The Duffing system applied to jacket type offshore structures

The Duffing system applied to jacket type offshore structures

Citation for published version (APA):

Leeuw, de, P. P. M. (1989). The Duffing system applied to jacket type offshore structures. (DCT rapporten; Vol. 1989.042). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1989

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THE DUFFING SYSTEM APPLIED TO

JACKET TYPE OFFSHORE STRUCTURES

Patrick de Leeuw

Supervision: Mr. G. Stewart, M.Sc.

Koninklijke/Shell Exploratie en Prod. Laboratorium Rij swi j k

Section: W.F.W.

Report : WFW 89.042

CONTENTS : 1. 2 . 3. 4 . LIST OF SYMBOLS ACKNOWLEDGMETJT S U M M A R Y INTRODUCTION

2.1 APPLIED WAVE LOAD

1.2 DERIVATION OF THE EQUATION OF MOTION DUFFING SYSTEM: ANALYTICAL

2.1 PARAMETERS OF THE DUFFING SYSTEM

2.2 PERTURBATION METHOD OF MULTIPLE SCALES

2 . 3 CONCLCTSIONS

DUFFING SYSTEM: NUMERICAL

3 . 1 SIMPMOD 3.2 TWO EXAMPLES

3 . 3 COMPARISON ANALYTICAL AND NUMERICAL DUFFING SYSTEM CONCLUSIONS REFERENCES TABLES FIGURES APPENDIX 1. APPENDIX 2. APPENDIX 3 . APPENDIX 4. APPENDIX 5 . listing STIFF.FOR listing ZEROPNT.FOR listing SIMPMOD.FOR

The input parameters of the SIMPMOD program listing NEWMARR.FOR

1

1 1 5 5 6 10 1 2 12 1 3 15 17

LIST OF SYMBOLS HW LW TW g CB Y X' X " Y L X O X 52 t m SrY SIY C qY e h cd,y'cD ' m

,

y "M caly'cA Y AY c ,CH,C HY U U' P N D msl 6 X Y M h M K Q

*

wave height wave length wave period gravity constant shape function

local jacket displacement local jacket velocity local jacket acceleration vertical jacket coordinate jacket length

platform deflection main wave frequency time

distibuted structural mass distributed structural damping distributed spring force

distributed wave excitation wave scale parameter

length increment

distributed, local drag parameter distributed, local inertia parameter distributed, local added mass parameter distributed, local, linearized hydrodynamic

damping parameter water particle velocity

water particle acceleration density of the water

number of hydrodynamic piles hydrodynamic diameter

mean sea water level

arbitrary displacement field top side mass

part of jacket above msl equivalent mass

equivalent linear, initial stiffness equivalent restoring force

F E lr 'cub 'fif" P E f Y 7 X X I X X w dYn dyn,max max statrxst O A A N r A *A D A F ~ ~ DAF 0

*

*

*

A r B r C

equivalent linear excitation equivalent wave loading relative damping

nonlinear stiffness factor for cubic term

nonlinear stiffness factor for fifth order term dimensionless nonlinear stiffness factor

dimensionless normalized damping factor dimensionless excitation amplitude frequency ratio

dimensionless time dynamic defection

maximum dynamic deflection deflection by static load case natural system frequency

half amplitude of the linear DAF

dimensionless amplitude of the free vibration Detuning parameter

amplitude of the nonlinear vibration component phase angle

displacement amplification factor upperbound of DAF

excitation frequency amplitudes

ACKNOWLEDGMENT

This report is a result of my second practical, which I had to fulfil for my study in Mechanical Engineering at the Eindhoven University of Technology. First I want to thank the KSEPL for giving me the opportunity to do my practical at their laboratory.

A special word of thanks is due to Mr. G . Stewart of Shell for introducing me in the world of offshore dynamics, showing me the theory learnt at the University really is essential for solving problems in practice.

SUMMARY

In December 1988 a student of Delft University of Technology has inves- tigated the nonlinear response of a jacket type offshore structure. A 2D nonlinear finite element model was analysed, using a specific software package. The result was a nonlinear load-displacement relationship, which was considered to be typical for fixed type offshore structures. To simplify the problem, the 2D model was reduced to a generalized single degree of freedom model (SDOF). To take the nonlinearity into account a cubic Duffing restoring force model was considered and also a hysteretic model was developed to include energy dissipation.

For the SDOF model a computer program was written. The program did not work perfectly. There were some figures obtained in report [ i ] that are suspect not to be right. Hence, it was necessary to check the program by solving some simple examples and to verify the figures.

Also in this report a fifth order Duffing model is developed analytically, using the perturbation method of multiple scales, and verified with the SIMPMOD program,

- 1 - Introduction

1 INTRODUCTION

For assessing the ultimate strength of a jacket-type offshore platform (fig. 1)r a simple degree of freedom model, which models the platform, is developed. The derivation of the generalized properties and the applied load is discussed shortly in the understanding. Further details can be found in Ill.

1.1 APPLIED WAVE LOAD

The load which is used in the single degree of freedom (SDOF) model i s based on the 100-year-design-wave. This wave is expected to occur on average, once in a hundred years. The wave attack is assumed from the east direction, for which the wave height H = 30.5 (m). The jacket operates in deep water (mean sea water level msl = 163 (m) above the mud line), so the deep water Airy theory can be applied. According to this theory:

W

Lw = 16 Hw

where Lw = wave length Tw = wave period g = 9.81 (m/s 2 )

= gravity constant

This results for the 100-years-design-wave in: Hw = 30.5 (m)

L~ = 4 8 9 . (m) Tw = 17.7 (S)

1.2 DERIVATION O F THE EQUATION OF MOTION

All the properties are derived from the principle of virtual work. Since the primary natural vibration mode dominates the response, the deck deflection can be modeled accurately using shape function Q (fig. 2):

Y

where X ( Y r t ) = local jacket displacement

Y = vertical coordinate

t = time

Q = shape function Y

- 2 - Introduction

O

X

L

52

= displacement amplitude at top of the jacket

= 184.5 (m) = jacket length 2a - - - TW = wave frequency

The wave load distribution is calculated from Morison's theory [2], yielding the equation of motion for length increment Ay:

X"+C _{x'+q Y (x,t)]Ay = Xe Y Ay}

ImSIY SrY (4)

where Iu-x'

I

(u-x')+Cm u t - c x"

- 'd,y rY aIY

-

e

X = wave scale multiplier on the design wave m = distribution function for structural mass

C

Y

StY

STY = distribution function for structural damping

= distribution function for structural

restoring forces qY

'a,Y

U = local water participle velocity

= 1/2pCDND = n/4pCMND = a/4pCAND 2 2 'm, y 'a,y

where C D' C M' C A = local drag, inertia and added mass N

D

= local equivalent number of piles

= equivalent diameter of the piles From the linear deep water wave theory follows the local water participle velocity :

where k = - 2a

LW

= wave number

The part of damping caused by structural damping is negligible to hydrodynamic damping, hence c S is neglected. Rewriting equation (4) gives:

[ m x"+Ac (tIx')x'+q (x,t)IAy = X[C, Iu-x'Iu+C mI Y u'lhy

- 3 - Introduction

where

The virtual work done by system (5) for an arbitrary displacement field 6x follows from: Y

*

2 M (l+h/L) x"6x +

4

[ m x"+hC (trX')x'+q (xft)1dy6x =

o o

y=o y HY Y Y where m l ]U-X' ~ u + C U' 1dy6xy 'y10 ['dry m f Y

*

M = top side mass

h = part of jacket above msl

( 7 )

Substituting ( 3 ) and it's derivatives into ( 6 ) and taking into account that

( 6 ) holds for arbritrary virtual displacements 6x yields the equation o f

motion:

O'

MX"+XCH(trX')~'+Q(Xort) O 0 = hE(tiX6)

O

where the generalized properties:

= m @2dy+M*(i+h/L) 2 M Y=o Y Y (mass ) ( damping ) (restoring force) E(t,Xg) = C \ U -x'IU +C U' d O O O m O - X I @

IU

@ +C U'@ ldy - m l

- &O [cdryluY 0 Y Y Y mrY Y Y

(drag/inertia forces)

It can be shown [i], that the generalized properties for the structure of

fig. 1 are:

6 M = 28.0*10 (kg)

C,(t) = 4.8031cosSltl (MNs/m) K = 134.0 (MN/m)

E(t) = 56.851cosOt IcosQt-11.5sinQt (MN)

( 9 )

In the next section the fifth order Duffing system will be considered analytically and therefore the damping is linearized. Also the load is modified using the Fourier series expansion, only considering the drag part

- 4 - Introduction

of the load (the first term of E(t)) and hence neglecting the inertia part (the second term).

- 5 - Duffing system: analytical

2 DUFFING SYSTEM: ANALYTICAL

In [ i ] the cubic Duffing system is worked out analytically and verified numerically. In this section the fifth order Duffing system will be solved analytically using the perturbation method [ 3 1 .

2.1 PARAMETERS O F THE DUFFING SYSTEM

The equation of motion (8) using the earlier made assumptions of linear damping and one term excitation appears to be:

Mx"+Cx'+K(x)x = XFcosQt

where M = generalised mass

C = generalised linear damping F = generalised force amplitude X = wave scale parameter

52 = excitation frequency

K(x) = nonlinear Duffing stiffness

It has been shown in report [ l ] that the linear damping term C derived from C (t) is C = 4.552 (MNs/m). Hence the dimensionless damping coefficient is:

H

C = Z/(MK) = 0.037

However in the offshore dynamics and therefore in report [I] the damping coefficient is assumed to be y = 0.025. In the following this value for the damping coefficient is used.

In fig. 3 the restoring force of the regarded structure is plotted, as well as the restoring force of the cubic and fifth order Duffing system, respec- tively defined as:

and :

4 Q(x) = K(l-qfifx )X

where K = initial stiffness

= small parameter of cubic Duffing system 'cub

"fif=

11 II _{fifth order}

I' II

The minus sign in front of q points to a softening spring. The frame model curve is obtained by the INTRA program (INelastic Tower Response Analysis). Using the SDOF model the curves of the restoring force must fit as good as

- 6 - Duffing system: analytical

possible (cubic: q = 1/6). Notice that the Duffing system doesn't take hysteresis in account. Therefore in [i] also a SDOF hysteresis model is developed.

A disadvantage of the fifth and higher order Duffing systems is that the perturbation expansions become more and more complex. The curves of both the Duffing systems are obtained by simply solving:

AF = Q(x)

where Q(x) = restoring force according to (12) resp. (13) So it is clear the fifth order model will fit better. Now the small parameter q has to be chosen. In fig. 3b for several values of q the restor-

= 0.12 ing force is plotted using program STIFF (app. i ) and it seems q

fits the best.

Substituting the nonlinear restoring force (13) in differential equation (10) the equation of motion is obtained:

f if

In the next subsection this equation will be solved using the perturbation met hod.

2.2 PERTURBATION METHOD OF MULTIPLE SCALES

An analytical treatment of (14) is best handled in dimensionless form:

u"+2€pu'+u-€u5 = fcosvr Xdyn where u = X stat r = o t O 0 v = - o O 4 € = VXst dimensionless deflection I1 _time II frequency

nonlinear stiffness parameter

u = O ( € )

dimensionless damping

I 1 force amplitude

natural system frequency

- 7 - Duffing system: analytical

The static deflection can simply be obtained by solving (10) for: x' =

o

x" = 0

and the right hand side the maximum force amplitude AF, hence: K(x)x = AF

The solution of this fifth order polynomial can be found with the program ZEROPNT (app. 2).

While the wave period T is known from ( 2 ) and hence the excitation frequency

S2 and the natural frequency o can be calculated using ( 9 ) ' the dimension- less frequency is:

w O 2x = - TW = 0.35 (rad/s) = d ( K / M ) O = 2.188 ( s ) 52 y = - O = 0.16 o

iience only attention is paid to the fifth superharmonic ( Y = 0.2).

The main use of the method of multiple scales is to separate slowly and rapidly varying parts of the solution, which is:

U(7rE) = U

o o'

(T T1r

...,

T,)+U ₁ (T o' T

1'

...,

Tn)+ ( 1 7 )

...

+un(TOI T l I

...'

Tn) where the independent variables:

n T n = e 7

The derivatives of (17) are:

2 d dTo

a

dT1

a

dT2

a

2

- - .

- + - - + - - t . . . = D + s D + E D dt dt i3To dt JT, dt aT2 O 1 d2 3 --2 = D

+

2eD D

+

s 2 ( D 2 + 2D D )

+

s ( 2 D o D 3 t 2D1D2) +

...

dt O

o 1

1 o 1

- 8 - Duffing system: analytical

An expansion up to e2 is accurate enough. For more accurate solutions higher order terms should be included, increasing the problem. Rewrite the right hand side of (15) using T = r to:

O

i vT fcosvr = 1/2fe O+cc.

where cc. stands for the complex conjugate

Substituting (17) and (18) in (15) using (19) and collecting terms of O( en)

,

n = O, 1, 2,

...,

yields the set of equations:

+cc. O i YT 2 O 0 2 5

o 1

1

o 1

O 0 2 2 2 o 3 3 o 3 1 2

o

1 o 2 1 o 1 2 D u +uo = 1/2fe D U +U = -2D (D + p ) U +U +CC. 2 4 = -(D +2D D )u -2D D u -2P(D u +D U )+SU U fCC. DOU2 +u2 1 o 2

o

o 1 1

1 0

o 1

o 1

D U +U = -2(D D +D D )U -(D +2D D )u -2D D U - 3 2 4 2@(D u +D u +D u )+5(2u u +u u )+Cc. o 2 1 1 2 0 o 1 o 2 From the first equation of (20) follows the solution for u *

O' iT i vT

U = A(T1, T2, T3)e

o

+he O+CC.

o

f 2(1-V 2 )

where A =

Substituting this solution in the second equation of (20) gives:

5

iT i vT iT i vT

D 2 u +u = -2i(D +P)Ae '-2fiivAe '+cc.+(Ae '+A, O+cC.)

o 1 1

1

Apparently the last term will be a sommation of 28 terms, but the only terms of interest for the fifth superharmonic are the ones, which may cause resonance if v 1/5. According to Poincaré [4] this terms must be set to

zero:

= o

O iT iT iT 5viT 3- 2 iT

- 9 - Duffing system: analytical

Define :

v = (l+~N)/5 Let the amplitude A be:

iQ, A = 1/2r e

A

where r and Q are both real. Substituting (24) and (25) in (23) and seperating the real and imaginary parts leads to a system of coupled non- linear differential equations:

A A 5 A D r = -Br +A siny 1 A A 2 5 4 5

= -[15/2A rA3+ï0/64r +15A r A +A cosyAl

A I A

r ~ D i Q ~ A

where y = NT -Q

For steady-state response (26) turns into: 5

A

OK,

= A siny

5

- [ 1 5 / 2 A 2 r ~ + 1 0 / 6 4 r 4 + i 5 A 4 + ~ ] A r A = A cosy A

Squaring (27) and summing both equations results in: [p2+{15/2A 2 2 rA+10/64rA+15A 4 4 +NI2 Ir: = A 10

Dividing the first equation of (27) by the second results in:

-

1

-1

4 2 2 4 y A = tan 10/64rA+15/2A rA+15A +N

Solving (28) for N yields the relation:

4 2 2 4

N = -[10/64rA+15/2A rA+15A I +

For increasing r 2 the positive or negative contribution of the root term in (30) is vanishing. This ultimately leads to a negative right hand side and hence a negative N, indicating a negative frequency shift.

- 10 - Duffing system: analytical

When the root term is zero:

Using (21) and (25) the solution leads to:

U ( 7 , E ) = u + O ( € ) O A A A A r and +y according to (28), (29) A A = r cos(.r+Q )+2Acos(v~)+O(~) = r cos(5v~-+y ) + ~ A c o s ( ~ . ~ ) + ~ ( G ) where 2.3 CONCLUSIONS

In the dimensionless parameters the maximum dynamic amplitude of system (14) is related to the static solution x This parameter is called the dynamic

s t '

amplification factor (DAF) and is defined as:

X

d yn

,

max DAF =

X

st

An upper bound for the DAF following from (32) is:

DAFUP = u _O = (2A1+lrAI

(33)

(34) The equation (28) can be rewritten to:

[ ( 1 0 / 6 4 ) 2 Ir~0+[150/64A 2lrA+[225/4A 4 8 +20/64(15A 4 +N)]rA+ 6

(35)

2 4 4 2 4 2 2 l o z o

[15A (15A +N)lrA+[@ +(15A +N) Ir -A A 2

While (35) is a fifth order polynomial in r n r it generally has five pairs of roots, which are conjugatives. Only the real positive components have physi- cal meaning and are solutions of f The program PRESPONS calculates the roots of (35). The program also can calculate the linear response, the roots of the prime, the third and the fifth superharmonic of the cubic Duffing system. N o t only DAFUP according to (34), but also r can be chosen as output (respectively resonance component and total resonance response). In case of the data of table 2 only three roots have physical meaning. With these data fig. 4-7 are obtained. In table 3 the data of the fig. 7 are compared with DAFUP using (34). The data seems to be right, while the upper- bound value is not exceeded. When h increases the response makes a shift

i-5

A'

- I1 - Duffing system: analytical

downwards. This shift is of order e , just as the error in the solution (32). Hence it is possible that the response of the Duffing system will be a

factor e higher. In fig. 5-6 the Cubic Duffing System responses as obtained in [l] are checked and found correct.

In the next section the response of the Duffing System will be checked numerically using a Newmark-Beta algorithm. Before doing this the SIMPMOD program is checked on validity.

-

1 2 - Duffing system: numerical

3 DUFFING SYSTEM: NUMERICAL

In this section first the SIMPMGD program (app. 3) will be discussed. The program is checked on validity by considering some simple examples. In the last subsection the results obtained in the latest section will be checked using the SIMPMOD program and some figures of report [ i ] will be checked. 3.1 SIMFMGD

The SIMPMGD program can solve a differential equation of form:

MX"+C(t)X'+Q(trX) = F(t) ( 3 6 )

where C(t) can be linear, userdefined or hydrodynamic Q(t,x) can be linear or according to (12,13) (Duffing) F(t) can be userdefined, hydrodynamic or a sommation

or with hysteresis E i ]

of cosine terms The SIMPMOD program consists of four files:

1 )

SIMPMOD.FOR: The program written in FORTRAN.

2 ) SIMPMOD.OBJ: Compilated program, obtained by giving the

3 ) SIMPMOD.EXE: Execute file, obtained by LINK SIMPMOD.

4) FOR001.DAT: Input file with comment. In this file the input command FOR SIMPMOD.

parameters can be changed. The apostrophe in the beginning of a line points to a comment line. When running the program (RUN SIMPMOD), the first output data to the screen

i s the input data as specified in FOROOl.DAT, followed by a mention that the static and dynarnic analysis are finished. The static solution is necessary to calculate the DAF according to (33). The last output is the maximum dynamic and static deflection.

The program generates seven files:

i ) SIMPMOD-DIS.OUT: Displacement history, fig. 8a 2 ) SIMPMOD-RMP.OUT: Ramp history, fig. 8b (IFREQ=l)

3 ) SIMPMODRFC,OUT: Restoring force history, fig. 8c (IFREQ=l) 4) SIMPMOD-LOD.OUT: Loading history, fig. 8d (IFREQ=l)

5) SIMPMODRUB.OUT: Iteration/integration information

(only generated if IRI(OUT=1 OR ITOUTP=l)

6 ) CIMPMOD-DAF.OUT: Frequency

-

DAF curve, fig. 11 (IFREQ=2) 7) FOR003.DAT: Input file without comment

(only generated if IFREQ=l; see app. 4)

-

13

-

Duffing system: numerical

the SIMPMOD program is:

28x"+3x'+134x = 87cosot The solution will be of form:

x = Acocwt+Bsinot The derivatives of (38) are:

x' = -Aosinot+Bocoswt x" = -Au 2 cosot-Bo sinot 2 Substituting (38) and (39) in (37) 3.2 TWO EXAMPLES

The first diffenrential equation which will be solved analytically and with

(37)

( 3 9 )

2 2

28(-Ao cosot-Bo sinwt)+3

This equation holds f o r all t, s o :

2 2 -28Bo -3A~+134B = O -28A0 +33~+134A = 87 -Aosinot+Bocosot)+ 134(Acosot+Bsinot) = 87coswt

Solving (41) for the unknown amplitudes A and B results in: 2 11658-2436w 4 2 A = 7840 -74950 +17956 2610 4 2 B = 7840 -74950 +17956 Take for example o = i then ( 4 2 ) :

A = 0.8201 B = 0.0232

The maximum deflection x reached at time t when x ' = O , s o :

max

(43)

Rewriting this equation results in: B tanwt = - A Using w = 1 and (42): t = 0.0283

and substituting this in (38):

X = 0.8204

max

-

14

-

Duffing system: numerical

According to the SIMPMOD program with input parameters according to (table 4):

X = 0.8322

max

(45)

(47)

The both solutions are shown in fig. 9a. The ramp function (see fig. 8b) which is multiplied by the generalised force, is to avoid transient responses as shown in fig. 9b. The deviation of the analytical maximum deflection and the maximum deflection obtained by the SIMPMOD program is only 1.4%.

The second differential equation to solve is: 2 Mx"+Cx'+Kx = Fcos ot where M = 28 C = 3.06 (corresponding with Y = 0.025) K = 134 F = 58 o = w v O where w = I/(K/M) O

Rewrite (49) using a geometric formula: Mx"+Cx'+Kx = 1/2F[l+cos(2ot)]

(49)

Now the solution can be obtained following the same procedure as above which results in:

*

*

*

x = A +B sin(2ot)+C cos(2ot) * F where A = - 2K

*

FCw 2 2 B =

+

4c o

- 1 5 - Duffing system: numerical

*

2

1/2F(K-4Ma )

(K-4Mw 2 2 2 2 ) +4C 0

c

=

In fig. 10 the DAF is plotted for different values of Y for the analytical solution and with the SIMPMOD program (input in table 5.). The deviation is less than 18, so it seems the SIMPMOD program is satisfactory (One can never prove a program to be right, only wrong). In fig. 10 also the results of another program, DNONLIN (written by G. Stewart) are plotted. These values differ very little from the results of SIMPMOD.

3.3 COMPARISON ANALYTICAL AND NUMERICAL DUFFING SYSTEM

In this subsection the results of report [i] are checked and the results of the latest section are checked with the SIMPMOD program.

In fig. 11 the results of the cubic Duffing system which considered in fig. 23 of report [i] are checked. The input parameters which are used to obtain this figure are in table 6 . The problems which occurred, regenerating this figure were:

i) The frequency loop was not yet implemented (IFREQ). Hence for every point on the curve the wave period had to be changed in the input- file. This has now been corrected.

2) The calculation of DAF was not implemented, so for every load case the x had to be calculated and the DAF had to be written manually in the CUECHART input file. (CUECHART is a drawing program, which can connect points,using straight lines or polynomial curves.) This has now been implemented.

stat

3) The x was determined in the interval TO

-

TEND, but as shown in fig. 9b the response is at time t = 20 (s) not yet a steady state response and the x is higher than it will be if the steady state response is reached. It is now possible to specify the time range over which the program will search for the maximum response.

dyn , max

dyn

,

max

4) In the SIMPMOD program was not a warning implemented that the stiff- ness got negative. Hence sometimes the program calculated a solution, while the stiffness was already negative.

5) In [i] the input files as used were not in the report and it was not very clear what input parameters were used.

In fig. 12 the SIMPMOD program is checked with the NEWMARK program (app. 5). The NEWMARK program is the same as the DNONLIN program, only there is a ramp function used to avoid transient responses.

-

16 - Duffing system: numerical

In fig. 13 the DAF for the cubic Duffing system is plotted for different values of A for the fifth superharmonic. The highest value used is A = 1.9, because at higher levels the SIMPMOD program will not converge to a solu- tion, while a negative stiffness i s detected.In fig. 14 for X = 1.9 the same plot is made as in fig. 13. Near the peak the stepsize of v is AV = 0.01, else A V = 0.1.

In fig. 15 the fifth order Duffing system is considered. The input parameters are in table 7. Comparing fig. 11 and fig. 15 i t can be seen the response of the cubic Duffing system is higher than the response of the fifth superharmonic.

In fig 16a the fifth superharmonic of the fifth order Duffing system is plotted for three values of X. The highest value reached is A = 1.85. In fig. 1 7 (input parameters in table 8.) the response for v = 0.173 - 0.180 the program mentioned a negative stiffness. In fig. 16b the numerical (fig. 16a) and analytical(fig. 7) response are compared. For A = 1.5 the analyti- cal fifth superharmonic has a peak of DAF = 1.047 and numerical a peak of DAF = 1 . 0 7 . The difference is only 7% and is caused by the truncation error ( e = 0.023). If more terms were included the analytical solution would probably closer to the numerical as it is now. For respectively A = 1.75 and

h = 1.85 the difference is 9% and 18%. This seems much larger, but as shown in table 2 the truncation error is also larger.

In fig. 15 (input parameters in table 9.) fig. 17 of report [i] is checked and found right. In fig. 19 with input parameters in table 10 the curves of fig. 25 of report [ i ] are verified. Only for the two-term excitation the curve differs much from the curve of [i]. In the last figure the two term excitation of fig. 19. is verified using the NEWMARK program. Hence, it seems the curve in this report is correct, but there can be a difference in

- 1 7 - Conclusions

4. CONCLUSIONS

The cubic Duffing system has been enlarged to a fifth order Duffing system and the results of both have been checked numerically.

The other task, verification of the SIMPMOD program has been done by solving two simple differential equations and comparing the solution with the analytical solution. A second verification consisted of comparison of solu- tions of nonlinear differential equations with the solutions obtained using another available program.

The (non)linear dynamic response as produced by considering a linear and a cubic Duffing system (developed in [i]) has been verified.

The program SIMPMOD is improved (see app. 3 : the lines ending with PDL). It now is easier to regenerate the (non)linear response of the different sys- tems. Also the input parameters of the SIMPMOD program are specified. There still are some problems with use of the SIMPMOD program:

1. The hydrodynamic differential equation (IWAVE = i ) is not yet 2. The Bouc-Wen hysteretic stiffness model does not work, while sub-

ver if ied.

routine DERIVS (see app. 3) is not implemented (see ref. 1 6 1 ) .

Finally, some remarks for enhancing further development of the program are:

1.

While the program used in [ i ] was, in first instance, written on a

P.C. and afterwards implemented on a VAX, it is not certain whether all features will work properly. For example the functions EXCIT and DAMP did not work, because one argument was missing. It now does work.

2 . The program has been developed by two students. Modifications to the program implemented by the second student, as part of this study, have been verified for the examples considered. However, problems can occur considering other differential equations, i.e. the fre- quency loop will need verification for the BOUC-WEN hysteretic or

REFERENCES 1. 2. 3 . 4 . 5. 6. Jansen, H.,

(Internal Shell Report) Fish, P., Rainey, R.,

The importance of structural motion in the calculation of wave loads on an offshore structure,

Atkins Research and Development, U.K.,

August 1979, Second International BOSS Conference. Clough, W.C., Perzien, J,

Dynamics of Structures,

McGraw-Hill International Book Company, 5th Printing 1982.

Campen, D.H. v.,

Lecture Notes: Niet-lineaire Dynamica (dictaatno. 4 6 6 1 ) , Department of Engineering Mechanics (T.U.Eindhoven), November 1988.

Press, H.W., Flannery, B.P., Teukolsky, A.S., Vetterling, W.T., Numerical Recipes, The Art of Scientific Computing,

Cambridge University Press, 1986.

Bathe, K.J.,

Finite Element Procedures in Engineering Analysis,

TABLE 1 . Dimensionless parameter values for Cubic Duffinq System.

Cubic Duffing System:

Parameters: Parameter values: A 1.00 1.50 1.75 1.85 1.90 Xayn u = X st XF f = - st Kx 77 = 1/6 @ = 0.025 F = 58. (MI?) K = 134. (MN/m) M = 28. (Mkg) X st O. 447802 O. 708538 O. 865530 0.938527 O. 978566 & O. 0334211 O . 0836710 O I 1248570 O -1468055 O. 1595986 f O -966579 O. 916329 O. 875143 O. 853195 O. 840401 P O. 748030 0.298789 0.200229 O. 170293 O. 156643

TABLE 2. Dimensionless parameter values for Fifth Order Duffing System.

Fifth Order Duffing System: u"+2€0u'+u-€u5 = fCOSVT

Parameters: u =

Xdyn

X

st

p = - N

E

=e = ox4 (This parameter differs st

from 17 of Cubic Duffing System)

Parameter values: 7 = 0.12 = 0.025 E' = 58. (MN) K = 134. (MN/m) M = 28. ( M k g )

x

1.00 1.50 1.75 1.85 1.90 X f st f

P

O. 434698 0.0042848 0.995716 5.834561 O. 664841 0.0234451 0.976555 1.066320 O . 795751 0.0481161 0.951884 0.519577 O . 855848 0.0643825 0.935617 O. 388304 O. 889034 O. 0749643 O. 925036 O. 333492

TABLE 3 . Comparison of DAFUP and DAF obtained by PRESPONS for Fifth Order Duffing System.

f 2(1-V )

- -

2 Parameters: A

(values of f and 6 see table 2 . )

D A F ~ ~ =

I

rA,max

I

+ 1 2 4 Fifth superharmonic: Y = 0 . 2 r DAF UP DAF(fig. 7 ) A I max

x

A 1 . 0 4 3 6 3 3 1 . 0 4 3 6 3 1 1 . 0 0 O . 518602 O . 006429 1 . 5 0 0.508622 O . 031922 1 . 0 4 9 1 6 6 1 . 0 4 6 9 9 5 O . 057645 1 . 0 4 9 1 9 1 1 , 0 4 4 5 6 2 1 . 7 5 O . 495773 O . 4 8 7 3 0 1 O . 070764 1 . 0 4 5 3 6 6 1 . 0 3 9 4 0 6 1 . 8 5

TABLE 4. Input file used to obtain fig. 8 + 9.

I * * * * * * * * Input data file assigned to channel FOR001 for PROGRAM SDOF * * * * * * *

'23456789012345678?01234567890123456789012345678?0123~5678?012345678?0123~567~?0 I 1 2 3 4 5 6 7 8 1 I I I I I I I 1 I I I 1 I I 1 I I I I I I ANALY 2 ISTIFF 1 TO O. TSAMPL 10 o NRA-PT 2 XMAS S 28. HALPHA 0.02 YJACLT 184.5 YMSL 163. BEG I NNU 0.18 NCOMPS L IFREQ 1

-

I DAMP 1 TEND 40. FACNEG 1 . TRAMP O. 20 XMSTOP 20. HETA 1. CDND 105. WAVHGT 30.5 ENDNU 0.20 EXAMP 87 e IDUF 5 IEXCIT 1 DT 0.05 SCLINC 0.02 RMPFAC O.

1.

XMS JAC 22.4 HA 1 . CMND2 170. WAVPER 17.7 STEPNU 0.01 IRKOUT ITOUTP O O IWAVE EP S H1 HMIN O l.E-5 0.1

.o

TOUTP GAMMNB BETANB ETOL MAXIT

30. 0.25 0.5

o

D 001 20 NCYC 1 XMSADD 20.0 A H N U 1 . NSTICK 1 XCREST O.

DMPFAC STIFF CUBFAC

0.0245 134 0.12 HBETA HGAMMA HN 0.1 1.2 4. WIDTH 40. REI0 1. EXPER(s) EXPHSE(deg) 6.2832 O.

TABLE 5 . Input file used to obtain fiq. 10.

I * * * * * * * * Input data file assigned to channel FOR001 for PROGRAM SDOF

*******

'23456789012345678901234567890123456789012345~789012345678901234567890123456789U I 1 2 3 4 5 6 7 8 1 1 I I I I I I I I 1 I I I I I I I IANALY 2 ISTIFF 1 TO O . TSAMPL 1 0 . NRAPT 2 XMASS 2 8 . HALPHA 0.02 YJACLT 1 8 4 . 5 YXCL 163. BEGINNU 0.16 NCOMPS 1 I IFREQ 2 I DAMP 1 TEND 150. FACNEG 1 . TRAMP O . 20. XMSTOP 20. HETA 1. CDND 1 0 5 . WAYdGT 3 0 . 5 ENDNU 0.25 EXAMP 5 8 . IDUF 5 IEXCIT 2 DT 0.05 SCLINC 0 . 0 2 RMPFAC O . 1 . XMS JAC 2 2 . 4 HA 1 . CMND2 170. WAVFER 17.7 STEPNU 0.01 IRKOUT O IWAVE O TOUTP 130. NCYC 1 XMSADD 2 0 . 0 HNU 1. NSTICK 1 XCREST O . ITOUTP O EPS 1 . E-5 GAMMNB 0 . 2 5 DMPFAC 0.025 HBETA 0.1 WIDTH 40. REG 1. EXPER(s) EXPHSE(deg) 6 . 2 8 3 2 O . H1 0.1 BETANB 0.5 STIFF 1 3 4 . HGAMMA 1.2 HMIN

. o

ETOL MAXIT

o.

0 0 1 20 CUBFAC 0 . 1 2 HN 4.

user defined force:

TABLE 6. Input file used to obtain fiç. 11, 1 = 1.5.

I * * * * * * * * Input data file assigned to channel FOR001 f o r PROGRAM SDOF

*******

I I 2 3 4 5 6 7 I I I I I I I I I I I I t I I I I I I IANALY 2 ISTIFF 2 TO O. TCAMPL 10. N M P T 2 XMASS 2 8 . HALPHA 0.02 YJACLT 1 8 4 . 5 YMSL 1 6 3 . BEGINNU 0 . 1 6 NCOMPS 1 I IFREQ 2 IDAMP 1 TEND 150. FACNEG 1. TRAMP O. 2 0 . XMSTOP 2 0 . HETA 1 . CDND 105. WAVHGT 3 0 . 5 ENDNU 0.36 EXAMP 8 7 . IDUF 3 IEXCIT 1 DT 0.05 SCLINC 0 . 0 2 RMPFAC O. 1. XMS JAC 2 2 . 4 HA 1 . CMND2 170. WAVPER 1 7 . 7 STEPNU 0.01 IRKOUT O IWAVE O TOUTP 130. NCYC 1 XMSADD . 2 0 . 0 HNU 1. NSTICK 1 XCREST O. ITOUTP O EPS l.E-5 GAMMNB 0.25 DMPFAC 0 . 0 2 5 HBETA 0.1 WIDTH 4 0 . RHO 1. EXPER(s) EXPHSE(deg) 6 . 2 8 3 2 O. H1 0.1 BETANB 0.5 STIFF 1 3 4 . HGAMMA 1 . 2 HMIN

.o

ETOL iMX.

o .

001 I CUBFAC O. 1667 HN 4 .

TABLE 7 , I n p u t f i l e Lised t o o b t a i n f i g . 1 5 , h = 1.5. I I I I 1 1 I 1 I I 1 I 8 I I 1 I I I IANALY 2 ISTIFF 2 TO O. TSAMPL 10. NRAPT 2 XMASS 2 8 . HALPHA 0 . 0 2 YJACLT 1 8 4 . 5 YMSL 1 6 3 . IFREQ 2 IDAMP 1 TEND 150. FACNEG 1. TRAMP O . 2 0 . XMSTOP 2 0 . HETA 1. CDND 105. WAVHGT 3 0 . 5

'

BEGINNU ENDNU 0.16 0.36 I NCOMPS 1 EXAMP 87. IDUF 5 IEXCIT 1 DT 0 . 0 5 SCLINC 0 . 0 2 RMPFAC O . 1. XMS JAC 2 2 . 4 HA 1. CMND2 170. WAVPER 17.7 STEPNU 0.01 IRKOUT O IWAVE O TOUTP 1 3 0 . NCYC 1 XMSADD 2 0 . 0 HNU 1. NSTICK 1 XCREST O . ITOUTP O EPS 1. E-5 GAMMNB 0 . 2 5 DMPFAC O . 025 HBETA 0.1 WIDTH 4Q. RHO I. EXPER(s) EXPHSE(deg) 6 . 2 8 3 2 O . H1 0.1 BETANB 0.5 STIFF 1 3 4 o HGAMMA 1 . 2 HMIN

. o

CUBFWC Q.12 H 4 ,

TABLE 8 . Inout file used t o obtain fic. 1 7 . I I ? I I I 1 I ? I I I I I I I I I IANALY 2 ISTIFF 2 TO O . TSAMPL 10. NRAPT 2 XMASS 28. HALPHA 0.02 YJACLT 184.5 YMSL 163. IFREQ 2 IDAMP 1 TEND 150. FACNEG 1. TRAMP O. 20. XMSTOP 20 e HETA 1. CDND 105. WAVHGT 30.5 BEGINNU ENDNU 0.15 0.24 NCOMPS EXAMP 1 110.2 I IDUF 5 IEXCIT 1 DT 0.05 SCLINC 0.02 RMPFAC O. 1. XMS JAC 22.4 HA 1. CMND2 1 7 0 . WAVPER 1 7 . 7 STEPNU 0.01 TRKOUT O IWAVE O TQUTP 130. NCYC 1 XMSADD 20.0 HNU 1. NSTICK 1 XCREST O. ITOUTP O EPS 1. E-5 GAMMNB 0 . 2 5 DMPFAC O. 025 HBETA 0.1 WIDTH 40. RHO 1. EXPER(s) EXPHSE(deg) 6.2832 O. H1 8.1 BETANB 0.5 STIFF 134. HGAMMA 1.2 HMIN

.o

ETOL MAXIT 0 . 0 0 1

20

CUBFAC 0.12 HN 4.

TABLE 9 . Input file used to obtain fiq. 18, u = C!.&

I * * * * * * * * Input data file assigned to channel FOR001 for PROGRAM SDOF

* * * * * * *

c

I ₁ 2 3 4 5 6 7

'2345678901234567890123456789012345678901234567890123456789012345678~0123456789~

I I FREQ IDUF IRKOUT ITOUTP

2 2 5 O O

I I DAMP IEXCIT IWAVE EPS

1

I

2 O l.E-5 1 IANALY I ISTIFF H1 0.1 HMIN .Q TOUTP 130. GAMMNB 0.25 BETANB 0.5 ETOL 0,OQl MAXIT 20 TO O. TEND 150. DT 0.05 SCLINC 0.02 NCYC 1 TSAMPL 10. FACNEG

I.

TRAMP O . 20. RMPFAC O . NRAPT 2 1. XMS JAC 22.4 XMSADD 20.0 DMPFAC O. 025 STIFF 134 e CUBFWC 0.12 XMASS 28. XMSTOP 20. HETA HA 1. HNU 1. HBETA 0.1 HGAMMA 1.2 HN 4. HALPHA 0.02 1. YJACLT 184.5 CDND 105. CMND2 170. NSTICK 1 WIDTH 40. RE8 1. YMSL 163. WAVHGT 30.5 WAVPER 1 7 . 7 XCRECT O . BEGINNU 0.16 NCOMPS 1 I ENDNU 0.24 STEPNU 0.01 EX?ER{s) EXPHSE(deg) 6.2832 O. EXAMP 58.

user defined force:

EXCIT = 58.*ABS(CCS(WF*T))*CoS(WF*T)

TABLE 10. Input file used to obtain fig. 1 9 . + 20.

' * * * * * * * *

Input data file assigned to channel FOR001 for PROGRAM SDOF

*******

'23456789012345678901234567890123456789012345678901234567890123456789012345678~ 1 3 4 5 6 7 1 2 I I I I I I I I I t I I I I I t I 1 I IWNALY 2 ISTIFF 2 TO O . TSAMPL 10. NRAPT 2 XMASS 2 8 e HALPHA 0.02 YJACLT 1 8 4 . 5 YMSL 1 6 3 .

'

BEGINNU 0 . 1 5

'

NCOMPS 1 8 IFREQ 2 I DAMP 1 TEND 1 5 0 . FACNEG

1.

TRAMP O. 2 0 . XMSTOP 2 0 * BETA 1. CDND 105. WAVHGT 3 0 . 5 ENDNU 0 . 2 6 EXAMP 87 e IDUF 3 IEXCIT 2 DT 0 . 0 5 SCLINC 0 . 0 2 RMPFAC O .

1.

XMSJAC 2 2 . 4 HA

1.

CMNDZ 170 WAVPER 1 7 . 7 STEPNU 0.01 IRKOUT O IWAVE O TOUTP 130. NCYC 1 XMSADD 20.0 HNU 1. NSTICK 1 XCREST O * ITOUTP O HMIN EPS H1

1.

E-5 0 . 1

.o

GAMMNB BETANB ETOL MAXI'

0.25 0.5 0 . 0 0 1 2t STIFF CUBFAC DMPFAC 1 3 4 . O e 1667 0 . 0 2 5 HBETA HGAPIMA HN 0 . 1 1 . 2 4 . WIDTH 4 0 . RHO

1.

EXPER(s) EXPBSE(deg) 6 . 2 8 3 2 O ,

user defined force ( t w o term excitation):

EXCIT = 87.*(0.85*COS(WE'*T)+O.l7*COS(3.*WF"T))

user defined force (three term excitation):

EXCIT = 87.*(0.85*COS(WF*T)+O~l7*COS(3.*WF*T)-0.02*COS(5.*WF*T))

user defined force (drag only):

EXCIT = 87.*COS(WF*T)*ABS(COS(WF*T))

user defined force (drag & inertia):

Figure 1. Typical Jacket-typ?

,

M *

$ ( y ) = 1

-

cos-

fly

2 L

I w VO H

:x Principle of Virtual Work:

:. Mass :

3e :

J

_o

eFRFlflE RODEL

ETA = 0.00 E A E - 0-0- ETA = 0.20

ETA = 0.05 ETA = 0.15

THIRD ORDER RESTORING FORCE

EA-: - 925. LAMBDA 7 / / , 3 2 1 O 0.5 1 1.5 2 O

DECK DEFLECTION (M) F i g u r e 3a.

ORDER RESTORING FORCE

~

ETA = 0.10 E A 3 - 044- ETA = 0.18- ETA = 0.12 ETA = 0.16 ET!?-=_ - -

!--?P.

2 1 . 5 1 0 . 5 O O 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 DECK DEFLECTION (M) F i g c r e 3'0.

CUBIC DUFFING SYSTEM: ANALYTICAL RESPONSE THIRD SUPERHARMONIC

MU = 0.01: OTHER DATA SEE TABLE 1.

NU X UNSTABLE

_ _ , - - - -

DECREASING FREQUENCY - - + - - - ó-- Figure 4 .

CUBIC DUFFING SYSTEM: ANALYTICAL RESPONSE THIRD SUPERHARMONIC

MU = 0.025; OTHER DATA SEE TABLE 1.

D.A.F. LINEAR RESPONSE LAMBDA = 1.9 g M E D A = LAMBDA = 1.- 1.5 - LAMBDA = 1.00 . 1 . 6 I 1 . 5 1 . 4 1 . 3 1 . 2 1 . 1 1 0 . 9 0.26 O .28 0.30 0.32 0.34 0.36 O .24 NU Figure 5 .

1 . 1 5 0.9- 1 . 1 I I I I I I 1 .O5 1 0 . 9 5

CUBIC DUFFING SYSTEM: ANALYTICAL RESPONSE FIFTH SUPERH AR M ON1 C

MU

=

0.025; OTHER DATA IN TABLE 1.

LINEAR LAMBDA-= -0 LAMBDA = 1.90 LAMBDA = 1.00 LAMBDA = 1.75 D.A.F. -__-- 0.10 0.12 0.14 0.16 0.18 NU 0.20 0.22 0.24 0.26 Figure 6.

FIFTH ORDER DUFFING SYSTEM: ANALYTICAL RESPONSE FIFTH SUPERHARMONIC

MU = 0.025; OTHER DATA IN TABLE 2.

LINE AR LAMBDA-= -0 LAMBDA = 1.85 LAMBDA = 1.00 LAMBA = 1.75

D.A.F. - _ _ _ -

1

i I I I I I I I

S.D.O.F.: LINEAR SYSTEM

28*XTT

+

3*XT

+

134*X

=

87*cos(t)

DISPLACEMENT 0 . 5 O - 0 . 5 - 1 1 0 . 8 0 . 6 0 . 4 0 . 2 O

S.D.O.F.:

LINEAR SYSTEM

28*XTT

+

3*XT

+

134*X

=

87*cos(t)

I I I I I I I I

O 5 10 15 20 25 30 35 40 45

S.D.B.F.:

LINEAR SYSTEM

28*XTT

+

3*XT

+

134*X

=

W*cos(t)

5 0 O -50 -100 150 1 O 0 50 O -50 -100 -150

S.B.O.F.:

LINEAR 5YST€M

28*XTT

+

3*XT

₊

134*X

=

87*cos(t)

GENERALISED FORCE (MN)

1 O 0

O 5 10 15 20 25 30 35 40 45

1 I I I I I I I I 0 . 5 O - 0 . 5

-1

SDOF:

LINEAR SYSTEM

28*XTT

+

3*XT

+

134*X

=

87*cos(t)

x2

(Ml

AN ALY TI CAL SO LUTI ON

SIMPMOD PROGRAM --____---

Figure 9a.

SDOF:

LINEAR SYSTEM

1 . 5 1 0 . 5 O - 0 . 5 - 1 - 1 . 5 -2 O 5 10 15 20 25 30 35 40 T ANALYTICAL RAMP _{_ _ _ -} ~ _ _ _ _ - NO RAMP F i g b z e Ob.

S.D.O. F.

LINEAR

SYSTEM

WITH SQUARE

COSINE

EXCITATION

. 2 D.A.F. SIMPMOD 0.16 0.18 0.20 NU 0.22 1 . 8 1 . 6 1 . 4 1 . 2

S.D.O.F.: CUBIC

DUFFING SYSTEM

MU=8.025

D.A.F. 0 LAMBDA=1.5 ~ _ _ _ 0.24 0.26 Figure 10. 1 0.15 0.20 0.25 0.30 NU 0.35 0.40 Figure 11.

S. D.

O.

F.

:CUB IC DUFFING SY ST€M

1

&1=0.025

I . SIMPMOD IJ DNONLIN D.A.F. 1 . 8 1 . 6 1 . 2 0.25 0.30 NU 0.35 0.40 Figure 12.

S. D.O.

F.

:CUB

IC

DUFFING SYSTEM

MU=0.025,

ETA=0.1667

LAMB DA=1.75 LAM4 DA= lL5

_ _ ~ _ _ - LAM BOA=1.9 D.A.F. ~ _ _ _ 1 . 4 1 . 3 1 . 2 1 . 1 1 1 1 1 1 0.14 0.16 0.18 0.20 NU 0.22 0.24 Fioure 13.

1 . 4

1 . 3

1 . 2

1 . 1

1

S.D.B.F.

:CUBIC

DUFFING SYSTEM

MU=0.025, ETA=0.1667

LAMBDA=1.9 D.A.F. I I 1 i i i i i i 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 NU F i g u r e 14.

S.D.B.F.:

FIFTH ORDER DUFFING SYSTEM

MU=0.025, EBA=0.12

LAMBDA=1.75 i AM B D A=l.5 _{~ _ _ - -} D.A.F. 1 . 4 - 1 . 3 - 1 . 2 ~ 1 . 1 - 1 - 0.20 0.25 0.30 0.35 0.40 0.15 NU F i g u r e 15.

S.D.O.F.:

FIFTH

ORDER DUFFING SYST€M

MU=O.O25, EV~0.12

1 1 i i i i i i i

0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 NU _{F i g u r e 16a.}

S.D.O.F.: FIFTH ORDER DUFFING SYSTEM MU=0.025, ETA=O.i2 COMPARISON NUMERICAL/ANALYTICAL 1 . 2 5 1 . 2 1 . 1 5 1 . 1 1 . O 5 1 0 . 9 5 NU F i g u r e 16b.

S.D.O.F.:

FIFTH ORDER DUFFING SYSTEM

MU=O.O25,

ETA=0.12

1 1 . 1 1 . 1 1 .

S.D.O.F.

1 . 6 1 . 5 1 . 4 1.3 1 . 2 1 . 1 1 15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 NU F i g u r e 18.

1 . 4

1 . 3

1 . 2

1 . 1

S.D.O.F.: DIFFERENT EXCITATION MODELS

BDA

= 1.5,

MU

=

0.025

1 . 6 1 . 5 1 . 4 1 . 3 1 . 2 1 . 1 1 0.14 0.16 0.18 0.20 0.22 0.24 NU Fig

.O.F.: CUBIC DUFFING SYSTEM

LAMBDA

= 1.5,

MU

=

0.025

D.A.F. SIMPMOD 0 1 1 1 1 1 0.14 0.16 0.18 0.20 0.22 02-4 NU FigUre 2 0 .

program restfor C C C C C C C C C C

This program calculates the restoring force of the fifth order

Duffing system. It can easily be modified to calculate another

order Duffing system.

In fact it can be used for obtaining the zeropoint of any other

n order polynomial:

n n-1

a(n)x +a(n-1)x

+

...

+

a(1)x

+

a ( 0 ) = O

The solution will be stored in stiff.out and is immediately ready for cuechart.

ex ter na1 zpolr

integer complex real real real real err, i, j, nl, n2

,

m z(6) k, f, eta, lambda, cl, c2, c3 begineta, endeta, stepeta

beginlambda, endlambda, steplambda

a(6), c(3), x1(1000), y1(1000), x2(1000), y2(1000)

open(unit=3, file='stiff.out', type='new')

write(3,*) 'GENERATE A PLOT.'

write(3,") 'EVERY CURVE TBICKNESS 1.'

write(3,*) 'X AXIS LABEL TEXT IS "DECK DEFLECTION (M)".' write(3,*) 'Y AXIS LABEL TEXT IS "LAMBDA".'

write(3,*) 'INDEPENDENT GRID ON.'

write(3,*) 'DEPENDENT GRID ON.'

write(3,*) 'TITLE TEXT IS "RESTORING FORCE"' write(3,*) 'LEGEND ON.'

write(3,*) 'INPUT DATA.'

k = 134.0 f = 58.0 beginlambda = 0.1 endlambda = 3.0 steplambda = 0.01 write(6,*) 'begineta =>

'

read(5,*) begineta read(5,") endeta write(6,*) 'stepeta =>

'

read(5,") stepeta write(6,") 'endeta =>

'

do 50 eta = begineta, endeta, stepeta

nl=0

n2=0

do 51 lambda = beginlambda, endlambda, steplambda

a(ï)=-eta*k/f a(2)=0. a(3)=0. a(4)=0. a(5)=k/f a(6)=-lambda

call zpolr(a, 5, z , err)

j=û

52 51 1 0 0 5 0

+

+

5 4 120 121

+

continue continue if ((aimag(z(i)) .eq. 0.0) .and. (real(z(i)f .gt. 0.0)) c(j)=real(z(i)) then j=j+l endif continue if ( j .eq. O) goto 100 if ( j .eq. i) then nl=nl+l xifnl) = c(1) yl(n1) = lambda endif if ( j .eq. 2) then cl = min(c(l),c(2)) c2 = nax(c(l),c(2)) nl=nl+l xl(n1) = cl yl(n1) = lambda n2=n2+1 x2(n2) = c2 y2(n2) = lambda endif if ( j .eq. 3) then cl = min(c(l),c(2),c{3)) c3 = max(c(l),c(2),c(i)) do 54 m=1,3

if ((c(m) .ne. cl).and.(c(m) .ne. c3))

c2=c(m) then endif continue nl=nl+l xi(n1) = cl yl(n1) = lambda n2=n2+1 x2(n2) = c2 y2(n2) = lambda endif unit=3

write(unit,lll)

'

"ETA =

' ,

ETA,

' '"

write(unit,l20) (xi(i), yi(i), i=l,nl) write(unit,l20) (x2(i), y2(i), i=n2,1,-1)

format(t2, f, 3x, f)

format(a,f9.2,a)

write(3,*)

'

END OF DATA. i

write(3,*)

'

GO.

stop end

program nulpunt

+

+

52

This program calculates the zeropoint of the equation: (-eta*k/f)*x**5

+

(k/f)*x

-

lambda = O .

lambda is a input parameter

Of course you can also calculate the roots of any other equation with this program.

ext er na1 zpolr

integer complex real real err, j,

m

z(6)

k, f, eta, lambda, zeropnt a(6), c ( 3 ) k = 1 3 4 . 0 f = 58.0 eta = 0.12 write(6,*) 'lambda =>

'

read(5,*) lambda a(l)=-eta*k/f a(2)=0. a(3)=0. a(4)=0. a(5)=k/f a(ó)=-lambda

call zpolr(a, 5, z, err)

j=O do 52 i = 1, 5 if ((aimag(z(i)) .eq. 0.0) .and. (real(z(i)) .gt. 0.0)) then j=j+i c(j)=reai(z(i)) endif continue if ( j .eq. O ) then

write(6,") Geen oplossing'

got0 1 0 0 endif if ( j .eq. 1) zeropnt=c(l) if ( j .eq. 2) zeropnt=min(c(lf,c(2)) if {j .eq. 3) zeropnt=min(c(l),c(2),~(3)) write(6,*)

'

nulpunt=' write(6,*) zeropnt 100 stop end

2 3 4 5 6 7 € C 1 C234567890123456789012345678901234567890123456789012345678901234567890123456789C PROGRAM SIMPMOD C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C

c

C C C C C C C C C C C C C C C C C C C

is, in its most sophisticated form, a generalized single degree of

freedom concept of a jacket-type offshore structure, the loading of of which is niodelled with a regular wave (Airy-theory) with relative velocity and free surface effects incorporated. The structure may be conceived as a "single-" or "double stick" model (NSTICK=l or 2), of which the latter case, where WIDTH specifies the interspace between the legs, seems to produce the most realistic results.

Linear, cubic (Duffing) and hysteresis (Bouc-Wen) stiffness

characteristics are optional by specification of ISTIFF=1,2 or 3

respectively. Generalized stiffness characteristics may be established in relation to the magnitude of the generalized FORCEG, as a result of which, deflection equivalent to the real structure will be produced. In subroutine HYDCOF, the calculated parameters result from integration along the jacket length (see report). Static or dynamic anaysis is

specified by IANALY=1,2 respectively. The specifications for IEXCIT and IDAMP, which specify the excitation and damping type, are followed

when IWAVE=O. When IWAVE=l, IEXCIT and IDAMP are set to 3 internally,

due to which wave loading and hydrodynamic damping are enabled. For

IWAVE=2, static wave loading is asplied, and relative velocity and inertia effects are disabled.

Also, more simple excitation models can be specified, such as (multi-

frequency) harmonic excitation (IEXCIT=l) or an arbitrary user-defined

*atime-dependent excitation (IEXCIT=2,FUNCTION EXCIT). Damping is linear

if IDAMP=l (DMPFAC= C/(2*SQRT(M*K))), user-defined if IDAMP=2 (FUNCTION

DAMP) or hydrodynamic if IDAMP=3.

DATA CHANNELS:

FOR001

-

Data input

UNIT20

-

Output file: SIMPMOD DIS.OUT Displacement history

UNIT21

-

SIMPMODRFC.OUT Restoring force history

UNIT22

-

SIMPMOD-LOD.OUT Loding history

UNIT23

-

SIMPMODRMP.OUT - Ramp history

VARIABLE LIST: NMAX

-

IRKOUT

-

EPS H1 HMIN

-

ITOUTP

-

TIME

-

T DX DXlST

-

IANALY

-

ISTIFF

-

IDAMP

-

IEXCIT

-

-

-

-

-

IWAVE

-

NCOMPS -

Asummed maximum number of samples

Supplies information on Runge-Kutta integration (=1/0)

Accuracy for Runge-Kutta

-

integration.

-

First increment

Mininmum increment

-

Supplies information on Newton-Rhapson iteration ( = 1 / 0 )

Time

Actual time

Displacement increment

Static(=l),Dynamic(=2)

Linear(=l),Cubic(=2),Hysteretic(=3)

Linear(=l),Time dependent(=2),Relative

-

velocity(=3)

-

Harmonic(=l),

Nonlinear: Rel.velocity and free surf. effect ingn.(=3)

Explicite definition of full nonlinear wave load.

IF.EQ.0 : Calculated IEXCIT and IDAMP as specified

-

-

-

-

-

due to integration (zero hit)

-

-

(=2) I

-

.EQ.l : Sets IEXCIT=3 and IDAMP=3 (hydrodyn. case)

.EQ.2 : Sets Newmark-Beta param's=O. (Static wave)

C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C

-

TO TEND

-

DT GAMMNB

-

BETANB

-

ETOL

-

MAXIT

-

ENRAT

-

NRAPTS

-

-

TRAMP

-

RMPFAC

-

FACNEG

-

SCLINC

-

NCYC

-

XMASS

-

XMSTOP

-

XMSJAC

-

XMSADD

-

DMPFAC

-

STIFF

-

XKTAN

-

CUBFAC

-

HALPHA...HN NSTICK

-

WIDTH

-

YJACLT

-

CDND

-

CMND2

-

YMSL

-

L WAVHGT

-

WAVPER

-

WAVLTH WAVETA

-

WK WF XCREST

-

RHO CA FI DRAMP1

-

DRAMP2

-

-

-

-

-

-

-

XINAMP

-

Begin time of dynamic analysis (if IANALY=2)

End

Time increment

Gamma factor in Newmark-Beta integration routine Beta

Energy tolerance for Newton-Rhapson iteration stop Maximum number of iterations.

Energy ration to measure against ETOL

Number of ramp points (TRAMP,RMPFAC) for multilinear ramp (NRAPTS<20)

Ramp time over which ramp must be applied

Continuous ramp factor after TRAMP (s)

Fraction of RMPFAC for load in reversed part of static analysis (if IANALY=l)

increment for static analysis (if IANALY=l)

Maximum number of static analysis (

Generalized mass

For physical dynamic base shear, masses below are required :

Physical top side mass

---

---

1

-

jacket mass (homogeneously distributed assumed:

added

-

-

-

-

Relative damping factor in perc. of critical damping (Initial) Linear stiffness

Tangent stiffness

Nonlinear stiffness part in cubic restoring force model

as fraction of linear stiffness (Duffing stiffness) Coefficients specifying Bouc-Wen hysteresic stiffness Number of sticks (If NSTICK=2 then WIDTH specifies the

interspace between the legs).

Interspace between the legs if NSTICK=2. Jacket length (called LJ in the program) Generalized drag coefficient

Mean sea water level

Instantaneous water height along jacket Wave height

-

period =GRAV*WAVPERk*2/(2.*PI)]

-

water elevation

-

number

-

frequency

-

crest position before ( < O ) or after ( > O ) jacket

-

inertia

-

Specific water density

Initial cyclus angle if XCREST different from zero

Proccess angle of excitation

Help cmplitude for drag force calculation (Don't confus (...with RAMP)

-

inert ia

Explanation of the next FORCEG variables:

F G/R DR/IN MX/CO Force Generalized/Realistic DRag/Inertia MaXimum/COrresponding FGDRMX

-

FGINMX

-

FGDRCO

-

FGINCO

-

FRDRMX

-

FRiNMX

-

Maximum generalized drag force inertia force

Generalized drag force occurring at FGINMX.

inertia FGDRMX

.

Maximum realistic drag force inertia force

C C C C C C C C C C C C C C C C C C C C C C C C C C C C

+

+

FRDRCO

-

Realistic drag FORCEG occurring at FRINMX.

FRINCO

-

inertia FRDRMX

.

BSRGMX

-

Maximum generalized base shear.

BSRRMX

-

_realistic

WLOGMX

-

generalized wave load.

WLORMX

-

_realistic

XMAX

-

realistic (=generalized) deflection.

DFACMX

-

realistic (=generalized) dimensionless

damping factor.

z1,z2

-

Auxilary variables for hysteresis

ISAMPL

-

Sample counter

CALLED SUBROUTINES:

ODEINT

-

Ordinairy differential equation solver based on

fourth-order Runge-Kutta integration.

FUNCTIONS:

EXCIT(T)

-

User defined function if the excitation is

DAMP(T)

-

User defined function if the damping is

a non-harmonic only time-dependent function (EXCIT=2).

nonlinear (a non-fixed value).

EXTERNAL ODEINT,DERIHY

REAL LJ

CHaRACTER*80 RECORD, QUOTE*l

DATA QUOTE/' ' I ' /

PARAMETER (NINCST=1000, NCOMPM=lO,

GRAV=9.815)

COMMON /PATH/ KMAX,KOUNT,DXSAV,DXRNGE(2OO),Z(10,200) COMMON /HYSPAR/ HALPHA,HETA,HA,HNU,HBETA,HGAMMA,HN

DIMENSION A(O:7),ZSTART(l),SCALE(NINCST),HYDARR(2,5),

+

DZDX(i),EXAMP(NCOMPM),EXPER(NCOMPM),EXPHSE(NCOMPM),

+

TRAMP(20),RMPFAC(20) OPEN(UNIT=l ,FILE='FOROOl',STATUS='OLD') OPEN(UNIT=3 ,FILE='FOROO3',STATUS='OLD') OPEN(UNIT=20rFILE='SIMPMOD DIS.OUT',STATUS='UNKNOWN') OPEN(UNIT=21,FILE='SIMPMOD~RFC.OUT',STATUS='UNKNOWN') OPEN(UNIT=22,FILE='SIMPMOD~LOD.OUT',STATUS='UNKNOWN') OPEN(UNIT=23,FILE='SIMPMOD~RMP.OUT',STATUS='UNKNOWN') OPEN(UNIT=24,FILE='SIMPMOD~DAF.OUT',STATUS='UNKNOWN') OPEN(UNIT=25,FILE='SIMPMOD-RUB.OUT',STATUS='UNKNOWN') - PI=4.*ATAN(1.) C KIN = 1 KTMP = 3 PDL PDL PDL PDL PDL

C 1 0 15 C PDL PDL DO J = 1, 1000 PDL READ(KINf1O,ERR=15) RECORD PDL FORMAT(A80) PDL

IF ( RECORD(1:l) .NE. QUOTE ) THEN PDL

WRITE(KTMP,lO) RECORD PDL XOUNT = KOUNT

+

1 PDL END IF PDL END DO PDL CONTINUE PDL PDL REWIND KTMP PDL PDL READ(3,*)IANALY,IFREQfIDUF,IRKOUT,ITOUTPfISTIFFfIDAMPf PDL IEXCIT,IWAVE,EPS,H1,HMINfTOfTENDfDTfTOUTP, PDL GAMMNB,BETANB,ETOL,MXIT,TSAMPL,FACNEG,SCLINC, PDL NCYC,NRAPTS,(TRAMP(I),RMPFAC(I),I=l,NRAPTS), PDL XMASS,XMSTOP,XMSJAC,XMSADD,DMPFAC,STIFF,CUBFAC, PDL HALPHA,HETA,HA,HNU,HBETA,HGAMMA,HN, PDL YJACLT,CDND,CMND2,NSTICKfW1DTH, PDL YMSL,WAVHGT,WAVPER,XCREST,RHO,BEGINNU,ENDNU,STEPNU, PDL NCOMPS,(EXAMP(I),EXPER(I),EXPHSE(I), PDL I=l,NCOMPS) PDL PDL WRITE(6,*)IANALY,IFREQ,IDUF,IRKOUT,ITOUTPfISTIFFfID~Pf PDL IEXCIT,IWAVE,EPS,H1,HMIN,TO,TEND,DT,TOUTPf PDL GAMMNB,BETANB,ETOL,MAXIT,TSAMPLfFACNEG,CCLINCf PDL NCYC,NRAPTS,(TRAMP(I),RMPFAC(I),I=l,NRAPTS), PDL XMASS,XMSTOP,XMSJAC,XMSADD,DMPFAC,STIFF,CUBFAC, PDL HALPHA,HETA,HA,HNU,HBETA,HGAMMA,HN, PDL YJACLTfCDND,CMND2,NSTICK,WIDTH, PDL YMSL,WAVHGT,WAVPER,XCRESTfRHO,BEGINNU,ENDNUfSTEPNUf PDL -.--- -

NCOMPS, (EXAkP( I) ,EXPER( I) ,EXPHSE( I)

,

I=l,NCOMPS) Make the output files ready for input in CUECHART.

DO IUNIT = 20,24,1

WRITE(IUNIT,*) 'GENERATE A PLOT.'

WRITE(IUNIT,*) 'EVERY CURVE THICKNESS 1.'

ENDDO

WRITE(24,*) 'EVERY CURVE INTERPOLATION POLY3.' WRITE(24,*) 'EVERY CURVE SYMBOL COUNT ON.' WRITE(24,*) 'X AXIS LABEL TEXT IS

WRITE(24,*) 'Y AXIS LABEL TEXT IS WRITE(20,*) 'X AXIS LABEL TEXT IS WRITE(21,*) 'X AXIS LABEL TEXT IS WRITE(22,*) 'X AXIS LABEL TEXT IS WRITE(23,*) 'X AXIS LABEL TEXT IS WRITE(20,*) 'Y AXIS LABEL TEXT IS WRITE(21,*) 'Y AXIS LABEL TEXT IS WRITE(22,*) 'Y AXIS LABEL TEXT IS WRITE(23,*) 'Y AXIS LABEL TEXT IS

IINUII 1 "D.A.F.

".

'

"TIME (S) I ' .

'

"DISPLACEMENT (M) "

.

'

"TIME (S)

".

'

"TIME ( S )

".

'

"DISPLACEMENT (M) I ' .

'

"RESTORING FORCE (MN

1

' I .

'

"GENERALISED FORCE (MN) "

.

'

"RAMP"

.

'

DO IUNIT = 20,24,1

WRITE(IUNIT,*) 'INDEPENDENT GRID ON.'

WRITE(IUNIT,*) 'DEPENDENT GRID ON.'

WRITE(IUNIT,*) 'TITLE TEXT IS "S.D.O.F.'"

WRITE(IUNIT,*)

'

WRITE(IUNIT,*) 'LEGEND ON.' WRITE(IUNIT,*) 'INPUT DATA.' WRITE(IUNIT,*) I' I ; 9 PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL

WRITE(IUNIT,*) ' I ' ' I ' WRITE(IUNIT,*) ENDDO CONTINUE TEND2=TEND IANALY=l IFREQ2=IFREQ IFREQ=l IF (IFREQ.EQ.2) CONTINUE 5002 IF (IFREQ.EQ.1) GOT0 5000

. . .

* *

FREQUENCY LOOP

. . .

**

XNATPER=2*PI/SQRT(STIFF/XMASS)

DO WHILE(SEG1NNU .LE. ENDNU)

IF (IEXCIT.EQ.1) EXPER(l)=XNATPER/BEGINNU

IF ((IEXCIT.EQ.2).OR.(IEXCIT.EQ.3)) WAVPER=XNATPER/BEGINNU

. . .

* *

NUMBER OF INCREMENTS FOR SELECTED ANALYSIS

* *

. . .

5 0 0 0 CONTINUE

IF((IANALY.EQ.2).AND.(IWAVE.NE.O))THEN

C...FORCEG to hydrodynamic case

IDAMP=3 IEXCIT=3 IF(IWAVE.EQ.2)THEN ENDIF TSAMPL=T O ELSE ENDIF IF((IANALY.EQ.2).AND.(.NOT.(IWAVE.EQ.2)))THEN

C...Damping and excitation as specified.

G=GAS?IMNB B=BETANB A(O)=l./(G*DT**2) A(l)=B/(G*DT) A{2)=l./(G*DT) A(3)=1./(2.*G)-l. A(4)=B/G-1. A(5)=DT/2.*(B/G-2.) A(G)=DT*(l.-B) A( 7 ) =B*DT DO I=0,7 ENDDO IF(IANALY.EQ.1)THEN T=-O. 5 TEND=REAL(NCYC) ELSE A(I)=O.O ENDIF ENDIF PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL DX=O. RAMP=O. DXl=O. KMAX=1 O O DXSAV=l

. . .

* *

INITIAL CONDITIONS

* *