Page 68 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

CONCEPTS, DEFINITIONS AND METHODS 51

2.3 LINEAR AND NONLINEAR DYNAMICS

Consider a one-degree-of-freedom linear system subjected to fluid loading, F(t); the
equation of motion is written as

mi + cx + kx = F(t), (2.158)

and F(t) may be expressed as
F(t) = -m’x - c’x - k’x, (2.159)

in which m’ is the added or virtual mass of the fluid associated with acceleration of the
body, c’ is the fluid damping term associated with the velocity of the body, and k’ is the
fluid added stiffness, as discussed in Section 2.2.l(g). Hence, the equation of motion may
be written as
(m + m’)i + (c + c’)x + (k + k’)x = 0. (2.160)

It is noted that the form of equation (2.159) implies that there is no external forcing
of the system: all fluid loading is associated with motion. In general, the coefficients
associated with the linearized forces in (2.159) are not constant, but depend on flow
velocity, amplitude and frequency of motion, fluid viscosity, and so on. For the purpose
of this introduction, however, let us neglect most of these effects and take m’ = const.,
c’ = c’(U), k’ = k’(U), where U is a characteristic flow velocity in the system. Hence,
equation (2.160) may be written as
x + 2<(U)L?,, (U)i + L?;(u)x = 0, (2.161)

where, as denoted, the damping factor, <, and the natural frequency, L?,, , are functions of
U, which is the only variable parameter of this system.
If c’(U) > 0 and k’(U) > 0 for all U, then the response of the fluid-loaded system
is qualitatively the same as that of the mechanical system: only damped oscillations
would be observed, with higher or lower frequency, depending on whether added mass or
fluid stiffness effects predominate [Le. whether (k + k’)/(m + m’) > or < k/m], and with
higher or lower damping (<), depending on whether (c + c’)/(m + m’) > or < c/m.
If, however, k’(U) can become negative, and Ik’(U)l = k for some critical value of
U, U,, then the overall stiffness of the system vanishes - and for U > U, may become
negative - which signifies that the system is then statically unstable. The premier example
of this (albeit for a system with more than one degree of freedom) is the static instability, or
divergence, of an articulated or continuously flexible pipe with supported ends conveying
fluid (see Chapter 3); it is similar to the divergence, or buckling, of a column subjected
to an end load. At that point, Le. when Ik’(Cr)l = k, x becomes indeterminate: i.e. the
static equilibrium position xSt = 0 is replaced by a condition where an infinite set of static
equilibria are possible (Ziegler 1968) according to linear theory.
Similarly, if <(U,) < 0 [Le. if c’(U,) < 0 and sufficiently large], this implies a negative
damping: instead of the oscillations dying out with time, they are amplified exponentially.
A good example of this is the oscillatory instability (in the linear sense), orjutter, of a
cantilevered pipe conveying fluid (see Chapter 3).
Mathematically, the evolution of a system towards divergence or flutter may be tracked
by plotting the complex eigenvalues or, equivalently, the eigenfrequencies in the complex