Passive heave compensation

Application

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. PHC may be used along with active heave compensation to form a semi-active system.

Calculation of PHC

Efficiency for a PHC used during offshore lifting operations

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 systems 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:

${\displaystyle {\eta }_{PHC}=1-T_R}$

Calculating PHC stiffness

The stiffness of a PHC device is given by:

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

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.