misalignment of the stinger with the shaker or the attachment site will cause a moment input. This is undesirable for two reasons. First, it is assumed, for the.
A DESIGN-OF-EXPERIMENTS APPROACH TO QUANTIFYING TEST-TO-TEST VARIABILITY FOR A MODAL TEST John A. Cafeo Robert V. Lust
Spencer Doggen Donald J. Nefske
Douglas A. Feldmaier Shung H. Sung
Body Engineering and Integration Department General Motors Research & Development Center Warren, Michigan
However, it is known that there is some variation in the results from a modal test. In order to assess whether the model is correlated to the test results, knowledge of the test variability is crucial. If test variability is not accounted for, much effort is expended updating models that may actually be correlated with the test results. Therefore, it is important to know the test variability so that design and development decisions can be made intelligently.
ABSTRACT Design and development decisions are often based on the predicted and/or measured values of the frequenck?s of the vibration modes of a vehicle. It is known that test variability can affect our ability to experimentally measure these frequencies. Consequently, it is important fo understand and quantify typical fesf-fo-test variations. One method to isolate test variability effects is to use a formal Design of Experiments (DOE) approach. This approach prescribes the number of tests which must be conducted and the number of vehicles that need to be tested and is based on some initial estimates of variaation between tests and between vehicles. It was found that in order to practica//y implement this approach for vehicle modal studies, a test procedure which minimizes test variab;/ity needed to be developed and used.
One method that can be used to isolate the effects of test variability is to use a formal Design of Experiments (DOE) methodology. To stari the experiment design, the important factors that need to be separated must be determined. In this case, the variability of the modal results was determined to be due to two general causes of interest; 1) the variability in the modal test/analysis procedure and 2) the variability inherent in the vehicles that are tested. In the terminology of DOE, this is a one random factor (i.e. vehicle) test design.
A study was conducted lo determine the test-fo-fest variability in vibration response for a typ;ca/ mid-sized automobile. The purpose was to validate the DOE approach along with the test procedure. Based on the outcome of the DOE, seven different vehicles were tested in a laboratory sefting using the test procedure that was developed lo minimize fest-to-test variability. Each vehicle was then tested nine different times. The results of this study are presented in the following ways: 1) Average frequency response functions (FRF) are examined for the total vehicle as we// as for individual components.; 2) Four vehicle modes of vibration were identified between 19 and 25 Hz. Typical values for the test-to+& variability were calculated for the modal parameters (frequencies and damping r-a Uos) corresponding to these modes. If is observed that the magnitude of the test-to-test variability depends on the vehicle being tested and the mode of vibration.
To complete the experiment design, the number of vehicles and the number of tests of each of these vehicles must be specified. In order to determine these parameters, a chart from [l] was used and is shown in Figure 1. This chart shows the minimum detectable difference between means (8) (normalized by the estimated standard deviation of the data (s)) versus the number of vehicles and number of test repetitions. It can be seen that if the test variability can be reduced, thereby decreasing s, then the ratio on the ordinate of Figure 1 is increased. The implication is that fewer tests can be run to obtain the same results or that a finer distinction between the means would be possible for the same number of tests. This observation led us to attempt to reduce test variability as much as possible. Every aspect of a typical vehicle modal test was scrutinized for ways to minimize test variability. These efforts are summarized in Section 2.
1. INTRODUCTION The modal characteristics of vehicles are used as a design tool and therefore, early in the design stage, a set of specifications are developed. Typically, then, there is a large modeling effort using finite element analysis, to achieve the desired modal characteristics. As the development process continues, modal test results are used to verify and tune the finite element model and to support design decisions. Often, the underlying assumption is that the test results represent the “true” answer.
Further examination of Figure 1 shows that at about the eight vehicle by eight repetition level, the plot starts to level out; implying that gains for any increase in testing are small. Originally then, an eight car by eight repetition test design was specified. However, this was subsequently modified to a seven car by nine repetition test design due to vehicle availability.
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In order to validate the DOE approach along with the test procedure, the proposed study was conducted. Seven nominally identical vehicles were tested in a laboratory setting using the test procedure that was developed to minimize test-to-test variability. Each vehicle was tested nine different times. Qualitative as well an quantitative results are presented in Section 3.
2. TEST PROCEDURE DEVELOPMENT In the first part of this study, a “typical” baseline test procedure was evaluated and then modified in order to minimize test variability. This part of the study was conducted using a separate vehicle that was not part of the official test matrix. This avoided the potential problem of biasing the test procedure toward a particular vehicle. To reduce test variability, the effects of nonlinearities must be minimized as well as variation in the test setup and data acquisition procedure. 2.1 Linearity Considerations. It is important to consider linearity issues when investigating variability because the basic assumption underlying modal analysis is that the system is linear. However, it is known that a vehicle is a nonlinear system. Because of these effects, errors are introduced into the results that can be misinterpreted as test or product variability. Judicious selection of the input force(s) used to excite the structure is necessary to reduce the effect of the nonlinearities on the analysis. Therefore, the force level was set as low as possible but still high enough so the response measurements were above the noise floor of the instrumentation. A burst random waveform was used for the input force excitation. It has the advantage that leakage errors are greatly reduced without the need for window functions. The disadvantage is that a random input can have high peak force levels compared to its mean value. In order to use the low force levels and aid the parameter extraction process, it was found that two shakers were sufficient. The two driving points selected were on the right front frame rail (inclined 30 degrees laterally) and the lefl rear frame rail (vertical direction). 2.2 Shaker Attachment Method. The method employed to attach the shakers to the structure can have a significant influence on the data. Any misalignment of the stinger with the shaker or the attachment site will cause a moment input. This is undesirable for two reasons. First, it is assumed, for the FRF estimations, that there are only two uncorrelated force inputs. Therefore any moment input(s) will lead to erroneously estimated FRFs. The second reason is that side loads on the force transducer will introduce inaccuracies in the force measurement. Furthermore, inaccuracies that are introduced during any test will eventually be interpreted as variability. Therefore, an easily repeatable method for connecting the shakers to the structure was desired. This method must also be able to impart the necessary force and
minimize moment inputs. The use of a pre-tensioned wire stinger meets all of these requirements [2]. 2.3 Measurement System. P C B Flexcel accelerometers, with a sensitivity of 1 volt/g were used. These accelerometers were mounted in triaxial mounting blocks at each of the 64 measurement locations on the vehicle. The accelerometers ware all calibrated at a level of 1 g (rms) at a frequency of 30 Hz, which was within the acquisition frequency range for the tests. The accelerometers were calibrated once a month during the testing. The calibration histories were maintained and if the sensitivity of an accelerometer changed by more than 2 percent from the initial calibration, that accelerometer was replaced. T h e f o r c e w a s m e a s u r e d w i t h P C B 206A force transducers, which have a nominal sensitivity of 50 mVllb. The measured sensitivity for each sensor never varied by more than 1.5 percent during the six months of testing. Data acquisition and analysis were conducted using LMS software operating on an Hewlett Packard (HP) workstation with a 32 channel HP front-end. In addition. a pair of PCB Data Harvesters (providing 224 channels) were used for conditioning and organizing the collected data before being sent to the HP front-end modules. This large channel count acquisition system allowed the use of non-roving accelerometers. This minimizes the frequency shifting commonly encountered when using roving accelerometers. 2.4 Accelerometer Placement Several concerns regarding accelerometer mounting and placement had to be addressed in order to minimize test-to-test variation. Accelerometers were positioned to provide good mode shape definition and also reflect the test structure (a vehicle) when viewing the mode shape animation. A means of attaching the accelerometers to the vehicle had to be evaluated. It is a common practice to secure the accelerometers by the use of modeling clay. The clay provides adhesion to the vehicle and also allows the accelerometer to be approximately aligned with the global axis of the vehicle. This provides an adequate mounting for the frequencies measured for a typical vehicle modal analysis test. However, because the purpose of this study was to investigate vehicle variability, it was felt that building up an accelerometer mounting pad location with clay would not be repeatable from vehicle to vehicle or test to test. The clay mounting would also likely cause alignment errors which would result in an unacceptable amount of test variability. To overcome this limitation, the mounting pads were secured to the vehicle surface with hot glue. This setup allowed the pads to remain on each vehicle throughout the experiment, thus maintaining measurement point locations, and allowing for accurate placement of the accelerometers.
After the mounting pads were installed and measured for the first vehicle, the individual accelerometer placement locations were copied to the other vehicles. In order to maintain accelerometer alignment from test-to-test and vehicle to vehicle, an alignment tool was fabricated. One end of this tool was machined to fit on the triaxial mounting block, and the opposite end had a pointer which was aligned to the permanent reference marks on each vehicle. This setup allowed the accelerometers to be quickly and accurately positioned for each test. 2.5 Signal Processing Considerations. Each burst random input signal had a bandwidth of 64 Hz. The length of the burst was set at 60% of the time window. This allowed the responses to decay c&e to zero. A 1% pre-trigger was used so that each time sample started at zero. The RMS level of the burst was set to 1.0 Ibf +/0.05 Ibf RMS. The anti-aliasing filters for the 32 acquisition channels were set at 51.2 Hz. This provided ample bandwidth for the modes that were of interest. The time resolution was 0.0078125 seconds and the blocksize used was 2046 yielding a 16 second acquisition duration for each average. Since 32 channels were gathered at any given time, seven different patches needed to be measured in order to complete each test. This yielded a total test duration of about 3.5 hours. An automated switching routine was used to switch between each patch and to autorange all of the channels after the switch. Finally, the H, estimator was used to estimate all of the FRFs with 100 averages. 3.0 TEST VARIABILITY As discussed previously in the Introduction. the test procedure described above was employed in a designed experiment to evaluate test-to-test variability. The results of the experiment are discussed below. First test variability is examined qualitatively by comparing FRFs for various tests, vehicles and vehicle components. Then, the modal parameters are used to quantify the variability. 3.1 Average frequency response function results. The first and most basic way to look at the results from an experimental modal test is to look at the FRFs before doing any parameter extraction. The modal parameter extraction process introduces another source of variability into the data due to the high level of analyst interaction in the process. Because there are a large number of FRFs to consider (63 tests x 64 measurement sites x 3 directions x 2 inputs = 24,192 FRFs), the average FRF was calculated for each of the 63 tests. The average FRF amplitude from 0 to 25 Hz for each of the 63 tests are shown overlaid in Figure 2. Some general observations can be made about Figure 2. The low frequency modes (O-7 Hz) seem to be the most consistent and least variable both in frequency and amplitude content. The mid-frequency modes (7-15 Hz) seem to be the most variable in frequency. The highfrequency region (15-25 Hz) also shows a fair amount of
variability. It should be noted, that from this figure, test variability cannot be separated from vehicle variability. A better understanding of the variations that are contained in the measured data can be gained by plotting the average FRFs for different groups of accelerometers. For instance, Figure 3 illustrates the variations present on a component basis for all 63 tests. Note that the frequency range is 0 to 50 HZ . The six components are the body, cradle, engine, wheels, exhaust, and steeting column. The range on all the plots is the same so that a visual comparison of variation is straight-forward. It is clear from this figure that the body component contains the least total variation while the exhaust and engine components contain the most. Another general observation is that at higher frequencies, the variations are larger than at lower frequencies. As noted above, it is not possible to directly observe the test-to-test variability from Figures 2 and 3 because this set of FRFs represents not only different tests but, also, different test vehicles. To isolate the test variability the nine different FRFs (corresponding to the nine tests) for three representative vehicles (Car #l, Car #3 and Car #5) have been plotted in Figures 4 and 5. In Figure 4 the average acceleration response is plotted for the body. Note that the trends observed in Figures 2 and 3 are, again, observed here. Also note that the amount of test variation is dependent on the car being tested. Clearly, the test-to-test variation for Car #3 is greater that that for either Car #l or Car #5. This result is also evident from Figure 5; where the average acceleration response is plotted for the exhaust system. Note also, by comparing Figure 4 with Figure 5, that the observed test variability is greater for the exhaust system than for the body. 3.2 Modal parameter results. The average response curves discussed in the previous section give a qualitative indication of variation. Typically, however, the extracted modal parameters (natural frequency, damping, and mode shape) are the actual quantities used for vehicle design and development. As a part of this experiment, modal parameters were extracted from 0 to 25 Hz. The variability analysis was concentrated in the 19-25 Hz region. While there were between five and seven modes extracted in this range during the modal parameter estimation process, there were four that appeared consistently in all of the tests. These are denoted sequentially, Modes l-4. Only the natural frequencies and modal damping values have been characterized at this time. Figure 6 summarizes the results for the natural frequencies. Overall, the extracted natural frequencies for Modes 1 and 2 exhibit the least test-to-test variability. For these modes that maximum observed range is +I.25 Hz. In contrast, the extracted natural frequency for Mode 3 is much more variable; with a maximum variation of cl- .65 Hz being observed for Car #4. The variability of the frequency for Mode 4 is less than that of
stingers is more difficult, they appear to be less sensitive to alignment errors that can introduce unmeasured bending moments into the structure under test.
Mode 3 but greater than that of either Modes 1 or 2. However, Figure 6 also shows that these overall trends have exceptions for certain vehicles. The test-to-test variability for Mode 3 of Car #l and Mode 4 of Car #5 is nearly equal to or better than that of Modes 1 and 2 for the same cars. This highlights the observation that the magnitude of the test variability depends not only on the mode being measured but also on the vehicle being tested. Similar results for the modal damping are shown in Figure 7. The modal damping typically has the highest variability of the three modal quantities that are extracted. Trends similar to those seen for the modal frequencies can be seen here for the damping values. Again. the magnitude of the test variability depends not only on the mode being measured but also on the vehicle being tested. A final, and important, observation should be made regarding the test-to-test variations of the extracted modal parameters. Up to this point all of the discussion of the results has been specific to this designed experiment. In order to use these results for correlation a n d d e v e l o p m e n t t e s t i n g a s d i s c u s s e d in the Introduction, a general value for test variability for the modal frequencies and damping ratios needs to be developed. Given the number of test repetitions conducted during the experiment and the emphasis that was placed on reducing test variability, the overall range of the raw data provides a reasonable metric for this purpose. Examining the range (maximum value minimum value for each vehicle) of natural frequencies for all four modes shows the maximum value of the range is 1.3 HZ . Likewise for the damping ratio, the maximum value of the range is 1.7 % critical. Furthermore, if we consider that these variations were observed as the result of a carefully controlled experiment using a test procedure which was specifically developed to reduce test variability, then it seems reasonable that these values for test-to-test variability, while ‘“worst case” for this experiment, could represent typical values of test variability for correlation and development testing.
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Accelerometers should remain attached to the vehicle for the duration of the test. This will eliminate frequency shifts which can be introduced by roving accelerometers between different measurement locations during the test. The lack of frequency shifts during the test enables more accurate extraction of modal parameters from the data.
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Accelerometers were mounted flat on the surface of the vehicle. This eliminates the need to build up a base for the accelerometer in order to orient the accelerometer in the global coordinate system of the vehicle.
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More variability is observed in the FRFs for the powertrain and exhaust response points than for points on the body.
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Variations in FRFs i n c r e a s e w i t h i n c r e a s i n g frequency. Variations above 25 Hz are significantly greater than those below 25 Hz.
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The maximum overall test variability for the frequencies of the studied modes was 1.3 Hz while the overall test variability for the damping ratios was 1.7 % critical. Because the reduction of test variability was a primary concern in these tests, this represents the minimum test variability one should expect. 5. ACKNOWLEDGMENTS
W e w o u l d l i k e t o a c k n o w l e d g e J e a n n e Polan Engineering Center, for her valuable support in conducting this work, Lynn Truss, Research and Development Center, for her guidance in designing the study and her assistance in analyzing the results, and the Mid-Sized Car Division for providing the vehicles used in this study.
4. SUMMARY &CONCLUSIONS
6. REFERENCES .
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[l] Lorenzen, T . J . a n d A n d e r s o n , V.L.. Desian o f Experiments., Marcel Dekker. Inc., 1993.
Vehicles are non-linear at even the lowest input force levels that maintain adequate signal to noise ratio. Therefore input force levels need to be kept as low as possible when conducting a vehicle modal analysis test.
121 Peterson, E.L., “Modal Excitation: A Comparison of Three Different Methods,” Proceedings of the 6th international Modal Analysis Conference, Orlando, FL, 1966.
The use of piano wire stingers for input force excitation should be considered for vehicle modal analysis tests. Although the implementation of these
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