DYNAMIC ANALYSIS OF THE NAVAL POSTGRADUATE SCHOOL OCEAN INSTRUMENT PLATFORM Mark Francis Crane United States aval Postqraduate School TMT^ SIR 1 ; DYNAMIC ANALYSIS OP THE NAVAL POSTGRADUATE SCHOOL OCEAN INSTRUMENT PLATFORM by Mark Francis Crane Thesis Advisor: E. B. Thornton September 1971 Appn.ove.cl faoh. puhtlc A.ele.abe.; dii>txibation uiitimi£.e.d. J Dynamic Analysis of the Naval Postgraduate School Ocean Instrument Platform by Mark Francis ,Crane Lieutenant Commander, United States Navy B.S., United States Naval Academy, 1963 Submitted in partial fulfillment of the requirements for the degree of MASTER OP SCIENCE IN OCEANOGRAPHY from the NAVAL POSTGRADUATE SCHOOL September 1971 C793V ABSTRACT The dynamic and static response of the proposed NPS ocean instrument platform is investigated by developing and solving linear differential equations of motion of the tower in surge, heave, and pitch. The motion is expressed as a response spectrum which is directly proportional to a wave spectrum as the exciting force. The analysis is made for various configurations of the lateral restraining cables using both a five point and nine point mooring system. For all configurations j the heave response of the tower is found to be less than one percent of wave height. The stability of the tower in pitch is found to be considerably improved after shifting 1'rom a five point mooring system zo a nine point mooring system and optimizing the location of the cable attachment points. Using this design, a significant wave height of 7-7 feet is found to produce a significant pitch of 5-^ degrees, a significant surge of .97 feet, and a 5.28 foot excursion of the lower platform on the tower. All os- cillations will be superimposed on any heel angle of the tower which may exist due to steady wind forces. The angle of heel for a wind of 20 mph is evaluated to be 1.9 degrees. TABLE OF CONTENTS I. INTRODUCTION 10 A. BACKGROUND 10 1. General H 2. Dimensions H 3. Weights of Tower and Appendages 13 4. Mooring Cables 13 5. Buoyancy 15 II. THEORETICAL DEVELOPMENT i6 A. GENERAL APPROACH i6 B. SPRING-MASS SYSTEM ANALOGY 17 C. ASSUMPTIONS 22 U . METHOD Or A j\i A L, i' o " . - .^i -r o l_ ^r E. DETERMINATION OF DRIVING FUNCTION (SPECTRAL APPROACH 2° F. GENERAL SOLUTION 28 III. SOLUTIONS TO EQUATIONS OF MOTION 32 A. SOLUTION TO SINGLE DEGREE OF FREEDOM EQUA- TION-SURGE 32 B. SOLUTION TO SINGLE DEGREE OF FREEDOM EQUA- TION-PITCH ^1 C. SOLUTION TO SINGLE DEGREE OF FREEDOM EQUA- TION-HEAVE 4b D. SOLUTION TO COUPLED SYSTEM-SURGE AND PITCH -- 53 IV. STATIC WIND FORCE ANALYSIS 62 V. DISCUSSION OF RESULTS AND CONCLUSIONS ^ LIST OF REFERENCES ■ 77 INITIAL DISTRIBUTION LIST ?8 FORM DD 1^73 79 LIST OF TABLES I. Weights of Tower Equipment and Appendages 1^ II. Wave Heights as a Function of Windspeed for a Fully Developed Sea 28 III. Significant Heave for Various Values of Significant Wave Height °7 IV. Significant Heave, Surge, Pitch, and Excursion for Various Values of Significant Wave Heights Using a Five Point Mooring System 73 V. Significant Heave, Surge, Pitch and Excursion for Various Values of Significant Wave Height Using a Nine Point Mooring System 73 LIST OF FIGURES 1. Tower Schematic 12 2. Modes of Oscillation 18 3. Spring-Mass System 18 4. Force Diagram-General 19 5- Force Diagram-Surge 21 6. Spectrums 30 7. Force Diagram-Surge 33 8. Force Diagram-Pitch 42 9. Force Diagram-Heave 49 10. Force Diagram-Coupled System 54 11. Force Diagram-Nine Point Mooring 59 ±2. Nine roinl Mooring Schematic — bO 13. Wind Force Diagram 63 14. Graph of Significant Response as a Function of Cable Connection Point for a Five Point Moor 69 15- Graph of Significant Response as a Function of Cable Connection Point for a Nine Point Moor 70 16. Total Excursion of Lower Platform as a Function of Cable Connection Point for a Five and Nine Point Mooring System 72 17. Angle of Heel of Tower as a Function of Wind Velocity 7^ TABLE OP SYMBOLS AND ABBREVIATIONS A Wave amplitude b.. Effective width of open tower bp Effective width of closed tower C Damping coefficient C Added mass coefficient a CD Drag Coefficient CnT Linearized drag coefficient CH Damping coefficient for heave C Inertia coefficient m C Damping coefficient for pitch C Damping coefficient for surge C Area constant for wire cable w d Diameter D-. Distance from MWL to top of buoyancy material Dp Distance from MV/L to bottom of tower E Modulus of elasticity F Force F, Vertical force due to lateral cable tension Fl Tension in vertical cable Fp Vertical force due to lateral cable tension FB Buoyance force Fn Drag force 'Py)y) Drag disturbing force Fp^ Linearized drag disturbing force FnT Linearized drag force F~R Restoring force F-j. Inertial force FTD Inertial disturbing force Ft-. Restoring force n F1TTWI Total wind force on mast of tower Fwc, Total wind force on tower structure g Acceleration due to gravity h Depth of water H-, ,~ Significant wave height J Added moment of inertia a J Moment of inertia about the center of gravity J Virtual moment of inertia D k Wave numoer K Stiffness coefficient K,, Equivalent stiffness coefficient for heave KT Stiffness coefficient for lateral cables K Stiffness coefficient for pitch K Stiffness coefficient for surge s L Distance from MWL to center of gravity M Mass of tower M Added Mass a M~ Moment due to drag MDD Disturbing moment due to drag MH Virtual mass for heave MT Moment due to inertial forces MTf Disturbing moment due to inertial forces M Virtual mass for surge s & MR Restoring moment R, Distance from MVJL to center of gravity Rp Distance between center of gravity and center of buoyancy R~ Distance between center of gravity and bottom of tower t Time variable u Horizontal water particle velocity U Windspeed V Volume w Vertical water particle velocity x,y.z Space coordinates ^ Angle which lateral ^ahlps make with horizontal Y Specific weight <(> Phase angle ri Instantaneous height of free water surface p Mass density of water a Radial frequency of waves 0 Roll or pitch of tower Q-i/o Significant roll or pitch w Natural frequency of oscillation of tower C CC/M) ACKNOWLEDGEMENTS Dr. Edward B. Thornton proposed the topic for this thesis and guided me during my investigation. I am especially indebted to him for his assistance, encourage- ment and professional advice. I am also very deeply thankful to my wife, Pat, without whose cooperation and understanding I could not have com- pleted this thesis. I. INTRODUCTION A. BACKGROUND The Naval Postgraduate School Is planning to install in approximately 240 feet of water in Monterey Bay a moored ocean instrument platform supported by a tower of approxi- mately ninety feet in length. Such a platform could be used for the mounting of instrumentation designed to measure oceanographic and meteorological data. The tower under consideration was obtained from the government as surplus and was originally designed for use as an umbilical tower prior to launching U.S. Air Force "Thor" missiles. It is the purpose of this thesis to determine the dynamic response of the proposed tower to wave action in Monterey Bay. The instrument package, appendages, proposed location, and further background is described in a Naval Postgraduate School thesis by Lt . H.H. Seibert [3]. In analyzing the dynamic response of the tower to the force of the waves, basic design parameters of the tower are varied so as to achieve an optimum design configura- tion which results in minimum motion of the tower due to wave action. The equations of motion are developed for the tower by direct analogy to a mechanical spring-mass system with damping and sinusoidal driving forces described by linear wave theory. The tower will be taut moored to the bottom using four one-half inch plow steel lateral cables 10 plus a one inch plow steel center cable. The buoyancy chambers will be below mean water level at all times so as to provide a constant buoyancy force. The tower's natural frequency of oscillation is high in comparison to the frequency around which most of the wave energy is centered. This high natural frequency results in a stiff system design thereby preventing conditions of resonance which would otherwise occur if it were a softer system with a lower natural frequency of oscillation. B. PLATFORM DESCRIPTION 1 . General Since a detailed description of the instrumentation or^ ?.ppe>""^giQ<5 to the tower was made by Lt . H.H. Seibert in his thesis, a description here is made of just the basic design of the tower, taking into consideration only those factors which affect its motion (see Figure 1). Although instrumentation and appendages are not shown in Figure 1, the location and mass of each item are considered in deter- mining the tower's center of gravity and moment of inertia. 2 . Dimensions The overall length of the basic tower is 90.5 feet. A bottom steel section extends over a distance of 60.5 feet, and an upper aluminum section has a length of 30.5 feet. A 30.0 foot aluminum mast which is planned for Installation at the top of the tower will extend the overall length to 120.5 feet. The cross section of the tower is square and has a constant width of 3.0 feet for the aluminum section. 11 Aluminum Section Aluminum Platform / Aluminum/Steel Interface Buoyancy Material MWL 1" Plow Steel Cable 1/2" Plow Steel Cable s ' s / / / s , y s / / s s y~7~7 s / '/ / y / FIGURE 1 12 At the aluminum/steel intersection it has a width of 3.0 feet and gradually increases to a width of 4.0 feet at the bottom of the tower. The lower platform is 18.0 feet above mean water level (MWL), and the buoyancy material rises from 7-0 feet above the bottom of the tower to 10.0 feet below MWL. The steel section located just below the buoy- ancy material will act as ballast. It extends over a length of 7.0 feet below the buoyancy material. 3 . Weights of Tower and Appendages The steel section has a total weight (all weights taken in air unless otherwise stated) of 12,780 lbs. The aluminum section has an overall weight of 720 lbs., and all other appendages and equipment have a total weight of 66lj ids. Table I Id a list or the instrumentation ana appendages giving wet and dry weights of each item and the distances of their center of gravity from MWL. The total weight of the tower including all equipment and appendages is 22,313 lbs. 4 • Mooring Cables There are four 680 ft \ inch plow steel cables each weighing 0.37 lbs/ft (in water) amounting to a total weight (in water) of 664.0 lbs for all of the \ inch mooring lines The 1 inch plow steel cable weighs a total of 1.5 lbs/ft (in water), amounting to a total weight (in water) of 252.0 lbs. The assumed depth of mooring is 240 feet. This will require four lengths of \ inch cable each extending out in directions 90 degrees apart and at an angle of 19 13 Item TABLE I WEIGHTS OP TOWER EQUIPMENT AND APPENDAGES Dist. of C.G.* from MWL (ft. ) Mast _ +50.0 Boat Fender + 2.0 Lwr Platform +18.0 Upper Boom +18.0 Rail -17.0 T wv> Room —67.5 Ladder -24.7 Wave Gauge -2 3-0 Misc. Equip. +20.0 Tackle -33.0 Buoyancy Mat '1. ** -29.8 1/8 Sheet Metal -29.8 Total Weight 8813 7907 *Plus sign indicates above MWL and negative sign indicates below MWL. **Buoyancy considered separately. Dry Weight (lbs) Wet Weight (lbs) 60 60 840 400 275 275 60 60 600 544 240 209 225 204 93 81 1520 1520 2000 1814 900 900 2000 1740 14 degrees below the horizontal for a distance (slant) of 680 feet. The center 1 inch vertical mooring cable will extend from the bottom of the tower to the sea floor over a total distance of 166 feet. Each of the h, inch cables will have a tension sufficient to exert equal vertical components of force, and the center cable will have a tension just equal to the vertical component of tension in each one of the \ inch cables. Thus, the vertical components of force in all of the cables will be equally distributed, and their sum will be equal and opposite to the net reserve buoyancy of the tower. 5 • Buoyancy The buoyancy chambers will be filled with salvage roam wnicn has a density of 2.0 lbs/ft and a tensile strength of 70 psi . The foam will be encased in 1/8 inch steel plating which will not be water tight. Hence, the foam will absorb water at its surface. It is assumed the foam will absorb 1.0 lb/ft of water which is a quite con- servative assumption. Since the density of sea water is 64.0 lbs/ft , the net reserve buoyancy provided by the foam will be 61.0 lbs/ft . This result in a total buoyancy force of 33,280 lbs, giving a net reserve buoyancy (excluding mooring cables) of 13,538 lbs. 15 II. THEORETICAL DEVELOPMENT A. GENERAL APPROACH In describing the dynamic response of the NPS platform to wave action, each degree of freedom is first examined separately. There are six degrees of freedom for the tower which supports the platform. These six degrees of freedom, or modes of oscillation, are roll, pitch, yaw, heave, surge, and sway (see Figure 2). Since the tower has symmetry with respect to the x and y axes, roll may be interpreted as pitch, and sway may be interpreted as surge. And it is assumed that any motion of the tower in yaw is negligible since the tower is symmetrical with respect to the z axle. So there now remains three degrees of freedom which must be analyzed, namely, surge, pitch, and heave. It is most logical to first analyze each degree of freedom separately, and using these results, it can be determined whether an analysis of a two or more degree of freedom system is necessary. The motion of the tower is analyzed as a linear system in order that superposition may be used in obtaining a general response spectrum which is directly proportional to a wave spectrum. The wave spectrum itself is generated by superimposing wave components of many different frequen- cies and amplitudes. 16 B. SPRING-MASS SYSTEM ANALOGY The simplest approach to the problem Is to describe each degree of freedom separately by an equation of motion analogous to a spring-mass system with viscous damping and a sinusoidal driving term. The linear differential equa- tion which describes the translatory motion of this system (see Figure 3) is Mx + Cx + Kx = F sin () FIGURE 3. J 18 AAAM- 1 /////// 7 / / ////// FIGURE H. Force Diagram-General 19 components of the four horizontal restraining cables. Other forces shown are the weight of the tower acting through the center of gravity, and the buoyancy force acting through the center of buoyancy. The distance from MWL to the top of the buoyancy material is represented by D-. , and the effective width of this same section is represented by b-, . Since the area above the buoyancy material is a void, the waves will tned to pass through and be impeded only by the cross supporting members of the tower. The effective width, b-, , represents the width of a solid surface which is equivalent to the surface area of the open tower structure for purposes of determining drag and inertial forces. Like- wise the width, b„, represents the average width of the section of tne tower which encdoes the buoyancy material. The geometry of the tower is idealized in this manner, the assumption being slightly conservative. In order to consider only one degree of freedom, the model shown in Figure 4 must be restrained so as to move in only one mode of oscillation at a time; only those forces affecting the motion of the tower in a particular mode should be considered. Take for example the motion in surge (see Figure 5). The equation of motion for this system is Mx + Cx + Kx = FTsin(-oT) + Fncos(~c-r) (2.2) s s s I D where M = the virtual mass in surge, or the mass, M, plus added mass C = damping or all influences which are velocity sensitive 20 FIGURE 5- Force Diagram-Surge 21 K = stiffness where Ks = 2K, or all influences which are position sensitive. FD = Exciting force due to drag F-j. = Exciting force due to the unsteady motion of the waves CT~ - radial frequency of exciting forces (waves) The equations of motion for pitch and heave may be obtained using a similar analogy to a spring-mass system. C. ASSUMPTIONS ' In order to derive the equations of motion mentioned above and to facilitate obtaining a complete analysis of the motion of the tower, it is necessary to make the follow- ing assumptions: 1, Basic Assumptions a. That the pressure field of the fluid is not affected by the tower. To satisfy this assumption it is assumed that the width of the tower is small compared to wavelength . b. The center leg (vertical) restraining cable has a negligible effect in restraining the surge and pitch motion of the tower due to the waves. c. All wave components are from ore direction. This is a conservative assumption. 2 . Assumptions Necessary for Linearization of the Problem a. All assumptions necessary for linear wave theory which include: 22 (1) Fluid is inviscid. However, this assumption is made only in describing particle motion due to waves, and it is not made when drag forces on the tower are con- sidered. (2) Fluid is incompressible and homogeneous. (3) Wave amplitude, A, is small compared to wavelength . b. In accordance with linear wave theory the wave profile ,^ , is h « A cos (kx -rt ) and water particle velocity in the x and z directions is where c is the radial frequency, k is the wave number and h the depth of water. c. The roll or pitch, &s of the tower sufficiently small such that : (1) sine.'^e (2) cosO^AO d. In accordance with linear wave theory, ftOt-M > is considered negligible and set equal to zero when inte- grating wave forces over the complete length of the tower. e. The drag coefficient may be linearized for insertion into a linear differential equation. f. The amount of catenary in the restraining cables is negligible, and the tension in the cables is sufficiently 23 high so as to justify using the relation Pr =-Kx where F = restoring force of cable due to stretch K = stiffness coefficient x = amount of stretch D. METHOD OF ANALYSIS Using the method of undetermined coefficients, the steady state solution to the differential equation of motion in surge is found to be v i ■ ■-* \^, where or FD = t-pfa^J = drag force FT = hxfafl) - inertial force C - C\ (^)^) = linear damping coefficient A = wave amplitude ^ = -v~W C - ■£- where 21) In terms of maximum surge where x(max) = X, the steady state solution reduces to 2 2 2 X = C^ + Cp (2.5) 2 2 2 - By factoring out the A from C, and Cp, the solution may be expressed as x*.[cr *<)/»■ (2.6) where now c2 = G(V) A = fl (r) or i . i (2.7) where xV) =[t.r^)|/?M 2 X (8) D = 2.0 x 10~5 sec -1 2 g = acceleration due to gravity = 32.2 ft/sec 0" = natural wave frequency, sec U = windspeed in ft/sec necessary to develop a given sea state 26 It has been found from observations that the probability distribution function of wave heights is described by a Rayleigh distribution. A Rayleigh distribution is com- 2 pletely defined by the variance, S , where the variance for waves is given by V S2= ^^[n^U (2.9) o where T is the record length. The variance can be deter- mined either directly from equation (2.9) or from the energy-density spectrum by applying Parseval's theorem. Parsevals theorem relates the variance to the area under the spectrum such that -2 -2\^ ^^' (2.10) J o It is now possible to obtain the significant wave height, 2 H-,/-,, which is directly related to the variance, S , by the Rayleigh distribution such that H1/3 = 2.83/s^ (2.11) The significant wave height is defined as the average of the highest one third waves. This is an important parameter because it is the wave height that one observes visually and is used continually in oceanography. Values of significant wave height for various wind speeds were generated by the author of this thesis using equations (2.8) through (2.10). These values are listed in Table II and agree to within one percent of the values listed in a table developed by Bretschneider [2] . A 27 TABLE II WAVE HEIGHTS AS A FUNCTION OP WINDSPEED FOR A FULLY DEVELOPED SEA U Hl/3 knots feet 10 1.4 15 3.7 20 7.6 25 13-5 30 21.4 significant wave height of 21.4 feet will be used as the maximuF! design wave fnv the NPS tower. F. GENERAL SOLUTION Knowing the transfer function for each degree of freedom, and describing the driving term as a "Neumann" wave spectrum, the response function can be expressed as a spectrum by using the previously derived expression: xV) = [t, f c*-)]/? V) The transfer function merely acts as a filter and is a function of wave frequency and the overall geometry and physical make-up of the tower. The driving function, A 00 , represents the energy-density of the waves, part of which is transferred to the tower resulting in pitch, surge, or heave motion. A typical set of curves representing a wave 28 spectrum (driving term), transfer function, and surge spec- trum (response) is shown in Figure 6. The area under the wave and surge spectra is propor- tional to the total energy-density of the waves and potential energy of the tower (in surge) respectively. The frequency at which the transfer function is a maximum corresponds to the natural frequency of oscillation,^, of the tower where as previously defined by equation (2.3) was : ^y = \/~7yhu (2.12) It is seen in Figure 6 that the response spectrum is a product of the wave spectrum and transfer function, and it ic obtained by multiplying uukchici-, ordinate by ordinate t the transfer function and wave spectrum. The wave spectrum in Figure 6 was generated by utilizing equation (2.8), and it represents a significant wave height of 21.3 feet. The transfer function in the same figure represents the surge transfer function of the tower. It can be observed that the response spectrum has two peaks. The peak which occurs at the higher frequency results from the fact that this is the natural frequency of oscillation, or resonant frequency, of the tower. The peak which occurs at the lower frequency is due to the high amount of wave energy at that particular frequency band as shown by the wave spectrum. It is then obvious that as the peak at the transfer func> tion approaches the peak of the wave spectrum, the response 29 120 ■- 100 .. 1 .15 -20 FREQUENCY (Hz) FIGURE 6. Spectrums 30 spectrum of the tower increases rapidly due to resonance. It is therefore necessary that the NPS tower be designed as a stiff system in order to have the peak of the transfer function occur at a high frequency so as to eliminate resonance . It is now assumed that the response heights will also have a Rayleigh distribution which has been found true for other similar systems. It is possible then to determine a value for significant response defined in the same manner as H, ,~ (analogous to wave heights) by X1/3 = 2.83y .5:~ (2.13) 2 where X, ,~ in this case represents significant surge. S found from the response spectrum in the same manner as for the waves such that S2(c-) = 2jX (r) + C>* + ks* -&#*£«$ + & = / ^£ 7^ (3-3) = disturbing force - restoring force. Considering first the disturbing force due to drag, F^r^ where Mo = fy.f*^ (3.4) It is apparent that this term must be linearized for inser- tion into a linear differential equation. As illustrated by- Thomson [4], this can be done by defining a new drag coeffi- cient, CDT , evaluated such that by using CDT , the same amount of work per wave cycle would be done as when using the non- linear drag term, Cn, where energy, E, per cycle is "T E =A=>,>"-C'X (3.5) and where JUL =l\s/*( And it is assumed that the linearized drag force, EDDT 5 is of the form 35 where CDT represents the linearized drag coefficient. Now substituting equation (3-7) into equation (3.5) 3 and inte- grating once more to obtain energy per wave cycle due to P, DDL' ^ //u // = — 7*^fgr** " (3.8) By requiring that the work cycle done by Fnn equal the work done per cycle by FnnT > equation (3.8) can be set equal to equation (3.6) to yield C - g C'pJ1* (3.9) 37/ From linear wave theory the horj 7ontal wat^r narticle velocity amplitude is t-J0 ~ .}///t c/// . _ C Pc y^^- the bottom of the tower Y C? /??-^ S, iSf/^w k Vc. - ?/*># n UA J/TZfc S/JV/f ■# k -*)) 7- (3.14) Referring again to equation (2.2), the virtual mass, Mg3 damping coefficient, Cs, and drag force, F^ have now been determined. The remaining terms must now also be expanded and evaluated. 4. Stiffness Coefficient Considering again the free body diagram in Figure 7, the equivalent stiffness coefficient is K = K = 2K eq s (3-15) 37 where it is assumed that the stiffness coefficient, K, for each cable is M - ~ecs oc (3.16) and ot- = angle of cables with respect to horizontal = 19° E = elastic modulus = 13 x 10 psi L = length of cable = 680 feet 2 A = C d = effective cross sectional area of cable = W " 0.1012in C = area constant for wire cable = 0.405 w d = diameter of cable. Various values for C have been developed by Wilson [1] . And in its final expanded form, equation (3-15) becomes /' = S) b W£ L (s/AJ// JkA- S//V* A (£-J>,j\ // SyAf(A4,\ 6 . Complete Solution Each term of the equation of motion for surge /Ms £ 'Cs^* /£<£ = ^ sw&rfy t Fp e°* (-**) (2.2) has now been evaluated completely, and by using the method of undetermined coefficients, the steady state solution to equa- tion (2.2) is found to be 40 X = - fr * — Sr Co3 ( — or CJ or r 7 ~ v, \ * . «rtr *- , -i ' (.3.22) = Q ecs (-*t) +■ CI s/AJ (-«*} (3.23) In terms of maximum surge, X, the solution becomes X2 = C2 + C2. (3.24) 2 2 2 And by factoring the A out of C- and C?, it becomes p p \ j> • *-J j where |t.F.(« )J = (C2 + C2)/A2. (3.26) B. SOLUTION TO SINGLE DEGREE OF FREEDOM EQUATION-PITCH Having solved the single degree of greedom equation for surge, the motion of the tower in pitch is now analyzed. As in surge, it is necessary to consider a free body diagram which shows the restoring forces which must be considered in the pitch problem (see Figure 8). In this analysis, the downward force, Fj., of the vertical cable will cause a re- storing moment, and the buoyancy force will cause a disturb- ing moment about the center of gravity. The lateral restraining cables have no effect on the motion of the tower In pitch since 41 FIGURE 8. Force Diagram-Pitch 42 they act through the center of gravity , and when the tower oscillates in pitch motion alone, the rotation is about the center of gravity. If they did not act through the center of gravity, the motion of pitch would be coupled to surge and it would no longer be considered a single degree of freedom problem. The coupled pitch-surge problem is con- sidered later. 1. General Equation A differential equation of motion for pitch is sim- ilar to the equation for surge and may be written as T0 rCpO ♦- Kpa ' fflx S//v(-«t)+-/fli>aos(-«z) (3.27) or Inertial , Moment due , spring ~ . , ,. .. , .. , + , „. _ + ^ , . = Disturbing Moments Moment to Viscous Restoring „ , tT„° ,„ , Due to Waves Drag Moment where J = virtual moment of inertia about center of gravity, p ° ° C = pitch damping coefficient. K = pitch stiffness coefficient. P MT = disturbing moment due to the unsteady motion of the waves . Mn = disturbing moment due to drag. Each term in the above equation must now be evaluated. 2 . Virtual Moment of Inertia Like the virtual mass, M , in equation (2.2), there is a virtual moment of inertia for pitch, J , such that f i p > J = J + J (3-28) p o a 43 where J = moment of inertia of tower about center of gravity J = added moment of inertia due to entrained fluid. a or J = J + J p o a = 231,500.0 + 400,000.0 = 631,500 slug-ft2. 3- Damping Coefficient It has already been determined from evaluation of equation (3.14) that the damping force, C x, on an elemental section, dz , of the tower, is given by k or ^Yz), -s^w jk/>,\+ *h + £ Lp s/w* akUpt ecs//A0t -s/Mjp, -Jlpze*s//£l>i *S/W&&}\ _ gSVLSAf**'** ^S/M&D, -2Jl% mftAh f-ZSfflH M -ifcstMkK <2J:l>x. s-//^^^ - £. at*// Jt Pi 45 4. Stiffness Coefficient Considering again Figure 8, it can be seen that the buoyancy force, FD , will cause a disturbing moment, FDR„A D ad. whereas the force, F^, will act as a restoring moment, F|.R^6, such that the total restoring moment, K ©, becomes Kp = (Fi|R3 - FBR2)e (3.39) and KP " F4R3"~ F3R2 (3.^0) 5 . Exciting Moment Due to Inertial Forces In evaluating the disturbing moment due to inertial forces, MTn, it is necessary to refer again to equation (3-20) to determine the relation dFzp = Cy^^i ^'U) Jv J* ^J.^-l> which defines the inertial force on a section, dz , of the tower. Now multiplying this by its moment arm, (L - z) , O and then integrating from the water surface to the bottom of the tower, we obtain Mz> -r^\ U^'^J^U^^ (3.42) "lb (^ = 4-^^) where /jl = ^I^A^Sl w{4Jj)M- cos// A A sw#M iL s,mAI(i- £P, _ 3//J//APS -t A S/A/(M±) ^ &*>#£i(s//u# 4A - C*s// 46 /l s;/-*'* 'Aj>?\ 6 . Disturbing Moment Due to Drag Likewise the disturbing moment due to drag, Mnn, may be determined by multiplying equation (3.11) by a vari' able moment arm, (L - z), to obtain Ad = t [ fa- "- (l* " ^ 7 c/i J J.C3 where ■) ■ — * /U -6 m eos/y j^p, - S//U/J j>&p, ~2£pL £a//3A#i. V- / 2. 2. ^7 J 7 . Complete Solution Now that each term of equation (3-27) has been evaluated, the steady state solution may be evaluated to give (o>--S / f J; ^ w -h £($■)* -yrw S/A,{-^) (3.46) (3.47) where the natural frequency oscillation in pitch is given by /",Y _ rj/j^O 7 J/> In terms of maximum roll or pitch, , the solution becomes or &*- ci^il &ti = lT.F:(-r)\t(«) where [T.P.(y)] = (cl + C^)/A2 3 ^ (3.48) (3.49) (3-50) C. SOLUTION TO SINGLE DEGREE OP FREEDOM EQUATION-HEAVE Considering first the free body diagram for heave as shown in Figure 9, it is apparent that all of the restraining 48 FIGURE 9- Force Diagram-Heave 49 cables will act as restoring forces in heave. All of the cables act through the center of gravity for the single de- gree of freedom problem. 1 . General Equation The differential equation of motion for heave may be written as *to £ -h Cfy H f- /( z- =z. /C^. s-y/ir^t) /- ^J G*s{-«z) / " (3.5D where MH = virtual mass in heave. CH = heave damping coefficient. K„ = heave stiffness coefficient. PT = exciting force due to unsteady motion of waves. Ft^ — exciting *.Oj ce d^^ to d^as;. Again each term above must be evaluated. 2. Virtual Mass The equation for virtual mass in heave is the same as for the surge analysis Mu = M + Mo (3-52) H a where M = mass of tower = 693-0 slugs M s = added mass = C pN a/ 1 and 830.0 slugs MR = 1523.0 slugs 50 3 . Heave Damping Coefficient The heave damping coefficient is evaluated in the same manner as the surge damping coefficient except that the vertical velocity and acceleration of water particles are involved instead of the horizontal water particle motion. Proceeding in a manner similar to the surge analysis, the equation for C„ is found to be •:' CH = /" ^j/7 (3.53) where CnT = linearized drag coefficient for heave A = cross sectional area of tower or jr/j Z7 /V - „,,, (I ft) n N C = 2-i \.fi ,rs ■ l). Stiffness Coefficient-Heave Considering the cables shown in Figure 93 the re- lation for Ku can be written as n Ku = f /C, s/*/cL *-■&.- fi t'i ) I ■>""'. - - ii (3.56) 6 . Drag Forces-Heave Using the linearized drag coefficient for heave, the drag disturbing force, Fnn, may be written as "5- = /£ Coj ^---r) (3-57) where F, - ae^SSs^UU-'jf^y*'-^ -A (3.58) 52 7 . Complete Solution-Heave With each term now evaluated, the steady state solu- tion for heave amplitude, Z, of the tower may be expressed exactly by equations (3.22) through (3.26) with the exception that the subscript "s" indicating surge be changed to "H" for heave . D. SOLUTION TO THE COUPLED SYSTEM-SURGE AND PITCH Referring once more tP a free body diagram as shown in Figure 10, it is apparent that the problem of tower mo- tion should also be considered for cases in which the lateral restraining cables are attached at some point above or below the center of gravity. By analyzing the motion of the tower for all possible configurations, it is possible to arrive at an optimum design configuration. If the cables are in fact attached above or below the center of gravity, any surge of the tower results in the lateral cables causing a moment about the center of gravity. This indicates that the surge motion is coupled to the motion of the tower in pitch. From the results obtained in the heave analysis which is shown later, the heave motion of the tower is found to be small even for large significant wave heights; it is there- fore considered negligible in comparison to pitch and surge and not accounted for in the coupled problem. With this in mind, the two degree of freedom problem for surge and pitch is now analyzed for the general case in which the lateral cables are attached at any point of dis- tance, R, , above or below the center of gravity. 53 FIGURE 10. Force Diagram-Coupled System 54 1 . General Equations-Coupled Motion The coupled motion of the tower in surge and pitch may be described by two simultaneous linear differential equations as follows: ~3~?0 -h QpOf kp^t- f1*- = t&j:S/fil(-<'c)-h/t[)C6s(-«Z) (3.60) The term which couples equations ( 3 • 59 ( and (3-60) together is F. As the lateral cables are attached more closely to the center of gravity, this term approaches zero. As would be expected, when these cables are attached exactly at the center of gravity, F equals zero and the motion is no longer COuuicu . 2 . Evaluation of Terms Each of the terms listed below have been evaluated previously by the corresponding equations listed: NL - (3-D J - (3-28) b p C - (3-14) C - (3.38) b p Ks - (3.17) Mj - (3.43) Pj - (3.21) MD - (3.45) FD - (3-12) It is now necessary to evaluate the coupling term, F, and stiffness coefficient, K . The value of K has changed 3 P P from previous calculations since moving the lateral cables up or down introduces new disturbing or restoring forces in pitch. In the case of surge, no new restoring or disturbing 55 forces are added, but 'those which did exist in the single degree of freedom problem are now modified by the coupling term, F. Therefore the value for K has not changed from s the uncoupled situation. Considering now the total restoring force, F , in surge (see Figure 10) which is found to be = -/(V^ t-^G (3.61) where F = o?KK, = coupling term. Proceeding one step further to evaluate the restoring moments in pitch t it can be shown bj again referring ^~o Fig11-100 1 n that the summation of restoring moments, M , is = -K/>& r ^ (3.62) where K = new stiffness coefficient for pitch F = 2kFL = coupling term. A useful check in the analysis is that the coupling term, F, must be equal in both differential equations, and in this case it checks . 3 . Solution to Coupled Equations Referring again to equations (3-59) and (3.60), a steady state solution may be assumed as 56 X n -5/a;{-<*t) + S Cos (-«"£) & 9 = C S/A/^l) +J>eeJ (-**T) (3.63) (3.64) Substituting the above equations back into equations (3.59) and (3-60), and using the method of undetermined coefficients a set of four simultaneous equations with four unknowns, A, B, C, and D is obtained such that fa /I + /*-*- B tj^ C +jzif D = F0 whpr-e thp elements e* . . are coefficients. The solution to ij the above set of equations may be expressed conveniently in matrix form as S = G 1F (3.65) where S = (3.66) G = 4x4 coefficient matrix of elements g.. (-msr\/(s) fa) (If) o (-E) o foAKP)M o (-f) C-eP^(v5kP) (3.67) 57 F = r- " pi PD Kj _MD_ (3.68) Now that the unknowns of the assumed solution have been determined, the values of matrix S may be substituted back into equations (3.63) and (3.64) for a complete steady state solution to the coupled problem. A convenient tool which may be used in obtaining the solution matrix S is the NPS computer subroutine GELG . This subroutine was used ex- tensively by the author of this thesis. As a check on the previous single degree of freedom solutions, the coupled solutions were compared to the un- coupled solutions alter setting me value of uhe coupling term, F, in the coupled system equal to zero. The results were identical. 4 . Nine Point Mooring System An analysis of coupled motion of the tower was also made for a nine point mooring configuration as shown in Fig- ures 11 and 12. The analysis and solution to the nine point mooring system is identical to that obtained for the five point mooring system with the exception that the values of K , K , and F are changed such that now p5 s 3 to /G -?/t fc (3-69) kp .aiC-f.Z-g*. <-$£ +jJZ-f£ (3.70) h =£/ = density of air = .00237 slugs/ft3 U = uniform wind velocity in ft/sec A = projected area of structure 62 FIGURE 13. Wind Force Diagram 63 F„„ = wind drag force on portion of tower struc- ture above MWL . W = weight of tower in water = 19,7^2.0 lbs & = angle of heel of tower using five point moor MR = summation of moments about the bottom of the tower FB = buoyancy force = 33,280.0 lbs d1 = 105.5 ft d2 = 73.0 ft d^ = 28.12 ft d5 = 25.7 ft. Solving equation (4.1) for 0 in terms of U gives B. NINE POINT MOORING SYSTEM Referring again to Figure 13 and this time including all cables and forces, the equilibrium equation may be written as where & = heel of tower using nine point mooring sys- tem F_ = vertical component of forces due to upper cable tension K = stiffness coefficient = 1825-0 lbs/ft d„ = distance from the bottom of the tower to -* the point of attachement of the upper cables 6H And again solving for & but expressing it this time in terms of cU such that Using equations (4.2) and (4.4), the heel of the tower for various windspeeds can be determined. 65 V. SUMMARY OF RESULTS AND CONCLUSIONS Utilizing the results obtained from each analysis, a series of response spectra (see Figure 6) for various modes of oscillation, sea states, and design configurations are developed from the following relations: X^-[TF/r)]^Y0 (3.72) ZU.^*(*)\fl%® (3.73) ©"A- [T, F, C^]/?» (3.71) Then the values of significant surge, pitch, and heave for each of the above situations are obtained using equation (3.13). A. MOTION OF THE TOWER IN HEAVE The motion of the tower in heave was found to be very slight, and almost negligible when compared to surge and pitch, thus justifying the assumption for a two degree of freedom analysis. The results of the heave analysis for various sea states for both the five and nine point mooring systems are tabulated in Table III. The results for heave are independent of cable location. B. MOTION OF TOWER IN SURGE AND PITCH-FIVE POINT MOORING SYSTEM The results for surge and pitch are presented together since they both depend on the lateral cable location. As- suming a five point mooring configuration, various distances, 66 Mooring H , Z . System ^ \Q 5 Pt. 1.3 0.024 5 Pt. 3-7 0.066 _ 5 Pt. 7.7 0.119 5 Pt. 13.5 0.177 5 Pt . 21.3 0.232 9 Pt . 1.3 0.019 9 Pt. 3-7 0.061 9 Pt . 7-7 0.111 9 Pt . 13-5 0.165 M Pt - ^.l. i 0 .215 TABLE III. SIGNIFICANT HEAVE FOR VARIOUS VALUES OF SIGNIFICANT WAVE HEIGHT. 67 d, of cable attachment point from the bottom of the tower were used to obtain corresponding values of significant surge and pitch as shown in Figure 13. A significant wave height of 7-6 feet was assumed. It becomes obvious from the results presented in Fig- ure 14 that the best point of attachment for the cable in minimizing tower motion is at the bottom of the tower, This was found to be the best point of attachment for all other values of significant wave height as well. The values of significant heave, surge, and pitch for various sea states using this design configuration are presented in Table IV. C. MOTION OF THE TOWER IN SURGE AND PITCH-NINE POINT MOORING SYSTEM If a nine point mooring system is utilized in bl 1C U.C — sign of the ocean platform, it is necessary to determine the optimum distance from the bottom of the tower for the lateral cable attachment point as in the case of the five point mooring system. Assuming that four of the lateral restraining cables would be attached to the bottom of the tower, an analysis was made to determine the response of the tower for various distances, d, to the points of attachment of the remaining four lateral cables. The results of this analysis are shown in Figure 15 • D. LOWER PLATFORM EXCURSION Although Figures 14 and 15 are useful in illustrating surge and pitch separately, it is also important to determine 68 hO C •H CO D P £ •H ^N O -P Ph Cm o ^^ £ vo O M •H CD P £ O O CD EH £ Cm O O O o LPi S CD O H • p £1 P p ctj CD O O CD pq Cm Cm £ O VO 0 • O M £ r- -=T Cm 0 •H II P p C 0 00 •H £ \ O 3 H Ph Ph c oj O 0 V. ( '""'. 5 V.J v-W 0 CD x: e C O CD c p p 0 •H CO 0 Ph >5 CO O (D T3 CM iH £ hO X) ctj £ cd •H 0 CD c-i hO O 0 U O p 3 S CO o c cd p en •H Q hO c •H CO D ■P c ■H O P-, c o ■H P O a> c c o o CD rH .Q cd o s C OQ cd hO CD C fcO-H M rM 3 o CO o S p c p cd c o •H •H o Cm Ph •H C CD 50 C •H •H CO S C\J in cd M hO •H 70 the total excursion of the lower platform due to surge and pitch. It is necessary that the platform excursion be known for various sea states in order to determine the type of measurements which can be validly taken from the platform. The graph is Figure 16 shows the lower platform excursion as a function of cable attachment point for both the five and nine point mooring system. It becomes obvious from the results shown in Figure 16 that the optimum distance from the bottom of the tower for attachement of the second set of restraining cables is 46 feet. This point of attachment is still 9-5 feet below MWL and therefore acceptable in terms of preventing small craft interference with the cables. This was also found to be the best point of attachment for all other values of signi- ficant wave height as well. The values of significant heave, surge, and pitch for various sea states using this design configuration are presented in Table V. E. HEEL DUE TO WIND FORCES Utilizing equations (4.2) and (4.4), the graph in Figure 17 was developed to illustrate the heel angle of the tower as a function of wind speed for both the five point mooring system and the nine point mooring system. It is ob- bious from equation (4.4) that the angle of heel 'decreases as the distance, d~, increases. 71 bO bO C £ •H •H U h O O O O S s -p -p c C •H •H- O O Cm PL, CD 03 > c PH £ co O 6 •H CD <»->. -P -P -p o co CD £ >> 0) 3 CO Cm bO cd £ m •H o •H CD -H O 3 fe P-, -d C 6 4J o M o •H O ,0 i *t Cm -__> cj ; -. " , m (D cd O c r-\ Cm c Cm o P o M £ CD tH 0) £ o H o cu ,Q Ml o cd c CM o Cm O O -H Cm P o £ O O CD CD •H C a co £ £ m O cd 3 o o p o <-{ CQ X CD •H W rH Q 16. Total of Cab (q.j) uij:oj^"B"[j jomot; jo uoxsjmoxg w o H 72 Hl/3 Xl/3 Zl/3 1/3 HORIZONTAL EXCURSION (ft) (ft) (ft) (°) (ft) 1.3 0.36 .024 0.78 0.98 3-7 1.78 .066 2.00 3-37 7.7 4.8 .119 6.33 9.82 13-5 6.37 .117 8.51 13-12 21.3 7.65 .232 10.47 15.94 TABLE IV. SIGNIFICANT HEAVE, SURGE, PITCH, AND EXCURSION FOR VARIOUS VALUES OF SIGNIFICANT WAVE HEIGHTS USING FIVE POINT MOORING SYSTEM H , X , Z . n/ HORIZONTAL 1/5 /5 1/J /5 EXCURSION (ft) (ft) (ft) (°) (ft) 1.3 .608 .019 1.3^ 1.67 3-7 .755 .061 3.04 3.16 7-7 -973 .111 5.43 5.28 13-5 1.27 .165 8.22 7.79 21.3 1-73 .215 11.51 10.84 TABLE V. SIGNIFICANT HEAVE, SURGE, PITCH, AND EXCURSION FOR VARIOUS VALUES OF SIGNIFICANT WAVE HEIGHTS USING NINE POINT MOORING SYSTEM 73 oo hO bO G G •H •H ?H m O o O o S S -p -P g c •H •H O o Ph cu 0) CD > G •H •H Ph S vo -=r CM O l-\ r-{ .H rH CO VD o >5 LTv P ■H o o H > O -3" T3 G •H 3: £ Cm ft O ^6 C O >, •H O -p P OO •H O O c o 3 rH Ph 0) > cd X5 CO o G cd CM ■H ^ U cu <: o Eh Cm O o H rH CD CD