Passive heave compensation

PHC is often used on offshore equipment that is at or linked to the seabed. Not requiring external energy, PHC may be designed as a fail-safe system reducing the wave impact on sub-sea operations.^[4] PHC may be used along with active heave compensation to form a semi-active system.^[5]

The PHC device is in this calculation connected to the crane hook. Newton's second law is used to describe the acceleration of the payload:

${\displaystyle (m+m_{A}){\ddot {y}}=-k_{c}(y+H\cos \omega t)}$

Where

${\displaystyle m}$ - is the mass of the load underneath the PHC device

${\displaystyle m_{A}}$ - is the added mass of the load underneath the PHC device

${\displaystyle {\ddot {y}}}$ - is the acceleration of the mass of the load underneath the PHC device

${\displaystyle k_{c}}$ - is the stiffness of the PHC device

${\displaystyle y}$ - is the vertical position of the mass underneath the PHC device

${\displaystyle H}$ - is the vessel motion amplitude

${\displaystyle \omega }$ - is the angular wave frequency

${\displaystyle t}$ - is time

Ignoring the transient solution, it is found that the ratio between the amplitude of the load and the wave amplitude is:

${\displaystyle {\frac {A}{H}}={\frac {\frac {k_{c}}{m+m_{A}}}{\omega ^{2}-{\frac {k_{c}}{m+m_{A}}}}}}$

To simplify the expression, it is common to introduce ${\displaystyle \omega _{0}}$ as the system's natural frequency, defined as:

${\displaystyle \omega _{0}={\sqrt {\frac {k_{c}}{m+m_{A}}}}}$

This leads to the following expression for the ratio:

${\displaystyle {\frac {A}{H}}={\frac {1}{({\frac {\omega }{\omega _{0}}})^{2}-1}}}$

The transmissibility ${\displaystyle T_{R}}$ is defined as:

${\displaystyle T_{R}=\left|{\frac {1}{({\frac {\omega }{\omega _{0}}})^{2}-1}}\right|}$

Finally, the efficiency is defined as:

${\displaystyle \eta _{PHC}=1-T_{R}}$

The stiffness of a PHC device is given by:^[6]

${\displaystyle k_{c}={\frac {p_{0}A}{S}}(C^{\kappa }-1)}$

Where

${\displaystyle p_{0}}$ - is the gas pressure at equilibrium stroke

${\displaystyle A}$ - is the piston area

${\displaystyle S}$ - is the stroke length

${\displaystyle C}$ - is the compression ratio

${\displaystyle \kappa }$ - is the adiabatic coefficient

The product ${\displaystyle p_{0}A}$ corresponds to the submerged weight of the payload. As can be seen from the expression, low compression ratios and long stroke length give low stiffness.

The natural period of a PHC is given by:^[3]

${\displaystyle T_{n}=\pi {\sqrt {{\frac {m+m_{A}}{m}}\,{\frac {\rho }{\rho -\rho _{w}}}\,{\frac {2S}{g\!\left(\left({\frac {R-0.5}{R-1}}\right)^{\gamma }-1\right)}}}}}$

Where

${\displaystyle S}$ - is the stroke length

${\displaystyle m}$ - is the payload mass

${\displaystyle m_{A}}$ - is the hydrodynamic added mass

${\displaystyle \rho }$ - is the mass density of the payload

${\displaystyle \rho _{w}}$ - is the mass density of the seawater

${\displaystyle g}$ - is the acceleration of gravity

${\displaystyle R}$ - is the gas to oil ratio

${\displaystyle \gamma }$ - is the adiabatic exponent