7.2.1 The Dry Mass Matrix

The mass matrix for the dry body can be written as:

$M=\left[\begin{array}{cccccc}M& 0& 0& 0& M{z}_G& 0\\ 0& M& 0& -M{z}_G& 0& 0\\ 0& 0& M& 0& 0& 0\\ 0& -M{z}_G& 0& M_{44}& 0& 0\\ M{z}_G& 0& 0& 0& M_{55}& 0\\ 0& 0& 0& 0& 0& M_{66}\end{array}\right].$([7.4])

Here, it assumed that the center of gravity (CG) is located at $\left(0,0,z_G\right)$ and that the $\left(x,z\right)$ and the $\left(y,z\right)$ -planes are planes of symmetry, which frequently is the case for floating bodies. $M$ is the mass of the body, and the moments of inertia are given by:

$\begin{array}{c}{M}_{44}\equiv I_{11}=\underset{M}{\int }\left(z^2+y^2\right)\mathrm{dm}=I_{11G}+z_G^{2}M\\ M_{55}\equiv I_{22}=\underset{M}{\int }\left(x^2+z^2\right)dm=I_{22G}+z_G^{2}M.\\ M_{66}\equiv I_{33}=\underset{M}{\int }\left(y^2+x^2\right)dm=I_{33G}\end{array}$([7.5])

Here, $I_{ii}$ refers to the mass moment of inertia about axis $i$ and $I_{iiG}$ refers to the mass moment when the axis has origin in CG.

In the more general case without symmetry and where the CG is located in $\left(x_G,y_G,z_G\right)$ , the mass matrix may be obtained by:

$M=\left[\begin{array}{cc}M{I}_{3*3}& -MS\\ MS& I_{bb}\end{array}\right],$([7.6])

where

$I_{3*3}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\text{, }S=\left[\begin{array}{ccc}0& -z_G& y_G\\ z_G& 0& -x_G\\ -y_G& x_G& 0\end{array}\right]\text{ and }{I}_{bb}=\underset{M}{\int }\left[\begin{array}{ccc}{y}^2+z^2& -xy& -xz\\ -yx& z^2+x^2& -yz\\ -zx& -zy& x^2+y^2\end{array}\right]dm.$

For further details, see Reference Perez and FossenPerez and Fossen (2007).

7.2.2 The Added Mass Matrix

In many of the proposed designs for offshore wind support structures, the floater is composed of slender horizontal pontoons and vertical columns. Both the pontoons and the columns may have a cross-sectional area that varies along its length. In addition, flat solid or perforated plates may be introduced to obtain the wanted dynamic characteristics of the floater.

There are two main options to obtain the added matrix for such structures: strip theory or 3D ideal fluid theory based upon, e.g., boundary element techniques, as discussed in Chapter 6. Strip theory approach will be addressed here.

7.2.2.1 Vertical Columns

Consider a slender, circular and vertical column of constant radius $R$ and extending from $z_b$ to $z_t$ , where $z_b<z_t\le 0$ . The cylinder axis is located at $\left(x_c,y_c\right)$ . The added mass for linear motion in the horizontal direction can then be approximated by:

$A_h=A^{\left(2D\right)}\left(z_t-z_b\right)=\pi \rho R^2{C}_{ah}L,$([7.7])

where the length of the column is $L$ and $C_{ah}$ is the 2D added mass coefficient for the cylinder. The added mass for oscillation in the vertical direction can similarly be written as:

$A_v=\left(C_{avb}+C_{avt}\right)\pi \rho R^3.$([7.8])

Here, the indices $b$ and $t$ refer to the bottom and top of the column respectively. If the column pierces the free surface, $C_{avt}=0$ , and if the column is sitting on top of a pontoon, $C_{avb}=0$ . If two columns are sitting on top of each other, an approximate value for the added mass contribution at the junction may be applied; see Section 7.4.2. A 3 x 3 added mass matrix for linear motions is obtained as:

$A_c=\left[\begin{array}{ccc}{A}^{\left(2D\right)}L& 0& 0\\ 0& A^{\left(2D\right)}L& 0\\ 0& 0& A_v\end{array}\right].$([7.9])

As compared to the dry mass matrix, it is observed that the mass values differ between the three directions. $A_c$ will now constitute the new submatrix corresponding to the upper-left part of [7.6]. Similarly, the submatrix $mS$ is replaced by $S_{Ac}=A_c*S_c$ , where:

$S_c=\left[\begin{array}{ccc}0& -\frac{1}{2}\left(z_b+z_t\right)& y_c\\ \frac{1}{2}\left(z_b+z_t\right)& 0& -x_c\\ -y_c& x_c& 0\end{array}\right].$([7.10])

The rotational coupling terms are obtained as:

([7.11])$\begin{array}{c}{I}_{11}=\underset{L}{\int }\left(y^2+z^2\right)dm=y_c^2{A}_v+\frac{1}{3}{A}^{\left(2D\right)}\left(z_t^3-z_b^3\right)\\ I_{12}=\underset{L}{\int }-xydm=-x_c{y}_c^{}{A}_v\\ I_{13}=\underset{L}{\int }-xzdm=-x_c\frac{1}{2}{A}^{\left(2D\right)}\left(z_t^2-z_b^2\right)\\ I_{22}=\underset{L}{\int }\left(x^2+z^2\right)dm=x_c^2{A}_v+\frac{1}{3}{A}^{\left(2D\right)}\left(z_t^3-z_b^3\right)\\ I_{23}=\underset{L}{\int }-yxdm=-y_c\frac{1}{2}{A}^{\left(2D\right)}\left(z_t^2-z_b^2\right)\\ I_{33}=\underset{L}{\int }\left(y^2+x^2\right)dm=\left(y_c^2+x_c^2\right)A^{\left(2D\right)}\left(z_t-z_b\right).\end{array}$

With $I_{ij}=I_{ji}$ , the rotational submatrix becomes:

$I_{bbc}=\left[\begin{array}{ccc}{I}_{11}& I_{12}& I_{13}\\ I_{21}& I_{22}& I_{23}\\ I_{31}& I_{32}& I_{33}\end{array}\right].$([7.12])

The full added mass matrix for one vertical column thus becomes:

$A_{col}=\left[\begin{array}{cc}{A}_c& -S_{Ac}\\ S_{Ac}& I_{bbc}\end{array}\right].$([7.13])

It should be kept in mind that in this derivation it has been assumed that the 2D added mass is equal at all sections, i.e., no end effects are accounted for when integrating the 2D added mass along the column. If end effects are to be accounted for, the various terms involved should be obtained from [7.7] and [7.11] by performing integration along the axis and accounting for variation in $A^{\left(2D\right)}$ .

The added mass related to the end surfaces of a long slender cylinder is frequently taken to be the mass of a half-sphere with the same radius as the column, i.e., $C_{av}=2/3$ . If two columns are located on top of each other, a rough estimate of the vertical added mass can be obtained by setting the vertical added mass for the surface of the column with the smallest diameter to zero and for the column with the largest diameter to the difference between two half-spheres. I.e., $A_v\simeq \frac{2\pi }{3}\left(R_2^3-R_1^3\right)$ , where the indices 2 and 1 refer to the largest and smallest radius respectively. Experience has shown that this approach may overestimate the vertical added mass; however, it provides the correct results in the limits of $R_1=R_2$ and $R_1=0$ .

7.2.2.2 Horizontal Pontoons

Consider a horizontal pontoon of rectangular cross-section extending from $(x_1,y_1,z_1)$ to $(x_2,y_2,z_2)$ , see Figure 7.8. As the pontoon is horizontal, $z_1=z_2=z_{\mathrm{p}}$ . To establish the added mass matrix in this case, we employ strip theory once more. The pontoon is split into short transverse sections over which the flow is assumed to be 2D. It is assumed that the 2D added mass in the horizontal and vertical direction differs, i.e., $A_h^{\left(2D\right)}\ne A_v^{\left(2D\right)}$ . Further, the added mass in the axial direction due to the end surfaces of the pontoon may be included. Consider a section of length $\Delta L$ of the pontoon. The mid-point of the center axis through the section is located at $\left(x,y,z\right)$ . The pontoon axis forms an angle $\alpha$ with the $x$ -axis. Considering an acceleration in $x$ direction ${\ddot{\eta }}_1$ , the forces acting on the fluid in direction 1, 2 and 3 due to this acceleration are:

$\begin{array}{c}\Delta F_{11}=a_n{A}_h^{\left(2D\right)}\sin \alpha ={\ddot{\eta }}_1\sin \alpha A_h^{\left(2D\right)}\sin \alpha ={\ddot{\eta }}_1{A}_h^{\left(2D\right)}{\sin }^2\alpha .\\ \Delta F_{21}=a_n{A}_h^{\left(2D\right)}\cos \alpha ={\ddot{\eta }}_1{A}_h^{\left(2D\right)}\sin \alpha \cos \alpha \\ \Delta F_{31}=0.\end{array}$([7.14])

The same procedure applies for the two other directions. Integrating over the length of the pontoon thus gives the following added mass matrix for linear translations in $\left(x,y,z\right)$ :

$A_p=\left[\begin{array}{ccc}{A}_{hx}& A_{hxy}& 0\\ A_{hxy}& A_{hy}& 0\\ 0& 0& A_v\end{array}\right],$([7.15])

where:

$\begin{array}{c}{A}_{hx}=A_h^{\left(2D\right)}L{\sin }^2\alpha +2A_e{\cos }^2\alpha .\\ A_{hy}=A_h^{\left(2D\right)}L{\cos }^2\alpha +2A_e{\sin }^2\alpha \\ A_{hxy}=\left(-A_h^{\left(2D\right)}L+2A_e\right)\cos \alpha \sin \alpha \\ A_v=A_v^{\left(2D\right)}L.\end{array}$([7.16])

Here, the end surfaces are also accounted for, the added mass due to an acceleration in axial direction is $2A_e$ . The angle of the pontoon relative to the $x$ -axis is given by $\alpha =\arctan \left(\frac{y_2-y_1}{x_2-x_1}\right)$ and $L$ is the length of the pontoon.

Considering rotational acceleration around the $x$ -axis, and the resulting moments about the other axes $\Delta L$ , the following contributions are obtained:

([7.17])$\begin{array}{c}\Delta M_{11}=-\Delta F_{y}z+\Delta F_{z}y=-\Delta F_h\cos \alpha z+\Delta F_{z}y\\ =-a_h{A}_h^{\left(2D\right)}\Delta L\cos \alpha z+a_v{A}_v^{\left(2D\right)}\Delta Ly\\ ={\ddot{\eta }}_{4}z\cos \alpha A_h^{\left(2D\right)}\Delta L\cos \alpha z+{\ddot{\eta }}_{4}y{A}_v^{\left(2D\right)}\Delta Ly\\ =\left[z^2{A}_h^{\left(2D\right)}{\cos }^2\alpha +y^2{A}_v^{\left(2D\right)}\right]{\ddot{\eta }}_4\Delta L.\end{array}$

Here, $\Delta M_{11}$ is the moment around the $x$ -axis from a small section of the pontoon of length $\Delta L$ located at $\left(x,y,z\right)$ due to an acceleration around the $x$ -axis, ${\ddot{\eta }}_4$ . Similar considerations are made for the other moments. The contributions from each section are integrated over the length of the pontoon. The result is a symmetric rotational inertia matrix, $I_{ij\left(L\right)}$ due to the sectional added mass of the pontoon (details are given in Appendix D):

$I_{11\left(L\right)}=A_h^{\left(2D\right)}L{z}_p^2{\cos }^2\alpha +A_v^{\left(2D\right)}L\left(y_p^2+\frac{1}{12}{L}^2{\sin }^2\alpha \right)$([7.18])

$I_{21\left(L\right)}=-A_v^{\left(2D\right)}L\left(x_p{y}_p^{}+\frac{1}{12}{L}^2\cos \alpha \sin \alpha \right)+A_h^{\left(2D\right)}L{z}_p^2\cos \alpha \sin \alpha$([7.19])

$I_{31\left(L\right)}=-A_h^{\left(2D\right)}L{z}_p\cos \alpha \left(y_p\sin \alpha +x_p\cos \alpha \right)$([7.20])

$I_{22\left(L\right)}=A_v^{\left(2D\right)}L\left(x_p^2+\frac{1}{12}{L}^2{\cos }^2\alpha \right)+A_h^{\left(2D\right)}L{z}_p^2{\sin }^2\alpha$([7.21])

$I_{32\left(L\right)}=-A_h^{\left(2D\right)}L{z}_p\sin \alpha \left(y_p\sin \alpha +x_p\cos \alpha \right)$([7.22])

$I_{33\left(L\right)}=A_h^{\left(2D\right)}L\left[{\left(x_p\cos \alpha +y_p\sin \alpha \right)}^2+\frac{1}{12}{L}^2\right].$([7.23])

$\left(x_p,y_p,z_p\right)$ is the volume center of the pontoon. The end surfaces will only experience pressure in axial direction and only if they are wetted. However, if the pontoon is attached to a column, as illustrated in Figure 6.5, one will normally ignore the effect of the pontoon when considering the added mass of the column. Thus, one should evaluate if the total added mass is better represented by considering the pontoon ends to be wet or dry. This may be done by using a 3D panel method. The contributions from a wetted end to the rotational inertia are given in Appendix D. The total rotational added mass matrix thus becomes:

$I_{p3*3}=I_{\left(L\right)}+I_{\left(e\right)}.$([7.24])

The total 6 x 6 added mass matrix for one horizontal pontoon becomes thus:

$A_{pon}=\left[\begin{array}{cc}{A}_p& -A_p{}^{*}{S}_p\\ A_p{}^{*}{S}_p& I_{p3{}^{*}3}\end{array}\right],$([7.25])

with:

$S_p=\left[\begin{array}{ccc}0& -z_p& y_p\\ z_p& 0& -x_p\\ -y_p& x_p& 0\end{array}\right]\text{.}$([7.26])

Added Mass of a Horizontal Pontoon

Consider a horizontal pontoon with length $L=30m$ , width $B=5m$ and height $H=3m$ . The axis of the pontoon is 8.5 m below the free surface. The 2D added mass in vertical and horizontal direction for the pontoon section is estimated to be $A_v^{\left(2D\right)}=1.67HB$ and $A_h^{\left(2D\right)}=0.72HB$ respectively. The added mass of the end sections, $A_e$ , is ignored in this example. The pontoon is located with one end of the axis at $\left(x=10m,y=0m,z=-8.5m\right)$ . The angle between the pontoon axis and the $x$ -axis is 30 deg; see Figure 7.2.

The 6 × 6 added mass matrix is computed using the strip theory approach as well as using a 3D boundary element method as described in Section 6.4. The distribution of the quadrilateral, constant potential boundary elements are shown by the black lines in Figure 7.2. Note that the panel sizes are reduced toward the edges of the pontoon. This is to improve the computational accuracy. The 3D method accounts for the free surface effect. The added mass thus becomes frequency-dependent. In Table 7.1, the added mass matrix as obtained by strip theory as well as 3D results at a low frequency (0.087 Hz) and a high frequency (0.5 Hz) are presented. The matrix is symmetric. In general, the strip theory method and 3D results do not differ much. One exception is $A_{11}$ , which is sensitive to the added mass related to the end surfaces. This effect was ignored in the strip theory method example.

Figure 7.2 The horizontal pontoon used in the example. Quadrilateral panels as used in the 3D boundary element method.

Table 7.1 Nondimensional added mass for the pontoon shown in Figure 7.2 as computed by strip theory and at a low frequency (0.087 Hz) and a high frequency (0.5 Hz) using a 3D panel method. The added mass values are made dimensionless in the following way: ${\tilde{A}}_{ij}=\frac{A_{ij}}{\rho B^{\gamma }}$ , with $\gamma =3$ for $(i,j)=(1:3,1:3)$ , $\gamma =4$ for $(i,j)=(1:3,4:6)$ and $(i,j)=(4:6,1:3)$ and $\gamma =5$ for $(i,j)=(4:6,4:6)$ . $B=5m$ .

	i / j	1	2	3	4	5	6
Strip	1	0.650	-1.126	0.000	-1.913	-1.105	-6.150
3D high	1	0.871	-0.959	0.000	-1.648	-1.448	-5.717
3D low	1	0.937	-1.015	0.000	-1.696	-1.647	-6.071
Strip	2		1.949	0.000	3.314	1.913	10.652
3D high	2		1.979	0.000	3.351	1.648	10.537
3D low	2		2.109	0.000	3.605	1.696	11.218
Strip	3			6.001	9.002	-27.594	0.000
3D high	3			5.740	8.610	-26.394	0.000
3D low	3			6.381	9.571	-29.340	0.000
Strip	4				23.637	-45.934	18.108
3D high	4				22.254	-42.810	17.878
3D low	4				24.382	-47.567	19.120
Strip	5					142.260	10.455
3D high	5					134.417	9.749
3D low	5					149.038	10.268
Strip	6						66.000
3D high	6						63.498
3D low	6						67.291

7.2.2.3 Horizontal Disks

In some cases, the substructures are equipped with horizontal plates of almost circular shape and with small thickness (as discussed in Section 4.4.1). The reason for using such plates is to tune the dynamic behavior of the platform. The plates will add inertia to the system, thus moving the natural periods in heave, roll and pitch to higher values. At the same time, plates with sharp edges will contribute to viscous damping and thus reduce the motion response in the resonant domain. To improve the damping properties, perforation of the plates is an option. A perforation will, however, reduce the added mass effect of the plate (Reference Molin and NielsenMolin and Nielsen, 2004).

The added mass of a circular disk with radius $R$ oscillating in infinite fluid is given by Lamb (1975, 144):

$A_n=\frac{8}{3}\rho R^3.$([7.27])

In most cases, the plate will be located at the bottom of a vertical column. In such cases the added mass will be somewhat smaller, depending upon the ratio of the disk radius to the column radius (see discussion on vertical columns in Section 7.2.2.1).

Figure 7.3 shows examples of the importance of the perforation to the added mass and linearized damping. The figures are from Reference Molin and NielsenMolin and Nielsen (2004). The nondimensional added mass and damping is presented as a function of the “porous Keulegan–Carpenter number”:

Figure 7.3 Added mass and linearized damping for a perforated disk as a function of the “porous Keulegan–Carpenter number,” KC_por. Period of oscillation 20 s, water depth 100 m, radius of disk 10 m and submergence of disk 20 m.

According to theory as described by Reference Molin and NielsenMolin and Nielsen (2004).

$K{C}_{por}=\frac{1-\tau }{2\mu {\tau }^2}\frac{A}{R}.$([7.28])

Here, $\tau$ is the perforation ratio (open area divided by total area of disk) and $\mu$ is the “discharge ratio”, relating the pressure drop over the disk and the relative fluid velocity through the disk. It is thus related to the flow resistance through the disk, which again is dependent upon the local geometry of the perforation. $\mu$ usually has a value between 0.5 and 1.0. Reference MolinMolin (2011) discusses various approaches to estimate the discharge ratio. It is observed from Figure 7.3 that for small $K{C}_{por}$ , the added mass as well as the damping tends to zero. This case corresponds to a situation with a very large perforation area, $\tau \to 1$ . On the one hand, as $\tau \to 0$ the added mass tends toward the solid disk value of [7.27]. The computed damping tends to zero because the damping due to the edge effect of the disk is not accounted for in this theory. Including the edge effect (see Reference MolinMolin, 2011), a better agreement with the experiments is obtained for the damping.

7.2.2.4 Transformation of the Added Mass Matrix to a New Coordinate System

Frequently the added mass matrix is computed in a local coordinate system, for example, as referred to the center axis of a column or pontoon. For further analysis a different platform coordinate system may be preferred. The transformation between the two coordinate systems may be done as follows. Denote coordinates in the original (local) coordinate system by $x_0=(x_0,y_0,z_0)$ and the new (platform) coordinate system by $x_1=(x_1,y_1,z_1)$ . Assume the two systems are parallel, so that:

$\Delta x=x_1-x_0=\left(\Delta x,\Delta y,\Delta z\right).$([7.29])

The kinetic energy in the fluid while oscillating the body in a certain direction must be independent of the coordinate system used. By considering the kinetic energy using the velocity potentials, it can be shown that the 6 x 6 added mass matrix in the new coordinate system, $A_1$ , is related to the added mass matrix in the original coordinate system, $A_0$ , by:

$A_1=K^T{A}_{0}K,$([7.30])

where:

$K=\left[\begin{array}{cc}{I}_{3*3}& K_1\\ 0_{3*3}& I_{3*3}\end{array}\right].$([7.31])

Here:

$K_1=\left[\begin{array}{ccc}0& -\Delta z& \Delta y\\ \Delta z& 0& -\Delta x\\ -\Delta y& \Delta x& 0\end{array}\right]\text{, }{I}_{3*3}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\text{, }{0}_{3*3}=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\text{.}$([7.32])

Details of the derivation as well as the more general form valid also when rotations are involved may be found in Reference KorotkinKorotkin (2008).

7.3.1 Radiation Damping

Radiation damping is considered to be a linear damping contribution. For a general, rigid floating structure the damping matrix will be a full 6 × 6 matrix with frequency-dependent coefficients. To establish this damping matrix, a 3D radiation-diffraction approach is needed (see Section 6.4). A structure’s capability to generate waves is reduced if the structure is deeply submerged. This implies that a surface-piercing vertical column generally contributes more to the wave radiation damping than, e.g., a horizontal pontoon. However, in a strip theory approach, the 2D damping of a pontoon section may be applied to establish an estimate on the damping for the complete pontoon. The horizontal, normal force on the pontoon due to a harmonic motion ${\eta }_n={\eta }_{An}{e}^{i\omega t}$ normal to a section of the pontoon may be written as:

$\begin{array}{c}{F}_{pn}\left(\omega \right)=\underset{L}{\int }\left[A_h^{\left(2D\right)}\left(\omega \right)\left(-{\omega }^2\right)+B_{rh}^{\left(2D\right)}\left(\omega \right)\left(i\omega \right)\right]dL{\eta }_n\\ =\underset{L}{\int }\left[A_h^{\left(2D\right)}\left(\omega \right)+\frac{1}{i\omega }{B}_{rh}^{\left(2D\right)}\left(\omega \right)\right]dL\left(-{\omega }^2{\eta }_n\right).\end{array}$([7.33])

The subscript r indicates radiation damping. In [7.33] it is indicated that both the added mass and damping are frequency-dependent. The radiation effect will only account for waves radiated perpendicular to the pontoon axis. The 6 by 6 damping matrix can now be established similarly as shown for the added mass matrix. A strip theory approach accounts neither for the interaction of the radiated waves from each of the pontoon strips, nor for the interaction between the pontoons. The interaction effects may in some cases be significant for some frequencies and directions of oscillation.

Within the context of ideal fluid flow and linear wave dynamics, there exists a reciprocity relation that relates the wave forces on a fixed body to the forces needed to oscillate the body in otherwise calm water. This is called the Haskind relation (for further discussion, see Reference NewmanNewman, 1977; Reference FaltinsenFaltinsen, 1990). The relation is valid for general 3D bodies. Applying the Haskind relation on a vertical column with a rotational symmetry, simple relations between the wave excitation forces and the diagonal of the damping matrix are obtained:

$B_{rii}\left(\omega \right)=\gamma \frac{k}{\rho g{c}_g}{\left|\frac{F_i}{{\zeta }_A}\right|}^2.$([7.34])

Here, $F_i$ is the wave force in direction $i$ , $i=\left(1,3,5\right)$ when the waves are propagating along the $x$ -axis. $\gamma =1/4$ for $i=1$ and $5$ and $\gamma =1/2$ for $i=3$ . In deep water, [7.34] may be written as:

$B_{rii}\left(\omega \right)=\gamma \frac{{\omega }^3}{\rho g^3}{\left|\frac{F_i}{{\zeta }_A}\right|}^2.$([7.35])

The computation of the wave force on a vertical column is addressed in Chapter 6. Note that for a substructure with several columns, there may be significant wave interaction between the columns, modifying the radiated waves and thus the damping. A summation of the damping contribution from each of the columns will thus cause errors. One should rather make a summation of the radiated wave fields, taking phases properly into account, and estimate the damping based upon the radiated energy. This is what is obtained by using 3D potential theory methods.

The Haskind relation may also be invoked to estimate the radiation damping for horizontal pontoons. Having established the wave excitation force on a segment $dL$ of the pontoon, the corresponding contribution to the damping may be obtained. Reference NewmanNewman (1962) derived a relation between the 2D wave force and damping for a long horizontal body in deep water and beam seas. For a segment of the pontoon this relation is identical to [7.35] using $\gamma =1$ and considering three degrees of freedom: the transverse horizontal direction, the vertical direction and rotation about an axis parallel to the body axis.

7.3.2 Viscous Damping

Viscous damping has contributions from all structural elements where flow separation occurs. Pure skin friction is in most cases so small that it may be disregarded. The viscous force is normally expressed as a quadratic quantity with respect to the relative velocity, i.e., on a short, 2D section of a vertical column, the viscous force may be written as:

$\Delta F_{visc}=\frac{1}{2}\rho C_{D}D{U}_{rel}\left|U_{rel}\right|\Delta z.$([7.36])

Here, $C_D$ is the drag coefficient, $D$ is the column diameter and $U_{rel}=v_h-\dot{x}\left(z\right)$ is the relative horizontal velocity between water and structure at the z-level considered. $\Delta z$ is the length of the short vertical section considered. It is observed that the viscous force contributes both to excitation via the $v_h^2$ term and damping via ${\dot{x}}_h^2$ . Further, there is a coupling term between the two that contributes to damping or excitation depending upon the phase between the wave particle velocity and the motion velocity.

7.3.3 Linearization of Viscous Damping

In linear dynamic analysis there is a need for linearization of the viscous effect. This is in particular the case when accounting for viscous damping in frequency domain analyses. Due to the nonlinear nature of the damping and the coupling to the fluid velocity, i.e., wave particle and current velocities, it is in general not possible to perform a consistent linearization of the viscous damping. However, disregarding the fluid velocities and considering a single-degree-of-freedom (SDOF) system, an equivalent linear damping can be derived as follows. Consider a long slender structure, e.g., a cylinder. Denote the 2D damping force acting normal to a short section of length, $dz$ by $F_{B}dz$ . The force is assumed to be composed of a linear and a quadratic contribution, i.e.:

$F_{B}dz=\left(B_1\dot{x}+B_2\dot{x}\left|\dot{x}\right|\right)dz.$([7.37])

The body velocity normal to the cylinder axis is assumed to be harmonic, i.e., $\dot{x}=-\omega x_A\sin \left(\omega t\right)$ . To find the equivalent linear damping $B_e$ , the dissipation of energy over one cycle of oscillation, $T=2\pi /\omega$ , is considered. By requiring the dissipated energy to be the same for the equivalent linear system and the quadratic system, $B_e$ is thus found from:

$\underset{T}{\int }{F}_B\dot{x}dt=\underset{T}{\int }\left[B_1\dot{x}+B_2\dot{x}\left|\dot{x}\right|\right]\dot{x}dt=\underset{T}{\int }{B}_e{\dot{x}}^{2}dt.$([7.38])

Inserting for $\dot{x}$ and working out the integrals, the equivalent damping is obtained as:

$B_e=B_1+\frac{8}{3\pi }\omega x_A{B}_2.$([7.39])

It is observed that the equivalent linear damping is proportional to the velocity amplitude, $\omega x_A$ . That implies that an iteration procedure usually must be implemented to establish a proper damping estimate. As the damping is of key importance to the resonant response, one will have to guess a resonant response amplitude, estimate the equivalent damping, then compute the response and correct the damping according to the computed response.

Viscous Damping

Consider the following simple 1D example. A small body is exposed to an oscillating flow given by $v=v_A\exp \left(i\omega t\right)$ . The body is moving harmonically in the same direction with a velocity $\dot{x}={\dot{x}}_A\exp \left(i\left(\omega t+\theta \right)\right)$ . The relative velocity is thus given by $U_{rel}=\mathrm{Re}\left\{v-\dot{x}\right\}$ . The viscous force is given from [7.36]. Considering one cycle of oscillation, the average dissipated power becomes:

$P=-\frac{1}{T}\underset{0}{\overset{T}{\int }}{F}_{visc}\dot{x}dt,$

where $T=2\pi /\omega$ . In Figure 7.4 the dissipated power is plotted as a function of phasing between the fluid velocity and the body velocity. It is observed that for cases with $v_A/{\dot{x}}_A<1$ , the damping (dissipated power) is positive independent of phasing between the fluid motion and the body motion. However, for $v_A/{\dot{x}}_A>1$ , the damping may become negative for certain phases, implying an excitation effect. For zero fluid velocity the average dissipated power amounts to $\frac{4}{3\pi }\left[\frac{1}{2}\rho C_{D}A{\dot{x}}_A^3\right]$ .

Figure 7.4 Average dissipated power as a function of phase between fluid velocity and body velocity. Amplitude ratio $v_A/{\dot{x}}_A$ ranging from 0 to 2.

The above procedure works fine for a SDOF and in cases where the various modes of motion are uncoupled or close to uncoupled. For most substructures the heave mode has little coupling to other modes, while, for example, the surge and pitch modes may have significant coupling. Frequently the surge motion is referred to the waterline level, while the eigenmode for pitch may have a center of rotation far below the waterline. This causes a significant coupling between the surge and pitch motion when viscous drag forces are accounted for.

To illustrate this point, consider a spar platform designed as a vertical cylinder with constant diameter and a pure surge motion. The drag forces in surge and pitch may then be written as:

$\begin{array}{c}{F}_1\left(t\right)=C\underset{z_b}{\overset{z_t}{\int }}{\dot{x}}_1\left|{\dot{x}}_1\right|dz=C{\dot{x}}_1\left|{\dot{x}}_1\right|\left(z_t-z_b\right)=C{\dot{x}}_1\left|{\dot{x}}_1\right|L\\ F_5\left(t\right)=C\underset{z_b}{\overset{z_t}{\int }}z{\dot{x}}_1\left|{\dot{x}}_1\right|dz=C{\dot{x}}_1\left|{\dot{x}}_1\right|\frac{\left(z_t^2-z_b^2\right)}{2}.\end{array}$([7.40])

Here, $z_t=0$ and $z_b=-L$ are the top and bottom coordinates of the cylinder. $C=1/2\rho C_{D}D$ , with $D$ being the diameter of the cylinder. Computing the dissipated energy as above, the linearized damping in surge is obtained as:

$B_{11lin}=\frac{8}{3\pi }{\dot{x}}_{1A}CL.$([7.41])

Similarly, integrating the pitch moment over one cycle of oscillation and comparing the quadratic and the linear process, a linearized coupling term between the surge motion and pitch moment is obtained as:

$B_{51lin}=\frac{8}{6\pi }C{\dot{x}}_{1A}\left(z_t^2-z_b^2\right).$([7.42])

The above approach may be repeated for a pure pitch motion, with the pitch motion referred to $z=0$ . The surge and pitch forces corresponding to [7.40] now become:

$\begin{array}{c}{F}_1\left(t\right)=C\underset{z_b}{\overset{z_t}{\int }}z{\dot{x}}_5\left|z{\dot{x}}_5\right|dz=C{\dot{x}}_5\left|{\dot{x}}_5\right|\frac{\left(z_t^3-z_b^3\right)}{3}\\ F_5\left(t\right)=C\underset{z_b}{\overset{z_t}{\int }}zz{\dot{x}}_5\left|{\dot{x}}_5\right|dz=C{\dot{x}}_5\left|{\dot{x}}_5\right|\frac{\left(z_t^4-z_b^4\right)}{4}.\end{array}$([7.43])

The linearized damping coefficients for the pure pitch motion are obtained as:

$\begin{array}{c}{B}_{15lin}=\frac{8}{9\pi }C{\dot{x}}_{5A}\left(z_t^3-z_b^3\right)\\ B_{55lin}=\frac{2}{3\pi }C{\dot{x}}_{5A}\left(z_t^4-z_b^4\right).\end{array}$([7.44])

From the above relations it is observed that the linearized damping depends upon the choice of surge and pitch velocity amplitude used as basis for the linearization. If one focuses on a good linearization of the pitch damping at the pitch natural period, the coupling effect will cause damping also in surge that may be unrealistic. To succeed in linearization of the damping, one should aim at reducing the coupling terms in the damping matrix as much as possible. This is normally obtained by using a coordinate system in which the modes of motions are close to the eigenmodes of the system.

Viscous Damping in Coupled Motion

Consider a vertical cylinder with length equal to draft 100 m and diameter 10 m. Center of gravity is at -70 m. The 2D added mass and drag coefficients are both set to 1.0. A horizontal mooring system with stiffness 50 kN/m is attached at the waterline level. The natural periods in surge and pitch are 118.6 and 17.70 s. The pitch eigenmode has a center of rotation at z = -61.5 m. The linearized coupled damping matrix has been established by assuming a surge amplitude of 0.7 m and a pitch amplitude of 0.5 deg. The system is set into free oscillations in calm water. The initial surge amplitude is 1.0 m, while the initial pitch and all initial velocities are set to zero. Two cases are considered, one using the quadratic damping and one using the linearized damping matrix. Figure 7.5 shows the results for the two cases.

It observed that the surge motion is well reproduced using the linearized damping (upper-left), even if the surge damping force contains large contributions from the pitch motion (lower-left). Initially, the pitch motion obtained by the linearized equations follows the motions obtained by using quadratic damping well (upper-right). This is because the inertia effects dominate initially. After a while, however, the pitch motion is more and more dominated by the surge natural period in the linearized case. Large differences are also observed in the pitch drag moment (lower right).

Figure 7.5 Motion decay in surge and pitch for a floating vertical circular cylinder using quadratic and linear damping. Upper figures: displacements after an initial surge of 1.0 m and zero pitch; lower figures: damping force in surge and moment in pitch.

7.3.4 The Drag Coefficient

In most practical cases, the viscous forces are related to the pressure distribution over the structure due to flow separation. That implies that the drag coefficient, $C_D$ , depends upon the body geometry, including surface roughness as well as flow conditions. The flow conditions are expressed via three nondimensional numbers: the Reynolds number, $\mathrm{Re}=\frac{UD}{\nu }$ ; the Keulegan–Carpenter number, $KC=\frac{U_{A}T}{D}$ ; and the relative current number, $=U_c/U_A$ . Here, $U$ is a characteristic flow velocity; $U_A$ is the amplitude of the oscillatory velocity, either of the body or the flow; $U_c$ is a steady current velocity; $D$ is a characteristic cross-sectional dimension of the body; $\nu$ is the kinematic viscosity of the fluid; and $T$ is the period of oscillation. Thorough discussions of the relations between these parameters and the drag coefficient are given in, e.g., Reference Sarpkaya and IsaacsonSarpkaya and Isacsson (1981) and Reference FaltinsenFaltinsen (1990). Recommended values to be used are found in, e.g., DNV (2021c).

For circular cylinders the drag coefficient is sensitive to where flow separation takes place, which again is sensitive to all the above parameters. For cross-sections with a rectangular shape, the drag coefficient is less dependent upon the flow conditions as flow separation occurs at the sharp corners. Classical results for the drag coefficient for a 2D circular cylinder in steady flow as a function of the Reynolds number are shown in Figure 7.6. A drop in the drag coefficient for the Reynolds number in the order of 10⁵ is observed. As the surface roughness of the cylinder increases, the drop occurs at a lower Reynolds number, and is less than for a smooth cylinder.

Figure 7.6 Drag coefficient for a 2D circular cylinder in steady flow as a function of the Reynolds number and surface roughness $k$ .

Reproduced from DNV (2021c).

7.4.1 Slender Bodies of General Shape

The estimation of wave excitation forces on floating substructures is now to be addressed. As for the discussion on the added mass coefficients above, structures composed of slender vertical cylinders and a horizontal pontoon using strip theory will be addressed. One of the advantages with this approach is that it is straightforward to use in a finite element analysis of the structure based upon beam elements. However, the global forces are focused upon here as these are needed for estimating the rigid-body motions. Some floating substructures may have a barge-like shape (see Section 4.4.4). To estimate the wave forces on such structures, 3D methods as discussed in Chapter 6 should be used.

As for the added mass, the forces need to be referred to a common point of reference. Further, by using the strip theory approach, it is assumed that the flow over any cross-section of the columns or pontoons may be considered to be 2D, even if the cross-sectional dimensions are changing. No hydrodynamic interaction is assumed between the various structural components.

In computing the six degrees of freedom of rigid-body wave forces, it may be convenient to refer to a coordinate system located at the mean sea surface, with $z=0$ at the surface level and positive upward.

7.4.2 Wave Forces on a Vertical Column

Consider regular waves propagating in direction $\beta$ relative to the $x$ -axis. The complex wave potential may, see Chapter 2, be written as:

$\varphi =\frac{ig{\zeta }_A}{\omega }\frac{\cosh \left[k\left(z+d\right)\right]}{\cosh \left(kd\right)}{e}^{i(\omega t-kx\cos \beta -ky\sin \beta )}.$([7.45])

There are two options to estimate the wave force on a vertical circular column. One may either assume a very slender column, with no diffraction effects, and apply the Morison equation or one may include diffraction effects and apply the MacCamy and Fuchs theory. Both these approaches are discussed in Chapter 6. However, the expressions need to be modified to account for the fact that the column does not extend to the sea floor. Using a strip theory approach, this implies that the sectional force is integrated from the bottom to the top of the column, i.e., from $z=z_b$ to $z=z_t$ . $\left(z_b<z_t<0\right)$ . It is assumed that the column axis is located in $\left(x_c,y_c\right)$ . Similarly as for the monopile, the surge and sway forces are now obtained as:

$\begin{array}{c}{F}_1=\pi \rho g{R}^2{\zeta }_A{C}_m\left\{\frac{\sinh \left(k{s}_t\right)-\sinh \left(k{s}_b\right)}{\cosh \left(kd\right)}\right\}{e}^{i\left(\omega t+\delta -k{x}_c\cos \beta -k{y}_c\sin \beta \right)}\cos \beta .\\ F_2=\pi \rho g{R}^2{\zeta }_A{C}_m\left\{\frac{\sinh \left(k{s}_t\right)-\sinh \left(k{s}_b\right)}{\cosh \left(kd\right)}\right\}{e}^{i\left(\omega t+\delta -k{x}_c\cos \beta -k{y}_c\sin \beta \right)}\sin \beta .\end{array}$([7.46])

Here, $C_m$ and $\delta$ are given in [6.15], $s_t=z_t+d$ and $s_b=z_b+d$ . It is observed that the forces have an extra phase shift as the column is offset from $x=y=0$ . The vertical force may be estimated using the pressures from the undisturbed wave, the Froude-Krylov pressure at the bottom and top surfaces of the column, i.e.:

$\begin{array}{c}{F}_3=-\pi \rho R^2\left\{{\gamma }_b\frac{\partial \varphi \left(z_b\right)}{\partial t}-{\gamma }_t\frac{\partial \varphi \left(z_t\right)}{\partial t}\right\}\\ =\pi \rho g{R}^2{\zeta }_A\left\{\frac{{\gamma }_b\cosh \left(k{s}_b\right)-{\gamma }_t\cosh \left(k{s}_t\right)}{\cosh \left(kd\right)}\right\}{e}^{i\left(\omega t-k{x}_c\cos \beta -k{y}_c\sin \beta \right)}.\end{array}$([7.47])

If the column is surface-piercing, $z_t=0$ , there is no wave pressure on the top end and ${\gamma }_t=0$ . Similarly, if the column is sitting on the bottom, ${\gamma }_b=0$ . For wetted end surfaces, $\gamma =1$ . Note that a bottom-fixed vertical cylinder piercing the free surface is not exposed to vertical wave forces.

The moments about the $x$ - and $y$ -axes are obtained similarly as in [6.15] and [6.16]; accounting for the horizontal offset, the direction of the waves and that the moment axis is now at the free surface level, the roll and pitch moments are obtained as:

$\begin{array}{c}{F}_4=\pi \rho g{R}^2{\zeta }_A{C}_m\frac{1}{k}\left\{\frac{-k{z}_t\sinh \left(k{s}_t\right)+k{z}_b\sinh \left(k{s}_b\right)+\cosh \left(k{s}_t\right)-\cosh \left(k{s}_b\right)}{\cosh \left(kd\right)}\right\}\cdot .\\ e^{i\left(\omega t+\delta -k{x}_c\cos \beta -k{y}_c\sin \beta \right)}\sin \beta +F_3{y}_c\\ \\ F_5=-\pi \rho g{R}^2{\zeta }_A{C}_m\frac{1}{k}\left\{\frac{-k{z}_t\sinh \left(k{s}_t\right)+k{z}_b\sinh \left(k{s}_b\right)+\cosh \left(k{s}_t\right)-\cosh \left(k{s}_b\right)}{\cosh \left(kd\right)}\right\}\cdot \\ e^{i\left(\omega t+\delta -k{x}_c\cos \beta -k{y}_c\sin \beta \right)}\cos \beta -F_3{x}_c\end{array}$([7.48])

The last term in the above expressions is due to the moment contribution from the vertical wave force on the column. Note that in the deep-water case, $d\to \infty$ , $\sinh \left(ks\right)/\cosh \left(kd\right)\to \cosh \left(ks\right)/\cosh \left(kd\right)\to e^{kz}$ .

The moment around the $z$ -axis, the yaw moment, is obtained from the horizontal forces:

$F_6=-F_1{y}_c+F_2{x}_c.$([7.49])

All the above expressions are valid for one single column. If several columns are present, the total force is obtained by summation over all the columns. If a column diameter is changing over the length of the column, a pragmatic approach is to split the column into, e.g., two parts and compute the force on each of the parts separately. This is illustrated in Figure 7.7. The split may be done into two or more parts. To obtain a realistic model, the body volume should be conserved. The vertical wave force at the conical part of the column may be modeled by the wave pressure at the area representing the difference between the cross-sectional area of the cylinders. The modeling of this force may be improved by representing the conical section by more cylinders.

Figure 7.7 Vertical column with conical section modeled by two cylindrical sections.

If the distance between the columns is not large compared to the diameter of the columns, the interaction effect may be important. In such cases, a full 3D analysis should be performed to obtain accurate estimates on the wave forces.

7.4.3 Wave Forces on a Horizontal Pontoon

Horizontal pontoons in most cases either have a circular or a rectangular cross-section. In the case of a rectangular cross-section the added mass coefficient in horizontal and vertical directions differs. Consider the horizontal pontoon illustrated in Figure 7.8. A slender body is assumed, implying that the length of the pontoon is much longer than the characteristic cross-sectional dimension. Further, long wavelength theory is used, implying that the wavelength is much longer than the characteristic width of the pontoon. Following the principles outlined in Reference FaltinsenFaltinsen (1990), the vertical and horizontal forces on a 2D section of length $\Delta L$ may be written as:

Figure 7.8 A horizontal pontoon. Notations used in deriving the wave forces. $\alpha$ is the direction of the pontoon axis relative to the coordinate system used for the body. $\left(x_1,y_1,z_p\right)$ and $\left(x_2,y_2,z_p\right)$ are the coordinates of the end points. $\beta$ is the direction of wave propagation. $\left(s,n,z\right)$ are the local pontoon coordinates, parallel and perpendicular to the pontoon axis. The $\left(x,y\right)$ and $\left(s,n\right)$ planes coincide.

$\begin{array}{c}\Delta F_n=\left[\rho A_p+A_n^{\left(2D\right)}\right]a_n\Delta L\\ \Delta F_v=\left[\rho A_p+A_v^{\left(2D\right)}\right]a_v\Delta L.\end{array}$([7.50])

Here, $A_p$ is the cross-sectional area of the pontoon; $A_n^{\left(2D\right)}$ is the 2D added mass in horizontal direction, normal to the pontoon axis; $A_v^{\left(2D\right)}$ is the 2D added mass in vertical direction; $a_n$ and $a_v$ are the acceleration in the water horizontally, normal to the pontoon axis and in vertical direction respectively.

To obtain the total forces on the pontoon, the forces in [7.50] have to be integrated over the length of the pontoon. To perform this integration, it is convenient to introduce the local $\left(s,n\right)$ coordinates, as illustrated in Figure 7.8. The relations between the two coordinate systems are:

$\begin{array}{c}s=x\cos \alpha +y\sin \alpha \\ n=-x\sin \alpha +y\cos \alpha .\end{array}$([7.51])

The coordinates of the end points of the pontoon axis are thus:

$\begin{array}{c}{s}_2=x_2\cos \alpha +y_2\sin \alpha \\ s_1=x_1\cos \alpha +y_1\sin \alpha \\ n_2=n_1=-x_1\sin \alpha +y_1\cos \alpha .\end{array}$([7.52])

Considering a pontoon of constant cross-sectional shape, it is only the normal component of the horizontal acceleration, $a_n$ , and the vertical acceleration, $a_v$ in [7.50], that vary along the pontoon length. The horizontal acceleration perpendicular to the pontoon axis may be written as:

$\begin{array}{c}{a}_n=-a_x\sin \alpha +a_y\cos \alpha =i{a}_{nA}\left[-\cos \beta \sin \alpha +\sin \beta \cos \alpha \right]e^{i\left(\omega t-kx\cos \beta -ky\sin \beta \right)}\\ =i{a}_{nA}\sin \left(\beta -\alpha \right)e^{i\left(\omega t-kx\cos \beta -ky\sin \beta \right)}\\ \text{with }{a}_{nA}=kg{\zeta }_A\frac{\cosh \left(k\left(z_p+d\right)\right)}{\cosh \left(kd\right)}.\end{array}$([7.53])

Integrating along the pontoon, the following result is obtained for the horizontal force on the pontoon: