
1000
Freq (Hz)
-1
-2.5
-4
1500
-0.5
-1.5
-3
-3
-4
500
1000
Freq (Hz)
0
-1
-2
-2
-6
0.5
1
1
-4
-1
1
2
3
2
0
x 10
2.5
4
2
-7
x 10
4
0.5
Real
-8
x 10
x 10
4
1
-6
-7
-7
-8
-4
-2
-1.8
-5
-5
-2
-9
-8
500
1000
Freq (Hz)
1500
500
1000
Freq (Hz)
1500
500
1000
Freq (Hz)
1500
-2.5
500
1000
Freq (Hz)
1500
500
1000
Freq (Hz)
1500
500
1000
Freq (Hz)
1500
Figure 43 Real, Imaginary Plot of Test 1 FRF Data
All of the imaginary components in Figure 43 have the same sign at the resonance
frequencies. The same components displayed a small sign change (became positive) at the
anti-resonant frequencies, where the response signal is small. Noise or leakage errors could
be dominating the signal at the nodes, thus causing the sign change in the imaginary data,
and the falling phase angle. The phase error could be one predictor of data quality going
into the DSM algorithm.
71
Next, the real and imaginary components of the DSM for both tests are examined. Three
features should exist in the data. First, both components should display continuity over the
entire frequency range. Assuming the system is linear (required for valid FRFs), the real
and imaginary DSM components must exist and have real values for
-∞<ω<+∞ :
imag ([ B (ω )]) = ωC + D,
(3.9a)
or the general form, modified from Equation 3.8:
N
imag ([ B(ω )] ) = ∑ [C r ]ω r −1
(4.7)
real ([ B(ω )]) = K − Mω 2
(3.9b)
r =1
and the real component:
Discontinuities can represent excessive levels of noise, or an insufficient number of spatial
degrees of freedom.
Figure 44 and Figure 45 display the driving point real and imaginary components of the
experimental DSM. Only driving point measurements are displayed since they contain most
of the significant observations. Non-driving point (off-diagonal) measurements will be
presented when unique features exist and require comment.
72
9
Real
8
9
x 10
9
x 10
8
8
9
x 10
8
9
x 10
8
6
6
6
6
6
6
4
4
4
4
4
4
2
2
2
2
2
2
0
0
0
0
0
0
-2
-2
-2
-2
-2
-2
-4
-4
-4
-4
-4
-4
-6
-6
-6
-6
-6
-6
-8
-8
-8
-8
-8
-8
-10
-10
-10
-10
-10
-10
0
500
1000
Freq (Hz)
1500
0
9
2.5
Imaginary
9
x 10
8
500
1000
Freq (Hz)
1500
0
9
x 10
1500
9
x 10
2.5
500
1000
Freq (Hz)
500
1000
Freq (Hz)
1500
9
x 10
2.5
0
2.5
0
500
1000
Freq (Hz)
1500
9
x 10
2.5
2.5
2
2
2
2
2
2
1.5
1.5
1.5
1.5
1.5
1
1
1
1
1
1
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
-0.5
-0.5
-0.5
-0.5
-0.5
-0.5
-1
-1
-1
-1
-1
-1
-1.5
-1.5
-1.5
-1.5
-1.5
-1.5
-2
-2
0
500
1000
Freq (Hz)
1500
-2.5
-2
0
500
1000
Freq (Hz)
1500
-2.5
-2
0
500
1000
Freq (Hz)
1500
-2.5
-2
0
500
1000
Freq (Hz)
1500
-2.5
0
500
1000
Freq (Hz)
1500
500
1000
Freq (Hz)
1500
9
x 10
1.5
-2.5
x 10
x 10
-2
0
500
1000
Freq (Hz)
1500
-2.5
0
Figure 44 DSM Driving Point Measurement for Test 1, Bungee Cord Boundary Conditions.
9
Real
8
9
x 10
8
8
9
x 10
8
9
x 10
8
9
x 10
8
6
6
6
6
6
6
4
4
4
4
4
4
2
2
2
2
2
2
0
0
0
0
0
0
-2
-2
-2
-2
-2
-2
-4
-4
-4
-4
-4
-4
-6
-6
-6
-6
-6
-6
-8
-8
-8
-8
-8
-8
-10
-10
-10
-10
-10
-10
0
500
1000
Freq (Hz)
1500
9
2.5
Imaginary
9
x 10
0
500
1000
Freq (Hz)
1500
9
x 10
2.5
0
500
1000
Freq (Hz)
1500
9
x 10
2.5
0
500
1000
Freq (Hz)
1500
9
x 10
2.5
0
500
1000
Freq (Hz)
1500
9
x 10
2.5
2.5
2
2
2
2
2
2
1.5
1.5
1.5
1.5
1.5
1
1
1
1
1
1
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
-0.5
-0.5
-0.5
-0.5
-0.5
-0.5
-1
-1
-1
-1
-1
-1
-1.5
-1.5
-1.5
-1.5
-1.5
-1.5
-2
-2
0
500
1000
Freq (Hz)
1500
-2.5
-2
0
500
1000
Freq (Hz)
1500
-2.5
-2
0
500
1000
Freq (Hz)
1500
-2.5
-2
0
500
1000
Freq (Hz)
1500
-2.5
0
500
1000
Freq (Hz)
1500
500
1000
Freq (Hz)
1500
9
x 10
1.5
-2.5
x 10
x 10
-2
0
500
1000
Freq (Hz)
1500
-2.5
0
Figure 45 DSM Driving Point Measurement for Test 2 Bungee Cord Boundary Conditions.
The DSM components in Figure 44 and Figure 45 appear continuous over the majority of
the frequencies. There does appear to be more noise in the Test 1 data, which is most
73
evident in the imaginary component. The next step is to examine the plots for patterns that
match Equations 3.9b and 4.8.
The real components of the DSM should represent an equation with a zero and second order
polynomial terms. Figure 44 and Figure 45 both display real component with positive
slopes between zero and 400 Hz. The curves appear to contain a first order property, in
addition to mass and stiffness. Intuitively, this is not possible; however, no easy explanation
could be discovered. Some low frequency error always exists in signals from piezoelectric
sensor, due to the poor DC response of the transducer system. However, the sensor error is
usually contained below 10 Hz. The same phenomenon could not be reproduced in the
analytical model of Section 3, regardless of the boundary conditions to ground (free-free or
fixed-fixed). The error was handled by selecting 400 Hz as the minimum frequency for
curve fitting the data.
The imaginary data component contains the contributions of the structure’s nonconservative mechanisms, which are complicated. The DSM algorithm assumes a
polynomial form for the data, but the equation’s order is determined by the qualitative
properties of the imaginary plots in Figure 44 and Figure 45. One observation from the data
is that the driving point imaginary plots are all negative for most of the frequency range.
This would imply that the non-conservative mechanisms are bringing energy into the
system, instead of removing it, which is impossible. From the real component plots, the
data from 400 Hz and below is considered error. If the imaginary plots from 400 Hz and
above, the plots appear to have a second order shape. The algorithm fit three coefficients to
74
the data. The first coefficient is independent of frequency, and is equal to the Y-intercept of
the plots. Unfortunately, all of the imaginary curves in Figure 44 and Figure 45 have
negative intercepts when extrapolated from 400 Hz to zero Hz. Any negative values for
driving point damping coefficients denotes an energy generating mechanism, which is
physically impossible. Because the regular modal analysis indicates valid test results,
excluding low frequencies < 10 Hz, it has to be further studied why the DSM shows such
deviant results up to 400 Hz. Further simulation study and related tests are believed
necessary.
4.4.2.2 Curve Fitting the DSM Data
The data was fit between 400 and 1600 Hz. The calculated matrix coefficients were used to
synthesize the real and imaginary DSM data. The synthesized DSM for Test 1 was plotted
in order to observe the accuracy of the model (Figure 46):
9
9
Real
x 10
9
x 10
9
x 10
9
x 10
x 10
6
6
6
6
6
4
4
4
4
4
4
2
2
2
2
2
2
0
0
0
0
0
0
-2
-2
-2
-2
-2
-2
-4
-4
-4
-4
-4
-4
-6
-6
-6
-6
-6
-6
-8
-8
-8
-8
-8
-10
0
500
1000
Freq (Hz)
1500
-10
9
0
500
1000
Freq (Hz)
1500
-10
9
x 10
Imaginary
9
x 10
6
0
500
1000
Freq (Hz)
1500
-10
9
x 10
0
500
1000
Freq (Hz)
1500
-10
9
x 10
-8
0
500
1000
Freq (Hz)
1500
-10
9
x 10
2
2
2
2
1.5
1.5
1.5
1.5
1.5
1.5
1
1
1
1
1
1
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
-0.5
-0.5
-0.5
-0.5
-0.5
-0.5
-1
-1
-1
-1
-1
-1
-1.5
-1.5
-1.5
-1.5
-1.5
-1.5
-2
0
500
1000
Freq (Hz)
1500
-2.5
-2
0
500
1000
Freq (Hz)
1500
-2.5
-2
0
500
1000
Freq (Hz)
1500
-2.5
-2
0
500
1000
Freq (Hz)
1500
-2.5
1500
500
1000
Freq (Hz)
1500
x 10
2
-2
500
1000
Freq (Hz)
9
x 10
2
-2.5
0
-2
0
500
1000
Freq (Hz)
1500
-2.5
0
Figure 46 Real-Imaginary DSM Plots of Synthesized ( | ) and Original ( | ) Data from Test 1.
75
The synthesized real component curve matches the test data between 550 and 1600 Hz.
Below 550 Hz, the test data’s curvature diverges from the second order polynomial model.
The synthesized imaginary component was able to fit the experimental data down to 400 Hz,
which was the lowest frequency included in the curve fit equation. The goal of synthesizing
the data was to reproduce the experimental FRFs. Good data synthesis will not guarantee
that the solution is unique, but poor results will signify inadequacies in the data or
algorithm. For reference, the third row of the experimental and synthesized DSM matrices
are plotted:
9
Real
8
9
x 10
8
8
Adjacent
Off-Diagonal
9
x 10
8
6
6
6
6
4
4
4
4
2
2
2
0
0
0
-2
-2
-2
-4
-4
-4
-6
-6
-6
-8
-8
-8
0
500 1000
Freq (Hz)
1500
9
3
Imaginary
9
x 10
0
500 1000
Freq (Hz)
1500
9
x 10
3
Driving
Point
0
500 1000
Freq (Hz)
1500
3
8
Adjacent
Off-Diagonal
8
6
4
4
2
2
2
0
0
0
-2
-2
-2
-4
-4
-4
-6
-6
-6
-8
-8
0
500 1000
Freq (Hz)
1500
9
x 10
9
x 10
6
9
x 10
9
x 10
3
0
500 1000
Freq (Hz)
1500
-8
9
x 10
3
3
2
2
2
2
2
1
1
1
1
1
1
0
0
0
0
0
0
-1
-1
-1
-1
-1
-1
-2
-2
-2
-2
-2
-2
0
500 1000
Freq (Hz)
1500
-3
0
500 1000
Freq (Hz)
1500
-3
0
500 1000
Freq (Hz)
1500
-3
0
500 1000
Freq (Hz)
1500
-3
0
0
500 1000
Freq (Hz)
1500
500 1000
Freq (Hz)
1500
9
x 10
2
-3
x 10
500 1000
Freq (Hz)
1500
-3
x 10
0
Figure 47 Real-Imaginary Plot, Third Row of the Experimental ( | ) and Synthesized ( | ) DSM Matrices,
Test 1
Figure 47 displays properties that are analyzed during the matrix discussion in Section
4.4.2.3. The real DSM plot of the driving point displays the greatest Y-intercept and
76
curvature; therefore the K and M entries should be greatest values in the third row of the
Test 1 coefficients. The two adjacent real plots in Figure 47 display large Y-intercept
magnitudes, but negative signs. The corresponding entries in the coefficient matrices should
also contain large negative values. The remaining real plots display smaller Y-intercepts
and curvature; likewise, their M and K matrix entries should be significantly smaller than
the driving point values. The damping matrix entries should correspond to the imaginary
plots of Figure 47 in an analogous manner as the real DSM components.
The DSM data in Figure 46 is inverted to produce synthesized FRFs. Figure 48 displays the
resulting synthesized driving point FRFs at location 1 for both Test 1 and Test 2:
Synthesized Driving Point FRF for Location 1, Test 1
Synthesized Driving Point FRF for Location 1, Test 2
0
500
-200
Phase
Phase
0
-500
0
200
400
600
800
1000
Freq (Hz)
1200
1400
-1000
1600
-6
0
200
400
600
800
Freq (Hz)
1000
1200
1400
1600
0
200
400
600
800
Freq (Hz)
1000
1200
1400
1600
-6
10
10
-8
10
Magnitude
Magnitude
-600
-800
-1000
-10
10
-12
10
-400
-8
10
-10
10
-12
0
200
400
600
800
1000
Freq (Hz)
1200
1400
1600
10
Figure 48 Synthesized ( | ) and Experimental ( | ) FRF Bode Plots for Test 1 and 2 Data.
The modes at 684 and 1339 Hz fit relatively well. The matrix coefficients from Test 1 and 2
accurately reproduce the second and third modal frequencies. Noticeable error does exist on
the synthesized magnitudes of the two modes in both sets of data. The first mode, on the
other hand, synthesized with incorrect frequencies and amplitudes. The algorithm was
unable to generate mass and stiffness matrices which simulated the rigid body modes. In
fact, the synthesized data resembles a system with fixed-fixed boundary conditions. The
77
rigid body mode is instead shifted up next to the first flexible mode. Next, note the phase
error in the Test 1 synthesized data. The matrix polynomials calculate a +180 degree shift at
the second mode. The error is also displayed in real-imaginary FRF plot in Figure 49. The
imaginary component has a positive value at 684 Hz, while being negative at all other
frequencies. Why the error exists in the Test 1 and not the Test 2 synthesized data is
unknown.
-7
x 10
-7
Location 1 Synthesized Driving Point FRF for Test 1
2
1
1
0.5
0
Real
Real
1.5
0
-2
0
200
400
600
-7
x 10
800
1000
Freq (Hz)
1200
1400
0
1600
200
400
600
800
1000
Freq (Hz)
1200
1400
1600
200
400
600
800
1000
Freq (Hz)
1200
1400
1600
-7
x 10
0
0
Imaginary
Imaginary
1
Location 1 Synthesized Driving Point FRF for Test 2
-1
-0.5
-1
x 10
-1
-1
-2
-3
-2
0
200
400
600
800
Freq (Hz)
1000
1200
1400
1600
-4
0
Figure 49 Real-Imaginary Plots of Synthesized ( | ) and Experimental ( | ) FRF Data from Test 1 and 2.
In Figure 48, both sets of synthesized data contained temporal errors, which are largely
attributed to M and K, or the real part of the DSM curve fit. The both tests are pseudosynthesized using experimental data for the real DSM component, and the calculated matrix
polynomials for the imaginary DSM component, as discussed in Section 3. The DSM is
inverted to create the pseudo-synthesized FRF, which is displayed in Figure 50:
78
Location 1 Pseudo-Synthesized Driving Point FRF for Test 2
Location 1 Pseudo-Synthesized Driving Point FRF for Test 1
500
1000
0
Phase
Phase
500
0
-500
-500
-1000
0
200
400
600
-6
800
1000
Freq (Hz)
1200
1400
1600
-1000
200
400
600
800
1000
Freq (Hz)
1200
1400
1600
0
200
400
600
800
1000
Freq (Hz)
1200
1400
1600
-6
10
-8
10
Magnitude
Magnitude
10
0
-10
10
-8
10
-10
10
-12
10
0
200
400
600
800
1000
Freq (Hz)
1200
1400
1600
-12
10
Figure 50 Bode Plot of Pseudo-Synthesized ( | ) and Experimental ( | ) FRFs from Test 1 and 2.
The new calculated curves are temporally much more accurate. All three flexible modes are
accurately synthesized. The rigid body modes are still poorly represented; however, the
extra mode in Figure 48 is absent. Error still exists in the predicted magnitude at each peak.
All three modes in Test 1 are under predicted, while only one of the three Test 3 modes
appears to be predicted correctly. Also, note the positive phase shift at Test 1’s second
mode remains in the data.
-4
10
CMIF Plot of Pseudo-Synthesized FRF Data from Test 2
-5
10
-5
10
-6
10
-6
10
-7
10
-7
10
-8
10
-8
10
-9
10
-9
10
-10
10
-10
10
-11
-11
10
0
200
400
600
800
Freq (Hz)
1000
1200
1400
1600
10
0
200
400
600
800
Freq (Hz)
1000
1200
1400
1600
Figure 51 CMIF Plot of Pseudo-Synthesized and Original FRF Data from Test 1 and 2
The plots of Figure 48 through Figure 50 displayed the synthesized data for a single degree
of freedom. The CMIF plots in Figure 51 give a better global estimate of the synthesized
damping data’s quality. Again, the actual and calculated modal frequencies plot on top of
each other. Some differences are observed between the magnitudes of the singular values.
79
The calculated residuals appear accurate, but the modal peaks appear under-estimated in
both tests, with the exception of one mode in Test 2. The second Test 2 mode appears to be
over-estimated. In order to quantify the magnitude error, both sets of pseudo-synthesized
data were fit using the poly-reference time domain algorithm. The resulting modal
frequencies and damping are displayed in:
PTD Modal Fit of Pseudo-Synthesized FRFs, Calculated from Bungee
Cord Tests
Location of
Bungee Cords
Mode Number
Frequency
(Hz)
Damping (%)
Synth.
Actual (Table 10)
1,2 / 5,6
Test 1
1
2
3
247
683.8
1339.3
1.8152
-0.1325
0.1719
0.4316
0.0637
0.0715
2,3 / 4,5
Test 2
1
2
3
246.4
685.6
1339.4
2.5538
0.0181
0.2298
0.1700
0.2175
0.0701
Table 11 Modal Properties of Pseudo-Synthesized FRF Data from Tests 1 and 2
The modal fit results signify an inaccurate fit of the damping characteristics of the FRF data.
In almost every case listed in Table 11, the pseudo-synthesized FRF data calculated a higher
level of damping than what actually existed in the data. The difference in numbers was
sometimes an order of magnitude off. The second mode in both test cases was a source of
variation from the pattern observed in the other two modes. For Test 1, the damping was
calculated as a negative value for the pseudo-synthesized data. The result was not surprising
after observing the phase error in Figure 48. The calculated damping for Test 2 was less
than the actual value, which also was not a surprise after observing the data fit in Figure 48
through Figure 51.
80
Beyond selecting a frequency range for processing, there were not many tools available for
handling noise in the experimental DSM data. One attempt at removing noise may utilize
the SVD decomposition. The CMIF plot was a good tool for identifying the number of
modes in the FRF data and for getting a qualitative feel for the spatial truncation in the data.
The plot also helped identify which singular values were dominated by signal noise. When
singular values displayed observable noise, the value and its corresponding {u} and {v}
vectors were removed from the data set. The dimensions of the three SVD matrices would
have smaller dimensions, however, the overall spatial dimensions of the FRF matrix was
preserved. Once the FRFs were “conditioned” using the SVD, the data was pseudo-inverted
(since the FRF matrix was now rank deficient) to generate the DSM data. The remaining
steps of the DSM algorithm were unchanged. The results from the SVD conditioning were
mixed. The pseudo-synthesized data appeared to represent the system with much less error,
as shown in Figure 52, where the first two modes appear to synthesize well.
Pseudo-Synthesized Driving Point FRF Calculated from Conditioned DSM Matrix; Location 1, Test 1
1000
Phase
500
0
-500
-1000
0
200
400
600
800
1000
Freq (Hz)
1200
1400
1600
1800
2000
0
200
400
600
800
1000
Freq (Hz)
1200
1400
1600
1800
2000
0
10
Magnitude
-5
10
-10
10
Figure 52 Pseudo-synthesized ( | ) and Experimental ( | ) FRF Plot for Driving Point 1, Test 1
81
The same types of results occurred with both sets of test data. Table 12 contains the
calculated modal damping values for the SVD conditioned data.
Modal Properties of SVD Conditioned Peudo-Synthesized Data
Test Num
Mode Num
1
1
2
3
2
1
2
3
Damping (%)
SVD Cond.
Actual
0.3681
0.4316
0.0846
0.0637
0.0660
0.0715
0.1483
0.1738
0.3308
0.1700
0.2175
0.0701
Table 12 Modal Damping for SVD Conditioned Pseudo-synthesized FRF Data
The Test 1 data was processed with only four singular values, and curve fit using a 250 to
800 Hz frequency range. The test two data was processed with five singular values, and
curve fit using a 300 to 1300 Hz frequency range. Comparing Table 11 and Table 12, the
SVD filtering appears to create much more accurate synthesized data. Almost all the
damping values are within the calculation variation of the PTD curve fit algorithm. Mode 3
in Test 2 is the only damping value that did not improve. The results were encouraging,
however, the SVD conditioning resulted in matrix coefficients whose form did not follow
physical intuitions. Evaluating matrix coefficient characteristics is covered in the next
section.
82
4.4.2.3 Evaluating Matrix Coefficients
Three main characteristics were examined when evaluating the quality of the identified
damping matrices in Lee and Kim’s work [1, 2]:
•
Exhibit positive diagonal or dominantly diagonal form
•
Symmetry of the matrices
•
Results explain the spatial properties of the experiment set-up
First, all of the diagonal elements of the matrices were expected to be positive. Next, the
diagonal elements should be symmetric along the diagonal if the set-up (sensors and
structure) is symmetric. Because of the structural properties and sensor layout, similar
matrix values should be observed at the first and last measurement points and at the second
and fifth points as well. The location of the sensors was also intended to remove, or
minimize, any coupling between non-adjacent measurement points. In other words, there
should not be a stiffness component between Points 1 and 4, since any energy passing
between the points must also pass through Points 2 and 3. The final main matrix polynomial
characteristic is an observable (or negligible, depending on the example and the structural
property) difference between the matrices of Test 1 and Test 2. Test 1 and Test 2 in this
experimental example are attempting to use the DSM algorithm to discriminate between two
different boundary conditions. The mass and stiffness matrices should be identical between
the two sets of test data. The boundary conditions should change how energy is removed
from the system; therefore, the damping matrices should have an observable difference
83
between Test 1 and Test 2. The matrices should also identify where the changes occurred
on the structure.
The non-SVD filtered real matrix polynomials for test 1 and 2 are displayed in Table 13 and
Table 14:
[M] Test1
64.2
18.8
-9.3
2.9
-0.5
0.6
18.8
133.0
8.6
-8.1
2.9
-0.5
-9.3
8.6
119.1
14.4
-8.6
3.1
2.9
-8.1
14.4
117.2
13.5
-11.1
-0.5
2.9
-8.6
13.5
130.2
22.0
0.6
-0.5
3.1
-11.1
22.0
61.6
-0.66
2.75
-9.41
14.47
128.90
21.43
0.47
-0.72
3.01
-11.16
21.43
61.65
[M] Test2
64.28
18.68
-9.26
2.95
-0.66
0.47
18.68
130.84
11.24
-8.56
2.75
-0.72
-9.26
11.24
117.64
14.67
-9.41
3.01
2.95
-8.56
14.67
117.10
14.47
-11.16
Table 13 Calculated Real Mass Coefficients, Test 1 and 2, 400 Hz- 1600 Hz.
The mass matrices appear to display all of the desired matrix characteristics. Both
coefficients are symmetric, and dominated by the diagonal entries. Ideally, the entries just
off of the diagonal should be zero; however, the spatially truncated analytical examples in
Section 3 displayed the same type of characteristics. The non-zero entries are probably just
a symptom of spatial truncation in the data set. The similarities of both mass matrices fulfill
the third main matrix characteristic. One other validation method exists. The mass matrices
can be compared to the actual mass of the bar, which is 635 grams. The diagonal entries of
the mass matrices are added, which results in an estimate of 622 grams and 625 grams for
the Test 1 and Test 2 matrices, respectively.
84
x10^5
9535
-17082
12673
-3533
814
-18
[K] Test1
-17082
52239
-47447
20413
-4653
760
x10^5
9661
-16947
12652
-3572
801
-53
12673
-47447
71119
-51294
19363
-3007
-3533
20413
-51294
70906
-45626
11618
814
-4653
19363
-45626
51012
-16031
-18
760
-3007
11618
-16031
8814
801
-4674
19132
-45344
50731
-16368
-53
732
-3117
11794
-16368
8936
[K] Test2
-16947
50991
-46159
20167
-4674
732
12652
-46159
70207
-51061
19132
-3117
-3572
20167
-51061
70961
-45344
11794
Table 14 Calculated K Coefficents, Test 1 and 2, 400 Hz-1600 Hz.
The [K] matrices appear to both be symmetric, and dominated by positive diagonal entries,
and negative entries just off of the diagonal. However, other off-diagonal entries are within
the same order of magnitude as the diagonal entries. Matrices from both tests appear very
similar, which was expected.
The same three characteristics were applied to the damping matrices; however, an attempt
was also made to correlate the matrix entries and the calculated modal damping for both
tests.
85
Table 15 contains the zero order imaginary matrix coefficients for Test 1 and Test 2:
x10^5
-2233
7191
-5937
2898
-770
228
[D] Test1
7191
-18899
18427
-10807
3983
-605
x10^5
-3505
10854
-9570
4510
-1106
167
-5937
18427
-22481
18415
-9960
2641
2898
-10807
18415
-22445
16317
-5029
-770
3983
-9960
16317
-14245
5048
228
-605
2641
-5029
5048
-1720
-1106
4994
-12208
20196
-16880
5697
167
-711
3050
-5803
5697
-2042
[D] Test2
10854
-25840
27062
-15397
4994
-711
-9570
27062
-32171
25404
-12208
3050
4510
-15397
25404
-29616
20196
-5803
Table 15 Calculated D (zero order) Coefficients, Test 1 and 2, 400-1600 Hz
The [D] matrices from both tests do not fulfill the first man characteristic. Both matrices
contain negative values along the diagonal, and strong non-negative values in the just-off
diagonal entries. The graphical analysis of the DSM data in Section 4.4.2.1 predicted the
signs of the results displayed in Table 15. The imaginary DSM plots appeared to have
negative Y-intercepts, which must result in negative structural damping coefficients. Thus,
the [D] matrices do not make physical sense. They appear to represent energy generating
mechanisms, instead of damping mechanisms. Setting aside the sign errors, the matrices
appear to have some encouraging properties. Both [D] matrices display a small amount of
asymmetry along the diagonal. This observation will be discussed in greater detail with the
[C] matrix results below. For the third characteristic, the Test 1 coefficient matrix is lower
than the Test 2 matrix. This property appears to agree with the modal damping calculation
listed in Table 9; however, the spatial distribution of damping appears the same in both
86
matrices. The bungee cords, and therefore a damping mechanism, were located at different
locations for both tests. The matrices in Table 15 fail to differentiate any localized changes
between the tests. One more step was performed in order to identify the importance of the
[D] matrix. Random values were generated and placed into the [D] matrix. The resulting
matrix and pseudo-synthesiszed FRF are displayed:
x10^3
5631
4535
10343
12161
3423
4766
[D] Random
5631
4535
10343
12161
3423
4766
5631
4535
10343
12161
3423
4766
5631
4535
10343
12161
3423
4766
5631
4535
10343
12161
3423
4766
5631
4535
10343
12161
3423
4766
Table 16 Randomly Generated Structural Damping Matrix
800
Phase
600
400
200
0
-200
0
200
400
600
800
Freq (Hz)
1000
1200
1400
1600
0
200
400
600
800
Freq (Hz)
1000
1200
1400
1600
-6
Magnitude
10
-8
10
-10
10
-12
10
Figure 53 Pseudo-Synthesized Location 1 Driving Point FRF using Randomly Generated Structural
Damping Matrix
The [D] matrix appears to have a significant influence on the synthesized FRF, as show in
the difference between Figure 50 and Figure 53. The random structural damping matrix
87
produces an FRF with heavily damped, or non-existent modes. The CMIF plot of the Figure
53 produces the same conclusions.
-4
10
-5
10
-6
10
-7
10
-8
10
-9
10
-10
10
-11
10
0
200
400
600
800
Freq (Hz)
1000
1200
1400
1600
Figure 54 CMIF Plot of Pseudo-Synthesized FRF Data Generated using a Random Structural Damping
Matrix
x10^2
581.9
-1947.9
1404.6
-677.6
171.1
-84.3
[C] Test1
-1947.9
4849.8
-4403
2559.2
-901
97.8
x10^2
890.1
-2959
2301.4
-1031.6
246.6
-53
1404.6
-4403
5356.1
-4523.4
2385.8
-652.7
-677.6
2559.2
-4523.4
5681.4
-4014.9
1253.7
171.1
-901
2385.8
-4014.9
3434
-1327.4
-84.3
97.8
-652.7
1253.7
-1327.4
439.6
246.6
-1049.9
2744.5
-4895.9
4143.9
-1400.5
-53
112.7
-683.6
1297.5
-1400.5
456.2
[C] Test2
-2959
6356.2
-6375.1
3519.1
-1049.9
112.7
2301.4
-6375.1
7364
-6017
2744.5
-683.6
-1031.6
3519.1
-6017
7292.3
-4895.9
1297.5
Table 17 Calculated Visous Damping (First Order) Coefficients, Test 1 and 2, 400-1600 Hz.
88
The [C] matrices fulfill the first characteristic, but fall short on the second and third. Both
matrices in Table 17 contain maximum values along the diagonal and adjacent off-diagonal
entries, as expected. Both the Test 1 and Test 2 matrices fail to display symmetry along the
diagonal. The upper left diagonal entries appear greater than the lower diagonal entries,
which is not expected. A similar, but less dramatic, asymmetry existed in the [D] matrices.
The matrix entries increased in magnitude between the two tests, while the distribution of
the values remained constant. Some unobserved difference may have existed between the
two bungee cord supports, and overshadowed properties of the structure.
A source of this deviation from assumptions is not obvious; however, it indicates the
difficulty in identifying damping properties due to the small magnitudes of the mechanisms’
forces. Whether the results improve using more accurate DSM data remains to be seen. For
example, a non-contact FRF measurement technique may be tried.
89
The second order damping coefficient is shown in Table 18. The [E] terms scale the ω2
properties of the imaginary DSM component.
[E] Test1
-3.7
12.9
-7.8
3.6
-0.8
0.7
12.9
-30.3
24.4
-13.8
4.5
-0.2
-7.8
24.4
-30.0
25.4
-13.0
3.7
-5.3
19.6
-12.9
5.2
-1.2
0.4
19.6
-38.0
35.3
-18.1
4.6
-0.2
-12.9
35.3
-39.1
32.6
-13.8
3.3
3.6
-13.8
25.4
-34.1
23.3
-7.3
-0.8
4.5
-13.0
23.3
-19.2
8.9
0.7
-0.2
3.7
-7.3
8.9
-2.2
-1.2
4.6
-13.8
28.3
-24.3
8.7
0.4
-0.2
3.3
-6.4
8.7
-1.8
[E] Test2
5.2
-18.1
32.6
-42.2
28.3
-6.4
Table 18 Calculated Second Order Coefficients, Test 1 and 2, 400-1600 Hz.
The [E] matrices differ from the [C] in magnitude, but share the same qualitative properties.
The largest entries are on the diagonal, or near-diagonal terms, but the entries are not
symmetric along the diagonal. The Test 2 entries are larger that those of the Test 1 matrix,
but the spatial distribution of terms appears to be similar.
90
In Section 4.4.2.2 some discussion was given to using the SVD to filter out small or noisy
singular values. The practice produced favorable synthesized FRF values, but a significant
amount of error was introduced into the matrix coefficients. Table 19 and Table 20 display
the mass and viscous damping coefficients which were calculated using SVD conditioned
data from Test 1.
[M]
62.1
21.0
-9.0
-4.7
6.7
0.6
21.0
67.3
51.8
-2.2
-23.1
5.3
-9.0
51.8
64.2
28.5
-2.4
-4.1
-4.7
-2.2
28.5
63.2
52.5
-7.7
6.7
-23.1
-2.4
52.5
68.6
20.0
0.6
5.3
-4.1
-7.7
20.0
61.6
Table 19.Mass Matrix Coefficient of SVD Conditioned Test 1 Data.
[C]
24015
-29924
-22414
14254
22088
-12964
-29924
26960
19486
-8494
-23620
15819
-22414
19486
-220
-13247
-11367
11926
14254
-8494
-13247
3327
18156
-19014
22088
-23620
-11367
18156
28929
-27595
-12964
15819
11926
-19014
-27595
14194
Table 20 Viscous Damping Matrix Coefficient of SVD Conditioned Test 1 Data
Both matrices are smaller than the coefficients calculated from the non-conditioned data
sets. The viscous damping matrix is two orders of magnitude smaller than the matrix in
Table 17. Additionally, the damping matrix no longer displays diagonally dominant
properties. The entries in the extreme off-diagonal locations of the matrix have the same
magnitude at the diagonal entries. The implications of reconditioning the FRFs using the
SVD will have to be further studied using numerical simulations.
91
4.4.3 Summary of the Free-Free Boundary Condition Experiment
This experiment was initiated with several objectives:
•
Create a measurable change in a structure’s damping properties through a change in
boundary conditions.
•
Evaluate the quality and effect of spatial truncation of the experimental data using
several pre-processing graphical methods.
•
Fit the data and use several post-processing graphical methods to validate the quality
of the polynomial fit.
•
Differentiate between the two test set-ups by identify the location and magnitude of
localized structural properties.
The first objective was fulfilled, as observed quantitatively in Table 10 and qualitatively in
Figure 40. The natural frequencies remained constant between test arrangements, while
modal damping ratios changed. The graphical analysis of the DSM data revealed relatively
clean curves and the existence of an identifiable polynomial pattern. However, a low
frequency error existed in the DSM data, which resulted in some erroneous results in the
damping matrix fit. More work is needed in order to identify the source of the low
frequency phenomenon. The CMIF identified noise in the data and flagged potential
problems with over-truncation of spatial degrees of freedom. The synthesized data did not
exactly match the experimental data. Evaluating the pseudo-synthesized data (using only
the calculated damping information) resulted in encouraging results qualitatively, but the
calculated modal damping values under predicted the experimental data. An SVD based
filtering method was presented as a possible pre-processing noise reduction method,
92
however, its use produced mixed results. Finally, the calculated matrix coefficients were
able to quantitatively differentiate between the Test 1 and Test 2 set-up, however, the
matrices were unable to spatially discriminate between the two set-ups.
93
4.5 Experimental Case Study 2: Cable Damping Test
The second experiment applied the DSM algorithm to a system whose structural properties
are modified between tests. The experiment in Section 4.4 was repeated while changing
boundary conditions, to see if a change could be observed in the identified matrices. The
cable damping test attempted to identify the structural changes, mass and damping, between
two tests.
The structure was a variation on the bungee cord set-up of Section 4.4. The test employed
an aluminum plate with the same overall dimensions Figure 37, but also included four thruholes drilled between Points 1-2, 2-3, 4-5, and 5-6, as shown in Figure 55. The plate was
suspended by low stretch woven fishing line, contacting the bar between Points 1-2 and 5-6.
The material was chosen in order to minimize the energy loss between the bar and the
supports. Sensor locations were identical to those used in Section 4.4. Test arrangement 1
consisted of the suspended plate without any additional components. Test arrangement 2
added a 1 Kg mass suspended from a steel cable which was threaded through the holes
between Points 1-2 and 3-4 (Figure 55).
Data was collected identical to Section 4.4. The same modal hammer and tip were used to
excite the structure at each measurement location. Again, driving points were measured at
each location, producing a 6x6xN FRF matrix.
94
Steel
Cable
Low
Stretch
Line
1
2
3
4
5
6
1 Kg
Mass
Figure 55 Cable Damping Set-up
4.5.1 Modal Analysis of Cable Damping Tests
Again, as a benchmark, the modal properties were calculated for both test conditions:
PTD Modal Fit of Experimental FRFs, Calculated from
Cable Damping Tests
Test Condition
Without Cable
With Cable
Mode Number
Frequency (Hz)
Damping (%)
1
2
3
232
634.8
1248.4
0.1270
0.0828
0.0428
1
2
3
232.4
634.9
1248.3
1.3584
0.1974
0.1030
Table 21 Modal Properties for Cable Damping Tests
95
An intriguing property of Table 21 is the negligible difference in modal frequencies between
both tests. The additional 1 kg mass almost triples the overall mass of the system. A
possible reason could be the additional mass’s proximity to the support. The mass may be
transferred through the support and is not allowed to participate in any of the modes. The
table does display an increase in modal damping between Test 1 and Test 2. Therefore, it is
expected to observe little difference between the stiffness and mass matrices of Test 1 and
Test 2; while the damping matrices should identify and increase in damping located near
measurement Point 2.
Figure 56Mode Shapes and Location of Additional Mass, Test 2.
4.5.2 DSM Algorithm Identification of Spatial Damping: Case 2
96
The algorithm should be able to discriminate between the two test conditions, and identify
the location of the cable damper and mass.
4.5.2.1 Graphical Evaluation of Experimental Data and Calculation of
DSM: Case 2
Graphical analysis of the experimental data begins with CMIF plots of the Test 1 and Test 2
FRFs:
MIFBar
Plot of
Undamped
Bar FRFs
CMIF PlotCof
FRF
without
Cable Damping
-4
10
-5
-5
10
10
-6
-6
10
10
-7
-7
10
10
-8
-8
10
10
-9
-9
10
10
-10
-10
10
10
-11
10
CMIF
Bar FRFs
CMIF Plot
ofPlot
Barof Damped
FRF with
Cable Damping
-4
10
-11
0
200
400
600
800
Freq (Hz)
1000
1200
1400
1600
10
0
200
400
600
800
Freq (Hz)
1000
1200
1400
1600
Figure 57 CMIF Plots of FRF Data from Test 1 and Test 2
The three flexible modes are easily observed in the plots. The increased damping level is
noticeable between the two plots; and agrees with the behavior calculated by the PTD curve
fit. The rigid body modes are more prevalent than in Case 1. In contrast to the first case
study, the stiff supports have raised the rigid body modes above zero Hz. Below 400 Hz,
most of the singular vales display some level of error. Above 400 Hz, noise appears to be
restricted to the first or second lowest singular values and all of the residuals are well
defined. Therefore, the plots in Figure 57 indicate using the data above 400 Hz for the curve
fit.
97
CMIF
PlotDSM
of Undamped
DSMs Damping
CMIF Plot of
Bar
withBar
Cable
12
10
11
11
10
10
10
10
10
10
9
9
10
10
8
8
10
10
7
7
10
10
6
6
10
10
5
5
10
10
4
10
CMIF
PlotDSM
of Damped
BarCable
DSMs Damping
CMIF Plot of
Bar
with
12
10
4
0
200
400
600
800
Freq (Hz)
1000
1200
1400
10
1600
0
200
400
600
800
Freq (Hz)
1000
1200
1400
1600
Figure 58 CMIF Plot of DSM Data from Cable Damping Tests 1 and 2
For reference, the CMIF plots of the DSM are plotted above. As expected, they are simply
the inverse of the Figure 57 plots.
400
-300
200
200
0
0
-400
300
-600
-400
-400
-700
Phase
300
200
-200
-200
100
-200
0
-600
100
-800
-600
-400
-200
-800
-1000
-1000
-1000
-200
0
500 1000
Freq (Hz)
1500
-6
-1300
0
500 1000
Freq (Hz)
1500
-6
10
10
500 1000
Freq (Hz)
1500
-500
0
500 1000
Freq (Hz)
1500
-6
10
-8
-1600
1500
-11
-12
-10
10
-13
10
500 1000
Freq (Hz)
1500
10
-11
10
10
-13
0
-13
0
500 1000
Freq (Hz)
1500
10
10
-11
10
-12
10
-10
10
-11
10
-12
10
1500
-9
10
-10
10
10
500 1000
Freq (Hz)
-8
10
-9
10
-10
-11
0
-7
-9
10
1500
10
10
-8
-9
-10
500 1000
Freq (Hz)
-8
10
10
0
-6
10
-10
10
-600
10
10
-7
10
10
-9
10
500 1000
Freq (Hz)
-7
10
-9
10
0
10
-8
10
-1000
-6
10
-7
10
-8
10
0
10
-7
-400
-1400
-6
10
-7
-1400
-300
-800
-1200
-1200
-600
-1200
-1100
-100
-100
-800
-900
0
Magnitude
400
0
-500
200
200
-12
0
500 1000
Freq (Hz)
1500
10
-11
10
-12
10
-13
0
500 1000
Freq (Hz)
1500
10
-12
0
500 1000
Freq (Hz)
1500
10
Figure 59 Driving Point FRF Plot of Plate Without Cable Damping
The FRF magnitude plots in Figure 59 appear clean and indicative of driving points. The
phase plots display the same properties as Figure 42 for Case 1.
98
Figure 60 and Figure 61 display the DSM data generated from the FRFs above.
9
2
9
x 10
4
1
10
x 10
1
2
0
9
x 10
x 10
1
2
0
0
-1
-2
-2
-4
-3
-6
-4
-8
-5
0
0.5
-2
-3
-0.5
-4
-4
0
-1
-5
-6
-6
-0.5
-1.5
-8
-7
0
500 1000
Freq (Hz)
1500
-10
9
1
4
0.5
1
-2
-8
9
x 10
0
-1
Real
10
x 10
1.5
0
500 1000
Freq (Hz)
1500
-1
0
9
x 10
6
500 1000
Freq (Hz)
1500
-2
9
x 10
500 1000
Freq (Hz)
1500
9
x 10
2
0
-10
1
0
500 1000
Freq (Hz)
1500
-6
0
8
1500
500 1000
Freq (Hz)
1500
8
x 10
x 10
500 1000
Freq (Hz)
x 10
3
8
1.5
4
0.5
Imaginary
6
0.5
1
2
0
1.5
2
1
0
0
0
-0.5
0.5
-2
-0.5
-0.5
2
4
0.5
0
2.5
0
-4
-2
-1
-1
-0.5
-6
-1.5
-1
-4
-1.5
-2
-1.5
0
500 1000
Freq (Hz)
1500
-6
0
500 1000
Freq (Hz)
1500
-2.5
0
500 1000
Freq (Hz)
1500
-2
0
500 1000
Freq (Hz)
1500
-8
-1
-10
-1.5
-12
0
500 1000
Freq (Hz)
1500
-2
0
Figure 60 DSM Driving Point Measurement for Plate without Damping
9
2
9
x 10
4
1
10
x 10
1.5
1
2
0
9
x 10
1
2
0
0
-1
-2
-2
-4
-3
-6
-4
-8
-5
x 10
0
0.5
-2
-3
-0.5
-4
-4
0
-1
-5
-6
-6
-0.5
-1.5
-8
-7
0
500 1000
Freq (Hz)
1500
-10
9
1
4
0.5
1
-2
-8
9
x 10
0
-1
Real
10
x 10
0
500 1000
Freq (Hz)
1500
-1
9
x 10
6
0
500 1000
Freq (Hz)
1500
-2
9
x 10
2
0
500 1000
Freq (Hz)
1500
-10
9
x 10
1
0
500 1000
Freq (Hz)
1500
-6
8
500 1000
Freq (Hz)
1500
500 1000
Freq (Hz)
1500
8
x 10
x 10
0
3
x 10
8
1.5
4
0.5
Imaginary
6
0.5
1
2
0
1.5
2
1
0
0
0
-0.5
0.5
-2
-0.5
-0.5
2
4
0.5
0
2.5
0
-4
-2
-1
-1
-0.5
-6
-1.5
-1
-4
-1.5
-2
-1.5
0
500 1000
Freq (Hz)
1500
-6
0
500 1000
Freq (Hz)
1500
-2.5
0
500 1000
Freq (Hz)
1500
-2
0
500 1000
Freq (Hz)
1500
-8
-1
-10
-1.5
-12
0
500 1000
Freq (Hz)
1500
-2
0
Figure 61 DSM Driving Point Measurement for Plate with Damping
The real components of both figures appear more accurate, and similar to the data from the
first case study. Above 400 Hz, the plots display the expected form of a second order
99
polynomial with a zero and second order terms. All the curves are continuous, and display
minimal noise compared to the overall shape of the data. Below 400 Hz, the curves exhibit
a negative slope, signifying the existence of a first order term, which is contrary to the
physical intuition. The same type of behavior observed in Case 1. The imaginary
components in Figure 60 and Figure 61 appear similar in overall form, but contain slightly
different amounts of noise. The error observed in both the real and imaginary components
below 400 Hz required that data to be excluded from the curve fit algorithm; and therefore,
the processed data will only contain inertance residuals from the first flexible mode and the
rigid body modes.
4.5.2.2 Curve Fitting the Case 2 DSM Data
As in Case 1, both imaginary data sets display a second order polynomial shape above 400
Hz and up to 1600 Hz. Two real polynomial terms and three imaginary polynomial terms
were fit to the DSM data using a frequency range of 400 to 1600 Hz (Figure 62 and Figure
63).
100
9
Real
2
10
x 10
1
1
0.8
0
0.6
-1
0.4
-2
0.2
-3
0
9
x 10
8
9
x 10
8
6
6
4
4
2
2
0
0
-2
-2
-4
-4
-6
-6
-0.2
-5
-0.4
-6
-0.6
-7
-0.8
-8
-8
-8
-1
-10
-10
500 1000
Freq (Hz)
1500
9
1
6
0.5
0
500 1000
Freq (Hz)
1500
0
500 1000
Freq (Hz)
1500
9
x 10
2
4
0
-2
-3
-5
-5
0
500 1000
Freq (Hz)
1500
2
1
1
0
0
-1
-1
-2
-2
Imaginary
1500
4
0
500 1000
Freq (Hz)
1500
-3
-3
-4
-4
-5
0
500 1000
Freq (Hz)
1500
-5
-6
0
500 1000
Freq (Hz)
1500
500 1000
Freq (Hz)
1500
8
x 10
10
2
x 10
8
0
6
4
-6
-4
-6
500 1000
Freq (Hz)
8
x 10
-2
1500
0
-4
-1.5
500 1000
Freq (Hz)
-10
9
x 10
0
0
x 10
0
-2
-1
-2.5
1
0
2
-0.5
-2
9
x 10
-4
9
x 10
5
-1
-4
0
9
x 10
2
-8
0
-10
0
500 1000
Freq (Hz)
1500
-12
0
500 1000
Freq (Hz)
1500
-2
0
Figure 62 Synthesized ( | ) and Experimental ( | ) DSM Driving Points, Plate without Damping
9
9
x 10
5
10
x 10
1
10
x 10
1
9
x 10
5
9
x 10
1
x 10
1
0.8
0
Real
0
-2
0
-4
-0.5
-3
-0.2
-5
-5
-1
-0.4
-6
-5
-4
-0.6
-7
-1.5
-5
-0.8
0
500
1000
1500
-10
9
1
0
0.2
-3
-8
-1
0.4
0
-2
0
0.5
0.6
-1
0
500 1000
Freq (Hz)
1500
-1
9
x 10
6
0.5
0
500 1000
Freq (Hz)
1500
-2
9
x 10
2
500 1000
Freq (Hz)
1500
9
x 10
2
1
4
0
-10
0
500 1000
Freq (Hz)
1500
8
x 10
5
Imaginary
-0.5
3
-1
0
0
-1
-1
1500
1500
0
-5
-2
-2
-3
-3
-4
-4
-5
-5
-10
-2
-3
-15
-4
1000
500 1000
Freq (Hz)
1
-2
500
x 10
-1
-1.5
0
1500
2
1
0
-6
500 1000
Freq (Hz)
8
x 10
2
-2.5
0
0
0
-2
-6
0
500 1000
Freq (Hz)
1500
0
500 1000
Freq (Hz)
1500
-4
0
500 1000
Freq (Hz)
1500
-20
0
500 1000
Freq (Hz)
1500
-5
0
Figure 63 Synthesized ( | ) DSM and Experimental ( | ) Driving Points, Plate with Damping
101
Like Case 1, the synthesized DSM data matches the experimental data between 500 and
1600 Hz. Below 500 Hz the experimental data displays curvature in the real and imaginary
components that the polynomial algorithm is unable to match.
Figure 64 displays the synthesized FRF data generated form the matrix coefficient
polynomial equation.
Synthesized Driving Point FRF for Measurement Location 1, Plate with Damping
Synthesized Driving Point FRF for Measurement Location 1, Plate without Damping
400
400
200
0
Phase
Phase
200
-200
-400
-400
-600
0
-200
0
200
400
600
800
Freq (Hz)
1000
1200
1400
-600
1600
-6
10
0
200
400
600
800
Freq (Hz)
1000
1200
1400
1600
0
200
400
600
800
Freq (Hz)
1000
1200
1400
1600
-6
10
-8
-8
10
Magnitude
Magnitude
10
-10
10
-10
10
-12
10
0
200
400
600
800
Freq (Hz)
1000
1200
1400
1600
-12
10
Figure 64 Synthesized ( | ) and Experimental ( | ) FRF Driving Point for Location 1
As in Case 1, error in the mass and stiffness matrices has resulted in incorrect synthesis of
the modal frequencies. The upper two modes are close in frequency, while the rigid body
modes and first flexible frequency are shifted upward by a significant amount. The
synthesized phase also does not effectively reproduce the experimental data. Only the upper
two flexible modes appear to be properly calculated.
102
200
200
0
0
P has e
Phase
Pseudo-Synthesized Driving Point FRF Plot for Measurement Location 1, Plate without Damping
400
-200
0
200
400
600
800
Freq (Hz)
1000
1200
1400
-600
1600
0
200
400
600
800
Freq (Hz)
1000
1200
1400
1600
0
200
400
600
800
Freq (Hz)
1000
1200
1400
1600
-6
-6
10
10
-8
10
-8
M agnitude
Magnitude
-200
-400
-400
-600
Pseudo-Synthesized Driving Point FRF for Measurement Location 1, Plate with Damping
-10
10
10
-10
10
-12
10
0
200
400
600
800
Freq (Hz)
1000
1200
1400
1600
-12
10
Figure 65 Pseudo-Synthesized ( | ) and Experimental ( | ) Driving Point FRF, Measurement Location 1
The data was re-synthesized using the experimental real DSM component and the calculated
imaginary component (Figure 65). Removing the real component error does not
dramatically improve the agreement between the experimental and synthesized data. The
second and third flexible modes line-up temporally, yet some error is observed around the
peaks. While the upper two peaks have improved, the first flexible mode and the rigid body
modes are now almost non-existent. The low frequency residual error observed in Figure 57
resulted in an inaccurate calculation of both the real and imaginary matrix coefficients.
103
CMIF of FRF Data, Plate without Damping
-5
-6
-6
10
10
-7
-7
10
10
-8
-8
10
10
-9
-9
10
10
-10
10
-10
10
-11
10
CMIF of FRF Data, Plate with Damping
-5
10
10
0
200
400
600
800
Freq (Hz)
1000
1200
1400
1600
-11
10
0
200
400
600
800
Freq (Hz)
1000
1200
1400
1600
Figure 66 CMIF Plot of Pseudo-Synthesized FRF Data, Case 2
The CMIF plots of Figure 66 identify the global effects of error in the damping coefficients.
As previously observed, the upper two modes display little magnitude error between the
experimental and pseudo-synthesized error. The first flexible mode resonance frequency
and the rigid body modes do not appear in the calculated data. The pseudo-synthesized
singular values match the experimental data until around 370 Hz; below that, the two sets of
data diverge. For comparison, the modal damping and frequencies of the pseudosynthesized data were calculated using the poly-reference time domain algorithm:
Table 22 Calculated Modal Properties of Pseudo-Synthesized FRF Data
PTD Modal Fit of Pseudo-Synthesized FRFs, Calculated from
Cable Damping Tests
Test
Condition
Without Cable
With Cable
Mode Number
Frequency
(Hz)
Damping (%)
1
2
3
232
634.8
1248.4
Synth
--------0.0843
0.1798
1
2
3
232.4
634.9
1248.3
------0.2438
0.1357
Actual (Table 21)
0.1270
0.0828
0.0428
1.3584
0.1974
0.1030
104
For both test cases, the first flexible mode was not identified. The other two flexible modes
were consistently identified for both test conditions. Excluding Mode 3 without damping,
the calculated damping for the pseudo-synthesized data agrees relatively well with the
experimental data. The damping variation between the experimental and calculated values
was within the variation observed between iterations of the PTD algorithm.
Several attempts were made to improve the DSM curve fit of the experimental data. The
initial data fit used a frequency band beginning at 400 Hz, which excluded the resonant
frequency of the first flexible mode. The data was reprocessed using a frequency band of
190-1600 Hz. The new frequency band included the first resonant frequency; however, the
data contains more noise, as show in all of the plots in Section 4.5.2.1. The synthesized
DSM data is presented in Figure 67:
9
9
x 10
5
10
x 10
1
10
x 10
1
9
x 10
5
9
x 10
1
x 10
1
0.8
0
Real
0
-2
0
-4
-0.5
-3
-0.2
-5
-5
-1
-0.4
-6
-5
-4
-0.6
-7
-1.5
-5
-0.8
0
500 1000
Freq (Hz)
1500
-10
9
1
0
0.2
-3
-8
-1
0.4
0
-2
0
0.5
0.6
-1
0
500 1000
Freq (Hz)
1500
-1
9
x 10
6
0.5
0
500 1000
Freq (Hz)
1500
-2
9
x 10
2
500 1000
Freq (Hz)
1500
-10
9
x 10
2
1
4
0
0
500 1000
Freq (Hz)
1500
8
x 10
5
Imaginary
-0.5
3
-1
0
0
-1
-1
1500
0
-5
-2
-2
-3
-3
-4
-4
-5
-5
-10
-2
-3
-15
-4
1500
500 1000
Freq (Hz)
1
-2
500 1000
Freq (Hz)
x 10
-1
-1.5
0
1500
2
1
0
-6
500 1000
Freq (Hz)
8
x 10
2
-2.5
0
0
0
-2
-6
0
500 1000
Freq (Hz)
1500
0
500 1000
Freq (Hz)
1500
-4
0
500 1000
Freq (Hz)
1500
-20
0
500 1000
Freq (Hz)
1500
-5
0
Figure 67 Synthesized ( | ) and Experimental ( | ) DSM Data, 190-1600 Hz Input to Algorithm, without
Cable Damping
105
As expected, the synthesized curve is unable to conform to the sub-400 Hz curvature in both
the real and imaginary components of the experimental DSM data. The overall fit of the
synthesized DSM data did not appear encouraging. Was the new frequency range effective
in pseudo-synthesizing the experimental FRF data? The results are mixed.
Synthesized
FRF
Data,
Processing
Frequencies
Pseudo-Synthesized
Driving
Point400-1600
FRF Plot for Hz
Measurement
Location
1, Plate without Damping
400
200
200
0
Phase
Phase
Synthesized FRF Data, 190-1600 Hz Processing Frequencies
400
-200
-400
-400
-600
0
200
400
600
800
Freq (Hz)
1000
1200
1400
-600
1600
0
200
400
600
800
Freq (Hz)
1000
1200
1400
1600
0
200
400
600
800
Freq (Hz)
1000
1200
1400
1600
-6
-6
10
-8
10
10
-8
Magnitude
10
Magnitude
0
-200
-10
10
-10
10
-12
-12
10
10
0
200
400
600
800
Freq (Hz)
1000
1200
1400
1600
Figure 68 Pseudo-Synthesized ( | ) and Experimental ( | ) Data using Two Different Input Frequency
Ranges, without Damping
Figure 68 displays the original pseudo-synthesized data along side the data generated from
the DSM synthesis of Figure 67 (imaginary component only). Only two small
improvements are observed. The first flexible mode resembles a peak in the new
calculation, but the estimated damping is excessive. Also, the wider frequency range
resulted in lower calculated magnitude for the second resonant frequency. The lower
frequency phase appears to have improved in the new calculation, however, the antiresonance near 1150 Hz diverges from the experimental data.
The experimental data was also fit using a frequency range of 700-1600 Hz, but did not
display an improvement. The lower frequency modes have a stronger contribution to the
106
DSM at higher frequencies, due to their residuals. The higher starting frequency appeared
promising since noise on the residuals decreased with increasing frequency. The resulting
pseudo-synthesis plots looked worse than the initial 400-1600 Hz results, thus the results are
not presented.
4.5.2.3 Analysis of Matrix Coefficients
Two main characteristics were used to evaluate the quality and usefulness of the matrix
coefficients:
•
Exhibit positive diagonal or dominantly diagonal form
•
Spatially descriptive and unique to the structural set-up
The characteristics are similar to those defined in Section 4.4, and the same reasoning
behind the characteristics applies. Unlike the matrices in the first case study, symmetry
along the diagonal is only expected for the non-cable damping matrices, due to the
asymmetric mounting of the suspended mass and cable. The damped plate should produce
matrices that identify the increased level of damping and its off-center location.
The mass and stiffness matrices are shown in Table 23 and Table 24, respectively. The
mass matrices appear similar to the calculated values in Case 1. Both matrices in Table 23
are dominantly diagonal, but also contain significant (within 10% of the diagonal values),
positive off-diagonal entries. A notable characteristic is the consistent values between the
mass matrices for the structure with and without the cable/mass attachment. The cable was
107
mounted off-center and increased the system’s mass by 200%. Yet, the matrices for both
test set-ups are almost identical and appear symmetric along the diagonal. The results do
agree with the modal analysis data, which did not detect a significant difference in the
system’s natural frequency with the addition of the mass.
[M]without cable
63.4
19.2
-9.0
3.5
-1.1
0.2
19.2
126.4
12.1
-10.5
3.3
-1.0
-9.0
12.1
112.7
18.0
-9.9
3.4
3.5
-10.5
18.0
109.5
13.4
-10.3
-1.1
3.3
-9.9
13.4
121.5
21.1
0.2
-1.0
3.4
-10.3
21.1
56.7
-0.9
2.9
-10.0
14.4
120.9
20.8
0.2
-0.9
3.6
-11.4
20.8
55.8
[M]with cable
63.2
19.7
-9.4
3.1
-0.9
0.2
19.7
130.2
8.9
-8.2
2.9
-0.9
-9.4
8.9
113.4
17.4
-10.0
3.6
3.1
-8.2
17.4
108.6
14.4
-11.4
Table 23 Case 2 Mass Matrices, Plate with and without Cable Damping
Evidently, the additional mass of the damping device does not change the amount of mass
participating in the system’s modes from 0 to 1600 Hz. This does not seem likely, since the
cable mounting location appears to have some kind of displacement during all three modes
(Figure 56), however, a better reason is not available.
As expected, the structure’s stiffness matrix did not change noticeably with the additional
mass and damping (Table 24). Both matrices, with and without the cable damper, appear
dominantly diagonal and relatively symmetric along the diagonal. Therefore, the stiffness
properties appear to be evenly distributed along the plate’s geometry, and do not identify
any localized characteristics positioned off-center of the plate. Again, the results are not
108
surprising, considering the negligible difference in the system’s natural frequency after the
addition of the cable damping mechanism.
x10^5
8655
-14183
10424
-2632
525
-90
[K]without cable
-14183
43646
-37166
15093
-3471
492
x10^5
8634
-14060
10463
-2832
590
-100
10424
-37166
55711
-37874
14365
-2251
-2632
15093
-37874
54400
-34928
8946
525
-3471
14365
-34928
40968
-12314
-90
492
-2251
8946
-12314
7401
590
-3598
14231
-34164
40235
-12127
-100
539
-2122
8507
-12127
7204
[K]with cable
-14060
44463
-38325
15829
-3598
539
10463
-38325
56871
-38318
14231
-2122
-2832
15829
-38318
53860
-34164
8507
Table 24 Case 2 Stiffness Matrices, Plate with and without Cable Damping
Table 25 contains the zero order damping matrices for the system with and without the cable
damping set-up. For both matrices, the largest entries are located on the diagonal, but
significant off diagonal terms exist. The algorithm calculated cross terms between nonadjacent points, which is physically unlikely. While the matrices are not identical, the two
sets of data in Table 25 are not definitively different. The observation leads to two
conclusions. The algorithm was unable to observe a change in structural damping between
the two tests. In some entries, the non-cable damping tests contained higher damping
magnitudes. The second conclusion is almost redundant. The algorithm was unable to
identify a localized source of structural damping in the second test. The [D] matrix for the
system with damping is semi-symmetric around its diagonal, just like the non-cable
damping [D] matrix.
109
x10^5
-2454
6060
-5287
3841
-748
-83
[D]without cable
6060
-15227
18219
-10414
3711
-482
x10^5
-2210
5867
-4707
3499
-581
-111
-5287
18219
-29001
15928
-8754
2401
3841
-10414
15928
-22510
12987
-3491
-748
3711
-8754
12987
-10888
3775
-83
-482
2401
-3491
3775
-1053
-581
3149
-7502
12399
-12264
4354
-111
-715
1506
-2932
4354
-643
[D]with cable
5867
-15771
16593
-8716
3149
-715
-4707
16593
-20868
13407
-7502
1506
3499
-8716
13407
-17686
12399
-2932
Table 25 Case 2 Structural Damping Matrices, Plate with and without Cable Damping
The [C] matrix calculations appeared ineffective in differentiating between the system with
and without the cable damping set-up. In a similar fashion as the [D] matrix calculations,
both [C] matrices in Table 26 have the largest entries located on the diagonal, but also have
significant values in the off diagonal entries. The [C] matrix from the cable damping test
does not show significantly higher damping in the upper left entries, which would identify
the existence of the cable/mass system. Instead, the non-cable system appears to display
unequally distributed damping, and greater levels of damping than the cable/mass equipped
bar. Note the negative value in the lower right entry (C6,6) of the cable damping matrix.
The cause can be observed in the imaginary component of the corresponding DSM plot
(Figure 63). The fit line is sensitive to the noise in the higher frequencies. The sharp jumps
around 1500 Hz cause the first order property to have a negative value. Curve fitting the
DSM data between 400 and 1500 Hz results in a C6,6 value of +535x101 N/m.
110
x10^1
6835
-15466
10610
-9432
1174
650
[C] without cable
-15466
34318
-42130
20673
-7081
644
10610
-42130
77423
-31003
18771
-5829
-15104
42003
-40313
18351
-6242
1624
9239
-40313
49731
-27335
15841
-2068
x10^1
5846
-15104
9239
-9102
755
782
-9432
20673
-31003
57234
-30155
7346
1174
-7081
18771
-30155
25729
-9799
650
644
-5829
7346
-9799
2224
755
-6242
15841
-28281
31073
-11078
782
1624
-2068
4378
-11078
-141
[C] with cable
-9102
18351
-27335
42046
-28281
4378
Table 26 Case 2 Viscous Damping Matrices, Plate with and without Cable Damping
The second order damping terms displayed the same qualitative properties as the structural
and viscous damping matrices (Table 27):
[E] without cable
-4.6
9.5
-4.2
5.6
-0.2
-0.7
9.5
-18.6
22.9
-8.4
2.5
0.1
-3.8
9.4
-3.7
5.9
0.1
-0.8
9.4
-26.8
23.4
-8.1
2.5
-0.9
-4.2
22.9
-51.1
11.6
-8.8
3.4
5.6
-8.4
11.6
-35.7
16.4
-3.2
-0.2
2.5
-8.8
16.4
-14.1
6.4
-0.7
0.1
3.4
-3.2
6.4
-0.6
0.1
2.5
-7.4
15.3
-19.0
7.2
-0.8
-0.9
0.1
-0.7
7.2
1.6
[E] with cable
-3.7
23.4
-28.1
11.4
-7.4
0.1
5.9
-8.1
11.4
-24.9
15.3
-0.7
Table 27 Case 2 Second Order Damping Matrices, Plate with and without Cable Damping
The largest terms are again located on the diagonals of both matrices. The terms are almost
all negative, due to the slightly negative concavity (negative second derivative of the
111
polynomial) of the imaginary DSM terms (Figure 62 and Figure 63). The E6,6 entry is the
only positive value on the diagonal, which is explained by the slightly positive concavity of
the corresponding imaginary DSM plot.
4.5.3 Summary of Cable Damping Experiment
The objectives of this experiment were fourfold:
•
Devise a structure which can exhibit a quantifiable change in damping due to a
localized structural change
•
Evaluate the quality and effect of spatial truncation of the experimental data using
several pre-processing graphical methods.
•
Fit the data and use several post-processing graphical methods to validate the quality
of the polynomial fit.
•
Differentiate between the two test set-ups by identifying the location and magnitude
of localized structural properties.
The cable damping mechanism produced a change in plate’s damping properties that was
qualitatively, and quantitatively observed. As listed in Table 21, the cable damping
mechanism increased the calculated modal damping by two to ten times the damping of the
unmodified plate. The CMIF plots of Figure 57 show an observable change in the
magnitude and sharpness of each modes’ peak. The measured DSM real and imaginary
plots (Figure 60 and Figure 61) both display a polynomial shape, between 400 and 1600 Hz,
which dominates over the measurement noise. The resulting curve fit of the data resulted in
112
qualitative agreement between the experimental and pseudo-synthesized DSM and FRF.
The modal properties of the pseudo-synthesized FRF data showed some quantitative
agreement between the estimated and the experimental damping values of the structure.
Unfortunately, examining the matrix coefficients produced no definitive differences
between the two test conditions, and also failed to identify the location of the cable damping
system on the plate. It is unclear if or how the DSM data and/or curve fitting algorithm can
be improved in order to effectively distinguish between the two test conditions.
113
4.6 Case 3: Fixed-Fixed Boundary Condition Test
The final experimental case study attempted to identify the distributed damping properties
of a beam with clamped ends. The work was a revisiting of tests performed by J. –H. Lee
et. al. [2].
Damper
Figure 69 Case 3 Experimental Structure
The test structure and damping mechanism were the same as those used by J. –H. Lee.
19mm
270mm
1
2
3
4
5
6
6.0mm
Figure 70 Clamped Beam Dimensions
114
The bar was clamped at both ends, and the entire fixture was clamped to ground. The same
six accelerometers from Case Studies 1 and 2 were mounted to the underside of the bar.
Likewise, an impact hammer was used to excite the top surface of the plate. One test set-up
collected data on the bar as shown in Figure 70. The second set-up is pictured in Figure 69,
with a small damper mounted between Points 4 and 5.
The objectives of this experiment were similar to the previous case studies:
•
Devise a structure which can exhibit a quantifiable change in damping due to a
localized structural change
•
Collect DSM data which has an identifiable frequency-dependent polynomial form
•
Observe and identify data characteristics which are unique to the clamped-clamped
boundary conditions
•
Fit the data with matrix coefficients which are able to distinguish between test
arrangements and identify localized structural properties.
The one unique objective of this case study was to observe the effects of clamped boundary
conditions on both the DSM data and the calculated polynomials. Case 1 and 2 used freefree boundary conditions, and both exhibited deviations from the expected DSM form below
200 Hz. The clamped boundary condition was expected to change the low frequency
features of the DSM data.
115
The measurement frequency range was less than the prior studies due to excitation
difficulties. The structure interacted with stiffer hammer tips, thus resulting in consistent
double hits in the data. The hardest, clean hitting tip produced frequencies out to 800 Hz.
Only one mode (around 400 Hz) exists within the sub 800 Hz range. The resulting DSM
data consisted almost entirely of high frequency mode residuals and noise.
4.6.1 Modal Analysis of Clamped Beam Tests
PTD Modal Fit of Experimental FRFs, Calculated from
Clamped Beam Tests
Test Condition
Mode Number
Frequency (Hz)
Damping (%)
Without
Damper
1
397.26 Hz
1.1557
With Damper
1
411.5 Hz
3.8612
Table 28 Modal Properties of Clamped Beam Tests
Table 28 contains the resulting modal properties of the clamped beam, with and without the
damper, for a frequency range of 0-800 Hz. The damper appears to add stiffness to the
system, and thus increases the natural frequency by 3.5 %. The damping is also increased
during the second test by over a factor of three.
4.6.2 Graphical Examination of the Clamped Beam Test data
Graphical analysis of the clamped bar FRF data begins with the CMIF plots:
116
CMIF Plot of Clamped Bar FRFs, No Damper
-4
10
-6
-6
10
10
-8
-8
10
10
-10
-10
10
10
-12
-12
10
10
-14
10
CMIF Plot of Clamped Bar FRFs, with Damper
-4
10
-14
0
100
200
300
400
Freq (Hz)
500
600
700
10
800
0
100
200
300
400
Freq (Hz)
500
600
700
800
Figure 71 CMIF Plot of Clamped Beam FRF, with and without Dashpot Damper
As with the previous experiments, several key observations are made from the CMIF plots.
Only one mode exists within the 0-800 Hz frequency range, and the mode displays an
increase in damping with the addition of the damper. The data set contains a significant
amount of noise in the lowest singular value. Since the bottom curves in Figure 71
dominate the DSM data (Figure 72), good quality DSM plots can not be expected.
CMIF Plot of Clamped Bar DSMs, No Damper
14
10
12
12
10
10
10
10
10
10
8
8
10
10
6
6
10
10
4
10
CMIF Plot of Clamped Bar DSM Data, with Damper
14
10
4
0
100
200
300
400
Freq (Hz)
500
600
700
800
10
0
100
200
300
400
Freq (Hz)
500
600
700
800
Figure 72 CMIF Plot of Clamped Beam DSM, with and without Dashpot Damper
The FRF plots for both test cases also exhibit an observable amount of noise (Figure 73 and Figure 74).
117
-340
250
-360
200
-380
0
-150
-540
-200
0
200 400 600
Freq (Hz)
800
-8
0
-100
300
-150
250
-200
200
-250
150
-200
-300
-400
200
-500
0
200 400 600
Freq (Hz)
800
-7
10
350
250
-100
-520
-50
-100
-50
-500
400
100
-440
-480
0
300
300
50
-460
450
200
100
-420
50
350
150
-400
Phase
400
150
0
200 400 600
Freq (Hz)
800
-6
10
-8
200 400 600
Freq (Hz)
800
-6
10
200 400 600
Freq (Hz)
800
-600
0
200 400 600
Freq (Hz)
800
0
200 400 600
Freq (Hz)
800
-8
10
-7
10
0
-7
10
-7
10
0
10
-8
10
10
-9
-9
10
Magnitude
10
-9
-8
10
-8
10
-9
10
10
-10
-10
10
10
-10
-9
10
-11
10
-11
0
200 400 600
Freq (Hz)
800
-9
10
-10
10
0
200 400 600
Freq (Hz)
800
10
-10
10
10
-10
0
200 400 600
Freq (Hz)
800
10
-11
0
200 400 600
Freq (Hz)
800
10
-11
0
200 400 600
Freq (Hz)
800
10
Figure 73 FRF Plot of Clamped Bar, without Damper
400
400
350
100
50
50
350
300
0
0
Phase
300
-50
250
-50
250
-100
200
-100
200
150
-150
-150
20
0
0
-200
-20
-400
-40
-600
-60
-800
-80
-1000
-100
-1200
-120
-1400
-140
-1600
-160
100
0
200 400 600
Freq (Hz)
800
-7
150
0
200 400 600
Freq (Hz)
800
-6
10
-200
0
200 400 600
Freq (Hz)
800
-7
10
-200
0
200 400 600
Freq (Hz)
800
-7
10
-180
-1800
0
200 400 600
Freq (Hz)
-2000
800
0
-4
10
200 400 600
Freq (Hz)
800
200 400 600
Freq (Hz)
800
-7
10
10
-5
10
-7
10
-8
-8
10
-8
10
10
-6
10
-8
10
-8
10
-7
10
-9
-9
10
-9
10
10
-8
10
-9
10
-9
10
-10
-9
10
-10
10
10
-10
10
-10
10
-10
10
-11
10
-11
0
200 400 600
Freq (Hz)
800
10
-11
0
200 400 600
Freq (Hz)
800
10
-10
0
200 400 600
Freq (Hz)
800
10
-11
0
200 400 600
Freq (Hz)
800
10
-11
0
200 400 600
Freq (Hz)
800
10
0
Figure 74 FRF Plot of Clamped Bar, with Damper
The resulting DSM plots in Figure 75 and Figure 76 do not look promising for a good
polynomial fit. Both sets of tests, with and without the damper, contain large discontinuities
118
in the real and imaginary components of their DSMs. Any resulting polynomial fit would
be highly sensitive to the frequency range chosen for the algorithm.
11
1.5
11
x 10
5
11
x 10
4
4
1
10
x 10
4
11
x 10
3
3
3
2
2
2
1
1
1
0
0
0
-1
-1
-1
-2
12
x 10
1.5
x 10
1
3
Real
0.5
2
0
0.5
1
0
0
-0.5
-1
-1
-1.5
-2
0
200 400 600
Freq (Hz)
800
-3
11
Imaginary
3
0
200 400 600
Freq (Hz)
800
-2
11
x 10
4
0
200 400 600
Freq (Hz)
800
-2
11
x 10
4
2.5
3
2
2
1.5
1
1
0
0.5
-1
0
-2
-0.5
-3
-2
-1
-4
-3
0
200 400 600
Freq (Hz)
800
-3
10
x 10
3
3
-0.5
0
200 400 600
Freq (Hz)
800
-1
11
x 10
2.5
2
2
1
0
200 400 600
Freq (Hz)
800
11
x 10
10
2
8
1.5
6
1
4
0.5
2
x 10
1
0
0
0
200 400 600
Freq (Hz)
800
-1
-1
0
200 400 600
Freq (Hz)
800
-2
0
200 400 600
Freq (Hz)
800
-3
0
200 400 600
Freq (Hz)
800
0
0
-0.5
-2
-1
-4
-1.5
0
200 400 600
Freq (Hz)
800
-6
0
200 400 600
Freq (Hz)
800
Figure 75 DSM Plot of Clamped Bar, without Damper
119
11
1.5
11
x 10
2.5
11
x 10
2.5
2
1
10
x 10
7
11
x 10
6
6
2
1.5
2
1.5
1
2
1
x 10
1.5
4
5
0.5
4
0.5
0
0.5
1
3
0
-0.5
0
0.5
2
-2
Real
12
x 10
0
-0.5
1
-1
0
-4
-1
-1.5
-2
0
200 400 600
Freq (Hz)
800
-2
-0.5
0
200 400 600
Freq (Hz)
800
2.5
16
0
200 400 600
Freq (Hz)
800
-2
0
200 400 600
Freq (Hz)
800
8
8
6
0.5
0
200 400 600
Freq (Hz)
800
-1
0
200 400 600
Freq (Hz)
800
800
2.5
x 10
2
1
0.5
0
0
0
-2
-0.5
-2
-4
-2
-4
200 400 600
Freq (Hz)
12
x 10
2
2
-0.5
0
1.5
0
0
-2
2
4
0
800
6
6
1
200 400 600
Freq (Hz)
4
8
2
0
4
10
1
-1.5
11
x 10
12
1.5
-8
10
x 10
14
3
Imaginary
-6
-1
10
x 10
2
4
-1
-1
11
x 10
-1
0
-1.5
11
5
-0.5
0
200 400 600
Freq (Hz)
800
-6
0
200 400 600
Freq (Hz)
800
-4
-1
0
200 400 600
Freq (Hz)
800
-1.5
0
200 400 600
Freq (Hz)
800
Figure 76 DSM Plot of Clamped Bar, with Damper
The experimental DSM data contain one notable trait. The imaginary components of both
tests displayed predominately positive magnitudes. The negative valued low frequency data
features of Case 1 and 2 were missing in the Case 3 data. It is not clear weather the
observation is due to the boundary conditions, or the experimental noise.
120
Figure 77 displays the B4,4 DSM data and synthesized DSM data from two different matrix
coefficient polynomial fit attempts.
10
10
3
3
2
2
1
1
0
0
-1
-1
-2
0
100
200
300
400
Freq (Hz)
500
600
700
-2
800
0
10
3
2
2
1
1
0
-1
-2
-2
0
200
300
400
Freq (Hz)
500
600
700
800
100
200
300
400
Freq (Hz)
500
600
700
800
100
200
300
400
Freq (Hz)
500
600
700
800
x 10
0
-1
-3
100
10
x 10
Imaginary
Imaginary
3
Synthesized DSM using 400-800 Hz
x 10
4
Real
Real
Synthesized DSM using 200-600 Hz
x 10
4
-3
0
Figure 77 Bar without Damping DSM, Entry B44, Synthesized ( | ) using Two Different Frequency
Ranges
Both synthesized data sets in Figure 77 used the same number of data points in the
calculation, but the starting frequencies are shifted by 200 Hz. The resulting polynomial fits
calculated significantly different Y-intercept magnitudes, curvature signs, and slopes. Due
to the poor quality of the experimental DSM data, synthesized FRF data and the
corresponding matrix coefficients will not be presented.
121
4.6.3 Summary of Clamped Beam Experiment
The objectives of this experiment were similar to the previous case studies:
•
Devise a structure which can exhibit a quantifiable change in damping due to a
localized structural change
•
Collect DSM data which has an identifiable frequency-dependent polynomial form
•
Observe and identify data characteristics which are unique to the clamped-clamped
boundary conditions
•
Fit the data with matrix coefficients which are able to distinguish between test
arrangements and identify localized structural properties.
Very few of the goals were accomplished. The modal fit data in Table 28 showed the
fulfillment of the first goal. The one observed mode between 0-800 Hz exhibited a change
in modal frequency and damping with the addition of the damper. Both data sets did display
new and potentially encouraging properties at low frequencies. The DSM data in Case
Studies 1 and 2 contained low frequency characteristics which deviated from the anticipated
form (presented by Section 3), and displayed negative values for the imaginary components.
While the Case 3 DSM data did contain error, the low frequency DSM data displayed
positive magnitudes. It still remains to be seen whether the observation is caused by the
clamped boundary conditions, or by experimental error. The second goal could not be
achieved, and therefore, the final goal could not be accomplished. The experimental DSM
measurements contained an excessive amount of noise, which resulted in significant
122
discontinuities in the real and imaginary data plots. The data did not exhibit any sort of
polynomial form, and any fit or discussion of matrix coefficients would be moot.
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5 Conclusion and Future Work
The objective of this work was to evaluate the validity of the DSM distributed damping
identification algorithm. The algorithm’s effectiveness was measured by how well it
exhibited the following features:
•
Minimize numerical error through computational simplicity.
•
Not rely on a priori information and be as independent of M and K properties as
possible.
•
Use several methods to identify data quality and to generate an intuitive feel for the
properties of the algorithm’s input data.
•
Straightforward mechanics in order to generate an intuitive feel for the algorithm’s
output.
•
Sensitive to the spatial properties of the structure’s damping.
•
Exhibit enough flexibility to differentiate between different damping mechanisms.
Some properties of the algorithm are readily apparent and have been demonstrated in this
work. As displayed in Section 2.1, the computational steps are minimal, and are arguably as
simple as can be. Therefore, the quality of results from the algorithm is primarily dependent
on the quality of the input data. A significant benefit of the algorithm is the independent
curve fitting of the real and imaginary component of the DSM. The damping calculations
become totally independent of the system’s mass or stiffness properties, thus no prior
knowledge is required.
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Several techniques were presented for validating experimental data and curve fit results.
CMIF plots of the experimental data allowed the user to identify the quantity of noise in the
data set, and also provided clues for establishing the optimal number of experimental spatial
degrees of freedom. The real and imaginary DSM plots allowed the user to inspect the data
for polynomial-like characteristics, and evaluate the optimal frequency ranges for the curve
fit. The final fit data was evaluated in the same manner as the experimental data, using the
real-imaginary and CMIF DSM plots, resulting in an intuitive understanding of the input
and output data of the algorithm. All of these features above were demonstrated using
multiple examples, analytical and experimental, and were effective at evaluating the quality
of the remaining algorithm features.
Unfortunately, the algorithm was unable to demonstrate its absolute effectiveness at
identifying a system’s damping properties. The analytical test cases produced favorable
results and helped identify some potential shortcomings of the algorithm. However, even
the simple structures of the experimental examples resulted in erroneous experimental
measurements and algorithm outputs. The experimental data of Case 1 and 2 contained low
frequency data with negative imaginary magnitudes; the data displays an increase in the
system’s energy due to non-conservative forces. The observed phenomenon does not satisfy
physical realities, and the reason for this deviation is not known. For the acceptable
frequency ranges of Case 1 and 2, the resulting curve fits appeared reasonable with certain
evaluation tools, but inconclusive using other methods. The pseudo-synthesized DSM and
FRF plots displayed encouraging levels of agreement between the experimental data and the
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algorithm output. However, analysis of the calculated matrix coefficient polynomials often
fell short of identifying differences between test set-ups and assigning localized damping
values which matched the test set-up. During Case 2, an asymmetric test structure was able
to produce symmetric damping matrices from the curve fitting algorithm. The DSM
calculation also appeared to be sensitive to complicated structures. The clamped boundary
conditions of Case 3 resulted in very poor DSM measurements. The observed data did not
contain any resemblance of the expected polynomial form.
More work is required in order to fully establish the effectiveness and usefulness of the
DSM spatial damping identification algorithm.
First, the source of the low frequency error of experimental Case 1 and 2 should be
identified. One potential source of the error is potentially due to the constant acceleration
input used for the experiments. The force hammer inputs a relatively flat frequency input
within the frequency range of interest. The effect is a constant acceleration (in a rigid body
sense), which is integrated to displacement by a factor of ω2. High bandwidth tests will
contain low and high ω values, and thus the integration can result in poorly conditioned
matrix inverse. Future work should examine ways to reduce the integration steps before
performing the inverse and applying a matrix coefficient fit. This could involve
reformulating the fit equations to handle F/V or even F/A data, or measure response velocity
instead of acceleration. A second potential source of error could be in the electronics of the
sensors. The amplifier/sensor crystal interaction can induce a second order error as a
function of ω. If the sensor error is too great, another sensor would be required. Regardless
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of the source, the observed DSM error is an inherent property of the experimental data, and
will always lead to erroneous matrix coefficients.
Next, signal processing techniques need to be developed in order to minimize the influence
of noise in the DSM data. The CMIF plots presented in this work were effective at
identifying experimental error, but few actions could be taken to attenuate the error.
Additionally, an effective on-line evaluation of experimental DSM measurements should be
developed. Currently, data quality cannot be fully evaluated until the complete FRF matrix
has been measured; requiring considerable time and equipment before any estimate of data
quality can be achieved.
Future tests should apply new excitation and measurement techniques. The DSM data
displayed a significant sensitivity to the sensor/structure interaction; therefore, non-contact
measurements could be promising. Impact hammer excitation was the only input method
applied in all three experiments. Shaker tests may be equipment intensive, but might offer
some good observations. Also, non-contact excitation may have some promise.
The DSM algorithm results should be compared to other damping identification methods.
The Rayleigh proportional damping identification method has produced good results on
lightly damped structures. The DSM method is intended to be used over a broad range of
structures, and ideally should perform as well as the Rayleigh method. The comparison
should serve as a good measurement of the algorithm’s performance compared to an
established benchmark.
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More work is needed to evaluate experimental DSM measurements and resulting curve fits
from a greater variety of structures. One test should contain a high level of damping (like a
constrained layer between two plates) in order to observe high damping forces and evaluate
the algorithm’s accuracy with heavily damped modes. Other structures should contain
complicated features, like joints or sharp changes in mass or stiffness. The structures
chosen for Case 1, 2, and 3, were intended to be easy to analyze using modal analysis and
the DSM algorithm. Some conclusions in this work are most likely confounded with
properties of the chosen analytical and experimental structures.
Finally, some additional steps should be added to the data analysis tools used in future
damping identification studies. The CMIF plots should track vectors across the frequency
range. Each unique vector in the CMIF plot should be displayed with one curve. In other
words, one color will follow a single vector through out the frequency range, instead of one
color being assigned to the highest singular value, and a second color to the next value and
so forth. Unsorted experimental CMIF data can display properties which are only anomalies
of the plot and do not exist in the data. Lastly, the matrix coefficients should be processed
using averaged (forced reciprocity) and un-averaged DSM data. Comparing the two
solutions offers another measurement of data quality, and can give the user more confidence
in the results.
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