112
SINGLE DEGREE OF FREEDOM STRUCTURES
of motion. Using first order perturbation theory, it will now be shown that
for structures with nonlinear restoring forces, the free vibration frequency will
be dépendent on the amplitude of motion. In both the linear and nonlinear
Systems considered here, this amplitude of free vibration is imposed as an initial
condition for the motion.
Consider the free, undamped motion of a virtual mass m with the indépendant coordinate v and with a nonlinear restoring force given by équation (2.69).
The corresponding équation of motion is
mv + k\V + ky? = 0
(5.43)
Recall that the restoring force term in the last équation was used to model the
line stiffness of a spread moored ship. See Example Problems 2.10 and 2.11.
Now rewrite this last nonlinear équation as
v + WqU + eu3 = 0
(5.44)
in which
— fcj/m is the square of the natural frequency if k3 — 0, and
e = ks/m. It is assumed that e is always small enough so that ev3
An approximate solution to équation (5.44) can be derived as follows using
classical perturbation theory (Cunningham, 1964). At time t = 0 the mass is
displaced by the amplitude A and then released. At the instant of release, the
initial velocity is zéro. These two initial conditions are expressed as
v(0) = A;
û(0) = 0
(5.45)
Now assume that the solution to équation (5.44) and its corresponding frequency
w are
v = v0(t) + evi(t)
(5.46)
û2 -
+ e/(A)
(5.47)
In équation (5.46), the quantity Vo(t) is a time-dependent displacement and
should not be confused with the Symbol vq used for the constant displacement
amplitude in Section 5.1. When the quantity «i(t) and the amplitude function f(A) are combined with the governing équation (5.43) and the resuit is
regrouped in ascending powers of e, the resuit is
(üo + w2uo)e0 + [Ü! + ù>^V1 - f(A)VQ +
+ o(g2) = Q
(5.48)
The coefficients of e° and e1 are each equated to zéro and terms of higher in
e or o(e-). are assumed to be small enough to be neglected. The resuit is the
following two linear differential équations:
••
~ 9
t’o + ûj v0 0
(5.49)
SINGLE DEGREE OF FREEDOM STRUCTURES
of motion. Using first order perturbation theory, it will now be shown that
for structures with nonlinear restoring forces, the free vibration frequency will
be dépendent on the amplitude of motion. In both the linear and nonlinear
Systems considered here, this amplitude of free vibration is imposed as an initial
condition for the motion.
Consider the free, undamped motion of a virtual mass m with the indépendant coordinate v and with a nonlinear restoring force given by équation (2.69).
The corresponding équation of motion is
mv + k\V + ky? = 0
(5.43)
Recall that the restoring force term in the last équation was used to model the
line stiffness of a spread moored ship. See Example Problems 2.10 and 2.11.
Now rewrite this last nonlinear équation as
v + WqU + eu3 = 0
(5.44)
in which
— fcj/m is the square of the natural frequency if k3 — 0, and
e = ks/m. It is assumed that e is always small enough so that ev3
An approximate solution to équation (5.44) can be derived as follows using
classical perturbation theory (Cunningham, 1964). At time t = 0 the mass is
displaced by the amplitude A and then released. At the instant of release, the
initial velocity is zéro. These two initial conditions are expressed as
v(0) = A;
û(0) = 0
(5.45)
Now assume that the solution to équation (5.44) and its corresponding frequency
w are
v = v0(t) + evi(t)
(5.46)
û2 -
+ e/(A)
(5.47)
In équation (5.46), the quantity Vo(t) is a time-dependent displacement and
should not be confused with the Symbol vq used for the constant displacement
amplitude in Section 5.1. When the quantity «i(t) and the amplitude function f(A) are combined with the governing équation (5.43) and the resuit is
regrouped in ascending powers of e, the resuit is
(üo + w2uo)e0 + [Ü! + ù>^V1 - f(A)VQ +
+ o(g2) = Q
(5.48)
The coefficients of e° and e1 are each equated to zéro and terms of higher in
e or o(e-). are assumed to be small enough to be neglected. The resuit is the
following two linear differential équations:
••
~ 9
t’o + ûj v0 0
(5.49)