FREQUENCIES FOR NONLINEAR STRUCTURES
113
üi + ü2vi = f(A)v0 - Vq
(5.50)
A particular solution to équation (5.49) that satisfies the initial conditions
of équations (5.45) is
vo = A cos ait
(5.51)
This latter solution is then inserted into the right side of équation (5.50) and
the following identity is used:
i -
3
.
1
cos ait — - cos wt 4— cos 3ût
4
4
(5.52)
This procedure leads to the general solution to équation (5.50), or
a3
t r
3
«i = By cos ait + B2 sin wt + ——? cos 3ût 4----- A fl A) - -A3
32ôj
2a;
4
(5.53)
where By and B2 are constants. In this solution, the last term on the right
is called the secular term, and this term is observed to become unbounded as
t becomes large. Unboundedness is not physically possible for this System so
restrained, and therefore the coefficient of t in the secular term must vanish.
Excluding the trivial case for A — 0, this reasoning leads to
/(A) = ?A2
(5.54)
When this amplitude function is substituted into équation (5.47), with e =
ka/m, then
^ + ^a2Ÿ/2
o+4mA J
(5.55)
This resuit clearly shows that the frequency of the nonlinear System increases
due to the addition of the cubic restoring force term, assuming that kj > 0, and
that this increase dépends on the initial System displacement A. However, if
ka < 0, then w < o;q.
Next
is calculated from équation (5.53) by imposing the zéro initial conditions Vi(0) = ûi(0) = 0, with which the constants are evaluated as By =
-A/(32ù>2) and B2 = 0. When this resuit and the solution for v0, équation
(5.51), are used with équation (5.46), the first-order perturbation solution becomes
ka 43
v = A cos ût----- ——^(cos ût - cos 3û'f)
32mw2
(5.56)
This resuit shows that the free oscillation amplitude for the counterpart linear
System (fc3 = 0) is distorted by a frequency component 3û when k3 > 0.
Example Problem 5.5.
Calculate wq and û for bot h surge and sway for
the spread-moored ship described in Example Problem 2.10. Assume that the