“Report 2151 HYDROMECHANICS O AERODYNAMICS O STRUCTURAL MECHANICS O APPLIED MATHEMATICS oO | ACOUSTICS AND ' @@, TION 03 no als! i! om \ ( POs pce xe \ LCOLLECTion / : Wer A ree eee ee, at PITCHING MOTIONS OF A MOORED SUBMERGED MINE IN WAVES by J. N. Newman P. Plaia E. E. Zarnick Distribution of this document is unlimited. HY DROMECHANICS LABORATORY RESEARCH AND DEVELOPMENT REPORT March 1966 Report 2151 DAVID TAYLOR MODEL BASIN WASHINGTON, D. C. 20007 PITCHING MOTIONS OF A MOORED SUBMERGED MINE IN WAVES by J. N. Newman Pee Piltania ls, 185 Aasaaal ele Distribution of this document is unlimited. March 1966 Report 2151 NOL WR 5-0044 TABLE OF CONTENTS ARVNINGETICG NL, DERI NOON: So ocugoancndddauo ob ad FOOD DObOH OGD DODD OOODDON DESCRIPTION OF MODEL AND EXPERIMENTAL TEST SET-UP .. eee ee ee oe ee ooo RIENCE) IPITITCSULNG MOWIOIN ION AN SIVAN Goeadooaocodgggnn05caDOD0d0000 PROBABILITY OF EXCEEDING A CERTAIN PITCH AMPLITUDE CPIBNK WALUE)) oddcdoesoooeuecooddevc0gs0 ds ooOoDbOUKdEODO UDO GOODOCND CONCHUSIONS soccedcedccos goog doo cod oo go KUOUOD UD OO OOOUDDOGDOS ODDO" NAMING Fdacacocovsd deo 0G dO DD UH GOD C DODD DODD DOOD DOO SOUOOD OU OOOOCS Figure Figure Figure Figure Figure Figure Figure Figure Co) SS ON Ch oS CY WW [2 LIST OF FIGURES Coordinate System and Geometric Set-Up 2... ec © rele lexekeL (re IWIN DO Won soaoocoovocagdccnbocasoocg0c00000 ecece eee eee oe eo ow SSE Os WOSE SSCHMD coocosbooccccscc00000000000 Predicted Pitch Response of Full-Scale Mines eee eee cc eeee Comparison of Measured and Predicted Pitch Response for Model 56 Mine at 20-Foot and 40-Foot Depths and of Altered (Long Mooring Arm) Mine at 20-Foot Depth ii ee eec ee 18 Table LIST OF TABLES Page Rese RESwulies (Wineukeerescl WherslSI)) yoo ocascoscbounooseodaonus 19 Tese Resuiltes (“Aulicerac! Moglel)): goécédcoguocnscccceboococne 20 Root-Mean-Square Pitching Motion in Degrees ............ 20 Probability (P) That the Pitch Amplitude 6 Exceeds a Value of ore =Veoml DEP GESS! MIEN OIE Can tat cocuehaney Gel onane aneustewenceetene rare 21 Probability (P) That the Pitch Amplitude 6 Exceeds a 21 Value of 85 = LOOP DE PIS CS mrs rater eriepis repleticrsoy st arre\avenenenemeilauesexowslatse siete tone iii YY NOTATION Incident wave amplitude (single-amplitude) Mooring cable length Gravitational acceleration Depth of submergence of body axis Pitch moment of inertia of body Pitch added moment of inertia Wave number, K = We fig = m/f Length of mooring arm, measured vertically from body axis Body mass Probability function Fluid pressure Sectional area of body Wave spectrum Mooring cable tension Time Normal velocity on body surface Volume of body Cartesian coordinates fixed in space Incident wave heading angle (8 = O for head waves) Angle of mooring cable from vertical Vertical Cartesian coordinate fixed in body Heave displacement Vertical coordinate of body center of gravity Transverse Cartesian coordinate fixed in body Pitch angle Wavelength Longitudinal coordinate fixed in body iv Surge displacement Longitudinal coordinate of center of buoyancy Longitudinal coordinate of center of gravity Fluid density Root-mean-square pitch amplitude Velocity potential of fluid Velocity potential due to pressure of body Velocity potential of incident wave Frequency Characteristic frequency based on wind speed at 19.5 meters i a | 7 ay ry TS at tb oie T ih | ih . re beat | bay ite! vm Lig! pa Wert (ies lee ree i | ABSTRACT This report describes a theoretical and experimental in- vestigation of the pitching motions of a moored, submerged mine. The theoretical predictions are based upon linearized- wave theory as well as the assumptions that the body is slender and axisymmetric and is ballasted to be at equi- librium in the horizontal plane. The mooring cable is assumed to be massless and inelastic; the fluid is assumed to be inviscid. The theory results in an equation of un- damped motion. Parallel experimental results were obtained on a 2-foot long model in wavelengths ranging from 15 to 55 feet, and these results confirm the theoretical predictions except in the vicinity of resonance, where viscous damping is important. Full-scale predictions are made for the root-mean-square pitching motions in Sea States 4 through 7 for two proposed mine configurations at various depths of submergence. The predicted values are from 1 to 9 degrees in Sea State 4, depending on depth and mine configuration, increasing to greater than 25 degrees in Sea State 7. ADMINISTRATIVE INFORMATION This work was requested by U.S. Naval Ordnance Lab ltr JM:JR:ich/ 3900 Ser 6343 to DATMOBAS dated 8 Sep 1964. INTRODUCTION In response to a request from the Naval Ordnance Laboratory, an investigation was undertaken to predict the pitching motion in waves of moored mines. The mines are elongated bodies of revolution, about 8 feet long, which are moored with an anchor cable from the nose so as to be in equilibrium at depths of from 50 to 200 feet below the free surface when the axis is horizontal. Experimental modeling of the problem is complicated because lab- oratory wavelengths are limited to a maximum of approximately 50 feet, thus implying a scale ratio of at least 1-to-10 between the model and full scale; but a model length of less than 1 foot is impractical, especially when a pitch-measuring gyroscope is incorporated inside the model. In view of this situation it was decided to adopt a scale ratio of 1-to-4, test in the resulting short-wavelength domain, and use a slender-body theory for the pitch motion to extrapolate to longer wavelengths. Thus a combined theoretical and experimental study is presented with a comparison of the two. The comparison is quite good for the original mine configuration at the short wavelengths (40 feet) where the experiments are reliable. At the longer wavelength (55 feet) there is more scatter of the experimental data, and comparison with the theory is less conclusive. However, the principal deficiency of the theory is its inability to account for the (predominantly viscous) damping near reso- nance, which occurs for the original mine configuration at much longer wavelengths. To illustrate this, the model was altered physically by in- creasing the length of the mooring arm, bringing the resonance frequency into the range of experimental wavelengths. The resulting data near reso- nance is substantially different from the theoretical prediction, so that motions at all wavelengths cannot be accounted for with the theory. To provide statistical predictions in a realistic seaway the reso- nance response is determined empirically, using as a rough guide the experimental data with the long mooring arm. The Pierson-Moskowitz- spectrum for fully-developed seas is then applied for Sea States 4 through 7 and yields predictions of the root-mean-square pitch amplitude for two mines at various depths of submergence. It is significant that, due to different values of the excess buoyancy of the two mines, the resonant frequencies correspond to very different wavelengths (450 and 1340 feet); thus the relative superiority of the two mines depends critically on the sea state and also on the depth of submergence. It is clear from this comparison that small changes in the volume or weight of such a body can have large effects on the motions. ANALYTICAL DERIVATION In this section we shall derive the equations of motion of a moored, submerged body in regular waves. The following assumptions are made: 1. The fluid is inviscid and incompressible. taeferences are listed on page 22. 2. The wave height is small so that all wave and body motions can be linearized. 3. The body is a rigid slender body of revolution. 4. The mooring cable is inelastic; its dynamics can be neglected, and it is attached to the body through a friction- less joint below the body axis. 5. The body is ballasted to lie in the horizontal plane. 6. The fluid is deep. We utilize two Cartesian coordinate systems, (x, y, z) and (é, n, t), with (x, y, z) fixed in space and (&, n, ¢) fixed in the body (Figure 1). The &-axis is the body axis, and when the body is in equilibrium this coincides with the horizontal x-axis. The z-axis is chosen to be vertical and positive upwards. The body is fastened to the mooring cable at the point (0, 0, -2%), where & is the length of the arm. The centers of buoyancy and gravity of the body are assumed to lie at (€ O, O) and (Epg» 9, Sg CB the small buoyancy of the arm.) Since the body is ballasted to rest in CB’ ) . zespectavelly,..) (Note that ee "= 101s) a result) of neglecting the horizontal position, it follows that M Eg = © ¥ Ecp [1] where M is the body mass; p is the fluid density; and ¥V is the body volume. The tension in the mooring cable is T = og ¥ - Mg [2] where g is the gravitational constant. Restricting ourselves to analysis of motions in the vertical plane, these will consist of a surge displacement Eo heave displacement Co» and pitch displacement 6. However, the restraint of the mooring cable (with the origin directly above the mooring-attachment point) will restrain the heave displacement to a second-order amplitude, which may be neglected in linearized analysis. As a consequence of the surge and pitch displace- ments BO and 6, the two coordinate systems may be related as follows: x - a = 6 cos Oo 6 Sim 0 Se > CO [3] BOG GOS 0 = & Sim 6) S 6 = €6 [4] Because of the motions EO and 6 the mooring cable will be inclined at a small angle y from the vertical. Assuming the cable length is c, the cable is straight; and the lower end of the cable is at a fixed point Gere tchiewanchon)/easebhien ey sp Le) Ihe Naat [5] yY = b/c - 26/c [6] We can now write the equations of motion, for the longitudinal force and pitch moment, as follows: => Bo = BG + Sfp cos(@,x)) dSi+ T (E/e = O/C) = 0 [7] =f) 00%) = fino | Cos(i,s)) = & cost.) cS STD (Cl + L/C)O = a fe) - Mg (& 6) = 0 [8] ae ~ Sae Here dots denote time-derivatives, p is the fluid pressure, cos(n,x) and cos(n,z) are the direction cosines of the (outward) normal to the body surface, and the surface integrals are over that surface, representing the negative of the pressure force and moment exerted by the fluid on the body. The last term in Equation [7], equal to T y, is the horizontal com- ponent of the force exerted on the body by the cable. In Equation [8], I is the body-pitch moment of inertia about the origin; the term involving T is the pitch moment due to the cable tension T and the moment arm 2; and the last term is the moment due to the weight of the body acting vertically downward. To proceed further we must know the pressure p, and it is here that the slender-body assumption becomes necessary. From the Bernoulli equation plus a constant, where $(x,y,z,t) is the velocity potential, whose gradient is the fluid velocity vector. This potential must satisfy the Laplace equation 2 2 2 do ¢ a9 ete ee SE gl | ee 2 anc ay Zz Io 2 70 [10] throughout the fluid; the linearized free surface of the boundary condition ea. 8d es Pg a SO Cin = in [11] 5t2 OZ where z = h is the plane of the free surface, and the kinematic boundary condition —=V [12] on the body, where ve is the normal velocity of the body surface. For a slender body of revolution, the surge contribution to the normal velocity will be small, compared to the pitch contribution, and Nee 2. § x €0S (m,)) (asi) The potential ¢ will consist of an incident wave potential d5 3 and a body potential by» due to the presence of the body. For plane progressive waves of circular frequency w, $, = 2 exp {K(z-n) +i K x cos 8 + i Ky sin # - iut [14] where the real part is understood. Here A is the wave amplitude; K = aie is the wave number; and 8 is the angle of incidence (f = O for head waves). Combining Equations [12] to [14], the body potential must satisfy 2%, 20 aah Se 6 Xxticosi(n,z)) = ee [15] Assuming the body is slender, the body potential for points near the body is then equal to by = OMG 1) tun Aue exp | Kh + i K & cos B - iot| [16] plus an odd function of n. Thus, adding Equation [14], the total potential on the body is ¢=0€ t+ (gA/w + 20 AZT) exp |- Kye KOS CoSias— iut | 2h ee? O(a [17] plus an odd function of n. Here we have expanded Equation [14] in a Taylor series for small values of the body transverse dimensions. Substituting Equation [17] in the Bernoulli Equation [9], we find that pressure on the body is u Peo OW Star Cpe AS Die At) exp ;-Kh + i K € cos 8 - iut [18] a oy Gr pes © where in Equation [18] and hereafter we consistently delete the nonlinear second order terms in EO 8, and the wave amplitude A. The surface integral in Equation [17] can be evaluated as follows from the divergence theorem: [Jo costo 65 « fff B ov = [ff(% o 0 Ba = {fc p 6% - og K A cos B (1 + 2Kz) exp (- Kh FRE RCOSmBe = Wty lav = - og K A cos 8 exp (- Kh - iut) | exp (iK— cos 8) S(&) dé [19] where S(£) is the cross-sectional area of the body. The volume integral is over the interior of the body, and the line integral is over the body length. - Similarly, for the surface integral in Equation [8], [fv [c cos (n,é) - & cos (n,z)] ds = [ff ee? = 2 iw“p A& exp (- Kh + ikKé cos B - iwt) + pg &] dv [20] « 2 Lb , : = [so [o6& - 2 iw p A& exp (- Kh + iKé cos 8B - iuwt) + og |) dé Substituting Equations [19] and [20] in the motion Equations [7] and [8], we obtain We > (ye) 6 = (uae) = og KA cos 8 exp (-Kh - iut) fs exp (iK— cos 8) dé [21] 6(I + of stare” dé) + T2 (1 + A/c) @ = (T2/e) &, - Mg Zag 6 = 2 ipg KA exp (- Kh - iut) [22] [se — exp (iKé cos 8) dé where in Equation [22] we have used the equilibrium condition Equation [1] and the fact that fe S(E) di = ¥ Eo, to cancel the nonoscillatory terms. Equations [21] and [22] are coupled equations of motion for the surge and pitch motions of the body. Two simplifications can be made. First, if the body length is small compared to the wavelength, the ex- ponential eiké cos B Ea and it follows that M 5. + (T/c)&, - (T2/c)® = pg KA ¥ cos B exp (-Kh - iut) [23] 6(1 + p | scene dé) + TR (1 + &/c)8 - (TR/c) EO [24] - Mg CG 6 = 2 ipg KA ¥ oR exp (- Kh - iwt) In this form the only integral involved is the geometric quantity , 2 Teve o { se dé 125) (i.e. the added moment of inertia), and computations are facilitated. In particular, solving the pair of coupled Equations [23] and [24] gives the pitch amplitude GRC) “Cos (Ss Za ECR (- ae M + T/c)] pgkKA¥exp (-Kh - iuwt) aS [26] (- nie M+ T/c) [T& (1 + &/c) - Mg CG - ie Gea aie (Te/c)? As a second simplification we may assume that the cable length c is very large compared to the arm length %. Then the equations of motion are uncoupled, and Bat ECR og KA*¥exp (-Kh - iwt) a= RTM d Se ee [27] TR - Mg ae =o (Qe I> We note that in this circumstance the pitch response is independent of the heading angle 8. Finally we recall that the cable tension T is given by Equation (Zip thus 21 ECR og KA ¥ exp (- kh - iwt) 2 eg ¥ 2 - Mg (2+ t..) - wu (I + 17) DESCRIPTION OF MODEL AND EXPERIMENTAL TEST SET-UP A 1/4-scale model of the MARK 56 mine was constructed, except for the tail fins, according to drawings provided by the Naval Ordnance Laboratory. The model was designed to accommodate an aircraft-type vertical gyro to measure pitch angle. Photographs of the model are shown in Figures 2 and 3. The model was ballasted to the corresponding full-scale weight-in- air, and the center of gravity was adjusted so that the main axis of the model was horizontal while moored beneath the water surface. The inertia of the model in pitch about the center of gravity was determined by oscillating the model in air, using bifilar suspension. The center of gravity was located by using a knife edge and by freely floating the model in water and noting the angle of inclination. Model characteristics such as the inertia, weight, and location of the center of gravity are pre- sented in Figure 4. Dynamic characteristics of the model were altered for some tests to provide for a more thorough corroboration of theoretical computation with experimental results. This was accomplished by lengthening the mooring arm 20 inches (7 feet full scale). Characteristics of the altered model are presented in Figure 5. In addition to pitch angle, measurements were made of the wave height and heading angle of the buoy relative to the wave direction. The wave height was measured with a resistance type probe. Heading angle was measured by manually rotating the shaft of a calibrated potentiometer to align with the axis of the model. The model was moored with soft stainless steel wire 0.031 inches in diameter. Cables from the gyro were looped to minimize interference with the model motions. A sketch of the test set-up is shown in Figure 6. TEST RESULTS Tests were conducted in waves on the unaltered model at mooring depths of 5 and 10 feet below the undisturbed free surface (20 and 40 feet full scale). Water depth in the test basin was 20 feet (80 feet full scale). The model was tested through a range of wave heights and for wavelengths of approximately 40 feet and 55 feet. These wavelengths are very close to the maximum that can be generated in the facility. The results of these tests are tabulated in Table l. Free oscillation of the model in pitch indicated that the resonant period of the model (4.45 seconds at 5 feet depth and 4.3 seconds at 10 feet depth) was much larger than the wave periods that could be generated in the facility. Consequently, to explore model behavior at a resonant condition, the dynamic characteristics of the model were altered as indicated previously. Tests were conducted on the altered model at 5 feet below the undisturbed free surface in a variance of wavelengths spanning the new resonant period in pitch (2.26 seconds). The results of these tests are presented in Table 2. FULL SCALE PREDICTIONS OF MARK 56 AND 57 MINES We shall now present a theoretical comparison of the pitch response of the MARK 56 and 57 mines. Pertinent data are as follows: MARK 57 Length inches 89.8 101.8 Diameter inches 23.4 20.8 Weight pounds 975 1,040 ECG inches 5) 62 68.0 L inches 27.6 24.1 gl pounds square inches } **3,550,000 IS) (00) 7/ OOO ¥ cubic inches e 33,900 cs 32,600 Displacement, sea water} pounds we 2510 8 1,210 T, sea water pounds 275 170 ECR inches 44.0 es 56.9 g I> sea water pounds square inches 3,030,000 4,860,000 * Bare hulls, without appendages. kk Based upon assumed radius of gyration about the center of gravity of 0.272 L, measured on the MARK 56 aluminum model. 10 From Equation [27] we define the following nondimensional pitch response parameter: || 2Ecp eg ¥ KA e TM} Sl tare ee I) where we assume Sg = 0. This parameter is the ratio of pitch angle to the maximum wave slope at the depth h. Using the relation for the wave- length i = aey ice it follows that |o| Aog ¥ I aces eer KA e IW Na Qi (Gls I>) or, from the above data, and with i in feet, pees Oiler ey als a, [MARK 56] KA e [454 - a| 33.6 2d epg sO Aan [MARK 57] | 10) =| A dimensional measure of pitch response is |e | An € ¥ ican radians/unit length he ~~ Th Rs An @ @ 2 17) 720 pg ¥ ECR ee degrees/unit length TON = Dae (Cl ap Il) where Ae *? is the effective wave height at the depth h. Thus for the two mines |6| 5200 ee a eee degrees £00 [MARK 56] fo SP hase eal 12,100 =) ———._ degrees|/ foot [MARK 57] [1340 - al ILA These two response functions are plotted in Figure 7, along with values of the exponential function a for various depths. Also shown by dashed lines are approximate expected values for the response near resonance, as inferred from the experiments with the altered (long mooring arm) model. Figure 8 shows the comparison of theoretical and experi- mental data for the MARK 56 mine and for the altered (long mooring arm) version of this mine. The agreement is generally good except for ex- cessive scatter of the experimental data at the longest wavelength, which is attributed to the limitations of the wavemaker and for the breakdown of the theory in the resonance domain of the altered model. PREDICTED PITCHING MOTION IN A SEAWAY Predictions have been made of the pitching motions of both mines in several idealized sea conditions and at several mooring depths by applying the principles of linear superposition. Using this procedure, the spectrum of pitching motion is obtained by multiplying the wave-height spectrum by the square of the amplitude response in pitch. The form of the wave spectrum used in these calculations is that proposed by Pierson and Moskowitz? for a fully developed wind generated sea and is given by =. 2 0.74 (@ tes)” BOs 1OTT Ee eS 0 5 Ww S(w) = where g is the acceleration due to gravity; ws g/V, where V is the wind velocity 19.5 meters above the sea surface. The calculations were made for Sea States 4 through 7, correspond- ing to significant wave heights of 6, 10, 15, and 30 feet, respectively, 2 > * These theoretical predictions differ from the above formulae insofar as the fluid density of fresh water is used; the only significant effect is on the cable tension. The theoretical prediction for the long arm is based upon the coupled equations of motion, the pitch amplitude being given by Equation [26]; in this instance, the ratio of 2/c was sufficiently large (0.157) to affect the motions. 12 and at mooring depths of 50, 100, 150 and 200 feet below the surface. The results are summarized in Table 3 which presents the root-mean-square values of pitching motion for the various conditions. Values greater than 25 degrees are not reported since this implies pitching motion in excess of 90 degrees which is definitely not within the limits of the linear theory used in these calculations. The data in Table 3 show that the pitch motion of the MARK 56 buoy is greater than the MARK 57 in the lower sea conditions. The resonant pitch period of the MARK 56 buoy is much shorter than the MARK 57 and, consequently, it responds more to the higher frequency-wave content in the seaway. As the sea state increases, the energy at the lower wave fre- quencies increases very rapidly, and the MARK 57 buoy with its longer resonant period responds with greater pitching motion than the MARK 56. Increasing the mooring depth attenuates the motion for both buoys since the wave energy is attenuated with depth. PROBABILITY OF EXCEEDING A CERTAIN PITCH AMPLITUDE (PEAK VALUE) The pitch spectra have sufficiently narrow bands that we may assume that the envelope, hence the peak values of the pitching motion, follow a "Rayleigh distribution." The Rayleigh distribution is given by where 6 is the envelope or a peak value of the pitch motion and on, is the mean-square value of pitch and may be obtained by squaring the values in Table 3. The probability that the envelope of the pitching motion exceeds a value OF is obtained by integrating the expression for the Rayleigh distribution from on to ©, which yields BED P@> oy se” OA Ela ARS The theoretical probability distribution would always indicate that a certain value of pitch amplitude (peak value) would be exceeded, even though this probability may be extremely small. In actuality the distri- bution will be truncated at some finite value of pitch amplitude. This cut-off point is not known with any certainty; however, it is reasonable to assign some low value of probability, for instance, 0.0001, beyond which no value of pitch motion is considered to exist. Thé probability of exceeding pitch amplitudes of 51 and 60 degrees for various sea states and buoy depths is presented in Tables 4 and 5. These results show that neither body will exceed these angles in Sea State 4. As the sea state increases these angles will be exceeded first at the shallow depths and, finally, in Sea State 7 at all depths for which the calculations were made. CONCLUSIONS 1. The theoretical predictions of the pitch motions of the MARK 56 and MARK 57 mines are reliable, except near resonance, which will occur at wavelengths of approximately 450 and 1340 feet, respectively. At resonance the motions are governed by viscous damping which can not be predicted with confidence from theory or experiments. 2. For wavelengths less than 700 feet the MARK 57 mine is superior with regard to minimization of pitching motions. But for larger wavelengths the MARK 56 is superior and with a larger difference involved. 3. In a seaway both the relative and absolute performance of the two mines depends on the sea state and the depth of submergence, the MARK 56 being better in situations involving higher sea states and deeper submergence depths. 14 x FREE SURFACE Figure 1 - Coordinate System and Geometric Set-up Figure 2 - Model of MARK 56 Mine Figure 3 - Model of Altered MARK 56 Mine 15 ORIGINAL MODEL WEIGHT =15.23 LB. au 26.8" WEIGHT OF ADDITIONAL ié i ~ MOORING ARM = OR7SiEB , TOTAL WEIGHT 15.96 LB 6.88" WEIGHT OF MODEL=15.23 LB ! RADIUS OF GYRATION IN PITCH ABOUT CG = 6.1 IN. GORING EST MOORING POINT Figure 4 - Characteristics of Model Figure 5 - Characteristics of Altered Model RECORDING INSTRUMENTATION CARRIAGE PLATFORM MODEL DEPTH= 5 FT OR 10 FT GYRO CABLES MOORING POINT WATER DEPTH 20 FT SOFT STAINLESS STEEL WIRE 0.031 IN. DIA. Figure 6 - Sketch of Test Set-Up 16 0 Ae Kh DEGREES/FOOT 100 80 40 20 at 4 200 400 Figure 7 - Predicted Pitch Response of Full-Scale Mines Je et 1 600 800 1000 1200 Xd IN FEET Ly 1400 IN DEG/FT FULL SCALE PITCH PEAK TO PEAK ON BOTH Wave qE\oGHT 24 | t 7 ai 20 palin N [ | ALTERED MODEL — | MOORED BUOY — Oreo BERTH | CIRCLED NUMBERS INDICATE —40 DEPTH EXTENSION ADDED ; : | | 10) 6 mack A oe s@ | 12 C) | 4@ ai IC) f : OF O® q f a ve TT af 2@ ; {be &© | 5 = ) fo) Ti SQ | ‘i | + Tr fo) ‘ . 50 100 150 eo ae A IN FEET FULL SCALE Figure 8 - Comparison of Measured and Predicted Pitch Response for Model 56 Mine at 20-Foot and 40-Foot Depths and of Altered (Long Mooring Arm) Mine at 20- Foot Depth Run numbers denote experimental measurements and solid lines denote theoretical predictions. 18 TABLE 1 Test Results (Unaltered Model) Depth of model beneath free surface { © feet,” model’ scale 20 feet, full scale Depth OF weer ie feet, model scale 80 feet, full scale Period of free oscillation in pitch | vot Seconds, model sealle 8.8 seconds, full scale Wave length Wave height Heading Pitch Model, Full scale, Model, Full scale angle, ** amplitude, REECE feet inches ECE degrees degrees 598 159 Uo@ 2.4 3 39.8 159 15.0 5.0 2 59) 83 159 Zoyyll 8.4 1 Sy, 219 6.4 Bod 0 54.7 219 WZ. 4.2 7 54.7 219 18.3 6.1 eaest(0) 10 feet, model scale Depth of model beneath free surface 40 feet, full scale 4.3 seconds, model scale Period of free oscillation in pitch 8.6 seconds, full scale Wave length Wave height Heading Model , Full scale, Model, Full scale angles, ** amplitude, feet feet inches feet degrees degrees 39.8 159 759) 6 39.8 159 14.8 9 39.8 159 ball 58) Bao 7 219 5.0 7 54.7 219 1355 5 54.7 219 19.0 3 * Double amplitude. K* Heading angle relative to wave direction. KKK Head seas. 19 TABLE 2 Test Results (Altered Model) 5 feet, model scale AY sre, savulil Scale Depth of model beneath free surface { 20 feet, model scale Depth of water { SORfeetey fullligscaile Period of free oscillations in pitch es BEGINS MLN SENG 4.6 seconds, full scale Wave length o Wave height Heading Pitch Full scalé, | Model, ] Full scale, | angle,**| amplitude, feet inches feet degrees degrees .8 159 Zia dh Vo 22 .0 7 219 20.1 Oo7/ iS 08) 0 120 Uo 2.6 .8 0) 100 8.3 28 .6 50) 80 9.0 S10) 55) 6 90.4 9.3 Soll ol 6 110 2.5 Sie 58) 0 100 4.8 1.6 4 0 60 9.0 3.0 .8 * Double amplitude. Heading angle relative to wave direction. Head Seas. TABLE 3 Root-Mean-Square Pitching Motion in Degrees 200 feet 100 feet 150 feet MARK 56 MARK 57 We) O57 ORY, 0.3 6.8 4.2 Dos) Bor 14 14 8.0 95 >25 >25 19 >25 20 TABLE 4 * Probability (P) That the Pitch Amplitude 6 Exceeds a Value of a = 51 Degrees 100 feet 150 feet MARK 56 | MARK 57 | MARK S56 | MARK 57 <0.0001 <0.0001 <0.0001 <0.0001 0.0046 <0.0001 <0.0001 <0.0001 0.1250 0.068 0.0013 0.0013 >0.1250 | >0.1250 | >0.1250 | >0.1250 50 feet MARK S6 | MARK 57 <0.0001 <0.0001 >0.1250 0.0013 >0.1250 | >0.1250 >0.1250 | >0.1250 200 feet MARK 56 | MARK 57 <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 0.0274 >0.1250 * LS Envelope or peak values of the pitch signal. TABLE 5 * Probability (P) That the Pitch Amplitude 6 Exceeds a Value of e. = 60 Degrees 50 feet 100 feet 150 feet MARK 56 MARK 56 MARK 57 MARK 56 MARK 57 MARK 56 MARK 57 <0.0001 <0.0001 <0.0001 >0.0562 <0.0001 <0.0001 >0.0562 0.0001 <0.0001 >0.0562 >0.0562 >0.0562 * Envelope or peak values of the pitch signal. 21 REFERENCE 1. Pierson, W. J. and Moskowitz, L., "A Proposed Spectral Form for Fully Developed Wind Seas Based on Similarity Theory of S. A. Kitaigorodskii," Journal of Geophysical Research, Vol. 69, No. 24, (1964) pages 5181-5190. 22 Copies iil NO FF Fe ee Be RP EP F INITIAL DISTRIBUTION CDR, USNOL, White Oak CHB Lt ee ct eS De ee oe CHB il 1 1 CHO 1 1 2 ONR, ONR ONR ONR ONR, DIR 1 CDR CDR CDR DIR At DDC Paul Aylor (Code FM4) Charles Bowersett (Code FM) F. McLean (Code UF) W. Chamberlin (Code JMO) Rosenberg (Code JMO) . Sullivan (Code JM) Baldwin (Code TX) Hightower (Code XU) Henschen (Code FS) S. 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DIR, WHOI O in C, PGSCOL, Webb DIR, Iowa Inst of Hydrau Res 1 Dr. Landweber DIR, St Anthony Falls Hydrau Lab Head, NAME, MIT 1 Prof Abkowitz 1 Prof Kerwin 1 Prof Brown 1 Mr. Herreshoff Inst of Mathematical Sci, NYU, New York Dept of Engin, Naval Arch., Univ of Calif 1 Dr. J. Wehausen Hydronautics, Inc., Pindell School Road, Laurel, Maryland 1 Mr. Eisenberg 1 Mr. Tulin 1 Mr. Webster Presi O.Ceanatesyy mimi Cee Plainview, N.Y. reg Watliewacl as IskersomN, die, Coll of Engin, NYU, N.Y. Copies 1 Dr. Finn Michelsen, Dept of Nav Arch, Univ of Michigan, Ann Arbor 1 Prof Richard MacCamy , Carnegie Tech, Pittsburgh 13 1 Dr. T. Y. Wu, Hydro Lab, CIT, Pasadena 1 Des We KoOwA<, ING. Rt. 110, Mellville, N.Y. 1 Prof Byrne Perry, Dept of Civil Eng, Stanford Univ, Palo Alto, California 1 Prof B. V. Korvin-Kroukovsky, East Randolph, Vermont 1 Prof L. N. Howard, Dept of Math, MIT, Cambridge 39, Mass 1 Electric Boat Div, Groton, Conn., Dept. 451 1 Lockheed Missiles and Space Div, Sunnyvale, Calif. Dr. Cuthbert 1 Dr. W. Kim, Boeing Sci Res Lab, Box 3981, Seattle 24, Wash. 24 Unclassified Security Classification DOCUMENT CONTROL DATA - R&D (Security classification of title, body of abstract and indexing annotation must be entered when the overall report is classified) 1. ORIGINATING ACTIVITY (Corporate author) 2a. REPORT SECURITY CLASSIFICATION Unclassified David Taylor Model Basin Washington D.C. 20007 3. REPORT TITLE PITCHING MOTIONS OF A MOORED SUBMERGED MINE IN WAVES 4. DESCRIPTIVE NOTES (Type of report and inclusive dates) Final Report 5. AUTHOR(S) (Last name, first name, initial) J. N. Newman; P. Plaia; E. E. Zarnick 6. REPORT DATE 7a. TOTAL NO. OF PAGES 76. NO. OF REFS 1 March 1966 29 8a. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S) NOL WR 5-0044 b. PROJECT NO. 2151 9b. OTHER REPORT NO(S) (Any other numbers that may be assigned this report. 10. AVAIL ABILITY/LIMITATION NOTICES Distribution of this document is unlimited. 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY U.S. Naval Ordnance Laboratory, White Oak 13. ABSTRACT This report describes a theoretical and experimental in- vestigation of the pitching motions of a moored, submerged mine. The theoretical predictions are based upon linearized-wave theory as well as the assumptions that the body is slender and axisymmetric and is ballasted to be at equilibrium in the horizontal plane. The mooring cable is assumed to be massless and inelastic; the fluid is assumed to be inviscid. The theory results in an equation of undamped motion. Parallel experimental results were obtained on a 2-foot long model in wavelengths ranging from 15 to 55 feet, and these results confirm the theoretical predictions except in the vicinity of resonance, where viscous damping is important. Full-scale predictions are made for the root-mean- Square pitching motions in Sea States 4 through 7 for two proposed mine configurations at various depths of submergence. The predicted values are from 1 to 0 degrees in Sea State 4, depending on depth and mine configuration, increasing to greater than 25 degrees in Sea State 7. DD ia 1473 Unclassified Security Classification Unclassified Security Classification KEY WORDS Mines Slender Body-Theory Moored Mines in Waves Submerged Mines in Waves Pitch Motions of Submerged Body INSTRUCTIONS 1, ORIGINATING ACTIVITY: Enter the name and address of the contractor, subcontractor, grantee, Department of De- fense activity or other organization (corporate author) issuing the report. 2a. REPORT SECURITY CLASSIFICATION: Enter the over- all security classification of the report. Indicate whether “Restricted Data’’ is included. Marking is to be in accord- ance with appropriate security regulations. 2b. GROUP: Automatic downgrading is specified in DoD Di- rective 5200.10 and Armed Forces Industrial Manual. Enter the group number. 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