Proceedings of the IMAC-XXVIII February 1 4, 2010, Jacksonville, Florida USA 2010 Society for Experimental Mechanics Inc. Modal Test Data Adjustment For Interface Compliance Ryan E. Tuttle, Member of the Technical Staff Jeffrey A. Lollock, Director Structural Dynamics Department The Aerospace Corporation P.O. Box 92957 Los Angeles, CA 90009-2957 Nomenclature d Total motion at reference sensor l Distance from interface to reference sensor µ t Tip displacement ratio θ Angular rotation of interface plane Circular frequency for system 2 (updated) mode ω 2 { q} Generalized (modal) coordinate vector { x} Physical degree of freedom vector [ K] System stiffness matrix [ M] System mass matrix [ ΔK] Differential stiffness for model update ΔM [ ] Differential mass for model update [ ] Differential boundary mass ΔM bb 2 [ Ω 1 ] Diagonal matrix of system 2 (updated) eigenvalues [ Φ 1 ] System 1 modal matrix [ Φ 2 ] System 2 (updated) modal matrix Φ fb [ ] Test modes adjusted to fixed-base configuration [ Φ rb ] Rigid body modes of test article relative to interface [ Φ t ] Test-measured modes { ψ} Generalized modal vector [ Ψ] Generalized modal matrix [ i] Matrix for ith system [ ] Updated generalized matrix [ ] Generalized matrix with appended rigid body degrees of freedom { } Second time derivative ABSTRACT. Modal tests are ideally performed in configurations that represent boundary conditions that are both appropriate (facilitate application of analytical methods) and easily modeled (fixed, free, or interfaced to hardware of known stiffness). Due to the constraints of the Ares 1-X program, test fixtures that would guarantee an idealized boundary were unavailable for the desired subsystem test configurations (cantilevered). The decision was made to test subsystems in a configuration as near cantilevered as could be achieved within program constraints. It was decided to test Super Stack 1 (SS1) and Super Stack 5 (SS5) the two tested subsystems resting on the test facility floor. In addition, the configuration for SS5 would include non-flight transportation / access hardware
between it and the floor. Since the facility floor and non-flight hardware were of unknown stiffness, analytical methods were developed to address the effects of this compliance on the measured data. A method was developed to identify interface motion from test measurements, judge when interface motion would affect test results significantly, and adjust the test results to represent those of a fixed-base configuration. Introduction Mode survey testing is used to gather data for validation of analytical models. During such testing careful consideration and measurement of boundary conditions must be undertaken as boundary condition variations can affect the dynamics of a system. Every effort is usually made to sufficiently approximate idealized boundary conditions for ease of modeling. Additionally, the type of boundary condition should be chosen to facilitate the intended analyses of the validated models. For the Ares 1-X modal test program, the desired boundary conditions are fixed at the base of both subsystems planned for test SS1 and SS5 as these were used in the traceability study [1]. Fixed boundary conditions for these subsystems eliminate numerous shell-type modes that only serve to unnecessarily complicate the tests and were not needed to meet the goals of this test program based on the traceability study performed. However, due to program constraints, development of fixturing capable of providing fixed boundary conditions could not be pursued. The tests were required to be performed in line with integration processes. Given these constraints, the decision was made to test the subsystems resting on the Vehicle Assembly Building (VAB) floor (with additional support equipment as became necessary) at the National Aeronautic and Space Administrationʼs (NASA) Kennedy Space Center (KSC), relying on the structureʼs weight to provide adequate preload for required reaction forces under test excitation. This configuration introduces the unknown stiffnesses of the floor and support hardware, unfortunately, which may need to be accounted for in subsequent analyses. One possibility of dealing with the interface stiffness is including this unknown stiffness explicitly in the analytical model and removing it after adjustment to the test data. Rather than include the boundary stiffness as a variable in model adjustment, the influence of interface compliance on the test data can be removed and the adjusted data used subsequently in model adjustment of a fixed interface model. This paper outlines such an analytical process and documents its application to the Ares 1-X subsystems. Adjustment Methodology The boundary motion adjustment method is based on a standard mass or stiffness update procedure. This procedure is used commonly throughout the industry to adjust reduced models to account for configuration changes. Consider two systems: system 1, the nominal system and system 2, the updated system. The undamped homogeneous equations of motion for each system are, [ M 1 ]{ x } + [ K 1 ]{ x} = { 0} (1a) [ M 2 ]{ x } + [ K 2 ]{ x} = { 0} (1b) We define the delta matrices as the differential mass or stiffness of the updated system relative to the nominal system. [ ΔK] = [ K 2 ] [ K 1 ] (2a) [ ΔM] = [ M 2 ] [ M 1 ] (2b) At this point, the mode shapes for the nominal system are used to transform the updated system equations. { x} = [ Φ 1 ]{ q 1 } (3a) [ Φ 1 ] T [ M 1 ][ Φ 1 ] = [ I] (3b) [ Φ 1 ] T 2 [ K 1 ][ Φ 1 ] = Ω 1 [ ] (3c) [ Φ 1 ] T [ M 2 ][ Φ 1 ]{ q 1 } + [ Φ 1 ] T [ K 2 ][ Φ 1 ]{ q 1 } = { 0} (4)
Substituting Equation 2 into 4, [ Φ 1 ] T M 1 ([ ] + [ ΔM] )[ Φ 1 ]{ q 1 } + [ Φ 1 ]([ K 1 ] + [ ΔK] )[ Φ 1 ]{ q 1 } = { 0} (5) ([ I] + [ Φ 1 ] T [ ΔM] [ Φ 1 ]) q 2 { 1} + Ω 1 [ M ]{ q 1 } + K [ ] q 1 ([ ] + [ Φ 1 ] T [ ΔK] [ Φ 1 ]) q 1 { } = { 0} (6) { } = { 0} (7) As can be seen by the delta terms in Equation 6, the nominal system modes do not diagonalize the updated system matrices. This is accomplished by performing a final eigensolution using the matrices of Equation 7 to determine the modes of the updated system. [ K ] ψ [ ] ψ { } ω 2 2 M { } = { 0} (8) Computing the solution at the reduced set of coordinates for the nominal system modes gives the updated system modes as a linear combination of nominal system modes. { q 1 } = [ Ψ] { q 2 } (9) [ Φ 2 ] = [ Φ 1 ][ Ψ] (10) The relationship expressed by Equation 9 defines the necessary condition for this method to be applied: the differential mass or stiffness must result in the updated system modes being linear combinations of the nominal system modes. To put it in other terms, the modal vectors of the nominal and updated systems must span the same space. For Ares 1-X, this method is intended to transform the modal data from the test configuration to that representing a fixed boundary configuration. This can be accomplished in either of two ways: (1) adding stiffness to the interface degrees of freedom (DOFs) to approach a cantilevered system or (2) adding inertia to the boundary DOFs to approach a seismic mass-type configuration. Prior to testing for Ares 1-X commenced, the planned adjustment method was run through a comprehensive example. Theoretical Example A simple beam model was used for the example problem. Each station along the length of the beam contained six degrees of freedom. The beam was free at one end and attached to ground at the other with springs in six DOFs. These springs are meant to represent the unknown stiffness of the test boundary. Of particular interest are the rotational spring stiffnesses about either axis perpendicular to the axial direction of the beam. These are the rocking DOFs and are the most influential to the bending behavior of the beam. Initial attempts at correcting boundary motion using this method required a prohibitively large number of modes to be kept in the updated eigensolution (Equation 8). As the number of modes retained was reduced, the results of the update process deviated further from the true fixed-interface configuration. Adding a significant number of modes to the target mode set for Ares 1-X tests was not feasible, so an analytical solution was pursued. Appending additional analytical flexible body vectors to the measured set was seen as problematic as they are products of the analytical model that is assumed to be in error. Generating vectors associated with rigid body motion of the test article requires only knowledge of the structureʼs geometry (assumed to be accurate). However, assigning a frequency to these vectors is still required, but that can be done using only the stiffness of the test boundary (estimation of which will be discussed later) and not that of the test article. Furthermore, for the case of a determinant interface, these vectors are more closely associated with the boundary motion that is contaminating the test data, as they are directly associated with the interface DOFs.
With the addition of the rigid body vector to the test vectors, the generalized matrices of Equations 3b and c take the form of those in Equations 11a and b. [ M ] = Φ t Φ rb I [ M ] = Φ rb [ ] T [ M] [ Φ t Φ rb ] [ ] [ Φ t ] T [ M] [ Φ rb ] [ ] T [ M] [ Φ t ] [ Φ rb ] T [ M] [ Φ rb ] [ K ] = Φ t Φ rb 2 Ω [ K ] = 1 Φ rb [ ] T [ K] [ Φ t Φ rb ] [ ] [ Φ t ] T [ K] [ Φ rb ] [ ] T [ K] [ Φ t ] [ Φ rb ] T [ K] [ Φ rb ] (11a) (11b) The update method for Ares 1-X is best illustrated by assuming the physical DOFs are partitioned between those corresponding to the interface and those that are part of the test article. Equation 12 shows the eigenproblem corresponding to the mass update problem. The delta matrix is made large enough such that the updated frequencies and modal vectors are converged. A simple diagonal matrix can be used with elements increased iteratively until the resulting shapes and frequencies converge. The frequencies resulting from the solution to Equation 12 are the fixed-interface modal frequencies with additional zero values corresponding to the rigid body (or suspension) vectors. The eigenvectors can be transformed back to physical space using the test modes and analytical rigid body vectors (Equation 13). 2 { } ω 2 M [ K ] ψ [ ] + Φ [ Φ] { ψ} = { 0} (12) 0 ΔM bb [ ] 0 0 [ Φ fb ] = [ Φ] [ Ψ] (13) An analogous form for the stiffness update option can be formed in a similar manner. Figure 1: Tip Displacement Ratio Metric In order to assess the affects of interface motion on the test measurements, the degree of interface motion must be quantified. As a comparison tool, a tip displacement ratio metric was devised. The total motion of the test article is assumed to be the sum of a flexible portion due to deformation of the structure and a rigid portion due to interface motion. The tip displacement ratio metric is defined as the ratio of the rigid portion of this motion to the total motion (Figure 1, Equation 14). It is noted that this metric is valid only for the pair of first bending mode shapes. This is appropriate as these are the modes most sensitive to the interface stiffness. d l µ t = θl d (14) This update process was performed using a range of interface stiffnesses resulting in corresponding amounts of interface motion. The method converges very near the θ θ
fixed-interface modes for smaller amounts of interface motion. As the interface motion increases, however, large errors in converged frequency relative to the fixedinterface values are present (Figure 2). It is concluded that there may exist some amounts of interface motion that cannot be corrected to a sufficient error bound using this update process. This sensitivity also can be used to define an acceptable amount of interface motion where no correction is needed based on an error budget. This acceptable value can then be compared to what is measured in test to determine if adjustments to the test results are necessary. These sensitivities are not absolute and require calculation for each new structure or configuration encountered. Figure 2: Adjustment Sensitivity To Interface Motion To better simulate a real world application of the method, errors were introduced into the analytical stiffnesses of the ground and test article. The update process was then applied to models with varying amounts of error. Stiffness and modal errors are tracked separately and the resulting differences between converged updated frequencies and the truth (cantilevered) model are recorded in a three-dimensional plot (Figures 3a and b). In both plots (mass or stiffness update) there is a valley of small frequency error (approximately two percent) around the intersection of no modeling errors. The valleys show that the proposed update method can converge to acceptable values (given some acceptable error) even in the presence of modeling errors. The steep gradient to the right in each plot shows that grossly overestimating the interface stiffness results in large errors for the converged values. This should be avoided by using the sensitivities defined in Figure 1. Figure 3: Update Method Results In the Presence of Noise (a) Mass update (b) Stiffness update
The process described in the Figure 4: Interface Stiffness Sensitivity preceding requires an estimate of the stiffness of the test article and the test boundary. The estimate for the test article is generally derived from a pre-test finite element model (FEM). Often such an estimate for the test boundary is not available. However, this stiffness can be estimated directly from the test data if the proper sensitivities are calculated beforehand. Figure 4 shows the relationship between interface stiffness and the tip displacement ratio (for the primary bending modes) for this example problem. The upper portion of this curve represents a very soft boundary as the test article motion is due almost entirely to rigid body motion relative to the interface. The far right is a very stiff boundary as the tip displacement ratio is very small. In the middle region deformation of both the boundary and test are exhibited. If the tip displacement ratio is calculated from the test measurements, this sensitivity curve can be used to estimate interface stiffness by reading across from the measured tip displacement ratio to the curve and then down to the interface stiffness value. Again, these sensitivities are problem dependent, and it is the responsibility of the analyst to define appropriate interface parameters. Adjustment of SS5 Data The mode survey test of SS5 was performed in a configuration that included non-flight support hardware below its interface to the Upper Stage Simulator (USS). This interface was chosen as that at which to perform the update procedure discussed in the preceding text. In this manner, all influence of non-flight hardware can be removed from the test data, resulting in parameters equivalent to those that would be estimated from a fixed-base test of SS5. Calculation of the tip displacement ratio metric from the SS5 test data resulted in values of 11 to 12 percent rigid body motion. This was significantly higher than the analytical values calculated using the FEMs of SS5 and the non-flight support hardware. Cross-orthogonality matrices were calculated for the test configuration and the cantilevered (fixed interface) configuration (Tables 1 and 2). Comparing the tables between the two configurations, it can be seen that the comparison between test and model is better when the non-flight support hardware is removed; the shape matches improve, and the modal frequency differences are reduced. This is particularly evident for the first pair of modes which are the most sensitive to interface flexibility. The second pair of corrected modes actually corresponds closest to the third pair in the test configuration. The fact that the test-tomodel correlation is worse for the test configuration is evidence that there are errors present in the model of the support hardware. This highlights the advantage of the adjustment method. By adjusting the modal test resutls, the modeling of the non-flight hardware can be ignored, and all effort for model adjustment can be focused on the flight hardware. The third pair of bending modes for the cantilevered configuration could not be estimated as there was an insufficient number of modes measured to correct this pair.
Table 1: SS5 Test-To-Model Cross Orthogonality (Test Configuration) Table 2: SS5 Test-To-Model Cross Orthogonality (Cantilevered Configuration) Adjustment of SS1 Data The configuration for the SS1 mode survey test consisted of the SS1 flight hardware resting on shims and pads to an I-beam frame structure, with the I-beam frame in turn resting on concrete piers built into the Vehicle Assembly Building (VAB) floor. The configuration resulted in a tip displacement ratio of approximately 85 percent. This degree of interface motion is beyond the region of application for the adjustment method and thus was not applied. Conclusions The Ares 1-X modal test program encountered significant technical challenges as a result of program constraints. One such challenge was incorporating modal testing into the normal integration progress without the use of dedicated test fixtures. An analytical method was proposed to adjust modal test data to negate effects of interface motion. This method was shown to work on an example problem, and subsequently applied with success to the SS5 modal test results. Reference 1. Tuttle, R. E., Lollock, J. A., and Hwung, J. S., Identifying Goals For Ares 1-X Modal Testing, International Modal Analysis Conference XXVIII.