COMPONENT MODE SYNTHESIS, FIXED-INTERFACE MODEL Revision A

Transcription

1 COMPONEN MODE SYNHESS, FXED-NERFACE MODEL Revision A By o rvine Eail: toirvine@aol.co February, ntroduction Coponent ode synthesis is a ethod for analyzing the dynaic behavior of a syste consisting of an assebly of substructures. t is particularly useful for analyzing a syste consisting of a launch vehicle and a spacecraft. he fied-interface odel is an approach for ipleenting this coponent ode synthesis. Reference eplains: Fied-interface odeling is the ost coonly eployed technique for spacecraft. t is coonly viewed as capable of producing lower odes siilar to those occurring in the coupled syste because the spacecraft interface points are attached to a relatively stiff structure. Furtherore, the well established base-fied odal test approach is applicable for odel verification purposes. he resulting spacecraft odel is referred to as a Craig-Bapton odel. Derivation A spacecraft is used as the subsyste. he dynaic response of the spacecraft is odeled as where M & K F () M K F is the ass atri is the stiffness atri is the force vector is the displaceent vector he ass and stiffness coefficients are deterined by a finite eleent odel.

2 he physical coordinates are separated into a set of interface coordinates and a set of noninterface (interior) coordinates N. he atrices and vectors in equation () are then partitioned as M M M N K M K K N F N K F () N () ypically, the non-interface forces are zero, F N. An eception would be the case where direct acoustic ecitation of the spacecraft surface was considered. he generalized eigenvalue proble is where K Ωn M Φn () Ω n is the diagonal atri of circular natural frequencies Φ n is the atri of ass-noralized ode shapes or eigenvectors where Φ equal the atri of fied-interface odes Φ equal the atri of constraint odes he constraint odes ay be rigid-body vectors.

3 ransfor the physical coordinate as N q N () he Craig-Bapton transforation atri is () As an iportant note, the proble size can be greatly reduced by including only the first few odes in the subatri Φ. ransfor the stiffness atri. K K K (7) K K K K (8) K K K [ ] [ ] K K K K K K K (9) K K

4 ~ K () K () [ K K ] () K K () [ K ] K K K () Note that K Ω n () where Ω n is the eigenvalue atri for the fied-interface odal analysis. Let [ K K ] () [ K K ] (7) [ K K ] (8) K K (9)

5 hus [ K K ] () Again, K K () Substitute equation (9) into (). [ K ] K K K () K K K K () K K () () [ K ] K K K () Substitute equation (9) into (). K K (7) hus Ω n (8) K K

6 where K K (9) ransfor the ass atri. Let ~ M () ~ M () M M M [ ] [ ] M M M M M M () he eigenvectors are ass-noralized. hus M () he other ters are [ M M ] () M M () [ M ] M M M ()

7 he transfored load atri is ~ F F N F (7) ~ F N F F N F (8) n suary, ~ M (9) ~ K () he Craig-Bapton transforation atri is () where is the boundary ass atri is the dynaic coupling atri is the interface stiffness atri if the boundary point is a single grid. 7

8 8 he transfored equation of otion including daping is Ω Ω ξ N N N n N n n N F F F q q q M ~ & & () f the non-interface forces are zero, then Ω Ω ξ N n N n n N F q q q M ~ & & () hus, the fied-interface coponent ode approach results in coponent equations of otion, which have only inertial coupling between the fied-interface noral odes and the constraint odes. he net step is to asseble equations of otion for all substructures. his is done by adding the atrices in equation (). Note that the substructure atrices ust be epanded with zeroes to becoe conforable for addition. he asseble equation ay be solved by either the noral ode ethod or direct nuerical integration. he physical displaceents for each substructure are then found using equation (). References. NASA-HDBK-7, Dynaic Environental Criteria,.. Scott Gordon, he Craig Bapton Method, FEMC Presentation, 999.

9 APPENDX A Consider an undaped, hoogeneous proble. he rows and coluns are interchanged for copatibility with the author s software progras. ~ q N M q N Ω n (A-) he Craig-Bapton transforation atri is (A-) where K K (A-) Recall Φ equal the atri of fied-interface odes Φ equal the atri of constraint odes And Ω n is the eigenvalue atri for the fied-interface odal analysis. he generalized eigenvalue proble can be represented as ~ Ω n Φn (A-) 9

10 K K K K (A-) M M M M (A-) A saple syste is shown in Figure (A-). Assue that the botto four asses represent a launch vehicle, and that the two asses represent a spacecraft. Retain the launch vehicle s four degrees-of-freedo in the reduced odel. Restore the spacecraft degree-of-freedo nuber in the fied-interface odel.

11 Spacecraft nterface Point Launch vehicle a, a, Figure A-.

12 he following values are used for the odel. English units: stiffness (lbf/in), ass (lbf sec^/in),,,,,, he unreduced equation of otion is: (A-7)

13 he atrices are partition as follows: M M M K K K

14 >> dof_reduced_cb dof_reduced_cb. ver.8 February, by o rvine Eail: toirvine@aol.co his progra solves the following equation of otion: M (d^/dt^) K where M and K are reduced according to the users specification. he discarded dof ay be brought bac in using the Craig-Bapton ethod. Enter the units syste English etric Assue syetric ass and stiffness atrices. Select input ass unit lb lbf sec^/in stiffness unit lbf/in Select file input ethod file preloaded into Matlab Ecel file Mass Matri Enter the atri nae: ass Stiffness Matri Enter the atri nae: stiff he ass atri is he stiffness atri is

15 - - - Enter the nuber of degrees-of-freedo to retain: Enter retained dof nuber: Enter retained dof nuber: Enter retained dof nuber: Enter retained dof nuber: Partitioned Matrices _partition _partition Restore any odes via the CB ethod? yes no How any odes to restore? Enter restored dof nuber Natural Frequencies (Hz) Modes Shapes (colun forat) ModeShapes

16 ransforation Matri C ransfored atrices q q.e * Natural Frequencies (Hz) Modes Shapes (colun forat) MS_ordered

17 A natural frequency coparison is given in able A-. he results agree out to two decial places for the first five odes. able A-. Natural Frequencies Mode Full Model, dof fn (Hz) Model Reduced to dof restored dof fn (Hz)