This document explains how the added inertia coefficients and linearized damping coefficients are calculated and reported in MOSES. Often asked questions are also discussed at the end of the document.

MOSES v10.13 was used in generating results for this work.

*Frequency Domain Analysis and Output Results*

A frequency domain analysis is usually the basis for generating transfer functions (RAOs) for frequency dependent excitation forces, added mass and damping. The output from a typical radiation-diffraction frequency domain analysis usually contains added mass coefficients, added radii of gyration and linearized values. The linearization is needed due to the nonlinear damping effects particularly in the low frequency response.

MOSES provided the **MATRICES** command to return an additional PPO file in the answer directory when the -file option is used as in

matrices -file

The **MATRICES** command is issued in the Frequency Response Menu after the response amplitude operators have been calculated. The added mass and damping matrices are in 6 x 6 format and are reported for the periods and headings defined in **G_PRESSURE**. Note that the reported damping values are the maximum values over all headings and not the ones really used. This is valid for a spectral linearization, but not for a wave steepness one.

The Added Inertia Coefficients report in the output file contains the added mass coefficients and the added radii of gyration components as a function of frequency. A typical output table is shown in Figure 1.

Figure 1 Added inertia coefficients (Click image)

In the output file the Linearized Damping Coefficients are also tabulated with reference to Damping / Mass ratio and Damping Radii of Gyration as a function of frequency again. An example is shown in Figure 2.

Figure 2 Linearized damping coefficients (Click image)

The values from Figure 1 and Figure 2 are fitted into the matrices found in the PPO file. For demonstration, this work will focus on the values reported when the encounter period is 20 seconds. The rest of the figures in the matrix are found in a similar manner.

Figure 3 is a snapshot of the corresponding added mass and added radii of gyration coefficients as found in the output file.

Figure 3 Normalized added mass and added radii of gyration coefficients (Click image)

The same values can be found in the PPO file with the necessary mathematical operation in the rotational degrees of freedom. The comparison is shown in Figure 4.

Figure 4 Added mass matrix for T=20 sec. (Click image)

The reported added radii of gyration values in the rotational degrees of freedom are simply the squared values found in the output file, i.e.

Roll Added Mass = (4.221)^{2} = 17.815

Pitch Added Mass = (11.345)^{2} = 128.719

Yaw Added Mass = (12.089)^{2} = 146.139

Similarly, the reported values of linearized damping coefficients at T=20sec. in the output file are found in PPO file (damping matrix). The values are reported diagonally in the matrix again as shown in Figure 5 and Figure 6.

Figure 5 Linearized damping coefficients (Click image)

Figure 6 Damping matrix for T=20sec. (Click image)

The reported damping values in the rotational degrees of freedom are simply the squared values found in the output file, i.e.

Roll Added Mass = (1.470)^{2} = 2.161

Pitch Added Mass = (6.357)^{2} = 40.417

Yaw Added Mass = (1.848)^{2} = 3.414

Each requested period in the **G_PRESSURE** command will result in two matrices, i.e. one for added mass and one for damping.

*Questions & Answers*

*Q: Do the reported damping values include the radiation component?*

A: The reported damping does include radiation. In addition, the -cs_current option will also include viscous terms.

*Q: How can I add, or control roll damping in my system?*

A: One can use the -roll_damping option in **&DESCRIBE PIECE** command. Alternatively, bilge keels can be modelled.

*Q: How can I add bilge keels in a model?*

A: You can use the **#PLATE **command and use it on the appropriate points along the hull. Look at the b-keels example found in the hdesk directory of the installation files.

*Q: The linearization of added mass and damping coefficient is done by 1/20 and is divided by the vessel weight. What is the nature of this ratio?*

A: The 1/20 is the wave steepness used to linearize the added mass and damping equations spectrally. This ratio is a remnant from the days when computers could not linearize on a wave spectrum. To override this ratio, the -spectrum option is now available in the **RAO** command to specify the wave spectrum of the environment which will be used to linearize the motions equations spectrally.

Also see -steep and -rod_steep options in the **RAO** command to alter the default wave steepness for roll damping and Morison’s drag on bodies, and constant wave height for drag on rod elements.

*Q: What is the #AMASS command doing and how is it used?*

A: This is a command used to define an added mass matrix at a point. This is normally used with a linear drag matrix in the **#DRAG** command. It is required that these two commands are included in a jacket launch analysis so that drag and added mass of the barge is included.

*Q: What are the units for Damping Radii of Gyration reported in the Linearized Damping Coefficients?*

A: The units for damping radii of gyration would be length/sqrt(time).

*Q: What are the units for radiation damping coefficients in MOSES output?*

A: The output table reports “Values Normalized by Mass with Weight = xxx”. MOSES reports the weight of the body in question and needs to covert this to mass for computing the damping coefficient for later use in third-party applications. The translational components have units of 1/time, while the rotational components have units of length^2/time.

*Q: Why is Damping divided by Mass in the linearized damping coefficients output?*

A: The output table reports the mass (displacement) that has been used to carry out this division. Without this operation the damping coefficients would be very large numbers; it is used solely for convenience. Note that in the frequency domain, the units for damping after this division will be .1/time

*Q: Is the added mass coefficient reported by MOSES the total added mass? In other words, will the total added mass be the displacement **× the added mass coefficient reported, or displacement **× (1 + added mass coefficient reported).*

A: What MOSES reports is the added mass. This differs from the hydrodynamic mass by 1; i.e. the hydrodynamic mass is the basic mass plus the added mass