1. SEAKEEPING Introduction (8.1)
Seaworthiness defines the operational limits of our vessels!
USCG 47’ MLB

2. SEAKEEPING Introduction (8.1)
The ship is a system excited by external moments and forces.
Excitations (inputs) are primarily wind and waves.
“RAO” : response amplitude operator
Responses (output) are motions in the six degrees of freedom, plus structural loading.

3. SEAKEEPING Introduction (8.1) Ship response depends on two things:
1. Size, direction, and frequency of the inputs.
2. The seakeeping and structural characteristics of the ship.

4. SEAKEEPING
Waves (8.2)
Wind and waves are both important but our study is limited to wave systems as this is the dominant input load.
Waves are created by energy supplied to water (from wind, ship’s bow, etc.).
Wave energy is related to wave height by:

5. The damage included bending the foremast 20 degrees and busting windows on the bridge 40 ft abv the deck! Taken off South Carolina in 1998.

6. SEAKEEPING How Waves Are Made (8.2) Forces that make waves:
Wind: Most common. Energy transfer through shear stresses.
Geological Events: Seismic activity on the sea bed (i.e. underwater volcanoes, landslides and earthquakes). “Tsunami’s”
Currents: Interaction of ocean currents. Greatly influenced by the coastline’s shape.

7. Wind Generated Wave Systems Factors for Wave Size (8.2)
SEAKEEPING
Wind Generated Wave Systems Factors for Wave Size (8.2)
Wind Strength: Faster wind = more energy transferred. Strong winds form large waves.
Wind Duration: Longer = larger waves.
Water Depth: Different relationships for deep and shallow water.
Fetch: Area influenced by wind Larger area = more energy transfer.

8. SEAKEEPING The Simplified Wave Equation for coastal waters!
This won’t be on the exam!

9. SEAKEEPING
Waves (8.2)
Energy transfer is constantly occurring in a wave.
Water viscosity dissipates wave energy by viscous friction. Dissipation increases with wave height.
To maintain the wave height, the energy lost to friction must be replaced.

10. SEAKEEPING Wave Life Cycle (8.2)
Birth - wind over water creates ripples; high frequency (f), low wave length ().

11. SEAKEEPING Wave Life Cycle (8.2)
Growing - freq  but length and ht  as wind continues and energy content of wave system grows.

12. SEAKEEPING Wave Life Cycle (8.2)
Fully Developed - sea stops growing with wave height and energy content maximized.

13. SEAKEEPING Wave Life Cycle (8.2)
Reducing - wave system no longer maintained as winds reduce. Waves dissipate (from high to lower freqs) as energy content drops.
Swell - Eventually the wave system consists of low freq, long  waves associated with an ocean swell.

14. SEAKEEPING
Wave Superposition (8.2)

15. SEAKEEPING Wave Superposition (8.2)
Confused seas are modeled as a destructive/constructive interference pattern.
The wave systems are modeled by superimposing sinusoidal wave components, each with their own wavelength, speed, and amplitude.
Bottom Line: We must look at spectral densities and statistical analysis methods to determine wave system and sea properties.

16. SEAKEEPING
Wave Superposition (8.2)

17. SEAKEEPING
Waves (8.2)
Wave Spectrum: analyzing the sea in the frequency domain.

18. SEAKEEPING
Waves (8.2)

19. SEAKEEPING
Waves (8.2)
Modal wave periods from the Sea Spectra Chart are easily converted to modal (or circular) wave frequencies by the following relationship:
Don’t confuse this with linear frequency, f=1/T!

20. SEAKEEPING
Waves (8.2)
Each sea state has a predominant modal frequency and significant wave height.
Direction of the seas is assumed to be the same as local, observed wind.
So, we now know the magnitude, direction, and frequency of the Excitation Forces!
The next step is understanding the ship motions...

21. Simple Harmonic Motion (8.3)
SEAKEEPING
Simple Harmonic Motion (8.3)
A harmonic motion is a system where a mass displaced from its “at rest” location experiences a linear restoring force resulting in an oscillating motion.
Linear - size of force or moment is proportional to displacement. “Non-linear” restoring forces work, too.
Restoring - force or moment opposes the direction of motion.

22. Simple Harmonic Motion (8.3)
SEAKEEPING
Simple Harmonic Motion (8.3)
Common Model:
Mass is displaced, the spring is either in compression or tension with a restoring force trying to return it to the original location.
The size of the force will be proportional to the amount of displacement - a linear force. (F=k·x)
Motion will continue indefinitely if no damping is in the system.

23. Simple Harmonic Motion (8.3)
SEAKEEPING
Simple Harmonic Motion (8.3)
The mathematics involves analysis of a 2nd order linear differential equation of motion with displacement (z) and time (t) and damping effects (c) =0:
the solution is a simple cosine.
where Z0 is the initial displacement and n is the natural (circular) frequency of the system.

24. Simple Harmonic Motion (8.3)
SEAKEEPING
Simple Harmonic Motion (8.3)
A plot of the displacement (z) against time (t):
The period (T) can be determined from the plot.

25. Simple Harmonic Motion (8.3)
SEAKEEPING
Simple Harmonic Motion (8.3)
From the period the natural frequency can be calculated and checked against the observed natural frequency calculated from the known system parameters, mass (m) and spring constant (k).

26. Simple Harmonic Motion (8.3)
SEAKEEPING
Simple Harmonic Motion (8.3)
Amplitude of spring, mass, damper system may reduce with time due to damping or dissipation effects.
Three conditions:
Under damped: continued oscillations.
Critically damped: one overshoot.
Over damped: no oscillations, slow recovery.

27. Simple Harmonic Motion (8.3)
SEAKEEPING
Simple Harmonic Motion (8.3)

28. Simple Harmonic Motion (8.3)
SEAKEEPING
Simple Harmonic Motion (8.3)
Forcing Function and Resonance
For spring- mass-damper system to remain oscillating, energy must be put into system (if damping  0).
This energy is required to overcome the energy being dissipated by the damper. In this system it would be applied as an external force, often called an external forcing function.

29. Simple Harmonic Motion (8.3)
SEAKEEPING
Simple Harmonic Motion (8.3)
To create maximum displacement, the forcing function has to inject its energy to coincide with the movement of the mass (i.e. be in phase).
So to maintain system oscillation, a cyclical force is required that is at the same frequency as the SHM system.
When this occurs, the system is at resonance and maximum amplitude oscillations will occur. If the forcing function is applied at any other frequency, the amplitude of oscillation is diminished.

30. Simple Harmonic Motion (8.3)
SEAKEEPING
Simple Harmonic Motion (8.3)
The differential equation for the mass, spring, damper (=0) system with forcing function becomes:
where F is the size of the forcing function and is the frequency at which it is applied.
The solution becomes
(still neglecting damping):

31. Simple Harmonic Motion (8.3)
SEAKEEPING
Simple Harmonic Motion (8.3)
When  << n Z = F/K F = forcing function
K = spring rate
When  >> n Z = 0
When  = n Z = 
System motion amplitude versus
the forcing function frequency.

32. Simple Harmonic Motion (8.3)
SEAKEEPING
Simple Harmonic Motion (8.3)
The figure below compares a system that is sharply tuned and one that is not.
Lightly damped systems are more “sharply tuned” and are more sensitive to forcing function frequency than those with high damping. Ships are often sharply tuned in some motions...

33. SEAKEEPING Ship Response (8.4)
As we saw in 8.1, the system output depends on the magnitude and frequency of the excitation force and the ship’s RAO’s.
Excitation force frequency depends on the wave frequency (from sea state table) and ship speed and heading.
w=input freq. (Vs=0)
wn=natural freq.
(if Vs>0) then w= we
Recall

34. SEAKEEPING Ship Response (8.4)
Encounter frequency (e) accounts for the relative velocity between ship and waves.
Where:
ww is the wave frequency
V is the ship speed in ft/s.
µ is the heading of the ship relative to the direction the waves are moving.

35. SEAKEEPING Ship Response (8.4)
For a given wave frequency (w), changing course or speed alters e. (Example?)

36. SEAKEEPING Ship Response (8.4)
Knowing encounter frequency, we can predict ship responses.
The 3 major sets of response can be grouped as:
1. Rigid Body Motions.
2. Structural Responses.
3. Non-oscillatory Dynamic Responses.

37. SEAKEEPING Ship Response (8.4)
A ship has 6 degrees of freedom about the xyz axis system, 3 rotational and 3 translational. All are rigid body motions.

38. SEAKEEPING Rigid Body Motions (8.4) Heave (Z axis translation)
Imbalance between displacement and the buoyant force creates a resultant force which attempts to restore the ship to its original waterline.

39. Rigid Body Motions - Heave (8.4)
SEAKEEPING
Rigid Body Motions - Heave (8.4)
The vertical motion is completely analogous to the mass-spring-damper system.
It is possible to predict the natural heave frequency (heave) of a ship.

40. Rigid Body Motions - Heave (8.4)
SEAKEEPING
Rigid Body Motions - Heave (8.4)
TPI depends heavily on area of the DWL.
Larger waterplane area for a given displacement equals greater restoring forces.
‘Beamy’ ships (e.g. tugs) will have short period oscillations and high accelerations (less comfortable).
‘Narrow’ hulls like frigates, catamarans and SWATH have more gentle heave motions.
Heave is heavily damped.

41. Rigid Body Motions - Roll (8.4)
SEAKEEPING
Rigid Body Motions - Roll (8.4)
External wave slopes create internal righting moments to realign “B” and “G”. Rotation is about the X axis.
Roll

42. Rigid Body Motions - Roll (8.4)
SEAKEEPING
Rigid Body Motions - Roll (8.4)
Righting moment depends on righting arm and ship displacement.
For small angles (in radians) this becomes:
This creates a linear restoring moment which is a rotational SHM.

43. Rigid Body Motions - Roll (8.4)
SEAKEEPING
Rigid Body Motions - Roll (8.4)
By rotational analogy to the mass-spring- damper system.
Similarly, the expression for the natural roll frequency (roll).

44. Rigid Body Motions - Roll (8.4)
SEAKEEPING
Rigid Body Motions - Roll (8.4)
Combining empirical knowledge and the relationship between natural roll frequency (roll) and period of roll Troll.
where B is the ship’s Beam
C is a constant whose value can range from s/ft½ when GMT and beam are measured in ft. (0.44 when damping unknown)
What happens if B is increased?
T stays about the same! Huh?

45. Rigid Body Motions - Roll (8.4)
SEAKEEPING
Rigid Body Motions - Roll (8.4)
GMT value is a compromise between good seakeeping (small GMT) and good stability (large GMT).
Naval Architects design for a GMT of between 5 - 8% of beam as a compromise.

46. Rigid Body Motions - Pitch (8.4)
SEAKEEPING
Rigid Body Motions - Pitch (8.4)
Pitch (about Y axis) wants to restore vertical alignment of “B” and “G”.

47. Rigid Body Motions - Pitch (8.4)
SEAKEEPING
Rigid Body Motions - Pitch (8.4)
Internal righting moment acting to restore the ship is linear and depends on MT1" value.
As in roll, rotational motion is analogous to the mass- spring-damper system.
Large MT1" = large moments & accelerations
Motions heavily damped in all cases.

48. Rigid Body Motions - Resonance (8.4)
SEAKEEPING
Rigid Body Motions - Resonance (8.4)
Resonance - if freq of the forcing function = natural freq of the system: then maximum amplitudes!
To minimize undesirable motions, resonance must not occur.
Since heave, pitch, and roll are SHM, it is important that they do not match with encounter frequency (e).

49. Rigid Body Motions - Resonance (8.4)
SEAKEEPING
Rigid Body Motions - Resonance (8.4)
Heave and pitch are well damped and as such are not “sharply tuned” (amplified).
Roll motion is sharply tuned, lightly damped, and very susceptible to the encounter frequency!

50. Ship Response - Structural (8.4)
SEAKEEPING
Ship Response - Structural (8.4)
Distinct from rigid body motion, waves can negatively impact ship structural components.
Primary structural loads:
1. Longitudinal bending: hogging and sagging
2. Torsion: twisting effect upon the ship structure
3. Transverse stresses: hydrostatic pressure of the sea

51. SEAKEEPING Ship Response (8.4) Non-Oscillatory Dynamic Response
Caused by the relative motions of the ship and sea.
Maximized when a movement of the ship due to heave, pitch, or roll superimposed with a wave peak or trough.

52. Non-Oscillatory Dynamic Response (8.4)
SEAKEEPING
Non-Oscillatory Dynamic Response (8.4)
Shipping Water - bow of the ship submerged, considerable loads on the ship structure.
Forefoot Emergence - bow unsupported, severe structural loads.
Slamming - severe structural vibration from forefoot emergence.
Racing - propeller leaves the water.

53. Non-Oscillatory Dynamic Response (8.4)
SEAKEEPING
Non-Oscillatory Dynamic Response (8.4)
Shipping Water!

54. SEAKEEPING Non-Oscillatory Dynamic Response (8.4)
Forefoot emergence and slamming!

55. Non-Oscillatory Dynamic Response (8.4)
SEAKEEPING
Non-Oscillatory Dynamic Response (8.4)
Large following seas at speeds close to the wave speed may cause undesirable responses:
Broaching - sudden and uncontrollable turning of a ship to a “beam on” orientation with a risk of capsize.
Loss of Stability - ship surfs, can adversely effect stability.

56. Non-Oscillatory Dynamic Response (8.4)
SEAKEEPING
Non-Oscillatory Dynamic Response (8.4)

57. Ship Response Reduction (8.5)
SEAKEEPING
Ship Response Reduction (8.5)
Historically, seakeeping has been less important than hull resistance, strength and space efficiency considerations. (Heck, who cares about the crew’s comfort?!)
DDG-51 hull form was the first to be created with seakeeping as a high priority. (In order to expand the mission envelope.)

58. SEAKEEPING DDG 51 Hull Advantages (8.5)
The hull shape was designed to reduce accelerations.
Forward and aft sections are V-shaped, giving nonlinear MT1”, reducing pitch accelerations.
Similarly, volume distributed higher (above DWL); limits Awp and TPI, reducing heave accelerations.
Wider water plane forward and higher G reduces the stiffness of the GZ curve giving reduced roll accelerations.

59. Ship Response Reduction (8.5)
SEAKEEPING
Ship Response Reduction (8.5)
Recall pitch and heave are well damped but roll motion is sharply tuned, lightly damped, and very susceptible to the encounter frequency!

60. Ship Response Reduction (8.5)
SEAKEEPING
Ship Response Reduction (8.5)
“Anti-Roll Devices” are used to damp roll motion more effectively.
Two categories of Anti-Roll Devices
Passive- no external input required
Active- require some kind of power or control system

61. Passive Anti-Roll Devices (8.5)
SEAKEEPING
Passive Anti-Roll Devices (8.5)
Bilge Keel - very common, dampens roll up to 35 %.
Tank Stabilizers - ‘throttled’ fluid flow across a transverse tank.
Others - tried w/o much success such as delayed swinging pendulums, shifting weights, and large gyroscopes.

62. Active Anti-Roll Devices (8.5)
SEAKEEPING
Active Anti-Roll Devices (8.5)
Fin Stabilizers - common systems found on many ships, use control hydraulics to move fin.
Others - again with only limited success such as pumping tanks and moving weights with hydraulics.

63. Passive and Active System Effects (8.5)
SEAKEEPING
Passive and Active System Effects (8.5)
Resonance can still occur, however roll amplitude at resonance is reduced.
Anti-roll devices have little impact on the motions of heave and pitch (which are heavily damped anyway).

64. Ship Response Reduction (8.5)
SEAKEEPING
Ship Response Reduction (8.5)
Responses are significantly influenced by the encounter frequency.
If e is near any n , angular and vertical accelerations may cause severe negative consequences!
Altering course and/or speed may be the easiest solution!

65. SEAKEEPING Ship Response Reduction (8.5)
A seaworthy design is only as good as the crew,
and you only appreciate the design when you need it!