A combined numerical empirical method to calculate finitetime Lyapunov exponents from experimental time series with application to vessel capsizing


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1 Ocean Engineering 33 (6) A combined numerical empirical method to calculate finitetime Lyapunov exponents from experimental time series with application to vessel capsizing Leigh McCue a,, Armin Troesch b a Aerospace and Ocean Engineering, Virginia Tech, Blacksburg, VA 461, USA b Naval Architecture and Marine Engineering, University of Michigan, USA Received 16 May 5; accepted 1 September 5 Available online 4 January 6 Abstract This paper presents a method to calculate finitetime Lyapunov exponents (FTLEs) for experimental time series using numerical simulation to approximate the local Jacobian of the system at each time step. This combined numerical experimental approach to the calculation of FTLE is applicable to any physical system which can be numerically approximated. By way of example, the method is applied to the problem of vessel capsize. r 5 Elsevier Ltd. All rights reserved. Keywords: Finitetime Lyapunov exponents; Capsize; Jacobian 1. Introduction The strong sensitivity of vessel capsizing to initial conditions has been a subject of research for decades (Paulling and Rosenberg, 1959; Thompson, 1997; Spyrou and Thompson, ; Lee et al., 6). This sensitivity is an inherent sign of a chaotic system (Theiler, 199), therefore an intuitive approach to the quantitative study of Corresponding author. Tel.: ; fax: address: (L. McCue) /$  see front matter r 5 Elsevier Ltd. All rights reserved. doi:1.116/j.oceaneng.5.9.8
2 L. McCue, A. Troesch / Ocean Engineering 33 (6) capsize is to employ Lyapunov exponents. The Lyapunov exponent is a measure of the rate of convergence or divergence of nearby trajectories with positive values indicating exponential divergence and chaos. However, Lyapunov exponents are by definition an asymptotic parameter, whereas capsize is a finitetime phenomena. Therefore, finitetime Lyapunov exponents (FTLE) must be used to investigate behaviors leading to capsize. To yield insight applicable to realistic vessels, a feasible and physical method to calculate FTLE from experimental time series in combination with a simplified numerical model is presented. The use of Lyapunov exponents to study capsize has been touched upon in the literature for both naval architecture and nonlinear dynamics. In recent years, the asymptotic Lyapunov exponent has been calculated from equations of motion for the mooring problem (Papoulias, 1987), singledegreeoffreedom capsize models (Falzarano, 199; Murashige and Aihara, 1998a,b; Murashige et al., ; Arnold et al., 3), and works studying the effects of rudder angle while surfriding as it leads to capsize (Spyrou, 1996). Additionally, the authors conducted a study of the use of Lyapunov exponents to investigate large amplitude vessel roll motions in beam seas for a multidegree of freedom numerical model in comparison to experimental results (McCue, 4; McCue and Troesch, 4). Based upon the results of this study, it was shown that the Lyapunov exponent can be used as a validation tool for large amplitude roll motion simulators. Through calculation of similar maximal Lyapunov exponent for experimental runs and numerically simulated runs, one can conclude that the numerical simulator likely captures the relevant, multidimensional physics of the problem. However, since the Lyapunov exponent is defined in the limit as time approaches infinity, it is ineffective for the study of the finitetime phenomena of capsize. This serves as the motivation for the present work in which capsize is studied using the finitetime measuregivenbytheftle. In order to be of use onboard a vessel and to make a sizeable improvement in safety, it is necessary to compute FTLEs from actual vessel time series in real time. While it is not impossible to approximate FTLEs through statistical methods (Lu, 1997; Lu and Smith, 1997) or approaches derived from those used for Lyapunov exponents (Wolff, 199; Yao and Tong, 1994; Sano and Sawada, 1985), it is preferable for this research that the Jacobian approximation be calculated rapidly and in such a manner as to be physically intuitive. For a system such as the capsize model discussed herein, a numerical approximation for the Jacobian is readily available via a simulation tool. Therefore, to estimate the FTLEs for the experimental time series, a combined numerical experimental approach is used. Rather than using statistical or dimensionally limited methods to approximate the Jacobian of the system in time, a validated numerical simulator can be used to model the Jacobian in an incremental manner. For example, if given a time series from experimental data, and a numerical simulator capable of accurately integrating the equations of motion for an approximation of the system, one can use the simulator in a stepwise manner to estimate the Jacobian about each point in the experimental time series. For the system discussed in this paper, sixstate variables are recorded in 1/3th of a second increments. Reading into the numerical simulator a row of experimental data containing these sixstate variables, and integrating the linearized
3 1798 L. McCue, A. Troesch / Ocean Engineering 33 (6) form of the equations of motion over the following 1/3th of a second increment yields the Jacobian of the system and from that the FTLE at each step in time. The experimental values of the sixstate variables are therefore treated incrementally as initial conditions in the numerical simulation. Details of the numerical simulator, experimental data, and methodology are contained in the following section. As an aside, the use of Lyapunov exponents should not be confused with Lyapunov s direct method. Lyapunov s direct method is an analytical stability approach studied heavily for the capsize problem by Odabasi and collaborators in the 197s 198s (Kuo and Odabasi, 1975; Odabasi, 1976, 198). An overview of this method is also given more recently by Fossen (1994). Spyrou and Thompson () note that at the time Odabasi published his ideas on the use of Lyapunov functions, Odabasi s work was possibly too mathematical for common acceptance in the naval architecture community. It is the opinion of the authors that such mathematical approaches paved the way for ongoing work in the field of naval architecture using nonlinear dynamics theory and various mathematical tools through the 199s and early 1st century. This includes, but is not limited too, works by Arnold et al. (3), Chen and Shaw (1997), Chen et al. (1999), Falzarano (199), Falzarano et al. (199), Nayfeh (1988), Bikdash et al. (1994), Soliman and Thompson (1991), Spyrou (1996), Spyrou and Thompson (), Spyrou et al. (), Thompson et al. (1987), as well as the authors, McCue and Troesch (4) and McCue (4).. Background Determination of the Lyapunov exponent from a numerical simulation is relatively straightforward with the primary nontrivial detail arising from accurately finding the linearized form of the equations of motion about each point in the simulation. For a system of equations written in state space form _x ¼ uðxþ, small deviations from the trajectory can be expressed by the equation d _x i ¼ðqu i =qx j Þdx j (Eckhardt and Yao, 1993). dx is a vector representing the deviation from the trajectory with components for each state variable of the system. Using this Jacobian, the Lyapunov exponent, defined by Eq. (1), can be calculated through a series of progressive Gram Schmidt reorthonormalizations which are then summed in Eq. () in which m represents the number of renormalization steps conducted and L denotes the length of each element (Wolf et al., 1985; Wolf, 1986). l 1 ¼ lim t t!1 1 ðl 1 Þ m ¼ 1 t X m j¼1 log kdxðtþk kdxðþk, (1) log Lðt jþ1þ Lðt j Þ. () Numerous algorithms are developed for calculating the Lyapunov exponent for an experimental time series. In the foundation work comparing the Lyapunov exponent for numerical and experimental noncapsize large amplitude roll motion data, the tangent space Sano and Sawada (1985) method included in the TISEAN (Hegger
4 et al., ) package was used to calculate Lyapunov exponents for the experimental time series. Mean values for the maximal Lyapunov exponent of noncapsize runs for the numerical simulation and experimental data were found to be 1.77 and /s, respectively (McCue, 4; McCue and Troesch, 4). This level of comparison is the basis for the conclusion that the numerical model likely captures the relevant multidimensional physics of the experiments. The FTLE from ordinary differential equations is calculated in much the same manner as the Lyapunov exponent. The n sets of differential equations linearized about the fiducial trajectory are calculated to measure incrementally stretching and shrinking principal axes. The n linearized sets, where n is the dimension of the phase space, are reorthonormalized after each step in the same manner as conducted for the asymptotic Lyapunov exponent. Eq. (3), which gives the definition of the FTLE is discretely represented by Eq. (4) (Eckhardt and Yao, 1993). Calculation of FTLE in using solely the numerical model demonstrates the usefulness of this finitetime quantity for detecting chaotic behaviors of the capsizing system (McCue and Troesch, 4). Yet it is of use to be able to calculate FTLE from experimental time series both to yield greater insight into the dynamics of the system as well as to enable the development of realtime predictive tools. l T ðxðtþ; dxðþþ ¼ 1 T kdxðt þ TÞk log, (3) kdxðtþk l 1 ðxðtþ; DtÞ ¼ 1 Lðt þ DtÞ log. (4) Dt LðtÞ Brief details of the numerical and experimental models are contained in the following two subsections. For further detail refer to Obar et al. (1), Lee et al. (6) or McCue and Troesch (3)..1. Experimental setup ARTICLE IN PRESS L. McCue, A. Troesch / Ocean Engineering 33 (6) One hundred sixtyfive separate experiments were conducted in which a box barge was excited in beam seas. The experiments were conducted by Obar et al. (1) in the Gravity Wave Facility (35 m long,.75 m wide, and 1.5 m deep) at the University of Michigan Marine Hydrodynamics Lab. A schematic is shown in Fig. 1. Those tests effectively modeled a threedegreeoffreedom (sway, heave, and roll) twodimensional freely floating y y 9 x 9 φ x Fig. 1. Coordinate system for capsize model.
5 18 L. McCue, A. Troesch / Ocean Engineering 33 (6) Fig.. Sketch of barge with dimensions (Obar et al., 1). rectangle with water on deck. The model used for the primary experiments was a simple box barge. Mounted on an aluminum platform were two infrared lights used to track the model via a Matlab motion collection system. The motions were determined by analyzing the locations of the infrared lights in time relative to fixed references as well as the location of the wave relative to the center of gravity of the body. The model had principle dimensions as follows: length, 66. cm; draft, 18.5 cm; freeboard, 1.1 cm; angle of vanishing stability y v,11:4 (see Fig. ). The deck became awash when the hull heeled approximately 5, port or starboard. Within certain critical wave amplitude and frequency ranges, the states of capsize or noncapsize were functions of how and when the model was released, therefore demonstrating a strong sensitivity to initial conditions. See Table 1 for numerical values of all coefficients. For further details of the experimental process refer to Obar et al. (1) or Lee et al. (6)... Numerical model A quasinonlinear time domain simulation is used to predict capsizing behavior of a twodimensional rectangular body. The model accounts for the hydrostatic effects of water on deck, including deck immersion and bottom emersion, timedependent roll righting arm and submerged volume, and an effective gravitation field which accounts for centrifugal forces due to the circular particle motion. The model is limited in that it makes use of a longwave assumption and added mass and damping values are calculated from a linear seakeeping program, SHIPMO (Beck and Troesch, 199), at a fixed frequency. While this longwave model is admittedly simplistic, it captures the essence of quasistatic 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. For details into this methodology and assumptions, see Lee (1) and Lee et al. (6). The equations of motion for this numerical model in the inertial coordinate system are written below in Eq. (5) where subscripts of, 3, and 4 represent sway, heave, and roll degrees of freedom, respectively.
6 L. McCue, A. Troesch / Ocean Engineering 33 (6) Table 1 Definition of coefficients Coefficients for numerical model from SHIPMO Experimentally determined coefficients m 5.31 kg/m T n.75 s a =m.3481 T =T n 1/3 a 4 =ðmbþ.6 l 1.3 m a 33 =ðmþ.891 B=l.3 a 4 =ðmbþ.6 T=l.137 a 44 =ðmb Þ.467 fb=l.84 I cg =ðmb Þ.1338 b =ðmoþ b 4 =ðmobþ.11 b 33 =ðmoþ.168 b 4 =ðmobþ.11 b 1 =ðmob Þ.46 b =ðmb Þ.83 jf D j=ðmgz Þ.97 jf D 3 j=ðmgz Þ.534 jf D 4 j=ðmgbz Þ m þ a a 4 x g b b 4 _x g 6 m þ a 33 7B y 4 g C A þ 6 b 33 7B _y 4 5 g A a 4 I cg þ a 44 f b 4 b 1 _f 3 1 rg e rþf D 1 þ 6 4 7B C A ¼ rg e3 r mg þ f D B 3 C b _fj fj A. rg e4 GZrþf D 4 ð5þ An explanation of terms is as follows (see Table 1 for numerical values): a ij, b ij : added mass and damping coefficients f D j : diffraction forces b 1 and b : linear and nonlinear roll damping coefficients g ei : timedependent sway and heave components of effective gravity x g : sway position of the center of gravity y g : heave position of the center of gravity f: roll angle r: timedependent volume of hull including possibility for deck immersion and bottom emersion GZ: timedependent roll righting arm The values of GZ, r, and g e are numerically determined for each time step. Therefore they implicitly depend on variations in the motion variables; for example
7 18 L. McCue, A. Troesch / Ocean Engineering 33 (6) 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. In regard to the roll equation specifically, rg e4 r represents a moment due to the nonlinear hydrostatic force and Froude Krylov exciting force, i.e. rg e4 r¼r ðrg e rþ (Lee, 1). The experimentally validated, blended model of Eq. (5) simulates hours of data in seconds allowing one to generate years of realtime data in a matter of days (McCue and Troesch, 3; Lee et al., 6). Calculating the Jacobian of the equations of motion used in the numerical simulator (Obar et al., 1; McCue and Troesch, 3; Lee et al., 6), given by Eq. (5) is relatively straight forward, though a few aspects are worthy of discussion as follows. While the mass and linear damping terms are easily treated, the quadratic damping and forcing terms require extra consideration. Two approaches to treat the quadratic damping term are as follows. One method is to replace, in the linearized model, the term fj _ fj _ with _f for f4, _ f _ for fo, _ and assume that the precise singularity at f _ ¼ will never be encountered due to double precision computational accuracy. The second approach is to use Dalzell (1978) treatment for quadratic damping. Dalzell (1978) fits an odd function series of the form fj _ fj¼ _ P k¼1;3;... a kð f _ k = f _ k c Þ. Solving for a k the truncated thirdorder fit becomes fj _ fj _ 5 _ 16 f f _ c þ ð f _ 3 = f _ c Þ over some range f _ c o fo _ f _ c (Dalzell, 1978). Basic testing indicated both treatments yield similar results, therefore the Dalzell treatment, with f _ c ¼ 1 degrees, was used for the results presented herein to avoid any difficulties due to the singularity associated with the first method. The linearized influence of the forcing side of the equation is calculated using a simple difference scheme. Forces are calculated as the difference between their values on the fiducial trajectory and their values at the offset from the trajectory. Due to linear superposition this can be calculated in a more computationally efficient manner for the differential at ðx þ dx; y þ dy; f þ df; tþ rather than conducting the summation of force differentials at ðx þ dx; y; f; tþ, ðx; y þ dy; f; tþ, and ðx; y; f þ df; tþ. Therefore, the linearized form of the equations of motion about the fiducial trajectory are written as Eq. (6). 3 1 m þ a a 4 d x g 6 m þ a 33 7B d y 4 5 g A a 4 I cg þ a 44 d f 3 b b 1 4 d _x g þ b B d _y g C 4 b 4 b 1 þ b ð 5 _ A 16 f c þ 35 _f 16 Þ _f c d f _ 1 rg e rþf D rg e rþf D ¼ B rg e3 r mg þ f D 3 C B rg e3 r mg þ f D rg e4 GZrþf D 4 rg e4 GZrþf D 4 ðxþdx;yþdy;fþdf;tþ 1 C A ðx;y;f;tþ. ð6þ
8 L. McCue, A. Troesch / Ocean Engineering 33 (6) From this linearized form of the equations of motion the Lyapunov exponent and FTLE can be calculated for the numerical simulation from Eqs. () and (4), respectively, with results presented in McCue and Troesch (4). 3. Finitetime Lyapunov exponents (FTLE) from experimental time series McCue and Troesch (4) presented calculations of asymptotic Lyapunov exponents from experimental results used in comparison to calculations of asymptotic Lyapunov exponents from numerical simulation. Specifically, consideration of noncapsize runs were used to demonstrate that the numerical model captures the physics of the experimental results as demonstrated by similar magnitude of maximal asymptotic Lyapunov exponent. Since the asymptotic Lyapunov exponent is defined in the limit as time approaches infinity, in McCue and Troesch (4) FTLE based upon the definition in Eq. (3) are calculated from numerical simulation to lend insight into the finitetime phenomena of capsize. This paper extends the method described in McCue and Troesch (4) to incorporate input of experimental time series into the numerical model to calculate a FTLE from experimental results coupled with numerical simulation Theory and implementation As discussed by Eckhardt and Yao (1993), it is relatively simple to calculate a FTLE from simulation of the equations of motion of the system. Calculation of the FTLE using solely the numerical model demonstrates the usefulness of this finitetime quantity for detecting chaotic behaviors of the capsizing system (McCue and Troesch, 4). In McCue and Troesch (4), comparison of numerically and experimentally calculated asymptotic Lyapunov exponents for noncapsize runs was used to argue that the numerical model presented in Eq. (5) likely captures the relevant underlying physics of the experimental problem. It is of use to be able to calculate FTLE from experimental time series both to yield greater insight into the dynamics of the system as well as to enable the development of realtime capsize prediction tools. Therefore, the numerical model was used in conjunction with the experimental time series to calculate the FTLE. In the experiments described in the previous section, data was measured at a rate of 3 frames per second. To calculate the FTLE, the values of each of the sixstate space variables of roll, roll velocity, sway, sway velocity, heave, and heave velocity, are entered into the numerical simulator. The equations of motion, along with the linearized form of the equations of motion are then integrated over the next 1/3th of a second to measure the rate of expansion or contraction of the principal axes of the infinitesimal sixdimensional sphere anchored to the fiducial trajectory defined by the experimental data. At the end of the 1/3th of a second integration step, the FTLE is calculated and new state variables are read into the simulation based upon the experimental time series. Therefore, the entire time series is read into the numerical simulator with the equations of motion simulated in 1/3th of a second
9 184 L. McCue, A. Troesch / Ocean Engineering 33 (6) intervals between experimental data points in order to yield values for FTLE defined by a combination of experimental data and numerical simulation. This is a fairly intuitive approach for engineering applications emulating real phenomena. However, the limitation of this approach is the inherent bias imposed upon the output data by the numerical model; using this methodology the dynamics inherent to the numerical simulator are also introduced. For this reason, it is of the utmost importance to verify that the numerical model appears to encompass the physics of the underlying chaotic system through qualitative comparison (McCue and Troesch, 3; Lee et al., 6) and quantitative validation calculations of the asymptotic Lyapunov exponent for a long, or, asymptotic benchmark. For this example the benchmark cases were those of large amplitude rolling motions not leading to capsize (McCue and Troesch, 4). The benefit to this methodology is in its potential application for the prediction of realtime fullscale vessel motions and instabilities. Roll Angle (deg) 6 4 θ =.13, dθ /dt=.7577 θ =.18, dθ /dt= Time (s) 1 5 θ =.13, dθ /dt=.7577 θ =.18, dθ /dt= Time (s) θ =.13, dθ /dt=.7577 θ =.18, dθ /dt= Time (s) Fig. 3. FTLE as a function of time for nearby capsize and noncapsize experimental cases released at time t ¼ 1:5667 s. Top panel shows roll experimental time series. Middle panel shows full time series for FTLE from t ¼ to 48.9 s. Bottom panel shows identical data over critical region from t ¼ 5 to 35 s. Initial roll and roll velocity for noncapsize and capsize runs equal to (.13 deg, :7577 deg =s), (.18 deg, :637 deg =s), respectively.
10 To use in an onboard sense, a combination of a realtime calculator with a numerical simulator could enable development of a numerical empiricaltype approach in which at every time step a FTLE is calculated and fed back into a numerical simulator along with the latest vessel parameters. Using a simulator that runs substantially faster than real time, such as that presented by the equations of motion 5, a system could be developed for detecting and warning of instabilities in vessel motions in real time onboard a ship. Similarly, this could be developed for any physical system which can be numerically modeled. The following subsection presents results demonstrating the potential for this form of application. 3.. Results ARTICLE IN PRESS L. McCue, A. Troesch / Ocean Engineering 33 (6) Consider Fig. 3 showing time series for nearby capsize and noncapsize experimental trajectories. The top panel presents roll time series for two cases which exhibit strong similarities up to capsize. There is no distinct phase difference in the motions as both roll trajectories are nearly identical until capsize. The middle 1 Noncapsize runs Time (s) 1 Capsize runs Time (s) Fig. 4. FTLE as a function of time for all runs released at time t ¼ 1:5667 s. Viewed over typical region from t ¼ 3 to 33 s. Top panel shows 6 time series leading to noncapsize. Bottom panel shows 8 time series leading to capsize. Initial conditions for all 14 time series in sixstate variables are given in Table.
11 186 L. McCue, A. Troesch / Ocean Engineering 33 (6) panel shows time series for the FTLE for the same neighboring cases generated using the combined numerical/experimental approach discussed in the previous subsection. The bottom panel gives the same data as the middle panel over the critical time region between 5 and 35 s. The FTLE, which is sensitive to the influence of all state variables, shows a phase difference between the two experimental time series. This serves as a means to demonstrate the sixdimensional phase space on a singledimensional plane. As is apparent on the bottom panel, the noncapsize run leads the capsize FTLE time series in phase. In the experiments, slight phase variations in all degrees of freedom can result in dramatically different end results. A sensitivity to initial conditions is a fundamental characteristic of a chaotic system; therefore, while the complicated dynamics of sixdimensional phase space are not discernable graphically, it is the hope that through further exploration of FTLE from experimental time series, a means to capture multidimensional effects in a single representation will be apparent. In Fig. 4, as in the bottom panel of Fig. 3, the FTLE for 14 runs, 6 noncapsize and 8 capsize, are plotted as a function of time. All 14 runs are released at the same time with different initial conditions given in Table. The phase of the FTLE for runs leading to capsize lag the phase of the FTLE for noncapsize runs. The time series are abridged over the period from t ¼ 3 to 33 s for visual clarity though this lag behavior is consistent throughout the time series. A brief investigation of all 165 experimental runs is presented in Figs It can be seen that in the time leading to capsize, those time series leading to capsize often lead or lag the majority of the noncapsize runs released at the same initial time. However, the distinctiveness and duration over which this lead/lag behavior occurs Table Initial conditions for runs plotted in Fig. 4 Roll (deg) Roll vel. (deg/s) Sway (ft) Sway vel. (ft/s) Heave (ft) Heave vel. (ft/s) Noncapsize runs Capsize runs All runs released at time t ¼ 1:5667 s.
12 L. McCue, A. Troesch / Ocean Engineering 33 (6) varies significantly. An area of future study is to conduct a more thorough examination of the lead/lag behavior between noncapsize and capsize runs at multiple release times and realistic hull geometry to determine if a consistent and quantifiable predictive measure can be distinguished. Unlike the asymptotic Lyapunov exponent where magnitude is used to verify the detection of the proper physics of the system, magnitude is a relatively trivial feature when considering a FTLE since it is dependent on the size of the time interval over which the finite exponent is calculated. The greater source of information lies in the relative behavior between different experiments of similar design. It is apparent in Fig. 8 that through this combined numerical experimental approach the peak FTLE for noncapsize cases is significantly smaller, with a narrower standard deviation, then the capsize cases. With further research, it is hoped that the detection of these peaks prior to capsizing and/or any potential consistent phase differences, are what will enable the use of FTLE as a predictive tool for the detection of capsize. Lastly, Fig. 9 illustrates the time period in which capsize occurs after encountering the largest finitetime Lyapunov exponent. The majority of cases capsize within Time (s) Fig. 5. FTLE as a function of time for all runs released at s (top), s (upper middle), s (lower middle), and.333 s (bottom). Capsize runs denoted with dotted lines, noncapsize solid lines. Viewed over typical region from t ¼ 5 to 35 s.
13 188 L. McCue, A. Troesch / Ocean Engineering 33 (6) Time (s) Fig. 6. FTLE as a function of time for all runs released at.333 s (top),.4333 s (upper middle),.6333 s (lower middle), and.8333 s (bottom). Capsize runs denoted with dotted lines, noncapsize solid lines. Viewed over typical region from t ¼ 5 to 35 s. onewave cycle of the maximum finitetime Lyapunov exponent. This is likely because the lowfreeboard vessel studied inherently capsizes rapidly. However, even for this simplistic example, some advance warning of instability is given via peaks of the finitetime Lyapunov exponent time series. Through use of the finitetime Lyapunov exponent as an indicator of instability, even in short time periods, some form of corrective measure can be undertaken. It is anticipated that through further studies on more realistic ship models, consistent phase and/or peak behavior will lead to an indicator that can be used to warn captains of impending danger with sufficient time to allow corrective measures. 4. Conclusions While the Lyapunov exponent is of use for longtime series, such as those not leading to capsize, for a finite event, such as capsize, a finitetime Lyapunov
14 L. McCue, A. Troesch / Ocean Engineering 33 (6) Time (s) Fig. 7. FTLE as a function of time for all runs released at s (top), s (upper middle), 3.5 s (lower middle), and 3.7 s (bottom). Capsize runs denoted with dotted lines, noncapsize solid lines. Viewed over typical region from t ¼ 5 to 35 s. exponent (FTLE) is necessary. Further development of the realtime numerical experimental FTLE approach presented in this paper could establish the method s potential for realistic vessel dimensions and hull forms in addition to countless other chaotic applications. A combined numerical experimental method for calculating FTLE from an experimental time series is a viable method for physical systems which can be reasonably accurately modeled by numerical integration of equations of motion. An important check of the validity of the model is comparison of the Lyapunov exponent for longtime simulations to ensure that the model does not fail to capture the relevant physics of the real system. Since the Lyapunov exponent is a system parameter indicating the rate of chaotic behavior it can be used in comparison between experimental and numerical runs to validate the physics of a numerical model thus justifying the use of a combined numerical empirical approach for the FTLE which is both intuitive and dimensionally unlimited. From the calculation of the FTLE further system information is gained through the capture of information in all sixstate variables. This information could
15 181 L. McCue, A. Troesch / Ocean Engineering 33 (6) Occurrences Mean=4.545, σ= Maximum short time Lyapunov exponent for experimental capsize time series 1 8 Occurrences 6 4 Mean=1.6395, σ= Maximum short time Lyapunov exponent for experimental noncapsize time series Fig. 8. Histograms indicating range of FTLE values for runs leading to capsize and noncapsize based upon experimental data. potentially be used in a predictive tool to indicate lost stability leading to capsize. A phase relationship was also detected between the FTLE of capsize and noncapsize runs for a small sampling of 165 time series. Greater investigation into this is necessary to determine if phasing could be used to predict capsize prior to a spike in FTLE time series. More work must be done to further identify the different types of capsize as characterized by small and large FTLE and the relationship between the phasing of noncapsize and capsize FTLE time series. Extending the principals and simulation tools presented in this paper towards time series for realistic vessels in random seas would provide useful data and could serve as a beneficial proof of concept. This work demonstrates preliminary data indicating that an onboard simulation tool paired with a predictive device such as the finitetime Lyapunov exponent can provide realtime warnings of chaotic behavior with the express purpose of saving cargo, ships, and lives. Further studies could quantify phase relations between capsize and noncapsize time series and/or define a practical and consistent means to detect peaks in the FTLE time series.
16 L. McCue, A. Troesch / Ocean Engineering 33 (6) Maximum short time Lyapunov exponent Maximum short time Lyapunov exponent Cycles from instant of maximum short time Lyapunov exponent to capsize Cycles from instant of release to capsize Fig. 9. (top) Peak value of largest local Lyapunov exponent as a function of the number of cycles from peak exponent to capsize based upon experimental data. (bottom) Peak value of largest local Lyapunov exponent as a function of the number of cycles from release to capsize based upon experimental data. Acknowledgments The authors wish to express their gratitude for funding on this project from the Department of Naval Architecture and Marine Engineering at the University of Michigan and the National Defense Science and Engineering Graduate Fellowship program. Additionally, the authors acknowledge Dr. YoungWoo Lee for the initial development of the numerical simulator used in this paper as well as the work of Lt. Michael Obar and Dr. YoungWoo Lee in conducting the experiments analyzed in this work. References Arnold, L., Chueshov, I., Ochs, G., 3. Stability and capsizing of ships in random sea a survey. Technical Report 464, Institut fu r Dynamicsche Systeme, Universität Bremen. Beck, R.F., Troesch, A.W., 199. Students Documentation and Users Manual for the Computer Program SHIPMO.BM. Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor.
17 181 L. McCue, A. Troesch / Ocean Engineering 33 (6) Bikdash, M., Balachandran, B., Nayfeh, A., Melnikov analysis for a ship with a general rolldamping model. Nonlinear Dynamics 6, Chen, S.L., Shaw, S.W., Phase space transport in a class of multidegreeoffreedom systems. In: Proceedings: DETC ASME Design Engineering Technical Conferences, September. Chen, S.L., Shaw, S.W., Troesch, A.W., A systematic approach to modeling nonlinear multidof ship motions in regular seas. Journal of Ship Research 43 (1), Dalzell, J.F., A note on the form of ship roll damping. Journal of Ship Research (3), Eckhardt, B., Yao, D., Local Lyapunov exponents in chaotic systems. Physica D 65, Falzarano, J.M., 199. Predicting complicated dynamics leading to vessel capsizing. Doctoral Dissertation, University of Michigan. Falzarano, J.M., Shaw, S.W., Troesch, A.W., 199. Application of global methods for analyzing dynamical systems to ship rolling motion and capsizing. International Journal of Bifurcation and Chaos (1), Fossen, T.I., Guidance and Control of Ocean Vehicles. Wiley, New York. Hegger, R., Kantz, H., Schreiber, T., et al.,. Tisean.1, nonlinear time series analysis. Kuo, C., Odabasi, A., Application of dynamic systems approach to ship and ocean vehicle stability. In: Proceedings: International Conference on the Stability of Ships and Ocean Vehicles, March. Lee, Y.W., 1. Nonlinear ship motion models to predict capsize in regular beam seas. Doctoral Dissertation, Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, MI. Lee, Y.W., McCue, L., Obar, M., Troesch, A., 6. Experimental and numerical investigation into the effects of initial conditions on a three degree of freedom capsize model. Journal of Ship Research, in press. Lu, Z.Q., Statistical estimation of local Lyapunov exponents: toward characterizing predictability in nonlinear systems. Preprint. Lu, Z.Q., Smith, R.L., Estimating local Lyapunov exponents. Fields Institute Communications 11, McCue, L.S., 4. Chaotic vessel motions and capsize in beam seas. Doctoral Dissertation, University of Michigan. McCue, L.S., Troesch, A.W., 3. The effect of coupled heave/heave velocity or sway/sway velocity initial conditions on capsize modelling. In: Proceedings: Eighth International Conference on the Stability of Ships and Ocean Vehicles, September. McCue, L.S., Troesch, A.W., 4. Use of Lyapunov exponents to predict chaotic vessel motions. In: Proceedings: Seventh International Ship Stability Workshop, November. Murashige, S., Aihara, K., 1998a. Coexistence of periodic roll motion and chaotic one in a forced flooded ship. International Journal of Bifurcation and Chaos 8 (3), Murashige, S., Aihara, K., 1998b. Experimental study on chaotic motion of a flooded ship in waves. Proceedings of the Royal Society of London A 454, Murashige, S., Yamada, T., Aihara, K.,. Nonlinear analyses of roll motion of a flooded ship in waves. Philosophical Transactions of the Royal Society of London A 358, Nayfeh, A.H., On the undesirable roll characteristics of ships in regular seas. Journal of Ship Research 3 (), 9 1. Obar, M.S., Lee, Y.W., Troesch, A.W., 1. An experimental investigation into the effects initial conditions and water on deck have on a three degree of freedom capsize model. Fifth International Workshop on the Stability and Operational Safety of Ships, Trieste, Italy, September. Odabasi, Y., Ultimate stability of ships. Transactions of the Royal Institution of Naval Architects 119, Odabasi, Y., 198. A morphology of mathematical stability theory and its application to intact ship stability assessment. In: Proceedings: Second International Conference on Stability of Ships and Ocean Vehicles, October. Papoulias, F.A., Dynamic analysis of mooring systems. Doctoral Dissertation, Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, MI.
18 L. McCue, A. Troesch / Ocean Engineering 33 (6) Paulling, J., Rosenberg, R., On unstable ship motions resulting from nonlinear coupling. Journal of Ship Research. Sano, M., Sawada, Y., Measurement of Lyapunov spectrum from a chaotic time series. Physical Review Letters 55 (1). Soliman, M.S., Thompson, J., Transient and steady state analysis of capsize phenomena. Applied Ocean Research 13 (). Spyrou, K., Homoclinic connections and period doublings of a ship advancing in quartering waves. Chaos 6 (). Spyrou, K., Thompson, J.,. The nonlinear dynamics of ship motions: a field overview and some recent developments. Philosophical Transactions of the Royal Society of London Spyrou, K., Cotton, B., Gurd, B.,. Analytical expressions of capsize boundary for a ship with roll bias in beam waves. Journal of Ship Research 46 (3), Theiler, J., 199. Estimating fractal dimension. Journal of the Optical Society of America A 7 (6). Thompson, J., Designing against capsize in beam seas: recent advances and new insights. Applied Mechanics Review 5 (5). Thompson, J., Bishop, S., Leung, L., Fractal basins and chaotic bifurcations prior to escape from a potential well. Physics Letters A 11 (3). Wolf, A., Quantifying Chaos with Lyapunov Exponents, Chaos. Princeton University Press, Princeton, NJ (Chapter 13). Wolf, A., Swift, J., Swinney, H., Vastano, J., Determining Lyapunov exponents from a time series. Physica D 16, Wolff, R.C., 199. Local Lyapunov exponents: looking closely at chaos. Journal of the Royal Statistical Society Series B 54 (), Yao, Q., Tong, H., Quantifying the influence of initial values on nonlinear prediction. Journal of the Royal Statistical Society Series B 56 (4),
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