5-Seakeeping - Ship Motions in Regular Waves (2012)

However, only heaving, pitching and rolling are purely oscillatory motions, as these motions are subject to a restoring force or moment when the ship is disturbed from its equilibrium po

SEAKEEPING –

SHIP MOTIONS IN REGULAR WAVES

Note: These notes are drawn from those issued by Dr Jonathan Duffy to students of JEE329 Seakeeping & Manoeuvring at

the Australian Maritime College Dr Duffy has used edited extracts from the main reference books for the subject and

which are listed at the end of the module

A ship in waves is almost always in oscillatory motion However, only heaving, pitching and rolling are purely oscillatory motions, as these motions are subject to a restoring force or moment when the ship is disturbed from its equilibrium position Therefore, heave, pitch and roll motions are particularly important as they possess natural response periods and the potential for resonance For surge, sway and yaw motions the ship does not return to its original equilibrium position after being disturbed unless the exciting forces or moments act alternately from opposite directions

Figure 1 The x, y and z axes of a ship (Bhattacharyya 1978)

In Figure 1 the ship motions are defined as follows:

 Translation along the x axis is surging

 Translation along the y axis is swaying

 Translation along the z axis is heaving

 Rotation around the x axis is rolling

 Rotation around the y axis is pitching

 Rotation around the z axis is yawing

MODULE 5

Although a ship experiences all 6 types of motion simultaneously, each motion will be dealt with separately here It is important to note that any one kind of motion is not independent of the others; however, to simplify the problem the coupling between the motions will be neglected in this course

The heave, pitch and roll motions of a ship can be compared to the oscillatory motion of a mass

on a spring, as these motions are subject to a restoring force or moment when the ship is disturbed from its equilibrium position If a mass attached to the end of a spring is disturbed from its equilibrium condition and the motion is considered to be undamped, the oscillation around the equilibrium position is a sine wave and is an example of simple harmonic motion The force exerted on the stretched spring is:

In addition to the two terms in the equation for simple harmonic motion there are two other factors that apply to a ship in waves, i.e damping and a forcing function Damping of a ship’s motion can be caused by many factors, including friction and wave energy dissipation The forcing function is due to the waves Therefore, the oscillatory behaviour of a ship in waves is fundamentally similar to the response of the classical damped spring mass system acted on by an oscillating force Therefore an understanding of the characteristics of such a system is a good basis for the study of ship motions Note that for a ship the motion will generally be in 6 degrees

of freedom, however, to explain the concept it is easiest to consider one degree of freedom only

Consider the damped spring mass system in Figure 1.In this case the mass is denoted by m, b

denotes the damping coefficient, c denotes the spring stiffness and F is an oscillatory force acting

on the structure The forces acting in the damped spring mass system are:

 The inertial force, 𝑚𝑑2𝑥

𝑑𝑡 2

 The damping force, 𝑏𝑑𝑥𝑑𝑡

 The spring restoring force, 𝑐𝑥

 The time dependent oscillatory force on the structure, F

The forces must be in equilibrium, thus the differential equation of motion of the system is:

Or

Note that a dot above symbols denotes differentiation with respect to time

In the following sections this fundamental differential equation will be applied to the oscillatory motions experienced by a ship and solutions will be provided to give the motion of the ship

Figure 1 Damped spring mass system with oscillating exciting force

Let us assume that a ship is forced down deeper into the water from its equilibrium position and suddenly released An oscillatory motion will occur, known as heaving This oscillatory motion occurs as when the ship is forced down deeper into the water the buoyancy force is greater than the weight of the ship, causing a restoring force The ship accelerates upward towards the equilibrium position due to the restoring force As the ship approaches the equilibrium position, the restoring force decreases and the acceleration toward the equilibrium position diminishes When the ship reaches its equilibrium position the restoring force and acceleration vanish, but by this time the ship has reached its maximum velocity and it will continue to rise due to momentum As the ship rises above its equilibrium position the ship weight is greater than the buoyancy force and hence the upward velocity will be reduced Eventually the ship will reach its extreme position where the velocity of the ship is zero At this point the weight of the ship exceeds the magnitude of the buoyancy force causing a restoring force moving the ship downward until it reaches an extreme position below the equilibrium position If there is no damping force the oscillatory motion will continue indefinitely This is free, undamped oscillation and is known as free oscillation This movement of the ship is a simple harmonic motion If damping is introduced we have a free, damped oscillation (sometimes termed damped oscillation)

If we now assume that the ship is being oscillated vertically up and down by a fluctuating force that is periodic in nature the motion will be irregular and is known as transient oscillation Due to the damping the irregularities disappear and a steady-state oscillation occurs This is a forced, damped oscillation (sometimes termed a forced oscillation), in which the amplitude and frequency of the motion are dependent on the amplitude and frequency of the exciting force The damping will also affect the amplitude of the forced oscillation

3.1 General Form of the Heave Equation

Applying the equation for a damped spring mass system to a ship heaving in a seaway we obtain:

Where

which tends to bring the ship back to its equilibrium position

mass of the ship

3.2 Heave in Calm Water

Now we will look at two simplified cases before moving onto the case with all forces acting, as follows:

Case 1 Free, undamped heaving motion (𝑭𝟎= 𝟎, 𝒃 = 𝟎)

Let us assume that a ship is forced down deeper into the water from its equilibrium position and suddenly released If damping is assumed to be negligible the resulting motion is a free, undamped oscillation (also termed free oscillation) In the case of free oscillation the distance above the equilibrium position is the same as the distance from the equilibrium position to the maximum downward distance travelled by the ship

The magnitude of motion on either side of the equilibrium position is termed the amplitude of the heaving motion The heaving period is defined as the time required for one complete cycle Since the free heaving motion is a simple harmonic motion, the period of oscillation is independent of the amplitude and is thus termed the natural period

Since both 𝐹0and 𝑏 =0 for this case the equation for the condition of equilibrium is:

Where

𝑎 = virtual mass of the ship

𝑐 = heave restoring force coefficient

The solution to this differential equation is:

or

Where

𝐴 is a constant that can be found from initial conditions

𝐵 is a constant that can be found from initial conditions

𝛽 is the phase angle

𝜔𝑧 is the natural frequency of the heaving motion 𝜔𝑧 = 2𝜋 𝑇 = 𝑐 𝑎𝑧

𝑇𝑧 is the heaving period, which may be considered to be constant for small and moderate motions and not dependent upon the amplitude of the motion

Therefore the natural frequency of the heaving motion is:

𝜔𝑧 = 𝜌𝑔 𝐴𝑤 𝑝

Case 2 Free, damped heaving motion (𝑭𝟎= 𝟎)

This is sometimes termed damped oscillation Since 𝐹0 is zero for this case the equation for the condition of equilibrium is:

𝜈 is the decaying constant (𝜈 = 𝑏 2𝑎 )

𝜔𝑑 is the circular frequency of the damped oscillation (𝜔𝑑 = 𝜔𝑧2− 𝜈2)

𝜔𝑧 is the natural circular frequency of the free, undamped oscillation

𝐶1, 𝐶2, 𝐴 𝑎𝑛𝑑 𝛿 are constants that can be determined from initial conditions

The damped heaving period is:

The damped heaving period is greater than the natural heaving period since 𝜔𝑑 < 𝜔𝑧

An example of free damped oscillation is given in Figure 2

Figure 2 Free damped oscillation (Bhattacharyya 1978)

Note that if 𝜈 > 𝜔𝑧 (damping is very large) the motion is no longer oscillatory and is known as aperiodic Such cases are not of interest to us as the magnitude of 𝜈 is always very small compared to the magnitude of 𝜔𝑧 for a ship in waves Damping acts in the opposite direction to the motion, which slows the motion The amplitude of the heaving motion therefore gradually decreases until the ship finally comes to rest at the equilibrium position The period of oscillation for the case with damping will be slightly larger than for a case without damping

3.3 Forced Damped Heave (in Regular Waves)

This case is analogous to a ship heaving in waves as it includes an oscillatory exciting force due

to waves Since all forces act for this case the equation for the condition of equilibrium is:

Where

𝑧𝑎is the amplitude of the forced motion

𝜀2is the phase angle of the forced motion in relation to the exciting force

Equation 17 describes the sum of two oscillations It can be seen that the first term is the same as the solution for free, damped heaving oscillation The second term describes a sinusoidal oscillation with constant amplitude with the same circular frequency as that of the exciting force for heaving and a phase angle of 𝜀2 to the exciting force At time t=0 the motion is said to be transient as both oscillations exist As t increases the first term diminishes (at a rate depending on

the value of 𝜈) and the motion reaches a steady state oscillation governed by the second term Therefore the solution for the steady-state condition (when the first term dies out with time) is:

Figure 3 shows an example of the forced response, natural response and the total response

Figure 3 Forced response, natural response and total response

An example of just the total response is shown in Figure 4 The damped forced vibration corresponds

to the transient oscillation, whilst the steady forced vibration corresponds to the steady state forced response

Figure 4 Example response for forced damped heave (steady amplitude = 𝑧𝑎)

Let us consider the steady forced oscillation If it is assumed that a wave of a certain length and amplitude passes along rather slowly, so that the ship is in position to balance itself statically on the wave at every instant of its passage, the ship will then rise and fall slowly with the encountering frequency so as to keep balance between weight and buoyancy, and a static

amplitude, z st, will result If the wave is now considered to move at its correct velocity, a

dynamic amplitude, z a, will be produced The ratio of the amplitude in the dynamic case to that

in the static case is called the magnification factor and is denoted by 𝜇𝑧:

𝜈 is the decaying constant (𝜈 = 𝑏 2𝑎 )

𝜔𝑧 is the natural circular frequency of the free, undamped oscillation (𝜔𝑧 = 𝑐 𝑎 )

𝜀2 is the phase angle of the forced motion in relation to the exciting force 𝜀2= 𝑡𝑎𝑛−1 2kΛ

1−Λ 2

It is important to know the phase angle when studying ship motions as this affects such things as slamming and deck wetness It can be seen that the phase angle depends on damping For an undamped system (k = 0) the phase angle is 0 for frequencies less than the natural frequency and 180 degrees for frequencies greater than the natural frequency The magnification factor can

be plotted as a function of tuning factor, which yields the figure for dynamic response, as illustrated in Figure 5

From Figure 5, for low damping, if the encountering frequency of the exciting force is close to the natural frequency of heaving for the ship the amplitude of the steady-state oscillation may be much larger than the static amplitude and the magnification factor, 𝜇𝑧, is much greater than one The maximum response is obtained by differentiating Equation 21 with respect to Λ and equating

to zero From this it can be shown that the magnitude of the magnification factor is maximum when Λ = 1 − 2k2

Hence, for very small values of k (low damping) the maximum magnification factor occurs when the tuning factor is very close to 1 Therefore, when designing a ship for a seaway when damping

is low Λ = 1 should be avoided to minimize motions It can be seen that for cases with larger damping the maximum value of magnification factor shifts towards the left

Figure 5 Magnification factor as a function of tuning factor (Bhattacharyya 1978)

From Figure 5 it can be seen that if the frequency of the exciting force is very low then the amplitude of oscillation in the steady state condition will be equal to the static amplitude, i.e the ship will rise and fall slowly with the encountering frequency with a balance maintained between weight and buoyancy, hence 𝑧𝑎 = 𝑧𝑠𝑡 and 𝜇𝑧 = 1 It can be seen that the natural frequency of a ship is an important parameter when designing for a given seaway, as this influences the tuning factor and this is important in finding the region of resonance In order to predict the amplitude

of the steady forced heave motion the terms a, b, c and 𝐹0 need to be predicted The terms are discussed in the following sections along with methods to predict them

3.4 Inertial Heave Force

The following description of the prediction of heave inertial force is taken from Bhattacharyya (1978)

A body having an accelerated motion in a continuous medium of fluid experiences a force that is greater than the mass of the body times the acceleration Since this increment of force can be

defined as the product of the body acceleration and a quantity having the same dimension as the

mass, it is termed added mass This concept is needed to discuss the inertial force of a ship The

inertial force is the body accelerating force (ship mass x acceleration) plus a liquid accelerating force:

𝑚 is the ship mass

𝑎𝑧 is the added mass in heave

𝑚𝑑2𝑧

𝑑 𝑡 2 is the body accelerating force

𝑎𝑧𝑑2𝑧

𝑑𝑡 2 is the liquid accelerating force

One should remember that the concept of added mass is introduced into fluid mechanics for convenience of evaluation and does not have any physical significance For example, one should not imagine that a body accelerating in an ideal fluid in a certain direction drags with it a certain amount of fluid mass

A method to predict the added mass of a ship section follows According to Lewis (1929), an inertial coefficient C is defined as:

Where breadth 𝐵𝑛 = 2𝑟 For shapes other than semi-circular ones, the added mass of a ship

section, a n, is found to be:

𝑎𝑛 = 𝐶𝜌𝜋 𝐵𝑛2

The coefficient C for Lewis-form sections for two dimensional floating bodies is obtained from Figure 6 as a function of:

 the draught/beam ratio 𝐵𝑛 T

 the sectional area coefficient 𝛽𝑛 = 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎 𝐵𝑛 ∗ 𝑇

 the circular frequency of oscillation

Figure 6 Added mass coefficients for two-dimensional floating bodies in heaving motion

(Bhattacharyya 1978)

It is assumed that the added masses for sections other than those of the mathematical Lewis form will not differ appreciably as long as the beam, draught, and area of each section are equal in both cases For conventional hull forms the Lewis-form representation of a section that has the correct beam, draught and area, but not shape, has been found to be quite satisfactory Therefore the ship may be divided into transverse sections and the added mass can be determined for each section The added mass for each section can then be added to give the total added mass of the ship, as follows:

It should be noted that, by computing the added mass as the sum of individual strips, the interaction between adjacent sections are ignored This will be discussed in further detail later

3.5 Damping Heave Force

The following description of the prediction of heave damping force is taken from Bhattacharyya (1978)

The damping in heave is caused mainly by the waves generated by the heaving motion of the ship The damping force always acts in the opposite direction to the motion of the ship and causes a gradual reduction in the amplitude of the ship motion A simplified prediction of heave damping force can be obtained from:

Where 𝑏 is the dampingforce coefficient in heaving and normally depends upon the following factors:

Type of oscillatory motion

Encountering frequency of the waves

Form of the ship

It can be seen that the damping force coefficient is assumed to be proportional to the velocity of oscillation, 𝑑𝑧𝑑𝑡, since the damping in heaving is mainly due to the waves generated by the ship; i.e the damping due to friction and eddy-making is quite small in comparison with that due to

wavemaking The coefficient b can also be calculated using strip theory The damping

coefficient per unit length of the ship is directly related to the wave amplitude since the damping

in heaving is caused mainly by the waves generated by the heaving motion of the ship The heave damping coefficient of a ship section can be predicted using Equation 31, as presented by Grim (1959):

The amplitude ratio 𝐴 can be obtained from Figure 7

Figure 7 Amplitude ratios 𝐴 for two-dimensional floating bodies in heaving motion, Grim (1959)

𝛽𝑛isthe sectional area coefficient , 𝐵𝑛 is the breadth of each section and T is ship draught

(Bhattacharyya (1978)

The damping coefficient for the ship can then be obtained by integrating over the entire length of the ship (summing the damping coefficient of each individual section):

From both experimental and theoretical investigations (Bhattacharyya 1978) concludes:

a The damping force coefficient is proportional to the square of the waterplane area

b Ships with V-type sections experience more damping than do U-type ships for the same load waterplane

c The non-dimensional damping coefficient, when plotted to a base of 𝜔𝑒 𝐿 𝑔 , reaches a maximum value and then decreases rapidly In the region of resonance, where damping plays

a very important role, this decrement in damping does not appear to have a large effect

d The wave damping is lower for high speed of advance

3.6 Heave Restoring Force

The restoring force in heave is the additional buoyancy force due to the draught being increased from the static equilibrium case If we assume that there is no significant change in the waterplane area when the ship draught is increased (i.e wall sided approximation) then the restoring force is simply equal to the specific weight multiplied by the additional submerged volume Therefore the restoring force can be given by:

Where

𝐴𝑤𝑝is the waterplane area

𝑧is the additional immersion of the ship

𝐴𝑤𝑝𝑧is the additional displacement volume due to the additional immersion

Hence, the restoring force coefficient is:

3.7 Exciting Force for Heave

The exciting force is generated by the waves Therefore, a simplified approach to predicting the exciting force is to assume the ship remains stationary in the vertical plane and sum the additional buoyancy along the ship due to the changes in free surface elevation Using this approach it is assumed that the waves pass along the ship gradually Assuming that the ship is wall sided in the region of the load water line the additional buoyancy on each section of the ship

is as follows:

Integrating over the length of the ship gives the exciting force for heaving:

We will assume that the direction of the ship’s heading and the direction of wave propagation are

at an angle μ with each other, and that the ship is symmetrical about the midship section, and that the surface wave profile is the effective wave profile Let ζ = ζ𝑎𝑐𝑜𝑠𝜔𝑒𝑡Substituting the wave profile expression and expanding the cosine term we obtain the exciting heave force:

𝑓𝑜 = 𝐹0

3.8 Heave Transfer Function

Using Equation 20 the steady forced heave amplitude may be predicted for a range of encounter frequencies to give a transfer function A response amplitude operator (RAO) may then be obtained by plotting the square of the y ordinates of the transfer function as a function of encounter frequency RAOs can then be used to predict the response of a ship in an irregular seaway, as we shall see later

An example heave RAO is given in Figure 8

Figure 8 Example Heave RAO

4.1 Pitch Radius of Gyration & Moment of Inertia

The pitch radius of gyration about a designated y axis is obtained by considering the ship as the sum of many small weights and then adding the products of each of these small weights and the square of its distance from the designated axis, as follows:

𝑘𝑦𝑦 is the pitch radius of gyration

𝑤𝑖 is the weight of the ith

element

𝑟 is the distance of the centre of gravity of the ith

element from the axis of rotation

∆ is the total weight of the ship

From Figure 9 the distance of each element can be expressed in terms of x and z coordinates,

thus:

𝑘𝑦𝑦 = 𝑤𝑖 𝑥𝑖2+𝑧𝑖2

Figure 9 Radius of gyration for pitching (Bhattacharyya 1978)

The moment of inertia of a ship depends on the distribution of mass of the body as well as the axis of rotation The moment of inertia for pitching is:

or

Where m is the total mass of the ship

4.2 General Form of the Equation for Motion of Pitch

Since pitching is an angular motion it involves moments rather than forces Recall that for heave there were four forces acting in a seaway Similarly there are four moments which act in pitch in

moment of inertia plus the added mass moment of inertia

o

𝑎 = 𝐼𝑦𝑦′ = 𝐼𝑦𝑦 + 𝛿𝐼𝑦𝑦

 𝑐𝜃 is the pitch restoring moment, which tends to bring the ship back to its equilibrium position

the mass of the ship

4.3 Pitch in Calm Water

As with heave we will look at the case of pitch in calm water to determine the natural frequency for pitching The equation of motion for pitching in calm water is:

Where 𝜔𝜃is the natural circular frequency for pitch without the presence of damping

The natural circular frequency (without damping present) for pitch can be obtained from Equation 52:

To avoid resonance 𝑇𝜃

𝑇𝑒 should be away from 1

4.4 Pitch in Regular Waves

The equation of motion for pitch in a wave is given by:

or