DYNAMIC RESPONSE OF AN OCEAN CONSTRUCTION BARGE TO VARIOUS SEA STATES Lawrence K. Donovan DYNAMIC RESPONSE OF AN OCEAN CONSTRUCTION BARGE TO VARIOUS SEA STATES by LAWRENCE K. DONOVAN B.S. , University of Notre Dame (1958) M.S. , University of California (1959) SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN OCEAN ENGINEERING at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September, 1970 - — i LIBRARY i^^/wr KAVAL POSTG-RAmJATB SCHOOL MONTEREY, CALIF. 93940 DYNAMIC RESPONSE OF AN OCEAN CONSTRUCTION BARGE TO VARIOUS SEA STATES by Lawrence K. Donovan Submitted to the Department of Ocean Engineering in partial fulfillment of the requirements for the degree of Master of Science in Ocean Engineering. ABSTRACT Load handling technology in the sea environment to place heavy and bulky oads accurately on the ocean floor is in its preliminary stages of development. ?he load handling system is usually mounted on a platform which interacts with lie sea surface, imparting dynamic loads to the lifting lines that are not ex- >erienced on land. This study evaluates the response of a specific load handling ystem composed of a crane and constant-tension winch mounted on AMMI pontoon arge to sea states 3, 4, and 5. The motions of heave, pitch, and roll were oupled by phasing at specific load suspension points and the rms values of the erticle displacement and acceleration at these points were calculated using pectral analysis techniques. It was found that the constant tension winch could ompensate for the motions of sea state 3 but that the probability of exceeding the /inch capability in sea state 4 was large enough to limit operations to missions of ery short duration. Thesis Supervisor: Martin A. Abkowitz Title: Professor of Naval Architecture ACKNOWLEDGEMENTS The author wishes to express his sincere appreciation to his thesis supervisor, Professor Martin A. £bkowitz, for his encouragement, understanding and invaluable assistance throughout this project. Also special thanks are extended to the Ocean Engineering staff of the Naval Civil Engineering Laboratory , Port Hueneme, California for invaluable reference material on previous experiments of this type and to the Ocean Engineering Project Office of the Naval Facilities Engineering Command, Washington, D. C. for suggesting this topic and providing data on the barge. TABLE OF CONTENTS Page Title Page 1 Abstract 2 Acknowledgements 3 Table of Contents 4 List of Figures 5 List of Tables 6 List of Symbols 7 1 Introduction 9 2 Scope of Thesis 11 3 Method of Analysis 16 3.1 Equations of Motion 16 3.2 Solution of the Equations 17 3. 3 Hydrodynamic Coefficients 18 3.4 Wave Excitation 22 3.5 Response Operators 23 3.6 Description of the Seaway 24 3.7 Response Spectra 26 4 Discussion of Results 36 5 Conclusions and Recommendations 38 References 39 Appendicies A Evaluation of the Hydrodynamic Coefficients 41 B Wave Excitation Forces and Moments 46 LIST OF FIGURES Figure Page 1 Project AFAR Crane Barge 12 2 Drawing of Project AFAR Crane Barge 13 3 System of Axes 21 4 Heave Response Operator for CG 29 5 Pitch Response Operator for CG 29 6 Roll Response Operator for CG 30 7 Example of Phase Diagram 30 LIST OF TABLES Table Page 1 Physical Characteristics of AMMI Crane Barge 14 2 Hydrodynamic Coefficients 20 3 Hydrostatic Coefficients 21 4 Excitation Amplitudes and Phase Angles 27 5 Motion Response Operators 28 6 Response Operators at Points of Interest 31 7 Pierson-Moskowitz Spectral Data for a Fully Developed Sea 32 l/N 8 Values for: Response = f (N) (rms) 32 9 Response Spectrum Values 33 10 Response rms Values 35 LIST OF SYMBOLS Wave amplitude Matrix element for verticle plane motion Area of the underwater section (B* x H) Local section beam Barge beam Verticle distance between CB and CG Section coefficient Center of buoyancy Center of gravity Total energy or integrated area under spectral curve Force Acceleration due to gravity Verticle distance between CG and the metacenter Barge draft Moment of inertia Roll moment Amplitude of roll excitation Verticle distance from the keel to CG Effective radius of gyration Barge length Boom length Pitch moment Amplitude of pitch excitation Mass of barge Roll angular velocity Pitch angular velocity Time Period Linear velocity in the x direction LIST OF SYMBOLS (Cont'd.) w Linear velocity in the z direction V Displacement volum- X,Y,Z Forces acting positively along the x, y, and z axes x, y,z Orthogonal axes system y Local sectional center of buoyancy distance from the z-axis CB x Distance of CG from origin of coordinate system along x-axis G a Boom verticle angle with positive x-axis A Displacement weight e. Phase lag between wave and barge motion (i = z, 0 or 0) Y Boom horizontal angle with positive x-axis 77 Wave surface elevation © Pitch angle X Wave length £ Dummy variable along x-axis p Mass density 0 Roll angle p Angle between barge heading and incident wave train ju Wave angular frequency ju Frequency of encounter between barge and wave. 1. INTRODUCTION Construction involves the lifting and transportation of heavy equipment and all kinds of materials. For dry land application, a whole spectrum of simple and complex load handling systems have been designed and built to achieve the comple- tion of construction projects of all kinds and magnitudes. As man now moves to a completely different environment for construction, the ocean floor and the sea above it, it is found that load handling technology is in its preliminary stages of development at best. To accomplish underwater construction there is a need to lift and lower and also recover heavy and bulky objects in the open seas. One of the conventional systems often used for this type of load handling consists of a winch and a line with the line being run out on some kind of boom. The winch and boom are mounted on a small floating platform and a load is sus- pended at the end of the line. As the platform interacts with the waves of the ocean surface, the platform motions cause the load to oscillate and this in turn generates oscillation in the line tension. In order to design a load handling system for a specific job, the ocean engineer must be able to predict how the dynamic tension in a line from a given platf' rm will vary with the sea state and specific loads at- tached to the end of the lin . Because of the U.S. Navy's interest in lifting and lowering loads to 6, 000 foot depths, a development program has been carried out at the Naval Civil Engi- neering Laboratory (NCEL) to predict the dynamic stress response of lifting lines 12 3 in Oceanic operations. These studies ' ' and others have resulted in a method of predicting lifting line stresses for various ideally shaped loads using the sea induced motions of the platform at the last point of contact with the line as one of the major inputs. Much work has been accomplished in hydrodynamics for the prediction of ship motions in both regular and irregular seaways since the introduction of the strip theory of ship motions by Korvin-Kroukovsky in 1955. Rightfully this re- search into the optimization of the hydrodynamic characteristics of transportation type ship structures belongs in the field of Hydrodynamics or Naval Architecture. 10 The Ocean Engineer on the other hand is most likely concerned with a working platform that is barge shaped and may be moored or remains fairly stationary in the seaway. Thus, this thesis will concern itself with the prediction of the motions of a fairly simple ocean engineering construction load handling system in a seaway. The goal of this work is to provide a simplified method of analysis that will give a good first estimate of platform vertical accelerations at points on the platform where lifting lines would receive the greatest effect from the barge motions. The data developed by this study should be accurate enough to allow sizing of lifting lines within engineering safety and prediction of the operational limitation of the system as a function of sea state. 11 SCOPE OF THE THESIS 5 In 1970, the Navy executed project AFAR which involved the implantment of an acoustical array on a sea mount in 1300 feet of water. It was determined that the most economical load handling system to support the project was to con- struct work barges from 90 feet x 28 feet x 5 feet AMMI pontoon sections or from 5 basic 5 feet x 5 feet x 7 feet pontoon cans. In the operations plan , the construc- tion was administratively limited to sea state 3 on the Beaufort scale, however no analysis was provided to indicate what magnitude of barge motions or line tensions that could be expected at that or higher sea states. This thesis will specifically analyze the interaction of the Crane Barge of Project AFAR shown in Figures 1 and 2 with various sea states starting with sea state three. The barge is composed of one AMMI pontoon section fitted with a Harbor Master, 180 HP, propulsion unit in each corner, a 30 ton capacity stand- ard construction crane, and a Leithiser constant -tension winch. The propulsion system can be controlled so that the barge can remain on a specific station, with- out mooring, within the accuracy required for the implantment. The physical characteristics of the barge are listed in Table 1. The method of analysis to be used will employ the linearized equations of motion for pitch, heave and roll for a regular unidirectional train of waves ap- proaching the barge from the starboard quarter. The wave frequency dependent coefficients and the wave excitation amplitudes and phase lags of the equations will be determined by comparing a method of calculation for the coefficients pub- lished by Muga of NCEL to the experimentally determined coefficients of Vugts for a rectangular cylinder with a beam-draft ratio of 8, and using the coefficients felt to be most accurate. The equations will then be solved for input waves of seven different wave lengths. A response operator for each of these wave lengths for the verticle acceleration will be determined with phase coupling of pitch, heave and roll at four points of interest: the deck amidships at both the bow and stern and the end of the crane boom with the boom at both the port beam position and GO degrees after the port bow. A smooth curve for the response operator 12 Figure 1 . Project AFAR Crane Barge 13 untni 0UH1IKUUIUN HOUSE SCUBA CHARGING AIR COMPRESSORS ' POSITIONING CONTROL SYSTEM HARBORMASTER - FAIRLEAD CRANE BARGE Figure 2. Drawing of Project AFAR Crane 3arge 14 TABLE 1 PHYSICAL CHARACTERISTICS OF AMMI CRANE BARGE Length 90 feet Beam 28 feet Calm water draft 3.5 feet Displacement 252 L. Tons Weight 564, 480 lbs GM 147.25 feet I 7.33 x 106 Slug-ft2 y ^ 7.125 x 10 Slug-ft Mass 17,640 Slugs 15 at each of the points will be drawn and using spectral analysis techniques, a re- sponse spectrum for several sea states will be determined. From the response spectrum, the R.M.S. value of the accelerations for each sea state will be calcu- lated and statistical information developed which can provide the basis for limiting operations above certain sea states. 16 3. METHOD OF ANALYSIS 3 . 1 Equations of Motion The linearized equations of motion for the barge will be developed along o the technique suggested by Abkowitz . The motions are separated into motions in the horizontal plane and motions in the verticle plane. Because of the difficulty and complexity in determining cross coupling coefficients and the uncertainty in the state of the art in handling all six degrees of freedom, only pitch and heave will be considered in the verticle plane and only roll in the horizontal plane. The excitation due to waves is the only dynamical response term that will be considered on the right hand side of the equations. It is assumed that the barge can maintain position with no forward speed and thus control and propulsion forces are neglected. The regular wave train is assumed to be unidirectional and approaches the barge from the starboard quarter or 45 off the bow. To simplify the analysis, the reg- ular wave is broken down into two equal components of wave length V2\, one from directly head-on and the other from beam-on. Pitch and heave will be affected by head-on waves while roll and heave will be affected by beam-on waves. The cen- ter of gravity is assumed to be at the water line, amidships. The notation to be used is that D = — and a symbol with a dot over it, 0 , is the first derivative with respect to time. A symbol with two dots over it, z is the second derivative with respect to time. The derivatives for the hydrodynamic coefficients, ( —r J will be written as Z • which is the partial derivative with re- \ow/o w spect to the vertical acceleration taken at the original conditions with the barge at rest. In this notation it should be remembered that p = 0 and q = 0. Thus the general linearized equations of motion for the barge system are: Heave (Z--m)D +Z D + Z . W W ZJ z + (Z- + mx )D +Z D+Z^ . q G' q 0J 0= -Z^e i(wt-ez) [an] . + Ca12] 0= Ax(t) (1) . 17 Pitch 2 l (M. + mx )D +M D + M z + . w G w zj (M. -I )D2 + M D + M-l 0 = -M el(a)t_eM) q y' q ©J 1 ^ z + ^ 0 = A2(t) (2) Roll "(K. - I )D2 + K D + K \ = -K ei(UJt"CK> Lv p x' p 0_T 1 (3) where Z M and K are the amplitudes of the excitation and e is the phase lag. Since there is no forward speed, uu was used in place of uu , the frequency of en- /2ttp" counter. Thus uu = uu = v~~ ~ , the wave frequency. The coefficients on the left e k hand side of the equations are independent of time but may be a function of uu. Due to the choice of the CG and the constant section of the barge, the coupling coeffi- cients will be zero. 3 . 2 Solution of the Equations Solution of the equations of motion is accomplished by solving for the roots of the determinant of the coefficient matrix for equations (1) and (2) or by just solving for the roots as in equation (3). The solution will be of the form: m £.(t)= Y C/ + (t.) ei(u,t-ei> y L k j'o (4) k=l where f .(t) is the chosen independent variable, m is the number of independent variables in the system, C are the constants of integration, u are the roots of the determinant, (f .) is the amplitude of the steady state response and e. is the phase lag. The solution of equations (1) and (2) in determinant form is: 18 z(t) = Al® ai2 V» a22 ail ai2 a21 a22 (5) and 0(t) = an V> a21 A2« ail ai2 a a 21 22 (6) The steady state solution of equation (3) is: 0(t)=- K1G m-€K-T) K -(K. -I )u> 0 P X N p 2 2 + (K UU) (7) where T = tan' K uu P K - (K. -I )uu 0 N p X' (8) 3.3 Evaluation of the Coefficients The hydrodynamic coefficients are independent of time because they are made up of various derivatives defined by reference only to the initial equilibrium condition. They may depend on the vehicle size, shape and inertial distribution and on the frequency of excitation since the motion of the vehicle on the free sur- face generates waves. The frequency of this response must be the same as that of the excitation in order for the coefficients to remain time independent. A sys- tem of axes is defined for this analysis as shown in figure 3 with the forces X, Y, and Z positive in the direction shown, pitch positive in the bow up direction and roll positive with the starbord side down. The origin is taken at the center of gravity of the barge which is assumed to be directly above the keel amidships at the calm water line. Thus x„ = y^ = z^ = 0. Each of the coefficients will G JG G 19 generally include derivatives which are either inertia!, damping, or hydrostatic in nature. In the case of damping, only damping due to wave generation will be considered. There has been significant research carried out in recent years to develop methods of theoretical calculation for the more important hydrodynamic coeffi- cients. References 7 through 12 are representative of the current state of the art and provide an excellent reference list. Coefficients for motion in the verticle plane are better defined and understood than those in the horizontal plane. Some of the coupling coefficients have never been measured due to the difficulty in de- vising an accurate experiment. Methods for calculating the coefficients were developed for the Navy by 13 Kaplan and Putz (1962) for the CUSS I Project MOHOLE Barge in a moored condition and by Muga (1966) for the missile recovery barge FISHOOK, also in a moored condition. Although the barges considered in these studies were slightly 14 more complex, the method of coefficient calculation is applicable. Frank (1967) has published section;: 3 coefficient values for the center section of a series-60 ship, however, with a smaller beam-draft ratio than the barge of this study. 7 Vugts (1968) has also published the sectional coefficients of long rectangular cylinders for various beam-draft ratios, some closely approximating the condi- tions of this study. /? To simplify this analysis, the method of calculations published by Muga 7 and the experimental data of Vugts will be used as best fitting a rectangular box shaped barge. The sectional derivatives calculated by Muga's equations will be compared with the experimental data of Vugts and the experimental derivative will be used whenever available. Because of the symmetry of the barge, it is assumed that the sectional coefficients are constant over the length of the barge. Details of the derivation of each coefficient is presented in Appendix A. Table 2 lists the nondimensional values of the frequency dependent coefficients calculated by Muga's equations compared to those determined by Vugts. 20 TABLE 2 HYDRODYNAMIC COEFFICIENTS (Calculated vs Experimental Data) (Note: p = sectional mass density) X 50 100 150 200 300 500 600 Heave Added Mass Z. w pA Muga6 2.51 2.76 3.77 4.27 5.01 5.27 5.41 Vugts7 2.75 2.75 3.08 3.70 3.90 4.50 5.00 Z w Heave Damping = -— J— pA /2g Muga 0.51 2.30 3.24 3.61 3.52 3.09 2.85 Vugts 1.45 2.13 2.25 2.22 2.20 2.10 1.95 M. Pitch Added Moment = — - — pAL2 Muga 0.209 0.230 0.312 0.356 0.418 0.439 0.451 Vugts 0.229 0.229 0.256 0.308 0.325 0.375 0.417 M Pitch Damping = ^— /— pAL2 <2* 2g Muga 0.043 0.191 0.270 0.301 0.293 0.257 0.237 Vugts 0.121 0.177 0.187 0.185 0.183 0.175 0.162 Roll Added Mass = - 2 pAB Muga (insufficient information to make the calculations) Vugts 0.098 0.122 0.139 0.146 0.148 0.146 0.145 Kp /b~ -2 Roll Damping = — )L-z /— (Note: Muga Values all x 10 ) pAB * g Muga 0.0153 0.0104 0.0081 0.0065 0.0050 0.0029 0.0023 Vugts 0.0300 0.0375 0.0270 0.0180 0.0080 0.0050 0.0030 21 Figure 3. System of Axes TABLE 3 HYDROSTATIC COEFFICIENTS Z - 1.615x 10 lbf/ft M 0 - 1. 086 x 108 lbf - ft K 0 - 8.34 x 10 lbf- ft 22 3.4 Wave Excitation The wave induced effects on the barge are composed of inertial, damping and displacement components of the wave induced force acting on individual ship sections of length. For this study it has been assumed that a unidirectional train of regular waves approaches the barge at a 45 degree angle with the bow at some wave length, X. The incoming wave length has been broken down into a head-on component and a beam-on component, each of wave length, J2\. The wave excita- tion forces and moments on the right hand side of equations (1), (2) and (3) will be defined in terms of the hydrodynamic coefficients previously derived in Section 3.3. The details of the definition of the wave excitation forces and moments are explained in Appendix B. It should be noted that the general form of the equations for the forces, F or moments, M , in Appendix B are of the general form: F or M = A cos uut + C sin uut (9) UU UU UL> UU x ' which can also be written as /a- F or M = J A +C cos (uut- e) (10) UU UU UU UU ' ' where „ e = tan — (11) uu If it is understood that only the real part of complex notation will be considered, then equation (10) can be written as = yA2+C2e1(tt*-C» F or M = VA +C e x ' (12) uu uu uu uu 23 Comparing this expression to the right hand side terms of the equation of motion, it can be seen that for heave as an example . = /a- -Zn = /A +C (13) 1 U) ou and e-v~- ^z, =e-v~ -/ ^14^ i(uut -€z) _i(uot - €) Evaluation of the excitation forces in this form gives the maximum ampli- tude of the excitation for one wave and the phase relationship of this amplitude com- pared to the wave with its crest exactly admidships at the center of gravity. The excitation amplitudes Z, , 1VL and K and the phase angles e , e _ and c have been 111 z 0 0 calculated and are tabulated in Table 4 for each of the frequencies of interest. For comparison, the values of maximum generated wave amplitudes and phase angles 7 measured by Vugts as a consequence of a known excitation of his model are also tabulal .1 in Table 4. 3.5 Response Operators The equations of motion can now be solved for the desired variables z, _ 45 degrees,- © and 0 for any wave length or frequency of regular wave. (Note: ip is the angle between the bow of the barge and the incoming component of the wave train. ) The solution vectors will be of the form; r(t) = R cos (tut + 6 ) (15) or r(t,=Rre^t + 6r) (16) where r represents any of the motions, R is the magnitude of that motion or am- plitude response operator, and 6 is the phasing of the maximum possible motion relative to maximum positive wave displacement at amidships. 6 is sometimes called the phase response operator. Thus for a unit wave amplitude, r(t) is the 24 response operator which is frequency dependent. The values of the motion re- sponse operators for heave, pitch and roll are listed in Table 5 and plotted as a function of uu in Figures 4, 5 and 6. Having calculated the motions, the vertical displacement, Z , velocity • • • Z , and acceleration, Z , of any point P = (x , y ) can be calculated by coupling the motions from each component of the wave. Thus the vertical re- sponse operator of any point can be written as : ZP=[Z,i=45°-e] <17> ZP = ilU hi =45° "6(XP) +*(yP)] (18) ^P=-^b=45»-e(XP)+* It must be remembered that each of the response operators is a vector and thus must be added keeping track of all the phase response operators 6 . Figure 7 is an example phase diagram of this type of vector addition. Since two of the four points of interest in de.jrmining maximum accelera- tion includes a boom, the boom can be included in the response operators of equa- tions (17), (18) and (19) by merely letting x = 1 cos a cos y and yp = 1 cos a sin y where 1 is the boom length, a is the elevation angle and y is the azimuth angle. Values for the response operator at the four points of interest are tabulated in Table 6. 3.6 Description of the Seaway Since the response to any given wave frequency can now be calculated, it is necessary to apply the frequency spectra of an irregular sea using the energy 15 density technique described by Loukakis (1970) to predict the statistical propel ties of ship responses in a seaway. The spectral family that will be used to 25 i p describe the seaway is the Pierson and Moskowitz (1963) family for a fully developed seaway. The general form of the spectral density function is given by -5 -4 Q(uu) =auu exp(-Bui) ) (20) 2 where Q(uu) is the one sided spectral density function with units (ft x sec). The parameters a and (3 for a fully developed sea are given by: a - 0.0081 g2 [ft/sec4] (21) P = 0. 0324 g2/(lP) 2 [sec"4] where H3 is the significant wave height which by definition equals four times the root mean square (rms) of the spectrum. Specific values for the spectral density function for three sea states at the frequencies of interest are tabulated in Table 7 along with corresponding wind speeds. 26 3 . 7 Response Spectra All absolute and relative barge motions, velocities and accelerations are nearly linear with respect to the wave. Thus the barge behaves approximately as a linear, time invariant filter and the output of the filter will have the same distri- bution as the input. If the sea spectral density function is plotted as proportional 2 to a /6uu where a is the wave amplitude and 6uu is an incremental frequency element 2 and the barge response operator is plotted as proportional to (z/a) , both as func- tions of uu, it is clear that by the superposition principle: 2 2 2 6^ X ~2 = 6^ (22) a 2 When z /6uu is plotted as a function of uu, a response spectrum results. A quantity is now defined which is the average value of the response average over the fre- quency range. This average is obtained by integration over frequency or is the area of the spectrum. The root mean square, rms, of the response amplitude is then, rms = ,/area, which represents a statistical average of the irregular random responses. From the rms value, other statistical averages such as the average value of the 1/3 or 1/10 highest response can be calculated by the relationship: 1/N Response ' = f (N) (rms) (23) The values of N and f (N) which are of interest in this study are listed in Table 8. The values for calculated responses at the points of interest for the seven wave lengths investigated are tabulated in Table 9. These values were plotted and the area under the curve determined graphically. The rms values for each of the responses are listed in Table 10. 27 TABIE 4 EXCITATION AMPLITUDES AND PHASE ANGLES w 2.01 1.42 1.16 1.00 0.82 0.63 0.58 3 3 Amplitudes (Forces - lbf x 10 , Moments Ibf-ft x 10 ) Z 7.84 34.5 73.1 83.3 108.5 128.8 130.3 unf= 45o Z .(Vugts) 51.5 82.1 96.6 107.8 116.0 124.0 128.7 M^ 291 1610 1573 1358 1250 724 624 Kw 96 265 287 255 196 125 106 KjVugts) 200 319 302 266 222 128 109 Phase Angles (Degrees) *Z,V»=45° 113 233 216 212 203 195 193 135 75 40 30 25 20 10 24 322 308 302 293 285 283 90 270 270 270 270 270 270 0 270 225 (remainder not measured) (z (Vugts) *M f* e K (Vugts) 28 TABLE 5 MOTION RESPONSE OPERATORS CO 2.01 1.42 1.16 1.00 0.82 0.63 0.58 Heave (ft/ft) z/a 0.063 0.497 0.755 0.840 0.943 0.999 0.999 Pitch (Radians/ft x 10 ) ©/a 4.36 18.42 16.75 14.0 12.42 7.11 5.97 Roll (Radians/ft x 10"3) 0/a 1.28 3.43 3.60 3.16 2.40 1.52 1.28 29 Figure 4. Heave Response Operator (0 Figure 5. Pitch Response Operator X G fO -r-l fO X 15.0 10.0 5.0 -L 1.0 CO -1 (sec ) 2.0 30 Figure 6. Roll Response Operator 1 3.0 •l-l Tt fO 7 2.0 o X (0 1.0 1.0 -l CO (sec) 2.0 Figure 7. Sample Phase Diagram x 0.63 0.71 0.89 0.97 0.97 0.85 0.71 0.60 0.39 0.33 31 TABLE 6 RESPONSE OPERATOR AT POINTS OF INTEREST CJ 2.01 1.42 1.16 1.00 0.82 0.63 0.51 Bow Amidships (x =45 ft) E z/a (ft ft) 0.22 0.92 z/a(ft/sec2-ft) 0.90 1.84 Stern Amidships (x=-45 ft) P z/a (ft/ft) 0.16 1.02 1.37 1.30 1.27 1.13 1.09 z/a (ft/sec2-ft)0.65 2.05 1.84 1.30 0.85 0.45 0.37 Boom at Port Beam (Boom = 60 ft ,oC=60o, y=-90°) z/a (ft/ft) 0.10 0.40 0.64 0.74 0.89 0.97 0.98 z/a (ft/sec2 -ft) 0.42 0.81 0.86 0.74 0.60 0.39 0.33 Boom Aft Port Bow (Boom = 60 ft, 0C= 60°, /=-60°) z/a (ft/ft) 0.10 0.23 0.54 0.66 0.82 0.93 0.95 z/a (ft/sec2-ft)0.41 0.47 0.73 0.66 0.55 0.38 0.32 32 TABLE 7 PIERSON-MOSKOWITZ SPECTRAL DATA FOR A FULLY DEVELOPED SEA Sea State Data Sea State 3 4 5 Wind Speed (KTS) 14.5 19.5 23.2 H* (ft) 4 7 10 a 8.294 8.294 8.294 B 2.07 0.677 0.332 (V 0.58 0.634 0.82 1.002 1.16 1.42 2.01 2 Spectral Density Function (ft -sec) Q3((ju) Q4(uj) o 0.0018 0.323 6.77 0.0191 1.252 10.50 0.228 5.02 10.75 1.06 4.18 5.88 1.26 2.725 3.30 0.865 1.22 1.33 0.223 0.242 0.247 TABLE 8 VALUES FOR: RESPONSE =f (N) (rms) N fx(N) 1 1.25 3 2.00 10 2.55 10 3.34 106 5.27 33 TABLE 9 RESPONSE SPECTRUM VALUES CO 2.01 1.42 1.16 1.00 0.82 0.63 0.58 Sea State 2 2 Bow Amidships - Displacement z /fro) (ft /sec) 3 0.011 0.73 0.50 0.55 0.20 0.018 0.017 4 0.012 1.02 1.09 2.13 4.48 1.18 0.31 5 0.012 1.12 1.31 2.99 9.60 9.87 6.37 2 2 5 Bow Amidships - A cceleration z /Sto (ft /sec ) 3 0.18 2.94 0.91 0.55 0.08 0.003 0.000 4 0.19 4.15 1.96 2.13 1.80 0.191 0.035 5 0.20 4.52 2.37 2.99 3.87 1.60 0.724 2 Stern Amidships - Displacement z /y* 3 0.006 0.89 2.37 1.80 0.37 0.024 0.002 4 0.006 1.26 5.12 7.10 8.06 1.57 0.39 5 0.006 1.37 6.20 9.97 17.30 13.27 8.10 Stern Amidships - Acce leration 'z /s&o 3 0.090 3.62 4.26 1.80 0.17 0.004 0.000 4 0.101 5.10 9.28 7.10 3.64 0.258 0.044 5 0.103 5.56 11.16 9.97 7.79 2.065 0.916 34 TABLE 9 (Cont'd.) RESPONSE SPECTRUM VALUES CO 2.01 1.42 1.16 1.00 0.82 0.63 0.58 2 Boom at Port Beam - Displacement z /Ju 3 0.092 0.14 0.51 0.55 0.18 0.018 0.002 4 0.003 0.20 1.11 2.19 3.94 1.19 0.311 5 0.003 0.22 1.34 3.08 8.45 9.97 6.52 Sea State Boom at Port Beam - Acceleration z /^u) 3 0.040 0.57 0.92 0.55 0.081 0.002 0.000 4 0.043 0.81 2.00 2.19 1.78 0.19 0.035 5 0.044 0.88 2.42 3.08 3.81 1.62 0.74 2 Boom Aft Port Bow - Displacement z /%oj 3 0.002 0.046 0.37 0.47 0.15 0.017 0.002 4 0.003 0.066 0.79 1.84 3.38 1.09 0.29 5 0.003 0.071 0.97 2.58 7.25 9.15 6.07 « .9 Boom Aft Port Bow - Acceleration z /^oO 0.037 0.19 0.67 0.47 0.069 0.003 0.000 0.040 0.26 1.43 1.84 1.53 0.178 0.033 0.041 0.29 1.74 2.58 3.27 1.49 0.68 35 TABLE 10 RESPONSE RMS VALUES Sea State Location Motion Z ft 0.75 1.35 2.16 Bow Amidships Stern Amidships Boom at Port Beam Z ft/sec 1.24 1.61 1.95 1.09 2.04 3.13 1.57 2.37 2.74 0.53 1.25 2.02 0.71 1.19 1.57 0.44 1.13 1.92 Boom Aft Port Bow 0.57 0.99 1.32 36 4. Discussion of Results 4.1 Response Operators The response operator amplitudes for heave, pitch and roll have been presented in Table 5 and Figures 4, 5 and 6. The heave operator behaves as expected approaching a maximum value of the wave ampli- tude. The pitch operator is slightly lower than the maximum wave slope at the low frequencies. If the maximum wave slope is given 2TTa v -2 by — r — , then at A= 600 ft, the slope would be only about 1.05x10 (Radians/Unit amplitude). The values of the operator are in general 13 agreement with the values found by Kaplan and Putz for CUSS I. The roll operator behaves as ecpected but is about an order of magnitude lower than values determined by Kaplan and Putz. It should be mentioned that CUSS I has a more ship type hull than the 7 crane barge. The roll coefficients of Vugts also had considerable spread in the data which may account for the difference. 4.2 Response Operators at Points of Interest The response operator amplitudes for vertical displacement and acceleration for the four points of interest have been presented in Table 6. The values for bow and stem amidships reflect the coupling of pitch and heave. The contribution of roll to the motion 37 of the crane boom was fairly small compared to pitch and heave due again to the small value of the response operator. 4.3 RMS Values The results of the spectral analysis were presented in Table 10 in the form of rms values for vertical displacements and accelerations. The value for displacement in sea states 3 and 4 is felt to be accurate, however sea state 5 had significant contribution between 1.42 and 2.01 radians where the shape of the curve was not calculated. The acceler- ations are more accurate however since the values at both ends of the frequency span chosen are negligible. All of the rms values would have been more accurate with more points, especially at the low end of the frequency scale and between ^= 2.01 and ^= 1.42. 4.4 Limitation on Operations Now that the rms amplitude values of verticle displacement and acceleration are known, the probability of having a total response of any magnitude can be calculated from equation (23). For example, for sea state 4 the rms of the displacement amplitude at the bow is 1.35 ft. Using Table 8, for 66.7% of the time, the displacement would be less than about 2.7 ft. Only one displacement in 100 would exceed 4 . 5 f t and one in one million would exceed 7.1 ft. Since the average period of sea state 4 in this study is about 5 seconds, this means that 38 the occurrence of an excitation which would produce a displacement response greater than 7 . 1 ft would be only once in a time period of about 58 days. Based on this type of analysis, the displacements and accelerations re- ceived as a result of sea state 4 are too large for anything but very short mission duration weight handling. Sea state 3 rms values of this study should be satisfactory for operations when synthetic line and the leithiser constant tension winch are used to compensate for part of the displace- ment and acceleration. 5. Conclusions and Recommendations It appears that the operational limitation on Project AFAR to sea state 3 on the Beaufort scale was a bit conservative since this corresponds to a wind speed of 10 knots whereas the Pierson-Moskowitz sea state 3 of this study corresponds to a wind speed of 14.5 knots. Operations in this latter sea state should be acceptable if the Leithiser constant-tension winch is used with synthetic line to compensate for the motions. For sea state 4 of this study, the probability of exceeding the capability of the weight handling system is great enough that operations would be limited to very short load lifting missions at best. It is recommended that this method of analysis be reduced to a computer program for all heading angles at more frequency points to pro- vide more accurate rms values. It is also recommended that more research be done on refining the hydrodynamic coefficients for roll for shallow draft box shaped hulls to eliminate the uncertainties in the data of Vugts. 39 REFERENCES 1. NCEL Report TR-543 "Mechanics of Raising and Lowering Heavy Loads in the Ocean: Experimental Results", Muga, B.J. Sept 1967. 2. NCEL Report TR-652 "Feasibility Study and Comparative Analysis of Deep Ocean Load Handling Systems" by Davis, D.A. and Wolfe, M.J. Dec 1969. 3. NCEL Report TR-703 "Dynamic Stress Response of Lifting Lines for Oceanic Operations" by Liu, C.L. Nov 1970. 4. Korvin — Kroukovsky, B. V. "Investigation of Ship Motions in Regular Waves, " Transactions of the Society of Naval Architects and Marine Engineers, Vol 63, pp. 386-435 (1955). 5. Naval Facilities Engineering Command, Project AFAR(AUTEC), Management/ Operation Plan 1/27/70. 6. NCEL Technical Report R-440, "Hydrodynamic Analysis of a Spread-Moored Platform in the Open Sea'.', Muga, B.J. Aug 1966. 7. Laboratorium Voor Scheepsbouwkunde Report No. 194 "The Hydrodynamic Co- efficients for Swaying, Heaving and Rolling Cyclinders in a Free Surface" by Vugts, J.H. Jan 1968, 8. Abkowitz, M. A. "Stability and Motion Control of Ocean Vehicles" MIT Press (1969). 9. Abkowitz, M.A., Vossilopolous, L.A., and Sellers, F.H. "Recent Develop- ment in Seakeeping Research and its Applications'', Trans SNAME, Nov 1966. 10. Mandel, P. "Principles of Naval Architecture —Ship Maneuvering and Control", SNAME 1967. 11. Korvin — Kroukovsky, B.V. "Theory of Seakeeping" SNAME 1961. 12. Lewis, E. V. "Principles of Naval Architecture — The Motion of Ships in Waves", SNAME 1967. 13. Kaplan, P and Putz, R.R. , "The Motions of a Moored Construction Type Barge in Irregular Waves and Their Influence on Construction Operations", NCEL Contract NBY -322 00, August 1962. 14. NSRDC Report 2375, "Oscillation of Cylinders in or Below the Free Surface of of Deep Fluids", Frank, W. , October 1967. 40 REFERENCES (Cont'd.) 15. MIT Report 70-3, "Computer Aided Prediction of Seakeeping Performance in Ship Design", Loukakis, T.A., August 1970. 16. Pierson, W.J. and Moskowitz, L., "A Proposed Spectral Form Based on the Simularity Theory of S. A. Kilaigorodski", Geophysical Sciences Laboratory, Re- port No 63-12, N.Y. University, (1963). 17. Vossers, G. (1960) "Fundamentals of the Behavior of Ships in Waves", Inter- national Shipbuilding Progress, Vol 7, No. 65, Jan 1960. 41 APPENDIX A EVALUATION OF THE HYDRODYNAMIC COEFFICIENTS Al Inertial Coefficients The coefficients of the acceleration terms are the result of the hydrody- namic forces and moments caused in reaction to the acceleration of a body through the water they are negative in value and thus add to the mass a inertia of the body. Al. 1 Heave Added Mass, Z . and Z. w q Z . is the added mass in heave which is in phase with the verticle acceller- w * ation in equation (1) while Z. is the added inertia in phase with the angular accel- 6 ^ leration of pitch. Muga defined these coefficients over a unit section of ship length as: t 2 **~\ 2 4 d^ <24> ^s Vlbzw«d« <25> ^s where B* is the sectional beam, C is a frequency dependent correction factor, and 4 is a dummy variable along the x-axis which is integrated from the stem, 4 , to the bow £ . The limits of integration will be deleted in future integrals s b unless they differ from the above. It should be noted that due to Z . being con- w stant for each section, the coupled pitch-heave added mass, Z. , will be zero for all frequencies. The values calculated by equation £4 ) for Z . are tabulated in Table 2 along with the comparable experimentally determined values of Vugts. A1.2 Pitch Added Inertia, M. and M. w q These derivatives are in phase with the verticle accelleration and pitch angular accelleration and have been defined by Muga as: 42 M. w = SZw^ (26) M. = [ Z. 42df (27) For the same reasons as expressed for Z. in equation (25), M . is also equal to q w zero. The values of M. calculated from equation (27) are listed in Table 2 along q 7 with the experimental values of Vugts . A1.3 Roll Added Inertia, K. P Muga calculated K. from the effective radius of gyration, k , by the rela- P e tionship: 2 mk =(I -K.) (28) e xx p v ' The effective radius of gyration was calculated from the experimentally deter- mined roll natural period of FISH HOOK by the relationship: 2nk *-T (29) Cg(GH)] 2 Since the natural roll period of the AFAR crane barge is not known with any accu racy, only the experimental values of K. determined by Vugts are listed in Table 2, A2 Dynamic Damping Coefficients The hydrodynamic damping forces and moments are in phase with the body velocity and are proportional to the dissapated wave energy. Viscous effects of the fluid have been neglected in this study. Thus the damping coefficients are the coefficients of the velocity terms of the equations of motions. 43 A2.1 Heave Damping, Z and Z w q Muga derived the heave damping coefficients as follows: 2 T 2 r pg (Az> Z =-C \ ^dS (30) w zz J 3 UD 2—2 r pg uu where A is the ratio of the amplitude of the heave generated two dimensional waves z to the heaving motion of the section and C and C are three dimensional damping factors. Again because of the constant sectional coefficient, Z is equal to zero in equation (31). The values of Z calculated by equation (30) are compared with the n experimental values of Vugts in Table 2. A2.2 Pitch Damping, M and M w q These coefficients in phase with the verticle velocity and the pitch velocity are defined by Muga as: M =C^ \ ~ £d£;C^ =C =1 (32) w 0z J 3 s s> 6z z0 y ' uu r PS(A ) 2 M =-C^^\ 7T— £ df;C^^=C (33) q 00 J 3 s ^' 00 zz K ' i is once again equal to zero due to the coefficient being constant over the length w f the body. The values of M calculated from equation (33) are compared to the Ixpcrimental values of Vugts in Table 2. 44 A2.3 Roll Damping, K Muga derives the roll damping coefficient as: K = -C (BG)2 °j- [ B*2 (d )2 d£ (34) P yy 4 J y ' * where d is a correction coefficient which is a function of sectional area and beam- y 17 draft ratio derived by Vossers (1960) . The values of K calculated by equation (34) are compared with those determined by Vugts in Table 2. The large differ- ence between the calculated and the experimental values should be noted. Muga's assumption of transverse damping force times the moment arm is not very valid. Therefore, the values determined by Vugts will be used in the solution of the equa- tion of motion. A3 Hydrostatic Coefficients The hydrostatic coefficients are easily calculable and are independent of frequency. A3.1 Heave, Z and Z~ z t) The hydrostatic heave and coupling coefficients are: Zz = "pgSB*d^ (35) z0 = pg5B*£d£ <36> Due to constant beam, the value of the coupling coefficient, Zft, is zero. 45 A3. 2 Pitch, M and M^ z © The hydrostatic pitch and coupling coefficients then are given by: M =pg ^B*£d£ (37) z M0=-pg ^B*42d4 where M is also equal to zero z A3. 3 Roll, K 0 The roll hydrostatic coefficient is given by the relation: or (38) K =-pgv|GM| (39) K =-a|gm| (40) 0 where V is the immersed volume of the barge and A is the barge displacement. A3. 4 Values Calculated values for the hydrostatic coefficients are listed in Table 3 in dimensional form for use in solving the equations of motions. 46 APPENDIX B WAVE EXCITATION FORCES AND MOMENTS Bl General Description The wave surface elevation of a unidirectional train of regular waves in deep water is defined as: T7(x, t) = a cos (— — -cut + c J (41) where : M uu = v — r5- Angular frequency of the wave A. a = Wave amplitude x = Distance along direction of wave travel \ = Wave lengtl in direction of wave travel e = Phase at origin when t = 0, (zero in this case) The wave induced excitation forces and moments are given by: F.e1** "'J)-/ -Jdx (42) J J . dx v 7 «/ stern Bow , Mei(a*-cj)= I -jJxdx (43) ' stern The general form of the sectional wave induced force is given by: dF. 1 dx pgB*(x)77(x,t) +N(x)7](x,t) |i.(x) ij(x,t)] exp (- J}y&-) (44) 47 where : 2_ N,(x) = Sectional damping [lbf - sec/ft ] U.(x) = Sectional added mass [slugs/ft] The specific forces and moments can now be developed in terms of the hydrodynamic coefficients derived in Appendix A. B2 Heave Excitation The heave wave forces will be broken down into two components, one ap- proaching the barge from head-on along the x-direction and the other from beam-on along the y-direction. The sectional hydrodynamic coefficient is found by dividing the total coefficient of Table 2 by the barge length since the sectional coefficients are constant over the length. The limits of all integrals will be from -L/2 to L/2 unless otherwise noted and the limits will be deleted in future equations where the meaning is clear. The general expression for the surface elevation of the waves, equation (41) has been modified to account for the two components of the incoming wave as follows : 4.x /2ttx cos >j) 2ttv sin ib \ .._. 77(x,y,t) = acos (^ — *+ ^ — *--