Frequency response is a linear approximation to the equations of motion with assumed harmonic input. In MOSES, a menu is devoted to computing frequency response and its post-processing. To enter the menu, one inputs:

FREQ_RESPONSE

When completed, the menu is exited with **END_FREQ_RESPONSE**.

The traditional way of obtaining frequency response is to consider a set of unit amplitude waves and to linearize the equations of motion for each wave. This is
exactly what is done with the **RAO** command. Results obtained in this way can later be easily combined with different spectra to obtain approximations to the extremes
in many different situations, and are thus quite popular. The major disadvantage of this approach is that one can only look at the response to wave frequency
excitation. In other words, with the response operator approach, one cannot investigate the effect of wind or slow drift wave excitation. To cope with these other
effects, one can no longer look at unit amplitude waves, but must consider the simultaneous effect of all environmental forces. This is what is done with the
**SRESPONSE** command. The major disadvantage with spectral response is that this response is applicable to a *single* environment and thus the post processing options are
limited.

Before getting into the details of these commands, a few general words are in order. The frequency response calculations depend not only on the sea pressures, but
also on the stiffness of the connectors and the mass properties of the system. Instead of requiring the user to directly specify the weight and radii of gyration of
the system, MOSES uses those computed from the element and weight data supplied in the structural model, and from currently active tanks, weights, etc. This is
necessary to insure that when structural loads are computed, there will not be seriously out of balance in the loads. It is therefore *important* to accurately model
the weights not only from a local sense, but also in a global one. Another important thing to remember is that the "response operators" produced with the **RAO** command
are not really independent of wave amplitude. For most cases, nonlinear damping (and perhaps forcing) is important. Thus, some *real* wave amplitude must be used to
linearize the system. If post processing results are obtained for an environment which is radically different from that used in the linearization, then the
applicability of the results is open to question.

MOSES provides two ways to linearize the equations for RAO computations: specified wave amplitudes, and a spectral linearization. With the specified wave height
method, one specifies a steepness, a period, and a wave height. This is the default method. MOSES will for each period and heading, use a constant wave steepness to
obtain a "real" wave amplitude for linearization for periods less than the specified period. For larger periods, the specified height will be used. The energy from
the linear system and the real system over a period are set equal to obtain the linearized results. For damping, this appears to be a rational approach. When,
however, one looks at the forcing from a Morison's equation element, it does not appear as attractive. Here one is saying that the linearized coefficient depends only
on the velocity for a single Fourier coefficient. The alternative is a "spectral" linearization. Here, it is assumed that the seas are unidirectional so that the RAOs
for a set of periods and given direction are linearized at one time. The spectrum is used to compute the RMS of the relative velocity at a point, and this is used to
compute the "drag" coefficient. The two approaches yield somewhat different results depending on the difference in the peaks of the response and the spectrum. For
**SRESPONSE**, the spectral method is always used. However, the linearization here is not only over periods, but also over headings.

In this light, it can be seen that frequency domain forces are no longer for zero speed, and the interpretation of wave heading is different than before. That is, the
frequency domain forces are those used to compute the response. Sometimes these are force/wave amplitude results (RAOs), and sometimes these are Fourier coefficients,
depending on the use of **SRESPONSE**.

Response operators are computed by simply issuing the command:

RAO, -OPTIONS

where the available options are:

-HEADING, H(1), H(2), ...., H(N)-PERIOD, T(1), T(2), ...., T(N)-SPEED, VR-ITER, MAXIT-SPECTRUM, ENV_NAME-STEEP, ST, PBCHEI, CHEI-ROD_STEEP, ST, PBCHEI, CHEI

When this command is issued, MOSES will compute a new set of response operators, for the same frequencies and headings as those of the basic pressure data (those
specified on the **G_PRESSURE** command), unless the options **-HEADING** or **-PERIOD** are exercised. The influence of any *one* speed can be
included in the statistic results by including the option **-SPEED**, where VR is the speed desired in knots.

If either **-PERIOD** or **-HEADING** are used, MOSES will interpolate values of the hydrodynamic forces, added mass, and damping from the values contained in the pressure
database. If no data for a quadrant exists in the pressure database, then MOSES will assume symmetry about the vessel centerline or about amidships when interpolating
hydrodynamic results.

The remaining options describe how MOSES deals with viscous damping. In general, there are three types of viscous damping which are considered: empirical roll
damping, Morison's drag for bodies, and Morison's drag for rod elements. Normally, MOSES will iterate up to thirty times to achieve a proper solution considering the
viscous damping. In some cases where there are many bodies connected by rigid connectors, there is a substantial computational effort involved with this iterative
solution. The **-ITER** option may be used to limit the number of iterations. Here, MAXIT is the maximum number of iterative steps which will be taken,
where 30 is the default value.

The **-SPECTRUM** option specifies the wave spectrum of the environment, ENV_NAME, which will be used to linearize the equations spectrally. Here, any
nonlinear dependence will be replaced by the RMS value times SMULT, where SMULT can be specified with the **-SPE_MULTIPLIER** option of an **&DESCRIBE** BODY command. If
**-SPECTRUM** is omitted, then an equivalent linearization will be performed, where the drag is linearized by either constant wave steepness or constant wave height. The
default is to use constant wave steepness for roll damping and Morison's drag on bodies and constant wave height for drag on rod elements. These assumptions can be
altered by using the **-STEEP** option for roll damping and body drag, and **-ROD_STEEP** for rod drag. When either of these options is
used, the wave steepness will be held constant at 1/ST for periods less than PBCHEI seconds and a constant wave height of CHEI (feet or meters) will be used for
larger periods. If neither PBCHEI nor CHEI are specified, then constant steepness will be used for all periods.

During the RAO computation, the dependence of drag coefficient with Reynolds Number is *not* considered. Instead, the drag coefficient corresponding to the value
specified with the **-F_CD_TUBE** option of **&DEFAULT** command is used. During the computation of nonlinear damping, the computed value is multiplied by a factor to obtain
that which is used in the computation. For a rod, a drag multiplier can be defined with **&DEFAULT -FM_ROD**. For bodies, the **-FM_MORISON** option of **&DESCRIBE BODY** is used
to define the multiplier.

Spectral frequency response is computed by issuing the command:

SRESPONSE, ENV_NAME, -OPTIONS

where the available options are:

-PERIOD, T(1), T(2), ...., T(N)-HEADING, H(1), H(2), ...., H(N)-FIX_TEN, YES/NO-ITER, MAXIT

Here, ENV_NAME is the name of the environment which will be used to compute the response, and the options operate in the same manner as with the **RAO** command. If
ENV_NAME is omitted, then the current environment will be used. Here, one will definitely want to specify the **-PERIOD** option since the objective is
to investigate the effect of non wave excitation frequencies. When this command is issued, MOSES will take the environment and expand the direct wave excitation, the
wave drift force, and the wind force in a Fourier series of the periods specified with **-PERIOD**. The direct wave force and drift force series will also be expanded
with headings specified with **-HEADING**. The wind series will be applied to the heading closest to the specified wind heading. These series of forces
will be used to obtain the Fourier coefficients of the response series. This is precisely what is called spectral response. The option **-FIX_TEN** can
be used to "fix the tensions". Since the connector tensions are normally a nonlinear function of the motions, the linearization necessary in the frequency domain will
not be nearly as effective for predicting the connector forces as it will be for predicting the motions. "Fixing the tensions" is a partial remedy for this problem.
If the option is used with a value of YES/NO of **YES**, then MOSES will compute the value of the tension at the maximum position and compute a ratio of this maximum to
that predicted by the force response. The force response is then scaled by the ratio so that the predicted maximum will be that computed for the extreme position.

After obtaining frequency response results, one can examine them with other commands in this Menu. Some of the commands produce response operators, others produce statistics for irregular seas, and others produce equation force data. At the conclusion of most of these commands, the user is placed in the Disposition Menu where he is given the option of reporting, viewing, or graphing the results of the command.

The behavior of the commands in this menu differ with the type of frequency response data to be examined. With response operators, (obtained with an **RAO** command) one
*must* specify an environment to obtain statistical or time synthesis results, while for nonlinear spectral results (obtained with a **SRESPONSE** command), an environment
*cannot* be specified. With RAOs, the commands in this menu which deal with sea-states have a final syntax which is identical to that of the **&ENV** command. With
nonlinear spectral results, the environment data must be omitted. In other words, with spectral results, no environmental data will be allowed. With RAOs, these
commands not only initiate the computation of quantities in an irregular sea, but are also **&ENV** commands. Thus, when one issues one of these commands with a non-blank
ENV_NAME, he is altering the definition of this environment within the database. If ENV_NAME is omitted, then the environment used will be totally defined by the
options specified.

To produce a time domain process from the frequency response and and an environment, one should issue:

FR_2TIME, ENV_NAME, -OPTIONS

where the available options are:

-SEA, SEA_NAME, THET, HS, PERIOD, GAMMA-SP_TYPE, TYPE-SPREAD, EXP-TIME, TOBSERV, DELTA_TIME, TTRA_SET, NCYCLES-T_REINFORCE, TB

This command generates a set of configurations of the system and the connector forces by summing the frequency response with the sea. One is not put into the
Disposition Menu. Instead, one can enter the Process Post-Processing menu where on can look at the position of points, the relative motion of points, connector
forces, etc. You cannot, however, use the **TRAJECTORY**, **POSITION**, **STABILITY** **TANK_FLD**, **TANK_BAL**, **HOLE_FLOODING**, **R_VIEW**, **R_ENVELOPE**, or **R_DETAIL** commands because data for
them is not generated. Also, notice that this command creates events for a process and it will overwrite any existing events you may have. Finally, this command is
not only useful for looking at a true sample but also for making movies.

While commands discussed later give the user complete control over the results he obtains, a single command has been provided to produce a set of "standard results" which suffice in many circumstances. The form of this command is:

FP_STD, X, Y, Z, -OPTIONS

where the options are

-WEIGHT, WEI, RX, RY, RZ-HEIGHT, WAVE_HEIGHT

This command is not applicable to nonlinear spectral results. If it is issued with no options, then the response operators will be computed at the point X, Y, and Z
(feet or meters) and these results will be reported and graphed. If the **-HEIGHT** option is used, then statistics of the motions will be computed in an
ISSC sea of height WAVE_HEIGHT (feet or meters) for periods from 4 to 18 seconds. Again, these results will be reported and plotted. Finally, if the **-WEIGHT** option is used, the response operators of the forces acting on a body of weight WEI (bforce) with radii of gyration RX, RY, and RZ (feet or
meters) located at this point will be computed. The response operators will be reported and graphed. If both the **-WEIGHT** and **-HEIGHT** options are used, then the
statistics of the forces will be computed, reported, and graphed for the same set of conditions as the motions.