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Analytical calculation of uncoupled heave and roll A parametric study of a barge

Tor Edvard Søfteland

OFF600 – Marine Operations Fall 2012

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ABSTRACT The theory of how vessels behave and their additional response at sea is called seakeeping and is of great importance in all kinds of marine operations. The response is normally obtained by different software using a finite element approach, but to understand and change the properties of a vessel, one should have a general understanding of how the response relates to different factors. The response of a vessel is divided into six degrees of freedom, three linear and three rotational. In this report, one linear (heave) and one rotational (roll) response will be in focus where the other four degrees of freedom can be calculated by similar approaches as what is shown in this report. After analyzing the heave and roll motion of a vessel with certain assumptions, the formulas for the motion of a barge is derived and the additional assumptions is discussed thoroughly. The resulting motion can be calculated by few easy obtainable parameters which are found in ship certification documents and weather statistics. The roll response where calculated to be 30% larger than the response obtained by a wave tank test of the same vessel due to necessary approximations described in this report. The obtained formulas can be used to show the impact of change in vessel or sea parameters and should be used to compare different vessel motions.

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Table of content 1. INTRODUCTION .............................................................................................................. 8 2. STATE OF ART .............................................................................................................. 10 3. METHODS .................................................................................................................... 12 3.1 VESSEL DEGREES OF FREEDOM .............................................................................................. 12 3.2 UNCOUPLING OF HEAVE AND ROLL ........................................................................................ 13 3.2.1 Linear theory .................................................................................................... 13 3.2.2 Coupled motions .............................................................................................. 13 3.2.3 Symmetric and anti‐ symmetric motions ......................................................... 14 3.3 EQUATION OF MOTION ........................................................................................................ 14 3.3.1 Solutions of equation of motion ...................................................................... 15 4.3.2 Particular solution ........................................................................................... 15 3.3.3 Dynamic amplification factor .......................................................................... 16 3.3.4 Homogeneous solution .................................................................................... 17 3.3.5 Phase angle...................................................................................................... 18 3.4 HARMONIC RESPONSE OF A VESSEL ........................................................................................ 19 3.4.1 Dynamic amplification of a vessel ................................................................... 19 3.4.2 Added mass force ............................................................................................ 20 3.4.3 Damping force ................................................................................................. 21 3.4.4 Restoring force................................................................................................. 22 3.4.5 Exciting force ................................................................................................... 22 3.4.6 Strip theory ...................................................................................................... 24 3.5 HEAVE MOTION ................................................................................................................. 25 3.5.1 Equation of motion .......................................................................................... 25 3.5.2 Added mass in heave ....................................................................................... 25 3.5.3 Damping in heave ............................................................................................ 27 3.5.4 Restoring force in heave .................................................................................. 28 3.5.5 Exciting force in heave ..................................................................................... 29 3.5.6 Natural period in heave ................................................................................... 34 3.5.7 Solution ............................................................................................................ 34 3.6 ROLL MOTION .................................................................................................................... 38 3.6.1 Equation of motion .......................................................................................... 38 3.6.2 Mass moment of inertia and added mass moment of inertia in roll ............... 38 3.6.3 Damping in roll ................................................................................................ 42 3.6.4 Restoring force in roll....................................................................................... 42 3.6.5 Exciting force in roll ......................................................................................... 45 3.6.6 Natural period in roll ....................................................................................... 46 3.6.7 Solution ............................................................................................................ 47

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4. RESULTS ....................................................................................................................... 50 4.1 THE CASE .......................................................................................................................... 50 4.1.1 Input................................................................................................................. 50 4.2 STABILITY .......................................................................................................................... 52 4.3 NATURAL PERIOD ............................................................................................................... 53 4.3.1 Natural period, heave ...................................................................................... 53 4.3.2 Natural period, roll .......................................................................................... 54 4.3.3 Comparison of natural period in roll and heave .............................................. 56 4.4 WAVE PARAMETERS ........................................................................................................... 58 4.5 EXCITING FORCE ................................................................................................................. 59 4.5.1 Exciting force, heave ........................................................................................ 59 4.5.2 Exciting force, roll ............................................................................................ 61 4.6 PARTICULAR SOLUTION ........................................................................................................ 63 4.6.1 Particular solution, heave ................................................................................ 63 4.6.2 Particular solution, roll .................................................................................... 65 4.7 DISCUSSION OF RESULTS ...................................................................................................... 67 4.7.1 Heave ............................................................................................................... 67 4.7.2 Roll ................................................................................................................... 69 5. CONCLUSION ............................................................................................................... 72 6. REFERENCES ................................................................................................................. 73

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List of symbols az

Heave, added mass

[kg]

Heave, added mass for each strip

[kg/m]

A

Roll, added mass moment of inertia

[kgm2]

A 2 D

Roll, added mass moment of inertia for each strip

[kgm2/m]

Aw bz

Waterline area Heave, damping coefficient

[m2] [kg/s]

b 2z D

Heave, damping coefficient for each strip

[kg/sm]

bzC

Heave, critical damping

[kg/s]

B

Roll, damping coefficient

[kgm2/s]

B 2 D

Roll, damping coefficient for each strip

[kgm2/sm]

B

Breadth of vessel

[m]

BM cz

Metacenter radius Heave, restoring force coefficient (stiffness)

[m] [N/m]

C

Roll, restoring force coefficient

[Nm2/m]

CA CB CW D DAFz DC FA FD Fz F0 g

Added mass coefficient Block coefficient Waterline coefficient Draft of vessel Heave, dynamic amplification factor Characteristic dimension Approaching wave force Diffraction wave force Heave, exciting force Heave, exciting force amplitude Gravity acceleration

[-] [-] [-] [m] [-] [m] [N] [N] [N] [N] [m/s2]

a

2D z

GM Metacenter height

[m]

I

Mass moment of inertia

[kgm2]

k

Wave number

[1/m]

KB

Center of bouyancy

[m]

KG

Center of gracity

[m]

L LW M Meq

Length of vessel Wave length Mass Equivalent mass

[m] [m] [kg] [kg]

M

Roll, exciting force

[Nm] VI

M A

Approaching wave force

[Nm]

M0 pD rz

Roll, exciting force amplitude Dynamic water pressure Heave, frequency ratio

[Nm] [N/mm2] [-]

r

Roll, frequency ratio

[-]

T Tnz

Wave period Heave, natural period

[s] [s]

Tn

Roll, natural period

[s]

U

Vessel forward speed

[m/s]

w z

Vertical particle acceleration

[m3]

z

Heave, vessel motion

[m]

z z

Heave, vessel velocity

[m/s]

Z Z0 zp zh

Heave, vessel acceleration Heave, amplitude of particular solution Heave, amplitude of homogeneous solution Heave, particular solution Heave, homogeneous solution

[m/s2] [m] [m] [m] [m]

Roll, vessel motion

[-]

Roll, vessel velocity

[1/s]

Roll, vessel acceleration

[1/s2]

Submerged volume

[m3]

Density of seawater

[kg/m3]

Vessel angle (deviation from x-axis)

[rad]

i

Roll added mass moment of inertia coefficient factor

[-]

z

Heave, phase amplitude particular solution

[-]

z0

Heave, phase amplitude homogeneous solution

[-]

a

Amplitude of radiating waves

[m]

0

Linear wave amplitude

[m]

z

Heave, damping frequency

[-]

Roll, damping frequency

[-]

Wave frequency

[1/s]

e

Encounter frequency

[1/s]

dz

Heave, damped (reduced) natural frequency

[1/s]

nz

Heave, natural frequency (eigen frequency)

[1/s]

n

Roll, natural frequency (eigen frequency)

[1/s]

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1. Introduction When doing operations at sea, it is important to understand and predict the huge environmental forces which are present. The forces are difficult to anticipate, and it’s even harder to calculate the response of an operating vessel. The vessel can be considered as an oscillating system, with six degrees of freedom. By using certain assumptions, the information needed to perform the calculations is narrowed to a few easy obtainable factors. Those factors are found in ship certification documents and weather statistics. The response of vessels at sea is described by three linear and three rotational degrees of freedom. In this report one linear (heave) and one rotational (roll) response will be in focus. The combination of heave and roll is important when calculating loads on cargo and additional seafastening when cargo is transported at sea. When operating at sea, there are three different areas that need to be taken into consideration. (TMC Marine consultants (2008))

Estimation of environmental conditions encountered by the vessel. This is based on hindcast or predicted weather data regarding wind, wind waves, and swell waves. Determination of the vessels response characteristics. Operation criteria based on allowable accelerations for cargo, people, deck wetness, etc.

In this report, focus is drawn towards the vessels response which is determined by choosing some wave parameters based on North Sea environment. Through a structured setup and well defined formulas, the analysis part of the report clearly identifies the similarity between the rotational and linear motion. When assuming that heave and roll is unaffected by each other, the dynamic response of heave and roll can be described with the equations of motion with similar terms in both motions. In addition to the analysis, the results are presented with numerical simulations and parametric studies. The values are discussed and compared with relevant data obtained from a wave tank test of a Standard North Sea Barge.

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2. State of art Dynamic vessel response is normally calculated using finite element software program like Maxsurf, VERES, Seaway or MOSES. These computer programs use the finite element approach called strip theory. The strip theory is an approach where the vessel is divided into certain amount of two-dimensional strips along the longitudinal axis, where the hydrodynamic loadings on each strip are considered and summarized over the length of the vessel. (Journée (2001) p.5) These programs have proven to give accurate results for how vessels behave at sea. When using finite element software to calculate allowable vessel response in terms of operation criteria, certain data is required to get desired results. (TMC Marine consultants (2008))

Principal particular of vessel (dimensions, displacement, draught, trim) Appendages (rudder, bilge keels, stabilizers) Loading conditions (GM) Centre of Gravity (KG, LCG) Radii of Gyration (for rotational degrees of freedom) Vessel speed through water Wave spectrum type Wave parameters (height, period, regular/irregular) Longitudinal weight distribution (for wave induced shear, moment and torsion)

By inserting the needed information into the relevant software, one is able to find the relevant vessel response and vessel accelerations. Limit criteria for different operations is normally given as maximum acceleration limits. The acceleration limits is chosen based on what kind of operation that is considered and what kind of work that are conducted on the vessel (light manual work, heavy manual work, etc.). Of the possible software outputs, the response amplitude operator (RAO) is considered the most important output of a normal vessel analysis. The RAO is also called the transfer function and describes how the response of the vessel varies with frequency. By using the given RAO for a vessel one is able to find out to what extent external forces with different frequencies affect the response of the vessel. (Formsys (2012)) The RAO peek tells us which frequencies that result in potential resonance of the system. The different computer programs are developed by using analytical approaches combined with model testing. The combination of surge, sway, heave, roll, pitch and yaw are affecting each other, and some of the motions cannot be considered with a linear theory approach. When assuming linear theory, it is possible to summarize all terms that is relevant for the motion to get the final response. Roll motion is best described as a non-linear motion with half empirical formulas as a result of model testing. (Journée J.M.J. (2001) p.5)

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When a vessel is moving at sea, the formulation of the resultant vessel motion of each of the degrees of freedom is considered as simple harmonic functions with an amplitude, a phase shift and a frequency. (Journée J.M.J. (2001) p.8)

Surge: x p X sin( e t x )

Sway: y p Y sin( e t y )

Heave: z p Z sin( e t z )

Roll:

p sin( e t )

Pitch:

p sin( e t )

Yaw:

p sin( e t )

Where the frequency term, e is the encounter frequency which describe how fast the waves are approaching a moving vessel. When obtaining the six harmonic functions, one can easily find the velocity and acceleration functions by differentiating the particular solution one and two times respectively.

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3. Methods When doing operations at sea, it is of great importance to understand how vessels are affected by environmental forces at sea. How the ship is affected is described by vessels response in six degrees of freedom system, where the response of each of these motions must be calculated to find the resultant motions of a vessel. The theory of how a ship is behaving is called seakeeping. The environmental loading affecting a ship is hard to analyze and predict, and it is even harder to correctly calculate the influence on a given vessel. There are many methods for calculating vessels motions, and the analytical results that come close to the real behavior of a given vessel are all numerical methods based on finite element methods. This chapter describes how vessel motions is calculated in general and presents analytical formulas for how to calculate the motions of a barge based on proper assumptions which are thoroughly described. The goal is to calculate vessels motions with use of easy obtainable data. By using certain formulas, one can quickly understand how the vessel is behaving without needing time consuming finite element approaches or programming. One rotational and one linear motion will be thoroughly discussed in this report.

3.1 Vessel degrees of freedom Vessels motions are explained by its six degrees of freedom which is indicated in figure 3.01. Sway, surge and heave are linear degrees of freedom and roll, pitch and yaw are rotational degrees of freedom. In this report, the result of heave and roll motions is highlighted and considered.

Figure 3.01. Degrees of freedom with coordinate system Source: Mitra et al. (2012) It is important to be consequent when using different coefficient for the different motions. The coefficient for the different motions will be used according to figure 3.01. 12

3.2 Uncoupling of heave and roll Vessels motions in sea water are divided into six degrees of freedom. The six degrees of freedom are sway, surge, heave, roll, pitch and yaw respectively as shown in figure (3.01). A vessel will behave differently regarding each degree of freedom where the response in each direction will affect each other through coupling. The resulting forces on a vessel due to heave and roll can be calculated by assuming that roll and heave is uncoupled. This means that heave and roll is calculated by obtaining the resulting motions when roll and heave is considered as two single degree of freedom systems. If one where to consider heave and pitch, the motions will be strongly coupled. Both forces are working in the longitudinal direction of the ship. By considering either the front or the back of the ship, we see that the vertical motions is the sum of the vertical displacement in heave and pitch and the motions will be highly influenced by each other.

3.2.1 Linear theory Linear theory is a necessary assumption for general finite element calculations which is normally used when calculating vessel motions. To use linear theory, the non-linear contributions in any motion must be assumed small and neglected. Linear theory let us calculate resultant forces by adding all contributions to the considered force together. This concept is important when calculating a vessels response using the equation of motions which is discussed later in section 4.3.1. All contribution related to mass, added mass, damping, stiffness and exciting forces is obtained by linear theory where all contributions related to the considered motion is added together.

3.2.2 Coupled motions At sea there is a dynamic pressure field affecting a general vessel from all sides. When a ship is oscillating in either direction, the vessel will influence and change this pressure field. This means that the dynamic pressure field is changed when the vessel is moving in one of the six degrees of freedom so that the other degrees of freedom will be affected due to that movement. The coupled motions in either direction can be found by assuming linear theory and harmonic motions by summarizing properties for mass, added mass, damping, stiffness and exciting forces in matrixes shown in eq. (3.01) where all terms consist of six times six matrixes. In comparison the uncoupled single degree of freedom equation of motions is shown in eq. (3.02).

M 6

k 1

jk

k B jk k C jk k Fj , j 1...6 A jk

j B j j C j j Fj ( M j A j )

(3.01)

(3.02)

Where Mjk are mass components, Ajk and Bjk are added mass and damping coefficients, Cjk are hydrostatic restoring coefficients, and Fj are the exciting force and moment components. 13

(Salvesen et al. (1970)) The numbers from one to six represents respectively surge, sway, heave, pitch, sway, roll and yaw.

3.2.3 Symmetric and anti- symmetric motions All degrees of freedom will affect each other to some extent, but some of the degrees of freedom will be less coupled. When a ship has a lateral symmetry, the degrees of freedom can be divided into symmetric and anti- symmetric motions. Surge, heave and pitch are the symmetric motions, since the motion response to either of these motions will be the same on either side of the ship. (Sway, roll and yaw is then anti- symmetric motions) The symmetric and anti- symmetric motions are unaffected by each other when assuming linear theory. (Journée (2001) p.26) By assuming linear theory and harmonic motions in case of long slender ships; roll and heave will be uncoupled since they are symmetric and anti-symmetric motions. The motion on a particular point on a vessel can then be found by adding the contributions due to heave and roll.

3.3 Equation of motion The equation of motion is a product of Newton’s second law of motion which states that the product of an objects mass and acceleration equals the sum of forces acting on the object. In case of a vessel, the applied forces are the wave forces, radiation forces and hydrostatic forces as shown in eq. (3.03). The general equation of motion for heave is given in eq. (3.04) and represents the formulation of all six degrees of freedom.

F Fw FR FS

(3.03)

The force F is the mass times acceleration, the wave-induced forces (Fw FA FD ) is the approaching wave forces plus the diffraction forces due to a vessels disturbance of the pressure field. The radiation forces due to harmonic motion ( FR Az Bz ) are the hydrodynamic forces related to added mass and damping which points in the opposite direction of the external forces. The hydrostatic restoring forces (FS Cz ) are also called the stiffness which is the buoyancy force that restores the vessel to its initial position. (Faltinsen (1990) p.41) The forces affecting a vessel can be written as the general equation of motion as eq. (3.04).

( M a z ) z bz z c z z Fz (t )

(3.04)

The form of the equation of motion will be the same for the motions in all six degrees of freedom for a vessel moving in water. The only difference will be that the mass terms are changed with mass moment of inertia, and the damping and stiffness term will relate to rotation and the induced wave force must be converted into momentum force.

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3.3.1 Solutions of equation of motion The equation of motion is applied for all six degrees of freedom and is treated the same for all cases. The solution for uncoupled heave is shown, but it the equation is treated the same in case of uncoupled roll motion. When solving a differential equation, two solution must be considered; the homogeneous solution zh(t), and the particular solution zp(t). The total solution is z(t)=zh(t)+zp(t). (Rao (2011) p.261) The particular solution zp(t) and homogeneous solution zh(t) is the solution of respectively eq. (3.05) and eq. (3.06).

( M a z ) z bz z c z z Fz (t )

(3.05)

( M a z ) z bz z c z z 0

(3.06)

4.3.2 Particular solution Let us consider eq. (3.05) as a general differential equation, where the equation equals a harmonic force with a frequency e so that Fz ( t ) F0 sin(e t ) . The systems mass are

(M a z ) M az , the damping is bz, and the stiffness is cz. The equation can be solved as response of a damped system under harmonic force. The following calculations are also illustrated in Rao. (RAO (2011) p.271)

(M az )

d 2zp dt

2

bz

dz p dt

c z z F0 sin(e t )

(3.07)

Since the force is harmonic, the solution of eq. (3.05), zp(t) is also assumed harmonic with a phase difference between the exciting force and the motion which is z .

z p Z sin(e t z ),

dz p dt

Ze cos(e t z ),

d2zp dt

2

Ze2 sin(e t z )

(3.08)

By substituting eq. (3.08) into eq. (3.05), eq. (3.09) is obtained:

Z c z M az e2 sin(e t z ) b z e cos(e t z ) F0 sin(e t )

(3.09)

By using the trigonometric relations:

sin(t ) sin t cos cos t sin cos(t ) cos t cos sin t sin Equation (3.10) and (3.11) is obtained by equating coefficients of sinuses and cosines respectively.

Z c z M az e2 cos( z ) b z e sin( z ) sin( e t ) F0 sin( e t ) 15

(3.10)

Z c z M az e2 sin( z ) b z e cos( z ) cos(e t ) 0

(3.11)

The amplitude Z is obtained from eq. (3.10) and the mass terms are inserted. ( (M a z ) M az ) The phase angle, z is obtained from eq. (3.11) and is given in the following equations.

Z

c

F0 z

( M a z )e2

(3.12)

b 2

2

z

b z e z tan 1 2 M a z e c z

e

(3.13)

3.3.3 Dynamic amplification factor To understand which frequencies that give the largest response of the vessel, eq. (3.12) can be showed in a different form giving a relation between the relative response and frequency rate. Let us divide the numerator and denominator in eq. (3.12) with the stiffness coefficient, cz.

F0 cz

Z

2

c z (M a z ) 2 b z e e cz cz cz

2

(3.14)

The following relations for natural frequency, nz (the coefficient number “z” is used to distinguish the natural frequency of heave and roll) damping ratio, and frequency ratio, r are given in eq. (3.15) and substituted into eq. (3.14) to obtain eq. (3.16).

nz

cz bz , , r e M a z nz 2(M a z ) nz

(3.15)

DAF

Zc z F0

(3.16)

1

1 r 2r 2 2

2

This relation will be the same for all single degree of freedom systems which is oscillating in harmonic motions. The dynamic amplification factor in eq. (3.16) is plotted for different damping frequencies in figure 3.02

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Figure 3.02. Dynamic amplification factor Figure 3.02 shows how much a general force will influence the response motion of a system related to the frequency rate. When the force frequency rate equals zero, the response is the same as the constant force divided by stiffness. And in case of relative low force frequencies, the motion is highly dependent on the restoring force cz. A case where force frequencies are close to the natural frequency is in general undesirable. This is where resonance occurs. This is called the damping controlled region since the value of damping is controlling how high the possible amplitude motion can be. In case of a damping frequency of 0.1, the response is five times larger than the external force divided by the stiffness of the system. Cases of large force frequencies related to natural frequency, is called mass dominated motions. This is because the response in general will decrease depending on the value of the mass system. How the dynamic amplification factor relates to ship motions will also be discussed in section 3.4.1.

3.3.4 Homogeneous solution The homogeneous solution is also called the transient motion. This motion is the result of an initial force where the system is left alone to oscillate. This motion will in all realistic cases die out do to damping since the solution is the result of a motion with external force equals zero. When the system is undergoing a change of motion, like in lifting operations or sudden wind increase, there will be a change in the system that must be accounted for. The motion will die out after a while, but if the vessel is not designed for this initial response, the ship or cargo will be damaged. The homogeneous solution, zh(t) is obtained by solving eq. (3.06) with given initial conditions for motion and velocity at time zero. 17

(M az )z b z z c z z 0

(3.17)

In most realistic cases of motion, the damping ratio, would be between zero and one which is called an underdamped case. The solution of the homogeneous solution would be in the form shown in eq. (3.18). (Rao (2011) p.160, p.270)

z h Z 0 e nz t sin(d t z 0 )

(3.18)

dz 1 2 nz

(3.19)

Where the amplitude, Z0 is multiplied by an exponential term which goes to zero in a slope that depends on the damping ratio, . The damping frequency dz is the reduced natural frequency which also depends on the damping ratio, shown in eq. (3.19). The amplitude and phase angle of the homogeneous solution is given in respectively eq. (3.20) and eq. (3.21). 1

2 1 2 2 Z 0 z 0 Z sin z 2 nz z 0 z 0 nz Z sin z e Z cos z dz

z0

z z 0 nz Z sin z e Z cos z tan 1 nz 0 d (z 0 Z sin z ) 2

(3.20)

(3.21)

Where Z is the amplitude of the particular solution, z is the phase angle of the particular solutions, z 0 and z 0 is the initial movement and speed when the time is zero.

3.3.5 Phase angle When considering the particular solution of equation of motion, the phase angle is defined as the phase difference between the external force and the motion of a single degree of freedom system. In case of ship motions, the exciting force will have a frequency equal the encounter frequency e . In case of a vessel, each of the six single degree of freedom systems will have its own phase angle, telling when each motion is occurring with respect to the force. How the phase angle relates to damping ratio, and frequency ratio, r is derived by dividing the nominator and denominator by the stiffness in eq. (3.13)

2 r z tan 1 2 1 r

(3.22)

The phase angle is dependent on the damping where a small damping results a smaller phase angle between the motion and the force frequency.

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3.4 Harmonic response of a vessel As shown in section 3.3, the equation of motion can describe the behavior of vessels. The accuracy of the solution depends on how the terms for radiation, restoring, mass and external forces are obtained. In this section the relation between these terms and a vessels behavior is discussed. In section 3.5 and section 3.6 different ways to calculate those terms in case of heave and roll is considered.

3.4.1 Dynamic amplification of a vessel The dynamic amplification factor for the equation of motions is calculated and described in section 3.3.3. Figure 3.03 shows which terms in the equation of motions that have most effect on the response of the system related to different frequencies. Where r= e / nz .

Figure 3.03. Dynamic amplification factor Figure 3.03 describes three different behaviors of vessel motions. (Journée and Massie (2001) p. 6-24) 1. In case of low relative force frequencies, e n .The motion of a vessel is dominated by the restoring forces of the system, the hydrostatic term. When this occurs, the vessel motions will follow the wave in a very stiff motion. In case of small vessels this occurs since the vessel length is relatively small compared to the wave length. The vessel will then follow the waves and feel all waves as “stiff movements”. See figure 3.04. 2. In case of force frequencies close to the eigenfrequency, e n , resonance occur. When there is little damping in the system, the motions in these force frequencies is critical. The damping dominated area should in cases of vessel motion be narrowed as much as necessary to be avoided. 19

3. In case of high relative force frequencies, e n .The motion of a vessel is dominated by the inertia or mass terms. When it comes to calculations, it is very important to understand and compute the hydrodynamic terms correctly since these terms control the motion of the vessel, and not the stiffness. (Added mass, damping and dynamic pressure are all hydrodynamic terms) In this case the movement of a vessel will behave more comfortably than in case of hydrostatic dominated movement. See figure 3.04.

Figure 3.04. Vessels with different lengths in waves When it comes to comfort and how people or cargo is affected, it is obvious that the stiff motions in case of e n are most uncomfortable. If this happens in case of small natural frequencies which are the case for small vessels, the cargo can be damaged due to fatigue and large accelerations.

3.4.2 Added mass force When a vessel is moving in water, it will interact and displace the fluid close to the vessel. When the vessel is moving with a velocity, the approaching fluid will be accelerated by the moving body. The affected water particles will affect other water particles creating a field of fluid moving along with the vessel. If the fluid has enough space to be accelerated freely, (in case of deep water) the resultant force created from the water particles will be in phase with the acceleration of the vessel. (Cormick et al. (2002) p. 13) The added mass in the equation of motion is actually a hydrodynamic force. Since the added mass force is in phase with the acceleration of the vessel, the force is taken into account by finding an equivalent mass to the added mass force. The added mass can then be considered as some additional fluid with a given mass moving along with the vessel. The added mass depends on a vessels ability to move additional fluid. The displacement of fluid is mostly because of the large body of the vessel, but mark that the viscosity plays a part in how much fluid that “sticks” to the vessel. Unlike the mass of the system, the added mass will change when the force frequency of the system is changing. In case of zero motion of a body, the additional added mass will be zero, proving that the added mass is dependent on frequency.

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3.4.3 Damping force The damping in an oscillating or vibrating system tells us how much energy that is dissipating while the system moves. By looking at a system induced with an initial force where no other external forces is working on the system, the response and displacement of the system will gradually decrease due to the reduction in energy. The damping term is telling us how much energy that is dissipated in the process. (Rao (2011) p.160) In case of a vessel moving in fluid, the damping of the system is the total of all contributions reducing the energy in the system. The real damping of a system is affected by viscosity, eddy currents and the energy in the waves that the vessel is able to create when moving. The contributions in roll damping are summarized to get the equivalent damping and are shown in eq. (3.23).

B eq B f B e B w B L B BK

(3.23)

Where Beq is the equivalent damping and is discussed for roll in section 3.6.3. The terms in the equation must be defined by semi empirical formulas. Bf is damping due to skin friction which is dependent on viscosity, Be is damping due to eddy making and vortexes created in the movement, BW is the radiation damping, BL is damping due to lift forces and BBK is the damping due to a bilge keel which has a significant effect on a systems damping. This damping is not in phase with the velocity of the system and can be multiplied by the amplitude to make it in phase with the velocity. The damping in case of roll is therefore given as in eq. (3.24). (Chakrabarti (2000))

B (Beq )

(3.24)

In case of a vessel, the ship will create waves and thus dissipate energy in each motion. When considering a linear system like surge, sway and heave, the damping will be proportional to the velocity. This is because the total damping is approximately equal to the dissipated energy in the radiation waves which is in phase with the speed of the system. (Journée and Massie (2001) p. 6-11) The concept of radiation damping is illustrated in figure 3.05.

Figure 3.05. Radiating waves Source: Journée and Massie (2001) p. 6-14

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Figure 3.05 illustrates the concept of radiation waves in case of a vessel moving up and down in heave. When the vessel is moving it will create additional waves with amplitude a . The energy dissipated in these waves is equal to the damping of the system. (Faltinsen (1990) p.46) The damping term and added mass terms are called radiation forces affected by a change in frequency of the system. They are also affected by viscosity which makes them impossible to calculate by purely analytical formulas (without more than few approximations). In most cases these terms are calculated by strip theory and a numerical approach described in section 3.4.6.

3.4.4 Restoring force The restoring force of a oscillating is defined as the system’s ability to restore itself to its original position. In case of a vessel it is dependent on hydrostatics and stability properties of the vessel. The relation for a spring in general is that an applied force equals the stiffness of the spring multiplied by the displacement of the spring as shown in eq. (3.25). The same relation follows here, and the stiffness relates to the restoring forces given from geometrical properties and buoyancy.

Fj C jk k

(3.25)

Where Fj is applied force, k is the related displacement and Cjk is the equivalent stiffness of the system. The stiffness is found by considering a static system affected by a given force which induces a displacement. The term is considered negative with respect to the force since the restoring force is countering the force. Every term in the equation is known except the stiffness.

3.4.5 Exciting force To understand how vessels behave at sea it is important to understand the exciting forces applied on a certain vessel. At sea there are three kind of environmental loads present at all times. Those are wave, current and wind forces. Forces from currant would be constant and must be taken into consideration when discussing forces on a drifting structure, but is in general neglected when considering moving vessels. Wind forces are also important in case of drifting vessels, and should be taken into consideration in case of roll motions of vessels with significant height. (Journée and Massie (2001)) The wave exciting forces are divided into two kinds of forces, diffraction forces and approaching wave forces. It is also important to understand the meaning of the encounter frequency, e which is discussed in this section. Approaching wave forces Approaching wave forces are found by considering wave forces on a rigid body at sea, restrained from oscillation. These forces can be found by using linear wave theory where the dynamic wave pressure is given as p D in eq. (3.26) (Faltinsen (1990) table 2.1) The total wave 22

force is found by integrating this pressure over the area it is working on, and this force is called the Froude-Kriloff force. (Faltinsen (1990) p.59)

p D g 0 e kz sin(e t kx )

(3.26)

Where and g is the density of seawater and the gravity constant, 0 is the wave amplitude,

e is the encounter frequency, t is the time, k the wave number, and x is the position relative to the center of gravity of the vessel. (Positive direction against the front of the ship) Diffraction forces The considered approaching wave forces form an undisturbed pressure field around the vessel, but since the vessel in truth is moving, the vessel will disturb this pressure field and this disturbance must be taken into account. In case of bodies with small breadth (diameter in case of a cylinder) compared to the wave length, the Morrison equation is applied. The force will then be related to the added mass and the acceleration of the vessel. (Faltinsen (1990) p.61) Encounter frequency When calculating forces on a vessel there will be two relevant wave frequencies and it is important not to mix them. Those frequencies are the frequency of the waves and the encounter frequency which tells us the frequency of the encountered waves depending on the speed and direction of the vessel. Let’s consider a point on a vessel with a constant forward velocity U [m/s] with an angle . The point will encounter two waves in a time Te. The phase velocity of the waves is c=g/ and the wave length is Lw as shown in figure 3.06.

Figure 3.06. Encountering periods, Te Source: Faltinsen (1990) figure 3.12

U cos()Te cTe L w Te

(3.27)

2 2g and L w 2 by assuming deepwater. When substituted into eq. (3.27), eq. e

(3.28) is obtained.

23

g 2g 2 U cos() 2 e

(3.28)

By solving eq. (3.28) with respect to e , eq. (3.29) is obtained.

e

2 U cos g

(3.29)

Eq. 3.29 is also given by DNV (DNV-RP-C205 (2010)).

3.4.6 Strip theory To use the equation of motion and calculate the response of a vessel, a numerical approach is used for calculating the forces on a vessel. The strip theory is a numerical theory where the cross section of the ship at a distance x from the center of gravity, is considered as one strip with infinite length. This means that each strip is unaffected by other strips. A ship is divided into as many strips as necessary where the hydrodynamic properties are added together by integrating the two-dimensional strips over the length. (Journée (2001) p.5) See figure 3.07.

Figure 3.07. Representation of strip theory Source: Journée (2001) figure 2.7 A few assumptions is necessary for using the strip theory on vessels in the computer code for “Seaway” which is used to calculate wave induced loads and motions on ships. (Journée (2001) p.5) -

The vessel is a rigid body floating in an ideal fluid The motions of the vessel is linear or can be linearized

That a vessel is floating in ideal fluid, means that all assumptions for linear wave theory is applicable so that the sea water is assumed homogeneous, incompressible, free of surface tension, irrotational and without viscosity.

24

To assume that the vessel motion is linear, one must neglect the effect of what is happening above water. Waves on deck is then neglected. Other than that, the strip theory has been proved effective for predicting ship motions with length divided by breadth larger than three.

3.5 Heave motion The heave motion of a vessel is the vertical response of the vessel. When assuming uncoupled motions which are described in section 3.2, the heave motion of the ship is considered as a single degree of freedom where all terms related to the equation of motion must be calculated for this specific motion. When finding the relevant equation of motion in heave, all necessary assumptions will be given and described in its respective section.

3.5.1 Equation of motion The equation of motion is defined in section 3.3 and eq. (3.04) is improved by considering a sinusoidal force with amplitude F0 moving in phase with the encounter frequency defined in section 3.4.5.

(M a z )z b z z c z z F0 sin( e t )

(3.30)

3.5.2 Added mass in heave When a vessel is moving in a fluid, additional water will move with the vessel. As described earlier the fluid will oscillate with the moving body and the added mass is in reality a mass equivalent to the hydrodynamic force related to the movement of the fluid. (Faltinsen (1990) p.42) The added mass is normally calculated by using strip theory. Where the vessel is divided into many elements, (depending on the shape of the vessel) the hydrodynamic properties for each strip are considered and integrated over the length (x), as shown in eq. (3.31). In section 3.2.3 the concept of added mass was discussed and concluded that the added mass goes to zero when the frequency goes to zero.

a z a 2z D ( x )dx

(3.31)

L

Added mass equals half a circle The added mass of each strip can be considered as half a circle with diameter equal to the breadth of the vessel as shown in figure 3.08 thus making the added mass independent of the wave frequency. By using this approximation one assume that the added mass is independent of the waves. (Sarkar and Gudmestad (2010))

25

Figure 3.08, Added mass equals half a circle The formula for area of half a circle is A

d 2 8

, where d is the diameter of the circle. By

using the strip theory, we find that the added mass is the sum off all strips that will have the final form of half a cone in case of a triangular shape or half a cylinder in case of a rectangular shape. We find the added mass by integrating the 2D sections over the length.

a 2z D

B 2 8

a z a z2 D ( x)dx L

a z , b arg e

(3.32)

8

L 2

B( x) dx 2

(3.33)

L 2

B 2 L 8

(3.34)

In case of a rectangular barge the added mass is the same as half a cylinder with diameter, B and length, L as seen in eq. (3.34). Added mass, DNV DNV uses the same equation as obtained in eq. (3.34) but advice that the added mass is multiplied by a factor, CA. (DNV-RP-H103 (2011) Table A-1) Figure 3.06 is the relevant information extracted from the relevant table where the added mass per unit length is given.

26

Table 3.01. Added mass coefficients from DNV Source: DNV-RP-H103 (2011) Table A-1

The total added mass of a rectangular shape oscillating under water is considered in table 3.01. In this case, the only interest is the equivalent added mass under the vessel, so the added mass is half the value given in the table. The added mass is then given as in eq. (3.35) and eq. (3.36). The values of 2a and 2b from the table 3.01 is respectively the breadth and draft of our vessel

a 2z D C A

a z ,b arg e

AR 2

(3.35)

AR B 2 L L C A C A 2 8

(3.36)

Where CA is the added mass coefficient. The added mass coefficient is found by interpolating the relation between breadth and draft with respect to values in table 3.01.

3.5.3 Damping in heave When the frequency of external forces working on the ship is the same as the natural frequency, the motion amplitude will go to infinity in case of no damping. As discussed in section 3.4.1, these motions are called damping controlled motions. The contributions of damping are shown in eq. (3.23). All contributions will affect the total damping of the system, but the radiation damping will be very large compared to the other contribution in case of heave. The radiation damping can be calculated without empirical formulas and is the only damping that gives satisfactory results for the single degree of freedom systems except roll. (Chakrabarti (2000)) When a vessel is moving vertically in water it will create its own waves. This process is called radiation, and vessels (especially barges) moving vertically in heave motion will displace a lot of water thus losing a lot of energy, making radiation damping the largest contribution to damping in heave as shown in figure 3.07. The total damping in heave is therefore assumed equal to the radiation damping.

bz BW

(3.37)

27

There will be other contributions such as damping due to eddy vortexes and viscosity but they will be very small compared to the radiation damping. So saying that radiation damping is the only damping term present in case of heave is a very good approximation for the motion. The radiation damping can be found using an energy relation where the total energy, E in a fluid volume, is the sum of kinetic and potential energy where is the displaced volume of the system as shown in eq. (3.38). (Newman (1977) p.266)

E ( t ) 0,5 2 gz d

(3.38)

Faltinsen has made a formula for heave damping using Newmans energy relation based on how much water a vessel will displace. (Faltinsen (1990) p.47) The formula is related to radiation wave amplitude a , vessel movement z, and frequency , of the waves as shown in eq. (3.39).

b

2D z

a z

2

g2 3

(3.39)

The radiation wave amplitude a , illustrated in figure 3.07, must be found by a forced motion test related to each force frequency. Eq. 3.39 takes the energy of each created wave into consideration making this a close to truth relation for calculating the damping of the system. The total damping is found by integrating over the length.

3.5.4 Restoring force in heave The restoring force of a body oscillating vertically in water is related to a vessels ability to restore itself to its original position. The original position is at z=0 at the waterline area in stillwater. The restoring force is based on the geometry of the vessel which defines a vessels restoring force. To find the restoring force we consider a static system with a static force, based on buoyancy and the additional displacement.

F czz

(3.40)

The force is equivalent to the mass of displaced water multiplied by the gravity constant, which equals the stiffness, c z times the depth of the displaced water, Ddisplace.

F displaced g Aw Ddisplaced g c z Ddisplaced

(3.41)

This gives that the stiffness, cz is proportional to the waterline area in heave motion of any vessel if the waterline area is constant during the vertical motion.

c z Aw g BLg

(3.42)

28

3.5.5 Exciting force in heave When considering the uncoupled heave of a vessel the vertical motions is considered alone and the only relevant external forces are the vertical wave forces. The external heave force, Fz(t) can be divided into approaching wave forces and diffraction forces as discussed in Section 3.2.6.

Fz (t ) FA (t ) FD (t )

(3.43)

These forces are calculated differently depending on the size of the structure relevant to the wave length. In case of large structures both diffraction and inertia forces must be accounted for. The relevant force regimes are presented in figure 3.09 where, 0 is the wave amplitude, DC is the characteristic dimension and LW is the wave length found by the deepwater relation presented in section 3.2.5.

Figure 3.09. Different wave force regimes Source: DNV-RP-H103 (2011) figure 2-3

In case of heave, the characteristic dimension will be either the length or the breadth of the barge. These dimensions are large compared to the wave length making it necessary to take diffraction forces into account. Two general assumptions is made for the calculations: -

Linear wave theory is applicable Deepwater relations is applicable

Formulas used in linear wave theory are found in table 3.02.

29

Table 3.02.Llinear wave theory Source: Faltinsen (1990) table 2.1

Approaching wave forces The approaching wave force corresponds to the undisturbed pressure field and is called the Froude-Kriloff force. This force is applicable for both large and small structures and is calculated by integrating the pressure over the surface of the vessel. The approaching wave force can be calculated as in eq. (3.44) where pD is the dynamic pressure found from table 3.02. (Faltinsen (1990) p.61)

FA ,z ( t ) p D ds

(3.44)

p D g 0 e kz sin(t kx)

(3.45)

S

By substituting eq. (3.45) into (3.44) a general formula for the Froude-Kriloff force is obtained.

FA ,z

g e 0

kz

sin(t kx )ds

(3.46)

S

30

To obtain the resultant force in heave, eq. (3.46) must be integrated over its bottom surface and the same assumptions as for strip theory is applied. In case of a barge, the resulting force is calculated by eq. (3.47), where z is the draft of the barge. B/2 L/ 2

FAz,b arg e

g e

kD

0

sin(t kx )dxdy

(3.47)

B / 2 L / 2

Diffraction forces The diffraction forces can be calculated using the Morison equation. This formula can only be applied for waves with wave lengths, LW>5DC where DC is the characteristic dimension as shown in figure 3.09.The Morison equation can be modified in terms of added and the z found from table 3.02 as “a3” in the table . (Faltinsen (1990) relevant wave acceleration, w formula 3.36) The usability of the formula is confirmed by DNV with the following statement: “In case inertia dominated, large volume structures where the diffraction loads are much larger than drag induced global loads, a Morison load model with predefined added mass coefficients can be added to the radiation/diffraction model.” (DNV-RP-H103 (2011) 2.3.4.5) (The radiation model refers to added mass and damping forces)

z FD,z a z w

(3.48)

w z kg 0 e kz sin(t kx)

(3.49)

To obtain the resultant force in heave, eq. (3.48) must be integrated over its length the same assumptions as for strip theory is applied by taking possible change in the added mass into consideration and integrate the 2D- added mass over the length.

FD,z a 2z D kg 0 e kz sin(t kx )dx

(3.50)

L

The resulting diffraction wave force is calculated by eq. (3.51), where z is half the draft of the barge. (Since the resultant force would be applied at a depth of the center of buoyancy where z=D/2) L/2

FD,z

a

2D z

kg 0 e

kD 2

sin(t kx )dx

(3.51)

L / 2

Minimum characteristic dimension, DC As mentioned earlier, the formula for the diffraction forces gives reliable answers in cases where the wave length is five times larger than the characteristic dimension. In case of a barge where the characteristic dimension of heave is the breadth, the formula is used for large wave periods for best results. Since wave length is related to wave period in deepwater as in eq. (3.52).

31

LW

g 2 T 2

(3.52)

Requirement for use of Morison equation:

DC

LW 5

(3.53)

Figure 3.10 shows which wave periods the formula for diffraction forces give best results relating to the characteristic dimension, DC.

Figure 3.10. Minimum characteristic dimension, where DC,min=LW/5 Total wave forces By substituting eq. (3.51) and (3.47) into eq. (3.43). The total force for a barge can be calculated in eq. (3.54).

Fz ( t )

B/2

L/2

B / 2 L / 2

g 0 e

kD

L/2

sin(t kx )dxdy

a

2D z

kg0e

kD 2

sin(t kx )dx

(3.54)

L / 2

The following trigonometric relation is used.

L/2

L / 2

sin(t kx )dx

2 kL sin sin(t ) k 2

(3.55)

By deriving the integral in eq. (3.54) and using the trigonometric relation in eq. (3.55) the harmonic wave force of a barge can be calculated by eq. (3.56) (Faltinsen (1990) p.89) Wave frequency is exchanged with encounter wave frequency, since we are interested in the motion response on the barge.

32

kD 2 kL Fz ( t ) Bg 0 e kD a 2z D kg 0 e 2 sin sin( e t ) F0 * sin( e t ) k 2

(3.56)

In case of a vessel moving in water the frequency of the waves, will be equal the effective frequency e which is derived in section 3.4.5. In eq. (4.54) B is the breadth of the barge, L is the length of the barge, is the density of sea water, g is the gravity constant o is the wave amplitude, k is the wave number, a z2 D is the added mass of a strip (half circle with breadth B) and D is the draft. Wave number The wave number and frequency are related through the dispersion relations shown in table 3.02 where k= 2/g in case of deepwater. The wave frequency must not be mixed with the effective frequency e shown in eq. (3.29) which tells us how fast the vessel is encountering the waves. The encounter frequency is already a function of wave number k, where k describes the ratio of waves per meter in the equivalent wave direction. In other words, the wave frequency is used to calculate the wave number when considering the force on the barge. Wave force when wave frequency goes to zero Since the resulting wave forces in heave is not applicable for all wave periods it interesting to look at the worst case values of the function. The largest value of force amplitude, F0 in eq. (3.56) is found when the wave frequency goes to zero. The dispersion relation is shown in eq. (3.57).

k

2 g

(3.57)

Remember that sin() in case of small angles. When eq. (3.57) is substituted into eq. (3.56), eq. (3.58) is obtained. D D 2D 2 g a z D 0 e 2g L Fo Bg 0 e 2

2

(3.58)

If the frequency is close to zero, the diffraction term in eq. (3.58) will be much smaller than the approaching wave, and the exponential term will be close to one so that the equation is simplified to a relation between the stiffness of the system, c z and wave amplitude, 0 .

Fo BLg0 c z 0

(3.59)

By calculating F0 by eq. (3.59) the largest force for the relevant wave height is obtained. This is also the basic relation for a spring in a static system where the force equals the stiffness multiplied by the displacement. (P=kx) 33

3.5.6 Natural period in heave The natural period is considered one of the most important parameters when analyzing motions of a vessel. The natural period is also called the resonance period, since induced vibrations with period equal to the natural period’s results in resonance. If a system, after an initial disturbance is left to vibrate on its own, the period of the oscillations without any external forces is known as the natural period. A vibratory system having a number of degrees of freedom will have as many distinct natural periods of vibration. (Rao (2011) p.62) To calculate the natural period the motion is assumed uncoupled from the other degrees of freedom. General formula of uncoupled natural period of heave, Tz is given provided by DNV, where the mass equals the added mass plus the mass (DNV-RP-H103 (2011) 2.3.3.4).

Tnz

M az 2 2 nz cz

(3.60)

By substituting values for the mass (eq. (3.61)) and the stiffness (eq. (3.42)) for a rectangular barge into eq. (3.60) the relation for natural beriod is given in eq. (3.62).

M BLD

Tnz,b arg e

(3.61)

a BLD1 z M 2 D 1 a z 2 gBL g M

(3.62)

Where D is the static draft of the barge, g is gravity acceleration constant, az is added mass given in section 3.3.3 and M is the total mass of the barge.

3.5.7 Solution The equation of motion for heave motion is given in eq. (3.63):

(M a z )z b z z c z z Fo sin( e t )

(3.63)

Where M is the total mass of the barge, az is the added mass, bz is the damping, cz the stiffness, F0 the force amplitude, e the equivalent wave frequency, t is time and z is the vertical motion of the barge. Table 3.03 shows examples of which formulas that can be used in the equation of motion and which assumptions that is needed. When for example strip theory is assumed, all relevant assumptions mentioned in section 3.4.6 is used.

34

Table 3.03. Terms in equation of motion

Term

az

Assumptions used for general formulas

General formulas in heave for a vessel

Additional assumptions for barge

2D az ( x)dx

Added mass is unaffected by wave frequency No viscosity Deepwater

Strip theory

L

Only damping because of radiation Energy relation in fluid from Newman Linear wave theory

a g2 L z 3 dx

cz

Constant waterline in range of motion

Aw g

Fz (t )

Strip theory Morison equation Linear wave theory Long relative wave length No other force contributions

bz

Barge formulas

CA

B2 L 8

2

2

g2 a 3 L z Rectangular waterline in range of motion

p D n z ds a z w z S

Sinusoidal waves B

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