Modal studies on a truck frame and suspension
B. Swaminathan, B. Sharma, R. J. Allemang
Structural Dynamics Research Laboratory,
Department of Mechanical Engineering, University of Cincinnati
Cincinnati, Ohio – 45221-0072, USA.
Email: [email protected]
S. Chauhan
Brüel & Kjær Sound & Vibration Measurement A/S,
Skodsborgvej 307, DK-2850 Nærum, Denmark
Email: [email protected]
Abstract
This paper presents a comparative study of conventional Experimental Modal Analysis (EMA) and
output-only Operational Modal Analysis (OMA) techniques for estimating the modal parameters of a
moderately damped truck chassis. The OMA, without its need to measure input forces, would serve as a
convenient methodology to test vehicle structures when response data alone is available. Data has been
acquired from the test structure with different excitation techniques, namely, shaker and impact hammer
for the EMA tests, and, shaker and random impact using hammer for the OMA tests. Re sponse powerspectral data has been processed for output-only OMA and the estimates obtained have been validated
against those based on processing frequency response function (FRF) data based on conventional EMA
methods. The results indicate a fairly consistent estimation of modal parameters for the test structure using
both OMA and EMA methods.
1
Introduction
Modal analysis [1] is defined as the study of dynamic characteristics of a system in terms of its natural
frequencies, damping, mode shapes, and modal scaling. Conventional modal analysis techniques are
commonly used to identify the modal parameters of physical systems based on frequency response
functions (FRFs), with applications ranging from validating/updating finite-element models to structural
health monitoring and design changes in physical systems, especially involving dynamic modifications to
the structure. These, however, have required the knowledge of both the input forces and response data to
compute the modal parameters. An alternative has emerged over the last few years in the form of
Operational Modal Analysis [2], where, under the validity of certain assumptions regarding the nature and
spatial distribution of excitation forces, modal parameters can be estimated purely on the basis of response
data, rendering the need to measure input forces unnecessary. However, additional input-output tests are
required to determine modal scaling. OMA techniques have been successfully implemented in civil [3-4],
aerospace [5] and industrial applications [6].
The application of OMA techniques to automotive structures is however, quite different from other
applications. The basic assumptions of broadband and spatially distributed excitations do not hold true in
real operating conditions for a vehicle, as it does for a civil structure. Reasons include the presence of
engine and other strong rotational harmonics, and the fact that road-induced operational forces are filtered
out by the suspension system. It is also to be noted that these road-induced inputs can excite the system
primarily through the four wheels only. This again is not a spatially well distributed excitation. Hence, a
response-only OMA test on the vehicle structure using excitation methods such as random impacts using
hammers would yield better results than testing the vehicle in operating conditions as on a test rig or on a
test track. The work done with the OMA approach in this paper is based upon response-only data, but not
truly operational in this sense. Operational response-only data will be evaluated in a later study.
This paper attempts to obtain modal parameters of a truck chassis based on specialized response data and
to validate the modal parameters with results obtained from well-established EMA methods. The
suitability of using OMA techniques for the truck frame is studied in this paper. This structure poses a few
challenges in that it is moderately damped by the suspension system and is known to have closely spaced
modes. Further, the study of suspensions involves non-linearities [8]. Focus is kept on the rigid body
modes of the suspension, such as pitching, yawing and rolling and the structural modes in the 0 - 30 Hz
spectral range. Power spectra obtained by processing response time histories have been used as the basis
for parameter estimation under the OMA framework [9] and validated with the FRF-based EMA methods.
The following section describes the test setup and sensor locations for the whole structure as well as for
the various sub-structures. Section 3 starts with describing the EMA based shaker test, its signal
processing parameters, modal estimates and a plot of the Modal Assurance Criterion (MAC) [1]
coefficient-matrix, and similarly follows for the EMA based impact hammer test, the OMA based shaker
test and finally the OMA based random impact hammer test. Results from various tests are compared
against each other in Section 3.3 and the OMA methods are validated with the EMA results. The final
mode shapes are presented in Section 4 along with a complete table of modal parameters from all tests.
Conclusions are drawn in Section 5 along with scope for future work.
2
Experimental Setup
A truck frame with engine and gearbox mounted is chosen for this study. The frame is supported by
independent double wishbone suspensions in the front and solid axle leaf springs at the rear. For the
purpose of this study, the effect of tire dynamics is not explored, considering the tires to be linear within
the purview of this study. A single tire pressure is maintained for all the tests.
Figure 1: Test Structure with sensors mounted
The overall structure is studied based on the constituent elements, viz. the frame, the suspensions, the
gearbox and the engine. Key positions on the suspension are considered and three points are selected on
the upper control arm (UCA) and lower control arm (LCA), and one point near the kingpin, for each side
of the front suspension, as shown in Figure 2(a). Sensors are similarly distributed on the rear leaf-springs
(Figure 2(b)), the frame, the engine (Figure 2(c)) and the gearbox. Eight points are chosen on the engine to
better understand the nature of its interaction with the frame and other components in each mode of
vibration. A total of 50 tri-axial accelerometers are distributed at various points across the structure, as
shown in Figure 3.
(a)
(b)
(c)
Figure 2: Front left wishbone, rear left leaf spring, and engine with mounted sensors.
Four tests are conducted for the purpose of this study. Two EMA-based tests - a conventional impact-test
and a shaker test - are carried out to identify the baseline set of parameters. Response time-histories are
collected on all channels for the output-only analysis, with excitations being random signals from two
shakers for one test and random impacts from hammers for the other test. Detailed explanations of the
tests are included in relevant sections.
Figure 3: Sensors and excitation locations on the test structure
3
Data Acquisition and Modal Parameter Estimation
3.1
Conventional FRF-based EMA tests
3.1.1
Shaker Test
Two shakers are used at point numbers 2 and 12 with random forces as excitation functions. Responses are
measured at 152 locations distributed over the structure (refer Figure 3). The data acquisition parameters
for this test have been summarized ahead.
•
•
•
•
•
Sampling Frequency : 125 Hz
Frequency Resolution : 0.0625 Hz
20 RMS averages with 4 cyclic averages [10]
Window : Hanning
Excitation degrees of freedom: 2
The FRF data so obtained is used as the basis for parameter estimation. The PTD (Polyreference Timedomain) algorithm [1], a time-domain algorithm, is used to estimate the modal parameters. A consistency
diagram for an estimate using this algorithm is shown in Figure 4. It can be observed that the system
modes consistently show up over varying polynomial model orders.
Figure 4: Consistency Diagram for an estimate for the EMA Shaker Test data using PTD
The Modal Assurance Criteria (MAC) plot for estimates computed using the PTD algorithm is shown in
Figure 5, and the linear independence of modes from one another can be observed by the presence of unity
coefficients along the diagonal, and their absence off the diagonal. A total of 19 modes are estimated
based on this test, which are summarized later in Table 1 in Section 4.
Figure 5: MAC plot for modes from the FRF-based shaker test
3.1.2
Impact Test
A second test is conducted with impact excitations at 7 locations on the structure (refer Figure 3). The data
acquisition parameters for this test are listed ahead.
•
•
•
•
Sampling Frequency : 125 Hz
Frequency Resolution : 0.125 Hz
RMS averages : 3
Excitation degrees of freedom: 14
With the system being moderately damped, response vibrations damp out well within the chosen time
period of 8 seconds, which explains a relatively coarser frequency resolution of 0.125 Hz. For the same
reason, the use of an exponential window is not needed. A sample consistency diagram for an estimate
using the PTD algorithm has been shown in Figure 6, and the consistency in the estimation of the modes
over varying model orders can be observed.
Figure 6: Consistency diagram for an estimate for EMA Impact test
The MAC plot shown in Figure 7 again highlights the linear independence of modes with one another. The
modal estimates have been summarized later in Table 1 in Section 4.
Figure 7: MAC plot for modes from the FRF-based impact test
3.2
Power-spectra-based (output-only) OMA tests
3.2.1
OMA based on response time histories from shaker excitations
Two shakers are employed at the same locations as used for the EMA test for exciting the structure (refer
Figure 3). The purpose of this test is to study the nature of estimates knowing the excitations to be
uncorrelated and random but with limitations on the spatial distribution and the directions of inputs.
Response time histories are collected over 150 channels, and processed to obtain power spectral data for
OMA. The following data acquisition and processing parameters are used.
•
•
•
•
•
Sampling Frequency : 160 Hz
Duration of data acquisition : 20 minutes (191488 time points)
Number of excitation locations : 2
Cyclic Averaging over 3 ensembles with 66.6% overlap processing employed for noise reduction
Hanning window employed for reduction of leakage errors
A sampling frequency deviant from the earlier tests is used since a different software package is used to
record time-histories, with 160 Hz being the nearest sampling frequency that could have been chosen
under the requirements of this study. Given the constraints on computing capabilities, only the first
102400 time points are used in processing power spectra using the Welch Periodogram Method [11].
The PTD algorithm is again employed to estimate the modal parameters for the structure, with the
algorithm using power spectra information instead of frequency response functions as the basis for
parameter estimation [9]. The references are chosen by observing their spectral content from the auto
power spectra plots of each channel and the nature of correlation of each channel time history with the
other channels. Parameters are estimated from different combinations of reference channels over narrow
frequency bands covering the entire frequency range of interest. The recurring presence of modes for
varying model orders can be observed from a sample consistency diagram shown in Figure 8.
Figure 8: Consistency diagram for an OMA estimate based on shaker excitations
The modal frequencies obtained for this test are shown alongside the EMA shaker test estimates in Table
1. From the MAC plot for this set of estimates shown in Figure 9, modes at 14 and 18.7 Hz might seem to
indicate partial linear dependence. But visual inspection of the corresponding mode shapes and the fact
that they are well-separated on the frequency scale confirm them to be distinct modes.
Figure 9: MAC plot for OMA estimates based on shaker excitations
3.2.2
OMA based on response time histories from random impact excitations
This test is conducted to study the nature of estimates knowing the excitation to be spatially welldistributed and assumed to be random and broadband in the absence of force measurements. Multiple
hammers are employed to excite the structure with random impact excitations covering most parts of the
structure in all directions. The data acquisition and processing parameters are similar to those described in
Section 3.2.1.
•
•
•
•
•
Sampling Frequency : 160 Hz
Number of excitation locations : Multiple locations uniformly spread across the structure.
Duration of data acquisition : 20 minutes (191488 time points)
Cyclic Averaging over 3 ensembles with 66.6% overlap processing employed for noise reduction
Hanning window employed for reduction of leakage errors
A sample consistency diagram for estimates from this test using the PTD algorithm has been shown in
Figure 10. The presence of modes over varying model orders highlights the consistency of the modes
estimates.
Figure 10: Consistency diagram from PTD estimates for OMA random impact excitations
The modal estimates for this test are again listed in Table 1 in Section 4. Figure 11 shows the MAC plot
for this set of estimates. Modes 13.5 Hz and 13.8 Hz seem to show a certain amount of linear dependence.
A study of the mode shapes also indicates a high level of similarity. These modes around 13-14 Hz are
predominantly engine modes, and might not have been excited well with the random impacts. Modes at
22.7 Hz and 24.58 Hz however are seen to be distinct physical modes in spite of a possible indication of
linear dependence by the MAC plot. The rest of the modes appear to be linearly independent.
Figure 11: MAC plot for OMA estimates based on random impact excitations
3.3
Comparison between estimates
The estimates obtained from the four tests described in the earlier sections have been compared for
consistency in the modal parameters. Figure 12 shows the cross-MAC plot comparing the estimates from
the two EMA-based tests. The low-frequency 3.8 Hz mode seen in the EMA shaker test has not been
estimated in the EMA impact test since impact tests have been known to have limitations with exciting
very low frequency modes. The 20.66 Hz, 23.7 Hz and 24.58 Hz modes are predominantly modes in the
lateral direction and are difficult to excite using shakers due to their directional limitations. While these
modes show up well in the EMA impact test and, as will be shown later, in the OMA test based on
random-impact excitations, they are estimated poorly in the tests involving shaker excitations. The 28 Hz
mode is a torsion mode that has been fairly difficult to excite using hammer impacts. Moreover, the
limited spatial information due to sensor locations affects the system’s observability. Hence this mode
does not figure in the EMA impact test estimates. The rest of the rigid-body and structural modes have
been observed to be consistently estimated in both the EMA tests.
Figure 12: EMA shaker test estimates vis-à-vis EMA impact test estimates
Estimates from the EMA and OMA tests with shaker excitations have been compared in the cross-MAC
plot in Figure 13. The low-frequency modes around 3-4 Hz and the higher order modes at 17 Hz and
between 20-25 Hz, being lateral modes, are poorly estimated due to the violation of the OMA requirement
of spatially well-distributed excitations in all directions. The closely lying modes around 14 Hz have not
been estimated very distinctly in the OMA methods and reasons for the same are being investigated. Other
prominent rigid-body and structural modes are estimated well across both tests.
Figure 13: EMA shaker test estimates vis-à-vis OMA shaker-excitation estimates
Figure 14 compares the modal estimates from the OMA test based on random impact excitations with the
EMA impact test. It can be readily seen that more modes match well with each other in the impactexcitation based tests since the random-impact excitations follow the OMA requirements more closely
than the tests with shaker excitations discussed earlier. As in the previous case, the two modes at around
14 Hz are not well estimated. The mode at 17.4 Hz is a lateral sway mode lying close to a very dominant
18.9 Hz mode and does not lend itself very well to estimation in the power spectral-based estimation
methods. The high-order complex torsion mode at 26.4 Hz might not have been well-excited with the
random impacts and hence does not show up in the OMA estimate. Most of the other modes from the
OMA-based estimates match up with corresponding modes from the EMA test with high modal
consistency.
Figure 14: EMA Impact test estimates vis-à-vis OMA random-impact excitation estimates
Figure 15 compares the estimates from the two response-only methods. A series of low cross-MAC
coefficients for modes at 3.8 Hz, 20.03 Hz, 23.4 Hz and 24.7 Hz can be attributed to the fact that one of
these methods does not fully conform to the OMA assumption of spatial distribution across the structure.
Again, barring the closely lying modes around 14 Hz, most modes that appear in both estimates compare
well with each other.
Figure 15: Cross-MAC between OMA (shaker-excitations) and OMA (random-impact excitations)
From these comparisons, it can be observed that most of the modes show a high degree of similarity and
consistency across the EMA and OMA estimates both in terms of the MAC coefficients and in terms of
the nature of the physical mode shapes, and that results obtained are better when OMA assumptions are
met more closely.
4
Summary of results
Using results from the EMA shaker test, modes at frequencies 4.9 Hz and 6.7 Hz are observed to be the
rigid-body pitching modes, with predominantly larger deflections at the front and rear end of the structure
respectively. Rigid-body yawing and rolling modes are observed to lie at 5.7 Hz and 10.0 Hz respectively.
The first torsion mode appears at 11.7 Hz and the first frame bending mode appears at 18.9 Hz. These
mode shapes have been shown in Figures 16.1 through 16.8 and are consistent for all tests.
Figure 16.1: Pitching mode (front) at 4.9 Hz
Figure 16.2: Pitching mode (rear) at 6.7 Hz
Figure 16.3: Yaw Mode at 5.7 Hz
Figure 16.5 Transaxle bending mode at 10.5 Hz
Figure 16.7 First frame bending mode at 18.9 Hz
Figure 16.4: Rolling Mode at 10.0 Hz
Figure 16.6 First Torsion Mode at 11.7 Hz
Figure 16.8 Lateral bending mode at 30.9 Hz
Table 1 lists the results from all estimates. The mean frequency and standard deviation from all tests have
been computed and tabulated in the last two columns. As has been the general case with OMA [12],
damping ratios are overestimated for a few modes in the OMA tests.
EMA Shaker ‐ PTD Freq Damp (Hz) (%) 3.794 2.461 3.929 2.175 4.984 2.205 5.746 2.014 6.703 2.252 10.004 2.385 10.518 1.828 11.784 2.495 13.924 2.733 14.154 2.225 16.309 2.412 17.025 2.257 18.946 1.213 20.660 1.705 23.700 1.663 24.580 1.919 26.402 1.520 28.747 1.869 30.935 0.634 EMA Impact ‐ PTD Freq Damp (Hz) (%) ‐ ‐ 3.885 1.281 4.991 1.569 5.793 1.806 6.754 2.445 10.009 2.276 10.565 1.759 11.479 3.174 13.946 2.316 14.171 1.915 16.307 2.076 17.472 1.355 18.900 1.233 20.563 1.288 23.681 1.788 24.664 1.357 26.364 1.493 ‐ ‐ 30.862 0.757 OMA Shaker ‐
PTD Freq Damp (Hz) (%) ‐ ‐ 3.879 2.285 4.921 3.702 5.701 2.429 6.574 3.917 9.858 2.521 10.344 2.277 11.360 4.052 13.799 2.533 13.997 1.570 16.178 2.540 ‐ ‐ 18.742 2.639 19.944 1.401 23.485 1.919 24.729 2.535 25.591 3.382 28.250 2.251 30.663 0.899 OMA Impact ‐ PTD Freq Damp (Hz) (%) ‐ ‐ 3.799 2.874 4.841 3.773 5.488 3.499 6.457 4.246 9.653 4.341 10.101 5.689 11.340 3.611 13.527 4.082 13.807 2.717 15.927 1.622 ‐ ‐ 18.787 1.924 20.034 2.116 22.869 2.743 24.578 2.535 ‐ ‐ ‐ ‐ 30.518 1.076 Average Std. Frequency Deviation (Hz) (%) 3.794 0.000 3.873 0.054 4.934 0.070 5.682 0.135 6.622 0.134 9.881 0.167 10.382 0.210 11.491 0.205 13.799 0.193 14.032 0.169 16.180 0.180 17.249 0.316 18.844 0.095 20.300 0.363 23.434 0.389 24.638 0.073 26.119 0.458 28.499 0.351 30.745 0.190 Table 1: Modal Estimates from the various testing and processing methods
5
Conclusions and scope for future work
The applicability of using OMA methods for modal analysis of a truck chassis has been presented
and discussed. Different excitation methods are employed with varying levels of adherence to the
OMA assumptions of uncorrelated randomness with temporal consistency and spatial coverage
across the structure, and the results so obtained are validated against conventional EMA-based
methods. While the results are reasonable, presence of localized inputs, as in the OMA shaker
test, hinders the estimation of certain modes. Likewise, limitations in spatial resolution affect the
quality of the estimated modal vectors as measured by the cross-MAC values between some
vector estimates. Complexities in system geometry and issues with suspension non-linearities are
other reasons why some modes are not excited or estimated well. Most modes are observed to be
estimated consistently across EMA and OMA tests as long as the input characteristics meet the
OMA assumptions. Excitation methods for OMA of this structure have been compared and
limitations for each method therein have been discussed. The shaker approach, with a limited
spatial distribution of inputs, does not match the input assumptions of OMA as well as the
randomized impact excitation.
For future work, the applicability of using a 4-post road simulator for exciting the structure
similar to operating conditions will be considered and compared with the random impact and
shaker excitations employed for this study. This will be operational modal analysis in a much
closer sense in that the vehicle's on-road behavior is simulated in the laboratory. However, the
spatial distribution of inputs will still be very limited and frequency content will be broadband
unlike the harmonic nature of road inputs when operating at a limited engine speed and vehicle
velocity. The purpose of the future study will be an incremental change from this study, to
observe the effectiveness of the excitation method given the lack of spatial distribution, with
inputs coming only from the four wheels, and with the suspension acting as a mechanical filter.
Also, further work needs to be completed in developing a selection process to identify the most
suitable reference channels of response that are utilized in OMA-based estimates, especially for
studies involving a large number of responses. If some sort of model is available, the optimal
response sensor locations can be selected using singular value decomposition of the modal
matrix, as is done in current EMA test methods. If no a priori model exists, a purely
experimental method based upon information theory and an initial test using all response sensors
will be considered to optimize the parameter estimation process based on power-spectra
information directly.
Acknowledgements
We would like to thank Dr. Allyn W. Phillips, SDRL, University of Cincinnati for his patience in
answering our numerous queries and for his expert guidance in matters of critical importance to this paper.
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