Understanding Ship Response to Ocean Waves

Understanding
Sea Waves and Ship
Response
Prof. Samirsinh P Parmar
Mail: samirddu@gmail.com
Asst. Professor, Department of Civil Engineering,
Faculty of Technology,
Dharmsinh Desai University, Nadiad-387001
Gujarat, INDIA
Lecture-32

Content of the presentation
• What is sea waves?
• Ocean waves nature/ behavior
• Wave characteristics
• Wave creation- causes
• Waves-superposition theorem
• Simple harmonic motion
• Ship response
• Heave motion
• Roll motion
• Pitch motion
• Frequency and amplitude
• Resonance
• Hull Shape effect
• Ship Response Reduction
• Fin Stabilizer

What Are Waves?
• A wave is a disturbance that carries energy through matter or
space without transferring matter.
• Wave Demo

Wave Characteristics
• Amplitude: The greatest
distance from equilibrium.
• The bigger the amplitude of
the wave, the more energy
the wave has.

• Crest: The top point of a wave.
• Trough: The bottom point of a wave.

• Wavelength: The shortest distance between points where the
wave pattern repeats itself.
Wavelength is
measured in
meters, the
symbol is 𝜆

• Phase: Any two points on a wave that are one or more whole
wavelengths apart are said to be “in phase”.

• Frequency: Is the number of complete oscillations a point on
that wave makes each second.
Frequency is
measured in Hertz
(Hz), its units are
1
𝑠

• Speed: The speed or velocity of a wave is how fast the energy
is moved. For most waves, wave speed does not depend on
amplitude, frequency, or wavelength.
• Speed depends only on the medium through which it moves.
• 𝑣 =
𝑚
𝑠
• 𝜆 = 𝑚 , 𝑓 =
1
𝑠
• 𝑣 = 𝑓𝜆

• Period: The time it takes a wave to go through one cycle, or the
time it takes a point to go through one phase of the wave.
• The period of a wave is measured in seconds, and it’s symbol is
“T”.
• 𝑇 =
1
𝑓

Waves at Boundaries
• Recall that a wave’s speed depends on the medium it passes
through:
• Water depth
• Air temperature
• Tension, and mass
• A “Boundary” is when a wave goes from one medium to another.

Waves at Boundaries
• There are three ways you’ll be interested in at boundaries.
1. Incident Wave: A wave pulse that strikes the boundary.
2. Reflected Wave: A wave that bounces backwards after
hitting the boundary.
3. Transmitted Wave: A wave that continue forward after hitting
the boundary.

Waves at Boundaries Demos
• Different Medium Demo:
• The incident energy is split between reflected and transmitted energy.
• Rigid Boundaries Demo:
• All of the incident energy is turned into reflected energy.

• Ship is so far assumed to be in calm water to determine,
- stability of ship
- EHP calculation through Froude expansion
• Ship usually, however, encounters waves in the sea.
• Ship will respond due to wave action.
Excitation
Wave
Wind
Response
Input
output
Motions
Structural load
Seakeeping

Waves
Wave Creation and Energy
Energy transfer to sea Wave Creation
High speed ship Large wave
Wave energy, E= f(wave height²)
- Doubling in wave height  quadrupling of Wave Energy
- Cw at hull speed rapidly increases due to higher wave creation.
2
8
1
gH
E 


Wave Energy Sources
 Wind : most common wave system energy source
 Geological events : seismic action
 Currents : interaction of ocean currents can create
very large wave system.
Waves

Wind Generated Wave Systems
The size of these wave system is dependent on the following factors.
• Wind Strength :
- The faster the wind speed, the larger energy is transfer to the sea.
- Large waves are generated by strong winds.
• Wind Duration :
- The longer wind blow, the greater the time the sea has to become fully
developed at that wind speed.
Waves

Wind Generated Wave Systems
• Water Depth :
 Wave heights are affected by water depth.
 Waves traveling to beach will turn into breaking wave by a depth
effect.
• Fetch
 Fetch is the area of water that is being influenced by the wind.
 The larger the fetch, the more efficient the energy transfer between
wind and sea.
Waves

Wave Creation Sequence
Energy Dissipation
due to viscous friction
Fully Developed Wave
(W. energy=Dissipation Energy)
Swell (low frequency long wave)
Ripples and Growing
(W. energy>Dissipation Energy)
Ripple
(high freq.)
Reducing
(W. energy<Dissipation Energy)
Waves

Ripples
Swells
Growing Seas
Fully Developed Seas
Reducing
Waves

• Ripple : high frequency, short wave
• Fully developed wave : stable wave with maximized wave
height and energy (does not change as the wind continues
to blow)
• Swell : low frequency, long wave, high frequency waves
dissipated
Definitions
Waves

z
t (sec)
o
z

T
o
z

Sinusoidal Wave- A wave pattern in the typical sine pattern
Period, T- Distance to complete one complete wave (sine) cycle, defined as 2p radians
(Here the period is 2/3 second, .667sec)
- Remember that p = 180o, so 2p is 360o, or one complete cycle
1
Waves

z
t (sec)
o
z
+
o
z

Frequency, w - The number of radians completed in 1 second (here
the wave completes 9.43 radians in 1 second, or 3p… = to 1.5 times
around the circle)
1
w = 2p
T
2p
p
3p
w is given in RADIANS/sec
Waves

wn = 2p
T
wn = k
m
These two formulas for frequency are also referred to as
the Natural Frequency, or the frequency that a system will
assume if not disturbed:
Where k = spring constant (force/ length compressed/ stretched)
Waves

z
t (sec)
T
Displacement, Z - The distance traveled at a given time, t
- Zo reflects the starting position
- Z will be cyclical…it will not be ever-increasing
1
…This will give you the height of the wave or the length of
the elongation / compression in a spring at a given time
Z = Zo Cos(wnt)
+ Zo
- Zo
Z
Waves

Wave Interference
• When we had two particles (carts) and pushed them into each
other they collided and then stopped.
• When waves collide they temporarily interfere with one another,
but they do not stop each other.

Wave Interference
• The Principle of Superposition states that the displacement of a
medium caused by two or more waves is the algebraic sum of
the displacements caused by the individual waves.

Superposition Theorem
The configuration of sea is
complicated due to interaction of
different wave systems.
(Irregular wave)
The complicated wave system
is made up of many sinusoidal
wave components superimposed
upon each other.
Fourier Spectral Analysis
Waves

Wave Spectrum
Frequency
o
0
m
4.0
height
t wave
Significan
curve
under the
Area
:
)
(
Energy
Total





dw
w
S
m
gm
o
o

Significant wave height :
- Average of the 1/3 highest waves
- It is typically estimated by observers of wave systems
for average wave height.
Waves

Wave Data
Modal Wave Frequency :
T
ww
p
2

Waves
Number Significant Wave
Height (ft)
Sustained Wind
Speed (Kts)
Percentage
Probability of
Modal Wave Period (s)
Range Most
Probable
Range Mean Range Mean
0-1 0-0.3 0.2 0-6 3 0 - -
2 0.3-1.5 1.0 7-10 8.5 7.2 3.3-12.8 7.5
3 1.5-4 2.9 11-16 13.5 22.4 5.0-14.8 7.5
4 4-8 6.2 17-21 19 28.7 6.1-15.2 8.8
5 8-13 10.7 22-27 24.5 15.5 8.3-15.5 9.7
6 13-20 16.4 28-47 37.5 18.7 9.8-16.2 12.4
7 20-30 24.6 48-55 51.5 6.1 11.8-18.5 15.0
8 30-45 37.7 56-63 59.5 1.2 14.2-18.6 16.4
>8 >45 >45 >63 >63 <0.05 15.7-23.7 20.0

Simple Harmonic Motion
Condition of Simple Harmonic Motion
+a
-a
- Linear relation :
The magnitude of force or moment must be linearly proportional
to the magnitude of displacement
- Restoring :
The restoring force or moment must oppose the direction of
displacement.
a
 
A naturally occurring motion in which a force causing displacement is
countered by an equal force in the opposite direction.
- It must exhibit a LINEAR RESTORING Force

0

z a
z 

a
z 

Tension
Compression
kz
f 

kz
f 

 If spring is compressed or placed in tension, force that will try to return the mass to
its original location Restoring Force
 The magnitude of the (restoring) force is proportional to the magnitude of
displacement  Linear Force
k

0

z o
z
z 

kz
f 

Mathematical Expression of Harmonic Motion
0
0
2
2






kz
dt
z
d
m
kz
ma
ma
-kz
ma
f
law
2nd
Newtons'
)
cos( t
z
z n
o 

Solution
frequency
Natural
:
nt
displaceme
Initial
:
constant
Spring
:
block
of
Mass
:
n

o
z
k
m
m
k

Mathematical Expression of Harmonic Motion
)
cos( t
z
z n
o 

- Equation
- Curve Plot
z
t
o
z
 T
o
z

m
k
T
m
k
T
n
n
p

p

2
1
Period,
or
,
2




- Natural frequency

Spring-Mass-Damper System
sprin
g
mass damper
m
c
)
cos(
,
0 )
2
/
(
2
2
t
z
e
z
kz
dt
dz
b
dt
z
d
m n
o
t
m
b






- Equation of motion (Free Oscillation) & Solution
C : damping
coefficient
The motion of the system is affected by the magnitude of damping.
 Under damped, Critically damped, Over damped
If left undisturbed, these systems will continue to oscillate, slowly
dissipating energy in sound, heat, and friction
- This is called free oscillation or an UNDAMPED system
k

- Under Damped : small damping, several oscillations
- Critically Damped : important level of damping, overshoot once
- Over damped : large damping, no oscillation
t
z No-Damping
Under damped
Critically damped
Over damped t
a
b
oe
z )
2
/
(

o
z

Roll
Motion source : exiting force or waves
Damping source : radiated wave, eddy and viscous force
Radiated wave
Eddy
Friction
Ship motion (Pitch, Roll or Heave)

Forcing Function and Resonance
Unless energy is continually added, the system will
eventually come to rest
An EXTERNAL FORCING FUNCTION acting on the
system
- Depending on the force’s application, it can hinder oscillation
- It can also AMPLIFY oscillation
When the forcing function is applied at the same frequency
as the oscillating system, a condition of RESONANCE
exists

External Force, Motion, Resonance
sprin
g
mass
m
)
cos(
2
2
t
F
kz
dt
z
d
m 


- Equation of motion (Forced Oscillation) & Solution
)
cos( t
F 
External force
)
cos(
1
1
2
t
k
F
z
n













)
(
:
when
,
when
,
0
when
,
resonance
z
z
k
F
z
n
n
n













freq.
force
external
:

k

Forcing Function & Resonance
Condition 1- The frequency of the forcing function is much
smaller than the system
Displacement, Z = F/k
Condition 2- The frequency of the forcing function is much
greater than the system
Z = 0
Condition 3- The frequency of the forcing function equals the
system
Z = infinity
THIS IS RESONANCE!

External Force, Motion, Resonance with damper
)
cos(
2
2
t
F
kz
dt
dz
b
dt
z
d
m 



Equation of forced motion
Amplitude of force motion
2
2
2
2
1
2
4
1
1



































n
n
n m
b
k
F
z





m b b : damping
coefficient
)
cos( t
F 
k

External Force, Motion, Resonance with damper
Frequency
n

Very low damped
:Resonance
Lightly
damped
Heavily damped

Ship Response Modeling
Ship Response
m b
)
cos( t
F 
• Heave of
ship
m
damping
• Spring-mass-damping
wave
to
due
force
exiting
:
)
cos( t
F w

modeling
k
Additional Buoyancy Force
z
gA
kz w



Encounter Frequency
- Motion created by exciting force in the spring-mass-
damper
system is dependant on the magnitude of exciting force
(F) and
frequency (w).
2
2
2
2
1
2
4
1
1



































n
n
n m
b
k
F
z





- Motion of ship to its excitation in waves is the same as one of
the spring-mass-damper system.
- Frequency of exciting force is dependent on wave frequency,
ship speed, and ship’s heading.
Ship Response

Encounter Frequency
V
V
180

 0


90


45


Wave
direction V
g
V
w
w
e




cos
2


direction
wave
the
to
relative
angle
heading
s
ship'
(ft/s)
speed
ship
V
freqency
wave
freqency
encounter







w
e
Ship Response

• Encounter Frequency Conditions
- Head sea : A ship heading directly into the waves will meet the
successive waves much more quickly and the waves will appear
to be a much shorter period.
- Following sea : A ship moving in a following sea, the waves will
appear to have a longer period.
- Beam sea : If wave approaches a moving ship from the broadside
there will be no difference between wave period and apparent
period experienced by the ship
Ship Response

Rigid Body Motion of a Ship
• Translational motion : surge, sway, heave
• Rotational motion : roll, pitch, yaw
• Simple harmonic motion : Heave, Pitch and Roll
surge roll
pitc
h
heave
sway
yaw
6 degrees of freedom
Ship Response

Heave Motion
Generation of restoring force in heave
z
z
Ship Response

= FB
Zero Resultant Force
DWL
Resultant
Force
FB >
DWL
Resultant
Force
C
L C
L C
L
•
•B
G
•
•
G
B
•
•
G
B
> FB



Heave Motion
Restoring force in heave
• The restoring force in heave is proportional to the additional
immersed distance.
• The magnitude of the restoring force can be obtained using
TPI of the ship.
in
lb
LT
in
in
ft
A
s
ft
g
ft
lbs
TPI wl
1
2240
1
12
1ft
1
)
(
)
(
)
( 2
2
4
2




 
• Restoring force
TPI
k
inch
z
TPI
z
k


 )
(
wl
A
TPI 
Ship Response

Heave Motion
: Natural frequency of spring-mass system
m
k
n 

• Heave Natural frequency
wl
heave
heave
wl
heave
A
T
A
TPI
g
TPI
g












p

2
/
m


TPI
heave

Ship Response

Roll Motion
Generation of restoring moment in roll
Creation of Internal Righting Moment
0
2
2

 kz
dt
z
d
m
G
S
B
FB
¸ B
FB
¸
G Z
S
•
•
•
•


0
2
2

 

k
dt
d
Ixx
Ship Response

Roll Motion
0
2
2

 

k
dt
d
Ixx
• Natural Roll frequency
m
k
n 
 0
2
2

 kz
dt
z
d
m
• Roll Period
T
roll
roll
M
G
B
C
T




p
2
(ft)
GM
ft
s
C
ft
B
T height
c
metacentri
transverse
unknown)
if
good
is
(0.44
)
/
55
.
0
35
.
0
(
constant
)
(
ship
of
beam
2
/
1




xx
T
roll
I
GM



Equation of spring mass
Equation of ship roll motion
Ship Response

Roll Motion
Roll motions are slowly damped out because small wave
systems are generated due to roll, but
Heave motions experience large damping effect.
xx
T
roll
I
GM



T
roll
roll
GM
B
C
T




p
2
Ship Response

Roll Motion
Stiff GZ curve; large GM
Tender GZ curve; small GM
Angle of heel (degree)
Large GM ; stiff ship  very stable (good stability)
 small period ; bad sea keeping quality
small GM ; tender ship  less stable
 large period ; good sea keeping quality
Ship
Response

Pitch Motion
yy
I
m
MT
k


)
(
Mass
1
)
(
Constant
Spring ''
)
ship
a
for
1
(
1
'
'
'
'
MT
I
MT
I
T yy
yy
pitch 

(Long and slender ship has small Iyy)
Pitch motions are quickly damped out since large waves
are generated due to pitching.
G
B
B
F
S

G
B
B
F
S

<Generation of pitch restoring moment>
12
3
BL

yy
I
:
barge
yy
L
pitch
I
GM
w


Pitch moment  ; Tpitch  ; pitch accel. 
Ship Response

Resonance of Simple Harmonic Motion
Heave Pitch Roll
e
 e
 e

heave
 pitch
 roll

• Resonance : Encounter freq.  Natural freq.
• Heave & Pitch are well damped due to large wave generation.
• Roll amplitude are very susceptible to encounter freq. And roll motions are not damped well
due to small damping.
• Resonance is more likely to occur with roll than pitch & heave.
• Thus anti-rolling devices are necessary.
Ship Response

Non-Oscillatory Dynamic Response
 Caused by relative motion of ship and sea.
 Shipping Water (deck wetness) : caused by bow submergence.
 Forefoot Emergence : opposite case of shipping water where the bow of
the ship is left unsupported.
 Slamming : impact of the bow region when bow reenters into the sea.
Causes severe structural vibration.
 Racing : stern version of forefoot emergence.
 Cause the propeller to leave the water and thus cause the whole ship
power to race (severe torsion and wear in shaft).
 Added Power : The effects of all these responses is to increase the
resistance.
Ship Response

Hull Shape
Ship Response Reduction
• Forward and aft sections are V-shaped
limits MT1” reducing pitch acceleration.
• Volume is distributed higher ;
limits Awl and TPI reducing heave
acceleration.
• Wider water plane forward :
limits the Ixx reducing the stiffness of GZ
curve thereby reducing roll acceleration.
Hull Shape

Passive Anti-Rolling Device
• Bilge Keel
- Very common passive anti-rolling device
- Located at the bilge turn
- Reduce roll amplitude up to 35 %.
• Tank Stabilizer (Anti-rolling Tank)
- Reduce the roll motion by throttling the fluid
in the tank.
- Relative change of G of fluid will dampen the roll.
Throttling
U-type tube
Bilge keel

Active Anti-Rolling Device
• Fin Stabilizer
- Very common active anti-rolling device
- Located at the bilge keel.
- Controls the roll by creating lifting force .
Lift
Anti-roll moment
Roll moment

Fin Stabilizer

Ship Operation
g
V
w
w
e




cos
2


• Encountering frequency
• Ship response can be reduced by altering the
- ship speed
- heading angle or
- both.
heave
w
roll
w
pitch
w

ship speed = 20 kts, heading angle=120 degree
wave direction : from north to south, wave period=12 seconds
Encountering frequency ?
V=20kts
60


Wave frequency : s
rad
s
T
w /
52
.
0
12
2
2



p
p

Encountering angle : o
60
120
180 



Encountering freq. :
s
rad
g
V
w
w
e
/
38
.
0
14
.
0
52
.
0
17
.
32
60
cos
)(33.78)
52
.
0
(
52
.
0
cos
2
2











)
/
78
.
33
1
/
689
.
1
20
( s
ft
kts
s
ft
kts
V 

120°
N
S
Example Problem

Example Problem
• You are OOD on a DD963 on independent steaming in the center of your box during supper.
You are doing 10kts on course 330ºT and the waves are from 060ºT with a period of 9.5 sec.
The Captain calls up and orders you to reduce the Ship’s motion during the meal. Your JOOD
proposes a change to course 060ºT at 12 kts. Do you agree and why/why not?
• The natural frequencies for the ship follow:
wroll = 0.66 rad/s wlongbend = 0.74 rad/s
wpitch = 0.93 rad/s wtorsion = 1.13 rad/s
wheave= 0.97 rad/s

Example Answer
• Your current condition:
ww = 2p/T = 2p/9.5 sec = .66 rad/s
Waves are traveling 060ºT + 180º = 240ºT
we = ww - (ww²Vcosµ) / g
= .66 rad/s – ((.66rad/s)² × (10 kt × 1.688 ft/s-kt) × cos(330º -
240º)) / (32.17 ft/s²) = .66 rad/s = wr
• Encounter frequency is at roll resonance with seas from the beam -
bad choice

Example Answer
• JOOD proposal:
= .66 rad/s – ((.66 rad/s)² × (12 kt × 1.688 ft/s-kt)
× cos(060º - 240º)) / (32.17 ft/s²) = .93 rad/s = wp
• Encounter frequency is at pitch resonance with seas from the bow -
another bad choice.
• Try 060º at 7kts:
= .66 rad/s – ((.66r ad/s)² × (7kt × 1.688 ft/s-kt)
× cos(060º-240º)) / (32.17ft/s²) = .82 rad/s
• This avoids the resonant frequencies for the ship - Good Choice.

Understanding Ship Response to Ocean Waves

Understanding Ship Response to Ocean Waves

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