CRAIG-BAMPTON METHOD FOR A TWO COMPONENT SYSTEM Revision C

Transcription

1 CRAIG-BAMPON MEHOD FOR A WO COMPONEN SYSEM Revision C By om Irvine tom@vibrationdata.com May, 03 Introduction he Craig-Bampton method is method for reducing the size of a finite element model, particularly where two or more subsystems are connected. It combines the motion of boundary points with modes of the subsystem assuming the boundary points are held fixed. he following tutorial provides an example for the Craig-Bampton fixed-interface method in Reference. Governing Equation of Motion he dynamic response of a system is modeled as M x K x F () where M K F x is the mass matrix is the stiffness matrix is the force vector is the displacement vector Matrix Partitioning he partitioned mass and stiffness matrices for each subsystem or component are respectively Mii M bi Mib M b b and Kii K bi Kib K bb he subscript i denotes an interior degree-of-freedom. he subscript b denotes an interface boundary degree-of-freedom.

2 Normal Modes he component fixed-interface normal modes are obtained by restraining all boundary degrees-offreedom and solving the generalized eigenvalue problem: K ii Mii i 0 j j () he complete set of N i fixed-interface (flexible) normal modes is ii. he assembled modal matrix is i N u x N i ii 0bi (3) Next, the modes are normalized so that Constraint Modes ii Miiii Iii (4) ii K ii ii ii diag (5) j A constraint mode is defined as the static deformation of a structure when a unit displacement is applied to one coordinate of specified set of constraint coordinates, C, while the remaining coordinates of that set are restrained, and the remaining degrees-of-freedom of the structure are force-free. he interface constraint mode matrix is calculated via Kii K bi Kib i b 0 ib K I b b bb R bb (6)

3 where ib is the interior partition of the constraint mode matrix R contains the reaction forces on the component due to its connection to adjacent components at boundary degrees-of-freedom he interface constraint mode matrix c is c N u x N b i b K ii Kib Ibb Ibb (7) Note that the constraint modes are stiffness-orthogonal to all of the fixed-interface normal modes, that is i K c 0 (8) he displacement transformation of the Craig-Bampton Method uses both fixed-interface normal modes and interface constraint modes. he physical coordinates u can be represented as u ui ik ib pk u b 0 I (9) bb pb where p k = interior generalized displacements p b = boundary generalized displacements ik = interior partition of the matrix of kept fixed-interface modes ib = interior partition of the constraint mode matrix 3

4 he Craig-Bampton transformation matrix is ik ib 0 I (0) bb Reduced Component Matrices he reduced component mass matrix for system s is Mˆ M s) ( () Mˆ ik 0 Mii Mib ik ib ib I bb Mbi M b b 0 I () bb Mˆ ik 0 Miiik Miiib Ibb Mib (3) M ib Ibb bi ik Mbi ib Ibb M b b Mˆ ik Miiik ik Miiib Ibb M ib (4) ib Mii Ibb Mbi ik Ibb Mbi ib Ibb M b b Mˆ Îkk Mˆ kb (5) Mˆ b k Mˆ b b 4

5 he reduced stiffness matrix for system s is Kˆ K s) ( (6) Kˆ ik 0 Kii Kib ik ib ib I bb Kbi K b b 0 I (7) bb Kˆ ik 0 Kiiik Kiiib Ibb Kib (8) K ib Ibb bi ik Kbi ib Ibb K b b Kˆ ik Kiiik ik Kiiib Ibb K ib (9) ib Kii Ibb Kbi ik Ibb Kbi ib Ibb K b b Again, i b Kii Kib (0) hus, the off-diagonal terms are each zero. Kˆ kk 0 kb () 0bk Kˆ b b 5

6 he reduced force vector for system s is Fˆ F Fi F () b Assembled Global Matrices he following assembled mass matrix is formed. Mˆ Î kk 0k k () Mˆ b k 0k Îk k k () Mˆ b k () Mˆ kb () Mˆ k b () () Mˆ b b Mˆ b b (3) Again, the subscript b denotes an interface boundary degrees-of-freedom. he numerical subscripts denote non-interface degrees-of-freedom. he following assembled stiffness matrix is formed. Kˆ () kk 0k k 0bk 0k k () k k 0bk 0 kb 0k b () () Kˆ b b Kˆ b b (4) Mˆ pk pk Kˆ Fˆ p b pb (5) 6

7 Examples Examples are given in Appendices A and B. References. R. Craig & A. Kurdila, Fundamentals of Structural Dynamics, Second Edition, Wiley, New Jersey, Irvine, Component Mode Synthesis, Fixed-Interface Model, Revision A, Vibrationdata, 00. 7

8 APPENDIX A Example m a,4 xa,4 m b,4 xb,4 ka,4 kb,3 m a,3 xa,3 m b,3 xb,3 ka,3 kb, ma, m a xa, ma, m b, xb, ka, kb, ma, m a xa, ma, m b, xb, ka, Figure A-. Form two separate models as an intermediate step. he system on the left represents a launch vehicle on a pad. he system on the right represents a spacecraft that is to be mounted on top of the launch vehicle. Note that mass m b, is to be connected to m a,4 via a rigid link. 8

9 he following values are used for the model. English units: stiffness (lbf/in), mass (lbf sec^/in) ka 900,000 ma 50 ka 600,000 ma 5 ka3 500,000 ma3 00 ka4 40,000 ma4 00 kb 00,000 mb 0 kb 90,000 mb 8 kb3 80,000 mb3 6 mb4 5 Complete Launch Vehicle & Spacecraft Model, Unreduced >> mass_stiffness_assembly mass_stiffness_assembly.m ver. Feb 6, 00 by om Irvine Assemble mass and stiffness matrices using transformation matrices. Enter total dof 7 Enter number of systems Enter system mass matrix name MLV Enter system stiffness matrix name KLV Enter system transformation matrix name ta Enter system mass matrix name MSC Enter system stiffness matrix name KSC Enter system transformation matrix name tb MG =

10 KG = Natural Frequencies (Hz) Modes Shapes (column format) ModeShapes =

11 he transformation matrices for the assembly were >> ta ta = >> tb tb = System A, Launch Vehicle, Matrix >> Craig_Bampton Craig_Bampton.m ver.0 April 30, 03 by om Irvine Enter the units system =English =metric Assume symmetric mass and stiffness matrices. Select input mass unit =lbm =lbf sec^/in stiffness unit = lbf/in Select file input method =file preloaded into Matlab =Excel file Mass Matrix

12 Enter the matrix name: massa Stiffness Matrix Enter the matrix name: stiffnessa he mass matrix is m = he stiffness matrix is k = Enter number of boundary dof Enter boundary dof : 4 ** Fixed Interface Flexible Natural Frequencies & Modes ** Natural Frequencies No. f(hz) Modes Shapes (column format) ModeShapes = Enter number of modes to keep 3

13 Craig-Bampton ransformation Matrix M = Partitioned Matrices m_partition = k_partition = ransformed matrices (reduced component matrices) mq = kq =.0e+05 * order vector 3

14 ngw = 3 4 System B, Spacecraft, Matrix >> Craig_Bampton Craig_Bampton.m ver.0 April 30, 03 by om Irvine Enter the units system =English =metric Assume symmetric mass and stiffness matrices. Select input mass unit =lbm =lbf sec^/in stiffness unit = lbf/in Select file input method =file preloaded into Matlab =Excel file Mass Matrix Enter the matrix name: massb Stiffness Matrix Enter the matrix name: stiffnessb he mass matrix is m = he stiffness matrix is k =

15 Enter number of boundary dof Enter boundary dof : ** Fixed Interface Flexible Natural Frequencies & Modes ** Natural Frequencies No. f(hz) Modes Shapes (column format) ModeShapes = Enter number of modes to keep 3 Craig-Bampton ransformation Matrix M = Partitioned Matrices m_partition = k_partition =

16 ransformed matrices (reduced component matrices) mq = kq =.0e+04 * order vector ngw = 3 4 Combined System >> mass_stiffness_assembly mass_stiffness_assembly.m ver. Feb 6, 00 by om Irvine Assemble mass and stiffness matrices using transformation matrices. Enter total dof 7 Enter number of systems Enter system mass matrix name mqa Enter system stiffness matrix name kqa Enter system transformation matrix name tqa Enter system mass matrix name mqb Enter system stiffness matrix name 6

17 kqb Enter system transformation matrix name tqb MG = KG =.0e+005 * Natural Frequencies (Hz) Modes Shapes (column format) ModeShapes =

18 he transformation matrices for the assembly were >> tqa tqa = >> tqb tqb = Note that the interface is set at degree-of-freedom number 7. Summary he natural frequencies match. able A-. Natural Frequencies Mode Full Model fn (Hz) Combined Systems fn (Hz)

19 APPENDIX B Example, Part II Repeat the example from Appendix A but only include the first fixed interface mode from the spacecraft. System A, Launch Vehicle, Matrix he matrices are the same as in Appendix A. System B, Spacecraft, Matrix >> Craig_Bampton Craig_Bampton.m ver.0 April 30, 03 by om Irvine Enter the units system =English =metric Assume symmetric mass and stiffness matrices. Select input mass unit =lbm =lbf sec^/in stiffness unit = lbf/in Select file input method =file preloaded into Matlab =Excel file Mass Matrix Enter the matrix name: massb Stiffness Matrix Enter the matrix name: stiffnessb 9

20 he mass matrix is m = he stiffness matrix is k = Enter number of boundary dof Enter boundary dof : ** Fixed Interface Flexible Natural Frequencies & Modes ** Natural Frequencies No. f(hz) Modes Shapes (column format) ModeShapes = Enter number of modes to keep Craig-Bampton ransformation Matrix M =

21 Partitioned Matrices m_partition = k_partition = ransformed matrices (reduced component matrices) mq = kq =.0e+03 * Combined System >> mass_stiffness_assembly mass_stiffness_assembly.m ver. Feb 6, 00 by om Irvine Assemble mass and stiffness matrices using transformation matrices. Enter total dof 5 Enter number of systems Enter system mass matrix name mqa

22 Enter system stiffness matrix name kqa Enter system transformation matrix name tqaa Enter system mass matrix name mqbb Enter system stiffness matrix name kqbb Enter system transformation matrix name tqbb MG = KG =.0e+05 * Natural Frequencies No. f(hz) Modes Shapes (column format) ModeShapes =

23 he transformation matrices for the assembly were >> tqaa tqaa = >> tqbb tqbb = Note that the interface is set at degree-of-freedom number 5. able B-. Natural Frequencies Mode Full Model fn (Hz) Combined Systems, with One Fixed-Interface Mode for the Spacecraft fn (Hz)