UNIVERSITY OF CINCINNATI
11/07/2005
Date:___________________
Dhanesh M. Purekar
I, _________________________________________________________,
hereby submit this work as part of the requirements for the degree of:
Master of Science
in:
Mechanical Engineering
It is entitled:
A study of modal testing measurement errors,sensor
placement and modal complexity on the process of finite
element correlation
This work and its defense approved by:
Dr.Randall J.Allemang
Chair: _______________________________
Dr.Teik Lim
_______________________________
Dr.Dong Qian
_______________________________
_______________________________
_______________________________
A Study of Modal Testing Measurement Errors,
Sensor Placement and Modal Complexity on the
process of FE Correlation
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
MASTER OF SCIENCE
in the Department of Mechanical Engineering of the College of
Engineering
2005
By
Dhanesh M. Purekar
B.S.M.E University of Pune [India]-2001
Committee Chair: Dr.Randall J.Allemang
ABSTRACT
This thesis describes studies of finite element (FE) model validation methods for structural
dynamics. There are usually discrepancies between predictions of the structural dynamic
properties based on an initial FE model and those yielded by experimental data from tests on the
actual structure. In order to make predictions from the model suitable for evaluating the dynamic
properties of the structure, and thus to optimize its design, the model has to be validated. Model
validation consists of: data requirements, test planning, experimental testing, correlation, error
location and updating.
This requirement for the optimum experimental data is coupled with test planning to design an
optimum modal test in terms of specifying the best suspension, excitation and response locations.
The Effective Independence algorithm developed by Dr. Kammer has been implemented for the
optimal placement of sensors on the test structure. The assumption that the test results represent
the true dynamic behavior of the structure, however, may not be correct because of various
measurement errors. The errors involved in the modal testing (mass loading, sensor
misalignment, modal parameter estimation, DSP errors) are investigated and their effects on
estimated frequency response functions (FRFs) and on the modal parameters extracted from the
FRFs are also investigated. Also, the sensitivity of these measurement errors on the correlation of
the test structure is studied by comparing the experimental data with the analytical data.
The correlation phase of the modal validation process also demands calculation of real modes
from complex modes of the experimental test. This last topic is particularly important for the
validation of FE models and for this reason; a number of different measures of modal complexity
are presented and discussed in this thesis.
i
ACKNOWLEDGEMENTS
I express sincere gratitude towards my Advisor, Dr.Randall J.Allemang, for his valuable advice,
interest and encouragement throughout the thesis work. I would also like to thank him for
helping me in choosing the right research topic at the beginning of the project. Definitely he has
been a great source of inspiration in my process of learning, research and writing.
I would like to thank Dr. Teik Lim and Dr. Dong Qian for being on my graduate thesis
committee.
I would also like to thank Ray Martell for helping me with my experimental modal analysis
testing and briefing me on some of the X-Modal tools. Finally, I would also like to thank my all
friends at UC for their kind suggestions.
ii
Nomenclature
Basic Terms, Dimensions and Subscripts
x, y, z
Translational degrees-of-freedom / coordinates
t
Time
ω
Frequency of vibration, in rad/s
m
Number of degrees-of-freedom (DOFs)
n
Number of modes
i, j, k, r
Integer indices
Ε
Young’s Modules of materials
ψx
Experimental Mode shape
ψA
Analytical Mode Shape
ψ*
Complex Conjugate of mode shape
ωx
Natural Frequency of Experimental Mode
ωA
Natural Frequency of Analytical Mode
φi'
Real Part of i th mode shape
φi"
Imaginary part of i th mode shape
Matrices and vectors
[]
Two Dimensional Matrix
{}
Column Vector
iii
Structural dynamic properties
[M ] A
Analytical Mass Matrix
[Κ]
Stiffness Matrix
[φ]
Mode Shape/eigenvector Matrix
[E]
Predictive Matrix
[A]
Fisher Information Matrix
iv
Table of contents
Chapter1
Introduction…………………………………………………………….. 1
1.1 Finite Element Model Validation: General Concepts………………………………....2
1.2 FE Modeling and Correlation………………………....……………....…………........5
1.3 FE Model Updating……………………………………………………………………6
1.4 Objectives of research…………………………………………………………………7
1.5 Outline of Thesis………………………………………………...…………….....……7
Chapter 2
Background and literature survey.......................................................... 9
2.1 Introduction………………………………………………………………………...….9
2.2 Pre-Test Planning………………………………………………………………….......9
2.3 Target Mode Selection….............................................................................................11
2.4 Optimal Sensor Location Methods..............................................................................12
2.5 Effective Independence Algorithm..............................................................................13
2.6 Summary of Literature Review....................................................................................15
Chapter 3
FE Modeling Considerations..................................................................16
3.1 Introduction…………………………………………………...……...………...…….16
3.2 Basic Finite Element Types and Usage……………………………………………...16
3.3 Beam Element ……………………………………………………………………….18
3.3 The Shell Element……………………………………………………………………19
3.4 The Solid Element……………………………………………………………………20
3.5 Modeling Considerations in Ansys…………………………………………………..21
3.6 Other Modeling Considerations……………………………………………………...24
3.7 Test Structure-Material Properties and Geometry Details…………………………...26
v
3.8 Test Set Up Consideration - FE Modeling…………………………………………...27
3.9 Test Structure Using Solid45 Elements Only..............................................................29
3.10 Test Structure Using Shell63 Element Only………………………………………..30
3.11 Test Structure Using Combination-Shell63 & Solid45 Elements…………………..31
3.12 Convergence/Mesh density…………………………………………………………32
3.13 QR-Decomposition Method-Optimal Sensor Placement…………………………...34
Chapter 4
Experimental Modal Analysis………………………………………….37
4.1 Introduction…………………………………………………………………………..37
4.2 Analytical results-EI Algorithm……………………………………………………...38
4.3 Testing Set up Considerations………….....................................................................39
4.4 Modal Test of H Frame Structure……………………………………………………42
4.5 Test Set-up…………………………………………………………………………...43
4.6 Test Geometry & Analytical Results………………………………………………...44
4.7 Visual Comparison of Experimental & Ansys mode shapes………………………...44
4.8 Visual Inspection of Modal vectors………………………………………………….48
4.9 Modal Parameters of H Frame……………………………………………………….49
4.10 Discussion of Results……………………………………………………………….50
Chapter 5
Correlation for Model Validation……………………………………...52
5.1 Introduction…………………………………………………………………………..52
5.2 Role of correlation in model validation……………………………………………...53
5.3 Effect of mass overloading on correlation…………………………………………...61
5.3 Effect of transducer misalignment on correlation……………………………………71
5.5 Discussion of Results………………………………………………………………...74
vi
5.6 Concluding remarks on correlation study……………………………………………76
Chapter 6
Effect of Damping on Correlation……………………………………..77
6.1 Introduction……………………………………………………………....…………..77
6.2 Modal Complexity of Experimental Mode Shapes…………………………………..78
6.3 Modal Phase Collinearity [MPC]………………………………………....…………85
6.4 Mean Phase Deviation [MPD] ………………………………………………………86
6.5. Realization of Complex Mode Shape……………………………………………….88
6.6. Effect of Random Phase Scatter on Correlation…………………………………….89
6.7 Discussion of results…………………………………………………………………94
Chapter 7
Conclusions……………………………………………………………...96
7.1 Suggestion for future work…………………………………………………………..99
References……………………………………………………………………………...102
Appendix A:
Instrumentation……………………………………………………...106
Appendix B:
Modeling Code………………………………………………………107
vii
List of Figures
1.1 Product Design Cycle-Dynamics………………………………………………………….......3
3.2 Beam Element Geometry…………………………………………………………………….17
3.3 Four Noded Shell Element Geometry………………………………………………………..18
3.6.1 Shell Element Distortion Error…………………………………………………………….24
3.7 H Frame Test Structure………………………………………………………………………25
3.8.2 Simulation of Bungee Cord Stiffness in Ansys……………………………………………27
3.9 Test Structure using Solid45 Elements………………………………………………………28
3.10 Test Structure using Shell63 Elements……………………………………………………..29
3.11 Test Structure using Shell63 & Solid45 Elements…………………………………………30
3.12 QR-Decomposition Results………………………………………………………………...34
4.4 H Frame Test Geometry……………………………………………………………………..37
4.5 Experimental & Ansys Mode Shapes……………………………………………………….41
4.6 EI Algorithm Results………………………………………………………………………..43
4.7 Modal Test Mode Shapes……………………………………………………………………47
5.2 Natural Frequency Difference Plot………………………………………………………….56
5.4 Auto-MAC Analytical Modes Test Case1…………………………………………………..63
5.6 Auto-MAC Experimental Modes Test Case1……………………………………………….64
5.8 Cross-MAC Test Case1……………………………………………………………………..65
5.2.4 Test Case2-Master DOF…………………………………………………………………..66
5.11 Auto-MAC Analytical Modes Test Case2…………………………………………………67
5.13 Auto-MAC Experimental Modes Test Case2………………………………………………68
5.15 Cross-MAC Test Case2……………………………………………….................................69
viii
5.3 2D Model of Single Transducer Misalignment……………………………………………...71
6.2.2 Modal Complexity Plots…………………………………………………………………...78
6.6.2 Auto-MAC- 5° random Phase Scatter……………………………………………………..89
6.6.4 Cross-MAC- 5 ° random Phase Scatter…………………………………………………….90
6.6.6 Auto-MAC 10 ° random phase scatter……………………………………………………...91
6.6.8 Cross-MAC 10 ° random phase scatter……………………………………………………..92
ix
List of Tables
3.1
Order of Basic 3D Structural Finite Element Types…………………………………….16
3.7.1
H-Frame material properties table…………………………………................................25
3.9.1
Natural Frequencies for solid45 element model………………………………………...28
3.10.1 Natural Frequencies for shell63 element model………………………………………...29
3.11.1 Natural Frequencies for shell63 element model………………………………………...30
4.9.1
Mode Shape Descriptions……………………………………………………………….48
4.9.2
Modal Parameters of H-Frame…………………………………………………………..48
5.1
Natural Frequency Table………………………………………………………………...55
5.3
Auto-MAC values for Analytical Modes Test Case1…………………………………...63
5.5
Auto-MAC values for Experimental Modes Test Case-1………….................................64
5.7
Cross-MAC between Analytical & Experimental Mode Shapes………………………..65
5.10
Auto-MAC Analytical Modes Test Case 2……………………………………………...67
5.12
Auto-MAC Experimental Modes Test Case 2…………………………………………..68
5.14
Cross-MAC Test Case 2………………………………………………………………...69
5.17
Cross-MAC for 5° misalignment error…………………………….................................71
5.18
Cross-MAC for 10° misalignment error………………………………………………...71
5.19
Cross-MAC for 15° misalignment error………………………………………………...71
6.2.3
Modal Complexity Factor………………………………………………………………83
6.4.1
MPC and MPD………………………………………………………………………….86
6.6.1
Auto-MAC- 5 ° random Phase Scatter………………………………………………….89
6.6.3
Cross-MAC- 5 ° random Phase Scatter…………………………………………………90
6.6.5
Auto-MAC- 10 ° random Phase Scatter………………………………………………….91
6.6.7
Cross-MAC- 10 ° random Phase Scatter………………………………………………...92
x
Standard Abbreviations
DOF(s)
Degree(s)-of-freedom
FE
Finite element
FRF
Frequency response function
MAC
Modal assurance criterion
NFD
Natural frequency difference
SVD
Singular value decomposition
CMP(s)
Correlated mode pair(s)
EI
Effective independence
CMIF
Complex mode indicator function
CAD
Computer aided design
CAE
Computer aided engineering
RTDOF
Rotational degrees of freedom
COMAC
Coordinate modal assurance criteria
MCF
Modal complexity factor
Fn
Natural Frequency
MAC
Modal assurance criteria
MSF
Mode scale factor
OR
Orthogonality
XOR
Cross orthogonality
MEM
Modal effective mass
NCO
Normalized cross orthogonality
SEREP
System equivalent reduction expansion process
xi
2-D
Two dimensions
3-D
Three dimensions
MRIT
Multi reference impact test
MIMO
Multi input multi output
SNR
Signal to noise ratio
ECP
Eigenvector component product
TAM
Test analysis model
CPU
Central processing unit
DSP
Digital signal processing
SIMO
Single input multiple output
xii
Chapter 1
Introduction
Using Finite Element (FE) models to predict the dynamic properties of structures has become
more and more important in modern mechanical industries, such as the aerospace industry and
the automobile industry. Whenever there is a new design or modification of an existing design,
the structural dynamic properties of the product must be examined to fulfill the criteria proposed
either by the industry itself and/or external agencies before the product can be launched into the
market.
The traditional way for evaluating the structural dynamic properties of a product is to perform a
series of dynamic tests on prototypes of the product and to demonstrate its capacity to withstand
these tests. Until the experimental results show that the prototypes can be in compliance with the
relevant criteria, the product has to be redesigned and another design-test loop is followed. In
this design-test-redesign loop, much time and money is spent on producing prototypes and
performing tests.
With the growing capabilities of computing techniques, and the strength of the competition
between companies, FE model predictions are used more and more to take the place of practical
dynamic test data and to limit the number tests required. Furthermore, the FE modeling
technique may also be used to predict the dynamic response of structures when working in an
over limit situation which is very difficult, if not impossible, to simulate by experiments. All of
1
these situations depend on the accuracy of FE model predictions. So, the model must be
validated.
The definition of FE model validation in this thesis refers to the process of creating an FE model
which has acceptably similar dynamic behavior to that of the actual structure under consideration
and demonstrating this similarity. It should also be able to predict, with certain accuracy, the
dynamic properties of the structure under situations different to those in which the experiment
was undertaken.
1.1 Finite Element Model Validation: General Concepts
In general, it cannot be guaranteed that the dynamic properties from the initial FE model of an
industrial structure will have an accuracy that is acceptable in order to use the predictions
directly to replace the experimental data – there are generally discrepancies between the model
predictions and the experimental data from tests on the actual structure. Some of the
discrepancies come from noise in the experimental data. However, most are the result of
uncertainties and inaccuracies in the model. It is these discrepancies that make the FE model
technique not used as widely as it would be if the model predictions were accurate.
A strategy for the process of model validation as a part of product design cycle is shown as a
block diagram in Figure 1.1 In this block diagram, there are a series of distinct steps, including
tests on a structure and numerical calculations based on the initial model.
2
Once a process of model validation on the initial FE model is in progress, some tests (usually,
modal tests) have to be undertaken in order to obtain the actual dynamic properties of the
structure which the model is expected to predict with a certain accuracy. In order to ensure that
the results from the tests provide the information necessary for the model to be validated, and
that the tests are undertaken efficiently, predictions from the initial model are used in the test
planning procedure. Although these predictions may not be entirely accurate, the output from the
test planning procedure that includes margins set for this inaccuracy can help the test engineer(s)
to determine efficient test settings.
Model verification and correlation provides the information relative to the initial FE model
about its consistency with the experimental data. Although model verification has hardly been
mentioned in the literature of model validation, it is an important step in the process. It serves to
debug the initial model and to check if it is suitable for being subjected to the model updating
procedure. After correlation and verification on the initial model, if the model is found not to be
suitable for application of the updating procedure, the model has to be modified significantly, not
only for its parameter values, but also for its configuration or even for its mesh size.
3
Product Design Stage
Dynamics Consideration
Prototype Building
FE Model Building
Target Modes
Sensor Location
Initial Test Model
TEST
Correlation
Update
Valid Dynamic
FE Model
Figure 1.1 Product Design Cycle-Dynamics
4
1.2 FE Modeling and Correlation
Two important tasks in the model validation process are (i) Creation of the initial FE model (ii)
the process for determining how well it correlates with measured data. The ability to improve or
update the initial FE model is strongly dependent on the effective completion of both of these
tasks. The initial FE model must at least be a reasonable representation of the physical structure;
that is, it must be able to predict the modes close enough to the test modes so that the initial
correlation is sufficient to guide the updating process. Further, the correlation tools must be
capable of assessing the degree of correspondence between analysis and test modes despite the
presence of noise and other errors inherent in the testing process.
Complex geometric shapes pose the problem of determining how best to model them as
effectively as possible. This might be the case for structures with joints, or cast components,
having variable thicknesses at different cross sections. In such cases, the objective of the
structural engineer is to produce a practical (least number of DOFs) linear FE model with the
proper modal characteristics.
The ability to determine how well the FE model correlates with the test data is a significant part
of the model validation process. There has been, and continues to be, a great deal of effort spent
on model correlation. The current techniques for assessing correlation can sometimes be
misleading by indicating poor numerical correlation when the modes visually appear to be
identical.
5
1.3 FE Model Updating
Model updating plays a central role in the process of model validation. By constructing and
solving an updating equation, the model updating step tries to find a group of values that modify
the parameters in the initial model in order to reduce the discrepancies between the experimental
data and the dynamic properties predicted by the updated model. The selection of the updating
parameters will affect significantly the results from the model updating procedure. Although it
cannot be guaranteed that all the parameters selected are those with incorrect values, the
selection of parameters based on certain techniques helps the updating procedure to have
convergent results.
When an initial model has been subjected to the model updating step, the updated model is one
that can produce the dynamic property predictions with discrepancies from the experimental data
minimized in the way defined in the updating equation or in the residual function. Because of the
different choices of updating methods, updating parameters, and different choices for the
experimental data, there can be many updated models. Thus, each updated model must be
assessed to check if it is optimized in a pre-defined way and is capable of representing the
dynamic properties of the structure. This procedure of assessing updated models is the final and
crucial procedure in the process of model validation. Only after an updated model is assessed in
some way and the assessment yields a satisfactory result, can the model be said to be a validated
model.
6
1.4 Objectives of research
The overall objective of the research reported here is to analyze complete model correlation
process, which includes provision of a balanced set of algorithms for all the procedures in the
process of FE model validation. According to the above summary of all steps in the process of
FE model validation, the specific objectives of the research in this thesis are as follows.
• To study and analyze the various optimal sensor placement algorithms for modal testing.
• To study the procedures of correlation in the context of model validation process.
• To determine the effect of measurement errors on the model correlation.
• To study the effect of complexity of the mode shapes on the process of correlation.
1.5 Outline of Thesis
Chapter 2 specifically prescribes a literature review of the topic of optimal sensor placement &
model validation. The focus is put on three aspects: (i) test planning, (ii) FE model verification
and (iii) FE model updating. On test planning, several techniques are discussed for optimal
sensor location selection. On model verification, the methods for estimating and reducing
discretization errors are described.
Chapter 3 addresses all the issues related with the FE modeling of the test structure. In this
chapter, the basic theory of finite element modeling is discussed. This discussion is then
followed by the results obtained from different test cases on the ANSYS model. The case studies
of using these methods show that they are useful and efficient in order to verify FE models in the
light of model validation.
7
The focus of Chapter 4 is the experimental modal analysis of the test structure, from the
viewpoint of model correlation as a whole. The different test set up considerations as well as
peculiar problems encountered when performing the test are studied and results of the tests
corresponding to the different test scenarios are tabulated.
Chapter 5 is concerned with the correlation procedure in the process of model validation as a
whole. What kind of information is needed from the tests for validating the FE model for
structural dynamics and how the information can be best obtained? With the case studies of
using clean measurement data and data with experimental errors, these questions are explored.
Chapter 6 presents issues related with the complexity of experimental modal vectors and their
effect on the correlation, in case of lightly damped structures. The chapter briefly discusses
different measures of the modal complexity, their causes and ways of normalizing the complex
modes to real modes.
Chapter 7 presents a discussion on the strategy of model validation. The results of test
correlation in the light of all test cases are summarized giving useful information about possible
sources of errors which could lead to a poor test correlation. For further studies in the area of
model validation, some general suggestions are given at the end of this chapter.
8
Chapter 2
Background and literature survey
2.1 Introduction
In order to make sure that the data from a test, especially a modal test undertaken for the process
of model validation, can give useful information about the structure and the test can be carried
out effectively, the test settings must be well planned. Among the considerations in the settings,
selecting suspension points, driving point(s) and response points are the most important. This
determines how and where to suspend the test structure, how to select the point(s) in the test
structure as excitation point(s), and most importantly how to choose the response DOFs that
must be measured.
Test planning can utilize results predicted by an initial FE model and, by performing a series of
calculations with a certain amount of margin in the light of the expected inaccuracy of the model,
give some guidance for the test. This will ensure linear independence of a set of target modes
which would be sought in ground testing, flight testing or on orbit testing.
2.2 Pre-Test Planning
Each time a modal test is undertaken on a structure, one of the first things which need to be
determined is how many and which responses should be measured, how the structure should be
suspended and which positions are most suitable as excitation locations. Usually, experience and
intuition are used to answer these questions, but in many cases, after the test is finished and when
9
reviewing the experimental results, it is realized that not all the best measurement points were
always selected or maybe that some of them need not have been measured at all, and the same
applies to suspension and excitation position.
One of the approaches used to select measurement points is to show visually a preliminary
estimate of the theoretical mode shapes and the corresponding natural frequencies of the
structure and, on the basis of these data, to select the optimum suspension, excitation and
measurement positions. This is not a strongly recommended approach since any prior knowledge
of the values for modal parameters of a theoretical model can influence measurements and, in
extreme cases, the experimentalist can end up altering the experimental set-up in order to
measure values of modal parameters which are close to the theoretically-predicted ones.
One major goal of planning a modal test is determination of the data requirements for a particular
test. Depending on the final purpose of the experimental results, such as: (i) measuring the
structure’s natural frequencies only; (ii) correlation with theoretical predictions, or (iii) updating
of a theoretical model of a structure, the number and position of the measurement points will be
decided, but in any case, it is desirable that an optimum set of points is selected. This means that
before any actual selection of the measurement points is made, an evaluation of all degrees-of
freedom has to be made in order to select the minimum number and the best choice for
suspension, excitation and measurement points.
There are limited numbers of measurement positions and modes that can be measured during a
modal test and, in general, test data tend to be incomplete by comparison to theoretical data.
There are two main sources of incompleteness in the experimental data obtained from modal
10
tests; the extent of the measured frequency range (or number of measured modes) and the
number of measurement degrees-of-freedom.
The number of the measured degrees-of-freedom mainly depends on the final application of the
experimental results and/or the number of measured modes, although most correlation and
updating methods require that the number of measured degrees-of-freedom is greater than the
number of measured modes.
2.3 Target Mode Selection
A very first important step in the pre-test analysis is the selection of the target modes. The
following methods are widely used. [18]
2.3.1 Modal effective mass method:-A mode with a large effective mass is usually a significant
contribution to the system response. The criteria generally used is that a mode with an effective
mass greater than 2% of the total mass are selected as target modes.
2.3.2 Kinetic/Strain energy fraction:-To predict local modes more effectively, the kinetic
and/or strain energy fraction of the subsystem is calculated. The criterion is to consider a
component mode as target mode set is an energy content of 50% of the total system energy.
2.3.3 Mode participation factors:-None of the previous methods take excitation into account.
The structural integration is also function of excitation frequency. Modal participation factor is
frequency dependent and its amplitude is determined by structure’s response and by the
excitation spectra.
11
2.4 Optimal Sensor Location Methods
2.4.1 Energy distribution method: [1], [12], [13]
It is assumed that by placing the sensors at points of maximum kinetic energy, the sensors will
have the maximum observability of the structural parameters. Those DOFs with the maximum
kinetic energy for a mode (or modes) of interest would be chosen as sensor locations.
2.4.2 Eigenvector Component Product (ECP) method [12]
A Product for a DOF is calculated by multiplying the eigenvector components on the DOF over
the frequency range of interest. The DOFs with maximum values of this product should be
retained as candidate sensor locations. This method will preclude all points at nodal points
because of the zero products on these points.
2.4.3 Flexibility shape method
Flanigan developed a search algorithm to perform the sensor location selection based on the
TAM concept [10]. They used the static flexibility shapes of the initial FE model to define an
optimum reduced model.
2.4.3.1. Test-Analysis Model [TAM]
A basic objective of the modal survey is to verify that the FE model of a structure is sufficiently
accurate to predict the structure’s response to operating environments. In general FE model will
have many more DOF than the test configuration will have accelerometers. In order to compare
the FEM with the test results directly, a reduced representation or Test-Analysis Model [4] must
12
be generated. The DOF of the TAM will correspond one for one with accelerometers in the
modal survey test configuration. The development of TAM serves following purposes:
TAM enables a quantitative comparison of the accuracy of the FEM during pretest correlation
activities in the form of orthogonality and cross-orthogonality checks. All of these require an
accurate reduction of the FEM mass and stiffness matrices down to the TAM DOF, or the TAM
will not be able to perform its functions.
2.4.4 Effective Independence (EI) – Fisher Information Matrix [5, 7, 13, and 17]
Effective Independence is based on an idea that the sensors in a modal test should be placed such
that the obtained mode shapes on the measurement DOFs are spatially independent of each other.
The basic logic of the technique for most of the methods is the same - choosing the sensor
location candidates by the use of the Fisher Information Matrix.
2.4.5 Effective Independence – QR decomposition
Schedlinski and Link developed a method using QR decomposition for sensor location selection
[14]. The method is based on the QR decomposition of the eigenvector matrix and to localize a
subset of structural DOFs as sensor locations such that the linear independence of the mode
shapes to be measured is maximized.
2.5 Effective Independence [EI] Algorithm
The Effective Independence transducer location optimization technique was first proposed by
Kammer [7] in 1991, and is based on the following iterative process. The initial mode shape
matrix, obtained from the finite element analysis of the test structure [ φ ]N x m, is used to calculate
the following so-called ’Fisher Information matrix’, [A]:
13
[Α]m x m = [φ ]TN×m [φ ] N×m
and this matrix is then used to calculate so-called ’Prediction Matrix’, [E], of the mode shape
matrix, as follows:
[E]N x N = [φ ] N×m [ A]−m1×m [φ ]'N×m
This Prediction Matrix is a symmetric and an idempotent matrix, i.e. it possesses the
following features:
Trace ([E]) = rank ([E]) = rank ([φ])=m
[E ]2 = [E]
The prediction matrix was originally developed in linear regression theory as a tool for selection
of influential observations and for examination of sensitivity of observations and variables.
Every diagonal element of the prediction matrix, [E], represents the contribution of its
corresponding degree-of-freedom to the global rank of the truncated mode shape matrix, [ φ ].
The diagonal element which has the smallest value in the prediction matrix determines the DOF
which has the smallest contribution to the representation of mode shape matrix, [ φ ]. That DOF
is removed and the prediction matrix is calculated again in order to check the rank of the reduced
mode shape matrix,
[ φ ].
This iterative process of removing the least-contributing DOF is
continued until the rank of the mode shape matrix ceases to be equal to the number of modes in
mode shape matrix
[ φ ].Chapter-3
briefly discusses different finite element modeling
considerations in order to obtain good initial estimates for the modal matrix. This procedure can
be CPU-time consuming for large systems (those with over 10000 DOFs) and to avoid this
14
drawback, there is variant of this algorithm which reduces several DOFs in each iteration [44].
The results obtained from these two algorithms do not necessarily match with each other.
The condition number and rank of the prediction matrix were monitored during the elimination
process in order to have confidence in the results. The smaller the condition number of the
truncated mode shape matrix, the better the selection of measurement DOFs from mode shape
linear independence consideration. There are different levels for the critical condition number
defined in the literature, but it will be accepted here that any value of the condition number
below 103 represents a well conditioned matrix; the values between 103 and 107 can be
considered as poorly conditioned matrix but are still acceptable in a numerical sense and any
value higher than 107 represents a near-singular or a singular matrix. The values of these two
parameters have to be balanced to obtain the optimum selection of measurement DOFs.
15
2.6 Summary of Literature Review
The literature on test planning, FE model verification and model updating has been briefly
reviewed in this chapter. For test planning; the most important issue is what should be provided
from experimental data when the data are used to validate an FE model for structural dynamics.
All methods for test planning mentioned in this chapter are for the purpose of ensuring that all
modes in the frequency range of interest should be measured and all measured mode shapes
should be linearly independent. With ever increasing complexity of the practical structures, these
methods could be helpful for structural dynamics engineer as well as for optimum test
instrumentation.
The model validation process is computationally extensive in the sense that with few million FE
degrees of freedom, the reduction of FE model to chosen master degrees of freedom becomes
tedious with huge computational memory requirement.
16
CHAPTER 3
FE MODELING CONSIDERATIONS
3.1 Introduction
The focus of this chapter is to present a brief overview of finite element modeling of the test
structure for linear vibration analysis. With the amount of research that has gone into developing
finite elements into successful theory, one might expect a significant amount of literature to be
available on FE modeling. However, the ability to generate an accurate FE model using
minimum number of degrees of freedom is still a challenging task! The modeling method
adopted for one structure does not necessarily suit other types of structures; hence there is very
little literature available on efficient geometric modeling techniques.
3.2 Basic Finite Element Types and Usage
There are various types of 2-D and 3-D elements that are used for FE modeling of different types
of components. The goal of this section is to present the fundamental guide to basic 3-D
structural element usage and properties.
The Table 3.1 lists the basic 3-D element types used for structural dynamic design. These
elements are generally divided into three major types i.e. beams, shells/plates and solids. The
different element formulations within each group can also be used in many cases but in general
the discussion here is presented to facilitate the usage of these groups in certain situations.
17
Table 3.2 Order of Basic 3D Structural Finite Element Types
18
3.3 Beam Element
The beam element as shown in Figure 3.3 is used to represent those features that have one
dominant dimension. The geometry for this element is specified by two nodes and a vector which
is used to determine major and minor axis of the beam. Each of the node has six degrees of
freedom- 3 translations in the respective orthogonal axis directions and 3 rotations, one about
each orthogonal axis. The input for this element is also shown in the figure. The cross sectional
area as well as moment of inertia is specified as the input. In order to model taper sections, each
end of the beam can be modeled with different properties so as to simulate the section.
Figure 3.3 Beam Element Geometry
Beam elements are primarily used to model part geometry, such as a boss or other rod like part
member. The other typical applications of beam elements includes reinforcing elements
connected to thin shell types of elements, bolt representation, ribs, thin slender part of casing,
shafts.
19
3.4 The Shell Element
The shell element is generally used to represent features that have one dimension that is much
smaller than the other two. The element is based on shell theory and is ideal for representing
shell like structures, thin plates, and structures with hollow cross section.
Figure 3.4 Four Noded Shell Element Geometry
The element geometry is specified by at least three apex nodes for triangular or four corner nodes
for quadrilateral, while higher order elements have mid side nodes along the edges and a node in
the center of plane for a total of 10 and 12 nodes for triangle and quadrilateral respectively. For
the triangle each node has 6 DOFs-3 translations in the respective orthogonal directions and 3
rotations, one about each orthogonal axis. For quad elements, there are 3 translations but 2
rotations. The rotations about an axis perpendicular to the plane of the element is not allowed,
which is also called as drilling DOF. The shell element thickness can be constant for all nodes or
could be specified for each node independently in order to model complex geometries.
20
3.4 The Solid Element
These elements are basically used to represent the features that have three dimensions of similar
order. However, they can also be used to model thin shell like structures with caution. The
element geometry is specified by at least 4 apex nodes for tetrahedron and by 8 nodes for
hexahedron. Each node has 3 DOFs for 3 translations in the respective orthogonal axis directions
Sometimes, there is a need to use combination of shell and solid elements for structures like
engine block or gas turbine parts. This kind of modeling is called as hybrid modeling. In these
situations, constraint elements are used to place shell elements on the surface of solid elements.
The commercial FE packages like MSC/Nastran and Ansys provides different ways which can
be used for hybrid modeling.
As mentioned earlier, FE modeling demands attention to details of the structure, kind of analysis
to be needed and accurate material properties. Subsequent parts in this chapter dwell around the
details of the test structure geometry and how it has been implemented in the Ansys FE package.
21
3.5 Modeling Considerations in Ansys [19]
Choice of elements: Ideally, structures would be represented for FEA by solid elements, for this
would eliminate the problem of positioning the mid-plane of shell elements, of exactly
representing the sectional properties of components, and positioning welds in their design
location. Unfortunately, there would have to be several solid elements through the thickness of
sheets of metal plates to capture local bending effects with any accuracy, and the other
dimensions of the elements would have to be kept small so that the aspect ratios of the elements
were acceptable. Consequently, the number of elements would be unbelievably large, which in
turn demands more computational power and time. It is not feasible to model many thin-wall
structures with solid elements.
Shell elements efficiently represent thin sheets or plates of steel or aluminum, both flat and
curved surfaces. They include out-of-plane bending effects in their fundamental formulation, as
well as transferring shear, tension, and compression in the plane.
Shell-solid element connectivity: In many practical applications, a combination of shell and
solid elements needs to be used. Developing an interface between a shell portion and a solid
element portion of a model has a difficulty: All solid elements included in the latest version of
Ansys do not include rotational degrees of freedom at the nodes, and this results in a rotational
"joint" if shell elements are connected to a solid (In Ansys, Version 5.6, there were solid
elements available capable of six degrees of freedom at each node, but in the subsequent
versions, they have been discarded). Even if a solid element with rotational degrees of freedom is
used, the rotational stiffness at a solid's edge node is not appropriate for connection to shell
elements-these solid elements were intended to be connected to each other. A modeling trick that
22
is often used is to overlap one shell element with the first element in a solid, and join the nodes
in two locations in order to imply continuity of rotations, as well as deflections. This is not a
perfect fix. Rigid regions with node pairs (rigid links with CERIG) may be used to enforce
connection, although high local stresses will result. The rigid regions only apply accurately for
small displacement analysis.
In our analysis, the test structure has been modeled using following approaches:
•
FE modeling using only 8 noded brick elements.
•
FE modeling using combination of shell63 and solid45 elements by creating rigid regions at
the interface of shell and solid element nodes.
•
FE modeling using combination of shell63 and solid45 elements by inserting one layer of
shell elements into solid layer and joining the two nodes at two locations in order to imply
continuity of rotations as well as deflections.
•
FE modeling using shell63 elements only.
Application of boundary conditions: In modal testing, test structures are often suspended using
soft springs to approximate a free-free state of the structure. Free-free modal tests are popular
because it is felt that the effect of soft springs on the test structure is negligible and this condition
is easily approximated in a test. In the free-free state of the structure, during the test, the six rigid
body modes no longer have zero natural frequencies, but they have values which are
significantly lower than that of the first elastic mode of the structure. However, for flexible
structures, the lowest elastic mode may interfere with the rigid body modes. Therefore the
support system needs to be designed to avoid interference of the rigid body modes on the lowest
elastic mode of the test structure. On the other hand, even soft springs introduce stiffness and
23
damping into the system. This added stiffness and damping of the suspension may significantly
alter the modal parameters of the elastic modes of the test structure, if the separation between
rigid body and deformation modes is not achieved.
The test structure is suspended using bungee (shock) cords at four corners in order to simulate
the free free boundary conditions for modal analysis. In order to simulate these conditions in the
analytical model, spring element (COMB14) is employed to represent the stiffness of the
bungee cords.
Use of Units: Vibration and transient analysis require that the mass of the structure be entered in
units consistent with the other units in the model. This requires that mass be represented as
pounds/in/sec^2. Pounds here mean "pounds force", the force with which 1.0 g of gravity pulls
on the mass. This means dividing the weight in "pounds force", or the density in pounds/in^3, by
the number 386.1 (more accurate than 32.2*12=386.4), which is the acceleration due to gravity
expressed in inches per second squared (in/sec^2).
In consequence, when mass and mass density have been defined this way (the density of steel,
which depends on the alloy, if given as 0.2836 lb/in^3 would be entered into ANSYS as
0.0007345) it is necessary to enter 1.0 g of gravity as 386.1 in/sec^2 to let ANSYS apply the
correct force due to gravity on the structure. It will be referred to as pounds per square inch.
ANSYS refers to these units as "BIN" (/UNITS command for "British system using inches",
noting that the /UNITS command is for annotation of the database, and has no effect on the
analysis or data).ANSYS does not care what units are used, nor does it issue warnings. The
analyst must be consistent in the set of units in one model, to avoid errors.
24
3.6 Other Modeling Considerations [20]
The selection of element type is one part of the building FE model; the other due considerations
would be to select mesh density and order of element. With the selection of a particular element
type, the other issues may arise during modeling complex regions or geometries. These include
unacceptable large element distortions from large aspect ratios, skew or warp. Similarly, proper
selection of element order is important when modeling with different types of elements, for
example shells and solids. In such cases one should choose elements of as close to the same
order as possible. In our application shell63 and solid45 elements were used in order to match
element order (linear elements).
Element distortion:
An ideal element is one that has equal dimension and angle along all edges. So ideal triangular
shell element would be equilateral and ideal quad element should be perfect square. This is also
true for solid elements like solid92 or solid95.Real applications rarely allows user to have all
elements to be equally shaped. The element distortion can cause significant degradation to the
accuracy of FE model.
There are four primary forms of element distortions:
•
Large aspect ratio-Ratio of maximum and minimum lengths across the element
•
In plane skew
•
Out of plane warp
•
Excessively acute or obtuse apex angles.
25
Figure 3.6.1 shows primary types of element distortions using four nodes shell element.
Fig.3.6.1 Shell Element Distortion Error
Ansys allows for maximum aspect ratio of 7; otherwise, it will warn the user about unpredictable
results. The in plane skew and excessively acute/obtuse angles can be checked by issuing the
CHECK command in Ansys. The apex angle has to be within a required range such as between
45 and 135 degrees for Ansys. Similarly, nodes of each face of the element are checked to make
sure that they are within certain tolerance of being in the same plane. The different finite element
packages will have in general different threshold values for these parameters and will give
warning messages if the elements exceed these limits and abort the solution if the values are
larger than threshold values.
Nature of Geometry
The nature of the geometry often dictates what type of element should be used for meshing. For
example, parts produced by casting lend themselves to meshing with solid elements, while sheetmetal structures are best meshed with shell elements.
26
3.7 Test Structure-Material Properties and Geometry Details
The test structure is two parallel long, straight, homogeneous hollow columns of rectangular
cross section separated from each other and held in position by a shorter piece of identical cross
section tubing welded perpendicularly to the two longer columns. It is constructed by two 72inch-longs and one 32-inch-long steel tubular section, welded together in an H-shape. The
tubular sections which are made of carbon steel have dimensions of 6′′ × 2′′ × 1/4′′ as the height,
width and thickness, respectively. Four small steel plates with dimensions of 6
1/2′′ × 4′′ × 1/2′′ are then welded at the four ends to which four aluminum plates with
dimensions of 11′′ × 10′′ ×1′′ are connected by bolts.
Figure 3.7 H Frame Test Structure
Property
Carbon steel
Aluminum
Modulus of Elasticity (Mpsi)
30
10.3
Modulus of rigidity(Mpsi)
11.5
3.8
Density(lb/ in 3 )
0.282
0.098
Poisson’s ratio
0.292
0.34
Table 3.7.1 H-Frame material properties
27
3.8 Test Set Up Consideration - FE Modeling
Test set up and instrumentation used for the modal testing may have effect on the results.
Especially in modal testing, application of boundary conditions and effect of transducer mass
may affect the output adversely. These two factors have been given due consideration in the FE
modeling and are discussed in the following sections.
3.8.1 Mass loading of Accelerometers
The instrumentation that is mounted on the structure can, in many instances, have an effect on
the measured frequency response functions. From a theoretical standpoint, the natural frequency
is related to the square root of the ratio of stiffness to mass. So it stands to reason that if the mass
of an accelerometer is ‘added’ to the structure to make a measurement, then the natural
frequency will be lowered: obviously, the larger the accelerometer mass (or number of
accelerometers), the more pronounced and obvious the shift of the frequency.
The mass of the accelerometer is compared with the modally active portion of the test structure
(modal mass), which might be significantly different than its actual total mass. Dummy sensors
(having mass same as that of the accelerometer) have been placed at all the measurement
locations, to get consistent data and avoid frequency shifts, caused by moving the sensors from
point to point. These hexagonal nuts (dummy sensors) are modeled using concentrated mass
elements (mass21) in Ansys and are shown by star symbol in Figure 3.11.
28
3.8.2 Boundary conditions
One of the fundamental assumptions for experimental modal analysis is often a free free
boundary condition, which can not be ideally observed in real testing situations. In the analysis
of the H-frame, the frame is suspended by means of bungee cords at the four corners of it to
simulate the unconstrained behavior. A combined shell and solid element model of the H-frame
(shown in Figure 3.8.2), which was clamped at the four corners using bunging cords, was run
using ANSYS to extract first 10 modes. The number of DOFs of the full model was 7032.
Figure 3.8.2 Simulation of Bungee cord stiffness in ANSYS
The spring element [COMB14] in Ansys is used to model this stiffness, so that real boundary
conditions can be applied for theoretical modal analysis of the structure. The upper end of the
element is constrained for all DOF (translation as well as rotations) and is attached to the four
corner nodes of the legs as shown in Figure 3.8.2.
29
3.9 Test Structure Using Solid45 Elements Only
Figure 3.9 Test structure using solid elements
RESULTS FOR NATURAL FREQUENCIES:
Mode No.
Frequency [ Hz]
1
15.652
2
24.954
3
37.454
4
56.625
5
76.459
6
130.38
7
162.8
8
166.22
9
173.94
10
*
Table 3.9.1 Natural Frequencies for solid45 element model
30
3.10 Test Structure Using Shell63 Element Only
Figure 3.10 Test Structure using shell elements
NATURAL FREQUENCIES PREDICTED BY MODEL:
Mode No.
Ansys [Hz]
1
14.724
2
23.969
3
35.744
4
57.002
5
76.764
6
153.000
7
163.010
8
173.620
9
188.820
10
210.250
Table 3.10.1 Natural Frequencies for shell63 element model
31
3.11 Test Structure Using Combination-Shell63 & Solid45 Elements
Figure 3.11 H Frame using shell-solid combination
NATURAL FREQUENCIES PREDICTED BY MODEL:
Mode No.
Frequency [Hz]
1
15.267
2
24.194
3
37.222
4
57.63
5
78.191
6
153.66
7
164.19
8
174.2
9
189.6
10
207.3
Figure 3.11.1 Natural Frequencies for shell63 element model
32
3.12 Convergence/Mesh density
Increasing the mesh density (number of elements per unit area/volume) until the element
becomes infinitesimally small will produce the real or correct solution. This is assured by the
element by being able to pass the patch test. Elements that the pass patch tests are guaranteed to
give convergence towards the correct solution. In FE modeling, the model quality is very often
associated with the mesh density. However, when the objective is to correct or to update the
initial finite element model of a structure, there are enormous computational advantages in
keeping this initial model small while retaining a basic correlation with the experimental model.
Indeed, large finite element models are often very difficult to update because of the relative
insignificance of the individual elements which must be corrected one by one. This is especially
true for both response and sensitivity-based methods which assign and compute correction
factors for each elemental matrix. One is then confronted with the dilemma of (i) updating a
large (and hopefully fully converged) model and accepting the computational consequences, or
(ii) starting with a small (and probably not fully-converged) model and reducing the
computational requirements by several orders of magnitude. If both approaches lead to
comparably updated models in terms of representing the dynamic behavior of the actual
structure, then there will be a very good argument in favor of choosing route (ii) because of its
obvious computational advantages. [21]
Table 3.12 shows the convergence study of the natural frequencies for different mesh sizes using
combination of shell and solid elements.
33
Mode Number
Nodes[1112]
Nodes[1212]
Nodes[1252]
Nodes[1412]
Nodes[1820]
1
15.064
15.02
14.995
14.981
14.981
2
24.036
24.033
24.031
24.03
24.03
3
36.664
36.519
36.434
36.386
36.387
4
59.126
59.057
58.991
58.986
58.991
5
78.506
78.337
78.219
78.172
78.166
6
147.5
147.3
147.22
147.1
147.09
7
161.96
161.81
161.66
161.61
161.56
8
174.12
174.03
173.96
173.9
173.89
9
184.16
183.96
183.85
183.69
183.77
10
189.52
189.4
189.51
189.49
189.56
11
201.26
201.12
201.16
201.13
201.03
12
291.24
290.09
289.28
289.12
289.19
13
302.7
302.58
302.52
302.52
302.88
14
316.65
316.56
316.56
316.58
317.02
15
404.37
403.69
403.49
402.99
403.88
16
411.7
410.53
410.16
409.46
410.09
17
418.17
417.5
417.11
416.54
416.86
18
471.66
471.11
470.64
469.96
470.33
19
500.03
498.86
497.78
497.7
500.38
20
562.38
561.7
561.08
560.88
564.27
21
573.12
572.5
572.3
572.83
573.68
22
595.84
595.23
595.22
596.14
597.13
23
625.42
624.96
624.87
625.71
625.99
24
705.82
704.5
704.35
703.8
708.54
25
725.23
723.99
723.33
723.53
723.12
26
733.39
732.35
732.35
731.76
732.11
27
796.89
796.53
797.76
796.7
797.92
28
817.84
816.13
815.99
815.74
815.22
29
819.9
818.69
818.91
818.97
818.43
Table 3.12 Convergence of Natural frequencies of H Frame
34
3.13 QR-Decomposition Method-Optimal Sensor Placement [14]
The approach presented by Dr.Schedlinski and Dr.Link [14], uses a QR-Decomposition
technique to determine the most effective subsets of the modal matrix. In order to determine an
optimal subset of structural degrees of freedom that can be used as measurement points, modal
data obtained analytically was used. The idea is that the linear independent rows of the modal
matrix indicate the degrees of freedom that should be chosen as a sensor location because they
form the smallest possible modal matrix which provides a MAC [12] matrix with minimized off
diagonal terms.
The QR-decomposition is applied to the transpose of modal matrix because the information for
one given degree of freedom is provided in one single row of the modal matrix and QRdecomposition does only sort the columns and not the rows of a matrix. QR-decomposition will
give most independent columns equal to the rank of the modal matrix, which is equal to the
smaller dimension of the matrix. So if the number of master degrees of freedom needs to be
chosen are greater than the smallest dimension of the modal matrix (i.e. number of target modes
selected),then additional sensor locations can be chosen in the vicinity of master degrees of
freedom which will not improve the measurement information.
For a modal matrix ‘A’ (size of ‘m’ by ‘n’) obtained analytically, the QR-decomposition
produces following results:
A.E = Q.R
Where, m= total degrees of freedom
n=number of modes
35
A ∈ R m,n - Modal matrix
Q∈ R
- An orthogonal matrix ( Q T .Q = I )
R ∈ R m ,n - An upper triangular matrix with decreasing diagonal elements.
E ∈ R m ,n - A permutation matrix that exchanges columns of ‘A’
Due to characteristics of the QR-decomposition, the first ‘n’ columns of (A.E) are those that are
most independent columns. The result of applying QR-Decomposition to the test structure for the
first fifteen deformable modes is shown in Figure 3.12
Figure 3.13 QR-Decomposition Results
The results obtained by QR-decomposition have very limited practical application because of the
limited number of master degrees of freedom. However, in pre-test planning, this might be a
useful tool for avoiding nodal points on the structure.
36
CONCLUSON
The modal analysis of the H- Frame was carried out in Ansys in order to determine the modal
frequencies and mode shapes. The mode shape matrix formed the basis for the EI algorithm
which was used to determine the optimal sensor locations for the modal test. The EI algorithm
results are based upon the accuracy of the finite element model data: obviously if the initial FE
model closely resembles to the actual test structure, results obtained would be more realistic. If
there are too many discrepancies between actual structure and FEM model, the results obtained
would be incorrect and will eventually lead to a poor correlation. Considering these things in
mind, the test structure FE modeling was done using various approaches and the model giving
best results was retained for further analysis.
37
CHAPTER 4
EXPERIMENTAL MODAL ANALYSIS
4.1 Introduction
The modal parameters of the H-Frame structure were estimated using experimental data obtained
from a multiple reference impact test (MRIT) performed on the structure. The H-Frame
structure, suspended in a free-free state by elastic shock cords, was excited at two locations using
electromagnetic shakers. The position and the direction of the excitation location are chosen in
order to best excite all the resonances in the concerned frequency range.
Prior to carrying out the FRF measurements at the points chosen on the structure, the basic
assumptions should be verified: linearity, time invariance, reciprocity and observability.
Moreover also the quality of response functions in the test points should be good. To consider a
system as linear it is necessary to verify that changing the force also the changes the response in
such a way the response/force ratio is constant. This means that the FRF of a perfectly linear
structure should not depend on the excitation level.
Since the analytical results were obtained by doing modal analysis in Ansys, the corresponding
locations were chosen on the actual structure for modal testing. The purpose of the modal test
was to compare the test, contaminated with the roving mass effect, with the test with negligible
mass loading, for the purpose of correlation.
38
4.2 Analytical results-EI Algorithm
Chapter-2 describes the EI algorithm developed by Dr. Daniel C. Kammer [7], for the optimal
placement of sensors for modal testing. The results were calculated for 77 master degrees of
freedom, (based upon the user’s intuition about the structure and past experiences of testing).The
X-modal-II software, developed by Structural Dynamics Research Laboratory at University of
Cincinnati was used for experimental modal analysis by using total 77 degrees of freedom, as
shown in the following figure.
Figure 4.2 EI Algorithm Results
The Figure 4.6 shows the analytical results obtained by Kammer’s EI algorithm for optimal
placement of sensors. Based upon these locations, the experimental test was planned to get the
modal data which was then compared with analytical results for different test cases.
39
4.3 Testing Set up Considerations
In multiple-reference testing, as many rows (or columns) of the FRF matrix [H] are measured as
the number of references used. Multiple references are required when it is not possible to find
one single reference DOF containing all modes of interest or when very close modal frequencies
are present. This is the case for complex structures exhibiting modes only found in local areas of
the structure, so-called local modes. Often a triaxial accelerometer is used to simultaneously
capture movements in all three translational directions. The rows of the FRF matrix [H] can be
measured sequentially by moving a uniaxial accelerometer from one reference DOF to the other,
but measurement time is significantly reduced and data quality increased by simultaneously
measuring all references (MIMO testing).MIMO can also be performed using two or more fixed
shaker positions. Again a single shaker could be used by measuring one column in the FRF [H]
matrix at a time, but significant advantages can be obtained by performing simultaneous multipoint excitation. There are two ways in which MIMO testing is carried out on the structure.
•
Roving Hammer Test
In this test, an accelerometer is fixed at particular location and impact is made at as many
locations as desired to define the mode shape of the structure. The drawback for this method is
that all points on the structure could not be impacted in all three directions, so 3D motion can not
be measured at every measurement point. The modal test with the impact hammer is the best way
to avoid the effect of mass loading. The method was not suitable in our case as it was impossible
to impact in the master direction at chosen measurement location.
40
•
Roving Tri axial Accelerometer Test
While implementing results obtained from EI algorithm [7], it was observed that at certain
master nodes, it would not be possible to have impact in the master direction. Because of this
limitation, the roving tri axial accelerometer test has been carried out on the structure. A shaker
has been attached to the structure using stinger rod (long slender rod), so that the shaker will only
impart the force to the structure along the axis of stinger, the axis of force measurement. A load
cell is attached in between structure and shaker to measure input force.
4.3.1 Excitation Signal Selection
The amount of damping and the number of modes within the structure can also dictate the use of
specific excitation. If the modes are closely coupled and/or lightly damped, an excitation
function that can be implemented in a leakage free manner is most appropriate. In order to have
input signal to be completely observed within the sampling window, a burst random signal [22]
was selected. For very lightly damped systems, the burst length may have to be shortened below
20%.This may yield an unacceptable signal to noise ratio (SNR).The number of power spectral
averages used in the burst random excitation approach is a function of the reduction of the
variance error and the need to have a significant number of averages to be certain that all
frequencies have been adequately excited.
4.3.2 Shaker Attachment Criteria
Since the frequency response function is a single input function, the shaker should transmit only
one component of force in line with the main axis of the load cell. In practical situations, when a
structure is displaced along a linear axis it also tends to rotate about the other two axes.
41
To minimize the problem of forces being applied in other directions, the shaker should be
connected to the load cell through a slender rod, called a stinger, to allow the structure to move
freely in the other directions. The ideal stinger rod should have strong axial stiffness, but weak
bending and shear stiffness. In effect, it acts like a truss member, carrying only axial loads but no
moments or shear loads. [45]
4.3.3 Leakage Error Considerations
Leakage is a signal processing bias error caused by violating an assumption of the discrete
Fourier transform that the signal being transformed is periodic within the sampling period. If
both input and output of a system are harmonic functions of sampling period or completely
observed transients, there will be no leakage error. Cyclic averaging [23] is a time domain signal
processing technique that reduces leakage errors when measuring frequency response functions
of a lightly damped system. It also has an advantage of reducing the noise in the trailing segment
of time record, if noise is not periodic in the sampling period.
4.3.4 Modal Parameter Estimation Considerations
Highly damped systems exhibit broad resonance peaks in the frequency response functions, but
short duration impulse response functions. Hence, frequency domain methods perform better for
estimating the system poles. Whereas in our case, since the test structure was lightly damped
(critical damping ratio < 5 %), time domain algorithms have been preferred.
42
4.4 Modal Test of H Frame Structure
The H Frame was tested in an environment simulating free free boundary conditions. Its mass is
approximately 220 pounds and has the dimensions as shown in Table 3.2.Two shakers were used
to excite the structure in both horizontal and vertical configuration as shown in the Figure 4.4.
The excitation sources consisted of two independent burst random signals generated by system.
All force and response transducers were calibrated before testing. The frequency response
functions were made using 16 channel data acquisition system.
Bungee Cords
Dummy weights
Shaker-1
Figure4.4 Experimental Test Set Up – H Frame
43
4.5 Test Set-up
The MATLAB program MRIT (Multi-Reference Impact Testing), was used to acquire all of the
data. This program controls the VXI hardware and processes all of the acquired time data into
FRF’s which can subsequently be saved out to a file. The program setup consists of inputting a
channel list and test parameters. The channel list consists of each transducers calibration,
orientation, and corresponding channel. The test parameters include frequency range, number of
spectral lines, window type, and number of averages.
The test parameters for the current test were as follows:
•
Frequency range: 0 - 400 Hz
•
800 spectral lines
•
Averages: 20
•
Window: rectangular
•
Cyclic Averages: 4
•
Signal: Burst Random
•
Burst Length: 20 %
44
4.6 Test Geometry & Analytical Results
Based upon the analytical results obtained from the EI algorithm, the following degrees of
freedom were chosen for testing purpose. The numbers of points taken were greater than the
number of master degrees of freedom chosen by EI algorithm. The correlation was based upon
the first ten deformable modes of the structure. In order to have the measurement data for
different frequency range, additional points were chosen for measurements.
Figure 4.5 H-Frame Test Geometry
4.7 Visual Comparison of Experimental & Ansys mode shapes
Following figures illustrate the mode shapes obtained from modal testing as well as from Ansys.
The mode shapes obtained from Ansys plotted here are the modes without concentrated mass
elements. This is just to verify that the shapes look to be the same at the corresponding natural
frequencies.
45
46
47
48
4.8 Visual Inspection of Modal vectors [77 DOF]
Figure4.8 Mode Shapes corresponding to Master DOF
49
4.9 Modal Parameters of H Frame
Mode Number
Frequency Fn[Hz]
Description
1
14.86
Rigid Body Motion of Legs in X-Z plane
2
23.64
Rigid Body Motion of Legs in Y-Z plane
3
35.4
First Bending of legs in X-Z plane(in phase)
4
54.22
First Bending of legs in X-Z plane(out of phase)
5
74.43
Bending of short legs in X-Z plane (in phase)
6
147.47
Bending of long legs in X-Z plane (out of phase)
7
153.89
Bending in Y-Z Plane (in phase)
8
162.21
Bending in Y-Z Plane (out of phase)
9
175.75
First torsion in X-Y plane of long legs(in phase)
10
182.95
Second Bending in X-Z plane (in phase motion)
Table 4.9.1 Mode Shape Descriptions
Mode Number
Algorithm
Frequency [Hz]
Damping [%]
Modal A
1
PTD
14.86
1.67
1.9424e-007+1.0415e-005j
2
PTD
23.64
0.18
5.289e-007-3.9688e-005j
3
PTD
35.4
0.726
-4.4304e-008+1.106e-006j
4
PTD
54.22
0.149
-9.7024e-007+3.14e-005j
5
PTD
74.43
0.175
-1.449e-007+1.2196e-005j
6
PTD
147.47
0.164
1.34e-008+4.36e-007j
7
PTD
153.89
0.183
-2.98e-008-1.04e-006j
8
PTD
162.22
0.248
-9.34e-007-1.03e-005j
9
PTD
175.75
0.423
-1.96e-006-1.11e-005j
10
PTD
182.95
0.156
-8.99e-008+3.92e-007j
Table 4.9.2 Modal Parameters of H Frame
50
4.10 Discussion of Results
The test case was carried out in order to have some effect of mass loading. The accelerometers
were moved from one location to another, without using the dummy weights at some of the
measurement locations. Because of this addition of mass to the structure, different points on the
structure showed slightly different frequencies and hence the data became inconsistent. The
complex mode indicator function [24] clearly demonstrated this fact. The modal parameters
estimated from this data were also affected because of the frequency shifts and it induced
additional complexity into the structure. The Modal-A estimated for some of the modes was not
accurate, indicating the presence of complex modes.
From the modal assurance criteria (MAC) of experimental and analytical modal vectors, mainly
modes at 14.8 Hz and mode at 147 Hz were found to be in large error (31%).This linear
dependency of the chosen DOF is most probably the result of similarity of the mode shapes. This
could be verified by seeing the modal complexity plot of the respective vectors. Also, visually,
both modes even though far apart in the frequency domain have same motion for chosen masters
selected by Kammer’s EI algorithm. Although the spatial aliasing due to the sensor selection can
not be ruled out, that effect has been magnified by the presence of additional complexity
introduced by mass roving of accelerometers.
The poor MAC values for some of the modes could also be associated with the inefficient
excitation of the test structure. This is largely due to the peculiar nature of the test structure. The
modes in all three principal directions are mostly uncoupled. Because of the absence of shaker in
z-direction, some of the modes which have motion in that respective direction were not
sufficiently excited and this could be verified from the modal participation factors.
51
Along with the error introduced by roving the sensor mass, there could be many errors which
might have a small impact on the results. The mass loading of shakers, elastic suspension of test
structure, non linearity, and modal parameter estimation errors also contribute towards the final
results.
Conclusion
The test was attempted with accelerometer masses distributed at all of the measurement locations
in an attempt to reduce the frequency shifts. This method allowed many simultaneous response
measurements. But this model with small mass added at the measurement points decreased the
modal frequencies, and introduced additional damping to the structure. Sometimes, the weights
may also cause additional local modes to appear in the structure, making it difficult to extract the
global modal parameters. Although these structures are relatively heavy relative to the mass of
the accelerometer, mass loading was still significant, especially for modes with high participation
from local areas with thin sections, such as ribs and plates.
52
Chapter 5
Correlation for Model Validation
5.1 Introduction
Correlation is a numerical manipulation in which two sets of dynamic properties are compared
quantitatively. The results from the correlation calculation quantify the similarity and the
difference between these two data sets. There are many ways in which the correlation could be
carried out and many different methods have been used in the area of structural dynamics,
especially in comparing experimental data and analytical predictions.
The aim of the model, while presenting the physical nature of the model is to bring the finite
element model in agreement with the experimental data. In order to evaluate the degree of
agreement some generally accepted criteria are needed. The correlation method includes a set of
techniques to compare the analytical model data with the experimental modal data. The outcome
of the correlation study aids the decision whether updating is required or not. In a global
updating procedure, the correlation step follows the model matching step and precedes the
selection of updating parameters and the correction of the analytical model.
There are many methods which are used for the correlation procedure, but each one highlights
only a limited aspect of the correlation between experimental and analytical data. Therefore, a
profound correlation study cannot be performed by only one or two correlation methods. As
many techniques as possible must be used in parallel to get a fair view of the degree of
agreement between analytical and experimental data, including both statics and dynamics.
53
This chapter describes the role of correlation in the process of model validation in general.
Furthermore, a case study is presented analyzing the effect of one of the possible frequent
measurement error on the numerical correlation of the test structure.
5.2 Role of correlation in model validation
The process of model validation, as discussed in the first chapter of this thesis, is to create an FE
model by modifying an initial model so that the dynamic predictions from the validated model
are similar to those of the actual structure under consideration. In this process, the correlation
procedure measures the similarity and difference between the dynamic properties predicted by
the initial or a modified FE model and the reference dynamic properties that are usually obtained
from an experiment performed on the actual structure.
Validation of theoretical structural dynamic models is usually performed by comparison of
theoretically predicted with experimentally measured, dynamic characteristics of structures.
When a successful modal test of a structure is completed, correlation between theoretical and
experimental sets of data(frequencies, mode shapes, mode scaling) is undertaken and if the
agreement is satisfactory, the theoretical model of the structure is validated and considered to be
suitable for use in further analysis (transient response, assembled structure analysis, fluidstructure interaction analysis, etc.). However, if satisfactory agreement is not obtained, then
some modifications must be made to the theoretical model in order to improve correlation with
the corresponding experimental results.
54
Several parameters can be used to correlate analytical and experimental modes [26]: modal scale
factor (MSF), orthogonality (OR), cross-orthogonality (XOR), modal effective mass (MEM)
[27], modal assurance criterion (MAC) [28], normalized cross orthogonality (NCO) [29], SEREP
based cross-orthogonality (SCO) [30].
.
5.2.1. Visual Comparison of Mode Shapes
Visual comparison between two sets of modal data is the first step which involves the analyst’s
non-quantitative visual assessment of any kind of graphically presented data. It usually consists
of simultaneous animation of one mode shape from each of the two sets and direct comparison of
their natural frequencies. This method consists of a visual comparison of the patterns of two
different mode shapes and a non-quantitative analyst assessment of differences or similarities
between two mode shape patterns. A problem arises when one experimental mode appears to
match two or more theoretical modes. Although this can happen for several reasons, as will be
discussed later, a more detailed inspection is necessary in order to identify the CMPs.
5.2.2 Natural Frequency Difference [NFD]
A Natural Frequency Difference (NFD) correlation coefficient gives an assessment of the
difference between the natural frequencies of a pair of modes in two modal data sets. For two
modal data sets with m1 modes and m2 modes respectively, the NFD coefficients form a matrix of
dimension (m1 × m2). Each element in the matrix represents the difference between the natural
frequency of a mode in first data set and that of a mode in the other data set. It is calculated by
the following formula,
55
NFD( X i , A j ) =
(ω X )i − (ω A ) j
.100% i = 1… m1; j=1… m2;
min ((ω x )i ,(ω A ) j )
Where, (ω X )i = Natural Frequency of i th mode in the experimental modal data set.
(ω A ) j
= Natural Frequency of j th mode in the analytical modal data set.
The natural frequencies obtained from FE model as well as from experimental test are tabulated
below in Table 5.1
Mode #
Frequency
Ansys [Hz]
% Error
1
14.862
14.904
0.28
2
23.64
23.604
-0.15
3
35.4
36.469
2.93
4
54.22
55.993
3.17
5
74.43
76.218
2.35
6
147.47
149.696
1.49
7
153.89
159.768
3.68
8
162.21
168.925
3.98
9
175.75
181.663
3.25
10
182.95
184.89
1.05
Table 5.1 Natural Frequency Table
56
Figure 5.2 Natural Frequency Difference Plot
A typical plot of the NFD matrix is given in Figure 5.2. It is important to note that this
correlation coefficient should be used only as an additional piece of information when selecting
CMPs, and not as the sole indicator. The larger the NFD value for two mode shapes is, the less
likely the pair is to be a CMP.
5.2.2 Correlated Mode Pairs (CMPs)
The results from correlating the modal data are not only used for assessing the initial FE model
and the modified models but are also used for identifying the CMP information which will be
used within the updating procedure. The determination of CMPs is made by examining the MAC
matrix and the NFD values. Two thresholds can be proposed on MAC and NFD values,
respectively. The pairs of modes in two data sets with MAC values greater than the MAC
threshold and with NFD values smaller than the NFD threshold can be considered as correlated
mode pairs
57
The thresholds on MAC and NFD values may be different in different situations. For example,
the thresholds for determining CMPs within an updating procedure should be different from
those for measuring the similarity of an FE model to a test structure. In order to include more
information in the updating procedure, more modes should be considered as correlated modes.
That means that the MAC threshold within the model updating procedure may need to be lower
than that for assessing the initial model and modified models and the NFD threshold within the
model updating procedure should be higher than that for assessing the initial model and the
modified models.
5.2.3 45 Degree plot
An experimental mode shape is plotted in a xy-plot versus the corresponding analytical mode
shape. If both mode shapes are identical and scaled in the same way, then all points lie on a 45
degree line through the origin. The distances from the experimental versus analytical mode shape
points to the 45 degree line give an indication about the correlation of the mode shapes.
58
5.2.4 Modal Assurance Criteria [22]
The modal assurance criteria compare different sets of estimated mode shapes and provide a
measure of the least square deviation of the points from the straight line correlation. It is defined
as:
2
n
∑ (ψ
MAC (A, X) =
j =1
x
) j (ψ )
*
A j
⎛ n
⎞⎛ n
⎞
⎜ ∑ (ψ x ) j (ψ x ) *j ⎟.⎜ ∑ (ψ A ) j (ψ A ) *j ⎟
⎜
⎟⎜
⎟
⎝ j =1
⎠ ⎝ j =1
⎠
5.2.4.1 Auto-Modal Assurance Criteria [Auto MAC]
The Auto MAC is a version of MAC in which a set of mode shape vectors are correlated with
themselves. The analytical DOF defined only at the DOFs which are to be used in correlation
with the experimental model are used to calculate the Auto MAC which gives following
observations:
1. All diagonal terms must be unity, by definition, because each mode shape must correlate
perfectly with itself.
2. Auto MAC matrix is symmetric
3. Non zero off diagonal terms indicates degree of correlation between some of the modes,
the result which is not immediately expected, since modes are supposed to be orthogonal
to each other. The orthogonality is only strictly applicable when mass matrix is used and
when all DOF are included in the calculations.
59
5.2.5 Co-ordinate modal assurance criteria [33]
This criterion correlates the global set of analytical and experimental mode shapes for each
individual degree of freedom. The COMAC value for the degree of freedom is calculated as
follows:
ψ pr .φ pr
L
COMAC p = ∑
r =1
L
∑ψ
r =1
2
………………………………………… (5.2.1-1)
L
pr
.ψ .∑ φ pr .φ
*
pr
r =1
*
pr
where:
• L = Number of matching pairs of modes in the two sets of modal vectors being compared.
•ψ pr = Modal coefficient from (measured) degree of freedom p and modal vector pair r from one
set of modal vectors.
• φ pr = Modal coefficient from (measured) degree of freedom p and modal vector pair r from a
second set of modal vectors.
A COMAC value close to ‘1’ indicates good correlation. COMAC is sensitive to the way mode
shapes are scaled. Great care should be taken to scale analytical and experimental mode shapes
in a consistent way.
5.6 Comparison of frequency response functions
If analytical frequency response functions are calculated with a reasonable approach (damping
assumption) and if they are plotted for the same frequency range and on the same scale as the
experimental ones, comparison of FRF’s is a good criterion to check the quality of the finite
element model.
60
5.2.7 Numerical Comparison of Mode Shapes
Numerical comparison between two sets of modal data consists of the calculation of different
correlation coefficients which give quantitative values of similarities between two mode shapes.
All numerical correlation methods use the orthogonal or independence properties of mode shapes
in order to exploit the differences between them. The most commonly-used one for indication of
CMPs is the MAC (Modal Assurance Criterion) [22], which calculates the least-squares
deviation about a straight line of the plot of two arbitrarily-scaled mode shapes, and the NCO
(Normalized Cross-Orthogonality) [8], which is basically the MAC coefficient weighted by a
partitioned global mass or stiffness matrix. A good correlation between two modes is observed
for MAC and NCO values close to 1 and a poor correlation corresponds to values close to zero.
Since mode shapes are orthogonal with respect to the full mass or stiffness matrix, and the
measured set of DOFs is never complete, it is impossible to specify absolute limits for indication
of good or poor correlation based on the MAC or NCO correlation coefficients.
MAC (i, j) =
| {ψ X }T {ψ A }* | 2
………………………………………. (5.2.1-2)
{ψ X }T .{ψ X }* .{ψ A }T .{ψ A }*
5.2.7.1 Problem of Complex Mode shapes
It can be seen that in expressions (5.2.1-1) and (5.2.1-2) it is assumed that both mode shapes are
complex. In most practical situations, the theoretical mode shapes are real if the damping has not
been included in the model, whereas the experimental mode shapes are always complex because
of inevitable damping effects in any real structure. If, however, correlation is performed between
real and complex mode shapes, this will in general decrease the correlation parameter values,
61
since the correlation coefficient is a simple scalar product between two vectors divided by their
magnitudes. The issue of modal complexity is further discussed in the next chapter.
5.2.8 The normalized cross orthogonality (NCO)
It is basically the MAC coefficient weighted by a partition of the global mass or stiffness matrix.
This is defined by the following Equation
| {ψ X }T [ M A ]{ψ A }* | 2
NCO (i, j) =
{ψ X }T .[ M ] A .{ψ X }* .{ψ A }T .[ M ] A .{ψ A }*
To calculate the NCO it is necessary to use the analytical mass matrix [ M ] A .Generally, the
measured degrees of freedom are less numerous than the analytical model contains, it is
necessary to adapt one data set. It is possible to reduce the mass matrix to the measured degrees
of freedom or to expand the measured modes to the degrees of freedom of the analytical model.
The distinct methods to carry out these operations (expansion or reduction) influence the result
of the NCO. MAC and NCO values close to one observe a good correlation between modes and
a poor correlation corresponds to values close to zero.
However, this technique has been found inappropriate for some applications and a SEREP
theoretical mass reduced based normalized cross orthogonality (SCO) correlation coefficient (5)
can be used instead [14].
For the present thesis work, the normalized cross orthogonality [NCO] could not be implemented
for the current correlation because of the computational memory limitations.
62
The Modal Assurance Criterion was mainly used to judge the correlation between experimental
and analytical modal vectors. Different test cases were simulated in order to study the effect of
frequent measurement errors on correlation process. The analytical modal vectors were
compared with the three sets of experimental data, each one containing different kind of
measurement error in it.
•
Experimental test carried out with mass loading of accelerometers
•
Experimental test with practically negligible mass loading of accelerometers. The results
obtained from first test case were compared with the second test case in order to study the
effect of mass overloading on correlation process. The test data used for this test was taken
from the impact excitation already done on the test structure for SDA-III (2004) class.
•
Experimental test simulating the effect of misalignment of transducers on the process of
correlation.
5.3 Effect of mass loading on correlation
Several common measurement errors which can yield complex modal vectors when “real” modal
vectors are anticipated are aliasing, leakage and mass roving of accelerometers. Aliasing which
is classified as a bias amplitude error that converts high frequency energy to lower frequencies
may not influence the frequency and damping estimates significantly: however major errors will
be present in the calculation of modal vectors. Leakage error converts energy at each frequency
into energy within a relatively close narrow band. Thus inaccurate estimates are calculated and
inaccurate modal vectors result. Mass loading the structure is a severe position dependent
measurement error.
63
The other errors in the parameter estimation phase may also produce complex modal vectors.
One such error occurs when too few or too many effective poles are chosen for a particular
frequency range. In this case inaccurate damping estimate and modal vector errors will result.
Another error will occur when a pole is selected too close to the lower or upper frequency limits
of a particular bandwidth. The frequency resolution is also important in the estimation of highly
coupled modes. In cases where the frequency resolution is too low, both the frequency and
damping estimates will be inaccurate, producing modal vectors which are complex.
When an accelerometer is moved on a structure, a small mass is being moved around the
structure. Essentially every time the transducer is moved to another point, the mass matrix of the
structure is changed slightly. The change is the addition of the mass of the accelerometer to the
original diagonal term of the corresponding point in the modal mass matrix. If the mass is very
small as compared to the original diagonal term, the change of the matrix is negligible.
However, when the mass is not small, the system properties are changed every time the
accelerometer is moved. As a result, the frequency for every pole changes from point to point
due to the inconsistent data set. Any global technique such as the polyreference time domain
[PTD] method [] is very sensitive to this kind of error. The technique interprets every frequency
shift as a different pole in the data set and will try to estimate as many poles as there are
frequency shifts. All these poles will have approximately same damping and frequency value, so
it is very difficult to define how many effective modes there exactly are. [34]
In case of the roving accelerometer test, while moving accelerometers from one location to
another; it alters the mass of the structure. This result in a slight decrease in the natural
64
frequencies of the structure depending upon the how much mass has been added to the structure
at each step and also on the sensitivity of the structure. The results show the Auto-Mac
calculation for experimental as well as analytical modal vectors, corresponding to the
measurement degrees of freedom selected by EI algorithm.
The cross correlation between the experimental and analytical modal vectors is done by CrossMAC [28] calculation. The test results obtained from this case were compared with the analytical
data and are presented in the following tables. For comparison purpose, analytical as well as
experimental modal vectors were scaled to unity max value.
65
Auto-MAC Calculation of Analytical Modes: Test Case-1
Mode No.
1
2
3
4
5
6
7
8
9
10
1
1.000
0.000
0.000
0.001
0.000
0.445
0.011
0.000
0.001
0.000
2
0.000
1.000
0.000
0.000
0.000
0.000
0.002
0.000
0.000
0.035
3
0.000
0.000
1.000
0.000
0.044
0.000
0.000
0.000
0.000
0.436
4
0.001
0.000
0.000
1.000
0.000
0.006
0.000
0.000
0.000
0.000
5
0.000
0.000
0.044
0.000
1.000
0.000
0.000
0.000
0.000
0.001
6
0.445
0.000
0.000
0.006
0.000
1.000
0.001
0.000
0.016
0.000
7
0.011
0.002
0.000
0.000
0.000
0.001
1.000
0.001
0.043
0.000
8
0.000
0.000
0.000
0.000
0.000
0.000
0.001
1.000
0.000
0.064
9
0.001
0.000
0.000
0.000
0.000
0.016
0.043
0.000
1.000
0.000
10
0.000
0.035
0.436
0.000
0.001
0.000
0.000
0.064
0.000
1.000
Table 5.3 Auto-MAC values for Analytical Modes Test Case1
Figure 5.4 Auto-MAC Analytical Modes-Test Case1
Observations:
•
Strong linear dependency observed between mode’1’ and mode ‘6’ as well as between
mode’3’ and mode’10’
•
The linear dependency could be either caused by failure of the EI algorithm or by the
inefficient FE modeling of test structure.
66
Auto-MAC Calculation of Experimental Modes: Test Case-1
Mode No.
1
2
3
4
5
6
7
8
9
10
1
1.000
0.000
0.016
0.000
0.004
0.314
0.019
0.000
0.001
0.012
2
0.000
1.000
0.003
0.000
0.000
0.000
0.007
0.012
0.000
0.045
3
0.016
0.003
1.000
0.000
0.039
0.000
0.000
0.000
0.000
0.313
4
0.000
0.000
0.000
1.000
0.006
0.006
0.001
0.001
0.000
0.002
5
0.004
0.000
0.039
0.006
1.000
0.000
0.003
0.000
0.000
0.001
6
0.314
0.000
0.000
0.006
0.000
1.000
0.003
0.001
0.052
0.017
7
0.019
0.007
0.000
0.001
0.003
0.003
1.000
0.000
0.105
0.005
8
0.000
0.012
0.000
0.001
0.000
0.001
0.000
1.000
0.000
0.141
9
0.001
0.000
0.000
0.000
0.000
0.052
0.105
0.000
1.000
0.013
10
0.012
0.045
0.313
0.002
0.001
0.017
0.005
0.141
0.013
1.000
Table 5.5 Auto-MAC values for Experimental Modes Test Case-1
Figure 5.6 Auto-MAC Experimental Modes Test Case1
Observations:
•
Similar to Auto-Mac of analytical modal vectors, strong linear dependency observed
between mode’1’ and mode ‘6’ as well as between mode’3’ and mode’10’
•
There was also some dependency observed between modes ‘7’ and ‘9’ as well as mode ‘8’
and mode ‘10’.By comparing with the Auto-Mac of analytical data, it is clear that the extra
error was caused by the mass roving effect.
67
MAC Calculation: Test Case-1
Mode No.
1
2
3
4
5
6
7
8
9
10
1
0.974
0.000
0.018
0.000
0.003
0.405
0.010
0.001
0.002
0.006
2
0.000
0.995
0.000
0.000
0.000
0.000
0.004
0.000
0.000
0.037
3
0.000
0.002
0.995
0.000
0.043
0.000
0.000
0.000
0.000
0.441
4
0.001
0.000
0.000
0.989
0.008
0.006
0.000
0.000
0.000
0.001
5
0.000
0.000
0.039
0.000
0.994
0.000
0.000
0.000
0.000
0.001
6
0.347
0.000
0.000
0.007
0.000
0.979
0.009
0.001
0.042
0.000
7
0.019
0.004
0.000
0.001
0.002
0.000
0.967
0.005
0.081
0.000
8
0.000
0.010
0.000
0.000
0.000
0.000
0.001
0.956
0.000
0.101
9
0.001
0.000
0.000
0.000
0.000
0.021
0.059
0.000
0.986
0.001
10
0.003
0.041
0.306
0.000
0.001
0.015
0.004
0.090
0.016
0.925
Table 5.7 Cross-MAC between Analytical & Experimental Mode Shapes
Figure 5.8 MAC Test Case 1
Observations:
•
The diagonal terms were higher than 0.9 indicating a good match between experimental
and analytical mode shapes.
•
The mass roving induced extra complexity into the modal vectors causing the correlation to
drop at the corresponding modal vectors.
68
5.2.4 Test correlation without mass loading
The test data from impact testing was used in this test case in order to study the cross correlation
between analytical modes and experimental modes with negligible mass overloading. Since the
data was taken from impact testing, all the measurement degrees of freedom selected by EI
algorithm were not available. In order to compare the same kind of data, only those nodes were
selected on the test structure which was specified by EI algorithm. The only difference was that
some of the master directions were missing in this test data. To compensate for this, some zdirection degrees of freedom were chosen and the total numbers of master DOF were 72.
Figure 5.2.4 Test Case-2 Master DOF
The master DOF are shown only for one aluminum plate. The rest of the plates (where a point
mark is placed) have the same distribution of master degrees. The corresponding analytical data
was then extracted from the analytical data set and the results are tabulated in the following
sections.
69
Auto-MAC Calculation of Analytical Modes: Test Case-2
Mode No.
1
2
3
4
5
6
7
8
9
10
1
1.000
0.000
0.000
0.000
0.000
0.134
0.004
0.000
0.000
0.000
2
0.000
1.000
0.000
0.000
0.000
0.000
0.009
0.003
0.002
0.021
3
0.000
0.000
1.000
0.000
0.001
0.000
0.000
0.000
0.122
0.002
4
0.000
0.000
0.000
1.000
0.000
0.002
0.000
0.000
0.000
0.000
5
0.000
0.000
0.001
0.000
1.000
0.000
0.000
0.000
0.003
0.000
6
0.134
0.000
0.000
0.002
0.000
1.000
0.004
0.000
0.000
0.000
7
0.004
0.009
0.000
0.000
0.000
0.004
1.000
0.008
0.000
0.000
8
0.000
0.003
0.000
0.000
0.000
0.000
0.008
1.000
0.016
0.182
9
0.000
0.002
0.122
0.000
0.003
0.000
0.000
0.016
1.000
0.046
10
0.000
0.021
0.002
0.000
0.000
0.000
0.000
0.182
0.046
1.000
Table 5.10 Auto-MAC Analytical Modes Test Case 2
Figure 5.11 Auto-MAC Analytical Modes-Test Case 2
Observations:
•
The linear dependency between mode ‘1’ and mode ‘6’ was found to be 13 % as compared
with 45 % in Test Case-1.
•
The maximum linear dependency [18 %] was observed between mode ‘8’ and mode ‘10’
which could be attributed to the SPATIAL ALIASING.
70
Auto-MAC Calculation of Experimental Modes: Test Case-2
Mode No.
1
2
3
4
5
6
7
8
9
10
1
1.000
0.000
0.000
0.000
0.000
0.094
0.008
0.000
0.000
0.000
2
0.000
1.000
0.001
0.000
0.001
0.001
0.010
0.008
0.003
0.022
3
0.000
0.001
1.000
0.003
0.001
0.002
0.000
0.002
0.084
0.004
4
0.000
0.000
0.003
1.000
0.000
0.004
0.000
0.000
0.000
0.000
5
0.000
0.001
0.001
0.000
1.000
0.000
0.000
0.000
0.004
0.001
6
0.094
0.001
0.002
0.004
0.000
1.000
0.001
0.000
0.002
0.000
7
0.008
0.010
0.000
0.000
0.000
0.001
1.000
0.011
0.013
0.001
8
0.000
0.008
0.002
0.000
0.000
0.000
0.011
1.000
0.004
0.148
9
0.000
0.003
0.084
0.000
0.004
0.002
0.013
0.004
1.000
0.031
10
0.000
0.022
0.004
0.000
0.001
0.000
0.001
0.148
0.031
1.000
Table 5.12 Auto-MAC Experimental Modes Test Case 2
Figure 5.13 Auto-MAC Experimental Modes-Test Case 2
Observations:
•
The linear dependency between mode ‘1’ and mode ‘6’ was found to be 10 % as compared
with 30 % in Test Case-1.
•
The maximum linear dependency [14 %] was observed between mode ‘8’ and mode ‘10’
which could be attributed to the SPATIAL ALIASING.
71
MAC Calculations-Test Case 2:
Mode No.
1
2
3
4
5
6
7
8
9
10
1
0.981
0.000
0.000
0.000
0.000
0.106
0.006
0.000
0.001
0.000
2
0.000
0.992
0.000
0.001
0.000
0.000
0.009
0.006
0.002
0.026
3
0.001
0.001
0.988
0.004
0.002
0.000
0.000
0.000
0.086
0.005
4
0.000
0.000
0.000
0.983
0.000
0.003
0.000
0.000
0.000
0.000
5
0.000
0.000
0.000
0.000
0.984
0.000
0.000
0.000
0.003
0.000
6
0.113
0.001
0.001
0.003
0.000
0.986
0.000
0.000
0.002
0.000
7
0.005
0.009
0.000
0.000
0.000
0.006
0.984
0.007
0.024
0.001
8
0.000
0.006
0.001
0.000
0.000
0.000
0.012
0.989
0.007
0.171
9
0.000
0.002
0.115
0.000
0.003
0.000
0.000
0.012
0.750
0.036
10
0.000
0.017
0.002
0.000
0.000
0.000
0.000
0.153
0.014
0.950
Table 5.14 MAC Test Case 2
Figure 5.15 MAC Test Cases 2
Observations:
•
The diagonal terms were higher than 0.9 indicating good match between experimental and
analytical mode shapes.
•
The lower correlation for mode ‘9’ might have caused by the error in the modal parameter
estimation process. Modal complexity could probably be the other reason for lower
correlation between experimental and analytical modes.
72
5.4 Effect of transducer misalignment on correlation
This section reviews the concepts of transducer errors, specifically accelerometer
misalignment, as well as presenting a mathematical model for these errors. Another commonly
found error associated with accelerometers is cross sensitivity error which refers to the amount
of off-axis measurement that is measured by the transducer. Accelerometers, for example,
typically are specified at having cross sensitivity of less than 5 percent.
Misalignment error, on the other hand is due to human error in mounting the transducer on the
structure of interest and aligning it in the appropriate coordinate system. This error can be
represented in two ways, one is where a single transducer is not placed parallel to the principal
axis of measurement, and the second is when a biaxial or tri axial mounted set of accelerometers
is not aligned with its principal coordinate axes. These misalignment errors can easily be as
much 15 degrees, and are typically as much as 5 degrees when carefully installed. [600]
For this test case, 3-D transducer misalignment error is simplified to the 2-D case. These
misalignment errors can be modeled as individual skewed axes and represented mathematically
at a biaxial location as:
⎧⎪ x∧ ⎫⎪ ⎡ cos(α ) sin(α ) ⎤ ⎧ x ⎫
⎨∧⎬=⎢
⎥.⎨ y ⎬
−
sin(
α
)
cos(
α
)
⎣
⎦⎩ ⎭
⎪⎩ y ⎪⎭
Figure 5.16 illustrates this error in X-Y plane with misalignment simulated for 5 ° ,10 ° and 15 °
respectively.
73
Figure 5.4 2D Model of Single Transducer Misalignment
The master degrees of freedom selected by EI algorithm did not have any z-directional master
degrees of freedom. So the above test case was carried out such that only the x and y directional
modal coefficients were affected, simulating transducer alignment error in X and Y direction.
The test case was carried out with 5 ° ,10 ° and 15 ° error and the corresponding MAC
calculations are tabulated in Tables 5.17, 5.18 and 5.19 respectively. The results were then
compared with the original MAC calculations in Table 5.17, 5.18 and 5.19 respectively.
74
Mode No.
1
2
3
4
5
6
7
8
9
10
1
0.972
0.001
0.017
0.000
0.003
0.405
0.009
0.001
0.002
0.005
2
0.001
0.992
0.000
0.001
0.000
0.000
0.004
0.000
0.000
0.036
3
0.000
0.002
0.992
0.000
0.043
0.000
0.000
0.000
0.000
0.440
4
0.001
0.000
0.000
0.986
0.008
0.006
0.000
0.000
0.000
0.001
5
0.000
0.000
0.039
0.000
0.991
0.000
0.001
0.000
0.000
0.001
6
0.346
0.000
0.000
0.007
0.000
0.974
0.009
0.001
0.042
0.000
7
0.020
0.003
0.000
0.002
0.005
0.000
0.960
0.005
0.080
0.000
8
0.001
0.010
0.000
0.002
0.000
0.000
0.002
0.953
0.000
0.101
9
0.001
0.000
0.000
0.000
0.000
0.021
0.060
0.000
0.983
0.001
10
0.004
0.041
0.305
0.000
0.001
0.015
0.003
0.089
0.016
0.920
Table 5.17 MAC for 5° misalignment error
Mode No.
1
2
3
4
5
6
7
8
9
10
1
0.965
0.005
0.017
0.000
0.003
0.402
0.009
0.001
0.002
0.004
2
0.004
0.981
0.000
0.002
0.000
0.001
0.004
0.000
0.000
0.035
3
0.000
0.002
0.986
0.000
0.043
0.000
0.000
0.000
0.001
0.439
4
0.001
0.002
0.000
0.977
0.008
0.006
0.000
0.003
0.000
0.001
5
0.000
0.000
0.039
0.000
0.982
0.000
0.004
0.000
0.000
0.001
6
0.341
0.001
0.000
0.007
0.000
0.962
0.010
0.001
0.042
0.000
7
0.020
0.003
0.000
0.003
0.011
0.001
0.945
0.005
0.079
0.000
8
0.001
0.010
0.000
0.004
0.000
0.000
0.002
0.944
0.000
0.099
9
0.001
0.000
0.000
0.000
0.000
0.021
0.060
0.000
0.976
0.001
10
0.004
0.040
0.303
0.000
0.001
0.014
0.003
0.087
0.015
0.907
Table 5.18 MAC for 10° misalignment error
Mode No.
1
2
3
4
5
6
7
8
9
10
1
0.954
0.010
0.017
0.000
0.003
0.397
0.008
0.001
0.003
0.003
2
0.009
0.961
0.000
0.005
0.000
0.002
0.005
0.000
0.000
0.033
3
0.000
0.002
0.975
0.000
0.043
0.000
0.000
0.000
0.001
0.435
4
0.001
0.004
0.000
0.961
0.008
0.006
0.000
0.006
0.000
0.001
5
0.000
0.000
0.038
0.000
0.966
0.000
0.010
0.000
0.001
0.001
6
0.334
0.002
0.000
0.007
0.000
0.942
0.010
0.002
0.041
0.000
7
0.020
0.002
0.000
0.003
0.020
0.001
0.923
0.005
0.076
0.000
8
0.001
0.010
0.000
0.008
0.000
0.001
0.002
0.930
0.000
0.098
9
0.001
0.000
0.001
0.000
0.001
0.021
0.059
0.000
0.964
0.001
10
0.005
0.040
0.298
0.000
0.001
0.014
0.003
0.084
0.014
0.887
Table 5.19 MAC for 15° misalignment error
75
5.5 Discussion of Results
Comparison of Test Case-1, 2 and 3 give some valuable points to be seen in the context of sensor
placement algorithm used, roving effect of accelerometers and correct measurement practices.
•
The error caused by mass loading was very high as compared to the measurements made
with negligible mass roving.
•
Even with the true measurement data, corresponding to the chosen master degrees of
freedom, the linear dependency between mode ‘1’ and ‘6’ as well as between mode’8’ and
mode ‘10’ is considerable. This means that a part of the deformation proceeds in the same
way in the structure at different modes of the structure. This clearly shows spatial aliasing
effect and indicates the inability of the algorithm to produce perfectly linear independent
modal vectors for a chosen number of DOF.
•
In order to get perfectly linear independent vectors correlation, some user judgment on
selecting the number of master degrees of freedom is still needed.
•
The poor correlation could be result of number of different factors such as non proportional
damping, measurement errors, poor FE modeling of the structure, and insufficient number
of degrees of freedom chosen for correlation. The effect of random phase scattering in case
of non proportional damping has been studied in the next chapter.
•
The cross-orthogonality is a mass-weighted measure of the linear independence of the test
mode shapes compared to the analysis mode shapes. It could give a better estimate of the
correlation if the analytical mass matrix at the reduced DOF is available. The crossorthogonality calculations could not be done due to computational memory imitations.
•
The modal assurance criterion is sensitive to large values (wild points?) and insensitive to
small values. The modal assurance criterion is based upon the minimization of the squared
76
error between two vector spaces. This means that the degrees-of-freedom involving the
largest magnitude differences between the two modal vectors will dominate the
computation while small differences will have almost no effect. Therefore, small modal
coefficients will generally not have much effect on the MAC [28] calculation and large
modal coefficients will potentially have the greatest effect. This also means that, if there
have been erroneous data included in the modal vectors due to calibration errors, modal
parameter estimation mistakes, etc., these wild points may dominate the MAC [28]
calculation.
•
The main diagonal terms of the MAC matrix were close to 1 indicating that the mode
shapes obtained from experimental test and from ANSYS FE modeling match very close to
each other.
•
The MAC [28] is used to determine the correlation between analytical and experimental
modal vectors. It is useful in identifying the common modes between two models. The
main advantage of the MAC is that the calculation is straight forward and that no mass
matrix is required. Therefore, the error due to the incompatibility of the mass matrix and
experimental model is eliminated. The disadvantage of MAC is that MAC can not identify
whether the models are orthogonal or incomplete. [30]
•
There can be situations where MAC will show correlation of vectors which are actually
independent. This can be due to variety of reasons, one of which is that the vectors may be
incompletely determined. This can occur when not enough measurement points are
included in the modal testing. In this is the case, the correlation of off diagonal terms in the
MAC matrix may be misleading. The remedy is to acquire additional data, so that mode
shapes will be independent.[40]
77
•
The frequent measurement errors include the misalignment of transducers. If the
transducers are not aligned perfectly with the global coordinate system, it might introduce
small errors. The errors could be very small as long as the alignment error is not more
than 10 ° .
•
For quantitative comparison of errors caused by roving of sensors and misalignment, it is
evident that the roving sensors could be very detrimental for correlation if the dummy
weights have not been used for measurements.
5.6 Concluding remarks on correlation study
Correlation is a tool that is used widely for numerically comparing analytical predictions and
experimental data. In this chapter, the role of modal data correlation in the model validation
process has been discussed. This section also studies errors in orthogonal for transducer
misalignment. These are not the only errors that are present in measuring and estimating modal
vectors. Other errors include effects such as quantization, leakage, FRF Function estimation,
modal parameter estimation, calibration etc. These other errors certainly merit study to
understand their contribution to errors in orthogonality.
Certainly in this thesis only one structure was analyzed, which may or may not be representative
of typical structures which are encountered. Other structures need to be analyzed to see if certain
classifications of structures are more sensitive than others. Methods and hardware for reducing
misalignment error should also be considered. Methods for selecting transducer locations and
TAM model methods should also consider these transducer errors. Finally until a better
understanding of the errors which affect correlation is gained, use of orthogonality computations
for modal vector correlation should be interpreted and used carefully.
78
CHAPTER 6
EFFECT OF DAMPING ON CORRELATION
6.1 Introduction
In most practical situations, the theoretical mode shapes are real if the damping has not been
included in the model, as it is not in most practical cases, whereas the experimental mode shapes
are at least slightly complex because of inevitable damping effects in any real structure. If,
however, correlation is performed between real and complex mode shapes, this will in general
decrease the correlation parameter values, since the correlation coefficient is a simple scalar
product between two vectors divided by their magnitudes.
Because of difficulties associated with the modeling of non-proportional damping and nonlinearity, such effects are generally omitted and theoretical modes come out of the analysis as
numerically real values. Since it is likely that some influence of damping or non-linearity
(material, geometrical) will exist in practice, these effects will influence the measurements and,
after modal analysis, the extracted modes will generally be complex.
This incompatibility of mode shapes can be overcome by real normalization of complex mode
shapes. This section describes a method which assesses the modal complexity of complex mode
shapes. The best way to assess the degree of modal complexity of an experimental mode shape is
to plot real and imaginary values for each DOF in the complex plane. Ideally, if the mode shape
is a normal mode, the vectors would all lie on a single line in the complex plane. The more these
vectors diverge from a single line, the more the mode shape is complex.
79
6.2 Modal Complexity of Experimental Mode Shapes
In order to determine the complexity of mode shapes, different criteria have been proposed and
are used in the current thesis to check the complexity of modal data obtained from experimental
test.
6.2.1 Maximum Complex Area (MCA) [35]: MCA of a mode is defined as the area of a polygon
generated by constructing the envelope around the extremities of the tips of the eigenvector
elements plotted in the complex plane, with only obtuse angles permitted outside the resulting
polygon. It is clear that the more complex a mode is, the larger the maximum complex area of
the mode (MCA) will be. In order to quantify the Modal Complexity Factor (MCF2) [35] of the
i th mode shape, the following coefficient is defined as the ratio of the maximum complex area
(MCA) of the mode to the maximum complex area of that mode when it is maximally complex
as defined by expression,
MCF2 (i) =
PolygonArea (i )
× 100%
CircleArea(i )
The modal complexity factor can vary between 0 and 1. Real modes would have modal
complexity factor 0 and a maximally-complex mode would have a MCF2 value of 1. A very high
value of the MCF2 coefficient of a lightly damped structure for some experimental modes
indicates probable error, either in the measurements or in the modal analysis. The calculation of
MCF2 for experimental modes is explained below with schematics of the mode shapes obtained
from Matlab(R 7.01). AutoCAD (R2004) is used to calculate the area of the polygon and circle
for each mode shape respectively.
80
6.2.2 Modal Complexity Plots:
Frequency Fn = 14.862 Hz
[Modal Complexity Factor] =
PolygonArea
3.0684
=
= 8.62%
CircleArea
35.5603
Frequency Fn = 23.64 Hz
[Modal Complexity Factor]Fn = 23.64 =
81
0.5919
= 1.19%
49.6941
Frequency Fn = 35.40 Hz
[Modal Complexity Factor]Fn =35.40 =
1.3513
= 2.28%
59.1356
Frequency Fn = 54.22 Hz
[Modal Complexity Factor]Fn =54.22 =
82
1.9723
= 5.4%
36.4821
Frequency Fn = 74.43 Hz
[Modal Complexity Factor]Fn =74.43 =
0.3119
= 3.38%
9.221
Frequency Fn = 147.47 Hz
[Modal Complexity Factor]Fn =147.47 =
83
1.6351
= 9.03%
18.0992
Frequency Fn = 153.89 Hz
[Modal Complexity Factor]Fn =153.89 =
2.6140
= 11.47%
22.8347
Frequency Fn = 162.21 Hz
[Modal Complexity Factor]Fn =162.21 =
84
0.3076
= 7.04%
4.3676
Frequency Fn = 175.75 Hz
[Modal Complexity Factor]Fn =175.75 =
1.5474
= 19.9%
7.7384
Frequency Fn = 182.95 Hz
[Modal Complexity Factor]Fn =182.95 =
85
2.2030
= 10.1%
21.7302
6.2.3 RESULT TABLE FOR MCF2 OF EXPERIMENTAL MODES:
Mode No.
Frequency [Hz]
MCF [%]
Cross-MAC
1
14.862
8.62
0.974
2
23.64
1.19
0.995
3
35.4
2.28
0.995
4
54.22
5.4
0.984
5
74.43
3.38
0.994
6
147.47
9.03
0.979
7
153.89
11.47
0.967
8
162.21
7.04
0.956
9
175.75
19.9
0.986
10
182.95
10.1
0.925
Table 6.2.3 Modal Complexity Factor
From the above table, it is clear that modes at 175.75Hz and 182.95 have maximum modal
complexity according to this criterion. It was also found that for the modes for which the high
MCF values were observed, the MAC of the corresponding modes was in error. This clearly
shows the effect of modal complexity on correlation process. This shows that the correlation
obtained by this inconsistent data set has little to do with the ability of the EI algorithm which
promises the linear independency of modal vectors. One might expect to improve the overall
MAC values with correct measurement data. The complexity factor obtained does not point
towards the errors in the modal parameter estimation, as in some cases choosing the wrong
modal parameter estimation algorithm also induces extra complexity into the modal parameters.
86
6.3 Modal Phase Collinearity [MPC]
Mode shapes of a proportional damped system are normal modes. A fixed phase exists for all
mode shape coefficients of a specific mode. Depending upon the scaling used in the modal
parameter estimation process, the phase of modal coefficients is either ± 90 ° or ± 180 ° .Modal
Phase Collinearity is an indicator that checks the degree of complexity of a mode. It is based
upon the concept of variance-covariance matrix calculations. The variance-covariance matrix
consists of variances of the variables along the main diagonal and the covariances between each
pair of variables in the other matrix positions. MPC evaluates the functional linear relation
between real and imaginary parts of the mode shape coefficients. Let φi' and φi'' be the real and
imaginary parts respectively, for the identified mode ‘i’ respectively. The variance and
covariance of the real and imaginary parts are calculated first, by the formula,
S xx = φi' .φi'
T
S yy = φi'' .φi''
T
S xy = φi' .φi''
T
Letting μ =
S yy − S xx
2.S xy
∴ β = μ + sgn( S xy ). μ 2 + 1
Where τ = tan −1 ( β )
The eigen values of the variance-covariance matrix are,
λ1 = S xx +
λ2 = S yy −
S xy (2( μ 2 + 1) sin 2 τ − 1)
μ
S xy (2( μ 2 + 1) sin 2 τ − 1)
μ
87
MPC for mode ‘i’ is then defined as follows:
2
⎡ ⎛ λ1
⎞⎤
MPCi = ⎢2.⎜⎜
− 0.5 ⎟⎟⎥ × 100%
⎠⎦
⎣ ⎝ λ1 + λ 2
MPC values range from 0 for a mode with completely unrelated phase angles to 100 percent for
a monophase result. The modal phase colinearity of the experimental modal vectors is tabulated
in Table 6.4.1 along with their mean phase deviation values.
6.4 Mean Phase Deviation [MPD]
MPD is another statistical indicator of the complexity of a mode shape. The mean phase
deviation expresses the phase scatter of each mode shape. For normal mode shapes its value
should be zero. The mean phase of a mode ‘r’ is given by,
N
MPH
r
=
∑ W .φ
i =1
N
i
∑W
i =1
ir
i
Where, Wi = Weighting factor = Ψir
φir = arctan(Re(Ψir ) / Im(Ψir ))if arctan(Re(Ψir ) / Im(Ψir )) ≥ 0
φir = arctan(Re(Ψir ) / Im(Ψir )) + π if arctan(Re(Ψir ) / Im(Ψir )) <0
88
The corresponding mean phase deviation is given by,
N
MPDr =
∑W .(φ
i =1
i
ir
− MPH r )
2
N
∑W
i =1
i
The combination of MCF2, MPC as well as MPD can easily tell how much effect the modal
complexity has on the process of correlation. The values are plotted below in Table 6.4.1.
Mode
Frequency [Hz]
MPC [%]
MPD [degree]
Phase Dispersion
Mode Type
1
14.86
97.158
8.4223
High
Complex
2
23.64
99.747
1.8867
Low
Real
3
35.4
81.461
6.1184
Comparatively High
Real
4
54.22
99.01
2.8418
Low
Real
5
74.43
99.112
3.3313
Low
Real
6
147.5
87.981
5.9159
Low
Real
7
153.86
94.664
7.9087
Comparatively High
real
8
162.21
95.91
6.0158
Low
real
9
175.75
81.17
13.032
High
complex
10
182.95
75.895
12.48
High
complex
Table 6.4.1 MPC and MPD
The results show that because of the mass loading, modes 1, 3, 9 and 10 are seriously affected.
The mean phase deviation below 5 degrees marginally affects the correlation whereas MPD
above 10 degrees greatly affects the correlation values.
The modal phase colinearity was below 80% for modes 9 and 10 indicating the adverse effect of
mass loading for higher frequencies. The results also show that the sensitivity of mode ‘3’ for a
89
slight change in mass addition was greatest and these results could be beneficial for FE model
updating studies. The phase scatter for modal vectors really causes big problems especially in the
process of modal vectors correlation. Real normalization of experimental mode shapes by
appropriate methods is must in order to get good correlation between two sets of modal data. The
following section briefly describes the methods generally used for this purpose.
6.5. Real Normalization of Complex Mode Shapes [36]
After checking the MCF2 coefficient values for the experimental mode shapes, and if these
values are not very high, the mode shapes can be real normalized using one of the existing
techniques which can be found in [36]. Most existing techniques tend to real normalize the
complex mode shapes in such a way that the correlation between complex and the corresponding
real mode shapes is retained as high as possible.
Another reason for questioning these approaches is that many tune the experimental mode shapes
to the corresponding theoretical mode shapes (if these are known at that stage), but these
theoretical mode shapes may not be completely accurate (that is why the correlation and
updating are being performed in the first place).
An extremely simple approach for realization of complex modes will be described here. Since
the test structure was very lightly damped, the real mode shapes were obtained by taking the
modulus of the complex values multiplied by the sign of the imaginary part of the modal
coefficients. This method generally gives reasonable results if the damping is very low in the
structure. In case of heavily damped structures, there are different approaches for real
normalizing the complex modal data.
90
6.6. Effect of Random Phase Scatter on Correlation
When experimentally measuring the modal parameters of a structure, one of two different types
of systems will exist, namely: a proportionally damped system or a non- proportionally damped
system. In a proportionally damped system, the damping is distributed proportionally to the mass
and/or stiffness. These systems will have normal modes. This means that all points will vibrate at
same frequency, reach a maximum value and pass through their equilibrium position
simultaneously.
In non-proportionally damped systems, the damping is not distributed proportionally to either the
mass or stiffness. Structures of this type have complex mode shapes. Thus, in describing a
complex mode, one must consider not only the amplitudes of the various degrees of freedom but
also the phase relationships.
Complex modes do exist in non-proportionally damped systems. However, it is possible for
mode shapes of proportionally damped systems to appear be complex valued, because of errors
that occur in the measurements (mass roving of accelerometers), data acquisition (aliasing,
leakage) or analysis phases of a modal test.
In order to study the effect of random phase scattering of modal vectors, two test cases were
carried out in which experimental results were simulated such that the resultant modal vectors
have 5 degrees and 10 degrees random phase scatter in the modal coefficients. The modal data
was then correlated with the analytical modal vectors in order to see their effect on correlation.
91
Mode No.
1
2
3
4
5
6
7
8
9
10
1
1.000
0.002
0.002
0.024
0.036
0.254
0.003
0.004
0.001
0.004
2
0.002
1.000
0.003
0.001
0.002
0.002
0.081
0.001
0.000
0.029
3
0.002
0.003
1.000
0.000
0.038
0.007
0.001
0.006
0.001
0.363
4
0.024
0.001
0.000
1.000
0.074
0.001
0.007
0.006
0.000
0.000
5
0.036
0.002
0.038
0.074
1.000
0.000
0.006
0.011
0.000
0.001
6
0.254
0.002
0.007
0.001
0.000
1.000
0.011
0.000
0.054
0.004
7
0.003
0.081
0.001
0.007
0.006
0.011
1.000
0.019
0.097
0.001
8
0.004
0.001
0.006
0.006
0.011
0.000
0.019
1.000
0.001
0.166
9
0.001
0.000
0.001
0.000
0.000
0.054
0.097
0.001
1.000
0.017
10
0.004
0.029
0.363
0.000
0.001
0.004
0.001
0.166
0.017
1.000
Table 6.6.1 Auto-MAC- 5 ° random Phase Scatter
Figure 6.6.2 Auto-MAC- 5 ° random Phase Scatter
92
Mode No.
1
2
3
4
5
6
7
8
9
10
1
0.854
0.000
0.003
0.002
0.013
0.364
0.008
0.005
0.002
0.003
2
0.001
0.623
0.000
0.000
0.001
0.002
0.022
0.053
0.000
0.019
3
0.000
0.004
0.966
0.002
0.072
0.000
0.000
0.000
0.000
0.418
4
0.007
0.000
0.000
0.762
0.011
0.026
0.000
0.004
0.000
0.001
5
0.007
0.000
0.018
0.035
0.746
0.003
0.001
0.002
0.001
0.002
6
0.323
0.000
0.007
0.005
0.004
0.812
0.009
0.002
0.051
0.024
7
0.003
0.030
0.001
0.004
0.001
0.002
0.869
0.033
0.079
0.001
8
0.003
0.021
0.003
0.003
0.004
0.002
0.001
0.828
0.007
0.129
9
0.001
0.000
0.002
0.001
0.001
0.020
0.056
0.001
0.959
0.000
10
0.000
0.044
0.355
0.000
0.000
0.032
0.002
0.088
0.009
0.838
Table 6.6.3 Cross-MAC- 5 ° random Phase Scatter
Figure 6.6.4 Cross-MAC- 5 ° random Phase Scatter
93
Mode No.
1
2
3
4
5
6
7
8
9
10
1
1.000
0.001
0.002
0.002
0.011
0.202
0.007
0.004
0.001
0.002
2
0.001
1.000
0.012
0.002
0.000
0.002
0.008
0.096
0.000
0.075
3
0.002
0.012
1.000
0.005
0.007
0.022
0.001
0.005
0.001
0.342
4
0.002
0.002
0.005
1.000
0.144
0.010
0.001
0.013
0.003
0.000
5
0.011
0.000
0.007
0.144
1.000
0.001
0.005
0.002
0.002
0.004
6
0.202
0.002
0.022
0.010
0.001
1.000
0.017
0.004
0.082
0.000
7
0.007
0.008
0.001
0.001
0.005
0.017
1.000
0.001
0.106
0.003
8
0.004
0.096
0.005
0.013
0.002
0.004
0.001
1.000
0.016
0.133
9
0.001
0.000
0.001
0.003
0.002
0.082
0.106
0.016
1.000
0.012
10
0.002
0.075
0.342
0.000
0.004
0.000
0.003
0.133
0.012
1.000
Table 6.6.5 Auto-MAC- 10 ° random Phase Scatter
Figure 6.6.6 Auto-MAC 10 ° random phase scatter
94
Mode No.
1
2
3
4
5
6
7
8
9
10
1
0.626
0.002
0.000
0.021
0.007
0.284
0.006
0.025
0.001
0.023
2
0.001
0.402
0.000
0.001
0.000
0.003
0.083
0.056
0.011
0.014
3
0.007
0.004
0.911
0.000
0.086
0.003
0.000
0.000
0.000
0.377
4
0.000
0.001
0.001
0.693
0.021
0.010
0.001
0.010
0.001
0.013
5
0.000
0.000
0.016
0.045
0.691
0.000
0.001
0.000
0.004
0.002
6
0.356
0.001
0.013
0.010
0.004
0.749
0.008
0.002
0.061
0.005
7
0.001
0.032
0.001
0.006
0.002
0.004
0.753
0.011
0.075
0.000
8
0.005
0.021
0.000
0.004
0.001
0.002
0.009
0.780
0.028
0.086
9
0.001
0.000
0.014
0.002
0.004
0.034
0.049
0.000
0.908
0.001
10
0.000
0.040
0.305
0.000
0.003
0.055
0.002
0.082
0.015
0.762
Table 6.6.7 Cross-MAC- 10 ° random Phase Scatter
Figure 6.6.8 Cross-MAC 10 ° random phase scatter
95
6.7 Discussion of Results
The objective of the chapter was to study different measures of modal complexity and its effect
on FE model correlation. This is primarily required because most of the structures are nonproportionally damped and hence the realization of complex modal vectors into real modal
vectors is required in order to calculate modal assurance criteria. Most of the FEM packages do
not consider effect of damping and hence give rise to real modal vectors.
From different complexity measures seen here, the modal complexity factor takes into account
the magnitude as well as phase of modal coefficients. Modal phase colinearity and mean phase
deviation can be used in conjunction with the MCF to interpret the results of modal complexity.
From table 6.4.1 it is clear that the modes at 14.86 Hz, 35.4 Hz, 153.86 Hz, 175.75 Hz and
182.95 Hz showed high values for these measures. Some of the important observations are listed
below:
•
The effect of mass loading increased at higher frequencies which were reflected in the
MCF, MPC and MPD.
•
Strong correlation was observed between the modes at 14.86 Hz and at 147.5 Hz. This
could partly be attributed to the errors caused by mass loading, as the frequency from point
to point was different, making the data inconsistent. But apart from that, it was also result
of spatial aliasing caused by the chosen insufficient number of degrees of freedom. Since
while implementing the effective independence (EI) algorithm, it is still required to choose
arbitrarily the number of degrees of freedom that would lead to the linear independency of
modal vectors.
96
•
The correlation between the mode at 35.4 Hz and 182.95 Hz could again be attributed to
the modal complexity caused by mass roving effect. In case of impact testing or
measurements without these effects, these errors could not be present.
•
The test case carried out above is an indication of how correlation could get worse with the
effect of modal complexity. The reasons for modal complexity could be either the complex
geometry or could be caused by mass roving, modal parameter estimation errors. In such
cases, complex modes needs to be properly converted to real modal vectors before the
correlation is carried out between FE modal vectors and experimental modal vectors.
97
Conclusion
This chapter mainly dealt with the complexity encountered in the experimental modal data and
its effect on correlation. The specific conclusions are listed below:
•
The real normalization of complex mode shapes is of practical importance for structures
with significant damping.
•
The mass loading of roving accelerometers induces additional complexity into the modal
vectors giving rise to poor correlation results. The frequency shifts caused by mass roving
make the frequency response function data base inconsistent: frequency responses of
different points show different resonance frequencies.
•
The modal parameter estimation may sometime lead to the incorrect estimates of damping
into the structures.
•
From the test cases, it is evident that with increase in phase scatter for modal coefficients,
correlation starts to decrease. The major effect of random phase scattering was seen on
diagonal terms giving poor values whereas the off diagonal terms were in comparison less
affected.
•
In most of the practical situations, it is necessary to pay attention to the errors caused by the
measurement process and the main aim of this chapter was to study the commonly made
errors in the measurement.
98
CHAPTER 7
CONCLUSIONS
The process of FE modeling and correlation outlined in this thesis is a very useful tool. The
correlation will give a first idea of the modifications on the dynamics of the structure, or will
give information about the best type and location of modifications needed to achieve a specified
dynamic behavior. During the next step the effect of the most promising modifications can be
estimated. The best one is then implemented: first in the finite element model then as a hardware
change on the prototype itself. In order to make sure that predictions from FE models have an
acceptable accuracy with respect to experimental data, more and more practice of model
validation is expected in the future. Some of the important facts which were evident from the test
cases are listed below:
Modal testing techniques can be very sensitive to even small changes in the structure’s dynamic
behavior, such as those that are frequently assumed to be negligible in the measurement. So there
is a need to investigate the effect of the small changes in the system relating to the mechanical
devices such as accelerometers, suspension springs and stingers.
Performing model correlation with three different kinds of measurement errors has been studied.
The results from the study show that correlation drops significantly with mass loading of roving
accelerometers.
99
The influence of complexity of mode shapes in a case of lightly damped structure has been
examined. Using various measures, modal complexity is tabulated which in turn directly point
towards the bad correlation for the corresponding modes. In case of lightly damped structures,
presence of modes with high damping values is an indication of the following things; The higher
damping could be due to mass roving of accelerometers, modal parameter estimation errors or by
the presence of joints in the structure.[40]
Two frequent measurement errors have been simulated to study the correlation of mode shapes.
One method checks the misalignment effect of accelerometers on the test correlation. The results
did not vary considerably when the misalignment was within ± 10 degree from the principal
°
coordinate system. The second method checks the mass loading effect of accelerometers, which
had severe effect on modal correlation as compared to the other errors simulated.
The Effective Independence (EI) test-planning method has been studied for application of
optimal sensor placement for modal testing. The result proved very effective in the sense that
with only 77 degrees of freedom in place, the maximum error among the mode shape was found
to be 17 %. Much more research is needed in knowing the exact number of sensors required for
complete linear independency of mode shapes of any structure.
It was shown that the complex modes may also exist due to errors that occur in the data
acquisition and/or parameter estimation phases of the analysis. As a result of mass loading, the
frequency values of the poles are underestimated and damping value estimates are unreliable. In
the parameter estimation step, it is critical not to miss any mode in the frequency range of
100
interest as well as the damping estimate. If the relative error of damping estimates is greater than
5%, then it could also give rise to complex mode shapes.
One important observation regarding modal damping was that in the case of inconsistent data
obtained by mass loading, increasing the number of degrees of freedom do not necessarily
improves the modal parameter estimates of the structure.[38]
Correlation tools are an aid to the analyst, but it is through experience and good engineering
judgment that an analyst must scrutinize the model to achieve its full potential as a predictive
tool. From the case study of the joint modeling, it is clear that for large scale structure models
like automobile frames made up of many parts, connectivity approximations between the parts
can be easily be the most important issue in the correlation study. These connectivity
assumptions can be very difficult to account for in the FE modeling of the structures and which
in turn could result in the poor correlation.
7.1 Suggestion for future work
The study undertaken in this thesis has covered the major steps in the model validation process.
There are some encouraging results for the successful practice of validating industrial models.
However, some results from the study indicate that model validation is not, at the moment, a
mature tool that every engineer can make the best use of. From the study results in the thesis,
some general suggestions for future work regarding this thesis are outlined below.
101
Joint Modeling: Although this thesis came across the joint modeling problem, it was not
addressed in depth. The effect of the joint interface on the modal characteristics clearly demands
for a more attention.
Number of Master degrees of freedom required. The effective independence algorithm though
successful in the thesis, could not prove to be complete. It still demands user judgment in
deciding the exact number of master degrees of freedom which would lead to the linear
independence of modal vectors.
For structures with high modal densities (large number of modes in limited frequency range
and/or modes close together in frequency with moderate to heavy damping), the creation of an
experimental modal model and the definition of an appropriate comparison criterion with the FE
model are often difficult although recent advances in identification procedures may sometime
resolve this difficulty. Since the conclusions presented in this thesis work apply for lightly
damped structures, the future work should focus on more complicated structures. The mode
shapes are more susceptible to error in structures with heavy damping.
To verify the effect of welded joint on the dynamic properties, the model needs to be designed
with more details, properties should be chosen accordingly and the results in that case would be
better representative of the system stiffness. Welding joint could be simulated by creating a
wedge region at the interface of two sections and applying appropriate material properties in
order to closely simulate the actual conditions. This approach might not be a feasible solution for
structures with many welding joints.
102
In the algorithms of model updating techniques, based on experimental FRFs, the analytical FRF
information from the finite element model is always required. Since the finite element damping
information is usually not available and it is necessary to include the frequency response
functions from the FE model based on proportional damping assumptions. The identified
complex modes are very sensitive to experimental errors and also to errors arising from fitting
algorithms. The FRFs obtained from FE model in this way will serve to identify/simulate other
errors encountered in measurements or modal parameter estimation. However, the limitation of
proportional damping for the analytical situation should also be considered as a source of error.
Sensitivity of modal vectors towards errors is directly proportional to the system damping. In
case of lightly damped structures, the assumption of proportional or non proportional damping
makes little difference. The study for a more complicated structure with non proportional
damping, will give more insight into the measurements errors and its effect on correlation, since
real normalization of complex modal vectors would be the key to obtaining good correlation.
The results presented in this thesis are based on the FE results obtained from Ansys. The results
might differ if another FEM package like MSC/Nastran is utilized for the analysis. The future
work should incorporate a more intense study with MSC/Nastran to be able to simulate different
modeling errors. All of these tools cannot substitute for sound engineering judgement in the
modeling and analysis process. They cannot salvage a model having insufficient detail or
unwarranted assumptions. But they can play a vital role in the analysis and design of complex
structural systems.
103
CLOSURE
The primary aim of this work was to improve the current measurement techniques and to
develop new ones which permit the acquisition of data in high quality and to devise methods to
specify the accuracy of the measured data in a conventional modal test. This was achieved by
critical investigation of existing methods and a recommended test strategy has been presented in
this thesis.
A summary of conclusions and contributions showed the remarkable variations among different
test case measurements. Not surprisingly, however, there is still scope for further study to
improve the accuracy of modal correlation activity.
104
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In The Proceedings of IMAC 6, pages 1039--1047, 1988
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[41] M. Mains and H. Vold. Investigation of the effects of the cross sensitivity and
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random excitation with cyclic averaging. In The Proceedings of IMAC 13, pages 891--899, 1998
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identification and model reduction. Journal of Guidance, Control and Dynamics, Vol.8, No.5,
Sept.1985, p620-627.
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(Chaptar-2)
109
Appendix-A Instrumentation
Calibration of load cells and accelerometers
Device/Make
Load cell
Load cell
Tri axial accelerometer
Tri axial accelerometer
Tri axial accelerometer
Tri axial accelerometer
Tri axial accelerometer
Model Number
UT333M07
UT333M07
333A32
333A32
333A32
333A32
333A32
SN
SN18528
SN16083
1868
1855
1872
1850
1869
Calibration
54.23mv/lb
55.17mv/lb
96.54mv/g
98.43mv/g
101.65mv/g
97.33mv/g
98.35mv/g
Data Acquisition System
Device/Make
Data Acquisition System
Computer
Name
VXI
Dell
Description
16 Input channels
Amplifiers
Device/Make
Modal Shop Inc Make
MB Dynamics
Name
2050E02 Dual Mode Amplifier
SS250VCF Amplifier
Description
Amplifier, selectable current/voltage control
Dual Mode Amplifier
Shaker
Device/Make
Electrodynamic type
MB Dynamics Make
Name
Modal 50 Exciter
o
o
o
o
o
Description
1 to 4000Hz Range
Moving mass < 0.4lb
Weight:55lb
"
Stroke:1 peak-peak
Force o/p:50lbs(peak)
Other
Device/Make
PCB Hand held Calibrator
HP Make
Model Number
394B06
583A
110
Description
0.994 rms at 159Hz
Signal Conditioners(ICP)
Appendix-B Modeling Code
%-----------------------------------------------------------------------------------------------------------------Kammer’s Effective Independence Algorithm
%-----------------------------------------------------------------------------------------------------------------clc;
clear all;
load (‘Modal_matrix.mat');
A=(Modal_matrix)'*(Modal_matrix);
E=(Modal_matrix)*inv(A)*(Modal_matrix)';
for i=1:7077
Diagonal_entry=(diag(E))';
[least,index1]=min(abs(Diagonal_entry));
index(1,i)=index1;
least(1,i)=least;
Modal_matrix(index(1,i),:)=[];
A=(Modal_matrix)'*(Modal_matrix);
E=(Modal_matrix)*(inv(A))*(Modal_matrix)';
end
%-----------------------------------------------------------------------------------------------------------------The above algorithm selects 75 master degrees of freedom from the total degrees of freedom of
7152.The calculated master degrees of freedom were then tracked to the original nodes of the
finite element model by writing another small code, shown on next page.
%-----------------------------------------------------------------------------------------------------------------Finding the index of the master degrees of freedom
%-----------------------------------------------------------------------------------------------------------------clc;
clear all;
load('index.mat');
Slave_dof(1,1)=index(1,1);
for i=2:7052
if index(1,i) < Slave_dof(1,:)
Slave_dof(1,i)=index(1,i);
end
if Slave_dof(1,1)<=index(1,i)&& index(1,i)<=Slave_dof(1,end)
index2=index(1,i);
for j=1:size(Slave_dof,2)
if index2 > Slave_dof(1,j)
index2=index2+1;
end
end
while find(index2==Slave_dof(1,:))
index2=index2+1;
end
Slave_dof(1,i)=index2;
end
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if index(1,i) > Slave_dof(1,end)
Slave_dof(1,i)=index(1,i)+size(Slave_dof,2);
end
Slave_dof=sort(Slave_dof);
end
%-----------------------------------------------------------------------------------------------------------------Finding the X,Y and Z directional master degrees of freedom
%-----------------------------------------------------------------------------------------------------------------clc; clear all; load ('M2_solid.mat');
for i=1:80
if mod(M2_solid(1,i)-5820,3)==0
Master_dir(1,i)=3;
Master_node(1,i)=[(M2_solid(1,i)-5820)/3]+970;
elseif mod(M2_solid(1,i)-5820,3)==1
Master_dir(1,i)=1;
Master_node(1,i)=[((M2_solid(1,i)-5820)+2)/3]+970;
elseif mod(M2_solid(1,i),6)==2
%
Master_dir(1,i)=2;
%
Master_node(1,i)=(M2_solid(1,i)+4)/6;
%
% elseif mod(M2_solid(1,i),6)==3
%
Master_dir(1,i)=3;
%
Master_node(1,i)=(M2_solid(1,i)+3)/6;
%
% elseif mod(M2_solid(1,i),6)==4
%
Master_dir(1,i)=4;
%
Master_node(1,i)=(M2_solid(1,i)+2)/6;
else %mod(M2_solid(1,i),6)==5
Master_dir(1,i)=2;
Master_node(1,i)=[((M2_solid(1,i)-5820)+1)/3]+970;
end
end
%-----------------------------------------------------------------------------------------------------------------The first 970 nodes were shell nodes having six degrees of freedom whereas the remaining nodes
belong to solid45 elements having three degrees of freedom.
%------------------------------------------------------------------------------------------------------------------
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QR-DECOMPOSITION
clc;
clear all;
nmode=15;
%----> Set of Target Modes.
%-------------------------------------------------------------------------Mode7=load('mode7.txt');
Mode8=load('mode8.txt');
Mode9=load('mode9.txt');
Mode10=load('mode10.txt');
Mode11=load('mode11.txt');
Mode12=load('mode12.txt');
Mode13=load('mode13.txt');
Mode14=load('mode14.txt');
Mode15=load('mode15.txt');
Mode16=load('mode16.txt');
Mode17=load('mode17.txt');
Mode18=load('mode18.txt');
Mode19=load('mode19.txt');
Mode20=load('mode20.txt');
Mode21=load('mode21.txt');
%-------------------------------------------------------------------------Modal_Matrix1=[Mode7 Mode8 Mode9 Mode10 Mode11 Mode12 Mode13 Mode14 ...
Mode15 Mode16 Mode17 Mode18 Mode19 Mode20 Mode21];
% temp1=1;
% for i=1:nmode
% Modal_Matrix1(:,temp1)=[];
% temp1=temp1+3;
% end
%-------------------------------------------------------------------------temp2=1;
Modal_Matrix=zeros((size(Mode7,1)*3),nmode);
for k=1:nmode
temp3=1;
for i=1:size(Mode7,1)
for j=temp2:temp2+2
Modal_Matrix(temp3,k)=Modal_Matrix1(i,j);
temp3=temp3+1;
end
end
temp2=temp2+3;
end
clear Modal_Matrix1 Mode7 Mode8 Mode9 Mode10 Mode11 Mode12 Mode13 Mode14...
Mode15 Mode16 Mode17 Mode18 Mode19 Mode20 Mode21 temp2 temp3;
%-------------------------------------------------------------------------% Basic Principle:-The most linear independent rows of the modal matrix
% indicate degrees of freedom that should be chosen as pick up locations
113
% because they form the smallest possible modal matrix which provides a MAC
% matrix with minimized off diagonal terms => ability to distinguish
% between similar mode shapes is enhanced.
%-------------------------------------------------------------------------% Origional modal matrix obtained from finite element model.
% A1=fix(27*rand(140,30));
% A=A1';
%-------------------------------------------------------------------------% The QR-decomposition is applied to the transpose of modal matrix because
% the information for one given degree of freedom is provided in one single
% row of the modal matrix and QR-decomposition does only sort the columns
% and not the rows of a matrix.
%-------------------------------------------------------------------------modal_matrix_transpose=Modal_Matrix';
%-------------------------------------------------------------------------% QR-decomposition in Matlab
% [Q,R,E] = qr(A); %produces unitary Q, upper triangular R and a
% permutation matrix E so that A*E = Q*R. The column permutation E is
% chosen so that ABS(DIAG(R)) is decreasing.
[Q,R,E]=qr(modal_matrix_transpose);
%-------------------------------------------------------------------------% Product=modal_matrix*E;
%-------------------------------------------------------------------------% QR-decomposition will give most independent columns equal to the rank of
% the modal matrix,which is equal to the smaller dimension of the matrix.So
% if (s>n) then choose the dof in the vicinity of the master_dof.
master_dof=zeros(1,min(size(modal_matrix_transpose)));
reduced_modal_matrix=zeros(min(size(modal_matrix_transpose)),min(size(modal_matrix_trans
pose)));
%-------------------------------------------------------------------------for i=1:min(size(modal_matrix_transpose))
for j=1:size(E)
if E(j,i)==1
master_dof(1,i)=j;
reduced_modal_matrix(:,i)=modal_matrix_transpose(:,master_dof(1,i));
modal_matrix_transpose(:,master_dof(1,i))=0;
end
end
end
%-------------------------------------------------------------------------Master_node=zeros(1,size(master_dof,2));
Master_direction=zeros(1,size(master_dof,2));
for i=1:size(master_dof,2)
if mod(master_dof(1,i),3)==0;
Master_node(1,i)=(master_dof(1,i)/3);
Master_direction(1,i)=3;
114
elseif mod(master_dof(1,i),3)==1;
Master_node(1,i)=(master_dof(1,i)+2)/3;
Master_direction(1,i)=1;
else
mod(master_dof(1,i),3)==2;
Master_node(1,i)=(master_dof(1,i)+1)/3;
Master_direction(1,i)=2;
end
end
master_set=[(master_dof') (Master_node') (Master_direction')];
% 1,2,3->x,y,z respectively
%-------------------------------------------------------------------------% %-------------------------------------------------------------------------% % The row position of '1' in first 'n'(rank of matrix)columns of matrix 'E'
% % indicates the maximum independent dof => chosen as master_dof.
% %-------------------------------------------------------------------------% % VALIDATION OF THE RESULTS
% % The MAC matrix of the modal matrix versus itself and the modal matrix
% % reduced to the selected pick up degrees of freedom versus itslef needs to
% % be calculated.
% modal_matrix1=modal_matrix';
% modal_matrix_mac=modal_matrix1*modal_matrix1';
% for i=1:size(modal_matrix_mac)
% for j=1:size(modal_matrix_mac)
%
MAC(i,j)=modal_matrix(i,j)*modal_matrix(i,j);
% end
% end
% %-------------------------------------------------------------------------% % MAC for the reduced modal matrix is calculated below.
% % modal_matrix1=modal_matrix';
% % modal_matrix_mac=modal_matrix1*modal_matrix1';
% for i=1:size(reduced_modal_matrix)
% for j=1:size(reduced_modal_matrix_mac)
%
MAC(i,j)=reduced_modal_matrix(i,j)*reduced_modal_matrix(i,j);
% end
% end
% %--------------------------------------------------------------------------
115