Strength Analysis - Finite Element Analysis for Satellite Structures - page 85

• Assemble the simplified substructure mass and stiffness matrices to form a
coupled model
• Calculate the response of the assembled structure model to the applied forces
• Back-solve to obtain displacements, strains, and stresses in each substructure.

4.2.1 Substructuring Analysis

The substructure analysis uses the technique of matrix reduction to reduce the
system matrices to a smaller set of degrees of freedom DOFs. Matrix reduction is
also used by the reduced modal, harmonic, and transient analyses. A superelement
(substructure) may be used in any analysis type. It simply represents a collection of
elements that are reduced to act as one element. This one (super) element may then
be used in the actual analysis (use pass) or be used to generate more superelements
(generation or use pass). To reconstruct the detailed solutions (e.g., displacements
and stresses) within the superelement, an expansion pass may be done. The static
analysis solution method is valid for all DOFs). Inertial and damping effects are
ignored, except for static acceleration fields. The overall finite-element equilib-
rium equations for linear structural static analysis are:

½ K f u g¼f F g ð 4 : 1 Þ

Equation ( 4.1 ) may be partitioned into two groups; the master (retained) DOFs,
here denoted by the subscript ''m'', and the slave (removed) DOFs, here denoted
by the subscript ''s''.

u fg
u fg

½ K mm ½ K ms F fg

F fg

¼ ð 4 : 2 Þ

½ K sm ½ K ss

Expanding:

½ K mm u fgþ K ms ½ u fg¼ F fg ð 4 : 3 Þ

½ K sm u fgþ K ss ½ u fg¼ F fg ð 4 : 4 Þ

The master DOFs should include all DOFs of all nodes on surfaces that connect
to other parts of the structure. If accelerations are to be used in the use pass or if
the use pass is a transient analysis, master DOFs throughout the rest of the
structure should also be used to characterize the distributed mass. Solving Eq. ( 4.4 )
for u fg;

1 1

u fg¼ K ss ½ F fg K ss ½ ½ K sm u fg ð 4 : 5 Þ

Substitution u fg into Eq. ( 4.3 )