Page 26 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

CONCEPTS. DEFINITIONS AND METHODS 9

where 6 denotes the variational operator and W is the work done by forces not included in
V. The use of Hamilton's principle is especially convenient in cases of unusual boundary
conditions, because the equation(s) of motion and boundary conditions are determined in
a unified procedure [see, for example, Meirovitch (1967)l.
Special forms or interpretation of (2.2) and (2.3) may be necessary for 'open systems',
where the mass is not conserved, e.g. with in-flow and out-flow of mass and momentum,
as is common in fluidelastic systems. These, however, will be discussed in the chapters
that follow (e.g. in Section 3.3.3).

2.1.2 Brief review of discrete systems

A system is conservative if all noninertial forces may be derived from a potential function,
i.e. if they are all functions of position alone; thus, if the system is displaced from a to
b, the work is not path-dependent (or, equivalently, if the system is returned to a by
whatever path, the total work done is null). For a conservative system, the equations of
motion may be written as
[MlIiil+ IKlIql = {Ql, (2.4)

a special form of (2.1); the matrices are of the same order as the number of degrees
of freedom, N. Provided that (i) the generalized coordinates are measured from the
(stable) equilibrium configuration, (ii) the potential energy is zero at equilibrium, and
(iii) the constraints are scleronomic - conditions that are not difficult to satisfy in many
cases - the [MI and [K] matrices are symmetric.
Constraints are auxiliary kinematical conditions; e.g. in Figure 2.l(a) the mass MI
cannot move freely in the plane but must remain at a fixed distance 11 from the point of
support. The two constraint equations that must implicitly be satisfied for the system of
Figure 2.l(a) are what makes this system have two and not four degrees of freedom. If
a constraint equation may be reduced to a form f (x, y, z, t) = 0, then the constraint is
said to be holonomic; a subclass of this is when the constraint equation does not contain
time explicitly, in which case the constraint is said to be sclerotzoniic (Meirovitch 1970;
Nelmark & Fufaev 1972). Thus, if 11 were a prescribed function of time, the constraint
would be holonomic but not scleronomic.'
The homogeneous form of equation (2.4), representing free motions of the system,

may be re-written as
I4 + [WIISI = {Ol,

in which [W] = [M]-'[K] - provided that [MI can be inverted, i.e. if it is nonsingular.
Oscillatory solutions are sought, of the form

(q} = [Ale'"', (2.7)

+These words derive from the Greek 6Aas = whole or total and ubpas = law, hence holonomic means
totally demarcated or defined; the first component of scleronomic is from UKA&S = hard, hence the word
denotes a hard and fast rule!