Wave loads and seakeeping

Question 1

Draw a ship in a ship-fixed coordinate system. Show and give the names of the 6 degrees of freedom.

Answer 1

• η1: surge, translation along X axis
• η2: sway, translation about along Y axis
• η3: heave, translation along Z axis
• η4: roll, rotation about X axis
• η5: pitch, rotation about Y axis
• η6: yaw, rotation about Z axis

Question 2

What is the main reason for using a ship-fixed coordinate system and not an earth-fixed coordinate system for the ship motion analysis?

Answer 2

The body xed coordinate system is better for ship motion analysis as the forces, moments, moments of inertia and products of inertia will not vary in time (which would be the case for the earth xed cooridnate system). On the downside the equations of motions gets more complicated for the body xed coordinate system

Question 3

Describe the Strip-theory approach for solving the Seakeeping problem?

Answer 3

The ship can be represented by 2D strips for which the coeficients for added mass, damping and restoring force can be determined by

• A simple 2D potential own solution for each strip.
• Analytical solotion after coordinate transformation to circular 2d cylinder, (Lewis forms)
• Experiments for hull segment shaped 2D cylinder

A summation over strips will then give the properties for the hull.

Question 4

The Strip-theory approach is based on a number of assumptions. Give 4 of them

Answer 4

• The ship is slender
• The hull is rigid
• Moderate speed, no planing
• The motions are small
• The ship hull sides are wall sided
• Deep water
• Waves are not disturbed by the hull

Question 5

The ship motion problem is often split into a Seakeeping problem and a Manoeuvring problem. Give a motivation for this split based on the forces involved.

Answer 5

The separation into seakeeping is due to that this are moving out from the xy-plane

• η3, Heaving
• η4, Pitch
• η5, Roll

and manouvering problem moves only inside the xy-plane

• η1, Serch
• η2, Sway
• η6, Yaw

This is motivated by the small coupling between the degrees of freedom in the two problems

There is no copuling between manov and seakeep.
There are no restoring power in manov

Question 6

Describe the physical effects behind the three forces that occur for forced heave in calm water.

Answer 6

Added mass (−a33η ̈3)

The added mass is in phase with the force necessary to accelerate the body and it depends on the frequency close to the water surface but far from it the added mass only depends on the shape of the body.

Radiation damping (−b33η ̇3)

The radiation damping is heave motion caused waves that are radioated out from the body. This is in phase with the velocity of the heave motion. The radiation damping coecient is zero for frequency of zero and innity and has its maximum somewhere inbetween.

Vertical restoring (−c33η3)
The vertical restoring force is caused by the displacement of the body and is in phase with the vertical position.

Question 7

Explain the “Small ship approximation

Answer 7

Small-body approximation

The body can be assumed small in comparison with the wavelength (B < λ/4)

Question 8

Explain the “Small ship approximation

Answer 8

Wave attenuating eect, describes how the wave vanishes with depth (the decrease of dynamic pressure).
e^−kT describes how the exitation forces vanishes with the draught

Question 9

What is a Response Amplitude Operator (RAO)?

Answer 9

A Response Amplitude Operator (RAO) is the ratio of a studied variable to the wave amplitude.

Question 10

Describe the two free-surface boundary condition for free-surface gravity waves.

Answer 10

• The free surface kinematic boundary condition states that a particle on the surface will stay on the surface.
• The free surface dynamic boundary condition derives from the Bernoulli equation on the assumption that the pressure is constant on the free sur- face.

Question 11

Write the expression for the wave form of a plane harmonic wave progressing in the x-direction. The wave amplitude is a.

Answer 11

The real progressive wave is given by
Re(ζ_c) = a cos(kx − ωt)

Question 12

Draw a sketch of the fluid particle motion beneath a wave in deep water and in shallow water.

Answer 12

See gure 3.2 in the compendium.

The motion does not fully vanish in shallow water towards the bottom. The motion is not circular, which is the case in deep water, but elliptical with a longer horizontal length.

• The real valued pressure for deep water is
  p = ρgaekz cos(kx − ωt) − ρgz
• The real valued pressure for shallow water is
  p = ρga cos(kx − ωt) − ρgz

Question 13

Potential ow description of free-surface waves

Answer 13

Assumptions Potential ow assumes an inviscid, incompressible and irrota- tional ow. The boundary conditions are non-linear following the moving free surface and the simplest way to deal with this is to apply linear or small wave theory (Airy theory) which is based on the assumption that the wave amplitude, a, is small in comparison with the wavelength ,λ.

Question 14

Show how to decompose the hydrodynamic problem for a floating body oscillating in an incident wave?

Answer 14

The problem can be decomposed into

• The reaction forces appearing when the body is forced to oscillate in calm water
• The wave excitation forces when the body is xed in regular waves

Question 15

Tell in words what added mass is

Answer 15

Added mass is water surrounding the body that will be accelerated together with the body and is therefore in phase with the acceleration of the body.

Question 16

Why is the linear hydrodynamic damping called radiation damping?

Answer 16

The hydrodynamic damping is called radiation damping because the heave motion causes waves that radiates energy away from the body and the force needed to maintain this oscillation is in phase with the velocity of the body.

Question 17

Describe the assumptions that must be valid in order to analyze ship motions in the frequency domain

Answer 17

• The wave system can be approximated by a sum of sine and cosine waves.
• Small wave amplitudes
• Small response amplitudes
• The response can be computed for one wave component at the time.
• The total response can be obtained summarizing the responses from the wave components

Question 18

For a turning test describe
Advance,
Transfer,
Tactical diameter
Turning

Answer 18

• Advance: The longitudinal distance from rudder deflection to the point where the ship has turned 9
• Transfer: The transverse distance to the point where the ship has turned 90 deg.
• Tactical diameter: The transverse distance to the point where the ship has turned 180 deg.
• Turning radius: The radius of the turn for steady turn conditions

Question 19

Explain how to carry out a Zig-Zag test, and explain the two parameters that are the result of the test

Answer 19

Constant speed on straight course

• 10 deg rudder angle to port
• When ship turn is 10 deg to port
• 10 deg rudder angle to starboard
• When ship turn is 10 deg to starboard
• 10 deg rudder angle to port

Overshoot angle:
The max course deviation
after change of rudder deflectio

Time for one period:
The time between first and second rudder to port command

Question 21

Describe two methods to determine the hydrodynamic coefficients for the Strip-theory approach.

Answer 21

Annalytical solution for damping and added mass or an experimental solution. The experemental solution can pick up small details in the shape that cant be don in the analytic solution. Added mass and damping is a hydrodynamic motion when

Question 22

Write the equations of motion for a ship in an earth fixed coordinate system.

Answer 22

(m+a)äta(..)+Bäte(.)+Cäta = F_e + F_re

F_re = reactionforce

F_e= exitation force, (energi tillförs vågorna)

Question 23

The heave coefficients for added mass, damping and restoring force are obtained thru forced heave motions. Describe the assumption for using the same coefficients also for forces due to a regular wave.

Answer 23

The same coe-cients can be used (a33; b33; c33) if the following is assumed

Small-body approximation

The body can be assumed small in comparison with the wavelength (B < =4)

Question 24

Explain the “Smith effect”.

Answer 24

Wave attenuating eect, describes how the wave vanishes with depth (the decrease of dynamic pressure).

e?kT

describes how the exitation forces vanishes with the draught.

Question 25

Give the stiffness coefficients in pitch expressed in vertical coordinates of the points of gravity of mass and buoyancy and hydrostatic stiffness.

Question 26

Describe one passive and one active roll damping device

Answer 26

• Passive anti roll tank, (decreases stability)
• Controlled passive anti roll, (controlled valve)
  (decreases stability)
• Active fin stabilization, (increases resistance and works only when relative velocity between fin and water is positive)

Question 27

Explain two of the physical effects that contribute to the roll damping

Answer 27

Two physical effects that contribute to the roll damping are the radiated waves and the turbulent skin friction between the hull and the water.

Question 29

Why is it more difficult to estimate the roll motion than motions in heave, pitch, sway and yaw?

Answer 29

Becaus in roll the coefficient are not constant.
(Added mass, damping and restoring forces).

Question 30

Give two reasons why it is difficult to manoeuvre a ship during a stopping manoeuvre

Answer 30

• de rodder effect vanish
• Assymetric flow in the stern region due to the proppeller can not be compensated by the rodder.
• Since the water is flowing upwards from the bottom to the ship to the propeller the blades on the starbord side will operat at larger angle of attack which moves the center of effort to starboard side.

Question 31

What is a stability derivative. Explain

Answer 31

The stability derivative is lineral relation betweem a force or amoment and a certain velocity or acc when all other velocities and acc are zero. M.a.o its telling how the force/moment are changin due to velocity or acc.

Question 32

Give the wave celerity in terms of wave period and wavelength. Tell the meanings of wave celerity and group velocity.

Answer 32

• Celerity the phase speed depends on the wave period (T)
  C = lamda/T = w/k
• A groupe of wave will propagate slower then the individual wave. Thus the rermost wave will progress through the wave group.
  C_g=C/2 in deep water
  C_g=C in shallow water

Question 33

The dispersion relation for a first-order (linear) wave is tanh(kd ) = w2/gk.

Explain what the notations w, k and d means. Give and motivate the corresponding simplified expression for deep water and shallow water.

Answer 33

• Omega is the wave frequensy
• k is the wave-nr telling the nr of wave length per 2*pi unit distance
• d is the deppt of the water

This showes that the wave celerity depends on the wave length

Question 34

Explain the physics behind the term .

Answer 34

The term represents the exponential decay of the water particle velocity and dynamic pressure as a function of the distance below the undisturbed free surface. Here the distance is T