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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
June 1989
CONTENTS :
1.
2 .
3.
4 .
LIST OF SYMBOLS
ACKNOWLEDGMETJT
SUMMARY
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 6X 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 rC
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~ = 489. (m)
Tw = 17.7 (S)
1.2 DERIVATION OF 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(Yrt) = 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 ut-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 2a k = - 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 Y HY Y rY
- 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 mf 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
-XI@ I U @ +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 [31.
2.1 PARAMETERS OF 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
nonlinear static deflection at load AF
- 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 (T T1r ..., T,)+U (T T ..., Tn)+ ( 1 7 ) o o' 1 o' 1' ... +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 (2DoD3 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 -2i(D +P)Ae O+6ûA2ÄA2e '+10A A e O+30u4e O+A5e 1
- 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 -[ 15/2A2r~+10/64r4+i5A4+~] 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 yA = 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 r2 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.
A'
- 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 )+~Acos(~.~)+~(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 = (2A1+lrAI O
(33)
(34)
The equation (28) can be rewritten to:
[ ( 1 0 / 6 4 ) 2 Ir~0+[150/64A 2 8 lrA+[225/4A 4 +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'
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 Ei]
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)
The input parameters are shortly discussed in 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 cosot-Bo sinot 2 2
Substituting (38) and (39) in (37)
3.2 TWO EXAMPLES
The first diffenrential equation which will be solved analytically and with
(37)
(39)
2 2 28(-Ao cosot-Bo sinwt)+3
This equation holds fo r all t, so:
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)
-Aosinwt+Bwcosot = 0 (44)
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
- 15 - Duffing system: numerical
* 2 1/2F(K-4Ma )
(K-4Mw ) +4C 0 2 2 2 2 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 it 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 input parameters.
- 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 t h e hydrûdynamic zodel.
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, 1982 by Prentice-Hall, Inc., Englewood Cliffs, New Jersey.
TABLES
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: Xdyn u =
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. (Mkg)
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 28 .
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 105 .
WAYdGT 3 0 . 5
ENDNU 0.25
EXAMP 58 .
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:
EXCIT = 58.*(COS(~W*T))*(COS(WF*T))
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.
NMPT 2
XMASS 28 .
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.
20 .
XMSTOP 20 .
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 . 20 .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 .
20.
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 0.05
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 0.25
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 2 3 4 5 6 7 1 '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) EXCIT = 58.*(ABS(CCS(WF*T))*CCS(-WF*T)-0.2*SIN('~*T))
drag o n l y drag+iner t ia
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 28 e
HALPHA 0.02
YJACLT 184 .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.
20.
XMSTOP 20 *
BETA 1.
CDND 105.
WAVHGT 3 0 . 5
ENDNU 0 . 2 6
EXAMP 87 e
IDUF 3
IEXCIT 2
DT 0.05
SCLINC 0 . 0 2
RMPFAC O . 1.
XMSJAC 22.4
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 0.001 2t
STIFF CUBFAC DMPFAC 1 3 4 . O e 1667 0.025
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 (two 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):
EXCIT = 87.*(0.98*COS(WE'*T)*ABS(COS(COS(WF*T))-0~2*âIN~WF*T~)
Figure 1. Typical Jacket-typ?
O f f s h o r e S t r x t i l r e
, M * $ ( y ) = 1 - cos- f l y
2 L
I w
VO H :x Principle of Virtual Work:
:. Mass :
3e :
J o
Figure 2, ShaDe function and qeneralized SDOF properties.
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) Figcre 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 gMEDA= 1.- - LAMBDA = 1.00 . LAMBDA = 1.9 LAMBDA = 1.5
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 . 1
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 TIME (S ) Figure 8b.
S.D.B.F.: LINEAR SYSTEM 28*XTT + 3*XT + 134*X = W*cos(t)
50
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 TIME (S) Figure 8d.
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 Figbze 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 Figure 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 Figure 16a.
S.D.O.F.: FIFTH ORDER DUFFING SYSTEM MU=0.025, ETA=O.i2 COMPARISON NUMERICAL/ANALYTICAL
1 . 2 5
1 . 2
1.15
1 . 1
1 . O 5
1
0 . 9 5
NU Figure 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 Figure 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 .
APPENDU
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
do 52 i = 1, 5
52
51
1 0 0
50
+ +
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
PPEN d
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
PPENBO B w
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) Number of harmonic component if IEXCIT=I
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 * *
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T=TO x1=0. XTl=O. XTTl=O. x2=0. XT2=0. XTT2=O. F1=0. F2=0. z1=0 z2=0. XKTAN=STIFF XCROSS=O. BSRGMX=O. BSRRMX=O. WLOGMX=O. WLORMX=O. xMAx=o . DFACMX=O . FGDRMX=O. FGINMX=O. FRDRMX= O FRINMX=O. FGDRCO=O . FGINCO=O. FRDRCO=O. FRINCO=O. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
* * STRUCTURAL INERTIA PROPERTIES ** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LJ=YJACLT FACLJ =1.-2./PI FACMSL=1.-2./PI*SIN(PI*YMSL/(2.*LJ)) XMSEFF=XMSJAC*FACLJ+XMSADD*FACMSL+XMSTOP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
* * WAVE CALCULATION (used if specified) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * *
WAVLTH=GRAV*WAVPER**2/(2.*PI) WK=2.*PI/WAVLTH WF=2.*PI/WAVPER CA=WK*XCREST WAVETA=O . DRAMP1=0.5*RHO*CDND*(PI*WAVHGT/WAVPER)**2*EXP(-2,*WK*YMSL) DRAMP2=0.5*RHO*CDND*PI*WAVHGT/WAVPER*EXP(-WK*YMSL) XINAMP=PI/4.*RHO*CMND2*(2.*PI**2*WAVHGT/WAVPER**2)*EXP(-WK*~MSL) IF(NSTICK.NE.2) WIDTH=O.
C ***DRAG AND INERTIA AMPLITUDES***
*****************R*********************************** ** ** **
** ** ** TIME LOOP
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C C
ISAMPL=O DO WHILE(T.LT.TEND)
ISAMPL=ISAMPL+l IF(IANALY.EQ.1)THEN RAMP=RAMP+SCLINC
* * * STATIC FORCEG APPLIED *** FORCEG=EXAMP(l)*RAMP c=o.
PRINT*,'**** RAMP=',RAMP,' * * * * I
ELSE
IF(IANALY.EQ.2)THEN T=T+DT CALL RAMPCA(T,IWAVE,NRAPTS,TRAMP,RMPFAC,RAMP) CALL DYNFRC(T,IEXCIT,NCOMPS,EXAMP,EXPER,EXPHSE,PI,
+ WAVHGT,CA,LJ,WF,WK,YMSL,NSTICK,WIDTH, + XINAMP,RAMP, -!- GUU,GXU,FXU,FUU,FUT,GXX,WAVETA,HYDARR,FGINER, + FRINER,FORCEG)
IF(IDAMP.NE.3) C...........Simple damping (Linear or time dependent user defined)
+ CALL SMPLDA(T,IDAMP,DMPFAC,XWSS,STIFF, + C)
ENDIF ENDIF
FlEFF=Fl+C*XTl+XMASS*XTTl
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * * * * * * NEWMARK-BETA / NEWTON-RAPHSON INTEGRATION / ITERATION LOOP * * * * * * . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
J=-1 DO WHILE((ABS(ENRAT).GE.ETOL).OR.(J.EQ.-1)) J=J+1
C...........J=O corresponds with integration hit. C...........J>O corresponds with following iteration hit(s).
C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C ** UPDATED VELOCITY AND ACCELERATION * *
IF(IEXCIT.EQ.3) THEN IF(NSTICK.EQ.2)THEN FGDRAG=O. FRDRAG= O . c=o. ANGHIT=WK*O.S*WIDTH DO K=l,NSTICK
IF(K.EQ.l) ANGL=CA-ANGHIT IF(K.EQ.2) ANGL=CA+ANGHIT GUU=HYDARR(K,l) GXU=HYDARR(K,2) GXX=HYDARR(K,3) FUU=HYDARR(K,4) FUT=HYDARR(K,5) FGDRAG=FGDRAG+0.5*ABS(DRAMPl*GUU*COS(WF*T+ANGL) -
FRDRAG=FGDRAG+0,5*ABS(DRAMPl*FUU*COS(WF*T+ANGL) - + DRAMPS*GXU*XT2)*COC(WF*T+ANGL)*RAMP/l.E3
+ DRAMP2*GXU*XT2)*COS(WF*T+ANGL)*RAMP/l.E3
C=C+0.5*ABS(DRAMP2*GXU*COS(WF*T+ANGL) - + 0.5*RHO*CDND*XT2*GXX)*RAMP/l.E3
ENDDO ELSE
C...............Assumed: NSTICK=l ANGL=CA FGDRAG=ABS(DRAMPl*GUU*COS(WF*T+ANGL) -
FRDRAG=ABS(DRAMPl*FUU*COS(WF*T+ANGL) - + DRAMP2*GXU*XT2)*COS(WF*T+ANGL)*RAMP/l.E3
+ DRAMPS*GXU*XT2)*COS[WF*T+ANGL)*RAMP/i .E3
+
C C C
C C C C
C 2000
C 2001
C 2002
+
2010
C C C
+
C
C=ABS(DRAMP2*GXU*COS(WF*T+ANGL) - 0.5*RHO*CDND*XT2*GXX)*RMP/l.E3
ENDIF
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * * GENERALIZED AND REALISTIC EXCITATIONS ** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FORCEG=FGDRAG+FGINER FORCER=FRDRAG+FRINER
ENDIF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * * CALCULATE TANGENT STIFFNESS DEPENDING * * ** ON SELECTED MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GOT0(2000,2001,2002) ISTIFF
* *
*** LINEAR STIFFNESS *** XKTAN=STIFF F2=XKTANkX2 GOTO 2010 *** RESTORING FORCEG MODEL (DUFFING) * * * XKTAN=STIFF*(l.-CUBFAC*FLOAT(IDUF)*X2**(1DUF-l)) F2=ST1FF*(X2-CUBFACkX2**IDUF) GOTO 2010 *** HYSTERETIC STIFFNESS: BOUC-WEN FORMULATION * * * CONTINUE NVAR=l ZSTART(l)=Z2 IF(DX.EQ.0.) DX=l.E-lO CALL ODEINT(ZSTART,NVAR,DXl,DX,EPS,Hl,HMIN,NOK,NBAD,
DERIHY,RKQC) Z2=ZSTART(1) F2=STIFFk(HALPHA*X2+(1.-HALPHA)*Z2) CALL DERIHY(DX,ZSTART,DZDX) XKTAN=STIFF*(HALPHA+(l.-HALPHA)*DZDX(l)) IF(IRKOUT.EQ.1)THEN WRITE(25,*) 'Number of successful R-K steps= ',NOK WRITE(25,*) ' - bad WRITE(25,*) 'Stored intermediate values = ',KOUNT
WRITE(25,*) ' '
= ' ,NBAD -
WRITE(25,") ' Z (integration) = ',Z2
ENDIF CONTINUE IF (XKTAN.LE.O.0) THEN WRITE(6,*) '***NEGATIVE TANGENT STIFFNESS***' IF (IFREQ.EQ.2) WRITE(6,") '***NU=',BEGINNU,'***' WRITE(6,*) '***T=',T,'***' GOTO 3020
ENDIF
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DX=l./(A(O)*XMASS+A(l)*C+XKTAN) * X2=X2+DX
** INTEGRATION/ITERATION **
(FORCEG-F2-XMASS*XTT2-C*XT2)
*** ENERGY TOLERANCE CHECK * * * IF(J.EQ.0) DXlST=DX ETOLDN=DXlST*(FORCEG-FlEFF) ETOLNM= DX * (FORCEG-F2-C*XT2-XMASS*XTT2) IF(ETOLDN.NE.O.)THEN
ELSE
ENDIF
ENRAT=ETOLNM/ETOLDN
ENRAT=100.
PDL PDL PDL PDL
PDL PDL PDL PDL PDL PDL
IF(ITOUTP.EQ.1)THEN WRITE(25,*) I * * * * TIME=',T,' * * * * ' WRITE(25,*) 'Iteration number = ',J WRITE(25,k) 'FORCEG= ',FORCEG,
WRITE(25,*) 'ETOLNM= ',ETOLNM,' ETOLDN= ',ETOLDN,
WRITE(25,*) ' ' WRITE(25,*) ' '
' Displacement= ',X2,' DX= ',DX
' ENRAT= ',ENRAT
ENDIF IF((J+l).GT.MAXIT)THEN PRINT*,'Number of iteration too large (>MAXIT): I ,
IF (IFREQ.EQ.2) THEN ' Execution aborted.'
PRINT*,'**** NU =',BEGINNU,' * * * * I
GOTO 3020 ENDIF
ENDIF ENDDO
+ +
+
C *** END OF ITERATION LOOP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * * REINITIALIZATION AND CALCULATION OF INERTIA EFFECTS * *
X1=X2 XTl=XT2 XTTl=XTT2 F1=F2 z1=z2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * * ANALYSIS CONTROL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3001 CONTINUE C *** STATIC ANALYSIS *** C WRITE(20,3900) X2,RAMP C WRITE(21,3900) X2,F2 C WRITE(22,3900) T,FORCEG
SCLMAX=RMPFAC(NRAPTS) IF(RAMP.LT.0.) SCLMAX=FACNEG*RMPFAC(NRAPTS) IF(ABS(RAMP).GE.SCLMAX)THEN
**
GOT0(3001,3002) IANALY
T=T+O. 5 SCLINC=-SCLINC
ENDIF IF ((IANALY.EQ.l).AND.(ABS(RAMP).LE.l.)) THEN XSTAT=ABS (x2 RMPXST=RAMP
ENDIF IF(T.GE.TEND)THEN
ENDIF GOTO 3010
PRINT*,'*** End of static analysis * * * I
3002 CONTINUE C *** DYNAMIC ANALYSIS ***
IF (IFREQ.EQ.1) THEN WRITE(20,3900) T,X2 WRITE(21,3900) X2,F2 WRITE(22,3900) T,FORCEG WRITE(23,3900) T,RAMP
ENDIF 3900 FORMAT(2F15.5)
PDL PDL PDL PDL
PDL PDL PDL PDL
C IF(IWAVE.EQ.2)THEN static wave load applied.
C
WRITE(*,'(/A,F10.3)')' WAVE HEIGHT = ',WAVETA WRITE(*,'(/12X,A)')' FDRAG FINER BSHEAR (MN)' WRITE(*,'(/A,3(FiO.3))') ' GENERALIZED ',FGDRAG,FGINER,F2 WRITE(*,'(/A,3(F10.3)//)')' REALISTIC ',FRDRAG,FRINER,BSRR
ENDIF
IF(T.GE.TSAMPL)THEN BSRR=FORCER-XMSEFF*XTT2 DFAC=2.*C*SQRT(XMASS*STIFF)
IF(ABS(BSRGMX).LT.ABS(F2)) BSRGMX=F2 IF(ABS(WLOGMX).LT.ABS(FORCEG)) WLOGMX=FORCEG IF((IEXCIT.EQ.2).OR.(IEXCIT.EQ.3))THEN IF(ABS(FGDRMX).LT.ABS(FGDRAG))THEN
C...........generalized parameters
C
FGDRMX=FGDRAG FGINCO=FGINER
ENDIF IF(ABS(FGINMX).LT.ABS(FGINER))THEN FGINMX=FGINER FGDRCO=FGDRAG
ENDIF IF(ABS(DFACMX).LT.ABS(DFAC)) DFACMX=DFAC
IF ((T.GE.TOUTP).AND.(ABS(XMAX).LT.ABS(X2))) THEN C. ............ realistic parameters
XMAX=ABS(X2) TXMAX=T ENDIF
IF(ABS(BSRRMX).LT.ABS(BSRR)) BSRRMX=BSRR IF(ABS(WLORMX).LT.ABS(FORCER)) WLORMX=FORCER IF(ABS(FRDRMX).LT.ABS(FRDRAG))THEN FRDRMX=FRDRAG FRINCO=FRINER
ENDIF IF(ABS(FRINMX).LT.ABS(FRINER))THEN FRINMX=FRINER FRDRCO=FRDRAG
ENDIF ENDIF
ENDIF
3010
IF(T.GE.TEND)THEN
ENDIF PRINT*,'*** End of dynamic analysis * * * I
CONTINUE ENDDO
IF (IFREQ.EQ.1) GOT0 5001 IF (IFREQ.EQ.2) CONTINUE
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * * END OF TIME DOMAIN ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
* *
C
3020
5 0 0 1
** DETERMINATION OF D.A.F. ** DAF=XMAX/XSTAT WRITE(24,39QO)BEGINNUI DAF BEGINNU=BEGINNU+STEPNU CONTINUE ENDDO
CONTINUE IF (IANALY.EQ.1) THEN IANALY = 2 IFREQ=IFREQ2
PDL PDL PDL PDL
PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL
TEND=TEND2 GOT0 5002
ENDIF
C Make output files ready for input in CUECHART
DO IUNIT = 20,24,1 WRITE(IUNIT,*) WRITE(IUNIT,*) 'EOD.' WRITE(IUNIT,*) 'GO.' WRITE(IUNIT,*) ENDDO CONTINUE
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * * END OF FREQUENCY LOOP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
* *
PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL PDL
C ** STORE DATA IN VECTOR ** PRINT*,' ' PRINT*,' ' PRINT*,'Data stored in data file assigned to channel FOR002' PRINT*,' ' PRINT*,'Recorderd for T > ',TSAMPL,' (S)'
WRITE(*,'(A)') ' GENERALIZED MAXIMUM (MN) : ' WRITE(*,'(/A,F10.3)') ' WAVE LOAD = ',WLOGMX WRITE(*,'(A,F10.3)') ' BASE SHEAR = ',BSRGMX WRITE(*,'(2(A,FlO.3))') ' DRAG FORCE = ',FGDRMX,
+ ' WITH CORRESPONDING IMER FORCE = ',FGINCO WRITE(*,'(2(A,FlO.3))') ' INER FORCE = ',FGINMX,
+ ' WITH CORRESPONDING DRAG FORCE = ',FGDRCO
WRITE(*,'(//A)') ' REALISTIC MAXIMUM (MN) : ' WRITE(*,'(/A,F10.3)') ' WAVE LOAD = ',WLORMX WRITE(*,'(A,F10.3)') ' BASE SHEAR = ',BSRRMX WRITE(*,'(2(A,FiO.3))') ' DRAG FORCE = ',FRDRMX,
+ ' WITH CORRESPONDING INER FORCE = ',FRINCO WRITE(*,'(2(A,FlO.3))') ' INER FORCE = ',FRINMX,
+ I WITH CORRESPONDING DRAG FORCE = ',FRDRCO
WRITE(*,'(/2(A,FiO.6))') ' MAXIMUM DISPLACEMENT = ',XMAX, PDL
I PDL WRITE(*,'(2(A,FlO.6))') XSTAT = ',XSTAT, 1 AT TIME = ' ,TXMAX PDL
+ I AT RAMP = ',RMPXST PDL
+
END
SUBROUTINE RAMPCA(T,IWAVE,NRAPTS,TRAMP,RMPFAC,RAMP) DIMENSION TRAMP(*),RMPFAC(*) LOGICAL RANGE
C...........If IWAVE=2 then static wave applied to f u l l scale. ENDRMP=TRAMP(NRAPTS) IF((T.GT.TRAMP(NRAPTS)).OR.(IWAVE.EQ.2))THEN
ELSE RAMP=RMPFAC(NRAPTS)
I=O
I=I+l 0 0 0 1 CONTINUE
+ + + +
+ C
1000 C
+
1001 C
1002 C
IF(T.GT.TRAMP(I))THEN
ELSE RANGE=.FALSE.
RANGE=.TRUE. TIMEl=TRAMP(I-l) TIME2=TRAMP(I)
XRMP2=RMPFAC(I) XRMPl=RMPFAC(I-l)
RAMP=XRMPl+(XRMP2-XRMPl)/(TIME2-TIMEl)*(T-TIMEl) ENDIF IF(.NOT.(RANGE))THEN
ENDIF GOTO 1
ENDIF RETURN END
SUBROUTINE DYNFRC(T,IEXCIT,NCOMPS,EXAMP,EXPER,EXPHSE,PI, WAVHGT,CA,LJ,WF,WK,YMSL,NSTICK,WIDTH, XINAMP,RAMP, GUU,GXU,FXU,FUU,FUT,GXX,WAVETA,HYDARR,FGINER,FRINER, FORCEG )
REAL LJ DIMENSION EXAMP(NCOMPS),EXPER(NCOMPS),EXPHSE(NCOMPS),
HYDARR(2,*) *** DYNAMIC FORCEG APPLIED * * * GOT0(1000,1001,1002) IEXCIT CONTINUE
* * * HARMONIC EXCITATION *** FORCEG=O DO I=l,NCOMPS FORCEG=FORCEG+EXAMP(I)*COS((T/EXPER(I)+
EXPHSE(I)/360.)*2.*PI) ENDDO FORCEG=FORCEG*RAMP GOTO 1010
* * * TIME DEPENDENT EXCITATION FUNCTION (user defined) * * * FORCEG=RAMP*EXCIT(T,WF,ANGL) GOTO 1010
*** WAVE F0RCEG:HYD. DAMP. + FREE SURF. EFF. + REL.VEL. * * * IF(NSTICK.EQ.2)THEN
CONTINUE
CONTINUE
FGINER=O. FRINER=O. ANGHIT=WK*0.5*WIDTH DO K=1,2
IF(K.EQ.l) ANGLzCA-ANGHIT IF(K.EQ.2) ANGL=CA+ANGHIT CALL HYDCOF(T,ANGL,WF,WK,YMSL,PI,LJ,WAVHGT,
GUU,GXU,FXU,FUU,FUT,GXX,WAVETA) FGINER=FGINER+0,5*XINAMP*FXU*(-CIN(WF*T+ANGL+PI))
*RAMP/l .E3 FRINER=FRINER+0.5*XINAMP*FUT*(-SIN(WF*T+ANGL+PI))
*RAMP/l . E3 HYDARR(K,l)=GUU HYDARR(K,2)=GXU HYDARR(K,3)=GXX HYDARR(K,4)=FUU HYDARR(K,S)=FXU
ENDDO
C...........Assumed: NSTICK=l ELSE
ANGL=CA CALL HYDCOF(T,ANGL,WF,WK,YMSL,PIrLJ,WAVHGT,
+
1 0 1 0
+ 1501 C
1502 C
1510
+
+ + + + + +
+
GUU,GXU,FXU,FUU,FUT,GXX,WAVETA) FGINER=XINAMP*FXU*(-SIN(WF*T+ANGL+PI))*RAMP/l.E3 FRINER=XINAMP*FUT*(-SIN(WF*T+ANGL+PI))*RAMP/l.E3
ENDIF CONTINUE RETURN END
SUBROUTINE SMPLDA(T,IDAMP,DMPFAC,XmSS,STIFF, C)
GOT0(1501,1502) IDAMP CONTINUE
* * * LINEAR DAMPING * * * C=2.*DMPFAC*SQRT(XMASS*STIFF) GOT0 1510
CONTINUE *** TIME DEPENDENT DAMPING (STATEMENT FUNCTION) *** C=DAMP (T)
CONTINUE RETURN END
FUNCTION EXCIT(T,WF,ANGL)
RETURN END
EXCIT=87.*(0.85*COS(WF*T)+O.l7*COS(3.*WF*T))
FUNCTION DAMP(T,WF,ANGL)
RETURN END
DAMP=ABS(COS(WF*T-ANGL))*COS(WF*T-ANGL)
SUBROUTINE HYDCOF(T,ANGL,WF,WK,YMSL,PI,LJ,WAVHGT, GUU,GXU,FXU,FUU,FUT,GXXTWAVETA)
REAL LJ WAVETA=O.5*WAVHGT*COS(WF*T+ANGL) XL=YMSL+WAVETA FI=(PI*XL/(S.*LJ)) GUU= 0,5/WK*(EXP(2.*WK*XL)-l.) - 1./((2.*WK)**2+
(PI/(2*LJ))**2) * ( EXP(2.*WK*XL)*( WK*COS(FI) + 0.5*PI/LJ*SIN(FI) ) -2.*WK )
GXU= 1.5/WK*(EXP(WKkXL)-1.)-2./(WK**2+(PI/(2.*LJ))**2)* (EXP(WK*XL)*(WK*COS(FI)+0,5*PI/LJ*SIN(FI) )-WK)+ 0.5/(WKk*2+(PI/LJ)**2)*(EXP(WK*XL)*(WK*COS(2.*F~)+ PI/LJ*SIN(2.*FI) ) -WK)
( EXP(WK*XL)*(WK*COS(FI) + 0.5*PI/LJ*SIN(FI) ) - WK) FXU= l./WK*(EXP(WK*XL)-1.) - l./(WK**2+(PI/(2.*LJ))**2)*
GXX= 2.5*XL+XL/PI*( 1.5*COS(2.*FI)-7.5*SIN(FI)-
HYDRODYNAMIC PARAMETERS FOR REALISTIC BASE SHEAR FUU= i./(2.*WK)*(EXP(2.*WK*XL)-l.) FUT= l./WK*(EXP(WK*XL)-1.) RETURN END
l./S.*COS(3.*FI))
SUEROUTINE DERIHY(DX,Z,DZDX) COMMON /HYSPAR/ HALPHA,HETA,HA,HNU,HBETA,HGAMMA,HN DIMENSION Z(l),DZDX(l) IF(DX.NE.O.O)THEN DZDX(l)=l./HETA*(HA-HNU*(HBETA*SIGN(l.,DX)*Z(l)*ABS(Z(l))
**(HN-1.) + HGAMMA*ABS(Z(l))**HN)) ELSE
ENDIF RETURN END
DZDX(l)=O.O
C C C C C C C C C C C C C
C
+ SUBROUTINE ODEINT(YSTART,NVAR,Xl,X2,EPS,Hl,HMIN,NOK,NBAD,DERIVS,
RKQC 1
page 559. Runge-Kutta driver with adaptive stepsize control. Integrate the W A R starting values YSTART from X1 to X@ with accuracy EPS, storinc intermediate results in the common block /PATH/. H1 should be set as a guessed first stepsize, HMIN as the minimum allowed stepsize (can be zero). On output NOK and NBAD are the number of good and bad (but retried and fixed) steps taken, and YSTART is replaced by values at the end of the integration interval. DERIVS is the user-supplied subroutine for calculating the right-hand side derivative, while RKQC is the name of the stepper routine to be used. PATH contains its own information about how often an intermediate value is to be stored. For convenient use, ODEINT is managed by driver ODEINTDR.
PARAMETER (MAXSTP~10000,NMAX~1O,TWO~2.O,ZERO~O.O,TINY~l.E-30) COMMON /PATH/ KMAX,KOUNT,DXSAV,XP(2OO),YP(10,200)
DIMENSION YSTART(NVAR),YSCAL(NMAX),Y(NMAX),DYDX(NMX) x=x1 H=SIGN[Hl,X2-X1) NOK=O NBAD=O KOUNT=O DO I=l,NVAR
ENDDO IF(KMAX.GT.0) XSAV=X-DXSAV*TWO DO NSTP=l ,MAXSTP CALL DERIVS(X,Y,DYDX) DO I=l,NVAR
ENDDO IF(RMAX.GT.O)THEN
DXSAV and KMAX preset.
Y(I)=YSTART(I)
YSCAL(I)=ABS(Y(I))+ABS(H*DYDX(I))+TINY
IF(ABS(X-XSAV).GT.ABS(DXSAV))THEN IF(K0UNT.LT.KMAX-1)THEN KOUNT=KOUNT+l XP(KOUNT)=X DO I=l,NVAR
ENDDO XSAV=X
YP(I,KOUNT)=Y(I)
ENDIF ENDIF
ENDIF IF([X+H-X2)*(X+H-Xl).GT.ZERO) H=X2-X CALL RKQC(Y,DYDX,NVAR,X,H,EPS,YSCAL,HDID,HNEXT,DERIVS) IF(HDID.EQ.H)THEN
ELSE
ENDIF IF((X-X2)*(X2-Xl).GE.ZERO)THEN
NOK=NOK+l
NBAD=NBAD+l
DO I=l,NVAR YSTART(I)=Y(I)
ENDDO IF(KMAX.NE.O)THEN KOUNT=KOUNT+l XP(KOUNT)=X DO I=l,NVAR
ENDDO YP(I,KOUNT)=Y(I)
ENDIF
RETURN ENDIF IF(ABS(HNEXT).LT.EMIN) PAUSE 'Stepsize smaller than minimum.' H=HNEXT
ENDDO PAUSE 'TOO many steps' RETURN END
SUBROUTINE RKQC(Y,DYDX,N,X,HTRY,EPS,YSCAL,HDID,HNEXT,DERIVS) C C page 558. C
PARAMETER (NMAX=10,PGROW=-0.20,PSHRNK=-0.25,FCOR=1./15., ERRCON=6.E-4) + ONE=l.,SAFETY=0.9
C
C EXTERNAL DERIVS
DIMENSION Y(N),DYDX(N),YSCAL
XSAV=X DO I=l,N
+ DYSAV(NMAX)
YSAV(I)=Y(I) DYSAV(I)=DYDX(I)
ENDDO H=HTRY
N) JTEMP(NMAX),YSAV(NMAX),
1 HH=O. 5*H CALL RK4(YSAV,DYSAV,N,XSAV,HH,YTEMP,DERIVS) X=XSAV+HH CALL DERIVS(X,YTEMP,DYDX) CALL RK4(YTEMP,DYDXtN,X,HH,Y,DERIVS) X=XSAV+H IF(X.EQ.XSAV) PAUSE 'Stepsize not significant in RKQC.' CALL RK4(YSAV,DYSAV,N,XSAVfH,YTEMP,DERIVS) ERRMAX=O. DO I=l,N YTEMP(I)=Y(I)-YTEMP(1) ERRMAX=MAX(ERRMAX,ABS(YTEMP(I)/YSCAL(I)))
ENDDO ERRMAX=ERRMAX/EPS IF(ERRMAX.GT.ONE)THEN H=SAFETY*H*(ERRMAX**PSHRNK) GOT0 1
HDID=H IF(ERRMAX.GT.ERRC0N)THEN
ELSE
ENDIF ENDIF DO I=l,N
ENDDO RETURN END
ELSE
HNEXT=SAFETY*H*(ERRMAX**PGROW)
HNEXT=4.*H
YfI)=Y(I)+YTEMP(I)*FCOR
SUBROUTINE RK4(Y,DYDX,N,X,H,YOUT,DERIVS) C C page 553. C
PARAMETER (WNAX=lO)
C
C DIMENSION Y(N),DYDX(N),YOUT(N),YT(NMAX),DYT(NMAX),DYM(NMAX)
HH=H*O. 5 H6=H/6. XH=X+HH DO I=l,N
ENDDO CALL DERIVS(XH,YT,DYT DO I=L,N YT(I)=Y(I)+HH*DYT(I
ENDDO CALL DERIVS(XH,YT,DYM DO I=l,N YT(I)=Y(I)+H*DYM(I) DYM(I)=DYT(I)+DYM(I)
ENDDO CALL DERIVS(X+H,YT,DYT) DO I=l,N
ENDDO RETURN END
YT(I)=Y(I)+HH*DYDX(I)
YOUT(I)=Y(I)+H6*(DYDX(I)+DYT(I)+2.*DYM(I))
PPENDIX 4 The input parameters sf
APPENDIX 4. Input parameters SIMPMOD
This appendix mdst not be considered being a complete user manual of the SIMPMOD program, it is only a short summation of the input parameters of the program. For more information app. 3 and report [i] need to be studied. First the switches, which define the differential equation, are considered, afterwards the other input parameters.
SWITCHES :
IANALY = 1 2
IFREQ = 1
2
IDAMP = 1 2 3
ISTIFF = 1 2
3
IDUF = 3 5 ..
IEXCIT = 1
2 3
IRKOUT = O 1
static response (does not work anymore) dynamic response (also the static response will be determined, necessary for determination of the DAF)
only for one frequency the response is calculated, specify NCOMPS, EXPER, EXAMP, EXPHSE if IEXCIT=l
WAVPER, XCREST if IEXCIT=2,3 for more frequencies the response is calculated, specify BEGIWJ, ENDNU, STEPNU
linear damping, initialise DMPFAC = @ = C/2r/(KM) user defined damping, defined in SIMPMOD.FOR hydrodynamic damping (report [ll)
linear stiffness, initialise STIFF Duffing restoring force model stiffness, initialise initial stiffness STIFF, IDUF (order of Duffing system), Duffing parameter CUBFAC = hysteretic stiffness, doesn't work yet (specify EPS,...,HN)
cubic Duffing system, only specify when ISTIFF = 2 fifth order Duffing system higher order system
Harmonic excitation:
11 1 EXPHSE ( I 1 -+ T NCOMPS FORCEG = [ EXAMP(I)*COS(SZ*(
1=1 EXPER ( I ) 360 if IFREQ = 1, specify NCOMPS = number of cosine terms
EXAMP(I)= amplitudes (I=1, ..., NCOMPS) EXPER(I)= excitation period (idem) EXPHSE(I)= phase shift (idem)
if IFREQ = 2, specify EXAMP(l), EXPHSE(1) user defined force, specify XCREST, WAVPER hydrodynamic excitation (report [i])
no information about Runga-Kutta integration information about Runga-Rutta (only if ISTIFF = 3), stored in SIMPMODRUB.OUT
ITOUTP = O 1
IWAVE = O 1
no information about Newton-Raphson iteration information about Newton-Raphson iteration stored in SIMPMODRUB.OUT
IEXCIT, IDAMP as specified IEXCIT, IDAMP are set to 3 (hydrodynamic) static wave, Newmark-Beta parameters set O
Runga-Kutta parameters (ref.[ll and [ 5 1 ) :
EPS H1 HMIN HALPHA HETA HA- HNU HBETA HGAMMA HN
Time input parameters:
TO TEND DT TOUTP
= begin time of integration/iteration = end time of integration/iteration = time stepsize = begin time of determination of x dyn , max end time is TEND
Newton-Beta integration parameters (ref.[á]):
GAMMANB BETANB
Convergence check:
ETOL MAXIT
= tolerance check = maximum allowed iterations
Static analysis parameters:
TSAMPL FACNEG SCLINC NCYC
only is IANALP = 1, doesn't work anymore #k
Hydrodynamic input parameters (ref. [i]):
XMSTOP XMS JAC XMS JAC YJACLT CDND CMNDZ NSTICK WIDTH YMSL WAVEGT WAVPER XCREST RHO
Dimensionless frequency parameters:
BEGINNU ENDNTJ
STEPNU
only if IFREQ = 2 : if IEXCIT = 2 , 3 : WAVPER = T / v , where T = 2 7 ~ / 4 ( K / M )
if IEXCIT = 1: EXPER(1) = T / V
O O H O
Physical parameters:
XMASS DMPFAC STIFF CUBFAC
1 )must be specified always = K = 7)--)only if ISTIFF = 2
Excitation parameters:
NCOMPS EXAMP EXPER EXPHSE
7 specified above (see IEXCIT) J-
Ramp parameters:
NRAPT TRAMP(1) RMPFAC ( I )
= number of ramps = end time of ramp (I=1, ..., NRAPT) = height of ramp (I=l, ..., NRAPT), a t TEND RAMPFAC must be 1
I istin
PROGRAM NEWMARK
This is a test program for a differential equation solver using the Newmark-Beta method. This method can be used for solving the equation:
M*x' 'SC*x'+K*x=F(t)
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
REAL A(O:7)
PI=4.*ATAN(1.)
READ(1,2000) T0,TEND,DT,T2,ROfR1,RT0,RT1,
WRITE(6,*) TO,TEND,DT,T2,RO,Rl,RTO,RTl, + RM,EPS,RK,ETA,F,XSTAT,PER,G,B,ETOL,ITOUTP,IDUF
+ RM,EPS,RK,ETA,F,XSTAT,PER,G,B,ETOL,ITOUTP,IDUF 2000 FORMAT(i8(/F9.5/),2(/11/))
C=2.*EPS*DSQRT(RM*RK) W=2.*PI/PER
WRITE(2,*) 'GENERATE A PLOT.' WRITE(2,*) 'EVERY CURVE THICKNESS 1.' WRITE( 2,*) 'X AXIS LABEL TEXT "TIME". ' WRITE 2, * 9 ' Y AXIS LABEL TEXT "DISPLACEMENTt'. ' WRITE(2,3000) RM,EPS WRITE(2,3001) RK,ETA WRITE(2,3002) F,W WRITE(2,*) 'DEPENDENT GRID ON.' WRITE(2,k) 'INDEPENDENT GRID ON.' WRITE(2,k) 'INPUT DATA.' WRITE(2,*) ' '
WRITE(3,") 'GENERATE A PLOT.' WRITE(3,*) 'EVERY CURVE THICKNESS 1.' WRITE( 3,*) 'X AXIS LABEL TEXT "TIME". ' WRITE(3,*) 'Y AXIS LABEL TEXT "FORCE". ' WRITE(3,3000) RM,EPS WRITE(3,3001) RK,ETA WRITE(3,3002) F,W WRITE(3,*) 'DEPENDENT GRID ON.' WRITE(3,*) 'INDEPENDENT GRID ON.' WRITE(3,*) 'INPUT DATA.' WRITE(3,*) ' '
3000 3001 3002
FORMAS(' TITLE TEXT IC "SDOF: M=',F9.2,f EPS=',F9.4,'''') FORMAT( ' FORMAT( '
11 K=',F9.2,' ETA= ' , F9.4, ' I ' ' 1 11 F=',F9.2,' OMEGA=',F9.2,"'.')
x1=0. XTl=O. XTTl=O. x2=0. XT2=0. XTT2=O.
xMAx=o.
A(O)=l./(G*DT**2) A(l)=B/(G*DT) A(2)=1.//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
STIFFl=A(O)*RM+A(l)*C
XLAMBDA=O . T=TO J = O
DO WHILE (T.LE.TEND)
T=T+DT CALL RAMP(T,DT,RTO,RTl,RO,Rl,XLAMBDA) FORCEG=XLAMBDA*F*(DCOS(W*T)) ITER=O DX=1 .E12
DO WHILE (DABS(DX).GE.ETOL)
ITER=ITER+l RKNON=RK*(1.-DFLOAT(IDUF)*ETA*X2**(IDUF-l)) FSTIFF=RK*(X2-ETA*X2**IDUF)
IF (RKNON.LE.O.0) THEN WRITE(6,") 'NEGATIVE STIFFNESS' GOTO 50
ENDIF
RKEFF=RKNON+STIFFl RMEFF=A(O)*(X2-X1)-A(2)*XT1-A{3)*XTTl CEFF=A(l)*(X2-Xl)-A(4)*XTl-A(5)*XTTl FORCEFF=FORCEG-FSTIFF-RM*RMEFF-C*CEFF DX=FORCEFF/RKEFF X2=X2+DX
IF (ITOUTP.NE.0) THEN WRITE(4,*) 'T=',T,'ITER=',ITER WRITE(4,*) 'FORCEG=',FORCEG WRITE(4,*) 'FSTIFF,RM,C', FSTIFF,RM,C WRITE(4,*) 'RKEFF=',RKEFF WRITE(4,*) 'RMEFF,CEFF,FORCEFF',RMEFF,CEFF,FORCEFF WRITE(4,*) 'X2,DXi,X2,DX
ENDIF
IF (ITER.GT.20) THEN WRITE(G,*) 'TOO MANY ITERATIONS' GOTO 50
ENDIF
ENDDO
X1=X2 XTl=XT2 XTTi=XTT2
IF (T.GE.T2) THEN IDUMMY=~./DT IPROB=MOD(J,IDUMMY) IF (J.EQ.0) THEN
WRITE(6,*) I * * * * BEGIN OUTPUT WRITE(G,*)
ENDIF IF (IPROB.EQ.0) WRITE(6,*) I * * * * T='#T,'****' WRITE(3,lOOO) T,XLAMBDA WRITE(2,lOOO) T,X2 IF (XMAX.LE.DABS(X2)) THEN
XMAX=DABS(X2) TXMAX=T
ENDIF J=J+l
ENDIF
1000 FORMAT(2F15.8)
ENDDO
WRITE(6,") WRITE(6,*) 'XMAX=',XMAX,' AT TIME=',TXMAX
DAF=XMAX/XSTAT WRITE(6,*) 'DAF=',DAF
WRITE(2,*) 'EOD.' WRITE(2,") 'GO. '
WRITE(3,") 'EOD.' WRITE(3,*) 'GO.'
50 CONTINUE STOP END
SUBROUTINE RAMP(T,DT,RTO,RTl,RO,R1,XLAMBDA)
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
IF (T.GT.RT1) GOT0 100
* * * * i
100 CONTINUE RETURN END
