Experimental and Numerical Investigation Into the Effects of Initial Conditions on a Three Degree of Freedom Capsize Model

Transcription

1 Journal of Ship Research, Vol. 50, No. 1, March 2006, pp Experimental and Numerical Investigation Into the Effects of Initial Conditions on a Three Degree of Freedom Capsize Model Young-Woo Lee,* Leigh McCue, Michael Obar, and Armin Troesch *Hyundai Heavy Industries Company, Ltd., Ulsan, South Korea Aerospace and Ocean Engineering, Virginia Tech, Blacksburg, Virginia, USA US Coast Guard Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, Michigan, USA The dynamics and hydrodynamics of ship capsizing include strong nonlinearities, transient effects, and physical phenomena that have not been fully identified or studied. This paper presents a study of some of the various mechanisms associated with this extreme behavior. A quasi-nonlinear three degree of freedom numerical model is employed to examine the effects of initial conditions on the ultimate state of a box barge model. The numerical results are then used to provide structure and understanding to otherwise seemingly inconsistent and ambiguous experiments. 1. Introduction OVER the last two decades a substantial effort has been devoted to better understand nonlinear motions in capsize via theoretical analysis (e.g., Melnikov functions), brute force numerical simulations, and experimental tests. An incomplete but representative list of references would include Soliman and Thompson (1991), Thompson (1997), Spyrou and Thompson (2000), Dillingham (1981), Dillingham and Falzarano (1986), Falzarano et al. (1992, 2002), Laranjinha et al. (2002), Vishnubhotla et al. (2002), Chen et al. (1999), and Belenky et al. (2002). The implications of previous works for this research are discussed in greater detail in Section 2.1. Roll dynamics are conceptually described by a nonlinear system whose response and eventual state (capsize or noncapsize) are frequently dependent upon initial conditions (Soliman & Thompson 1991). Additionally, water on deck produces variable loads that may have a large impact on the motions. While there has been significant work investigating water on deck dynamics, little attention has been given to the initial state of the vessel subject to these variable loads. Based on the work of Soliman and Thompson (1991), the experiment described herein was devised to investigate the effects of varying initial roll angle and roll velocity in a 3DOF system. A series of experiments were conducted where a box barge was excited in beam seas. See the schematic shown in Fig. 1. Those tests effectively modeled a three degree of freedom (i.e., sway, heave, and roll) two-dimensional freely floating rectangle with water on deck. It became apparent that in certain critical wave Manuscript received May 30, 2003; accepted August 20, amplitude and frequency ranges, the states of capsize or noncapsize were functions of how and when the model was released. An example of some of the experimental results are shown in Fig. 2, where initial roll angles and roll velocities at the instant of model release are plotted. Those runs that capsized are denoted with + and those runs that remained upright after 20 wave encounters are marked with o. The test parameters for those series of tests are o /B , T o /T n 0.33, and /B 4.32, where o is the incident wave amplitude, B is the beam, T o is the incident wave period, T n is the small amplitude roll natural period, and is the incident wave length (see Table 1). The experimental results, of Fig. 2, which at first seem inconsistent and nonrepeatable, provide the motivation for the work presented here. This paper is organized as follows. We begin with a description of the quasi-nonlinear three degree of freedom numerical model used to predict vessel capsize for a rectangular box barge. Next, the methodology and results of experiments conducted at the University of Michigan Gravity Wave Facility are presented. Structure is then given to the seemingly random experimental results via extensive numerical simulation. From the insight this structure provides, the paper closes with conclusions and suggestions for future work. 2. Numerical model 2.1. Background The work of Soliman and Thompson (1991) suggested analyzing the motions of vessels exposed to a short pulse of regular waves. While sea states are random processes, a short pulse of regular waves can excite a vessel at a natural frequency causing MARCH /06/ $00.67/0 JOURNAL OF SHIP RESEARCH 63

2 Fig. 1 Coordinate system for capsize model. Plotted in this orientation for consistency with laboratory experiments resonant motions. This was viewed as the worst-case scenario in simulating vessel capsize (Soliman & Thompson 1991). For a one-dimensional (i.e., a single degree of freedom) model, they simulated initial conditions through a grid of roll angle and roll velocity pairs mapping safe versus capsize regions. This method yields a safe basin for the vessel at a given frequency and wave amplitude. From such safe basin calculations one can consider a range of wave heights and thus plot the ratio of safe area to capsize area. The resulting plot is known as an integrity curve. On a typical integrity curve, there is some region of heights in which the integrity values drop dramatically, thus indicating critical wave heights at which the vessel has a rapid loss of dynamic stability (Soliman & Thompson 1991). It is frequently assumed that a two degree of freedom roll-sway model can be reduced to a one degree of freedom model through use of a pseudo roll center. However, this often neglects hydrodynamic coupling effects (Jiang et al. 1996). Alternatively, Chen et al. (1999) argued that the use of a wave fixed coordinate system enables one to nondimensionalize the capsize problem in such a way as to allow reduction of the heave, sway, and roll dimensions to a single roll degree of freedom. This model reduction requires that the ratio of roll natural frequency to heave natural frequency be less than 1. Fully nonlinear ship motion computations are not sufficiently mature for use in assessing all possible parameter ranges of large amplitude motion leading to capsize. Dynamical models with wa- Fig. 2 Roll and roll velocity initial condition pairs leading to capsize or noncapsize of a rectangular barge in regular beam waves 64 MARCH 2006 JOURNAL OF SHIP RESEARCH

3 Table 1 Coefficients for Numerical Model From SHIPMO Definition of coefficients Experimentally Determined Coefficients m kg/m T n 2.75 sec a 22 /m T o /T n 1/3 a 24 /(mb) m a 33 /(m) B/ 0.23 a 42 /(mb) T/ a 44 /(mb 2 ) fb/ I cg /(mb 2 ) b 22 /(m ) b 24 /(m B) b 33 /(m ) b 42 /(m B) b 1 /(m B 2 ) b 2 /(mb 2 ) f D 2 /(mg) f D 3 /(mg) f D 4 /(mgb) ter on deck have been investigated numerically and experimentally, e.g., Belenky et al. (2002) or Huang et al. (1999). Most recently, Belenky et al. (2002) used a computational simulation to evaluate the effects of water on deck for a full six degree of freedom ship model using the LAMP system. This model, however, is not validated and proves to be prohibitively time consuming for investigations involving many conditions. Dillingham used Glimm s method for a linear two degree of freedom model with sway-roll coupling to analyze dynamic effects of water on deck under the assumption of shallow water theory (Dillingham 1981). Later works, such as Laranjinha et al. (2002), extend the method outlined by Dillingham to a six degree of freedom linearized ship motion model. While this method is quite interesting for investigating the effects of higher-order dynamics of water on deck, for the study contained herein it proves too computationally intensive to incorporate with the nonlinear ship motions model. Solving the two coupled problems of the nonlinear ship motions with the dynamics of the flow of water on deck would prohibit generating the volume of data needed to complete this study. Thus, the principal aim of this study was to focus upon the nonlinear motions of the vessel, including the firstorder influence of water on deck. In order to achieve reasonable computer run times while retaining relevant physics, blended hydrodynamic models, similar in concept to metamodels or surrogate models, have been developed and calibrated. See Beck and Reed (2000) for a recent review of several such methods. The blended hydrodynamic model shown in this paper has the significant advantage of computational efficiency. For example, the work in the next section alone has ranges in the parameters of initial conditions for six state variables, excitation frequency, and wave amplitude that produce simulated Fig. 3 Heave displacement integrity curve normalization factors. Heave velocity, sway displacement, and sway velocity zero at time of release. T o /T n 1/3. o = o 20, 15 o 15 MARCH 2006 JOURNAL OF SHIP RESEARCH 65

4 Fig. 4 Heave velocity and sway velocity integrity curve normalization factors. For heave velocity curve: heave displacement, sway displacement, and sway velocity zero at time of release. For sway velocity curve: heave displacement, heave velocity, and sway displacement zero at time of release. T o /T n 1/3. o =0. 20 o 20, 15 o 15 time series totaling more than 4.5 years, real time. This is based on setting the integration time steps to 0.01 second. More complex hydrodynamics models would not be capable of calculating the sheer volume of data presented here Modeling details In this paper a quasi-nonlinear time domain simulation based on the effective gravitational field and the long wave assumption (Chen et al. 1999) are used to predict capsizing behavior of a two-dimensional rectangular body. This approximation fully accounts for the effects of deck immersion in the computation of restoring forces and Froude-Krylov forces. It is assumed that the length of incident beam waves are much longer than the beam of the body. The forcing functions that dictate the body motion are determined from the wave excitation acting on the body. The wave contour can be prescribed by a single or series of cosine waves of arbitrary frequency, amplitude, and phase. Each wave component travels at its own phase velocity, from an initial position given by its specified phase at time zero. Thus, the wave elevation is completely defined at any lateral position, x, and instant of time, t. See Fig. 1. At each subsequent time, the wave contour is determined at any position of the global coordinate system (x o, y o ), and the instantaneous displacement, center of buoyancy, and local wave slope across the body are computed. From these parameters, the corresponding accelerations in sway, heave, and roll are computed. The accelerations are integrated at each time step to compute the velocity in sway, heave, and roll. In similar fashion, body velocities are integrated to determine the position and orientation of the body at each time step. It is expected that under the long wave assumption, the nonlinear large-amplitude motions of a floating body in regular beam seas can be calculated with reasonable engineering accuracy Effective gravitational field For periodic linear beam seas in deep water, the water particles can be shown to be moving in a circular path. The water particle will experience a centrifugal acceleration as well as gravitational acceleration. The gravitational acceleration can then be combined vectorially with the centrifugal acceleration to give an effective gravitational field normal to the instantaneous wave surface. The resulting acceleration is called the effective gravitational acceleration and is denoted by g e (t). This effective gravitational acceleration can be approximately expressed (Chen et al. 1999) as equation (1), where w denotes the wave frequency, 0 is the wave amplitude, and is a phase angle. Note there is no kx term because this equation is defined for the local wave-fixed coordinate system. 66 MARCH 2006 JOURNAL OF SHIP RESEARCH

5 Fig. 5 Sway integrity curves. 0.0 x o / 1.0. Sway velocity, heave, and heave velocity 0 at time of release. T o /T n 1/3. Note significant change with respect to sway in location of cliff on curve. Additionally, curve for x o = 0 is identical to x o =. 20 o 20, 15 o 15 g e (t) g w 2 0 cos( w t + ) (1) The wave body intersection points are determined from the mathematical wave contour and the body geometry using a bisection method at each time instant. At each time step a wave elevation is defined at any lateral point and the translational and rotational displacement of the body is also given in the global coordinate system. For a two-dimensional box barge, the coordinates of four corner points of the barge model are known at the present time step. With these known values of the body corner points and wave elevation, iterations based on the bisection technique are used to determine the exact position of the wave body intersection, including the complete submersion of the body under the water surface. This exact configuration of the wave body intersection can approximate the extreme behavior of the floating body in regular beam seas deck immersion and bottom emersion. These exact intersection points then facilitate displacement and center-of-buoyancy calculations. In the calculation of displaced water, it is assumed that the wave length, compared to the beam of the barge model, is long enough such that the water surface has a flat plane in the vicinity of the barge. Accordingly, the wave configuration between the two intersection points is considered linear. The displacement, center of buoyancy, and righting moment arm with respect to the center of gravity of the body (x G, y G ) can be calculated in an analytical way for any wave body interaction. The center of buoyancy in terms of a local body-fixed axis system is in turn transformed into the global coordinate system for use in the equations of motion. While this long wave model is admittedly simplistic, it captures the essence of quasi-static water on deck and extreme roll angle dynamics. The model has the significant benefit of being computationally efficient allowing for extensive searches of the parameter space Wave elevation The elevation of a sinusoidal wave at any time, t, and lateral position, x, is given by: (x, t) 0 cos(kx t + ) (2) where k is the wave number, 0 is the wave amplitude, and is the phase of the wave system. When applied to a floating body at rest, this wave system (2) will cause an abrupt excitation of wave forcing at time zero. To avoid this extreme form of loading, which can never be produced in a laboratory wave tank, a ramp function, which is a smooth increase of wave amplitude from zero to its maximum amplitude, can be added to the description of the wave system: (x, t) 0 (1 e t2 ) cos (kx t + ) (3) where is a parameter to adjust the rate of increase of wave amplitude. With this description of the wave system, the forcing magnitude would be increased gradually with the same rate of increase. In a laboratory wave tank, waves are not generally generated in the way described by the above two formulas. Lee (2001) and Obar et al. (2001) experimentally determined an envelope curve to match a cosine wave to the waves produced in the laboratory. The results are given here. When the wave maker starts, a train of growing transient waves are initially generated. This wave train reaches the ship model MARCH 2006 JOURNAL OF SHIP RESEARCH 67

6 Fig. 6 Comparison of sway integrity curves based on grid size. In legend first line denotes curve for which sway displacement initial value is zero and mesh values range from 30 30, Similarly, second line denotes zero sway displacement initial value and 20 20, Third line indicates initial sway displacement of.37 and 30 30, Fourth line denotes initial sway displacement of.37 and 20 20, Both meshes have fixed o = 1 deg and o = 1 deg/s. Sway velocity, heave, and heave velocity zero at time of release. T o /T n 1/3. Note steepness of cliff is affected somewhat by size of grid, but location is not. Fig. 7 Sway velocity integrity curves x o /( n B) Sway, heave, and heave velocity 0 at time of release. T o /T n 1/3. Note little change with respect to sway velocity in location of cliff on curve , MARCH 2006 JOURNAL OF SHIP RESEARCH

7 Fig. 8 Heave integrity curves y o /B Sway, sway velocity, and heave velocity 0 at time of release. T o /T n 1/3. Note significant change with respect to heave in location of cliff on curve but not in x-axis intercept (location where integrity is zero) , Fig. 9 Heave velocity integrity curves y o /( n B) Sway, sway velocity, and heave 0 at time of release. T o /T n 1/3. Note significant change with respect to heave velocity on general shape of curve , MARCH 2006 JOURNAL OF SHIP RESEARCH 69

8 x, t = 0 (4) Fig. 10 Sketch of barge with dimensions located in the middle of the tank at some later time. From a dynamical systems point of view, the loading produced in the laboratory is intermediate between the stepped sinusoid and slow ramped function of (2) and (3). A formula to account for the characteristics of the laboratory wave tank is proposed: where and are parameters taken from experimental wave measurements to adjust the rate of increase of the wave amplitude, c is the wave phase velocity, and T s is a time duration for the transient wave to reach its steady state. This form has the advantage of eliminating the sensitivity to the startup phase, which is a feature of the results under the stepped sinusoidal loading. The wave tank surface will remain flat if the wave train is not propagated both temporally and spatially. This description/model of wave propagation proved to simulate the same incident wave profile as that measured in the physical wave tank test (Obar et al. 2001). For the sake of consistency with previous capsize calculations, e.g., Soliman and Thompson (1991), a simple cosine wave is used for wave forcing in the integrity curve plots presented in this paper (Figs. 5 to 9). However, when numerical results are compared to the laboratory experiments (Figs. 17 to 24), the envelope fit forcing function, that is, equation (4), is used to more closely model experimental results. x, t = 0 e ct x c cos kx t + otherwise if ct x T c s 2.5. Equations of motion Based on the instantaneous wave body intercepts, pressure forces are determined from the instantaneous displacement, center Fig. 11 Experimental repeatability tests comparing runs 17 and 18. Both runs released at time t = seconds. Run 17 ic = 0.43 deg, ic = 1.06 deg/s, x ic = 0.42 cm, ẋ ic = 0.76 cm/s, y ic = 0.27 cm, ẏ ic = 0.34 cm/s. Run 18 ic = 0.27 deg, ic = 0.09 deg/s, x ic = 0.99 cm, ẋ ic = 8.22 cm/s, y ic = 0.31 cm, ẏ ic = 1.38 cm/s 70 MARCH 2006 JOURNAL OF SHIP RESEARCH

9 m + a22 0 a24 0 m + a 33 0 a 42 0 I cg + a 44 ẍg b22 0 b b 33 0 ÿ g 1 ẋg b 42 0 b D + f ge2 = D 0 D g e3 mg + f 3 (5) 0 0 b g e4 GZ + f 4 An explanation of terms is as follows (see Table 1 for numerical values): ẏ g Fig. 12 A view of the test section during a wave surface calibration run. The camera s iris was opened slightly more than during a regular run. of buoyancy, and local wave slope across the body (Lee 2001). The equations of motion in the inertial coordinate system are given in equation (5), where subscripts of 2, 3, and 4 represent sway, heave, and roll degrees of freedom, respectively. a ij, b ij : added mass and damping coefficients f j D : diffraction forces b 1 and b 2 : linear and nonlinear roll damping coefficients g ei : time-dependent sway and heave components of effective gravity : time-dependent volume of hull, including possibility for deck immersion and bottom emersion GZ: time-dependent roll righting arm. The values of GZ,, and g e are numerically determined for each time step. Therefore, they implicitly depend on variations in the motion variables; for example, an instantaneous change in heave alters the calculated submerged volume and center of buoyancy. In this sense the model allows for nonlinearities in sway and heave. Fig. 13 Sample wave profile (top) with enlarged region showing definition of points relative to t o (bottom). MARCH 2006 JOURNAL OF SHIP RESEARCH 71

10 Fig. 14 (a) Images taken every half wave period after release (T W = 23.64) for run leading to capsize. (b) Images taken every half wave period after release (T W = 23.24) for run leading to noncapsize. (c) Images taken every half wave period from T W = for run leading to noncapsize. T R and T W are normalized by the incident wave period. All three columns have initial conditions given by o 10 deg, o 0 deg/s, x o 0 cm, y o 0 cm, ẋ o 0 cm/s, ẏ o 0 cm/s, o / =.01, and = 6.8 rad/s. Recall, waves traveling from right to left as drawn in the schematic (Fig. 1) In regard to the roll equation specifically, g e4 represents a moment due to the nonlinear hydrostatic force and Froude-Krylov exciting force, i.e., g e4 r ( g e ) (Lee 2001). Testing was conducted on the numerical model to insure numerical convergence of the solution. A variable time-step Runga- Kutta fourth-fifth order integrator was used to numerically integrate the equations of motion. A time step of.01 second was used to determine the frequency with which the time-dependent coefficients are reevaluated and the integrator is called. However, with the variable time step integrator in each.01 increment, there can be multiple integrations performed. This procedure allowed for the greatest combination of numerical stability and accuracy. The intercept points of the body and wave were calculated to within.03 cm precision. Added mass, damping, and wave diffraction forces are assumed constant at each wave frequency and are numerically calculated 72 MARCH 2006 JOURNAL OF SHIP RESEARCH

11 Fig. 15 Vessel noncapsize roll motion (local roll angle). Released at seconds after wave maker start with an initial roll angle of 10 deg using the linear seakeeping program SHIPMO. The values of these coefficients are tabulated in Table 1. The SHIPMO program employs the slender-body strip theory approach of Salvesen et al. (1970) modified to account for the effects of blunt edges on such vessels as the barge presented in this work (Beck & Troesch 1990). With this blended model, dynamics of greater complexity involving three degrees of freedom can be studied; however, the hydrodynamics associated with water on deck, such as fluid sloshing and water egress, are not addressed (Obar et al. 2001) Numerical results for integrity curve analysis Based on the work of Soliman and Thompson (1991, Thompson 1997) in one degree of freedom, three degree of freedom results are examined using safe basin and integrity curve analysis. In essence, one can imagine that to truly represent the safe basin for a model with three degrees of freedom, one would need to consider a six-dimensional phase space with axes for roll, sway, heave, and their corresponding velocities. Additionally, this sixdimensional object would need to be generated for every wave MARCH 2006 JOURNAL OF SHIP RESEARCH 73

12 Fig. 16 Vessel noncapsize roll motion (global roll angle). Released at seconds after wave maker start with an initial roll angle of 10 deg height/frequency pair of interest. As this is not practical or feasible, the results are presented instead using the concept of integrity curves (Soliman & Thompson 1991). Each point on an integrity curve represents the ratio of safe area for a given wave amplitude in one phase space normalized by the safe area for the given set of initial conditions with a wave amplitude of zero. Each point then contains initial conditions that vary for roll and roll velocity, with all other initial conditions held to zero excepting the one condition that is the focus of the plot. Thus, each curve starts at an initial value of 1; however, a series of curves may have differing normalization factors. Recall that for the simulations comprising these results a pure cosine wave was used for the form of the wave excitation. Logically, for sway displacement, each curve is normalized by one as for zero wave amplitude, sway displacement initial conditions have no effect on integrity. Similarly, a numerical study has shown that sway velocity initial conditions are all normalized by values close to one. Figures 3 and 4 present the normalization 74 MARCH 2006 JOURNAL OF SHIP RESEARCH

13 Fig. 17 Comparison of numerical and experimental capsize basin boundaries at t o 2. Heave and sway displacements and velocities initially set equal to zero factors for heave, heave velocity, and sway velocity that will be used later in the data presented in Figs. 5 to 9. By means of this analysis, the six-dimensional space can be represented in twodimensional graphical form. It is not truly necessary for each initial condition to have its own normalization factor. However, one must consider that various effects alter normalization. Since the integrity point is defined as the ratio of safe area to capsize area at a given wave height normalized by the ratio of safe area to capsize area for zero wave height, normalization will change based upon the size of the initial condition grid. In that regard normalization factors are somewhat arbitrary. To not normalize every initial condition to 1.0 could then make some integrity curves collapse to a flatter line and thus appear less steep or dangerous than the effects of variations in other degrees of freedom. This could cause one to falsely trivialize the effects of some degrees of freedom, and so each initial condition curve is normalized to 1.0 with normalization factors also presented graphically. Consider Fig. 3, where the normalization factor is plotted versus initial heave displacement. The curve passes through (0,1), indicating that zero heave is defined as the reference condition. If the model starts with an initial heave of y o/b 0.04, the normalization factor increases to 1.06, indicating that the safe area associated with that heave initial condition has 6% more safe area. Conversely, for a heave displacement of y o/b 0.04, the integrity factor decreases to 0.84, indicating that the safe area has decreased by 16% relative to the zero heave reference condition. Figure 5 represents a series of sway integrity curves over one wave length, i.e., 0 x o, where x o is the initial sway displacement. That is, the sway displacement is the only nonroll initial condition varied; the heave displacement and the heave and sway velocities are all initially zero. In sway each curve is normalized by the same factor, because a sway of any amount, with a wave amplitude of zero, yields identical results. Additionally, the integrity curves were generated with a simple cosine wave forcing ensuring that x o 0 and x o represent identical curves. This would not be true for the time-varying envelope fit of the forcing. Figure 5 clearly demonstrates the influence of sway initial condition on the ultimate state of the vessel. Capsize simulation starting with x o 0to.68 produces the smallest range of wave amplitudes where the integrity curve is nonzero, while simulations starting with x o.74 to appear to be significantly safer over a wider incident wave amplitude range. Slight changes in sway initial position (or conversely in release time, altering one s location on the wave profile) can be the difference between capsize and safe behavior. In Fig. 6, it is shown that the size of the grid used for each phase space makes small differences primarily in the steepness of the cliff (Soliman & Thompson 1991) with which the safe basin degrades. Unless otherwise noted, the safe basins that result in the included integrity curves are based on a phase space defined as ranging from roll angles of 20 to 20 deg and roll velocities of 15 to 15 deg/s with increments of 1 deg and 1 deg/s, respectively. For MARCH 2006 JOURNAL OF SHIP RESEARCH 75

14 Fig. 18 Comparison of numerical and experimental capsize basin boundaries at t o 1.5. Each point has nonzero values for heave and sway displacements and velocities due to motion of vessel after release at t o 2 with zero initial conditions in heave, sway, heave velocity, sway velocity the comparison in Fig. 6, the larger grid consists of roll angles from 30 to 30 deg and roll velocities from 30 to 30 deg/s with increments of 1 deg and 1 deg/s, respectively. For consistency with experimental measurements, these numerical simulations are completed at a frequency of rad/s, approximately three times the small amplitude natural frequency of the box barge. In the previous two figures, wave amplitude ranges produce an integrity value above one. These values confirm what intuition implies, namely that there are initial condition pairs that under forcing are in fact more stable than with an absence of forcing. Imagine physically in a static case the barge will capsize for any initial roll angle beyond its angle of vanishing stability (11.4 deg). However, a factor from one of the other initial conditions, be it a sway or heave displacement and/or velocity, can provide a stabilizing force on the system. The other initial conditions (i.e., ẋ o, y o, and ẏ o ) are considered in the other integrity curve plots. Figure 7 demonstrates the relatively small influence of sway velocity on the stability of the vessel with heave, heave velocity, and sway displacement initial conditions set equal to zero. Figures 8 and 9, however, demonstrate the substantial influence heave and heave velocity have upon the ultimate stability of the vessel. Thus, the importance of the three degree of freedom study becomes apparent. It is clear that in an experiment, a vessel that may be marginally stable, but given a certain set of heave, heave velocity, sway, and to a lesser extent sway velocity initial conditions, may become unstable. The converse also holds: a seemingly unstable vessel can be driven to stability based upon the values of the six initial state variables. This proves crucial in analysis of the experimental results, as the experimental forcing wave amplitude per wave length was.01 (or a wave amplitude of cm). As seen in Figs. 5 to 9, this wave height corresponds to the steepest region of the integrity curves in the numerical simulation Experimental setup 3. Experiments The model used for the primary experiments was a simple box barge with a Plexiglas main deck and an aluminum platform supported by four threaded rods. Mounted on the platform were two infrared lights. The length of the model was 66.0 cm. The model had a draft of cm with a freeboard of 1.12 cm. (see Fig. 10). The center of gravity was adjusted to give an angle of vanishing stability, experimentally determined, of 11.4 deg. The deck became awash when the hull heeled approximately 5 deg, port or starboard. See Table 1 for numerical values of all coefficients. The experiments were conducted in the Gravity Wave Facility (35 m long, 0.75 m wide, and 1.5 m deep) at the University of 76 MARCH 2006 JOURNAL OF SHIP RESEARCH

15 Fig. 19 Comparison of numerical and experimental capsize basin boundaries at t o 1. Each point has nonzero values for heave and sway displacements and velocities due to motion of vessel after release at t o 2 with zero initial conditions in heave, sway, heave velocity, sway velocity Michigan Marine Hydrodynamics Lab. In order to synchronize the data collection and motion collection systems, much of the process was automated. Two computers were used to control the data collection and activation of the plunger style wave maker. The model was placed in the tank perpendicular to the wave maker. The model was fixed in an initial position by electromagnets suspended from the top of the tank. The magnets held the model by brackets mounted on the platform. The electromagnets were on a pivoting arm, which allowed for the vessel to be set at an initial roll angle and enabled movement of the magnets out of the way of the platform after model release. It was critical to the analysis of this experiment to develop a system that could capture the motion of the vessel without itself affecting that motion. Two infrared LED lights were mounted on the platform of the model. A COHU 4915 High Performance Monochrome CCD Camera was mounted, in plane with the lights, with the entire test section in view. The manual iris on the camera was then closed until only the IR lights were picked up. The rest of the test section was blacked out. The movie was captured at 30 frames per second in an MPEG-1 format, decomposed into individual frames and analyzed using Matlab to scan each image to find the locations of the model and indicator lights. From this information the relevant six state variables could be determined for the center of gravity on each of the 181 experimental runs conducted. Due to the 30 frames per second limitation, each wave cycle comprises approximately 27 frames, thus serving as the primary source of experimental error (i.e., this allows up to a 3% to 4% margin in the calculation of sway position). An additional source of experimental error arises from the pixel resolution of the images. If one were to assume a maximum of one pixel potential error in estimating the locations of the indicator lights, errors in roll could be as large as 0.8 deg. However, due to an averaging scheme to find the centroid of the illuminated lights, e.g., subpixel resolution, the errors in the indicator light location may range from 0.1 to 0.25 pixels, which corresponds to a maximum 0.2 deg error in roll angle measurement. Therefore, on the time series presented herein the roll angle is accurate to within 0.2 deg. Figure 11 demonstrates the repeatability of two similar runs released at the same time with release roll angles differing by 0.16 deg. While there are some phasing differences in the roll motion likely due to subtle differences in initial conditions of the six state variables, the motion is similar with an identical end result. A box barge was chosen as the vessel for this experiment due to its hydrostatic simplicity. The location of the center of gravity and expected angle of vanishing stability were numerically determined. Tests were then conducted manually heeling the model to one side to experimentally verify the angle of vanishing stability. Roll decrement tests were conducted in order to determine the roll natural period of the model. The nature of the decrement time history changed significantly when a deck edge would submerge. Therefore, the roll natural period was determined for angles of roll less than 5 deg. MARCH 2006 JOURNAL OF SHIP RESEARCH 77

16 Fig. 20 Comparison of numerical and experimental capsize basin boundaries at t o.5. Each point has nonzero values for heave and sway displacements and velocities due to motion of vessel after release at t o 2 with zero initial conditions in heave, sway, heave velocity, sway velocity The primary wave probe used to capture the wave profile of each run was mounted 8.86 m down-tank from the wave maker. Four reference IR lights were mounted on the exterior of the test section glass in each corner of the video frame to allow measurement of camera twist as well as to indicate both wave maker activation and model release. For further details of the experimental setup, refer to Obar et al. (2001). 3.2 Determination of wave conditions The physical dimensions of the Gravity Wave Tank and the wave maker design limited the available frequency of the waves and wave heights. Being unable to excite the vessel at or near its roll natural frequency of Hz, the model was excited at a super-harmonic of Hz, or three times the small-amplitude roll natural frequency; small here means roll amplitudes that excluded deck immersion. Experimental limitations prevented excitation of the barge at its roll natural frequency. Thus, the 1:3 ratio of excitation frequency to roll natural frequency was chosen to excite a superharmonic resonance condition. A further discussion of the implications of this choice are presented in Section 4. The appropriate incident wave height was determined experimentally. Several wave runs were conducted with no artificial initial conditions imparted to the vessel. The vessel was set motionless at the release location and a test wave train was generated. Capsize or no capsize was recorded, and a second wave height was attempted. Once the smallest wave height that consistently capsized the model was identified, the height setting was then slightly reduced. The goal was to find the largest wave height where the vessel did not consistently capsize, indicating behavior similar to the high slope region of the integrity curves shown in Figs. 5 to 9. That height, 2.67 cm ( o /.01) for the particular set of barge particulars and incident wave frequency described here, became the excitation wave height. The intent of this process was to determine an experimental critical wave height (Soliman & Thompson 1991) where initial conditions imparted upon the vessel would create the likely conditions needed for capsize. During these early tests, it was noticed that a slight longitudinal variation in the release point of the vessel (i.e., the location of the vessel relative to the wave maker) led to significant changes in the wave height needed to capsize the vessel. This coincides with the numerical results presented in the previous section, which imply the significant dependence of capsize on initial sway (e.g., Fig. 5). Once the wave frequency and height were determined, the repeatability of the wave generation capabilities of the tank was verified. The waves were found to be repeatable to within 5% of the wave amplitude and 0.05% of the wave period. Three wave probes allowed verification of wave repeatability as well as measurement of a critical cutoff time to the time series to guarantee vessel capsize occurs prior to the arrival of reflected waves off the far end of the tank (Obar et al. 2001). 78 MARCH 2006 JOURNAL OF SHIP RESEARCH

17 Fig. 21 Comparison of numerical and experimental capsize basin boundaries at t o. Each point has nonzero values for heave and sway displacements and velocities due to motion of vessel after release at t o 2 with zero initial conditions in heave, sway, heave velocity, sway velocity Since the model was allowed to translate down the tank during a run, the complete spatial description of the wave profile was required. However, determining the exact wave elevation at every point in the test section using wave probes was impractical. To capture the spatial behavior of the wave elevation at every point in the test section, a fluorescein solution was dissolved into the tank. The water surface was then illuminated with a laser sheet. Three runs were conducted without the model in the tank and a video time history was recorded and analyzed. Figure 12 presents an example video frame. Given the undisturbed wave profile as a function of time and space, it was possible to synchronize time and location of the desired wave crest (or trough) with the barge center of gravity, thus defining the starting reference phase or time. Here t o+ is defined as the time reference from the maximum wave crest. See Fig. 13 where the definitions used throughout this document for the location of the wave crests are defined. t o is the time of maximum wave crest, and ( 0) is the fraction of wave cycles before or after t o. For example, t o 1 refers to the wave crest prior to the maximum crest at t o. Attached to the model were teflon bumpers approximately 3/16 in. wide. While the model was not in constant contact with the tank, if yaw was encountered the teflon bumpers restricted the motion to create an idealized two-dimensional section in which any effects of yaw are negligibly small and a beam-sea condition can be assumed Poincaré mapping details The method of Poincaré mappings is a useful tool to understand global system behavior. Often this technique involves the dimensional reduction of a problem while enhancing conceptual clarity (Wiggins 1990). Typically, the construction of a Poincaré map consists of data points sampled at some logical frequency or event based upon the physics of the data being sampled. For a periodically forced oscillator, it is customary to base the Poincaré sampling time on the period of the forcing. For a discontinuous system, the sampling time might be the point of discontinuity (Shaw & Holmes 1983). For the capsize process studied here, the sample time was referenced to t o, the onset of approximately constant incident wave period and amplitude. Sampling was then conducted at the location of intersection of the center of gravity of the model with the projected location of a wave peak or trough in the absence of the model. This Poincaré sampling strategy accounts for the sway and drift motion of the model. Consequently, sampling is correlated to a physical event, namely, the occurrence of a peak or trough, rather than at a fixed temporal moment. If the process is diffeomorphic, the qualitative description of the map is independent of the sampling phase, that is, the initial time when the map construction starts. When analyzing the experimental results presented in this paper, the authors had two choices for when to begin sampling and two choices for sampling period. One could either begin sampling based upon the time from model release, MARCH 2006 JOURNAL OF SHIP RESEARCH 79

18 Fig. 22 Comparison of numerical and experimental capsize basin boundaries at t o+.5. Each point has nonzero values for heave and sway displacements and velocities due to motion of vessel after release at t o 2 with zero initial conditions in heave, sway, heave velocity, sway velocity which assumes that the transients due to the roll natural period T n are most important. Alternatively, time could be referenced to the excitation, e.g., sampling based upon the relative location to a wave crest. This second approach assumes incident wave forcing is the most important time scale. In either case, sampling based on the roll natural period or sampling based on the incident wave encounter period will not likely be independent of starting phase. As implied earlier by Fig. 5, a change in the start time of map construction is equivalent to a change in sway initial condition. Initial examination of several capsize/noncapsize sequences strongly suggested that a Poincaré map based upon a sampling strategy of a wave crest coinciding with the longitudinal location of the model center of gravity yielded the most insight to the process. Columns a and b of Fig. 14 show the comparison of two test runs, capsize and noncapsize, referenced to time of model release. Columns a and c of Fig. 14 show the same two experiments referenced to wave maker start time and thus approximately referenced to wave crest location. In these figures, T R is the time from model release normalized by T o, and T W is the time from the start of the wave maker also normalized by T o. The two experimental runs shown in Fig. 14 started from the same initial conditions for heave and roll displacements and sway, heave, and roll velocities, but were released at different times relative to the start of the wave maker. The time difference was approximately 0.40 of a wave period corresponding to a difference in the sway displacement initial condition of The similarities in the roll motion for the two runs quickly disappear following the release of the models. For example, compare (a, T R 0.0) and (b, T R 0.0) with (a, T R 1.02) and (b, T R 1.02). The capsize run a experiences deck immersion while the noncapsize run b has deck dry. Comparing the two runs relative to wave maker start (i.e., a compared to c), however, shows gross similar behavior for water on deck in that both runs have deck immersion at corresponding values of T W. The difference between capsize and noncapsize can be seen in (a, T W 25.64) and (c, T W 25.64). Both runs at this point have waves breaking across the deck similar to shallow water bores. However, due to the increased roll angle of a, the bore is more severe than that for c. The water-on-deck dynamics combined with the increased roll angle and larger roll velocity lead to subsequent capsize (a, T W 26.65), while the vessel of (c, T W 26.65) recovers, establishing a precarious equilibrium between the various forces. 4. Discussion In order to impart various initial conditions, the model was placed at five different roll angles and released at 12 different release times for each angle. By releasing the model at set intervals prior to t o, the model experienced a range of initial values for its six state variables, i.e., sway, heave, and roll motions. Figure 15 displays examples of captured motion plots. The upper plot shows roll angle versus time. All time-based position plots 80 MARCH 2006 JOURNAL OF SHIP RESEARCH

19 Fig. 23 Comparison of numerical and experimental capsize basin boundaries at t o+1. Each point has nonzero values for heave and sway displacements and velocities due to motion of vessel after release at t o 2 with zero initial conditions in heave, sway, heave velocity, sway velocity reference the energizing of the wave maker at time t 0. In Fig. 15, the model was released about seconds after the wave maker was energized and is notated on the time series plots with a square. The model was held at a release angle of 10 deg (starboard). The roll angle in Fig. 15 is displayed as the local roll angle (Chen et al. 1999). This is the roll angle relative to the wave slope at the location of the center of gravity of the model. The circles in the time history denote the Poincaré sampling points, i.e., the instants at which wave peaks or troughs overlap the center of gravity of the model. The time interval of the Poincaré samples corresponds approximately to twice the wave encounter period (due to sampling at both peaks and troughs) after adjusting for the mean drift of the model. The angle of vanishing stability is also marked. Note that there are many instances where the local roll angle exceeds but does not capsize. The motion displayed is typical of the general trend for every noncapsize run. Runs that were released with a starboard roll angle rolled down-tank initially, but eventually settled in to the motion seen in the figure where the mean heel angle was toward the incoming waves, i.e., heel to starboard. Runs that were released with a port roll angle rolled up-tank immediately, settling into a similar motion. Figure 16 contains the same test run, only the analysis is presented with the global roll angle. The global roll angle is referenced to the earth fixed horizontal axis. The Poincaré points are again sampled at the instants when wave peaks and troughs intersect the vessel s center of gravity. As in Fig. 15, there are periods where the global roll angle exceeds but does not capsize. Both Figs. 15 and 16 are based on the same data set but presented in different ways. By comparing the two, insight into the relevant physics may be gained. First, it is clear that both presentation methods show the roll angle exceeding the static angle of vanishing stability,. This is more pronounced for the local roll angle, Fig. 15, due to an asynchronism between the roll angle and local wave slope. The exceedence beyond the static stability limit demonstrates that a purely static analysis of capsize, either for a relative or absolute roll angle, will miss important dynamic features. Next, a comparison of the two figures shows that the global roll angle, Fig. 16, emphasizes the roll natural period more than the local roll angle. The roll natural period has increased from three to approximately four times the wave encounter period, reflecting the increased inertia and loss in restoring moment associated with water on deck. (Recall that the ratio T o /T n is based on small-amplitude rolling where the deck is not immersed.) Also note that the response at large-amplitude roll resonance is increasing rather than decaying, suggesting superharmonic excitation. And finally, the phase of the Poincaré samples appears to be more consistent relative to the global roll angle rather than the local. In Fig. 15, since the first Poincaré point corresponds to a peak and continues alternatively between troughs and peaks from then on, the Poincaré points map consistently lags 90 to 180 deg out of phase with the local roll motion. The global motion in Fig. 16 MARCH 2006 JOURNAL OF SHIP RESEARCH 81

20 Fig. 24 Roll trajectories for constant wave amplitude ( ft), wave frequency ( = ), sway velocity (0 ft/s), heave (0 ft), heave velocity (0 ft/s), roll ( o = 14 deg), and roll velocity ( o = 10 deg/s). Released at t 0 1 with varying sway initial conditions demonstrates that the local roll motion typically leads the wave motion by 0 to 60 deg. For reference, a linear prediction of roll for this rectangular hull form (e.g., Beck & Troesch 1990), which excludes water-on-deck effects and the large mean heel angle, has the roll maximum leading the wave crest by 55 deg. The preceding discussion on the effects of initial conditions and phase of excitation has provided the necessary foundation for understanding the apparent inconsistencies and randomness shown in Fig. 2. That figure is a projection of a sixdimensional phase-space surface onto a two-dimensional rollroll velocity plane. The definition of the starting phase, referenced to a particular point in the incident wave profile, is exactly analogous to setting a corresponding value to the sway displacement initial condition. To provide more structure and therefore a better understanding of the dynamics, the results have to be reevaluated or replotted with common initial conditions in as much as it is possible. Specific points in the wave cycle are compared with numerical results by an iterative method to determine the location of the wave relative to the center of gravity at particular instants (i.e., phases) in time. Using this method of numerically overlaying the experimental wave profile atop the center of gravity position, one can determine the conditions in heave, heave velocity, sway velocity, roll, and roll velocity for the center of gravity at the maximum wave crest, trough, or any other point of interest (note: sway, by definition, becomes zero). This was done for each experimental run to determine a set of initial conditions for each peak and trough in the wave cycle from t o 2 to t o+1 as defined in Fig. 13. These results are presented graphically in Figs. 17 to 23. Figure 17 is essentially a flattened six-dimensional space. The numerical results presented in that plot are purely for variations in roll and roll velocity initial conditions; that is, the other state variables are set equal to zero. The figure does not account for the variations in heave, heave velocity, and sway velocity initial conditions. It was found that once sway (i.e., starting sampling phase) was defined, the greatest impacts upon capsize are caused by heave and heave velocity (as can be seen in Figs. 7 to 9). However, to more closely simulate experimental results, rather than generating a numeric phase space portrait at each peak and trough, a sweep of the phase space is made at t o 2 and then each experimental or numerical point leading to capsize or noncapsize is mapped through time until t o+1 (Figs. 18 to 23). In essence, these plots to some degree account for the presence of the other degrees of freedom by releasing the numerical model at a time of t o 2 and allowing coupling in the other degrees of freedom to influence the shape of the safe basin. Subsequent maps of these capsize points extend out into the tendrils (e.g., Fig. 21), while the noncapsize region collapses to a central area as shown. Again, this is not a completely rigorous comparison, as the experimental runs were not all released with 0 initial conditions in heave, heave velocity, and sway velocity at t o 2. Instead they were released at a variety of release points up to t o 2 with a variety of nonzero initial conditions. It should be noted that the numerical model often capsized in the opposite direction as the experimental runs. It is believed that this is due, in some part, to significant drift forces not ac- 82 MARCH 2006 JOURNAL OF SHIP RESEARCH