2.1. PART 1: VARIATIONAL FOUNDATIONS OF MODERN STRUCTURAL DYNAMICS 13
following general form:
f
F
N
.t/; p.t C dt/
g
D
f
f .u
N
.t/; Pu
N
.t/; p.t /
g
: (2.17)
For the more restricted cases in which the parameters are not path dependent, the localized
nonlinear forces are “algebraic,” rather than “hysteretic.”
2.1.12 MATRIX STRUCTURAL ANALYSIS
Matrix Structural Analysis predates the Finite Element Method by several decades. Histori-
cally, matrix formulations for structural systems developed along two paths, namely (a) the force
method and (b) the displacement method. e displacement method, based primarily on Hamil-
ton’s principle and the Ritz method, ultimately eclipsed the force method due to the advent of
the finite element method. Matrix structural analysis procedures, applied to damped systems
that include localized nonlinear forces, most often employ an efficient transformation defined
by free vibration modal analysis of an undamped system. [Note: In the following exposition on
modal analysis, the “a” set subscript employed in Equations (2.12)–(2.16) is dropped.]
2.1.13 FREE VIBRATION AND MODAL ANALYSIS
Free vibration of an undamped structural system is described by solutions of the type
f
u.t/
g
D
f
ˆ
n
g
sin
.
!
n
t
/
or using complex exponentials
f
u.t/
g
D
f
ˆ
n
g
e
i!
n
t
; (2.18)
which define the real eigenvalue problem
Œ
K
f
ˆ
n
g
Œ
M
f
ˆ
n
g
n
D
f
0
g
n
D !
2
n
; (2.19)
which has as many independent eigenvectors,
f
ˆ
n
g
(normal modes) and eigenvalues .
n
D !
2
n
/,
as the system matrix order. e collection of all or a truncated subset of normal modes (the
modal matrix) defines the modal transformation
f
u
g
D
Œ
ˆ
f
q
g
: (2.20)
e modal matrix has the following mathematical properties (for unit mass normalized
modes):
Œ
ˆ
T
Œ
M
Œ
ˆ
D
Œ
I
;
Œ
ˆ
T
Œ
K
Œ
ˆ
D
Œ
; (2.21)
where
Œ
is a diagonal matrix of eigenvalues.
2.1.14 UNCOUPLED STRUCTURAL DYNAMIC EQUATIONS
Application of the modal transformation to Equations (2.15) and (2.16) results in
f
Rq
g
C
ˆ
T
Bˆ
f
Pq
g
C
Œ
f
q
g
D
ˆ
T
e
f
F
e
g
C
ˆ
T
N
f
F
N
g
;
f
u
N
g
D
T
N
ˆ
f
q
g
;
f
Pu
N
g
D
T
N
ˆ
f
Pq
g
:
(2.22)
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