2.1. PART 1: VARIATIONAL FOUNDATIONS OF MODERN STRUCTURAL DYNAMICS 9
2.1.3 MATHEMATICAL PHYSICS AND HAMILTON’S PRINCIPLE
Application of Hamilton’s principle to a dynamic system described as a continuum yields a vol-
ume integral of the type
t
2
Z
t
1
Z
R
.
ıT
R
ıU
R
C ıW
R
/
dR dt D 0; (2.1)
where T
R
, U
R
, and ıW
R
are the kinetic energy, potential (or strain) energy, and virtual work
functions per unit volume, R, respectively. Analysis of any particular dynamic system, described
in terms of displacement variables, u.x; y; z; t/, which may be vectors, results in the following
type of functionals:
t
2
Z
t
1
Z
R
.
P.D.E.
/
ıu dR dt C
t
2
Z
t
1
Z
S
.
N.B.C.
/
ıu dS dt D 0; (2.2)
“P.D.E.” represents the particular partial differential equation(s) within the system’s volume.
“N.B.C.” represents the natural boundary conditions, which are mathematically and physically
admissible on the system’s boundary surface(s), S. e general process for derivation of a system’s
partial differential equations and natural boundary conditions has provided a consistent basis
for the development of technical structural theories for prismatic bars, beams, rings, plates, and
shells.
2.1.4 THE CONTRIBUTIONS OF RITZ, GALERKIN, AND TREFFTZ
ree outstanding contributions that led to the development of approximate analysis techniques
date back to the early part of the twentieth century. e methods bearing the names of Ritz,
Galerkin, and Trefftz are all consequences of Hamilton’s principle and the assumption of ap-
proximate solution functions.
2.1.5 THE RITZ METHOD
A monumental contribution to approximate analysis was introduced by Ritz (1908) [7], who
described the displacement field in variable separable terms
u.x; y; z; t / D
N
X
nD1
‰
n
.x; y; z/ q
n
.t/; (2.3)
where ‰
n
.x; y; z/ are assumed shape functions and q
n
.t/ are temporal generalized coordinates
(displacements). In addition, the strain field,
".x; y; z; t / D
N
X
nD1
‰
";n
.x; y; z/ q
n
.t/; (2.4)
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