for
LIST OF SYMBOLS
Greek Symbols
INTRODUCTION
DESCRIPTION OF THE MODEL
DESCRIPTION AND RESULTS OF THE FOUR TESTS
Calm Water Tests
Wave Tests (HPS dynamometer)
Radiation Tests
Diffraction Tests
Analysis of Added Resistance in Waves
THE EQUIPMENT
Model Construction
Acrylic Windows
Force Blocks
Vertical Motions Mechanism (VMM)
Video Cameras and Multiplexer
Pitch and Heave
Wave Probes
Accelerometers
Data Acquisition System
ERROR ANALYSIS
SECOND ORDER EFFECTS
Wave Tests (HPS dynamometer)
Radiation Tests
Diffraction Tests
ACKNOWLEDGEMENTS
APPENDIX A: CROSS - REFERENCES
Index 1. Time Histories Available for Each Run Number.
Index 2. Run Number for a Given Type of Experiment and Wavelength.
Index 3. Radiation Run Number for a Given Wavelength and Wave Slope.
Index 4. Diffraction Run Number for a Given Wavelength and Wave Slope.
Index 5. HPS Wave Run Number for a Given Wavelength and Wave Slope.
Index 6. Format of ASCII Time History Files.
Index 7. Instructions to Download the ASCII Time Histories from the Internet.
APPENDIX B: TABLES
Table 1. Ship and Model 1661 Hull Characteristics.
Table 2. Model Window Center Positions, based on Forward Perpendicular at 19.5
foot Waterline.
Table 3. Seakeeping Test Matrix for HPS Free in Surge Wave Runs.
Table 4. HPS Wave First Order Coefficients Vk = 10 knots.
Table 5. HPS Wave Higher Order Coefficients Vk = 10 knots.
Table 6. HPS Wave First Order Coefficients Vk = 20 knots.
Table 7. HPS Wave Higher Order Coefficients Vk = 20 knots.
Table 8. Radiation First Order Coefficients Vk = 10 knots.
Table 9. Radiation Higher Order Coefficients Vk = 10 knots.
Table 10. Radiation First Order Coefficients Vk = 20 knots.
Table 11. Radiation Higher Order Coefficients Vk = 20 knots.
Table 12. Diffraction First Order Coefficients Vk = 10 knots.
Table 13. Diffraction Higher Order Coefficients Vk = 10 knots.
Table 14. Diffraction First Order Coefficients Vk = 20 knots.
Table 15. Diffraction Higher Order Coefficients Vk = 20 knots.
Table 16. Model 1661 Offsets (Half-Beam in inches) for a Scale Ratio of
37.786.
Table 17. Quantification of Errors.
APPENDIX C: ILLUSTRATIONS
Figure 1. Slot Positions for Acrylic Windows.
Figure 2. Schematic of Heave - Pitch - Surge (HPS) Dynamometer.
Figure 3. Ctm vs. Fnm (Calm Water, free to Surge).
Figure 4. Rtm vs. Vm (Calm Water, free to Surge).
Figure 5. Pitch Angle vs. Vm (Calm Water, free to Surge).
Figure 6. Heave Displacement vs. Vm (Calm Water, free to Surge).
Figure 7. Calm Water Sectional Forces.
Figure 8. Relative Water level vs. Froude Number (Calm Water, free to
Surge).
Figure 9. HPS Waves; Vk = 10 knots.
Relative Water level, Force, Motions and Acceleration Response in Waves.
Figure 10. HPS Waves; Vk = 20 knots.
Relative Water level, Force, Motions and Acceleration Response in Waves.
Figure 11. HPS Waves; Vk = 10 knots.
Relative Water level, Force, Motions and Acceleration Phases With Respect to
Waves.
Figure 12. HPS Waves; Vk = 20 knots.
Relative Water level, Force, Motions and Acceleration Phases With Respect to
Waves.
Figure 13. Radiation; Vk = 10 knots.
Relative Water level, Force and Acceleration Response to Heave.
Figure 14. Radiation; Vk = 20 knots.
Relative Water level, Force and Acceleration Response to Heave.
Figure 15. Radiation; Vk = 10 knots.
Relative Water level and Force Phases With Respect to Heave.
Figure 16. Radiation; Vk = 20 knots.
Relative Water level and Force Phases With Respect to Heave.
Figure 17. Diffraction; Vk = 10 knots.
Relative Water level and Force Response in Waves.
Figure 18. Diffraction; Vk = 20 knots.
Relative Water level and Force Response in Waves.
Figure 19. Diffraction; Vk = 10 knots.
Relative Water level and Force Phases With Respect to Waves.
Figure 20. Diffraction; Vk = 20 knots.
Relative Water level and Force Phases With Respect to Waves.
Figure 21. Determining Resistance in Waves; Total Resistance Coefficient vs.
Wave Steepness.
Figure 22. Determining Resistance in Waves; Total Resistance Coefficient vs.
[[lambda]]/Lwl.
Figure 23. Added Resistance in Waves Vk = 10 knots.
Figure 24. Added Resistance in Waves Vk = 20 knots.
Figure 25. CG-47 Body Plan.
Figure 26. Typical Camera Arrangement and Video Details.
Figure 27. Diffraction; Vk = 10 knots.
Ratio of second order Relative Water level to second order wave vs.
[[lambda]]/Lwl.
Figure 28. HPS Waves; Vk = 10 knots.
Relative Water level, Force, Motions and Acceleration second order response
in Waves.
Figure 29. HPS Waves; Vk = 20 knots.
Relative Water level, Force, Motions and Acceleration second order response
in Waves.
Figure 30. Radiation; Vk = 10 knots.
Relative Water level, Force, and Acceleration second order response in
Heave.
Figure 31. Radiation; Vk = 20 knots.
Relative Water level, Force, and Acceleration second order response in
Heave.
Figure 32. Diffraction; Vk = 10 knots.
Relative Water level and Force second order response in Waves.
Figure 33. Diffraction; Vk = 20 knots.
Relative Water level and Force second order response in Waves.
APPENDIX D: GRAPHICAL TIME HISTORIES
(Appendix D is a separate Report 028047-B)
LIST OF TABLES
Index 1. Time Histories Available for Each Run Number.
Index 2. Run Number for a Given Type of Experiment and Wavelength.
Index 3. Radiation Run Number for a Given Wavelength and Wave Slope.
Index 4. Diffraction Run Number for a Given Wavelength and Wave Slope.
Index 5. HPS Wave Run Number for a Given Wavelength and Wave Slope.
Index 6. Format of ASCII Time History Files.
Index 7. Instructions to Download the Report, Tables, Graphs, Curves of Time
Histories, or ASCII Tables of Time Histories from the Internet.
LIST OF ILLUSTRATIONS
1. Radiation at 10 knots - Runs 200-213, 216, 222-226.
2. Radiation at 20 knots - Runs 227-240.
3. Diffraction at 10 knots - Runs 301-302, 309-310, 312-321, 323-327.
4. Diffraction at 20 knots - Runs 328, 330-337, 339-341, 344-347, 349-350, 352,
354.
5. HPS at 10 knots - Runs 380, 384-385, 387-388, 392, 395-397, 400-403,
406-408.
6. HPS at 20 knots - Runs 360-368, 371, 374-378.
LIST OF SYMBOLS
Acln* = Acceleration at Window #n / 1 local Gravity (non-dimensional),
(n = 1,2,3,4,5)
An = Cosine Fourier Series Coefficient #N, Ao = Mean.
A/D = Analog to Digital Conversion
B = Beam (ft. or m)
Bn = Sine Fourier Series Coefficient #N, Bo = 0.
Cb = Block coefficient = Displaced volume/(Lwl . B . T)
Ctm = Model Total Resistance Coefficient (non-dimensional), Rtm/(0.5 [[rho]]
Vm2 Sm)
Drag* = Model Total Resistance Coefficient (non-dimensional), 1000 *
Rtm/([[rho]] g Lwl2 [[zeta]]o)
d = Distance from Baseline to Maximum Sheer (1.296 ft)
Dwl = Design Waterline (ft. or m)
Flatness = [[lambda]] / (2 [[zeta]]o)
Fnm = Froude number = Vk * 1.6889/(g . LWL)1/2
(dimensionless)
Fn = Force Measured at Window n, (n = 1,2,3,4,5)
Fn* = Non-dimensional Force at Window n, (n = 1,2,3,4,5), 1000 * Fn/([[rho]] g
Lwl2 [[zeta]]o)
FP = Forward Perpendicular, intersection of actual waterline with the stem
Heave* = Non-dimensional Heave Displacement, [[zeta]]3 / [[zeta]]o
HPS = Heave-Pitch-Surge measurements in waves
Lwl = Actual Length of the Waterline (ft. or m)
n = Window Number (n = 1,2,3,4,5), Locations listed in Table 2.
N = Number of Points in a Time History, regression curve, or Fourier Series
index
NaN = Not a Number, value too small to compute
Pitch* = Non-dimensional Pitch Angle, Lwl * [[zeta]]5 / (2 * [[pi]] *
[[zeta]]o)
Ra* = Added Resistance Coefficient (non-dimensional), (Rtm-Rto)/([[rho]]
Vm2 [[zeta]]o2)
Rtm = Model Total Resistance (lbsf. or N)
Rtm* = Model Total Resistance Coefficient (non-dimensional), Rtm/([[rho]]
Vm2 [[zeta]]o2)
Rto = Calm Water Resistance determined from faired Ctm vs. Fnm curve, Figure
3.
Rts = Ship Total Resistance (lbsf. or N)
Sm = Model Wetted Surface (ft2 or m2)
T = Draft (ft. or m)
VCG = Vertical Center of Gravity (ft. or m)
Vk = Ship Speed (knots)
Vm = Model Speed (ft./sec. or m/s)
Wave* = Non-dimensional Wave Amplitude, 100 * [[zeta]]o / Lwl
(non-dimensional)
Wtn = Wetted Surface Relative Water level measured from the base line (ft. or
m) at window n
Wtn* = Non-dimensional Wetted Surface Relative Water level / Maximum sheer at
window n,= Wtn / d, (n = 1,2,3,4,5)
Greek Symbols
[[lambda]] = Wavelength (ft. or m).
[[nu]] = Kinematic Viscosity, (ft2/sec or m2/s).
[[rho]] = Mass Density, (slugs/ft3 or kg/m3).
[[zeta]]o = Wave Amplitude (ft. or m), (i.e., wave height/2).
[[zeta]]3 = Heave Displacement (ft. or m).
[[zeta]]5 = Pitch Angle (degrees).
[[omega]]e = Wave Encounter Frequency (radians/sec).
[[sigma]] = Standard Deviation.
[[tau]] = Transverse tank interference when [[omega]]V/g <= 0.25
(non-dimensional).
INTRODUCTION
Seakeeping tests were performed on a 37.7857 scale model of a Navy CG-47 class Cruiser to determine not only the usual heave and pitch motions, but also sectional forces and sectional wetted surfaces in regular head seas. The ship and model characteristics are shown in Table 1. This vessel was chosen because it has "V" sections, a transom stern, sonar dome and strongly flared bow so non-linearity in some of the measured values was expected. In addition, results for a similar model tested for its structural behavior are available from David Taylor Research Center (DTRC), and seakeeping tests on the same hull was thought to be an important addition to a complete data base for this ship. The data presented here includes the traditional analysis based on [[lambda]]/Lwl, with tables of Fourier Series non-dimensional coefficients to third order. In addition, it also includes the measured time histories, available in tables, graphs and on electronic media, intended to be used to validate time-domain numerical models.
Four types of tests were done:
a) Calm water tests at speeds corresponding to 10 and 20 knots full scale.
b) Wave tests using a dynamometer that allowed the model to respond free in
surge.
c) Radiation tests in which the model was forced vertically in pure heave.
d) Diffraction tests in which the model was held rigidly while responding to
regular waves.
The results are available in several formats:
a) Tables of Fourier series components are in Appendix B of this Report.
b) Graphs of first order results based on [[lambda]]/Lwl are in Appendix C of
this Report.
c) The time histories are shown in graphical form in Appendix D of a separate
Report.
d) Tables of the time histories are stored electronically in ASCII form.
e) This Report with tables, graphs and time histories are stored
electronically in two separate retrieval systems.
To navigate around this large data set, several indices, cross references and tables, found in Appendix A, were prepared. The key to finding any specific data is the Run Number. A complete list of the time histories can be found in Index 1. Additional indices giving cross references of wave amplitude, wavelength and vessel speed comprise Appendix A. Appendix D, published separately as MHL Report 028047-B, contains graphs of each time history and duplicates of the cross-index tables for the convenience of the reader. Instructions are provided in Index 7 of both Appendices A and D to obtain the Report, tables, graphs, curves of time histories, or ASCII tables of time histories directly from the Internet so the data will be compatible to any computer system.
DESCRIPTION OF THE MODEL
The model was fabricated from fiberglass with ten slots separated by water tight bulkheads. Each slot had an individual clear acrylic section that was separately fitted to attain the correct hull shape. Each section represents a strip for comparison with strip theory calculations, and is about 2.1 inches (53 mm) wide. Only five of the ten acrylic windows were instrumented with pairs of force blocks on the top to measure vertical sectional forces and with a lightweight video camera inside to measure sectional water levels. Detailed descriptions of the model construction, windows, load cells, video camera and other equipment are in the "Equipment" section below. The model was instrumented for rigid body motions of heave, pitch, vertical acceleration and drag, and sectional heave forces and sectional water levels for windows numbered 1 through 5 located at stations approximately at 1.0, 2.5, 3.5, 4.5 and 19 on 20 stations as shown in Figure 1. The exact locations are listed in Table 2.
DESCRIPTION AND RESULTS OF THE FOUR TESTS
Calm Water Tests
Anticipating the use of our Heave-Pitch-Surge (HPS) dynamometer that allows the model to be free in surge during wave tests, we used the same dynamometer for the calm water tests. This device measures pitch angle, heave displacement, and indirectly, drag forces. The heave staff is mounted at the model midship position. It can be fitted with a rigid coupling to measure drag forces directly when the model is not free to surge. A sketch of the HPS system is shown in Figure 2.
To allow freedom in surge, there are two sub carriages. The "Rigid" subcarriage moves relative to the main towing carriage while a "Light-weight" carriage moves relative to the Rigid carriage. In calm water tests, the velocity of the subcarriage (if any) is added to the towing carriage speed to get the total model speed. A constant force, applied by the hysteresis clutch to the subcarriage, is set by hand before the test begins, based on the experience of the directly previous runs. Since the subcarriage speed is not known until the test is over, drag force adjustment during the test run is impossible. In many cases the total speed is very nearly equal to the towing carriage speed. In this case, the drag force is equal to the preset value. However, because of the limited tank length, the variation in the drag force is larger than if the measurements were done with the model fixed in surge.
To facilitate calculations of added resistance in waves, a faired curve of total resistance coefficient, Ctm, from Figure 3 was used to determine the ideal calm water resistance, Rto. By taking many runs, the variance of the drag is reduced by the square root of the number of runs and sufficient accuracy is attained. The curve shows the expected downward curve at low speeds similar to the flat plate skin friction line, and rises as the wave resistance begins to rise. Subsequent, the values level off, presumably due to the beneficial effect of the bulb near design speeds. The scatter of the coefficient values at slow speed is larger than at higher speeds because the absolute error due to instrumentation noise rises slowly with speed. The levels of error are quantified in Table 17. Ctm is non-dimensionalized by Vm2 so small increases in speed make large changes in resistance. Therefore, the scatter and the percentage error due to noise falls. The calm water resistance of the model is shown in Figure 4. The scatter in the measured values of drag is smaller at low speeds than high, as expected, since the absolute error is small at low speeds. The solid curve represents the ideal calm water drag, Rto.
The dimensional values for the calm water pitch and heave measured at amidships are shown in Figures 5 and 6. The pitch is essentially zero, while the heave shows a very slight sinkage over the speed range tested. The axes on these graphs are highly magnified so that the results can be used for error analysis. For the remainder of the tests, this slight sinkage at no trim angle was taken as the normal trim. The calm water non-dimensional sectional forces are shown in Figures 7 and the non-dimensional sectional wetted surface water levels shown in Figure 8.
Wave Runs (HPS dynamometer)
In wave tests, the mean oscillatory velocity of the subcarriage is added to the towing carriage speed to get the total model speed. Since the mean speed cannot be known until many surge cycles have passed, adjustments of the force supplied by the hysteresis clutch during wave tests are not done. Similarly to the calm water runs, the total speed is often very nearly equal to the towing carriage speed and the drag force is equal to the preset value. However, because of the limited number of oscillations, the variation in the wave drag force is larger than if the measurements were done with the model fixed in surge.
The general test matrix for wave runs is shown in the outlined area in Table 3. No zero speed tests were done due to the narrowness of the towing basin and the consequential wave reflections from the side walls. The values in the table are the wave amplitude, [[zeta]]o, for a given wavelength and desired wave slope. An attempt was made to follow the test matrix in DTRC Report SHD-0829-38, "Seakeeping Performance Results of the TICONDEROGA Class Cruiser as Represented by Model 5481" but some wave heights produced green water over the deck and were not repeated. A wave probe was located on the centerline 157.5 inches (4000.5 mm) in front of the heave staff such that the Kelvin wave did not directly impinge on the model. A 40 second sample (the period of the first longitudinal slosh mode of the tank) was taken before each run to establish the zero values. Time histories of the encountered wave amplitude, the five vertical sectional forces, the heave displacement, pitch angle and bow and stern accelerations were measured.
After each run, the first three Fourier series coefficients were calculated based on a truncated series starting and ending with a zero crossing. The mean of the truncated series is equivalent to Ao. If the generated wave was a perfect sinusoid and the measured signal was noiseless and linear, only the first sine coefficient (B1) would be non zero. For an ideal linear system the cosine coefficients (A1, A2, A3) are a measure of the error in the signal. Any higher order sine coefficients (B2, B3) are a measure of non-linear higher harmonics. For a non-linear system the An's and Bn's are a measure of the non-linearity. Considering a particular measurement, it is not known whether the An's are measurement error or non-linearity. Unless the An's were significantly larger than the uncertainty of measurement, we treated them as if they were errors in a linear system. This assessment seems good for much of the low speed, low wave height tests, and is conservative in that the reported response (Bn) is smaller than the non-linear response (An2 + Bn2)0.5. The tables contain complete listings of An and Bn and their ratios so each researcher can decide at what level the "error" becomes "non-linearity".
Tables 4 through 20 contain the sine and cosine components for all tests, where B1 is the linear component, the ratios A1/B1, A2/B1 and A3/B1 are first, second and third order "errors", and B2/B1 and B3/B1 are the second and third order non-linear components. In these tables, data not taken is denoted as "NaN" whereas data that is defective is denoted by a blank shaded area. In other words, the time histories in ASCII format do not exist if the data is marked "NaN", however the information marked by shaded area shows up in the ASCII files but is not correct. A column marker "ph" is the raw phase (-[[pi]] to +[[pi]]) relative to the wave crest at midship, in degrees. Graphs of the Response Amplitude Operators (RAO) and the phase angles for all runs are shown in Figures 9 through 20. The phase angles in these graphs run from -2[[pi]] to +2[[pi]] to form continuous curves and therefore do not correspond exactly to the format in the time history files. The Bn components are quite small, but from past experience, non-linearities in the previous structural test data are expected to be larger than the non-linearities in either the forces or the motion measurements in the present work.
Tables 4 through 7 are specifically for the HPS wave runs. Figure 9 shows the Response Amplitude Operators for the four measured quantities at Vk=10 knots and Figure 10 for Vk=20 knots. The non-dimensional relative water level amplitudes show a peak at [[lambda]]/Lwl=1.6 for Vk=10 that shifts to [[lambda]]/Lwl=1.2 for Vk=20. The force coefficient RAO does not change with speed. The force coefficients for window 5 show much more variation than the others for two reasons. First, the incident waves are diminished as they reach the stern, and second, the stern does not touch the baseline and therefore the submerged depth is less. Since all the load cells are identical, the smaller signals at window 5 have a smaller signal-to-noise ratio than the other windows. The non-dimensional heave and pitch show a minor effect at [[lambda]]/Lwl=1.6 at Vk=10 and a very large peak at [[lambda]]/Lwl=1.2 for Vk=20. At both speeds, the heave coefficient asymptotes to 1.0. The pitch coefficient is expected to tend to zero for long waves, and clearly does so for the higher speed Vk=20. For Vk=10 the value at [[lambda]]/Lwl= 2.5 seems high, but the time history is very clean and the measurement appears to be valid. The value at [[lambda]]/Lwl= 2.0 is a little low and [[lambda]]/Lwl= 2.5 is a little high when compared to the higher speed runs. Tests at longer wavelengths would confirm the downward trend. The accelerations are highest at the bow for both speeds and show a marked peak at [[lambda]]/Lwl=1.2 for Vk=20. There is no peak for the stern or the slower speed.
The small peaks at [[lambda]]/Lwl ~ 1.6 show up occasionally in other graphs. This may be connected to a transverse slosh mode for the towing tank when testing at forward speed. The problem is most acute when [[tau]] [[equivalence]] [[omega]]V/g <= 0.25. For Figure 9c, [[omega]]o = 3.00 r/s, Vm = 2.75 fps, [[omega]]e = 3.77 r/s giving [[tau]] = 0.32 which is not in the range of concern. Much longer wavelengths of [[lambda]]/Lwl = 2.5 are needed for [[tau]] = 0.25. However, the transverse standing wave occurs when a half-wave just fits between the walls. If the tank walls act like a wave guide, a wave could propagate longitudinally down the tank. The corresponding longitudinal wave has half the transverse frequency so the value of [[tau]] decreases until [[tau]] <= 0.25. Mathematically, this happens at [[lambda]]/Lwl ~ 1.5 and may be the cause of the peaks in the curves. For the faster speed, [[tau]] >= 0.25 even when considering the longitudinal wave and Figure 10c for Vk=20 does not show such a peak. The test procedure was designed to minimize transverse wave effects by starting the model movements in surge (HPS runs) and in heave (radiation runs) only after the carriage began accelerating. However, in hindsight it may have been better to wait even longer until the carriage attained constant speed before freeing the model.
It is not universally agreed what a second order phase calculation demonstrates, so only the linear, first order phase angle is reported for all the runs. The phase calculations are referred to the wave crest at amidships. Figures 11 and 12 show the phase angle for Vk = 10 and 20 respectively. In all cases the phase calculations show much more scatter at small values of [[lambda]]/Lwl. Small spatial errors in converting the wave elevation from the wave probe to amidships and determining the initial and final positions of the rigid subcarriage are a larger proportion of a short wave. Unlike traditional curves, these scales have been extended to -360 degrees through +360 degrees so the continuous increase in phase angle is apparent. Some values of phase had 360 degrees added or subtracted from the raw values reported in the tables to make a smooth curve. The water level phase peak seems to be shifted from [[lambda]]/Lwl=1.8 to a value beyond [[lambda]]/Lwl=2.5 for an increase of speed from Vk=10 to 20. The trough seems shifted an equivalent amount. The non-dimensional forces show the opposite trend: for Vk = 10 the trough is at [[lambda]]/Lwl=1.2 and for Vk=20 it is at [[lambda]]/Lwl=0.8. The heave, pitch and acceleration phases do not vary in shape with speed, but are 90 degrees higher at Vk=20 than at the lower speed. The phases themselves change with slot position, but the scale is too large for the approximately 7 degree variation. Since the error in the phase calculation is on the same order of magnitude, it would be necessary to fair each phase curve to see the small variation between them.
Radiation Tests
The radiation tests were done by attaching the model below the Vertical Motions Mechanism (VMM) which can oscillate the model in the vertical direction with any prescribed motion. For this test, only regular harmonic heave motions were used. The frequencies and amplitudes chosen were ones that approximately corresponded to those in Table 3 used for the HPS wave tests for a constant slope of 1:50. Results from Tables 8 through 12 are shown in Figures 13 - 16. The relative water level amplitude RAOs in Figure 13 have a peak at [[lambda]]/Lwl=1.6 for Vk=10 but are otherwise linear curves. This linearity of Wt* is a function of the non-dimensionalization since the wave slope was held constant so the wave heights increase linearly with wavelength. For Vk=20 the slopes of the lines are about double that of the slower speed, but without a peak. Similarly, the non-dimensional forces show a slight upward slope with a large peak for Vk=10 and none for Vk=20. Since the model was constrained to a prescribed vertical motion, the non-dimensional accelerations were essentially constant for a change in [[lambda]]/Lwl values, but were about doubled due to the doubling of forward speed. The phase results, shown in Figures 15 and 16, were approximately constant for all measurements in all the radiation tests with the values being zero for the water levels and accelerations and -90 degrees for the forces. No variation in phase between the windows is expected, so the outlying points in Figure 16a are obviously in error.
Diffraction Tests
The diffraction tests, in which the model is held rigidly while waves impinge on it, were done by fixing the VMM to constrain the model to its calm water line. During calm water tests the model showed a slight sinkage with speed, so the initial position was chosen to correspond to the equilibrium position at forward speed. Since the model was held tightly by the VMM, there were no vertical accelerations, although the accelerations were measured to verify this. The results listed in Tables 12 to 15 are shown in Figures 17 - 20. The relative water level amplitudes show a linear rise in RAO with increasing [[lambda]]/Lwl that does not change with speed. The non-dimensional Force RAO is approximately constant with increasing [[lambda]]/Lwl, also independent of speed.
The phase results are shown in Figures 19 and 20. The results show a minimum at [[lambda]]/Lwl=0.8 for both the water level and the force coefficients that is unchanged with speed. Like the HPS tests, the phases themselves change with position, but the scale is too small for the approximately 7 degree variation. However, the phase of window 5 at the stern shows a reduction in value of about 90 degrees for Vk=20.
Analysis of Added Resistance in Waves
Added resistance is a second order effect of the wave height, that is, it depends on [[zeta]]o2 and the square of the motions. It is always present whenever there are incident waves or vessel motions regardless of whether there are higher order motions or not. The higher order motions and forces are discussed separately in a subsequent section. This section discusses the variations in total resistance in waves, which made added resistance calculations unreliable. The majority of the wave tests were run at nominal wave slopes of 1:100 and 1:50 with a range of steepness for the [[lambda]]/Lwl=0.83 wave only. Small variations in [[zeta]]o make large variations in the wave steepness, so the total resistance coefficient, Rtm* = Rtm/([[rho]] Vm2 [[zeta]]o2) was used to display the resistance measured in waves. Rtm* was used rather than Drag* = 1000 * Rtm/([[rho]] g Lwl2 [[zeta]]o) or the usual coefficient, Ctm = Rtm/(0.5 [[rho]] Vm2 Sm) since Rtm* has both V2 and [[zeta]]o2 .
Figures 21 and 22 show Rtm* plotted against flatness2. For the shortest waves (except for [[lambda]]/Lwl=0.416) where [[lambda]]/Lwl is less than 1.5 the curves are represented by linear least square curve fits. For longer waves that had no steep wave measurements, straight lines through zero were used. The short wavelength data converge at a [[lambda]]/Lwl value of about -2500, crossing the vertical axis between values of 2 and 4. This, indirectly, is a measure of the calm water resistance so it would show up with the longer wavelengths if measurements had been made using steeper waves. The lack of the abscissa intercept results in values of added resistance at low flatness that are slightly too small.
Using the faired curves, points were interpolated for three nominal values of wave steepness, i.e., flatness values of 35, 50, and 100, and plotted in Figure 23 against a base of [[lambda]]/Lwl. For constant wave slope small wavelengths have small wave heights, so the forces and wave amplitudes at small [[lambda]]/Lwl have a larger percentage measurement uncertainty. In addition, the values for [[lambda]]/Lwl=0.416 were tested at only one wave steepness. The least squares line for this wave length goes through zero at a very sharp angle so interpolation at various wave slopes is very uncertain. This line gives a small negative value for added resistance at both speeds that is not reasonable. The uncertainty in force measurements shown in Table 17 is larger than the added resistance value.
Figure 24 shows the calm water resistance Rto plotted with Rtm*. The added resistance coefficient, Ra*, is found by subtracting the measured total resistance points from the calmwater resistance points or by subtracting the faired total resistance points taken from the curves of Figure 21, and normalizing by ([[rho]] Vm2 [[zeta]]o2). Due to the scatter in the plots, these did not give equivalent values of Ra*. Ra* is expected to peak at [[lambda]]/Lwl of about 1.2 and fall to a small positive value for short waves, but the measured data show a small negative resistance. Waves of flatness=100 have a small amplitude and even with extreme bow flare and a sonar bulb, are expected to generate approximately linear responses. The calculated values of Ra* vary from about 4 at the peak to near zero at [[lambda]]/Lwl of 0.5 and 2.0. However the uncertainty is over 50% of the calculated value and is therefore not reliable enough to display.
THE EQUIPMENT
Model construction
The ship offsets were supplied by DTRC, Code 1561 in SHCP format. Model lines, entitled "Litton-DD963" of DTRC Model 5265 with a scale ratio of 24.824 based on a LBP of 529.00 feet, were supplied to locate the bilge keels. These offsets and lines correspond to a nominal hull of the CG-47 and DDG-963 classes with a nominal draft of 19.5 feet. For these tests, the CG-47 class average was needed to compare with previous results. The Class average has the same body plan as the nominal hull, but has an operating draft of 22.34 feet. Since the hull has a large rake forward, the length on the water line changes greatly for small changes in draft. Ship and model characteristics are shown in Table 1. The hull has similar bow lines as a Series 60 ship where the forward "V" sections are nearly vertical but has a much wider transom than Series 60. At present, commercial liners have bow lines with 45 degree "V" sections and full width sterns to prevent both bow and stern slamming. The CG-47 has a small transom submergence and the stern width is about 25% less than maximum beam so stern non-linearities were expected to be observed.
The SHCP offsets did not contain complete values of waterlines at even increments, so it was not possible to simply scale the offsets or drawings as given. An additional body plan of the CG-47 was made available but only in full scale dimensions as shown in Figure 25. Therefore, a new body plan was constructed from the given offsets, and the waterlines and half-breadths faired to a scale ratio of 37.786 for a model LBP on 20 Stations of 14.00 feet with the design waterline deepened to 22.34 feet. The water line length of 14.074 was determined from the drawing at the deeper draft. The table of model offsets is given in Table 16.
The model was build by Trimode Corporation, Walled Lake MI from fullsize model blueprints. Initially, the entire model was laid up as a single female mold. The windows were hot formed in the female mold and after window bulkheads were installed, hull bulkheads were placed around the windows and the fiberglass hull was hand laid up. This posed some interesting construction problems at the upper bow and sonar bulb where the beam is less than the width of a person's hand, but aluminum angles were welded between a plate fiberglassed to the structure beyond Station 2 and a plate in the deck along the forepeak to insure bending strength between Stations 1.5 and 2.5.
The model was made with forward bulwarks and a stepped-in shearline so that water flow up the hull past the deck line was accurately modeled. To provide the strength needed for the radiation tests, the center quarter of the model contained a plywood box and a pair of longitudinal steel rails to attach to the base of the VMM. Plywood bulkheads were fiberglassed into the hull forward and aft of each window location and the hull between the bulkheads was removed to make the slots into which the windows would go. Pairs of 2 inch by 4 inch (51 mm by 102 mm) fiberglass box beams were stacked vertically above the sheer line but offset by 1/2 inch (13 mm) toward the centerline to tie all the windows together and to give longitudinal strength. Because the VMM force block mounts are wider than the open area, four slots were made in the tubes to allow the VMM to be lowered into the model.
When the swing table was used to set the moment of inertia, it was found that the VCG was much too high. The slot and window bulkheads, which do not exist in the full-scale ship, have their centers of gravity much higher than the corresponding part of the hull. By the time all 20 slot bulkheads and the center box were installed, the model was locally much too rigid. In hindsight, the bulkheads could have been made of cardboard or another lightweight material just strong enough to insure that the transverse fiberglass in the slots would maintain their position while being laid up. Major parts of the 2 inch by 4 inch box beam were cut away while checking for longitudinal stiffness. Large circular holes were cut into all bulkheads and flat surfaces that would not directly touch the water. Even so, the VCG was slightly higher than desired and the model was slightly more flexible than designed.
Cylindrical stud turbulence stimulators, 3 mm diameter, 2 mm high were glued to the bow at station 0.5 spaced 2 cm apart. Waterlines and station lines were marked on the model to determine where the windows actually were, and to provide a grid to calibrate the cameras for water level. Table 3 shows the actual positions of the window centerlines. The transom is rotated about 1.4 degrees clockwise, so all centerlines were found from the average of the forward and aft edges of the slots on the port side only based on the Aft Perpendicular on the centerline.
Acrylic Windows
The windows were made from 1/4 inch (6 mm) clear acrylic plastic by gluing the two flat window bulkheads to a narrow strip that formed the sides. The strip had been heated and inserted into the original female fiberglass mold that was used to make the hull. That way, the outer shape of the window was exactly the same as the outer shape of the model. The top was formed from a piece of 1/4 inch (6 mm) aluminum that was attached with flathead screws through the window bulkheads. No matter how carefully glued and sealed, moisture entered the windows and condensed on the inside. Subsequently, no attempt was made to make them waterproof at the top, and a 1/2 inch (13 mm) hole was drilled through the 2 aluminum plates (the one in the model hull and one in the window top) to allow water to be removed with a small pump. Figure 26 shows a schematic of the camera arrangement and details for window 1 which was the most constrained and hardest to install.
Only five of the ten acrylic windows were instrumented with the two load cells and one video camera each. The other five windows were filled with lightweight foam and covered with thin plastic film for waterproofing. For all but the last window, an exterior bracket was required to hold the video camera a minimum distance from the centerline so the full view of the window could be seen. The position of all windows is given in Table 2. Station 1.5 is so narrow that the camera in window 1was mounted outside the hull, and the view was restricted to the area above the baseline. This was expected to work satisfactorily since water levels below the sonar dome were not anticipated. In some extreme cases, the water level may have reached the baseline. Since the video cannot distinguish heights below the baseline, there is some slight error in the water level for these cases.
Initially, to insure that water could not enter the hull at the bulkhead tops where the force block wires passed, tests were done with and without material designed to keep water out of the slots. Rubber and two types of plastic were used. When attached to the hull and to a narrow strip along the edge of each window, the wave forces quickly tore the material away from the hull. Next, thin polyethylene sheets were stretched from side to side over each window. The 3/4 mil thick transparent sheets were held down with double-faced tape. No optical degradation was noticed. No matter how carefully done, eventually water would get inside the slot and accumulate at the bottom under the acrylic window. Heave motions then would slosh the fluid around giving measurable errors. Tests were made with and without the plastic sheets before they filled with water. Since no errors could be determined from leaving all sealing material off the windows, we concluded that water in the slots was either basically entrained with the model and stayed put, or moved in directions that were not affecting the load cell force measurements. All further tests were done with no sealing between slot and window.
Force blocks
Each window contained flat disc (pancake style) force blocks fitted in pairs to prevent roll. The custom made load cells were designed by A. L. Design, Buffalo, NY. Each was 2.00 inches (50.8 mm) outer diameter and 0.500 (12.7 mm) inches thick. Six #8 machine screws in a 1.6 inch (40.6 mm) bolt circle held the cell to the 0.25 inch (6 mm) aluminum top plate. A single 3/8 inch (9.5 mm) bolt threaded into the center of the load cell held it to an aluminum plate fiberglassed to the thwart between each bulkhead. Since the load cell center was only 0.500 inches (12.7 mm) in diameter, tightening the center bolt sometimes tipped the window enough to touch the bulkhead sides. Special shims were used to keep the windows in line, and a feeler gage was used each time the load cells were checked, adjusted or calibrated to insure clearance between the windows and the slot.
Each window had a small cap screw inserted horizontally near the top. Weights could be suspended from the screw to check the load cell frequently. This was not used as an accurate calibration, but was a good daily check to see that the entire system of force block, connectors and data acquisition system was working correctly.
Special amplifiers were designed both to give two channel amplification for precise calibration and the sum of two channels as output for accurate filtering and data collection. Each force block was checked to verify its natural frequency and cross talk rejection before being hooked to its window. Each load cell was rated for 50 pounds, even though the expected measured forces were about 5 pounds, because they needed to be strong enough to support the much larger forces during launching and storing. In addition, preliminary tests showed that the weight of the window lowered the natural frequencies of the force measuring system and, to get sufficient high natural frequencies, the additional stiffness of the higher rated load cells was required.
Vertical Motions Mechanism (VMM)
The VMM comprises a control system, two servo motors each driving a ball screw platform, a load cell connected to each platform, and all components aligned by two hardened shafts passing through linear bearings. The control system input can be a combination of any amplitude, frequency, phase difference or spectrum. Only regular oscillations in pure heave (phase = 0) were used. The values of frequency and amplitude used in the tests are in Table 16. Although the control system has a feedback voltage for position, the heave displacements were independently measured with precision rotary potentiometers. The initial zero position can be dialed in mechanically, but was also zeroed electronically before each run as part of the data acquisition sequence.
Video Cameras and Multiplexer
The CV500 CCD full frame video cameras were provided by Audio Video Supply, San Diego, CA. They comprised a remote head approximately 1 inch by 1.5 inches (25 by 38 mm) mounted to the port side of each instrumented window, a 4.8 mm focal length wide angle lens and a control box that was located on the carriage. At the MHL, the video cameras were installed by constructing aluminum mounting brackets, tapping additional holes in the cameras, and cutting oval holes in the windows to allow the cameras to partially fit inside. Due to glare, all inside surfaces of the windows, except for the starboard half along the hull, were painted with black primer paint.
The 5 video outputs were connected to a custom video multiplexer invented for the project by Automation Control of Ann Arbor, MI. The system consisted of a hardware board, computer control board and software that combined the five images (one for each narrow strip visible in each window) into one video frame. Since the cameras were not interlaced, a single full frame image containing the five video water levels could be written directly to a Panasonic Q3038 laser disc WORM drive. Originally, in-house software was written to control the disc and to retrieve and analyze the water level data, but although the hardware and software worked in principle, the variations in lighting as the carriage moved down the tank made automatic water level location impossible, and it was eventually entirely done by hand. The multiplexer made the otherwise overwhelming amount of video data easy to control and to archive.
Water level calibration was done very easily using the VMM as an electronic ruler by shifting the zero point mechanically. Points were marked on the model showing the Design waterline, and the model was moved vertically from completely out of the water until the midship windows were in danger of flooding, in 0.100 inch increments. At each step, the video cameras took one frame and when finished a pixel-to-waterline calibration curve was made. When one of the cameras was replaced due to water damage, the entire calibration was done again, and satisfyingly, the calibration curve was virtually identically duplicated. Since the video cameras were bolted and sealed into place, they could not move inside the windows and one calibration was used for all the tests.
Pitch and Heave
Pitch was determined by use of a Rotary Differential Variable Transformer (RVDT) calibrated to a range of +/- 8 degrees full scale using a precision wedge. The HPS pitch gimbal was installed at the midship point at the base of the HPS heave staff. The heave was measured with a rotary encoder that rolled with a friction drive along the flat side of the heave staff and was calibrated to a range of +/- 5 inches full scale using precision blocks. The heave encoder and pitch gimbal are standard parts of the HPS dynamometer as shown in Figure 2.
Wave Probes
The wave probe was located 157.5 inches (4000.5 mm) in front of the model for the wave tests, and 165.15 inches (4194.8 mm) for the diffraction tests measured from amidships. These distances were used to offset the wave in the phase calculations. The wave probes, manufactured at the MHL, are small glass tubes that measure capacitance, and were calibrated each day even though the calibration results were virtually identical. This was done because the small diameter tubes, used to minimize meniscus effects, are easily damaged and regular calibrations are a quick check for breakage before tests are begun.
Accelerometers
Two accelerometers were located at Stations 1 and 19 so that rigid body motions could be estimated and the mO(x,[[dieresis]]) term subtracted from the window force measurements if desired. Values of the mass of each window are in Table 2. The linear 1.5 g accelerometers were calibrated simply by turning them sideways and measuring one local gravity. All the measurements were non-dimensionalized on the basis of one gravity.
Data Acquisition System
The MHL used a Macintosh IIfx computer System 6.0.7 with LabView 2.1.1 software programmed in-house for data acquisition. All channels were filtered with 4 pole Butterworth filters set at a cutoff frequency of 5.6 hertz. The filters were checked to see that the phase shift was linear with frequency as predicted by theory. With the chosen cutoff frequency, less than 0.1% phase shift was measured for the expected wave encounter frequencies. Since all channels were filtered the same way, in principle there should be no phase difference in any case.
The water level video data was acquired automatically beginning when the data acquisition began. A trigger signal, automatically begun at the same time and position in the tank, was written to the video data, where a segment of one frame, chosen to be 1/100 second was displayed as a white band. Thus, precise identification of the time of data recording and the video frame was made. The force and motions sample rate was 30 times each second, corresponding to the video rate of one full frame each 1/30 second.
ERRORS
Table 17 describes the error levels for the various tests and measurements listed in column 1. The second column gives the actual units with the metric equivalents in parentheses. Column 3 lists the "Scale Error" which is the absolute limit that the measuring device can detect. Sometimes this is called the "Precision" or "Bias" error. For digital systems it is the quantizing error or the least significant bit. For analog systems, it is the minimum level of resolution. Column 4 lists the error after it has been digitized by the computer system. The accuracy of analog-to-digital conversion is nominally 16 bits over the full -10 to +10 volt range, but some measuring systems did not use all the bits or the full voltage range. Since the model was measured directly with steel scales, there was no digitizing error.
Column 5 lists the random errors due to noisy signals or to variations in the system under test, sometimes referred to as the "Accuracy". For most scientific institutions, management pressure tends to keep the Scale errors and the Noise errors roughly the same since higher precision equipment can identify which area needs testing technique improvement. The Noise error in the table is defined as +/- 1 standard deviation, however other definitions are possible. The standard deviation is calculated from a series of tests nominally done under identical conditions.
The Final Error in column 6 is the uncertainty from many tests done while varying one parameter. Using statistical theory, it can be shown that if a process follows a known law (a linear assumption is frequently made, but a higher order curve is also acceptable) then the variance of a value from a curve fit of N points decreases as N, that is, the standard deviation decreases as (N)1/2. Of course, if the Scale error is larger than the [[sigma]]/(N)1/2 value, then the Scale error represents the final error, but this is rarely the case. As an estimate of the importance of the errors, the last column is the percentage error for the hull characteristics.
SECOND ORDER EFFECTS
Several effects contribute to second order relative water level amplitudes, forces, accelerations and motions, including the bow flare, sonar dome, transom stern and the second order wave excitation. The effects, however small, are usually coupled and the coupling terms become non-negligible when compared with the primary effects that are themselves small.
Second and higher order effects are illustrated in Figures 28 to 33 in which the plots are displayed in the same order as the first order is presented, i.e., HPS wave runs at 10 and 20 knots, radiation at 10 and 20 knots, diffraction at 10 and 20 knots. The abscissa of all graphs is [[lambda]]/LwL. The ordinate of each graph is the ratio of second order magnitude to first order B1, that is, considering relative water level amplitude at station 1 (e.g. wt1), we define
Wt1+ = [ B2(wt1*) 2 + A2(wt1*) 2] 1/2 / B1(wt1*)
Except for radiation at 20 knots (Figure 31), all graphs have three lines connecting points with flatness in the range of 30, 50, and 100 respectively. The lines are intended to show a trend and may not mean more than that since, in some cases, the line denoting approximate flatness includes nearby values, e.g., ~ 100 may also connect points with flatness around 75. For radiation at 20 knots, only flatness curves of 75, 100 and 200 are shown, there are no values at 20 and 50.
Water waves as they exist in a test facility always have a second order component due to the non-linear properties of the free surface, mechanical imprecision in the wavemaker and hydrodynamic instability of sinusoidal waves. Any second order excitation would create a second order response in a linear system, so in a non-linear analysis it is very important to quantify the second order wave excitation component. Figure 27 (diffraction at 10 knots) is a comparison graph to help identify the levels of precision. It shows ratios of relative water level amplitude to wave amplitude, i.e., wt1+ / wave+ through wt5+ / wave+ where the super script "+" is defined above. It was chosen as a worst case since the relative water level amplitudes in the diffraction case are small and very sensitive to the wave heights. The wave+ graph shows very little contamination except for a region around [[lambda]]/Lwl = 1.5. The second order response is 2 to 5 times larger than the second order wave component and is distributed across the [[lambda]]/Lwl range indicating that the second order analysis should be valid for the diffraction tests.
This analysis can be applied to the other tests to reach the same conclusion. However, too much should not be read into this. The second order wave component is less than 10% of the first order for HPS runs, and the diffraction runs. For the radiation runs, the second order heave is essentially zero, showing the efficiency of the VMM. In contrast, the waves seem to contain some small second harmonics. Since the second order RAO's are formed by division of quantities that can vary by 10%, the result can be somewhat unstable. The water heights in Figure 27 show this effect. The general trends are evident. but each point cannot be analyzed out of context with its neighbors.
Wave Tests (HPS dynamometer)
The HPS cases are more complex than the others. For all the relative water level amplitudes in Figure 28 and 29, we see that apart from an increasing trend based on [[lambda]]/2[[zeta]]o, there is also an increase in the second order effect with increasing [[lambda]]/Lwl. In addition, F1+ is large, with peak values one-third to one-half of first order. F5+ is very moderate, and even lower than other forces at both speeds. Accelerations show a similar trend on the forces, while the second order motions are very small.
Radiation Tests
The radiation runs of Figure 30 at 10 knots shows wt1+ to wt4+ typically small, in the range of 4-5%, while wt5+ varies over 10-20% except for one point at 30%. Most graphs show no variation with wave flatness. However, wt5+ with flatness ~100 is slightly higher than flatness of about ~50 possibly due to wt5+ at first order being small at flatness of about ~100. The higher value of wt5+ in the 10 - 20% range indicates stern activity. Accordingly, F5+ shows a high trend 10 - 15% like wt5+, whereas F1+ is quite high compared both to other forces and the relative water level amplitudes, the reason presumably being the effect of the underwater bow bulb rather than the above water flare.
Figure 31 at 20 knots shows similar all over trends, however, due to a video failure no wt5+ measurements are available. At either speed there is no strong indication that second order effects are more pronounced at lower [[lambda]]/Lwl values. The two accelerations are very low and constant which indicates little second order rigid body motion. At higher speed (20 knots) the forces show much the same trend including F5+, which has become even more significant. The effect at the stern seems to have become important in the sectional force at higher speed.
Diffraction Tests
The diffraction runs in Figures 32 and 33 at both speeds follow similar trends, with a rather clear dependence on flatness. The second order effect generally increases as flatness decreases. Wt5+ and F5+ are again higher than the rest, with F1+ having the strongest dependence on wave flatness.
We performed some numerical experiments to identify trends in the flatness dependence. For the diffraction 10 knot cases, we found that multiplying all values wt1+ to wt5+ by the factor ([[lambda]]/2[[zeta]]o)0.75 generated curves that are almost constant versus [[lambda]]/Lwl and [[lambda]]/2[[zeta]]o although moderate slopes for wt3+ and wt5+ may be due to the variation in wave slopes that are grouped under one flatness category. The significance of the exponent (0.75) is not clear other than it shows a non-linear trend exists. A similar analysis can be conducted for diffraction at 20 knots.
ACKNOWLEDGEMENTS
This work was funded by Office of Naval Research as part of the Nonlinear Unsteady Ship Hydrodynamics Program, Contract N00014-90-J-1818. Many thanks go to Guenther Kellner for conducting the experiments, Bryan Johnson for frame grabbing and digitizing the water levels from the video measurements, Luis Garza-Rios for converting the binary data to non-dimensionalized time histories, to Krish Thiagarajan for analyzing and plotting the data and to Ralph Seguin for setting up the ftp and World Wide Web sites for archiving this Report and the time Histories.