UNIVERSITY OF CINCINNATI
September 22
03
_____________
, 20 _____
Douglas Morgan Coombs
I,______________________________________________,
hereby submit this as part of the requirements for the
degree of:
Master of Science
________________________________________________
in:
Mechanical Engineering
________________________________________________
It is entitled:
Detection of Structural Nonlinearities Using Combined
________________________________________________
Coherence
________________________________________________
________________________________________________
________________________________________________
Approved by:
________________________
Dr. Randall J. Allemang
________________________
Dr. David L. Brown
________________________
Dr. Allyn W. Phillips
________________________
Dr. Jay Kim
________________________
Detection of Structural Nonlinearities Using Combined Coherence
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
2003
by
Douglas Morgan Coombs
B.S.M.E. University of Portland 1999
Committee Chair: Dr. Randall Allemang
Abstract
A frequency domain method of detecting and spatially locating structural nonlinearities using
measurements made during a typical modal test is further developed and evaluated in this thesis.
The combined coherence function contains drops due to digital signal processing (DSP) errors,
however it is unaffected by nonlinearities. By comparing combined coherence to ordinary
coherence (or multiple combined coherence to multiple coherence) one can determine whether or
not a drop in ordinary coherence is due to a nonlinearity or a DSP error. Previous work was
done by Tom Roscher and Dr. Randy Allemang with theoretical data generated from a lumped
parameter (M, K, C) model [1, 2]. This paper expands on that work in two ways:
1) By applying the formulation to a “real-world” test structure.
2) By expanding the theoretical background to enable the use of post-processed data for
multiple input testing with Multiple Combined Coherence (MCCOH)
In applying combined coherence to two physical substructures with linear and/or nonlinear
connections, the results of several testing scenarios are examined to see how they affect the
ability of combined coherence to spatially locate nonlinearities. These include:
1) Cases with and without leakage errors
2) Varying the number of spectral averages
3) Reducing the number of nonlinear paths
4) Varying the input force locations
5) Changing the spatial density of the responses
Acknowledgements
There are many people I would like to thank for making this work possible.
First, I would like to thank Dr. Randy Allemang and Dr. Allyn Phillips for the hours spent
working with me on this, supplying needed direction and feedback. Without them this project
would not have been possible. I would also like to thank Dr. David Brown and Dr. Jay Kim for
the insight they would offer on various problems throughout my education here. I came to UC to
learn about experimental vibrations and could not have found a better group of people to do so
under.
Second, I would like to thank my wife for allowing me to turn down a good job offer and move
over 2300 miles away from family and friends in order to pursue further studies at SDRL. For
all of her love, encouragement and support, I am most grateful.
I would also like to thank Dr. Miroslav Rokos, retired professor of the University of Portland, for
seeing my potential when I could not and always pushing me to go to graduate school. Without
his impetus I would have remained in the working world and missed out on one of the greatest
experiences of my life.
Last of all, I would like to thank God for all the gifts he has given me and Mother Mary for
keeping me in her prayers.
Table of Contents
List of Figures ................................................................................................................................ iii
List of Tables ................................................................................................................................ vii
Nomenclature................................................................................................................................. ix
Abbreviations.................................................................................................................................. x
Introduction and Background ......................................................................................................... 1
Definition and Application of Combined Coherence and Multiple Combined Coherence ........ 1
Current Methods of Detecting and Identifying Nonlinearities ................................................... 3
Definition and Application of Ordinary and Multiple Coherence.............................................. 5
Combined Coherence Equation Development............................................................................ 7
Theoretical Basis for Equation................................................................................................ 7
Mathematical Development of Combined Coherence ............................................................ 8
Mathematical Development of Multiple Combined Coherence ............................................. 9
Theoretical Example ................................................................................................................. 11
Applying the Theory to a Real Structure ...................................................................................... 16
Test Subject............................................................................................................................... 16
Case 1: 4 Nominally Linear Connection Points Between Structures ....................................... 20
Case 2: Effect of Cyclic Averaging on Nominally Linear Case 1............................................ 25
Cases 3-5: Effect of Varying the Force Input with Four Nominally Nonlinear Connection
Points Between Structures ........................................................................................................ 30
Case 6: Effect of Cyclic Averaging on Nominally Nonlinear System ..................................... 41
Case 7: Spatial Resolution of the Sensors ................................................................................ 47
Case 8: Effect of Spatial Variation of the Input Forces............................................................ 49
i
Case 9: Effect of Reducing the Number of Connection Points Between the H-Frame and
Square Frame to One ................................................................................................................ 53
Cases 10-11: Effect of Spectral Averaging............................................................................... 66
Discussion ..................................................................................................................................... 73
Summary and Conclusions ........................................................................................................... 77
Future Work .................................................................................................................................. 79
Appendix 1: Derivation of Confidence Intervals for Ĥ (ω ) and θˆ(ω ) (i.e., ∠Ĥ (ω ) ) ............... 83
ii
List of Figures
Figure 1: Three DOF Lumped Mass System .................................................................................. 7
Figure 2: Theoretical Lumped Parameter (M, K, C) System used in example............................. 11
Figure 3: Comparison of FRF and COH for Linear and Nonlinear Theoretical Systems at a DOF
directly affected by the nonlinearity ..................................................................................... 13
Figure 4: Comparison of FRF and COH for Linear and Nonlinear Theoretical Systems at a DOF
directly affected by the nonlinearity ..................................................................................... 13
Figure 5: Comparison of FRF and OCOH Magnitudes for Linear and Nonlinear Theoretical
System at a DOF not directly affected by the nonlinearity................................................... 14
Figure 6: OCOH and CCOH for a Theoretical Hardening Stiffness Case ................................... 15
Figure 7: OCOH vs. CCOH for a Theoretical Hardening Stiffness Example .............................. 15
Figure 8: H-Frame and Square Frame Test Structures ................................................................. 16
Figure 9: Pictures of Shaker Setup showing skew input directions............................................. 17
Figure 10: Accelerometer and force input locations using a spread out spatial distribution....... 18
Figure 11: Accelerometer locations using a more concentrated spatial distribution ................... 19
Figure 12: Layout of connections between the H-frame and the square frame for Cases 1 and 2 21
Figure 13: Picture and schematic of the Bolted Joint for Cases 1 and 2 ...................................... 21
Figure 14: FRF and coherence functions showing the 54.25 Hz resonance peak at which
combined coherence improves over multiple coherence ...................................................... 23
Figure 15: FRF and coherence functions showing the 38.25 Hz resonance peak at which
combined coherence improves over multiple coherence ...................................................... 24
Figure 16: Overlay of FRF and Coherence Plots Showing the Effect of Cyclic Averaging on the
Nominally Linear System ..................................................................................................... 26
iii
Figure 17: Overlay of FRF and Coherence Plots Showing the Effect of Cyclic Averaging on the
Nominally Linear System ..................................................................................................... 26
Figure 18: Case 2 FRF and coherence functions showing the 177.75 Hz antiresonance valley at
which combined coherence improves over multiple coherence ........................................... 28
Figure 19: Case 2 FRF and coherence functions showing the 21.5, 25.25 and 772 Hz
antiresonance valleys at which combined coherence improves over multiple coherence .... 29
Figure 20: Layout of connections between the H-frame and the square frame for Cases 3-8...... 30
Figure 21: Schematics of the two connection types between the structures for Cases 3-8 .......... 31
Figure 22: Case 3: MCCOH for all DOFs at 24.25 Hz................................................................. 36
Figure 23: Case 4: MCCOH for all DOFs at 53.75 Hz................................................................. 37
Figure 24: Case 5: MCCOH for all DOFs at 396 Hz.................................................................... 37
Figure 25: Case 4 FRF and coherence functions showing the 24.25 Hz small resonances and
100.75 Hz antiresonance valleys at which combined coherence improves over multiple
coherence .............................................................................................................................. 38
Figure 26: Case 4 FRF and coherence functions showing the 53.75 Hz resonant peak at which
combined coherence improves over multiple coherence ...................................................... 39
Figure 27: Case 4 FRF and coherence functions showing the 395 Hz antiresonance valley at
which combined coherence improves over multiple coherence ........................................... 40
Figure 28: Case 6: MCCOH for all DOFs at 24.25 Hz................................................................. 43
Figure 29: Case 6: MCCOH for all DOFs at 52 Hz...................................................................... 43
Figure 30: Case 6: MCCOH for all DOFs at 53.5 Hz................................................................... 44
Figure 31: Case 6: MCCOH for all DOFs at 397 Hz.................................................................... 44
iv
Figure 32: Overlay of FRF and Coherence Plots Showing the Effect of a Burst Random Input
and Cyclic Averaging on the Nominally Nonlinear System................................................. 45
Figure 33: Overlay of FRF and Coherence Plots Showing the Effect of a Burst Random Input
and Cyclic Averaging on the Nominally Nonlinear System................................................. 45
Figure 34: Overlay of FRF and Coherence Plots Showing the Effect of a Burst Random Input
and Cyclic Averaging on the Nominally Nonlinear System................................................. 46
Figure 35: Overlay of FRF and Coherence Plots Showing the Effect of a Burst Random Input
and Cyclic Averaging on the Nominally Nonlinear System................................................. 46
Figure 36: Case 7: MCCOH for all DOFs at 396.75 Hz............................................................... 48
Figure 37: Case 7 FRF and coherence functions showing the 123.25 and 175 Hz antiresonance
valleys at which combined coherence improves over multiple coherence ........................... 50
Figure 38: Case 7 FRF and coherence functions showing the 175 Hz antiresonance valley at
which combined coherence improves over multiple coherence ........................................... 51
Figure 39: Overlay of H(45x,*x) before and after moving shaker #1 from the H-Frame to the
Square Frame ........................................................................................................................ 52
Figure 40: Pictures of the setup for Cases 9-11 showing the removal of 3 connection points and
the suspension of the square frame by shock cord................................................................ 53
Figure 41: Layout and schematic of the connection point between the H-frame and the square
frame for Cases 9-11 ............................................................................................................. 54
Figure 42: Case 9: MCCOH for all DOFs at 21.5 Hz................................................................... 57
Figure 43: Case 9: MCCOH for all DOFs at 401.5 Hz................................................................. 57
Figure 44: Case 9: MCCOH for all DOFs at 512 Hz.................................................................... 58
Figure 45: Case 9: MCCOH for all DOFs at 632.5 Hz................................................................. 58
v
Figure 46: Case 9: MCCOH for all DOFs at 800 Hz.................................................................... 59
Figure 47: Case 9 FRF and coherence functions showing the 21.5 Hz combined coherence
improvements over multiple coherence ................................................................................ 60
Figure 48: Case 9 FRF and coherence functions showing the 45 Hz combined coherence
improvements over multiple coherence ................................................................................ 61
Figure 49: Case 9 FRF and coherence functions showing the 512 Hz combined coherence
improvements over multiple coherence ................................................................................ 62
Figure 50: Case 9 FRF and coherence functions showing the 400 Hz combined coherence
improvements over multiple coherence ................................................................................ 63
Figure 51: Case 9 FRF and coherence functions showing the 400 and 635 Hz combined
coherence improvements over multiple coherence............................................................... 64
Figure 52: Case 9 FRF and coherence functions showing the 800 Hz combined coherence
improvements over multiple coherence ................................................................................ 65
Figure 53: Case 12: MCCOH for all DOFs at 638.25 Hz (25 spectral averages)......................... 68
Figure 54: Case 12: MCCOH for all DOFs at 638.25 Hz (100 spectral averages)....................... 69
Figure 55: Case 12: MCCOH for all DOFs at 800 Hz (25 spectral averages).............................. 70
Figure 56: Case 12: MCCOH for all DOFs at 800 Hz (100 spectral averages)............................ 71
Figure 57: Overlay of FRF and Coherence Plots Showing the Effect of a Spectral Averaging for
DOF 43 +X ........................................................................................................................... 72
vi
List of Tables
Table 1: Various Excitation Types and Their Ability to Mask or Highlight Nonlinear Effects in
FRF’s....................................................................................................................................... 2
Table 2: Force, Mass, Stiffness and Damping Values of Theoretical 4 DOF System.................. 12
Table 3: Test Setup for Case 1...................................................................................................... 20
Table 4: Table of Improvements in Combined Coherence for Case 1 ......................................... 22
Table 5: Test Setup for Case 2...................................................................................................... 25
Table 6: Table of improvements of 2% or more in Combined Coherence for Case 2 for adjacent
points..................................................................................................................................... 27
Table 7: Test Setup for Cases 3-5 ................................................................................................. 30
Table 8: Summary of Results for Cases 3-5 ................................................................................. 32
Table 9: Case 3 improvements of 20% or more in Combined Coherence for Case 3 for adjacent
points in the same direction .................................................................................................. 33
Table 10: Case 4 improvements of 20% or more in Combined Coherence for Case 4 for adjacent
points in the same direction .................................................................................................. 34
Table 11: Case 5 improvements of 20% or more in Combined Coherence for Case 5 for adjacent
points in the same direction .................................................................................................. 35
Table 12: Test Setup for Case 6.................................................................................................... 41
Table 13: Table of improvements of 10% or more in Combined Coherence for Case 5 for
adjacent points in the same direction (consecutive frequencies enumerated but not listed). 42
Table 14: Test Setup for Case 7.................................................................................................... 47
Table 15: Table of improvements of 20% or more in Combined Coherence for Case 7 for
adjacent points in the same direction (consecutive frequencies enumerated but not listed). 48
vii
Table 16: Test Setup for Case 8.................................................................................................... 49
Table 17: Table of improvements of 20% or more in Combined Coherence for Case 8 for
adjacent points in the same direction .................................................................................... 49
Table 18: Test Setup for Case 9.................................................................................................... 53
Table 19: Case 9 improvements of 20% or more in Combined Coherence for Case 9 for adjacent
points in the same direction (closely spaced frequencies enumerated but not listed)........... 56
Table 20: Test Setup for Cases 10-12 ........................................................................................... 66
Table 21: 10 Greatest Differences between MCCOH and MCOH for Case 10 (25 Spectral
Averages) .............................................................................................................................. 67
Table 22: 10 Greatest Differences between MCCOH and MCOH for Case 12 (100 Spectral
Averages) .............................................................................................................................. 67
Table 23: Similarities Highlighted Between the 638 Hz and 120 Hz MCCOH Improvements .. 67
viii
Nomenclature
[]-1
Matrix operation denoting inverse
[]+1
Matrix operation denoting pseudo inverse
[]T
Matrix operation denoting transpose
m
Mass
[M] or M
Mass matrix
[K] or K
Stiffness matrix
[C] or C
Viscous damping matrix, first order coefficient
[H] or H
Frequency response function matrix
Hij
Entry row I, column j of frequency response matrix
ω
Rotational velocity, radians/second
[I] or I
Identity matrix
X(ω)
Displacement response in the frequency domain
F(ω)
Input force in the frequency domain
GFF(ω) or GFF
Autopower spectrum of force in the frequency domain
GXX(ω) or GXX
Autopower spectrum of displacement in the frequency domain
GXF(ω) or GXF
Crosspower spectrum of force and displacement in the frequency domain
ix
Abbreviations
SDOF
Single Degree of Freedom (system)
MDOF
Multiple Degree of Freedom (system)
DOF(s)
Degree(s) of Freedom
DSP
Digital Signal Processing
COHp
Ordinary Coherence at output DOF p
CCOHp+r
Combined Coherence for output DOFs p and r
FRF
frequency response function
MCOHp
Multiple Coherence at output DOF p
MCCOHp+r
Multiple Combined Coherence for output DOFs p and r
MIMO
Multiple Input Multiple Output
SIMO
Single Input Multiple Output
SISO
Single Input Single Output
MISO
Multiple Input Single Output
x
Introduction and Background
Definition and Application of Combined Coherence and Multiple
Combined Coherence
The goal of this study is to further evaluate a frequency domain method for detecting
nonlinearities using measurements already made during a typical modal test. Drops in ordinary
coherence and multiple coherence are caused by one of two factors: nonlinearities and digital
signal processing (DSP) errors. If a coherence function could be formulated that was unaffected
by nonlinearities, but still was affected by digital signal processing (DSP) errors or vice versa,
then one could determine whether a drop in ordinary coherence was due to a nonlinearity or a
DSP error by comparing the two coherence functions. The coherence function proposed for
accomplishing this is Combined Coherence (CCOH). Previous work was done by Tom Roscher
with theoretical data generated from a lumped parameter (M, K, C) model [1, 2]. This paper
expands on that work in two ways:
1) By applying the formulation to a “real-world” test structure.
2) By expanding the theoretical background to enable the use of post-processed data for
multiple input testing with Multiple Combined Coherence (MCCOH)
Combined coherence is a method of detecting the presence of nonlinearities between two given
degrees of freedom (DOFs). To further identify the nonlinearity type and parameters values,
other analysis methods must be used. The advantage of combined coherence is that it gives a
quick and efficient way to spatially locate nonlinearities using data already taken during a typical
modal test. No special test setup or data collection are required.
1
The type of excitation has a great effect on whether nonlinear effects are masked or whether they
are highlighted. [3-5]. It also affects which type of analysis one should consider when doing
linear and nonlinear analysis. For example, nonlinearities tend to show specific types of
distortions after the Hilbert transform is applied with stepped-sine force input. However, random
excitation tends to mask these effects and even when FRF’s show some distortion due to
nonlinearities, the Hilbert transform will not be seriously distorted [3]. Table 1 is given below
which characterizes many types of excitation and their tendency to reduce or highlight distortion
due to nonlinearities [3-5]. Because the most common modal parameter estimation theories in
use are based on the assumption of linearity, random excitation is most commonly used in actual
testing situations. Because of this, any practical method for detecting nonlinearities using data
already measured must work with random excitation. For this reason, pure random excitation
has been used for all analyses involving combined coherence.
Table 1: Various Excitation Types and Their Ability to Mask or Highlight Nonlinear Effects in FRF’s
Excitation Type
Nonlinear Distortions Remain?
Swept Sine
Yes, Best at Highlighting Distortions
Steady State Sine
Yes
Periodic Chirp
Yes
Impact
Yes
Pseudo Random
Yes
Periodic Random
No
Burst Random
No
Pure Random
No, Best at Removing Nonlinear Distortions
2
Current Methods of Detecting and Identifying Nonlinearities
There are many methods used today to detect and identify nonlinearities. Most methods tend to
focus on parameter identification, and in some way assume a model for the nonlinearities. Many
methods also rely on data that is not readily available in experimental applications. A sample of
the a few methods in use today is given below to give a flavor of the current analysis procedures.
Piecewise analysis involves looking at measurement data under various conditions (forcing
levels, time periods) and making comparisons to see if the data is consistent. One of the easiest
nonlinear detection methods is simply to compare the FRF’s at various force levels throughout
the range the structure would experience in actual use checking for differences. Another way
would be to break a transient time history up into two or three sections, calculate FRF’s from
each time block and compare the results to see if there are frequency or amplitude variations
between them [6].
Bendat, Coppolino and Palo developed the Reverse MI/SO technique for identifying nonlinear
parameters of SDOF systems based on the conversion of nonlinear SISO models with or without
feedback into linear MISO models without feedback [7, 8]. It involves looking at the force as
the output and the acceleration response as the input and then connecting them by various linear
and nonlinear paths. It tends to be applied to systems with few degrees of freedom where the
gross behavior of the structure is nonlinear. Examples involving an automotive shock and the
stability of an ocean barge are given [7].
3
Storer and Tomlinson present a method of detecting the presence of nonlinearities using a sine
wave input and higher order FRF’s using a NARMAX model [9]. They comment on the
limitations of higher order FRF’s using random excitation and the limited success in their
application to real world structures. They present a nonlinear beam experiment in which they are
able to determine the frequencies at which the nonlinearities act by measuring higher order
FRF’s up to order three. Characterization of these nonlinearities still relies on an a priori model.
Chen and Tomlinson proposed a method for identifying nonlinear parameters using a time series
model based on acceleration, velocity and displacement (AVD) data [10]. This data need not be
independently measured. The acceleration data may simply be integrated to get the needed
velocity and displacement data. They demonstrate that using AVD data was more accurate than
simply using displacement or acceleration data. The AVD model results of an experiment
involving a pendulum with nonlinear supports are compared to NARMAX model results of the
same experiment. The argument is then made for the superiority of the AVD model especially
for higher (3rd) order FRF’s.
Ghanem and Romeo have proposed a method for identifying parameters of an a priori known
nonlinear system [11]. When the type of nonlinearity is unknown, a method of selecting the type
from various classes of nonlinearities is presented. The method makes use of wavelets and their
ability to focus on specific time/frequency bands to identify systems with nonlinearities whose
effects would be averaged out using other techniques. Theoretical noise free examples are given.
4
Worden and Tomlinson present a method of detecting and locating nonlinearites that makes use
of the Hilbert transform in identifying the part of the FRF that is due to artificial non-causality
caused by nonlinearities [3]. They define the non-causal power ratio (NPR) as the ratio of the
power of the non-causal part of the impulse response to the power of the total impulse response.
Calculating the NPR for each impulse response in the FRF matrix can give an indication of
where the nonlinearites are spatially located.
A method for detecting nonlinearities using only experimental data was developed by Chong and
Imregun using reciprocal modal vectors [12]. This involves going through the modal parameter
identification process, selecting the DOFs of interest and comparing the deviation of reciprocal
modal vectors from the unit matrix to see which modes are affected by nonlinearities. This
method, relying on the results of a mode fit seems to be contaminated by errors resulting from
closely spaced modes. Theoretical examples with and without noise are given.
Definition and Application of Ordinary and Multiple Coherence
Ordinary coherence (COH, Eqn. 1) is a statistical indicator of whether a given input and a given
output are correlated (i.e., linearly related) [5, 13-15]. It is bounded by zero and one (Eqn. 2)
with one indicating a perfectly linear relationship between the input and output and zero
indicating that the input and output are completely uncorrelated. Being a statistical indicator of
linearity, the more averages are taken, the more confidence one can have that a high value is a
true indication of a linear relationship between the input and output. A discussion of how
statistical confidence levels vary with the number of averages is given in Appendix 1.
Coherence is not an indicator of causality but of correlation. Because of this, a drop in
5
coherence does not necessarily mean that the response is not being caused by the input (though
that may be the case). A few of the more common errors causing drops in coherence are listed
below [13-15]. One can see that the errors may be broadly broken down into two categories,
DSP errors (1-3) and errors due to nonlinearities (4-8). All errors causing drops in coherence fall
into one of these two categories.
1) Leakage
2) Aliasing
3) Measurement noise
4) Friction Forces
5) Saturation, dead-zone
6) Unmeasured inputs
7) Harmonic distortion
8) Hardening or softening stiffness/damping
COH p (ω ) =
GXF ps (ω )
2
(1)
GXX pp (ω )GFFss (ω )
0 ≤ COH p (ω ) ≤ 1
(2)
Ordinary coherence indicates whether a single input is linearly related to a single output. Often
though, multiple inputs are used during testing so that energy can be more evenly distributed
throughout a structure and so that repeated roots can be separated. In the case of two
uncorrelated inputs both of which have equal output spectra at a given DOF, the ordinary
coherence would have a value of 0.5 for both input-output combinations. Multiple coherence
(MCOH) was developed to take all inputs into account (Eqn. 3). It is a statistical indicator of
whether all inputs taken together are linearly related to a given output [5, 13-15]. Because of
6
this, it can be used in MIMO testing scenarios similarly to the way ordinary coherence is applied
and interpreted in SISO or SIMO testing scenarios. MCOH is bounded by zero and one. A
value of one indicates that all the inputs together are linearly related to the output, while a value
of zero indicates that none of the inputs are correlated with the output (Eqn. 4).
Ni
Ni
MCOH p (ω ) = ∑∑
H ps (ω )GFFst (ω )H *pt (ω )
s =1 t =1
(3)
GXX pp (ω )
0 ≤ MCOH p (ω ) ≤ 1
(4)
Combined Coherence Equation Development
Theoretical Basis for Equation
If one considers the lumped mass system in Figure 1 below, there are both internal and external
forces acting on it. The internal forces, due to the springs and dampers may or may not be
nonlinear. If there was a nonlinear spring between two specific DOFs and the motion of those
DOFs was summed together, the nonlinear internal forces would then cancel out leaving only the
remaining external force terms.
F1
X1
K01
F2
C01
X2
K12
M1
F3
K23
X3
K30
M3
M2
C12
C23
C30
Figure 1: Three DOF Lumped Mass System
The equations of motion for DOFs 1 and 2 are given below.
&
&
&
m1 &
x&
1 = −(k 01 + k12 + k13 )x1 + k12 x 2 + k13 x3 − (c01 + c12 + c13 )x1 + c12 x 2 + c13 x3 + F1
(5)
&
&
&
m2 &
x&
2 = −(k12 + k13 )x 2 + k12 x1 + k 23 x3 − (c12 + c13 )x 2 + c12 x1 + c 23 x3 + F2
(6)
7
If one were to sum the motion of the two DOFs together, one arrives at a new equation.
1
[− (k 01 + k12 + k13 )x1 + k12 x2 + k13 x3 ] + 1 [− (k12 + k13 )x2 + k12 x1 + k 23 x3 ]
m1
m2
&
&
&
x&
1 + x2 =
(7)
+
1
[− (c01 + c12 + c13 )x&1 + c12 x&2 + c13 x&3 ] + 1 [− (c12 + c13 )x&2 + c12 x&1 + c23 x&3 ] + F1 + F2
m1
m2
m1 m2
If one makes the assumption that the masses of the two DOFs are equal, then the information
about the internal forces drops out.
&
&
&
x&
1 + x2 =
1
[− (k01 + k13 )x1 + k13 x3 − (k13 )x2 + k23 x3 ] + 1 [− (c01 + c13 )x&1 + c13 x&3 − (c13 )x&2 + c23 x&3 ] + F1 + F2
m
m
m
(8)
Thus, if a coherence function is derived, based on the sum of the motions of two DOFs
containing a nonlinearity between them, the drop in coherence due to that nonlinearity will go
away. However, drops in coherence due to digital signal processing errors such as leakage and
noise will remain [1, 2]. If the masses are not equal, scaling the motion of the two DOFs will be
necessary to completely cancel out the effect of internal forces.
Mathematical Development of Combined Coherence
The standard ordinary coherence equation is given below for input point q and output point p [5,
13-15].
∑ (X
Nave
COHpq =
GXF pq
2
GFFqq GXX pp
=
1
Nave
∑ (F
q
p
Fq*
*
q
F
1
)∑ (X
Nave
Fq
p
*
p
1
Nave
)∑ (X
)
*
p
X
1
(9)
)
Since the combined coherence function is based on the sum of the motion between two DOFs,
Xp+Xr is substituted for Xp.
CCOH ( pr )q
∑ ((X + X )F )∑ ((X + X ) F )
=
∑ (F F )∑ ((X + X )(X + X ) )
r
q
p
*
q
*
*
q
r
p
q
*
r
p
r
p
8
(10)
CCOH ( pr )q =
∑ (X
∑ (F
q
Fq*
r
Fq* + X p Fq*
)∑ (X
r
)∑ (X
Fq + X p Fq
r
)
*
X r* + X r X *p + X p X r* + X p X *p
(11)
)
Expanding and consolidating terms, one is left with the equation below [1, 2].
CCOH ( pr )q =
GXF pq + GXFrq
2
(12)
GFFqq (GXX rr + GXX rp + GXX pr + GXX pp )
Mathematical Development of Multiple Combined Coherence
The standard equation for Multiple Coherence is given by the equation below [5, 13-15].
Ni
Ni
MCOH p = ∑∑
H ps GFFst H *pt
s =1 t =1
(13)
GXX pp
Looking at H for a 2 input, 3 output system,
 GFF ss
GFF = 
 GFF ts
[GFF ]−1
GFF st 
GFF tt 
GFF tt

 GFF GFF − GFF GFF
ss
tt
st
ts
=
GFF
ts

 GFF ss GFF tt − GFF st GFF ts
 GXF 1 s
GXF =  GXF 2 s
 GXF 3 s
GXF 1t 
GXF 2 t 
GXF 3 t 
H 1 = [GXF ][GFF
]
H 1t = −
(14)
−1
 H 1s
=  H 2 s
 H 3 s
GFF st
GFF ss GFF tt − GFF st GFF ts
GFF tt
GFF ss GFF tt − GFF st GFF ts





(15)
(16)
H 1t 
H 2 t 
H 3 t 
(17)
GFF st GXF 1 s
GFF ss GXF 1t
+
GFF ss GFF tt − GFF st GFF ts GFF ss GFF tt − GFF st GFF ts
9
(18)
H 1s =
GFF tt GXF 1 s
GFF ts GXF 1t
−
GFF ss GFF tt − GFF st GFF ts GFF ss GFF tt − GFF st GFF ts
(19)
Since MCCOH is based on the summation of two degrees of freedom, let X1=Xp+Xr in H1s.
H 1s =
GFFtt GXF1s
GFFts GXF1t
−
GFFss GFFtt − GFFst GFFts GFFss GFFtt − GFFst GFFts
(20a)
∑ (F F )∑ (X F )
∑ (F F )∑ (X F )
−
)∑ (F F ) − ∑ (F F )∑ (F F ) ∑ (F F )∑ (F F ) − ∑ (F F )∑ (F F )
(20b)
∑ (F F )∑ ((X + X )F )
∑ (F F )∑ ((X + X )F )
=
−
∑ (F F )∑ (F F ) − ∑ (F F )∑ (F F ) ∑ (F F )∑ (F F ) − ∑ (F F )∑ (F F )
(20c)
*
=
t
∑ (F F
s
*
s
t
t
*
*
s
1
*
t
s
t
t
*
t
s
=
t
*
s
p
s
GFFtt (GXF ps + GXFrs )
GFFss GFFtt − GFFst GFFts
s
*
s
t
t
−
t
*
s
s
*
s
*
1
t
*
t
s
*
s
t
*
t
*
s
*
s
r
*
t
*
s
*
t
t
*
s
*
p
r
*
t
t
t
GFFts (GXF pt + GXFrt )
GFFss GFFtt − GFFst GFFts
= H ps + H rs
t
*
s
t
t
*
s
(20d)
(20e)
Also, let X1=Xp+Xr in GXX11.
GXX 11 =
∑ (X X )
1
*
1
(21a)
[
= ∑ (X p + X r )(X p + X r )
*
[
(
]
(21b)
)]
(21c)
= ∑ (X p + X r ) X *p + X r*
= ∑ X p X *p + ∑ X p X r* + ∑ X r X *p + ∑ X r X r*
(21d)
= GXX pp + GXX pr + GXX rp + GXX rr
(21e)
Thus,
Ni
Ni
MCCOH p + r = ∑∑
s =1 t =1
(H
(GXX
+ H rs )GFFst (H pt + H rt )
*
ps
pp
+ GXX pr + GXX rp + GXX rr )
10
(22)
Theoretical Example
Roscher gave several theoretical examples of the application of combined coherence to a test
structure [1]. One is included here to show how combined coherence can be applied to a
theoretical lumped mass system.
M4
K3
C3
K2
M3
C2
K1
K2
C2
C1
K1
M2
C1
K1
C1
M1
K0
C0
Figure 2: Theoretical Lumped Parameter (M, K, C) System used in example
A hardening stiffness is applied between DOFs 1 and 3 of the form k(x+εx3). The stiffness,
damping, mass and ε values are given in Table 2 below. Since the nonlinear stiffness is between
11
DOFs 2 and 3, one would expect combined coherence to show improvement when summing
those two DOFs, while not improving for other DOF combinations.
Table 2: Force, Mass, Stiffness and Damping Values of Theoretical 4 DOF System
Ground
Mass 1
12 kg
Mass 2
7 kg
Mass 1: 12 kg
K10=22000 N/m
C10=6 N-s/m
Random Input:
F1=30 N rms
Mass 2: 7 kg
Mass 3: 9 kg
Mass 4: 14 kg
K12=19000 N/m
C12=8 N-s/m
No Input Force
K13=19000 N/m
C13=8 N-s/m
K23_lin=24000 N/m
ε = 50000
K23=K23_lin(1+ε∆x2)
C23=9 N-s/m
No Input Force
K14=24000 N/m
C14=7 N-s/m
K24=20000 N/m
C24=4 N-s/m
Mass 3
9 kg
Mass 4
14 kg
K34=20000 N/m
C34=5 N-s/m
No Input Force
Comparing the frequency response and ordinary coherence values for the linear and nonlinear
systems, one can see that there are several significant drops in coherence for the nonlinear
system (Figure 3 through Figure 5). For all DOFs, there is a significant drop due to leakage at
the 3.5 Hz resonant peak. For DOFs 2 and 3, which are directly affected by the nonlinearity,
there are other drops in coherence at the 13 Hz antiresonance, from 15-19 Hz and at high
frequency. There is also a shift in the resonant peak at 17 Hz (Figure 3 and Figure 4). For DOF
4, which is not direcly affected by a nonlinearity, there is a slight drop in coherence at around 17
Hz and some shifting in the resonant peak there as well (Figure 4).
12
H(2,1) = X/F, normally distributed random excitation 50% overlapping blocks, 79 averages & Hanning window applied
-2
10
H1 lin(2,1)
H1 nonlin(2,1)
-4
10
F/
X
=
H
-6
10
-8
10
0
5
10
15
20
Frequency (hz)
25
30
35
40
1
e
c
n
er
e
h
o
C
f
o
e
d
ut
i
n
g
a
M
COH lin(2,1)
COH nonlin(2,1)
0.8
0.6
0.4
0.2
0
0
5
10
15
20
Frequency (hz)
25
30
35
40
Figure 3: Comparison of FRF and COH for Linear and Nonlinear Theoretical Systems at a DOF directly
affected by the nonlinearity
H(3,1) = X/F, normally distributed random excitation 50% overlapping blocks, 79 averages & Hanning window applied
-2
10
H1 lin(3,1)
H1 nonlin(3,1)
-4
10
F/
X
=
H
-6
10
-8
10
0
5
10
15
20
Frequency (hz)
25
30
35
40
1
e
c
n
er
e
h
o
C
f
o
e
d
ut
i
n
g
a
M
0.8
COH lin(3,1)
COH nonlin(3,1)
0.6
0.4
0.2
0
0
5
10
15
20
Frequency (hz)
25
30
35
40
Figure 4: Comparison of FRF and COH for Linear and Nonlinear Theoretical Systems at a DOF directly
affected by the nonlinearity
13
H(4,1) = X/F, normally distributed random excitation 50% overlapping blocks, 79 averages & Hanning window applied
-2
10
H1 lin(4,1)
H1 nonlin(4,1)
-4
10
F/
X
=
H
-6
10
-8
10
0
5
10
15
20
Frequency (hz)
25
30
35
40
1
e
c
n
er
e
h
o
C
f
o
e
d
ut
i
n
g
a
M
0.8
COH lin(4,1)
COH nonlin(4,1)
0.6
0.4
0.2
0
0
5
10
15
20
Frequency (hz)
25
30
35
40
Figure 5: Comparison of FRF and OCOH Magnitudes for Linear and Nonlinear Theoretical System at a
DOF not directly affected by the nonlinearity
As can be seen below in Figure 6, when adding the motions of DOFs 2 and 3, the combined
coherence plot shows significant improvement at 13 Hz, 17 Hz and at high frequency where the
nonlinear effect was quite noticeable. This indicates that a nonlinearity is acting between DOFs
2 and 3. Looking at Figure 7 below, while combined coherence may improve compared to the
ordinary coherence of one of the DOFs, it never improves over the ordinary coherence of both of
the DOFs. This leads to the expected conclusion that a nonlinearity is not located between DOFs
3 and 4.
14
COH 2/1, COH 3/1, CCOH(2+3)/1, F=30N
1
0.9
)
1,
2(
H
O
C
0.8
0.7
0.6
0.5
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
0
5
10
15
20
Frequency (f, Hz)
25
30
35
40
1
0.9
)
1,
3(
H
O
C
0.8
0.7
0.6
0.5
1
0.9
)
1,
3
+
2(
H
O
C
C
0.8
0.7
0.6
0.5
Figure 6: OCOH and CCOH for a Theoretical Hardening Stiffness Case
COH 3/1, COH 4/1, CCOH(3+4)/1, F=30N
1
)
1,
3(
H
O
C
0.9
0.8
0.7
0.6
0.5
)
1,
4(
H
O
C
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
1
0.9
0.8
0.7
0.6
0.5
)
1,
4
+
3(
H
O
C
C
1
0.9
0.8
0.7
0.6
0.5
Frequency (f, Hz)
Figure 7: OCOH vs. CCOH for a Theoretical Hardening Stiffness Example
15
Applying the Theory to a Real Structure
Test Subject
The test structure used is a welded metal frame of 2”x6”x.25” steel tubing. Attached to this
frame was another welded structure made from 2”x2”x.125” steel tubing. These were connected
at 4 discrete points, giving various options for linear/nonlinear connections. The test frames are
shown below in Figure 8.
Figure 8: H-Frame and Square Frame Test Structures
Two shakers were used as inputs in a MIMO type test. Each shaker was connected to the HFrame in a skew direction in order to get energy into the structure in all three principal
directions. The shakers were 45o skew to all three principal directions at Points 101 and 102,
16
while shaker #1 was only about 20o skew to the principal x direction at Point 100, due to
space/angle constraints. The setup of the shakers is shown below in Figure 9 through Figure 11.
Shaker #1 at Point 100
Shaker #1 at Point 100 - Closeup
Shaker #1 at Point 102
Shaker #1 at Point 102 - Closeup
Shaker #2 at Point 101
Shaker #2 at Point 101 - Closeup
Figure 9: Pictures of Shaker Setup showing skew input directions
17
The spatial distribution of the accelerometers varied depending on the test conducted. Some
tests had a “spread out” spatial distribution, giving an overall view of the structure and allowing
for a determination of structural mode shapes (Figure 10 below). Other tests used a more
“concentrated” spatial distribution focusing on the interactions across two specific connection
points between the structures (Figure 11 below).
Figure 10: Accelerometer and force input locations using a spread out spatial distribution
18
Figure 11: Accelerometer locations using a more concentrated spatial distribution
19
Case 1: 4 Nominally Linear Connection Points Between Structures
The first experimental case presented is that of the square frame bolted to the H-Frame at each
corner with a wood spacer separating the two. This is meant to simulate a hard “linear”
connection as much as possible, without welding the two structures together (see Figure 12 and
Figure 13 below). Since the connections are linear, there should not be large improvements in
multiple combined coherence (MCCOH) compared to multiple coherence (MCOH) across the
connection points. Random excitation was used with 150 spectral averages in order to get a
good estimate of the FRF’s. For a complete description of the test setup, see Table 3 below.
Table 3: Test Setup for Case 1
Test Setup Description
Accelerometer
Concentrated,
Excitation Type
Random
Distribution
Figure 11
Input Voltage
3V
Peak Force
10 lbf
Shaker #1 Location
Pt. 100
Shaker #2 Location
Pt. 101
# of Spectral Averages 150
# of Cyclic Averages 0
Attachment Method: Bolted connection with 2x4 wood spacer at each corner of the square
frame (Figure 12 and Figure 13).
20
Figure 12: Layout of connections between the H-frame and the square frame for Cases 1 and 2
Figure 13: Picture and schematic of the Bolted Joint for Cases 1 and 2
In comparing combined coherence to multiple coherence, it is useful to compare points located
next to each other in the same direction to determine if a nonlinearity is being excited between
21
them (e.g., comparing 44x to 48x). Table 4 below shows the DOF combinations with at least a
10% improvement in MCCOH over MCOH as well as whether the improvement occurred at a
resonance or antiresonance. As can be seen from Figure 14 and Figure 15, the largest
improvements all came at resonant peaks. This led to the question of whether leakage was
contaminating the results at all, contrary to earlier theoretical studies which showed that leakage
did not affect combined coherence [1, 2].
Table 4: Table of Improvements in Combined Coherence for Case 1
freq
pt1
dir1
pt2
dir2
(Hz)
MCCOH MCOH1 MCOH2
13
13
52
51
51
5
13
13
13
54
54
53
53
43
43
+X
+Z
-Y
-X
-X
+Y
+Y
+Z
+Z
+Y
+Z
-X
+Y
+Y
+Z
15
15
51
46
44
52
54
54
54
53
53
47
45
47
47
-X
+Z
+Y
+X
+X
-Y
+Y
+Z
+Z
+Y
+Z
+X
+Y
+Y
+Z
30.5
38.5
54.25
54.25
54.5
54.25
54.25
38.25
49.25
11
30.75
11
11
38.25
49.25
0.9823
0.9794
0.954
0.7375
0.9554
0.9925
0.9887
0.9642
0.9983
0.9834
0.9768
0.9552
0.9961
0.898
0.9836
0.8724
0.8663
0.8227
0.6301
0.6864
0.8366
0.838
0.7113
0.7997
0.8388
0.8803
0.8463
0.8736
0.7367
0.8395
22
0.8897
0.8871
0.766
0.5481
0.483
0.8227
0.8837
0.8153
0.835
0.8736
0.8814
0.7797
0.8405
0.7186
0.8713
%difference
10.4063
10.4094
15.9539
17.0329
39.1917
18.6288
11.8851
18.2648
19.559
12.57
10.8262
12.8657
14.0137
21.8927
12.8882
Improvement at
(anti)resonance
resonance
resonance
resonance
resonance
resonance
resonance
resonance
resonance
resonance
resonance
resonance
resonance
resonance
resonance
resonance
FRF Magnitude vs. Frequency for Various Input/Output Combinations
10
)f
bl
/
n(i
e
d
ut
i
n
g
a
M
F
R
F
10
10
FRF
FRF
FRF
FRF
-1
-2
-3
45
50
55
60
65
Frequency (Hz)
70
75
80
85
Multiple Coherence for Points 5 and 52 directions +Y and -Y.
1
0.8
MCOH 5+Y
MCOH 52-Y
0.6
H
O
C
M
5 +Y,100 +X
52 -Y,100 +X
5 +Y,101 +Z
52 -Y,101 +Z
0.4
0.2
0
0
100
200
300
500
600
700
800
Multiple Combined Coherence for Point 5 and 52 directions +Y and -Y.
1
0.8
H
O
C
C
M
400
Combined Coherence
5 Frequencies with 5% Improvement above 10 Hz.
0.6
0.4
0.2
0
0
100
200
300
400
Frequency (Hz)
500
600
700
800
Figure 14: FRF and coherence functions showing the 54.25 Hz resonance peak at which combined coherence
improves over multiple coherence
23
FRF Magnitude vs. Frequency for Various Input/Output Combinations
10
10
b
n
e
d
u
n
g
a
M
F
R
F
10
10
-1
FRF
FRF
FRF
FRF
47 +Y,100 +X
43 +Y,100 +X
47 +Y,101 +Z
43 +Y,101 +Z
-2
-3
-4
30
32
34
36
38
40
42
44
46
48
50
Frequency (Hz)
Multiple Coherence for Points 43 and 47 directions +Y and +Y.
1
0.8
MCOH 43+Y
MCOH 47+Y
0.6
H
O
C
M
0.4
0.2
0
0
100
200
300
500
600
700
800
Multiple Combined Coherence for Point 43 and 47 directions +Y and +Y.
1
0.8
H
O
C
C
M
400
Combined Coherence
2 Frequencies with 5% Improvement above 10 Hz.
0.6
0.4
0.2
0
0
100
200
300
400
Frequency (Hz)
500
600
700
800
Figure 15: FRF and coherence functions showing the 38.25 Hz resonance peak at which combined coherence
improves over multiple coherence
24
Case 2: Effect of Cyclic Averaging on Nominally Linear Case 1
Since the majority of improvements occurred at resonant peaks for the previous case, a test was
run to see what would happen if steps were taken to reduce the amount of leakage error. For this
purpose, 10 cyclic averages were used (Table 5).
Table 5: Test Setup for Case 2
Test Setup Description
Accelerometer
Concentrated,
Excitation Type
Random
Distribution
Figure 11
Input Voltage
3V
Peak Force
10 lbf
Shaker #1 Location
Pt. 100
Shaker #2 Location
Pt. 101
# of Spectral Averages 150
# of Cyclic Averages 10
Attachment Method: Bolted connection with 2x4 wood spacer at each corner of the square
frame (Figure 12 and Figure 13)..
As was expected, cyclic averaging vastly improved the multiple coherence function almost
universally. Of noted interest is that MCCOH no longer improved over MCOH at the
aforementioned resonant peaks, indicating that either both leakage and a nonlinearity had
affected the FRF estimate in Case 1, or that an improvement in combined coherence can be an
indication of a DSP error and not just a nonlinearity.
25
10
Overlay Plot of FRF's and Multiple Coherence for DOF 5 +Y for Cases with and without Cyclic Averaging
0
No Cyclic Averaging: 5 +Y
Cyclic Averaging: 5 +Y
e
d
ut
i
n
g
a
M
F
R
F
10
-5
0
50
100
150
1
0.8
e
d
ut
i
n
g
a
M
H
O
C
M
No Cyclic Averaging: 5 +Y
Cyclic Averaging: 5 +Y
0.6
0.4
0.2
0
0
50
100
150
Frequency (Hz)
Figure 16: Overlay of FRF and Coherence Plots Showing the Effect of Cyclic Averaging on the Nominally
Linear System
10
Overlay Plot of FRF's and Multiple Coherence for DOF 47 +Y for Cases with and without Cyclic Averaging
0
No Cyclic Averaging: 47 +Y
Cyclic Averaging: 47 +Y
e
d
ut
i
n
g
a
M
F
R
F
10
-5
0
10
20
30
40
50
60
70
80
1
0.8
e
d
ut
i
n
g
a
M
H
O
C
M
No Cyclic Averaging: 47 +Y
Cyclic Averaging: 47 +Y
0.6
0.4
0.2
0
0
10
20
30
40
50
60
70
80
Frequency (Hz)
Figure 17: Overlay of FRF and Coherence Plots Showing the Effect of Cyclic Averaging on the Nominally
Linear System
26
After cyclic averaging to remove leakage errors, MCCOH still improved over MCOH at certain
frequencies, though the improvements were much smaller in magnitude. The locations of these
improvements in the FRF’s changed in nature from resonances to antiresonances (Figure 18 and
Figure 19). Table 6 below lists DOFs next to each other which improved 2% or more. The
absence of any improvement in combined coherence at the same frequencies as before for the
same setup makes the insensitivity of combined coherence to leakage effects as noted in earlier
theoretical studies questionable for distributed mass systems [1, 2].
Table 6: Table of improvements of 2% or more in Combined Coherence for Case 2 for adjacent points
pt1 dir1 pt2 dir2 freq (Hz)
MCCOH
MCOH1 MCOH2 % Difference (anti)resonance
3 -Y
5 +Y
102.5
0.9920
0.6141
0.9683
2.4432 antiresonance
13 +Y
15 -Y
282.25
0.9922
0.9276
0.7788
6.9654 antiresonance
52 -Y
51 +Y
21.5
0.9934
0.8719
0.9714
2.2585
antiresonance
52 -Y
51 +Y
25.25
0.9941
0.6735
0.8594
15.6682
5 +Y
52 -Y
25.25
0.9975
0.9435
0.6735
5.7227 antiresonance
13 +X
54 -X
177.75
0.9896
0.9651
0.9302
2.5389 antiresonance
13 +Y
54 +Y
23.25
0.9712
0.9474
0.9461
2.5158
antiresonance
13 +Y
54 +Y
23.5
0.9762
0.9505
0.9555
2.1700
54 -X
53 -X
177.75
0.9577
0.9302
0.8926
2.9643 antiresonance
15 -X
54 -X
24.75
0.9611
0.9363
0.9282
2.6486
15 -X
54 -X
25
0.9514
0.9121
0.9228
3.1018 antiresonance
15 -X
54 -X
25.5
0.9050
0.7546
0.8788
2.9714
27
FRF Magnitude vs. Frequency for Various Input/Output Combinations
10
)f
bl
/
n(i
e
d
ut
i
n
g
a
M
F
R
F
10
FRF
FRF
FRF
FRF
-1
54 -X,100 +X
13 +X,100 +X
54 -X,101 +Z
13 +X,101 +Z
-2
165
170
175
180
185
Frequency (Hz)
190
195
Multiple Coherence for Points 13 and 54 directions +X and -X.
1
0.8
MCOH 13+X
MCOH 54-X
0.6
H
O
C
M
0.4
0.2
0
0
100
200
300
400
500
600
700
800
Combined Coherence for Point 13 and 54 directions +X and -X.
1
Combined Coherence
1 Frequencies with 1.5% Improvement above 10 Hz.
0.8
0.6
H
O
C
C
0.4
0.2
0
0
100
200
300
400
Frequency (Hz)
500
600
700
800
Figure 18: Case 2 FRF and coherence functions showing the 177.75 Hz antiresonance valley at which
combined coherence improves over multiple coherence
28
FRF
FRF
FRF
FRF
10
-2
10
)f
bl
/
n(i
e
d
ut
i
n
g
a
M
F
R
F
)f
bl
/
n(i
e
d
ut
i
n
g
a
M
F
R
F
10
10
51 +Y,100 +X
52 -Y,100 +X
51 +Y,101 +Z
52 -Y,101 +Z
-1
-2
-3
10
15
20
25
30
Frequency (Hz)
35
40
-3
720
730
740
750
760
Frequency (Hz)
770
780
790
Multiple Coherence for Points 51 and 52 directions +Y and -Y.
1
MCOH 51+Y
MCOH 52-Y
0.8
0.6
H
O
C
M
0.4
0.2
0
0
100
200
300
400
500
600
700
800
Combined Coherence for Point 51 and 52 directions +Y and -Y.
1
Combined Coherence
9 Frequencies with 1.5% Improvement above 10 Hz.
0.8
0.6
H
O
C
C
0.4
0.2
0
0
100
200
300
400
Frequency (Hz)
500
600
700
800
Figure 19: Case 2 FRF and coherence functions showing the 21.5, 25.25 and 772 Hz antiresonance valleys at
which combined coherence improves over multiple coherence
29
Cases 3-5: Effect of Varying the Force Input with Four Nominally
Nonlinear Connection Points Between Structures
In order to create nonlinear connections between the Square Frame and H-Frame, some vehicle
body mounts sandwiched between a foam with memory were inserted at the four connection
points where the wooden blocks had been (Figure 20 and Figure 21). The force was then varied
to see at what forcing level the nonlinearities between the structures were most excited.
Table 7: Test Setup for Cases 3-5
Test Setup Description
Accelerometer
Spread Out,
Excitation Type
Random
Distribution
Figure 10
Case 3 Input Voltage 1 V
Case 3 Peak Force
4 lbf
Case 4 Input Voltage 3 V
Case 4 Peak Force
10 lbf
Case 5 Input Voltage 5 V
Case 5 Peak Force
18 lbf
Shaker #1 Location
Pt. 100
Shaker #2 Location
Pt. 101
# of Spectral Averages 150
# of Cyclic Averages 0
Attachment Method: Four Vehicle Body Mounts at each corner of the square frame. Two
bolts barely hand tightened. Two bolts removed altogether. See Figure 20 and Figure 21.
Figure 20: Layout of connections between the H-frame and the square frame for Cases 3-8
30
Figure 21: Schematics of the two connection types between the structures for Cases 3-8
As can be seen in Table 9 through Table 11, when the force level was varied, the number and
nature of improvements in combined coherence varied. Figure 25 through Figure 27 below give
examples of the FRF and COH plots at the resonances and antiresonances for which MCCOH
improved over MCOH. At low forcing levels (Case 3), combined coherence changed more in
the 24.25 Hz range at some very small peaks/valleys (Figure 25). At this frequency the DOF
combinations that showed the most improvement in MCCOH over MCOH were between DOFs
on the lower structure and between DOFs spanning the nonlinear connection (Table 9 and Figure
22). At high forcing levels, there were still changes a couple resonant peaks with MCCOH
improvements, but the dominant changes occurred at a very wide frequency range around the
395 Hz antiresonance on the upper frame (Table 11, Figure 24 and Figure 27). This
improvement did not appear at the lower force levels. For the 10 and 18 lbf forcing levels, at the
53 Hz resonance peak combined coherence improved throughout both structures (Figure 23 and
Figure 26). Similar to the 25 Hz improvements, these DOF combinations were spanning the
nonlinear gap, within the square frame and within the H-frame. Table 8 below summarizes these
observations.
31
Table 8: Summary of Results for Cases 3-5
Frequency
Range
Resonant Peak/
Antiresonance
Excited More at
Low, Medium or
High Forcing
Levels
DOF Combinations with MCCOH Improvements
Within HFrame
Within Square
Frame
Across
Connection
Points
24.25 Hz
Resonant Peak
Low to Medium
Yes
Yes
Yes
53.5 Hz
Resonant Peak
Medium to High
Yes
Yes
Yes
~385-400 Hz
Antiresonance
Medium to High
No
Yes
No
One thing to note is that for Case 3 with the 4 lb/shaker input, the largest changes for adjacent
points between multiple combined coherence and multiple coherence occurred when combining
Points 21 and 12 as well as Points 21 and 13 (Figure 22). At 400%, the changes across these
DOFs dwarfed other changes. It is interesting to note that the shortest and most direct path
between the two DOF combinations is perfectly linear.
Also, oftentimes improvements in MCCOH over MCOH are present not only between various
DOFs in the same direction (e.g., 44X and 45X) but also between DOFs in orthogonal directions
(e.g., 42Y and 44X). In fact, Figure 24 shows that cross directional improvements are sometimes
more common than improvements in the same direction.
32
Table 9: Case 3 improvements of 20% or more in Combined Coherence for Case 3 for adjacent points in the
same direction
pt1 dir1
pt2 dir2 Freq (Hz) MCCOH MCOH1 MCOH2 % Difference (anti)resonance
not at
2
+Z
3
+Z
111.75
0.7952
0.6431
0.6614
20.2148
resonance
21
-Y
13
+Y
21.5
0.9729
0.0761
0.798
21.9237
21
-Y
13
+Y
24
0.951
0.2195
0.2365
302.0953
21
-Y
13
+Y
24.25
0.956
0.1791
0.1869
411.3762
21
-Y
13
+Y
24.5
0.9575
0.7518
0.7063
27.3636
small peaks
21
-Y
13
+Y
25
0.9643
0.6023
0.5271
60.1085
21
-Y
13
+Y
25.25
0.9604
0.4645
0.4174
106.7339
21
-Y
13
+Y
26.75
0.9537
0.6867
0.7139
33.5941
21
-Y
13
+Y
27
0.9557
0.6055
0.6553
45.8423
21
-Y
13
+Y
100.75
0.9267
0.6184
0.4771
49.8671
antiresonance
21
-Y
13
+Y
101
0.9326
0.7702
0.586
21.093
21
-X
12
+X
47.5
0.6358
0.4788
0.5266
20.735
antiresonance
21
-X
12
+X
48.5
0.4374
0.2988
0.357
22.508
21
-Y
12
+Y
21.5
0.9763
0.0761
0.7661
27.4387
21
-Y
12
+Y
24
0.9706
0.2195
0.2167
342.1419
21
-Y
12
+Y
24.25
0.9688
0.1791
0.1783
440.9924
21
-Y
12
+Y
24.5
0.9706
0.7518
0.7106
29.1004
small peaks
21
-Y
12
+Y
25
0.972
0.6023
0.5149
61.3782
21
-Y
12
+Y
25.25
0.9653
0.4645
0.4061
107.7982
21
-Y
12
+Y
26.75
0.9763
0.6867
0.7641
27.7582
21
-Y
12
+Y
27
0.9787
0.6055
0.7203
35.872
antiresonance/
21
-Y
12
+Y
100.75
0.8463
0.6184
0.3673
36.8678
peak for least
excited FRF
resonance
45
+Z
44
+Z
53.5
0.9477
0.6981
0.6596
35.7501
12
+Y
42
+Y
24
0.3484
0.2167
0.2494
39.7216
small peaks
12
+Y
42
+Y
24.25
0.2937
0.1783
0.1942
51.265
12
+Y
42
+Y
25.25
0.5398
0.4061
0.393
32.924
12
+Z
42
+Z
24
0.9086
0.2901
0.3573
154.3206
12
+Z
42
+Z
24.25
0.9088
0.3965
0.2953
129.2423
small peaks
12
+Z
42
+Z
25
0.95
0.7556
0.5166
25.7258
12
+Z
42
+Z
25.25
0.9123
0.6609
0.5631
38.0488
small peak
13
+Y
42
+Y
25.25
0.5085
0.4174
0.393
21.827
13
+Z
42
+Z
24
0.7111
0.4818
0.3573
47.592
small peak
13
+Z
42
+Z
24.25
0.6659
0.4688
0.2953
42.0393
3
+Z
44
+Z
25
0.7353
0.5973
0.4868
23.0976
small peak
3
+Z
44
+Z
25.25
0.6467
0.4372
0.465
39.067
4
+Z
44
+Z
24
0.6864
0.3511
0.0665
95.4814
small peak
4
+Z
44
+Z
24.25
0.7416
0.4718
0.0394
57.1842
13
+Y
45
+Y
24.25
0.241
0.1869
0.1839
28.9065
small peaks
13
+Y
45
+Y
25.25
0.5577
0.4174
0.3641
33.626
14
+Y
45
+Y
24
0.3211
0.2354
0.2414
33.0263
14
+Y
45
+Y
24.25
0.3296
0.1985
0.1839
66.0124
small peaks
14
+Y
45
+Y
25
0.7249
0.5518
0.5052
31.3702
14
+Y
45
+Y
25.25
0.6444
0.4386
0.3641
46.9463
33
Table 10: Case 4 improvements of 20% or more in Combined Coherence for Case 4 for adjacent points in the
same direction
(anti)resonance
pt1 dir1 pt2 dir2 Freq (Hz) MCCOH
MCOH1 MCOH2 % Difference
21
21
21
21
21
11
45
45
45
45
45
45
45
45
45
45
45
45
45
45
45
45
12
12
12
13
13
14
-Y
-Y
-X
-Y
-Y
+Z
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+Z
+Y
+Z
+Z
+Z
+Y
+Y
13
13
12
12
12
12
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
42
42
42
42
45
45
+Y
+Y
+X
+Y
+Y
+Z
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+Z
+Y
+Z
+Z
+Z
+Y
+Y
24
24.25
53.75
24
24.25
53.75
392.5
392.75
393
393.75
394
394.25
394.75
395
395.25
395.5
395.75
396
396.25
396.75
397
53.5
24.25
24
24.25
24.25
24.25
24.25
0.9939
0.9942
0.8059
0.9964
0.9965
0.9211
0.8623
0.8583
0.8777
0.8699
0.8591
0.8641
0.8808
0.8618
0.8342
0.8538
0.8278
0.7992
0.8131
0.809
0.8304
0.9346
0.775
0.9787
0.9771
0.8925
0.7784
0.8237
0.7672
0.687
0.6533
0.7672
0.687
0.7504
0.6961
0.689
0.7164
0.7112
0.6823
0.6944
0.7186
0.6657
0.5947
0.6063
0.5757
0.5524
0.5685
0.6171
0.6157
0.7615
0.6281
0.7642
0.76
0.7106
0.64
0.6521
34
0.7244
0.64
0.5347
0.7174
0.6281
0.5504
0.6431
0.6078
0.6772
0.6591
0.6337
0.6383
0.6468
0.6228
0.5823
0.5823
0.6167
0.6229
0.6631
0.6738
0.6621
0.6906
0.5719
0.7813
0.6502
0.6502
0.5558
0.5558
29.5463
44.7169
23.3632
29.8729
45.0495
22.7411
23.8802
24.5721
22.5216
22.3078
25.9134
24.4415
22.5627
29.4516
40.2745
40.8342
34.2281
28.2962
22.6324
20.061
25.4282
22.7206
23.4036
25.2672
28.57
25.583
21.6374
26.3235
small peak/valley
resonance
small peak/valley
resonance
antiresonance
resonance
small peak/valley
small peak/valley
small peak/valley
small peak/valley
small peak/valley
Table 11: Case 5 improvements of 20% or more in Combined Coherence for Case 5 for adjacent points in the
same direction
(anti)resonance
pt1 dir1 pt2 dir2 Freq (Hz) MCCOH
MCOH1 MCOH2 % Difference
small peak
21
-Y
13 +Y
24.25
0.9979
0.8245
0.7946
21.0337
21
-X
12 +X
53.5
0.8404
0.6745
0.6303
24.5933
resonance
21
-X
12 +X
53.75
0.8642
0.7039
0.6831
22.7749
21
45
45
45
45
45
45
45
45
45
45
45
45
45
45
45
45
45
45
45
45
2
-Y
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
-X
12
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
41
+Y
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
24.25
391.5
391.75
392.75
393
393.25
393.5
393.75
394
394.25
394.5
394.75
395
395.25
395.5
395.75
396
396.25
396.5
396.75
397
53.5
0.9987
0.8481
0.8499
0.8367
0.831
0.8641
0.8498
0.8451
0.8338
0.8458
0.8218
0.7964
0.8044
0.8051
0.7866
0.8188
0.8169
0.8177
0.802
0.7823
0.7833
0.7479
0.8245
0.6905
0.7065
0.665
0.6576
0.7029
0.6792
0.6478
0.6365
0.6654
0.6028
0.5458
0.5504
0.5367
0.5354
0.5901
0.5456
0.583
0.5543
0.5127
0.5201
0.6217
35
0.7843
0.6765
0.6822
0.6648
0.6661
0.668
0.6392
0.6163
0.577
0.6054
0.6336
0.6344
0.6344
0.5851
0.5763
0.5552
0.5312
0.5612
0.5855
0.639
0.6209
0.591
21.1232
22.8245
20.2848
25.8344
24.7628
22.9372
25.1206
30.4624
31.0061
27.1037
29.7134
25.5227
26.7836
37.5845
36.4775
38.7574
49.7291
40.2581
36.9719
22.4381
26.1532
20.2894
small peak/valley
antiresonance
resonance
Percent Difference Between MCCOH and the Maximum MCOH at f = 24.25 Hz.
Plot Min/Max = -0.05% and 440.9924%
45 +X
45 +Z
45 +Y
44 +X
44 +Z
44 +Y
43 +X
43 +Z
43 +Y
42 +X
42 +Z
42 +Y
41 +X
41 +Z
41 +Y
14 +Y
14 +X
14 +Z
13 +Y
13 +X
13 +Z
12 +Y
12 +X
12 +Z
11 +Y
11 +X
11 +Z
21 -X
21 -Y
21 +Z
4 -Y
4 -X
4 +Z
3 -Y
3 -X
3 +Z
2 -Y
2 -X
2 +Z
1 -Y
1 -X
1 +Z
101 +Z
100 +X
100 +X
400
350
300
250
200
150
100
50
2 +Z
4 +Z
11 +Z
13 +Z
Output DOF #1
41 +Y
43 +Y
45 +Y
0
Percent Difference Between MCCOH and the Maximum MCOH at f = 24.25 Hz.
Plot Min/Max = -0.05% and 5%
5
45 +X
45 +Z
45 +Y
44 +X
44 +Z
44 +Y
43 +X
43 +Z
43 +Y
42 +X
42 +Z
42 +Y
41 +X
41 +Z
41 +Y
14 +Y
14 +X
14 +Z
13 +Y
13 +X
13 +Z
12 +Y
12 +X
12 +Z
11 +Y
11 +X
11 +Z
21 -X
21 -Y
21 +Z
4 -Y
4 -X
4 +Z
3 -Y
3 -X
3 +Z
2 -Y
2 -X
2 +Z
1 -Y
1 -X
1 +Z
101 +Z
100 +X
100 +X
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
2 +Z
4 +Z
11 +Z
13 +Z
Output DOF #1
41 +Y
Figure 22: Case 3: MCCOH for all DOFs at 24.25 Hz
36
43 +Y
45 +Y
Percent Difference Between CCOH and the Maximum OCOH at f = 53.75 Hz.
Plot Min/Max = -0.05% and 5%
2
#
F
O
D
t
u
pt
u
O
5
45 +X
45 +Z
45 +Y
44 +X
44 +Z
44 +Y
43 +X
43 +Z
43 +Y
42 +X
42 +Z
42 +Y
41 +X
41 +Z
41 +Y
14 +Y
14 +X
14 +Z
13 +Y
13 +X
13 +Z
12 +Y
12 +X
12 +Z
11 +Y
11 +X
11 +Z
21 -X
21 -Y
21 +Z
4 -Y
4 -X
4 +Z
3 -Y
3 -X
3 +Z
2 -Y
2 -X
2 +Z
1 -Y
1 -X
1 +Z
101 +Z
100 +X
100 +X
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
2 +Z
4 +Z
11 +Z
13 +Z
Output DOF #1
41 +Y
43 +Y
45 +Y
Figure 23: Case 4: MCCOH for all DOFs at 53.75 Hz
Percent Difference Between MCCOH and the Maximum MCOH at f = 396 Hz.
Plot Min/Max = -0.05% and 5%
5
45 +X
45 +Z
45 +Y
44 +X
44 +Z
44 +Y
43 +X
43 +Z
43 +Y
42 +X
42 +Z
42 +Y
41 +X
41 +Z
41 +Y
14 +Y
14 +X
14 +Z
13 +Y
13 +X
13 +Z
12 +Y
12 +X
12 +Z
11 +Y
11 +X
11 +Z
21 -X
21 -Y
21 +Z
4 -Y
4 -X
4 +Z
3 -Y
3 -X
3 +Z
2 -Y
2 -X
2 +Z
1 -Y
1 -X
1 +Z
101 +Z
100 +X
100 +X
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
2 +Z
4 +Z
11 +Z
13 +Z
Output DOF #1
41 +Y
Figure 24: Case 5: MCCOH for all DOFs at 396 Hz
37
43 +Y
45 +Y
FRF Magnitude vs. Frequency for Various Input/Output Combinations
FRF
FRF
FRF
FRF
10
)f
bl
/
n(i
e
d
ut
i
n
g
a
M
F
R
F
10
10
21 -Y,100 +X
13 +Y,100 +X
21 -Y,101 +Z
13 +Y,101 +Z
-2
-3
-4
20
40
60
80
100
120
140
160
Frequency (Hz)
Multiple Coherence for Points 13 and 21 directions +Y and -Y.
1
0.8
MCOH 13+Y
MCOH 21-Y
0.6
H
O
C
M
0.4
0.2
0
0
100
200
300
500
600
700
800
Multiple Combined Coherence for Point 13 and 21 directions +Y and -Y.
1
0.8
H
O
C
C
M
400
Combined Coherence
7 Frequencies with 5% Improvement above 10 Hz.
0.6
0.4
0.2
0
0
100
200
300
400
Frequency (Hz)
500
600
700
800
Figure 25: Case 4 FRF and coherence functions showing the 24.25 Hz small resonances and 100.75 Hz
antiresonance valleys at which combined coherence improves over multiple coherence
38
FRF Magnitude vs. Frequency for Various Input/Output Combinations
10
10
)f
bl
/
n(i
e
d
ut
i
n
g
a
M
F
R
F
10
10
FRF
FRF
FRF
FRF
-1
11 +Z,100 +X
12 +Z,100 +X
11 +Z,101 +Z
12 +Z,101 +Z
-2
-3
-4
40
60
80
100
120
140
160
180
200
Frequency (Hz)
Multiple Coherence for Points 11 and 12 directions +Z and +Z.
1
MCOH 11+Z
MCOH 12+Z
0.8
0.6
0.4
0.2
0
0
100
200
300
400
500
600
700
800
Multiple Combined Coherence for Point 11 and 12 directions +Z and +Z.
1
0.8
Combined Coherence
2 Frequencies with 5% Improvement above 10 Hz.
0.6
0.4
0.2
0
0
100
200
300
400
500
600
700
800
Frequency (Hz)
Figure 26: Case 4 FRF and coherence functions showing the 53.75 Hz resonant peak at which combined
coherence improves over multiple coherence
39
FRF Magnitude vs. Frequency for Various Input/Output Combinations
10
)f
bl
/
n(i
e
d
ut
i
n
g
a
M
F
R
F
10
10
10
FRF
FRF
FRF
FRF
-2
44 +X,100 +X
45 +X,100 +X
44 +X,101 +Z
45 +X,101 +Z
-3
-4
-5
320
340
360
380
400
420
440
460
Frequency (Hz)
Multiple Coherence for Points 44 and 45 directions +X and +X.
1
0.8
MCOH 44+X
MCOH 45+X
0.6
0.4
0.2
0
0
100
200
300
400
500
600
700
800
Multiple Combined Coherence for Point 44 and 45 directions +X and +X.
1
0.8
Combined Coherence
44 Frequencies with 5% Improvement above 10 Hz.
0.6
0.4
0.2
0
0
100
200
300
400
500
Frequency (Hz)
600
700
800
Figure 27: Case 4 FRF and coherence functions showing the 395 Hz antiresonance valley at which combined
coherence improves over multiple coherence
40
Case 6: Effect of Cyclic Averaging on Nominally Nonlinear System
The mounting of the square frame to the H-frame was the same for Case 6 as in Cases 3-5. Since
Case 4 seemed to best balance whether the combined coherence improvements were located at
resonances or antiresonances, 10 lbf peak input per shaker was used. The only setup
modifications from Case 4 were that 75% burst random excitation and cyclic averaging were
used. This was done to see if reducing leakage errors drastically modified the frequency and
DOF combinations for which MCCOH improved (as in Case 2 compared to Case 1).
Table 12: Test Setup for Case 6
Test Setup Description
Accelerometer
Spread Out,
Excitation Type
75% Burst Random
Distribution
Figure 10
Case 4 Input Voltage 3 V
Peak Force
10 lbf
Shaker #1 Location
Pt. 100
Shaker #2 Location
Pt. 101
# of Spectral Averages 150
# of Cyclic Averages 10
Attachment Method: Four Vehicle Body Mounts at each corner of the square frame. Two
bolts barely hand tightened. Two bolts removed altogether. See Figure 20 and Figure 21.
As can be seen from Table 13 below it appears that cyclic averaging not only reduced leakage,
but changed the DOFs for which combined coherence detects changes (similar to Case 2). In the
case of the small resonance peaks around 25 Hz, which before registered very large changes,
virtually no change in MCCOH is seen relative MCOH (Figure 28). This paralleled the expected
reduction of leakage at all DOFs at that frequency (see Figure 32 through Figure 35). In the case
of the strong resonance at 53.5 Hz, combined coherence again detected no improvement at the
resonant frequency (Figure 30). It should be noted, however, that just to either side of the
resonant frequency, there was still some improvement when combining DOFs from just the
square frame (Figure 29). This again paralleled the expected reduction of leakage on all DOFs
for which combined coherence improved (see Figure 32 through Figure 35). In comparing Case
41
6 to Case 4 at the 390-400 Hz antiresonance valley, there was very little change. Virtually the
same DOFs registered improvement in MCCOH over MCOH (Figure 31). It should be noted
that when comparing MCCOH for the off-peak 55 Hz and for the 396 Hz antiresonance, the
DOF combinations which improved, while not exactly the same as in Case 4, are in the same
spatial vicinity and are isolated to the square frame. As in Cases 1 and 2, MCCOH for Case 6 no
longer improves at frequencies where leakage effects are greatly reduced.
Table 13: Table of improvements of 10% or more in Combined Coherence for Case 5 for adjacent points in
the same direction (consecutive frequencies enumerated but not listed)
Freq
Percent
Resonance/
#
Pt. 1 Dir. 1 Pt. 2 Dir. 2 (Hz) MCCOH MCOH1 MCOH2 Difference Antiresonance
1
2
3
38
39
40
42 +X
42 +X
45 +X
45 +X
44 +Y
44 +Y
45 +X
45 +X
44 +X
44 +X
41 +Y
41 +Y
338.75
339.00
389.00
398.25
204.75
206.00
0.9744
0.9728
0.8807
0.8075
0.9763
0.9763
42
0.8696
0.8741
0.7988
0.4967
0.8808
0.8817
0.8808
0.8837
0.7959
0.6535
0.8813
0.8873
10.6270 antiresonance
10.0823
10.2587
antiresonance
23.5625
10.7755
antiresonance
10.0333
Percent Difference Between MCCOH and the Maximum MCOH at f = 24.25 Hz.
Plot Min/Max = -0.05% and 5%
5
45 +X
45 +Z
45 +Y
44 +X
44 +Z
44 +Y
43 +X
43 +Z
43 +Y
42 +X
42 +Z
42 +Y
41 +X
41 +Z
41 +Y
14 +Y
14 +X
14 +Z
13 +Y
13 +X
13 +Z
12 +Y
12 +X
12 +Z
11 +Y
11 +X
11 +Z
21 -X
21 -Y
21 +Z
4 -Y
4 -X
4 +Z
3 -Y
3 -X
3 +Z
2 -Y
2 -X
2 +Z
1 -Y
1 -X
1 +Z
101 +Z
100 +X
100 +X
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
2 +Z
4 +Z
11 +Z
13 +Z
Output DOF #1
41 +Y
43 +Y
45 +Y
Figure 28: Case 6: MCCOH for all DOFs at 24.25 Hz
Percent Difference Between MCCOH and the Maximum MCOH at f = 52 Hz.
Plot Min/Max = -0.05% and 5%
5
45 +X
45 +Z
45 +Y
44 +X
44 +Z
44 +Y
43 +X
43 +Z
43 +Y
42 +X
42 +Z
42 +Y
41 +X
41 +Z
41 +Y
14 +Y
14 +X
14 +Z
13 +Y
13 +X
13 +Z
12 +Y
12 +X
12 +Z
11 +Y
11 +X
11 +Z
21 -X
21 -Y
21 +Z
4 -Y
4 -X
4 +Z
3 -Y
3 -X
3 +Z
2 -Y
2 -X
2 +Z
1 -Y
1 -X
1 +Z
101 +Z
100 +X
100 +X
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
2 +Z
4 +Z
11 +Z
13 +Z
Output DOF #1
41 +Y
Figure 29: Case 6: MCCOH for all DOFs at 52 Hz
43
43 +Y
45 +Y
Percent Difference Between MCCOH and the Maximum MCOH at f = 53.5 Hz.
Plot Min/Max = -0.05% and 5%
5
45 +X
45 +Z
45 +Y
44 +X
44 +Z
44 +Y
43 +X
43 +Z
43 +Y
42 +X
42 +Z
42 +Y
41 +X
41 +Z
41 +Y
14 +Y
14 +X
14 +Z
13 +Y
13 +X
13 +Z
12 +Y
12 +X
12 +Z
11 +Y
11 +X
11 +Z
21 -X
21 -Y
21 +Z
4 -Y
4 -X
4 +Z
3 -Y
3 -X
3 +Z
2 -Y
2 -X
2 +Z
1 -Y
1 -X
1 +Z
101 +Z
100 +X
100 +X
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
2 +Z
4 +Z
11 +Z
13 +Z
Output DOF #1
41 +Y
43 +Y
45 +Y
Figure 30: Case 6: MCCOH for all DOFs at 53.5 Hz
Percent Difference Between MCCOH and the Maximum MCOH at f = 397 Hz.
Plot Min/Max = -0.05% and 5%
5
45 +X
45 +Z
45 +Y
44 +X
44 +Z
44 +Y
43 +X
43 +Z
43 +Y
42 +X
42 +Z
42 +Y
41 +X
41 +Z
41 +Y
14 +Y
14 +X
14 +Z
13 +Y
13 +X
13 +Z
12 +Y
12 +X
12 +Z
11 +Y
11 +X
11 +Z
21 -X
21 -Y
21 +Z
4 -Y
4 -X
4 +Z
3 -Y
3 -X
3 +Z
2 -Y
2 -X
2 +Z
1 -Y
1 -X
1 +Z
101 +Z
100 +X
100 +X
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
2 +Z
4 +Z
11 +Z
13 +Z
Output DOF #1
41 +Y
Figure 31: Case 6: MCCOH for all DOFs at 397 Hz
44
43 +Y
45 +Y
Overlay Plot of FRF's and Multiple Coherence for DOF 45 +X for Cases with and without Burst Random Input and Cyclic Averaging
0
10
Pure Random Input and No Cyclic Averging: 45 +X
Burst Random Input and Cyclic Averaging: 45 +X
e
d
ut
i
n
g
a
M
F
R
F
10
10
10
-2
-4
-6
0
100
200
300
400
500
600
700
800
1
0.8
e
d
ut
i
n
g
a
M
H
O
C
M
0.6
0.4
0.2
0
Pure Random Input and No Cyclic Averging: 45 +X
Burst Random Input and Cyclic Averaging: 45 +X
0
100
200
300
400
500
600
700
800
Frequency (Hz)
Figure 32: Overlay of FRF and Coherence Plots Showing the Effect of a Burst Random Input and Cyclic
Averaging on the Nominally Nonlinear System
Overlay Plot of FRF's and Multiple Coherence for DOF 45 +Y for Cases with and without Burst Random Input and Cyclic Averaging
0
10
Pure Random Input and No Cyclic Averaging: 45 +Y
Burst Random Input and Cyclic Averaging: 45 +Y
e
d
ut
i
n
g
a
M
F
R
F
10
10
10
-2
-4
-6
0
100
200
300
400
500
600
700
800
1
0.8
e
d
ut
i
n
g
a
M
H
O
C
M
0.6
0.4
Pure Random Input and No Cyclic Averaging: 45 +Y
Burst Random Input and Cyclic Averaging: 45 +Y
0.2
0
0
100
200
300
400
500
600
700
800
Frequency (Hz)
Figure 33: Overlay of FRF and Coherence Plots Showing the Effect of a Burst Random Input and Cyclic
Averaging on the Nominally Nonlinear System
45
Overlay Plot of FRF's and Multiple Coherence for DOF 12 +Y for Cases with and without Burst Random Input and Cyclic Averaging
0
10
e
d
ut
i
n
g
a
M
F
R
F
10
10
-2
-4
Pure Random Input and No Cyclic Averaging: 12 +Y
Burst Random Input and Cyclic Averaging: 12 +Y
10
-6
0
100
200
300
400
500
600
700
800
1
0.8
e
d
ut
i
n
g
a
M
H
O
C
M
0.6
0.4
0.2
0
Pure Random Input and No Cyclic Averaging: 12 +Y
Burst Random Input and Cyclic Averaging: 12 +Y
0
100
200
300
400
500
600
700
800
Frequency (Hz)
Figure 34: Overlay of FRF and Coherence Plots Showing the Effect of a Burst Random Input and Cyclic
Averaging on the Nominally Nonlinear System
Overlay Plot of FRF's and Multiple Coherence for DOF 12 +Z for Cases with and without Burst Random Input and Cyclic Averaging
0
10
Pure Random Input and No Cyclic Averaging: 12 +Z
Burst Random Input and Cyclic Averaging: 12 +Z
-1
10
e
d
ut
i
n
g
a
M
F
R
F
10
10
10
-2
-3
-4
0
100
200
300
400
500
600
700
800
1
0.8
e
d
ut
i
n
g
a
M
H
O
C
M
0.6
0.4
0.2
0
Pure Random Input and No Cyclic Averaging: 12 +Z
Burst Random Input and Cyclic Averaging: 12 +Z
0
100
200
300
400
500
600
700
800
Frequency (Hz)
Figure 35: Overlay of FRF and Coherence Plots Showing the Effect of a Burst Random Input and Cyclic
Averaging on the Nominally Nonlinear System
46
Case 7: Spatial Resolution of the Sensors
Case 7 is similar to Case 6, except the accelerometers have been moved to give greater spatial
resolution in a particular part of the structure (Figure 11, p. 19). Table 14 below gives details of
the entire setup with major changes highlighted. The goal of this test is to determine if it is
simply an anomaly that 44X and 45X showed the greatest increase in combined coherence for
Case 6, or whether that trend holds as well with a finer spatial resolution on the upper frame.
The question arises because points 44X and 45X are DOFs with what appear from visual
inspection to be the same mass weighting with no possible nonlinearities in what is by far the
shortest, most direct path between them.
Table 14: Test Setup for Case 7
Test Setup Description
Accelerometer
Concentrated,
Excitation Type
Random
Distribution
Figure 11
Input Voltage
3V
Peak Force
10 lbf
Shaker #1 Location
Pt. 100
Shaker #2 Location
Pt. 101
# of Spectral Averages 150
# of Cyclic Averages 8
Attachment Method: Four Vehicle Body Mounts at each corner of the square frame. Two
bolts barely hand tightened. Two bolts removed altogether. See Figure 20 and Figure 21.
As Table 15 below shows, the results are very similar to Case 6, especially in respect to the
improvement of combined coherence in the upper frame. The improvement of combined
coherence in the “linear” section from 44 to 48 to 45 is confirmed. In comparing Figure 36
below with Figure 31 (p. 44) one can see that combined coherence improves throughout the
upper structure in the 390-400 Hz frequency range and is not isolated to any particular DOF.
This leads to questions about whether paths other than the most direct one can play a significant
role in the combined coherence calculation. This would make combined coherence much more
47
difficult to use as a tool to spatially locate nonlinearities. The largest improvement in MCCOH
over MCOH occurs at 120 Hz. This is very suspect, being a multiple of 60 cycle noise, but is
included because the improvement was so much larger than other improvements (e.g., 13Y and
15Y in Table 15).
Table 15: Table of improvements of 20% or more in Combined Coherence for Case 7 for adjacent points in
the same direction (consecutive frequencies enumerated but not listed)
Freq
#
Pt 1
Dir 1
Pt 2
Dir 2
(Hz)
MCCOH MCOH1
MCOH2
% Difference
1
3
-Y
5
+Y
21.5
0.9076
0.3652
0.5746
57.9527
2
3
-Y
53
+Y
21.5
0.9623
0.3652
0.7341
31.0792
3
13
+Y
15
-Y
120
0.9745
0.3109
0.1668
213.4371
4
54
+Y
15
-Y
120
0.9929
0.8178
0.1668
21.4048
5
45
+Z
48
+Z
389.75
0.9389
0.7703
0.708
21.8744
37
45
+Z
48
+Z
398.25
0.7429
0.5414
0.5818
27.6789
38
48
+Z
44
+Z
394
0.6656
0.4648
0.5442
22.3172
48
48
+Z
44
+Z
397.25
0.6052
0.4859
0.4767
24.5447
49
52
+X
45
+X
389.5
0.9489
0.6942
0.7838
21.077
90
52
+X
45
+X
400.25
0.7489
0.4977
0.5566
34.5369
91
52
-Y
45
+Y
28.25
0.9776
0.7797
0.795
22.9631
102
52
-Y
45
+Y
31
0.9721
0.7789
0.4678
24.8047
103
52
-Y
45
+Y
55
0.943
0.6572
0.7116
32.5248
105
52
-Y
45
+Y
55.5
0.9418
0.7789
0.5354
20.903
Percent Difference Between MCCOH and the Maximum MCOH at f = 396.75 Hz.
Plot Min/Max = -0.05% and 5%
5
45 +X
45 +Z
45 +Y
44 +X
44 +Z
44 +Y
43 +X
43 +Z
43 +Y
47 +X
47 +Z
47 +Y
46 +X
46 +Z
46 +Y
15 -Y
15 -X
15 +Z
13 +Y
13 +X
13 +Z
54 -X
54 +Y
54 +Z
52 -Y
52 +Z
52 +X
48 +X
48 +Z
48 +Y
5 +Y
5 +X
5 +Z
3 -Y
3 -X
3 +Z
53 -X
53 +Y
53 +Z
51 +Y
51 +Z
51 -X
101 +Z
100 +X
100 +X
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
53 +Z
5 +Z
52 +X
13 +Z
Output DOF #1
46 +Y
43 +Y
Figure 36: Case 7: MCCOH for all DOFs at 396.75 Hz
48
45 +Y
Case 8: Effect of Spatial Variation of the Input Forces
Case 8 is similar to Case 7, except shaker #1 has been moved to the upper frame (see Figure 11,
p. 19). Table 16 below gives details of the entire setup with major changes highlighted.
Table 16: Test Setup for Case 8
Test Setup Description
Accelerometer
Concentrated,
Excitation Type
Random
Distribution
Figure 11
Input Voltage
3V
Peak Force
10 lbf
Shaker #2 Location
Pt. 101
Shaker #1 Location
Pt. 102
# of Spectral Averages 150
# of Cyclic Averages 10
Attachment Method: Four Vehicle Body Mounts at each corner of the square frame. Two
bolts barely hand tightened. Two bolts removed altogether. See Figure 20 and Figure 21.
The first noticeable result of moving the shaker is that the number of DOFs with improvements
in combined coherence has been reduced (Table 17 below). Second, combined coherence no
longer detects any nonlinearities in the upper structure. Moving Shaker #1 from point 100 to
point 102 on the square frame has eliminated the detection of nonlinearities in that section of the
structure. In contrast, Shaker #2 is located at point 101 between DOFs 3 and 5 and a nonlinearity
is consistently detected in this part of the structure. These MCCOH improvements are
concentrated around the 123 Hz antiresonance valley (Figure 38).
Table 17: Table of improvements of 20% or more in Combined Coherence for Case 8 for adjacent points in
the same direction
pt1
dir1
pt2
dir2
Freq (Hz)
MCCOH MCOH1 MCOH2 % Difference
3
13
13
13
13
13
-X
+Z
+Z
+Z
+Z
+Z
5
54
54
54
54
54
+X
+Z
+Z
+Z
+Z
+Z
123.25
113.50
113.75
114.00
114.25
114.50
0.7135
0.7391
0.7146
0.6890
0.6412
0.7266
49
0.5469
0.5146
0.5695
0.5367
0.4825
0.5963
0.5185
0.6090
0.5382
0.5073
0.4458
0.4763
30.4685
21.3541
25.4825
28.3700
32.8984
21.8531
FRF Magnitude vs. Frequency for Various Input/Output Combinations
10
10
)f
bl
/
n(i
e
d
ut
i
n
g
a
M
F
R
F
10
10
10
FRF
FRF
FRF
FRF
-1
3 -X,100 +X
5 +X,100 +X
3 -X,101 +Z
5 +X,101 +Z
-2
-3
-4
-5
60
80
100
120
140
160
180
Frequency (Hz)
200
220
240
260
Multiple Coherence for Points 3 and 5 directions -X and +X.
1
MCOH 3-X
MCOH 5+X
0.8
0.6
H
O
C
M
0.4
0.2
0
0
50
100
150
200
250
300
Multiple Combined Coherence for Point 3 and 5 directions -X and +X.
1
0.8
H
O
C
C
M
0.6
0.4
Combined Coherence
6 Frequencies above 10 Hz with 5% Improvement
0.2
0
0
50
100
150
Frequency (Hz)
200
250
300
Figure 37: Case 7 FRF and coherence functions showing the 123.25 and 175 Hz antiresonance valleys at
which combined coherence improves over multiple coherence
50
FRF Magnitude vs. Frequency for Various Input/Output Combinations
FRF
FRF
FRF
FRF
10
)f
bl
/
n(i
e
d
ut
i
n
g
a
M
F
R
F
10
10
10
13 +Y,100 +X
15 -Y,100 +X
13 +Y,101 +Z
15 -Y,101 +Z
-2
-3
-4
-5
50
100
150
Frequency (Hz)
200
250
Multiple Coherence for Points 13 and 15 directions +Y and -Y.
1
0.8
0.6
H
O
C
M
0.4
0.2
0
MCOH 13+Y
MCOH 15-Y
0
100
200
300
400
500
600
700
800
Multiple Combined Coherence for Point 13 and 15 directions +Y and -Y.
1
0.8
H
O
C
C
M
0.6
0.4
Combined Coherence
10 Frequencies with 5% Improvement above 10 Hz.
0.2
0
0
100
200
300
400
Frequency (Hz)
500
600
700
800
Figure 38: Case 7 FRF and coherence functions showing the 175 Hz antiresonance valley at which combined
coherence improves over multiple coherence
51
Since all of the frequencies with significant multiple coherence improvements are at
antiresonances, the question arises as to why combined coherence did not register an
improvement at the 390 Hz antiresonance as in Case 6. The only change is the location of one
input force. From Figure 39 below it can be seen that MCOH greatly improves for the square
frame in this range because dominant FRF for this DOF is now linear. This would lead one to
believe that where the inputs are located has a significant effect on the ability of combined
coherence to detect nonlinearities. This also leads to more questions about how critical the
energy path is in this detection process. For instance, if most of the energy is coming from a
linear path, is combined coherence able to detect a nonlinearity from another less dominant path?
Also, if all of the energy comes through a nonlinearity some distance away, is combined
coherence able to distinguish that fact when combining two DOFs in very close proximity with
no nonlinearity between them? Also, does the number of nonlinear paths reduce the ability of
combined coherence to spatially locate nonlinearities?
10
e
d
ut
i
n
g
a
M
F
R
F
10
10
10
Overlay Plot of FRF's and Multiple Coherence for DOF 45 +X for Various Tests
0
-2
-4
-6
0
100
200
300
400
500
600
Test A: H2(45X, 100 skew)
Test A: H2(45X, 101 skew)
Test B: H2(45X, 102 skew)
Test
B: H2(45X, 101
700
800skew)
1
0.8
e
d
ut
i
n
g
a
M
H
O
C
M
Test A: MCOH(45X)
Test B: MCOH(45X)
0.6
0.4
0.2
0
0
100
200
300
400
500
600
700
800
Frequency (Hz)
Figure 39: Overlay of H(45x,*x) before and after moving shaker #1 from the H-Frame to the Square Frame
52
Case 9: Effect of Reducing the Number of Connection Points
Between the H-Frame and Square Frame to One
Case 9 is similar to Case 8, except shaker #1 has been moved and mounts were removed. Figure
40 shows the removal of the 3 mounts, while Figure 11 (p. 19) shows the location of the inputs.
Table 18 below gives details of the entire setup with major changes highlighted. The goal of this
test is to determine whether reducing the number of paths between the H-frame and square frame
improves the ability of combined coherence to detect and spatially locate nonlinearities.
Table 18: Test Setup for Case 9
Test Setup Description
Accelerometer
Concentrated,
Excitation Type
Random
Distribution
Figure 11
Input Voltage
3V
Peak Force
10 lbf
Shaker #1 Location
Pt. 100
Shaker #2 Location
Pt. 101
# of Spectral Averages 200
# of Cyclic Averages 10
Attachment Method: Three vehicle body mounts have been removed and the square frame is
supported by shock cord at each of those corners. The one original body mount that remains
is not modified in any way. See Figure 40 and Figure 41.
Figure 40: Pictures of the setup for Cases 9-11 showing the removal of 3 connection points and the suspension
of the square frame by shock cord
53
Figure 41: Layout and schematic of the connection point between the H-frame and the square frame for
Cases 9-11
Table 19 below shows that by isolating the square frame and by reducing the number of paths
that couple the two structures together, the number of combined coherence improvements greatly
increases. Depending on the frequency, combined coherence improvements are spread
throughout both structures (Figure 42) or isolated to the square frame (Figure 43 through Figure
46). Of the few improvements in MCCOH from combining a DOF from both structures, none
involve adjacent DOFs (e.g., Pt. 52 and Pt. 54), but DOFs with several other DOFs in between
the two (e.g. 51X and 15Z at 21.5 Hz in Figure 42). When looking at the frequencies and FRF’s
for which combined coherence improves, they are at both antiresonances (Figure 47, Figure 49,
Figure 50 and Figure 51) and resonant peaks (Figure 48, Figure 51 and Figure 52). Some of the
resonant peaks at which MCCOH improves are atypical in shape compared to other resonant
peaks around them. One example would be an extremely small resonant peak in the FRF relative
to the surrounding resonant peaks which only shows up in the FRF’s for a single input (e.g. see
54
21.5 Hz FRF in Figure 47). However, combined coherence does not seem to be able to spatially
locate these nonlinearities. Another interesting thing to note about combined coherence is that
often the largest increases come about from combining two DOFs in orthogonal directions at the
same spatial point (e.g., 44Y and 44Z in Figure 46). Off axis sensitivity may also be the cause of
some of these improvements in MCCOH over MCOH. The amplititude of some of the
improvements, however it unlikely that it is the only cause of cross axis MCCOH improvements.
55
Table 19: Case 9 improvements of 20% or more in Combined Coherence for Case 9 for adjacent points in the
same direction (closely spaced frequencies enumerated but not listed)
#
Pt 1 Dir 1 Pt 2 Dir 2 Freq (Hz) MCCOH MCOH1 MCOH2 % Difference
1
2
4
11
12
45
47
48
51
52
55
56
57
59
62
63
65
66
126
128
180
181
246
247
251
278
279
280
281
284
285
291
292
293
333
334
3
13
52
52
52
52
52
51
51
51
51
51
15
44
44
44
46
46
46
43
43
47
47
47
45
45
45
45
48
48
48
48
48
48
48
48
-Y
+Y
-Y
-Y
-Y
-Y
+Z
+Z
+Z
-X
-X
+Z
-Y
+X
+X
+Z
+X
+Z
+Z
+X
+X
+X
+X
+X
+Y
+Y
+Z
+Z
+X
+X
+X
+X
+Y
+Z
+Z
+Z
5
15
51
51
51
51
51
46
46
44
44
44
54
46
46
46
43
43
43
47
47
45
45
45
48
48
48
48
44
44
44
44
44
44
44
44
+Y
-Y
+Y
+Y
+Y
+Y
+Z
+Z
+Z
+X
+X
+Z
+Y
+X
+X
+Z
+X
+Z
+Z
+X
+X
+X
+X
+X
+Y
+Y
+Z
+Z
+X
+X
+X
+X
+Y
+Z
+Z
+Z
21.5
21.5
400
402
628.25
637
800
799.25
800
799.25
800
401.25
21.5
799.25
800
512
408.75
624
639.5
626.5
641.25
626.5
643.75
799.25
631.75
640
799.75
800
401
401.75
798.5
800
60
626.5
637
800
0.9074
0.8853
0.916
0.942
0.4678
0.392
0.4277
0.5254
0.3864
0.6923
0.666
0.8723
0.9028
0.761
0.6021
0.8729
0.7683
0.7712
0.8424
0.9378
0.6287
0.9171
0.8435
0.6998
0.856
0.8066
0.3253
0.266
0.8575
0.9045
0.795
0.6192
0.3473
0.6021
0.5697
0.2535
56
0.6074
0.7078
0.7579
0.4908
0.3152
0.2856
0.3035
0.4327
0.2923
0.5536
0.4868
0.5555
0.671
0.3508
0.3313
0.5823
0.625
0.6207
0.6832
0.2972
0.4954
0.7519
0.7023
0.5614
0.5432
0.5618
0.2675
0.2131
0.6553
0.6745
0.6187
0.2388
0.2737
0.4939
0.4552
0.1738
0.6191
0.671
0.6809
0.7799
0.3657
0.172
0.2923
0.3533
0.2579
0.3508
0.3313
0.7103
0.5984
0.609
0.4265
0.6831
0.2912
0.5581
0.5085
0.7519
0.5206
0.3413
0.6937
0.3492
0.7069
0.6634
0.2293
0.1738
0.6813
0.7256
0.576
0.3313
0.2218
0.4511
0.4442
0.1854
46.5721
25.0707
20.8619
20.7799
27.9134
37.25
40.9337
21.403
32.1966
25.0432
36.8184
22.8126
34.5451
24.9547
41.1844
27.7883
22.9373
24.2417
23.3002
24.7151
20.7683
21.9647
20.0991
24.6688
21.0935
21.5931
21.603
24.8584
25.856
24.6483
28.5094
86.8891
26.8904
21.8952
25.1508
36.7231
Percent Difference Between MCCOH and the Maximum MCOH at f = 21.5 Hz.
Plot Min/Max = -0.05% and 5%
5
45 +X
45 +Z
45 +Y
44 +X
44 +Z
44 +Y
43 +X
43 +Z
43 +Y
47 +X
47 +Z
47 +Y
46 +X
46 +Z
46 +Y
15 -Y
15 -X
15 +Z
13 +Y
13 +X
13 +Z
54 -X
54 +Y
54 +Z
52 -Y
52 +Z
52 +X
48 +X
48 +Z
48 +Y
5 +Y
5 +X
5 +Z
3 -Y
3 -X
3 +Z
53 -X
53 +Y
53 +Z
51 +Y
51 +Z
51 -X
101 +Z
100 +X
100 +X
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
53 +Z
5 +Z
52 +X
13 +Z
Output DOF #1
46 +Y
43 +Y
45 +Y
Figure 42: Case 9: MCCOH for all DOFs at 21.5 Hz
Percent Difference Between MCCOH and the Maximum MCOH at f = 401.5 Hz.
Plot Min/Max = -0.05% and 5%
5
45 +X
45 +Z
45 +Y
44 +X
44 +Z
44 +Y
43 +X
43 +Z
43 +Y
47 +X
47 +Z
47 +Y
46 +X
46 +Z
46 +Y
15 -Y
15 -X
15 +Z
13 +Y
13 +X
13 +Z
54 -X
54 +Y
54 +Z
52 -Y
52 +Z
52 +X
48 +X
48 +Z
48 +Y
5 +Y
5 +X
5 +Z
3 -Y
3 -X
3 +Z
53 -X
53 +Y
53 +Z
51 +Y
51 +Z
51 -X
101 +Z
100 +X
100 +X
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
53 +Z
5 +Z
52 +X
13 +Z
Output DOF #1
46 +Y
Figure 43: Case 9: MCCOH for all DOFs at 401.5 Hz
57
43 +Y
45 +Y
Percent Difference Between MCCOH and the Maximum MCOH at f = 512 Hz.
Plot Min/Max = -0.05% and 5%
5
45 +X
45 +Z
45 +Y
44 +X
44 +Z
44 +Y
43 +X
43 +Z
43 +Y
47 +X
47 +Z
47 +Y
46 +X
46 +Z
46 +Y
15 -Y
15 -X
15 +Z
13 +Y
13 +X
13 +Z
54 -X
54 +Y
54 +Z
52 -Y
52 +Z
52 +X
48 +X
48 +Z
48 +Y
5 +Y
5 +X
5 +Z
3 -Y
3 -X
3 +Z
53 -X
53 +Y
53 +Z
51 +Y
51 +Z
51 -X
101 +Z
100 +X
100 +X
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
53 +Z
5 +Z
52 +X
13 +Z
Output DOF #1
46 +Y
43 +Y
45 +Y
Figure 44: Case 9: MCCOH for all DOFs at 512 Hz
Percent Difference Between MCCOH and the Maximum MCOH at f = 632.5 Hz.
Plot Min/Max = -0.05% and 5%
5
45 +X
45 +Z
45 +Y
44 +X
44 +Z
44 +Y
43 +X
43 +Z
43 +Y
47 +X
47 +Z
47 +Y
46 +X
46 +Z
46 +Y
15 -Y
15 -X
15 +Z
13 +Y
13 +X
13 +Z
54 -X
54 +Y
54 +Z
52 -Y
52 +Z
52 +X
48 +X
48 +Z
48 +Y
5 +Y
5 +X
5 +Z
3 -Y
3 -X
3 +Z
53 -X
53 +Y
53 +Z
51 +Y
51 +Z
51 -X
101 +Z
100 +X
100 +X
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
53 +Z
5 +Z
52 +X
13 +Z
Output DOF #1
46 +Y
Figure 45: Case 9: MCCOH for all DOFs at 632.5 Hz
58
43 +Y
45 +Y
Percent Difference Between MCCOH and the Maximum MCOH at f = 800 Hz.
Plot Min/Max = -0.05% and 305.6667%
45 +X
45 +Z
45 +Y
44 +X
44 +Z
44 +Y
43 +X
43 +Z
43 +Y
47 +X
47 +Z
47 +Y
46 +X
46 +Z
46 +Y
15 -Y
15 -X
15 +Z
13 +Y
13 +X
13 +Z
54 -X
54 +Y
54 +Z
52 -Y
52 +Z
52 +X
48 +X
48 +Z
48 +Y
5 +Y
5 +X
5 +Z
3 -Y
3 -X
3 +Z
53 -X
53 +Y
53 +Z
51 +Y
51 +Z
51 -X
101 +Z
100 +X
100 +X
300
250
200
150
100
50
53 +Z
5 +Z
52 +X
13 +Z
Output DOF #1
46 +Y
43 +Y
45 +Y
Percent Difference Between MCCOH and the Maximum MCOH at f = 800 Hz.
Plot Min/Max = -0.05% and 5%
5
45 +X
45 +Z
45 +Y
44 +X
44 +Z
44 +Y
43 +X
43 +Z
43 +Y
47 +X
47 +Z
47 +Y
46 +X
46 +Z
46 +Y
15 -Y
15 -X
15 +Z
13 +Y
13 +X
13 +Z
54 -X
54 +Y
54 +Z
52 -Y
52 +Z
52 +X
48 +X
48 +Z
48 +Y
5 +Y
5 +X
5 +Z
3 -Y
3 -X
3 +Z
53 -X
53 +Y
53 +Z
51 +Y
51 +Z
51 -X
101 +Z
100 +X
100 +X
0
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
53 +Z
5 +Z
52 +X
13 +Z
Output DOF #1
46 +Y
Figure 46: Case 9: MCCOH for all DOFs at 800 Hz
59
43 +Y
45 +Y
FRF Magnitude vs. Frequency for Various Input/Output Combinations
10
10
FRF
FRF
FRF
FRF
-1
13 +Y,100 +X
15 -Y,100 +X
13 +Y,101 +Z
15 -Y,101 +Z
-2
M
R
10
-3
10
15
20
25
30
35
40
45
Frequency (Hz)
Multiple Coherence for Points 13 and 15 directions +Y and -Y.
1
0.8
MCOH 13+Y
MCOH 15-Y
0.6
H
O
C
M
0.4
0.2
0
0
100
200
300
500
600
700
800
Multiple Combined Coherence for Point 13 and 15 directions +Y and -Y.
1
0.8
H
O
C
C
M
400
Combined Coherence
2 Frequencies above 10 Hz. with 5% Improvement
0.6
0.4
0.2
0
0
100
200
300
400
Frequency (Hz)
500
600
700
800
Figure 47: Case 9 FRF and coherence functions showing the 21.5 Hz combined coherence improvements over
multiple coherence
60
FRF Magnitude vs. Frequency for Various Input/Output Combinations
FRF
FRF
FRF
FRF
10
)f
bl
/
n(i
e
d
ut
i
n
g
a
M
F
R
F
10
10
48 +Y,100 +X
44 +Y,100 +X
48 +Y,101 +Z
44 +Y,101 +Z
-3
-4
-5
45
50
55
60
65
Frequency (Hz)
70
75
80
Multiple Coherence for Points 44 and 48 directions +Y and +Y.
1
0.8
0.6
H
O
C
M
0.4
MCOH 44+Y
MCOH 48+Y
0.2
0
0
50
100
150
200
250
300
Multiple Combined Coherence for Point 44 and 48 directions +Y and +Y.
1
0.8
H
O
C
C
M
0.6
0.4
Combined Coherence
3 Frequencies above 10 Hz. with 5% Improvement
0.2
0
0
50
100
150
Frequency (Hz)
200
250
300
Figure 48: Case 9 FRF and coherence functions showing the 45 Hz combined coherence improvements over
multiple coherence
61
FRF Magnitude vs. Frequency for Various Input/Output Combinations
10
)f
bl
/
n(i
e
d
ut
i
n
g
a
M
F
R
F
10
10
10
FRF
FRF
FRF
FRF
-2
46 +Z,100 +X
44 +Z,100 +X
46 +Z,101 +Z
44 +Z,101 +Z
-3
-4
-5
480
490
500
510
520
530
Frequency (Hz)
540
550
560
570
Multiple Coherence for Points 44 and 46 directions +Z and +Z.
1
0.8
0.6
H
O
C
M
0.4
0.2
0
MCOH 44+Z
MCOH 46+Z
0
100
200
300
400
500
600
700
800
Multiple Combined Coherence for Point 44 and 46 directions +Z and +Z.
1
0.8
H
O
C
C
M
0.6
0.4
0.2
0
Combined Coherence
5 Frequencies above 10 Hz. with 5% Improvement
0
100
200
300
400
Frequency (Hz)
500
600
700
800
Figure 49: Case 9 FRF and coherence functions showing the 512 Hz combined coherence improvements over
multiple coherence
62
FRF Magnitude vs. Frequency for Various Input/Output Combinations
10
)f
bl
/
n(i
e
d
ut
i
n
g
a
M
F
R
F
10
10
10
FRF
FRF
FRF
FRF
-2
46 +X,100 +X
47 +X,100 +X
46 +X,101 +Z
47 +X,101 +Z
-3
-4
-5
300
320
340
360
380
400
420
Frequency (Hz)
440
460
480
500
Multiple Coherence for Points 46 and 47 directions +X and +X.
1
0.8
0.6
H
O
C
M
0.4
MCOH 46+X
MCOH 47+X
0.2
0
0
100
200
300
400
500
600
700
800
Multiple Combined Coherence for Point 46 and 47 directions +X and +X.
1
0.8
H
O
C
C
M
0.6
0.4
Combined Coherence
84 Frequencies above 10 Hz. with 5% Improvement
0.2
0
0
100
200
300
400
Frequency (Hz)
500
600
700
800
Figure 50: Case 9 FRF and coherence functions showing the 400 Hz combined coherence improvements over
multiple coherence
63
10
FRF Magnitude vs. Frequency for Various Input/Output Combinations
-1
FRF Magnitude vs. Frequency for Various Input/Output Combinations
10
10
)f
bl
/
n(i
e
d
ut
i
n
g
a
M
F
R
F
10
-2
10
)f
bl
/
n(i
e
d
ut
i
n
g
a
M
F
R
F
-3
10
10
10
FRF
FRF
FRF
FRF
-1
51 +Y,100 +X
52 -Y,100 +X
51 +Y,101 +Z
52 -Y,101 +Z
-2
-3
-4
-4
360
370
380
390
400
410
420
Frequency (Hz)
430
440
450
460
580
600
620
640
660
Frequency (Hz)
680
700
720
740
Multiple Coherence for Points 51 and 52 directions +Y and -Y.
1
0.8
0.6
H
O
C
M
0.4
0.2
0
MCOH 51+Y
MCOH 52-Y
0
100
200
300
400
500
600
700
800
Multiple Combined Coherence for Point 51 and 52 directions +Y and -Y.
1
0.8
H
O
C
C
M
0.6
0.4
0.2
0
Combined Coherence
73 Frequencies above 10 Hz. with 5% Improvement
0
100
200
300
400
Frequency (Hz)
500
600
700
Figure 51: Case 9 FRF and coherence functions showing the 400 and 635 Hz combined coherence
improvements over multiple coherence
64
800
10
10
)f
bl
/
n(i
e
d
ut
i
n
g
a
M
F
R
F
10
10
FRF Magnitude vs. Frequency for Various Input/Output Combinations
-1
FRF
FRF
FRF
FRF
44 +Y,100 +X
44 +Z,100 +X
44 +Y,101 +Z
44 +Z,101 +Z
-2
-3
-4
700
710
720
730
740
750
760
Frequency (Hz)
770
780
790
800
Multiple Coherence for Points 44 and 44 directions +Y and +Z.
1
0.8
0.6
H
O
C
M
0.4
0.2
0
MCOH 44+Y
MCOH 44+Z
0
100
200
300
400
500
600
700
800
Multiple Combined Coherence for Point 44 and 44 directions +Y and +Z.
1
0.8
H
O
C
C
M
0.6
0.4
Combined Coherence
57 Frequencies above 10 Hz. with 5% Improvement
0.2
0
0
100
200
300
400
Frequency (Hz)
500
600
700
800
Figure 52: Case 9 FRF and coherence functions showing the 800 Hz combined coherence improvements over
multiple coherence
65
Cases 10-11: Effect of Spectral Averaging
Spectral averaging with random excitation is known to reduce the effects of nonlinearities,
sometimes making it possible to even get good estimates of FRF’s at frequencies affected by
nonlinearities. The setup for Cases 10-11 is the same as Case 9, except the number of spectral
averages has been varied and cyclic averaging has been eliminated. Table 20 below gives details
of the entire setup with major changes highlighted. The purpose of this test is to see if reducing
the number of spectral averages increases the ability of combined coherence to spatially locate
the nonlinearities.
Table 20: Test Setup for Cases 10-12
Excitation Type
Random
Test Setup Description
Accelerometer
Distribution
Peak Force
Shaker #2 Location
Concentrated,
Figure 11
10 lbf
Pt. 101
Input Voltage
3V
Shaker #1 Location
Pt. 100
Case 10:
25
# of Cyclic Averages 0
# of Spectral Averages
Case 11:
100
# of Cyclic Averages 0
# of Spectral Averages
Attachment Method: Three vehicle body mounts have been removed and the square frame is
supported by shock cord at each of those corners. The one original body mount that remains is
not modified in any way. See Figure 40 and Figure 41.
The 10 greatest percentage differences between combined coherence and coherence are given in
Table 21 and Table 22 below. Greater differences are seen with less averaging, indicating that
some of the effect of nonlinearities may be averaged out. Combined coherence improvements
for all DOF combinations are shown in Figure 53 through Figure 54 (638 Hz) and in Figure 55
through Figure 56 (800 Hz). An FRF showing the change in “smoothness” of the FRF and COH
plots at 638 Hz and 800 Hz is given in Figure 57. For the most part, the MCCOH improvements
still seem to be isolated primarily to DOF combinations either within the square frame or within
66
the H-frame. Thus the effect of averaging on the ability of combined coherence to spatially
locate nonlinearities is minimal. The MCCOH improvements over MCOH were completely
restricted to the square frame at the 638 and 800 Hz antiresonance areas (Figure 55 and Figure
56). These spatial trends were not affected by spectral averaging. Table 23 below summarizes
the similarities in features for the 638 Hz and 800 Hz antiresonance areas.
Table 21: 10 Greatest Differences between MCCOH and MCOH for Case 10 (25 Spectral Averages)
#
Pt 1
Dir 1 Pt 2
Dir 2 Freq (Hz)
MCCOH MCOH1 MCOH2 % Difference
1
2
3
4
5
6
7
8
9
10
44
43
47
47
52
43
47
48
47
44
+Z
+X
+X
+X
+Z
+X
+X
+X
+X
+Z
46
47
45
45
45
47
45
44
45
46
+Z
+X
+X
+X
+Z
+X
+X
+X
+X
+Z
800.00
638.25
638.25
634.75
800.00
634.75
638.00
800.00
638.50
799.00
0.5334
0.6408
0.8545
0.8053
0.3343
0.7723
0.9156
0.7882
0.8470
0.7383
0.1052
0.1862
0.1609
0.2788
0.1167
0.0969
0.2916
0.1777
0.3277
0.2923
0.1220
0.1609
0.2733
0.1719
0.0769
0.2788
0.3363
0.3020
0.2658
0.2241
337.2935
244.1921
212.6885
188.8842
186.4567
177.0420
172.2692
160.9655
158.4943
152.5539
Table 22: 10 Greatest Differences between MCCOH and MCOH for Case 12 (100 Spectral Averages)
#
Pt 1
Dir 1 Pt 2
Dir 2 Freq (Hz)
MCCOH MCOH1 MCOH2 % Difference
1
2
3
4
5
6
7
8
9
10
44
44
47
47
43
43
43
44
44
43
+Y
+Z
+X
+X
+X
+X
+X
+Y
+Z
+X
46
46
45
45
47
47
47
46
46
47
+Y
+Z
+X
+X
+X
+X
+X
+Y
+Z
+X
800.00
799.75
638.00
638.25
638.00
633.00
634.75
799.75
799.50
638.25
0.7604
0.5018
0.8696
0.8533
0.7735
0.8327
0.8382
0.7338
0.5813
0.7629
0.1587
0.1768
0.3107
0.3206
0.2343
0.2692
0.2873
0.2811
0.2430
0.2403
0.2379
0.1775
0.3073
0.3159
0.3107
0.3427
0.3466
0.3066
0.2297
0.3206
Table 23: Similarities Highlighted Between the 638 Hz and 120 Hz MCCOH Improvements
Frequency
638 Hz
800 Hz
Frequency Band
More Broad (>10 Hz)
More Broad (>10 Hz)
Structures Affected
Square Frame Only
Square Frame Only
Percent Improvement of
MCCOH over MCOH
Large (~250% max)
Large (~330% max)
Trends Hold for Both 25
and 100 Spectral Averages
Yes
Yes
Description of FRF at
Frequency Band of Interest
Ratty but distinct
antiresonance valley
Ratty but distinct
antiresonance valley
67
219.5513
182.6535
179.8830
166.1219
148.9619
142.9733
141.8571
139.3334
139.1734
137.9200
Percent Difference Between MCCOH and the Maximum MCOH at f = 638.25 Hz.
Plot Min/Max = -0.05% and 244.1921%
45 +X
45 +Z
45 +Y
44 +X
44 +Z
44 +Y
43 +X
43 +Z
43 +Y
47 +X
47 +Z
47 +Y
46 +X
46 +Z
46 +Y
15 -Y
15 -X
15 +Z
13 +Y
13 +X
13 +Z
54 -X
54 +Y
54 +Z
52 -Y
52 +Z
52 +X
48 +X
48 +Z
48 +Y
5 +Y
5 +X
5 +Z
3 -Y
3 -X
3 +Z
53 -X
53 +Y
53 +Z
51 +Y
51 +Z
51 -X
101 +Z
100 +X
100 +X
200
150
100
50
53 +Z
5 +Z
52 +X
13 +Z
Output DOF #1
46 +Y
43 +Y
45 +Y
0
Percent Difference Between MCCOH and the Maximum MCOH at f = 638.25 Hz.
Plot Min/Max = -0.05% and 5%
5
45 +X
45 +Z
45 +Y
44 +X
44 +Z
44 +Y
43 +X
43 +Z
43 +Y
47 +X
47 +Z
47 +Y
46 +X
46 +Z
46 +Y
15 -Y
15 -X
15 +Z
13 +Y
13 +X
13 +Z
54 -X
54 +Y
54 +Z
52 -Y
52 +Z
52 +X
48 +X
48 +Z
48 +Y
5 +Y
5 +X
5 +Z
3 -Y
3 -X
3 +Z
53 -X
53 +Y
53 +Z
51 +Y
51 +Z
51 -X
101 +Z
100 +X
100 +X
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
53 +Z
5 +Z
52 +X
13 +Z
Output DOF #1
46 +Y
43 +Y
45 +Y
Figure 53: Case 12: MCCOH for all DOFs at 638.25 Hz (25 spectral averages)
68
Percent Difference Between MCCOH and the Maximum MCOH at f = 638.25 Hz.
Plot Min/Max = -0.05% and 166.1219%
45 +X
45 +Z
45 +Y
44 +X
44 +Z
44 +Y
43 +X
43 +Z
43 +Y
47 +X
47 +Z
47 +Y
46 +X
46 +Z
46 +Y
15 -Y
15 -X
15 +Z
13 +Y
13 +X
13 +Z
54 -X
54 +Y
54 +Z
52 -Y
52 +Z
52 +X
48 +X
48 +Z
48 +Y
5 +Y
5 +X
5 +Z
3 -Y
3 -X
3 +Z
53 -X
53 +Y
53 +Z
51 +Y
51 +Z
51 -X
101 +Z
100 +X
100 +X
160
140
120
100
80
60
40
20
53 +Z
5 +Z
52 +X
13 +Z
Output DOF #1
46 +Y
43 +Y
45 +Y
0
Percent Difference Between MCCOH and the Maximum MCOH at f = 638.25 Hz.
Plot Min/Max = -0.05% and 5%
5
45 +X
45 +Z
45 +Y
44 +X
44 +Z
44 +Y
43 +X
43 +Z
43 +Y
47 +X
47 +Z
47 +Y
46 +X
46 +Z
46 +Y
15 -Y
15 -X
15 +Z
13 +Y
13 +X
13 +Z
54 -X
54 +Y
54 +Z
52 -Y
52 +Z
52 +X
48 +X
48 +Z
48 +Y
5 +Y
5 +X
5 +Z
3 -Y
3 -X
3 +Z
53 -X
53 +Y
53 +Z
51 +Y
51 +Z
51 -X
101 +Z
100 +X
100 +X
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
53 +Z
5 +Z
52 +X
13 +Z
Output DOF #1
46 +Y
43 +Y
45 +Y
Figure 54: Case 12: MCCOH for all DOFs at 638.25 Hz (100 spectral averages)
69
Percent Difference Between MCCOH and the Maximum MCOH at f = 800 Hz.
Plot Min/Max = -0.05% and 441.6431%
45 +X
45 +Z
45 +Y
44 +X
44 +Z
44 +Y
43 +X
43 +Z
43 +Y
47 +X
47 +Z
47 +Y
46 +X
46 +Z
46 +Y
15 -Y
15 -X
15 +Z
13 +Y
13 +X
13 +Z
54 -X
54 +Y
54 +Z
52 -Y
52 +Z
52 +X
48 +X
48 +Z
48 +Y
5 +Y
5 +X
5 +Z
3 -Y
3 -X
3 +Z
53 -X
53 +Y
53 +Z
51 +Y
51 +Z
51 -X
101 +Z
100 +X
100 +X
400
350
300
250
200
150
100
50
53 +Z
5 +Z
52 +X
13 +Z
Output DOF #1
46 +Y
43 +Y
45 +Y
0
Percent Difference Between MCCOH and the Maximum MCOH at f = 800 Hz.
Plot Min/Max = -0.05% and 5%
5
45 +X
45 +Z
45 +Y
44 +X
44 +Z
44 +Y
43 +X
43 +Z
43 +Y
47 +X
47 +Z
47 +Y
46 +X
46 +Z
46 +Y
15 -Y
15 -X
15 +Z
13 +Y
13 +X
13 +Z
54 -X
54 +Y
54 +Z
52 -Y
52 +Z
52 +X
48 +X
48 +Z
48 +Y
5 +Y
5 +X
5 +Z
3 -Y
3 -X
3 +Z
53 -X
53 +Y
53 +Z
51 +Y
51 +Z
51 -X
101 +Z
100 +X
100 +X
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
53 +Z
5 +Z
52 +X
13 +Z
Output DOF #1
46 +Y
43 +Y
45 +Y
Figure 55: Case 12: MCCOH for all DOFs at 800 Hz (25 spectral averages)
70
Percent Difference Between MCCOH and the Maximum MCOH at f = 800 Hz.
Plot Min/Max = -0.05% and 297.2975%
45 +X
45 +Z
45 +Y
44 +X
44 +Z
44 +Y
43 +X
43 +Z
43 +Y
47 +X
47 +Z
47 +Y
46 +X
46 +Z
46 +Y
15 -Y
15 -X
15 +Z
13 +Y
13 +X
13 +Z
54 -X
54 +Y
54 +Z
52 -Y
52 +Z
52 +X
48 +X
48 +Z
48 +Y
5 +Y
5 +X
5 +Z
3 -Y
3 -X
3 +Z
53 -X
53 +Y
53 +Z
51 +Y
51 +Z
51 -X
101 +Z
100 +X
100 +X
250
200
150
100
50
53 +Z
5 +Z
52 +X
13 +Z
Output DOF #1
46 +Y
43 +Y
45 +Y
0
Percent Difference Between MCCOH and the Maximum MCOH at f = 800 Hz.
Plot Min/Max = -0.05% and 5%
5
45 +X
45 +Z
45 +Y
44 +X
44 +Z
44 +Y
43 +X
43 +Z
43 +Y
47 +X
47 +Z
47 +Y
46 +X
46 +Z
46 +Y
15 -Y
15 -X
15 +Z
13 +Y
13 +X
13 +Z
54 -X
54 +Y
54 +Z
52 -Y
52 +Z
52 +X
48 +X
48 +Z
48 +Y
5 +Y
5 +X
5 +Z
3 -Y
3 -X
3 +Z
53 -X
53 +Y
53 +Z
51 +Y
51 +Z
51 -X
101 +Z
100 +X
100 +X
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
53 +Z
5 +Z
52 +X
13 +Z
Output DOF #1
46 +Y
43 +Y
45 +Y
Figure 56: Case 12: MCCOH for all DOFs at 800 Hz (100 spectral averages)
71
10
e
d
ut
i
n
g
a
M
F
R
F
10
10
10
0
Overlay Plot of FRF's and Multiple Coherence for DOF 43 +X for Various Tests
25 Avg: H2(43 +X,100 +X)
25 Avg: H2( 43 +X,101 +X)
100 Avg: H2(43 +X,100 +X)
100 Avg: H2(43 +X,101 +X)
-2
-4
-6
550
600
650
700
750
800
1
0.8
e
d
ut
i
n
g
a
M
H
O
C
0.6
0.4
0.2
0
550
25 Avg: MCOH(43 +X)
100 Avg: MCOH(43 +X)
600
650
700
750
800
Frequency (Hz)
Figure 57: Overlay of FRF and Coherence Plots Showing the Effect of a Spectral Averaging for DOF 43 +X
72
Discussion
The theoretical lumped parameter study shows that combined coherence improves over ordinary
coherence for two DOFs with a nonlinear connection between them. In contrast, when the
nonlinearity was not directly between the two DOFs being combined, combined coherence did
not improve over the ordinary coherence of both of the original DOFs.
When this was tested for a real system with distributed parameters, Case 1 showed that even a
nominally linear system had large improvements at resonant peaks (Table 4, p. 22). Case 2
showed that for the linear structure the dominant combined coherence improvements all
disappeared when steps were taken to eliminate leakage errors (Table 6, p. 27). This showed up
not only as improvements in MCOH but in a lack of improvement in MCCOH over MCOH at
the frequencies which previously had been affected. There were still some much smaller
improvements in multiple combined coherence over multiple coherence. These no longer
occurred at resonant peaks but instead at antiresonances, suggesting that combined coherence
may be affected by leakage errors.
Cases 3-5 showed the effect of varying the input force level without cyclic averaging. While
there was some change in the dominant frequencies showing improvement and in what part of
the structure was affected the most, the largest changes for both high and low forcing levels
occurred between DOFs which had no possible nonlinearity in the most direct path between
them. Examples of this are the improvements between points 12 and 21 for low forcing levels
(Table 9, p. 33) and the improvements between points 44 and 45 for high forcing levels (Table
11, p. 35).
73
Case 6 was the application of cyclic averaging and burst random input to reduce leakage errors.
As with the linear structure, everywhere that MCOH greatly improved (indicating a reduction in
leakage), MCCOH was no longer greater than MCOH (Figure 30 and Figure 32 through Figure
35). Unlike the nominally linear structure, though, there were frequency bands where the large
drops in coherence did not seem to be affected by leakage (Figure 31 through Figure 35). At
those frequencies combined coherence still improved over regular coherence. As in Case 2, it
seemed that leakage was having an effect on the ability of combined coherence to detect
nonlinearities.
Case 7 examined the earlier noted pattern of large improvements in combined coherence
occurring between DOFs with very linear direct paths between them by changing the density of
the accelerometers to see if the effect still occurred with even more direct linear paths between
the DOFs. It was discovered that there was basically no effect from doubling the density of the
accelerometers in certain areas in question (Figure 11, p. 19). The combined coherence
improvements seemed to affect all of the DOFs in the upper structure at certain frequencies
instead of primarily affecting those DOFs nearest to the nonlinear connection point, making
process of locating a nonlinearity using combined coherence alone impossible in this case.
Case 8 looked at how changing the location of the input forces would affect the ability of
combined coherence to detect and spatially locate nonlinearities. Moving one of the shakers to
the square frame eliminated any combined coherence improvements in the upper structure.
Where before the largest improvements in combined coherence occurred between DOFs on the
74
square frame away from the inputs, now there were no improvements seen (Table 17, p. 49).
Figure 39 shows that this does not mean that all the FRF’s on the upper frame were clean and
that multiple coherence never dropped. For FRF’s relating the response of the square frame to
the input on the H-frame, there were still very large drops in MCOH and ratty FRF’s, indicating
that a nonlinearity was still affecting the structure there. However, the dominant FRF was linear
and combined coherence would no longer improve.
Case 9 examined whether reducing the number of nonlinear paths would have an effect on
combined coherence. It was discovered that by removing three of the connection points between
the H-frame and the square frame that the number of MCCOH improvements actually increased
relative to other similar test setups (e.g., Case 7). Thus, it appeared that reducing the number of
paths between the two structures did little to improve the ability of MCCOH to spatially locate
the nonlinearities. In fact the number of MCCOH improvements and their magnitude both
increased between DOFs with very linear direct paths between them (Table 19, p. 56). One thing
that was not considered was the structure’s interaction with the supporting shock cord. If energy
was being transmitted through them, then the number of nonlinear paths may not have been
sufficiently reduced at affected frequencies. Another thing that was not considered was whether
the energy distribution was sufficient to excite the nonlinearities.
Cases 10 and 11 looked at the effect of spatial averaging to see if the nonlinearities were getting
averaged out at all. It was seen, when comparing cases with 25 and 100 spectral averages and no
cyclic averaging, that there did seem to be a slight increase in the magnitude of MCCOH
improvements when fewer averages were done, but there was no qualitative improvement in the
75
ability of combined coherence to detect and locate nonlinearities. The same parts of the structure
saw improvements in MCCOH regardless of the number of averages (Table 21 and Table 22, p.
67).
Cases 9-10 also highlight the fact that MCCOH improvements were primarily at antiresonances.
This highlights the fact that noise can be significant in these parts of FRF. In retrospect, a test
should have been done to determine whether there truly were nonlinearities being excited in
those frequency ranges. Assuming that nonlinearities are the cause of the improvements in
combined coherence, MCCOH seems unable to distinguish between energy transmitted through
a nonlinearity some distance from two DOFs located adjacent to each other and energy
transmitted through a nonlinearity very close to or even between the two adjacent DOFs. Often
there even seems to be more of an improvement in MCCOH among DOFs located in the upper
structure where all of the energy comes through a nonlinearity rather than across the nonlinear
connection itself where some energy comes directly from the shaker along a linear path and
some comes through the nonlinear connection.
76
Summary and Conclusions
In conclusion, it does not seem that combined coherence, as it is currently applied, is particularly
useful in spatially locating nonlinearities in a structure for a variety of reasons. Several
observations are highlighted below.
1) Leakage errors seem to give false improvements and throw off the calculation whether in
a nominally linear structure or a nominally nonlinear structure (Case 2 vs. Case 1 and
Case 6 vs. Case 4).
2) Varying the magnitude of force input seems to vary the types of nonlinearities detected,
but does not affect the ability of combined coherence to spatially locate nonlinearities
(Cases 3-5).
3) Doubling the density of the response accelerometers does not seem to improve the ability
of combined coherence to spatially locate nonlinearities (Case 7 vs. Case 6).
4) There seems to be a great sensitivity to the location of the input and the amount of force
coming from a direct linear path (Case 8 vs. Case 7).
5) Reducing the number of nonlinear paths does not seem to improve the ability of MCCOH
to spatially locate nonlinearities. In fact, it can get worse (Case 9 vs. Case 7).
6) Spectral averaging does not seem to qualitatively affect the ability of combined
coherence to spatially locate nonlinearities, although overall larger improvements are
seen with fewer averages (Cases 10-11).
In addition one unexpected and unusual result noted was that sometimes the greatest
improvement in combined coherence came when combining DOFs at the same point, but in
orthogonal directions (e.g., 44Y and 44Z in Figure 46, p. 59). In some ways this might be
77
considered an extreme example of Case 7 changing the density of the responses. Off-axis errors
in the accelerometers may also be the cause of some of these MCCOH improvements over
MCOH.
Scaling the motions of the DOFs by their respective masses was not done and cannot be easily
done for this distributed parameter case. Perhaps if correct mass scaling factors were obtained
and used, the greatest improvements would be seen across the nonlinear connections rather than
between DOFs on the same linear structure. It is doubtful whether scaling the masses would
produce the desired results, though, due to the fact that even when efforts were made to even out
the differences in mass scaling by placing accelerometers on the brackets on either side of the
nonlinearity, MCCOH improvements were still not seen between those DOFs.
Whether or not a nonlinear situation was even excited is another question. Other test procedures
were not used which would tell definitively whether or not a nonlinear situation was excited, and
if so, where the nonlinearities were located. It was not determined whether a nonlinearity was
sufficiently excited and if so where it was (e.g., which mount, tire behaved nonlinearly).
78
Future Work
Combined coherence, though a useful tool in small lumped parameter models, will be more
difficult to apply usefully to real structures. Work still needs to be done to understand its
properties when applied to real structures before it is widely used. To that end suggestions for
future work are listed below.
1) Before further evaluation of combined coherence on a real structure, testing should be
done using other methods (such as swept sine input) to check that nonlinearities are in the
expected locations and that they are being excited sufficiently.
2) It would also be useful to test each structure separately prior to connecting them and to
perhaps hang each structure vertically. Testing each structure separately before
combining them will ensure that any new MCCOH improvements over MCOH are due to
the nonlinear connections. Also, hanging the frame will ensure that when a mount is
removed, any change in the MCCOH calculations is a result of the missing mount and not
the additional shock cord supports.
3) A theoretical model of a continuous system with a full mass matrix should be used to see
if there is something inherent in continuous systems that make it difficult or impossible to
use combined coherence to detect nonlinearities.
4) The path dependence of combined coherence should be looked into more. One start
might be a theoretical lumped mass model with hundreds of interconnected DOFs. Other
experiments with real structures would also be useful.
5) There seemed to be a great sensitivity of the structure to the location of the input force.
Further testing regarding the sensitivity of the formulation to this aspect, may lead to a
testing method that makes use of this aspect to locate nonlinearities.
79
6) A sensitivity analysis of the effect of the spatial density of the responses on MCCOH
calculations would be useful as well, since it is unclear how close the responses need to
be in order to distinguish between nonlinear internal forces and nonlinear external forces.
7) The apparent sensitivity of MCCOH to leakage needs to be examined in greater detail to
see if the MCCOH improvements may have been due to nonlinearities or if they were
truly due to leakage.
80
References
[1]
T. Roscher, Detection of Structural Non-Linearities using the Frequency Response and
Coherence Functions, Master’s Thesis, University of Cincinnati, 2000
[2]
T. Roscher, R. J. Allemang, A. W. Phillips, A New Detection Method for Structural NonLinearities, Proceedings of the International Conference on Noise and Vibration
Engineering, September 13-15, 2000, Katholieke Universiteit, Leuven, Belguim pp. 695702
[3]
K. Worden and G. R. Tomlinson, Nonlinearity in Structural Dynamics: Detection,
Identification and Modeling, Philadelphia, PA: Institute of Physics Publishing, 2001
[4]
D. L. Brown, Excitation Signals for Modal Analysis, University of Cincinnati, 1996
[5]
R. J. Allemang, Vibrations: Experimental Modal Analysis, University of Cincinnati, 1999
[6]
R. J. Allemang, Nonlinearities: A Modal Analysis Perspective, University of Cincinnati,
2003
[7]
J. S. Bendat, Nonlinear System Techniques and Applications, John Wiley and Sons, Inc.,
New York, 1998
[8]
Bendat, Julius S., Robert N. Coppolino, Paul A. Palo, Identification of Physical Parameters
with Memory in Non-Linear Systems, International Journal of Nonlinear Mechanics, Vol:
30, Issue: 6, pp. 841-860
[9]
D. M. Storer, G. R Tomlinson, Recent Developments in the Measurement and
Interpretation of Higher Order Transfer Functions from Non-Linear Structures,
Mechanical Systems and Signal Processing, Vol: 7, Issue: 2, March 1993, pp. 173-189
81
[10] Q. Chen, G. R. Tomlinson, A New Type of Time Series Model for the Identification of NonLinear dynamical Systems, Mechanical Systems and Signal Processing (1994) 8(5), pp.
531-54
[11] R. Ghanem, F. Romeo, A Wavelet Based Approach for Model and Parameter Identification
of Non-Linear Systems, International Journal of Non-Linear Mechanics, Vol: 36, Issue 5,
July 2001, pp. 835-859
[12] Y. H. Chong, M. Imregun, Use of Reciprocal Modal Vectors for Nonlinearity Detection,
Archive of Applied Mechanics 70 (2000) pp.453-462
[13] J. S. Bendat, A. G. Piersol, Engineering Applications of Correlation and Spectral Analysis,
2nd Ed., John Wiley and Sons, Inc., New York, 1993
[14] J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures, 2nd Ed.,
John Wiley and Sons, Inc., New York, 1986
[15] D. L. Brown, Frequency Response Function Estimation Concepts, University of Cincinnati,
1996
82
Appendix 1: Derivation of Confidence Intervals for
θˆ(ω ) (i.e., ∠Ĥ (ω ) )
Ĥ (ω )
and
The estimate of H after nd averages is given by
ˆ
Hˆ (ω ) = Hˆ (ω ) e − jθ (ω )
The normalized error term is given by
εr =
[]
σ φˆ
where φˆ = Ĥ or φˆ = θˆ
φ
[ ]
The error term for Ĥ is given by ε r Hˆ =
[1 − γ ]
2
xy
γ xy 2nd
Confidence intervals are then given by
(
)
(
φˆ 1 − C f σ u ≤ φ ≤ φˆ 1 + C f σ u
)
where the C f constant is found by relating the desired confidence level to the area under the
standard normal distribution curve (e.g., C f =2 for 95% confidence levels and C f =1.64 for
90% confidence levels).
Also, σ u is the estimate of the confidence interval, not the estimate of the standard deviation of
the magnitude or phase (according to Doebling and Farrar, though Bendat and Piersol do not
seem to differentiate between the two from my reading of them). σ u is given below for Ĥ and
θˆ .
83
[ ]
σ u Hˆ =
[1 − γ ] Hˆ = ε [Hˆ ]Hˆ
2
xy
γ xy 2nd
[
] 
 1− γ 2
xy
−1 
ˆ
σ u θ = sin 
γ
2nd
 xy
[]


( [ ])
= sin −1 ε Hˆ
Thus the 90% confidence levels for Ĥ and θˆ are given by
(
[ ])
(
[ ]) and θˆ(1 − 1.64σ [θˆ]) ≤ θ ≤ θˆ(1 + 1.64σ [θˆ])
Hˆ 1 − 1.64σ u Hˆ ≤ H ≤ Hˆ 1 + 1.64σ u Hˆ
u
u
Using the above equations, the 90% confidence limits are given below for various numbers of
averages and coherence values.
84
Upper and Lower bounds on |H| (dB magnitude) and bounds on θ (degrees) using Bendat’s equations
γxy2 \ nd
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
16
32
64
128
256
5.44
4.16
3.14
2.33
1.71
-17.70
60.4
-8.29
38.0
-4.96
25.8
-3.19
17.9
-2.13
12.6
3.97
2.98
2.21
1.62
1.18
-7.53
35.4
-4.58
24.2
-2.97
16.9
-1.99
11.8
-1.36
8.3
3.18
2.37
1.74
1.26
0.91
-5.08
26.3
-3.26
18.2
-2.17
12.8
-1.48
9.0
-1.02
6.4
2.64
1.95
1.42
1.03
0.74
-3.81
20.8
-2.51
14.5
-1.70
10.2
-1.17
7.2
-0.81
5.1
2.21
1.62
1.18
0.85
0.61
-2.97
16.9
-1.99
11.8
-1.36
8.3
-0.94
5.9
-0.65
4.2
1.85
1.34
0.97
0.70
0.50
-2.35
13.7
-1.59
9.6
-1.09
6.8
-0.76
4.8
-0.53
3.4
1.51
1.09
0.79
0.56
0.40
-1.83
10.9
-1.25
7.7
-0.87
5.4
-0.60
3.8
-0.42
2.7
1.18
0.85
0.61
0.43
0.31
-1.36
8.3
-0.94
5.9
-0.65
4.2
-0.46
2.9
-0.32
2.1
0.80
0.57
0.41
0.29
0.21
-0.88
5.5
-0.61
3.9
-0.43
2.8
-0.30
2.0
-0.21
1.4
85
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
The same table taken from the HP literature is given below
Upper and Lower bounds on |H| (dB magnitude) and bounds on θ (degrees) from HP’s literature
γxy2 \ nd
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
16
32
64
128
256
5.20
3.80
2.80
2.10
1.50
-14.60
54.0
-7.10
34.0
-4.20
23.0
-2.70
16.0
-1.80
11.0
4.20
3.10
2.20
1.60
1.20
-8.40
38.0
-4.80
25.0
-3.00
17.0
-2.00
12.0
-1.40
8.0
3.50
2.60
1.80
1.30
1.00
-6.00
30.0
-3.60
20.0
-2.30
14.0
-1.60
10.0
-1.10
7.0
3.00
2.10
1.50
1.10
0.80
-4.50
24.0
-2.80
16.0
-1.90
11.0
-1.30
8.0
-0.90
5.0
2.50
1.80
1.30
0.90
0.70
-3.50
19.0
-2.20
13.0
-1.50
9.0
-1.00
6.0
-0.70
4.0
2.10
1.50
1.00
0.70
0.50
-2.70
15.0
-1.70
10.0
-1.20
7.0
-0.80
5.0
-0.60
4.0
1.60
1.10
0.80
0.60
0.40
-2.00
12.0
-1.30
8.0
-0.90
6.0
-0.60
4.0
-0.40
3.0
1.10
0.80
0.50
0.40
0.30
-1.30
8.0
-0.80
5.0
-0.60
4.0
-0.40
3.0
-0.30
2.0
86
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
The differences between these two tables are highlighted below.
Diffference Between the Tables compiled from Bendat’s equations and HP literature
γxy2 \ nd
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
16
32
64
128
256
1.23
0.82
0.59
0.48
0.32
-7.07
18.6
-2.52
9.8
-1.23
6.1
-0.71
4.2
-0.44
2.7
1.02
0.73
0.46
0.34
0.29
-3.32
11.7
-1.54
6.8
-0.83
4.2
-0.52
3.0
-0.38
1.6
0.86
0.65
0.38
0.27
0.26
-2.19
9.2
-1.09
5.5
-0.60
3.8
-0.43
2.8
-0.29
1.9
0.79
0.48
0.32
0.25
0.19
-1.53
7.1
-0.81
4.2
-0.54
2.7
-0.36
2.1
-0.25
0.8
0.65
0.46
0.33
0.20
0.20
-1.15
5.3
-0.61
3.4
-0.41
2.2
-0.24
1.2
-0.17
0.6
0.59
0.41
0.21
0.14
0.10
-0.87
4.1
-0.45
2.3
-0.33
1.6
-0.20
1.2
-0.18
1.3
0.42
0.25
0.19
0.17
0.09
-0.64
3.7
-0.36
2.1
-0.25
1.8
-0.14
1.1
-0.08
0.9
0.30
0.23
0.09
0.11
0.09
-0.42
2.5
-0.19
1.1
-0.17
1.2
-0.10
1.0
-0.09
0.6
Hupper
H
θ (degrees)
Hlower
θ (degrees)
0 < |H| <.1
.1 < |H| <.2
0 < |θ| < .5
.5 < |θ| < 1
Hupper
.2 < |H| <.3
1 < |θ| < 2
Hlower
θ (degrees)
.3 < |H| <.5
2 < |θ|
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
Assuming the same process is used for deriving the two tables, the differences seem to arise
because of a
2 term in Bendat’s equation for the error of Ĥ which is not used in the
[ ]
equations HP used to compile their table. If ε r Hˆ =
error term, the following table is calculated.
87
[1 − γ ]
2
xy
γ xy nd
is used as the equation for the
Upper and Lower bounds on |H| (dB magnitude) and bounds on θ (degrees) using Bendat’s equations
(except
[ ] is multiplied by
ε r Hˆ
γxy2 \ nd
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
2)
16
32
64
128
256
6.97
5.44
4.16
3.14
2.33
-17.70
60.4
-8.29
38.0
-4.96
25.8
-3.19
17.9
5.20
3.97
2.98
2.21
1.62
-14.89
55.1
-7.53
35.4
-4.58
24.2
-2.97
16.9
-1.99
11.8
4.22
3.18
2.37
1.74
1.26
-8.55
38.8
-5.08
26.3
-3.26
18.2
-2.17
12.8
-1.48
9.0
3.53
2.64
1.95
1.42
1.03
-6.06
30.1
-3.81
20.8
-2.51
14.5
-1.70
10.2
-1.17
7.2
2.98
2.21
1.62
1.18
0.85
-4.58
24.2
-2.97
16.9
-1.99
11.8
-1.36
8.3
-0.94
5.9
2.51
1.85
1.34
0.97
0.70
-3.54
19.6
-2.35
13.7
-1.59
9.6
-1.09
6.8
-0.76
4.8
2.07
1.51
1.09
0.79
0.56
-2.71
15.6
-1.83
10.9
-1.25
7.7
-0.87
5.4
-0.60
3.8
1.62
1.18
0.85
0.61
0.43
-1.99
11.8
-1.36
8.3
-0.94
5.9
-0.65
4.2
-0.46
2.9
1.11
0.80
0.57
0.41
0.29
-1.28
7.9
-0.88
5.5
-0.61
3.9
-0.43
2.8
-0.30
2.0
#NUM!
#NUM!
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
Hupper
Hlower
θ (degrees)
When this is used, the table matches quite nicely with the table compiled by HP. The differences
are given below. As one can see, there are slight differences which tend to get larger as the error
terms get larger. Overall, though, the tables match quite nicely.
88
Diffference Between the Tables compiled from modified Bendat’s equations and HP literature
γxy2 \ nd
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
16
32
64
128
256
0.00
-0.17
-0.18
-0.11
-0.12
0.29
-1.1
0.43
-1.4
0.38
-1.2
0.27
-0.9
0.19
-0.8
-0.02
-0.08
-0.17
-0.14
-0.06
0.15
-0.8
0.28
-1.3
0.26
-1.2
0.17
-0.8
0.08
-1.0
-0.03
-0.04
-0.15
-0.12
-0.03
0.06
-0.1
0.21
-0.8
0.21
-0.5
0.10
-0.2
0.07
-0.2
0.02
-0.11
-0.12
-0.08
-0.05
0.08
-0.2
0.17
-0.9
0.09
-0.8
0.06
-0.3
0.04
-0.9
Hlower
θ (degrees)
-0.01
-0.05
-0.04
-0.07
0.00
Hupper
0.04
-0.6
0.15
-0.7
0.09
-0.6
0.09
-0.8
0.06
-0.8
Hlower
θ (degrees)
0.03
-0.01
-0.09
-0.09
-0.06
0.01
-0.6
0.13
-0.9
0.05
-0.7
0.07
-0.4
0.00
0.2
-0.02
-0.08
-0.05
-0.01
-0.03
-0.01
0.2
0.06
-0.3
0.04
0.1
0.05
-0.2
0.06
0.1
Hlower
θ (degrees)
-0.01
0.00
-0.07
-0.01
0.01
Hupper
-0.02
0.1
0.08
-0.5
0.01
0.1
0.03
0.2
0.00
0.0
Hlower
θ (degrees)
Hupper
Hupper
θ (degrees)
Hlower
θ (degrees)
0 < |H| <.1
.1 < |H| <.2
0 < |θ| < .5
.5 < |θ| < 1
Hupper
.2 < |H| <.3
1 < |θ| < 2
Hlower
θ (degrees)
.3 < |H| <.5
2 < |θ|
Hupper
Hlower
θ (degrees)
Hupper
Hupper
Hlower
θ (degrees)
Hupper
It seems from the brief overview above that HP is using a slightly different equation from
Bendat. The origin of and reason for the
2 term in the equations used by HP is unknown at
this point.
89