Hydrodynamic loads on bodies (Part I)
=====================================

First order loads
~~~~~~~~~~~~~~~~~

Hydrodynamic forces and moments are obtained from direct pressure integration over the body surface. The body gravity center will be used as a reference for body movements and moments acting on it.

.. math::
   \begin{aligned}
      & \boldsymbol{F}^1 = \int_{S_B^0} P_p^1 \boldsymbol{n}_p^0 ds = \boldsymbol{F}_H^0 + \boldsymbol{F}_H^1 + \boldsymbol{F}_D^1
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \boldsymbol{M}^1 = \int_{S_B^0} P_p^1 \left( \boldsymbol{x}_G - \boldsymbol{x}_p \right) \times n_p^0 ds = \boldsymbol{M}_H^0 + \boldsymbol{M}_H^1 + \boldsymbol{M}_D^1
   \end{aligned}
   :label:


where sub-index :math:`H` stands for hydrostatic loads and :math:`D` stands for dynamic loads. The hydrostatic loads are split as follows:

.. math::
   \begin{aligned}
      & \boldsymbol{F}_H^0 = -\int_{S_B^0} \rho gz\boldsymbol{n}_p^0 ds = \rho g\forall
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \boldsymbol{M}_H^0 = -\int_{S_B^0} \rho gz \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0 ds
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \boldsymbol{F}_H^1 = -\int_{S_B^0} \rho gr_{pz}^1 \boldsymbol{n}_p^0 ds = \overline{\overline{\boldsymbol{K}}}_H \boldsymbol{\Delta }^1
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \boldsymbol{M}_H^1 = -\int_{S_B^0} \rho gr_{pz}^1 \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0 ds = \overline{\overline{\boldsymbol{K}}}_H \theta^1
   \end{aligned}
   :label:

where :math:`\forall` is the body displacement, and :math:`\overline{\overline{\boldsymbol{K}}}_H` is the hydrostatic restoring matrix, which are obtained as follows:

.. math::
   \begin{aligned}
      & ∀ = -\int_{S_B^0} z_p n_{pz} ds
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & x_B = -\frac{1}{2∀} \int_{S_B^0} x_p^2 n_{px} ds
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & y_B = -\frac{1}{2∀} \int_{S_B^0} y_p^2 n_{py} ds
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & z_B = -\frac{1}{2∀} \int_{S_B^0} z_p^2 n_{pz} ds
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & K_H (3,3) = \rho g \int_{S_B^0} n_{pz} ds 
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & K_H (3,4) = \rho g \int_{S_B^0} (y_p-y_G) n_{pz} ds
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & K_H (3,5) = -\rho g \int_{S_B^0} (x_p - x_G) n_{pz} ds
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & K_H (4,4) = \rho g \int_{S_B^0} (y_p - y_G)^2 n_{pz} ds + \rho g∀ (z_B - z_G)
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & K_H (4,5) = -\rho g \int_{S_B^0} (x_p -x_G) (y_p - y_G)  n_{pz} ds 
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & K_H (4,6) = -\rho g∀ (x_B - x_G) 
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & K_H (5,5) = \rho g \int_{S_B^0} (x_p - x_G)^2 n_{pz} ds + \rho g∀(z_B - z_G)
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & K_H (5,6) = -\rho g∀ (y_B - y_G)
   \end{aligned}
   :label:

where :math:`B` stands for the body center of buoyancy, and :math:`G` for the body center of gravity. The dynamic loads are computed as:

.. math::
   \begin{aligned}
      & F_D^1 = -\int_{S_B^0} \rho \frac{\partial \varphi^1}{\partial t} \boldsymbol{n}_p^0 ds
   \end{aligned}
   :label:


.. math::
   \begin{aligned}
      \boldsymbol{M}_D^1 = -\int_{S_B^0} \rho \frac{\partial \varphi^1}{\partial t} \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0 ds
   \end{aligned}
   :label:

Second order loads
~~~~~~~~~~~~~~~~~~

Up to second order loads can be split as:

.. math::
   \begin{aligned}
      & \boldsymbol{F}^{1+2} = \boldsymbol{F}_H^0 + \boldsymbol{F}_H^1 + \boldsymbol{F}_H^2 + \boldsymbol{F}_D^1 + \boldsymbol{F}_D^2 
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \boldsymbol{M}^{1+2} = \boldsymbol{M}_H^0 + \boldsymbol{M}_H^1 + \boldsymbol{M}_H^2 + \boldsymbol{M}_D^1 + \boldsymbol{M}_D^2
   \end{aligned}
   :label:

Where the hydrostatic loads are:

.. math::
   \begin{aligned}
      & \boldsymbol{F}_H^{1+2} = -\int_{S_B^0} \rho g (z_p + r_{pz}^{1+2} ) \boldsymbol{n}_p^1 ds = \boldsymbol{F}_H^0 + \boldsymbol{F}_H^1 + \boldsymbol{\theta}^1 \times \boldsymbol{F}_H^1 + \overline{\overline{\boldsymbol{K}}}_H \boldsymbol{\delta }^2 \\
      & - \int_{S_B^0} \rho g \left( \overline{\overline{H}} \overrightarrow{\boldsymbol{R}^0 \boldsymbol{P}^0} \right)_z \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0 ds
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \boldsymbol{M}_H^{1+2} = -\int_{S_B^0} \rho g \left( z_p + r_{pz}^{1+2} \right) \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^1 ds = \\
      & \boldsymbol{M}_H^0 + \boldsymbol{M}_H^1 + \boldsymbol{\theta}^1 \times \boldsymbol{M}_H^1 + \overline{\overline{\boldsymbol{K}}}_H \boldsymbol{\theta}^2 - \int_{S_B^0} \rho g \left( \overline{\overline{\boldsymbol{H}}} \overrightarrow{ \boldsymbol{R}^0 \boldsymbol{P}^0 } \right)_z \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0 ds
   \end{aligned}
   :label:

Then:

.. math::
   \begin{aligned}
      & \boldsymbol{F}_H^2 = \overline{\overline{\boldsymbol{K}}}_H \boldsymbol{\delta }^2 + \boldsymbol{\theta}^1 \times \boldsymbol{F}_H^1 - \int_{S_B^0} \rho g (\overline{\overline{\boldsymbol{H}}} \overrightarrow{\boldsymbol{R}^0 \boldsymbol{P}^0})_z \boldsymbol{n}_p^0 ds
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \boldsymbol{M}_H^2 = \overline{\overline{\boldsymbol{K}}}_H \boldsymbol{\theta}^2 + \boldsymbol{\theta}^1 \times \boldsymbol{M}_H^1 - \int_{S_B^0} \rho g (\overline{\overline{\boldsymbol{H}}} \overrightarrow{\boldsymbol{R}^0 \boldsymbol{P}^0})_z \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0 ds
   \end{aligned}
   :label:

On the other hand, dynamic loads up to second order are split in four components:

.. math::
   \begin{aligned}
      & \boldsymbol{F}_D^{1+2} = \boldsymbol{F}_D^1 + \boldsymbol{F}_D^2 = \boldsymbol{F}_D^1 + \boldsymbol{F}_{D1}^2 + \boldsymbol{F}_{D2}^2 + \boldsymbol{F}_{D3}^2 + \boldsymbol{F}_{D4}^2
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \boldsymbol{M}_D^{1+2} = \boldsymbol{M}_D^1 + \boldsymbol{M}_D^2 = \boldsymbol{M}_D^1 + \boldsymbol{M}_D1^2 + \boldsymbol{M}_D2^2 + \boldsymbol{M}_D3^2 + \boldsymbol{M}_D4^2
   \end{aligned}
   :label:

where 

.. math::
   \begin{aligned}
      & \boldsymbol{F}_{D1}^2 = -\rho \int_{S_B^0} \frac{\partial \varphi^2}{\partial t} \boldsymbol{n}_p^0 ds + \boldsymbol{\theta}^1 \times \boldsymbol{F}_D^1
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \boldsymbol{F}_{D2}^2 = -\rho \int_{S_B^0} \left( \boldsymbol{r}_p^1 \cdot \nabla \left( \frac{\partial \varphi^1}{\partial t} \right) \right) \boldsymbol{n}_p^0 ds
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \boldsymbol{F}_{D3}^2 = -\frac{1}{2} \rho \int_{S_B^0} \left( \nabla \varphi^1 \cdot \nabla \varphi^1 \right) \boldsymbol{n}_p^0 ds
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \boldsymbol{F}_{D4}^2 = -\frac{1}{2} \rho g \int_{\Gamma_B^0} \left( \xi^1 - r_{pz}^1 \right)^2 \frac{\boldsymbol{n}_p^0}{\sqrt{1 - {n_{pz}^0}^2}} dl
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \boldsymbol{M}_{D1}^2 = -\rho \int_{S_B^0} \frac{\partial \varphi^2}{\partial t} \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0 ds + \boldsymbol{\theta}^1 \times \boldsymbol{M}_D^1
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \boldsymbol{M}_{D2}^2 = -\rho \int_{S_B^0} \left(r_p^1 \cdot \nabla \left( \frac{\partial \varphi^1}{\partial t} \right) \right) \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0 ds
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \boldsymbol{M}_{D3}^2 = -\frac{1}{2} \rho \int_{S_B^0} \left( \nabla \varphi^1 \cdot \nabla \varphi^1 \right) \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0 ds
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \boldsymbol{M}_{D4}^2 = -\frac{1}{2} \rho g \int_{\Gamma_{B}^0} \left( \xi^1 - r_{pz}^1 \right)^2 \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \frac{\boldsymbol{n}_p^0}{\sqrt{1 - {\boldsymbol{n}_{pz}^0}^2}} dl
   \end{aligned}
   :label:

Mean drift loads
~~~~~~~~~~~~~~~~

Being the first and second order responses harmonic and taking time average, the following relations holds when:

.. math::
   \begin{aligned}
      \Big<F^1\Big>=0
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      \Big<M^1\Big>=0
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \Big<F_H^2\Big> = \Big<\boldsymbol{\theta}^1 \times \boldsymbol{F}_H^1\Big> - \Big<\int_{S_B^0} \rho g \left( \overline{\overline{\boldsymbol{H}}} \overrightarrow{\boldsymbol{R}^0 \boldsymbol{P}^0} \right)_z \boldsymbol{n}_p^0  ds \Big>
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \Big<M_H^2\Big> = \Big<\boldsymbol{\theta}^1 \times \boldsymbol{M}_H^1\Big> - \Big<\int_{S_B^0} \rho g \left( \overline{\overline{\boldsymbol{H}}} \overrightarrow{\boldsymbol{R}^0 \boldsymbol{P}^0} \right)_z \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0  ds\Big>
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \Big<F_{D1}^2\Big> = \Big<\boldsymbol{\theta}^1 \times \boldsymbol{F}_D^1\Big>
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \Big<F_{D2}^2\Big> = - \rho \Big<\int_{S_B^0} \left( \boldsymbol{r}_p^1 \cdot \nabla \left( \frac{\partial \varphi^1}{\partial t} \right) \right) \boldsymbol{n}_p^0  ds\Big>
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \Big<F_{D3}^2\Big> = - \frac{1}{2} \rho \Big<\int_{S_B^0} \left( \nabla \varphi^1 \cdot \nabla \varphi^1 \right) \boldsymbol{n}_p^0  ds\Big>
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \Big<F_{D4}^2\Big> = - \frac{1}{2} \rho g \Big<\int_{\Gamma_B^0} \left( \xi^1 -r_{pz}^1 \right)^2 \frac{n_p^0}{\sqrt{1 - {\boldsymbol{n}_{pz}^0}^2}} dl\Big>
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \Big<M_{D1}^2\Big> = \Big<\boldsymbol{\theta}^1 \times \boldsymbol{M}_D^1\Big>
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \Big<M_{D2}^2\Big> = - \rho \Big<\int_{S_B^0} \left( r_p^1 \cdot \nabla \left(\frac{\partial \varphi^1}{\partial t} \right) \right) \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0  ds\Big>
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \Big<M_{D3}^2\Big> = - \frac{1}{2} \rho \Big<\int_{S_B^0} \left( \nabla \varphi^1 \cdot \nabla \varphi^1 \right) \overrightarrow{\boldsymbol{}G^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0  ds\Big>
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \Big<M_{D4}^2\Big> = - \frac{1}{2} \rho g \Big<\int_{\Gamma_B^0} \left( \xi^1 - r_{pz}^1 \right)^2 \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \frac{ \boldsymbol{n}_p^0 }{ \sqrt{ 1 - {\boldsymbol{n}_{pz}^0}^2}} dl\Big>
   \end{aligned}
   :label:

Second order terms depending with non-zero time average depends on first order quantities. Hence, second order drifting loads only depends on the first order problem solution.