Faculty of Mechanical Engineering and Marine Technology Chair of Modelling and Simulation
Ship dynamics in waves (Ship Theory II)
Prof. Dr.-Ing. habil. Nikolai Kornev
Rostock 2012
2
Contents 1 Ship motion in regular sea waves 1.1 1.1 Co Coup upli ling ng of diff differen erentt ship ship osci oscilllati lation onss 1.2 Classification of forces . . . . . . . . 1.3 Radiation force compo pon nents . . . . . 1.3.1 Hydrod ody ynamic damping . . . 1.3.2 Added mass compo pon nent . . . 1.4 Hydrostatic compo pon nent . . . . . . . . 1.5 Wave exciting force . . . . . . . . . . 1.6 Motion equations . . . . . . . . . . . 1.7 Haskind’s relation . . . . . . . . . . . 1.8 Exercises . . . . . . . . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
11 11 13 17 18 20 24 24 25 25 28
2 Free oscillations with small amplitudes 31 2.1 Intro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3 Ship Ship osci oscill llat atio ions ns in smal smalll tran transv sver erse se waves (bea (beam m see) see) 3.1 Hydros rostati atic forces rces and moments . . . . . . . . . . . . . . . . . 3.2 Hydrody rodyna nam mic Kryl rylov - Frou roude force rce . . . . . . . . . . . . . . 3.3 3.3 Full ull Kry Krylov lov - Froud roudee forc forcee an and d mome momen nt . . . . . . . . . . . . . 3.4 Force and mome moment nt actin actingg on the the ship ship frame frame in in acceler accelerated ated flow flow 3.5 Full wave ind ndu uced ced force rce an and d moment . . . . . . . . . . . . . . 3.6 3.6 Equa Equati tion onss of ship hip hea heave an and d roll roll osci oscilllati lation onss . . . . . . . . . . 3.7 Analysis of the formula (3.27) . . . . . . . . . . . . . . . . . . 3.8 Sway ship osc oscillation tionss in beam beam sea . . . . . . . . . . . . . . . . 3.9 Ship oscill oscillations ations at finite finite beam to wa wave ve length ratio and and draught draught to length ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Effect of ship spee peed on rolling . . . . . . . . . . . . . . . . . . 3
37 38 40 42 42 43 43 45 48 49 52
4 Ship oscillations in small head waves 55 4.1 Exciting forces and ship oscillations . . . . . . . . . . . . . . . 55 4.2 Estimations of slamming and deck flooding . . . . . . . . . . . 58 5 Seasickness caused by ship oscillations
61
6 Ship oscillations in irregular waves 6.1 Representation of irregular waves . . . . . . . . . 6.1.1 Wave ordinates as stochastic quantities . . 6.1.2 Wave spectra . . . . . . . . . . . . . . . . 6.2 Calculation of ship oscillations in irregular waves
65 65 66 68 72
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
7 Experimental methods in ship seakeeping 75 7.1 Experiments with models . . . . . . . . . . . . . . . . . . . . . 75 7.2 Seakeeping tests with large scale ships . . . . . . . . . . . . . 82 8 Ship oscillation damping (stabilisation) 8.1 Damping of roll oscillations . . . . . . 8.1.1 Passive means . . . . . . . . . . 8.1.2 Active stabilizer . . . . . . . . . 8.1.3 Passive Schlingerkiel . . . . . . 8.1.4 Active rudders . . . . . . . . . 8.1.5 Damping of pitch oscillations . 9 Parametric oscillations
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
85 85 85 88 88 89 89 95
10 Principles of Rankine source method for calculation of seakeeping 101 10.1 Frequency domain simulations . . . . . . . . . . . . . . . . . . 101 10.2 Time domain simulation . . . . . . . . . . . . . . . . . . . . . 106
4
List of Tables 2.1 2.2
Frequencies and periods of different oscillation types . . . . . . 33 Referred damping factors for different oscillation types . . . . 34
5
6
List of Figures 1.1 Ship motion with 6 degree of freedom (from [1]) . . . . . . . . 1.2 Displacement of the center of effort due to change of the ship draught (from [2]) . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Illustration to derivation of damping coefficient . . . . . . . . 1.4 Added mass and damping coefficient of the semi circle frame at heave oscillations. Here A is the frame area. . . . . . . . . . 1.5 Added mass and damping coefficient of the box frame at heave oscillations. Here A is the frame area. . . . . . . . . . . . . . . 1.6 Added mass and damping coefficient of the semi circle frame at sway oscillations. Here A is the frame area. . . . . . . . . . 1.7 Added mass and damping coefficient of the box frame at sway oscillations. Here A is the frame area. . . . . . . . . . . . . . . 1.8 Added mass and damping coefficient of the box frame at roll (heel) oscillations. Here A is the frame area. . . . . . . . . . . 1.9 Mirroring for the case ω 0. . . . . . . . . . . . . . . . . . . 1.10 Mirroring for the case ω . . . . . . . . . . . . . . . . . .
22 23 23
3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12
. . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of hydrostatic force . . . . . . . . . . Illustration of hydrostatic moment . . . . . . . . . Ship as linear system . . . . . . . . . . . . . . . . Response function versus referred frequency . . . Phase displacement versus referred frequency . . . Ship oscillations in resonance case . . . . . . . . . Oscillation of a raft with a big metacentric height Illustration of the frame in beam waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reduction coefficient of the heave oscillations . . Sea classification . . . . . . . . . . . . . . . . . .
38 39 39 45 46 47 47 48 48 50 52 53
4.1 4.2
Illustration of the ship in head waves . . . . . . . . . . . . . . 55 Position of ship at different time instants in a head wave . . . 58
→ →∞
7
. . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . .
12 12 19 20 21 21 21
4.3 4.4
Curves y = z max and y = z (x) . . . . . . . . . . . . . . . . . 59 Sample for a real ship . . . . . . . . . . . . . . . . . . . . . . . 60
±
5.1
Influence of the vertical acceleration on the seasickness depending on the oscillation period . . . . . . . . . . . . . . . . 5.2 Influence of the vertical acceleration on the seasickness depending on the oscillation period . . . . . . . . . . . . . . . . 5.3 Number of passengers suffering from seasickness on a cruise liners depending on vertical accelerations . . . . . . . . . . . . 5.4 Adaption to seasickness . . . . . . . . . . . . . . . . . . . . . . 6.1 Irregular seawaves, 1- two dimensional, 2- three dimensional. (Fig. from [3]) . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Profile of an irregular wave. (Fig. from [3]) . . . . . . . . . . . 6.3 Representation of irregular wave through the superposition of regular waves. (Fig. from [3]) . . . . . . . . . . . . . . . . . . 6.4 p.d.f. of the wave ordinate . . . . . . . . . . . . . . . . . . . . 6.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62 63 63 64 65 66 67 68 71
7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9
Determination oft he inertia moment I zz . . Determination of I xx and z g . . . . . . . . . Heel test . . . . . . . . . . . . . . . . . . . . Method of forced rolling . . . . . . . . . . . Seakeeping test at MARIN ([4]) . . . . . . . Scetch of the MARIN Seakeeping basin ([4]) Wave generator of MARIN Seakeeping basin Method of wave detection . . . . . . . . . . Ship motion during large scale tests . . . . .
. . . . . . . . . . . . . . . . . .
77 78 79 80 81 82 82 82 83
8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13
U-tube stabilization system of Frahm of the second type . . . Free surface Type passive Roll stabilization systems of Flume . Free surface Type passive Roll stabilization systems of Flume . Active stabilizer . . . . . . . . . . . . . . . . . . . . . . . . . . Passive Schlingerkiel . . . . . . . . . . . . . . . . . . . . . . . Active rudders . . . . . . . . . . . . . . . . . . . . . . . . . . . Damping of pitch oscillations . . . . . . . . . . . . . . . . . . Damping of pitch oscillations . . . . . . . . . . . . . . . . . . [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86 87 87 88 89 89 89 90 90 91 91 92 92
8
. . . . . . . . . . . . . . . . . . ([4]) . . . . . .
. . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . .
8.14 [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 8.15 [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 8.16 [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 9.1 9.2
Ship oscillations during parametric resonance . . . . . . . . . . 97 Conditions for parametric resonance appearance . . . . . . . . 97
9
10
Chapter 1 Ship motion in regular sea waves 1.1
Coupling of different ship oscillations
The ship has generally six degrees of freedom which are called as surge ζ , sway η, heave ζ , heel (or roll) ϕ, yaw ϑ and pitch ψ (see Fig.1.1) for explanation of each oscillation motion). In this chapter we consider first the case of the ship with zero forward speed. Generally, different ship oscillations are strongly coupled. There are three sorts of coupling: •
hydrostatic coupling
•
hydrodynamic coupling
•
gyroscopics coupling
The hydrostatic coupling is illustrated in Fig.1.2. If the ship draught is changed, the center of effort of vertical hydrostatic (floating) force is moving usually towards the ship stern because the frames in the stern are more full than those in the bow region. The displacement of the center of effort towards the stern causes the negative pitch angle. Therefore, the heave oscillations cause the pitch oscillations and vice versa. With the other words, the heave and pitch oscillations are coupled. Hydrodynamic coupling can be illustrated when the ship is moving with acceleration in transverse direction (sway motion). Since the ship is asymmetric with respect to the midships, such a motion is conducted with appearance of the yaw moment. Therefore, the sway and yaw oscillations are hydrodynamically coupled.
11
According to gyroscopic effect, rotation on one axis of the turning around the second axis wheel produced rotation of the third axis. This rule can be applied to the ship. For instance, if the ship performs rolling motion and the transverse force is acting on the ship, it starts to perform the pitch oscillations. The gyroscopic effects are present in the equation system (1.13). They are represented in ”’i-th”’ force equation by products V j=i ωm= j =i and by products ω j =u ωm= j =i in the ”’i-th”’ moment equation. In this chapter we consider the ship oscillations with small amplitude. For such oscillations the coupling mentioned above can be neglected.
Figure 1.1: Ship motion with 6 degree of freedom (from [1])
Figure 1.2: Displacement of the center of effort due to change of the ship draught (from [2])
12
1.2
Classification of forces
According to the tradition used in ship hydrodynamics since almost a hundred years, the forces acting upon the ship are subdivided into hydrostatic forces, radiation and diffraction forces. This subdivision can be derived formally utilizing the potential theory. The potential theory is still remaining the theoretical basis for the determination of wave induced forces, since the most contribution to these forces is caused by processes properly described by inviscid flow models. Let us consider the plane progressive waves of amplitude A and direction ψw are incident upon a ship, which moves in response to these waves. The ship oscillation caused by waves can be written in the form ζ j = ζ j0 sin ωt,
j = 1, 2,...6.
The corresponding speeds of ship oscillations U j , U j =
dζ j = ζ j0 ω cos ωt, dt
(1.1)
j = 1, 2,...6 are:
j = 1, 2,...6.
(1.2)
and accelerations: dU j = ω2 ζ j0 sin ωt, j = 1, 2,...6. (1.3) dt Here ζ j0 are small ship oscillations amplitudes and ω is the frequency. Within the linear theory the ship oscillation frequency is equal to the incident wave frequency. In what follows we use the linear theory and assume that the both waves and ship motion are small. The total potential ϕ can be written, using the superposition principle, in the form: a j =
−
6
ϕ(x,y,z,t) =
U j ϕ j (x,y,z ) + AϕA (x,y,z ) cos ωt =
j=1
6
=
ζ j0 ωϕ j (x,y,z ) +
AϕA (x,y,z ) cos ωt
j=1
(1.4)
where •
ϕ j (x,y,z ) is the velocity potential of the ship oscillation in j-th motion with the unit amplitude ζ j0 = 1 in the absence of incident waves, 13
•
ϕA (x,y,z ) is the potential taking the incident waves and their interaction with the ship into account.
The first potentials ϕ j (x,y,z ) describes the radiation problem , whereas the second one the wave diffraction problem . The potentials ϕ j (x,y,z ) and ϕA are independent only in the framework of the linear theory assuming the waves and ship motions are small. Within this theory ϕA is calculated for the ship fixed in position. The potentials must satisfy the Laplace equation ∆ϕ j = 0, ∆ϕA = 0 and appropriate boundary conditions. The boundary conditions to be imposed on the ship surface are the no penetration conditions (see also formulae 3.18 in the Chapter 3 [6]): •
for radiation potentials ∂ϕ 1 ∂n ∂ϕ 4 ∂n ∂ϕ 5 ∂n ∂ϕ 6 ∂n
•
= cos(n, x);
∂ϕ 2 = cos(n, y); ∂n
∂ϕ 3 = cos(n, z ); ∂n
= (y cos(n, z )
− z cos(n, y)); = (z cos(n, x) − x cos(n, z )); = (x cos(n, y) − y cos(n, x)).
for wave diffraction potentials ∂ϕ A =0 ∂n
(1.5)
(1.6)
where n is the normal vector to the ship surface, directed into the body, (x,y,z ) are the coordinates of a point on the ship surface. The r.h.s. of the conditions (1.5) is the normal components of the ship local velocities caused by particular oscillating motions. The diffraction potential ϕA is decomposed in two parts ϕA = ϕ∞ + ϕ p
(1.7)
which ϕ∞ is the potential of incident waves not perturbed by the ship presence and ϕ p is the perturbation potential describing the interaction between 14
the incident waves and the ship. The potential of regular waves ϕ∞ is known (see Chapter 6 in [6]). The boundary condition for ϕ p on the ship surface is ∂ϕ p ∂ϕ ∞ = (1.8) ∂n ∂n Away from the ship the radiation potentials ϕ j and the diffraction perturbation potential ϕ p decay, i.e. ϕ p 0, ϕ j 0.
−
−−−→
−−−→
r→∞
r→∞
On the free surface the linearized mixed boundary condition (see formula (6.17) in [6]) reads ∂ 2 ϕ ∂ϕ + g = 0 on z = 0. ∂t 2 ∂z Substituting (1.4) in (1.9) yields for ϕ j (x,y,z ) and ϕA (x,y,z ): ω2 ∂ϕ j ϕ j + = 0 on z = 0. g ∂z ω2 ∂ϕ A ϕA + = 0 on z = 0. g ∂z
− −
(1.9)
(1.10)
It is obvious from (1.10) that ϕ j (x,y,z ) and ϕA (x,y,z ) depend on ω. Additionally in the wave theory the radiation condition is imposed stating that the waves on the free surface caused by the potentials are radiated away from the ship. The potentials introduced above can be found using panel methods. The force and the moment on the ship are determined by integrating the pressure over the wetted ship surface. The pressure can be found from the Bernoulli equation written in the general form: ρu2 ∂ϕ p + + ρgz + ρ = C (t) (1.11) 2 ∂t Here the potential is the potential of the perturbed motion. The constant C(t) which is the same for the whole flow domain is calculated from the condition that the pressure on the free surface far from the ship is constant and equal to the atmospheric pressure: pa = C (t) Substituting (1.12) in (1.11) gives: 15
(1.12)
p
− p
a
=
−
ρu2 2
− ρgz − ρ ∂ϕ ∂t
(1.12a)
Remembering that the ship speed is zero and perturbation velocities as well as the velocities caused by incident waves are small we neglect the first term in (1.12a): p
− p
a
=
Together with (1.4) it gives
−
∂ϕ ρ + gz ∂t
6
p
− p
a
=ρ
ζ j0 ωϕ j (x,y,z ) + AϕA (x,y,z ) ω sin ωt
j=1
(1.13)
(1.14)
− ρgz
The forces and the moment are then calculated by integration of p pa over the wetted ship area
−
= F
= M
pndS,
S
p(r
S
× n)dS.
(1.15)
The normal vector direction in (1.15) is into the body. The vertical ordinate z of any point on the wetted area can be represented as the difference between the submergence under unperturbed free surface ζ and free surface elevation ζ 0 . Substituting (1.14) in (1.15) one obtains = F
−ρg
nζdS + ρg
S
+ρ
ζ j0 ω2 sin ωt
j=1
+ ρ(Aω sin ωt)
nζ 0 dS
S
6
nϕ j dS +
(1.16)
S
n(ϕ∞ + ϕ p )dS
S
Four integrals in (1.16) represent four different contributions to the total force: •
the hydrostatic component (the first term) acting on the ship oscillating on the unperturbed free surface (in calm water), 16
•
the hydrostatic component arising due to waves (the second term),
•
the damping and the added mass component (the third term) and
•
the hydrodynamic wave exciting force (the fourth term).
The moment is expressed through similar components. The third term describes the force acting on the ship oscillating in calm water. The last term arises due to incident waves acting on the ship. Within the linear theory keeping only the terms proportional to the amplitude A and neglecting small terms of higher orders proportional to An, n > 1 one can show that the integration in the last term can be done over the wetted area corresponding to the equilibrium state. Thus, the last term describes the force induced by waves on the ship at rest.
∼
1.3
Radiation force components
Let us consider the second term of the force 6
2 = ρ F
ζ j0 ω 2
sin ωt
j=1
nϕ j dS
(1.17)
S
Each component of this force is expressed as 6
F 2i = ρ
ζ j0 ω2
sin ωt
j=1
6
ni ϕ j dS =
− j=1
S
c ji
dU j dt
(1.18)
As shown by Haskind [17], the hydrodynamic coefficient c ji is represented as the sum of two coefficients: c ji = µ ji
− ω1 λ
ji
The term ω1 λ ji has been introduced to take the fact into account that the force due to influence of the free surface depends not only on ω2 but also on ω. The force is then
−
6
F 2i =
− j=1
µ ji
−
1 λ ji ω
dU j = dt
6
− j=1
17
6
µ ji a j
−
s
λ ji U j t = t
j=1
−
π 2ω
(1.19)
As seen from (1.19) the first component of the force is proportional to the acceleration a j whereas the second one is proportional to the velocity U j . The first component is called the added mass component, whereas the second one- the damping component. Using the Green’s theorem, Haskind derived the following symmetry conditions for the case zero forward speed: c ji = cij
1.3.1
⇒
µ ji = µij λ ji = λij
Hydrodynamic damping
There are two reasons of the hydrodynamic damping of the ship oscillations on the free surface. First reason is the viscous damping which is proportional ρU 2 to the square of the ship velocity C Dj 2j S . Within the linear theory this term proportional to the amplitude (ζ j0 )2 is neglected. The main contribution to the damping is done by the damping caused by radiated waves. When oscillating on the free surface the ship generates waves which have the mechanic potential and kinetic energy. This wave energy is extracted from the kinetic energy of the ship. Ship transfers its energy to waves which carry it away from the ship. With the time the whole kinetic energy is radiated away and the ship oscillations decay.
∼
Similarly to the added mass one can introduce the damping coefficients. The full mechanic energy in the progressive wave with the amplitude A is (see chapter 6.4 in [6]) 1 E = ρgA 2 2
(1.20)
per wave length. The energy transported by waves through sides 1 and 2 (see Fig. 1.3) per time unit is 1 δE = 2 ρgA 2 U 2
(1.21)
where U is the wave group velocity. The damping coefficient is defined as δE = λij U j2 18
(1.22)
Figure 1.3: Illustration to derivation of damping coefficient where λij is the coefficient of damping in i-th direction when the ship oscillates in j-th motion. U j2 is the time averaged square of the ship oscillations 2 (ωζ j0 ) 2 speed. Obviously, U j = 2 and
2
ωζ j0 δE = λij 2 The group velocity (see formulae (6.39) and (6.40) in [6]): U =
c 1 = 2 2
(1.23)
g g = k 2ω
(1.24)
since kg = ω2 (see formula (6.21) in [6]). Equating (1.21) and (1.23) one obtains with account for (1.24)
ρgA 2U = λij
g ρgA = λij 2ω 2
2 ωζ j0
2
⇓ ⇒
ωζ j0 2
2
ρg2 λij = 3 ω
(1.25)
A ζ j0
2
The damping coefficient λij depends on the square of the ratio of the wave amplitude to the ship oscillation amplitude causing the wave. The damping coefficient of slender body can be found by integration of damping coefficients of ship frames along the ship length 19
L/2
B22 =
L/2
λ22 dx,
−L/2
−L/2
L/2
λ33 dx,
B44 =
−L/2
L/2
B55 =
B33 =
−L/2
L/2
x2 λ33 dx,
B66 =
λ44 dx, (1.26)
x2 λ22 dx
−L/2
The damping coefficients of different frames are shown in Fig. 1.4, 1.5, 1.6, 1.7 and 1.8 taken from [2]. Solid lines show results obtained from the potential theory. Generally, the results show the applicability of the potential theory for calculation of damping coefficients. The accuracy of prediction is not satisfactory for the box B/T=8 in heave and B/T=2 in sway because of the flow separation at corners which has a sufficient impact on hydrodynamics in these two cases. The agreement for λ44 is not satisfactory (see Fig. 1.8) because of dominating role of the viscosity for this type of damping. For the semi circle frame the damping coefficient in roll is zero λ44 = 0 within the inviscid theory. One hundred per cent of the roll damping is due to viscosity. Usually λ44 are determined using viscous flow models. It is remarkable, that the damping coefficients depend on the frequency and amplitude (see Fig. 1.8).
Figure 1.4: Added mass and damping coefficient of the semi circle frame at heave oscillations. Here A is the frame area.
1.3.2
Added mass component
When the ship oscillates, the force acting on the ship contains the component associated with the added mass like in every case of accelerated body motion. 20
Figure 1.5: Added mass and damping coefficient of the box frame at heave oscillations. Here A is the frame area.
Figure 1.6: Added mass and damping coefficient of the semi circle frame at sway oscillations. Here A is the frame area.
Figure 1.7: Added mass and damping coefficient of the box frame at sway oscillations. Here A is the frame area. The difference with the case of the motion in unlimited space is the presence of the free surface. The added mass µij have to be calculated with account for the free surface effect. For their determination the panel methods can be 21
Figure 1.8: Added mass and damping coefficient of the box frame at roll (heel) oscillations. Here A is the frame area. used. The problem is sufficiently simplified in two limiting cases ω 0 and ω . The boundary condition (1.10) can be written in the form:
→
→∞
∂ϕ j ∂z
= 0 for ω
ϕ j = 0 for
→ 0, ω → ∞.
(1.27)
on z = 0.
The conventional mirroring method can be used for the caseω 0 (Fig. 1.9). The mirroring frame is moving in the same direction for surge, sway and yaw. For the heave, roll and pitch the fictitious frame is moving in the opposite direction. At the free surface, these tricks make the normal components of the j total velocity induced by the actual and the fictitious frames zero, i.e. ∂ϕ =0 ∂z on z = 0.
→
In the case ω
→ ∞ the tangential component of the total velocity should be zero, since ϕ = 0 ⇒ ϕ = dx = 0 ⇒ = 0 on z = 0. The modified mirroring method is implemented for the case ω → ∞ (Fig. 1.10). x
j
j
−∞
∂ϕ j ∂x
∂ϕ j ∂x
The fictitious frame is moving in the opposite direction for surge, sway and yaw. For the heave, roll and pitch the fictitious frame is moving in the same direction as that of the original frame. These tricks make the normal components of the total velocity induced by the actual and the fictitious j frames zero, i.e. ∂ϕ = 0 at the free surface on z = 0. ∂z Using mirroring method the added mass can be found using the panel without explicit consideration of the free surface since it is taken into account by fictitious frames. The added mass of slender body can be found by integration of added mass 22
Figure 1.9: Mirroring for the case ω
→0
Figure 1.10: Mirroring for the case ω
→∞
of ship frames along the ship length L/2
A22 =
L/2
µ22 dx,
−L/2
−L/2
L/2
µ33 dx,
A44 =
−L/2
L/2
A55 =
A33 =
−L/2
L/2
x2 µ33 dx,
A66 =
µ44 dx, (1.28)
x2 µ22 dx
−L/2
The added mass of different frames are shown in Fig. 1.4, 1.5, 1.6, 1.7 and 1.8 taken from [2]. Like in case of damping coefficients the results of the potential theory are not acceptable for roll added mass because of dominating role of the viscosity. As seen from Fig. 1.4 - Fig. 1.8 the added mass depend on the 23
frequency ω.
1.4
Hydrostatic component
Let us the ship is in the equilibrium state. The ship weight is counterbalanced by the hydrostatic lift. Due to small heave motion the equilibrium is violated and an additional hydrostatic force appears. The vertical component of this additional hydrostatic force can be calculated analytically from (1.16) for the case of small heave motion
∆F ζ =
− ρg
− −
cos(nz )zdS
S
ρg
cos(nz )zdS
S
T +ζ
=
−ρgA
W P ζ
T
(1.29) where ζ is the increment of the ship draught, AW P is the waterplane area and T is the ship draught in the equilibrium state. The roll and pitch hydrostatic moments for small change of the roll and pitch angles are M ϕ =
−ρg∇ GM ϕ,
(1.30)
M ϑ =
−ρg∇ GM ψ,
(1.31)
0
γ
0
L
where ϕ and ψ are the roll and pitch angle respectively, GM γ
is the transverse metacentric height,
GM L
is the longitudinal metacentric height and
∇
is the ship displacement.
0
1.5
Wave exciting force
per = ρωA sin ωt The wave exciting force F
n(ϕ∞ + ϕ p )dS contains two com-
S
ponents. The first component, determined by the integration of the incident potential ϕ∞ , ρωA sin ωt nϕ∞ dS is referred to as the hydrodynamic part of
S
the Froude-Krylow force. This force called as the Smith effect is calculated by the integration of wave induced pressure as if the ship is fully transparent for 24
incident waves. The full Froude-Krylow force contains additionally the hydrostatic force arising due to change of the submerged part of the ship caused by waves (the second term in (1.16)). The second component ρωA sin ωt nϕP dS
S
takes the diffraction effect (the contribution of the scattering potential ϕ p to pressure distribution) into account. As shown by Peters and Stokes the Froude Krylov force is a dominating part of the wave induced forces for oscillations of slender ships in directions j=1 (surge), 3 (heave) and 5 (pitch).
1.6
Motion equations
The linearized decoupled motion equations of the ship oscillations are written in the form added mass damping force forces ¨ ¨ mξ = A11 ξ B11 ξ ˙ m¨ η= A22 η¨ B22 η˙ ¨= ¨ mζ A33 ζ B33 ζ ˙ I xx ϕ¨ = A44 ϕ¨ B44 ϕ˙ I yy ψ¨ = A55 ψ¨ B55 ψ˙ I zz ϑ¨ = A66 ϑ¨ B66 ϑ˙
− − − − − −
− − − − − −
hydrostatic forces
wave exciting forces +F ξ,per (t), +F η,per (t), ρgAW P ζ +F ζ,per (t), ρg 0 GM γ ϕ +M ϕ,per (t), ρg 0 GM L ψ +M ψ,per (t), +M ϑ,per (t).
− − ∇ − ∇
(1.32)
The weight is not present in the second equation of the system (1.32) because it is counterbalanced by the hydrostatic force at rest. The additional hydrostatic force ρgA W P ζ is the difference between the weight and the full hydrostatic force. The system (1.32) is written in the principle axes coordinate system [7].
−
1.7
Haskind’s relation
One of the most outstanding results in the ship oscillations theory is the relation derived by Max Haskind who developed in 1948 the famous linear hydrodynamic theory of ship oscillations. Haskind shown how to calculate the wave induced hydrodynamic force utilizing the radiation potentials ϕ j and the potential of incident waves ϕ∞ . The determination of the diffraction potential ϕ p what is quite difficult can be avoided using this relation which is valid for waves of arbitrary lengths.
25
The Green’s formula for two functions Φ and Ψ satisfying the Laplace equation is
∂ Φ Ψ ∂n
S w
−
∂ Ψ Φ dS = 0, ∂n
(1.33)
where S w is the flow boundary (wetted ship surface plus the area away from the ship, see the sample in Chapter/Section 3.2). Particularly, the relation (1.33) can be applied to radiation potentials ϕ j . Since the potential ϕ p satisfies the Laplace equation and the same boundary conditions as the radiation potentials ϕ j , the Green’s formula (1.33) can also be applied to ϕ j and ϕ p
∂ϕ j ϕ p ∂n
−
S w
∂ϕ p ϕ j dS = 0 ∂n
(1.34)
The last term in (1.16) is the wave induced force ζ,per = ρ(Aω sin ωt) F
sin ωt n (ϕ∞ + ϕ p ) dS = XA
S
= ρω where X
n (ϕ∞ + ϕ p ) dS . Taking (1.5), (3.34) and (1.8) into account
S
we get
X j = ρω
(ϕ∞ + ϕ p )
∂ϕ j dS ∂n
(1.35)
S w
S w
∂ϕ ϕ p ∂nj dS =
S w
p ϕ j ∂ϕ dS ∂n
∂ϕ p ∂n
⇒ X = ρω j
=
−
∂ϕ ∂n
S w
∂ϕ ϕ∞ ∂nj
X j = ρω
dS
∞
χ = 90◦
⇓
+
p ϕ j ∂ϕ ∂n
(1.36)
∂ϕ j ϕ∞ ∂n
S w
−
∂ϕ ∞ ϕ j ∂n
dS
(1.37)
The formula (1.37) is the Haskind’s relation. As seen the wave induced force can be calculated through the radiation and free wave potentials avoiding 26
the determination of the diffraction potential ϕ p . The calculation of the integral (1.37) is a complicated problem because the incident waves don’t decay away from the ship and the integral (1.37) should be calculated over both the surface far from the ship and the ship wetted surface. Note that the potential ϕ∞ does not decay away from the ship. The method of the stationary phase [8] allows one to come to the following force expression using the Haskind’s relation (1.37): F i,per = Bii ζ ˙i ,
k where Bii = 8πρg(c/2)
2π
|
X i (χ) 2 dχ
|
0
Here c is the phase wave velocity (celerity) and χ is the course angle. Let us consider the slender ship B(x, z ) 0 in a beam wave (χ = 90◦ ). The wetted area is approximately equal to the projection on the symmetry plane y = 0, S wetted = [0, L] [0, T ].
∼
X j = ρω
∂ϕ j ϕ∞ ∂n
S w
−
∂ϕ ∞ ϕ j ∂n
∂ϕ 3 ∂B = cos(n, z ) = ∂n ∂z
dS
≈ ρω
ϕ∞
∂ϕ j dS ∂n
S w
⇒ X ≈ 2ρωω 3
ϕ∞
∂B dS ∂z
S w
The coefficient 2 arises due to the integration over two boards y = +B(x, z ) and y = B(x, z ). Using the potential of an Airy wave (see formulae (6.18) in ([6]) estimated at y = 0 one can find the potential ϕ∞ :
−
ϕ=
Ag kz e sin(ky ω
− ωt) ⇒ ϕ
∞
=
−ge
kz
/ω
For the case of a vertical cylinder for which the vertical force does not depend on the wave course angle χ the damping coefficient B33 takes a very simple form [8]:
B33 =
2ρg (c/2)
0
ekz
−T
27
∂B dz ∂z
2
(1.38)
1.8
Exercises
1. Schwimmender Balken
(a) Ist die Schwimmlage des Balkens stabil? (b) Bleibt die Schwimmlage stabil, wenn die H¨ ohe h des Balkens 0, 15 m betr¨ agt? (c) Ab welcher Balkenh¨ ohe wird die Schwimmlage instabil? 2. ([8]) Berechnen Sie die maximale elektrische Leistung einer Turbine, die die ganze mechanische Energie einer Welle umwandelt. Die Welle hat eine H¨ohe von 1 m, eine L¨ange von 100 m und eine Breite von 1 km in Richtung des Wellenkammes. 3. Hinter dem bildet sich ein station¨ ares Wellensystem. Wie groß ist die Geschwindigkeit der Querwellen, wenn die Schiffsgeschwindigkeit 10 m/sek betr¨ agt? 4. Welche L¨ange haben die Querwellen hinter einem Schiff, das sich mit der Geschwindigkeit von 10 m/sek bewegt? Wie groß ist die Wellenfrequenz? 5. Die Querwelle hinter einem Schiff hat die Amplitude 1 m. Sch¨ atzen Sie den Widerstand des Schiffes! Benutzen Sie das Bild 6.12 aus dem Buch von Newman. 28
6. ([8])Das Modell eines Schiffes wird in einer sehr breiten Schlepprinne mit der Geschwindigkeit 1 m/s 100 geschleppt. Die Modell¨ange betr¨ agt 5 m. Nach 100 Metern wird das Modell gestoppt. Wie viel Querwellen befinden sich in der Schlepprinne, wenn die Reflektion von Schlepprinnenseiten nicht auftritt? Hinweis: Das Schiff wird als Superposition von zwei Punktst¨orungen betrachtet: Bug und Heck. Der Bug erzeugt die Welle. Das Heck erzeugt die Welle. Gesamtes Wellenbild wird als Summe betrachtet: Benutzen Sie die Formel (6.21) und die Aufgaben 2 und 3. 7. F¨ ur eine fortschreitende Welle mit der Amplitude 6 m und der L¨ange 200 m berechnen Sie die Phasengeschwindigkeit und die maximale Geschwindigkeit der Wasserteilchen. In welchen Punkten ist diese Geschwindigkeit maximal? 8. Task: Develop the theory of vertical oscillations of a very sharp cone with the draught T=10m and the diameter of 1m in regular and irregular waves using the Haskind’s relation (1.38). The added mass A33 can be neglected.
29
30
Chapter 2 Free oscillations with small amplitudes 2.1
Introduction
Let us consider a ship in the equilibrium position at calm water condition. The ship has zero forward speed. If a perturbation acts on the ship, it performs oscillating motions in three directions: •
heave,
•
roll (heel),
•
pitch.
Yaw, surge and sway motions did not arise at calm water conditions. The reason is the presence of restoring hydrostatic forces in heave, roll and pitch directions. The motion equations of the free oscillation read: ¨ B33 ζ + ˙ ρgA W P ζ = 0, (m + A33 ) ζ + (I xx + A44 ) ϕ¨ + B44 ϕ˙ + ρg 0 GM γ ϕ = 0, (I yy + A55 ) ψ¨ + B55 ψ˙ + ρg 0 GM L ψ = 0.
∇ ∇
(2.1)
In ship theory the equations (2.1) are written in the normalized form: ¨ + 2ν ζ ζ ˙ + ω2 ζ = 0, ζ ζ ϕ¨ + 2ν ϕ ϕ˙ + ωϕ2 ϕ = 0, ψ¨ + 2ν ψ ψ˙ + ω 2 ψ = 0, ψ
31
(2.2)
where ν ζ =
B33 , 2 (m + A33 )
ν ϕ =
B44 , 2 (I xx + A44 )
ν ψ =
B55 2 (I yy + A55)
(2.3)
ωψ =
∇
(2.4)
are damping coefficients and ωζ =
ρgA W P , m + A33
ωϕ =
∇
ρg 0 GM γ , I xx + A44
ρg 0 GM L I yy + A55
are the eugen frequencies of non damped oscillations. The equations (2.2) are fully independent of each other. The solutions of the equations (2.2) written in the general form: ¨ 2ν ξ + ˙ ω 2ξ = 0 ξ +
(2.5)
ξ = Ce pt
(2.6)
is given as: Substitution of (2.6) into (2.5) yields the algebraic equation p2 + 2νp + ω2 = 0 which solution is p1,2 =
−ν ±
√
ν 2
−ω
(2.7) 2
(2.8)
If the system has no damping the solution is p1,2 = iω
→ ξ = Ce
iωt
= C (cos ωt + i sin ωt)
(2.9)
The system oscillates with the constant amplitude and frequency ω. That is why the frequency ω is referred to as the eigenfrequency. For real ships the damping coefficient is smaller than the eigenfrequency ν < ω and the equation (2.8) has two solutions: p1 p2
√ = −ν + i ω − ν = −ν + i¯ ω, √ = −ν − i ω − ν = −ν − i¯ ω, 2
2
2
2
(2.10)
In turn, the solution of the differential equation is ξ = Ce−νt e±i¯ω = Ce−νt (cos ω ¯t 32
± i sin ω¯t)
(2.11)
It describes damped oscillations with decaying amplitude which decrease is governed by the factor e−νt , e−νt 0. The rate of the decay is charact→∞ terized by the damping coefficient ν . The frequency of damped oscillations is ω ¯ = ω2 ν 2 < ω (2.12)
−−−→
√
−
Due to damping the frequency of the oscillations is shorter whereas the period is longer: 2π 2π T = = (2.13) ω ¯ ω 2 ν 2 Since ω ν 2π 2π 2π T = = (2.14) ω ¯ ω ω2 ν 2
√ −
∼
√ − ≈
From (2.12) and (2.14) we obtain the frequencies for different types of oscillations which are listed in the table below: Table 2.1: Frequencies and periods of different oscillation types Oscillation
Eigenfrequency
Heave
ωζ =
Rolling
ωϕ =
Pitch
ωψ =
Frequency of damped oscillations
ρgAW P m+A33
ω¯ζ =
ρg ∇0 GM γ I xx +A44
ω¯ϕ =
ρg∇0 GM L I yy +A55
ω¯ψ =
− − −
Period of oscillations
m+A33 ρgAW P
ωζ 2
ν ζ 2
T ζ = 2π
ωϕ2
ν ϕ2
T ϕ = 2π
I xx +A44 ρg∇0 GM γ
ωψ2
ν ψ2
T ψ = 2π
I yy +A55 ρg ∇0 GM L
The damping is characterized by the logarithmic decrement which is the logarithm of the ratio of the oscillation amplitude at the time instant t to that at the time instant t+T, i.e. ξ (t) e−νt = −ν (t+T ) = eνT ξ (t + T ) e
(2.15)
The logarithm of the ratio (2.15) is ξ (t) e−νt 2πν ln = ln −ν (t+T ) = lneνT = ¯ ξ (t + T ) e ω
(2.16)
The ratio ων ¯ is called as the referred damping factor ν¯ . The decay of the oscillation amplitude is equal to this factor multiplied by 2π. Referred damping factors for different types of oscillation can be found from this definition. The results obtained under assumption ω ν ω ¯ ω are listed in the table 2.2.
∼ → ≈ 33
If the metacentric heights GM γ and GM L are getting larger, the periods T ϕ = 2π
I xx +A44 and T ψ = ρg ∇0 GM γ B44 and (I xx +A44 )ρg∇0 GM γ
√
2
2π
I yy +A55 ρg∇0 GM L
ν¯ ψ =
as well as the damping factors ν¯ ϕ =
B55 (I yy +A55 )ρg ∇0 GM L
√
2
decrease. Therefore, the
smaller are the metacentric heights the larger are the oscillations periods and the less oscillations are necessary to decay. The time of decay depends only on damping and doesn’t depend on the metacentric height. Table 2.2: Referred damping factors for different oscillation types Oscillation Heave
Referred damping factor B33 ν¯ ζ =
Rolling
ν¯ ϕ =
Pitch
ν¯ ψ =
2.2
√ √ √
2
2
2
(m+A33 )ρgAW P B44 (I xx +A44 )ρg∇0 GM γ B55 (I yy +A55 )ρg ∇0 GM L
Exercise
1. ϕ0 is the roll angle at t = 0. Find the number of periods N of free roll oscillations necessary to reduce the amplitude oscillations by factor e−a . What is influence of the metacentric height on N? 2. The period of undamped oscillations is T. The referred damping factor ν¯ is 0,2. Calculate the period of damped oscillations! Calculate the reduction of the amplitude within the period of damped and undamped oscillations! 3. Typical periods of roll and pitch oscillations for different ships are [9]: Ship tanker ice breaker trawler big cruise liner container ships (20000
− 30000 t)
Explain why T ϕ > T ψ . 34
T ϕ , sec 9 ... 15 8 ... 12 6 ... 8 20 ... 28 16 ... 19
T ψ , sec 7 ... 11 3 ... 5 3 ... 4 10 ... 12 7 ... 9
4. The periods of oscillations can be estimated from the following simple empiric formulae [9] T ζ
√ ≈ 2.5 T ,
T ψ
√ ≈ 2.4 T ,
T ϑ
≈ cB/
GM γ ,
where T and B are draught and beam. The empiric coefficient c is equal approximately 0, 8...0, 85 for big cruise liners. Calculate the change of the period of roll oscillations of a cruise liner if a load with mass 1 ton is elevated in vertical direction from 10 m from the keel line to the 1 meters from the keel line! The ship displacement is 20 000 t and the beam is 30 m.
35
36
Chapter 3 Ship oscillations in small transverse waves (beam see) The formalism developed in this chapter is based on the following assumptions: •
•
waves are regular, waves amplitudes related to the wave lengths are small. Wave slope is small.
•
wave length is much larger than the ship width,
•
The ship has zero forward speed.
From the first two assumptions it follows, that the collective action of waves on ship can be considered through the superposition principle. Therefore, the theory can be developed for the interaction of the ship with a single wave with given length and amplitude. The effects of different waves are then summed. For the case of small waves the oscillations are decoupled. The hydrodynamic, hydrostatic and gyroscopic coupling effects are neglected. The perturbation forces (see the last column in the equation system (1.32)) arise due to wave induced change of the hydrostatic forces and due to hydrodynamic effects caused by orbital motion in waves. The orbital motion causes the hydrodynamic pressure change which results in the wave induced hydrodynamic forces. In each frame, the pressure gradient induced by waves is assumed to be constant along the frame contour and equal to the pressure gradient at the centre A on the free surface. When considering the roll and pitch oscillations 37
in transverse waves it is additionally assumed that the ship draught change ζ and the ship slope relatively to the free surface are constant along the ship. The wave ordinate is given by the formula derived for the progressive wave (see Chapter 6 in [6]) ζ 0 = A sin(ωt + kχ) (3.1) where A is the amplitude, χ is the wave propagation direction and ω is the frequency. In this section the incident waves are perpendicular to the ship (see Fig.3.1). The wave propagation direction is in η direction, i.e. χ = η. The waves induce roll and heave oscillations. The curvature of the free surface is neglected, the free surface is considered as the plane performing angular oscillations and translational oscillations in vertical direction.
Figure 3.1:
3.1
Hydrostatic forces and moments
The hydrostatic forces during the heave oscillations are calculated neglecting the wave surface slope. The hydrostatic force acting on the ship with draft increment ζ in the wave with the ordinate ζ 0 (see Fig.3.2) is F ζ hydr =
−ρgA
W P (ζ
− ζ ) = −ρgA 0
W P (ζ
− A sin ωt)
(3.2)
where η = 0 at the point A. Since the wave slope is neglected, the dependence of ζ 0 on ζ is not considered. The first part ρgA W P ζ is the
−
38
Figure 3.2: Illustration of hydrostatic force restoring hydrostatic force which is already present in (1.32). The second part ρgA W P A sin ωt is the wave induced hydrostatic force. The additional hydrostatic moment is determined from the analysis of the Figure 3.3.
Figure 3.3: Illustration of hydrostatic moment The relative slope of the ship to the free surface is ϕ 39
− α, where α is the wave
surface slope dζ 0 α= dη
η−0
ω2 = Ak cos ωt = A cos ωt = αA cos ωt g
(3.3)
αA is the amplitude of the angular water plane oscillations. The hydrostatic pressure increases linearly in direction perpendicular to the free surface plane. Therefore, the restoring moment is the same as in the case if the free surface is horizontal and the ship is inclined at the angle ϕ α. The restoring moment is known from the ship hydrostatics
−
M ϕhydr =
(3.4) −ρg∇ GM (ϕ − α) = −ρg∇ GM (ϕ − α cos ωt) The first part −ρg ∇ GM ϕ is the restoring hydrostatic moment, whereas the second part ρg∇ GM α cos ωt is the wave induced hydrostatic moment. 0
γ
0
0
3.2
0
γ
A
γ
γ A
Hydrodynamic Krylov - Froude force
Hydrodynamic forces arise due to wave induced hydrodynamic pressures. From the Bernoulli equation the pressure is (see (1.13)). 2
p =
+p − ρu2 − ρgz − ρ ∂ϕ ∂t
(3.5)
a
The constant pressure pa does not need to be considered since being integrated over the ship wetted surface results in zero force and moment. The first term in (3.5) is neglected within the linear theory under consideration. The second term results in force and moment considered above in the section 3.1. The remaining term punst = ρ ∂ϕ is responsible for hydrodynamic ∂t effects caused by waves. If the interaction between the ship and incident waves is neglected (Krylov - Froude formalism) the potential can be written as the potential of uniform unsteady parallel flow:
−
ϕ = uζ (t)ζ
(3.6)
where uζ (t) is unsteady velocity of the flow in the wave uζ (t) =
dζ 0 dt
(3.7) ∂u
Since the unsteady pressure punst = ρ ∂ϕ = ρζ ∂tζ is zero at ζ = 0 the ∂t total pressure is equal to the atmospheric pressure p = pa (see (3.5)). The gradient of the hydrodynamic pressure in vertical direction reads:
−
∂p unst = ∂ζ
−
∂ ρ ∂ζ
∂ϕ ∂t
=
∂ ρ ∂t
−
∂ϕ ∂ζ
40
−
=
−ρ ∂u∂t(t) = −ρζ ¨ = ρω A sin ωt ζ
0
2
(3.8)
The unsteady pressure at the point ζ < 0 is then: ζ
p
unst
(ζ ) = p
unst
−
(ζ = 0)
∂p unst dz = ∂z
0
0
∂p unst dz ∂z
ζ
The force caused by punst on each frame is calculated by the integration of the pressure over the frame wetted area
dF ζ dyn
p
=
0
unst
cos(nζ )dC =
∂p unst dz cos(nζ )dC = ∂z
ζ
0
=
ρω2 A sin ωtdz cos(nζ )dC = ρω2 A sin ωt
ζ
(3.9)
ζ cos(nζ )dC
Here the normal vector is the inward normal vector. Since the integral ζ cos(nζ )dC is equal to the frame area taken with the opposite sign, i.e. Af , the hydrodynamic force caused by waves takes the form: dF ζ dyn = ρω2 A sin ωtA f (3.10)
−
−
Being integrated along the ship length this force gives the force acting on the whole ship length L
F ζ dyn =
L
−
dF ζ dyn dξ =
0
ρω 2A sin ωtA f dξ =
0
(3.11)
L
=
2
−ω A sin ωt
ρAf dξ =
0
−mω A sin ωt = mζ ¨ 2
0
The hydrodynamic moment acting on the ship frame dM ϕdyn =
punst (η cos(nζ )
= ρω2 A sin ωt
− ζ cos(nη))dC = ζ (η cos(nζ ) − ζ cos(nη))dC
(3.12)
Within the linear theory considering small ship slopes the last integral in (3.12) is zero, i.e. L
dM ϕdyn = 0
dyn ϕ
→ M
41
=
dM ϕdyn = 0
0
(3.13)
3.3
Full Krylov - Froude force and moment
The full Froude Krylov force takes the form: F ζ l = ρgA W P A sin ωt
2
− mω A sin ωt = ρgA
W P A sin ωt
¨0 + mζ
(3.14)
The first term is caused by hydrostatic effect, whereas the second one by hydrodynamic effects. The second term is referred in the literature to as the Smith effect. The full Froude Krylov moment contains only the wave induced hydrostatic component: M ϕl = ρg 0 GM γ αA cos ωt (3.15)
∇
3.4
Force and moment acting on the ship frame in accelerated flow
These forces are determined using the concept of the relative motion. Let us ζ j is a ship displacement in j-th direction. As it has been explained in pre¨ j vious chapters, the force acting on the body moving with the acceleration ζ in a liquid at rest is equal to the product of added mass with the acceleration ¨ j . If the liquid moves with the accelerataken with opposite sign, i.e. A jj ζ ¨ jL relative to motionless body, the force acting on the body is towards tion ζ ¨ jL . If both body and liquid move with acthe acceleration direction, i.e. A jj ζ ¨ j ζ ¨ jL ). Similarly, the damping force can celerations the total force is A jj (ζ introduced being proportional to the relative velocity B jj (ζ ˙ j ζ ˙ jL ). The ¨ j and B jj ζ ˙ j are already represented first components of both forces A jj ζ by the first and the second columns in the motion equations (1.32). The sec¨ jL and B jj ζ ˙ jL represent the hydrodynamic forces due ond components A jj ζ to interaction between the incident waves and floating body. Remembering ¨ jL = ω2 A sin ωt we obtain the lift force caused that ζ ˙ jL = ωA cos ωt and ζ by the interaction between the ship and incident wave:
−
−
−
−
−
−
−
−
F ζ 2 =
−A
33 ω
2
A sin ωt + B33ωA cos ωt
(3.16)
In roll oscillations the ship moves with the angular velocity ϕ˙ and angular acceleration ϕ. ¨ The free surface oscillates with the angular velocity α˙ and acceleration α. ¨ Taking α from (3.3) we obtain the roll moment caused by the interaction between the ship and incident wave: M ϕ2
=
−A
44 ω
2
αA cos ωt
−B
44 ωα A
sin ωt =
42
−
ω4 A44 A cos ωt g
−
ω3 B44 A sin ωt g (3.17)
3.5
Full wave induced force and moment
In the section 1.5 we divided the wave induced forces into the Froude Krylov part and the interaction force. Commonly, the Froude Krylov force is the dominating part of the wave induced forces. To calculate the full wave induced force we have to note that the Smith effect ¨ jL + B jj ζ ˙ jL . All hydrodynamic effects is already represented in the force A jj ζ are taken into account. Only the hydrostatic part of the Froude Krylov force ¨ + B jj ζ ˙ j L to get the full wave induced force: should be added to A jj jL
F ζ,per = ρgA W P
2
−ω A
33
A sin ωt + B33 ωA cos ωt
(3.18)
The full moment is the sum of (3.15) and (3.17):
M ϕ,per = ρg
3.6
∇ GM − A 0
γ
44 ω
2
ω2 A cos ωt g
−
ω3 B44 A sin ωt g
(3.19)
Equations of ship heave and roll oscillations
Substitution of all forces derive above into the original differential equations results in two following decoupled ordinary differential equations
− −
¨ (m + A33 ) ζ (I xx + A44 ) (ϕ¨
¨0 + B33 ζ ˙ ζ
− α¨) + B
44
ζ ˙0 + ρgA W P (ζ
− ζ ) = −mζ ¨ (ϕ˙ − α) ˙ + ρg∇ GM (ϕ − α) = −I α ¨. 0
0
0
γ
xx
(3.20) (3.21)
The solution of both equations can be represented as the sum ζ = ζ inh + ζ f ree ϕ = ϕinh + ϕfree, where ζ f ree and ϕfree are free heave oscillations: ϕfree = Ce−ν ϕt (cos ω ¯ϕ t i sin ω ¯ ϕ t), ζ f ree = Ce−ν ζ t (cos ω ¯ζ t i sin ω ¯ ζ t) satisfying the homogeneous equations:
±
±
¨ B33 ζ + ˙ ρgA W P ζ = 0 (m + A33 ) ζ + (I xx + A44 ) ϕ¨ + B44 ϕ˙ + ρg 0GM γ ϕ = 0.
∇
When the free oscillations decay ϕfree , ζ f ree
−−−→ 0, the solutions of the t→∞
equation (3.20) and (3.21) tend to the solutions of inhomogeneous equations: ¨ B33 ζ ˙ + ρgA W P ζ = A33 ζ ¨0 + B33 ζ ˙0 + ρgA W P ζ 0 (m + A33 ) ζ + (3.22) (I xx + A44 ) ϕ¨ + B44 ϕ˙ + ρg 0 GM γ ϕ = A44 α ¨ + B44 α˙ + ρg 0 GM γ α.
∇
∇
43
The inhomogeneous equation (3.21) is written in terms of relative roll angle ϕ(r) = ϕ α in the normalized form:
−
(r)
ϕ¨
(r)
+ 2ν ϕ ϕ˙
ωϕ2 ϕ(r)
+
ω2 = αA cos ωt, 1 + kϕ
(3.23)
where kϕ = A44 /I xx . The solution of (3.23) is seeking in the form (r)
ϕ(r) = ϕA cos(ωt
− δ )
(3.24)
ϕ
Substituting (3.24) into (3.23) and separating terms proportional to cosωt and sin ωt gives two equations: ϕ(r) A
(ωϕ2
ϕ(r) A
−
−
ω2 ω )cos δ ϕ + 2ν ϕ ω sin δ ϕ = αA 1 + kϕ 2
2ν ϕ ω cos δ ϕ + (ωϕ2
2
− ω )sin δ
It follows from (3.25) and (3.26)
(r)
ϕA
(r) ϕA
2
2
ωϕ2
ωϕ2
−ω −ω
(r) ϕA
2
ωϕ2
ϕ
=0
cos2 δ ϕ + 4ν ϕ2 ω 2 sin2 δ ϕ + 4ν ϕ ω sin δ ϕ
sin2 δ ϕ + 4ν ϕ2 ω2 cos2 δ ϕ
2 2
ωϕ2
2 2
−ω
2
ϕ
−ω
2 2
(3.26)
ω4 2 cos δ ϕ = 2 αA (1 + kϕ )
ϕ
+ 4ν ϕ2 ω 2
(3.25)
−−
−4ν ω cos δ
The sum of two last equations
ωϕ2
2
cos δ ϕ
ω 2 sin δ ϕ = 0.
ω4 2 = 2 αA (1 + kϕ )
(r)
allows one to find the ratio ϕA /αA ϕ(r) A = αA
ω ˆ ϕ2 / (1 + kϕ )
−
2 ω ˆ ϕ2
1
,
(3.27)
+ 4ˆ ν ϕ2 ω ˆ ϕ2
where ω ˆϕ = ωωϕ and νˆ ϕ = ων ϕϕ . Eigenfrequency ωϕ and damping coefficient ν ϕ are given by formulae (2.3) and (2.4). The phase of the response relative to that of the input (phase displacement) is found from (3.26): δ ϕ = arctg
44
2ˆ ν ϕ ω ˆϕ 1 ω ˆ ϕ2
−
(3.28)
Similar solutions are obtained for the heave oscillations: (r)
ζ A = A
ω ˆ ζ 2 /(1 + kζ )
− ω ˆ ζ 2)2
(1
2ˆν ζ ω ˆ ζ 1 ω ˆ ζ 2
δ ζ = arctg with kζ = A33/m,
3.7
ω ˆ ζ =
ω ωζ
and νˆ ζ =
+ 4ˆ ν ζ 2 ω ˆ ζ 2
(3.29)
(3.30)
−
ν ζ . ωζ
Analysis of the formula (3.27)
The formula (3.27) can be rewritten as follows: (r)
ϕA ϕA αA = = αA αA or
−
ϕA = 1+ αA
ω ˆ ϕ2 / (1 + kϕ )
− 1
ω ˆ ϕ2
2
+ 4ˆ ν ϕ2 ω ˆ ϕ2
ω ˆ ϕ2 / (1 + kϕ )
− 1
2 ω ˆ ϕ2
(3.31)
+ 4ˆ ν ϕ2 ω ˆ ϕ2
The physical meaning of terms in (3.31) is obvious from the following expression amplitude of ship roll oscillations = 1 + enhancement (3.32) amplitude of wave angle oscillations 2
2
ω ˆ /(1+k ) ω ˆ /(1+k ) Since the function ϕ 2 ϕ is positive ϕ 2 ϕ > 0 the ship (1−ωˆϕ2 ) +4ˆν ϕ2 ωˆϕ2 (1−ωˆϕ2 ) +4ˆν ϕ2 ωˆϕ2 roll amplitude is larger than the the amplitude of the angular water plane oscillations αA , i.e. ϕαAA > 1.
Figure 3.4: Ship as linear system 45
The ship can be considered as a system with the waves as input and the resulting motion as the output (Fig.3.4). As seen from (3.31) this system is linear for small amplitude oscillations. In terms of linear system theory the formula (3.31) reads output = 1 + enhancement input
(3.33)
The linear system is time invariant. The output produced by a given input is independent of the time at which the input is applied. The function 1 + enhancement which characterizes the system response in the frequency domain is called the frequency response function. ω ˆ 2 /(1+k ) The enhancement function ϕ 2 ϕ goes to zero if referred frequency (1−ωˆϕ2 ) +4ˆν ϕ2 ωˆϕ2 ω ˆ 2 /(1+k ) becomes zero. At very large frequencies ω , ϕ 2 ϕ 1/ (1 + kϕ ). (1−ωˆϕ2 ) +4ˆν ϕ2 ωˆϕ2 1 The enhancement is maximum in the resonance case ωˆϕ = ωωϕ = 2
→∞
ω =
√
1−2ˆ ν ϕ
ωϕ
√
→
1−2ˆ ν ϕ
. Strictly speaking the resonance frequency ω =
ωϕ 2 1−2ˆ ν ϕ
√
⇒
is not
equal to the eigenfrequency ωϕ , i.e. ω > ωϕ . Since νˆ ϕ is small, this discrepancy can be neglected ω ωϕ . Typical dependence of the ratio referred frequency is presented in Fig. 3.5.
≈
(r)
ϕA αA
on the
Figure 3.5: Response function versus referred frequency Typical dependence of the phase displacement on the referred frequency is presented in Fig. 3.6. 46
Figure 3.6: Phase displacement versus referred frequency The phase displacement is equal to π/2 in the resonance case ωˆϕ = 1 for every damping. For ω ˆϕ = ωωϕ 0 the phase displacement disappears. For ω ω ˆ ϕ = ωϕ the phase displacement tends to π. The largest relative roll angle occurs in the resonance case either at wave crests or at wave troughs (Fig. 3.7). Indeed, the magnitude of the relative roll angle in the (r) (r) resonance case ϕ(r) = ϕA cos ωt π2 = ϕA sin ωt attains the maximum (r) ϕA at sin ωt = 1. It corresponds to wave crest ζ 0 = A sin ωt = A and wave trough ζ 0 = A.
→∞
−
±
→
−
Figure 3.7: Ship oscillations in resonance case At very large metacentric height GM γ
→ ∞ the eigenfrequency is also get-
0 GM γ ting large ωϕ = ρgI x∇x+A . The referred frequency for a limited wave 44 frequency ω tends to zero ωˆϕ = ωωϕ 0. The relative roll angle amplitude and phase displacement are zero. The floating body moves together with the free surface as shown in Fig. 3.8 like a raft.
∼∞
→∞
→
Similar results are obtained from analysis of the heave oscillations formu47
Figure 3.8: Oscillation of a raft with a big metacentric height lae (3.29) and (3.30).
3.8
Sway ship oscillations in beam sea
The equation describing the sway oscillations is (see the second equation in the system (1.32)): m¨ η=
−A η¨ − B 22
22 η˙
+ F η,per (t)
(3.34)
The wave exciting force F η,per (t) consists of two components of hydrostatic and hydrodynamics nature. As seen from Fig. 3.9 the hydrostatic force is F ηhyd =
2
2
−ρg∇ α = −ρ∇ ω A cos ωt = −mω A cos ωt 0
0
(3.35)
Figure 3.9: Illustration of the frame in beam waves The horizontal oscillations of the wave surface can be presented in harmonic 48
form (see formula (6.23) in [6]): η˙0 (y, z ) = Aωe kz cos(ky ωt), η˙0 (0, 0) = Aω cos(ωt) η0 (0, 0) = A sin(ωt) η¨0 = Aω 2 cos ωt
− ⇒
−
(3.36)
The hydrodynamic component of the wave induced force is written in the similar form as (3.16): F η2 = A22 η¨0 + B22 η˙ 0 (3.37) Substitution of (3.37) and (3.35) into (3.34) gives: m¨ η=
−A η¨ − B 22
22 η˙
+ m¨ η0 + A22 η¨0 + B22 η˙ 0
(3.38)
or (m + A22 )(¨ η
− η¨ ) + B
22 (η˙
0
− η˙ ) = 0 0
(3.39)
The solution of the equation is written in the form: (η
− η ) = Ce 0
λt
(3.40)
which substitution into (3.39) allows one to find λ (3.41) − m B+ A The parameter λ is positive. Therefore, (η − η ) = Ce −−−→ 0 ⇒ η −−−→ η . 22
λ=
22
0
λt
t→∞
t→∞
0
As soon the transitional process is finished, the ship oscillates together with the wave η = η0 = A sin ωt (3.42)
3.9
Ship oscillations at finite beam to wave length ratio and draught to length ratio
The analysis presented above was carried out for the case of a very long wave, i.e. both the beam to length ratio B/L and the draught to length ratio T /L are small. The results for roll oscillation obtained for the case B/L 0, T /L 0 are extended to the case B/L 0(1), T /L 0(1) using reduction coefficients. According to this traditional in shipbuilding approach the wave amplitude is multiplied with the reduction coefficient κ, i.e.
≈
∼
∼
Ared = κA The ship oscillations at B/L 0(1), T /L B/L 0, T /L 0 due to two reasons
≈
≈
∼
49
≈
(3.43)
∼ 0(1) are smaller than these at
•
•
Hydrostatic force is smaller because the submerged volume is smaller due to wave surface curvature, Hydrodynamic force is smaller because the velocities caused by the orbital motion are not constant as assumed above. They decay with the increasing submergence as exp( kz ).
∼
−
The first reduction factor is mainly due to the finite beam to length ratio B/L 0(1).
∼
Let us consider first the reduction coefficient for the heave oscillations. The factor κBζ considers the reduction of the hydrostatic force due to the finite beam to length ratio. To estimate κBζ the fixed ship is considered at the time instant ωt = π/2 when the wave crest is in the symmetry plane (Fig. 3.10).
Figure 3.10: The free surface ordinate ζ 0 = A sin
π + kη = A cos kη 2
The hydrostatic force obtained in the previous analysis is R0 = ρgAAwp 50
(3.44)
whereas the actual one is calculated by the integral:
Rtrue = ρgA
L/2 B(ξ)/2
cos kηdξdη = 2ρgA
Awp
cos kηdηdξ =
0
−L/2
(3.45)
L/2
=
2ρgA k
sin
kB(ξ ) dξ 2
−L/2
Using the Taylor expansion for sin kB(ξ ) kB(ξ ) kB sin = 2 2
−
(kB)3 + ... 48
the final formula for Rtrue takes the form: L/2
2ρgA Rtrue = k
L/2
kB(ξ ) sin dξ 2
≈
2ρgA k
−L/2
= ρgAAwp
−
− − (kB)3 48
kB/2
dξ =
−L/2
L/2
ρgAk 2 24
3
B dξ = ρgAAwp 1
−L/2
where
k2 I 2 Awp
L/2
Awp =
(3.46)
,
L/2
1 I = 12
Bdξ,
−L/2
B 3 dξ,
−L/2
The reduction of the hydrostatic force can be taken by the following coefficient into account: κBζ =
−
ρgAAwp 1
k2 I 2 Awp
ρgAAwp
=1
−
k2 I 2 Awp
(3.47)
The second reduction factor is mainly due to the finite draught to length ratio T /L 0(1). The factor κT ζ considers the reduction of the hydrodynamic force due to the finite draught to length ratio. The reduction coefficient is given here without derivation:
∼
κT ζ = 1
−
−
T χ T χ 2π + 2π L 2(2 χ) L
−
51
2
χ 6(3
−
T 2π 2χ) L
3
(3.48)
where χ is the coefficient of the lateral area χ = ALA/(LT ). LT ). The total reduction coefficient κζ is calculated as the product of κ of κBζ and κT ζ neglecting their mutual influence: κζ = κBζ κT ζ
(3.49)
L The formula (3.49) is valid at BL > 4, T > 8. For heave calculations one can use the formula (3.29) with Aκζ instead of A.
Reduction coefficient of the roll oscillations can be calculated from the expression gained from regression of experimental data: κϕ = exp
−
4.2 (Rω ˆ ϕ )2 ,
R = χωϕ
BT χrγ /GM γ γ 2πg
1/2
(3.50)
Here rγ is the metacen metacentric tric radius. radius. Amplit Amplitude ude of roll roll oscilla oscillation tionss is found from (3.31) with αAκϕ instead of αA . A sample of the reductio reduction n coefficien coefficientt for a real ship is presented in Fig. 3.11.
Figure 3.11: Reduction coefficient of the heave oscillations
3.10 3.10
Effec Effectt of shi ship p speed speed on on roll rollin ing g
In the previous chapters the ship speed was assumed to be zero. If the ship moves in waves with speed v in x-direction the following modifications should be made in theory 52
Figure 3.12: Sea classification •
the added mass Aij and damping coefficients Bij depend on the encounter counter frequency frequency v cos ϕwave ωe = ω ω 2 g
−
where χ is encounter angle (Fig. 3.12) •
•
reduction coefficient coefficient x (3.43) depends on ω In the hydrodynamic hydrodynamic theory one uses time derivative derivative in ship coordinate ∂ϕ system ∂t which can be expressed through the time derivative in the ∂ϕ inertial reference system ∂t as 0 ∂ϕ ∂ϕ = ∂t 0 ∂t
− v∇ϕ
The formulae (3.27 - 3.30) can be used also in the case v = 0 with the substitution ωe instead of ω of ω .
53
54
Chapter 4 Ship oscillations in small head waves 4.1
Exciting forces and ship oscillations
Let us consider the ship oscillations in small head waves coming from the stern (ψwave = 0◦ ), where ψwave is the wave course angle. The wave ordinate, wave orbital motion velocity and acceleration are:
− kξ ) ζ ˙ = ωA cos(ωt − kξ ) ¨ = −ω A sin(ωt − kξ ) ζ 0
ζ 0 = A sin(ωt
(4.1)
0
(4.2)
2
(4.3)
Figure 4.1: Illustration of the ship in head waves Within the linear theory of ship oscillation the ship is considered at rest. The perturbation force acting on the section AB (Fig. 4.1) can be represented as the sum of •
the hydrostatic Froude Krylov force ρgB(ξ )ζ 0 (ξ ),
•
¨0 (ξ ) + B33 ζ ˙0 (ξ ) the hydrodynamic force A33 ζ 55
dF ζ,per ¨0 (ξ ) + B33 ζ ˙0 (ξ ) + ρgB(ξ )ζ 0 (ξ ) = A33 ζ dξ
(4.4)
Here we used the principle of relative motion (see section 3.4) for a ship dF ζ,per frame. Integrating dξ over the ship length we obtain the whole wave induced force F ζ,per . If the ship is symmetric with respect to the midship B(ξ ) = B( ξ ), Af (ξ ) = Af ( ξ ), A33 (ξ ) = A33 ( ξ ), B33 (ξ ) = B33 ( ξ ), the terms with sin(kξ ) are neglected and the formula for F ζ,per is simplified to:
−
F ζ,per = +A
−
2
−ω A
−
A33 cos kξdξ sin ωt + ωA
·
L
ρgB(ξ ) cos kξdξ sin ωt = A
·
L
+ ωA
B33 cos kξdξ cos ωt+
·
L
ρgB(ξ )
L
B33 cos kξdξ cos ωt = F ζ,per sin(ωt
·
L
−
2
−ω A
33
cos kξdξ sin ωt+
·
ζ,per )
− δ
(4.5) where
− − −
F ζ,per = A
ω 2A33 )cos kξdξ
(ρgB(ξ )
ω
B33 cos kξdξ
(ρgB(ξ )
+ ω2
2
B33 cos kξdξ
L
L
arc tan
2
L
δ ζ,per =
ω2 A33 )cos kξdξ
L
;
(4.6)
The wave exciting moment is calculated by multiplication of arm ξ :
− −
,
dF ζ,per dξ
with the
dF E,ζ ξ dξ dξ
M ψ,per =
L
=A
ρgB(ξ )
L
2
−ω A
33
ξ sin kξ cos ωt
·
−
B33 ξ sin kξ sin ωt = M ψ,per sin(ωt
ωA
L
·
56
(4.7) ψ,per )
− δ
where
M ψ,per = A
2
(ρgB(ξ )
L
δ ψ,per =
2
−ω A
B33 ξ sin kξdξ
L
− (ρgB(ξ )
−π + arc tan
+ ω2
33 ) ξ sin kξdξ
2
ω2 A33) ξ sin kξdξ
L
ω
;
B33 ξ sin kξdξ
L
,
(4.8)
Substitution (4.5) and (4.7) in the third and sixth equations of the system (1.3.2) gives: ¨= mζ
I yy
−A ζ ¨ − B ζ ˙ − ρgA ζ + F ψ¨ = −A ψ¨ − B ψ˙ − ρg ∇ GM ψ + M ⇓ 33
33
55
55
0
− δ sin(ωt − δ
sin(ωt
W P
ζ,per
L
ψ,per
∇ GM ψ = M 0
L
ψ,per
(4.9)
ψ,per )
¨ B33 ζ + ˙ ρgA W P ζ = F ζ,per sin(ωt (m + A33 ) ζ + (I yy + A55 ) ψ¨ + B55 ψ˙ + ρg
ζ,per )
(4.10)
− δ ) sin(ωt − δ
(4.11)
ζ,per
ψ,per )
(4.12)
Dividing both equations by the coefficient of the first term one obtains: ¨ + 2ν ζ ζ ˙ + ω 2ζ = f ζ sin(ωt δ ζ,per ) ζ ζ ψ¨ + 2ν ψ ψ˙ + ωψ2 ψ = f ψ sin(ωt δ ψ,per )
−
(4.13)
−
where f ζ =
F ζ,per , m + A33
f ψ =
M ψ,per , I yy + A55
ωζ =
ν ζ =
ρgA W P , m + A33
B33 , 2(m + A33 )
ωψ =
ν ψ =
B55 , 2(I yy + A55 )
∇
ρg 0 GM L . I yy + A55
Solution of (4.13) is seeking in the form
ζ = ζ A sin ωt
per ζ
− δ − δ ζ,per
,
ψ = ψA sin ωt
− δ
ψ,per
per ψ
− δ
.
(4.14)
After some simple manipulations the amplitudes of the heave and pitch oscillations as well as the phase displacements are obtained from (4.13) and (4.14): ζ A =
− (ωζ 2
f ζ ω 2 )2
+ 4ν ζ 2 ω2
,
δ per ζ = arc tan 57
2ν ζ ω ωζ 2 ω2
−
(4.15)
ψA =
f ψ
(ωψ2
−
ω 2 )2
+ 4ν ψ2 ω2
,
δ per ψ = arc tan
2ν ψ ω ωψ2 ω2
−
(4.16)
In the resonance case the phase displacement is equal to π/2, i.e. per δ per ζ = π/2 in case ωζ = ω and δ ψ = π/2 in case ωψ = ω.
4.2
Estimations of slamming and deck flooding
Results of ship oscillations obtained in the previous section can be used for practically useful estimations. For instance, we can estimate the slamming and deck flooding. Using the relations derived above ζ 0 = A sin(ωt
− kξ ), ζ = ζ
A
sin(ωt
per ζ ), ψ
− δ − δ ζ,per
= ψA sin(ωt
per ψ )
− δ − δ ψ,per
(4.17)
one can display the ship positions in head waves as shown in Fig. 4.2
Figure 4.2: Position of ship at different time instants in a head wave Let us represent the formulae (4.17) in the form: ζ 0 = A sin ωt cos kξ
ζ = ζ A sin ωt cos δ ζ,per + δ per ζ ψ = ψA sin ωt cos δ ψ,per + δ per ψ The local change of the draft is: z (x) = ζ 0
−− −
A cos ωt sin kξ
ζ A cos ωt sin δ ζ,per + δ per ζ
(4.18)
ψA cos ωt sin δ ψ,per + δ per . ψ
− ξ + xψ = f (x)cos ωt + f (x)sin ωt 1
2
(4.19)
where f 1 (x) = A cos kξ + ζ A sin(δ ζ,per + δ per ζ ) f 2 (x) = A sin kξ
− xψ
A sin(δ ψ,per
+ δ per ψ )
per per A cos(δ ζ,per + δ ζ ) + xψA cos(δ ψ,per + δ ψ )
− ζ
58
(4.20)
Figure 4.3: Curves y =
±z
max
and y = z (x)
A sample of the curve y = z (x) is shown in Fig. 4.3. The maximum draft is then z max(x) = f 12 (x) + f 22 (x) (4.21)
The curve y = +z max (x) shows the contour of maximum wave elevations along the ship board in the symmetry plane whereas the curve y = z max (x) the minimum wave elevations. Both curves are symmetric with respect to the equilibrium water plane.
−
•
Deck flooding takes place if z max (x) > H , where H is the board height.
•
Slamming takes place if z max (x) > T
There three zones limited by curves y = the ship board (see Fig. 4.3): •
Allways dry area (white),
•
Allways wetted area (red),
•
Intermediate area (orange).
±z
max (x)
can be distinguished along
A sample of flooding curves for a real ship is given in Fig. 4.4
59
Figure 4.4: Sample for a real ship
60
Chapter 5 Seasickness caused by ship oscillations Symptoms of the seasickness are giddiness (Schwindelgef¨ uhl), headache (Kopf¨ schmerz), sickness (Ubelkeit) and vomiting (Erbrechen). The seasickness is the reason of work capacity reduction, memory decline (R¨ uckgang der Ged¨achtnisleistung), motion coordination (Bewegungskoordinierung), reduction of muscular strength, etc. Diagram of Sain Denice (Fig. 5.1) shows the influence of the vertical acceleration on the seasickness depending on the oscillation period. For the irregular sea state the similar diagram was proposed by Krappinger (Fig. 5.2) who estimated the percentage of people suffering from the seasickness depending on the root mean square deviation and frequency. According to standards developed in US Navy the oscillations have no significant effect on the work capacity if the amplitude of the roll oscillations is under eight degrees, the amplitude of pitch oscillation is below three degrees, the vertical accelerations does not exceed 0.4 g whereas the transversal accelerations 0.2 g. The upper limit of the roll angle for the deck works is 20 degrees which corresponds to the reduction of the work capacity of about 50 percent. At present the seasickness has insufficiently been studied in medical science. As shown in the study by Vosser, the seasickness is developed at a certain level of overloads and then can remain even the ship oscillations decay. It is shown in diagram 5.3 presenting the number of passenger on a cruise liner nδ /n suffering from the seasickness during eighty hours of the journey. ¨ was less than 0.1 and At the journey beginning the vertical acceleration ζ/g only 16 percent of passenger were sick. As soon as the vertical acceleration 61
Figure 5.1: Influence of the vertical acceleration on the seasickness depending on the oscillation period attained 0.4g more than 80 percent of passengers were sick. In spite of the ship oscillation decay after 30 hours of the way the number of sick passengers is not reduced. On the contrary this number is slightly increased during the next 24 hours. Only after 36 hours the seasickness retreated. The next diagram 5.4 illustrates the fact that the adaption to seasickness is relatively weak. Diathesis to seasickness depends on the individual properties of organisms. There are many people who had never had problems with seasickness. However there are experienced seamen who suffers from this sickness the whole professional life.
62
Figure 5.2: Influence of the vertical acceleration on the seasickness depending on the oscillation period
Figure 5.3: Number of passengers suffering from seasickness on a cruise liners depending on vertical accelerations
63
Figure 5.4: Adaption to seasickness
64
Chapter 6 Ship oscillations in irregular waves 6.1
Representation of irregular waves
The irregular waves can be both two dimensional and three dimensional (Fig. 6.1).
Figure 6.1: Irregular seawaves, 1- two dimensional, 2- three dimensional. (Fig. from [3]) A feature of the irregular waves distinguishing them from regular ones is the non-recurrence of their form in time (Fig. 6.2). The following relations between wave lengths L and wave heights h are recommended in practical calculations for swell: Within the linear theory the irregular waves can be represented as the superposition of regular waves with different amplitudes, frequencies and course angles, as shown in Fig. 6.3. 65
Figure 6.2: Profile of an irregular wave. (Fig. from [3]) h = 0.17L 17L3/4 h = 0.607L 607L1/2 h = 0.45L 45L0.6
6.1.1
Zimmermann, British Lloyd, det Norske Veritas.
Wave ave ordinate ordinatess as stochastic stochastic quantitie quantitiess
The wave ordinate is the stochastic function with a certain probability density functio function n (see Fig. 6.1). The p.d.f. p.d.f. distrib distributio ution n of the real irregula irregularr wave wave ordinates is Gaussian. i.e, p.d.f. =
1 2 (2σ 2 ) e−(ζ −ζ 0 ) /(2σ , 2πDζ
(6.1)
where ζ 0 is the mathematical expectation (in our case ζ 0 = 0), σ is the standard deviation: deviation: σ2 = (ζ ζ 0 )2 = Dζ (6.2)
−
Dζ is the dispersio dispersion. n. Probabil Probabilit ity y P ( P (ζ 1 < ζ < ζ2 ) of the event, that the ordinate lies in the range between ζ 1 and ζ 2 is then ζ 2
P ( P (ζ 1 < ζ < ζ 2 ) =
p.d.f.(ζ p.d.f.(ζ )dζ =
ζ 1
√
ζ 2 /
ζ 2
1 2πD ζ
2
2
(2σ ) e−ζ /(2σ dζ =
ζ 1
√ 1π
2Dζ
2
e−t dt
√
ζ 1 /
2Dζ
(6.3)
The last integral is known as the probability integral x
√
2 ϕ(x) = π
0
66
2
e−t dt
(6.4)
Figure Figure 6.3: Represen Representati tation on of irregul irregular ar wa wave ve through the superposi superposition tion of regular waves. (Fig. from [3]) satisfying the following properties ϕ( x) =
−
−ϕ(x),
ϕ(
−∞) = −1,
ϕ( ) = 1
∞
(6.5)
Using the probability integral, the probability P ( P (ζ 1 < ζ < ζ 2 ) takes the form
− ∞ −
P ( P (ζ 1 < ζ < ζ 2 ) =
ζ 2 2Dζ
1 ϕ 2
ϕ
ζ 1 2Dζ
(6.6)
The probability P ( P ( < ζ < ζ 2 ) = P ( P (ζ < ζ 2) is the probability of the event that ζ does not exceed exceed ζ 2:
−∞
P ( P (ζ < ζ 2 ) = The probability P ( P (ζ 1 < ζ < that ζ larger than ζ 1 :
1 1+ϕ 2
ζ 2 2Dζ
(6.7)
) = P ( P (ζ 1 < ζ is the probability of the event
P ( P (ζ 1 < ζ ) =
1 1 2
67
ϕ
ζ 1 2Dζ
(6.8)
Figure 6.4: p.d.f. of the wave ordinate In the probab probabil ilit ity y theory theory is show shown n that that the p.d.f. p.d.f. of the the ampl amplit itude ude of a stochastic quantity having the Gaussian p.d.f. distribution satisfies the Raleigh law: ζ a −ζ a2 /(2D (2Dζ ) p.d.f.(ζ p.d.f.(ζ a ) = e (6.9) Dζ The probability that the amplitude is larger than ζ ∗ is ∞
P ( P (ζ a > ζ ∗ ) =
ζ
ζ a −ζ a2 /(2D 2 (2D ) (2Dζ ) ζ e dζ a = e−ζ /(2D Dζ ∗
(6.10)
∗
When evaluating the wave height an observer determines the middle height of one third of the highest waves. This height is referred to as the significant wave height and designated as h1/3 . Dependence between the dispersion and the significant wave is Dζ = 0.063h21/3
6.1. 6.1.2 2
(6.11)
Wave spect spectra ra
Irregular waves are considered as the superposition of infinite number of regular waves of different frequencies, amplitudes and course angles (Fig. 6.3). 68
According to this concept the wave elevation ζ (x,y,t) is represented in form of Fourier - Stieltjes integral: ζ (x,y,t) = Real
dA(ω, χ)exp[ ik(x cos χ + y sin χ) + iωt + δ (ω, χ)]
−
(6.12) Here ω is the wave frequency, k is the wave number k = ω /g, χ is the wave course angle and δ (ω, χ) is the phase angle. The quantity dA(ω,χ,t) is the function of the amplitude corresponding to the wave propagating at the course angle χ < χ < χ + ∆χ with the frequency ω < ω < ω + ∆ω. The mean square elevation is obtained from time averaging the quadrat of the elevation: 2
1 ζ 2 (x, y) = limT →∞ T
T
ζ 2 (x,y,t)dt =
0
=
dA(ω, χ)exp[ ik(x cos χ + y sin χ) + iωt + δ (ω, χ)]
−ik (x cos χ 1
−
1
+ y sin χ1 )
− iω t − 1
1 δ (ω, χ)] = 2
dA∗(ω1 , χ1 )exp [
−
dA(ω, χ)dA∗ (ω, χ) (6.13)
Here the superscript stands for the complex conjugate amplitude function. Rigorous derivation of the formula (6.13) can be found in [10]. Multiplying ζ 2 (x, y) with ρg
∗
ρg ρgζ 2 (x, y) = 2
dA(ω, χ)dA∗ (ω, χ)
(6.14)
and comparing the result with the expression for the energy (6.34) derived in [6] ρgA 2 E = T F l + E p = L 1m 2
×
One can conclude that ρgζ 2 (x, y) is the time averaged energy per surface unit. Using the representation of the integral 2π ∞
dA(ω, χ)dA∗ (ω, χ) = 2
S ζ (ω, χ)dωdχ
0
(6.15)
0
we introduce the spectral density of the irregular waves S ζ (ω, χ) which is the contribution of the wave with the frequency ω < ω < ω + ∆ω and the 69
course angle χ < χ < χ + ∆χ to the irregular wave energy. Commonly the function S ζ (ω, χ) is called shortly the wave spectrum. At present there is no much information on the energy distribution both on the frequency and the course angle. The typical measurements with buoy do not provide information about the dependency of wave elevations on the course angle. In the ship theory is assumed that the irregular waves have a preferential propagation direction and the wave have long wave crest. The waves are approximately two dimensional. Such rough sea can fully be characterized by the frequency spectrum S ζ (ω) defining as 2π
S ζ (ω) =
S ζ (ω, χ)dχ
(6.16)
0
The spectrum of the wave state S ζ (ω) shows the distribution of the wave energy on frequencies. The two dimensional spectrum S ζ (ω, χ) can be restored from the one dimensional one S ζ (ω) using the following simple approximation: 4 S ζ (ω, χ) = S ζ (ω)cos4 χ 3π To determine the spectrum, the wave ordinates are measured and represented in Fourier series. The energy ∆E (ω < ω < ω + ∆ω) is calculated as the squared wave ordinate for each interval of the frequencies ∆ω. The spectral density of waves is calculated as ∆E (ω < ω < ω + ∆ω) ∆ω→∞ ∆ω
S ζ (ω) = lim
(6.17)
From the probability theory: ∞
Dζ =
S ζ (ω)dω
(6.18)
0
One of the most popular wave spectral densities is the spectrum of Pierson and Moskowitz (PM): αg 2 S ζ (ω) = 5 exp ω
− g β Uω
4
,
(6.19)
where α = 0.0081, β = 0.74, U is the wind velocity at the height of 19.4 m over the free surface. The spectrum (6.19) has been obtained by approximation of data measured in 1964 in the North Atlantic region. Fig. 6.5 70
Figure 6.5: illustrates the PM spectra depending on the wind velocity U. The mean wave height is ∞
¯h = 2
∞
ζ a p.d.f.(ζ a )dζ a = 2
0
ζ a
0
ζ a −ζ a2 /(2Dζ ) e dζ a = (2πDζ )1/2 Dζ
(6.20)
The significant wave height is: ∞
2 h1/3 =
ζ ap.d.f.(ζ a )dζ a
ζ 1
(6.21)
∞
p.d.f.(ζ a )dζ a
ζ 1
where the amplitude ζ 1 is chosen from the condition ∞
p.d.f.(ζ a)dζ a = 1/3
ζ 1
−ζ 12 /(2Dζ )
⇒e
= 1/3
⇒ ζ = 1
2Dζ ln3
(6.22)
It follows from (6.9) and (6.21):
h1/3 = 4 Dζ 71
(6.23)
The middle frequency is defined as
∞
2
ω S ζ (ω)dω
ω ¯=
0
∞
S ζ (ω)dω
0
1/2
∞
2
ω S ζ (ω)dω
=
0
Dζ (ω)
1/2
(6.24)
Substitution of (6.19) into (6.18), (6.23) and (6.24) results in h1/3
6.2
U 2 α =2 g β
1/2
,
ω ¯ = (πβ )1/4 (g/U )
(6.25)
Calculation of ship oscillations in irregular waves
Using the assumption of small waves we can substitute the superposition of regular waves into equations (3.20) and (3.21) describing the heave and roll oscillations. Since the equations are linear the responses of the ship to each regular wave can be calculated separately. In this case one can obtain the history of oscillations in time. However, from point of view of practical applications only the statistical parameters of oscillations are of importance. To determine them, the ship is considered as the dynamic system. The seaway is the input which is transformed by the ship into oscillations considered as the output. In the statistical theory shown, that if the input signal has the Gaussian p.d.f. distribution the output signal has also the Gaussian p.d.f. distribution. With the other words, the ship oscillation parameters (roll angle, etc) obey the normal Gaussian law whereas the amplitudes of oscillation parameters satisfy the Raleigh law. The only unknown value in these distributions laws is the dispersion D. Let us consider the roll oscillations of a ship with the zero forward speed. As shown in the previous lectures the ratio of the roll oscillations amplitude (output signal) to the wave slope amplitude (input signal) is given by the formula (r)
ϕA ϕA αA = = αA αA
−
=
ω ˆ 2 /(1 + kϕ ) ϕA = αA (1 ω ˆ 2 )2 + 4ˆ ν 2 ω ˆ2 ω ˆ 2 /(1 + kϕ ) + 1 = Φ(ω) (1 ω ˆ 2 )2 + 4ˆ ν 2 ω ˆ2
− −
72
⇒
where
ω ˆ 2 /(1 + kϕ ) +1 (6.26) 2 2 2 2 (1 ω ˆ ) + 4ˆ ν ω ˆ is the so called response function. Since the wave spectral density is proportional to the wave ordinates squared and taking the superposition principle into account, we obtain the following relation between the spectral density of the seaway and the spectral density of oscillations Φ(ω) =
−
S ϕ (ω) = Φ2 (ω)S ζ (ω) or S ϕ (ω) = S ζ (ω)
(6.27)
2
ω ˆ /(1 + kϕ ) +1 (1 ω ˆ 2 )2 + 4ˆ ν 2ω ˆ2
−
2
(6.28)
Dispersion of the roll oscillations and the standard deviation are found from the definitions (6.2) and (6.18) ∞
Dϕ =
∞
S ϕ (ω)dω =
0
Φ2 (ω)S ζ (ω)dω,
σϕ =
0
Dϕ
(6.29)
Similar formulae can be obtained for angular roll velocity and acceleration ∞
Dϕ˜ =
2
ω S ϕ (ω)dω,
σϕ˜ =
ω 4 S ϕ (ω)dω,
σϕ˜ =
0
∞
Dϕ˜ =
0
Dϕ˜
(6.30)
Dϕ˜
(6.31)
The dispersions obtained from (6.29), (6.30) and (6.31) determine fully the irregular ship oscillations in heavy seaway. Using them the following further parameters can be calculated •
Most probable amplitude of oscillations corresponding to the maximum of the p.d.f.(ζ a ) distribution ϕm = σϕ
(6.32)
The probability that the roll amplitude exceeds ϕm is 60.6 %. •
Averaged amplitude of roll oscillations (mathematical expectation) ϕ¯ =
π σϕ 2
≈ 1.25σ
ϕ
The probability that the roll amplitude exceeds ϕ¯ is 45.6 %. 73
(6.33)
•
The probability that the roll amplitude exceeds the value ϕ∗ : p(ϕA > ϕ∗) = e−0.5(ϕ
∗
•
(6.34)
Averaged frequency and averaged period of oscillations: ω ¯ϕ =
•
/σϕ )2
σϕ˜ , σϕ
¯ϕ = 2π = 2π σϕ T ω ¯ϕ σϕ˜
(6.35)
Number of ship inclinations (semi periods) within the time interval t: 2t N t = ¯ T ϕ
•
(6.36)
Number of ship inclinations within the time interval t provided the roll angle amplitude is larger than ϕ∗ : 2t N ϕ = N t P (ϕA > ϕ∗) = ¯ e−0.5(ϕ T ϕ
∗
∗
/σϕ )2
(6.37)
The formulae (6.32) - 6.37) are derived under assumption that the oscillations obey the Gaussian distribution law.
74
Chapter 7 Experimental methods in ship seakeeping 7.1
Experiments with models
The seakeeping experiments are performed under condition that the Froude number of the model F nm and the large scale ship F ns are equal: F nm = F ns
⇒
V m
=
gD m 1/3
V s
gD s 1/3
Since the periodic motion are considered, the similarity of Strouhal numbers should also be satisfied: Sh m = Sh s
⇒ ω V L m
m
=
m
ωs Ls , V s
Where ω is the frequency. Unfortunately the similarity with respect to Reynolds (viscosity effects) and Weber (spray effects) numbers are not fulfilled: Rem = Res ,
W em = W es .
It is recommended to choose the model length from the condition Re If λ is the scale factor: Ls = λLm
The following relations derived from the similarity conditions are valid 75
6
≥ 10 .
ϕs = λ0 ϕm ¨s = λ0 ξ ¨m ξ ξ ˙s = λ1/2 ξ ˙m τ s = λ1/2 τ m ϕ˙ s = λ−1/2 ϕ˙ m ωs = λ−1/2 ωm ϕ¨s = λ−1 ϕ¨m F s = λ3 F m I s = λ5 I m
angles linear accelerations linear velocities periods angular velocities frequencies angular acceleration displacement, mass, forces inertia moments
(7.1)
The following parameters of the model are to be determined before experiment: •
weight G
•
positions of the center of gravity xg , z g
•
metacentric heights GM γ , GM l
•
inertia moments I xx , I yy , I zz
They should satisfy the similarity conditions: Gs = λ3 GM (xgs , z gs ) = (xgm , z gm ) λ GM γ,ls = λGM γ,lm I xxs = λ5 I xxm , I yys = λ5 I yym I zzs = λ5 I zzm .
·
Since the draught and beam are approximately equal B T one assumes that I yy I zz . For the determination of I zz the model is hanged out as shown in Fig. 7.1. The model is oscillating in horizontal plane about the vertical axis as shown in Fig. 7.1. The period of oscillation τ is measured. The inertia moment is calculated then from formula:
∼
≈
Gm a2 τ 2 I zz = I yy = 4τ 2 l The additional loads are placed on the ship and they are shifted along the x-axis as long as the condition 76
Figure 7.1: Determination oft he inertia moment I zz
I zzs = λ5 I zzm Is fulfilled. The inertia moment I xx and the center of gravity z g is determined using the setup shown in Fig. 7.2. To determine z g only one load is used, which causes the model heeling. If ϕ is the heel angle, the distance a calculated from moment equilibrium equation: a Gm ϕ = P l
·
·
⇒ a = GP lϕ m
The gravity center ordinate is then: z g = z n
−a
The inertia moment I xx is determined when the model is forced to roll with the period τ of free roll oscillations. The moment I xx is then calculated as I xx = Gm ((
τ 2 ) 2π
− ag )
The similarity conditions I xxs = λ5 I xxm and zg s = λz gm 77
Figure 7.2: Determination of I xx and z g are fulfilled by vertical and horizontal shifts of loads P . The metacentric height GM γ is determined from heel tests (see Fig. 7.3). GM γm =
Pl Gm ϕ
The forces arising in roll oscillations are found from free roll oscillations without waves. The model is brought from the equilibrium state and experiences free decaying oscillations ϕ = ϕ0 e−νϕ cos ω1 t,
−
where ω1 = ωϕ2 ν ϕ2 . Since ν ϕ2 ωϕ2 the frequencies ω1 and ωϕ are approximately equal, i.e. ω1 ωϕ . 2π Once the period of free oscillations T ϕ is measured, the frequency ωϕ = T ϕ and the sum I xx + A44 are calculated (see table 2.1):
≈
I xx + A44 =
g
0m GM γm ωϕ2
∇
(7.2)
Since the inertia moment is known, the added mass A44 is found from (7.2). 78
Figure 7.3: Heel test
A44 =
g
0m GM γm ωϕ2
∇
− I
xx
The damping factor ν ϕ is determined from the definition formula (2.16). ln
ϕ(t) 2πν ϕ = ϕ(t + T ϕ ) ω ¯
≈ ν T
ϕ ϕ
where the ϕ(t) and ϕ(t + T ϕ ) are measured. The damping coefficient B44 is also calculated from its definition B44 = 2ν ϕ (I xx + A44 ). The results are represented depending on the magnitude of ϕ. Since the measurements are performed only at ωϕ , the results at ω = ωϕ are not obtained in these measurements. For ships with large damping the free oscillations decay very quickly. It causes big error in data analysis. For this case the method of forced oscillations is applied. The model is forced to roll using horizontally oscillating load as shown in Fig. 7.4. The load produce the perturbation moment
M per = M 0 sin ωt The roll oscillations without incident waves are described by the ordinary differential equation of the second order: 79
Figure 7.4: Method of forced rolling
ϕ¨ + 2ν ϕ ϕ˙ + ωϕ2 ϕ = ωϕ2
M 0 sin ωt GGM γ
which has the solution ϕ = ϕ0 sin(ωt
−ε ) ϕ
where ϕ0 =
M 0 GGM γ
ν¯ ϕ =
1
−
ν ϕ
− ωϕ2
(1
ω ¯ 2 )2
¯= ω
, ν ϕ2
εϕ = a tan
(7.3)
+ 4¯ ν ϕ2 ω ¯2 ω ωϕ
2¯ ν ϕ 1 ω ¯2
(7.4)
−
The measured quantities are M per and roll angle ϕ. The damping factor is calculated from (7.3) at different frequencies ω . The method of forced roll oscillations is very accurate only at ω ωϕ , i. e. in the resonance case.
≈
For reliable determination of inertial and damping forces depending on frequency ω it is necessary to apply more complicated setups than these described above. A description of these setups can be found in [11]. 80
The main purpose of seakeeping measurements in regular waves is the ex(r) (r) perimental determination of dependencies ϕαAA (ω), δ ϕ (ω), ζ AA (ω) and δ ζ (ω) (see formulae (3.27 - 3.30) and Fig. 3.5, 3.6). Knowledge of response functions spectra of ship oscillations.
ϕA (r) (ω) αA
and
ζ A (r) A
allows to calculate the
The seakeeping tests in regular and irregular waves Fig. (7.5) are performed in seakeeping and manoeuvring basin. The seakeeping basins can be open or closed. In the closed basin the irregular waves are generated using segmented wave generators consisting of hinged flaps. Each flap is controlled separately by a driving motor. The seakeeping basin of MARIN (Fig. 7.6) has dimensions 170 x 40 x 5m. The wave generator (Fig. 7.7) produces waves with significant wave height of 0,45 m and a peak period of 2 sec. The irregular waves have a prescribed spectrum. The model is either self propelled (free running test) or carried by the carriage with the speed up to 6m/sec. Model length range is from 2m to 8m. Additionally to waves the wind is generated by an adjustable 10m wide platform with electrical fees. Free running tests are performed such that the model follows an arbitrary pre-defined track through the basin. The seakeeping tests in open basin are performed under condition that the free waves, generated naturally, have desirable heights and periods.
Figure 7.5: Seakeeping test at MARIN ([4]) One of the most important aims of seakeeping tests in irregular waves is the evaluation of slamming (Fig. 7.5) and flooding.
81
Figure 7.6: Scetch of the MARIN Seakeeping basin ([4])
Figure 7.7: Wave generator of MARIN Seakeeping basin ([4])
7.2
Seakeeping tests with large scale ships
The most reliable evaluation of ship seakeeping performances can be gained on the base of tests with large scale ships. The first task within framework of such measurements is the determination of sea state. The wave heights are measured using bues, hydrostatic pressure sensors, or by stereo photography from airplanes (see Fig. 7.8).
Figure 7.8: Method of wave detection The aim of these measurements is the wave spectra S ζ (ω, χ). The ship performs tack motions as long as the one hundred full oscillations 82
occur (Fig. 7.9).
Figure 7.9: Ship motion during large scale tests The time history of all kinematic parameters is documented by different sensors. Fig. 7.9 illustrates the dependence of the roll angle on the time ϕ(t). The time signals of kinematic parameters are evaluated using Fourier analysis. After that the spectra of ship kinematic parameters are calculated. Knowing the spectra, the repones function can be determined from (6.27) as the final aim of the large scale tests: Φ(ω) =
83
S ϕ (ω) S ζ (ω)
84
Chapter 8 Ship oscillation damping (stabilisation) 8.1 8.1.1
Damping of roll oscillations Passive means
U-tube Passive Roll stabilization system
ωϕ eigenfrequency of the ship ωz eigenfrequency of fluid oscillations in tank The Frahm systems ([12]) belong to the resonance type. The duct and air channel ore selected to fulfill the conditions: ωϕ /ωz δ ϕ
≈1
− δ = π2 z
85
(8.1)
Disadvantage: The draught of ship is changed in operations. ωϕ is changed. But ωz remains constant The system becomes non efficient.
→
U-tube stabilization system of Frahm of the second type
Figure 8.1: U-tube stabilization system of Frahm of the second type ωϕ ωz
≈1
Free surface Type passive Roll stabilization systems of Flume
Simplified mathematical model. Equation without stabilizer: A44 ϕ¨ + B44 ϕ˙ + g∆0 GM j ϕ = F ϕϕ A44 - effective mass moment of inertia 86
Figure 8.2: Free surface Type passive Roll stabilization systems of Flume
ωz is changed in a wide range due to change of water level in tank.
Figure 8.3: Free surface Type passive Roll stabilization systems of Flume B44 - damping coefficient F ϕϕ - wave induced forces
Equation with stabilizer: (A44 + mT ϕ )ϕ¨ + (B44 + N T ϕ )ϕ˙ + (BT ϕ + g∆0 GM j )ϕ = F ϕϕ mT ϕ - effective mass moment of inertia of water in stabilizer unit N T ϕ - damping coefficient of stabilizer unit BT ϕ - restoring moment coefficient of stabilizer unit The values mT ϕ , N T ϕ and BT ϕ are related only to the difference between an active stabilizer and a condition where the fluid is replaced by a solid mass corresponding to the frozen liquid. A44 ϕ¨ + B44 ϕ˙ + (tg∆0 GM j )ϕ = F ϕϕ mT ϕ ϕ¨ + N T ϕ ϕ˙ + BT ϕ ϕ = F T ϕ
·
87
− F
Tϕ
Solution: ϕ = ϕA cos ωt F T ϕ = F T ϕA cos(ωt εF T ϕ ) F T ϕA = (BT ϕ mT ϕ ω2 )2 + (N T ϕ ω)2 ϕA N T ϕ ω εF tϕ = α tan BT ϕ mtϕ ω 2
−
−
−
ϕA destabilized ϕA =
N 44
(g∆0 GM j
ϕA stabilized ϕA =
44 ω
2 )2
2 ω2 + B2 ω2 + B44 44
N 44
(g∆0 GM j
8.1.2
−A
−A
44 ω
2
+ BT ϕ
−m
T ϕω
2 )2
Active stabilizer
Figure 8.4: Active stabilizer
8.1.3
Passive Schlingerkiel p =
S k ; LB
r0 =
bk = 0.01...0.02 lk 88
rk B
+ (N T ϕ + B44 )2 ω 2
Figure 8.5: Passive Schlingerkiel
Figure 8.6: Active rudders
8.1.4
Active rudders
8.1.5
Damping of pitch oscillations
Figure 8.7: Damping of pitch oscillations
Q˙ α = k1 + k3 ωQ
•
Q˙ ωQ
3
Utilizes Hoppe/Flume data with data from > 300 vessels seakeeping test and > 2000 tank model test 89
Figure 8.8: Damping of pitch oscillations
Figure 8.9: [5] •
Database allows unique analysis on vessel-tank combination of existing vessels high accuracy for initial tank design consulting
⇒
90
Figure 8.10: [5]
Figure 8.11: [5]
91
Figure 8.12: [5]
Figure 8.13: [5]
92
Figure 8.14: [5]
Figure 8.15: [5]
93
Figure 8.16: [5]
94
Chapter 9 Parametric oscillations Parametric oscillations arise when one of parameters characterizing the oscillating system depends periodically on time. Parametric ship oscillations arise due to periodic change of the metacentric height: GM γ = GM γ 0 + AGM cos ωt
(9.1)
which is in previous chapters assumed to be constant, i.e. GM γ = GM γ 0 . The motion equation of free roll oscillations with variable metacentric height is: I xx ϕ¨ + A44 ϕ¨
−B
˙ 44 ϕ
+ ρg
0 γ +
∇ (GM 0
AGM cos ωt)ϕ = 0
or: ϕ¨ + 2ν ϕ ϕ˙ + ωϕ2 ϕ = µϕ cos ωtϕ, where B44 , 2(I xx + A44 ) AGM ρg 0 µϕ = I xx + A44 ν ϕ =
ωϕ2 =
0 0 GM γ
(9.2)
ρg I xx + A44
∇
∇
Analyzing (9.2) one can state that change of the metacentric height can cause the perturbation moment resulting in parametric oscillations. Physical reason for the appearance of the additional perturbation moment is the effect of the ship submergence change during roll and vertical oscillations. The change of the roll angle causes the moment. 95
ρ
∇ gGM ϕ 0
(9.3)
γ
The change of the ship draught ξ results in an additional moment ρg
∇A 0
GM cos ωtϕ
which is proportional to ζ and ϕ.
The equation (9.2) has no analytic solution in elementary functions. However, properties of its solution are well known. It can be shown that if the frequency of perturbation moment ω wave frequency is twice as large as the free roll oscillation frequency, i.e. ω = 2ωϕ , the parametric resonance takes places. Since the perturbation moment depends on heave ζ , the parametric resonance takes place when the frequency of vertical oscillations is approximately twice as large as the frequency of roll oscillations. With the other words, during semi period of roll oscillations the heave change performs the full period oscillation as shown schematically on Fig. 9.1. The parametric resonance is typical for ships with big distance between the center of gravity and water plane surface (Fig. 9.2). For parametric oscillation it is necessary that ship is brought from the equilibrium state by a certain perturbation. The natural reason of such perturbation for roll oscillations is beam seaway. However, the parametric roll oscillation can arise also in head waves. If the ship has a certain roll angle in head waves, the hydrostatic lift force becomes larger at the wave crest and smaller at wave valley. This results in a perturbation moment depending on roll angle. This kind of parametric oscillations depends on the wave lengths and ship altitude. Periodic change of the metacentric height results in asymmetry of roll oscillations. Due to phase displacement between heave oscillations and waves δ χ the averaged additional moment is different for positive and negative ϕ. In the resonance case of vertical oscillations ωχ ωϕ , the middle roll angle is [9]
≈
2
ϕ0 = κϕ
ω z g T A g GM γ
−
Aκζ 2ν ζ κT
where
zg - the position of the center of gravity with respect to keel, 96
(9.4)
T - ship draught, κ=
v Awp ·T
- are reduction factors (see 3.9)
v - ship volume, Awp - water plane surface.
The larger z g T is, the bigger is the middle roll angle ϕ0 . For beam sea ϕ0 is in the direction of waves.
−
Figure 9.1: Ship oscillations during parametric resonance
Figure 9.2: Conditions for parametric resonance appearance 97
Vertikale Schwingungen und Rollschwingungen ˙ ϕ − B ϕ˙ − g∆ GM − A cos ωt
I xx ϕ¨ = Auu ϕ¨ u = GM j0 GM j
−
uu
0
j
GM
Ursache der parametrischen Rollschwingungen
I xx ϕ¨ = +Auu ϕ¨ + Buu ϕ˙ + g∆0 (GM j 0 + AGM cos ωt)ϕ = 0 ϕ(I ¨ xx + Auu ) + Buu ϕ˙ + g∆0 (GM j 0 + AGM cos ωt)ϕ = 0 ϕ¨ + 2ν ϕ ϕ˙ + ωϕ 2ϕ = µϕ cos ωtϕ ν ϕ = Buu /2(I xx + Auu )ω 2ϕ = g∆0 GM j0 /(I xx + Auu ) keine L¨osung in Elementenfunktionen. Gleichung wurde gr¨ undlich untersucht.
Wichtige Bedingungen f¨ ur parametrische Resonanz:
Parametrische Resonanz Frequenz der vertikalen Schwingungen > 2 Frequenz der Rollschwingungen 98
Q0 = αe
Zg
− T
h
αe = xα0 V X = ST
99
Ax 2ν ζ XT
100
Chapter 10 Principles of Rankine source method for calculation of seakeeping 10.1
Frequency domain simulations
Generally there are two basic methods of simulation of time dependent processes. In the first method called the simulation in frequency domain the unsteady process is considered as the sum of the mean part which is time independent and the periodical part. The mean part can be calculated using nonlinear strategy, i.e. the seeking parameters depends on ship kinematic parameters in non linear manner. The periodic part is considered as small and found from linear theory. This formalism is based on Fourier analysis of unsteady processes. Each unsteady quantity q is represented in form: q = q 0 + qˆeiωt where q 0 q ω
(10.000)
is the mean value, is amplitude and is frequency.
The potential around the ship is represented as the sum of four terms [13] ϕ = ( V x + ϕs ) + (ϕw + ϕI ) where
−
101
(10.1)
−V x V ϕw ϕI ϕs
potential of incident uniform flow, ship speed, potential of incident wave, remaining unsteady potential, potential of the steady flow disturbance.
The terms in the first parenthesis describes the steady flow around the ship with account of free surface effects. The second parenthesis represent the periodic flow due to waves. determination of the potential ( V x + ϕs ) is discussed in the wave resistance potential theory. This problem can be solved utilizing either full nonlinear or linear formalism. Numerical linear method for two dimensional case is described in Chapter 6.7 (Kornev N., Ship Theory I, Manuscript, 2008). The boundary conditions for ϕI are linearized. The following boundary conditions should be satisfied [13]:
−
•
no penetration on the hull,
•
kinematic boundary condition on the free surface,
•
dynamic boundary condition on the free surface,
•
decay of disturbances far away from the ship,
•
radiation condition.
The Laplace equation and decay condition are automatically satisfied within the Rankine source method. The unsteady potential is decomposed into radiation and diffraction components: 6
I
ϕ =
ϕi ui + ϕd
(10.2)
i=1
where ui - ship velocities. For the sake of simplicity we consider fully linear formalism (ϕs and ϕI are small). The kinematic and dynamic boundary conditions are used to derive mixed boundary condition on the free surface. The dynamic boundary condition written in the inertial reference system ist (see page 110 in [6]). ∂ϕ 1 + ∂t 0 2
∂ϕ ∂x
2
+
∂ϕ ∂y
Its linear version is 102
2
+
∂ϕ ∂z
2
+ gz = 0
(10.3)
∂ϕ + gz = 0 (10.4) ∂t 0 The relation between the derivative on time in inertia ∂ϕ/∂t0 and ship fixed coordinate system ∂ϕ/∂t is ∂ϕ ∂ϕ = ∂t 0 ∂t
− V ∇ϕ
(10.5)
is the speed of the reference system substituting (10.5) into (10.4) where V we obtain: ∂ϕ V ϕ + gz = 0, z = ζ (10.6) ∂t The linearized version of the kinematic boundary conditions (see page 111 in [6]) reads:
− ∇
∂ζ ∂ϕ = ∂t 0 ∂z
(10.7)
∂ 2 ϕ ∂ζ +g =0 2 ∂t 0 ∂t 0
(10.8)
Differentiating (10.5) on time
and substituting (10.7) into (10.8) one obtains ∂ 2 ϕ ∂ϕ +g =0 2 ∂t 0 ∂z
(10.9)
using relation ∂ ∂ = ∂t 0 ∂t
− V ∇
we get
− ∇ − ∇ − − ∇
∂ 2 ϕ ∂ = ∂t 20 ∂t 0 ∂ = ∂t ∂ 2 ϕ = 2 ∂t ∂ 2 ϕ = 2 ∂t
∂ϕ ∂ = ∂t 0 ∂t 0 ∂ϕ V ϕ ∂ ∂ϕ V V ∂t ∂ϕ 2V ∂t
∂ϕ ∂t
v ∇
ϕ V
∂ϕ ∂t
=
ϕ = V
∇ ϕ = + V − ∇ − ∇ ∂ϕ ∂t − ∇ 103
2
2
The mixed boundary condition in ship fixed system reads ∂ 2 ϕ ∂ϕ + g ∂ϕ = 0 2 V (10.10) ∂t 2 ∂t ∂z Substituting ϕ = ϕ0 + ϕ1 , where ϕ0 = V x + ϕs and ϕ1 = ϕω + ϕI into 10.10 one obtains
− ∇
−
∂ 2 ϕ1 ∂ϕ 1 ∂ϕ 1 ∂ϕ 0 2V +g +g =0 ∂t 2 ∂t ∂z ∂z 10.11 takes the form:
− ∇
Since ϕ1 = ϕˆ1 eiωt
2
1
1
−ω ϕˆ −
(10.11)
0
ϕ ∂ϕ −iωt ϕˆ1 + g ∂ ˆ 2iω V +g e =0 ∂z ∂z
∇
0
within the fully linear formalism developed here ∂ϕ = 0 at the free surface. ∂z Therefore the problem is reduced to the problem with respect to amplitudes ϕˆI since ϕω is given ˆ −ik(x cos µ−y sin µ) eiωt ) ϕω = Re( icAe
− ϕ − ω ϕˆ − 2iωV · ∇ϕˆ + g ∂ ˆ ∂z ϕ ∇ϕˆ − g ∂ ˆ + 2iωV ; ∂z 2
I
I
I
= +ω 2 ϕˆω +
(10.12)
ω
ω
The no penetration condition yields on the ship hull n ϕˆ(1) + uˆ (m
∇
+n where m = (n )
(0)
× ∇ϕ (0)
∇ ∇ϕ ω
ϕ = Re
−
− iω n) + α ˆ [x × (m − iω n) + e
e
=0
ˆ −kz e−ik(x cos µ−y sin µ) eiωe t icAe
u = (u1 , u2 , u3 )T
(10.13)
describes the translations,
α = (u4 , u5 , u6 )T = (α1 , α2 , α3 )T
the rotations.
The panels are distributed on the hull and on the free surface around the ship. To get the boundary conditions for diffraction potential, the potentials ϕi are set to zero in equations (10.12) and (10.13). If the radiation potential, say ϕ(3), is to be found, the diffraction potentials ϕd and ϕi(i=3) are set to zero in boundary conditions (10.12) and (10.6). The boundary conditions (10.12) 104
and (10.13) are fulfilled at centres of each panel. This results is a system of linear algebraic equations with respect to the source strengths, which can be solved by Gauss method. Once the source strengths are known, all potentials and derivatives (velocities) can be calculated. The unknown motion amplitudes ui are computed from momentum equations:
= a
ag + a (x U
× a )
ndS
−x
α
¨+α ¨ m u
× x
g
−
× −
x y G = mg I
g
¨ + I α ¨= u
m x g +
− × G +
(1)
p
g
g
p(1)
S
× × × ×
(10.14)
+ G
g
[ua + α (x
−
α )]
(10.15) (¯ x
× n¯) dS
center of gravity ship weight matrix of the moments of inertia of the ship
M =
m 0 0 0 mz g 0
0 m 0 mz g 0 mxg
0 0 m 0 mxy 0
−
Qxx = (y2 + z 2 )dm,
−
Qxz =
I =
0 mz g 0 Qxx 0 Qxz
−
mz g 0 mxg 0 Qyy 0
−
−
0 mxg 0 Qxz 0 Qzz
− −
xzdm
−
Qxx 0 0 Qyy Qxz 0
−Qxz 0 −Q zz
The harmonic pressure p(1) is decomposed into parts due to incident wave, due to diffraction, and due to radiation. 6
p
(1)
ω
d
=p +p +
pi ui
i=1
105
These components can be calculated from the linearized Bernoulli equation: pi =
−
∂ϕ i + ∂t
(0)
∇ϕ
∇ ϕi
Two momentum vector equations (10.14) and (10.15) form a linear system of six equations for the six motions ui which is easily solved.
10.2
Time domain simulation
Consideration of nonlinear effects. Seaway is computed as superposition of elementary waves. •
•
The wave frequencies ω j are chosen such that the area under the sea spectrum between ω j and ω j+1 is the same for all j. This results in constant amplitudes for all elementary waves regardless of frequency. The frequency interval for simulation is divided into subintervals. These subintervalls are larger where S ζ or the important RAOs are small and vice versa. In each subinterval a frequency ω j is chosen randomly (based on constant probability distribution). Encounter angles are chosen randomly.
The frequencies, encounter angles, and phase angles chosen before the simulation must be kept during the whole simulations.
106
