chennagowni suresh babu

8/8/2019 CHENNAGOWNI SURESH BABU

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UNIVERSITY OF CINCINNATI

Date: __04/14/06 ________

I, ___Suresh Babu Chennagowni _______________________________,

hereby submit this work as part of the requirements for the degree of:

Master of Science

in:

Mechanical EngineeringIt is entitled :

Study of the effect of Mass Distribution, Pathof Energy and Dynamic Coupling on Combined Coherence (A Non-linerarity Detection Method)

This work and its defense approved by:

Chair: _Dr. Randall J. Allemang ______ _Dr. Allyn W. Phillips ________

_Dr. Ronald L. Huston _________ ______________________________


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Study of the Effect of Mass Distribution, Path of Energy and Dynamic

Coupling on Combined Coherence (A Non-linearity Detection Method)

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

2006

by

Suresh Babu Chennagowni

Bachelor of Engineering, Osmania University, Hyderabad, India, 2002

Committee Chair: Dr. Randall J. Allemang


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ABSTRACT

Almost all practical systems are non-linear to some extent with the non-linearity being

caused by one or a combination of factors. If the system is non-linear, errors are

introduced in the data analysis and are observed during the modal tests of a structure. For

example, high forcing levels may cause the frequency response function estimates to

show non-coherent behavior over certain frequency bands. A new coherence function

(Combined Coherence) provides a method to separate the effects of structural non-

linearities and the digital signal processing errors.Thomas Roscher [1] applied the combined coherence formulation to theoretical data

generated from a lumped parameter (M, K, C) with static coupling. The results showed

improvement in the combined coherence function over the ordinary coherence, but when

Doug Coombs [2] applied combined coherence to a real world structure, it did not show

improvement. In this thesis, as an extension of previous work, study is done on

theoretical data generated from a lumped mass model with dynamic coupling. The effects

of mass distribution, spatial density, forcing level, location of forcing function, path of

energy and the dynamic coupling on the combined coherence are studied. The testing

cases include SIMO and MIMO cases for a MDOF simulink model with a cubic

hardening type of nonlinearity applied at different locations. Combined Coherence is

calculated for a non-linear model and effects on the combined coherence are studied for

the following cases.

Effect of varying the force input

Effect of dynamic coupling


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Effect of location of input and path of energy

Effect of mass distribution

Effect of spatial density of masses

Effect of scaling of motions


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ACKNOWLEDGEMENTS

I would like to express my gratitude towards all who were involved in the

completion of this thesis. First of all, I would like to thank Dr. Randall Allemang for

providing me with this opportunity to work under his guidance. I would also like to thank

Dr. Allyn Phillips who helped me out through the research. Dr. Randy and Dr. Allyn

have always been a source of support and encouragement. Their inputs and advice have

contributed substantially to the completion of my work.

I would like to thank Dr. Ronald Huston for serving as member on my thesis

committee.

I express my thanks to my colleagues at Structural Dynamics Research

Laboratory for their helpful discussions in various matters during the course of this work.

I would also like to thank all those who helped me with the style and grammar of the

writing.

Finally, I would like to thank my parents and family for constantly supporting my

academic pursuits.


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I

Table of Contents

1. Introduction 1

2. Theoretical Background 3

2.1 Linear Model..3

2.2 SDOF Mechanical System.....4

2.3 Frequency Response Function...5

2.4 Theory of Ordinary and Multiple Coherence....7

2.5 Excitation Techniques8

2.6 Overview of Non-Linearity9

2.7 Non-Linearity Detection Techniques...12

3. Non-Linear Detection Method (Combined Coherence Function) ...15

3.1 Theory of Combined Coherence..15

3.2 Development of Combined Coherence17

3.3 Application of Combined Coherence to Rocher Analytical Model.18

3.4 Application of Combined Coherence to Real world system22

3.5 Theoretical Model used to study the Combined Coherence....24

4. Application of Combined coherence to Analytical Model 29

4.1 Effect of Varying the Force Input....29

4.2 SIMO situations with Dynamic Coupling33

4.3 MIMO situations with Dynamic Coupling..42

4.4 Effect of Dynamic Coupling on Combined Coherence...51

4.5 Effect of Location of Input and Path of Energy on Combined Coherence..59

4.6 Effect of Mass Distribution on Combined Coherence.66


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II

4.7 Effect of Spatial Density of Masses on Combined Coherence74

4.8 Effect of Scaling of Motions of DOF on Combined Coherence..82

5. Conclusions ...89

6. Future Work .....92

7. References .....93

8. Appendix ..95

8.1 Simulink Model when the non-linearity is between DOFs 1 and 2....95

8.2 Simulink Model when the non-linearity is between DOFs 1 and 3.96



8.5 Simulink Model when the non-linearity is between DOFs 2 and 4.99

8.6 Simulink Model when the non-linearity is between DOFs 3 and 4...100


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III

LIST OF FIGURES

Figure 2-1: SDOF4

Figure 2-2: Single Input System..6

Figure 2-3: Cubic Stiffness11

Figure 2-4: FRF and Coherence of nonlinear system....12

Figure 3-1: a) Lumped mass structure system b) Force system.....15

Figure 3-2: 2 DOF model with rotary inertia.....16

Figure 3-3: Roscher Theoretical Model.....19

Figure 3-4: FRFs and Coherences for Case 1...20

Figure 3-5: Comparison of Coherence and CCOH for Case 1......21

Figure 3-6: FRFs and Coherences for Case 2...21

Figure 3-7: Comparison of Coherence and MCCOH for Case 2...22

Figure 3-8: Line diagram of Doug Coombs model ...23

Figure 3-9: Theoretical 4 DOF lumped model..............................................................25

Figure 3-10: Comparison of Analytical and Simulation Results...28

Figure 4-1: FRFs, Coherences and MCCOH for Case 4.1.1........................................31

Figure 4-2: FRFs, Coherences and MCCOH for Case 4.1.2....32

Figure 4-3: FRFs and Coherences of Case 4.2.1..... .. . ..... .35

Figure 4-4: Coherence and CCOH of Case 4.2.1.. ... ....36






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IV


Figure 4-10: FRFs, Coherences and MCCOH of Case 4.3.1...45

Figure 4-11: FRFs and Coherences of Case 4.3.246




Figure 4-15: FRFs, Coherences and MCCOH of Case 4.3.5.. 49


Figure 4-17: FRFs, Coherences and MCCOH of Case 4.4.1...53Figure 4-18: FRFs, Coherences and MCCOH of Case 4.4.2...54















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V









Figure 4-39: FRFs, Coherences and MCCOH of Case 4.8.1...84Figure 4-40: FRFs, Coherences and MCCOH of Case 4.8.2...85





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VI

LIST OF TABLES

Table 1-1: Sample test cases of combined coherence applied to Roscher model..19

Table 4-1: MIMO situations for different force exciting levels30

Table 4-2: System with Dynamic Coupling SIMO situations...33

Table 4-3: MIMO situations of system with Dynamic Coupling..43

Table 4-4: MIMO situations of system with no Dynamic Coupling.51

Table 4-5: MIMO situations to study effect of Path of Energy.59

Table 4-6: MIMO situations to study effect of Mass Distribution66

Table 4-7: MIMO situations to study effect of Spatial Densities of Masses.74

Table 4-8: MIMO situations to study effect of Scaling of Motions..82


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VII

NOMENCLATURE

NOTATION

m MassM Mass Matrix

k Stiffness

K Stiffness Matrix

c Viscous Damping

C Damping Matrix

q Input Location

p Output Location

H pq Frequency Response Function at output p and input q

)(..

t x Acceleration

)(.

t x Velocity

)(t x Displacement

F Force input in frequency domain

)(t f Force input in time domain

2,1 Eigen Value

Circular Frequency

Noise on output

Noise on input

X`( ) Measured input of the system

F`() Measured output of the system

)(2 pq Coherence Function

)( qpGFX Cross Power Spectrum of Input q and output p

)( qqGFF Input Power Spectrum at input q

)( ppGXX Output Power Spectrum at output p

Non-linear Scaling Factor


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VIII

j Rotary Inertia

r Radius

t Sample time

ABBREVIATION

DOF Degree of Freedom

SIMO Single Input Multiple Output

MIMO Multiple Input Multiple Output

SDOF Single Degree of Freedom

COH Coherence Function

MCOH Multiple Coherence Function

CCOH Combined Coherence

MCCOH Multiple Combined Coherence


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1

1. Introduction

Experimental modal analysis is often used for checking the accuracy of an analytical

approach such as finite element analysis and verification/correction of the results of the

analytical approach (model updating). During the modal analysis procedure, there are

four basic assumptions (linearity, time invariance, reciprocity and observability) made

concerning any structure. Because these assumptions are assumed to be valid, errors

accumulate at the modal parameter estimation phase. Among these errors are the errors

due to nonlinearities in the structure and the errors due to digital signal processing. The

errors due to nonlinearities are visible in the measured data as slight distortions in the

frequency response function (FRF) plots, but they are also responsible for significant

discrepancies in the modal analysis process. Some of the algorithms used to extract

modal parameters can be surprisingly sensitive to the small deviations (from linear

characteristics), which accompany the presence of slightly nonlinear elements in the

structures. Understanding these effects and detecting their presence, means that

alternative test procedures can be used so that the nonlinear effects are not only prevented

from contaminating the measurement and analysis processes but can actually be

quantified and included in the model. In this thesis, a further study is done on the

Combined Coherence Method, which is a frequency domain method of detecting

structural nonlinearities. It is the method of detecting the presence of nonlinearities

between degrees of freedom by separating the errors due to digital signal processing and

nonlinearities. It is a quick and efficient method to detect structural nonlinearities

between the degrees of freedom from the data taken during the modal test. Thomas


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Roscher applied combined coherence to the theoretical data generated from lumped mass

model [1] and Doug Coombs applied combined coherence to the data measured from a

practical nonlinear structure [2]. Combined coherence was able to locate the

nonlinearities in the first case for theoretical lumped model whereas in the second case it

could not locate the nonlinearities spatially. In this thesis, further study is done on

theoretical data generated from a lumped model with dynamic coupling similar to the real

world system used by Doug Coombs. A study is done on how different parameters such

as mass distribution, spatial density, forcing level, location of forcing function, path of

energy and the dynamic coupling effects the combined coherence. The second chapter inthis thesis gives an introduction to non-linear vibration and methods in detecting the

nonlinearities. Chapter 3 gives introduction to combined coherence method, its derivation

and the previous work of Roscher and Coombs. In Chapter 4 combined coherence is

applied to a non-linear model and effects on combined coherence are discussed for the

following cases.

Effect of varying the force input.

Effect of dynamic coupling

Effect of location of input and path of energy

Effect of mass distribution

Effect of spatial density of masses

Effect of scaling of motionsSummary and conclusions are given in Chapter 5.


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3

2. Theoretical Background

2.1 Linear Systems

A clear understanding of the concept of a degree of freedom is required for understanding

the concept of modal analysis. The number of degrees of freedom is the minimum

number of coordinates required to specify completely the motion of a mechanical system .

There exist six degrees of freedom at each point, the motion in each direction and the

rotational motion of each axis. A mechanical system has an infinite number of degrees of

freedom, because the system is continuous. The observed degrees of freedom are in

reality, of course, a finite number, limited by different physical causes. The following

parameters reduce the effective number of degrees of freedom: the frequency range of

interest and physical points of interest.

There are four assumptions made during the modal analysis procedure [11]. The first

basic assumption is that the structure is linear. This means that the structure obeys the

superposition principle, which states that the response of the system to a combination of

forces applied simultaneously is equal to the sum of the responses due to the individual

forces. The second assumption is that the structure is time invariant. This means that the

properties of the system such as mass, stiffness and damping do not change with time

(i.e., they remain unchanged for any two different testing times). The third assumption is

that the structure obeys Maxwell reciprocity. This principle states that the response of the

function at a degree of freedom q due to the input at p is equal to the response at p

due to the input at q i.e., H pq = H qp. The fourth assumption is that the structure is


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4

observable. The response points are chosen such that the complete structure is observed.

For example, structures with loose components, which have degrees of freedom that

cannot be measured, are not completely observable.

2.2 SDOF Mechanical System

Simple systems can be modeled as a mass-damper-spring system at a single point in a

single direction. These are referred as single degree of freedom (SDOF) systems. A

SDOF mechanical system is described by Newtons equation as shown in equation

below.

)()()()(...

t f t kxt xct xm =++ (2.1)

Figure 2-1: SDOF


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This equation has two solutions, a transient solution and a steady state solution. The

equation can be solved using the Laplace transform, assuming initial conditions to be

zero. The equation of the above system can be written as

Ms 2 X(s) + csX(s) + kX(s) = F(s) (2.2)

where: s is a complex-valued frequency variable (Laplace variable).

The above equation can be rewritten as

1/(ms 2+cs+k)=X(s)/F(s) = H(s) (2.3)

and finally the homogeneous equation is solved using f (t) = 0 which yields the natural

frequencies as

mk

mc

mc = 22,1 )2

(2

(2.4)

2.3 Frequency Response Function

The relation between the input to the system and its response is determined by the

frequency response of the system, which is a characteristic feature of the system. The

response of a system to an output is completely determined by its frequency response

function.


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6

Consider a single input system shown below:

Figure 2-2: Single Input System

Ideally the frequency response of the system is calculated by

H () = X ( )/F ( ) (2.5)

where: H ( ) = Frequency response function of the system

F () = Frequency Domain information of the input signal with no noise on signal

X ( ) = Frequency Domain information of the output signal with no noise on the

signal

But, due to measurement errors, the actual frequency response function is given by

X` ( ) - = (F` ( ) -) H ( ) (2.6)

where: = Noise on the input signal.

= Noise on the output signal.

F` () = Measured input of the system.

X` ( ) = Measured output of the system.


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The three most common types of frequency response algorithms are based on the least

squares model: the H 1 algorithm which minimizes the noise on the output, the H 2

algorithm which minimizes the noise on the input and H v algorithm which minimizes the

noise on both the input and output. In this thesis, the H 1 algorithm is used.

2.4 THEORY OF COHERENCE

Ordinary and Multiple Coherence :The ordinary coherence function (COH) is computed as [11]:

)()(

)()(

)()(

|)(|)()(

22

ppqq

qp pq

ppqq

pq pq pq GXX GFF

GFX GXF

GXX GFF

GXF COH === (2.7)

This function is frequency dependent and is a real value between zero and one. The value

1 indicates that the measured response power is totally correlated with the measured input

power. The value zero indicates the output is totally correlated with the sources other

than the measured input. A coherence value less than unity at any frequency is due to

variance and bias errors. The low coherence due to a variance error like random noise can

be significant provided sufficient averaging had occurred. Since coherence is a statistical

indicator, the more ensembles averaged, the more reliable is the result (smaller standard

deviation). The bias errors can be broadly classified into two categories, digital signal

processing errors and the errors due to nonlinearities. All errors causing drops in the

coherence fall into one of these two categories. The frequencies where the coherence is

low are often the same frequencies where the FRF is maxima in magnitude (resonance) or

minima in magnitude (anti-resonance), which may be an indication of leakage. The drop


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8

in coherence at any other frequency is more clearly due to other errors such as noise or

nonlinearities. Multiple inputs are often desired during testing so that the energy is more

evenly distributed throughout a structure and as a result the vibratory amplitudes across

the structure will be more uniform, with a consequent decrease in the effect of

nonlinearities. Coherence is not an appropriate measure of linear dependency between

input and output when there is more than one input. The multiple coherence function

(MCOH) that determines the linear dependency of input and output is computed as [12]

= =

=i i N

q

N

t pp

pt qt pq p GXX

H GFF H MCOH

1 1

*

)(

)()()()(

(2.6)

The value of MCOH varies between zero and one. A value of one indicates an output is

correlated with all known inputs, while a value less than unity indicates unknown

contributions such as measurement noise and nonlinearities.

2.5 Excitation Techniques

For a linear system the dynamic characteristics will not vary according to the choice of

the excitation technique used to measure them. However, the effects of most kinds of

nonlinearities, encountered in structural dynamics are generally found to vary with the

external excitation. Hence, the first problem of a nonlinearity investigation is to decide

the type of excitation so that the nonlinearity is exposed and identified. There are

currently many types of excitation methods widely used in vibration study practice.

These excitation techniques are broadly classified as sinusoidal, transient and random

excitation. Sinusoidal excitation is widely regarded as the best excitation technique for

the identification of nonlinearities. The advantage of a sinusoidal excitation is, it is easy


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9

to accurately control the input signal level and hence, enables a high input force to be fed

into the structure. However, the drawback of this type of excitation is, it is relatively slow

compared to many of the other techniques used in practice. Since the excitation is

performed frequency by frequency and at each step, time is required for the system to

settle to its steady-state value, sinusoidal methods are very time consuming. On the other

hand, with the random excitation technique, the system can be excited at every frequency

simultaneously within the range of interest. This wide frequency band excitation enables

it to be much faster than the sinusoidal excitation. Also, random excitation in general

linearizes the nonlinear structure due to randomness of input force amplitude. Thistechnique is the best match for modal analysis, as most of the modal parameter estimation

methods are based on linearity. Due to the above stated factors, random excitation is very

commonly used in actual testing conditions. Hence, test engineers need a nonlinear

detection method that is compatible with normal modal analysis methods employing

random excitation. Therefore, in this thesis, random excitation is used in detecting the

structural nonlinearities using combined coherence method.

2.6 Overview of Non-linearity

Most practical engineering structures exhibit a certain degree of nonlinearity due to

nonlinear dynamic characteristics of structural joints, nonlinear boundary conditions and

nonlinear material properties. For practical purposes, in many cases, they are regarded as

linear structures because the degree of nonlinearity is small and therefore, insignificant in

the response range of interest. Most theories, upon which structural dynamic analysis is


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10

founded, rely heavily on this assumption of linearity (superposition principle). The

superposition principle states that, the deflection due to two or more simultaneously

applied loads is equal to the sum of the deflections caused, when the loads are applied

individually. But for some cases, the effect of nonlinearity may become so significant

that it has to be taken into account in the analysis of dynamic characteristics of the

structure. The present thesis focuses on the location of nonlinearity based on the

measurement of input and output using combined coherence function.

Nonlinear structures are often divided into three main types: zero memory, finite memoryand infinite memory systems. The zero memory type of system is the most simple of the

three types, as it only applies the nonlinear operator at system input, whereas the infinite

memory type of system applies nonlinearity to the system response as well. A typical

infinite memory type of system for a MDOF system can be written as [12] [3]

[M ] x(t ) +[ C ] x(t ) +[ K ] x(t ) +[ K n] x3(t ) = f (t ) (2.7)

The common types of nonlinearities are displacement type nonlinearities (hardening,

softening, hardening/softening and dead zone) and velocity related nonlinearities

(quadratic damping, softening/hardening damping and coulomb friction). In this study

the effect of a cubic stiffness non-linearity on the combined coherence is studied by

applying it to the MDOF system. The mathematical model of a cubic stiffness element

can be expressed as

f(x) = k( x+ x3) (2.8)


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where the coefficient k represents spring stiffness, and the coefficient represents the

degree of nonlinearity. The Figure 2.3 below represents both the linear and the nonlinear

behavior of a cubic stiffness element. It can be seen that the overall stiffness changes with

the displacement x, while the stiffness coefficients k and remain constant.

Figure 2-3: Cubic Stiffness

Cubic stiffness is applied to the simulation model used in this study to observe its

nonlinear characteristics by exciting the system at five forcing levels. The FRF and

coherences of a nonlinear system can be seen in the Figure 2.4. It can be seen from the

FRF and COH function plots that the anti-resonances and resonances are changed as the

excitation force level changes and thus it can be assumed that the system is non-linear.


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Figure 2-4: FRF and Coherence of nonlinear system

2.7 Non-linear detection techniques

A linear time-invariant system is relatively well understood and theoretically well

developed. The same is not true for the case of a nonlinear system. In most of the

situations, it is necessary to first detect the presence of nonlinearity. A lot of work is done

in this direction and quite a number of procedures are suggested. A brief review of some

of the detection methods is presented here.

M. Simon and G. R. Tomlinson [4] proposed a Hilbert transform technique to detect and

quantify structural nonlinearities. The basis that the Hilbert transform technique can be

used to identify nonlinearity is due to the fact that for a linear structure, the real and

imaginary parts of a measured FRF constitute a Hilbert transform pair, whereas for the

FRF of a nonlinear structure, the Hilbert transform relationships do not hold. By

calculating the Hilbert transform of the real part (or the imaginary part) of a measured


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FRF and comparing it with the corresponding imaginary part (or real part), the existence

of nonlinearity can be identified based on the difference of the transform pair.

M. Mertens, H.Vander, P. Vanherck, R. Snoeys [6] proposed a complex stiffness method,

which is based on the mapping of different estimates of stiffness and damping for each

measured frequency as a function of magnitude of displacement and the velocity

respectively. The equivalent stiffness and damping of a linear system are constant while

for a nonlinear system stiffness and/or damping vary. This method gives an idea of

degree and type of nonlinearity.

He J. and D.J. Ewins [7] proposed Inverse Receptance method in which nonlinearity is

detected as whether it exists in the stiffness or damping, by displaying the FRF data in

inverse form. For a linear system a plot of real part of inverse FRF against 2 and the

imaginary part against yields a straight lines while for non-linear systems the plots are

not straight lines. The nonlinearities associated with stiffness show up in the real part

while in the imaginary part the nonlinearities due to damping show up.

Vanhoenacker K., T. Dobrowiecki, J. Schouskens [8] proposed a multisine excitation

method to detect nonlinearities. In this method, the system is excited at only a few chosen

set of frequency lines. It is shown that by exciting the system only at a selected set of

frequency lines, the even nonlinear disturbances can be determined at the even frequency

lines while at unexcited odd frequency lines the odd nonlinear distortions can be

determined.


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Kim W-J and Y-S Park [10] proposed non-causal power ratio (NPR) method. It is a

causality check method that quantifies the non-linearity. The NPR value grows with the

increase in nonlinearity and is a function of excitation amplitude. NPR function detects

the non-linearity and also the type of nonlinearity by examining the variation of the NPR

values with excitation force. The advantages of this method are

1. It takes less computation time

2. This method does not require prior information of the system

3.

It can be applied without any limitations to the nonlinearities


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3. Non-linear Detection Method (Combined Coherence

Function)

In this chapter the theory of the combined coherence is discussed and mathematical

equations for both the ordinary and multiple combined coherence (MCCOH) are derived.

3.1 THEORY OF COMBINED COHERENCE

In general structures are represented by assuming lumped masses as node elements with

mass and no stiffness, and are connected by stiffness and damping terms. The distribution

of mass is important in dynamic analysis. The general representation of the structure and

the force system is shown in figure below [1].

Figure 3-1: a) Lumped mass structure system b) Force system

At any node point if Newtons law is applied and an equation of motion is developed,

then the acceleration is the sum of both the internal force terms caused by stiffness and


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damping terms, and the external force terms. Considering a 2 DOF model shown in the

Figure 3-2, the equations of motion can be written as

Figure 3-2: 2 DOF model with rotary inertia

121212012012

.

121

.

201201

..

22

22

..

12

221 )()()/()/( f xk xk k k xc xccc xr j xr jm +++++++=+

(3.1)

2112220121

.

122

.

2012

..

12

22

..

22

222 )()()/()/( f xk xk k xc xcc xr j xr jm +++++=+ (3.2)

When motions of two degrees of freedom are combined under the condition of equal

mass i.e., (m 1 = m 2), the contribution of motions due to internal forces between degrees

of freedom will disappear. The equation obtained by combining the motions of DOF is

m f f xk xk m xc xcm x x /][][/1][/1 211012201.

012

.

20

..

2

..

1 +++=+ (3.3)

If a coherence function is calculated for a virtual coordinate created by combining the

motions between these DOFs, the drops in coherence due to non-linearity would go

away but the low coherence values due to digital signal processing errors would not

improve. The critical condition for this method is the equality of masses between the

degrees of freedom between which the motions are combined. If the masses are not equal,


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the detection method can still be applied, if the motions are scaled according to the mass

ratio. In this thesis, a test case is run to see if scaling the masses would improve the

combined coherence in detecting the nonlinearities.

3.2 Development of Ordinary and Multiple Combined Coherence functions [2]:

The standard equation for ordinary coherence function is given by

)()(

)()(

)()(

|)(|)()(

22

ppqq

qp pq

ppqq

pq pq pq GXX GFF

GFX GXF

GXX GFF

GXF COH === (3.4)

Since the CCOH function is based on the sum of the motion between two DOFs.

Substituting X p + X r for X p we get

**

**

)( ))((

)()(

pr pr qq

q pr q pr q pr X X X X F F

F X X F X X CCOH

++++

= (3.5)

)(

))((*****

***

)( p pr p pr r r qq

q pqr q pqr q pr X X X X X X X X F F

F X F X F X F X CCOH

+++++

= (3.6)

)(

|| 2

)( pp pr rprr qq

rq pqq pr GXX GXX GXX GXX GFF

GXF GXF CCOH

++++

= (3.7)

The standard equation for multiple coherence function is given by [11]

= =

=i i N

q

N

t pp

pt qt pq p GXX

H GFF H MCOH

1 1

*

)(

)()()()(

(3.8)

after following the similar steps as for CCOH, MCCOH can be derived as

= =

+ +++++

=i i N

s

N

t rr rp pr pp

rt pt qt rs psr p GXX GXX GXX GXX

H H GFF H H MCCOH

1 1 )()()()(

*))()()(())()(()(

(3.9)


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3.3 Applying CCOH formulation to the Roscher Theoretical Model

Roscher applied the combined coherence function to the data generated from the

theoretical lumped parameter (M, K, C) model with static coupling. The model used by

the Roscher is shown in the Figure 3.3. Roscher had applied the combined coherence

formulation for various testing conditions for different types of displacement and velocity

related nonlinearities. There was complete improvement in the combined coherence for

some of the cases and in some cases, for some frequency ranges, the combined coherence

did not show improvement. Only a few cases were tested for different kinds of nonlinearities. The mass distribution, which is a critical parameter for combined

coherence in determining the nonlinearities, was not extensively studied. In this thesis, a

study is done on how the mass distribution affects the combined coherence by simulating

cases with mass equality between the DOFs.


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Figure 3-3: Roscher Theoretical Model

A few cases simulated by Roscher are shown in the Table 1-1. As it can be seen from the

combined coherence (CCOH) plot, it is not improved completely. There is a drop in

CCOH in the range of 16 to 18 Hz and this can be due to the nonlinear motion entering

through other paths. This drop in CCOH still needs to be studied, before CCOH can be

applied to any real world structure.

Case Location of

Non-Linearity

Force M 1, M 2, M3

and M 4 (Kg)

1 1 and 3 F 3 = 30 N 12, 7, 9 and 14 50000

2 1 and 3 F 1 = 50 N and F 3 = 50 N 12, 7, 9 and 14 50000

Table 1-1: Sample test cases of combined coherence applied to Roscher model


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Figure 3-4: FRF and Coherence for Case 1


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Figure 3-5: Comparison of Coherence and CCOH for Case 1

Figure 3-6: FRF and Coherence for Case 2


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Figure 3-7: Comparison of Coherence and MCCOH for Case 2

3.4 Application of CCOH to Real world structure

Doug Coombs applied combined coherence to a real world structure. The system

consisted of an H-frame with (2x6x0.25) with another square frame (2x2x0.125) steel

tubing. These two frames were connected at 4 discrete points giving various options for

linear/non-linear conditions. Two shakers were connected in a skew direction at an angle

of 45 0 in order to get energy in all three directions. The following testing scenarios were

examined to check the ability of combined coherence to spatially locate nonlinearities.

The line diagram of testing structure is shown below in Figure 3.8.


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Figure 3-8: Line diagram of Doug Coombs model

Different testing cases such as

Cases with and without leakage errors Varying the number of spectral averages

Reducing the number of nonlinear paths

Varying the input force locations

Changing the spatial density of the responses

on combined coherence were studied. For a nominal linear connection between the

connections, the improvement in combined coherence was near the resonances instead at

the anti-resonances raising a question if leakage is affecting the combined coherence. For

many of the testing cases the improvement in the combined coherence was small when

compared to multiple coherence. In one case, when the combined coherence is examined


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by changing the location of input to the square frame, the previous large improvements in

the combined coherence away from the input locations were not seen.

In this thesis, a study is done on a theoretical model with dynamic coupling similar to the

real world system used by Coombs to study the behavior of combined coherence for

various testing conditions.

3.5 Theoretical Model used to study Combined Coherence

A 4 DOF model with rotary inertia is used to study combined coherence. Figure 3.9shows a near real time 4 DOF model, similar to that used by Doug Coombs, which is

dynamically coupled. The mi, cij, and k ij variables denote the mass, linear viscous

damping, and linear stiffness parameters; the f i variables denote the applied external

forces. The independent coordinates, xi, are defined with respect to an absolute

coordinate system. The idea of this type of model is to study the effect on combined

coherence when the path of energy is across the boundary and to get dynamic coupling

between the degrees of freedom. As can be seen from the equations of motion, degrees of

freedom 1, 2 and 3, 4 are dynamically coupled.


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Figure 3-9: Theoretical 4 DOF lumped model

The equations of motion of the model are expressed in terms of a set of coordinates that

are defined with respect to the unique static equilibrium point of the linear system:


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=

+++++++++

+

+++++++++

+

++

++

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

)()(

)(

)(

)(

)(

//00

//00

00//

00//

4

3

2

1

4

3

2

1

04342414342414

343423132313

242324231212

14131214131201

4

3

2

1

04342414342414

343423132313

242324231212

14131214131201

4

3

2

1

4444

444

444

4443

2222

222

222

2221

t f

t f

t f

t f

t x

t x

t x

t x

k k k k k k k

k k k k k k

k k k k k k

k k k k k k k

t x

t x

t xt x

ccccccc

cccccc

ccccccccccccc

t x

t x

t x

t x

r jmr j

r jr jm

r jmr j

r jr jm

&&&&

&&&&&&&&

(3.10)

Frequency response function and coherence are evaluated using the dynamic stiffness

method, where the FRF matrix was computed by inverting the system impedance matrix

at each frequency of interest. This provided a means of checking the simulink model,

which used time domain integration to obtain the responses. It can be seen from the FRF

plots below that the dynamic stiffness method results matched the results obtained

through simulink model perfectly. Figure 3-10 below shows the FRFs of all 4 DOFs for

an input applied at DOF 1.


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Figure 3-10: Comparison of Analytical and Simulation Results


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4. Application of Combined Coherence to Analytical model

In this chapter, the simulation results obtained for various cases from a 4 DOF system

MATLAB Simulink model are presented. The 5 th order fixed-step Dormand-Prince ODE

method is used. The sample time, t, is set at 0.005 seconds and 2 16 time steps are

computed, resulting in 327.68 seconds of signal for each simulation. Data is processed in

the Fourier frequency domain and FRFs are determined for each simulation using the H 1

FRF calculation [11] with F jk ( ) as the input and X ik ( ) as the output. The H 1 calculation

seeks to minimize noise on the output.

4.1 Effects of Varying the Force Input

In this section, simulations are done to verify whether the system is linear or non-linear,

by exciting the system with five different force-exciting levels. Further, the effect on the

combined coherence in detecting the structural non-linearities, for different exciting

levels is studied. The following MIMO cases shown in Table 4-1 are simulated.


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Table 4-1: MIMO situations for different force exciting levels

It can be seen from the FRF and coherence function plots (Figures 4-1 and 4-2), that the

anti-resonances and resonances are changed as the excitation force level changes. Thus, it

can be assumed that the system is non-linear. The drops in coherence can be attributed to

digital signal processing errors as well as to the non-linearity. For example, from the

coherence plot (coherence 1) of case 4.1.1, it can seen that the drop in coherence value at

3 Hz is due to digital signal processing error and drops at 8 11 Hz, 14 19 Hz, 21 25

Hz are due to non-linear motion.

It can be seen from the MCCOH of case 4.1.1 (Figure 4-1), at lower forcing levels the

MCCOH showed improvement while at higher forcing levels it still showed improvement

but with more distortion. The distortion at a forcing level of 70 N is more when compared

to a forcing level of 30 N. Hence, it can be concluded that the nature of the improvement

in the MCCOH is inversely proportional to the forcing level, i.e., at lower force levels the

Case Location of

Non-Linearity

Force (Increased in steps

of 10 N)

M 1, M 2, M 3 &

M 4 (Kg)

4.1.1 1 & 3 F 1 = 30 to 70 N &

F3 = 20 to 60 N

12, 10, 8 &14 100000

4.1.2 2 & 4 F 1 = 30 to 70 N &

F3 = 20 to 60 N

12, 10, 8 & 14 100000


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ability of the MCCOH to detect non-linearities is greater when compared to higher force

levels.

Figure 4-1: FRFs, Coherences and MCCOH for Case 4.1.1


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4.2 SIMO Situations for a system with Dynamic Coupling

In this section, the following cases shown in Table 4-2 are simulated for a system close to

the real world testing conditions where the mass distribution between the DOF is uneven

and also, there is mass coupling between the degrees of freedom.

Case Location of

Non-Linearity

Force M1 M2 M3 M4

4.2.1 1 and 2 F1 = 50 N 12 10 8 14 100000

4.2.2 1 and 3 F3 = 50 N 12 10 8 14 100000

4.2.3 1 and 4 F1 = 50 N 12 10 8 14 100000

4.2.4 2 and 3 F3 = 50 N 12 10 8 14 100000

4.2.5 2 and 4 F2 = 50 N 12 10 8 14 100000

4.2.6 3 and 4 F4 = 50 N 12 10 8 14 100000

Table 4-2: System with Dynamic Coupling SIMO situations

It can be seen from the FRF plots that there are distortions at both resonances and anti-

resonances. From the coherence function plots, it can be seen that the drops in the

coherence value can be attributed to digital signal processing errors as well as to the non-

linearity. In the coherence function plot of Case 4.2.1, one could see the drops between 5

to 6 Hz and 10 to 14 Hz which are not associated with either resonance or anti-resonance

but are due to non-linearity. Also, one could see the drops at 3.5, 6 and 8 Hz that are at

resonances or anti-resonances and the drops in the higher frequency range (above 20 Hz).


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As a next step, the CCOH for responses 1 and 2 is compared with ordinary coherence. It

can be seen that the drops in the higher frequency range and the drops associated with

non-linearity (5-6 Hz) are completely eliminated and one could only see the drops at

resonances. The drop in coherence over the frequency range of 10 to 14 Hz is not

completely eliminated but has shown improvement over the ordinary coherence. The

complete clear up of the CCOH is not seen because of one or a combination of three

factors:

1.

Dynamic coupling2. Due to the non-linear motion entering the system from other paths

3. Mass difference between the DOFs between which the CCOH has been

computed

Similar results have been observed for all other combination of cases i.e., when the non-

linearity is located between DOFs 1 and 3, 1 and 4, 2 and 3, 2 and 4, and 3 and 4 that

there are drops in coherences due to non-linearity, leakage and at higher frequencies.

CCOH has shown improvement at anti-resonances but not at resonances and complete

clear up the CCOH has not been registered.


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Figure 4-4: Coherence and CCOH of Case 4.2.1


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Figure 4-5: FRFs, Coherence and CCOH for Case 4.2.2


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Figure 4-7: FRFs, Coherences and CCOH for Case 4.2.4


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4.3 MIMO Situations for a system with Dynamic Coupling

In this section, the MIMO situations shown in the Table 4-3 below are simulated. The

testing conditions, severity of non-linearity, locations of non-linearity, mass and all other

conditions are similar to that of the previous case (4.2) except for the input given at two

DOFs between which the non-linearity is located. Multiple inputs determine if the

structure responds in a non-linear regime. More often, most modal analysis procedures

involve the application of multiple inputs in order to get more uniform energydistribution. Whereas, the SIMO situation induces non-linear behavior in the vicinity of

the input location and structure might not be excited well at remote points, therefore,

further study is done only for MIMO situations.


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Case Location of

Non-Linearity

Force M1 M2 M3 M4

4.3.1 1 & 2 F1 = 50 N &

F2 = 40 N

12 10 8 14 100000

4.3.2 1 & 3 F1 = 50 &

F3 = 40N

12 10 8 14 100000

4.3.3 1 & 4 F1 = 50N &

F4 = 40 N

12 10 8 14 100000

4.3.4 2 & 3 F2 = 50 N &

F3 = 40 N

12 10 8 14 100000

4.3.5 2 & 4 F2 = 50 N &

F4 = 40 N

12 10 8 14 100000

4.3.6 3 & 4 F3 = 50 N & F4 =

40 N

12 10 8 14 100000

Table 4-3: MIMO situations of system with Dynamic Coupling

The FRF estimation, the MCOH and the MCCOH obtained are as shown in the Figures

(4-10 to 4-16) below. As seen from the plots below, the results obtained in this case are

similar to that of the previous case. There are frequency shifts in the FRFs at resonances

and anti-resonances. Also, there are low coherence values due to non-linearity and digital

signal processing errors, like leakage at resonances and anti-resonances. One can see the

complete improvement of the MCCOH values at higher frequencies and at anti-


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resonances. At some frequencies, the MCCOH has not registered complete improvement.

In the coherence plot of Case 4.3.1, one can see the drop in coherence over the frequency

range of 5 7 Hz, which is at anti-resonance, is completely improved. The drop over the

frequency range of 10 14 Hz, which is near the resonance, is not improved. It can be

concluded from this observation that the MCCOH is sensitive to anti-resonance.

Similar results have been observed for all other combination of cases i.e., when the non-

linearity is located between DOFs 1 and 3, 1 and 4, 2 and 3, 2 and 4, and 3 and 4 that

there are drops in coherences due to non-linearity, leakage and at higher frequencies.MCCOH has shown improvement at anti-resonances, but not at resonances, and complete

improvement of the MCCOH has not been registered.

It can be seen from the figures that the MCCOH has shown improvement over the

MCOH in all of the above situations but for some frequency ranges the complete

improvement in the MCCOH is not accomplished. As mentioned in the previous SIMO

situations, this can be due to one or combinations of the three factors:

1. Dynamic coupling.

2. Due to the non-linear motion entering the system from other paths.

3. Mass difference between the DOFs between which the MCCOH has been

computed.

But, it is not clear from this case whether the incomplete improvement in the MCCOH is

due to either dynamic coupling or due to the mass difference or because of the non-linear

motion entering from other paths.


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Figure 4-10: FRFs, Coherences and MCCOH of Case 4.3.1


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Figure 4-12: Coherence and MCCOH of Case 4.3.2


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4.4 Effect of Dynamic Coupling on Combined Coherence

In the previous cases, it is seen that the complete improvement of the combined

coherence is not observed and reasons for it are attributed to dynamic coupling, mass

difference and/or path of energy. In this section, the following cases are simulated to

study the effect of dynamic coupling on the MCCOH. The rotary inertia term has been

reduced by 100 times (i.e., making the dynamic coupling between DOFs negligible.)

Sl. No. Location of

Non-Linearity

Force M 1, M 2, M 3

& M 4

J 2, J 4

4.4.1 1 and 2 F 1 = 50 N &

F2 = 40 N

12, 10, 8 &14 J 2=M*R 22

/200

J4=M*R 42/200

100000

4.4.2 1 and 3 F 1 = 50 &

F3 = 40N

12, 10, 8 & 14 J 2=M*R 22/200

J4=M*R 42/200

100000

4.4.3 1 and 4 F 1 = 50N &

F4 = 40 N

12, 10, 8 & 14 J 2=M*R 22/200

J4=M*R 42/200

100000

4.4.4 2 and 3 F 2 = 50 N &

F3 = 40 N

12, 10, 8 & 14 J 2=M*R 22/200

J4=M*R 42/200

100000

4.4.5 2 and 4 F 2 = 50 N &

F4 = 40 N

12, 10, 8 & 14 J 2=M*R 22/200

J4=M*R 42/200

100000

4.4.6 3 and 4 F 3 = 50 N &

F4 = 40 N

12, 10, 8 & 14 J 2=M*R 22/200

J4=M*R 42

/200

100000

Table 4-4: MIMO situations of system with no Dynamic Coupling


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The FRF estimation, the MCOH and the MCCOH obtained are shown in Figures (4-17 to

4-22) below. It is concluded from last case that dynamic coupling is one of the reasons

why the MCCOH has not shown complete improvement. Therefore, it is expected from

this case, that the MCCOH will show improvement, as the dynamic coupling is made

negligible. It can be seen from the MCCOH plot of Case 4.4.1 the improvement in

MCCOH is complete whereas in all other cases (4.4.2 to 4.4.6) there is not complete

improvement in MCCOH. In the MCCOH plot of Case 4.4.2, it can be seen that over the

frequency range of 15 18 Hz, the MCCOH has not shown improvement. Though, the

effect of dynamic coupling is made negligible, the MCCOH has not shown completeimprovement in all the cases. Therefore, it can be concluded from this case that the

dynamic coupling has no effect on the MCCOH. So, the incomplete improvement of

MCCOH might be due to either the mass difference or the non-linear motion entering

from other paths.


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4.5 Effect of Location of Input and Path of Energy on Combined Coherence

This case is simulated for the MIMO situations as above but the forcing function is not

placed directly on the DOF that is associated with the non-linearity. These cases are

simulated to study the ability of the combined coherence to detect the non-linearity when

the energy comes from the linear path and also when input is placed some distance away

from the DOFs between which non-linearity is present.

Sl. No. Location of Non-

Linearity

Force M 1, M 2, M 3 &

M 4

4.5.1 1 & 2 F 3 = 50 N &

F4 = 40 N

12, 10, 8 and 14 100000

4.5.2 1 & 3 F 2 = 50 &

F4 = 40N

12, 10, 8 and 14 100000

4.5.3 1 & 4 F 2 = 50N &

F3 = 40 N

12, 10, 8 and 14 100000

4.5.4 2 & 3 F 1 = 50 N &

F4 = 40 N

12, 10, 8 and 14 100000

4.5.5 2 & 4 F 1 = 50 N &

F3 = 40 N

12, 10, 8 and 14 100000

4.5.6 3 & 4 F 1 = 50 N &

F2 = 40 N

12, 10, 8 and 14 100000

Table 4-5: MIMO situations to study effect of Path of Energy


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The FRF estimation, the MCOH, and the MCCOH obtained are shown in Figures (4-23

to 4-28) below. It can be seen that for Cases 4.5.1 and 4.5.6 where the forcing function is

away from the DOFs between which the non-linearity is located, the MCCOH has

registered a drastic improvement. For these cases, when compared with Cases 4.3.1 and

4.3.6 respectively for the same level of excitation, the FRFs and coherence functions are

not distorted as much as when the forcing function is directly placed at the DOFs where

the non-linearity is located. Whereas in Cases 4.5.2 to 4.5.5 the result is reversed, the

FRF and MCCOH are distorted more than Cases 4.5.1 and 4.5.6 and also there is notcomplete clear up of the MCCOH function. The improvement in the MCCOH in Cases

4.5.1 and 4.5.2 can be because the forcing function is away from the DOF where the non-

linearity is being located and the energy is entering through a more linear path. However,

it has been concluded in the previous case that the dynamic coupling has no affect on the

MCCOH, so from this case it can be concluded that the location of inputs and energy

path are critical in determining the ability of the MCCOH in detecting the non-linearities.


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4.6 Effect of Mass Distribution on Combined Coherence

In this section, MIMO situations are simulated by considering equal mass at all DOFs.

This is to see how the mass difference of the DOFs between which the non-linearity is

associated, affects the MCCOH. The masses at all the DOFs are made equal. The

following cases have been simulated.

Sl. No. Location of

Non-Linearity

Force M 1, M 2, M 3 &

M 4

4.6.1 1 & 2 F 1 = 50 N &

F2 = 40 N

15,15,15 & 15 100000

4.6.2 1 & 3 F 1 = 50 &

F3 = 40N

15,15,15 & 15 100000

4.6.3 1 & 4 F 1 = 50N &

F4 = 40 N

15,15,15 & 15 100000

4.6.4 2 & 3 F 2 = 50 N &

F3 = 40 N

15,15,15 & 15 100000

4.6.5 2 & 4 F 2 = 50 N &

F4 = 40 N

15,15,15 & 15 100000

4.6.6 3 & 4 F 3 = 50 N &

F4 = 40 N

15,15,15 & 15 100000

Table 4-6: MIMO situations to study effect of Mass Distribution


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It can be seen from the plots of the MCCOH, that it has shown complete improvement in

Case 4.6.6 and in all other cases from 4.6.1 to 4.6.5, the MCCOH is improved but still

exhibits drops over some frequency ranges. For example, from the MCCOH plot of Case

4.6.2, it can be seen that over the frequency ranges of 7 8 Hz and 13 18 Hz there is no

complete improvement in the MCCOH. By comparing the MCCOH of Case 4.6.6 and

Case 4.3.6 it can be concluded that the mass inequality between the DOFs can be a

possibility for the MCCOH to detect non-linearities. Even though the mass difference

between the degrees of freedom 3 and 4 in Case 4.3.6 is small (6 kg, this difference is

significant when compared to original masses of 14 kg and 8 kg), by eliminating thismass difference, the MCCOH has shown great improvement. In all other cases, the

MCCOH has shown improvement when compared to Cases 4.3.1 to 4.3.6 but of much

smaller values. It is expected that when the mass inequality between the degrees of

freedom is eliminated, the combined coherence should show greater improvement. But

from these cases, it can be concluded that besides the mass inequality, the path of energy

is also critical in detecting the non-linearities. This can be seen from Cases 4.6.1 to 4.6.5,

in which the improvement in the MCCOH is not complete.


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4.7 Effect of Spatial Density of Masses on Combined Coherence

In this section, MIMO situations similar to the real world testing situations, where the

system consists of more than one component, with a difference in mass densities are

simulated. As an example, it can be seen from the Doug Coombs model there are two

frames, one being lighter than the other.

Case Location of

Non-Linearity

Force M 1, M 2, M 3 and

M 4

4.7.1 1 & 2 F 1 = 50 N &

F2 = 40 N

100, 80, 10 & 14 100000

4.7.2 1 & 3 F 1 = 50 &

F3 = 40N

100, 80, 10 & 14 100000

4.7.3 1 & 4 F 1 = 50N &

F4 = 40 N

100, 80, 10 & 14 100000

4.7.4 2 & 3 F 2 = 50 N &

F3 = 40 N

100, 80, 10 & 14 100000

4.7.5 2 & 4 F 2 = 50 N &

F4 = 40 N

100, 80, 10 & 14 100000

4.7.6 3 & 4 F 3 = 50 N &

F4 = 40 N

100, 80, 10 & 14 100000

Table 4-7: MIMO situations to study effect of Spatial Densities of Masses


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It can be seen from the MCCOH plots of Cases 4.7.1 and 4.7.6 that the MCCOH has

shown improvement where the mass difference between the DOFs for which the

MCCOH is computed is negligible. In other cases, the MCCOH has not shown any

improvement at all due to the huge mass difference between the DOFs. In Cases 4.7.1 to

4.7.6, the improvement in the MCCOH is not complete due to the relative motion

entering from other paths.


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Cases 4.8: Effect of Scaling of Motions of DOF on Combined Coherence

In this section, MIMO situations are simulated by scaling the motions at the DOFs at

which the non-linearity is located. The crucial condition for the combined coherence

technique in determining the location of the non-linearity is the mass equality; however

in real world testing conditions, mass equality between the DOFs is not achieved. This

case is tested for MIMO situations to study if scaling the motions would improve the

ability of the combined coherence in determining the structural non-linearities spatially.

The following MIMO cases are simulated.

Sl. No. Location of

Non-Linearity

Force M 1, M 2, M 3 &

M 4

4.8.1 1 & 3 F 1 = 50 N &

F3 = 40 N

100, 80, 10 & 14 100000

4.8.2 1& 4 F 1 = 50 &

F4 = 40N

100, 80, 10 & 14 100000

4.8.3 2 & 3 F 2 = 50N &

F3 = 40 N

100, 80, 10 & 14 100000

4.8.4 2 & 4 F 2 = 50 N &

F4 = 40 N

100, 80, 10 & 14 100000

4.8.5 3 & 4 F 3 = 50 N &

F4 = 40 N

100, 80, 10 & 14 100000

Table 4-8: MIMO situations to study effect of Scaling of Motions


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It can be seen from the MCCOH plots that the MCCOH has shown improvement over the

ordinary coherence, but not improved completely. The complete improvement in the

MCCOH is not seen as it can be affected by non-linear motion entering from other paths.

Case 4.8.5 is simulated similar to Case 4.3.6 and it can be seen from the MCCOH of case

4.8.5 that there is not much improvement when compared to Case 4.3.6. From this

comparison, it can be concluded that non-linear motion entering from other paths has an

effect on the combined coherence in detecting the non-linearities.


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5. Conclusions

It is observed from all of the situations simulated above that, the combined coherence is

able to separate the nonlinearities from digital signal processing errors but that there is no

complete improvement in the combined coherence over some frequency ranges. The first

case is simulated for MIMO situations to study how the increase in the force affects the

combined coherence. It is concluded from this case that the nature of the improvement in

the MCCOH is inversely proportional to the force level, i.e., at lower force levels the

ability of the MCCOH to detect non-linearities is more when compared to that of higher

force levels.

The second and third cases are simulated for SIMO and MIMO situations respectively to

study the nature of the improvement of the combined coherence. It is concluded from

these cases that it is unclear whether the incomplete improvement of the combined

coherence is due to dynamic coupling, mass inequality and/or path of energy. The fourth

case is simulated to study the effect of dynamic coupling on the combined coherence by

making it negligible. This case has showed that the dynamic coupling has no effect on the

combined coherence in detecting nonlinearity. The fifth case is simulated for MIMO

situations to study the path of energy. The combined coherence has shown complete

improvement when the location of input is away from the DOFs between which

nonlinearity is located. It is concluded from this case that the path of energy is critical in

detecting the nonlinearities. The sixth case is simulated for the MIMO situations with

equal masses. It was expected from this case that the combined coherence would improve


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significantly over the Case 3, which is simulated with the same testing conditions except

for the difference in masses at DOFs. However, the results obtained have not shown the

complete improvement over Case 3. It is concluded that besides the mass inequality, the

path of energy is also critical in detecting the non-linearities.

The seventh case is simulated for MIMO situations similar to real world testing

conditions with a difference in the mass distribution, i.e., with one of the masses lighter

than the other. The combined coherence has not shown improvement when the

nonlinearity is located between the DOFs with large mass differences. The eighth caseis studied for MIMO situations by scaling the motions of the DOFs that are associated

with nonlinearity. Scaling the motions of the DOFs that are associated with nonlinearity

has shown improvement over the same cases where no scaling of motion is done. Scaling

the motions did not completely improve the combined coherence and that can be due to

the nonlinear motion entering from other paths.

It can be concluded from the above discussion that the dynamic coupling has no effect on

combined coherence in detecting the nonlinearities. The equality of mass, which is a

crucial condition, and the path of energy are the primary elements affecting the ability of

the combined coherence in detecting the non-linearity. Scaling the masses can improve

the combined coherence, but even when the scaling is done, because of the nonlinear

motion entering from other paths, combined coherence could not improve completely.

Because the equality of mass between degrees of freedom cannot be achieved in the real


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world systems and also energy can enter through non-linear paths, it is difficult to detect

non-linearities in the real world structures using combined coherence.


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6. Future Work

Though the combined coherence, in all the cases studied, is able to separate the

nonlinearity from the digital signal processing errors, drops are observed over some

frequency ranges. Work still need to be done to study the ability of combined coherence

in detecting the structural nonlinearities in the following areas.

1. It was observed that, in most of the situations high frequency distortions in the

multiple combined coherence were improved over the ordinary combined

coherence. Work can be done to study whether this improvement of high

frequency distortions can be used to detect the structural nonlinearities.

2. The system with single location of nonlinearity was studied, whereas in real

structures nonlinearities are located at more than one location. Therefore, this

work can be extended to study the effect of multiple locations of non-linearity in

the system.

3. Location of input and path of energy are critical in detecting the structural

nonlinearities using combined coherence, so a theoretical model with more

number of connections would give better picture of these effects on combined

coherence.

4. Other area can be the study on the ability of combined coherence in detecting

non-linearities in the presence of measurement noise.


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7. References

[1] Detection of Structural Non-Linearities using the Frequency Response andCoherence Functions, Masters Thesis,T. Roscher, University of Cincinnati, 2000

[2] Detection of Structural Non-Linearities using Combined Coherence, MastersThesisDouglas M. Coombs, University of Cincinnati, 2003

[3] Nonlinear Systems Techniques and ApplicationsJulius S. BendantJohn Wiley and Sons Inc, 1998

[4] Use of Hilbert transform in modal analysis of linear and non-linear structuresM. Simon and G. R. Tomlinson

Journal of Sound and Vibration, volume 96, Issue 4, 22 October 1984, Pages 421-436

[5] Introduction to the Theory of Fourier IntegralsTitchmarsh, W.C.Oxford, The Clarendon Press [1948]

[6] The complex stiffness method to detect and identify non-linear dynamic behavior of SDOF systemsMertens M., H. Van Der Auweraer, P. Vanherck and R.SnoeysMechanical Systems and Signal Processing 3 (1), pp37-54

[7] A simple method of interpretation for the modal analysis of nonlinear systemsHe J. and D.J. Ewins: 1987,Proceedings of the 5th International Modal Analysis Conference, London(England), pp 626-634

[8] An explanation of the cause of the distribution in the transfer function of aduffing oscillator subject to sine excitationStorer D.M. and G.R. Tomlinson: 1991,Proceedings of the 9th International Modal Analysis Conference, pp.1197-1205

[9] Recent developments in the measurement and interpretation of higher order transfer functions from non-linear structuresStorer D.M. and G.R. Tomlinson: 1993,Mechanical Systems and Signal Procession 7(2), pp173-179


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[10] Non-linearity identification and quantification using an inverse Fourier TransformKim W-J and Y-S Park: 1993,Fourier Transform, Mechanical Systems and Signal Processing 3, pp 239-255

[11] Vibrations: Experimental Modal AnalysisR.J. AllemangUniversity of Cincinnati, 1999

[12] Non-linear Vibrations: Course LiteratureR. J. AllemangUniversity of Cincinnati, 2000


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8. Appendix

8.1 Simulink Model when the non-linearity is between DOFs 1 and 2


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