Collaboration diagram for Ship with advanced hull:

The FhSim ship model with hull rendered from analysis input

The FhSim ship model implements a time-domain realization of frequency domain hydrodynamics from frequency dependent added mass and dampening. The main source of frequency domain hydrodynamic data is computer codes based on potential theory such as VERES and WAMIT. The current model implementation only supports VERES data sets, but the implemented approach can be extended to WAMIT upon request.

The added mass and dampening of the vessel is represented as a linear time independent (LTI) companion system in a feedback loop with the rigid body dynamics of the vessel. The input to the companion system is the velocities of the vessel and the output is the dampening force from wave generation by the hull as the vessel oscillates. The time-domain realization is generated directly from result files of the frequency domain analysis by application of the "Vector fitting" system identification algorithm. The vector fitting algorithm is implemented internally in the ship model and the conversion from frequency to time-domain is computed on-line. The order of the LTI approximation of the fluid memory effect is chosen by comparing added mass and dampening reconstructed from the frequency response of the LTI-approximation with the original input data.

Definition of body local axes and rotations on the ship. The hull is sectioned for calculation of both viscous forces in the FhSim ship model and potential forces by VERES

The structure of the seakeeping problem is used to generate a time-domain model. The seakeeping problem is studied by decomposing the forces acting on a harmonically oscillating vessel in harmonic waves into

• The diffraction problem: The excitation force experienced if the vessel was fixed relative to the incoming wave
• The radiation problem: The force experienced by the vessel during forced harmonic motions generation of waves on a flat ocean surface The force from the wave radiation ios decomposed into components in phase with the velocities (dampening) and accelerations (added mass), while the diffraction problem is represented by response amplitude operators.

The equations of motion of the vessel including the radiation force is derived by considering the equations of motion of the rigid body

\[ M_{rb}\ddot{\xi} = \tau \]

The excitation force is presumed to vary as a harmonic function and to be linearly independent. This results in linearly independent harmonic motions \(\xi=|\xi|sin(\omega t)\). The assumption of linearity allows the separation of the force \( \tau \) into the force from wave radiation, the restoring force and the excitation force from the waves

\[ \tau = \tau_{radiation} + \tau_{restoring} + \tau_{excitation}\]

The seakeeping problem can now be solved independently for the independently for the excitation force (the diffraction problem) and the radiation force while the restoring force acts as a linear spring force.

The radiation force can be expressed as:

\[ \tau_{rad}(\omega) = -A(\omega)\ddot{\xi} - \int_0^t K(t - t')\dot{\xi}dt' \]

The linear restoring force is expressed as

\[ \tau_{res} = G\xi \]

Substitution of the above expressions into the original equation and transferring it to the time domain yields the following equation named the "Cummins equation"

\[ (M+A_{\inf})\ddot{\xi} + \int_{0}^{t} K(t-t')\dot{\xi}dt'+G\xi \]

The Cummins equation contains a convolution integral which is cumbersome (but not impossible) to evaluate numerically as the time \(t\) increases. The convolution integral can be approximated by computing the frequency response of the \(K\) function and constructing an LTI system which approximates the frequency response.

\[ K(j\omega) = B(\omega) + j\omega (A(\omega) - lim_{\omega \to \inf} A(\omega)) = \frac{P(j\omega)}{Q(j\omega} \]

The doefficients of the rational transfer function \(\frac{P(j\omega)}{Q(j\omega)}\) is computed and transferred to a state-space representation which yields the final system of equations.

\[ (M+A_{\inf})\ddot{\xi}+X+G\xi = \tau_{excitation} \]

\[ \dot{\chi}=A_d\chi+B_d\dot{\xi} \]

\[ X = C_d\chi+D_d\dot{\xi} \]

Forces

The ship model includes calculation of external forces and internal dampening forces. All forces can be enabled/disbaled and scaled throught the appropriate interfaces.

Excitation

The excitation forces included are mainly wave forces as well as hydrostatic restoring forces

Restoring forces

The hydrostatic restoring force on the body

Wave forces

caption

First order wave forces

The first order wave force, or froude-krylov force, as experienced in the body frame of the vessel The first order wave loads are defined as a sum of harmonic functions with the same frequency distribution as the wave elevation.

\[ \tau_{1. order} = \sum_{n=0}^{n_{waves}}\zeta_n A(\omega_n,\psi_n,U) \cdot cos( \omega_{n} - k_n x + \epsilon_n) \]

Wave drift forces

The mean value of the higher order wave forces as experienced in the body frame of the vessel. The mean value of the heave, roll and pitch direction equals zero.

\[ \tau_{wave-drift} = \sum_{n=0}^{n_{waves}} {\zeta_{n}}^2 A(\omega_n,\psi_n,U) \]

Potential dampening

The dampening force from wave generation by the vessel motions. This is the output of the "fluid memory" model

Viscous forces

The viscous forces represent friction and vortex shedding which is not captrured by the seakeeping problem which defined the potential dampening. The viscous forces are dampening forces and represents the effect of ship resistance in surge, cross flow drag in sway and yaw and vortex shedding in roll.

Viscous forces on body

Resistance

The calm water resistance of the hull in the surge direction The resistance force is the defined as the calm water resistance of the hull. The calm water resistance is divided into the Reynolds number ( \( R_n\)) dependent friction contribution, and the Froude number ( \( F_n\)) dependent wave generation from the flow pattern around the hull. The resistance is calculated by scaling the dimensional total resistance coefficient, \( C_{ts}\), by the wet surface, \( S\), on the current waterline. The total resistance coefficient is both \( R_n\) and \( F_n\) dependent and is defined as.

\[ C_{ts} = (1+k) (C_f(R_n) + \Delta C_f(R_n)) + C_r(F_n) \]

Where

• \(C_{ts}\) is the total resistance coefficient
• \(k\) is a form factor modification of the friction resistance due to the difference between the a flat plate and a hull form defined as k= \((0.6+145(\frac{C_b}{L_{wl}}\sqrt{B\cdot(T_a+T_f)})^(2.5))\frac{C_b}{L_{wl}}\sqrt{B\cdot(T_a+T_f)}\)
  • \({C_b}\) is the block coefficient
  • \(B\) is the breadth
  • \(T_a\) and \(T_f\) is the aft and fore draught
• \(C_f\) is the \(R_n\) dependent friction coefficient of a flat plate defined as \(C_f = \frac{0.0075}{(log_{10}R_n-2)^2}\)
• \(\Delta C_f\) is the additional friction resistance due to roughness of the hull
• \(C_r\) is the \(F_n\) dependent resistance from generation of waves The resistance due to wind is neglected in this resistance calculation to separate the wind force from the resistance. The total resistance is calculated from

  \[ R_{ts} = \frac{1}{2}\rho U^2 \cdot S \cdot C_{ts}\]

• \(\rho\) is the density of sea water
• \(U\) is the surge speed of the vessel

Cross flow drag

Cross flow are calculated by summation of section wise cross flow drag along the hull. Dampening in yaw is computed by calculating the moment contribution around the center of gravity for each sections cross flow drag force

\[ v_r = v-x\cdot r\]

\[ Y_{cross-flow} = -\frac{1}{2}\rho\int_{-Lpp/2}^{Lpp/2}T(x)C_d^2d(x)|v_r|v_r dx \]

\[ N_{cross-flow} = -\frac{1}{2}\rho\int_{-Lpp/2}^{Lpp/2}T(x)\cdot x\cdot C_d^2d(x)|v_r|v_r dx \]

User defined dampening

The model supports additional user defined dampening forces. The forces are calculated from dimensionless dampening coefficients made dimensionless by \(m\) and by \(m\cdot L_{pp}\) for linear and rotational degrees of freedom respectively. The dampening force contains a linear and a quadratic modulus term which are controlled by separate dampening coefficients.

\[ F = C_{d_linear} \cdot m \cdot v + C_{d_quadratic} \cdot m \cdot v|v|\]

\[ M = C_{d_linear} \cdot m \cdot L_{pp} \cdot v + C_{d_quadratic} \cdot m \cdot L_{pp}\cdot v|v|\]

The dampening force decreases with increasing forward velocity by weighting the force with \(exp{-\frac{1}{2}|U|}\)

Configuration parameters

The input ports

The output ports