Assembling Reduced-Order Substructural Models

Transcription

1 . 21 Assembling Reduced-Order Substructural Models 21 1

2 Chapter 21: ASSEMBLNG REDUCED-ORDER SUBSTRUCTURAL MODELS NTRODUCTON The previous chapter has devoted to the reduced-order modeling o a single vibrating substructure. Once all o the substructures in a total system are approximated by their corresponding reduced-order models, the next task is to assemble the reduced-order substructural models. Third, the assembled reduced-order total system is either used or perormance evaluation and/or design improvements. n practice, the size o the assembled total structural model that consists o reduced-order substructural models is oten considdred too large. For such a case, it is customary to carry out additional reduction via total system modal analysis. Figure 21.1 illustrates the sequence o model development in large-scale vibrating structural systems. Total structural system Partition into substructures Substructure 1 Substructure Substructure s-1 Substructure s Reduce each substructure Reduced-Order Substructure-1 Model Reduced-Order Substructure-2 Model... Reduced-Order Substructure-s Model Assemble into Totoal Reduced- Order System Reduced-Order Total structural system Model development, Perormance analysis, Shock isolation design, Model updates,... Control synthesis Fig Sequence o Reduced-Order Modeling and Applications n the ollowing, the variational ormulation o partitioned equations o motion will be discussed irst. n particular, two treatment o interace constraints will be discussed: classical (or global) λ-method (which reads as Lagrange multiplier method), and localized λ-method. The partitioned equations o motion employing the two λ-methods are then derived. we will then ocus on one o the most widely used component mode synthesis method, the Craig-Bampton method. Finally, a component mode synthesis technique based on the localized λ-methoid will be described VARATONAL FORMULATON OF PARTTONED STRUCTURAL SYSTEMS Consider a structure that consists o two substructures as shown in Fig When the structure is partitioned into two structures, and, interactions orces, λ and λ (or λ (12) ), are developed along the interace boundaries o and. n addition, the displacement or substructure 1 consists o the interior ones u and along the partition boundary u. Similarly, or substructure 2 we have u and u. These can be expressed as u [ u u, u [ u u Using these notations, the energy unctionals or substructures 1 and 2 may be written as 21 2 (21.1)

3 VARATONAL FORMULATON OF PARTTONED STRUCTURAL SYSTEMS (a) Total Structural System Partition (12) u λ 1 2 u Substructure 1 Substructure 2 Constraint: c 12 u u (b) Partition modeled by classical λ-method u Substructure 1 Constraints: λ 1 u 2 (c) Partition modeled by localized λ-method λ c 1 u u c 2 u u u Substructure 2 Fig Partitioning o a Structure into Two Substructures Substructure 1: δ (δu ) T {K u ( M ü )} Substructure 2: δ (δu ) T {K u ( M ü )} (21.2) where M and K are mass and stiness matrix, resprectively, or a substructure, and the superscripts, (1, 2), denote substructure. While the virtual energy is completely contained in the preceding energy expressions, the interace conditions between these two substructures are needed or partitioning as well as assembly. The kinematic interace compatibility conditionmay be described inone o the two possible ways: Classical (or Global) orm: u u Localized orm: [ u u [ L L u (21.3) which states that the interace displacement along substructure 1, u, must be equal to that o substructure 2, u ; and, L is the interace displacement displacement operator. The constraint unctional that incorporates the above constraints canbe expressed as 21 3

4 Chapter 21: ASSEMBLNG REDUCED-ORDER SUBSTRUCTURAL MODELS 21 4 Classical (or Global) orm: π classical (λ (12) ) T (u Localized orm: [ λ π localized λ u ) T u {[ u [ L L u } (21.4) Finally, the total energy unctional is simply the sume o two substructural energy expressions, (21.2), plus one o the the constraint unctionals, (21.4): Classical interace orm: Localized interace orm: δ system δ + δ + δπ classical (21.5) δ system δ + δ + δπ localized We now derive the partitioned equations o motion or the two interace treatment cases PARTTONED EQUATONS OF MOTON EMPLOYNG CLASSCAL λ-method The total energy o the system or this case which is the one given by the irst o (21.5), consists o the two substructural energy expressions (21.2) plus the interace constraint unctional, viz., the irst expression in (21.4), as δ total (δu ) T {K u ( M ü )} + (δu ) T {K u ( M ü )} + (δλ (12) ) T (u u ) + (δu δu )T λ (12) (δu ) T {K u ( M ü ) + (B ) T u } + (δu ) T {K u ( M ü ) (B ) T u } + (δλ (12) ) T (B u B u ) (21.6) u B u, u B u where B (k) is the Boolean matrix that extracts the interace degrees o reedom at the interace o substructure k. The stationarity o the above variational equation, viz., δ total, yields the ollowing partitioned equations o motion: 21 4

5 PARTTONED EQUATONS OF MOTON EMPLOYNG CLASSCAL λ-method [ M M [ [ ü K ü + λ (12) (B ) T K (B ) T B B [ u λ (12) [ u [ [ [ M ü K B T [ cl u + λ (12) B cl λ (12) [ (21.7) n the above equation set, the irst row is the equations o motion or substructure 1, the second row or substructure 2, and the third is the interace constraint equation. To illustrate the compositions o the partitioned equation urther, we express each o the three equations in terms o the interior degrees o reedom, u, and the interace degrees o reedom, u as ollows: For substructure 1: [ M M M M [ [ ü K K [ [ u + ü K K u λ(12) (21.8) For substructure 2: [ M M M M [ [ ü K K [ [ u + ü K K u + λ(12) (21.9) Ω Ω Partitioning Ω u u u λ λ u Ω Fig Partitioning o a Structure into Two Substructures one is interested in constructing the equations o motion or the entire system, then all one has to do is to assemble the coeicient matrices that are associated with u and u into the same rows and the columns. This corresponds to an explicit enorcement o the second o the above constraint condition, e.g., u u u. The assembled equations o motion thereore becomes M M ü K K u M M + M M ü + K K + K K u (21.1) M M ü K K 21 5 u

6 Chapter 21: ASSEMBLNG REDUCED-ORDER SUBSTRUCTURAL MODELS 21 6 This assembly process is precisely the assembly procedure o a typical inite element sotware system, except it is repeated several hundreds or thousands times. The modes and mode shapes o the total system can, in principle, be extracted rom the eigenproblem associated with the above assembled equations o motion PARTTONED EQUATONS OF MOTON EMPLOYNG LOCALZED λ-method The total energy o the system or this case which is the one given by the second o (21.5), consists o the two substructural energy expressions (21.2) plus the interace constraint unctional, viz., the second expression in (21.4), as δ total (δu ) T {K u ( M ü )} + (δu ) T {K u ( M ü )} [ δλ T u [ + {[ δλ L u L u } [ u [ L [ +{δ u L δu } T λ λ (δu ) T {K u ( M ü ) + (B ) T λ } + (δu ) T {K u ( M ü ) + (B ) T λ } [ δλ T [ B [ u + δλ { L B u L u } [ L T [ δu T λ L λ u B u, u B u (21.11) The stationarity o the above variational equation, viz., δ, yields the ollowing partitioned equations o motion: M M [ [ M ü λl + ü ü ü λ λ ü + [ K B T l B l L L T K (B ) T K (B ) T B L B L (L ) T (L ) T [ u λl u [ u u λ λ u (21.12) 21 6

7 EGENVALUE PROBLEM USNG REDUCED-ORDER PARTTONED EQUATONS OF MOTON 21.5 EGENVALUE PROBLEM USNG REDUCED-ORDER PARTTONED EQUATONS OF MOTON Model reduction o a substructure has been presented in the previous chapter. n this section we will use the reduction procedure based on the constrained interace modes or assembling the substructures into a total system. To this end, let s express the reduction orm or substructure k as u (k) (k) q (k) (21.13) where u (k) is approximated rom the previous chapter as: [ [ u K 1 u (k) K u [ q u (2.6) Substituting the above reduction ormula into (21.7), one obtains [ M q [ K (B ) T M q + K (B ) T λ (12) B B [ p q λ (12) p [ q M ( ) T M, M ( ) T M K ( ) T K, K ( ) T K B ( ) T B, B ( ) T B p ( ) T, p ( ) T (21.14) Vibration analysis o the total system based on the above reduced-order model is carried using the ollowing equation: ˆK total ˆΨ total ˆM total ˆΨ total Λ total ˆK total ˆM total [ M M [ K (B ) T K (B ) T B B (21.15) t is noted that ˆ total is not the eigenvectors o the assembled model. The correct eigenvectors (mode shapes) o the assembled model are obtained by [ u [ [ q u q λ 12 λ 12 [ ˆΨ total ˆq total (21.16) [ Ψ total ˆΨ total 21 7

8 Chapter 21: ASSEMBLNG REDUCED-ORDER SUBSTRUCTURAL MODELS 21 8 which includes not only the modes that span the substructures but also the interace modes pertaining to the interace orce λ (12). However, the eigenvalues Λ total represent the assembled structural system. n other words, (Ψ total, Λ total ) constitute the mode shapes and modes o the assembled system even though we have obtained them rom the partitioned equations o motion EGENVALUE PROBLEM USNG REDUCED-ORDER ASSEMBLED EQUATONS OF MOTON n the preceding section the reduced-order partitioned equations o motion has been directly utilized or the ormulation o eigenvalue problem. While computationally equivalent, the resulting eigenvalue problem given by (21.15) involves non-deinite matrices, thus requiring a special care. One way to circumvent the non-deinite matrices is to assemble the partitioned equations o motion into the assembled orm akin to the equation given in (21.1). This can be accomplished in the ollowing way. First, we note that the constraint condition u u (21.3) implies that the interace Boolean matrices B and B yield the ollowing relation: [ B u [ u u [ B u [ u u (21.17) This means that the assembled degrees o reedom, (q, u, q ), can be related to the partitioned degrees o reedom, (q, u, q, u ),according to q u q u q q u q part L a q a (21.18) where the superscripts, (part, a), denote the partitioned and assembled degrees o reedom, and L a is an assembly Boolean matrix. Thereore, teh complete transormation relation can be expressed as q u q u λ (12) [ q part λ (12) [ [ T total q a λ (12), T total L a (12) (21.19) Substituting the above assembly transormation into (21.14) and ater some simpliications, one arrives at 21 8

9 THE CRAG-BAMPTON METHOD the ollowing reduced-order assembled equations o motion: M ˆq + K ˆq ˆp [ M (L a ) T M M L a [ K (L a ) T K K L a (21.2) [ ˆp (L a ) T p p [ ˆq (L a ) T q q As one can see, the above reduced-order assembled equations o motion is diicult to ollow through. We will examine a step-by-step derivation o the above equation below, which is konwn in the literature as the Craig-Bampton component mode synthesis or subtructuring method method THE CRAG-BAMPTON METHOD Equation(21.2) may be considered a generic component mode synthesis as it can accommodate several possible substructural reduction methods. One o its specializations was proposed by R.R Craig M.C.C. Bampton in As the Craig-Bampton component mode synthesis technique is perhaps the most widely used substructuring method, we present a step-by-step ormulation o their method below Step 1: Approximate the substructural displacements t approximates the displacement o each substructure by a set o ixed-interace normal modes plus a set o constraint modes. Speciically, or substructure 1, u is approximated by (see Eq. (2.6)) u [ u u [ φ ψ [ q Similarly, the substructural displacement u or substructure is approximated by u [ u u [ φ ψ u [ q u, ψ (K ) 1 K (21.21), ψ (K ) 1 K (21.22) 21 9

10 Chapter 21: ASSEMBLNG REDUCED-ORDER SUBSTRUCTURAL MODELS Step 2: Obtain the approximate substructural kinetic and strain energy The approximate substructural strain energy and kinetic energy derived in the previous chapter (see 2.13) and (2.14)) are restated below. U 1 2 (u ) T K u T 1 2 u M u [ q [ q u u T [ T [ T [ K K K K T [ M M M M [ [ [ q [ q u u (21.23) The reduced-order mass or substructure 1, M, is thus given by T [ M M M [ M M where M M M T [ M M M M T M T (M (due to massnormalization) + M ) T (M + M ) + M + M The reduced-order stiness or substructure 1, K A, is thus given by where K K T K Λ [ K K K K ω ω 2 (21.24) (21.25) K T (K K K + + K ) T (K + K ) + K K K (K ) 1 K T (K K (K ) 1 K ) Observe that K can be written as K [ Λ K 21 1 (21.26)

11 THE CRAG-BAMPTON METHOD which consists o the diagonal interior substructural modes and the Guyan-reduced matrix K. For substructure 2, a similar procedure employed or substructure A can be repeated to yield: M K [ M M M M [ Λ K (21.27) where it is understood that the substructural displacement u is approximated by the ixed-interace interior modes plus the constrained modes given by (2.6) Step 3: Sum up the substructural kinetic and strain energy expressions The strain energy and the kinetic energy o the total structure can be approximated by T T + T U U + U [ q T [ U 1 Λ 2 T 1 2 U 1 2 T 1 2 u [ q u [ q u [ q u K T [ M M M M T [ Λ K T [ M M M M [ q u [ q [ q u u [ q u (21.28) Step 4: Derive the reduced equations o motion or the total system The Lagrangian o the total system is given by L T U + (λ (12) ) T (u u ) (21.29) where the last term involving the Lagrange multiplier λ (12) is introduced to enorce the interace displacement compatibility constraint u u (21.3) The equations o motion or ree vibration (, ) can be derived rom (21.29) as 21 11

12 Chapter 21: ASSEMBLNG REDUCED-ORDER SUBSTRUCTURAL MODELS [ M [ q M q B [ q q, q u C T [ [ K + [ q [ q K q C T λ (12) u (21.31) desired, the interace orce λ (12) can be eliminated by expressing the interace displacement u in terms o u or vice versa. This can be accomplished by the reduction q u q u La q q u, L a, u u u (21.32) Substituting (21.32) into (21.31) and premultiplying the resulting equation by (L a ) T we obtain the ollowing reduced-order ree vibration equation: M q + K q, q q q u M M M M M M M M, M M + M (21.33) K Λ Λ, K K + K K Step 5: Perorm eigenanalysis o the total system First, we perorm an eigenanalysis o (21.33): K Φ M ΦΛ g, Φ Φ Φ (21.34) Φ Second, once (Φ, Λ g ) are obtained, the global eigenvector Φ g is obtained by the ollowing expression 21 12

13 THE CRAG-BAMPTON METHOD Φ g φ Φ Φ (21.35) Observe that the eigenvalues are preserved under a similarity transormation. Thus, the global eigenvalues and eigenvector pairs are given by (Φ g, λ g ). The component mode synthesis or other techniques due to Benield and Hruda, Hurty, MacNeal, Rubin, Hintz, Dowell and Klein, and Craig and Chang may be similarly constructed. Φ Remark 1: Note that rom (21.33) equation, in carrying out the component mode synthesis by the Craig- Bampton method, viz., ω 2 M K (21.36) the mass matrix M becomes dense even i the original substructural-level mass matrices are diagonal. Remakr 2: The stiness matrix at the interace is given by K K + K K K K (K ) 1 K K K K (K ) 1 K (21.37) Note that both K and K are the Schur complements (or in structural mechanics known as Guyanreduced matrices) that preserve the strain energy content o each substructure. Hence, no approximation is introduced at the interace strain energy contents. On the other hand, the same cannot be said regarding the kinetic energy. This can be seen by examing the interace mass traix M : M M + M M T (M M + M ) + M + M T (M + M ) + M + M (21.38) n other words, the constraint modes play the role o augmenting the interace kinetic energy by inusing the interior masses M unto the interace nodes

14 Chapter 21: ASSEMBLNG REDUCED-ORDER SUBSTRUCTURAL MODELS Reerences 1. Craig, Jr., R.R., Structural Dynamics: An ntroduction to Computer Methods, John Wiley & Sons (1981). 2. Hurty, W.C., Dynamic Analysis o Structural Systems Using Component Modes, AAA Journal, v.3, (1965). 3. Craig, Jr., R.R. and M.C.C. Bampton, Coupling o Substructures or Dynamic Analysis, AAA Journal, v. 6, (1968). 4. MacNeal, R.H., A Hybrid Method o Component Mode Synthesis, Comp. and Struct., v. 1, (1971). 5. Benield, W.A. and R.F. Hruda, Vibration Analysis o Structures by Component Mode Substitution, AAA Journal, v. 9, (1971). 6. Rubin, S., mproved Component-Mode Representation or Structural Dynamic Analysis, AAA Journal, v. 13, (1975). 7. Klein, L.R. and E.H. Dowell, Analysis o Modal Damping by Component Modes Method Using Lagrange Multipliers, J. Appl. Mech. Trans. ASME, v. 41, (1974). 8. Hintz, R.M., Analytical Methods in Component Modal Synthesis, AAA Journal, v. 13, (1975). 9. Craig, Jr., R.R. and C-J. Chang, A review o Substructure Coupling Methods or Dynamic Analysis, NASA CP-21, National Aeronautics and Space Admin., Washington, DC, v. 2, (1976). 1. Craig, Jr., R.R., Methods o Component Mode Synthesis, Shock and Vib. Digest, Naval Research Lab., Washington, DC, v. 9, 3-1 (1977). 11. Craig, Jr., R.R. and C-J. Chang, On the Use o Attachment Modes in Substructure Coupling or Dynamic Analysis, Paper 77-45, AAA/ASME 18th Struct., Struct. Dyn, and Materials Con., San Diego, CA (1977). 12. M. Baruch, Optimization Procedure to Correct Stiness and Flexibility Matrices Using Vibration Tests, AAA J., 16 (11), (1978) 13. B. Caesar, Update and dentiication o Dynamic Mathematical Models, Proc. 1st ntl. Modal Anal. Con., (1983) 14. J.C. Chen, C.P. Kuo, and J.A. Garba, Direct Structural Parameter dentiication by Modal Test Results, AAA/ASME/ASCE/AMS Proc. 24th Struc. Dynam. and Materials Con., (1983) 15. J.D. Collins, G.C. Hart, T.K. Hasselman, and B. Kennedy, Statistical dentiication o Structures, AAA J., 12, (1974) 16. J.C. Chen and B.K. Wada, Criteria or Analysis-Test Correlation o Structural Dynamics Systems, J. Appl. Mech., (1975) 17. J.E. Mottershead, Theory or the Estimation o Structural Vibration Parameters rom ncomplete Data, AAA J., 28 (3), (199) 18. N.G. Creamer and J.C. Junkins, dentiication Method or Lightly Damped Structures, AAA J. Guidance, Control, and Dynamics, 11 (6), (1988) 19. A.M. Kabe, Stiness Matrix Adjustment Using Mode Data, AAA J., 23 (9), (1985) 21 14

15 THE CRAG-BAMPTON METHOD 2. H. Berger, R. Ohayon, L. Barthe, and J.P. Chaquin, Parametric Updating o FE Model Using Experimental Simulation: A Dynamic Reaction Approach, Proc. 8th ntl. Modal Anal. Con., (199) 21. S.R. brahim and A.A. Saaan, Correlation o Analysis and Test in Modeling o Structures, Assessment and Review, Proc. 5th ntl. Modal Anal. Con., (1987) 22. W. Heylen and P. Sas, Review o Model Optimization Techniques, Proc. 5th ntl. Modal Anal. Con., (1987) 23. R.J. Guyan, Reduction o Stiness and Mass Matrices, AAA J., 3, 38 (1965) 24. M. Paz, Dynamic Condensation Method, AAA J., 22 (5), (1984) 25. H.P. Gysin, Comparison o Expansion Methods o FE Modeling Error Localization, Proc. 8th ntl. Modal Anal. Con., (199) 26. Kennedy, C.C. and Pancu, C.D.P., Use o Vectors in Vibration Measurements and Analysis, J. Aero. Sci., Vol. 14(11), Nov. 1947, Lewis, R.C. and Wrisley, D.L., A System or the Excitation o Pure Natural Modes o Complex Structures, J. Aero.Sci., Vol 17(11), Nov. 195, , Ewins, D.J., Modal Testing: Theory and Practice, John Wiley and Sons, nc., New York,