Extracting Fixed Base Modal Models from Vibration Tests on Flexible Tables

Transcription

1 Proceedings of the IMAC-XXVII February 9-12, 29 Orlando, Florida USA 29 Society for Experimental Mechanics Inc. Extracting Fixed Base Modal Models from Vibration Tests on Flexible Tables Randy L. Mayes and L. Daniel Bridgers Experimental Mechanics, NDE and Model Validation Department Sandia National Laboratories * P.O. Box 8 - MS7 Albuquerque, NM, 8718 rlmayes@sandia.gov dlbridg@sandia.gov Nomenclature a dof H f FE FRF Φ c Subscripts c f t s i acceleration degree of freedom Frequency response function Applied force Finite Element Frequency Response Function The mode shape matrix of the bare table at dof to be constrained degree of freedom at locations to be constrained dof at free locations dof at tip of the slip table dof at the classic shaker input location on the slip table any other dof not including t or s 1) Abstract Traditionally modal and vibration tests have been performed separately because their classical purposes require different inputs and outputs. However, motivation exists in some instances to be able to perform a modal test on a shaker table, if the boundary conditions could be accounted for appropriately. This is especially a concern for large test articles mounted on large tables because the table has flexible dynamics in the frequency range of interest for the modal test. For the past thirty years various attempts have been made to develop a method that would allow the two tests to both be conducted on a shaker table requiring only one setup. However, in most cases the table is assumed to be rigid. When the table cannot be assumed rigid the remaining approaches usually require that all six forces and all six degrees of freedom of motion at every attachment points be measured. Most approaches neglect moments and rotation measurements. Even measuring the translational forces and accelerations is rarely done. In the method employed here, the boundary condition is constrained mathematically. However, a measure of the shaker force is required. In addition, the classical mathematical constraints to produce a *Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the U.S. Department of Energy under Contract DE-AC4-94AL8.

2 fixed base result are augmented in a way that alleviates the ill conditioning that almost always results when using the classical constraint equations. The two major advances here are a method to estimate the shaker force, and improved conditioning of the constrained equations. The effect of improving the conditioning is demonstrated with a modal test of hardware on a base that is not fixed. The full process is demonstrated with a random vibration test on a simple flexible horizontal slip table with a cantilevered beam mounted as the test article. A general outline of the method proceeds as follows: 1) characterize the modes of the bare shaker table attached to the shaker; 2) mount and instrument the test article; 3) attach a portable shaker to the tip of the shaker table with a force gage and measure a specific frequency response function (FRF); 4) detach the portable shaker and run the typical random vibration test; ) calculate transmissibilities to the tip accelerometer; 6) create acceleration/force FRFs from reciprocity by multiplying the FRF in step 3 times every transmissibility; 7) extract modal parameters from FRFs; 8) finally apply augmented constraint equations with FRFs synthesized from the modal parameters and extract the fixed base modes. 2) Motivation Motivation for obtaining modal parameters for model validation from tests on flexible vibration tables flows from three driving concerns. First, executing a modal test in conjunction with a vibration test would save considerable time in that there would be no time lost in moving the object between facilities, and some duplication would be eliminated in tear down and setup. Second, because only one setup is required, handling is decreased thus reducing risk of damage to delicate and/or hazardous test articles. Finally, since damping tends to be nonlinear and vary with response amplitude, damping could be extracted at operational levels, instead of just the low level usually imparted in modal tests. Typically, damping is higher at operational levels, which is favorable from a qualification perspective. The two difficulties are: 1. to determine the modal parameters, including modal mass, from data for which no force measure exists; and 2. account for the flexible dynamic effects of the vibration table so that the modal parameters can be related to the FE model. Several attempts have been made to extract modal parameters and/or account for the vibration table dynamics in driven base tests. However all these methods assume one of the following: 1. that the vibration table is rigid and moves in only one translational direction; 2. the vibration table is rigid and the 6 dof of acceleration and force applied to the table are measured (requiring special force measurement devices); 3. the vibration table (or attachment fixture) is flexible, and forces and accelerations at every attachment degree of freedom must be measured (and in some cases additional quantities must be measured such as shaker current or voltage). Frequency response functions (FRFs) have been derived from transmissibilities for the simple case of a rigid table that moves only in one translation direction. Carne [1] follows Beliveau [2] in his derivation of the fixed base inertial force FRFs from transmissibility measurements. Standard modal parameter extraction algorithms can operate on these derived FRFs to obtain frequency, damping and mode shape, but the modal mass scaling is lost. Several methods expand the single translation assumption to a rigid table moving in more than one direction if the acceleration of the 6 dof of the rigid table and the 6 dof force is measured between the vibration table and the structure that is attached to it. One of these is by Fullekrug [4] [] who derives a method with the assumption that the table and a force measurement device are perfectly rigid. With his transformations imposed, Fullekrug s method can be used to extract modal parameters with traditional modal analysis algorithms. Additional equations are used to determine the modal mass. Sinapius [6] and Schedlinski [7] have similar developments assuming a rigid shaker table or fixture attached to the test structure along with an infinitely stiff force measuring device. Razeto [8] employs a method with similar purpose, but models the shaker to determine the connection forces instead of using the force measuring devices directly. He also assumes the shaker table is rigid and moves only in a single translational direction. A solution proposed by Brinker and Anderson [3] entails a method to obtain the modal mass by adding known masses to a system in subsequent test runs, however their goal was focused on obtaining modal parameters (especially modal mass) for as-built civil structures. None of these address the flexible base characterization which is one of our goals.

3 A different approach to obtain fixed base modes does not use a shaker table but characterizes the free modes and residual flexibility to obtain the fixed modes with the correlated FE model. Blair [9] [1] attempts to correlate a model with free modes, mass loaded modes and then a reduced set of residual flexibilities which were problematic to extract with the methods they had available. Problems with the suspension system, which was not modeled, hampered the results. Admire, et al, [11] developed a method to derive modes constrained at certain dof (not required to be a single node) using mass additive mode shapes. The method requires matrices from a verified FE model, and appears to be limited if a large number of dof need to be fixed. Examples indicate that only a few fixed base modes are accurate even if 3-4 mass additive modes are retained in the model. These techniques require FE models which are verified, but FE verification is the purpose of the modal test in this paper. In [13] Smallwood uses two port networks to characterize acceleration and force at any shaker/test article interface in terms of the current and voltage applied to the shaker. To characterize the shaker effects for one attachment dof on the table, a measure of force, acceleration, armature current and voltage requires 4 quantities to be measured at each frequency line. The number of force and acceleration measurements grows by two times the number of attachment dof. Once this is done for all possible attachment points, the characterization is complete, and any test article that can be attached at the chosen points can be tested. His goal is focused on determining whether the shaker can physically meet the forcing and acceleration requirements, but could be used to extract the effects of the flexible table. The methods which involve measuring the force/acceleration at each attachment point usually neglect rotations and moments, because the moments at attachment points are not easily measured. Indeed, care has to be taken to obtain three accurate translational forces at each attachment point, and such measurements usually perturb the geometry or the rotational stiffness of the bolted-to-the-table boundary condition significantly. The approach taken here serves the need for a FE model that needs to be validated for the fixed base condition, typical for many payloads that ride on some sort of delivery system. If the modal test on the vibration table could provide fixed base modes it would be very convenient for the FE analyst for the purpose of model correlation. The authors' goal is to provide these fixed base modes. To do this we apply constraints to the synthesized FRFs obtained from the vibration table modal test. In order to do this, the modal mass of each mode is required, and the authors chose to attack this problem by obtaining an estimate of the shaker force input. First let us assume that acceleration/force FRFs may be obtained from a shaker table modal test and consider the constraints that need to be applied to these data to produce fixed base FRFs. 3) Structural Modification Theory and Modified Constraints Crowley, et al presented the original structural modification using FRFs (sometimes called SMURF) [12], and it is repeated here as the basis for this method. The FRF matrix equations can be partitioned as a H H f ff fc f f = (1) a H cf H c cc fc where a is acceleration, H is a frequency response function and f is force. The subscripts associated with partitioning are f for free dof and c for dof which will be constrained mathematically at a later time. Only a few (perhaps just one) columns of equation (1) are measured, therefore the other columns must be reproduced from the modal parameters extracted from the measured columns. This method is limited by the effects of modal truncation. By constraining a c to zero, equation (1) can be rearranged 1 { a } = [ H H H H ]{ f } f ff fc cc cf f (2)

4 However, H cc is usually poorly conditioned on inversion and gives uncertain results. Poor results are probably the reason why this constraint theory has not been more popular. In the case study which follows, equation (2) provided poor results. By reducing the physical c dof in equation (2) to the modal coordinates of the table, the conditioning is much improved. Equation (3), which utilizes modal connection dofs, can be much more robust and is given by T T + + { a } [ [ Φ c ] 1 f = H ff H fcφ + c Φ + c H cc Φ c H cf ]{ f f } (3) where Φ c is the mode shape matrix of the flexible table without a test article, and the superscript + indicates the pseudo-inverse. To ensure the inverse in (3) can be calculated there must be more modes used in the construction of the synthesized FRFs in equation (1) than there are in Φ c, or results are not robust. (This is generally not a problem, since the addition of the test object increases the number of modes in the same frequency band to a number larger than the number of modes of the bare shaker table). 4) Hardware Example Demonstrating Improvements From Modified Constraints First, the augmented theory was applied to a physical system in attempt to constrain modal results obtained on a base that was not fixed. The physical system was a lightly damped steel test article bolted at its three feet to a 32, pound seismic mass which was supported by air springs. This hardware has come to be known as the "Gazilla Bird". The six rigid body modes of the entire system were below 4 Hz. In this case, the authors assumed that the rigid body modes of the bare seismic mass spanned the space of the motion of the feet, so these mode shapes were calculated analytically and used in the Φ c mode shape matrix in equation (3). A modal test was performed using three references. Burst random shaker excitation was applied to the nose of Gazilla Bird in the vertical, lateral and fore-aft directions, and the modes were extracted on the seismic mass supported by the air springs. The SMAC modal analysis code[14] was utilized to perform the modal analysis. The test setup is shown in Figure 1. Nineteen modes were extracted from the modal test. The analytical FRF matrix in equation (1) was synthesized from the modal parameters. Then fixed base FRFs were extracted utilizing the classic constraint equation (2) as well as the augmented equation (3). Then the fixed base modes were extracted Figure 1 - Gazilla Bird Test Setup

5 from both fixed base sets of FRFs. The results for the lowest fixed base modes are given in Table 1. The first column applies to the unconstrained modes of the test article with the seismic mass on air bags. Column two gives results from the application of the classic constraint equation (2) to mathematically achieve fixed base results. The results were poor, as the frequencies rose instead of falling as they should. Column three represents the application of constraint equation (3) which is more robust, and the frequencies at least move in the right direction. Table 1- Gazilla Bird Modal Frequencies (Hz) Seismic Mass Fixed eqn (2) Fixed eqn (3) No experimental data were available to provide a "truth measurement" of the fixed base frequencies. To understand uncertainty on frequency associated with equation (3), two analyses were performed to attempt to estimate the drop in frequency associated with fixed base vs seismic mass system on air springs. First, a simple four dof model was developed modeling the air springs, seismic mass, and a spring and mass to represent the lateral Gazilla Bird motion. Second, a finite element model of the Gazilla Bird on the seismic mass with air bags was generated, as well as a model of Gazilla Bird fixed at the feet (see Figure 2). Figure 2-Seismic Mass and Constrained Gazilla Bird Models Table 2 shows the comparisons in fixed base frequency drops for the constrained test, 4dof model and FE model for the first two elastic modes. Although the true fixed base frequency is not known, the equation (3) constrained test result is practically within the uncertainty of the two analytical estimates of the fixed base frequency drop. Table 2-Comparison of Experimental Constrained Frequency Differences with Models Mode (Hz) % change to fixed base from test data constraint % change in simple 4dof model % change in FE model Now, consider the effects of the constraints on mode shape. In Figure 3 the fixed base FE mode shape for the first elastic mode of the Gazilla bird is shown, which is a fore aft motion with some bending of the large plate. In Figure 4 the constrained mode shape results, as calculated utilizing the experimentally extracted mode shapes, are shown. Utilizing the classic equation (2),

6 the ill conditioning affects the mode shape producing unreasonable motion of the plate and legs. However, the result with the augmented modal constraint of equation (3) is qualitatively similar to the corresponding FE result and the experimental mode shape unconstrained on the seismic mass. The benefit of the augmented constraint in equation (3) is significant for both frequency and shape results. Figure 3 - First FE Constrained Elastic Mode of Gazilla Bird in Fore-Aft Direction Mode 1 Frequency: Hz Damping:.94 %Cr IDLine 1: Generated from reference 16X Y axis (in) Mode 1 Frequency: Hz Damping:.74 %Cr IDLine 1: Generated from reference 16X Y axis (in) Figure 4 - First Constrained Elastic Mode Shape of Gazilla Bird Utilizing Classic Constraint (Left) and Augmented Modal Constraint (Right) ) Estimating the Shaker Force In order to synthesize the FRFs from the mode shapes of the test article on the table, either the modal mass or a measurement of the input force (to obtain the modal mass) is required. The authors chose to approach this by attempting to estimate the shaker force. We cannot easily measure the shaker force, but an estimate of the shaker force can be inferred using reciprocity. Reciprocal FRFs can be written as H st = H ts (4) where H is the accelerance FRF and subscript s represents the shaker location and subscript t represents the tip of the table farthest from the shaker. Now the relation of the acceleration at the tip due to the force at the shaker is a t = H ts fs () which can be rewritten to determine the shaker force, if a and H are known, as f s = at / H ts (6). Substituting (4) into (6) gives f s = at / H st (7). Define location i as any location of interest. Then H is = ai / f s (8) and substitute (7) into (8) to estimate any accelerance FRF as

7 H = H a / a (9) is st i where a i /a t is simply the transmissibility to the tip accelerometer which is usually collected in a random vibration test for any response of interest. Theoretically, it does not have to be an accelerometer at the tip of the table; it may be anywhere; but practically, an accelerometer at the tip of the shaker table farthest from the shaker input force is well conditioned in that it has the fewest anti-resonances of any location on the table. For this reason, this is often the location of the control accelerometer in random vibration tests. By using equation (9), the classic accelerance FRF form is obtained for which any modal analysis package can estimate the modal parameters including the modal mass. There are some assumptions considered here. 1. The system is linear, which is the basis for reciprocity; 2. A measurement of the acceleration at the big shaker due to a force at the tip of the table can be obtained. The second assumption has at least two implications. One is that another shaker or exciter must be brought in and mounted at the tip of the table with a force gage at the tip location. This H st measurement must be performed when the test article is on the table (not bare table), so this is an extra step beyond the standard random vibration test, and all the classic precautions to obtain a good force measurement from the force gage must be employed. The second implication is associated with the assumption that the accelerometer measuring the shaker acceleration is colocated with the point the shaker force is applied. Logistically, this is never quite achievable because the "point" is inside the shaker armature, but if the motion of the accelerometer is the same as the motion where the force is applied, this assumption is valid. At low frequency, an accelerometer location close enough to the location where the shaker force is applied can usually be found, but at some high frequency, this assumption will eventually break down. 6) Experimental Simulation of Test Article on Flexible Shaker Table 6.1) Fixed Base Truth Experimental Simulation With the modified constraints and a theoretical method to measure the force in hand, experiments were performed to exercise the process in a severe test for the constraint theory. The first experiment was to provide an estimate of "truth" for a fixed base boundary condition. The test article chosen was a nylon beam one foot in length. The "truth" test for the fixed base boundary condition was performed by attaching the beam to the 32, pound seismic mass and measuring FRFs exhibiting the resonance of the first bending mode. The beam was one foot long and was designed with a base that could be attached to the shaker table as well as to the seismic mass. Although, it is difficult to achieve a truly fixed condition, the small inertia of the beam in comparison with the seismic mass was assumed to make this difference negligible. The beam was designed to have a 1 Hz mode, and when the test was performed on the seismic mass, the modal frequency and damping were extracted at 1.1 Hz and.9 percent of critical. The beam on the seismic mass is shown in Figure. t Figure Truth Modal Test of Beam on 32, Pound Seismic Mass

8 6.2) Hardware Implementation A small elastic slip table simulator was designed and fabricated. The design of this slip table was to attempt to have the first bending mode of the bare slip table near the same frequency (1 Hz) as the fixed base mode of the beam. This ensures maximum interaction between the elastic mode of the table and the beam, providing a severe test of previously explained method to eliminate the dynamics associated with the slip table. The hardware included a 3/4 thick aluminum plate approximately 3 by 16 supported by 3/8 steel rods to simulate a horizontal slip table on bearings. The modal test of the slip table without the beam showed that it had a 96 Hz bending mode near the desired 1 Hz. The bare slip table simulator is shown in Figure 6. Although producing a fixed base result requires only standard modal testing capability and software such as MATLAB to synthesize FRFs and apply the constraints, there are several steps in the process for the flexible table proof of concept. The implementation steps for obtaining the fixed base FRFs are: 1. The bare table with the large shaker attached must be instrumented to extract the modes, Φ c, and have independent vectors in all the mode shapes extracted. The number of modes extracted depends on the frequency band of interest. 2. The test article is then mounted to the shaker table and instrumented. 3. An external shaker is attached to the tip of the table with a force gage. The FRF of the acceleration near the big shaker armature due to the force at the tip shaker is measured. 4. After detaching the tip shaker, the random vibration test is performed, and transmissibilities to the tip accelerometer are measured. Transmissibilities of the table instrumentation of step 1 must be included in the data set.. Accelerance FRFs are created from the transmissibilities as shown in equation (9). 6. Modal parameters are extracted from FRFs in step in the traditional way. 7. The modal parameters are used to synthesize the full FRF matrix of equation (1). 8. Fixed base FRFs are obtained from synthesized FRFs and constraint equation (3), and fixed base mode shapes may be extracted from these FRFs. The experimental steps involving hardware are described below. Step 1: Characterizing the Bare Table The setup for the bare table modal test can be seen in Figure 6. The small shaker was moved to three different locations to excite all the modes of the bare table attached to the large shaker up to about 4 Hz. Data were taken using burst random. 3 averages with 16 lines were acquired. Some of the mode shapes from the bare table modal test are shown in Figure 7. These were the mode shapes used to condition equation (3) as Φ c. In this case, the authors utilized the first, fourth and fifth modes shown here, which span the motions most strongly excited by the shaker pushing in the X (longitudinal) direction.

9 Figure 6 - Bare Table Modal Mode 1 Frequency: Hz Damping: 1.88 %Cr IDLine 1: Generated from reference 1X- Mode 2 Frequency: Hz Damping: 1.48 %Cr IDLine 1: Generated from reference 3Y+ Z axis (in) Y axis (in) Z axis (in) 1 Y axis (in) Hz rigid body longitudinal mode 21 Hz rigid body lateral mode Mode 3 Frequency: Hz Damping:.79 %Cr IDLine 1: Generated from reference 3Y+ Mode 4 Frequency: Hz Damping:.393 %Cr IDLine 1: Generated from reference 3Z- Z axis (in) 1 Y axis (in) Z axis (in) - Y axis (in) Hz rigid body twist mode 96 Hz first bending mode

10 Mode 6 Frequency: Hz Damping:.18 %Cr IDLine 1: Generated from reference 3Z- Z axis (in) - Y axis (in) Hz second bending mode Figure 7 - Bare Table Mode Shapes Figure 8- Driving the Table from the Small Tip Shaker With the Large Shaker Attached Steps 2 and 3: Using the Tip Shaker to Estimate Armature Force After the table had been characterized, the beam was instrumented and mounted to the table. The smaller tip shaker was attached while keeping the large shaker attached as shown in Figure 8. The FRFs were measured out to Hz. Acceleration on the table where the large shaker attaches was measured due to the force of the tip shaker. By reciprocity, this FRF is assumed to be equal to the FRF of the acceleration at the tip of the table from a force at the large shaker as in equation (4). Step 4: Running the Vibration Test With the Large Shaker A random vibration test was run using the large shaker as shown in Figure 9. Data were acquired out to Hz with 3 averages to calculate the transmissibilities of all beam and slip table accelerometers to the table tip accelerometer. The reciprocity FRF from step 3 along with the transmissibilities were used to estimate accelerance FRFs as in equation

11 (9). Six modes were extracted from the big shaker modal test out to nearly Hz. The two modes that most affected the beam bending are shown in Figure 1. The beam was instrumented at the tip and mid-span, thus no apparent attachment to the table in the stick figures. This brings up one advantage of this method. Once the table is characterized with the modal test out to the frequency band of interest, no additional measurements need be taken at actual attachment points of the test article to the table. This is because the modal coordinates of the table are constrained, which can be obtained from the accelerometers mounted to the table in the bare table locations. The accelerometer locations on the table must yield independent mode shapes of the bare table, since the pseudo-inverse of the bare table mode shape matrix is required. This also assumes that the bare table mode shapes span the space of the actual motion during the random vibration test up to the highest frequency of interest. Figure 9- Large Shaker Driving The Table Mode 2 Frequency: Hz Damping:.399 %Cr IDLine 1: Generated from reference 1X+ n) Z axis (in) Mode 3 Frequency: Hz Damping:.48 %Cr IDLine 1: Generated from reference 1X n) Z axis (in) Figure 1- Two Modes With Large Interaction Between Beam and Table (87 and 11 Hz)

12 7) Results The results of the constrained vs. truth FRFs are given in Figure 11. The FRF plotted in black is the truth measurement of the driving point FRF at the tip of the beam while it was mounted to the seismic mass showing about 1 Hz resonance associated with the first bending mode. The FRF plotted in blue is the tip drive point FRF reconstructed from the modal parameters extracted from the random vibration slip table test. The red curve is the tip driving point FRF derived using the blue data and the augmented modal constraint from equation (9). Therefore the red curve was produced by applying the constraints to the data associated with the blue curve. As can be seen, the multiple resonances in the blue curve have been constrained to the single resonance near the frequency of the "truth" resonant frequency. The modal parameter extraction from the red FRF data gave a frequency within one percent of the truth frequency. One other observation is that the constrained red curve has lower damping than the truth curve. In retrospect, this may be due to the fact that the aluminum slip table is only damped about.3 percent, as compared with the nylon beam at about one percent. The modes from the blue data were.4 to.6 percent damping, which is between the damping of the bare table and the fixed beam. Therefore it appears that the damping of the table affected the damping of the constrained result, which was extracted at. percent. Now that this phenomenon has been identified, it may be possible to improve the damping estimate by taking into account the known bare table damping. This is not pursued in this paper. Frequency Response Function Comparisons Constrained Original Table FRF Beam on Seismic Mass Frequency - Hz Figure 11- Comparison of Constrained FRF (red) to Truth FRF (black) 8) Difficulties That Were Overcome It is instructive to list some of the difficulties that were overcome in implementing this work. There were some difficulties encountered when performing the reciprocity measurement with the tip shaker. First, a metal stinger rod.6" diameter was utilized to adapt the tip shaker to the force gage. Initially, this stinger was about four inches long, and the stinger went into lateral resonance below the Hz bandwidth, which invalidated the initial set of reciprocity FRFs. When the stinger was shortened to two inches, the lateral stinger resonance was above the bandwidth of interest, improving the Hz bandwidth force measurement. A second difficulty was that during the tip shaker data acquisition, the big shaker was not aligned properly, so that it had large side loads on the flexures. This caused extra friction that produced poor noisy FRFs. When the big shaker was aligned properly, the FRFs were clean. Finally, because the test setup relied on air bags and hydraulics in a table for proper alignment, the shakers were disconnected overnight. It was found that great care had to be exercised to reconnect the big shaker exactly in the manner it had previously been connected, or the modal frequencies would shift slightly. This change affects the accuracy of the FRF calculation in equation (9) if the transmissibilities and the H st

13 come from slightly different systems. Clearly, it is best if the transmissibilities are gathered shortly after the reciprocity FRF, so the system is time invariant. Another set of choices is which modes to use in the Φ c conditioning mode shape matrix. The first fore-aft rigid body mode, first and second bending modes gave the best results, shown in Figure 11. It has been found that if there are as many or more modes in the Φ c matrix than there are modes in the equation (1) FRFs, the results become very ill conditioned and the constraint process can fail. Adding mode shapes to Φ c which are of no significant value in spanning the space of the table motion in the random vibration test seems to degrade the results, sometimes only slightly and sometimes severely. The fewer modes, the better, and the better the condition number of Φ c, the better the result. It is interesting to note that adding in extractions of weakly excited modes in the unconstrained test on the table seems to reduce the sensitivity of the results to the number of modes included in Φ c. In one case, 16 modes were used to synthesize the FRFs for the beam and table, including several that were very weak in the data. In that case, Φ c contained six modes and the result was within two percent of the truth frequency. It appears that if the number of modes in Φ c is significantly less than the number of modes in the data, the resulting constrained FRFs are robust. 9) Uncertainty Quantification Uncertainty quantification can be performed with this methodology. Specifically, the effects of leaving out different modes from the Φ c matrix can be assessed. In this case, if the second bending mode of the bare table was eliminated (a mode 3. times the frequency of interest) it did not change the result even one Hz, so this mode is not extremely important to span the shape for the 1 Hz bending mode. In future work, sensitivity of the final result to changes in each shape, frequency or damping could be evaluated since the FRFs are synthesized from modal parameters. 1) Concluding Remarks As can be seen in Figure 11 this method shows promise as a practical means for eliminating the dynamics of the vibration table. Two major advances in this work are the application of robust modal constraints (instead of physical dof constraints which causes ill conditioning) and an approach to estimate the shaker force through reciprocity so that modal mass may be obtained. An additional advantage of the method is that the forces and motions at the locations where the test article attaches to the shaker table do not have to be measured. Only one modal characterization of the shaker and bare shaker table is required for all test articles. The accelerometers on the table during the bare table test must also be measured during the random vibration test with the test article. Although the effort here demonstrates that the basic ideas appear promising, this approach still needs to be applied to a large shaker to address other practical issues. Some of the questions the authors have to answer are: 1. Will the modal test on the bare slip table attached to the large shaker be practical? 2. Should the field coils of the shaker be energized or not during the bare table and the tip shaker tests? 3. Can a reasonable location for the large shaker accelerometer be found? This method is dependent on the accuracy of reconstructing FRFs from modal parameters, and so is subject to modal truncation errors. Typically, this will affect the higher frequency dynamics more than the lower frequency dynamics. The uncertainty in the modal parameters will produce corresponding uncertainty in the FRFs which will also affect the final results. The estimate of the shaker force is required so that the modal mass is accurately extracted for use in synthesizing the full FRF matrix from modal parameters. This requires an additional step beyond the normal random vibration test because the tip shaker must be brought in to measure the reciprocal FRF when the test article is mounted to the shaker table. The reciprocity approach used here will break down at high frequencies when the location of the "large shaker" accelerometer no longer provides a good estimate of the shaker armature motion at the "point" where the electric field forces are applied to the armature.

14 Although it was not pursued in this work, once the modal parameters of the bare table and the full system are extracted, an equivalent fixed base modal result should be achievable by setting up an eigenvalue problem with the modal coordinates of the full table and applying constraints forcing the bare table modal coordinates to be zero. One observation from this work is that the fixed base damping estimate of the test article was highly influenced by the damping in the nearby bending mode of the table. Future work needs to be done to account for the damping in the table. References 1 Carne, Thomas G., Martinez, David R. and Nord, Arlo R., "A Comparison of Fixed-Base and Driven Base Modal Testing of an Electronics Package", Proceedings of the Seventh International Modal Analysis Conference, Las Vegas, Nevada, February 1989, pp Beliveau, J.G., Vigneron, F.R. and Soucy, Y., and Draisey, S., "Modal Parameter Estimation from Base Excitation", Journal of Sound and Vibration, Vol. 17, January 1986, pp Brinker, Rune and Anderson, Palle, "A Way of Getting Scaled Mode Shapes in Output Only Modal Testing", Proceedings of the 21st International Modal Analysis Conference, Kissimmee, Florida, February 23, pp Fullekrug, U., "Determination of Effective Masses and Modal Masses from Base-Driven Tests", Proceedings of the 14th International Modal Analysis Conference, Dearborn, Michigan, February 1996, pp Fullekrug, Ulrich, and Sinapius, Michael, "Force Measurement in Vibration Testing - Application to Modal Identification", Seventh International Congress on Sound and Vibration, Garmisch-Partenkirchen, Germany, July 2, pp Sinapius, J.M., "Identification of Fixed and Free Interface Normal Modes by Base Excitation", Proceedings of the 14th International Modal Analysis Conference, Dearborn, Michigan, February 1996, pp Schedlinski, Carsten, and Link, Michael, "Identification of Frequency Response Functions and Modal Data from Base Excitation Tests Using Measured Interface Forces", Proceedings of the ASME Conference on Noise and Vibration, Boston, Massachusetts, Razeto, M., Golinval, J.C., and Geradin, M., "Modal Identification and Determination of Effective Mass Using Environmental Vibration Testing on an Electro-Dynamic Shaker", Proceedings of the 12th International Modal Analysis Conference, Honolulu, Hawaii, February 1994, pp Blair, Mark A., "Space Station Module Prototype Alternative Modal Tests: Convergence to Fixed Base", Proceedings of the 11th International Modal Analysis Conference, Kissimmee, Florida, February 1993, pp Blair, Mark A., "Space Station Module Prototype Alternative Modal Tests: Fixed Base Alternatives", Proceedings of the 11th International Modal Analysis Conference, Kissimmee, Florida, February 1993, pp Admire, John R., Tinker, Michael L. and Ivey, Edward W., "Mass-Additive Modal Test Method for Verification of Constrained Structural Models", AIAA Journal, Vol. 31, No. 11, November 1993, pp Crowley, John R., Klosterman, Albert L., Rocklin, G. Thomas and Vold, Havard, "Direct Structural Modification Using Frequency Response Functions", Proceedings of the Second International Modal Analysis Conference, Orlando, Florida, February 1984, pp Smallwood, David O., "Characterizing Electrodynamic Shakers as a Two-Port Network", Journal of the Institute of Environmental Sciences, September/October 1997, pp Hensley, Daniel P., and Mayes, Randall L., Extending SMAC to Multiple References, Proceedings of the 24 th International Modal Analysis Conference, pp.22-23, February 26.