Motion Transfer Functions

Figure 1. Motion Characteristics of Floating Vessel Motion Transfer Function Specification, Diagnostics and Transformation

By Ivar J. Fylling, MARINTEK

1. Introduction

Motion transfer functions have proved to give an efficient description of the motion characteristics of floating vessels. The transfer functions are calculated for the 6 motion degrees of freedom for a specified reference point on the vessel, either from potential theory or from model test results. In the interpretation of motion characteristics, we normally focus on the amplitudes and often neglect the phase angles.

In practical use one will often have to calculate motions of other points on the vessel, motions relative to other vessels, or motions referring to other coordinate systems. This will require a combination of two or more degrees of freedom, and the result will depend on both amplitudes and phase angles.

A lot of trouble has been caused by erroneous interpretation of transfer functions and erroneous transformation. In many cases a simple checking of asymptotic phase angles can be useful in verifying the actual interpretation. The present note is an effort to describe the asymptotic phase angles, the different coordinate systems that are often used, and to show a recipe for transformation of phase angles.

2. Terminology and Conventions

Phase angle - If nothing else is said this means forward phase shift. Backwards phase shift, delay, may be denoted phase lag.
Coordinate system - All coordinate systems are right-handed Cartesian systems.
Transfer function - Physical relation between harmonic wave and linear response \(\mathrm {X(t)=R_x\zeta_a\sin(\omega t+\phi _x)}\). The transfer function consists of an amplitude ratio, \(\mathrm {R_x=\displaystyle \frac{X_a}{\zeta_a}}\), and a phase angle, \(\mathrm {\phi _x}\).

Table 1. Naming conventions
Vectorial components	\(\mathrm {x_1}\)	\(\mathrm {x_2}\)	\(\mathrm {x_3}\)	\(\mathrm {x_4}\)	\(\mathrm {x_5}\)	\(\mathrm {x_6}\)
Names of axes	X	Y	Z
Wave particle motions	\(\mathrm {\xi }\)	\(\mathrm {\eta }\)	\(\mathrm {\zeta}\)	\(\mathrm {\gamma _1}\)	\(\mathrm {\gamma _2}\)	\(\mathrm {\gamma _3}\)
Names of motions	Surge	Sway	Heave	Roll	Pitch	Yaw

Observe that the naming of motions refers to the coordinate system, regardless of whether the X-axis is pointing forwards or backwards in a vessel.

3. Wave Potential

The velocity potential for a regular wave according to Airy’s theory can be expressed as follows:

\[\phi _0=\frac{\zeta_ag}{\omega }C_1\cos\big(-\omega t+kx\cos(\beta )+ky\sin(\beta )-\phi _\zeta\big)\]

where

\(\mathrm {\zeta_a\qquad}\) is the wave amplitude
\(\mathrm {g\qquad\:\:}\) is the acceleration due to gravity
\(\mathrm {k\qquad\:\:}\) is the wave number
\(\mathrm {\beta \qquad\:}\) is the direction of wave propagation (\(\mathrm {\beta =0}\) corresponds to wave propagation along the positive x-axis)
\(\mathrm {\phi _\zeta\qquad}\) is the wave component phase angle .

\(\mathrm {C_1}\) is given by

\[C_1=\frac{\cosh\big(k(z+d)\big)}{\cosh(kd)}\]

where \(\mathrm {d}\) is the water depth.

In deep water, \(\mathrm {C_1}\) can be approximated by

\[C_1\approx e^{kz}\]

We then obtain the following relations for the particle velocities and accelerations in the undisturbed wave field:

\[\begin{array}{l}\displaystyle v_x=\zeta_a\omega \cos(\beta )\:C_2\sin(\alpha )\\\\\displaystyle v_y=\zeta_a\omega \sin(\beta )\:C_2\sin(\alpha )\\\\\displaystyle v_z=\zeta_a\omega C_3\cos(\alpha )\\\\\displaystyle a_x=\zeta_a\omega ^2\cos(\beta )\:C_2\cos(\alpha )\\\\\displaystyle a_y=\zeta_a\omega ^2\sin(\beta )\:C_2\cos(\alpha )\\\\\displaystyle a_z=-\zeta_a\omega ^2C_3\sin(\alpha )\end{array}\]

where

\[\alpha =\big(\omega t-kx\cos(\beta )-ky\sin(\beta )+\phi _\zeta\big)\]

Using the deep water approximation we have

\[C_1=C_2=C_3=e^{kz}\]

Taking into account finite water depth we have

\[\begin{array}{l}\displaystyle C_1=\frac{\cosh\big(k(z+d)\big)}{\cosh(kd)}\\\\\displaystyle C_2=\frac{\cosh\big(k(z+d)\big)}{\sinh(kd)}\\\\\displaystyle C_3=\frac{\sinh\big(k(z+d)\big)}{\sinh(kd)}\end{array}\]

The surface elevation is given by

\[\zeta=\zeta_a\sin(\alpha )\]

It is important to notice these definitions.

Similarly the linearized dynamic pressure is given by

\[p_d=-\rho g\zeta_aC_1\sin(\alpha )\]

4. Phase Angles of Wave Particle Motions

4.1. Particle motions

Particle motions are obtained by integrating the velocity functions, Equation (4)-Equation (5).

\[\begin{array}{l}\displaystyle \xi =\int^t_0(v_x\mathrm {d}t)+x_0=\zeta_a\cos(\beta )C_2\big(-\cos(\alpha )\big)\\\\\displaystyle \eta =\int^t_0(v_y\mathrm {d}t)+y_0=\zeta_a\sin(\beta )C_2\big(-\cos(\alpha )\big)\\\\\displaystyle \zeta=\int^t_0(v_z\mathrm {d}t)+z_0=\zeta_aC_3\sin(\alpha )\end{array}\]

\(\mathrm {C_2}\) and \(\mathrm {C_3}\) are depth-dependent functions, according to Equation (7).

The integration constants \(\mathrm {x_0}\), \(\mathrm {y_0}\) and \(\mathrm {z_0}\) have been selected so that the average value is zero.

Thus, the particle motions are expressed by

\[\begin{array}{l}\xi =\zeta_a\cos(\beta )C_2\big(-\cos(\alpha )\big)\\\\\eta =\zeta_a\sin(\beta )C_2\big(-\cos(\alpha )\big)\\\\\zeta=\zeta_aC_3\sin(\alpha )\end{array}\]

4.2. Phase angles of particle translations

In nearly all contexts, the surface elevation is selected as the reference process when describing waves and wave-induced responses. According to the previous section, the particle translations can be written

\[\begin{array}{l}\zeta=\zeta_aC_3\sin(\alpha )\\\\\xi =\zeta_a\cos(\beta )C_2\big(-\cos(\alpha )\big)=\zeta_a\cos(\beta )C_2\sin(\alpha +\phi _\xi )\\\\\eta =\zeta_a\sin(\beta )C_2\big(-\cos(\alpha )\big)=\zeta_a\sin(\beta )C_2\sin(\alpha +\phi _\eta )\end{array}\]

where

\[\phi _\xi =\phi _\eta =-\frac{\pi }{2}\]

Differentiation with respect to time gives an increase of phase angle of \(\mathrm {\displaystyle \frac{\pi }{2}}\).

4.3. Transfer functions of wave angular motions

The wave angular motions of a wave particle rod that at rest is parallel with the x-axis (right handed rotations positive) are defined by

\[\begin{array}{l}\displaystyle \gamma _1=\frac{\partial \zeta}{\partial y}=\zeta_aC_3k\sin(\beta )\big(-\cos(\alpha )\big)\\\\\displaystyle \gamma _2=-\frac{\partial \zeta}{\partial x}=\zeta_aC_3k\cos(\beta )\big(+\cos(\alpha )\big)\\\\\displaystyle \gamma _3=\frac{\partial \eta }{\partial x}=\zeta_aC_2k\sin(\beta )\big(-\sin(\alpha )\big)\end{array}\]

This gives phase angles

\[\begin{array}{l}\displaystyle \phi _{\gamma 1}=-\frac{\pi }{2}\\\\\displaystyle \phi _{\gamma 2}=+\frac{\pi }{2}\\\\\displaystyle \phi _{\gamma 3}=\pi \end{array}\]

These will be the asymptotic phase angles of roll, pitch, and yaw motion of a slender ship when the wave length becomes large compared with the ship and the ship is oriented parallel with the x-axis.

4.3.1. Other motions

All other motions or other responses, \(\mathrm {r}\), that are linearly dependent on the waves are expressed in the following way:

\[r=r_a\sin\big(\omega t+\phi _\zeta+\phi _p+\phi _r\big)\]

where

\(\mathrm {r_a\quad }\) = harmonic amplitude
\(\mathrm {\phi _p\quad }\) = phase angle relative to the surface elevation.

4.4. Summary of phase angles of wave particle motions

This overview can be used to check asymptotic behaviour of floating structures (long wavelength) and can serve as a basis for converting transfer functions from one reference system to another.

4.4.1. Coordinate system

Figure 2. Wave profile

The wave profile is at \(\mathrm {t=0}\) for \(\mathrm {\phi _\zeta=0}\).

4.4.2. Surface elevation

\[\zeta=\zeta_aC_3\sin\big(\omega t+\phi _\zeta+\phi _p\big)\]

where

\(\mathrm {\phi _p\quad }\) = \(\mathrm {-kx\cos(\beta )-ky\sin(\beta )}\)
\(\mathrm {\phi _\zeta\quad }\) = defines the wave state at the origin at time \(\mathrm {t=0}\)
\(\mathrm {\zeta_a\quad }\) = wave amplitude at the surface
\(\mathrm {C_3\quad }\) = is a depth- and frequency-dependent function

Table 2. Summary of phase angles, \(\mathrm {\phi _r}\)
Motion		Displacement	Velocity	Acceleration
Surge	\(\mathrm {\xi }\)	\(\mathrm {-\displaystyle \frac{\pi }{2}}\)	\(\mathrm {0}\)	\(\mathrm {+\displaystyle \frac{\pi }{2}}\)
Sway	\(\mathrm {\eta }\)	\(\mathrm {-\displaystyle \frac{\pi }{2}}\)	\(\mathrm {0}\)	\(\mathrm {+\displaystyle \frac{\pi }{2}}\)
Heave	\(\mathrm {\zeta}\)	\(\mathrm {0}\)	\(\mathrm {+\displaystyle \frac{\pi }{2}}\)	\(\mathrm {+\pi }\)
Roll	\(\mathrm {\gamma _1}\)	\(\mathrm {-\displaystyle \frac{\pi }{2}}\)	\(\mathrm {0}\)	\(\mathrm {+\displaystyle \frac{\pi }{2}}\)
Pitch	\(\mathrm {\gamma _2}\)	\(\mathrm {+\displaystyle \frac{\pi }{2}}\)	\(\mathrm {+\pi }\)	\(\mathrm {+\displaystyle \frac{3\pi }{2}}\)
Yaw	\(\mathrm {\gamma _3}\)	\(\mathrm {+\pi }\)	\(\mathrm {+\displaystyle \frac{3\pi }{2}}\)	\(\mathrm {0}\)

Table 3. Summary of motion `amplitudes`
Motion		Displacement	Velocity	Acceleration
Surge	\(\mathrm {\xi }\)	\(\mathrm {\cos(\beta )C_2\zeta_a}\)	\(\mathrm {\omega \cos(\beta )C_2\zeta_a}\)	\(\mathrm {\omega ^2\cos(\beta )C_2\zeta_a}\)
Sway	\(\mathrm {\eta }\)	\(\mathrm {\sin(\beta )C_2\zeta_a}\)	\(\mathrm {\omega \sin(\beta )C_2\zeta_a}\)	\(\mathrm {\omega ^2\sin(\beta )C_2\zeta_a}\)
Heave	\(\mathrm {\zeta}\)	\(\mathrm {C_3\zeta_a}\)	\(\mathrm {\omega C_3\zeta_a}\)	\(\mathrm {\omega ^2C_3\zeta_a}\)
Roll	\(\mathrm {\gamma _1}\)	\(\mathrm {k\sin(\beta )C_3\zeta_a}\)	\(\mathrm {\omega k\sin(\beta )C_3\zeta_a}\)	\(\mathrm {\omega ^2k\sin(\beta )C_3\zeta_a}\)
Pitch	\(\mathrm {\gamma _2}\)	\(\mathrm {k\cos(\beta )C_3\zeta_a}\)	\(\mathrm {\omega k\cos(\beta )C_3\zeta_a}\)	\(\mathrm {\omega ^2k\cos(\beta )C_3\zeta_a}\)
Yaw	\(\mathrm {\gamma _3}\)	\(\mathrm {k\sin(\beta )}\) \(\mathrm {\cdot \cos(\beta )C_2\zeta_a}\)	\(\mathrm {\omega k\sin(\beta )}\) \(\mathrm {\cdot \cos(\beta )C_2\zeta_a}\)	\(\mathrm {\omega ^2k\sin(\beta )}\) \(\mathrm {\cdot \cos(\beta )C_2\zeta_a}\)

\(\mathrm {C_2}\) is a depth- and frequency-dependent function.

Observe that \(\mathrm {\sin(\beta )}\) and \(\mathrm {\cos(\beta )}\) enter the expression for amplitudes and include a sign.

When changing coordinate system this can be taken care of in the phase angles if \(\mathrm {\beta }\) goes out of the first quadrant and one wishes to have positive amplitude expressions.

5. Complex Notation of Harmonic Wave Field, Transfer Functions and Responses

We define a complex harmonic wave component by:

\[\begin{array}{l}\displaystyle Z=\zeta_aC_3\mathrm {exp}\big(i(\omega t+\phi _\zeta+\phi _p)\big)\\\\\displaystyle |Z|=\zeta_aC_3\\\\\displaystyle \mathrm {Arg}(Z)=+\omega t+\phi _\zeta+\phi _p\\\\\displaystyle \mathrm {exp}\big(i\mathrm {Arg}(Z)\big)=\cos\big(\mathrm {Arg}(Z)\big)+i\sin\big(\mathrm {Arg}(Z)\big)\end{array}\]

Thus, the surface elevation is

\[\begin{array}{l}\zeta=\mathrm {Im}\big\{Z\big\}=\zeta_aC_3\sin\big(\omega t+\phi _\zeta+\phi _p\big)\\\\\displaystyle =\mathrm {Re}\{Z\mathrm {exp}(-i\frac{\pi }{2})\}=\zeta_aC_3\cos(\omega t+\phi _\zeta+\phi _p-\frac{\pi }{2})\end{array}\]

All other responses, \(\mathrm {r}\), are related to the surface elevation by complex transfer functions, \(\mathrm {H_r}\), and can be derived from a complex harmonic function, \(\mathrm {R}\):

\[\begin{array}{l}R=H_rZ\\\\r=\mathrm {Im}\{R\}\\\\\displaystyle |H_r|=\frac{r_a}{\zeta_a}\\\\\mathrm {Arg}(H_r)=\phi _r\end{array}\]

6. Change of Coordinate Systems and Sign Conventions

6.1. Basis

A right-handed Cartesian coordinate system is used with the Z-axis pointing upwards. The sea surface elevation is:

\[\zeta=\zeta_a\sin\Big(\omega t-k\big(X\cos(\beta )+Y\sin(\beta )\big)+\phi _\zeta\Big)\]

where

\(\mathrm {\beta \quad \:\:}\) is the propagation direction
\(\mathrm {\phi _\zeta\quad }\) is the phase angle, interpreted as forward phase shift

The transfer function of any response \(\mathrm {j}\) for a given frequency is defined by an amplitude ratio, \(\mathrm {\big|H_j(\beta )\big|}\) and a phase angle, \(\mathrm {\phi _j(\beta )}\).

\[X_j=\zeta_a\big|H_j(\beta )\big|\sin\Big(\omega t-k\big(x\cos(\beta )+y\sin(\beta )\big)+\phi _\zeta+\phi _j(\beta )\Big)\]

Denoting the base case as case no. 1, the following alternatives are discussed: -# 180 deg. rotation about the Z-axis. This is often used in order to obtain a head wave condition with \(\mathrm {\beta =0}\).
180 deg. rotation about X-axis. This may be done in order to get positive rotation with the bow turning starboard, which is a nautical convention. This requires that the Z-axis points downwards in a right-handed system. -# Change of direction convention of the waves, so that \(\mathrm {\beta }\) is the direction from which the wave originates (coming-from direction) instead of the propagation direction. This is also according to nautical practice.
Change of phase angle convention so that \(\mathrm {\phi }\) denotes phase lag instead of phase addition. This is purely a matter of taste and habit.
Utilization of X-Z symmetry by mirroring the transfer functions about the X-Z plane.

Figure 3 gives an overview of these alternatives. For the coordinate selections, cases 1, 2, 3, directions and phase angles are written fully. For the alternative sign conventions, cases 4, 5, 6, only the changes are given. These can apply to any one of the cases 1, 2, 3.

Figure 3. Wave directions and phase angles

Notes:

A change of \(\mathrm {+\pi }\) is equivalent to a change of \(\mathrm {-\pi }\).
All cases except 6 represent the same physical system and responses.
The cases 4, 5, 6 can be superimposed on any of the coordinate system alternatives 1, 2, 3.
If relative motions are to be calculated, the same modifications must be carried out both for vessel transfer functions and for wave particle motion transfer function.

7. Examples

7.1. Example 1

The NSRDC program has been run (case 2, 5). The resulting transfer functions are to be used as MOSSI input (case 3, 4).

According to Figure 3:

Table 4. Example 1
MOSSI (M), (Case 3, 4)	NSRDC (N), (Case 2, 5)	MOSSI with \(\mathrm {\phi _N}\) as basis
\(\mathrm {\beta _M=-\beta +\pi }\)	\(\mathrm {\beta _N=\beta +\pi }\)	\(\mathrm {\beta _M=-\beta _N}\)
\(\mathrm {\Phi_1}\)	\(\mathrm {\Phi_2}\)	\(\mathrm {\Phi_3}\)

where \(\mathrm {\Phi_1=\phi _M=\begin{cases}\phi _1\\\phi _2+\pi \\\phi _3+\pi \\\phi _4\\\phi _5+\pi \\\phi _6+\pi \end{cases}}\), \(\mathrm {\qquad\Phi_2=\phi _N=\begin{cases}-\phi _1-\pi \\-\phi _2-\pi \\-\phi _3\\-\phi _4-\pi \\-\phi _5-\pi \\-\phi _6\end{cases}}\), \(\mathrm {\qquad\Phi_3=\phi _M=\begin{cases}-\phi _{N1}-\pi \\-\phi _{N2}\\-\phi _{N3}-\pi \\-\phi _{N4}-\pi \\-\phi _{N5}\\-\phi _6-\pi \end{cases}}\).

In order to use the result for the same wave directions (nominal values), the result is mirrored and \(\mathrm {\pi }\) is added to all phase angles, giving:

\(\mathrm {\begin{array}{l}\beta _M\:\:=\beta _N\\\\\phi _{M1}=-\phi _{N1}\\\\\phi _{M2}=-\phi _{N2}\\\\\phi _{M3}=-\phi _{N3}\\\\\phi _{M4}=-\phi _{N4}\pi \\\\\phi _{M5}=-\phi _{N5}\pi \\\\\phi _{M6}=-\phi _{N6}+\pi \end{array}}\)

The addition of \(\mathrm {\pi }\) is done in order to refer to wave downwards, see note 4 to Figure 3.

7.2. Example 2

Results from the NSRDC program (case 2, 5) are to be compared to those of WADIF (case 2). The difference is simply a sign change of all phases, so the comparable quantities are:

\(\mathrm {\begin{array}{l}\beta _N\quad \:=\beta _W\\\\-\phi _{N1}=\phi _{W1}\\\\-\phi _{N2}=\phi _{W2}\\\\-\phi _{N3}=\phi _{W3}\\\\-\phi _{N4}=\phi _{W4}\\\\-\phi _{N5}=\phi _{W5}\\\\-\phi _{N6}=\phi _{W6}\end{array}}\)

7.3. Example 3

WAMOF (case 1) is to be run to create input to MOSSI (case 3, 4) for wave directions \(\mathrm {\beta _M=0^\circ,30^\circ,60^\circ,90^\circ,120^\circ,150^\circ,180^\circ}\).

Table 5. Example 3
WAMOF	MOSSI
\(\mathrm {\beta _W=\beta }\)	\(\mathrm {\beta _M=-\beta \pi \rightarrow\beta =-\beta _M\pi }\)
\(\mathrm {\Phi_4}\)	\(\mathrm {\Phi_5}\)

where \(\mathrm {\Phi_4=\phi _W=\begin{cases}\phi _1\\\phi _2\\\phi _3\\\phi _4\\\phi _5\\\phi _6\end{cases}}\), \(\mathrm {\qquad\Phi_5=\phi _M=\begin{cases}\phi _1\\\phi _2+\pi \\\phi _3+\pi \\\phi _4\\\phi _5+\pi \\\phi _6+\pi \end{cases}}\).

Table 6. Corresponding wave directions
MOSSI	\(\mathrm {\beta _M}\)	\(\mathrm {0^\circ}\)	\(\mathrm {30^\circ}\)	\(\mathrm {60^\circ}\)	\(\mathrm {90^\circ}\)	\(\mathrm {120^\circ}\)	\(\mathrm {150^\circ}\)	\(\mathrm {180^\circ}\)
WAMOF	\(\mathrm {\beta _W}\)	\(\mathrm {180^\circ}\)	\(\mathrm {150^\circ}\)	\(\mathrm {120^\circ}\)	\(\mathrm {90^\circ}\)	\(\mathrm {60^\circ}\)	\(\mathrm {30^\circ}\)	\(\mathrm {0^\circ}\)

Table 7. Conversions
Conversion of phase angles carried out by WAMOF when results are written to a MOSSI file:	For relative motion problems, the \(\mathrm {180^\circ}\) phase shift of the wave from case 1 to case 3 has to be included:
\(\mathrm {\phi _{M1}=\phi _{W1}}\)	\(\mathrm {\phi _{M1}=\phi _{W1}+\pi }\)
\(\mathrm {\phi _{M2}=\phi _{W2}+\pi }\)	\(\mathrm {\phi _{M2}=\phi _{W2}}\)
\(\mathrm {\phi _{M3}=\phi _{W3}+\pi }\)	\(\mathrm {\phi _{M3}=\phi _{W3}}\)
\(\mathrm {\phi _{M4}=\phi _{W4}}\)	\(\mathrm {\phi _{M4}=\phi _{W4}+\pi }\)
\(\mathrm {\phi _{M5}=\phi _{W5}+\pi }\)	\(\mathrm {\phi _{M5}=\phi _{W5}}\)
\(\mathrm {\phi _{M6}=\phi _{W6}+\pi }\)	\(\mathrm {\phi _{M6}=\phi _{W6}}\)

For problems involving only absolute motions these two sets of phase angles are equivalent.

7.4. Sample Transfer Functions

7.4.1. Sample 1: A ship

Coordinate system according to case 1, Figure 3.

Figure 4. Transfer functions for a ship.

Figure 5. Transfer functions for a ship.

7.4.2. Sample 2: A deep draught floater

Draught \(\approx 150\mathrm {m}\), mass \(\approx 250000\mathrm {t}\), 4 columns.

Figure 6. Transfer functions for a deep-draught floater.