Page 329 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

310 SLENDER STRUCTURES AND AXIAL FLOW

Consider first the trivial equilibrium position, wo = 0, i.e. the origin {q, i) = (0, 01, in
which case we can use (5.82) rather than (5.88). The function
+
+
{
H, = ;1,12 + ;{lw1/12 - ~ 1 ~ ’ 1 ~ );4w’14 + (w, w)} (5.89)
~
u
is a suitable Lyapunov function (Movchan 1965; Parks 1967), in which u is to be chosen
subsequently. It is noted that, essentially, (5.89) has the form of kinetic plus potential
energy (the Hamiltonian), as related to (5.82); however, the extraneous last bracketed
term is essential in rendering Ha a Lyapunov function. To prove that Ha is positive
definite, we start with the inequality
Ha 2 ;1WI2 + ;{n41wl2 - u2n2lwI2} + iw{alw12 + (w, W)), (5.90)

in which (a) the first of equations (5.87) is used to show that Iw”I2 = n21w’I2, and (b) the
fact that I#I2 2 n219l2 for any continuous function with $(O) = +(l) = 0 - as easily
ascertained for trigonometric functions. Then, re-writing lWI2 + 2u(w, W) = I(uw + W)I2 -
w21wI2, inequality (5.90) may be written as

Ha p i[u(a - u) + n2(n2 u2)]1w12 + il(ww + W)I2,
~
which is globally positive definite provided that u < n and 0 5 w 5 a. Therefore, for
given u and u, as (lw’I2 + lW12}1/2 increases, so does Ha, monotonically.
Differentiating Ha with t and using (5.82) with q = w, and then applying the boundary
conditions in the resulting integrations by parts,
+
-- - -(a - u)lWI2 + 2wB’/*u(w’, 6) - ~lw’’1~ - ;~.dlw’1~ (5.91)
ma
dt
is obtained. Then, making use of the inequality ? n21@I2 again, this may be written as

ma 5 -{(a - u)(WI2 - 2~B’/~u(w’, + u(n2 - u2)1w’I2} - ;~dlw’1~. (5.92)
W)
dt
~
Provided that u < n and v < a this may be made negative definite if u is chosen positive
and sufficiently small. For example, letting u = a/k/3u2, equation (5.92) is re-written as

{(kBu2 - l)lWI2 - 2B’/2u(w’, W) + (n2 - ~~)1w’1~). (5.93)

BY utilizing the expansion of I(B’/~uw/&TP - w’&?TP)~~ a similar way as in
in
the foregoing, inequality (5.93) may be re-written as

The bracketed quantity is clearly positive definite if (kBu2 - l)(n2 - u2) 2 Bu2; hence, if
k is large enough (i.e. u small enough), we have the required behaviour: dHa/dt < 0 for
all w’ or W # 0, i.e. dHa/dt is globally negative definite. Hence, the trivial equilibrium
point wo = 0 is globally asymptotically stable if u < n. It is of interest that if a = 0, u
0;
must be set to zero also, and then it can only be proved that dHa/dt I hence only
stability, but not asymptotic stability may be proved.