12 2. DEFINITION OF TEST ARTICLE FINITE ELEMENT MODELS
2.1.10 ASSEMBLY OF LINEAR STRUCTURAL SYSTEM MODELS
Assembly of a linear system structural dynamic model involves the allocation and superposition
(overlapping) of individual finite elements onto a system degree-of-freedom (DOF) map. is
process defines sparse system mass and stiffness matrices that are positive semi-definite and
symmetric. A system composed of an assembly of “h” elements is defined by “grid” set mass
and stiffness matrices denoted by ŒM and ŒK, respectively. Additional quantities complete the
ingredients for a structural dynamic model, namely: (1) allocation of excitation forces to system
grid points, (2) formation of an assumed viscous damping matrix (which unfortunately does not
resemble physical reality in most commercial finite element codes), and (3) allocation of local
nonlinear internal forces. e grid set equations for the structural dynamic model are of the for
Œ
M
˚
R
U
C
Œ
B
˚
P
U
C
Œ
K
f
U
g
D
Œ
e
f
F
e
g
C
Œ
N
˚
F
N
U
N
;
P
U
N
; p
;
f
U
N
g
D
Œ
N
T
f
U
g
;
˚
P
U
N
D
Œ
N
T
˚
P
U
:
(2.11)
Constraints and boundary conditions, collectively described by transformations of the
form,
f
u
g
D
Œ
G
f
u
a
g
; (2.12)
are applied in a symmetric manner as a consequence of the quadratic forms and integrals defined
by the Ritz method, resulting in the “analysis” set equations and matrices (for a linear system)
Œ
M
aa
f
Ru
a
g
C
Œ
B
aa
f
Pu
a
g
C
Œ
K
aa
f
u
a
g
D
Œ
ae
f
F
e
g
; (2.13)
where the reduced order matrices are
Œ
M
aa
D
G
T
ga
M
gg
G
ga
;
Œ
B
aa
D
G
T
ga
B
gg
G
ga
;
Œ
K
aa
D
G
T
ga
K
gg
G
ga
;
Œ
a
D
G
ga
T
g
:
(2.14)
2.1.11 SYSTEM MODELS WITH LOCALIZED NONLINEARITIES
Structural assemblies sometimes display locally nonlinear behavior (primarily at structural joints
that connect subassemblies). e “analysis” set equations for such situations are augmented by
localized nonlinear forces, fF
N
g, allocated in accordance with the nonlinear force distribution
matrix, Œ
aN
, are of the form
Œ
M
aa
f
Ru
a
g
C
Œ
B
aa
f
Pu
a
g
C
Œ
K
aa
f
u
a
g
D
Œ
ae
f
F
e
g
C
Œ
aN
f
F
N
g
: (2.15)
e nonlinear forces are defined as functions of localized displacements and velocities that
are defined as
f
u
N
g
D
T
aN
f
u
a
g
;
f
Pu
N
g
D
T
aN
f
Pu
a
g
: (2.16)
In the most general case, localized nonlinear forces are “hysteretic” functions of localized
displacements, velocities, and (path dependent) parameters, fpg, according to algorithms of the
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