Theoretical prediction of springing of ships

Yanlin Shao1 & Odd M. Faltinsen2

1 Ship Hydrodynamics & Stability, DNV 2 CeSOS, NTNU

Springing is described as a periodic resonant excitation of structural

vibration

Example: Vertical two-node vibration of a ship

Ship bending in heave sea

Our springing analysis

2 3

0 1 2 3

Steady Linear Second Thirdflow order order

,

characteristic wave amplitude

A

A A

A

Perturbation scheme with the incident wave slope as a small parameter

Potential flow theory

5

Traditional Formulation of Boundary Value Problem (BVP) in

inertial coordinate system

, ,Z X X Y YU on Z X Y tt

Free-surface conditions

Kinematic:

1

0 , ,2

U g on Z X Y tt

Dynamic:

Taylor expansion about Z=0 (i.e. Calm water surface)

( ) ( )( )

1 0m m

mF on Zt Z

( )

( )

2 0m

m mg F on Zt

Known forcing terms

m=1: First-order

m=2: Second-order

……

6

(1)(0) (1) (0 () (1) ( )0 1 () 0)+n n u x n Ui

1st order

2nd order

2(1) (0(0) (2) (0) (2)

(1) (1) (1

(1) (1) (2) (0)

(1) ()

(2) (0)

)

0)

1

2n n u

n u

n Ui

xx x

x

0

2

(

1

)

( )

:

:

: Normal vector o

velocity induced by ri

n the mean body sur

gid-body motions

displacement induc

face

: Change of no

ed by rigid-body m

rmal vector due to body rotati

otio

, o

ns

ns

k

k

u

x

n

n n U forward speed

Double Gradient

Triple Gradient

Approximated body-boundary condition on mean position SB0

7

Reason of double & triple gradient:

Taylor expansion of the body boundary condition about the mean position of the body

Associated difficulties:

Difficult to calculate the higher-order derivatives accurately

on the body surface with high curvatures, e.g. ships.

Resulting Boundary Integral Equation (BIE) is not integrable

for bodies with sharp corners.

Summary of traditional formulation

• Use of inertial coordinate system causes unphysical singularities at sharp corners

• Reason: failure of Taylor expansion of body boundary conditions

• Occurs due to interaction of steady and unsteady flow

• Occurs due to body motions in higher-order problems

• The consequence can be divergent results

Formulation in body-fixed coordinate system

• Considering deep-water cases

U

X

Z

O

U

x

z

o

OXYZ:

Inertial,

Moving with U

OXY plane on calm water surface

oxyz:

Body-fixed

• Free-surface conditions approximated by Taylor expansion about oxy plane,

i.e. z=0 plane

U

x

z

o

• The perturbation scheme assumes that the wave amplitude and body

motions are asymptotically small

• The distance S is always smaller than the dimensions of the ship S

• Body-bounary condition on instataneous position. No Taylor expansion.

SB

SF

Sc

Formulation of the Boundary Value Problem

0 in water domain

m

z

x

0 is obtained by a double-body solution

1 2 and is obtained by a time-domain HOBEM

Numerical method • Time-domain HOBEM based on cubic shape functions

• Incident waves: Stokes 2nd order waves described in body-fixed coordinate system

U

x

z

o

Incident wave in inertial coordinate system: ( ) ( ), , 1,2k

i

k

i k

(1) (1) (1) (1)1st order : ,b i b i

Incident wave in body-fixed coordinate system: ( ) ( ), , 1,2k

b

k

b k

(2) (2) (1)

(1) (1)(2) (2)

1

1 1

1 2

,

2nd order : ,

b i i

i ib i

x

xy

xx

1x

1 1 1 1

1 2 3, ,x x x x

P0(x,y,z)

P

Summary of body-fixed formulation

• Use body-fixed coordinate system

• No Taylor expansion of body boundary conditions

• Only higher-order derivatives on free surface (much easier to calculate)

Other possible nice properties of body-fixed formulation

• Solving slowly-varying motions in a consistent way

• Solving a combined seakeeping and maneuvering

• No needed for ‘soft springs’ needed in time-domain seakeeping

Linear seakeeping

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.20.0

0.4

0.8

1.2

1.6

3a /

A

L

Journee (1992), A=0.018m

Present, DB

0.4 0.8 1.2 1.6 2.00.0

0.5

1.0

1.5

2.0

2.5

3

a/A

L

Experiment

Zhang et al (2010), DB

Present, DB

0.4 0.8 1.2 1.6 2.00

2

4

6

8

10

12

14

5aL / A

L

Journee (1992), A=0.018m

Present, DB

0.4 0.8 1.2 1.6 2.00

2

4

6

8

10 Experiment

Zhang et al (2010), DB

Present, DB

L

5

aL

/ A

Heave Pitch

Heave Pitch

Wigley I

Series 60

(CB=0.7)

Added Resistance

16

0.4 0.8 1.2 1.6 2.0

0

10

20

30

40

F(0

) /gA

2B

2/L

/ L

Journee (1992),

A=0.018m

Present, DB

Present, NK

Wigley I, Fr = 0.2

2 3 4 5 60

2

4

6

8

10

12

14

eL/g

F(0

) /gA

2B

2/L

Storm-Tejsen et al. (1973), Exp

Kim&Kim (2010), DB

Joncquez et al. (2009),DB

Present, DB

Series 60 (CB=0.7), Fr = 0.222

2 3 4 5 60

2

4

6

8

10

eL/g

F(0

) /gA

2B

2/L

Fujii (1975), Exp.

Nakamura(1977), Exp.

Present, DB

Present, NK

S175, Fr = 0.15

2 3 4 5 60

2

4

6

8

10

12

eL/g

F(0

) /gA

2B

2/L

Fujii (1975), Exp.

Nakamura(1977), Exp.

Present, DB

Present, NK

S175, Fr = 0.2

Second-order generalized excitation of ship springing

Springing is described as a periodic resonant excitation of structural

vibration

Example: Vertical two-node vibration of a ship

nT s

L m

Resonant excitation wlinear spring heing n 2 /e n nT

Example on ship length dependence

sum-frequency springResonant excitation whe /ng n 2i e n

triple-frequency spriResonant excitation w n henng /gi 3e n

Natural period for 2-noded vertical vibrations

• When does ship springing occur?

: encounter frequencye

: natural frequencyn

Natural period for two-noded vertical vibrations

Example: 300 , 2nL m T s 1Head sea, 10 0.18)U ms Fn

Resonant excitation for wavlinea elengr thspring /L=ing 0.12

Resonant and springing excitation for wavelengths

and , respect

sum-frequency

/ 0.2 iv

triple-frequency

el0. 1 y/ 59L L

The rigid-body vertical motion is small at linear, sum-frequency and triple-frequency resonance.

Resonant and harmonic springing excitation f4th

/ 0.7

or wavelengths

an

5th

/ 1.0d , respe8 8 ctivelyLL

A weakly-nonlinear theory may be applied

Large-amplitude ship motion, slamming, water exit…

The state-of-the-art ship springing analysis

• Based on the linear solution

• With nonlinear corrections,

e.g. Froude-Krylov forces, restoring forces,

quadratic effects of linear solution

• Slamming-type of loads may also be added

• Jensen and Pedersen’s quadratic strip theory with bi-chromatic second-order incident waves

What is missing ?

Nonlinear wave radiation and diffraction

The importance of them remains as mystery

This study investigates the relative importance of the quadratic effect and the second-order wave radiation/diffraction effect on the second-order excitation

Modified Wigley Hull

Miyake et al. (2008) found experimentally for a modified Wigley model that super harmonic (n-th) springing occurred, although the model is a simple mathematical hull form without bulbous bow.

Journee’s Wigley Hull I is scaled to L=300m

0

22 0 2 0(2) (0) (1)

7 3 3, , , , 0.5SB CW

F p x y z n x y xz ds p g n dlx

Generalized excitation of 2-noded vertical mode is studied

1L

x x dx is the 2-node dry modex

0.25 0.30 0.35 0.40 0.45 0.500.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

F(2

) 7,G

E /

gA

2L

0.5

F(2)

7,p2

F(2)

7,q

F(2)

7

Fr=0.18

/L 0.25 0.30 0.35 0.40 0.45 0.500.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

F(2

) 7,G

E /

gA

2L

0.5

/L

Fr=0.20

F(2)

7,p2

F(2)

7,q

F(2)

7

0.25 0.30 0.35 0.40 0.45 0.50

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

F(2

) 7,G

E /

gA

2L

0.5

/L

Fr=0.22

F(2)

7,p2

F(2)

7,q

F(2)

7

2nd-order velocity potential gives dominant contribution to quadratic effects

2

7

2

7,

2

7, 2

2

7,

: of 2-node vertical mode

: total

: due to 2nd-order velocity potentia

generalized e

l

: due to qua

xit

dratic effect

atio

s

n

a

p

q

F

F

F

F

2

7F

/ L

0.18Fr

/ L

2

7F

0.20Fr

/ L

2

7F

0.22Fr

Strong dependency on Froude numbers in short wave region

L=300m, B=30m, D=18.75m

0.25 0.30 0.35 0.40 0.45 0.500.00

0.02

0.04

0.06

0.08

0.10

/L

F(2

) 7,G

E /

gA

2L

0.5

Fr=0.18

F(2)

7,p2

F(2)

7,q

F(2)

7

0.25 0.30 0.35 0.40 0.45 0.500.00

0.02

0.04

0.06

0.08

0.10

/L

F(2

) 7,G

E /

gA

2L

0.5

Fr=0.18

F(2)

7,p2

F(2)

7,q

F(2)

7

0.25 0.30 0.35 0.40 0.45 0.500.00

0.02

0.04

0.06

0.08

0.10

F(2

) 7,G

E /

gA

2L

0.5

F(2)

7,p2

F(2)

7,q

F(2)

7

Fr=0.18

/L

L=300m B=30m D=18.75m

L=300m B=48m D=18.75m

L=300m B=48m D=11m

/ L

2

7F

/ L

2

7F

/ L

2

7F

• Blunt ship suffers higher second-order springing excitation • Higher second-order springing excitation in ballast condition • Relative importance of quadratic velocity terms increases in ballast condition • Dominant effects of second-order velocity potential in all studied cases

2

7 , ,

1 1Second-order transfer Second-order transferfunction a

,

mplitude function phase

,, , co, ,s +N N

i j e i e j i j

i j

e i e e jj e iF A A tT

, ,

, , ,

, ,

,

Symmetry properties

,

, ,

,

:

,e i e j

e i e j e j e i

e j e iT T

0.5 2

0

Wave encounterspectrum

Sum frequency spectral density:

8 0.5 0.5 0.5 , 0.5e eS S S dT

T

L

Amplitudes of second-order sum-frequency transfer functions on modified Wigley hull at Fn=0.178 in irregular waves

/L g

/ 0.002 L g

/ 1.112 L g

/ 1.662 L g

is difference frequency

is sum frequency

Sum-frequency force in time domain:

Yanlin Shao

Third and higher-order problems

Becomes increasingly difficult with increasing order due to antecipated singularity problems at the intersection between the free surface and the body surface

Difficult to deal with for instance large changes in wetted area at the transom occurring in the ship-motion range

Fourth and higher-order springing ought to be considered simultaneously with whipping (slamming) as a strongly nonlinear problem

A challenge is to derive an accurate method that can account for different sea states within realistic computational time

Linear (frequency-domain & time-domain)

Weakly-nonlinear 2nd order: mean, sum-frequency, slow-drift wave forces 3rd order: Triple-frequency Higher order: Impractical

Fully-nonlinear potential-flow Capable of describe higher-order harmonics

Navier-Stokes Equations Viscous flow separation, local wave breaking…

Co

mp

uta

tio

nal

Eff

ort

(C

PU

tim

e)

Numerical Hydrodynamics Analysis

Can we simulate the springing & whipping phenomenon simultaneously by using a fully-nonlinear potential-flow solver for a reasonably long period ?

References

• Storhaug, G. (2007). Experimental investigation of wave induced vibrations and their effect on the fatigue loading of ships, Ph.D Thesis, Dept. Marine Technology, NTNU.

• Jensen JJ and Pedersen PT. Wave-induced bending moments in ships – a quadratic theory. The Royal Institution of Naval Architects, pp151-165, 1978.

• Shao Y.L. and Faltinsen O.M. (2012). A numerical study of the second-order wave excitation of ship springing with infinite water depth. Journal of Engineering for the Maritime Environment, 226(2), 103-119.

• Shao, Y.L. and Faltinsen, O.M. (2012). Linear seakeeping and added resistance analysis by means of body-fixed coordinate system. Journal of Marine Science and Technology. DOI: 10.1007/s00773-012-0185-y.

• Shao Y.L. and Faltinsen O.M. (2012), Second-order diffraction and radiation of a floating body with small forward speed, Journal of Offshore Mechanics and Arctic Engineering, Volume 135, Issue 1.

• Shao Y.L. and Faltinsen O.M. (2010), Use of body-fixed coordinate system in analysis of weakly-nonlinear wave-body problems. Applied Ocean Research, 32, 1, 20-33.

• Shao Y.L. and Faltinsen O.M. (2011), Numerical study of the second-order wave loads on a ship with forward speed. 26th Workshop on Water Waves and Floating Bodies, Athens, Greece.

• Shao Y.L. and Faltinsen O.M. (2010), Numerical study on the second-order radiation/diffraction of floating bodies. 25th Workshop on Water Waves and Floating Bodies, Harbin, China.

• Shao Y.L. and Faltinsen O.M. (2008), Towards development of a nonlinear perturbation method for analysis of springing of ships, 23rd International Workshop on Water Waves and Floating Bodies, Jeju, Korea.

Thank you