1272 -101 - 'EPORT DOCUMENTATION EORT O. 2. R3. Recipienrts A fs N. PAGE MA-RD-940-80056 Title and Subtitle S Report Date February 1983 Non-Linear Ship Springing Experiments Author(s) 8. Performing Organization Rept. No. Scott Slocum and Armin Troesch 266 Performing Organization Name and Address 10. Project/Task/Work Unit No. Department of Naval Architecture and Marine Engineering 1. Contrct(C) o Grnt(G) No. The University of Michigan ( Ann Arbor, MI 48109 (G) L Sponsoring Organization Name and Address 13. Type of Report & Period Covered The American Bureau of Shipping, 65 Broadway, New York, NY 10006 an d Final U.S. Department of Transportation, Maritime Administration 14. Office of Research and Development, Washington, DC 20590 S. Supplementary Notes L Abstract (Limit: 200-words) The results of an experimental study invvestigating the main hull girder vibrations of Great Lakes bulk carriers are presented: The source of excitation is the non-linear excitation due to two wave trains of different frequencies. Large resonance responses are recorded when the sum of the two encounter frequencies match the natural frequency of the hull. Document Analysis a. Descriptors Wave induced bending moments Springing Non-linear ship dynamics Experimental springing results b. Identifiers/Open-Ended Terms c. COSATI Field/Group Availability Statement 19. Security Class (This Report) 21. No. of Pages Approved for Release ________ Unclassified 125 National Technical Information Service 2. Security Class (This Page) 22. Price Sprin field, 'Virinia 2215_1 ____Unclassified ANSIZ39.18) See Instructions on Reverse OPTIONAL FORM 272 (4-77) (Formerly NTIS-35) Department of Commerce

MARAD Report No. MA-Rn-q40 —O056 ITM/NAEF Report No. 266 February 1983 MON-LINEAR SHIT SPRIMG IMNG EXXPPFIMENTS Prepared by: Scott Slocum anti Armin W, ITroesch September 9, 1982 Co-sponsored by The American Bureau nf Shippinq U.S. Oepartment of Transportation, Maritime Alm:inistration Office of Research and IDevelopment Department of Naval Architecture and Marine EnIqineerinq Col].eqe of Enqineerinq The rniversity of Michiqan Ann Arbor, Michiqan 41 09

TABLE OF CONTENTS paqe List of Fiqures............................................. iv Nomencla ture.............................** *...***.........* ix Execution Summarymr........................** xii T. Introducc tion............................... 1 II. Instrumentation and FRiuipment................................. 6 III. Experimental Results.....****................ 18 Introduc tion to the Experimental Results.********** 1R Presentation of Experimental Data..................*........ 23 Comparison of Non-Linear Exci tat-ion and Response............ 29 Conclusions................................. ***********30 IV. Calculation of the Non-Linear Sprinqinq Excitation Spectrum.... 73 Development of a Computationral Formula....*................. 73 Alqorithm for Calculation of the Non-Linear Excitation Spectrum in Head Seas........................... 78 Example........................................... 79 V. Recommendations for Futu-re Research........................... 93 References............................................... 95 Appendix: Data Analysis Techniue.............................. 96 -iii

LIST OF FIGURES page 1. Linear and Non-Linear Transfer of Enerqy....... *......*......... 3 2. Body Plan of the S.J. Cort (with frame numbers) m b e rs............ 5 3. Schematic of Model for Springing Experiments..................... 6 4. Time History of a Single Wave Group and the Resulting Midship Bending Moment........................................... 13 5. Frequency Domain Representation of a Single Wave Group and the Resulting Midship Bending Moment...............................* 14 6. Time History of a Two Component Wave Group and the Resulting Midship Bending Moment........**.*.........................*** 15 7. Frequency Domain Representation of a Two Component Wave Group and the Resulting Midship Bending Moment..**...........*........ 16 8. Lines of constant we+ and Lines of Constant w- Plotted in the l,w2 Plane.................................................... 20 9. Line Segments in the wl,w2 Plane for which Experimental Data was Obtained................................................. 21 10. Normalized Non-linear Excitation vs. Encounter Frequency Sum, w+ (model scale). w_ = 0. Fn = 0.132. Model Restrained..*.. 32 11. Normalized Non-linear Excitation vs. Ship Length/Wavelength. w_ = 0. Fn = 0.132. Model Restrained......................*.. 33 12. Normalized Non-linear Excitation vs. Encounter Frequency Sum, a)+ (model scale). w. = 1.424 rad/sec. Fn = 0.132. Model Restrained..*******. *.*o*.********** ** 34 13. Normalized Non-linear Excitation vs. Encounter Frequency Difference, w_ (model scale). w+ = 13.949 rad/sec. Fn = 0.132. Model Restrained..*............*.....* *............ 35 14. Speed Dependence Test, Normalized Non-linear Excitation vs. Encounter Frequency Sum, w+ (model scale). w_ = 0. Fn = 0.086. Model Restrained...*............................... 36 15. Speed Dependence Test, Normalized Non-linear Excitation vs. Encounter Frequency Sum, w+ (model scale). w_ = 0. Fn = 0.109. Model Restrained...........*......**............... 37 16. Speed Dependence Test, Normalized Non-linear Excitation vs. Encounter Frequency Sum, e+ (model scale). w_ = 0. Fn = 0.132. Model Restrained...........................*...... 38 - v~

page 17. Speed Dependence Test, Normalized Non-linear Excitation vs. Encounter Frequency Sum, w+ (model scale). w_ = 0. Fn = 0.155. Model Restrained.............**...* *............ 39 18. Speed Dependence Test, Normalized Non-linear Excitation vs. Encounter Frequency Sum, w+ (model scale). w_ = 0. Fn = 0.086, 0.109, 0.132, 0.155. Model Restrained............. 40 19. Speed Dependence Test, Normalized Non-linear Excitation vs. Ship Length/Wavelength. w_ = 0. Fn = 0.086. Model Restrained,*.................................................. 41 20. Speed Dependence Test, Normalized Non-linear Excitation vs. Ship Length/Wavelength. w_ = 0. Fn = 0.109. Model Restrained..........................*.......................... 42 21. Speed Dependence Test, Normalized Non-linear Excitation vs. Ship Length/Wavelength. w_ = 0. Fn = 0.132. Model Restrained..................................................... 43 22. Speed Dependence Test, Normalized Non-linear Excitation vs. Ship Length/Wavelength. w_ = 0. Fn = 0.155. Model Restrained............................................ 44 23. Speed Dependence Test, Normalized Non-linear Excitation vs. Ship Length/Wavelength. w_ = 0. Fn = 0.086, 0.109, 0.132, 0.155. Model Restrained......................................* 45 24. Amplitude Dependence Tests, Normalized Non-linear Excitation vs. Encounter Frequency Sum, w+ (model scale). w-= 0. Fn = 0.132. Low, Medium, and High Amplitudes. Model Restrained......*.*..**............................... 46 25. Amplitude Dependence Tests, Normalized Non-linear Excitation vs. Encounter Frequency Sum, w+ (model scale). w_ = 1.424. Fn = 0.132. Low, Medium, and High Amplitudes. Model Restrained.................................................. 47 26. Normalized Non-linear Excitation vs. Encounter Frequency Sum, w+ (model scale). w_ = 0. Fn = 0.132. Model Restrained and Model Free to Heave and Pitch.................................. 48 27. Normalized Non-linear Excitation vs. Ship Length/Wavelength. w_ = 0. Fn = 0.132. Model Restrained and Model Free to Heave and Pitch............................................. 49 28. Normalized Non-linear Excitation vs. Encounter Frequency Sum, w+ (model scale). w_ = 1.424. Fn = 0.132. Model Restrained and Model Free to Heave and Pitch.............................. 50 29. Normalized Non-linear Response vs. Encounter Frequency Sum, w+ (model scale). wa = 0. Fn = 0.132. Model Restrained.......... 51 -vi

page 30. Normalized Non-linear Response vs. Ship Length/Wavelength. w_ = 0. Fn = 0.132. Model Restrained........................ 52 31. Normalized Non-linear Response vs. Encounter Frequency Sum, w+ (model scale). wo = 1.424. Fn = 0.132. Model Restrained..... 53 32. Normalized Non-linear Response vs. Encounter Frequency Difference, w_ (model scale). w+ = 13.949 rad/sec. Fn = 0.132. Model Restrained.............*..**...........*...... * **.***** 54 33. Speed Dependence Test, Normalized Non-linear Response vs. Encounter Frequency Sum, w+ (model scale). w_ = 0. Fn = 0.086. Model Restrained.................................. 55 34. Speed Dependence Test, Normalized Non-linear Response vs. Encounter Frequency Sum, w+ (model scale). w_ = 0. Fn = 0.109. Model Restrained.......................... 56 35. Speed Dependence Test, Normalized Non-linear Response vs. Encounter Frequency Sum, w+ (model scale). w_ = 0. Fn = 0.132. Model Restrained *.............................. 57 36. Speed Dependence Test, Normalized Non-linear Response vs. Encounter Frequency Sum, w+ (model scale). w_ = 0. Fn = 0.155. Model Restrained.......*.............*..*....... 58 37. Speed Dependence Test, Normalized Non-linear Response vs. Encounter Frequency Sum, w+ (model scale). o_ = 0. Fn = 0.086, 0.109, 0.132, 0.155. Model Restrained............. 59 38. Speed Dependence Test, Normalized Non-linear Response vs. Ship Length/Wavelength. w_ = 0. Fn = 0.086. Model Restrained ** ****** o********** o*** ooe**o** o*o*** e** **** e**** 60 Restraine................................................60 39. Speed Dependence Test, Normalized Non-linear Response vs. Ship Length/Wavelength. w_ = 0. Fn = 0.109. Model Restrained **............. **................******** 61 40. Speed Dependence Test, Normalized Non-linear Response vs. Ship Length/Wavelength. w_ = 0. Fn = 0.132. Model Restrained *o** *** *o*.........**.............................. 62 41. Speed Dependence Test, Normalized Non-linear Response vs. Ship Length/Wavelength. w_ = 0. Fn = 0.155. Model Restrained..................................................... 63 42. Speed Dependence Test, Normalized Non-linear Response vs. Ship Length/Wavelength. w_ = 0. Fn = 0.086, 0.109, 0.132, 0.155. Model Restrained............................. 64 -vii

page 43. Amplitude Dependence Tests, Normalized Non-linear Response vs. Encounter Frequency Sum, w+ (model scale). w_ = 0. Fn = 0.132. Low, Medium, and High Amplitudes. Model Restrained.**************....................................... 65 44. Normalized Non-linear Response vs. Encounter Frequency Sum, w+ (model scale). w_ = 0. Fn = 0.132. Model Restrained and Model Free to Heave and Pitch......................... 66 45. Normalized Linear Excitation vs. Ship Length/Wavelength. Fn = 0.132. Model Restrained.................................. 67 46. Amplitude Dependence Test, Normalized Linear Excitation vs Ship Length/Wavelength. Fn = 0.132. Low, Medium, and High Amplitudes. Model Restrained*...*...**...................*.... 68 47. Speed Dependence Tests, Normalized Linear Excitation vs Ship Length/Wavelength. Fn = 0.086, 0.109, 0.132, 0.155. Model Restrained............................................... 69 48. Normalized Linear Excitation vs. Ship Length/Wavelength. Fn = 0.132. Model Restrained and Model Free to Heave and Pitch...................................................... 70 49. Comparison of Calculated and Measured Non-Linear Response. Model Restrained............................................. 71 50. Comparison of Calculated and Measured Non-Linear Response. Model Free to Heave and Pitch............................ 72 51. Incident Wave Spectrum Used in the Example. Measured at Eagle Harbor, 28 Nov. 1966. Significant Waveheight 18.29 ft. (Taken from Ploeg (1971))............................. 81 52. Bending Moment Spectrum Calculated in the Example. Effects of Heave and Pitch Neglected.................................... 92 53. Bending Moment Spectrum Calculated in the Example. Effects of Heave and Pitch Included............................*...... 93 -viii

NOMENCLATUTRE Aij = qeneralized hydrodynamic added mass coefficient aii = generalized ship mass coefficient Bj j = generalized hydrodynamic damping coefficient B = ship beam at midship b22 = internal ship springing damping coefficient Cj = generalized hydrostatic restoring coefficient c22 = internal sprinq constant (k) E2 = amplitude of the kth order sprinqing excitation ei(t) = ith qeneralized wave excitation periodic in time (k) e2 (t) = kth order of the generalized springinq excitation periodic in time Fn = Froude number f1 = external force applied at Xcq f2 = external force applied at xccT q = qravitational constant H = total springing excitation transfer function H(k) = kth order springing excitation transfer function (k) HR = kth order sprinqinq response transfer function hl = linear impluse response function h2 = quadratic impulse response function I = mass moment of inertia of the model about midship I^ A = mass moment of inertia of the after part of the model about xcq IF = mass moment of inertia of the forward part of the model about Xcq Ks = model midship spring constant k = wave number of the incident wave (=2wn/X) -ix

L = ship length hA = maqnitrlde of distance from midship to the center of gravity of the after model section Q F = maqnitude of distance from midship to the center of qravitv of the forward model section M = mass of the model MA = mass of the after model section MF = mass of the forward model section (k) MO = amplitude of the kth order midship bending moment m0(t) = midship bending moment periodic in time (k) m0 (t) = kth order midship bending moment periodic in time qi(t) = the ith generalized coordinate i = 0: vertical rigid body motion i = 1: riqid body rotation about the y axis i = 2: two noded body motion S+( ) = one sided incident wave spectrum S+(o) = one sided springing excitation spectrum E S+(w) = one sided midship bendinq moment spectrum M U = ship speed Um = model speed x = the lonqitudinal ship coordinate axis, positive towards the bow xcg = x coordinate of the center of gravity of the after model section A (= -~A) xcg = x coordinate of the center of gravity of the forward model section (= )F) y = transverse ship coordinate axis, positive to port zi(x) = ith vertical mode shape z = vertical ship coordinate axis, positive up = ith phase angle of the gi(t) mode of motion (t) = time dependent water surface elevation due to waves -X

0 -= damping ratio in springing r = incident wave amplitude 0 = midship deflection angle due to springing motion X = incident wave length 3 -= rigid body heave amplitulde 5 = rigid body pitch amplitude a = incident wave frequency as seen from a stationary observer, 1, W02 = wave frequencies of encounter w_ = difference frequency (=()1 - W2) 'W+ = sum frequency (=W1 + W2) wo = natural frequency of main hull girder -xi

Executive Summary Under the joint sponsorship of the American Bureau of Shipping and the Maritime Pdministration, a Research program investigating Great lakes sprinqing was funded at the University of Michiqan. The emphasis of the project was placed on determining whether non-linear long wave sprinqinq excitation is important. Based upon the model tests conducted in the Ship Hydrodynamic Laboratory's towing tank and based upon the analytical work done, it is clear that the methodology used to analyze ship springing must include both a linear and a non-linear part. Consider a bulk carrier operating on the Great Lakes. If it encounters an incident wave of frequency w, then there will be a springing excitation at w. This is the so-called "linear" excitation and will produce a linear response in the form of a midship bending moment. The term linear implies that if the incident wave height is doubled, then the bending moment is also doubled. Preliminary springing excitation and response experiments conducted by Troesch (1980) have shown that there is also a significant transfer of energy from the fundamental frequency of encounter, w, to higher harmonics such as 2w and, sometimes 3w. Should 2w or 3w equal 0o, the natural frequency of the hull, then there will be a substantial increase in the springing response, and, consequently, the midship bending moment. The sea is not composed of a single wave component, but rather a complete spectrum. The steepness of the water surface and the interaction of different wave components introduce non-linearities. Wave components at one frequency will interact with components at other frequencies. The result will be waves with frequencies wi-wj and wi+wj, where wi and wj are the frequencies of the first order wave components. The implication of this for ship sprinq-xiii

inq is that, in addition to the lonq wave excitation resulting from 2w and 3w, there will also be ionq wave excitation from wi+-j. This report describes how the non-linear behavior of Great Lakes ship sprinqing was investigated through model tests. Both the harmonic (W0 = 2i ) and the sum frequency (w = w1 + w2) conditions were investigated. A description of the experimental equipment and data collection procedures is included. An empirical form of the equation describinq the non-linear springing response is given and verification offered. The use of 'he model test results in statistically evaluating the importance of the non-linear sprinaina is shown through an example. Based upon the results of this report, it is possible to draw the following conclusions: - The non-linear excitation and consequently the response to the nonlinear excitation are dependent upon ship speed, wave frequencv of encounter, and wave height. - The non-linear effects tend to be quadratic in wave amplitude. (The distinction between linear and quadratic systems is the following: As stated earlier, a linear system with an incident wave of encounter frequency w will produce a response at that same frequency, and if the wave height is doubled, the response will also double. In a quadratic system, an incident wave of encounter frequency w will produce a response at twice that frequency, or 2w. Of equal importance, however, is that if the wave height is doubled the response at 2w will increase by a factor of four.) -xiv

- For Great Lakes bulk carriers, particularly the "thousand footers," heave and pitch motions are considered to be small. However, both the non-linear excitation and non-linear response are influenced by the maqnitulde of these relatively small motions. In addition to the above conclusions, an example is qiven in the toxt where the non-linear effects are inclluded in estimatinq the area under the sprinqinq spectrum. It is found that the non-linearities account for approximately 25% of the total area. -xv

I. INTRODUCTION Under the joint sponsorship of the American Bureau of Shippinq and the Maritime Administration, a Research proqram investiqatinn Great Lakes sprin(inq was funded at the University of Michiqan. The emphasis of the project was placed on determininq the answer to two questions: (1) Can a linear theory predict sprinqinq response? and (2) Is non-linear lona wave excitation important? The unexpected answer to both is yes! Based upon the model tests conducted in the Ship Hydrodynamic Laboratory's towincr tank and haserl upon the analytical work done, it is clear that the methodoloqy used to analyze ship sprinqinq must include both a linear and a non-linear part. Most current ship motions and wave loadinq proqrams use a linear theory. (An exception is the work described by Jensen and Pedersen (1981)). This implies that any of the quantities of interest, i.e., the heave amplitudre or midship bendinq moment, are directly proportional to the incident wave heiqht. Also implied is that the response has the same frequency as the encounter frequency of the incident wave. Specifically omitted from the model is the possibility that low frequency, lonq waves will excite hiqh frequency ship springinq. While the linear model has worked well in past applications, the work done here at the University demonstrates that it will only nartially describe the sprinqinq excitation and response of the Great Lakes ore carriers. An analytical linear theory has been developed by Bishop and Price (1977) and Maeda (1980). Parts of this linear model have been validated by towinq tank experiments; see, for example Troesch (1980). One of the theory's deficiencies, however, is its inability to accurately predict a spriniqinq dampinq coefficient. While dampinq prediction is an important problem, this report will address another aspect of ship sprinqinq. If all the waves on the Great -1 -

-2 -Lakes were short, then a linear theory, with perhaps an empirical dampinq coefficient, would be sufficient. This has been demonstrated with the short wave experiments described by Troesch (1980). However, an ore carrier will encounter lonq waves in addition to short waves. Preliminary sprinqing excitation and response experiments conducted by Troesch (1980) have shown that there could be a significant transfer of energy from the fundamental frequency of encounter to hiqher harmonics. If the incident wave elevation was given as r(t), where -(t) = rl cos (t) then there was a measurable springing excitation at 2w and, sometimes 3. Here C is the encounter frequency and n is the incident wave amplitude. Should 2w or 3w equal w), the natural frequency of the hull, there will be a substantial increase in the springinq response. The sea is not composed of a single wave component, but rather a complete spectrum. The steepness of the water surface and the interaction of different wave components introduce non-linearities. If we wish to describe the nonlinear behavior of water waves, there are a number of approaches available. For example, if a perturbation technique is used, wave components at one frequency will interact with components at other frequencies. The result will be waves with frequencies wi-wj and wi + wj, where )i and wj are the frequencies of the first order wave components. A good description of this theory is given by Lonquet-Higqins (1963). The implication of this for ship springing is that, in addition to the long wave excitation resulting from 2w and 3w, there will also be long wave excitation from wi + wj. This is shown schematically in Figure 1.

-3 -^ ^ ^^V\a~~2wl=awo | /I w^-^ i+wj=wo Ai 1 j o o developed y Bishop and Price (1977) or aed (1979 i 1=j0 0 iii) Waves with encounter freguencies of ui and Wj where 0 0 Fiqure 1: Linear and Non-Linear Transfer of Enerqy In Fiqure 1, the spectraco of the incident waves and the resultinq sprin-. inq response are indicated. The huill has a natural frequency of wo. The response at o is the result of the followinq sources of excitation: i) Waves with an encounter frequency of.de (Thsn i is the linear case. We are ahle to estimate this excitation usin the thve-or developed by Bishop and Price (1977) or Maeda (1979)). ii) Waves with an encounter frequency of w1, where 2w1 = o. (This is the non-linear excitation due to harmonics of lonq waves). iii) Waves with encounter frequencies of wi and wj where C1 + Wj = o ~ (This is also a non-linear excitation caused by the interaction of two different wave components). A recent attempt to analyze the transfer of enerqy from low frequencies to hiqh frequencies is described in articles by J. Jensen and P.T. Pedersen (1978) and (1981). In those articles the authors calculate the wave-induced

-4 -bending moments in ships by a quadratic theory. There are a number of difficulties associated with their theory, however, one beinq the improper handling of the free surface condition. This problem, thouqh, only effects their ability to predict the magnitude of the non-linear bending moment. In principle, their procedure is correct. Specifically, they give the total hending moment response as follows: m(x,t) = m(1)(x,t) + m(2)(x,t) +.. where m(1)(x,t) is the linear moment due to wave excitation as previously described in case i) and m(2)(x,t) is the non-linear moment due to wave excitation described by ii) and iii). While it is extremely difficult to analytically estimate m(2)(x,t), it should be possible to determine its value experimentally. Jensen and Pedersen (1981) also show how a spectral density function of the bending moment can be calculated if the spectrum of the incident wave is known and if the bending moment response amplitude operator (RAO) is given or can be determined. The RAO is the bending moment due to waves of unit height and may include the effects due to non-linearities. The bending moment spectrum would then include both linear and non-linear parts. This report will describe how the non-linear behavior of Great Lakes ship springing was investigated through model tests. Both the harmonic (wo = 21 ) and the sum frequency (wo = w1 + w2) conditions were investigated. A description of the experimental equipment and data collection procedures is included. An empirical form of the equation describing the non-linear sprinqinq response is given and verification offered. The use of the model test results in statistically evaluating the importance of the non-linear springing is shown in an example.

II. INSTRUMENTATION AND EQUIPMENT The fiberglass model used in the experiments described by Troesch (19R0) was used aqain for the non-linear experiments described in this report. A description of the model and test apparatus will be included here for comple teness. A 15 foot fiberglass model of the S.J. Cort was constructed for the earlier project. In a manner similar to that described by Hoffman and van Hooff (1976), the model was made in two halves connected by a spring at midships. See Table I for the model characteristics and Figure 2 for the body plan. See Figqlre 3 for a description of the quantities measured. z M; - f - -~t3 f2 x A Figure 3: Schematic of Model for Sprinqinr Experiments In Figure 3, f1 and f2 are external forces, Mo is the midship bending moment, hA = -xcq where xcq is the x coordinate of the center A A of gravity of the after part of the model and QF = Xcq where Xcq is the F F x coordinate of the center of gravity of the forward part of the model. The model has three degrees of freedom. The normalized linear coupled equations of motion can be written in the following form. -5-,

FOC S L DECK AT SIDE MAIN DECK AT SIDE 0 1 3 5 10 15 20 I10 _/ / 0 DWL 135 2LAN OF THE 1 28 FIGURE 2: BODY ~UA ' OF THE S*-3 * COT (P!P F IAME FNUMBERS')

-7 -Table I: Model Characteristics Characteristics - Total Model scale 66.67:1 Length Overall, Tn(ft) 4.572 (15.00) Length between Perpendiculars, m(ft) 4.569 (14.99) Beam, m(ft) 0.4783 (1.569) Draft (mean), m(ft) 0.1179 (0.387) Displacement, N(lbs) 2,245. (504.8) Longitudinal Center of gravity (% L fwd midships) 0.1% Moment of Inertia about Midships, kg-m2 (slugs-ft2) 331.6 (244.6) Characteristics - Fore Body Length Overall, m(ft) 2.286 (7.50) Displacement, N(lbs) 1,142. (256.7) Longitudinal Center of Gravity (% L fwd midships) 22.5% Moment of Inertia about Forebody LCG, kq-m2 (sluqs-ft2) 40.86 (30.14) Characteristics - After Body Length Overall, m(ft) 2.286 (7.50) Displacement, N(lbs) 1.103 (248.1) Longitudinal Center for Gravity (% L aft midships) 23.1% Moment of Inertia about Afterbody LCG, kg-,m2 (slugs-ft2) 41.86 (30.88)

-R2 qo(t)aoo + I [A0j qj(t) + RBj qj(t) + C0j qj(t)] j=0 = fl + f2 + e0 (1) 2 ql(-t)a11 + [ [Alj qj(t) + Blj qj(t) + C j qj(t)] j=0 xcqA + Xcq f + el (2) A 1 A 2 q2(t)a22 + q2(t)b22 + [A2j qj (t) + R2j j (t) + C2j qi (t)] j=0 + + m 1 - + e2 (3) XC(TF The generalized coordinates are aiven as qo(t) = C3 e iWt+ia ql1(t) = 55 e 1 iot+ia q2(t) = [00/(1+XA/XF)1 e 2 where E3 is the heave amplitude, E5 is the pitch amplitude, and 80 is the midship deflection anqle. The mode shapes, zi, are defined as follows: z0 = 1 z% = x

-9 -IA+X for x<0 z2 = ( A/F) ( F-X) for x>0 The midship bendinq moment, m0, is given as mo(t) = -Ks(1 + A/QF)q2(t) where Ks is a sprinq constant. The coefficients have the followinq forms: a00 = M, the mass of the model, al1 = I, the mass moment of inertia of the model about midships, 9LA a22 = IA + - IF where IA and IF are the mass moments of QF inertias about the centers of qravity of the after part and forward part respectively. ei = f dx zi(x) fe(x,t) where fe is the sectional hvdrodynamic L exciting force and periodic in time, and Cij = f dx zi(x) zj(x)pq R(x) where pq is the weight density of L the fluid and B is the beam of the ship. Also Aij represents the i-th generalized hydrodynamic force due to an acceleration in the j-th mode, Bij represents the i-th qeneralized hydrodynamic force due to a unit velocity in the j-th mode, and b22 represents the spr ingin internal mechanical damping.

-10 -Note that fe is a result of the integration of the hydrodynamic pressures on a hull section and may include forward speed effects if the model is moving forward. For the complete theoretical expressions of these coefficients, see the corresponding formulas given by Maeda (1980). As explained in the linear theory section of Troesch (19R0), the vertical displacements are expanded in terms of the dry modes. For our particular case this is heave, pitch, and springing. These mode shapes must satisfy a condition of orthoqonality that requires MA + MF = M, MA2A = MFZF r and IA = (XA/pF)IF. here MA and IA and MF and IF are the mass and mass moment of inertia of the after part and forward part respectively. The rationalization of using dry mode expansions instead of the more common wet mode expansions will be briefly explained. The hydrodynamic pressures are treated as external incremental forces allowing the hydrodynamic coefficients to be expanded in terms of the orthoqonal dry mode shapes. This permits both the added mass and damping coefficients to be expressed in the same rigorous, consistent manner. When the frequency of oscillation become sufficiently high and the damping forces go to zero, either the wet or dry mode expansions are correct. However, the relatively low springing frequency (that is relative to typical ship vibration frequencies) causes hydrodynamic reactive forces that are in phase with both the acceleration and velocity of the displacement. The non-conservative nature of the external forces suggests the use of the dry mode shapes. The derivation of the general equations of motion for an elastic ship's hull is given in detail by Bishop and Piece (1977) or Maeda (1979).

-11 -In the experiments, measurements were made of the midship bending moment and the incident wave. The motions were not measured since the incident waves were typically shorter than the model and thus did not noticeably excite heave or pitch. If the sprinqinq excitation was to be measured, then the spring constant, KS, must be large enough so that the contribution of the dynamic terms in the equation of motion, equation (3), be effectively zero. This was accomplished by using a stiff 4450N (1000 lb) load cell. The actual experimental confiquration is detailed by Troesch (1980). When measurinq the sprinqinq response, the load cell was replaced with a relatively soft sprinq that qave the model a full scale, two noded, natural frequency of approximately 0.30 cycles per second. The wave elevation was measured usinq a Wesmar TM7000 sonic wave probe. The probe was attached to the carriage approximately 2m (7 ft) in front of the model's bow. The output from the instruments were in analog form. These signals were converted to digital form on a Tektronix 4052 minicomputer equipped with a Trans Era A/D converter, Memory module, and special purpose FFT ROM pack. The record length was 4096 sample points per channel, and all record processinq was done on the Tektronix. Each signal was Fourier transformed and the spectral peaks in the wave record were located. Given the incident wave transform and the convolution of that transform, the first and second order transfer function could be determined. The actual data reduction method is described in detail in the Appendix. As described earlier, the non-linear experiments consisted of both harmonic and sum frequency tests. The harmonic tests followed a format similar to

-12 -Troesch (1980). In those tests, the wave maker produced waves ofT a single frequency with the non-linear excitation or response occurring at twice the encounter frequency. In order to conduct the sum frequency tests, two wave trains were generated. Since waves are dispersive (i.e., the speed at which they travel depends upon their frequency), the wave maker was driven by a predetermined signal that had the two wave groups meet at a particular location in the tank. By properly coordinating the start of the cowi;nrl icircLaJTe and the start of the wave generator, the model and the wave gror)f)l -ir.rLv; at tlhe same place at the same time. This produced a springing excitation of five distinct frequencies. If the frequencies of the two wave groups are given as w1 and w2, then the linear responses were at wl and ()2. The three nonlinear responses were at l1 + w2, 2wl, and 2w2. The location of nonlinear responses could be accurately determined by simply viewing the convolution of the Fourier transform of the incident wave signal. The graphics screen of the Tektronrli 4052 was used to monitor the inplit signals from the load cells/strain gages and wave probe. It also was used to display various quantities of interest at selected steps in the data reduction procedure. Examples of the traces shown on the screen are reproduced in Figures 4 through 7. The time, frequency, and magnitude axes are indicated. Figure 4 shows the time histories of the incident wave and resulting bending moment for the harmonic response test. The fundamental and harmonic content of the response are clearly visible. The relative magnitudes can be seen in Figure 5 where the Fourier Transform of the bending moment is graphed. The transform of the incident wave and the convolution of that transform are also presented. The use of the convolution is described in the Appendix. Figure 6 shows the time histories of the wave and bending moment for the sum frequency

-13 -Harmonic Springing Response Test (2w1 = wo) Encounter Frequency w1 = 7.55 rad/sec Froude Number Fn = 0.132 04. -.-I Time 4j ra Time FIGURE 4: TIME HISTORY OF A SINGLE WAVE GROUP AND THE RESlULTING MIDSHIP BENDING MOMENT.

-14 -Harmonic Springing Response Test (2w1 = o0) E f~ I ~ Encounter Frequency w1 7.55 rad/sec o J Froude Number F = 0.132 05k - ' c I ~4 Uo ~C E* UH (, I C ~-4. —4 — 2w1 Frequency o C Tn ar h) r -3 2w1 Frequency FIGURE 5: FREQUENCY DOMAIN REPRESENTATION OF A SINGLE WAVE GROUP AND THE RESULTING MIDSHIP BENDING MOMENT.

-15 -Sum Frequency Springing Response Test (Wl + w2 = w0) Encounter Frequencies w1 8.80 rad/sec )2 6 6.30 rad/sec Froude Number Fn 0.132.4 r-4 Time 4 -0' Time FIGURE 6: TIME HISTORY OF A TWO COMPONENT WAVE GROUP AND THE RESULTING MIDSHIP BENDING MOMENT.

-16 -Sum Frequency Springing Response Test J1 + w2 = o0) Encounter Frequencies wl 8.80 rad/sec E | W~2 6.30 rad/sec 4 | T Froude Number Fn = 0.132 r { cc o l (1 "2 Frequency EU I *O-4 t$ rc >. I I 0 2 1 + Frequency WAVE GROUP AND ING MIDSHIP BENDING MOEN ert, w2 wl u2 Frequency 7. E:%f n 2 -f -KJ rI^K Il AK ICN

-17 -response test. The beat phenomenon that occurs when waves of two slightly different frequencies interact is present. Figure 7 contains the Fourier Transforms of the incident waves and bending moment and also the convolution of the wave transform. The doubly spiked graph of the wave transform demonstrates the two wave component nature of the incident wave. The bending moment transform shows the linear response at wl and w2 and the non-linear sum frequency response at 01+W2. The difference in the magnitude of the response at wl relative to that at w0 again demonstrates the hump-hollow behavior of the linear excitation c(lnrve as snhown by Troesch (1980).

III. EXPERIMENTAL RESULTS Introduction to the Experimental Results Let the springing excitation transfer function, H, be the sum of the first order excitation, H(1), and the second order excitation, H(2) Since the transfer function H(2) (W1,2) is a function of two frequency arguments, its maqnitude might be presented in the form of a three-dimensional graph. For example, if we take the wl-axis to lie in the x-direction, and the w2-axis to extend in the v-direction, we can plot the transfer function magnitude in the z-direction as a function of w1 and 2. Alternatively, we can represent the transfer function by a series of twodimensional curves, each showing a sectional vertical cut of the three-dimensional surface H(2) (1,W2). This was the method used here. The directions in which the sectional cuts were taken were chosen to provide the most useful quantitative and qualitative information in view of the computational formulas presented in section IV. Let us define the encounter frequencv sum w+ by 0+ = e1 + )2, (4) and the encounter frequency difference w by ). = e1 - )2. (5) We can then write H(2)(o)l,o2) = H( 2)-(+ + W-), -(a+- ) (6) As can be seen from the formulas in section IV, the calculation of the non-linear excitation spectral density at a given frequency on requires a knowledge of the behavior of the transfer function as e_ varies and 0+ is -18 -

-19 -non-linear excitation spectral density at several different frequencies n,w we are also interested in the behavior of the transfer function for constant w_ and varying w+. In order to illustrate the relationship between the arguments wl and 02 and the newly defined parameters w+ and w_, lines of constant w+ and lines of constant w_ are plotted on the w1,w2 plane in Figure 8. We emphasize here that the arguments w+ and w_ are used merely as an alternative (and more appropriate) set of coordinates with which to specify a location on the w1,w2 plane. We are in all cases interested in non-linear effects at the "sum frequency" w+, and use of the frequency difference parameter w_ does not in any instance imply that we are investigating non-linear "difference frequency" phenomena. The non-linear transfer function was determined experimentally by testing the model in wave systems composed of two wave components, one at encounter frequency w1 and the other at encounter frequency 02. (A description of the analysis procedure used to calculate the transfer function is given in the appendix). The behavior of H() -(w+ + w_),-(+ - X3 for varying )+ was investigated by changing the wave frequencies w1 and w2 for each test run such that w_ was held constant at some chosen value and w+ was systematically varied. The results of these tests are plotted versus w+, and the constant value of o_ for a given set of runs is indicated in the caption to the corresponding data plot. Conversely, an investigation of w, dependency was conducted by allowing o_ to vary while w+ was held constant. These results are plotted versus o_, and the constant value of w+ is given in the caption. Line segments shown in Figure 9 indicate the regions of the 1, 02 plane for which experimental data was obtained.

CO2 18 I~ ~~0~ ~0 16-, 14 — / q, 12 -0 * FIGURE 8: LINES OF CONSTANT W+ AND LINES OF CONSTANT (PLOTTED IN THE 'lJi)2? PLANE

(A)2 18 16 14 12 0 q r go,~ t,~ 8 6 4 *19 2 '9 0 I I I I I (tW I 0 2 4 6 8 10 12 14 16 18 FIGURE 9: LINE SEGMENTS IN THE,1Lw2 PLANE FOR WHICH EXPERIMENTAL DATA WAS OBTAINED.

-22 -Several tests were conducted for the important special. case in which w_ = - 02 = 0 and o+ varies. In this situation, the transfer function H(2)(1,i1 ) was determined by testinq the model in a monochromatic wave system. Since only one wave component was present in these tests, the results can be plotted against the ship lenqth to wavelength ratio as well as the frequency sum (or, in this special case, the "harmonic" frequency) w+ = 21w. The linear transfer function H(1)() was determined from measurements taken in these "single wave" tests, and is in all cases plo)tted as a function of ship lenqth to wavelenqth ratio. Both the linear and ron-linear excitation transfer functions are normalized by factors suqqested by zero-speed Froude-Krylov calcullations. The linear excitation data is given as F( 1) (,) 2 [H(1)(w)]norm = -- (7) pqBL2 in which E(i)(w) is the amplitude of the linear generalized springinn exci2 tation measured at encounter frequency w and rn is the wave amplitude measured at the same frequency. The mass density of water is denoted by p, gravitational acceleration by q, B is the ship beam, and L denotes ship lenqth. The non-linear excitation data is given as E(2) (W1+w2) 2 [H(2)(Wl,W2)]norm = - (8) pqBL2 /klk2 (l(1 )rn2(W2) where E(2) (1+w2) is the amplitude of the non-linear excitation at the 2 frequency sum + = 1 + W2, nt1l(w) and n2(w2) are the measured wave amplitudes at encounter frequencies W1 and )2, respectively, and k1 and k2 are wave numbers correspondinq to o1) and w2.

-23 -The relationship between the actual time history of the sprinqing excitation, e2(t), and the amplitudes of the various components of the excitation, E(1), and E(2), is described in Section IV and the Appendix. 2 2 The non-linear response is plotted in the form R(2) (W1+w2) 2 [H(2)(W1,2) ]norm = (9) R pqBL2/kl1k2n1n2 in which R(2) (1+w2) is defined by the expression 2 R(2)(w1+w2) = (1+A/ZF)M(2) (w1+2) (10) 2 0 with M(2) (1-+W2) denoting the amplitude of the non-linear hendinq moment at 0 the frequency sum )+ = 1w + w2. Presentation of Experimental Data (1) Figures 10 through 28: Experimental Data from Measurements of the Non-linear Sprinqinq Excitation Figures 10 throuqh 13 show the results of the primary thrust of the investiqation. The normalized non-linear transfer function as defined by equation (8) is shown for each of the three line seqments in the 1, W2 plane which are indicated in Figure 9. The data for all four plots was obtained from tests at Froude number Fn = 0.132 (model speed Ut = 2.90 ft/sec), with the model restrained in heave and pitch by rigid attachment at the node points. Figure 10 shows the data taken in the "single wave" tests, in which w_ was held constant at w_ = 0, and w+ was systematically varied. [H(2) (wl,2)]norm is plotted versus encounter frequency sum e+ (model scale).

-24 -Fiqure 11 shows the same data, plotted as a function of ship to wavelength ratio. This data agrees well with the non-linear excitation data (iven by Troesch (1980). The data in Figure 12 was obtained in tests in which w_ was held constant at w_= 1.424 radians/sec (model scale) and w+ was varied. [H(2) (w,2)]norm is plotted versus w+ (model scale). Note that there is little difference between the curves in Figures 10 and 12. The purpose of taking the data in Fiillre 13 was to show the variation in the transfer function magnitude at constant w+ for different values of w)_ [H(2) (,w2)]norm is plotted versus w_, for w+ = 13.949 radians/sec. Unfortunately the quality of this data is very poor, for reasons whichl have not yet been determined. To investigate the effects of forward speed, the "single wave" tests (w_ 0) were repeated for different Froude numbers, again with the model restrained. Figures 14 through 17 show the results of these tests plotted against w+. Figure 14 corresponds to tests at Froude number Fn = 0.0R6 (model speed Um = 1.90 ft/sec). In Figure 15, Fn = 0.109 (rTm = 2.40 ft/sec). Figure 16 is a reproduction of the data in Figure 10, with Fn = 0.132 (Um = 2.90 ft/sec). In Figure 17, Fn = 0.155 (rJm = 3.40 ft/sec). A curve has been faired through the data in each of these plots, and all four faired lines are shown toqether in Figure 18. Figures 19 through 23 show the same data as Figures 14 through 1R, but this time [H(2) (w,w2) norm is plotted on a ship lenqth/wavelenqth scale. We note that plotting the data on this scale seems to collapse the curves (compare Figures 18 and 23), indicating that the maqnitude of the transfer

-25 -function is predominantly dependent upon the ship lenqth/wavelength parameter. There is clearly a speed dependence, but it is not severe over small changes in Froude number. In order to show that the transfer function itself is not dependent lupon the wave amplitude, tests were conducted with reduced and increased wave amplitudes. Figure 24 shows the results of amplitude variation tests for -_= 0 and varying w+. Figure 25 shows the data from amplitude variation tests for o_= 1.424 radians/sec and varying w+. In both plots "medium amplitude" data was measured in waves with amplitudes of approximately 0.8-1.0 inches. This is the amplitude level maintained in all other tests. The "medium amplitude" data in Figures 24 and 25 is in fact the same as that shown in Figures 10 and 12, respectively. It is reproduced here for purposes of comparison. "High amplitude" data was measured in waves of approximately 2.0-2.3 inches in amplitude, a 130% increase. Data from "low amplitude" tests was measured in waves with amplitudes of approximately 0.5-0.7 inches, a 35% reduction. Pigures 24 and 25 do not seem to support the statement that the transfer function is amplitude independent. We remark, however, that the results shown in these two plots are suspect. Since the non-linear excitation is small, there were inevitably some noise related problems in all of the excitation measurements. Waves of "medium" amplitude produced excitation signals which could be measured with reasonable accuracy. "Low" amplitude waves, however, produced signals which could not be analyzed with the same degree of confidence. In "high" amplitude waves, on the other hand, the model experienced a definite lonqitudinal impact as the bow entered each successive wave crest. This most likely caused a shudder in the support structure which would appear in the load cell signal as contaminant noise. In any event, the amplitude tests were repeated in the response experiments (in which the signal to noise problem was much

-26 -less severe) with a hiqh degree of success (see Fiqure 43). In order to investigate possible coupling effects between vertical rigidbody motions and the non-linear springing excitation, tests were conducted in for tests with w_ = 0, and w+ varying. Data from the restrained model test (Figure 10) is also plotted for purposes of comparison. Unlike the linear transfer function, which was shown by Troesch (1980) to be independent of heave and pitch, the non-linear transfer function appears to depend strongly upon vertical riqid-body motions. In Figure 27 the data in Figure 26 is plotted against the ship length to wavelength ratio. Figure 28 shows the results of tests with the model free to heave and pitch for o_= 1.424 radians/sec and w+ varying. The normalized transfer function is plotted versus w+. The quality of the data is poor for unknown reasons. (2) Fiqures 29 through 44: Experimental Data from Measurements of the Non-linear Springing Response Figures 29 throuqh 32 show plots of the normalized non-linear response transfer function as defined by equation (9). These correspond to the excitation data in Figures 10 through 13. The data was obtained in tests at Froude number Fn = 0.132 (Um = 2.90 ft/sec), with the model restrained in heave and pitch. The data shown in Figure 29 was obtained in "sinqle wave" tests, in which e_ = 0 and w+ varied. [H(2) (wl,2)]norm is plotted versus )+ (model R scale).

-27 -Fiqure 30 shows the same data, plotted versus ship lenqth/wavelength. The data in Fiqure 31 was obtained in tests in which w_ was held constant at w- = 1.424 radians/sec and w+ varied. [H(2) (l,w12)]norm is R plotted aqainst w+ (model scale). Note that there is no significant difference between the curves in Figures 29 and 31. Figure 32 shows the results of tests conducted with w+ = 13.949 radians/sec and varying w. [H(2) (wl,2)]norm is plotted versus w. R As was the case in the corresponding excitation tests (Figure 13), the data quality is poor for unknown reasons. Speed variation tests equivalent to those conducted in the excitation measurements (Fiqures 14 through 23) were run to determine the effects of forward speed on the non-linear response. Figures 33, 34, 35, and 36 show the results for Froude numbers Fn = 0.086, 0.109, 0.132, and 0.155 respectively, with w_ = 0 in all cases. [H(2) (Wl,2)]norm is plotted against w+, and R faired lines are shown in each plot. The four faired curves are replotted for comparison in Figure 37. The data from Figures 33 through 36 is plotted versus ship lenqth/wavelength in Figures 38 through 41. Faired lines are shown on a ship length/wavelenqth scale in Figure 42. Since the dynamic non-linear response is dependent primarily upon the proximity of the frequency sum w+ to the sprinqing natural frequency (15.08 radians/sec for the model), plotting the response data on a ship lenath/wavelength scale does not collapse the data as was the case with the excitation data.

-28 -Amplitude variation tests were conducted to show that the normalized response transfer function is independent of the wave amplitude. Figure 43 shows the data obtained in tests with o_ = 0 and varying w+. "Low," "medium," and "hiqh" amplitudes are defined in the same manner as in the excitation measurements. Since signal quality was not a problem in the response tests, the quality of the data for "high" and "low" amplitulde waves is much better than that for the corresponding excitation tests. Fiqure 43 supports the statement that the normalized response transfer function tends to be independent of wave amplitlude. Equation (9) implies that the non-linear response at the srum frequTency 01 + 02 = w+ is proportional to the product. nllr12[HR(2) (W1Wl2) norm. If [HR(2) (W,2)]norm is itself independent of the wave amplituldes nrl and n2, the non-linear response is wave amplitude dependent only throuqh the quantity n1nl2 and therefore is the result of non-linear effects are "quadratic" in wave amplitude. In the special "sinqle wave" case, in which (1 = "2, this quantity becomes n2, and the response obeys an "amplitude squared" law. To investigate possible coupling effects between vertical rigid-body motions and non-linear sprinqinq response, tests were conducted in which the model was allowed to heave and pitch. Figure 44 shows the results for _ = 0, with w+ varyinq. Data from the restrained model test (Fiqure 29) is also shown for comparison. The non-linear response appears to be stronqly dependent upon vertical riqid-body motions. (3) Fiqures 45 throuqh 48: Experimental Data from Measurements of the Linear Springinq Excitation The linear excitation data, obtained from "single wave" tests (w_= 0) is

-29 -shown in Fiqures 45 throuqh 48 as [H(1)()) Inorm versus ship lenqth/wavelength. This data compares well with that qiven by Troesch (1980). Figure 45 shows the data obtained in tests at Froude number Fn = 0.132 (Um = 2.90 ft/sec) with the model restrained in heave and pitch. Figure 46 is a plot of linear excitation data from the amplitude variation tests. "Low," "mediulm," and "high" amplitudes are defined as before. Clearly, the linear transfer function seems to be independent of wave amplitude. Fiqure 47 shows the data obtained in speed variation tests, in which the linear transfer function was determined for Froude numbers Fn = 0.086, n.109, 0.132, 0.155. This figure shows that small to moderate changes in forward speed have minor effects on the linear excitation, at least in the range of ship lenqth/wavelength shown. Figure 48 is a comparison which shows the effects of heave and pitch on the linear springing excitation. Data from Figure 45 (model restrained) is shown along with data obtained in tests in which the model was allowed to heave and pitch. The effects of vertical rigid-body motion are noticeable but tend to be small. Comparison of Non-Linear Excitation and Response Given the definitions (8), (9), and (10), we can show that the normalized excitation and response transfer functions are related by [H(2) (W1,2)1norm 22 [H(2)(Dl,{i2)]norm = -— 1 -.2 l+ R 1/21 w02(a22 + A22)] L1 - (_)2) + (2-) j (11)

-30 -(See equation (16) in Section IV.) This expression holds under the assumption that the second order excitation is the only source of non-linear effects. In order to investiqate the validity of this hypothesis, the normalized response transfer function [H(2) (Wa,2)]norm was calculated from equation R (11) usinq experimentally measured valules of the normalizedl excitation transfer function [H(2)(wl,w2)]norm as shown in equation (8). The calculated values were then compared to the measturerd values from the non-linear response tes ts. Fiqure 49 shows this comparison for w_ = 0 and Fn = 0.132 with the model restrained in heave and pitch. Fiqure 50 illustrates the analoqous comparison for the case in which the model is free to heave and pitch. Note that in the reqion near the resonance the measured response is in both cases lower than that predicted by equation (8). This indicates that there exists siqnificant non-linear effects which are associatel with the response, possibly influencinq the motion throuqh non-linear dampinq. Conclusions The followinq conclusions can be drawn from Fiqures 10 throuqh 50: 1. The non-linear excitation is depdedent upon both ship speed and wavelenqth. See Fiqure 23. 2. The non-linear excitation is only weakly depededent upon ship speed. For small chanqes in Froude number the speed dependence can be neqlected. See Fiqure 23.

-31 -3. Both the non-linear excitation and non-linear response show only a slight dependence upon the encounter frequency difference parameter o_, at least for w_ < 1.424 radians/sec model scale. In some region near _ = 0 both can be treated as a function only of the frequency sum parameter +. Compare excitation plots in Figures 10 and 12, and compare response plots in Figures 29 and 31. 4. The non-linear effects are "quadratic" in wave amplitude. See Figure 43 and the correspondinq discussion. 5. Both the non-linear excitation and non-linear response are influenced by heave and pitch motions. See Figures 26 and 44. 6. Although second order excitation is the dominant source of non-linear effects, it appears that response dependent non-linearities are also significant. See Figures 49 and 50. 7. The linear excitation data agrees well with that obtained by Troesch (1980). 8. The linear excitation is independent of speed in the range of wavelengths tested. See Figure 46. 9. The linear transfer function is independent of wave amplitude. See Figure 47. 10. The linear excitation is not greatly influenced by heave and pitch motions. See Figure 48.

0.10 Zo. e + 0.06 U 10. 12. 140 16. 18. 20. F ++ + Lu + +g ^ +I Z; + + o + +^ + + 0.02 + —4~ — 0. + 10. 12. 14. 16. 18. 20. ENCOUNTER FREQENCY SUM CRAD/SECJ FIGURE 10: NORMALIZED NON-LINEAR EXCITATION VS. ENCOUNTER FREQUENCY SUM/ o"+ (MODEL SCALE). _- = 0 ~ FN = 0.132. MODEL RESTRAINED.

0. 1. 0. Z + FI< 0.06... X + LJ ++ + S I+ ++ ++ + o ++ + + + z 0. 2 ---— * + --- - 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 SHIP LENGTH / WAVE LENGTH FIGURE 11: NORMALIZED NON-LINEAR EXCITATION VS SHIP LENGTH/WAVELENGTH. w- = o. FN =.132. MODEL RESTRAINED

0.10[ H O+ L) 0 Lu I — 0... O~~~~~~ + X + ENCOUNTER FREQENCY SUM CRAD/SEC) __ -I - GURE 12: NORMALZED NON NEAR EXCTATON VS z 0.02 10. 12. 14. 16. 18. 20. ENCOUNTER FREQENCY SUM CRAD/SEC3 FIGURE 12: NORMALIZED NON-LINEAR EXCITATION VS. ENCOUNTER FREQUENCY SUM, (+ (MODEL SCALE). - = 1.424 RAD/SEC. FN = 0.132. MODEL RESTRAINED.

0.20, Zo 0.15'=I-+-I +. I... JI- + + + 0 1 0.0._,. 4..., ENCOUNTER FREQUENCY DIFFERENCE CRAD/SECD FIGURE 13: NORMALIZED NON-LINEAR EXCITATION VS. ENCOUNTER FREQUENCY DIFFERENCE, M- (MODEL SCALE). + 13-949 RAD/SEC. FN = 0.132. MODEL RESTRAINED Ld + 0~ 1. 2. 3. 4. 5. 8. ENCOUNTER FREQUENCY DIFFERENCE [RAD/SECD FIGURE 13' NORMALIZED NON-LINEAR EXCITATION vS. ENCOUNTER FREQUENCY DIFFERENCE, ~- (MODEL SCALE)~ (.o+ = 13.949 RAD/SEC. FN = 0.132. MODEL RESTRAINED.

0.1 0.0 Z 0 I — " 0. Lij ENCOUNTER FREQENCY SUM C RAD/SEC) FI GURE 14: SPEED DEPENDENCE TEST, NORMALIZED NON-LINEAR W- = 0. FN = 0.086. MODEL RESTRAINED10. 12. 14. 16. 18. 20. ENCOUNTER FREQENCY SUM CRAD/SEC) FIGURE 14: SPEED DEPENDENCE TEST, NORMALIZED NON-LINEAR EXCITATION VS- ENCOUNTER FREQUENCY SUM, o+ (MODEL SCALE)- = O- FN = 0.086. MODEL RESTRAINED

0.10 0. —, 10. 14 1 20O FIGURE 15: SPEED DEPENDENCE TEST NORMALIZED NON- I NEAR 0.0- ------— 0- FN — - MODEL RES-T-RA I NEDz < - z 0.02 0 O 10. 12. 14. 16. 18. 20. ENCOUNTER FREQENCY SUM CRAD/SEC) FIGURE 15: SPEED DEPENDENCE TEST, NORMALIZED NON-LINEAR EXCITATION VSo ENCOUNTER FREQUENCY SUM, L+ (MODEL SCALE). - = 0 FN = 0.109. MODEL RESTRAI NED.

0.10 0z 08 -o.c + 0 ui C-) 1 12. 14. 16- 18. 20. N o0.0 X ++ U + O\ O 10. 12. 14. 16. 18. 20. ENCOUNTER FREQENCY SUM CRAD/SEC) FIGURE 16: SPEED DEPENDENCE TEST, NORMALIZED NON-LINEAR EXCITATION VS. ENCOUNTER FREQUENCY SUM, W+ (MODEL SCALE). N- = 0 FN = 0.132. MODEL RESTRAINED.

0.10 0.08 z 0 I0. 06 X ILl 10. 12 14 16 18 20 ENCOUNTER FREQENCY SUM CRAD/SECJ FIGURE 17: SPEED DEPENDENCE TEST, NORMALIZED NON-LINEAR I 004.3 0.0 EXCITATION VS. ENCOUNTER FREQUENCY SUM, W+ (MODEL SCALE). - = 0. FN = 0.155. MODEL RESTRAINED.

0.1 II. F ____ n = 0.086 -- _ Fn = 0.109 Fn = 0.132 Fn = 0.155 Z0.0I I- 0.0 >< I.l N — 0.04 x........ ____..___ 0 0. a o / \ 1 \,- I --- —-,,< \ \ I,/ x ' 1 00 / ~/.o I N. / I I FIGURE 18: SPEED DEPENDENCE TEST, NORMALIZED rON-LINEAR EXCITATION VS. ENCOUNTER FREQUENCY SUM, w+ (MODEL SCALE).,- = 0. FN = 0.086, 0.109, 0.132, 0.155. MODEL RESTRAINED.

0.10 Z < 0.06 U x LJ FD N 0.04 _J III 0 O Z 0.0. 0.0 - ------ 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 SHIP LENGTH / WAVE LENGTH FIGURE 19: SPEED DEPENDENCE TEST, NORMALIZED NON-LINEAR EXCITATION VS. SHIP LENGTH/WAVELENGTH. w- = 0. FN = 0.086. MODEL RESTRAINED.

0.10 0 t0 L — I — i. 0 2 0.0 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 SHIP LENGTH / WAVE LENGTH FIGURE 20: SPEED DEPENDENCE TEST, NORMALIZED NON-LINEAR EXCITATION VS. SHIP LENGTH/WAVELENGTH. m- = 0. FN = O.lnq. MODEL RESTRAINED.

0.1 0.08. +..... Ii: 00 L-J 0.06 -I I L / Z 0.024 Iv + 0.0 02- 0.0 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 SHIP LENGTH / WAVE LENGTH FIGURE 21: SPEED DEPENDENCE TEST, NORMALIZED NON-LINEAR EXCITATION VS- SHIP LENGTH/WAVELENGTH. D- = 0. FN = 0.132. MODEL RESTRAINED.

0.10 0.08 z 0 idNfr-I 0 06 -QL LU j \ /' 33 zo 3 0. 02. 0.0 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 SHIP LENGTH / WAVE LENGTH FIGURE 22: SPEED DEPENDENCE TEST, NORMALIZED NON-LINEAR EXCITATION VS. SHIP LENGTH/WAVELENGTH. '- = O. FN = 0.155. MODEL RESTRAINED

0,.. I 0. —1 Fn = 0.086. —Fn = 0.109 Fn 0.132 Fn = 0.155 '-z 00 -I J — 0, SHIP LENGTH / WVE LENGTH 0.109, 04132, 0155. MODEL RESTRAINED/ i. 0.0 0 --- —--------------------------- 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 SHIP LENGTH / WAVE LENGTH FIGURE 23: SPEED DEPENDENCE TEST, NORMALIZED NON-LINEAR EXCITATION VS- SHIP LENGTH/WAVELENGTH- w_ = 0. FN = 0.096i 0.109, 0.132, 0.155. MODEL RESTRAI NED

0.10 L Low Amplit ude + Medium Ampli tude H High Amplitude x +++ N O. _ _ + L z ~.08 ----? ---------- X ~ f + u + + N 0. 04 0. 0 L L+ L - < LL + r + -+ o WHI+ L + + z 0.02 -..+ ++H+,i + -' 0.0 'I' 10. 12. 14. 16. 18. 20. ENCOUNTER FREQENCY SUM CRAD/SEC) FIGURE 24: AMPLITUDE DEPENDENCE TESTS, NORMALIZED NON-LINEAR EXCITATION VS. ENCOUNTER FREQUENCY SUM, w+ (MODEL SCALE). w-= 0. FN = 0.132. Low, MEDIUM, AND HIGH AMPLITUDES. MODEL RESTRAINED.

0.1 L Low Amplitude + Medium Amplitude H High Amplitude 0 0.+ L H LU L H 0 H N 0. 04+1 1 0.0 ___UR H 2+ AMLIUD DEC + + + z HH L + +H H 0. 0 0_4_ _10. 12. 14. 16. 18. 20. ENCOUNTER FREQENCY SUM CRAD/SEC) FIGURE 25: AMPLITUDE DEPENDENCE TESTS, NORMALIZED NON-LINEAR EXCITATION VS. ENCOUNTER FREQUENCY SUM, W+ (MODEL SCALE). = 1.424. FN = 0.132. Low, MEDIUM, AND HIGH AMPLITUDESMODEL RESTRAINED. MODEL RESTR&[ NED

0.1 + Model Restrained 1 Model Free 0 08 I< 5 0.06-11 x ++ 11 L J+ + 1 1 uI + 0. 0 1J^ 12 14. 1 61 1+ | ^ll0^,^ 1 + 0.02 --- —- + 0. 0 10. 12. 14. 16. 18. 20. ENCOUNTER FREQENCY SUM CRAD/SEC) FIGURE 26: NORMALIZED NON-LINEAR EXCITATION VS. ENCOUNTER FREQUENCY SUM, (1+ (MODEL SCALE). w- = 0. FN = 0.132. MODEL RESTRAINED AND MODEL FREE TO HEAVE AND PITCH.

0.10 + Model Restrained 1 Model Free 0 Z + I0. 1. 1 1.4 1.6 18 2 22 2.4 26 x +.. 11 m 1+ + i 1 Q, ' SH I P LENGTH / AVE LENGTH FIGURE 27: NORMALIZED NON-LINEAR EXCITATION VS. SHIP LENGTH/ WAVELENGTH. FN = 0.132. MODEL RESTRAINED AND MODEL FREE TO HEAVE AND PITCH.

0.1 + Model Res tra ined 2 Model Free 0 i0.06. X x + L 2 +- 22 < 2 04+2_____- ++ +___ 5= + + + +2+ + + + (~ + ~ + + + 0 L + + + 1 I I + I + 0. 0 10. 12. 14 16 18. 20. ENCOUNTER FREQENCY SUM CRAD/SEC) FIGURE 28: NORMALIZED NON-LINEAR EXCITATION VS. ENCOUNTER FREQUENCY SUM,.+ (MODEL SCALE). - = 1.424. FN = 0.132. MODEL RESTRAINED AND MODEL FREE TO HEAVE AND PITCH.

0.8 -LJ 0 L 0.6 uL 0o 0.42; + Q2 + -+ ++ +++++ + ++++++ 0.0 10. 12. 14. 16. 18. 20. ENCOUNTER FREQUENCY SUM CRAD/SEC) FIGURE 29: NORMALIZED NON-LINEAR RESPONSE VS. ENCOUNTER FREQUENCY SUM, "+ (MODEL SCALE) w- = 0. FN 0.132. MODEL RESTRAINED.

0. uJ (n z 0 LLJ a-N ( 0O. 7_ 0.4- 1- - - —.8., < ++ o + 0.2 + -t +++ + + -++ +0.+ +1~ +F++ + ++ I +-t+ + + ++ 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 SHIP LENGTH / WAVE LENGTH FIGURE 30: NORMALIZED NON-LINEAR RESPONSE VS. SHIP LENGTH/ WAVELENGTH. "- = 0. FN = 0.132. MODEL RESTRAINED.

1.0 CD ( 0. (fl LO 0 Q 0 + 0 + + + + 0. 0 10. 12. 14. 16. 18. 20. ENCOUNTER FREQUENCY SUM CRAD/SEC) FIGURE 31: NORMALIZED NON-LINEAR RESPONSE VS. ENCOUNTER FREQUENCY SUM, w+ (MODEL SCALE). m_ = 1.424. FN = 0.132. MODEL RESTRAINED.

+ + u(0.15 + + z + L. i 0 1 o +~~~~~~~~~~ (t) + ~~~+ ++ (+ + O. 0 N t 0(:: + + + 0 0__ + _ z 0a 0. 0 ----- 0. 1. 2. 3k 4 5. 6. ENCOUNTER FREQUENCY DIFFERENCE CRAD/SEC) FIGURE 32: NORMALIZED NON-LINEAR RESPONSE vs. ENCOUNTER FREQUENCY DIFFERENCE, w- (MODEL SCALE). w+ = 13.949 RAD/SEC. FN = 0.132. MODEL RESTRAINED.

1.0 0.8 z 0 U_ 0.6 CJ ENCOUNTER FREQUENCY SUM CRAD/SEC) FIGURE 33: SPEED DEPENDENCE TEST, NORMALIZED NON-LINEAR n0.4 ________ 0:: r0 10. 12. 14. 16. 18. 20. ENCOUNTER FREQUENCY SUM CRAD/SEC3 FIGURE 33: SPEED DEPENDENCE TEST, NORMALIZED NON-LINEAR RESPONSE VS. ENCOUNTER FREQUENCY SUM, w+ (MODEL SCALE). - = 0. FN = 0.086. MODEL RESTRAINED.

1.0 0.8 -(0 z 0 0.6 0.2 0. 0 10. 12. 14. 16. 18. 20. ENCOUNTER FREQUENCY SUM CRAD/SECJ FIGURE 34: SPEED DEPENDENCE TEST, NORMALIZED NON-LINEAR RESPONSE VS. ENCOUNTER FREQUENCY SUM, w+ (MODEL SCALE). m- = 0. FN = 0.109. MODEL RESTRAINED

0.8 z 0 aUJ 05 N _o 0.4, 0 z 0.2 -0. 1 140 16 1 10. 12. 14. 16. 18. 20. ENCOUNTER FREQUENCY SUM CRAD/SEC) FIGURE 35: SPEED DEPENDENCE TEST, NORMALIZED NON-LINEAR RESPONSE VS. ENCOUNTER FREQUENCY SUM, W+ (MODEL SCALE)=- = 0. FN = 0.132. MODEL RESTRAINED

1.0 -0.86 -u ) ( ~M z a_ on 06; uJ N i 0.4 -__. I:: 0 z 0.2 0. 0 10. 12. 14. 16. 18. 20. ENCOUNTER FREQUENCY SUM CRAD/SEC) FIGURE 36: SPEED DEPENDENCE TEST, NORMALIZED NON-LINEAR RESPONSE VS. ENCOUNTER FREQUENCY SUM, W+ (MODEL SCALE)W- = 0. FN = 0.155. MODEL RESTRAINED

1.01.....1r ---- Fn = 0.086 F- =_ 0.109 Fn = 0.132 Fn = 0.155 0.8....._ ---LuO I. 0, Z) I 0 I L-0.6, I I, _ MOELRETRIN, / \ '. Lo,I \ 0.2- "?I — 0\ A 10. 12. 14. 16. 18. 20. ENCOUNTER FREQUENCY SUM CRAD/SEC) FIGURE 37: SPEED DEPENDENCE TEST, NORMALIZED NON-LINEAR RESPONSE VS. ENCOUNTER FREQUENCY SUM, (o+ (MODEL SCALE). W- = 0. FN = 0.086, 0.109, 0.132, 0.155. MODEL RESTRAINED.

1.0 0.8 UJ z 0 -" N o --- 0. - ------ C --- NED. — f1 0.02_ 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 SHIP LENGTH / WAVE LENGTH FIGURE 38: SPEED DEPENDENCE TEST, NORMALIZED NON-LINEAR RESPONSE VS- SHIP LENGTH/WAVELENGTH. 0- = 0. FN = 0.086. MODEL RESTRAINED.

1. 0 0.8......... — Lu () Z 0 Q_ LLJ 2= S (0 ED LiI N 0:: z 0. 0. ___'___2 A_ 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 SHIP LENGTH / WAVE LENGTH FIGURE 39: SPEED DEPENDENCE TEST, NORMALIZED NON-LINEAR RESPONSE VS. SHIP LENGTH/WAVELENGTH. W- = 0. FN = 0.109. MODEL RESTRAINED

1.i LO z 0 LO C, 0. LU N 0.2 0. 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 SHIP LENGTH / WAVE LENGTH FIGURE 40: SPEED DEPENDENCE TEST, NORMALIZED NON-LINEAR RESPONSE VS- SHIP LENGTH/WAVELENGTH. w- =0. FN = 0132. MODEL RESTRAINED

1.0 — l-l-l-l -lIl 0. 8_ uJ Cr) z 0 n11 uJ rY N '""0. z - Lg 0, 2 0. 0 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 SHIP LENGTH / WAVE LENGTH FIGURE 41: SPEED DEPENDENCE TEST, NORMALIZED NON-LINEAR RESPONSE VS. SHIP LENGTH/WAVELENGTH. * - = 0. FN = 0.155. MODEL RESTRAINED.

1.0.......... Fn 0.086.. = 0.109.Fn = 0.132 Fn =0.155 0.8 --- LJ I i z I o I I 0~ (n 0.6 Io I LJ I/ 0.4 --- — II jI0~ / 0. 0. 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 SHIP LENGTH / WAVE LENGTH FIGURE 42: SPEED DEPENDENCE TEST, NORMALIZED NON-LINEAR RESPONSE VS. SHIP LENGTH/WAVELENGTH. w- = 0. FN = n.086, 0.109, 0.132, 0.155. MODEL RESTRAINED.

1.00 -L Low Amplitude + Medium Ampli tude H High Amplitude 0.8 LU (n z 0 U L ECUT FREQUE Y L+ = HtLH H ~~~0.2~MODEL RESTRAINED. ++*++ ++++ 0.0. 10, 12. 14. 16. 18. 20. ENCOUNTER FREQUENCY SUM (RAD/SEC] FIGURE 143: AMPLITUDE DEPENDENCE TESTS, NORMALIZED NON-LINEAR RESPONSE VS. ENCOUNTER FREQUENCY SUM, w+ (MODEL SCALE). o~ = O. FN = 0-132. Low, MEDIUM, AND HIGH AMPLTUDESMODEL RESTRAINED.

1.C 1 ' + Model Restrained F F Model Free 0.8 F O0 z F 0a o 0. F LI. N _j 0. + +F 0 F + Z F F 0.2 F +.^:+ +F F++F+F. + ++++-++++ +++ 0.0.... 10. 12. 14. 16. 18. 20. ENCOUNTER FREQUENCY SUM CRAD/SEC) FIGURE 44: NORMALIZED NON-LINEAR RESPONSE VS. ENCOUNTER FREQUENCY SUM, to+ (MODEL SCALE).~;J- = 0. FN = 0.132. MODEL RESTRAINED AND MODEL FREE TO HEAVE AND PITCH.

0. + ++ 0.05 o + + 1- 4 H do +S — I^+ _, 01 }. + s U + E30.03 0 0+.0 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 SHIP LENGTH / WAVE LENGTH FIGURE 45: NORMALIZED LINEAR EXCITATION VS. SHIP LENGTH/ WAVELENGTH- FN = 0.132. MODEL RESTRAINED

0. L Low Amplitude.+ +L 4.. + Medium Ampli tude I H High Ampli tude O. - 0 +< 004 + IL, + 0 UJ LU 4' 0. 0,01- 0.0 ------- I --- SHIP LENGTH / WAVE LENGTH FIGURE 46: AMPLITUDE DEPENDENCE TEST, NORMALIZED LINEAR EXCITATION VS SHIP LENGTH/WAVELENGTH. FN = 0.132. Low, MEDIUM, AND HIGH AMPLITUDES. MODEL RESTRAINED.

I Fn = 0.086 '+[ 2 Fn = 0.109 +. + Fn = 0.132 c0.0 --------------------— 3 F = 0.155 0, z - 10 + 0 t FIGURE 47: SPEED DEPENDENCE TESTS, NORMALIZED LINEAR EXCITATION 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 SHIP LENGTH / WAVE LENGTH FIGURE 47: SPEED DEPENDENCE TESTS, NORMALIZED LINEAR EXCITATION VS SHIP LENGTH/WAVELENGTH. FN = 0.086, 0.109, 0.132, 0.155. MODEL RESTRAINED.

0., +- + Model Restrained ++ M o F Model Free 0.05 - ~ ------------ z o +<H.0 + r-+ + x o F rF FF 0N _WAVELENGTH- FN = 0-13- MODEL + 0.....0 --- —----------— TO HEAVE AND PITCH. FIGURE 48: NORMALIZED LINEAR EXCITATION VS. SHIP LENGTH WAVELENGTH. FN = 0.132. MODEL RESTRAINED AND MODEL FREE TO HEAVE AND PITCH.

2.0 * Measured Response 1 Calculated Response uJ 1.5 (n z 0 aQ_) CO LUJ Qr o 1.0 LU N -J 11 or it 0 0 1 *c 1 1 *1^ *NU^ ________^____ 0. 10. 12. 14. 16. 18. 20. ENCOUNTER FREQUENCY SUM CRAD/SEC.) FIGURE 49: COMPARISON OF CALCULATED AND MEASURED NON-LINEAR RESPONSE- MODEL RESTRAINED

24I * Measured Response 1 Calculated Response 0 r) o 1.10 FIGURE 50: COMPARISON OF CALCULA-K ED AND MEASURED NON-LINEAR 0 _ 1 1 Z 0.5 — 1 * 0. 10. 12. 14. 16. 18. 20. ENCOUNTER FREQUENCY SUM CRAD/SEC) FIGURE 50: COMPARISON OF CALCULATED AND MEASURED NON-LINEAR RESPONSE. MODEL FREE TO HEAVE AND PITCH

IV. CALCULATION OF THE NON-LINEAR SPRINGING EXCITATION SPECTRUM Presented in this section is a method for calculating the second order springing excitation through use of the experimental results given in section III. This non-linear excitation, given as a one-sided spectral density SE(2)+ versus encounter frequency, w, can be combined with the linear excitation spectrum SE(1)+ to give the total springing excitation spectrum, SE+, as a function of encounter frequency. The total excitation can in turn be used to calculate the total linear and nonlinear springing induced bending moment at midships. It is assumed that all the nonlinear behavior is in the excitation. As seen from Figures 44, 49 and 50 this is not an entirely valid assumption. Development of a Computational Formula As discussed in the introduction, assume that the major contribution to the non-linear springing behavior of a Great Lakes bulk carrier comes from the time dependent excitation term e2. Here e2 is the total normalized 2 noded springing excitation written as e2(t) = e2(l)(t) + e2(2)(t), (12) where e2(1) and e2(2) are the first and second order excitations respectively. If e2(t) has a sinusoidal time dependence of exp(iwt), where w is the frequency of encounter, then E2(1) is the amplitude of the excitation due to incident waves of frequency w and E2(2) is the amplitude of the excitation due to all pairs of waves whose sums are equal to w. The uncoupled differential equation describing the normalized springing response, q2, for a hinged ship is given from equation (3) as q2(a22+A22) + q2(b22+B22) + q2C22 = mo(l+1A/QF) + e2. (13) -73 -

-74 -The left hand side of equation (13) follows a linear relationship. For an explanation of the various coefficients see the comments following (3). It is only important here to note that mO = -Ks(1 + QA/QF)q2 (14) where m0 is the midship bendinq moment and Ks is the internal spring cons tant. Now define an internal spring constant, c22, in the Followinq manner: c22 = Ks(1 + -A/F)2 and also the damping ratio, D, as (h22+B22)w)0 22 + B22 2(c22+C22) 2(22+C22)(a22+A22) where w0 is the natiiral frequency in springing. By using the above definitions and equation (14), the solution for the sprinqinq response to sinusoidal excitation is q2 =..r ~ (15) r2 (a2 2+A22) (1-w2/2 )+i 2Cr)-w /w0oJ ) 0 Here i is used as an imaqinary notation and is equal to V-1. By manipulatinq the above expressions, the bendina moment at midship can be shown to be [ (1- 2/2 ) +i 2CDW/WO] L -2 (a 2~A?2+ ) ) L + A/QF 0 0 Since e2(t) is shown by equation(12) to have a linear and non-linear component, mo(t) can also be expressed as mo(t) = m(1)(t) + m(2)(t) (17) 0 0 where m(1)(t) is the bending moment due to e(1)(t) and m(2)(t) is the 0 2 0 bendinq moment due to e(2)(t). If e(1)(t) represents the excitation due to 2 2 an incident wave of unit amplitude and frequency w, then m(l)(t) has an 0

-75 -amplitude that can be written as M(1)(we) and thus represents the first order 0 transfer function. In a similar fashion, let E(2) (1,W2) be the amplitude 2 of the second order excitation due to incident waves of unit amplitudes and encounter frequencies of w1 and W2. Then M(2) becomes the second order 0 transfer function. Strictly speaking, M(2) should be written as M(2) (1, 2) 0 0 where the frequencies w1 and w2 must satisfy the relation o1 + ' 2 = w If there exists a one sided sea spectrum of S+(w), then we can use equations (12) and (16) to calculate the total response spectrum composed of + both first and second order parts. Let SM(a) be the total midship bending moment response spectrum at some encounter frequency w. Then Neal (1974) or Yamanouchi (1974) have shown that S+(w) = S(1)+(W) + S(2)+(W) (18) MI M M where S(1)+(w) is the usual first order response spectrum given as M S(1)+(t) = IM(1)() 12 S+()) (19) M 0 and S(2)+(W) is the second order response spectrum qiven as M 00 S(2)+(w) = f dt IM(2)(-_.U)I)2 S+( |w-p) S+(Ipl). (20) M 0 If certain assumptions are made about the behavior of the inteqrand of equation (20), simplification is possible. First, equation (20) can be separated into two reqions of integration, -oo<p0 and (04p<. When p<0, the integral represents the contribution of the non-linear bending moment at wo due to two waves, one at a frequency I1 and another at a frequency wo+ | 1 It is unlikely that this contribution is significant and so the integral over the negative range of p will be considered small.

-76 -Followinq equation (16), we may write M(2)(_-,p) = -H(2)(W-P,p) * q( ) (21) 0 where C22 1 q(w)) 1 -.....22 1 ------ (22) (1- A2/W2) + i2D/woJ L W2(a22+A22) L1+A/2P o o It follows that S+()) = -q() j2[S(1)(w) + S(2)(W)l (23) M E E where S(1)(w) = IH(1)(w)12 S+()) E 00 and S(2)(w) = f dp |H(2)(a)-zp) 2 S+( I-wl)-)S+(- ). F O It is clear that S( 1 ) () and S(2)(w) are the first and second order E F spectral densities of the sprinqinq excitation. If we restrict our attention to frequencies whose sum equals w and if we note that the inteqrand in the above equation is symmetric about p = w/2, then S(2)(W) can be expressed alternatively as We/2 S(2)(w) = 2 f dplH(2)(w-_, ) 12 S+( l-ij)S+(( ) (24) E 0 An experimental investiqation of the linear component $(1)(w) was conducted E by Troesch (1980) in an earlier study. Here the subject of interest is the non-linear component S(2)(). E In order to calculate the non-linear excitation component over a ranqe of encounter frequency values w, the maqnitude of the second order transfer function H(2)(w-_,4) must be known for wide ranqes of its arquments p and 03-p. However, the experimental results suqqest an approximation which limits to a considerable extent the amount of information required to calculate S(2)(). E

-77 -Consider Figures 10 and 12 in section III. The first is a normalized plot of H(2)(w-i,p) as a function of w (model scale), with p specified by p=-. The second is a normalized plot of H(2) (-p,1p) versus, 2 with p specified by p = - - 0.712. A comparison of the two curves shows 2 that, for a qiven w, there is little chanqe in the magnitude of the transfer function as p varies from - - 0.712 to -. This indicates that the 2 2 magnitude of H(2)(w-_,1) is essentially constant for a ranqe of p near = -, i.e. over the high end of the interval of integration in equation 2 (24). For the typical single-peaked narrow band wave spectrum, the product S+(p)S+(w-p) will be largest for values of p near p =-. The siqnifi2 cant contribution to the inteqral in equation (24) will therefore come from a reqion in which H(2) (o-i,p) varies only slightly. For this reason the following approximation can be made w/2 S(2)(w) I= H(2)(w/2,w/2) 2 ~ 2 f dVS+(w-u)S+(U) (25) E 0 in which H(2) ( _w-U,) has been assumed to be constant over the entire interval of integration, taking its value at p = - 2 Since the encounter frequency A is a function of Froude number Fn and ship length to wavelength ratio L/X we can write H(2)(- -) = H(2)(L/XL/X,Fn) (26) 2 2 Now consider Figure 23. This is a plot of the normalized excitation transfer function versus L/A with Froude number as a parameter. It is clear that H(2) varies slowly over the broad range of Froude numbers tested. It is therefore reasonable to neglect the speed dependence over small changes of Froude number, or H(2)(L/X,L/X,Fn) - H(2)(L/X,L/X). (27)

-78 -Equation (25) can then be written 2/2 s(2)(w) IH(2)(L/,L/X)I2 2 f d2S+( I-P)S+(p). (2R) E 0 For purposes of numerical computation equation (22) can be rewritten in the discrete frequency form of n-1 S(2)(nAw) =|IH(2)(L/X,L/X) 2 I Aw S+(nAw-mAw)S+(mAw) (29) E m-O m=0 in which Aw is some selected increment of encounter frequency, and X is 1 the wavelenqth correspondinq to the frequency of encounter w = -nAw. A 2 trapezoidal inteqration formula is implied. Fquation (29) is the basis for the method of calculation outlined below. Algorithm for Calculation of the Non-linear Excitation Spectrum in Head Seas The second order excitation component can be calculated by the followinc alqorithm. The spectral density of the incident sea is assumed to be given as a function of absolute frequency a. The result of the computations will be a set of values of the spectral density of non-linear sprinqinq excitation S(2)(w) for evenly spaced values of encounter frequency w E 1. Select an increment of encounter frequency Aw. Values of S(2) () will be calculated for frequencies = nAw. Smaller values of Aw will result in improved accuracy and an increased amount of computational work. 2. For each value of n calculate the absolute frequency an. -1 + /1 + (4JnAw/q) n = - (30) 2U1J/q The ship's speed Hr, the qravitational constant q, and Aw must be in consis tant units.

-79 -3. Obtain values of the wave spectral density function, S+(nA), for each n where S+(nAw) = S+(an)/(1+2anU/q) (31) 4. Calculate the excitation spectral density S(2)(nAw) for each n by the E following steps: a. Calculate In where n-1 In = E Aw S+(nAw-mAw)S+(mAw) (32) m=O b. Calculate L/Xn where L/Xn = L[-/q + Vq + 2UnAw12/8rIU2 (33) c. Enter Fiqure 16 with L/Xn and read the normalized transfer function (H(2))norm n d. Calculate H(2) n L H(2) = 2rpqBL(-)(H(2))nor T (34) n Xn n e. Calculate the spectral density of non-linear excitation S() (nAw) by S(2)(nAw) = [H(2)]2In (35) E n Example As an example, a non-linear bending moment spectrum is determined here for the wave spectrum shown in Fiqure 51 (taken from Ploeq (1971)). The calculations are carried out for a ship speed of 21.56 ft/sec (14.7 mph). The increment of encounter frequency is taken to be Aw = 0.1 radians/sec, and the effects of heave and pitch are neglected.

-80 -Table II shows the results of steps 2 and 3 of the computations. Tables III, IV, and V show calculations of In from step 4.a for n = 18, 19, and 20, respectively. Calculations of In for other n values are not shown. Table VI shows calculations of S(2)(nAw) for each n. The parameter L/Xn E is found usinq equation (33), and is then used to determine H norm from n Fiqure 16. The transfer function H(2) is then computed from equation (34). n Finally, equation (35) is used to find S(2)(nA). F The linear excitation sperctrum S(l)~(n~w) is calculatefd usinq experime-ntal data qiven by Troesch (1980). The details of the computations are not shown here. Once the linear and non-linear excitation spectral components S(l)+(nAw) E and S(2)+(nAw) are known, the correspondinq components of the bendinq moment E spectrum can be calculated usinq equations (21), (22), and (23), with o -= 1.904 radians/sec, rD = 0.0154 1 -,A/,F = 2.025 C22 1 = 0.857. wo2(a22 + A22) Computed values of the non-linear bendinq moment spectrum S(2)+(nAw) are qiven M in Table VII. Table VIII shows the values obtained for the linear component S(1)+(nAw) M the non-linear component S(2)+(nAw), and the total bendinq moment spectrum M S+(nA03). Fiqure 52 shows the results of tie computations: a spectral density M

70. 60. ' u Co LO 50. 40. >. 3 0.__ Lu o 20. ~-4 10.0 0.1.52.02. 0. 0.0 0.5 1.0 1.5 2.0 2.5 ABSOLUTE FREQUENCY CRAD/SEC] FIGURE 51: INCIDENT WAVE SPECTRUM USED IN THE EXAMPLEMEASURED AT EAGLE HARBOR, 28 NOV. 1966. SIGNIFICANT WAVEHEIGHT = 18.29 FT. (TAKEN FROM PLOEG (1971)).

-82 -TABLE II: Transformation of tlhe Incident Wave Spectrum onto the Encounter Frequency Axis. n nAw an S+( n) S+(nAw) 0 0 0 0 0 1 0.1 0.9 0 0 2 0.2 0.18 0 0 3 0.3 0.26 0 0 4 0.4 0.33 0 0 5 0.5 0.40 0 0 6 0.6 0.46 0 0 7 0.7 0.52 3.5 2.06 8 0.8 0.58 17.5 9.85 9 0.9 0.63 28.0 15.18 10 1.0 0.69 56.5 29.35 11 1.1 0.74 57.5 28.87 12 1.2 0.79 33.0 16.03 13 1.3 0.83 18.5 8.76 14 1.4 0.88 9.8 4.50 15 1.5 0.93 9.9 4.41 16 1.6 0.97 9.3 4.04 17 1.7 1.01 8.0 3.40 18 1.8 1.05 6.3 2.62 19 1.9 1.10 6.2 2.51 20 2.0 1.14 4.5 1.78 21 2.1 1.17 3.0 1.17 22 2.2 1.21 2.0 0.76 (cont. )

-83 -n nAw n S+(an) S+(nAw) 23 2.3 1.25 2.0 0.75 24 2.4 1.29 2.1 0.77 25 2.5 1.32 2.1 0.76 26 2.6 1.36 1.7 0.60 27 2.7 1.40 1.3 0.45 28 2.8 1.43 1.6 0.55 29 2.9 1.46 1.7 0.57 30 3.0 1.50 1.1 0.37 31 3.1 1.53 1.0 0.33 32 3.2 1.56 1.0 0.32 33 3.3 1.60 0.8 0.25 34 3.4 1.63 0.8 0.25 35 3.5 1.66 0.7 0.22 36 3.6 1.69 0.6 0.18 37 3.7 1.72 0.6 0.18 38 3.8 1.75 0.6 0.18 39 3.9 1.78 0.6 0.18 40 4.0 1.81 0.6 0.18 41 4.1 1.84 0.6 0.17 42 4.2 1.87 0.5 0.14 43 4.3 1.89 0.5 0.14 44 4.4 1.92 0.4 0.11 45 4.5 1.95 0.3 0.08 46 4.6 1.98 0.2 0.05 47 4.7 2.01 0 0

-84 -TABLE III: Calculation of In for n = 18 m n-m S+(mAw) S+(nA)-mAw) S+(mAw)S+(nA(wj-mAw) 0 18 0 2.62 0 1 17 0 3.40 0 2 16 0 4.04 0 3 15 0 4,41 0 4 14 0 4.50 0 5 13 0 8.76 0 6 12 0 16.03 0 7 11 2.06 28.87 59.47 8 10 9.85 29.35 289.10 9 9 15.18 15.18 230.43 10 8 29.35 9.85 289.10 11 7 28.87 2.06 59.47 12 6 16.03 0 0 13 5 8.76 0 0 14 4 4.50 0 0 15 3 4.41 0 0 16 2 4.04 0 0 17 1 3.40 0 0 2 = 927.57 I18 = Aw x = 92.8

-85 -TABLE IV: Calculation of In for n = 19 m n-m S+(mAw) S+(nAw-mAw ) S+(mAw)S+(nAw-mAw) 0 19 0 2.51 0 1 18 0 2.62 0 2 17 0 3.40 0 3 16 0 4.04 0 4 15 0 4.41 0 5 14 0 4.50 0 6 13 0 8.76 0 7 12 2.06 16.03 33.02 8 11 9.85 28.87 284.37 9 10 15.18 29.35 445.53 10 9 29.35 15.18 445.53 11 8 28.87 9.85 284.37 12 7 16.03 2.06 33.02 13 6 8.76 0 0 14 5 4.50 0 0 15 4 4.41 0 0 16 3 4.04 0 0 17 2 3.40 0 0 18 1 2.62 0 0 = 1525.84 I19 = Aw x =- 152.6

-86 -TABLE V: Calculation of In for n = 20 m n-m S+(mAw) S+(niAw-mAw) S+(mAw)S+(nAw-mAw) 0 20 0 1.78 0 1 19 0 2.51 0 2 18 0 2.62 0 3 17 0 3.40 0 4 16 0 4.04 0 5 15 0 4.41 0 6 14 0 4.50 0 7 13 2.06 8.76 18.05 8 12 9.85 16.03 157.90 9 11 15.18 28.87 438.25 10 10 29.35 29.35 861.42 11 9 28.87 15.18 438.25 12 8 16.03 9.85 157.90 13 7 8.76 2.06 18.05 14 6 4.50 0 0 15 5 4.41 0 0 16 4 4.04 0 0 17 3 3.40 0 0 18 2 2.62 0 0 19 1 2.51 0 0 = 2089.82 I20 = Aw x I = 209.0

-87 -TABLE VI: Calculation of Non-linear Excitation from Eagle Harbor Spectrum (Model Restrained) n L/Xn (Hn(2))norm Hn(2) In SE(2)+(nAw) 1 0.012 0 0 2 0.044 0 0 3 0.093 0 0 4 0.158 0 0 5 0.235 0 0 6 0.324 0 O 7 0.424 0 0 8 0.532 0 0 9 0.649 0 0 10 0.773 0 0 11 0.904 0 0 12 1.042 0 0 13 1.185 0 0 14 1.334 0.036 1.966 x 106 0.4 1.546 x 1012 15 1.488 0.022 1.340 x 106 4.1 7.362 x 1012 16 1.647 0.022 1.483 x 106 16.0 3.519 x 1012 17 1.810 0.030 2.223 x 106 42.0 2.076 x 1014 18 1.977 0.024 1.943 x 106 92.8 3.503 x 1014 19 2.148 0.025 1.319 x 106 152.6 2.655 x 1014 20 2.323 0.025 2.378 x 106 209.0 1.182 x 1015 21 2.502 0.020 2.049 x 106 237.3 9.963 x 1014

TABLE VII: Calculation of Non-linear Rending Moment Spectrum n SE (2)+(nA) q(nAw) SM(2)+(nAw) SM(2)+(nA)) (ft-lbs) 2sec (ft-Im) 2sec 1 0 0 0 2 0 0 0 3 0 0 0 4 00 0 5 0 0 0 6 0 0 0 7 0 0 0 8 0 0 0 9 0 0 0 10 0 0 0 11 0 0 0 12 0 0 0 13 0 0 0 14 1.546 x 1012 0.920 1.309 x 1012 2.609 x 105 15 7.362 x 1012 1.113 9.102 x 1012 1.818 x 106 16 3.519 x 1013 1.434 7.236 x 1013 1.442 x 107 17 2.076 x 1014 2.067 8.R70 x 1014 1.768 x 108 18 3.503 x 1014 3.839 5.163 x 1015 1.029 x 109 19 2.655 x 1014 13.636 4.937 x 1016 9.839 x 109 20 1.182 x 1015 3.905 1.802 x 1016 3.591 x 109 21 9.963 x 1014 1.930 3.711 x 1015 7.396 x 108

-89 -TABLE VIII: Calculation of Combined Linear and Non-linear Bending Moment Spectrum n SM( )+(nAw) SM(2)+(nAw) SM+(nAw) (ft-LT)2sec (ft-LT)2sec (ft-L) 2sec 1 0 0 0 2 00 0 3 00 0 4 0 0 0 5 0 0 0 6 0 0 0 7 8 1.639 x 1010 0 1.639 x 1010 9 1.659 x 109 0 1.659 x 109 10 1.393 x 1010 1.393 x 1010 11 8.147 x 1090 8.147 x 109 12 3.528 x 109 0 3.528 x 109 13 8.367 x 1090 8.367 x 109 14 1.041 x 109 2.609 x 105 1.042 x 109 15 3.378 x 1091.818 x 106 3.380 x 109 16 5.083 x 109 1.442 x 107 5.098 x 109 17 5.162 x 109 1.768 x 108 5.339 x 109 18 2.491 x 1010 1.029 x 109 2.594 x 1010 19 8.004 x i010 9.839 x 109 8.988 x 1010 20 6.281 x 109 3.591 x 109 9.872 x 109 21 1.135 x 109 7.396 x 108 1.875 x 109

-qocurve of the combined linear and non-linear springinq induced bendinq moment as a function of encounter frequency. Calculations which take into account the effects of heave and pitch can he undertaken in a similar manner by using Fiqure 26 instead of Fiqure 16 in step 4.c. The resulting bending moment spectrum is shown in Fiqure 53. In this example, when the effects of vertical riqid-body motion are neqlected, the linear component of the combined bending moment spectrum accounts for 91% of the peak value, and the non-linear component contributes the remaining 9%. Of the area under the curve in the region near resonance (i.e., in the frequency range between 1.7 and 2.1 radians/sec), linear analysis accounts for 86%, and 14% is due to non-linear effects. When heave and pitch effects are included, linear analysis accounts for only 70% of the peak spectral value, and the non-linear spectral component contributes 30%. The linear component accounts for only 76% of the area under the "combined" curve in the frequency ranae between 1.7 and 2.1 radians/sec, and the remaining 24% arises from non-linear effects. It must be noted that the calculations leading from the springinq excitation spectra to the bending moment spectra shown in Figures 50 and 51 are based upon the hypothesis that the second order excitation is the only source of nonlinear effects. The use of equation (23) is correct only if this assumption holds. However, as discussed in section III, comparisons of experimentally measured excitation and response indicate that the assumption is cquestionable at frequencies near resonance, and will result in over-prediction of the bending moment maqnitudes at frequencies near the natural springing frequency. See, for example, Fiqures 49 and 50.

14. LU ----- - Linear Bending Moment S( 1)+ (n M 04 12. _ --------- Non-Linear Bendinq Moment S(2)+ tf)~~~~~~~~~(n M F ~ ----— 0. --- Combined Bendinq Moment S+ Z M 17 I I o ---IL C -~ 1. 0. / x 6. z,I LJu o 4. —ICDEFFECTS OF HEAVE AND PITCH NEGLECTED LI 2..j _ LLJi / \ 0.0 0.5 1.0 1.5 2.0 2.5 ENCOUNTER FREQUENCY7 RAD/SEC FIGURE 52: BENDING MOMENT SPECTRUM CALCULATED IN THE EXAMPLEEFFECTS OF HEAVE AND PITCH NEGLECTED

14. l l I U (a _Linear Bending Moment S ) M 04 12. Non-Linear Bending Moment S(2)+ M Z _- - Combined Bending Moment S+ o M 10. U^ I I o~8.I 10................1........ _ / 00 1 x 6 z I o 4. —1 --- —----- LU 0., 0.0 0.5 1.0 1.5 2.0 2.5 ENCOUNTER FREQUENCY7 RAD/SEC FIGURE 53: BENDING MOMENT SPECTRUM CALCULATED IN THE EXAMPLE. EFFECTS OF HEAVE AND PITCH INCLUDED*

V. RECOMMENDATIONS FOR FUTURE RESEARCH The results presented in this work indicate the siqnificance of nonlinear springinq. However, various questions raised by the experiments have vet to be answered. As a consequence, the followinq areas are recommended for future research: - Theoretical calculations should be performed to compare with experiment. The most straiqhtforward approach would be to try and model the nonlinear excitation on a body fixed in waves. This would eliminate the effect of vessel motions and simplify considerably the proposed boundary value problem. - Experiments and theory should be used to explain why the excitation for the restrained and free conditions were different. See, for example Fiqures 26 and 27. (The same type of behavior was also demonstrated in the response tests as shown in Fiqure 44.) It is clear, that even with small vessel motions, heave and pitch couple with nonlinear springinq to increase the qeneralized sprinqinq excitation. - The off diaqonal behavior, characterized by chanqinq w_ and holding w+ fixed, should be explored. The results as shown in Fiqure 13 exhibit a larqe amount of scatter making it difficult to identify any significant trends. The study of the off diaqonal term can be done using either theory or experiments. - The assumption that the reactive hydrodynamic forces, A2j the sprinqing added mass and B2j the sprinqinq dampinq, follow a linear relationship does not appear to be valid. Fiqures 49 and 50 clearly show this. The measured response is considerably reduced over the calculated response. -93 -

-94 -This suqqests an important nonlinear damping force. The research investigating this nonlinear dampinq should he both experimental and theoretical.

REFERENCES Bishop, R., et. al. (1977), "A Unified Dynamic Analysis of Ship Response to Waves," Roy. Inst. Nav. Arch. Sprinq Meetinq. Hoffman, D., and Van Hoof, R. (1976), "Experimental and Theoretical Evaluation of Sprinqinq on a Great Lakes Bulk Carrier," IJSCCr Renort CG-D-74. Jensen, J.J., and Pedersen, T. (1978), "Wave-Induced Bendinc Moments in Ships A Quadratic Theory," Roy. Inst. Nav. Arch. Jensen, J.J., and Pedersen, T. (1981), "Bendinc Moments and Shear Forces in Ships Sailinq in Irreqular Waves," JSR, Vol. 25, No. 4, pp. 243-251. Lonquet-Hiqqins, M.S. (1963), "The Effects of Nonlinearities on Statistical Distributions in the Theory of Sea Waves," JFM, 1, 1 pp. 459-48O. Maeda, H. (1980), "On the Theory of Coupled Ship Motions and Vibrations," Department of Naval Architecture and Marine Fnineering, The rniversity of Michiqan. Neal, E. (1974), "Second-Order Hydrodynamic Forces Due to Stochastic Excitation," Tenth Naval Hydrodynamics Symposium, Cambridge, Massachusetts. Ploeq, J. (1971), "Wave Climate Study - Great Lakes and nGulf of St. Tawrence," SNAME T&R Bulletin 2107. Troesch, A.W. (1980), "Ship Springinc - An Experimental and Theoretical Study," Department of Naval Architecture and Marine Enqineering, The University of Michiqan, Ann Arbor, Michiqan. Yamanouchi, Y. (1974), "Ship's Behavior on Ocean Waves as a Stochastic Process," International Symposium of the Dynamics of Marine Vehicles and Structures in Waves, London, 178-192. -95 -

APPENDIX: DATA ANALYSTS 'TECHNIQUE The method of analysis leading to the evaluation of the linear and nonlinear transfer functions is presented in this Appendix. Given first is a development of the underlying theory. Followinq is a discussion of the application of this theory to the model test situation. Preliminary Theory Let e2(t) denote the springinq excitation as a function of time. We wish to relate this excitation to the undisturbed incident wave elevation C(t), also measured as a function of time at some specified location which is fixed with respect to the moving ship. we can express the excitation in terms of the wave elevation by making use of a second order formula given by Yamanouchi (1974). 00 00 00 e2(t) = f h1(T))(t-T)dT + f/ h2(T1,T2)S(t-T1)C(t-T2)dTldT2 (A-1) -00 -00 -00 Here hl(-) is the impulse response function and h2(T1,T2) is the cuadratic impulse response function. Le t e2(t) = e(l)(t) + e(2)(t), (A-2) 2 2 where e(1) is the linear (first order) excitation 2 00 e(1) = / hl(T)C(t-T)dT (A-3) 2 -00 and e(2) is the non-linear (second order) excitation 2 -96 -

-97 -00 00 -00 -0 Suppose that the wave elevation C(t) can be written as a Fourier series 00 (t) = rn(nAw)einAt, (A-5) n=1 where Aw is some incremental frequency of encounter ancd the coefficients rn(nAw) are complex. Substitu-tion of the series (A-5) into the expression for the linear excitation (A-3) gives 00 00 e(1)(t) = f h (T). n(nAw)einAw(t-T)dT -1o n=1 00 00 -= n(nAw)einAwt f hl (T)e-inAWTdT n=1 -0 00 - n (nAw)H( 1 ) (nAw))ein At (A-6) n=1 where H( )(nAw) is the complex first order transfer function 00 -00 Substitution of the Fourier series (A-5) into the expression for the nonlinear excitation (A-4) gives 2 -_0 -|o |n=1 I m=1 00 00 00 00 = I n(nAw') f(mAw)ei(n+m)Awt / f h2(T1,T2,)e-inAmT1 e-imAuT2 dTldT2 n=1 m=1 -0 - -0

-98 -00 0o = ) nl(nAw)n(mAw)H( l)(nAw,mAw)ei(n+m)Awt, (A-R) n=l m=1 where H(2 )(nAw,mA) is the complex second order transfer function defined by H(2)(nAw,mAw) = f f h2(T1,T2)e-inTAWT eimAWT2 dTldT2. (A-9) -00 -00 The total springinq excitation can now be written to second order as 00 e2(t) = n (kAw)H(()(kAw)eikAwt k=1 00 00 + f n(nAw))n(mAw)H(2) (nAw,mA()ei (n+m) At. (A-10) n=1 m=1 We next express e2(t) as a Fourier series. 00 e2(t) = E2(kAw)eikAwt (A-11) k=1 Combininq the expressions (A-10) and (A-11) we obtain 00 00 X E2(kAw)eikAt = y n(kAw)H(1)(kAw)eikAwt k=l k=1 00 00 + y n(nAw)n(mAw))H(2) (nA,,TnAw)ei(n+m)Awt. (A-12) n=1 m=1 The double sum can be re-ordered 00 00 y E nq(nA)q(mAL)H(2) (nAL,mA)ei (n+m) At n=1 m=1 oo k-1 = C n(nAw)fl(kA-nAw)H (2) (nAk,k A-nA()e)e ikAwt (A-13) k=2 n=l

-99 -to give 00 00 E E2(kAw)eikAwt =. n(kAw)H(1) (kAw)eikAwt k=l k=l o k-1 +. n(nAw)n(kAw-nAw)H(2)(nAw,kAw-nAw)eikAwt (A-14) k=2 n=l or 00 E E2(kAw)eikAwt = n(Aw)H(1)(Aw)eiAWt k=l 0o - k-1 + k= [n(kA)H(l)(kAw) + k n(nAw)n(kAw-nAw)H(2)(nAw,kAw-nAw) eikAwt. k=2 _ n=1 (A-15) From this it follows that E2(kAW) = r(kAw)H(1)(kAw), k=1 k-1 -- (kAw)H(l)(kAw) + n(nAw)n(kAw-nAw).(2)(nAw,kAw-nAw), k>2 n=1 (A-16) Defininq E()(kAw) = n(kAw)H(1)(kAw) (A-17) 2 and E(2)(kAw) = 0 k=1 2 k-1 = ( n(nAw)n(kAw-nAw)H(2)(nAw,kAw-nAw), k)2, (A-1R) n=l we can decompose the Fourier coefficient F2(kAw) into its linear and nonlinear components. E2(kcA) = E() (kAw) + E(2)(kAw) (A-19) 2 2

-1 00 -Experimental Determination of the Linear and Non-Linear Transfer Functions The primary purpose of the experiments is the determination of the nonlinear transfer function H(2)(nAw,kAw-nAw) in equation (A-16). A secondary purpose is the evaluation of the linear transfer function H(1)(kAw) for comparison with Troesch (1980). Discussed here is the method by which these transfer functions are determined from recorded time histories of the incident wave elevation and the sprinqinq excitati.on. The same method is used to treat measurements of sprinqinq response. Consider a case in which the model encounters an incident wave system composed of two sinusoidal waves with encounter frequencies w1 = pAw and W2 = qAw. The Fourier series (A-5) representinq the wave elevation would then have only two non-zero terms. C(t) = r(pAw)eiPAwt + n(qA^)eicAwt (A-20) From equation (A-16) we find that the Fourier series (A-11) representinq the sprinqinq excitation would, in qeneral, have five non-zero terms. Two of these would represent the linear excitations caused by the two wave components. E2(pAw) = E(1)(pAw) = n(pAW)H(1)(pAw) (A-21) 2 E2(qAw) = E(l)(qAw) = n(qAW)H(2)(qAW) (A-22) 2 Two more would arise throuqh non-linear effects at "harmonics" of the wave frequencies, i.e. at w = 2pAw and t = 2qA. E2(2pAw) = E(2)(2pAw) = n(pAw)n(pAW))H(2)(pAW),pAw), (A-23) E2(2qAw) = E(2)(2qAo) = n(q^A)n(q4A))n(2) (qfAa,cA), (A-24) 2

-101 -the last would arise as a non-linear excitation at the sum of the wave frequencies, i.e. at w = pAw + qAw E2(pAw+qAw) = E(2)(pAw+qAw) = 2n(pAw)n(qAw)H(2) (pAwqA) (A-25) 2 If the wave elevation and springing excitation were recorded as functions of time, and the Fourier coefficients were calculated for each time history, we could use equations (A-21) throuqh (A-25) to obtain the following values of the linear and non-linear transfer functions: H(1)(pAw) =,, (A-26) nr (pAw) E2(PAw) H(1)(qAw) = -)-, (A-27) rn(rAw) E2 ( 2pA ) H(2)(pAw,qAw) = E22pA), (A-2R) (pAw) n(pAw) E2(2qAgw) H( 2) (qA w,qAw ).= -., (A-29) rT (qAw) rn (qAw) E2(pAwo+qAw) H(2)(pAw,qAw) = 2 -(pAw+Aw). (A-30) 2r(pAw))(qAw) Further data could he obtained throuqh this method hv testinq at several different combinations of wave encounter freauencies pAw and qAw. Unfortunately, this method is overly idealistic. It is not possible in practice to produce perfectly sinusoidal waves in the towing tank. If an attempt is made to qenerate an incident wave pattern composed of sinusoidal

-1 02 -waves at two discrete frequencies pAo and qAw, creating the situation analyzed above, the Fourier analysis of the surface elevation will show that nonzero Fourier components will be present at frequencies adjacent to w = pAw and w = qAow in addition to the components at the intended frequencies. As will be shown, this energy "leakaqe" is of no consequence in the evaluation of the linear transfer function, but must be taken into account in the determination of the non-linear transfer function. In order to simplify the discussion as much as possible we consider a case in which enerqy leakaqe is minimal. We will assume that an attempt has been made to generate a wave system with components at encounter frequencies w = pAw and w = qAo, but that Fourier analysis has shown that non-zero Fourier components were present at frequencies W = (p-1)AO,pAw),(p+1 )Aw and = (q-1 )Aw,qA, (q+1)Ao. From equation (A-16) we find that the linear excitation component at w = pAo is given by E2(pAW) = E(2)(pAw) = n (pAw)H(1)(pAO) (A-31) This is the same as equation (A-21), and equation (A-26) is aqain a valid expression for the linear transfer function at w = cAw. Likewise, equations (A-22) and (A-27) aqain apply at - = qAw * In qeneral, we find that E2(p'Ao) = E(l)(p'Aw) = n(p'Aw)H(1)(p'Aw) (A-32) 2 and E2(p' Aw) H( ) (p'IA ) = --- (A-33) n(p'Aw) are valid in treatinq the linear behavior for p' = p and p' = q

-103 -However, in this more realistic case, equation (A-23) is no lonqer a valid expression for the "harmonic" excitation at w = 2Ap. We have instead E2(2pAw) = E(2)(2pAw) = 2n [(p-1 )Aw n[ ( p+ )Aw]H(2) [ (p-1)Aw, (p+ ) Aw] 2 + n[pAw]n[pAw]H(2)[pAw,pAw], (A-34) and it is obvious that equation (A-28) cannot be used. In order to extract data from equation (A-34) we must employ the followinq approximation. We assume that for small chanqes in its arauments the transfer function H(2) is nearly constant. Then, if Aw is sufficiently small, E2(2pAw) = E(2)(2pAw) = [2n[(p-1)Aw] [p+1)Aw + n [pAw] [pAw11 H(2)[pAw,rAw] 2 2p-1 H(2)[pAw,pAw] - n(nAw)n(2DnA-nA), (A-35) n=1 and we obtain an approximate value for H(2) from E2(2DAw) H(2) [pApAw] (A-36) 2p-1 n ((nAw)n( 2pAw-nAw) n=l Similar treatment of the non-linear excitation components at frequencies w = 2qAw and o = pAw + qAw results in expressions analoqous to equation (A-36). We give the qeneral formula E2(P'Aw + q' A) H(2) [p'Ao,q'1A] -. (A-37) p'+q' y n(nAw)n(p ' Aw+q' Aw-nA) n=l "Harmonic" excitation is analyzed by using equations (A-35) and (A-36) with

-104 -p' = q' = p. "Sum frequency" excitation is treated with p' = p, q' = q As implied in the development, the approximations are valid only when the siqnificant wave enerqy is concentrated at encounter frequencies near the intended frequencies = pAo and w = qA(o as shown in Fiqures 5 and 7. lEquations (A-33) and (A-37) are the basis for the data analysis procedure used to produce the experimental results qiven in section III. ourinq each test run the surface elevation r and sprinqinq excitation e2 were measured as functions of time, and both time histories were stored in diqitized form in the memory of a Tektronix 4052 computer. Fourier coefficients were calculated for each record by a Fast Fourier Transform tFT technique. Fvaluation of the linear transfer function at frequencies ) = pAw and w = qAO) was then possible usinq equation (A-33). A convolution stubroutine was used to compute the S 1Irp' P I+q' S = n(nAw)n(p'Aw+q'w A-nAw) (A-3R) n=1 for p' = q' = p; p' = q' = q; and p = q. The values of the non-linear transfer function for the three non-linear excitation components were then calculated by equation (A-37).

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