VIEWS: 77 PAGES: 28 POSTED ON: 5/16/2011
A MIXED-BOUNDARY METHOD IN COMPONENT-MODE SYNTHESIS THE 2002 S/C & L/V DYNAMICS ENVIRONMENTS WORKSHOP Arya Majed, Ph.D. Applied Structural Dynamics (ASD) Ed Henkel Associate Tech. Fellow Boeing Company A PPLIED S TRUCTURAL D YNAMICS Objectives • Present a mixed-boundary method of Component-Mode Synthesis (CMS) that avoid the problem of rank-deficiency associated with Hintz’s mixed-boundary CMS method • Show that the mixed-boundary CMS method used in MSC NASTRAN (referred to in the briefing as the modified Hintz method) is directly derivable from Hintz’s ‘method of constraint modes’ • Examine the characteristics of the proposed and modified Hintz mixed-boundary CMS methods for the bounding cases of all fixed and all free boundaries A PPLIED S TRUCTURAL 3 D YNAMICS Objectives • Examine the cause for the error associated with the modified Hintz CMS method for the case of all free boundary • Examine the consequence of correcting the modified Hintz CMS method for the general case of mixed boundary A PPLIED S TRUCTURAL 4 D YNAMICS Abstract (paper to be published in AIAA J.) • A mixed-boundary Component-Mode Synthesis (CMS) method is presented1. Component generalized coordinates are defined using normal modes, constraint modes, and residual flexibility. The normal modes may be computed with any set of boundary conditions. The method is a generalization of both Hurty’s and Rubin’s methods and reduces to Hurty’s for all fixed-boundary and Rubin’s for all free- boundary CMS. Unlike the mixed-boundary CMS method of Hintz, the present mixed-boundary method is formulated such that linear independence of different mode sets is guaranteed. (1) The authors have derived this method from basic notions and thought-experiments. A literature search on the methods of CMS has been conducted and given in the reference section. A review paper on the existing methods of CMS is given in [ref. 11]. The authors have derived the present method a second way from Hintz’s ‘method of constraint modes’ [ref. 9] by (a) replacing the inertia-relief attachment mode set with residual flexibility (not inertia-relief) and (b) transforming to a final set of physical and generalized coordinates. This ‘mechanical’ way of deriving the proposed method may have been done by other authors. A PPLIED S TRUCTURAL 5 D YNAMICS Historical Review • Hurty [ref. 1] developed the first method in CMS for analysis of components with redundant boundaries. Component generalized coordinates were defined using fixed-boundary normal modes, rigid-body modes, and redundant boundary constraint modes. • Craig and Bampton [ref. 2] and Bajan and Feng [ref. 3] noted that Hurty’s method may be simplified by considering rigid-body modes a special case of constraint modes leading to the well- known Craig-Bampton method – The value-added of such a simplification is unclear. The case of the under-determinate boundary set becomes physically unrealizable in the Craig-Bampton method while perfectly treatable in Hurty’s method that uses both rigid-body coordinates as well as redundant boundary constraint modes. A PPLIED S TRUCTURAL 6 D YNAMICS Historical Review – continued • Goldman [ref. 4] and Hou [ref. 5] developed methods which use free-boundary normal modes • Benfield and Hruda [ref. 6] develop the concept of the loaded component boundary – Component mode set no longer independent of the inertial and stiffness properties of adjacent components – a draw-back of Benfield-Hruda method • MacNeal [ref. 7] developed the first free-boundary method with residual flexibility correction • Rubin [ref. 8] modified MacNeal’s method by adding residual inertia effects A PPLIED S TRUCTURAL 7 D YNAMICS Historical Review – continued • Hintz [ref. 9] developed the first mixed-boundary CMS method (combination of fixed and free boundary degrees of freedom). Hintz developed two methods: – Method of constraint modes – Method of attachment modes – The two methods are exactly equivalent since both mode sets span the same space • Herting [ref. 10] implemented a modification of Hintz’s method of constraint modes mixed-boundary CMS as a solution sequence in MSC NASTRAN (Superelement C-set method). The modification circumvents rank-deficiency problems in Hintz’s method by removing the linearly dependent constraint modes from the normal modes. A PPLIED S TRUCTURAL 8 D YNAMICS Historical Review – Summary • Pioneering works in CMS: – Hurty – first fixed-boundary CMS method – MacNeal – first free-boundary CMS method – Hintz – first mixed-boundary CMS method • All other methods of CMS are variations of the above original contributions A PPLIED S TRUCTURAL 9 D YNAMICS Hintz Mixed-Boundary CMS Method – The Method of Constraint Modes • Hintz [ref. 9] developed the first mixed-boundary CMS method published in literature – In the method of constraint modes, Hintz defines a statically complete ‘interface mode set’ using a combination of rigid-body modes, redundant constraint modes (displacements due to unit displacement at the redundant boundary) and inertia-relief attachment modes (displacements due to unit rigid-body accelerations with the boundary fixed) – Hintz warns that once this statically complete interface mode set is augmented with a large fraction of component normal modes, these different modes sets may no longer be linearly independent • Other researchers have observed that so long as the truncated set of normal modes is only a small fraction of the total flexible modes, it may be possible to remove the contribution of the normal modes from the other mode sets and form a linearly independent basis A PPLIED S TRUCTURAL 10 D YNAMICS Hintz Mixed-Boundary CMS Method – The Method of Constraint Modes continued u = φ r φ c φ i φ n q Notes: 1. The superscripts r, c, i and n denote rigid-body modes, constraint (redundant) modes, inertia-relief attachment modes and normal modes, respectively. 2. Care must be taken if this method in the above form is to be used as a computational method. Rank-deficient generalized stiffness can result due to the linear dependence of these different mode sets. 3. Hintz’s CMS method, without further modifications, is of more a symbolic rather than practical (computational) value due to the problem of rank-deficiency of resulting generalized matrices. A PPLIED S TRUCTURAL 11 D YNAMICS Modified Hintz Mixed-Boundary CMS Method • Start with the Hintz transformation partitioned into the t-set (t = b + c, where b is fixed and c is free in the component eigenvalue problem) and i-set representing the interior degrees of freedom: ut \ I \ φ t n qt = c n u i φit φ i q • As it stands, the transformation may lead to a rank-deficient problem, so … A PPLIED S TRUCTURAL 12 D YNAMICS Modified Hintz Mixed-Boundary CMS Method continued • The constraint modes are subtracted from the normal modes to ensure linear independence: ut \ I \ 0 ut = c c n u i φit φi − φ it φ t q n A PPLIED S TRUCTURAL 13 D YNAMICS Modified Hintz Mixed-Boundary CMS Method continued • Expand the t-set into the b-set and c-set to achieve the Modified-Hintz Mixed-Boundary CMS method coordinate transformation: ub \ I \ ub uc = \ I\ u c u φ c φ ic c n φ i − φ ic φ c q n c i ib Notes: 1. Note that we are not accounting for inertia-relief attachment modes in the above derivation. The consequence will be examined later. 2. This transformation (the modified Hintz CMS) is implemented in MSC NASTRAN mixed-boundary CMS formulation. A PPLIED S TRUCTURAL 14 D YNAMICS Hintz and Modified Hintz Mixed-Boundary CMS Method - Summary • The Hintz mixed-boundary CMS method can lead to a rank- deficient problem and is therefore more of a symbolic value rather than a computational method of CMS • The modified Hintz mixed-boundary CMS method circumvents this rank-deficiency of Hintz’s method by subtracting the linear dependent constraint modes from the normal modes as shown in slide 14 A PPLIED S TRUCTURAL 15 D YNAMICS Proposed Mixed-Boundary CMS Method - continued • Some advantages of the CMS method not dictating the boundary conditions for the computation of normal modes are: – Component may be represented with a minimum number of generalized dof if the boundary conditions on the normal modes are the same as for the coupled component in the operating environment – Truncation criterion for convergence will be simple to define – A solution to the problem of highly over-constrained Hurty (Craig- Bampton) component – Component reduction may employ correlated normal modes with modal test boundary conditions • As a consequence, component damping may be defined in a simple manner, consistent to test A PPLIED S TRUCTURAL 17 D YNAMICS Proposed Mixed-Boundary CMS Method – continued • A generalization of both Hurty’s and Rubin’s CMS methods – Hurty’s and Rubin’s methods are special cases of the proposed mixed-boundary method (speaks to the soundness of the development) • Highly accurate – See numerical study A PPLIED S TRUCTURAL 18 D YNAMICS Proposed Mixed-Boundary CMS Method – continued ub ub \ I\ uc = \ I\ u c u φ c − G R G R −1φ c R R −1 R −1 n φi − Gic Gcc φ c q n R i ib ic cc cb Gic Gcc Notes: R R 1. Residual Flexibility matrices G ic and G cc are displacements due to unit forces on the c-set minus the retained flexibility of the normal modes. 2. For an under-determinate b-set, the residual flexibility calculations utilize inertia-relief. A PPLIED S TRUCTURAL 19 D YNAMICS Numerical Study • Part 1 of the study compares the accuracy of the two mixed- boundary methods (modified Hintz and the proposed method) relative to Hurty’s and Rubin’s method for the normal mode boundary conditions of all fixed (Hurty) and all free (Rubin) – For this part of the exercise, the two mixed-boundary methods will use identical sets of normal modes as used in Hurty’s and Rubin’s methods to ensure a consistent comparison basis • Part 2 studies the accuracy and convergence of the proposed method for two different sets of mixed boundary conditions and retained component modes A PPLIED S TRUCTURAL 20 D YNAMICS Benfield Truss Problem • Classical problem to gauge the accuracy of CMS methods – Each joint has 2 in-plane dof – Interface is 6 dof (3 dof redundant) – Combined system is 60 dof 6 4 5 3 5 4 3 2 1 4 EQUAL BAYS 5 EQUAL BAYS COMPONENT B COMPONENT A 30 dof 36 dof A PPLIED S TRUCTURAL 21 D YNAMICS Numerical Study – Part 2 Table 2 - Percent frequency error and convergence for the proposed method with 2 different sets of mixed boundary conditions using 24 and 30 dof Elastic Boundary dof Boundary dof Mode 1-3 fixed 1-4 fixed Number 4-6 free 5-6 free 24 dof 30 dof 24 dof 30 dof 1 0.000284 0.000106 0.000801 0.000500 2 0.000777 0.000312 0.000695 0.000299 3 0.015952 0.005636 0.017492 0.005926 4 0.005645 0.001481 0.006613 0.003850 5 0.012911 0.005295 0.010797 0.004766 Notes: 6 0.010992 0.002571 0.008063 0.003651 • 24 dof coupled system 7 0.722158 0.250560 0.760019 0.322906 is 11 normal modes for 8 0.059582 0.015813 0.064663 0.034954 component A plus 7 normal 9 0.255054 0.084566 0.245404 0.110280 modes for component B plus 10 0.132848 0.031005 0.098563 0.036602 6 boundary dof 11 0.142715 0.033808 0.135268 0.063455 12 0.474264 0.036441 0.360215 0.043915 • 30 dof coupled system 13 0.840657 0.117165 0.835679 0.115787 is 14 normal modes for 14 1.196414 0.312854 1.128539 0.336670 component A plus 10 normal 15 6.354274 0.865436 7.361097 1.005541 16 5.139335 0.399398 4.291459 0.609718 modes for component B plus 17 10.549672 0.257136 4.958601 0.350082 6 boundary dof 18 16.157189 0.309611 15.154960 0.339006 19 24.732812 1.641815 24.565258 1.431521 20 34.414436 1.448679 33.079856 0.902953 A PPLIED 21 93.695742 1.648637 93.625911 1.899394 S TRUCTURAL 23 D YNAMICS Concluding Remarks • A mixed-boundary CMS method was presented that employs residual flexibility and a subset of constraint modes that are guaranteed to be linearly independent relative to the kept normal modes • As is the case of the Hintz mixed-boundary CMS method, the proposed method allows for any boundary conditions for the calculation of the normal modes – Circumvents truncation issues related to highly over-constrained Hurty (Craig-Bampton) type components • The proposed method is a generalization of both Hurty’s and Rubin’s methods, reducing to the former for the all fixed boundary case and the latter for the all free boundary case • Numerical example shows excellent accuracy and convergence characteristics A PPLIED S TRUCTURAL 26 D YNAMICS References – continued 9. Hintz, R. M., “Analytical Methods in Component Modal Synthesis”, AIAA J., vol. 13, no. 8, Aug. 1975, pp. 1007-1016. 10. Herting, D. N., “A General Purpose, Multi-Stage, Component-Mode Synthesis Method”, presented at AIAA/ASME 20th Structure Dynamics Conference, 1979. 11. Craig, R. R., Jr., and Chang, C. J., “A Review of Substructure Coupling Methods for Dynamic Analysis”, 13th Annual Meeting, Soc. For Eng. Sci., Advances in Engineering Science, 2, NASA CP-2001, 393-408, 1976. A PPLIED S TRUCTURAL 28 D YNAMICS
"A MIXED-BOUNDARY METHOD IN COMPONENT-MODE SYNTHESIS"