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Passive [[Degrees of freedom (mechanics)|heave]] compensation is a technique used to reduce the influence of [[Wind wave|waves]] upon lifting and drilling operations.<ref>Passive and Active heave Compensation, Albers, TU Delft</ref> A simple passive heave compensator (PHC) is a soft spring which utilizes [[Vibration isolation|spring isolation]] to reduce [[Transmissibility (vibration)|transmissibility]] to less than 1.<ref>Bob Wilde and Jake Ormond: ''Subsea Heave Compensators'', Deep Offshore Technology 2009</ref> PHC differs from [[Active heave compensation|AHC]] by not consuming external power.

==Principle==
The main principle in PHC is to store the energy from the external forces ([[waves]]) influencing the system and dissipate them or reapply them later. [[Shock absorber]]s or [[drill string compensator]]s are simple forms of PHC, so simple that they are normally named [[heave compensator]]s, while "[[Passivity (engineering)|passive]]" is used about more sophisticated hydraulic or mechanical systems.

A typical PHC device consists of a hydraulic cylinder and a gas accumulator. When the piston rod extends it will reduce the total gas volume and hence compress the gas that in turn increases the pressure acting upon the piston. The [[compression ratio]] is low to ensure low stiffness. A well designed PHC device can achieve efficiencies above 80 percent.<ref>[http://www.safelink.no/ The Engineers Guide] Safelink AS</ref>

==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.<ref>Passive Heave Compensation, www.huismanequipment.com/en/products/heave_compensation/passive_heave_compensation</ref> PHC may be used along with [[active heave compensation]] to form a semi-active system.<ref>Passive Heave Compensation of Heavy Modules, Sten Magne Eng Jakobsen, 2008, University of Stavanger [http://brage.bibsys.no/uis/bitstream/URN:NBN:no-bibsys_brage_7771/1/Jakobsen%2c%20Sten%20Magne.pdf]</ref>

==Calculation of PHC==

===Efficiency for a PHC used during offshore lifting operations===
[[File:PHC sketch.png|thumb|Sketch of system]]
The PHC device is in this calculation connected to the crane hook. [[Newton's laws of motion|Newton`s second law]] is used to describe the acceleration of the payload: 
<math> (m+m_A) \ddot y =-k_c(y+H \cos \omega t) </math>

Where 
<math> m </math> - is the [[mass]] of the load underneath the PHC device 
<math> m_A </math> - is the [[added mass]] of the load underneath the PHC device 
<math> \ddot y </math> - is the [[acceleration]] of the mass of the load underneath the PHC device 
<math> k_c </math> - is the [[stiffness]] of the PHC device 
<math> y </math> - is the vertical position of the mass underneath the PHC device 
<math> H </math> - is the wave amplitude 
<math> \omega </math> - is the angular wave frequency 
<math> t </math> - is time 
 
If we ignore the transient solution we will find that the ratio between the amplitude of the load and the wave amplitude is: 
<math> \frac {A}{H} = \frac{ \frac {k_c}{m+m_A}} { \omega^2 - \frac {k_c}{m+m_A}} </math> 
 
To simplify the expression it is common to introduce <math> \omega _0 </math> as the systems natural frequency, defined as: 
<math> \omega _0 = \sqrt {\frac {k_c}{m+m_A}}</math> 
 
We then get the following expression for the ratio: 
<math> \frac {A}{H} = \frac {1}{({\frac {\omega}{\omega_0}})^2-1}</math>

The transmissibility <math> T_R </math> is defined as: 
<math> T_R= \left | \frac {1}{({\frac {\omega}{\omega_0}})^2-1} \right |</math> 
 
Finally the efficiency is defined as: 
<math> \eta_{PHC}= 1-T_R </math>

===Calculating PHC stiffness===
The stiffness of a PHC device is given by:<ref>Peter Albers: ''Motion Control in Offshore and Dredging'', Springer, 2010. ISBN 978-9048188024</ref> 
<math> k_c= \frac {p_0 A}{S}(C^\kappa-1) </math>

Where 
<math> p_0 </math> - is the gas pressure at equilibrium stroke 
<math> A </math> - is the piston area 
<math> S </math> - is the stroke length 
<math> C </math> - is the compression ratio 
<math> \kappa </math> - is the adiabatic coefficent

The product <math> p_0 A </math> corresponds to the submerged weight of the payload. As we can see from the expression it is clear that low compression ratios as well as long stroke length gives low stiffness.

==References==
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[[Category:Springs (mechanical)]]
[[Category:Petroleum production]]
[[Category:Lifting equipment]]

Polynomial kernel - Revision history

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