100-Year Analysis of Roll Decay for Surface-Ship Models Free

Typically, roll decay tests in calm water are an initial part of a seakeeping test for surface-ship scale models at the Naval Surface Warfare Center Carderock Division (NSWCCD), the naval hydrodynamic research facilities known as David Taylor model basin (DTMB). A model in this paper refers to a physical model manufactured from a scale ratio, λ = L_S/L_M, for a full-scale ship. The seakeeping test of R/V Melville as DTMB model 5720 is an example in Minnick et al. [1]. Roll decay was performed at three speeds in calm water, Froude number, Fr = 0.0, 0.15, and 0.22. The important results are the damping coefficient and roll period. Historically, damping coefficient has been computed by the logarithmic decrement or log decrement. The damping coefficient was computed from sequential peak values of the roll angle in a strip chart recording of the time series, and the period from the zero crossing or time between peaks. The history of roll damping at DTMB and a comparison of computational methods and results including uncertainty estimates are discussed in the following sections.

One of the earliest experiments in roll decay was experimental model basin (EMB) report 69 [2] in 1923. The model for the test was a linear-scale ratio λ = 18.4 of the U.S. crane ship no. 1. The model number was 2375. The report simply states “the model was rolled in still water” but does not explicitly state how roll was initiated. Presumably, the model was rolled by pushing down on one side at the center of gravity to a predetermined roll angle and then released, which is the current method for roll initiation at DTMB. Such a method does not likely generate pure roll. A more precise method of roll initiation is demonstrated by Oliva-Remola et al. [3]. The model is lifted by a weight on a pulley system to a precise angle, and roll is initiated when the electromagnet is de-energized. Additional details are in Figs. 1–4 of from Oliva-Remola et al. [3].

Rolling experiments of EMB [2] in 1923 produced one of the earliest examples of the time history of roll decay. An example of time history is in Fig. 1 from sheet II of EMB [2]. The roll-damping coefficient is indicated graphically by a curve fit to the peaks. The roll period of 1.51 s for bilge keels off was computed from 10 cycles over a time interval of 15.1 s, while the time interval for bilge keels on is 16.3 s for a roll period of 1.63 s. The time interval is indicated by the square wave in the figure with a period of 1.0 s. More recently, damping coefficient was determined quantitatively by an exponential curve fit of the peaks by Mehr et al. [4]. The period in Mehr et al. [4] was determined from the frequency peak in the fast Fourier transform (FFT) spectrum.

where k is the coverage factor, u is the standard uncertainty, s_x is the standard deviation of the series x, and n are the number of samples. The initial peak is 14.034 deg for the curve fit from the offset correction of 0.477 deg, where the curve fit is indicated by the short dashed in Fig. 2(a). A plot of the offset correction for the roll angle is in Fig. 2(c).

In Mehr et al. [4], the exponential curve fit is terminated at an amplitude of 2 deg and 23 s. Small roll angles persisted to 100 s. The conclusion was that the analysis at small angles should be avoided, as other physics may be influencing the result such as wave reflections generated by the model. The water surface may never be still, or the duration of the measurement may not be long enough. Additional discussion on small roll angles is in the Zeroes section, and examples are in Figs. 2(c), 6, and 7. A comparison of Figs. 6 and 7 indicates that the wait time between tests is insufficient for the data in Fig. 6 as an offset correction for roll angle.

where φ_p is the peak roll angle relative to zero and the subscripts j and j + 1 are successive peaks from Park [5]. By the definition in Eq. (6), the damping coefficient may be computed from a minimum of two peaks, one positive and one negative.

By this method, a minimum of four (4) successive peaks are required to compute the damping coefficient by log decrement. The symbol definitions are indicated in Fig. 3.

The results of damping coefficient are compared in Table 1. The comparison includes the following methods:

• Log-decrement from successive roll amplitudes in Eq. (6)

• Log-decrement by peak-to-peak values in Eq. (8)

• Exponential curve fit of peaks in Eq. (5) from Mehr et al. [4]

• curve fit exponential cosine function from Eq. (2) in Park [5].

For the example run, the damping coefficient solution is the same for all methods within the uncertainty estimates. The uncertainty is the lowest by curve fit of exponential cosine Eq. (2) and highest by log decrement Eqs. (6) and (8).

Lack of correction offset causes a large uncertainty in log decrement for successive amplitudes. The effect is minimal in the curve fit of the exponential cosine function in Eq. (2). From Table 2, the uncertainty, ±0.013 (±20%), is the same for the successive roll amplitudes corrected for offset in comparison to the peak-to-peak method. The offset correction from Fig. 2(c) of +0.477 deg ± 0.016 deg by Type A method in Eq. (1) is from the average of the uncorrected time series for the roll angle before roll is initiated. The measured initial amplitude is +14.0 deg in comparison to the analytical value of +13.9 deg from the curve fit, where the uncertainty in the gyroscope calibration is ±1.0 deg.

The differences in the various calculation methods for the damping coefficient are listed in Table 2. The data are derived from the information in Table 1, where the reference is the exponential cosine function from Eq. (2) corrected for offset. The largest difference is the log-decrement method. The smallest difference is with the exponential curve fit of the peaks from Eq. (5) at 4.4%.

The period is calculated independently by one of three methods: zero crossing time, time between peaks, or frequency peak in an FFT of the time series, Mehr et al. [4]. The results for the period calculation by zero crossing are similar to the damping coefficient by log decrement. That is, the uncertainty is significantly larger for the time series uncorrected for offset from the zero-crossing calculation, but the mean values are the same within the uncertainty estimate.

The peak-to-peak calculation does not require correction for offset. The peak-to-peak method for the period is similar to that of Fig. 1. For seven (7) cycles from Fig. 2 for the experimental data, the time interval is 15.375 ± 0.042 s, where the time resolution for 0.042 s is the estimated uncertainty. The resulting period is 2.1964 ± 0.0060 s (±0.27%). From the average of the time interval between successive peaks, the result is 2.198 ± 0.042 s (1.9%), where the uncertainty is computed by the Type A method in Eq. (1) from a standard deviation s_x = 0.07979 s and n = 16 samples with a coverage factor of 2.13 from student t for ν = 15. Again, the uncertainty is lowest from the exponential cosine curve fit of Eq. (2). The different methods agree within the uncertainty estimates.

Calculation of the period from the FFT does not require correction for offset. The result of the data over the curve-fit range of Fig. 2(a) is in Fig. 4. Offset is indicated by the magnitude and zero frequency. The peak frequency is f_p = 0.4570 Hz, or the period T_φ = 1/f_p is 2.188 s. The frequency interval is computed from Δf = f_s/2ⁿ, where the sample frequency f_s = 24 Hz and n =11. The resulting frequency interval is Δf = 0.01172 Hz.

The uncertainty in the period is estimated by a uniform probability density function in the Guide to the expression of uncertainty in measurement (GUM) from JCGM [6]. The standard deviation is $σ=a/3$ and a is ½ the interval. For this example, the standard deviation is 0.0034 Hz, and the expanded uncertainty for the period is ±0.032 s (±1.5%) with a coverage factor of k = 2 for Gaussian.

The differences in the roll period results are compared to the exponential cosine curve fit of Eq. (1) in Table 3, derived from the data in Table 1. The differences of the three methods for period in time are about the same at –0.01 s or –0.5%. The FFT method differs by –0.019 s or –0.85%.

The damping coefficient by log decrement is compared to the exponential cosine curve fit and the exponential curve fit of the peaks in Fig. 5. In comparison to the results of Jones and Cox [8], the data are randomly scattered about the value from the curve fit and do not have the downward trend with decreasing roll angle of Jones and Cox [8]. In Figure 5a, the log-decrement result is plotted as a function of time, where time is the zero-crossing time between the two-peaks from the amplitudes. Time is the true independent variable, while the average peak is another dependent variable. Average peak abscissa in Fig. 5(b) is essentially time in reverse. In the peak-to-peak result, the range is less since the average is for four amplitudes, while amplitude method is an average of two values.

An example of roll angle at zero speed in calm water or float zero is in Fig. 6. The data are acquired near the time of Run #0193 in Fig. 2. The roll oscillations are near the period of run #0193, and roll decay tests just before and after the float zero as summarized in Table 4 from Park et al. [10], Appendix C. The mean value in Fig. 6 is +0.697 deg ± 0.013 deg (±1.9%) by the Type A method in Eq. (1) for uncertainty, where the standard deviation s_x = 0.212 deg, and the number of samples is 961 samples for 40 s and coverage factor is 1.96 for Gaussian. The data are approximated by a cosine function with a period of 2.2395 ± 0.0017 s (±0.077%) from a standard error s_b = 0.000863 s from the regression analysis and coverage factor of 2. The constant amplitude is 0.2789 deg ± 0.0069 deg (±2.5%) from the standard error s_a = 0.00345 from the regression analysis with a coverage factor k = 2.

The purpose of the float zero is offset correction. For run #0193, offset before roll initiation is a more accurate correction and is near the value obtained from the curve fit. The difference of the float zero value of 0.697 deg from the applied value of 0.477 deg is 0.220 deg or 46% and is fifteen (15) times larger than the uncertainty of ±0.016 deg by the type A method in Eq. (1).

Prior to roll initiation in run #201 at zero speed, the float zero was recorded as run #0200 in Fig. 6. Prior to roll initiation in run #0201, the run included data of 34.77 s in Fig. 7 at zero speed, where the roll decay part of the run has been excluded. The initial peak of –17.8 deg occurs at 41 s or 6.2 s after the end of the trace in Fig. 7. From the random nature of the signal, the signal is at the instrument noise level. The random character of Fig. 7 in comparison to Fig. 6 indicates that sufficient time had not elapsed between runs for the float zero recording in run #200. The data of Fig. 7 may better as the offset correction than the float-zero data from Fig. 6.

where s is the standard deviation, the subscript s is signal, and n is noise.

From Fig. 2 of Conolly [9], the mean damping coefficient computed from Eq. (9) increases linearly with ship speed in knots. By the curve-fitting method in Park [5], damping coefficient, N, increases linearly with both speed in Froude number, Fr, and initial roll angle, a, in Fig. 8 from Park [5], where the initial amplitude a is computed from Eq. (2). Figure 8 is autogenerated from a commercial code tablecurve 3D [12], where the colors are proportional to the value of the damping coefficient N. The data points are residuals relative to the curve fit of the plane. The correlation coefficient for the curve fit r = 0.9929, and the standard error of estimate SEE = 0.00268. From Park [5] and Table 2, the average value of the damping coefficient by log decrement is the same as the single value from the curve fit within the uncertainty estimate.

Roll decay data for the EMB [2] model in Fig. 1 are re-analyzed with current methods from the measured peak values. The data are rescaled for the roll angle and time with the ruler function in a commercial portable document format (PDF) Windows program with a resolution of 0.40 mm (1/64 in.). The measurements are listed in Appendices  A and  B for bilge keels off and bilge keels on, respectively. The resulting resolution for bilge keels off in angle is 0.29 deg, and in time is 0.048 s with similar results for bilge keels on at 0.29 deg and 0.049 s. The resolution is applied as the uncertainty estimate. The measured time interval in Fig. 1(a) for bilge keels off is 15.096 ± 0.048 s, which is consistent with the original measurement of 15.1 s. For the time interval of ten (10) cycles, the period is 1.5096 ± 0.0048 s (±0.32%). For the average period, the result is 1.510 ± 0.045 (±3.0%) from a standard deviation of 0.096 s and 20 samples with a coverage factor of 2.09 from Student t with ν = 19.

The measured time interval in Fig. 1(b) for bilge keels is 16.543 ± 0.049 s in comparison to the original value of 16.3 s. For ten (10) cycles, the new measured period is 1.6543 ± 0.0049 s (±0.30%) or a difference of 0.02 s from the original value in Fig. 1(b). For the average period, the result is 1.654 ± 0.042 s (±2.5%) from a standard deviation of 0.090 s and 20 samples with a coverage factor of 2.09 from Student t with ν = 19. The difference from the period of 1.63 s in Fig. 1(b) is 0.01 s, which is within the uncertainty estimate.

With the initial time measurement in Fig. 1(a) at the third peak as time zero (0.0 s), the curve fit to the peaks is presented in Fig. 9(a) for bilge keels off, where the data were offset by 17.06 deg from Fig. 1(a). The resulting damping coefficient and period are 0.071 ± 0.012 (±17%) and 1.540 ± 0.098 s (±6.4%), respectively. The period by the curve fit method differs from Fig. 1(a) by 0.03 s. By log decrement, the damping coefficient is 0.078 ± 0.021 (±27%) from peak-to-peak roll angle calculation and 0.070 ± 0.045 (±64%) from the amplitudes. The average damping-coefficient from the curve fit of the peaks is 0.0853 ± 0.012 (±14%). The results are summarized in Table 5(a).

The authors of EMB [2] ignored the first cycle in Fig. 1 in their analysis. The curve fit from the peaks for bilge keels off provides a good estimate with a correlation coefficient of 0.9986 in Fig. 9(a). The initial measured roll angle is 15.59 deg ± 0.29 deg in comparison to the curve fit value of 11.98 deg or a difference of 3.6 deg. For bilge keels in Fig. 9(b), the correlation coefficient is 0.9925. The initial measured angle is 13.68 deg ± 0.29 deg in comparison to the curve fit value of 8.56 deg or a difference of 5.1 deg. For these calculations, the data before time zero (0.0 s) and peak angles smaller than 0.8 were removed and are indicated by the yellow symbols in Fig. 9. The differences between the experimental and analytical values at the initial roll angle are similar to those in Park [5], where the range of the differences is ±6 deg.

The residual plots of the data in Fig. 9 are located in Fig. 10. The dashed lines are the 95% prediction limit from the curve fit. As indicated by the prediction limit, the quality of the bilge off data is significantly better than the bilge keels data. The difference is also demonstrated by a comparison of the SEE. For the bilge keels off the value is 0.268 deg, while the value for bilge keels on is 0.559 deg. As a comparison with modern data acquisition, the data quality has significantly improved as measured by SEE. As an example in Fig. 2, SEE = 0.128 deg for the R/V Melville model data.

With the initial time measurement in Fig. 1(b) at the second peak as time zero (0 s), the curve fit to the peaks is presented in Fig. 9(b) for bilge keels, where the data were offset by 14.56 deg from Fig. 1(b). The resulting damping coefficient and period are 0.090 ± 0.047 (±52%) and 1.66 ± 0.53 s (±32%), respectively. The period by the curve-fit method differs from 1.63 s in Fig. 1(b) by 0.03 s. By log decrement, the damping coefficient is 0.080 ± 0.052 (65%) from peak-to-peak roll angle calculation. The average damping-coefficient from the curve fit of the peaks is 0.102 ± 0.019 (±19%). The results are summarized in Table 5(b).

An early report of roll decay of a surface-ship model in calm water was EMB [2] in 1923. An example roll-decay time-history is in Fig. 1. The roll damping is indicated graphically by a curve fit of the peaks in the time series. Roll period is documented in the figure as 1.51 s for bilge keels off and 1.63 s for bilge keels on but not the damping coefficients. The roll-damping coefficient has been computed by an analogous method with an exponential curve fit of the roll angle peaks by Mehr et al. [4] by Eq. (5).

Historically at DTMB, the damping coefficient has been computed from the log decrement in Eq. (6). An early report is Jones and Cox [8] from 1976. The basis of the damping coefficient by log decrement is from Conolly [9] in 1969. However, Conolly [9] states that the appropriate value of the damping coefficient is the average value. The average value was not reported in Jones and Cox [8] or most NSWCCD reports since 1976.

A better method with lower uncertainty estimates is a curve fit of an exponentially decaying cosine function in Eq. (2) from Park [5]. The function is the solution of a second-order ordinary differential equation with constant coefficients for an under-damped roll motion (η < 1.0). By this method, the initial amplitude, the damping coefficient, period, and offset are determined in a single calculation. The damping coefficient by log decrement is derived from the same ODE; consequently, the damping coefficient has a single value. When the log decrement is applied, only the average value should be reported from Conolly [9] and Park [5]. Likewise, the average value of the period should be reported from multiple zero crossings and peaks. In another method, the period is computed from the peak value from the power spectrum by FFT in Mehr et al. [4] and Fig. 4. When the amplitude is applied in Eq. (6) for the damping coefficient, the time series should be corrected for offset to reduce the uncertainty estimate. Offset correction is not necessary in the peak-to-peak method for damping coefficient or period. When corrected for offset, the uncertainty estimates are the same for the amplitude, Eq. (6), and peak-to-peak, Eq. (8), calculations for the example in Table 1. Offset correction is not needed for the period calculated from the peak of the power spectrum of the roll angle in Fig. 4.

Offset correction is usually from float-zero measurements, that is, recording roll angle in calm water at zero speed. For roll decay experiments at speed, a better method is to record roll angle for a few seconds before roll initiation as the correction as demonstrated in Fig. 2.

Additionally, the time interval between the initiations of model roll should be sufficient so that the roll signal for float zero is indicative of instrument noise exemplified in Fig. 7 at the end of the run. As a comparison, the wait time in Fig. 6 for the float zero is insufficient as the signal is periodic. In the future, roll decay tests should have a longer runtime to determine the time for a float zero as demonstrated in Fig. 7. Extended runs similar to the run of 100 s in Mehr et al. [4] should be executed to determine the time interval between roll initiations. Two runs should be sufficient with one at a small initial roll angle and a second at a large roll angle.

Since the initial roll angle is fundamental to the ODE solution, the initial roll angle should always be documented in a roll decay test. One of the problems is the initiation of pure roll. The differences between the analytical and experimental values are documented in Park [5]; consequently, runs should be repeated at near the same initial roll angle. A roll initiation method of Oliva-Remola et al. [3] may produce more repeatable results. The differences between the analytical and experimental values at the initial roll angle may be caused by the lack of pure roll in the roll initiation method. That is, the effect of pitch, sway, and heave may be included.

Care should be taken in the evaluation of roll decay at small angles and high damping coefficient. Potentially, the results at small angles are impacted by waves generated by the model. In the analysis of Mehr et al. [4], data for peak angles less than 2 deg were excluded. The wait time in the float-zero measurements may be insufficient from a comparison of the examples in Figs. 6 and 7. The roll period of 2.2395 ± 0.0017 s in Fig. 6 is near that of the roll decay test in Fig. 2. Calm water tests should include wave probe measurements. In future tests, wave probe measurements should be compared to the float zero results for roll.

The roll-decay time history in Fig. 1 from EMB [2] was reevaluated with current methods by an exponential cosine function generated from the peak values. In the original paper, only the roll periods were calculated. The curve-fit method provides both the damping coefficient and period. The quality of the data for bilge keels off is significantly better than bilge keels on as indicated by uncertainty estimates and the SEE. Both the damping coefficient and period increase with the bilge keels on. Essentially, the resolution of modern measurement methods is significantly smaller. The results are questionable for the log decrement method from the peak amplitudes as a consequence of the poor resolution from the strip charts in Fig. 1 in comparison to modern digital methods. As a result, some values of the damping coefficient are zero and negative in Appendices  A and  B. The peak-to-peak values are always positive. However, the data trends are similar. Large differences occur between the analytical and experimental initial roll angles, and the roll peaks less than 1 deg (one degree) are removed from the analysis.

In the future, the following experimental methods should be considered in a roll decay test:

• Initial roll angle: a range of initial angles should be performed at both port and starboard from small to large initial roll angles. The suggested nominal angles are as follows: 5 deg, 10 deg, 15 deg, 20 deg, and 25 deg for a minimum of ten (10) runs. An improved roll-initiation method should be developed such as described in Oliva-Remola et al. [3].

• Model speed: a range of model speeds should be performed from zero to maximum ship speed as defined by Froude number. Typically, Fr = 0.0, 0.1, 0.2, 0.3, and 0.4.

• Float zero: the float zero should be part the roll decay run with float zero computed from the beginning of the run prior to roll initiation with Fig. 2 as an example. The run time interval between runs should be sufficient for float zero to be random as indicated in Figs. 2(c) and 7. Another alternative criterion would be a maximum roll peak of 0.2 deg.

Daniel R. Park created an updated version of Fig. 1 from EMB [2] in 1923 with modern graphics. His contribution is gratefully acknowledged.

A =

first peak-to-peak roll angle pair, Eq. (6), deg

B =

second peak-to-peak roll angle pair, Eq. (6), deg

f =

frequency, Hz

Fr =

Froude number, $Fr=V/gL$

g =

acceleration of gravity, m/s²

k =

coverage factor, usually k = 2

L =

length, m

n =

number of samples

r =

correlation coefficient

s_x =

standard deviation of x

SEE =

standard error of estimate

SNR =

signal-to-noise ratio, Eq. (9), dB

t =

time, s

T_φ =

roll period, s

u =

standard uncertainty

U =

expanded uncertainty, U = ku

V =

velocity, m/s

η =

damping coefficient, Eq. (3)

λ =

scale ratio, λ = L_S/L_M

ν =

number of degrees of freedom, ν = n – 1

φ =

roll angle, deg

avg =

average

j =

index for peak number

M =

model

n =

noise

p =

peak

p–p =

peak to peak

s =

sample or signal

S =

ship