Appendix D
Vessel Motion Response Phase
Angles
The correct understanding and definition of phase angles are very important for best riser
dynamics and vessel motions analysis results. However, the definition of vessel motion
phase angles in vessel response amplitude operators (RAO), or transfer functions (TF),
is not always clearly specified.
After considerable review of available public domain literature and reports, it is concluded that vessel phase angles are a very nebulous topic, In each instance, a clear
definition is necessary for proper interpretation and use of the vessel motion response
phase angles.
The following text presents the definition and understanding of the vessel motion response phase angle used in the STARIS theoretical formulation and computer program
implementation.
For discussion purposes, the following relationships are used:
η(x,t) = Aw cos(kx − ωt)
ξ(x,t) = A p cos(kx − ωt + φ)
where
η(x,t) = wave elevation, ft or m, excitation, for wave traveling along the positive x–axis,
at location x at time t.
ξ(x,t) = horizontal vessel motion response, ft or m, due to wave excitation, at location
x at time t.
x = horizontal displacement, ft or m, of “wavestaff” of the vessel within the wave
field, as measured from zero reference.
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φ = vessel motion response phase angles, radians or degrees.
Aw = single amplitude of excitation wave, ft or m.
A p = single amplitude of floating platform response, ft or m.
k = wave number.
T = wave period, sec.
t = time, sec.
ω = 2π/T , wave frequency, rad/sec.
f = 1/T , wave frequency, Hz.
Using the definitions above and considering a fixed point in space, such as a platform
wavestaff, the vessel response phase angle is defined as the time duration after the
passage of the wave crest that the next vessel motion response crest is reached. The
wave is assumed to be going along the positive x–axis direction. This time duration
(seconds) between the passage of the wave crest and the next maximum vessel motion
response crest is suitably expressed in radians or degrees.
Using cosine terms for η and ξ, as shown above, with the phase angle subtracted, as
shown above in ξ, the phase angle is usually referred to as a phase lag, because the
maximum vessel motion response (at a fixed wavestaff) lags in time, i.e., occurs after,
the passage of the wave crest.
Also, the picture, we have found, may be greatly confused if one is looking at: (1) a
spatial snapshot of the total wave field and vessel response trace at a given instant of
time, rather than looking at (2) values at a fixed point for different values of time.
Numerous variations on the above definition are found in the literature, but this definition appears to be consistent with results in Society of Naval Architects and Marine
Engineers (SNAME) publications:
• Principles of Naval Architecture, Society of Naval Architects and Marine Engineers, New York, 1975.
• B. V. Korvin-Kroukovsky, Theory of Seakeeping, Society of Naval Architects and
Marine Engineers, New York, 1961.
Depending upon whether cosine or sine functions are used and whether the phase angle
inside the cosine or sine term is prefixed by a ’+’ or ’−’ sign, the terms phase lag or
phase lead are frequently seen referenced. Clearly, the exact definition must be clear
before proper interpretation of the phase angles can be made.
Vessel motion response phase angles are difficult to obtain reliably, either in the model
basin or by computation. Review of available published phase angle data indicates wide
disagreements. Therefore, when in doubt, check and recheck the phase angle definition
and data. As a final strategy, phase angle sensitivity studies can be easily made with
STARIS to bound the results for riser dynamic behavior.
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To add further confusion or clarification, the “problem” of phase angles was discussed
in the ITTC Proceedings, “19th International Towing Tank Conference”, Volume 1,
Section 6.3.7, page 489, 16-22 September 1990, Madrid, Spain. The text presented is
quoted below:
6.3.7. Fourier Analysis, Definition of Phase Angle.
After the above points have been considered, the amplitude of the lead
signal and the amplitude and phase angle of the responses should be
determined by a standard Fourier analysis.
When presenting the results the sign of the phase angle is defined by the
requirement that the response shall be expressed by:
A = An cos(nωt + φn )
i.e., response lagging behind lead signal gives negative phase angle.
Until the phase angle issue settles somewhat, STARIS will continue to use the phase
angle convention presented above for η and ξ.
Beware of phase angles!