Arya Majed, Ph.D.
                                  Applied Structural Dynamics (ASD)

                                  Ed Henkel
                                  Associate Tech. Fellow
                                  Boeing Company

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• 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
• Examine the characteristics of the proposed and modified Hintz
  mixed-boundary CMS methods for the bounding cases of all
  fixed and all free boundaries

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• 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

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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

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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.

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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

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 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.

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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

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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

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Hintz Mixed-Boundary CMS Method –
The Method of Constraint Modes

      u = φ               r
                                     φ    c
                                                   φ    i
                                                                 φ   n
 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.

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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 …

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Modified Hintz Mixed-Boundary CMS Method

• 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 

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Modified Hintz Mixed-Boundary CMS Method

• Expand the t-set into the b-set and c-set to achieve the
  Modified-Hintz Mixed-Boundary CMS method coordinate

     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                                                       

       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.
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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

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Proposed Mixed-Boundary CMS Method -

• 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
   – 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

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Proposed Mixed-Boundary CMS Method –

• 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
• Highly accurate
   – See numerical study

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Proposed Mixed-Boundary CMS Method –

ub                                                                                          ub 
                                                                                            
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                                       

                                             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.

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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

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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

             3 5



                                                4 EQUAL BAYS
       5 EQUAL BAYS
                                                 COMPONENT B
                                                   30 dof
          36 dof
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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
                    Boundary dof             Boundary dof
                      1-3 fixed                1-4 fixed
                      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
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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

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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.

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