Feasibility of Use of Four-Post Road Simulator for
Automotive Modal Applications
A thesis submitted to the
Graduate School
of the University of Cincinnati
in partial fulfillment of the
requirements for the degree of
Master of Science
in the Department of Mechanical Engineering
at the College of Engineering
2010
by
Balaji R. Sharma
Bachelor of Engineering
Vishweshwaraiah Technological University, Karnataka, India
July 2006
Committee Chair: Dr. Randall J. Allemang
Abstract
Modal analysis is a critical part of the automotive development process. Identification of the
vehicle's modal signature, especially in the low-frequency end of the spectrum, is essential for
tuning the dynamic performance of the vehicle structure for optimal ride and handling comfort.
Traditional methods to characterize the system - in terms of its natural frequencies, associated
damping values and mode shapes - have typically employed conventional impact and shaker
tests. While these tests are able to accurately study the modal behavior of the vehicle under static
conditions, they are not truly reflective of the real-world operating conditions of the vehicle
A four-post road simulator is used in automotive development to simulate on-road conditions in
the laboratory primarily for durability, transmissibility, noise and vibration studies, etc. Some of
these studies often involve a similar setup of response sensors across the automotive structure as
conventional modal tests. Utilization of the road-simulator for modal analysis can potentially
reduce the duration of the automotive development cycle in the testing phase, allowing for a
faster time-to-market, in addition to improved accuracy of estimation of the vehicle's dynamic
performance under simulated operating conditions.
This thesis work explores the feasibility of using a four-post road simulator for experimental
modal analysis (EMA) of automotive structures. The MTS 320 Road Simulator in the Structural
Dynamics Research Laboratory, University of Cincinnati, is employed for the study, with a truck
frame being the test structure. Frequency response functions (FRFs) are estimated with
displacement and pressure measurements at the hydraulic excitation posts of the simulator,
provided by transducers built into the four-poster, replacing force measurements as inputs. The
applicability of these non-conventional FRF formulations for modal parameter identification
i
using existing parameter estimation algorithms is studied. Additionally, response-only data based
on random excitations from the simulator is processed under the framework of Operational
Modal Analysis (OMA) for parameter estimation. Modal estimates from these tests are compared
with one another and with those from conventional EMA-based impact tests, and a summary of
results is presented based on the findings therein.
The thesis begins with a general overview of the automotive testing process, and the role of
experimental modal analysis and the four-post road simulator therein. Thesis objectives are
presented in terms of utilizing the four-post road simulator for estimation of the modal
parameters of automotive structures in the absence of force measurements, and the motivation
for the same is discussed. Fundamental principles of experimental and operational modal
analysis are presented further, in addition to the theory behind the use of non-force
measurements in the measurement of modified FRFs and consequently the estimation of modal
parameters. The document then proceeds to describe the experimental setup and the various tests
conducted as a part of the study, with a summary of results from each test. A detailed discussion
follows on the comparison of results between the tests, and an overall summary of results is
presented. Conclusions from this research work are presented along with recommendations for
future work in this area.
ii
iii
Acknowledgements
I wish to thank my thesis advisor, Dr. Randall Allemang, for his support and guidance through
my graduate studies. Through a fairly engaging and highly organized coursework with a strong
emphasis on real-world applications, his academic inputs have been great learning aids. Nights
spent grueling with what students of Vibrations at UC would know as the ‘Randy Cart’, were
nights of great rigor, bundles of paper with scribblings all over, and a whole lot of fun. “You are
permitted to make mistakes, as long as you learn from them” is a line often emphasized in his
courseware, and over mistakes, over successes, his constant support and encouragement has been
a major driving force for this work. Thanks are also due for his inputs and feedback on this work
at short notice. His encouragement for research publications and support for travel to conferences
are gratefully acknowledged, as also are all the lighthearted moments in the corridors of Baldwin
and the smile that comes my way every time I run into him.
I would like to thank Dr. David Brown for his review and feedback on this work. The fascination
for this area of research began sometime during his lectures on Fourier Transforms, and has
continued over the course of my graduate studies here at UC. His experience and inputs have
added much to my academic experiences at UC.
Special thanks to Dr. Allyn Phillips for helping me at various stages of this work, and for
providing crucial inputs with the use of software and hardware around the SDRL. His inputs in
the development and successful completion of this work are invaluable, and I am very grateful
for his time and support in reviewing and refining this work.
iv
Thanks also to Shashank Chauhan, a very good friend and an esteemed mentor, for initiating us
into the field of operational modal analysis. There has been a lot to learn while working with
him, and his dedication and passion for research is inspiring. Many thanks to Balakumar, my
partner in crime of sorts in my research endeavors, and a valued friend, for making the time at
the SDRL and Cincinnati in general memorable. Thanks also to Ravi Mantrala and Udayan
Godbole for telling me about the SDRL even before I got here, and for pointing me in the right
directions when I was lost.
The academic works would not have been possible if life outside classes got monotonous. A
special thanks to my friends, and I resist the temptation to name them all, for having endured me
and for all the fun moments with them I have come to cherish in my stay at UC so far. I hope we
keep in touch no matter where life takes us ahead.
Thanks to Pa and Ma, for believing their son would eventually finish his master’s work someday,
and for their love and encouragement across the seas. Thanks also to my little sister Akhila,
whose light-hearted chatter over the phone and web has kept me going. Thanks to my uncle,
KNS Acharya, for initiating me to the field of experimental dynamics and vibrations, and for his
mentoring and encouragement all my life. Thanks also to my grandparents who always believed
their little grandson would turn out just fine in life. A special thanks to my cousin Kannan whose
words of wisdom, and the occasional nagging, spice up my life every now and then.
v
Contents
1
2
3
4
Introduction .......................................................................................................................1
1.1
Motivation for the Thesis ..............................................................................................3
1.2
Thesis Objectives ..........................................................................................................4
Background ........................................................................................................................8
2.1
Principles of Experimental Modal Analysis ...................................................................8
2.2
Non-conventional Formulations of Frequency Response Functions ............................. 11
2.3
Principles of Operational Modal Analysis ................................................................... 18
Test Details ....................................................................................................................... 21
3.1
Test Setup ................................................................................................................... 21
3.2
Tests ........................................................................................................................... 25
3.2.1
Conventional Impact Tests ................................................................................... 25
3.2.2
Modified FRF Based Analysis with Pressure as Input (EMA-Pressure) ................ 28
3.2.3
Modified FRF Based Analysis with Displacement as Input (EMA Displacement) 32
3.2.4
Response-only Modal Analysis ............................................................................ 35
Comparison of results ...................................................................................................... 42
4.1
Comparison of results from conventional impact tests ................................................. 42
4.1.1
EMA 4P vs EMA Ground 1 ................................................................................. 42
4.1.2
EMA 4P vs EMA Ground 2 ................................................................................. 43
4.1.3
EMA Ground 1 vs EMA Ground 2 ....................................................................... 44
4.2
Comparisons for results from the modified FRF based tests. ....................................... 45
vi
4.2.1
EMA 4P vs EMA-Displacement. ......................................................................... 45
4.2.2
EMA 4P vs EMA-Pressure. .................................................................................. 46
4.3
Comparisons for Results from the OMA Based Test ................................................... 47
4.3.1
OMA-SSI vs OMA-PTD...................................................................................... 47
4.3.2
EMA 4P vs OMA-SSI.......................................................................................... 48
4.3.3
OMA-SSI vs EMA-Displacement ........................................................................ 50
4.4
Modified MAC Computations ..................................................................................... 52
5
Summary .......................................................................................................................... 56
6
Conclusions and Scope for Future Work ........................................................................ 59
Bibliography ……………………………..……………….………………………………….... 62
vii
List of Figures
Figure 1-1 Dynamic Analysis of Automotive Systems .................................................................1
Figure 2-1: Experimental Modal Analysis ...................................................................................9
Figure 2-2. A post of the MTS-320 road simulator at UC-SDRL. .............................................. 12
Figure 2-3. Lumped-mass model for system representation ....................................................... 15
Figure 2-4. Comparison of force-based and displacement-based FRFs ...................................... 17
Figure 3-1 The MTS-320 Road Simulator at UC-SDRL. ........................................................... 21
Figure 3-2. Sensor distribution and excitation locations on the test structure.............................. 23
Figure 3-3: Sensors (L:R) on (a) (Front) Wishbones (b) (Rear) Leaf Springs (c) Engine ............ 24
Figure 3-4. Consistency diagram for an estimate for the EMA impact test on the simulator ....... 27
Figure 3-5. Auto-MAC plot for modal estimates from EMA 4P test .......................................... 28
Figure 3-6. Consistency diagram for an estimate for EMA Pressure Test ................................... 30
Figure 3-7. Driving-point Modified FRF curves for EMA Pressure Test .................................... 31
Figure 3-8. Auto-MAC plot for EMA Pressure .......................................................................... 32
Figure 3-9. Driving point FRF Curves for the EMA Displacement Test ..................................... 33
Figure 3-10. Consistency Diagram for the EMA Displacement test ........................................... 34
Figure 3-11. Auto-MAC plot for EMA displacement test .......................................................... 35
Figure 3-12. Auto-MAC plot for the OMA estimates using the SSI-Data algorithm................... 38
Figure 3-13. Consistency Diagram for an OMA estimate using the PTD algorithm ................... 39
Figure 3-14. Auto-MAC plot for the OMA estimates using PTD ............................................... 40
Figure 4-1. Cross-MAC of EMA 4P vs EMA Ground 1 estimates ............................................. 43
Figure 4-2. Cross-MAC of EMA 4P vs EMA Ground 2 estimates ............................................. 44
Figure 4-3. Cross-MAC of EMA Ground 1 vs EMA Ground 2 estimates................................... 45
viii
Figure 4-4. Cross-MAC of EMA 4P vs EMA-Displacement estimates ...................................... 46
Figure 4-5. Cross-MAC of EMA Pressure vs EMA 4P estimates ............................................... 47
Figure 4-6. Cross-MAC of OMA-PTD vs OMA-SSI ................................................................. 48
Figure 4-7. Cross-MAC of EMA 4P vs OMA-SSI estimates ..................................................... 50
Figure 4-8. Cross-MAC of OMA-SSI vs EMA-Displacement estimates .................................... 51
Figure 4-9. Modal Vector (16.4 Hz) - EMA 4P ......................................................................... 53
Figure 4-10. Modal Vector (16.4 Hz) - EMA Displacement ...................................................... 53
Figure 4-11. EMA 4P versus EMA-Displacement ..................................................................... 54
Figure 4-12. EMA 4P versus OMA SSI ..................................................................................... 54
Figure 5-1. 5.44 Hz: Yawing ..................................................................................................... 58
Figure 5-2. 4.93 Hz: Front-Axle pitching................................................................................... 58
Figure 5-3. 6.4 Hz: Rear-axle pitching ....................................................................................... 58
Figure 5-4 9.94 Hz: Rolling ....................................................................................................... 58
Figure 5-5. 10.4 Hz: Transaxle Bending .................................................................................... 58
Figure 5-6. 11.28 Hz: First torsion mode ................................................................................... 58
Figure 5-7. 18.77 Hz: First frame bending mode ....................................................................... 58
Figure 5-8 31.41 Hz: Higher order lateral bending mode ........................................................... 58
ix
List of Tables
Table 5-1. Mode shape descriptions .......................................................................................... 56
Table 5-2. Summary of modes obtained from all tests ............................................................... 57
Table 5-3. Rigid-body and prominent deformation mode shapes ............................................... 58
x
FEASIBILITY OF USE OF FOUR-POST ROAD SIMULATORS FOR
AUTOMOTIVE MODAL APPLICATIONS
1
Introduction
Experimental testing is a crucial aspect of the automotive development process. While modern
product development processes endeavor to lay more emphasis on virtual simulations and
computer-aided engineering methods, there is no denying the need for experimental testing to
both validate theoretical predictions and evaluate the completeness of such predictions beyond
simplifying assumptions in real-world behavior. Testing on a physical prototype often reveals
aspects of the system's performance that theoretical estimates, within their limitations, fail to
foresee. Figure 1-1 represents some of the conventional processes specific to the dynamic
analysis of automotive systems, and the role of vehicle testing therein. The list is by no means
exhaustive, and is merely indicative of the various aspects of modern automotive development.
Figure 1-1 Dynamic Analysis of Automotive Systems
1
Understanding the structural dynamics of a system is critical to tuning the dynamic performance
of the vehicle for optimal ride and handling. A starting point in predicting the dynamic response
of a vehicle is the experimental determination of the modal signature of the vehicle in terms of
its resonant frequencies, and corresponding damping values and mode shapes, with the process
referred to as Experimental Modal Analysis (EMA). The system response can then be predicted
as a function of the weighted superposition of these modes of vibration. Typically, EMA for
automotive applications employs the use of impact hammers or electro-dynamic shakers to excite
the vehicle, with accelerometers or electro-optical sensors recording the system response to these
excitations. The response-to-force relation thus measured forms the basis for the estimation of
the modal parameters of the system under the conventional scheme of modal testing.
A four-post road simulator is an indispensable part of the vehicle testing process, finding
multiple applications in noise-vibration-harshness (NVH) performance evaluation including ride
comfort studies and buzz, squeak and rattle (BSR) analysis, in addition to structural durability
studies, assessment of the non-linear behavior of the structure and end-of-the-line testing of
vehicle structures. Conventionally, the four-post test rig is not employed for modal analysis of
vehicle structures due to lack of availability of force input measurements with such rigs, or due
to the high costs involved with force-measurement sensors with such systems. The present work
explores alternate testing methods to evaluate the feasibility of using a road simulator for modal
analysis of automotive structures, under realistic operating conditions and in the absence of force
measurements.
2
1.1 Motivation for the Thesis
Test setup under the conventional modal analysis framework is typically not representative of the
real operating conditions of a vehicle. Classically, modal testing on structures is performed using
impact hammers (impact tests) or electro-dynamic shakers (shaker tests) to excite structures.
Tests are performed under the assumption that the boundary conditions for the test structure
usually conform to a free-free, fixed, operating or arbitrary boundary condition case
[1]
. In real
testing conditions, however, it is difficult for the actual boundary conditions to meet one of the
aforementioned requirements in entirety, and depending on the applications, approximations are
made to replicate the desired boundary conditions. For automotive modal testing, tests are
conducted in laboratories with the automotive structure usually subjected to an adjusted static
support on the ground. Such supports do not account for the dynamic boundary conditions that
come into play at the road-vehicle interface when the vehicle is in operation. Further, differences
in the estimation of modal parameters under various simulated boundary conditions at the roadtire interface for automotive systems is demonstrated in
[9]
, reiterating the need for a realistic
replication of the operating conditions for a vehicle during testing for accurate estimation of the
system’s dynamic behavior in real-world operation. Additionally, variations in the system over
time owing to reasons such as suspension stiffening are not evident when the vehicle is not
excited through the tires, as is typically the case with conventional modal testing for automotive
structures.
Short of an actual on-road test, a four-post road simulator is able to successfully replicate some
of the operating conditions for a vehicle in the low-frequency range - within certain limitations
imposed by the absence of non-vertical forces, wind and engine excitations, and variations
induced due to mechanical straps at the tire-post interface. In spite of these limitations, testing
3
with the road-simulator remains fairly well-suited to gain insights regarding the vehicle's
operational behavior. Also, a successful utilization of the road simulator for modal testing could
mean significant cost and time benefits, eliminating the need for a dedicated experimental set-up
for modal analysis and increasing the spectrum of applications of the road simulator for
automotive testing.
1.2 Thesis Objectives
This work explores the feasibility of using a four-post road simulator for modal analysis of
automotive structures, with a moderately damped truck chassis being used as the test structure
for this study. Benchmark estimates of the modal parameters of the chassis are obtained from
impact tests based on traditional experimental modal analysis techniques. The chassis is tested
initially using a conventional impact test for modal analysis with the structure resting on the
ground. The test is repeated with the vehicle hoisted onto the four-post road simulator with its
tires strapped firmly onto the posts. A third such test is performed at the end of the study with the
vehicle dismounted from the simulator back on the ground. These tests, based on traditional
response-over-force type frequency response function (FRF) formulations, are aimed at
establishing a set of benchmark estimates against which all other tests would be compared. These
tests also allow for studying variations in the system, if any, over the length of the testing period.
Due to the lack of availability of force measurements on the road simulator, tests are conducted
on the chassis with displacement and pressure measurements from the simulator (replacing force
spectra) leading to the formulation of modified FRFs. With the vehicle hoisted on to the
simulator and strapped firmly in place at the tires, random excitations at the four posts of the
simulator are employed as inputs. The first and second tests utilize pressure and displacement
4
measurements at the inputs, provided by transducers built into the simulator system, replacing
force measurements in the estimation of FRFs. These modified FRFs are then processed for
modal parameter estimation. A third test is conducted for evaluation of the test setup for
Operational Modal Analysis (OMA), with response time-histories recorded at sensors across the
structure. Results from these three tests are compared with one another, and with results from the
benchmark tests, and their validity is evaluated based on the nature of comparisons, using
conventional and modified validation techniques.
The main objective of this work is to evaluate the use of the road simulator in allowing for the
modal testing of the vehicle structure under more realistic operating conditions. To the best of
the author’s knowledge, there has been no documented work to have successfully established the
usability of the road simulator for modal analysis in the absence of force measurements. The
possibility of using displacement or pressure measurements as linear (proportional) equivalent
representations of the forces generated at the posts of the simulator, in the synthesis of FRFs and
subsequently in the estimation of the modal parameters, is evaluated in this work. Additionally,
the use of the simulator for OMA utilizing response-only data is explored. Such an analysis
requires that excitation forces are random, uncorrelated and spatially well-distributed. The
adherence of excitations from the simulator to these requirements and its effect on modal
parameter estimation is studied.
Specific issues under evaluation include the effect of dynamic boundary conditions during the
testing process, the effect of system non-linearity on the tests, and the effect of system variances
over time, due to reasons such as changes in suspension stiffening, on modal parameter
estimation. Considering that the structure is excited at the tires, the effect of the suspension
system acting as a mechanical filter on the nature and quality of excitations transmitted to the
5
structure, and the consequent effect on successful excitation of all modes of vibration of the
vehicle in the frequency range of interest, are investigated. Limitations imposed by the absence
of lateral forces during testing, and their influence on determination of the system’s modal
parameters are acknowledged.
Specific aims
1. Modal parameter estimation of a truck chassis based on a modified FRF formulation
utilizing pressure measurements from the road simulator as inputs.
2. Modal parameter estimation based on a modified FRF formulation utilizing displacement
measurements from the road simulator as inputs.
3. Validation of estimates from modified FRF tests against conventional impact tests on the
test structure.
4. Validation of estimates from modified FRF tests against estimates obtained from a
response-only modal test for the same structure.
5. Discussion of the successes and shortcomings of either formulation in comparison with
conventional and operational modal testing methodologies.
Chapter 1 describes the role of modal analysis in the automotive development process, and an
overview of conventional methods employed for the same. The motivation for this thesis work is
presented in terms of the benefits of using a four-post road simulator for modal applications, and
the main objectives of the work are laid out. Chapter 2 discusses in brief fundamental concepts
in Experimental and Operational Modal Analysis and additional mathematical formulations
employed in this study. Chapter 3 describes the experimental setup and details on the tests
conducted for this study, with a brief discussion on the nature of modal estimates from each test.
Comparisons between estimates across tests are discussed in detail in Chapter 4. Conclusions
6
from this work and the scope for future work are summarized in Chapter 5, followed by a list of
references.
7
2
Background
2.1 Principles of Experimental Modal Analysis
The process of determining the modal parameters of a system by experimental test methods
employing force and response measurements is referred to as Experimental Modal Analysis
(EMA). EMA is a well-established testing methodology, with the various aspects thereof
reprsentatively presented in Figure 2-1. The structure under study is excited by means of known
measurable forces and corresponding responses are measured at specific locations on the
structure. The response-to-force ratios for all input-output combinations are formulated
mathematically as Frequency Response Functions (FRFs), which in turn form the basis for modal
parameter estimation using various processing algorithms. Modal parameters include the
system’s natural frequencies and corresponding damping values, mode shapes and modal scaling
information. In conventional testing practices, inputs are typically provided by impact hammers
or electro-dynamic shakers, and responses are measured using accelerometers. FRFs are
processed using spatial, time or frequency domain algorithms. A brief overview is presented
ahead. A more detailed exposition on the principles of EMA can be found in References [1,2].
In its simplest form, the frequency response function (FRF) can be expressed by Equation (2.01).
𝐻𝑝𝑞 𝜔 =
𝑋𝑝 𝜔
𝐹𝑞 𝜔
(2.01)
where Xp ω represents the response spectrum of the system at point p, and Fq ω represents the
force spectrum at point q.
8
Figure 2-1: Experimental Modal Analysis
The FRFs measured for all input-output combinations for a multi-input-multi-output (MIMO)
system can be expressed in the partial-fraction form as the superimposition of individual modes
in a form given by Equation (2.02).
𝑁
𝐻 𝜔
=
𝑟=1
where
𝐴𝑟
𝐴∗𝑟
+
𝑗𝜔 − 𝜆𝑟 𝑗𝜔 − 𝜆∗𝑟
𝐴𝑟 = 𝑄𝑟 𝜓𝑟 𝜓𝑟𝑇
(2.02)
(2.03)
Q r represents the modal scaling coefficient and ψr the modal vector for the rth mode. λr and λ∗r
represent the conjugate pair of complex-valued natural frequencies for the rth mode, with N
representing the total number of modes.
9
Impulse response functions (IRFs), which are the time-domain equivalent of the FRFs, are given
as a linear superposition of modes by Equation (2.04).
𝑁
∗
𝐴𝑟 𝑒 𝜆 𝑟 𝑡 + 𝐴∗𝑟 𝑒 𝜆 𝑟 𝑡
ℎ(𝑡) =
𝑟=1
(2.04)
These IRFs form the basis for modal parameter estimation using time-domain algorithms such as
the Poly-reference Time-Domain (PTD) algorithm used for this study.
The PTD algorithm is well-suited for data configurations where the number of response degreesof-freedom No far exceeds the number of input degrees of freedom (Ni ), as in the present
study. It is a high-order parameter estimation algorithm, with the polynomial order given by
m≥
2N
Ni
. Frequency response functions obtained from the tests are used to compute impulse
response functions via the Fast Fourier Transform in the matrix-polynomial form represented by
Equation (2.05).
ℎ 𝑡𝑜
ℎ(𝑡1 ) .. ℎ(𝑡𝑚 −1 )
𝛼0
𝛼1
..
= − ℎ 𝑡𝑚
(2.05)
.. ℎ𝑝𝑞 𝑡𝑖+𝑘
(2.06)
𝛼𝑚 −1
where
ℎ 𝑡𝑘
𝑁𝑜 𝑥𝑁𝑖
= ℎ𝑝1 𝑡𝑖+𝑘
ℎ𝑝2 𝑡𝑖+𝑘
The coefficients αk , and subsequently the modal frequencies, damping, modal vectors and
modal scaling, are determined based on Equation (2.05) in a two-stage process described in [1]. A
more detailed description of the PTD algorithm can be found in [6].
For purposes of validation, modal vectors for each natural frequency, determined using different
testing/estimation methods, are compared using the Modal Assurance Criterion (MAC), given by
10
Equation (2.07). More details on validation techniques, and specifically the MAC, can be found
in [1].
𝑀𝐴𝐶
𝑐𝑑𝑟
=
𝜓𝑐𝑟
𝜓𝑐𝑟 𝐻 𝜓𝑑𝑟
𝐻 𝜓
𝑑𝑟 𝜓𝑑𝑟
2
𝐻
𝜓𝑑𝑟
(2.07)
ψcr and ψdr are the modal vectors being compared with each other.
The MAC coefficients indicate the degree of linear independence between modal vector
estimates. Two modal vectors are linearly independent if the MAC coefficient between them is
close to zero. A MAC value of unity is indicative of the vectors depicting the same mode within
limitations of observability. More discussion on interpreting MAC plots would follow during the
discussion of results in Section 4.
2.2 Non-conventional Formulations of Frequency Response Functions
On a road simulator such as the MTS-320 four-post simulator used in this work, there is
generally no provision for measurement of forces that excite the mounted structure at the four
posts, preventing FRF measurements in their conventional form. The simulator, however, does
offer the measurement of input pressure and displacement via transducers built into the hydraulic
actuators, shown in Figure 2-2, and alternate forms of FRF based on these inputs can be
formulated as expressed in Equations (2.08) and (2.09), in contrast to the form represented by
Equation (2.01). In the event that these inputs - pressure and displacement - are perfectly linear
functional (proportional) representations of the force generated at the posts, within the frequency
range of concern, application of conventional modal analysis theory for parameter estimation
would still be practical with the exception of estimation of modal scaling information. It is
11
consequently necessary that the dynamics of the actuator and the oil column within are beyond
the frequency range of interest for the study. Estimation of the system’s natural frequencies and
corresponding damping coefficients and mode shapes should then in theory still be possible
under this assumption, and studies in assessing the modal behavior of the system and
applications in validation of modal estimates from finite-element analysis (FEA) would remain
unhindered. Applications such as FE model-updating would not however be possible in the
absence of modal scaling information.
The tests with non-force measurements for modal analysis are unique and significant in many
ways. One of the main reasons to attempt these measurements is to analyze the possibility of
using road simulator facilities available in labs and industries to perform modal test on
automotive structures. This would enable obtaining modal frequencies and mode shapes while
using the very same facility that is used for various other vehicle development tests.
Figure 2-2. A post of the MTS-320 road simulator at UC-SDRL.
12
For the test utilizing pressure measurements, the pressure-based FRF is given by
Hxp ω =
X(ω)
P(ω)
(2.08)
where P(ω) represents the frequency spectra of the pressure entering the column of each
cylinder (P1, P2, P3, P4) of the simulator. It is to be noted that this is not the pressure acting at
the surface of the wheelpan but at the bottom of the column where the pressure transducer is
located. There is no direct method to compute the forces from the pressure measurements
because of the inertial effect caused by the weight of the wheelpan, the moving components of
the actuator of the simulator and the dynamics of the oil column and the wheelpan. These govern
how the pressure measurements would relate to the forces transmitted at the wheel pans, and if
the relation is indeed linear and proportional. As would be demonstrated further in this work,
these are the primary reasons for the modified FRFs, constructed using pressure measurements,
to have numerical characteristics that differ from conventional FRFs. A detailed discussion on
results obtained from a test based on the pressure-based FRF formulation follows in Section
3.2.2.
The second modified FRF measurement considered in this study is based on displacement
measurements of the rigid moving elements of the shaker posts of the simulator driving the
wheel pans. Linear variable differential transducers (LVDT), built into the simulator system, are
used to obtain these measurements. The displacement-based FRF is given by Equation (2.09).
Hst ω =
13
Xs (ω)
Xt (ω)
(2.09)
A conventional FRF, if force were measurable, would take the form
Hst ω =
Xs (ω)
Ft (ω)
(2.10)
Subscripts s and t refer to measurements on the structure and shaker/transducer respectively.
Xs (ω) refers to the response spectra measured using accelerometers mounted at various points on
the vehicle structure. Xt (ω) represents displacement measurements from the displacement
transducer built into the shaker post, with Ft (ω) representing the spectra of unmeasured forces
generated at the posts of the road simulator and transmitted to the vehicle structure at the tires.
Hst (ω) represents the ideal FRF measurements possible if forces were measurable. Hst ω
represents modified FRF measurements based on displacements at the posts as input.
Considering the fact that force inputs cannot be measured for this test, Hst ω is not available.
Xs (ω) and Xt (ω) measurements are used for direct computation of Hst ω , as represented in
Equation (2.09). For the given experimental setup, Hst ω is expected to contain the same
information regarding the poles of the system as Hst ω . This is analytically verified by
examining a three-degrees-of-freedom lumped mass system shown in Figure 2-3, which is a
simplified representation of the physical setup being tested.
14
Figure 2-3. Lumped-mass model for system representation
For this system, two forms of FRF are computed - one with conventional force measurement (f1)
as input, and the other with the displacement of the actuation element as input. The endeavor is
to demonstrate that the use of either input yields functions that, in essence, contain similar
system poles and would be expected to yield similar modal parameters for the system.
With the force at the wheelpan considered as input in the FRF formulation, the equations of
motion of the system can be represented by Equation (2.11).
𝑚𝑡
0
0
0
𝑚𝑏
0
0
0
𝑚𝑎
𝑥𝑡
𝑐𝑡𝑤 + 𝑐𝑏𝑡
−𝑐𝑏𝑡
𝑥𝑏 +
0
𝑥𝑎
−𝑐𝑏𝑡
𝑐𝑏𝑡 + 𝑐𝑎𝑏
−𝑐𝑎𝑏
0
−𝑐𝑎𝑏
𝑐𝑎𝑏
𝑘𝑡𝑤 + 𝑘𝑏𝑡
𝑥𝑡
𝑥𝑏 +
−𝑘𝑏𝑡
𝑥𝑎
0
15
−𝑘𝑏𝑡
𝑘𝑏𝑡 + 𝑘𝑎𝑏
−𝑘𝑎𝑏
0
−𝑘𝑎𝑏
𝑘𝑎𝑏
𝑥𝑡
𝑓𝑡
𝑥𝑏 = 0
𝑥𝑎
0
(2.11)
It can be shown that in the frequency-domain, the force-based FRF relationship for response xa is
given by Equation (2.12).
𝑋𝑎 (𝜔)
−𝜔2 . 𝐷𝑏𝑡 𝜔 . 𝐷𝑎𝑏 (𝜔)
=
𝐹𝑡 (𝜔) 𝐴 𝜔 . 𝐵 𝜔 . 𝐶 𝜔 − 𝐶 𝜔 . 𝐷𝑏𝑡 𝜔 2 + 𝐴 𝜔 . 𝐷𝑎𝑏 𝜔
2
(2.12)
where
𝐴 𝜔 = −𝑚𝑡 𝜔2 + 𝑗𝜔 𝑐𝑡𝑤 + 𝑐𝑏𝑡 + 𝑘𝑡𝑤 + 𝑘𝑏𝑡
(2.13)
and
𝐵 𝜔 = −𝑚𝑏𝑡 𝜔2 + 𝑗𝜔 𝑐𝑏𝑡 + 𝑐𝑎𝑏 + 𝑘𝑏𝑡 + 𝑘𝑎𝑏
(2.14)
and
𝐶 𝜔 = −𝑚𝑎 𝜔2 + 𝑗𝜔 𝑐𝑎𝑏 + 𝑘𝑎𝑏
(2.15)
and
𝐷𝑥𝑦 𝜔 = 𝑗𝜔 𝑐𝑥𝑦 + 𝑘𝑥𝑦
(2.16)
Similarly, with the displacement at the wheelpan xw as the input, the system equations can be
expressed in a form given by Equation (2.17).
𝑚𝑡
0
0
0
𝑚𝑏
0
0
0
𝑚𝑎
𝑥𝑡
𝑐𝑡𝑤 + 𝑐𝑏𝑡
−𝑐𝑏𝑡
𝑥𝑏 +
0
𝑥𝑎
−𝑐𝑏𝑡
𝑐𝑏𝑡 + 𝑐𝑎𝑏
−𝑐𝑎𝑏
𝑘𝑡𝑤 + 𝑘𝑏𝑡
𝑥𝑡
𝑥𝑏 +
−𝑘𝑏𝑡
𝑥𝑎
0
0
−𝑐𝑎𝑏
𝑐𝑎𝑏
−𝑘𝑏𝑡
𝑘𝑏𝑡 + 𝑘𝑎𝑏
−𝑘𝑎𝑏
0
−𝑘𝑎𝑏
𝑘𝑎𝑏
𝑥𝑡
𝑥𝑏 =
𝑥𝑎
𝑐𝑡𝑤 𝑥𝑤 + 𝑘𝑡𝑤 𝑥𝑤
0
0
(2.17)
The modified FRF relationship for this system, with xw as input and xa as response, is given by
Equation (2.18).
𝑋𝑎 (𝜔)
−𝜔2 . 𝐷𝑡𝑤 𝜔 . 𝐷𝑏𝑡 𝜔 . 𝐷𝑎𝑏 (𝜔)
=
𝑋𝑤 (𝜔) 𝐴 𝜔 . 𝐵 𝜔 . 𝐶 𝜔 − 𝐶 𝜔 . 𝐷𝑏𝑡 𝜔 2 + 𝐴 𝜔 . 𝐷𝑎𝑏 𝜔
16
2
(2.18)
Comparing Equations (2.17) and (2.18), it can be observed that the two FRF representations for
the system differ by a factor of 𝑗𝜔 𝑐𝑡𝑤 + 𝑘𝑡𝑤 , but essentially contain the same information
regarding the poles of the system. This is further evident from Figure 2-4, and it is observed that
with the exception of the scaling, both FRFs share similar characteristics and hence would be
expected to yield similar modal estimates.
Conventional (Force-based) and Modified (Displacement-based) FRFs
5
10
Force-based FRF
Displacement-based FRF
Magnitude
0
10
-5
10
-10
10
0
5
10
15
20
25
30
Frequency
35
40
45
50
200
Force-based FRF
Displacement-based FRF
Phase angle
100
0
-100
-200
0
5
10
15
20
25
30
Frequency
35
40
45
50
Figure 2-4. Comparison of force-based and displacement-based FRFs
Extending the argument to the vehicle structure, the displacement-based modified FRF for the
structure would be expected to yield similar modal parameters as a conventional force-based
FRF would yield under the conventional EMA framework. This expectation is examined through
a displacement-based modified FRF test, the results of which are detailed in Section 3.2.3.
17
2.3 Principles of Operational Modal Analysis
Operational Modal Analysis (OMA) is an emerging field of research involving estimation of
modal parameters based on measured responses only. This technique does not require the
measurement of input excitation forces, with the requirement being that the forces are random
and uncorrelated in nature, and have a good spatial distribution across the structure being tested.
The unmeasured excitation forces in the general case of OMA are often the naturally acting
ambient forces which are representative of actual operating conditions but may have
shortcomings in exciting all modes of vibrations. Artificial methods of exciting the structure are
often incorporated to improve the excitation characteristics to meet OMA requirements, and the
accuracy of the estimation process largely depends on how well the excitations meet the
aforementioned assumptions. While OMA methods have largely been employed for civil
structures such as bridges and wind turbines, automotive applications
[16,17]
have also been
emerging areas of interest.
For modal parameter estimation based on OMA methods, a number of algorithms have been
proposed. The extension of conventional parameter estimation algorithms such as the Polyreference Time-Domain (PTD) algorithm under the Unified Matrix Polynomial Approach
(UMPA) framework for OMA applications has been proposed by Reference
[12]
, and the same
has been employed for modal parameter estimation vis-à-vis OMA for this work. A more
rigorous exposition on OMA test methods and estimation algorithms can be found in
brief overview is presented ahead.
18
[11,19]
, and a
Equation (2.01) can be rewritten for all input-output combinations in its matrix form given by
Equation (2.19).
𝑋 𝜔
= 𝐻 𝜔
𝐹 𝜔
(2.19)
The Hermitian of the FRF given by Equation (2.19) is given by Equation (2.20).
𝐻
𝑋 𝜔
= 𝐹 𝜔
𝐻
𝐻 𝜔
𝐻
(2.20)
Multiplying Equations (2.19) and (2.20),
𝑋 𝜔
𝑋 𝜔
𝐻
= 𝐻 𝜔
𝐹 𝜔
𝐹 𝜔
𝐻
𝐻 𝜔
𝐻
(2.21)
After signal processing and averaging, the expressions in Equation (2.21) can be expressed in a
form given by Equation (2.22)
𝐺𝑋𝑋 𝜔
= 𝐻 𝜔
𝐺𝑋𝑋 𝜔
𝐺𝐹𝐹 𝜔
∝ 𝐻 𝜔
𝐻 𝜔
𝐼 𝐻 𝜔
𝐻
𝐻
(2.22)
(2.23)
Under the assumptions that the forces are random, broadband and smooth in the frequency range
of interest, the response power spectra are proportional to the frequency response functions as
shown by Equation (2.23), and can form the basis for modal parameter estimation based on
output-only data. Further, it is required that the excitations are spatially well-distributed to excite
all possible modes of vibration from multiple excitation points in the frequency range of interest.
19
Response correlation functions, represented by R xx , are determined from the measured response
power spectra, and form the basis for parameter estimation with time-domain algorithms for
OMA instead of the impulse response functions used by time-domain algorithms in the
conventional EMA framework. For use with high-order time-domain algorithms, the response
correlation functions, in a form represented by Equation (2.24), form the basis for modal
parameter estimation using the PTD algorithm in this work. This representation is analogous to
Equation (2.05) used by time-domain parameter estimation algorithms for EMA. It is to be noted
that only the positive lag components of the correlation functions are used in parameter
estimation [12].
𝛼1
𝛼2 .. 𝛼𝑚
𝑁𝑟𝑒𝑓 ×𝑚 𝑁𝑟𝑒𝑓
𝑅𝑥𝑥 (𝑡𝑖+1 )
𝑅𝑥𝑥 (𝑡𝑖+2 )
..
𝑅𝑥𝑥 (𝑡𝑖+𝑚 )
= − 𝑅𝑥𝑥 (𝑡𝑖+0 )
𝑁𝑟𝑒𝑓 ×𝑁0
(2.24)
𝑚𝑁 𝑟𝑒𝑓 ×𝑁0
In the present study, the excitations, while being random, broadband and smooth, are not
spatially well-distributed and are restricted to vertical excitations only. Further, the excitations
are not imparted to the structure directly but through the tires and the suspension system, which
act as a mechanical filter for the excitations, thus restricting a complete excitation of all modes of
vibration. Section 3.2.4 discusses the effect of these limitations on modal parameters estimated
under the OMA framework in detail.
20
3
Test Details
3.1 Test Setup
The test structure used for this study consists of a small truck frame with engine and gearbox
mounted. The frame is supported by independent double wishbone suspensions in the front and
solid axle leaf springs at the rear. The overall structure is composed of the frame, the suspensions
and tires, the gearbox and the engine. The simulator used for this study is an MTS 320 four-post
road simulator, located at the Non-linear Dynamics Testing Facility at the Structural Dynamics
Research Laboratory (SDRL), University of Cincinnati (UC) and is shown in Figure 3-1.
Figure 3-1 The MTS-320 Road Simulator at UC-SDRL.
The MTS Model 320 system
[3]
is a tire-coupled road simulator, used typically for automotive
squeak-and-rattle analysis, noise-vibration-harshness (NVH) evaluations, structural durability
21
and ride comfort tests. The overall system comprises of the road simulator, a rotary forklift
system for hoisting vehicles onto the simulator, a control room, a pump room and distribution
towers, all located at the Non-Linear Dynamics Testing facility at UC-SDRL.
simulator itself consists of four hydraulic actuators (MTS Model 248.03
[4]
The road
), a hydraulic pump
(MTS Model 506.52C) and controller (MTS 498.22 Test Processor and MTS 497.05 Hydraulic
Control Unit). The hydraulic actuators are capable of vertical excitations with a force rating of
5500 lbs and a stroke length of ±3 inches about the mean position (corresponding to voltage
signals between ±10 V measured by an internal LVDT). The actuators have an operating
frequency range of 2-80 Hz – the lower limit owing to limitations on the wheel pan and piston
assembly, and the higher limit restricted by the natural frequency of the actuator and the oil
column within. The hydraulic pump has a working pressure of 3000 psi with a flow rate of 55
gpm. The software interface with the hardware is provided for by FlexTest II® and allows for the
control of the test rig using a commercially available PC.
For measurement of the system responses, fifty ICP-powered piezoelectric tri-axial
accelerometers (PCB XT356B18) are distributed across each component of the vehicle structure.
These sensors have a nominal sensitivity around 1V/g and are capable of measurements in the
frequency range of 0.5 – 3000 Hz. Additionally, four single uni-axial accelerometers (PCB
UT333M07) are placed on each wheel-pan of the road simulator, adjacent to the tire. Responses
from these uni-axial sensors are indicative of the nature of excitations being imparted by the
simulator, and serve the purposes of calculating modified driving point FRFs which are desirable
from algorithmic point of view while carrying out modal parameter estimation and to evaluate
the validity of certain test-specific assumptions.
22
The distribution of the sensors across the vehicle structure is summarized ahead, and shown in
Figure 3-2. The locations were chosen based on estimates from previous studies so as to
maximize the observability of all modes of vibration of interest within hardware limitations
governed by the number of sensors available for the tests while retaining the ability to potentially
resolve modes close to one another in frequency. Details on the sensor distribution across the
structure are given ahead, with Figure 3-3 representatively showing sensor locations across select
components of the test structure.
Chassis and transaxle: 18 accelerometers - 54 response degrees-of-freedom.
Front Suspension: 14 accelerometers - 42 response degrees-of-freedom.
Rear Suspension: 8 accelerometers - 24 response degrees-of-freedom.
Engine: 8 accelerometers - 24 response degrees-of-freedom.
Gearbox: 2 accelerometers - 6 response degrees-of-freedom.
Figure 3-2. Sensor distribution and excitation locations on the test structure
23
Figure 3-3: Sensors (L:R) on (a) (Front) Wishbones (b) (Rear) Leaf Springs (c) Engine
The accelerometers are hooked to seven 16-channel breakout-boxes and six 8-channel ICP boxes
via BNC-Microdot cables. The breakout-boxes in turn are connected to signal conditioners via
ribbon cables, and in turn to digitizer cards (HP E1432A). The ICP boxes connect directly to the
digitizer cards, and do not need an external dedicated signal conditioner. The digitizer cards
connect to a VXI Mainframe (HP 75000 Series C) which in turn connects to a commercially
available PC via a Firewire interface (IEEE 1394). Considering the large number of channels
acquiring data, channels are selectively grouped for data gathering and monitored repeatedly for
functionality. Extensive labeling and tracking procedures ensure for easy identification of
sources of hardware failures for faster troubleshooting. The accelerometers are individually
calibrated using a 1-g RMS handheld calibrator at a frequency of 159.2 Hz. For data acquisition,
the software used for this study includes VXI DAC Express and UC-SDRL software, MRIT and
X-Acquisition. DAC Express is well-suited for capturing input and response time-histories over
a large number of channels, and saves files to a Standard Data Format (SDF). MRIT, an acronym
for Multiple-Reference Impact Testing, is designed for conventional EMA-based impact tests. XAcquisition on the other hand is used to acquire processed FRFs from the input-output
measurements in a multiple-input-multiple-output (MIMO) setup. The latter two save the data to
24
a Universal File (binary) Format (UFB). Data processing and parameter estimation are
performed using MATLAB and X-Modal II (another UC-SDRL software package).
3.2 Tests
The focus of this study is on the estimation of rigid-body modes and the low-frequency
deformation modes of the structure. Since the modes of interest have been known to be within
the 2-30 Hz range based on a preliminary test conducted on the structure
[15]
, the preferred
frequency range of interest for this study is the 2-30 Hz range. The road simulator is not capable
of generating excitations below 2 Hz owing to stroke limitations of the wheel pan and piston
assembly, or above 80 Hz owing to the natural frequency of the hydraulic actuator and the oil
column within, and thus fits perfectly with the requirements for the tests. The tests conducted for
this study are compared with one another, and are additionally compared with baseline results
obtained from conventional EMA-based impact tests.
3.2.1 Conventional Impact Tests
Baseline tests are conducted on the test structure under different configurations and at different
stages of the testing process to establish a set of benchmark estimates for comparison with test
results, and to evaluate variations in the system over the test length. An initial impact test
performed with the vehicle on ground prior to all tests (referred to further in the document as the
‘EMA Ground 1’ test) is documented in
[15]
. A similar test (referred to as the ‘EMA Ground 2’
test) is performed in this study with the vehicle brought back to the ground after all tests
involving the simulator are completed. Additionally, an impact test is conducted with the vehicle
strapped onto the posts of the road simulator (referred to as the ‘EMA 4P’ test) to evaluate the
effect of variations in boundary conditions on estimates. Unlike the test with the vehicle on the
25
ground, the structure now sits on the wheelpans of hydraulic actuators of the road simulator
which are bolted to the ground. The wheels are strapped firmly in place using nylon straps. These
straps alter the boundary conditions from the EMA Ground 1 and EMA Ground 2 tests, and their
effect on variations in the modal parameters estimated would be discussed in Section 4.1.1. The
data acquisition parameters (listed below) and excitation locations (shown in Figure 3-2) for
these tests remain unchanged across all impact tests.
Sampling Frequency: 125 Hz
Frequency Resolution: 0.125 Hz
RMS averages: 3
Window: Uniform
Excitation degrees of freedom: 14
Response degrees of freedom: 150
Excitation signal: Force
Response signal: Acceleration
Results from the EMA Ground 1 test can be found in a study documented in
[15]
, and are used in
this work for comparative purposes. A summary of the natural frequencies and corresponding
damping estimates obtained from this test is presented in Table 5-1.
For the EMA 4P test, Figure 3-4 shows the consistency diagram using the PTD algorithm for the
0 - 30 Hz frequency range.
26
Figure 3-4. Consistency diagram for an estimate for the EMA impact test on the simulator
A total of eighteen modes are obtained in the frequency range of interest. A high degree of linear
independence can be observed from the auto-MAC
[1]
plot shown below in Figure 4. Modal
frequencies and damping estimates are listed in Table 2 in Section 5. Estimates from this test are
to be used as a comparison for all tests performed on the simulator. It is important to note that
though this test is done with the structure on the simulator, it still has static boundary conditions
albeit different from those on the ground. This would differ from the forthcoming tests involving
simulator excitations due to dynamic boundary conditions caused by motion of wheelpans of the
simulator.
27
Figure 3-5. Auto-MAC plot for modal estimates from EMA 4P test
A final impact test is conducted with the vehicle taken off the road simulator after completion of
all other tests, to evaluate any potential changes in the modal characteristics of the system owing
to reasons such as variations in suspension stiffening and wheel camber angle changes, between
the vehicle being hoisted onto and off the simulator. The results are presented in Table 5-2, and
comparisons with other impact tests are documented in Section 4.1.
3.2.2 Modified FRF Based Analysis with Pressure as Input (EMA-Pressure)
This test is one of the two tests performed for experimental evaluation of the theories presented
in Section 2.2. Here, pressure measurements from transducers built into the hydraulic actuators
of the road simulator are used as inputs in the formulation of FRFs for modal parameter
estimation.
28
The following data acquisition parameters have been used for this test:
Sampling Frequency : 125 Hz
Frequency Resolution : 0.0625 Hz
50 RMS averages with 3 cyclic averages
Window : Hanning
Excitation degrees of freedom : 4
Response degrees of freedom : 154 (150 on the structure, 4 on the wheelpans)
Excitation signal : Pressure
Response signal : Acceleration
RMS averaging and cyclic averaging [8] are signal processing techniques employed for improved
statistical characteristics of the estimated FRFs, for reduction of unbiased and leakage errors
respectively. Figure 3-6 below shows the consistency diagram for one of the estimates using the
PTD algorithm for the EMA pressure dataset. Good estimates, with near complete modal
information in terms of frequency, damping and mode shapes, are represented by blue diamonds.
It can be seen from the diagram that only a few modes have been estimated consistently across
increasing model orders. Multiple estimates using narrow frequency bands, with varying
combinations of reference channels, are essential in order to extract maximum possible number
of modes of the structure.
29
Figure 3-6. Consistency diagram for an estimate for EMA Pressure Test
The pressure-based modified FRF does not appear to share similar characteristics as a forcebased conventional FRF. A modified driving-point FRF measurement, with pressure as input and
acceleration response at the wheelpan as the output, is shown in Figure 3-7. The mechanical and
hydraulic elements of the actuator and the oil film alter the pressure-force relation, hindering the
use of pressure measurements from the posts as substitutes for force in the FRF formulation. This
is further observed from the poor estimation of modal parameters based on pressure-based FRFs,
as can be seen in the consistency diagram in Figure 3-6.
30
Figure 3-7. Driving-point Modified FRF curves for EMA Pressure Test
The auto-MAC plot in Figure 3-8 shows thirteen modes obtained in the frequency range of 2-30
Hz. As this is not a conventional FRF measurement, it is difficult to extract some of the modes of
the system. It can also be noted that a few modes seem to have relatively high auto-MAC values
in the off-diagonal elements. On the basis of this analysis, the results from pressure-based FRFs
are not very encouraging. However, more insights will be available when results are compared
with those from other datasets in Section 4.2. Additionally, alternate methods for validation of
results from this test have been attempted and described in Section 4.4. A complete list of modal
frequencies and damping values are provided in Table 5-2 in Section 5.
31
Figure 3-8. Auto-MAC plot for EMA Pressure
3.2.3
Modified FRF Based Analysis with Displacement as Input (EMA Displacement)
This test utilizes displacement measurements from LVDTs built into the hydraulic actuators as
inputs, and are used in FRF estimation in consonance with the theory discussed in Section 2.2.
The data acquisition parameters for this test have been summarized below:
Sampling Frequency : 125 Hz
Frequency Resolution : 0.0625 Hz
50 RMS averages with 3 cyclic averages
Window : Hanning
Excitation degrees of freedom: 4
Response degrees of freedom: 154 (150 on the structure, 4 on the wheelpans)
Excitation Signal : Displacement
Response Signal : Acceleration
32
To estimate modified driving point FRFs for this test, responses are measured on the wheelpans
of the simulator rather than on the test structure. These modified driving point FRFs are essential
to ascertain the dynamics of the column to be rigid within frequency range of concern for this
study. It is important to note that these measurements do not provide any useful information
regarding the test structure itself nor do they allow estimation of modal scaling as in the case of
conventional driving point FRF measurements. The modified driving point FRFs for this test,
shown in Figure 3-9, are smooth curves in the frequency range of interest devoid of any peaks or
valleys. This plot is reflective of the dynamics of the mechanical elements between the actuator
elements and the wheelpan, and demonstrates that the posts of the shaker contain no poles in the
frequency range of interest.
Figure 3-9. Driving point FRF Curves for the EMA Displacement Test
The PTD algorithm has been used for estimation, and a consistency diagram for one estimate is
shown in Figure 3-10. It can be observed that this consistency diagram is significantly better than
33
the consistency diagram for the EMA-Pressure test, indicating possibly good parameter
estimation.
Figure 3-10. Consistency Diagram for the EMA Displacement test
Seventeen modes are determined in the 2-30 Hz range, and have been summarized in Table 5-2.
The auto-MAC plot for these modes has been shown in Figure 3-11, and the linear independence
of the modes can be observed.
34
Figure 3-11. Auto-MAC plot for EMA displacement test
3.2.4 Response-only Modal Analysis
Tests conducted for OMA in a related study [15] involved the use of random impact excitations in
one case and shaker excitations in the other. Shaker excitations, as used in that study, violated
the OMA requirement of uniform spatial distribution, and limited the ability to identify all modes
of the system. Random impact excitations, though conforming to the OMA assumptions and
providing better results, were not representations of true operational conditions. This was due to
the fact that both excitation methods were used directly on the frame and hence did not account
for excitations coming from the ground through the suspension system. The use of the simulator
in this study is expected to more closely replicate operational conditions such as the effects of
vehicle ground interactions and the role of the suspension system in isolating the vehicle
structure from these excitations.
35
It is important to note that this test is not fully operational in the exact sense, as the effects of
strong rotational harmonics (unbalanced tires, etc) and wind excitations are absent in the lab
environment. This condition is favorable for this study, as OMA requirements of broadband
excitations could be violated by the presence of sub-component interactions such as engine
harmonics
[17][18]
. The simulated forces thus satisfy the spectral requirements of excitation for
OMA. Given that these excitations are limited to forces coming through the four wheels and also
being mitigated by the suspension system’s mechanical filtering, the OMA requirements of
spatially well distributed forces are partly violated. The effects of these violations on the quality
of the estimates would be discussed in detail in Section 4.3.
Time histories recorded from 150 response channels are processed to obtain power spectra for
use in OMA methods
[12]
. The data acquisition and processing parameters have been listed
below:
Sampling Frequency : 160 Hz
Duration of data acquisition : 12 minutes (114892 time points)
Number of excitation locations : 4
Response degrees of freedom : 150
Cyclic Averaging over 3 ensembles with 66.6% overlap processing employed for noise reduction
Number of averages after cyclic averaging and overlap processing : 97
Hanning window employed for reduction of leakage errors
Response Signal : Acceleration
36
The power spectra data computed from the recorded response time histories are processed using
two different algorithms; the data-dependent SSI algorithm (SSI-Data)
[13]
and the PTD
algorithm, both being time domain algorithms.
The PTD algorithm is sensitive to the selection of the reference channels in the parameter
estimation process. In the EMA framework, these reference channels are readily available as the
input degrees of freedom. However, in the case of OMA, unless the excitation is spatially welldistributed, a judicious selection of reference channels is critical to successful parameter
estimation. In a test configuration such as the current one, the presence of just four input
locations implies that the rank of the response power spectral matrix is at best four, even though
150 channels of response data are measured, making the parameter estimation process strongly
sensitive to the choice of the reference responses. This strongly hinders the parameter estimation
process using an algorithm such as the PTD.
A need was felt to employ a more well-established algorithm vis-à-vis OMA methods that did
not suffer from the kind of aforementioned shortcomings in the context of the current study. The
data dependent SSI algorithm (SSI-Data)
[13]
, as implemented in B&K Type 7760, was
considered since it works directly on measured output responses without processing the time
histories to calculate correlation functions (used by most time domain OMA algorithms
including PTD). The author, not being familiar with the use of the SSI-Data algorithm for
parameter estimation, gratefully acknowledges Dr. Shashank Chauhan (Bruel & Kjaer) for the
estimation of the modal parameters of the system using the SSI-Data algorithm based on the
response data from this study. The results from the SSI-Data algorithm are used in order to
compare a commercial OMA algorithm. The modal parameters estimated from each algorithm
have been listed below and compared in detail in Section 4.3.
37
OMA results using SSI-Data algorithm (OMA-SSI)
With the use of the SSI-Data algorithm, choosing five projection channels, fourteen modes are
found to be present between 2-30 Hz. The MAC plot for the estimates obtained has been shown
in Figure 3-12. Engine and lateral modes are not well excited given the nature of the test, and
appear to have relatively high off-diagonal MAC values, as observed between the 16.4 - 17.64
Hz modes and the 22.85 - 24.32 Hz modes. The rest of the modes show a high degree of linear
independence.
Figure 3-12. Auto-MAC plot for the OMA estimates using the SSI-Data algorithm
OMA results using PTD algorithm (OMA-PTD)
The PTD algorithm is used to process the response data to extract modal parameters, in addition
to SSI-Data. PTD is a high order algorithm and generally a few response channels are selected as
reference responses for PTD to work satisfactorily with OMA data. Autopower spectral plots of
38
the response channels are studied, and reference channels are chosen so as to maximize the
combined spectral information between the selected channels. Various combinations of response
channels are used as references and their results are combined to choose the best estimates.
Starting with two and choosing up to twenty reference responses, modes are obtained by
consolidating estimates from various combinations of response channels used as references. A
consistency diagram for one of the estimates using PTD algorithm on the response power spectra
is shown in Figure 3-13.
Figure 3-13. Consistency Diagram for an OMA estimate using the PTD algorithm
Fourteen modes are found in the 2-30 Hz frequency range. The auto-MAC plot for the estimates
obtained has been shown in Figure 3-14. There is a clutter of off-diagonal MAC values observed
in the higher frequency range. These are again the closely lying lateral modes, predominantly in
the 20-31 Hz range. One probable explanation for this could be that the combinations of
39
reference responses chosen for parameter estimation did not have sufficient information
pertaining to these lateral modes.
Figure 3-14. Auto-MAC plot for the OMA estimates using PTD
It is observed that results from PTD are very sensitive to selection of reference response
channels. The fact that OMA assumptions of spatially well distributed input excitations are also
not true in this case adds more complexity. While such issues are not encountered in EMA, they
are critical in case of OMA as there are no definite guidelines about choosing reference
responses, except that these responses should be able to extract all modes of interest. One rule of
thumb in this regard is to check the auto spectra of selected references and see if they have all
modal information, i.e., peaks corresponding to most system modes. This is often a tedious task,
particularly when a large number of sensors are involved and the system being analyzed is
complex. Thus it is pertinent to find more effective ways of choosing suitable responses as
references, especially in case of high order algorithms like PTD, Rational Fraction Polynomial
40
(RFP)
[5]
, etc. One way of utilizing maximum reference data would be the use of singular-value
decomposition methods to develop virtual reference responses for optimal parameter estimation.
41
4
Comparison of results
4.1 Comparison of results from conventional impact tests
The first comparison is made between EMA impact test results with the structure raised on the
simulator (EMA 4P) and results from the EMA Ground 1 test documented in [15]. This is done to
validate the modes obtained by the impact test on the simulator. Results are also compared with
those from the impact hammer test conducted after bringing down the truck frame back on
ground at the end of all experiments (EMA Ground 2). There are various factors like spring
stiffening, wheel camber angle change, lateral deflection of tires, etc., that can affect the test
structure once it is lifted on to the simulator for tests, excited with the simulator and finally
brought down to the ground at the end. These three tests and comparisons among them would
help in better understanding of the changes the system undergoes across various tests. This
would also help in ascertaining the time-invariant aspect of the structure, considering that all
tests were conducted over a span of one month.
4.1.1
EMA 4P vs EMA Ground 1
From the cross-MAC plot shown in Figure 4-1, it can be observed that most of the modes
obtained by the EMA 4P test have good cross-MAC values with those obtained by the EMA
Ground 1 test. Some of the low cross-MAC values can be attributed to the fact that boundary
conditions differ between the two tests, as mentioned earlier in Section 3.2.1. This might also be
a reason for the slight shifts in frequencies across the two tests. Additionally, a mode at 28.03 Hz
which could not be earlier estimated in the ground 1 test has been determined in the EMA 4 P
test. Overall, there are not significant changes in the dynamic characteristics of the structure
when it is on ground and when it is raised on the simulator.
42
Figure 4-1. Cross-MAC of EMA 4P vs EMA Ground 1 estimates
4.1.2 EMA 4P vs EMA Ground 2
In this case also, most modes obtained in EMA 4P have been determined in the EMA Ground 2
test and the corresponding cross-MAC values are high (Figure 4-2). The low frequency mode at
3.76 Hz is missing from the EMA Ground 2 estimates, probably due to the low energy of impact
going into the test structure. It has also been observed from the previous study
[15]
that it is
difficult to determine the closely lying engine modes at 13.95 -14.17 Hz, given the nature of the
modes and limitations on the observability of engine modes. The higher frequency lateral modes
at 23.07 Hz, 26.27 Hz and 28.03 Hz are also difficult to fully excite consistently in each test, as
will be seen in further comparisons.
43
Figure 4-2. Cross-MAC of EMA 4P vs EMA Ground 2 estimates
4.1.3 EMA Ground 1 vs EMA Ground 2
As seen in the cross-MAC plot shown in Figure 4-3, most modes are determined consistently in
the EMA tests with the vehicle on the ground before and after the use of the simulator, barring
the modes at 3.89 Hz, 14.17 Hz, and some higher frequency modes beyond 20 Hz hitherto
discussed. It is to be noted that the mode at 17.47 Hz estimated by the EMA Ground 1 test has
shifted to around 16.54 Hz. These factors might indicate a change in the dynamics of the system
itself after running it on the simulator. The vehicle suspension system was not fully relaxed once
hoisted off the simulator, and variations in the suspension stiffness, wheel camber angle changes
when placing the vehicle on the ground, etc are probable reasons for the system not being fully
restored to its pre-testing conditions, and hence the variations in the modal estimates. However,
based on these comparisons in total, it can be concluded that the system dynamics are not
affected much before and after the tests. This indicates the time-invariant nature of the system
44
and also the fact that changing boundary conditions (structure on ground and structure on
simulator with tires strapped) do not have a significant effect on its overall dynamics.
Figure 4-3. Cross-MAC of EMA Ground 1 vs EMA Ground 2 estimates
4.2
Comparisons for results from the modified FRF based tests.
4.2.1 EMA 4P vs EMA-Displacement.
The EMA-Displacement test differs from the EMA 4P test fundamentally in the nature of
excitations. Unlike the direct excitation of the frame in the EMA 4P case, the EMA displacement
test involves the frame being excited by forces coming through the suspension system. The
engine modes around 14 Hz are again poorly excited, as also the lateral modes beyond 20 Hz.
Despite these limitations, some modes match well as can be seen from Figure 4-4. It has been
observed that some modal vectors for this test have a high level of complexity, and hence result
in low cross-MAC values for those modes. Visual inspection, however, indicates similarity
between these mode shapes and the expected shapes from conventional tests, and there is a need
45
for more robust estimation algorithms that handle modal vector complexities better. The effects
of these complexities in modal vectors on the MAC values have been discussed in relevant detail
in Section 4.4.
Figure 4-4. Cross-MAC of EMA 4P vs EMA-Displacement estimates
4.2.2 EMA 4P vs EMA-Pressure.
A cross-MAC plot (Figure 4-5) between estimates from the EMA-Pressure test and the EMA 4P
test indicates that most of the modes do not have a high cross-MAC value. As explained in
Section 3.2.2, parameters obtained using this method are inconsistent and not very accurate. This
does not instill high level of confidence in the results obtained using pressure-based EMA.
However, visual inspection of some mode shape animations indicates a similar underlying
pattern in both estimates (EMA 4P and EMA-Pressure), with the EMA-Pressure modes tending
to display greater complexity in motion. An approach based on separating the components of the
modal vectors and recalculating MAC values has been made in Section 4.4. This approach
46
results in increasing the MAC values by a certain level thus being more indicative of the
similarity between mode shapes. However it can be said at this point of time that the modified
FRFs based on displacements offer more potential than those based on pressure measurements.
Figure 4-5. Cross-MAC of EMA Pressure vs EMA 4P estimates
4.3 Comparisons for Results from the OMA Based Test
Modal parameters extracted using the SSI-Data algorithm are considered over those from the
PTD algorithm - for comparisons with the conventional and modified EMA methods – due to its
relative superiority in parameter estimation using OMA methods in the context of this study.
4.3.1 OMA-SSI vs OMA-PTD
Response data collected for the OMA studies are processed using two time domain algorithms,
PTD and SSI-Data. Some modes determined by the use of the PTD algorithm for the OMA test
match with those by the SSI-Data algorithm, as shown in Figure 4-6. As mentioned in Section
47
The high order PTD algorithm is highly sensitive to the choice of reference channels, especially
when OMA requirements for the excitations are not fully met and hence present inherent
challenges in the estimation of modes. It will be interesting to find an optimal method for
determining the reference channels for use with this algorithm, and to evaluate if successful
parameter estimation is possible with PTD in test configurations where the inputs are severely
limited spatially.
Figure 4-6. Cross-MAC of OMA-PTD vs OMA-SSI
4.3.2 EMA 4P vs OMA-SSI
One important aspect of the structure being tested using the simulator is the variation in
boundary conditions. The conventional EMA tests had the vehicle being tested on the ground or
on the simulator when the boundaries were for practically rigid. Subsequent tests have the posts
of the simulator exciting the structure and involve dynamic boundary conditions. This in turn
induces moderate to significant shifts in modal frequencies of the structure.
48
A significant observation has been the reordering of the sequence in which the low-frequency
rigid-body modes appear in the estimates. Conventional tests with static boundary conditions
have the front-axle pitching mode at 4.93 Hz preceding the yaw mode at 5.44 Hz. The tests with
the simulator in action, however, consistently see the yaw mode preceding the front-axle pitching
mode, as observed in the cross-MAC plot shown in Figure 4-7. This clearly indicates changes in
the dynamics of the structure in operation – a phenomenon that could not be observed in tests
where boundaries are static. Note that the same behavior was observed in case of the
displacement-based EMA test which shares the same excitation scenario as the OMA tests
(Figure 4-4).
Also, as emphasized earlier, excitations generated by the simulator violate the OMA requirement
of good spatial distribution over all degrees of freedom. The fact that the frame and engine are
not directly excited, but through the suspension system further affects the energy that goes into
the response degrees of freedom of those sub-structures. This explains the poor cross-MAC
coefficients for some of the modes - for instance, the engine mode at 13.87 Hz. The simulator
generates excitations primarily in the vertical direction, and hence most lateral modes such as the
frame modes between 26-29 Hz are poorly excited as well. In view of these limitations, not all
modes determined by the OMA test have high MAC values against EMA 4P results. But the
low-frequency rigid-body modes compare very well with corresponding modes from the EMA
4P test, as also some of the dominant deformation modes at higher frequencies.
49
Figure 4-7. Cross-MAC of EMA 4P vs OMA-SSI estimates
4.3.3 OMA-SSI vs EMA-Displacement
The EMA-Displacement test utilizes modified FRF computations, while the OMA test relies on
response-only data for parameter estimation. However, these tests are very similar to each other
with respect to the nature of the boundary conditions and the physical system setup and
excitation scenario. The modal estimates by these two methods are found to be in strong
agreement with each other, as shown by the cross-MAC plot in Figure 4-8. The low frequency
rigid body modes match very well both in terms of frequencies and cross-MAC values. The
mode at 10.62 Hz, being a transaxle mode, has not been excited well due to the nature of input
forces. Unlike the displacement-based test which is closer to conventional FRF-based methods,
OMA estimates using the SSI-Data algorithm are affected to a greater extent by violations in
terms of the nature of excitations, and hence certain modes such as the rigid body mode around 7
Hz, and modes at 11.42 Hz and 28.14 Hz have not been estimated with the SSI-Data algorithm
for the OMA test.
50
Modal vectors from both methods have a significantly complex nature for some of the modes,
especially the deformation modes. There is a good similarity between the mode shapes obtained
in these tests from visual inspection. The overall results from this comparison are encouraging as
they indicate positively that displacement readings from the simulator can be utilized for finding
dynamic characteristics of an automotive structure while simulating true road conditions.
Figure 4-8. Cross-MAC of OMA-SSI vs EMA-Displacement estimates
As discussed in Section 4.2.2, pressure-based EMA test results do not have high cross-MAC
values in comparison with EMA 4P test results. The various issues involved with this particular
method at present require a deeper understanding to enable its usage for successful parameter
estimation. Comparison between modal estimates from the OMA-SSI test and the EMA-Pressure
test has predominantly low cross-MAC values across most of the modes and is not presented in
this work.
51
4.4 Modified MAC Computations
Conventional validation methods have generally utilized computation of MAC coefficients
(Equation (2.07)) for comparing modes obtained between multiple estimates. This is known to
work well for traditional EMA tests where the measurement of force consistently normalizes the
data. However, modal vectors estimated by OMA methods and the unconventional tests
sometimes have a significant complex character and hence there is a need for an altered
validation methodology to compare modes with greater reliability. Note that this complex
character is not simply a complex scalar times an underlying similar mode shape; MAC is not
sensitive to this complex scaling issue when computed properly.
For modified MAC computations, modal vectors obtained from various tests are studied in terms
of their real and imaginary components. Modes obtained from conventional EMA tests do not
exhibit a strongly complex character unlike modal vectors obtained from the OMA test and from
the tests based on modified FRF measurements. MAC computations are performed to compare
modal vectors from the EMA 4P test with real and imaginary components of modal vectors from
the other tests separately to evaluate if there is a marked improvement in the MAC values
between the modal vectors. This essentially is a real normalization of the vector or a truncation
of the vector. The modes chosen for comparison are the ones which have been observed to have
a high similarity in their mode shapes when visually inspected, yet having a relatively low crossMAC coefficient. As an example, modal vectors for the 16.4 Hz mode from the EMA 4P test
and the EMA Displacement test are plotted on the complex plane and shown in Figure 4-9 and
Figure 4-10. Higher complexity in the modal vector corresponding to the EMA Displacement
test is evident.
52
Figure 4-9. Modal Vector (16.4 Hz) EMA 4P
Figure 4-10. Modal Vector (16.4 Hz) - EMA
Displacement
Figure 4-11 displays traditional and recomputed cross-MAC values between estimates from the
EMA 4P test and from the EMA-Displacement test for certain modes. It can be observed that
cross-MAC values between complex modal vectors from the EMA 4P test and the dominant
imaginary component of the modal vectors from the EMA-Displacement test have a marginal
improvement over cross-MAC values computed between complex modal vectors from both tests.
The use of imaginary components of modal vectors from both datasets further improves the
MAC computations, and is perhaps more reflective of the observed visual similarity in mode
shapes. It is to be noted that these modified MAC values might still be much lower than the high
values expected for similar mode shapes. It is clear in Figure 4-10, that the scatter about the
mean slope of this modal vector is much greater when using non-conventional EMA methods.
53
Figure 4-11. EMA 4P versus EMA-Displacement
In the case of the OMA estimates obtained using the SSI-Data algorithm, modal vectors are
found to be normalized along the real axis. Hence the real components of these vectors are used
in recalculation of cross-MAC values against vectors from the EMA 4P test. As seen in Figure
4-12, there is a moderate improvement in the recalculated cross-MAC values in comparison with
the traditional cross-MAC coefficients.
Figure 4-12. EMA 4P versus OMA SSI
Tests conducted based on response-only data and the two modified FRFs differ from
conventional EMA tests in terms of their numerical characterisitics, and consequently, their
modal vectors have relatively higher levels of complexities. As has been demonstrated in this
54
study, calculation of MAC coefficients after examination of these complexities, or real
normalizing the modal vectors, might be a better validation methodology than conventional
MAC computations. The reasons for higher complexities in the modal vectors obtained from the
modified EMA Displacement and OMA methods need further investigation.
55
5
Summary
Estimates from the displacement-based EMA test has been the most promising results across this
study, both in comparison with a conventional impact test (EMA 4P) and the response-only test
(OMA-SSI) test. Visual inspection of mode shapes also confirms similarities across these tests
for the rigid-body modes and the dominant deformation modes. Estimates from the
displacement-based EMA test are used for the description of prominent mode shapes in Table
5-1. Prominent rigid-body modes have been found below 10 Hz. The first torsion mode appears
at 11.42 Hz and the first frame bending mode is observed at 18.46 Hz. These mode shape
descriptions are consistent with corresponding modes across all tests.
Mode
Description of mode shape
3.76
Rocking in the lateral direction
5.16
Yawing
5.43
Front axle pitching
7.05
Rear axle pitching
9.1
Rolling
9.6
Transaxle bending
11.42
First torsion
14.07
Engine rolling with transaxle lateral movement
16.4
Engine pitching out of phase with frame
18.46
First frame bending
20.16
Complex mode with leaf springs bending
22.74
Complex mode with lateral frame bending
24.02
Engine lateral swaying with frame twisting
30.21
Higher order lateral frame bending
Table 5-1. Mode shape descriptions
56
Rigid Body
Modes
Deformation
Modes
Table 5-2 lists the complete set of modal frequencies and damping values for all the tests
performed in this study. It can be observed that not all modes have been estimated across the
various tests, reasons for which have been discussed earlier in Sections 3 and 4. Due to the
modified forms of the FRFs used for modal parameter estimation, reasons for high damping
estimates observed in Table 5-2 are not known at present, and need further investigation. Mode
shapes for select modes, based on the EMA 4P test, have been presented in Table 5-3.
EMA
EMA
EMA
OMA
EMA
EMA
Ground 1
4 Poster
Ground 2
SSI
(pressure)
(displacement)
Freq Damp
Freq Damp
Freq
Damp Freq
Hz
%
Hz
%
Hz
%
3.89
1.28
3.76
1.96
-
4.99
1.57
4.93
1.99
5.79
1.81
5.44
6.75
2.45
10.01
Damp Freq
Std.
Freq.
Dev.
Damp
Freq
Hz
%
Hz
%
Hz
%
Hz
-
-
3.73
7.59
3.96
0.81
3.76
4.17
3.82
0.10
4.75
2.03
5.43
8.58
-
-
5.43
8.40
5.19
0.31
2.02
5.51
1.94
5.17
5.52
5.96
14.02
5.16
5.57
5.50
0.32
6.41
2.29
6.38
2.54
9.11
5.81
9.02
7.17
7.05
16.28
7.67
1.27
2.28
9.94
2.59
9.78
2.54
9.65
6.84
10.09
8.18
9.10
6.55
9.76
0.36
10.57
1.76
10.40
1.65
10.41
1.55
10.63
3.07
10.64
1.72
9.60
5.45
10.37
0.39
11.48
3.17
11.28
3.00
11.21
2.99
-
-
11.04
10.62
11.42
14.64
11.30
0.18
13.95
2.32
13.87
3.00
13.66
2.81
14.00
8.23
-
-
14.07
8.08
13.97
0.16
14.17
1.92
14.54
1.75
-
-
-
-
-
-
14.36
0.26
16.31
2.08
16.31
2.34
16.06
2.83
16.42
5.26
16.54
5.23
16.40
6.42
16.39
0.16
17.47
1.36
16.57
1.47
16.54
1.66
17.67
5.72
17.65
9.66
17.33
7.06
17.34
0.52
18.90
1.23
18.77
1.71
18.81
0.94
18.39
2.97
18.62
2.70
18.46
2.89
18.63
0.20
20.56
1.29
20.96
1.34
20.22
1.26
20.12
2.81
20.17
4.18
20.16
3.19
20.39
0.33
23.68
1.79
23.07
1.82
22.85
1.14
22.90
6.29
-
-
22.74
5.84
23.10
0.37
24.66
1.36
24.48
1.91
24.31
1.85
24.35
5.19
24.70
0.79
24.02
6.54
24.44
0.25
26.36
1.49
26.27
1.78
26.74
1.46
-
-
-
-
27.66
1.47
26.76
0.63
-
-
28.03
1.65
28.17
1.57
-
-
-
-
28.14
5.88
28.08
0.08
30.86
0.76
31.41
1.17
30.95
0.65
30.50
2.36
30.57
1.16
30.21
1.77
30.71
0.42
Table 5-2. Summary of modes obtained from all tests
57
Damp
Avg.
Figure 5-1. 5.44 Hz: Yawing
Figure 5-2. 4.93 Hz: Front-Axle pitching
Figure 5-3. 6.4 Hz: Rear-axle pitching
Figure 5-4 9.94 Hz: Rolling
Figure 5-5. 10.4 Hz: Transaxle Bending
Figure 5-6. 11.28 Hz: First torsion mode
Figure 5-7. 18.77 Hz: First frame bending
mode
Figure 5-8 31.41 Hz: Higher order lateral
bending mode
Table 5-3. Rigid-body and prominent deformation mode shapes
58
6
Conclusions and Scope for Future Work
A series of tests have been conducted for the evaluation of the feasibility of using a four-post
road simulator for modal analysis of automotive structures. In a related study [15], OMA methods
using simulated excitations were successfully evaluated against conventional EMA methods for
automotive applications. However, a need was felt to evaluate OMA methods under more
realistic operational conditions. This is achieved in this study with the use of a four-post road
simulator for exciting the vehicle structure through the wheels and the suspension system.
For this study, conventional FRF-based impact tests have been conducted on the structure at
specific stages of this exercise to observe variations in the system over the length of the study.
An initial test performed with the vehicle on the ground, characterizes the dynamics of the
system in terms of its modal parameters. The second test, with the structure mounted on and
strapped to the simulator, studies the effects of variation in boundary conditions induced by the
straps and the simulator posts on the modal parameters of the system. A third test is conducted at
the end of the study to identify changes in the system such as suspension stiffness variations and
wheel camber angle changes. Between them, these conventional tests evaluate the timeinvariance of the structure as it is subjected to a variety of excitations with differing boundary
conditions and shifts in frequencies have been discussed.
Non-conventional tests have been conducted as a part of this work to explore the potential of the
road simulator for modal applications. Two tests have been conducted with pressure and
displacement measurements from the simulator instead of force in the estimation of FRFs. These
are subjected to parameter estimation using conventional EMA methods.
59
The pressure-based EMA test has inherent limitations in its suitability for experimental modal
analysis using conventional methods. The numerical characteristics of the pressure-based FRFs
limit the extent to which pressure can be used as an alternative input measurement to force.
Estimates from the displacement-based EMA test have been found to be in very good agreement
with benchmark estimates from the EMA 4P test. Estimates from this test are also in substantial
agreement with estimates from an OMA test using the SSI-data algorithm, and also indicate the
reordering of rigid-body modes under operational conditions. Of significance is the similarity in
the order of modes between the estimates from this test and those from the OMA test, and
reiterates the influence of dynamic boundary conditions on the modal behaviour of the system.
The displacement-based test thus makes a strong case for further research in the applicability of
the four-post road simulator for modal analysis.
Excitations for the OMA test, although limited spatially and by the action of the suspension
system, are closer representations of true operational conditions for the automotive structure. The
OMA test, similar to the modified EMA test with displacement measurements, highlights
variations in the modal signature of the vehicle when in operation due to the influence of
dynamic boundary conditions. It is significant to note that such shifts in the pattern of modes are
difficult to determine using conventional EMA tests with static boundary conditions. The OMA
test using the simulator has thus been found to be fairly successful in estimating modal
parameters of the structure, as also in understanding variations in the system dynamics under true
operational conditions. A more efficient method to identify the most suitable reference channels
for use with time and frequency domain algorithms such as PTD, RFP, etc, which are sensitive to
selection of references, would however need to be evaluated based upon an initial singular-valuedecomposition-based reformulation to virtual references.
60
Considering the presence of complexities in the modal vectors determined by the tests on the
simulator, future work could involve the development of better validation techniques that would
take into account these complexities. The possibilities of improving estimates from the pressurebased test by compensating for the inertial effects of the moving components of the simulator
column can be explored. Exploration into the dynamics of the actuators for a better
understanding of the effect they have on modal analysis using the four-poster system is
recommended. For the OMA studies, a future test could be done with the vehicle running on a
test track in true operational conditions, and estimates obtained could be compared with the
results from the road simulator tests.
A test that permits the actual measurement of forces exciting the structure at the posts would be
strongly useful in validating the results obtained in this study and in evaluating the potential of
these methods for successful experimental modal analysis, and could be an area of further work
in this regard. On a similar note, the development of cost-effective transducers, for measurement
of forces being transmitted to the structure at the posts, could also be an area of potential
academic and commercial interest.
61
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