Page 25 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

8 SLENDER STRUCTURES AND AXIAL FLOW

where [MI, [C] and [K] are, respectively, the mass, damping and stiffness matrices, {q)
is the vector of generalized coordinates, and [Q} is the vector of the imposed forces; the
overdot denotes differentiation with time.
On the other hand, the form of PDEs tends to vary much more widely from one
system to another. Although helpful classifications (e& into hyperbolic and elliptic types,
Sturm-Liouville-type problems, and so on) exist, the fact remains that the equations of
motion of distributed parameter sytems are more varied than those of discrete systems,
and so are the methods of solution. Also, the solutions are generally considerably more
difficult, if the equations are tractable at all by other than numerical means. Furthermore,
the addition of some new feature to a known problem (i.e. to a problem the solution
of which is known), is not easily accommodated if the system is continuous. Consider,
for instance the situation of the articulated pipe system which can be described by an
equation such as (2.1), and the ease with which the addition of a supplemental mass at
the free end can be accommodated. Then, contrast this to the difficulties associated with
the addition of such a mass to a continuously flexible pipe: since the boundary conditions
will now be different, this problem has to be solved from scratch, even if the solution of
the problem without the mass (Le. the solution of the simple beam equation) is already
known. Hence, it is often advantageous to transform distributed parameter systems into
discrete ones by such methods as the Galerkin (or Ritz-Galerkin) or the Rayleigh-Ritz
schemes (Meirovitch 1967).
In this section, tirst the standard methods of analysis of discrete systems will be
reviewed. Then, the Galerkin method will be presented via example problems, as well
as methods for dealing with the forced response of continuous systems. Along the way,
a number of important definitions and classifications of systems, e.g. conservative and
nonconservative, self-adjoint, positive definite, etc., will be introduced.

2.1.1 The equations of motion
The equations of motion of discrete systems are generally derived by either Newtonian
or Lagrangian methods. In the latter case, for a system of N degrees of freedom and
generalized coordinates qr, the Lagrange equations are

d aT aT av
(G) - + ag, =Qr, r = 1,2, ..., N,

where T is the kinetic energy and V the potential energy of some or all of the conservative
forces acting on the system, while Qr are the generalized forces associated with the rest
of the forces (Bishop & Johnson 1960; Meirovitch 1967, 1970).
For continuous (distributed parameter) systems, the equations of motion may be
obtained either by Newtonian methods (by taking force and moment balances on an
element of the system) or by the use of Hamilton's principle and variational techniques,
i.e. by using

S I" (T - V+ W)dr = 0, (2.3)