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R 433Technical Report MECHANICS OF RAISING AND LOWERING
.... ....... HEAVY LOADS IN THE DEEP OCEAN:
CABLE AND PAYLOAD DYNAMICS
April1 1966
BUREA' OF YAMS5 Ati lO'(
IFOR FEDERAL SCIENTI1FIC ANDTECHNICAL INFtRMATION
Hardcopy Niaorof sb
U. S. Lg 9 VBRTORY
Port Hueneme, California~frc
Distribution of this documnton is unlimitted.
MECHANICS OF RAISING AND LOWERING HEAVY LOADS IN THE DEEPOCEAN: CABLE AND PAYLOAD DYNAMICS
Technical Report R-433
Y-F015-01-01-001(b)
by
P. Holmes, Ph D
ABSTRACT
Based on a theoretical analysis of the cable and payload dynamics duringlowering or raising heavy loads in the deep ocean given in Project Trident TechnicalReport No. 1370863, further calculations of the maximum dynamic stresses expectedin the lowering cable are presented covering a wide range of cable and payloadparameters. The theoretical analysis is adapted to a proposed design procedure, andtwo typical design examples are given, the results of which are discussed in terms ofthe safety of the lowering or raising operations.
In order to make the design procedure applicable with a greater degree ofconfidence, it is considered necessary to make measurements of cable tensions andload and ship motions during a full-scale operation to fill in deficiencies of dataand provide a basis for verification of theory and calculations. In particular, dataare needed on the coefficients of drag and mass, which at this stage must of necessitybe estimates.
Distrbution of this docujm*is v unldmted.
The Loborototy invites common? On .Ih i report. particu|Otly o- the
results obto-ned by trhose who hove oppled the informotion.
CONTENTSpage
INTRODUCTION ...................................... 1
THEO RY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1. .
RESULTS ............................................ 5
DiSCUSSION OF RESULTS ................................ 7
PROPOSED DESIGN PROCEDURE ............................. 9
APPLICATION OF PROPOSED DESIGN PROCEDURE TO TWOPROTOTYPE EXAMPLES .................................. 11
Design Example Using Polypropylene Cable ................... 11
Design Example Using Steel Cable ....... ................... 13
DISCUSSION OF RESULTS OF APPLICATION OF THE PROPOSEDDESIGN PROCEDURE ......... ............................. 15
FINDINGS ........ ....................... ........... .. 22
CONCLUSIONS ........... ............................... 22
ACKNOWLEDGMENTS ......... ............................ 22
APPENDIXES
A - Analysis of Cable ond Payload Dynamics for Raising andLowering Loads in the Deep Ocean ...... ................ 58
B - Evaluation of the Normalized Amplitude of the MaximumDynamic Stress as w' - nir ........ ..................... 62
C - Computer Program for the Evaluation of the NormalizedAmplitude of the Maximum Dynamic Stress, IZx. ........... 66
D - Summary of Drag Coefficients ......................... 69
E - Summary of Added Mass Coefficients ...... ............. 73
REFERENCES .......... .................................. 78
MATHEMATICAL NOTATIONS ....... ........................ 79
-
INTRODUCTION
The work described herein was carried out Gs part of BuDocks Task No.Y-F015-01-01-001, "Structures in Deep Ocean, " which originated from the require-ment of the Bureau to attain a deep ocean engineering capability in keeping with theincreased emphasis on the deep ocean as an operating environment for naval forces.This report is the result of work performed under Task N.. Y-F015-01-01-001(b),"Mechanics of Raising and Lowering Heavy Loads in the Deep Ocean. "
The objectives of this work were to analyze and report on the results ofpredictions of the forces in lines and the acceleration and displacements of loads ofvarious shapes euring raising and lowering operations in the deep ocean. The studyis based on, and an extension of, a theoretical analysis by Arthur D. Little, Inc.,given in Project Trident Technical Report No. 1370863 of the Bureau of Ships,entitled "Stress Analysis of Ship-Suspended Heavily Loaded Cables for Deep Under-water Emplacements. "1
THEORY
The problem considered in this report is that of a load suspended from a ship ormoored platform by means of a single cable, as shown in Figure 1.* The maxi..Ium depthfor such a lowering or raising operation is assumed to be on t+e order of 20, 000 feet.
The requirements are (1) to analyze this problem so as to predict the cable andthe load dynamics, in particular the maximum dynamic stress induced in the cable asa result of the motions of the suspension point, and (2) to provide a design procedurefor evaluating such stresses for a given load under specified conditions of sea-surfaceoscillations.
Solution of this problem is recognized to be difficult in view of the nonlinearitiesintroduced in the damping due to drag forces of vertical oscillations of the load. Asimplified solution has been obtained by Arthur D. Little, Inc., as given in the ProjectTrident report. A brief resume of the analysis presented in that report is given here,details of which may be found in Appendix A.
The equation ot motion of an element of the cable initially located at a distancex from the support point is given by
2 2Su + c u 6u=+ ~ u- (:'1)ige1toh a2 pixo)2
I * Figure I through 35 are presented immediately after the main body of text.
The boundary conditirt eit the toad is given in the form
u + 6 + B ! u 0 (2)
(6 t,)2 6 X, t' at,
A secondary boundary condition is that the displacement of the suspension point isknown for all time.
In the above equations, u(x, t) is the displacement at time t of a point on thecable originally located at a distance x from the suspension point. Hence the secondboundary condition is that
u(o, t) is known for t > 0. (3)
The following nomenclature applies:
K L (4a)
Pc S L-M (4b)
a
2 Ec - (4 c)
Pc
x1 x t tc (4d,e)L L
and B - (4f)2M a
where* pc = density of the cable
S = material cross-sectional area of the cable
E = modulus of elasticity of the cable
K = constant of friction on the cable due to the surrounding water
* Notations are defined where they first appear and are summarized for convenience
on a foldout page at the back of the report.
2
Ti
c = velocity of sound in the cable
Ma = virtual mass of the load
CD = drag coefficient appropriate to the load
A = projected area of the load in the direction of motion
L = length of cable
p = density of sea water
The difficulty in obtaining an exact solution to Equation 1 subject to theboundary conditions, Equations 2 and 3, arises from the nonlinear term I u/at' j(du/6t')in Equation 2. This difficulty is avoided in the A. D. Little report1 by an approxi-mation which is described in Appendix A.
Defining a normalized displacement amplitude U' equal to U divided by I Uoand noting that Uj is the value of U' at the load, a solution for U' as a function ofx' is given by
U' =U1 cosWOY' + C sinw'y' (5)
where C is a complex constant and
y' = 1 - x, (6a)
S= - (6b)c
Hence the maximum value of the dynamic stress in the cable is given by
2 = 1)2 )2r(1a)2 (U ) L- + tanp (tan 'k + sac k) 1 (7)
max1
2 cos2 W'o + p) 02 sin 2 w' sin 2 2o /2)
where (U)21+ 4 -1 (8)2 2 sin ( sin Cos (W' +P)
i= arc tan 0S (9)
-arc tan [ a J tan •. o
= rcan±9U)tan( - cot 2,] - (10)L01 2 2
3
4 C DPAU (11)
3 T Ma
and I:- L d (12)maxl UoIUE
In the A. D. Little report, Equation 7 together with Equations 8, 9, and 10was solved by use of a digital computer to give the maximum dynamic stress as afunction of the nondimensional frequency w' for various values of 0 and $4. Asimilar procedure was adopted ir this report for two reasons, firstly to investigatethe variation of stress over a wide range of w', 0, and p, and secondly as a meansof providing the basis of a design procedure for cables used for lowering or raisingheavy loads to or from the deep ocean floor.
The cable and load system considered herein is a part of the overall loweringsystem consisting of the vessel from which the operation is performed, its responseto the wave action present during the lowering or raising process, and the resultingoscillations of the cable and load. Within existing theoretical limitations of knowl-edge about waves and ship motions, and under the restrictions of a linear theory, theproblem of the response of a ship or platform to a particular sea state has been solvedin terms of certain probabilistic models by Kaplan and Putz. 2 Pierson and Hc.lmes 3
in a note on the engineering applications of the Kaplan and Putz report outlined aprocedure for the determination of the response of a drilling barge to sea states 3, 4,and 5. The results are obtained in terms of the probability of occurrence of variousamplitudes of motion in heave, surge, sway, yaw, pitch, and roll. The Cuss-Iocean-bottom drilling barge was used as an example, but the calculations as carriedout by Kaplan and Putz may be applied to other ships or moored platforms, given theuse of a digital computer.
Details of the program, which was written for an IBM 1620 computer, aregiven in Appendix C. For the purposes of this analysis, the calculations weredivided ', 1to sections based on the relative values of W' required for prototypecomputations.
Equation 6b relates the required range of w' to the length of and velocity ofsound in the cable. It is assumed that w, the frequency of oscillation of the cable-suspension point, has a maximum value on the order of 2.00 radians per second andthat the maximum length of the cable is 20,000 feet. Then the required range of w'is determined by the velocity of sound, c, in the cable. For steel and polypropylenecables, c is approximately 12, 000 and 2, 000 feet per second respectively, resultingin maximum nondimensional frequencies of 3. 33 and 20. 00.
4
¶ Initially the range of 0 was chosen as 0. 10 to 7.00, and values of o equal to0.10, 0.50, 1.00, 2.00, 5.00, and 10.00 were used. Preliminary computationsfor a typical design problem indicated that lower values of A would also be required,and corresponding additional computations were carried out as shown in the resultswhich follow.
Pierson and Holmes indicate the methods whereby the root-mean-square (R.AS)values of motion in each mode for each sea state may be determined. For the purposesof this report, the oscillation of most concern is that in heave, and knowing the RMSvalue of heave motions in sea state 4, for example, estimates can be made of theextreme value of heave to be expected in a given time. This procedure thus providesa basis for specifying the range ofI U01 to be used in determining .:.s for )se inthe design computations. The correlation between sec, ship, and cable stresses isdiscussed further later on in the text in the application of the results obtained hereinto two hypothetical prototype cases.
RESULTS
As illustrated above in the theoretical analysis, the parameters influencing
the dynamic stresses can be tabulated as follows:
Cable Parameters
L = maximum length of cablemax
S = material cross-sectional area of cable
w = weight of cable per unit length
E modulus of elasticity of cable
Id = allowable maximum dynamic stress in the cable
Load Parameters
M mass of the load
A = cross-sectional area of the load
Cm coefficient of mass
CD coefficient of drag
p density of sea water
Ship or floating Platform Motions in Heave
IU0o = amplitude of heave
w = frequency of heave (rod/sec)
5
These specific variables are combined as follows:
4CDpA
3 ifCmM iui1
w L (4b)S Maa
W - (6b)C
and E Id (12)
Values of the normalized amplitude of the maximum dynamic stress, I E.ax I,were calculated for four ranges of nondimensional frequency, w'. The ranges wereas follows:
".1/5 s u,,' <- 7.Or in increments of IT/5
"iT/10 S w' s 1.41T in increments of Tr,/10
0 T w' ! ff/2 in increments of ir/10
0 • w' s 0. 10 in increments of 0.01
The specific values of ; used in the computations were 0. 005, 0.01, 0. 03,0.05, 0. 10, 0.50, 1.00, 2.00, and 5.00. For low ranges of w', additional valuesof I were used as shown on the appropriate graphs. At each value of I, calculationswere performed for 0 = 0. 10, 0.30, 0.50, 0.70, 1.00, 3.00, 5.00, and 7. 00 overthe two higher ranges of wa', and values of 0 = 0.25, 0.50, 1.00, 3.00, 5.00, and7. 00 over the two lower ranges of ud'. The results of these computations are presentedin Figures 2 through 7 for the lowest range of uw' and in Figures 8 through 13 for thesecond lowest range. For the range w/10 % w' •I. 1.4f, it was found that the stresscalculated at values of w' near w were zero. Figure 14 illustrates this discontinuity.In order to investigate this behavior, Equations 7, 8, 9, and 10 were combined and 4evaluated as w' approached n f, where n = 1, 2, 3,... Details of this evaluation J
are given in Appendix B, the result being given in the form
6
4)2 tr2 1 K
Ilim( (V (n ) + K{ +K -. Kmax 2 K 2 -
! 2 1 2]1K+ 1+ + + ! I2 (13j
where is the parameter defined previously, Equation 11, and K n w/1. The soliddot in Figure 14 is the stress calculated from Equation 13 at that particular value of
Sand j. The values of I11m x1 as determined from Equation 13 were used ;n plottingFigures 15 through 22 as anc when necessary. The apparent discrepancies in thecomputer calculations are considered to be due to rounding-off errors inherent inthe computational procedures.
Figure 23 is included to indicate the variation of Rýnaxj with 0 for a particularvalue of ). Figure 24 shows the variation of maximum dynamic stress for the highestfrequency range fr/5 S 2 % 7. Of for particular values of 0 and p. As can be seen,the maximum dynamic stress is highly dependent upon the nondimensional frequency.Since in any application of these curves the peak values of 11ýaxl must be consid-ered, the results for this range of frequencies are presented in a simplified form inFigures 25 through 32, where each curve is drawn through the maximum values ofV I-xl in the same manner as the dashed line in Figure 24. Each graph is drawn for
various values of u at a given 0. In certain instances, it was again found necessaryto determine I-naxl at frequencies near 17 and its multiples by use of Equation 13.
DISCUSSION OF RESULTS
The results obtained from the computer program as outlined above are generallyin agreement with those quoted in the A. D. Little report] except for values of thenormalized maximum dynamic stress, i corresponding to nondimensionalfrequencies, u:', near or equal to 3. 142. The computation of jT.x at ' 1C
according to Equation 13 gave results which compare favorably with those obtained"J •from the computer program. A comparison of three typical results is given in Table I.
! It should be noted that the value of Inoxi computed from Equation 13 is notnecessarily the peak value, since resonance will occur at nondimensional frequenciesother than ff depending upon the values of 0 and ;. Wien ;A approaches zero, theresonant frequencies approach 17, 211, 3!, etc, and the vclue of T.MxI as determinedfrom Equation 13 may then be interpreted as the maximum value. This con be seen tobe true by inspection of Figures 15 through 22.
7
1able 1. Comparison of Typical Values of 11-axl as Computed by 'gQt9taLCompui-er Program and as Calculated From Equatio., 1t3
SLo. Dx maxi Percentage-C rntei yDiitl Cuicvlaiej aj rp~a ~iac.~e .m Difference I
Compo~er Progrorn i Eqkation 13- Difrec1.80
0.50 C.10 112 i 1i0.1 1.801.00 0.10 142.32 139. 55 2.015.00 0. 10 514.09 503.5 2.10
The parameters 0 and 14 are representative of the damping and the eatio of thz Iweight of the cable to the virtual mass of the load respectively. As was noted in theA. D. Little report, the variations of I xwith w', 0, and a are in agreementwith known results for simpler systems. As the mass of the load is decreased, i. e.,p I co, the system reduces to that of a free-ended spring, -with the resonant frequenciesapproaching r/2 and 37r/2, etc. As thL ,•,ass is increased, p - 0, the resonant fre-quencies approach v, 2;, etc., which agrees with the case of a fixed-ended spring.
The .,amping parameter, /, has a slight effect on the resonant frequencies buta far m 3 important effect on the amplitude of I 1 at resonance. A conci.sioncax)in t•,e A. D. Little report indicated that the maximum dynamic stress amplitude a~tres. iance increases when the damping is increased beyond a zertain vclue. Fromthe above calculations it can be seen that the amplitude at resonance increasesgene-ally with increased damping; i.e., there is no minimum amplitude as impiieJby the above conclusion. This result is compatible with the concept that as thedamping increases, the system becomes equivalent to a fixed-ended spring givingresonances at IT, 27r, etc., and amplitudes tending to infinity, restricted only byinternal and external damping of the cable. This argument considers the dampingeffects, a function of 0, to be divorced from the inertial effects, which aredependent u,-on IA.
In view of the dependence of the maximum dynamic stress ampltude on A,and since A depends on the parameters of the load - ;.e., the cross-sectionalarea, the mass, and the density of sea water, which are fixed - and on the coef-ficient of drag, th value of C: assumed for a given load C.onfiguration is ofpanrcular importane. This can be seen from the results given above where acl'ange of 0 from 1.00 to 3.00 results in a change in 1ýnax at resonance! from142 to 370. if If axj is interpreted as an allowable stress which when exceededresults in an unsafe condition for the operation, as implied in the design procedurewhich follows, then the value of CD used in the calculation of A becomes critical.For poor hydrodynamic shapes such as blunt bodies or open frameworks, it is notpossible within the present state of development of theoretical fluid dynamics tocalculate CD from the basic equations of fluid flow. The alternative, therefore, is
8
A -
to resort to an experimental determination of the coefficients of drag for the particularload conFiguration ;n question, for both steady and oscillatory motions. There existslittle expef-nentol data on the oppropriate coefficients of drag applicable to typical
mad sihapes being lowered to the deep ocean floor. A seriez of experiments directedtoward obtaining such data thus appears justified.
Sir, ilar arguments also apply to the values assumed for the coefficient of addedmess, Cm. Summaries of information pertinent to the determination of the drag andadded mats coefficients are presernted in Appendixes D and E respectively.
The shape of each curve determine . from Equations 7, 8, 9, ond 10 differsfrom thcse obtained for linear systems in that a second peak in the normalized max-ituum dynam.ic stress occurs at nondimenslonaI frequencies on the order of 0. 10 to 0.40."The significance of such c secondary peak, which is termed herein a subharmonicresponse, is more ea.-dy dicussed in relation to a specific design example. Two suchexamples are given below following a proposed design procedure.
PROPOSED DESIGN PROICEDURE
The parameters required in the design procedure are those applicable to theload and the cable. The,' may be tabulated as follows:
Load Parameters
M = mass of the load
A = cross-sectional area of the load in the direction of motion (ft 2 )
CD = coefficient of drag applicable to the load
C = coefficient of added mass applicable to the loadm
Cable Parameters
L = maximum length of cable (ft)max
w = weight of cable per unit length (lb/ft)
E = modulus of elasticity for the cable (lb/in. 2 )
Iuft = ultimate tensile strength of the cable (lb/in. 2 )
F = safety factor for maximum operating stress in the cable
p = density of sea water (lb/ft3 )
9
From the above parameters, the following may be determined:
I ult
d F static
where ýstatic equals the static stress in the cable, including that due to the cableitself at L and Id equals the operational maximum dynamic stress allowable witha cable of ultimate tensile stress, lult, and a safety factor of F.
(2) c Ii
the velocity of sound in the cable in feet per second; and
4 CDpA
(3) k 4 C Dpm
31Y Cm M
a constant. Since the cable length varies from zero to Lmax and because the designmust be valid for all lengths, several values of L should be chosen (L = I, wheren = 1, 2, 3,...) between zero and Lmax* Hence, values of 14n = wLn/C;n M maybe calculated.
Values of 1U01, the amplitude of the cable support-point oscillation, may beselected as IU01 = 1.0, 2.0, 5.0, 10.0 feet. A table may then be set up, asillustrated by Table II, for each gn corresponding to the selected Ln-
Table I. Outline of Table for Computctior. of Relationship BetweenFrequency and Amplitude of Oscillations, Given A, L, c,and k
(For IA =1' L = L1, c/L = c/L 1 )
Column 1 Column 2 Column 3 Column 4 Column 5
L IdUo , (ft) :kIUoj -W W
max IU01IE L1.02.0
5.010.015.0
10
L- _ - -, ~
The use of the table is as follows. Column 1 consists of the amplitudes of thecable support-point oscillations as selected above, and from this, Column 2 may becalculated by multiplying by k derived previously. Column 3 is computed for thegiven L, Ed, 1U01, and E values appertaining to the cable. For polypropylene ornylon cables, it may not be possible to determine Ed and E directly. Manufacturers'tables generally allow the computation of S Ed and S E, where S is the material crosssection of the cable. Hence, the constant ratio Id/E may be determined and used inColumn 3.
Column 4 is obtained from the curves of Figures 2 to 7, 8 to 13, 15 to 22, and25 to 32 for the particular values of p and A given in the table. Column 5 is eval-uated from Column 4 for the particular ratio of c/Ln. Thus, from Columns 1 and 5 acurve relating the allowable amplitude of oscillation to the frequency of that oscilla-tion may be drawn for the particular cable length used. The process may then berepeated for other cable lengths. A judicious choice of cable length can serve toreduce the numerical calculation to a minimum.
It may be assumed that operating conditions lying on or below the curve aresafe, with the safety factor, F, as defined, and that operating conditions lying above,to the right of, the curve are unsafe.
In line with comments raised in the Discussion of Results, the unknown parametersare CD and Cm. With existing deficiencies ;n data giving CD and Cm for various loadconfigurations, they must of necessity be estimated. See Appendixes D and E.
If it should occur in a design problem that the ranges of w', 0, or g are notcovered in the graphs developed in this study, then the appropriate curves may becalculated for specific values, or ranges of values, of those parameters by use ofthe digital computer program used in obtaining the results quoted herein. Detailsof this program are given in Appendix C.
Two examples of the application of the above design procedure are given below.
APPLICATION OF PROPOSED DESIGN PROCEDURE TO TWO PROTOTYPEEXAMPLES
The application of the proposed design procedure to two hypothetical prototypeexamples is demonstrated below for polypropylene and steel cable respectively. Theparameters used in these examples are ,uch as to enable the design procedure to becarried out using the curves presented previously in the results.
Design Example Using Polypropylene Cable
F The parameters of the load are given as
M : 5.0tons = 10,0001b
A : 12ft x 12 ft 144ft2
IL. I
CD = 2.0
Cm = 1.5
The cable parameters chosen are
L -= 20, 000 ftmax
w = 0.90 lb/ft
S E = 240, 000 l b
SId = 10,000 lb
c = = = 2, 930 ft/secc
4 CDPAk=- - 0.50
31tCmM
and, therefore, = 0.50 1Uo1.The cable daiu used herein was obtained from the August 1964 "Braided Rope
and Cordage Catalog" of the Samson Cordage Works, Boston, Mass. It is nowconvenient to select cable lengths such that values of p, where g = wL/Ma,coincide with those used in deriving the curves presented previously in the results.These values are given in Table Ill.
Table Ill. Cable Lengths Used in Design Example for
Polypropylene Cable
L (ft) __ c/L
16,660 1.00 0. 17588,330 0.50 0.35161,660 0.10 1.7580
833 0.05 3.5160166 0.01 17.5800
It is now possible to set up Table IV corresponding to Table II given in theproposed des~gn procedure by specifying that input amplitudes of oscillation, lUo 0Jof 1.0, 2.0, 6.0, 10.0, and 14.0 feet will be considered. Knowing that 0- k JUo1,Column 2 of Table IV may be calculated. The values of ll xl = L ' 1Uo01 E canbe derived for appropriate values of L and lUol as shown in Column 3 of the table.
12
.T
From inspection of the table, it can be seen that in view of the relationship betweenthe various cable lengths chosen, values of lITaxI at these different lengths are quitesimply related - thus facilitating the design calculations.
Values of the nondimensional frequency, u', are then entered in Column 4by the use of Figures 2 to 13, 15 to 22, and 25 to 32. That is, for a particular ksand 0, the value of Le corresponding to R4iaxI may be found. Hence, the circularfrequency, w, can be calculated and entered in Column 5. Fron; the completedtable, the relationship as a function of cable lenEth, can be drawn between inputamplitude or oscillation, l ul, and the allowable cirrular frequency, w, of thatamplitude - i.e., the circular frequency at which the oscillction can occur suchthe' the maximum dynamic stress in the cable is less than or equal o the designdynamic stress. Figure 33 shows this relationship for the computation given inTable IV. The significance of these results is discussed below together with that ofthe following design example.
Design Example Using Steel Cable
The relevant load parameters are given as follows:
M = 20 tons = 40,000 1b
A = 600 ft 2
CD = 2.0
Cm = 1.5
The appropriate cable parameters were chosen to be
Lmax = 20,000 ft
PC = 550 Ib/ft3
w =- 7.64 lb/ft
E = 15xl06 psi
Ird = 40, 000 psi
c 11,200 ft/sec4 CDPA
k 3TCmM 0.50
and, therefore, 0 0. 50 'Uj
13
-0V 66666 0-6666%0
~~~% 0 N.eino -D Ln CV Oý OD C
oo (LIc C%4-6
-o LaO
0. qt ýo
0
0000 ce66 -0ý* 6666
C; C; v-o LO N Lo.&nO
%o aOD0 00 Ct 0'000
~J 0 u
14 n v)NC4L
qt~ ~ ~ C-00-P
As in the first example, cable lengths are chosen to give values of p coincidentwith those used in the calculation of Figures 2 to 13, 15 to 22, and 25 to 32. Theselengths and the corresponding values of 14 and c/L are given in Table V.
Table V. Cable Lengths Used in Design Examplefor Steel Cable
L (ft) A c/L
15, 700.0 2.000 0.7137,850.0 1.000 1.4263,925.0 0.500 2.8501,963.0 0.250 5.600
785.0 0.100 14.260393.0 0.050 28.500236.0 0.030 47.500
78.5 0.010 142.60039.3 0.005 285.000
A similar table to that derived in the previous design example may now beset up and the numerical computations performed as above. These calculations aresummarized in Table IV, and the relationship between the input amplitude ofoscillation, JU01, and the aliowable circular frequency, w, of that amplitude isillustrated in Figure 34.
DISCUSSION OF RESULTS OF THE APPLICATION OF THE PROPOSEDDESIGN PROCEDURE
Figures 33 and 34 show the results obtained in the application of the proposeddesign procedure to the two hypothetical cases described above.
In any load-lowering operation, the cable used will have a certain knownultimate load at which the cable could be expected to break. With repeated useof a particular cable, this ultimate load will decrease. Hence, for any loweringoperation a safety factor must be chosen defining a load, or corresponding stress,which should not be exceeded. This allowable working stress is assumed Mo includestatic stress due to both the load and cable - the latter being negative in the caseof a buoyant cable such as polypropylene - arid the dynamic stress.
This discussion and the design procedure proposed earlier in the report arebased on the assumption that a particular maximum dynamic stress is given whichshould not be exceeded during the lowering or raising operation.
15
Figures 33 and 34 indicate the allowable frequency of an input oscillation atvarious amplitudes of this oscillation as a function of the cable length, such that thestipulated maximum dynam*c stress is not exceeded. Alternatively, they indicatethe maximum amplitude of an input oscillation at a particular frequency such that amaximum dynamic stress is not exceeded.
One of the most important problems in the interpretation of these graphs is theselection of appropriate input conditions, I Uo I and w, corresponding to the responseof the vessel used for the lowering or raising operation of the sea state in which theoperation is carried out.
It is apparent that the frequency range given in Figures 33 and 34 is far greaterthan that which would be expected under operational conditions. Data is availablein the literature (Kaplan and Putz, 2 Pierson and Holmes3 ) on the response of theCuss-I drilling barge to various sea states. In sea state 5, the range of frequenciesof oscillation in heave is 0.40 to 1.40 radians per second with root-mean-squarevalues of heave of 1.8 and 1. 3 feet at headings of 90 degrees and 0 degrees to thewind respectively. These values imply expected maximum amplitudes of oscillationof 7.2 and 5. 2 feet during a 4-hour period on station (Pierson and Holmes). It shouldbe noted, however, that combinations of heave and roll oscillations could easily pro-duce oscillations greater than this if the load-lowering operation is carried out usingthe boom over the s~de of the vessel. The above values are used here to llustratethe interpretntion of Figures 33 and 34.
Referring to Figure 33 for the design example using a polypropylene cable,it can be seen that for a heave amplitude of 7. 2 feet at frequencies of 1.40 and0.40 radians per second - to cover the entire frequency range - the design dynamicstress will be exceeded at a cable length of less than 200 feet for L = 1.40. Forfrequencies less than 1.40 radians per second down to 0.40 radian per second, it isestimated from the curves that the design dynamic stress will be exceeded at cablelengths which gradually decrease from 200 feet. Thus the operation will be unsaferelative to the prescribed maximrm dynamic stress for a cable length of ;ess than200 feet. This condition has been fully recognized in the design of various loweringoperations conducted by this laboratory. The dynamic stress will then be less thanthe design stress until the length of the cable reaches a value of approximately5, 000 feet. According to the results obtained above the operation will tken becomeunsafe, if the frequency is 1.40 radians per second and the amplitude is 7.2 feet. *For a frequency of 1. 00 radian per second the operation becomes unsafe at a depthof 7,000 feet. The reasons for this result ore difficult to visuclize, but may beexplained from both the mathematical and physical points of view. In tenrns of thenumerical computations carried out as shown in Table VI, the regoression of the curvesgiven in Figures 33 and 34 is due to the fact that when the maximum dynamic stress,
* These comments ore based on an approximate interpolation beween tl-e curvelabeled 6 feet and 10 feet in Figure 33 for IU j- 7.2 feet.
16
• r-- • 'I . . . ..1
I .naincreases, say doubles, due to an increase in the cable length of L to twiceL, the appropriate nondimensional frequency, w', at the corresponding value of A.is not twice the w,' at the shorter ;ength. This results in a lower circular frequency,W, at the longer cable length. This applies to the larger values of I ox I correspond-ing to these lengths. A similar argument applies at cable lengths lower thaoo 200 feet.For intermediate cable lengths a relatively small variation in .axi results in asignificant increase in wi', resulting in an increase in the circular frequency, W.This applies to a range of w' from 0. 4ff to 0.8ff - i.e., to values of Ij axj largerthan thor• corresponding to the first peak in Figures 2 to 13 and 15 to 22, but lessthan those associated with the peak responses occurring at w' approximately equalto 3. 142. As w' values tend toward ihis value, significant changes in I .naxI resultin larger variations in circular frequency for different cable lengths due to the influ-ence of the c/L ratio in computing the latter. Hence the variation of cable stresswith cable length depends upon the shapes of the computed curves relating Irmaxlto W1.
The physical behavior of the cable assembly may be described by reviewingthe significant results obtained by Little 1 and by Whicker. 4 The latter is a rathersimplified (i.e., no damping) theoretical analysis of the effect of ship motion onmooring cables in deep water. On Figure 33, values of w' corresponding to ir/2 andIT are indicated, as well as the roots of the equations tan w' = p/u' and ton w' = -Ww'.The case of the fixed-ended spring for -ero damping (,3 = 0.0) is shown by Whickerand by Little to have resonant frequencies of ,.' = ff, 2ff,,... n ir when the relativemass, 0•, of the cable and payload is decreased infinitely (iC e., the payload massincreases indefinitely). The case of the free-ended spring for zero damping (8 = 0. 0)is represented by:
I. Values of w' equal to w/2, 31r/2... .nf/2 when the relative moss is increasedindefinitely. When the damping P = 0.0, it is shown by Little that values ofW' equal to fT/2, 3ff/2... nfT/2 result in infinite dynamic stress for infinitelylarge values of M.
2. Values of u:' corresponding to the roots of the equation ton w' -- -.j/12.Whicker shows that for finite values of p, the least root of the above equa-tion results in infinite total stress.
In addition, Little shows that for finite values of #A, and for zero damping,infinite total stress will result when the roots of the equation tan w' = Ww' oresatisfied. This is in contrast to the results given by Whicker, yet both formulationsIi appear to be correct. Thus, it would appear that the Little and Whicker results arecompatible for long cable lengths, but that the Whicker results are not applicablefor shaovt lengths, since all of recorded data supports Little's conclusions. In anyevent, the least of the roots of the above equation lies between 0 and ir/2. As p is
P increased indefinitely, the roots of the above equation approach 'f/2, 3-N/2... n ?f/2.There is no comparable analogy with the well-known results of simpler systems.
17
-- / l • ., .. ,•-'- "
0 0 lNczg R14
44-0.9 co 0l 0 QVu n0oqun
C; ~ ~ cv C;(jC 6C : C
4- u
C14 0" N 14N N
0 '
N m N N
-V V
12. N LN n-: It No N.04 C-4 N
CLAJ
u uD0 0 90 0
0o I-1 9CAIOA ;c,0tN
*o0*-o 0 c) NOOItC;C 6C
CI
_ _ _ _ 1 _ _
18t
Beginning with large lengths, the cable assembly behaves as a fixed-endedspring, and the role of damping, which is dependent on the input amplitudes, is toprovide an additional margin of safety. Thus higher frequencies for the same lengthat constant input amplitudes can be tolerated without danger of breaking the cable.
As the length decreases, a transition occurs and the cable assembly takes onthe characteristics of a free-ended spring. Again the role of damping is to providean additional margin of safety against failure.
Finally at very short cable lengths, the cable appears to behave like a rigidconnection between source and load rather than as a "spring." In this case, therole of damping (drag) is reverseJ in that, for constant input amplitudes, smallerfrequencies car be tolerated than for the case when g = 0. 0.
All of the above trends are confirmed by the calculations for both the dampedand undamped cases as the length L approaches zero, as indicaled by Figure 33.For comparative purposes, a few results obtained by applying the method developedby Whicker for a free-ended cable are included. The mathematical formulation byWhicker does not include the damping term. Thus a direct comparison of the effectof damping is available.
It is recognized that the analysis used for this investigation - that given inthe report by A. D. Little, Inc. - does not accurately describe the prototype situ-ation by virtue of the linearization renuired in the drag term I u/at'l ('u/'t'), whichis necessary in order to solve the basic equation of motion. In the present state ofthe art concerning nonlinearly damped oscillations there appear,. to be no alternativeto the linearization. Further theoretical analysis is considered necessary and justifiedin order to resolve the question of safety of lowering systems as ihe length of thecable increases. Such an analysis would consider the use of analogue computationsto allow retention of the nonlinear I ý u/'t'l (1 u/, t0) term.
Similar interpretations regarding the safety of a load-lowering operation maywell apply at greater depths for lower frequencies if the curves presented in Figure 33were extended. From the general shape of the curve shown and for a given inputamplitude oscillation there is a limiting input frequency below which the operationis safe (the maximum dynamic stress is not exceeded) down to a certain depth. Fora particular lowering operation it may be possible to use a working vessel which haslittle or no response to excitations above this limiting frequency. Alternatively, theoperation may be carried out in a lower sea, but this does not necessarily imply amaximum frequency of input oscillation, althoLgh it does imply a diminished ampli-tude of oscillation which would render the operation safe.
A similar discussion may also be applied to Figure 34 for the case of a steelcable, although on unsafe condition is not likely to occur until depths of 20, 000 feeteven with input amplitudes of 14 feet. In this case, however, an additional factormust be considered, namely the very significant increase in static stress due to theweight of the cable. As for as the dynamic stress is concerned, it may be concludedthat the operation may be safe for cable lengths greater than 100 feet when inpxjt
19
amplitudes are 'ess than 6 feet at a maximum frequency of 1.40 radians per second.As the cable length diminishes to zero it would appear that the design dynamic stresswill be exceeded by an increasingly large amount. From Figures 33 and 34 it is seenthat for both examples at shorter lengths of cable, a greater amplitude of oscillationis relatively less safe than a smali amplitude, as expected. Further calcuiations arerequired to determine the furm of the curves as the cable length approaches zero.
On 13 April 1965, a Submersible Test Unit (STU) loaded with racks ofspecimens was lowered to a depth cf 2,500 feet by the Deep Ocean EngineeringDivsiion of NCEL. The record of cable tensions diring the lowering operation ':shown on Figure 35. The weight of the load in water wos approximotely 5.,500 pounds,the structure being in the form of an open truss, its base consisting of two flat plateswith a total cross-sectional area of approximately 150 square feet. The :oad vjslowered on a 1.3-inch-•diameter polypropylene cable. From a depth of 450 feet thedescent was carried out at a steady• rate of 132 feet per minute. At depths shallowerthan 450 feet the lowering operation was intermittent. The wave excitation wasestimated to be in the form of a 6- to 10-foot swell with periods of M1 to 12 seconds.A brief discussion of various properties of the record in the light of the calculationsgiven above is pertinent. The pararieters of the cable and load were approximatelythose used for the first design examFle, although values of the coefficients of dragand virtual mass do not necessarily agree.
The record shows an expected decrease of dynamic stress with increasing depth.The immediate reduction in the mean load at point B of approximately 2,000 poundscorresponds to a drag force on the structure with a coefficient of drag equal to 2. 76at a vertical velocity of 132 feet per minute. Since the load consisted of racks ofspecimens orientated pa'alei to the flow, thiW drag coefficient is not necessarily thatcorresponding to form drag alone; however, the ccntribution of tangential drag onthe cable can be shown to be negligible. The steady reduction of mean load from450 to 2,500 ;eet is due to the gradual removal of the weight of a 2,500-foot-longsteel cable suspended from the base of the STU to the ocean floor.
From the design cxampie given previously the expected maximum dynamicstresses in the lowering cable were calculated at cable lengths of 83, 830, and1,660 feet assuming that JUoIequals 6.0 feet. Table VII summarizes these compu-tations which were carried out for each cable length by determining the period ofthe cyclic stress from the record. Values of •-Qnoxj were then found from the curvesgiven In the results, for ihe appropriate 3 and ;A values. Hence, the maximumdynamic stresses were determined a- shown in Table Vii and superposed on the stressrecord as given in Figure 35.
In view of the uncertainty with respect to values of CD, Cm, and 1UoI for theoperation represented by Figure 35, the comparison of the calculated dynamic stresseswith those determined experimentally ;s ..onsidered to be fair. It is noted that at830 and 1, 660 feet the stresses recorded exceed those calculated by a factor of 2.As the load is lowered, only one variable chunges, that being the cable length.
20
10
However, the frequency and amplitude of the cyclic stress change with cable length.This may well result in a significant change in the coefficient of drag and mass sincethe amplitude and velocities of vertical oscillation will vary. This is possibly onecause of the discrepancies noted.
Table VII. Summary at Calculations for STU Lowering Operationto 2,500 Feet
Cable Length, L 83.0 ft 830.0 ft 1,660.0 ft
Parameter, A 0.005 0.05 0.10
Parameter, • 3.00 3.00 3.00
c/L 0. )283 0.283 0.566
Period of Oscillation, T 5.45 sec 8.00 sec 9.23 sec
Frequency of Oscillation, w, 1. 153 rads/sec 0.786 rad/sec 0.681 rad/sec
WL 0.0326 0.222 0.385c
11'ax I From Curves 0.660 1. 165 1.225
SId - Dynamic Load ±3,752.0 lb ±707.4 lb ±371.9 lb
Static Load 5,500 lb 3, 400 lb 3, 100 lb
Static + Max Dynamic Load 9,252 lb 4, 107 lb 3,472 lb
Static - Max Dynamic Load 1,748 lb 2,693 lb 2, 728 lb
From the record given in Figure 35 there also appears to be a somewhatirregular, long-pericd reinforcement of the dynamic stress amplitude. During thelowering operation it was not possible to record the actual motions of the load, andthus any departure from purely vertical motions are unknown. At a lowering velocityof 132 feet per minute it is quite possible that the load motion will become unstableand that sidewise oscillations will occur, giving rise to complex lift forces on thestructure with resulting variations in the cable tensions. There is no evidence in
j ~ Figure 35 to support the result shown in Figure 34 that dynamic stresses may increaseat greater depths, because of the limited depth of 2,500 feet to which the STU waslowered. There is a pressing need for a load-lowering operation to be carried out inwhich both the cable tensions and the load motions are recorded, thus allowing amore definite analysis of the results. Preferably this full-scale experiment would becarried out with a formalized body shape so that coefficients of drag and mass couldbe more accurately estimated.
21
FINDINGS
As a result of the investigation described above, it was found that:
1. The theoretical analysis presented by Arthur D. Little, Inc., in Project TridentTechnical Report No. 1370863 of the Bureau of Ships1 was readily adaptable to adesign procedure for heavily loaded cables for deep ocean emplacement or recoveryoperations.
2. This design procedure is relatively straightforward, but the results of its applica-tion require confirmation by prototype measurements.
3. The rrcost important unknowns in the input parameters for the det.4n procedure arethe coefficients of drag and virtual mass, which at this stage must of necessity beestimates, again requiring confirmation by prototype measurements.
4. For the two prototype examples given, the operations are unsafe with respect tothe specified maximum allowable dynamic stresses when the load is between zero andapproximately 200 feet below the ocean surface.
5. Using polypropylene cable, there exists a possibility that under certain conditonsof motion of the working vessel the operation will again become unsafe at depths onthe order of 5,000 feet and greater. The reason is not difficult to visualize. As thelength increases the natural period also increases, eventually corresponding to periodsof the exciting waves. If the damping is small, very large stresses may be induced byrelatively small input amplitudes.
CONCLUSIONS
It is concluded that the proposed design procedure is applicable to heavilyloaded cables in deep ocean emplacement or recovery operations. However, theresults of its application may not be considered rigorous in view of the estimates ofvalues of coefficients of drag and mass required for thie calculations. In order tomake the design procedure applicable to a prototype situation with a greater degreeof confidence, it is considered necessary to make measurements of cable tensionsand load and ship motions during a full-scale operation to provide a basis for com-parison between theory and prototype.
ACKNOWLEDGMENTS
Acknowledgment is made to NCEL personnel for their assistance in thepreparation of this report: to Mr. W. L. Wilcoxson for assistance in the analysisgiven in Appendix B; to Mr. R. E. Jones for valuable discussions in relation to thepracticalities involved in lowering heavy loads to the deep ocean floor; to Mr. B. J.Muga for overall guidance of the task effort and the preparation of Appendix D; andto Mr. Richard Tu for preparation of Appendix E.
22
-A
E
uJ a
IF 0 C
- a 'a* C~1 --
a C aL a
-0 E
0 C 0 -
It
-u
0~0
-~23
2.0 0.005
.00.01 11.8
1.6
o 1.4
X
aE0. 1.2
Ea 1.0C
EE
IE 0.8
E
0.6
=0.03
0.4
0.0.0'
0 0.02 0.04 0.06 0.08 0.10
Nondimensional Frequency, w'u (.4..)
Figure 2. Variation of normalized maximum dynamic stress, I.axj, withnondimensional frequency, U, for 0 = 0.25 over the range0 5 w' £ 0.10 for values of as indicated.
24
A.].
2.0
S=0.0051.8-
) 0.01
1.6
1.4. /
III
x 1.2
,• !0a
a
E.A 0.8
M
0 0.6 u 0.03
0.4
"3; .0- 0.05
0. /A U 0.10
0 0.02 0.04 OL06 0.o,0.,
Figure 3. Variation of normalized maximurn dynamnic stress~x, withnondimenstonal Frequency, w:, for 0 = 0.50 over the range,0 T- w' 1 0.10 for values of m• as indicated.
75
2.0
1.6
0 A' 0.005
1.4 , 0.01
o
EtN
"E 1.00
a~l 1.003
0.6
0.4
a 00.5
0.2 "M0.10
0 0 0.02 0.04 0.06 0.0a 0.10
Nondimensionel Froquency, /O (Y.L)
Figure 4. Variation of normalized maximum dynamic stress, NEO2 I , withnondimensional frequency, w', for 0 = 1.00 over the range0 s w' T 0.10 for values of p as indicated.
26
1.01 0.005
= 0.01
0.9
4 M 0.03
-U
_i 0.7
II
X 0.05E
V4 0.6
E
S0.5
E
V 0.4
o 00.10Z
01.3 -
,00.20.3
0.2X
0 0.02 0.04 0.06 0.08 0.10
Nondlmenslniel Frequevcy, L u ("4)
Figure 5. Variation of normalized maximum dynamic stress, i• l, withnondimenuional frequency, W', for 0 = 3.00 over the range0:9' ' 0.10 for values of p as indicated.
27
1.0, 0.005 0.01
0U..00.9- 0.03
as
-- 0.10
E
- 0.6
E
Ca
" 0.4
0
S0.5
0 0.02 0.0,4 0,06 0,01 0.1!0Nendmmleln.a.! Frequensc,. (A' (010
Fiue,,,. 6. Va,.iation, of,,o,,,oi~z., maximum,., ,,,,,<,m, stress, Iz-,,o,1, wit,nondimensional frequency, w', for .0 = 5.00 over the range0 9 W' :9 0.10 for values of si as indicated.
28
*11
1.0 - 1 T
0.6005 0.01
S-0.03
0.9 _
0.8-• -1. 0.05
"u 0.7
-I
_0
. • 0.6vJX~L 0.13
0.5
S 0.4
0.3
0. 1"
00
0 0.02 0.04 0..6 0.08 0.10
Figure 7. Variation of normalized maximum dynamic sftr,ýI=,, withnondlmimlional frequency, w', for 0 = 7.00 over the range0 !& w' S• 0.1 for va1u• of A as indicated.
I29
3.0
2.8lUJ
2.4 ....
NN
*2.0 C"_ -"_
- 0
E• 1.6
i00
E° '
z u = 0.005
0.8
0"
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Nond-.qlsone. Ff.evfc,. -
Figure 8. Variation of normalized maximum dynamic stress, II. ax, withnondimensional frequency, -;', for ,8 = 0.25 over the range0 S w' 5 ff/2 for values of 1 at indicated.
30
3.0
'U
0j 2.4 .
it
mKma
2.0
• 1.6a0a0
00.
S 0.8 ..
0.I
IK
0 0.2 0.4 0.6 0,1 1.0 1.2 1.4 L6
Figure 9. Variation of nomynlized maximum dynamic stress, iit.w•hnondlmensional frequency, w,', for ig:0.50 over the range0 • '•W/2 for values of A as indicated.
• 31
I,,.
1.5- t
1.4
1 . 3 ___
Uj
i IU = 0.01
o = 0.005
E 0. 9..= 0 0
0
"0. 16 . 0
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Nondimnea'siona! Frequency, W1 CL
Figure 10. Variat~ion of normalized maximum dynamic stress, IE1cIxiI withnondimensional frequency, W', for A = 1.00 over the rangeE w T/2 for values of U as indicated.
32
1.6
1.5
"1.4
LU 1.3H 0
S 1.2
0
. 1
'"A 0.905
000
-J.0
II,
0..0
a
.In
E
*0
0.6
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Nondimensional Froquency, W'" (-'L)
Figure 11. Variation of normalized maximum dynamic stress, ImIaxj, withnondimensional frequency, w', for 0 = 3.00 over the range0 ' w' ' ff/2 for values of / as indicated.
33
1.6 F
1.5
1.4
1.3'U
-j
1.2 --..00
E
S0.9
0.8.04
°..7,
0.6"0.03
1. 02.00.0 2 1.___.6
E
C
E
EX
0.
E
z
0.60 0.2 0.4 0.6 0.e 1.0 1.2 1.4 1.6
Nondimensional Frequency, w' x (.a.L)
Figure 12. Variation of normalized maximum dynamic stress, J1ýxj, withnondimensional frequency, w', for 0 = 5.00 over the range0 f w' s ff/2 for values of p as indicated.
34
'IpA -
1.6 1
1.5
1.4 -__"
1.3
S 1.2
"- E
S•.1 •• -• = 0.10S= 0.05
i • . =0.04S=0.03
11C AA ' 0.01
0 1.0 -= 0.005
-- ~0.7
0.60 0 .2 0 .4 0.6 0.8 1 .0 1.2 1.4 1.6
Noendimensionel Frequency, a ,.' C L,)'
i Figure 13. Variation of norma~ized maximum dynam•ic st:ress, lZ;.ol,, With
Ill nondimensionol frequency, w', for P = 7.00 over the range
S~0 1- w:' :9 ff/2 for values of/•u as indicated.
I35
f
'CI
1010
10
'C
- - - -- - -
00 U
'c '
CE
*00 0~-
cn
* C
1,00 0ri
100
M 0.05
- =0.1 1- - -I
CO 0.5
_1.02.0
E
X, 1" 0 -01 1 1 IL\
nodmnsoa freqency w5.0 .1 vrth ag
E / [OzO
700 s ' 1.41f for values of i as indiated.37
J.\ II,1;/1V - ;'
1,000
lt.
wi
0!
0 C~M0.050 A = 0.1
"-•100) A = 0.5 _1 00 =• = 1.0
E0
- ___ - - -~~~ =l.0 - _ _ _ _ _ _ -
~= 5.0 -
EiCa
Ez0E10
I
0 0.277 0.47T 0.677 0.8 7 1.0•7 1277 1.47
Nondimrnenional F,.qu.rc€, W'" (a!-L)
Figure 16. Variation of normalized maximum dynamic stress, 11ýxfwtnondimensional frequency, w', for 0 = 0.30 over terangeff/10 S d :6 1.411' for values of g as indicated.
38
0*]0
1,000 -
1..JLU
0
S100
IMI
a ,=0.05E 0 0.1
. 0.5E 0 ~1.0E- 2.0
:0 - 5.0
M
E0
10
iii
0 0.2r 0.4• 0.6r• O1 TT 1.0?? 1.2- 1 4 "
Nondimensionel Frequency, • u ('•1
Figure 17. Variation of normalized maximum dynamic stress, Jlaxl' withnondimensional frequency, w', for 0 = 0.50 over the rangeT/IO S J• s 1.4w for values of /A as indicated.
39
m7
" I I1,000 - --
100
-2 0
/10 : w , 1.0 f 0.0 -- - - 'i
S1 .0- - - - - = 5.0
-P
NA
0 0.2" 0.4 ' 0.6 -O.8 10!2" e" ;
NondimenslonoI Frequency. *, ••
Figure 18. Variation of normalized maximum dynamic stress, IIL 1 , withnondimensional frequency, wi', for • = 0.70 over the rangev/lO '' •' 1.4fr for values of 4 as indicated.
~-..--
1,000
a = 0.05
0 0=0.1- 0.i 5LU
"• 1"i:o . - 1.0J • u -- 2.0
A .5.0
III
-100 - - _ _ - _ _ _ _ _ _ _ __ _ _ -
N - ___ - -
• o _
aE,
E
011
_-- -\I,--- - _
U -. -y-/- - .
PIE
z10
%
"0 2 0.d- 0.6 0 " 80 2"0
Notndmensionu. Fiequncy, .
Figure 19. Variation of nornmlized maximum dynomic stress, a Xe withnondimensional frequency, w', for A = 1.00 over t range17/10 1 wl s 1.4w for values of p as indicated.
41
1,000
10 0.,, 1
- -- .- -- 1
C) M0.5
1 0 1.0=- . - -0.U; - . os -4 11 -
-. n A .o _ _• ....5.0 I JEo I
E
01
0 0.U- 0,4' ~ 0.6'- ~ i0 ~ 1.4-
NondiwnsoonI Freqlnc 1. ^
Figure 20. Variation of normalized maximum dynamic stress, jr; , wit'hnondimensionol frequency, w' for 0i3.00 over the range/101 w 1 1.4ff for values of 's indicated.
4? , -i
1,000 ... .. i
"100-
N ------- O'a 0.5--- - _
0 1.I- , ' O --i' o
V.I0 ;--2.0---- , V1-I--- - - ---'-
CI ___ -- K--S--5.0---1----
rI'
0 A I
" 100
0 0.2- 0.4 0.6_ "",2
NondjlownsionalFou.¢.. "( )Figure 21 Variation of normalized maximum dynam0ic stress, 2ax 'with
nondimensional frequency, w',, for • 5.00 over the range//10 w' '1.4w for values of/ as indicated.
43
1,000 ~ 1 -T - - -r- -
_.oo ' - --T• -I- __ ,_ ___
100 0 0.=1.
E- 0 •'-0.5 -__ ____-
CO =2.0 -,
7 =5.0
__ _1_ -4/1\-\E 43
Z0 ____o ____ ________ ..... _ ____ ____
O.2?047 . 70.87?7 1.0 77 1.2 77 1.477
No,,dimensional Frequency, ,)' S (-•.)
Figure 22. Variation of normalized maximum dynamic stes m~ax ,' withnondimensional frequency, w•', for 3 = 7.00 over the range !"7/10 'wu' < i.41nr for values of p as indicated. l
44
E.. . !40e. ,,
1.000
oL .0 3.0
A ~=5.0
100 ____ _ _ _ _--_ _ _ -
NN _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
E _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _- ---
E
E0
If-i
0 0.5-'T 1.0'7 1.5 71 2.07 2.5
Nondimensional Frequency, c
Ficure 23. Variation of IL1, 0 xI with for a particular A'.
1~ 45
1,000 T
_00__
//
-J ~ /
I-I
W j u 2 0 .5 0/
n• 0. 10
_100_ /
E
0/
E
t I
Nadmniya Fr! ~cW2(aCL
Figure 24. Variation of Iý'axj over a wide range of w'. Dashed line
illustrates approximation used.
461
10,000 z 0.10
'U
0
II
01
".•~~~ ~ .,, 50 °••n 1,000 --
C
E ,,-
E -
X
2.0
E
z
100 - Iz 'Air
10
0 IT 277 37T 477 57T 67T 777
Nondimenslonal Frequency, GL) a (-4)
Figure 25. Variation of normalized maximum dynamic stress, withnondimensional frequency, U, for t = 0.10 over thCe rangeir/5 6 W' S 7.0C* for values of ps as Indicated.
47
100,000 _____ _____ _____ _____ _____ __ ___
10,000 0.10,______ _____ _____
0
-J x_ _
'~1,000 _ _ _ _ _ __ _ _
E
E
E
10
10 IT 2 7T 377 47n 577 6777TI
Nondimensicnal Frequency, W'
Figure 26. Variation of normalized maximum dynamic stress, withnondimensionol frequency, w'9, for 0 = 0.30 over the rangeff/5 'w' d 7.01r for values of As as indicated.
48
100,000
10,000 _ _ _ _ _ _ _ _ _ __ _ __ _ _ 0.10
iU
-u
= 0.50
S1,000U4
E
0=.00
E
E
20
E
z
100•
100 _______ _ _ _ _ __ __ _ _ _ __ _ _ __
0 77 2- 31 477 577 6" 7U 8'
Nondimensional Frequency, •)(•")
Figure 27. Variation of normalized maximum dynamic stress, m , withnondimensional frequency, w', for 0 = 0.50 over the rangeff/5 : w' : 7.0i for values of A as indicated.
49
100,000
10,000 0.__._ [ .-- 0o. lO
'UIJ a"0
S] . - 0,50
E 1,000
E 0 1 .00O
-E
V
E• LI
0
1,000a
NadmnsoC Frequency,_ (A)'___
Fiur 28. __ Vaitonooraiedmxmu_ yamcsres__j wt
Eodmnsoa frqunc,_',fo_070ovr__erag
If5-ew .f o auso s;dctd
N5
10C,000
10,000
00
0.0
0.5
J; 1,000 _______ ___
E ______.00_____ ______
E
2.0
0
0
100
0 "2" 3- 4 5' 6 7
Nandimonsional Frequency,~ (~Figure 29. Variation of normalized maximum dynamic: stress,, PVIEaxI' with
nondlimensional frequency, w', for 0 = 1.0G avF the rangeff/5 7.01r for values of ;A as indicated.
51
100,000
10,000
k 0. 1 o 0s
V4 1000
100 i 0•0opw = 2 .° ° .C
00EI
10 0
z4
100 77 2 3 '67
NondimonsionaI Frequencyv, .(..)
Figure 30. Variation of normalized maximum dynamic stress, j I'Withnondimensional frequency, w •', for = 3.00 over t ,e rangeII/5 r' % 7.Nf for values of Am as indicated.
52
_----~---'T
100,0001
10,000 _____ _____ ___
-U _
vJ 0
005
E 1,000o
E
o N 5.001100
E
0z ""0
10 -_ _ _ _
Nondmii atonsl Imuny
Figure 31. Variation of nomlzd=aiu 5.00mi ovrerOIn 0wtnonimesioal reqenc, w, fr 0= 500 verthe range
W/5 'd is7.01 for values of ps as indicated.
53
10�,,000 _________
_________________ ___________-J
10,000
w'U __
0-J � ____ ____ ____ ____ ___ ____
__________ __________ __________ 2.00NU
-. 5.00aa0
� 1,000US
C
0
I2I ____________
NUI _______
'U0N
0I0
z
100
_____ ____ ____ - -�
32. Nondm.n,.aeiol Frequency . * (�A�L)3'
Variation of nor�ol�zed mcxmum dynamic stress, I2 '�OE 1 withnondmensionol frequei�cy, u.'� for � 7.00 over tF�e range
cd I 7,0w for values of �s as indicated.
5'
100,000 1 1 __ -- I I. , .....
I*strot f onC U0 I f. 050o
freeddspring) 0 LO 2f: 10U 6 f0 2t 3 00
Il U0 10 ft: 500o
0 U0 1 lft: 7.00
10,000 Dampi ng cons ditred
10,000 - -- Damping not cansidered
1000
0
CIN
100spigwe o
C~ A M I Kr I
Figue 3. Rsult fo deignexamle sin poyproylee cble
100,000
o U. =it,: 3=0.50
_____U =__ 2- 9 0 ft: i3 =1.00
U._ = 6U 6ft: 14 =3.00
U. = 10 ft: /5.00
qA C- U0 14 ft: 8 7.00
10,000 I
1,000
0 in sea stat_
.C
B - lowrfrequency limit
100 = -w ifrqec i tin sea state 3
- - D = upper frequency liint
in sea states 3, 4, and 5
10I II
110
Frequency, CJ(rad/s~c)
Figure 34. Results for design example using steel cable.
56
T T T TI T
9-- 4--.9- +-r
WI RI IViNýtf1 fII 1ý Iý
I,, .1TI,.
IPeriod of intermittent Isu mer,,rg -0 -L... _ _ _ - - --- l iL 7
Time From Start ofLowering Operation: 10 min
Depth Below Surface- 450 ft
Figure 35. Record ofCornparisci
T - T 10 -T---- T - -- T 1----- . .1--
84 7 -I- 74
I I I
; V - -'-5-±-----..... - ....... -- J- -- ... I " -I --- -.... i -lr.. .
6- +
r----- y---- I---- -- - *. . 6-• -r - {-- 4-- 4 ~o~ I .. ,,o- o , , .. , ,-, , >
5 5~
I - - -{ -+ 2
115 niin 20 min
1,130 ft i,RIO It
Figure 35. Record of cable tension during STU lowering operation.Comparison with calculated maximum dynamic stresses.
B §7
I .... --F, F- t. -- r- T-r v 1 0
, 4 !1 , __ . 1 __ I ____
t I
8 - 4 ------ -
7 - 1 - - --- -- -4, -6 6
5 5
I _I _ _'i 120 m I In
4_ _ - - Impact o oca fl r, 4
at 2,500 ft
LSte.~dy descent ot ivelocity of 2.27. ft -ec ~ ~ - 7 1i5 nin 20 minm 25 min
1,130 ft 1,810 ft 2,500 It
wring STU lowering operation.*d maximum dynamic stresses.
C. .,I .... 5
Appendix A
ANALYSIS OF CABLE AND PAYLOAD DYNAMICS FORRAISING AND LOWERING LOADS IN THE DEEP OCEAN
The analysis presented here is essentially that given in BuShips TechnicalReport No. 1370863, "Stress Analysis of Ship-Suspended Heavily Loaded Cables forDeep Underwater Emplacements," by Arthur D. Little, Inc. 1
The cable and load are considered to be in a vertical position as shown inFigure 1; i.e., the deflection of the cable due to currents is small. Any displace-ment of the cable support point will cause the dynamic displacement u(x, t) at timet of an eiement of cable originally located at x; see Figure 1. Displacementu(x, t) is of a form which satisfies the equation for the propagation of waves in thecable:
12 •2c U u C)up S - SE Ku (A-I)
c 2 - 2 atct ciX
where pc = density of the cable
S = material cross section of the cable
E = modulus of elasticity of the cable
K = constant of friction on the cable due to the surrounding water
In an actual lowering or raising operation, the length of the cable, L, varieswith time. However, it is assumed that L may be considered constant over short
periods of time; i.e., the net vertfcal motion of the load and cable does not influ-ence the dynamic displacements due to the cable support-point oscillations.
The following nondimensional variables and parameters may then be defined:
x5 X tc (4d, e)L t L
2 E KLc PP
P c PCS (4 c, a)
c
and Equation A-1 becomes
2 2+ ¾u __t'_ (1)
(L t') 2 (6x')2
58
T--
The boundary conditions at the upper end of the cable is the specification thatu at x' = 0. The boundary condition at the load, when x = L, is given by*
•2M u+ ES +u -1 p ul au _0 (A -2)
a ýt2 - x 2 DPA t at
where Ma = dynamic mass of the array
CD = drag coefficient of the array
A = horizontal cross section of the load
By defining the parameters
PC 4Z L CDPA
Ma B - 2Ma (4b, f)
Equation A-2 may be reduced to
32 u ýu au Iul 0+ (2)
0 t')2 X
at x' = 1.0.The difficulty of applying this boundary condition, Equation A-2, arises from
the nonlinear term B 16u/ t' I (au/6t'), which represents the drag on the load. Toavoid the complexities arising from this .nonlinear term, an approximation was madein the reference report by replacing the I bu/lt'I term by (8/31r)w U1, where U1 isthe amplitude of the load displacement, which is assumed to be sinusoidal. Thisselection results in the same energy dissipation when u is sinusoidal in the third termof Equation 2. It is demonstrated in the report that this approximation leads toerrors on the order of 20% ;n the drag term.
Defining a normalized displacement amplitude U' equal to U divided by I U01and noting that U' is the value of U' at the load, a solution for U' as a function ofx' is given byA
n U' = U1cos w'y' + Csin w'y' (5)whet. e Y' I x' and
w,' L (6 b )C
59
This solution satisfies the governing equation, Equation 1, provided thefriction of wctc: on the cable may be neglected. In Appendix A of the referencereport it is shown that this assumption is valid for the frequency range of interest.
Substituting Equation 5 into Equation 2 and incorporating the boundaryconditions, the unknown complex constant C is determined as
C-- U1 (-I + i U1) (A-3)
and hence, Equation 5 reduces to
U' = U 1 seccQCos(U."y' + () + ii(U)2 tanqPsin 'y' (A-4)
where
tan : , 0 :rh (A-5)
In requiring that IU'J at y' = 1 be equal to 1, Ul is determined to be
01)2 = co2 ( sin 2wesin + ... s . - 1 (8)
2/32 si n2 sin2 , Cosi4(' 1+/) 2If the amplitude of the dynamic stress is denoted by I and a normalized stress
amplitude, V', is defined equal to L I/IuoU E, the distribution of V" is given by
S' w'U1 secQ sin(e'y' +o) - iwA(U,) 2 tanPcosw'y' (A-6)
Hence the normalized amplitude of the maximum dynamic stress 1,no., is of the form
S(W') 2 (U)2 [1 + tan o (ton ýV +sec 4 )2 '7)
where
Cos (2, + Q) ., 2.,i 2 11/2(Ul) 2 = 2 sinL20 snn2 s4 +) - (8)
S= arc ton-, 0 (9)
60
S.. ... 4m= ' -" • 1----
F12 U12 1f4' arc tan -Y U k tano - cct2d , 2 4, (10)
Equation 7 together with Equations 8, 9, and 10 was evaluated by use of acomputer program as given in Appendix C, in which I-noax was determined as afunction of w' for various ranges of • and u.
61
Appendix B
EVALUATION OF THE NORMALIZED AMPLITUDE OF THEMAXIMUM DYNAMIC STRESS AS w'- n IT
As noted in the main text in the Discussion of results, the normalized amplitudeof the maximum dynamic stress, I I ax1, is particularly sensitive to variations of *•'For particular values of w', given specific values of A andli, it is desirable to eval-uate the peak in I ]axl more precisely than by an interpolation of the computeroutput. This essentially requires the derivation of dInax/d ' from Equation 7,equating this to zero, and solving for w' as a function of 0 and s. Inspection ofEquation 7 indicates the complexity of this derivation, which is unilluminating interms of a proposed design procedure. As an alternative, Equation 7 was evaluatedir the limit as w' approaches nfT, where n = 1, 2, 3,... In certain cases, thiscorresponds to the peak in the maximum dynamic stress I ax*. The analysis wascarried out in order to determine the inaccuraries involved in interpolating thecomputer output, and is repeated here for comp!etenass.
Given Equations 7, 8, 9, and 10 below, it is required to determine the valueof! xlas W' approaches nit, where n = 1, 2, 3,...
2 = 2 2 7,(.nax) = (we), (U01)1 + tan o (ton 4\ + sec'k) (7)
2/ ,2 1-1(8
(U) sin2 Qsin2 W,' cos4 (w' +0) ]-arc tan-, 0:1-- (9)S2
" arc ton g2I2 (U,1)2 tan (• co20
to o cot 2o]s- (10)
From Equation 9, when w' n ir,
62 n
62
2()cot 2 to
2 tan 2K)
From Equations 10, B-1, and B-2,
(q n~ 1 ("~
1 2 .2 n~ IT (ntant' - -2 (U1 ) - . . .
L 2(2i-
As ' n I sin 2 ' -" 0
2 2cos (.' + 0) cos -
4 4cos (w' + )-cos -
Bv ~fn n C (,2s2 2¢"o43,' de:fin ng C 2 sin 22 `cos 4', Equation 8 becomes
12 Cos C.22 2
) 24 in2 sin
In Equation B-6,
2 1.,,2 "/-2 . C 2 sin.' C O ;A;(I "C sin2 } - 1:I
I.i.'- ' si2,.; 2. Sn in 4?•2 ~ co ssin n C sin,,21
iClim .. - C-"nr[ 2
Therefore, from Equotion B-6,
S, / 2
2 2-n 26d2 sin3
63
sin2 2o
•, 4k2 cs4€l) 2= 1-n4 ( CO2psn ) _= cos2•s•2
That is, lim(Ul) 2 = 1 (B-7)
W-0 n i
Thus, from Equation B-3,
tan 1 2 n - A (B-8)
But sec 4 = V1 + tan24, and since -7T/2 ff 4 . iT/2,
sec 4 = + ý1 + tan2 4 (B-9)
From Equations 7, B-1, B-7, and B-9,
I'mr(ý-ax)2 =(nIT) 21 + [ ang' + (I+tan 1/) (B-10)
w n ifft
or (nff)2 1+ n 1V 1 -)-
t t ~L A--•I
2 1/2
1 ^2 n i li
2 i2
64
*i
and letting K = n Tr/p,lim(" ax2 (nT)2 1+K WK
2] 1/2
+!I + 2 K + K 13
. 2 2K 2
Values of I 1'.xI, evalunted from Equation 13, are given in the Resultssection of the main report for various ranges of 0 and s.
65
65
Appendix C
COMPUTER PROGRAM FOR THE EVALUATION OF THE NORMALIZEDAMPLITUDE OF THE MAXIMUM DYNAMIC STRESS,
The program evaluates the normalized amplitude of the maximum dynamicstress,, I Tx, according to Equations 7, 8, 9, and 10 given in Appendix A andis written or use on an IBM 1620 digital computer. A flow chart for the programis given in Figure C-1, followed by the FORTRAN source program.
Input parameters are read as follows in format (2F 10.2, 2F:10.8, 2F5.2, F 10.8):
CAY = parameter equal to 4CD3 A - k3ff M0wi
UM = parameter equal to -L
SFR = increment on nondimensional freqlency scale, Aw'
FMAX = maximum nondimensional frequency, 4nax
STU = increment on input displacement amplitude I UoJ, 1 uo
UMAX = maximum input displacement ampli'.ude, I U0omax
FZER = initial nondira6nsional freciuency, wo
The cornputod cJtput is presented on punched cords in the following form. Thevalues of k IA, P, andI UoI are given fo1lowed by the input parameters as definedabove. The computed values of I .,,I are thenr tabulated at each value of thenondimensional frequency, according to formnt (E15. 8, 3X, F8. 4) for a particularI nUIj. The process is then repeated for each value ofI U0 up toI UoImax, at whichpoint a new set of input data is required.
The initial frequency is used as an input parameter "n order that specific rangeson the nondimensional frequency axis may be investigated; e.g., relatively smallincrements in frequency may be used over a range of frequency corresponding to peakvalues in I EaxI. There is a limit to the smallest allowable increment in frequencyresulting from the rouJnding-off er,'ors inherent to the program, which results in .oxjequaling 0 at w' near to IT. These errors are discubsec' in the main text underDiscussion of Results.
66
66
Start
Read
ZILY
.Uamx, Ao 6U
Set
Uo = 1.0
SSet
PunchS- K.K AtReUo
Set
I K, JA, Awl',
S ComputeUooma
U0 m UmaxSet <:VM..eI
Uo~U Z: Uomax
Figure C-1. Flow chart for computer program to evaluate max '
67 0 StL
F'ORTRAN SOURCE PROGRAM
C STRESS IN CABLE LOWERING LOADS TO DEEP OCEAN.C CAY=CONSTANTK, UM=PARAMETER9 SFRZSTEP IN FREQUENCY.C FMAX=MAXIMUM FREQUENCY, STUmSTEP IN U-ZERO,C UMAX=MAX!MUM U-ZERO, FZER=INITIAL FREQUENCY.
1 READ 100,CAYUMSFRFMAXSTUUMAXFZERUO:1.0
8 BETA=CAY*UOPUNCH 101PUNCH 102,CAYUMBETAUOFREQ-FZERPUNCH 111,CAYUMSFRFMAXSTUUMAXFZERPUNCH 109
5 PHIxATAN(FREQ/UM)ALPH=2 .*PHIDELT=FREQ+PHIUONSZ(COS(DELT))**2/(2.**3ETA**2*SIN(PHI)**2*SIN(FREQ)**2)TERM=(1.+((BETA**2*SIN(FREQ)**2*SIN(ALPH)**2)/COS(DELT,**4))TERMU TERM**095TERM= TERM-ioUONsUONS*TERMU=UON**0. 5TPSIz((Oe5*BETA**2o*UON)*SIN(PHJ)/COS(PHI))-ICOS(ALPH)/SIN(ALPH))PSI =ATAN( TPSU)STRS=d1.+SIN(PHIU/COS(PHU*(SIN(PSI/COS(PSI)+leCOS(PSIr)STRS=STRS*UON*(FREQ** 2.)STR-STRS**0.5PUNCH 11OSTRtFREOIF(FREQ-FMAX)39494
3 FREQ=FREQ+SFRGO TO 5
4. IF(UO-UMAX)691916 UOzUO+STU
GO TO 8100 FORMAT(2F10.2,2F10.8,2F5.2,FlO.8)101 FORMAT(SX,9H CONSTANTv4X93H UM,1OX95H BETA,8Xv,7H U-ZERO)102 FORMAT(3XF1O.2,3XF1U.2,3XF1O.2,3XF1O.2)109 FORMAT(7H STRESS*1OX91UH FREQUENCY)110 FORMAT(E15e893X9F8*4)Ill FORMAT( ZF1O.2,2F10.5,2F5.2,F1O.5)
END
68
Appendix D
SUMMARY OF DRAG COEFFICIENTS
INTRODUCTION
In an analysis of the motions of a body through water, whether the body isfalling freely or being lowered by cable, one of the most important effects whichmust be considered is the resistance, or drag, experienced by the body.
The purpose of this appendix is to summarize existing information on dragforces and indicate areas of work which must be covered in order that such forcesmay be included in calculating the motions of a load being lowered to the deepocean.
DRAG IN UNIFORM FLOW
On the front of every solid body moving through water, there is at least onepoint where there is no relative motion between the water particles and the body;i.e., there is a stagnation point. The pressure at this point, termed the dynamicpressure, is given as
P 1 2 (D-1)stag 2p(
where p is the density of water and V is the relative velocity of the body to thewater. It is convenient to express the total drag due to pressure forces relative tothis stagnation pressure by defining
D =CD(I PV2)S (D -2)
where D is the drag force due to pressure, CD is the coefficient of drag, and Sis a representative area of the body - either its frontal or cross-sectional area.Equation D-2 is essentially a definition of CD.
The total drag on any body consists of the "pressure drag," defined above,plus drag forces due to skin friction. However, for angular bodies such as thoseenvisaged as loads to be lowered to the deep ocean floor, the skin friction dragmao1 be assumed small compared to the pressure drag.
69
Accord:-. , •,;ynolds' Similarity Law, the flow pattern around the dragcoefficients on two similar bodies (identical in shape but dissimilar in size) movingthrough a body of water are similar if their Reynolds numbers, Re, are identical:
R Vdp Vd (D-3)e A V
where V is the velocity of the body relative to the water, p is the density of thewater, 14 is its absolute viscosity, V is its kinematic viscosity (p/p), and d is acharacterizing dimension of the body.
Hence it is possible to determine the appropriate CD for a body moving throughwater from the results of experiments performed on an identically shaped body of adifferent scale and possibly in a different fluid, provided the Reynolds numbers areequal.
The kinematic viscosity of sea water at normal temperatures and pressures ison the order of 1.5 x 10-5 square feet per sec,)nd. If a load to be lowered to thedeep ocean has a typical dimension of 15 feet, and moves at a velocity on the orderof 1 foot per second, the Reynolds number, Re, equals 106. It appears that relativelylittle information is available from the literature on the variation of coefficients ofdrag at Reynolds numbers greater than 106 to 107 . Figure D-1 with inserts show thevariation of CD with Re for spheres and cylinders respectively, and summarizes some,though by no means all, existing data on the coefficients of drag applicable to bodiesof different shapes.
Although objects to be dropped or lowered to the deep ocean floor may not bespherical or cylindrical, a brief investigation of the dynamics of a sphere is illu-minating. Consider a body held stationary in water and which is then allowed tofall freely. During the initial motions, the velocity is small and the body willaccelerate under its own weight minus a buoyancy force due to the weight of waterdisplaced, the drag force being negligible at this stage. This net vertical force actson the mass of the body plus a certain frcction of its mass which is included toaccount for the water contained in the body, if any, and an effective mass of waterto which accelerations are imparted due to the motion of the body. The latter termsare usually called the "apparent added mass"; the total mass (body mass and apparentadded mass) being termed the "virtual mass." Values of the apparent added mass varyfrom 40% to 150% of the mass of the body.
As the velocity increases from zero, the drag force opposing the motion becomessignificant, and at a particular time t = tj this force is given by
FD CD (PVI2)S (D-4)
70
-4 - t,£r'
where Va is the velocity at t = t 1 , and CD is the approprinte coefficient of drag.After a given time the velocity of the falling body attains a terminal velocity, VT,in which condition the drag force balances the body weight minus the buoyancy force;i.e. ,
(WB - FB) FD (C PV 2 S (D-5)
where (CD)T is the coefficient of drag at the Reynolds number :,is.,eponding to a
velocity of VT. Equation D-5 may be rewritten as
VT L 2(WB ~FB FD f( P )± V ) (D -6)
The function f (VT L/g) is not known and cannot be defined analytically, andEquation D-6 cannot be solved explicitly for the terminal velocity, VT, withoutthe prior assumption of a particular CD.
However, starting from zero initial velocity, it is possible to determine themotion of a particular body by considering the acceleration and velocities attainedover small increments of time. A simple computer program was written to accomplishthis. At t = 0 the velocity is zero, there is no drag force, and the body will accel-erate under the force (WB-FB). At t = t 1 the velocity is finite and the appropriateReynolds number may be calculated together with the corresponding CD. For thepurpose of these calculations, CD was specified at increments of Re, and a simpleinterpolation was made to determine the specific CD corresponding to Re at t = t1Hence the drag and out-of-balance force may be calculated at t = t1 together withthe instantaneous acceleration at this point. The process can be repeated to deter-mine the velocity of the body at time increments from t = 0 to t = T, where T is thetime taken to attain a terminal velocity.
For an up-to-date complete treatment of hydrodynamic drag, the excellenttreatise prepared and published by Dr. Sighard F. Hoerner 5 should be consulted.
71
intin
rI.
enn v
' m K\\(qlCoco 0 0-;
0 \ N VP Ic8:
M 3t
C.1 I
+4W
II I- --x -0
U, o ,
iil
72°
0 -of+,0wwolU! !'or'-I
o I -C
Appendix E
SUMMARY OF ADDED MASS COEFFICIENTS
The concept of added mass is well known in fluid mechanics. The physicalexplanation of this phenomenon is that when a body is subjected to an unbalancedforce, not only must the mass of the body be accelerated, but also that of theadded fluid mass surrounding the body. The ratio of this added fluid mass to thebody mass is :'..c.-') mass coefficient, Cm.
The added nass depends on the dimensions and shape of the body and thedensity and viscosity of the fluid. In general, measurements of the apparent addedmass have been obtained under two fundamentally different flow situations. In onethe motion is oscillatory in that an immersed body is vibrated. In the other themotion is unidirectional in that an immersed body is cccelerated rectilinearly. Theexact analytical description of fluid resistance to the acceleration of an arbitrarilyshaped submerged solid is not known, hence the exact added mass coefficient forvarious shapes of objects is not known. The following reports are the results ofdifferent experiments under different conditions, but they are quite consistent:T. E. Steison and F. T. Mavis, 6 E. Silberman, 7 T. Sarpkaya 8 N. L. Ackemannand A. Arbhabhirama, 9 0. C. Zienkiowicz and B. Noth. 10 A summary of most ofthe important results obtained from these references is presented in Figure E-1.
The coefficients, Cm, have been arranged in terms of a common dimension,namely the ratio of the added mass to the mass of fluid displaced. The results obtainedfrom oscillatory motion are as follows:
1. Spheres: Cm =- 0.51. This compares with a value of 0.50 obtained fromideal fluid theory for rectilinear mot;on.
2. Cubes: Cm = 0.67 ("broadside-on" or "edge-on").
3. Cirulor Cylinders: See Figure E-1. The abscissa is the ratio of length todiameter. The motion is in the direction perpendicular to the circularcross section.
4. Rectangular Plates: See Figure E-1. The abscissa is the ratio of length to"width. The motion is in the broadside-on direction. The ratio of thicknessto width is limited to values less than 0. 04.
5. Square Prisms: See Figure E-1. The abscissa is the ratio of length to widthof the square sides. The motion is in the direction perpendicular to thesquare cross section.
73
6. Symmetrical Lenses: See Figure E-1. These lenses are two intersecting orseparated spheres:
SI , I I t t I ' , = ..I• I "
The abscissa is the ratio of B/R. The motion is ;n the direction shown.7. Two Parallel Rectangular and Square Plates: See Figure E-1. The abscissa
is the rato of spacing between two plates to width, where the spacing ismeasured from center to center of the plates. The ratio of length to widthof the plates ari over 17 to 1. The motion is in a direction parallel to the
thickness of the plates.
For till of the oscillatory motion cases the experiments were conducted at lowvelocities and high accelerations. Thus the total resistance force to the movingobject is largely due to the added mass which is dependent on the acceleration. Athigher velocities, the total resistance force is due to a velocity-dependent drag termas well as to the added mass term. That part of the resistance to motion due to viscousand form drag and that part due to added mass are difficult to separate. S ilson andMavis6 and Silberr.an 7 realized the difficulties. From experiments on a sphere themeasured added mass increased by approximately 1% above the values obtained fromideal fluid theory. Thus it was concluded that viscosity did not seriously affect theexperimental values for the added mass.
A recent method (Zienkiewicz and NathlO) of measuring the added mass isworth mentioning here. Using an electric analogy method, the virtual mass as wellas the pressure distribution around a rigid body accelerating in on incompressibiefluid can be determined. In the following table, the results are compared with theknown added man coefficients obtained from other sources. Agreement is excellent.
Added Moss Coefficient
Object Obtained by From Indicated Souce I
Zienkiewicz 10
Infinitely long vertical plate 1. 03 1.04 (Riabouchinsky 10 )
Infinitely long cylinder 0.98 1. 00(H. Lamb ))Thin circular disc 0.61 0.636 (H. Lambil)Cube 0.62 0. 67 (Stetson6 )
74
These values are measured for an infinitely large submergence depth. At smalldepths the measured added mass decreasea. This agrees with the physical explanationof the added mass phenomenon. The experiments were conducted for small-amplitudemotions and no separation occurred, hence the boundary effect is not considered.This method can be used for rotary acceleration of a body. This enperr•rntal methodhas important implications since it can be set up in a deep ocean simulating tank tomeasure the added mass as well as the pressure distribution of any orbitrari!ly shapedobject under translational or rotary motion.
For the unidirectional motion, the viscous and boundary effects must beconsidered. Experiments for this type of motion have been conducted for a numberof bodies, but only that for spheres will be cited. Arbhabhirama 9 found that whenthe ratio of the diameter of a sphere to the diameter of a fluid filling a concentricspherical shell is 0. 259, which is similar to a sphere oscillating in an infinite fluid,the added mass is found to be 1.03 times the added mass obtained from potential flow.
In summary, although some data is available on added mass coefficients inoscillatory flow, most of the experiments have been conducted at small scale andwithin the low Reynolds number regime. As an example, the following estimate ofthe added mass of a complicated frame structure such as the STU described below, iscited. Theoretically the oscillatory motion of a load being lowered to the oceanfloor and suspended by a cable is a damped simple harmonic motion. If the cable isconsidered to be elastic, the equation of motion is
M x + C> + kx = 0
a
where M is the virtual mass of the load, x is the elongation of t0e cable, k is theratio of t~e restraining force to the elongation of the cable, and C is the coefficientof damping. The solution of this equation is quite complicated, but the period ofoscillation is the same as for simple undamped harmonic motion:
M x + kx -- 0
Hence, the period T = 2 1? (M/A).On 13 April 1965, a Submersible Test Unit (STU) was lowered by this Laboratory
to the ocean floor to a depth of 2,500 feet using a 1.3-inch-diameter polypropyleriecable. The cable tensions were recorded as a function of time from the start oflowering operation. The curve of the graph (Figure 35) shows the oscillatory motionof the STU, consequently the average period of 9.8 seconds was obtained while theaverage tension of the cable in this interval (between 10 and 12 minutes as marked onthe figure) was 3,400 pounds. The breaking strength of the cable is 45,000 pounds("Braided Rope and Cordage Catalog, " Samson Cordage Works, Boston, Moss.). Fromthe percent load of breaking strength versus the percent elongation curve of polypro-pylene cables, the corresponding percent elongation of 4.0 is obtained. Since the
75
S""-
average cable length in this interval is 586 feet, the elongation of the cable is0.04 x 586 = 23.4 feet. The restoring-force constant was determined to be 145 poundsper foot (i. ,, 3,400/23.4). The virtual mass of the caL!1 assembly was fourd to be11,360 pounds; i. e., IM = (T2/4 2)g = [(9.8)2 (145)/423 32.2. Of this 11,350 pounds,950 poo.nds is tlue to the static weight of the suspended cable. The net weight of theSTU in woer is 5,500 pounds. Thus the added mass coefficient, Cm, is detenrmnedto be slighily less than 2.0 (i.e., 10,700/5,500).
76
.- pot -
-1 if 0 Cylinder- Rectangular plate
__ i t! t .~oo,,-°oouo ,t -
___ I - - 0 Square prisms' 1... 0 Symmetrical lenses
I -' 1i I A Two parallel rectangular plates
t 4j Two parallel square plates
__________Stelson and Ma~vis 6
-- •--Sarpkaya8
10 -zz ____zz _
a - 0 .1
I .7 = 0.2
10
_ _lii•i
a = Characteristic length dimension normal to direction of motion __ , i
t = Characteristic spacing dimension parallel to direction of motion
0.1 [ _ I_._ I I I_ I_ III0.01 0.1 1,0 10
(See text listing in Appendix E for abscissa units)
Figure E-1. Added mass coefficient as a function of circular frequencyfor constant geometric ratios.
77
REFERENCES
1. Arthur D. Little, Inc. Technical Report no. 1370863 on Project Trident, Bureauof Ships Contract NObsr-81564: Stress analysis of ship-suspended heavily loadedcables for deep underwater emplacements. Cambridge, Mass., Aug. 1963.AD-418028.
2. Marine Advisers, Inc. Report on Contract NBy-32206: The motions of a mooredconstruction-type barge in irregular waves and their influence on constructionoperation, by P. Kaplan and R. R. Putz. La Jolla, Calif., Aug. 1962.
3. U. S. Naval Civil Engineering Laboratory. Technical Note N-604: Theengineering applications of a report entitled "The motions of a moored construction-type barge in irreguiar waves and their influence on construction operation," byW. J. Pierson and P. Holmes. Port Hueneme, Calif., Aug. 1964.
4. U. S. Navy, David Taylor Model Basin. Report 1221: Theoretical analysis ofthe effect of ship motion on mooring cables in deep water, by L. F. Whicker.Washington, D. C., Mar. 1958.
5. S. F. Hoerner. Fluid-dynamic drag, 2nd ed. Author, 148 Bustead Drive,Midland Park, N. J., 1965.
6. T. E. Stelson and F. T. Mavis. "Virtual mass and acceleration in fluids,"American Society of Civil Engineers, Transactions, vol. 122, 1957, paper no. 2870,pp. 518-525.
7. E. Silberman. Discussion of the paper, "Virtual mass and acceleration in fluids,"by T. E. Stelson and F. T. Mavis. American Society of Civil Engineers, Transactions,vol. 122, 1957, pp. 527-529.
8. T. Sarpkaya. "Added mass of lenses and parallel plates," American Society ofCivil Engineers, Proceedings, Journal of the Engineering Mechanics Division,vol. 86, EM3, June 1960, pp. 141-152.
9. N. L. Ackermann and A. Arbhabhirama. "Viscous and boundary effects onvirtual man," American Society of Civil Engineers, Proceedings, Journal of theEngineering Mechanics Division, vol. 90, EM4, Aug. 1964, pp. 123-130.
10. 0. C. Zienkiewicz and B. Nath. "Analogue procedure for determination ofvirtual mass," American Society of Civil Engineers, Proceedings, Journal of theHydraulics Division, vol. 90, HY5, Sept. 1964, pp. 69-81.
11. H. Lamb. Hydrodynamics, 6th ed., 1932. London, Cambridge UniversityPress; New York, Dover Publications.
78
ASOW
MATHEMATICAL NOTATIONS
A Cross-sectional area of the load in the direction of motion Uo Input displacement
CDPA w Weight per unit ler
2 Mu
c Velocity of sound in the cable E Damping parameter
C KKL
CD Coefficient of drag applicable to the load PcCS
C Coefficient of mass applicable to the load W. Ratio of cable weicm
E Modulus of elasticity for the cable P Density of seawate,
F Safety factor for maximum operating stress in the cable P Density of cable mc
4 CD pA I" Amplitude of dynank A constant 3= C
31TC Mm V" Normalized dynam'
K Constant of friction on the cable due to the surrounding water"Ld Allowable dynamic r h
L Length of the cable
M Mass of the load i mraxi Maximum normaliz
M Virtual mass of the loada
1 tatic Static stress in theS Material cross-sectional area of the cable
u u(x, t) - displacement of element from support point Iult Ultimate tensile st l
U Displacement amplitude o Constant defined b
U' Normalized displacement amplitude U 'k Cons:ant defined I-
UUOI %A; Frequency of oscil
U1 Normalized displacement amplitude at the array W' Normalized freque
A79
MATHEMATICAL NOTATIONS
!ction of motion U0 1 Input displacement amplitude
w Weight per unit 1,ngth of cable
4CDPA Uo
SDamping parameter -- 4C D P 0
S KL•c Pc c CS
p SL
ps Ratio of cable weight to virtual mass of load - c _ w LMa Ma
p Density of seawater
in the cable p Density of cable material
I Amplitude of dynamic stress
V' Normalized dynamic stress
the surrounding water ut
hw Allowable dynamic stress in the cable -211- -E
d Fd stti
JL
1ýaxj I Maximum normalized dynamic stress in the cable -E
i Uoi
lstatic Static stress in the cablele
ipport point Iult Ultimate tensile strength of the cable
c Constant defined by Equation 9
U ' Constant defined by Equation 10
Jo' Frequency of oscillation 'cL
he array A' Normalized frequency Lc
79 E
Unc lass fiedSecurity Classification
DOCUMENT CONTROL DATA- R&D('Security classification of title, body of abstract and indexinif annotation mnuat be entered when the overall report• i c loasitted)
I Oq|GINATIN G AC'TIVy (Corp~orate author) 2s,2 REPORIT $£CU~qrTy C L.ASSI"FICATION' -
U. S. Naval Civil Engineering Laboratory - Unclassified
Port Hueneme, California 93041 2bG,,ou,,
3 REPORT TITLE
Mechanics of Raising and Lowe.ring Heavy Loads ;n the Deep Ocean: Cable and
Payload Dynamics4 DESCRIPTIVE NO'ES (Type of report and inclusive dates)
Not Final 1 November 1964 - 15 August 1965S AUTHOR(S) (Lost name. firt, name. initial)
Holmes, P., Ph D
* REPORT DATE 7r TOTAL NO OF PAGIES 7b NOr Kk's
April 1966 87 11Be CONTRACT Oft GRANT NO 90 ORIGINATOR'S REPORT NUhnUER(S)
b, PROJc", ,No Y-F015-01-01-001(b) TR-433
C 9b OTHE PORT NO(S) (Any othe.rnumb.er WIat may b.e aaal.dthis report)
d
1d0 A VAIL AIIILITY/LIMITATION NOTICES
Distribution of this document is unlimited.
"11 SUPPLEMENTANY NOTES 12 SPONSORING MILITARY ACTIVITY
BUDOCKS
13 ABSTRACT
Based on a theoretical analysis of the cable and payload dynamics during
lowering or raising heavy loads in the deep ocean given in Project Trident Technical
Report No. 1370863, further calculations of the maximum dynamic stresses expected
in the lowering cable are presented covering a wide range of cable and payload
paraoneters. The theoretical analysis is adapted to a proposed design procedure, and
two typical design examples are given, the results of which are discussed in terms of
the safety of the iowering or raising operations.in order to make the design procedure applicable with a greater degree of
confidence, it is considered necessary to make measurements of cable tensions and
load and ship motions during a full-scale operation to fill in deficiencies of data
and provide a basis for verification of theory and calculations. In particular, data
ore needed on the cuefficients of drag and mass, which at this stage must of necessity
be estimates.
DD fl0als. 1473 U1o,,o0.,,oo UnclassifiedSecurity Classification
. LINK A LINK 1 LINKCKEY WORDS ROLS WY ROLE WT nOLE WT
Mechanics
EmplacementLoads
Deep oceanAnalysisCablePayloadDynamicsComputationStressesDragMassCoefficients
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