6.2.1
The PIT technique needs to
be implemented in a suitable computer code, and it is not amenable
to hand calculations. A block diagram for the implementation of the
PIT is reported in figure 6.2. As it can be seen, the procedure is
based on two main components:
-
.1 a differential equation solver used to determine
the roll response predicted by the model for different trial sets
of parameters; and
-
.2 a suitable minimization algorithm used to achieve
the optimum set of parameters by minimizing the sum of the squared
differences between experimental and predicted roll amplitudes.
6.2.2
The differential equation solver
could be basically of two types:
-
.1 Exact time domain solver: it numerically solves
the general differential equation (N-6.1) by using discrete time step
algorithms (like the Runge-Kutta) for a certain number of forcing
periods, until the roll steady state is achieved. Finally, each time
history is analysed in order to get the steady state roll amplitude;
and
-
.2 Approximate frequency domain solver: it uses
an analytical approximate solution of the differential equation (N-6.1)
in order to determine the nonlinear roll response curve in frequency
domain. Typically used analytical methods are the harmonic balance
technique, the multiple scale method and the averaging technique [3].
6.2.3
The two approaches have different
pros and cons.
6.2.4
Time domain integration requires
more computational time, but it solves the original differential equation
without approximations (apart from numerical accuracy). On the other
hand, in case of strong bending of the response curve, when multiple
solutions are possible for the same forcing frequency, then care must
be taken in the numerical determination of the roll amplitude in order
to correctly deal with all the present solutions (see figure 6.1).
Example of experimental and numerically fitted nonlinear
response curve in the case of softening 
6.2.5
A typical numerical method
that could be used for dealing with this problem is based on the "frequency
sweep" idea, where the forcing frequency is slowly changed in the
time domain integration from the highest value to the lowest one,
and then vice-versa, in order to detect jumps due the presence of
bifurcations (see figure 6.1).
6.2.6
Analytical approaches are approximate
solutions, and this is the biggest drawback. However, the agreement
between numerical simulations and analytical solutions is often surprisingly
good, and more than sufficient for practical applications. In addition,
if the fitting of the experimental data is based on an analytical
method, and the same analytical method is used for the extrapolation,
i.e. a consistent methodology is used without mixing the analytical
and the numerical approach, good agreement is expected between numerical
and analytical approaches. The analytical methods are usually much
faster than the direct time domain integration, and they are able
to determine multiple stable solutions in region where more than one
solution is present, making the dealing with this type of problem
easier.
6.2.7
The differences in the final
predicted roll peak φ
1r between the application
of the numerical and of the analytical approach are expected to be
below the usual experimental uncertainty (that could be considered
of the order of ± 2°).
6.2.8
The minimization algorithm
could be any reliable minimization procedure (e.g., Levenberg-Marquardt
method, or any more advanced stochastic/deterministic method).
Block diagram for the PIT procedure