Experimental Based Substructuring Using a Craig-Bampton Transmission Simulator Model

Transcription

1 Experiental Based Substructuring Using a raig-bapton ransission Siulator Model Mathew S. llen, Daniel. Kaer Departent of Engineering Physics University of Wisconsin Madison, WI 376 kaer@engr.wisc.edu, sallen@engr.wisc.edu & Randy L. Mayes Structural Dynaics Sandia National Laboratories 1 lbuquerque, NM rlayes@sandia.gov Keywords: experiental substructures, substructuring, raig-bapton BSR Recently, a new experiental based substructure forulation was introduced which reduces ill-conditioning due to experiental easureent noise by iposing the connection constraints on substructure odal coordinates instead of the physical interface coordinates. he experiental substructure is tested in a free-free configuration, and the interface is exercised by attaching a flexible transission siulator. n analytical representation of the fixture is then used to subtract its effects fro the experiental substructure. he resulting experiental coponent is entirely odal based, and can be attached in an indirect anner to other substructures by constraining the odal degrees of freedo of the transission siulator to those substructures. his work explores a different alternative in which the transission siulator is odeled with a raig-bapton odel, a odel that ay be ore appropriate when the interfaces are connected rigidly. he new ethod is copared to the authors previous approaches to evaluate the errors due to odal truncation using finite eleent odels of several bea systes including one in which the transission siulator is connected to the coponent of interest at two points, potentially producing an ill-conditioned inverse proble. 1. Introduction Experiental-analytical substructuring is attractive in any applications where there is otivation to replace one subcoponent with an experiental odel. For exaple, a coponent ay be difficult to odel if it contains aterials with unknown properties, intricate geoetry that would require any eleents to approxiate or joints or interfaces with unknown stiffness and daping properties. In other cases the coponent ay be produced by an external supplier, in which case it ay be inefficient to expend effort odeling a coponent that one cannot change. In any case, it will be necessary to test the coponent at soe point in order to validate the coputational odel. In these scenarios, one can perfor a careful dynaic test and extract an experiental-based odel for the subcoponent(s) of interest. his odel can then be coupled to finite eleent (or other) odels for the rest of the structure to predict the response. Substructuring can be used to couple physical or odal odels of each subcoponent to create a odel (e.g. a set of ordinary differential equations) for the assebly, or, for linear systes, one can also operate on the frequency responses directly to predict those of the assebled syste. he latter approach is called frequency based substructuring or ipedance coupling and an excellent review is provided in [1]. his work focuses on the forer, which is coonly referred to as 1 Sandia National Laboratories is a ulti-progra laboratory anaged and operated by Sandia orporation, a wholly owned subsidiary of Lockheed Martin orporation, for the U.S. Departent of Energy National Nuclear Security dinistration under ontract DE-4-94L8. llen, Kaer & Mayes IM, Feb. 214 Page 1 of 1

2 odal substructuring, although the authors have found that frequency based substructuring and odal substructuring give siilar results when the data sets used are identical [2, 3]. In [4] llen, Mayes & Bergan presented a specific substructuring strategy, now known as the ransission Siulator (S) ethod, in which the coponent of interest is tested with a fixture known as the transission siulator attached in order to iprove the experiental odel for the coponent. he transission siulator allows one to capture the copliance and daping of the bolted joints at the interface, provides a eans for dealing with continuous, copliant interfaces where one cannot easily reduce the interface to a few connection points, and iproves the odal basis of the experiental coponent by exercising the interface. he disadvantage to this approach is that the transission siulator ust be odeled very accurately and its odel ust be subtracted fro the experiental odel, potentially introducing negative ass or stiffness [, 6]. In the authors previous works, the transission siulator has always been odeled with a free interface. More specifically, the odal odel for the S was described by up to six rigid body odes and a collection of elastic odes where all of the connection points were free. his basis ay not be ideal in soe applications, especially when the S is rigidly connected to a stiff structure of interest. his work explores a new alternative where a raig-bapton odel of the S is created and substracted fro the experiental easureents. Hence, the odal basis is described by the fixed-interface odes of the S rather than its free odes, and this has been found to iprove the results in a few test cases. he following section reviews the theory underlying the transission siulator ethod and how the transission siulator odel is replaced with a raig-bapton odel. he ethod is evaluated on a -shaped bea structure in Section 3 and conclusions are presented in Section heoretical Developent 2.1 Free-Free Modal ransission Siulator (FF-S) he transission siulator ethod is explained in detail in [4, 7, 8] so only a cursory review will be presented here. One is presued to have a finite eleent odel of the transission siulator, denoted substructure, and this can be used to copute its ode shapes, Φ, natural frequencies, r, r 1... n and its odal daping ratios, r are also presued to be known (e.g. they are typically taken to be half of the loss odulus of the aterial). In the traditional approach, these are coputed with the interface degrees of freedo free, and hence the first few odes are rigid body odes. n actual physical transission siulator is also constructed and attached to the substructure of interest, denoted B, and the assebly, denoted, is tested to identify its odal paraeters. he equations of otion of and ( ) can then be concatenated as follows I q q q \ \ 2 2 r r\ r \ Φ F I q \ \ 2 2 r r\ q q r \ Φ F u Φ q u Φ q where the DOF, q, are the generalized odal coordinates of each substructure. Let the set of degrees of freedo where the S joins the substructure be denoted as the c-set (or connection point set). hen one would ideally enforce equal otion between substructure and the S at these points. c c u u (2) However, this leads to several difficulties and hence llen, Mayes & Bergan, instead defined a set of easureent points that copletely characterize the transission siulator (e.g. so that Φ is full rank). hen the following constraints are enforced, which constrain all of the odes of the transission siulator to their projection onto the otion of, (1) q Φ Φ I q (3) where, (), denotes the pseudo-inverse of the atrix (). hese constraints are enforced using the transforation q q Bq where in Eq. (3). B I Φ Φ is in the null space of the constraint atrix, the atrix on the left llen, Kaer & Mayes IM, Feb. 214 Page 2 of 1

3 his ethod will only succeed in proportion to how well the odes of the transission siulator, Φ, span the space of the otions observed on, or Φ. While the free-interface odes have worked well in the cases studied so far, the authors have noted cases where these odes did not adequately span the space of the observed otions (see, e.g. []), leading to less accurate results or negative ass in the assebled syste. Furtherore, it would be preferable to obtain a raig-bapton odel for the subcoponent, with its constraint odes that capture the interface stiffness and allow it to accurately capture the otion of the subcoponent when it is connected to a variety of structures. 2.2 raig-bapton ransission Siulator In order to create a raig-bapton (B) odel of the transission siulator, the physical displaceent vector for substructure is partitioned into two sets, the c-set, corresponding to the connection degrees of freedo, and the o-set, corresponding to the interior degrees of freedo (DOF). he resulting partitioned ass and stiffness atrices can be written as M M oo M co M oc M cc K he displaceent vector for coponent can be transfored into B [9] coordinates using the relation u u Φˆ ψ qˆ K oo K co o BuB uc Ic uc K oc K cc (4) () where ˆ Φ is a atrix o fixed-interface odes, 1 ψ K K, I c is an identity atrix of order equal to the nuber of interface DOF, ˆq is the vector of displaceents of the fixed-interface odes, and u c represents the physical displaceent of the interface. he B representation describes the deforation of the transission siulator in ters of dynaic fixed-interface odes, and static shapes, called constraint odes ψ I c, in which each colun gives the static response of coponent due to a unit deflection of one of the interface DOF holding the others fixed. he corresponding B ass and stiffness atrices are then given by oo oc M B B M B I M cq M qc M S (6) K B B K B \ 2 ˆ r\ K S (7) where it has been assued that the fixed-interface odes have been ass noralized, \ 2 ˆ r\ is a diagonal atrix of fixedinterface odal frequencies, and M S and K S are the coponent ass and stiffness atrices statically reduced to the interface DOF, respectively. s entioned previously, the advantage of this transission siulator representation over a free-free odal representation is that the fixed-interface odes do a better job of representing the shape of the siulator in the coupled syste being tested, and the physical interface DOF are directly retained for connection to other substructures. he B representation of coponent ust now be subtracted fro the experiental odal representation of the cobined syste. he authors have explored two ethods for doing this, which will be outlined in the following two subsections Method 1: Free Modes of raig Bapton Model (B-S) he B representation of substructure has the following equation of otion (oitting the daping ters for siplicity) B B B B B M u K u F (8) his odel can then replace the odel for in Eq. (1), so the concatenated equations becoe, llen, Kaer & Mayes IM, Feb. 214 Page 3 of 1

4 \ 2 I q r \ q Φ F MB u B B B K u B F Φ q B ub u u In keeping with the odal constraint philosophy, all of the odes of the transission siulator will be coupled to their projection on, using the following constraints, where once again the easureent set,, includes enough sensors to fully characterize the otion of the transission siulator. B, q Φ I (1) ub Since the odel for substructure is known, after assebling the syste one can recover the response at any point on (including the connection point u c, both displaceent and rotation!) using Eq. (). Hence, one can estiate the fixedinterface odes of the B syste by applying the constraints u c by defining an appropriate transforation atrix, B, and then using the procedure outlined above. It is also interesting to note that one could obtain the sae result by using the B odel to estiate the free-interface odes of the S and then using those in Eq. (1) together with the usual transission siulator approach. he free-interface odes thus estiated would not be the true free-interface odes but siply another basis for the B odel, and hence this would produce the exact sae result as the procedure outlined in Eqs. (9) and (1). In fact, this approach was used in this coputations described in Section Method 2: Motion Relative to the Interface (B-IP) he ethod presented in the previous section is clearly very different fro the way in which raig Bapton odels have been eployed in the analytical doain over the past several decades. Specifically, no effort was ade to retain the structure of the B odel nor the partition of interface degrees of freedo as unique coordinates. Hence, an alternative is presented here, which will be denoted B-IP for Interface Preserving. Once again, following the odal constraint approach [4], coupling will be accoplished by enforcing physical displaceent copatibility between the two coponents at the easured locations u u (11) where it is assued that none of the connection DOF are in the set of easured locations. ransforing to B coordinates for coponent and odal coordinates for experiental coponent, Eq. (11) becoes ˆ ˆq u c q (12) in which the subscript indicates a row partition corresponding to the easured DOF. Equation (12) can be solved for the transission siulator fixed-interface odal response (9) ˆq ˆ q u c (13) 1 ˆ where ˆ ˆ ˆ is the generalized inverse of the fixed-interface odes row partitioned to the easureent locations. he existence of this inverse requires that atrix ˆ is full colun rank, iplying that n n. Notice that the constraints here can be viewed as projecting the otion of relative to the interface onto the fixed-interface odes of. he odal constraint can be applied by defining the coordinate transforation u G q ˆq u c u r I ˆ ˆ I q u c (14) llen, Kaer & Mayes IM, Feb. 214 Page 4 of 1

5 pplying the constraint to the uncoupled syste in Eq. (9) gives the approxiation of the equation of otion for the experiental based substructure B M B u B K B u B B F F in which the reduced displaceent vector u r in Eq. (14) has been renaed u B where (1) u B q B u c B (16) and (17) (18) with and. he substructure representation for coponent B, given by Eqs. (17) and (18), is not precisely a B representation because the stiffness atrix is not block diagonal. However, it is B-like in that the displaceent vector u B contains odal degrees of freedo q B, and the physical degrees of freedo u B c at the points where coponent B will connect to other substructures. he experiental based representation for coponent B can then be easily connected to other FEM based substructures using traditional assebly ethods. However, unlike the B representation, the odal degrees of freedo cannot be truncated based on frequency due to the stiffness coupling between the odal and interface DOF in Eq. (18). n advantage of the proposed coponent B substructure representation is that the fixed-interface odes and frequencies can be deterined directly by just crossing out the block row and colun associated with the interface DOF. he resulting constrained interface ass and stiffness atrices, Mc B and Kc B, just correspond to the expressions in the upper-left quadrants of the atrices in Eqs. (17) and (18). herefore, the fixed-interface odes and frequencies of coponent B can be derived fro a free-free vibration test of the coposite syste including the transission siulator. If the proper nuber of freefree experiental odes for coponent and fixed-interface odes fro the transission siulator are retained in the analysis, the atrices Mc B and K B c should both be positive definite. Further details regarding this approach can be found in [1]. 3. Nuerical Exaples his new approach was explored by applying it to two systes, one of which was studied in a previous work [1]. he second syste contains two connection points (statically indeterinate interface) and was developed specifically for this work Bea Syste fro [1] he first syste studied is the -bea syste shown in Figure 1, which was used in [1] to deonstrate the ethod that will be denoted the B-IP ethod here. he transission siulator is a 24-in long steel bea with. by.-in cross section while the structure of interest, coponent B, is a 36-in long steel bea with.3 by.-in cross section. Note that this syste s S is significantly ore flexible than that used in [4-6], so it is ore difficult to subtract fro the assebly. Further details regarding the setup are provided in [1]. s in the cited works, finite eleent odels for these beas will be used to siulate substructuring by truncating the odal basis. llen, Kaer & Mayes IM, Feb. 214 Page of 1

6 Figure 1: Scheatic of the -bea substructuring proble. he goal is to experientally derive a odel of coponent B, the horizontal (cyan) bea, fro easureents of the assebly. s in [1], the frequency range of interest is to 2 Hz, so 24 odes of are retained with natural frequencies ranging fro Hz (3 rigid body odes) to 478 Hz, or over about twice the frequency range of interest. In [1], 13 fixed-interface odes were used to describe substructure, with the highest ode at 1.9 khz, and after applying 13 constraints a odel for B was obtained. s a first step in evaluating the success of the substructuring calculations, the free odes of coponent B were calculated, and they are copared to those of the FE truth odel in able 1 in the colun labeled B-IP. he sae process was repeated using the traditional transission siulator ethod with free-interfaces odes, the FF-S ethod, and the result of that calculation is copared with the proposed ethod which uses a raig-bapton odel to describe the transission siulator, the B-S ethod, and those results are also shown in able 1. Both of these odels used 1 odes to describe the transission siulator, 3 rigid body odes plus 7 free-interface elastic odes for the FF-S ethod and 3 constraint odes plus 7 fixed-interface odes for the B-S ethod. hese odes spanned a frequency range of to 13.7 khz for the FF-S ethod and to 8.4 khz for the B-S ethod. Note that because the interface (connection point) is at the center of the transission siulator, its fixed interface odes cae in pairs in which either the top or botto vibrated as a cantilever while other side reained stationary. s a result, a larger nuber of odes would be required to span the sae frequency range. able 1: Free-Interface Modes of coponent B estiated by each substructuring approach. ctual B-IP (raig Bapton) FF-S (Free-Interface) B-S (raig Bapton) N FE Freq (Hz) % Error N Freq (Hz) % Error N Freq (Hz) % Error % % % % 12.1.% 12.1.% % % % % 7 4..% % % % % % % % % % % % % % % 13* % % % % % % % % % % % * % 17* % % % % % % % % % *Spurious odes were oitted fro the table. he results show that, while any of the ethods produces good estiate of the B syste s free-free natural frequencies in the frequency range of interest, the traditional FF-S ethod is soewhat ore accurate than the others. his is, perhaps, to be expected since our goal is to obtain a raig-bapton-like odel for syste B, and while a raig-bapton odel would be excellent at describing B in an assebly, it ight not produce the best results when the interface is free or lightly coupled. In those cases another subcoponent odel, such as a raig-hang odel [11], would probably be preferred and the FF-S ethod produces a odel that is ore along those lines. On the other hand, the table also shows the ode nuber of each llen, Kaer & Mayes IM, Feb. 214 Page 6 of 1

7 ode that is being copared in the colun labeled N. he odes were atched using a M and so only odes that were siilar are copared in the table. he FF-S and B-S ethods both show gaps in the ode nuber. he odes that are not shown had purely iaginary natural frequencies and hence were not included. he one exception was ode 12 in the FF-S ethod which appeared to be reasonable. he gaps reveal that the first spurious ode occurred above 14 Hz (after the 11 th ode) for the FF-S ethod and not until above 266 Hz (after the 1 th ode) for the B-S ethod. his could suggest that the B-S ethod captures the deforation of the transission siulator ore effectively. When the coponent of interest will be connected to a fairly rigid substructure, it ay be ore iportant for the substructure odel to accurately predict the coponent s fixed-interface odes rather than its free odes. he fixed-interface odes of each of the substructure odels were derived by constraining the three interface DOF as entioned in Sections and hen the fixed-interface natural frequencies were found and are copared in able 2. able 2: Fixed-Interface Modes of coponent B estiated by each substructuring approach. ctual B-IP (raig Bapton) FF-S (Free-Interface) B-S (raig Bapton) Freq % N FE N Freq (Hz) % Error N (Hz) % Error N Freq (Hz) Error % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 13* % % % % % * % 16* % 1* % % % % % % % % % % % % % % % % % he results in able 2 show that the ethods that use a raig Bapton basis to describe the transission siulator produced considerably ore accurate predictions of the fixed-interface odes. Indeed, both the B-IP and B-S ethods estiated all of the natural frequencies below khz (twice the bandwidth of interest) with less than.2% error. he errors for the B- IP ethod are soewhat lower, although any of these odal truncation errors would probably be negligible relative to easureent errors when the ethods are applied in practice. In contrast, while the natural frequencies predicted by the FF- S ethod are quite reasonable for ost of the odes (typically below 1%), the errors in odes 8 and 17 were quite large and the odel contains a larger nuber of spurious odes. o diagnose this, the nuber of odes used to describe the S in the FF-S ethod was varied between 3 and 11. In all cases the FF-S ethod predicted a reasonable nuber of freeinterface odes correctly (e.g. with errors in the natural frequencies below 1%). It also gave reasonable estiates for the lower fixed-interface odes, with about 2% axiu error over the first eight odes for n =3 decreasing to less than 9% axiu error (1.% average error) for the first 1 odes for n =8. s ore odes were added to the S odes 8 and 17 began to degrade as seen above while the other natural frequencies becae even ore accurate. In contrast, the B-S ethod gave excellent predictions for the fixed-interface odes for n >, with all of the lower natural frequencies having less than 1% errors. llen, Kaer & Mayes IM, Feb. 214 Page 7 of 1

8 hese results are soewhat surprising. While one would expect that, if one could create a raig Bapton odel for subcoponent B, that it would do an excellent job of predicting the fixed interface odes (actually the result would be perfect since the B odel contains the fixed-interface odes). However, here we have used a relatively sall nuber of the free odes of the assebly and a B odel for to derive an approxiate B odel for B by coputing B=(-), and the result is excellent! s has been entioned in the authors previous works [, 6], when one substructure is subtracted fro another it is possible for the ass atrix to becoe indefinite (contain negative eigenvalues), and these in turn tend to produce additional spurious odes in the substructuring predictions. he authors presented two algoriths in [6] to address this issue. he raig- Bapton ethod presented in this work is essentially interchangeable with the FF-S ethod in the authors previous works, so the sae algoriths can be used to force positive ass when substructuring. his was explored here by using the dded Mass ethod in [6] to add ass to the substructure odel for B until its ass atrix becae positive definite. his algorith was used to correct the odels for B for the case presented in ables 1 and 2 by adding the iniu aount of ass which caused the ass atrix to becoe positive definite. For the FF-S ethod the ass added was.6% of the total ass of substructure B, while for the B-S ethod only.1% ass had to be added. he added ass didn t change the B-S results appreciably. On the other hand, for the FF-S ethod it was surprising to see that the added ass draatically iproved the predictions of 8 th and 17 th fixed interface odes, reducing the errors in those natural frequencies just a few percent. ll of these results suggest that a raig-bapton odel is a ore accurate basis for the transission siulator in this application. In the authors previous works [4-6], they have also studied a -bea syste which is siilar to that described here (and in [1]) although it had a stiffer transission siulator. he B-S ethod was also applied to that syste and the results obtained showed siilar trends to those presented above, so for brevity those results will not be included here. 3.2 wo-diensional Structure with Indeterinate Interface In this next case study, a syste is considered which could lead to an ill-conditioned substructuring proble. he syste of interest is the -shaped assebly highlighted with a dash-dot (blue) box in Figure 2. In the assebly of interest (on the left in Figure 2), syste B is connected to another -shaped bea creating a box structure with an additional appendage that is 3 long. s shown in Figure 2, the beas that coplete the box in the assebly of interest are thicker than those in B, having a height of 2 where the beas coprising B have a height of 19. he box in both and in the assebly of interest is 47 square and all of the beas had a width of 2. transission siulator was designed to replicate the -shaped bea in the assebly of interest, although the cross section of the transission siulator was 19 high by 2 thick, so it is less stiff than the syste in the assebly of interest. ll of the other diensions were identical to those in the assebly of interest. he interface between the S and syste B consists of two separate ars and hence is reiniscent of the syste studied by P. Ind in his PhD thesis [12]. Ind s work showed that a syste such as this can exhibit draatic sensitivity to noise because the easureents are likely to contain very little relative otion of the ars and yet that otion provides the only available inforation about how the structure will ove once the transission siulator is reoved. However, in this case the proble should not be as poorly conditioned as the syste studied by Ind because the transission siulator does allow soe otion of the interface DOF in its lower odes of vibration. a.) Full ssebly E b.) Experiental ssebly Substructure, D oponent of Interest, B ransission Siulator, oponent of Interest, B llen, Kaer & Mayes IM, Feb. 214 Page 8 of 1

9 Figure 2: Scheatic of two-diensional structure with a statically indeterinate interface. a.) Full ssebly E=B+D, b.) ssebly. We desire to predict the otion of the assebly on the left but wish to avoid creating an FE odel for B. transission siulator,, is created and attached to B and the odes of the assebly are found and used to derive the odel for B. One diensional FE odels were created for each bea, with 2 long eleents, and the resulting nodes are shown in Figure 3. Duplicate nodes were erged when assebling the syste. hen, to siulate a odal test the odes of the FE odel for assebly were coputed out to 3kHz, resulting in a 16 ode odel for that included three rigid body odes. raig Bapton odel was then created of the transission siulator, with six constraint odes and two elastic (fixedinterface) odes. hose two elastic odes were the only fixed-interface odes below 1kHz. Figure 4 shows twelve of the thirteen elastic odes of ssebly. (he first elastic ode, which is not shown, was a first ovaling ode along the diagonal of the frae.) he higher odes show considerable bending of the frae suggesting that a odal odel based on these odes should be able to capture relatively coplicated otion of coponent B. he fixed interface and constraint odes of the transission siulator are shown in Figures and 6 respectively. his truncated odel for the S was evaluated by coputing its free-interface odes and coparing their natural frequencies with the true freeinterface natural frequencies. his revealed that the raig-bapton odel for the S accurately predicted the first three odes of the free-free transission siulator (up to 6 Hz) to within 1.2%, while the next few odes (beginning at 1.2 khz) were not predicted accurately, although their shapes were vaguely siilar Figure 3: Finite Eleent Mesh of the ssebly of Interest. llen, Kaer & Mayes IM, Feb. 214 Page 9 of 1

10 Mode = Hz Mode 6 = Hz Mode 7 = Hz Mode 8 = Hz Mode 9 = Hz Mode 1 = Hz Mode 11 = 16.9 Hz Mode 12 = Hz Mode 13 = Hz Mode 14 = Hz Mode 1 = Hz Mode 16 = Hz Figure 4: Elastic odes of ssebly in the frequency range of interest Mode 1 = Hz Mode 2 = Hz Figure : Fixed-interface odes of the transission siulator. llen, Kaer & Mayes IM, Feb. 214 Page 1 of 1

11 onstr. Mode 1 onstr. Mode 2 onstr. Mode onstr. Mode 4 onstr. Mode onstr. Mode Figure 6: onstraint odes of the transission siulator. hese easured odes for were cobined with the B odel for the transission siulator and Eq. (9) was used with the constraint in Eq. (1) to copute a odel for subcoponent B. In practice the odel for B would then be used to predict the response of the assebly of interest, or of various design iterations of that assebly. In this work, the true assebly is already known (via the full finite eleent odel), and so its odes will be copared with those predicted by substructuring to evaluate the B procedure. Before doing this, two other interediate cases were considered to seek to explain what inforation is captured in the experientally derived odel for B. First, both of the legs of B were constrained to ground at the connection points. In this case the odel for B accurately predicted the first three natural frequencies, with errors of % or less, and the next few odes were also quite reasonable with about 1% error. Next, only one leg of B was constrained to ground leaving the other leg free. In this case the odel produced qualitatively accurate estiates of the first three odes although their natural frequencies were about 3% too high. his reveals that, while the B procedure does produce a raig-bapton-like odel for substructure B, it will not necessarily have the sae resolution as a real analytical raig-bapton odel with the sae nuber of odes. he authors have previously suggested that the best practice is to use a transission siulator that iics the interface in the assebly of interest, and the S used here clearly does this so one would hope that the odel for B will be accurate in that assebly Results for ssebly of Interest he true odes of the syste in the application of interest are show in Figure 7. Each of the odes estiated by the substructuring procedure was atched to the syste odes using the M atrix shown in Figure 8. hen, the natural frequencies for the odes that corresponded were copared in able 3. he substructuring procedure was repeated using the B-S ethod as well as the original FF-S ethod fro [4]. Both ethods used 8 odes to odel the transission siulator. In the case of the FF-S ethod, the highest ode was at 2.2 khz, considerably higher than the highest freeinterface ode that the B-S odel predicted accurately. llen, Kaer & Mayes IM, Feb. 214 Page 11 of 1

12 Mode 4 = 139. Hz 1 Mode = 28. Hz Mode 6 = Hz Mode 7 = 414. Hz Mode 8 = 47. Hz Mode 9 = Hz Mode 1 = 986. Hz Mode 11 = Hz Mode 12 = Hz Mode 13 = 136. Hz 1 Mode 14 = Hz 1 Mode 1 = Hz Figure 7: Mode shapes of the truth odel for the application of interest. llen, Kaer & Mayes IM, Feb. 214 Page 12 of 1

13 ruth Model Plot of M Matrix Substructuring ruth Model Plot of M Matrix Substructuring Figure 8: M between the Modes of the ruth Model and those predicted by substructuring using the (left) B-S and (right) FF-S ethods. he results in able 3 show that the B-S ethod produced considerably ore accurate estiates for this assebly in the application of interest, at least in the lower frequency range. For the odes below 1kHz (or one third of the frequency range over which the assebly was tested), the B ethod accurately estiated all of the odes with less than.% error in any natural frequency and alost all of the M values above.999. On the other hand, the FF-S ethod also provides reasonable predictions although the frequencies were less accurate, soeties in error by as uch as 16%, and the ode shapes were also soewhat less accurate as shown by the M. On the other hand, the FF-S ethod provided reasonable estiates of all of the odes below 2kHz while the B-S ethod predicted several spurious odes beginning at 1.2kHz. able 3: Natural Frequencies (Hz) of the ssebly of Interest Estiated by Substructuring. Mode ruth N B-S B Error M N FF-S FF-Error M % % % % % % % % % % % % % % % % % % % % %.9923 hese odels were found to contain soe negative ass, so, as was done in Section 3.1, the dded Mass ethod fro [6] was used to add ass to the substructure odel for B until its ass atrix becae positive definite. Figure 9 and able 4 show the results when positive ass is forced onto the substructuring results. he ratio of the nor of the ass added to that of the ass atrix in each case was.766 and.226 for the B-S and FF-S ethods respectively. While the B-S ethod required three ties as uch ass to obtain a positive ass atrix, this siple addition of ass iproved the results rearkably. With added ass the B-S ethod produced only one spurious ode and predicted all of the odes below 2 khz quite accurately. On the other hand, the odel predicted by the FF-S ethod did not iprove significantly, except perhaps in its estiates of the rigid body odes. llen, Kaer & Mayes IM, Feb. 214 Page 13 of 1

14 ruth Model Plot of M Matrix Substructuring ruth Model Plot of M Matrix Substructuring Figure 9: M atrices for the cases entioned in Figure 8, when positive ass is enforced for each ethod: (left) B-S and (right) FF-S. able 4: Natural Frequencies (Hz) of the ssebly of Interest Estiated while enforcing positive ass. Mode ruth N B-S B Error M N FF-S FF-Error M % % % % % % % % % % % % % % % % % % % % % % % onclusion his work has presented an alternative to the ransission Siulator ethod in which the S is odeled with a raig- Bapton odel. his is expected to produce a ore accurate odel of the S in applications where the S is rigidly connected to the subcoponents of interest. he ethod was validated by applying it to a few siple bea structures and the results showed that this new ethod did indeed give ore accurate estiates of the assebly s natural frequencies. However, if one s goal was to estiate the free odes of the structure of interest (e.g. if it will not be assebled to a rigid coponent after reoving the transission siulator) then the B ethod was soewhat less accurate than the authors previous approach, which used free-interface odes to odel the S. he last application presented here explored a potentially ill-conditioned proble of reoving a S that connects to the subcoponent of interest at two points. In this case, one already requires six odes for the constraint odes of the B odel; if the interface was odeled with ore points then it would alost certainly be necessary to reduce the interface in soe way or there ight be ore S odes than there are in syste aking this subtraction ipossible. In this regard, the traditional FF-S ethod is convenient as it siply uses the free odes of the transission siulator to approxiate the interface so this does not becoe an issue unless the syste is such that too any free odes are required to adequately odel the S. llen, Kaer & Mayes IM, Feb. 214 Page 14 of 1

15 cknowledgeents his work was supported by Sandia National Laboratories. Notice: his anuscript has been authored by Sandia orporation under ontract No. DE-4-94L8 with the U.S. Departent of Energy. he United States Governent retains and the publisher, by accepting the article for publication, acknowledges that the United States Governent retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published for of this anuscript, or allow others to do so, for United States Governent purposes. References [1] D. de Klerk, D. J. Rixen, and S. N. Vooreeren, "General fraework for dynaic substructuring: History, review, and classification of techniques," I Journal, vol. 46, pp , 28. [2] M. S. llen and R. L. Mayes, "oparison of FRF and Modal Methods for obining Experiental and nalytical Substructures," presented at the 2th International Modal nalysis onference (IM XXV), Orlando, Florida, 27. [3] R. L. Mayes and E.. Stasiunas, "Lightly Daped Experiental Substructures for obining with nalytical Substructures," presented at the 2th International Modal nalysis onference (IM XXV), Orlando, Florida, 27. [4] M. S. llen, R. L. Mayes, and E. J. Bergan, "Experiental Modal Substructuring to ouple and Uncouple Substructures with Flexible Fixtures and Multi-point onnections," Journal of Sound and Vibration, vol. 329, pp , 21. [] M. S. llen, D.. Kaer, and R. L. Mayes, "Metrics for Diagnosing Negative Mass and Stiffness when Uncoupling Experiental and nalytical Substructures," Journal of Sound and Vibration, vol. 331, pp , 212. [6] R. L. Mayes, M. S. llen, and D.. Kaer, "orrecting indefinite ass atrices due to substructure uncoupling," Journal of Sound and Vibration, vol. 332, pp , 213. [7] R. L. Mayes and M. rviso, "Design Studies for the ransission Siulator Method of Experiental Dynaic Substructuring," presented at the International Seinar on Modal nalysis (ISM21), Lueven, Belgiu, 21. [8] R. L. Mayes and M. R. Ross, "dvanceents in hybrid dynaic odels cobining experiental and finite eleent substructures," Mechanical Systes and Signal Processing, vol. 31, pp. 6-66, 212. [9] R. R. raig and M... Bapton, "oupling of Substructures for Dynaic nalysis," I Journal, vol. 6, pp , [1] D.. Kaer, M. S. llen, and R. L. Mayes, "Forulation of a raig-bapton Experiental Substructure Using a ransission Siulator," presented at the 31st International Modal nalysis onference (IM XXXI), Garden Grove,, 213. [11] R. R. J. raig, Structural Dynaics. New York: John Wiley and Sons, [12] P. Ind, "he Non-Intrusive Modal esting of Delicate and ritical Structures," PhD Ph.D., Iperial ollege of Science, echnology & Medicine, University of London, London, 24. llen, Kaer & Mayes IM, Feb. 214 Page 1 of 1