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

Size: px
Start display at page:

Download "A combined numerical empirical method to calculate finite-time Lyapunov exponents from experimental time series with application to vessel capsizing"

Transcription

1 Ocean Engineering 33 (6) A combined numerical empirical method to calculate finite-time 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 finite-time 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: Finite-time 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: mccue@vt.edu (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 finite-time phenomena. Therefore, finite-time 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), single-degree-of-freedom 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 surf-riding 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 multi-degree 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, multi-dimensional 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 finite-time phenomena of capsize. This serves as the motivation for the present work in which capsize is studied using the finite-time measuregivenbytheftle. In order to be of use on-board 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, six-state variables are recorded in 1/3th of a second increments. Reading into the numerical simulator a row of experimental data containing these six-state 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 six-state 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 non-trivial 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 re-orthonormalizations 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 non-capsize 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 non-capsize 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 multi-dimensional 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 real-time 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 sixty-five 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 three-degree-of-freedom (sway, heave, and roll) two-dimensional 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 non-capsize 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 quasi-nonlinear time domain simulation is used to predict capsizing behavior of a two-dimensional rectangular body. The model accounts for the hydrostatic effects of water on deck, including deck immersion and bottom emersion, time-dependent 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 long-wave 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 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. 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 5@ 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 5@ 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 : time-dependent 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: time-dependent volume of hull including possibility for deck immersion and bottom emersion GZ: time-dependent 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 real-time 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 third-order 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 _ 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. Finite-time 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 non-capsize 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 finite-time 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 non-capsize 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 real-time 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 six-dimensional 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 real-time full-scale 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 non-capsize 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 non-capsize and capsize runs equal to (.13 deg, :7577 deg =s), (.18 deg, :637 deg =s), respectively.

10 To use in an on-board sense, a combination of a real-time calculator with a numerical simulator could enable development of a numerical empirical-type 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 on-board 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 non-capsize 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 Non-capsize 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 non-capsize. Bottom panel shows 8 time series leading to capsize. Initial conditions for all 14 time series in six-state 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 six-dimensional phase space on a singledimensional plane. As is apparent on the bottom panel, the non-capsize 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 multi-dimensional 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 non-capsize 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 non-capsize 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 non-capsize 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) Non-capsize 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 non-capsize 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 non-capsize 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 finite-time 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, non-capsize 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, non-capsize solid lines. Viewed over typical region from t ¼ 5 to 35 s. one-wave cycle of the maximum finite-time Lyapunov exponent. This is likely because the low-freeboard vessel studied inherently capsizes rapidly. However, even for this simplistic example, some advance warning of instability is given via peaks of the finite-time Lyapunov exponent time series. Through use of the finite-time 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 long-time series, such as those not leading to capsize, for a finite event, such as capsize, a finite-time 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, non-capsize solid lines. Viewed over typical region from t ¼ 5 to 35 s. exponent (FTLE) is necessary. Further development of the real-time 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 long-time 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 six-state 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 non-capsize time series Fig. 8. Histograms indicating range of FTLE values for runs leading to capsize and non-capsize 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 non-capsize 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 non-capsize 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 on-board simulation tool paired with a predictive device such as the finite-time Lyapunov exponent can provide real-time warnings of chaotic behavior with the express purpose of saving cargo, ships, and lives. Further studies could quantify phase relations between capsize and non-capsize 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. Young-Woo Lee for the initial development of the numerical simulator used in this paper as well as the work of Lt. Michael Obar and Dr. Young-Woo 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 multi-degree-of-freedom 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 multi-dof 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 non-linear prediction. Journal of the Royal Statistical Society Series B 56 (4),

Melnikov s Method Applied to a Multi-DOF Ship Model

Melnikov s Method Applied to a Multi-DOF Ship Model Proceedings of the th International Ship Staility Workshop Melnikov s Method Applied to a Multi-DOF Ship Model Wan Wu, Leigh S. McCue, Department of Aerospace and Ocean Engineering, Virginia Polytechnic

More information

Trajectory Tracking of a Near-Surface Torpedo using Numerical Methods

Trajectory Tracking of a Near-Surface Torpedo using Numerical Methods ISSN (Print) : 2347-671 An ISO 3297: 27 Certified Organization Vol.4, Special Issue 12, September 215 Trajectory Tracking of a Near-Surface Torpedo using Numerical Methods Anties K. Martin, Anubhav C.A.,

More information

THE INVESTIGATION OF THE SAFE BASIN EROSION UNDER THE ACTION OF RANDOM WAVES. Xianglu Huang, Shanghai Jiao Tong University Shanghai (China)

THE INVESTIGATION OF THE SAFE BASIN EROSION UNDER THE ACTION OF RANDOM WAVES. Xianglu Huang, Shanghai Jiao Tong University Shanghai (China) 8 th International Conference on 539 THE INVESTIGATION OF THE SAFE BASIN EROSION UNDER THE ACTION OF RANDOM WAVES Xianglu Huang, Shanghai Jiao Tong University Shanghai (China) Abstract The safe basin eroded

More information

SIMPLIFICATION BY MATHEMATIC MODEL TO SOLVE THE EXPERIMENTAL OF SLOSHING EFFECT ON THE FPSO VESSEL

SIMPLIFICATION BY MATHEMATIC MODEL TO SOLVE THE EXPERIMENTAL OF SLOSHING EFFECT ON THE FPSO VESSEL European International Journal of Science and Technology Vol. 3 No. 5 June, 2014 SIMPLIFICATION BY MATHEMATIC MODEL TO SOLVE THE EXPERIMENTAL OF SLOSHING EFFECT ON THE FPSO VESSEL LuhutTumpalParulianSinaga

More information

TECHNICAL NOTE: PREDICTION OF THE THRESHOLD OF GLOBAL SURF-RIDING BY AN EXTENDED MELNIKOV METHOD

TECHNICAL NOTE: PREDICTION OF THE THRESHOLD OF GLOBAL SURF-RIDING BY AN EXTENDED MELNIKOV METHOD 10 th International Conference 441 TECHNICAL NOTE: PREDICTION OF THE THRESHOLD OF GLOBAL SURF-RIDING BY AN EXTENDED MELNIKOV METHOD Wan Wu, Virginia Polytechnic Institute and State University, wanwu@vt.edu

More information

Requirements for Computational Methods to be sed for the IMO Second Generation Intact Stability Criteria

Requirements for Computational Methods to be sed for the IMO Second Generation Intact Stability Criteria Proceedings of the 1 th International Conference on the Stability of Ships and Ocean Vehicles, 14-19 June 15, Glasgow, UK Requirements for Computational Methods to be sed for the IMO Second Generation

More information

NONLINEAR ROLLING MOTION OF SHIP IN RANDOM BEAM SEAS

NONLINEAR ROLLING MOTION OF SHIP IN RANDOM BEAM SEAS Journal of Marine Science and Technology, Vol., No. 4, pp. 73-79 (4) 73 NONLINEAR ROLLING MOTION OF SHIP IN RANDOM BEAM SEAS Jia-Yang Gu* Key words: nonlinear roll, melnikov function, phase space flux,

More information

WAMIT-MOSES Hydrodynamic Analysis Comparison Study. JRME, July 2000

WAMIT-MOSES Hydrodynamic Analysis Comparison Study. JRME, July 2000 - Hydrodynamic Analysis Comparison Study - Hydrodynamic Analysis Comparison Study JRME, Prepared by Hull Engineering Department J. Ray McDermott Engineering, LLC 1 - Hydrodynamic Analysis Comparison Study

More information

Figure 1: Schematic of ship in still water showing the action of bouyancy and weight to right the ship.

Figure 1: Schematic of ship in still water showing the action of bouyancy and weight to right the ship. MULTI-DIMENSIONAL SYSTEM: In this computer simulation we will explore a nonlinear multi-dimensional system. As before these systems are governed by equations of the form x 1 = f 1 x 2 = f 2.. x n = f n

More information

Seakeeping Models in the Frequency Domain

Seakeeping Models in the Frequency Domain Seakeeping Models in the Frequency Domain (Module 6) Dr Tristan Perez Centre for Complex Dynamic Systems and Control (CDSC) Prof. Thor I Fossen Department of Engineering Cybernetics 18/09/2007 One-day

More information

Towards a theory of surf-riding in two-frequency and multi-frequency waves

Towards a theory of surf-riding in two-frequency and multi-frequency waves Proceedings of the 15 th International Ship Stability Workshop, 13-15 June 2016, Stockholm, Sweden 1 Towards a theory of surf-riding in two-frequency and multi-frequency waves K.J. Spyrou, k.spyrou@central.ntua.gr,

More information

Mooring Model for Barge Tows in Lock Chamber

Mooring Model for Barge Tows in Lock Chamber Mooring Model for Barge Tows in Lock Chamber by Richard L. Stockstill BACKGROUND: Extensive research has been conducted in the area of modeling mooring systems in sea environments where the forcing function

More information

Department of Aerospace and Ocean Engineering Graduate Study Specialization in Ocean Engineering. Written Preliminary Examination Information

Department of Aerospace and Ocean Engineering Graduate Study Specialization in Ocean Engineering. Written Preliminary Examination Information Department of Aerospace and Ocean Engineering Graduate Study Specialization in Ocean Engineering Written Preliminary Examination Information Faculty: Professors W. Neu, O. Hughes, A. Brown, M. Allen Test

More information

A coupled roll sway heave model for analysing ship capsize in beam seas on the basis of a nonlinear dynamics approach

A coupled roll sway heave model for analysing ship capsize in beam seas on the basis of a nonlinear dynamics approach 16 th International Conference on Hydrodynamics in Ship Design 3 rd International Symposium on Ship Manoeuvring Gdansk Ostrόda, Poland 7 1 September 5 N. Themelis and K. J. Spyrou A coupled roll-say-heave

More information

Time domain assessment of nonlinear coupled ship motions and sloshing in free surface tanks

Time domain assessment of nonlinear coupled ship motions and sloshing in free surface tanks Time domain assessment of nonlinear coupled ship motions and sloshing in free surface tanks 1 outline 1.Motivation & state-of-the-art 2.Simulation approach 1.SHIXDOF: nonlinear ship motion TD 6DOF 2.AQUAgpusph:

More information

NONLINEAR DYNAMICS ON PARAMETRIC ROLL RESONANCE WITH REALISTIC NUMERICAL MODELLING

NONLINEAR DYNAMICS ON PARAMETRIC ROLL RESONANCE WITH REALISTIC NUMERICAL MODELLING 8 th International Conference on 81 NONLINEAR DYNAMICS ON PARAMETRIC ROLL RESONANCE WITH REALISTIC NUMERICAL MODELLING Naoya Umeda*, Hirotada Hashimoto*, Dracos Vassalos**, Shinichi Urano* and Kenji Okou*

More information

The Behaviour of a Mobile Robot Is Chaotic

The Behaviour of a Mobile Robot Is Chaotic AISB Journal 1(4), c SSAISB, 2003 The Behaviour of a Mobile Robot Is Chaotic Ulrich Nehmzow and Keith Walker Department of Computer Science, University of Essex, Colchester CO4 3SQ Department of Physics

More information

Quantitative Description of Robot-Environment Interaction Using Chaos Theory 1

Quantitative Description of Robot-Environment Interaction Using Chaos Theory 1 Quantitative Description of Robot-Environment Interaction Using Chaos Theory 1 Ulrich Nehmzow Keith Walker Dept. of Computer Science Department of Physics University of Essex Point Loma Nazarene University

More information

Synchronization of two chaotic oscillators via a negative feedback mechanism

Synchronization of two chaotic oscillators via a negative feedback mechanism International Journal of Solids and Structures 40 (2003) 5175 5185 www.elsevier.com/locate/ijsolstr Synchronization of two chaotic oscillators via a negative feedback mechanism Andrzej Stefanski *, Tomasz

More information

ROLL MOTION OF A RORO-SHIP IN IRREGULAR FOLLOWING WAVES

ROLL MOTION OF A RORO-SHIP IN IRREGULAR FOLLOWING WAVES 38 Journal of Marine Science and Technology, Vol. 9, o. 1, pp. 38-44 (2001) ROLL MOTIO OF A RORO-SHIP I IRREGULAR FOLLOWIG WAVES Jianbo Hua* and Wei-Hui Wang** Keywords: roll motion, parametric excitation,

More information

Steady State and Transient Dynamical Systems Analysis of Uncoupled Roll Response for a Traditional Versus Advanced Hull Form

Steady State and Transient Dynamical Systems Analysis of Uncoupled Roll Response for a Traditional Versus Advanced Hull Form University of New Orleans ScholarWorks@UNO University of New Orleans Theses and Dissertations Dissertations and Theses 8-9-2006 Steady State and Transient Dynamical Systems Analysis of Uncoupled Roll Response

More information

One dimensional Maps

One dimensional Maps Chapter 4 One dimensional Maps The ordinary differential equation studied in chapters 1-3 provide a close link to actual physical systems it is easy to believe these equations provide at least an approximate

More information

A Preliminary Analysis on the Statistics of about One-Year Air Gap Measurement for a Semi-submersible in South China Sea

A Preliminary Analysis on the Statistics of about One-Year Air Gap Measurement for a Semi-submersible in South China Sea Proceedings of the Twenty-sixth (2016) International Ocean and Polar Engineering Conference Rhodes, Greece, June 26-July 1, 2016 Copyright 2016 by the International Society of Offshore and Polar Engineers

More information

SEAKEEPING AND MANEUVERING Prof. Dr. S. Beji 2

SEAKEEPING AND MANEUVERING Prof. Dr. S. Beji 2 SEAKEEPING AND MANEUVERING Prof. Dr. S. Beji 2 Ship Motions Ship motions in a seaway are very complicated but can be broken down into 6-degrees of freedom motions relative to 3 mutually perpendicular axes

More information

Assessing the stability of ships under the effect of realistic wave groups

Assessing the stability of ships under the effect of realistic wave groups Assessing the stability of ships under the effect of realistic wave groups Panayiotis A. Anastopoulos, Department of Naval Architecture and Marine Engineering, National Technical University of Athens,

More information

Wuchang Shipbuilding Industry Co., Ltd. China Shipbuilding Industry Corporation

Wuchang Shipbuilding Industry Co., Ltd. China Shipbuilding Industry Corporation Safety Assessments for Anchor Handling Conditions of Multi-purpose Platform Work Vessels Reporter:Yu Wang Wuchang Shipbuilding Industry Co., Ltd. China Shipbuilding Industry Corporation 2009.12.04 0 Outline

More information

PROOF COPY [EM/2004/023906] QEM

PROOF COPY [EM/2004/023906] QEM Coupled Surge-Heave Motions of a Moored System. II: Stochastic Analysis and Simulations Solomon C. S. Yim, M.ASCE 1 ; and Huan Lin, A.M.ASCE Abstract: Analyses and simulations of the coupled surge-and-heave

More information

On the evaluation quadratic forces on stationary bodies

On the evaluation quadratic forces on stationary bodies On the evaluation quadratic forces on stationary bodies Chang-Ho Lee AMIT Inc., Chestnut Hill MA, USA June 9, 006 Abstract. Conservation of momentum is applied to finite fluid volume surrounding a body

More information

Dessi, D., D Orazio, D.

Dessi, D., D Orazio, D. CORRELATION OF MODEL-SCALE AND FULL-SCALE DATA: SENSOR VALIDATION AND ELASTIC SCALING EVALUATION Dessi, D., D Orazio, D. INSEAN-CNR Rome - Italy 1 Project structure hydroelastic side This work was funded

More information

Handling Roll Constraints for Path Following of Marine Surface Vessels using Coordinated Rudder and Propulsion Control

Handling Roll Constraints for Path Following of Marine Surface Vessels using Coordinated Rudder and Propulsion Control 2010 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 30-July 02, 2010 FrB15.5 Handling Roll Constraints for Path Following of Marine Surface Vessels using Coordinated Rudder and

More information

Study on Motions of a Floating Body under Composite External Loads

Study on Motions of a Floating Body under Composite External Loads 137 Study on Motions of a Floating Body under Composite External Loads by Kunihiro Ikegami*, Member Masami Matsuura*, Member Summary In the field of marine engineering, various types of floating bodies

More information

Lyapunov exponent calculation of a two-degreeof-freedom vibro-impact system with symmetrical rigid stops

Lyapunov exponent calculation of a two-degreeof-freedom vibro-impact system with symmetrical rigid stops Chin. Phys. B Vol. 20 No. 4 (2011) 040505 Lyapunov exponent calculation of a two-degreeof-freedom vibro-impact system with symmetrical rigid stops Li Qun-Hong( ) and Tan Jie-Yan( ) College of Mathematics

More information

CHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD

CHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD CHALMERS, GÖTEBORGS UNIVERSITET EXAM for DYNAMICAL SYSTEMS COURSE CODES: TIF 155, FIM770GU, PhD Time: Place: Teachers: Allowed material: Not allowed: April 06, 2018, at 14 00 18 00 Johanneberg Kristian

More information

STABILITY ANALYSIS OF DAMPED SDOF SYSTEMS WITH TWO TIME DELAYS IN STATE FEEDBACK

STABILITY ANALYSIS OF DAMPED SDOF SYSTEMS WITH TWO TIME DELAYS IN STATE FEEDBACK Journal of Sound and Vibration (1998) 214(2), 213 225 Article No. sv971499 STABILITY ANALYSIS OF DAMPED SDOF SYSTEMS WITH TWO TIME DELAYS IN STATE FEEDBACK H. Y. HU ANDZ. H. WANG Institute of Vibration

More information

Final Exam TTK4190 Guidance and Control

Final Exam TTK4190 Guidance and Control Trondheim Department of engineering Cybernetics Contact person: Professor Thor I. Fossen Phone: 73 59 43 61 Cell: 91 89 73 61 Email: tif@itk.ntnu.no Final Exam TTK4190 Guidance and Control Friday May 15,

More information

SCALE MODEL TESTS OF A FISHING VESSEL IN ROLL MOTION PARAMETRIC RESONANCE

SCALE MODEL TESTS OF A FISHING VESSEL IN ROLL MOTION PARAMETRIC RESONANCE N. Perez Síntesis Tecnológica. V.3 Nº 1 (26) 33-37 SCALE MODEL TESTS OF A FISHING VESSEL IN ROLL MOTION PARAMETRIC RESONANCE NELSON A. PEREZ M. Instituto de Ciencias Navales y Marítimas, M.Sc, nperez@uach.cl,

More information

... it may happen that small differences in the initial conditions produce very great ones in the final phenomena. Henri Poincaré

... it may happen that small differences in the initial conditions produce very great ones in the final phenomena. Henri Poincaré Chapter 2 Dynamical Systems... it may happen that small differences in the initial conditions produce very great ones in the final phenomena. Henri Poincaré One of the exciting new fields to arise out

More information

Anisochronism of John Harrison s First Sea Clocks, H.1 and H.2

Anisochronism of John Harrison s First Sea Clocks, H.1 and H.2 Anisochronism of John Harrison s First Sea Clocks, H.1 and H. Douglas S. Drumheller 1 Introduction John Harrison built his sea clocks, H.1 and H., in the early part of the 18th Century. They were his first

More information

Coupled Heave-Pitch Motions and Froude Krylov Excitation Forces

Coupled Heave-Pitch Motions and Froude Krylov Excitation Forces Coupled Heave-Pitch Motions and Froude Krylov Excitation Forces 13.42 Lecture Notes; Spring 2004; c A.H. Techet 1. Coupled Equation of Motion in Heave and Pitch Once we have set up the simple equation

More information

Escape Trajectories from Sun Earth Distant Retrograde Orbits

Escape Trajectories from Sun Earth Distant Retrograde Orbits Trans. JSASS Aerospace Tech. Japan Vol. 4, No. ists30, pp. Pd_67-Pd_75, 06 Escape Trajectories from Sun Earth Distant Retrograde Orbits By Yusue OKI ) and Junichiro KAWAGUCHI ) ) Department of Aeronautics

More information

Fluid-Induced Nonlinear Dynamic Tensioning of Cables

Fluid-Induced Nonlinear Dynamic Tensioning of Cables Fluid-Induced Nonlinear Dynamic Tensioning of Cables Noel C. Perkins Department of Mechanical Engineering and Applied Mechanics University of Michigan Ann Arbor, MI 48109-2125 phone: (734) 936-0403 fax:

More information

Influence of yaw-roll coupling on the behavior of a FPSO: an experimental and numerical investigation

Influence of yaw-roll coupling on the behavior of a FPSO: an experimental and numerical investigation Influence of yaw-roll coupling on the behavior of a FPSO: an experimental and numerical investigation Claudio Lugni a,b, Marilena Greco a,b, Odd Magnus Faltinsen b a CNR-INSEAN, The Italian Ship Model

More information

Suppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber

Suppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber Suppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber J.C. Ji, N. Zhang Faculty of Engineering, University of Technology, Sydney PO Box, Broadway,

More information

INFLUENCE OF TETHER LENGTH IN THE RESPONSE BEHAVIOR OF SQUARE TENSION LEG PLATFORM IN REGULAR WAVES

INFLUENCE OF TETHER LENGTH IN THE RESPONSE BEHAVIOR OF SQUARE TENSION LEG PLATFORM IN REGULAR WAVES INFLUENCE OF TETHER LENGTH IN THE RESPONSE BEHAVIOR OF SQUARE TENSION LEG PLATFOR IN REGULAR WAVES 1 Amr R. El-gamal, 2 Ashraf Essa, 1 Faculty of Engineering, Benha Univ., Egypt, 2 Associated prof., National

More information

DREDGING DYNAMICS AND VIBRATION MEASURES

DREDGING DYNAMICS AND VIBRATION MEASURES DREDGING DYNAMICS AND VIBRATION MEASURES C R Barik, K Vijayan, Department of Ocean Engineering and Naval Architecture, IIT Kharagpur, India ABSTRACT The demands for dredging have found a profound increase

More information

Control design of fin roll stabilization in beam seas based on Lyapunov s direct method

Control design of fin roll stabilization in beam seas based on Lyapunov s direct method POLISH MARITIME RESEARCH 2(73) 2012 Vol 19; pp. 25-30 10.2478/v10012-012-0011-9 Control design of fin roll stabilization in beam seas based on Lyapunov s direct method Safak C. Karakas, M. Sc. Erdem Ucer,

More information

Overview of BV R&D activities in Marine Hydrodynamics

Overview of BV R&D activities in Marine Hydrodynamics Overview of BV R&D activities in Marine Hydrodynamics Special attention to hydro-structure interactions Šime Malenica Bureau Veritas Marine & Offshore Division Research Department Harbin, 29th of June

More information

1 POTENTIAL FLOW THEORY Formulation of the seakeeping problem

1 POTENTIAL FLOW THEORY Formulation of the seakeeping problem 1 POTENTIAL FLOW THEORY Formulation of the seakeeping problem Objective of the Chapter: Formulation of the potential flow around the hull of a ship advancing and oscillationg in waves Results of the Chapter:

More information

Optimal Design of FPSO Vessels

Optimal Design of FPSO Vessels November 2, 201 Optimal Design of FPSO Vessels Ezebuchi Akandu PhD, MTech, BTech, COREN, RINA, MNSE Department of Marine Engineering, Rivers State University, Port Harcourt, Nigeria akandu.ezebuchi@ust.edu.ng

More information

Boundary element methods in the prediction of the acoustic damping of ship whipping vibrations

Boundary element methods in the prediction of the acoustic damping of ship whipping vibrations ANZIAM J. 45 (E) ppc845 C856, 2004 C845 Boundary element methods in the prediction of the acoustic damping of ship whipping vibrations D. S. Holloway G. A. Thomas M. R. Davis (Received 8 August 2003) Abstract

More information

A Discussion About Seakeeping and Manoeuvring Models For Surface Vessels

A Discussion About Seakeeping and Manoeuvring Models For Surface Vessels A Discussion About Seakeeping and Manoeuvring Models For Surface Vessels Tristan Perez, Thor I. Fossen and Asgeir Sørensen Technical Report (MSS-TR-001) Marine System Simulator (MSS) Group (http://www.cesos.ntnu.no/mss/)

More information

SHIP BUOYANCY AND STABILITY. Lecture 03 Ship initial stability

SHIP BUOYANCY AND STABILITY. Lecture 03 Ship initial stability SHIP BUOYANCY AND STABILITY Lecture 3 Ship initial stability 1 Literature J. Matusiak: Laivan kelluvuus ja vakavuus Biran A. B., Ship Hydrostatics and Stability, 23 J. Matusiak: Short Introduction to Ship

More information

The Rationale for Second Level Adaptation

The Rationale for Second Level Adaptation The Rationale for Second Level Adaptation Kumpati S. Narendra, Yu Wang and Wei Chen Center for Systems Science, Yale University arxiv:1510.04989v1 [cs.sy] 16 Oct 2015 Abstract Recently, a new approach

More information

K. Pyragas* Semiconductor Physics Institute, LT-2600 Vilnius, Lithuania Received 19 March 1998

K. Pyragas* Semiconductor Physics Institute, LT-2600 Vilnius, Lithuania Received 19 March 1998 PHYSICAL REVIEW E VOLUME 58, NUMBER 3 SEPTEMBER 998 Synchronization of coupled time-delay systems: Analytical estimations K. Pyragas* Semiconductor Physics Institute, LT-26 Vilnius, Lithuania Received

More information

Chapter 2 Chaos theory and its relationship to complexity

Chapter 2 Chaos theory and its relationship to complexity Chapter 2 Chaos theory and its relationship to complexity David Kernick This chapter introduces chaos theory and the concept of non-linearity. It highlights the importance of reiteration and the system

More information

HEAVE DAMPING EFFECTS DUE TO CIRCULAR PLATES ATTACHED AT KEEL TO SPAR HULL

HEAVE DAMPING EFFECTS DUE TO CIRCULAR PLATES ATTACHED AT KEEL TO SPAR HULL HEAVE DAMPING EFFECTS DUE TO CIRCULAR PLATES ATTACHED AT KEEL TO SPAR HULL P.Uma 1 1 M.TECH Civil Engineering Dadi Institute of Engineering and Technology College Abstract Single point Anchor Reservoir

More information

STABILITY AND TRIM OF MARINE VESSELS. Massachusetts Institute of Technology, Subject 2.017

STABILITY AND TRIM OF MARINE VESSELS. Massachusetts Institute of Technology, Subject 2.017 STABILITY AND TRIM OF MARINE VESSELS Concept of Mass Center for a Rigid Body Centroid the point about which moments due to gravity are zero: 6 g m i (x g x i )= 0 Æ x = 6m i x i / 6m i = 6m i x i / M g

More information

Abstract: Complex responses observed in an experimental, nonlinear, moored structural

Abstract: Complex responses observed in an experimental, nonlinear, moored structural AN INDEPENDENT-FLOW-FIELD MODEL FOR A SDOF NONLINEAR STRUCTURAL SYSTEM, PART II: ANALYSIS OF COMPLEX RESPONSES Huan Lin e-mail: linh@engr.orst.edu Solomon C.S. Yim e-mail: solomon.yim@oregonstate.edu Ocean

More information

Quasipatterns in surface wave experiments

Quasipatterns in surface wave experiments Quasipatterns in surface wave experiments Alastair Rucklidge Department of Applied Mathematics University of Leeds, Leeds LS2 9JT, UK With support from EPSRC A.M. Rucklidge and W.J. Rucklidge, Convergence

More information

Dynamics and Control of the GyroPTO Wave Energy Point Absorber under Sea Waves

Dynamics and Control of the GyroPTO Wave Energy Point Absorber under Sea Waves Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 99 (7) 88 8 X International Conference on Structural Dynamics, EURODYN 7 Dynamics and Control of the GyroPTO Wave Energy Point

More information

= w. These evolve with time yielding the

= w. These evolve with time yielding the 1 Analytical prediction and representation of chaos. Michail Zak a Jet Propulsion Laboratory California Institute of Technology, Pasadena, CA 91109, USA Abstract. 1. Introduction The concept of randomness

More information

Renewable Energy: Ocean Wave-Energy Conversion

Renewable Energy: Ocean Wave-Energy Conversion Renewable Energy: Ocean Wave-Energy Conversion India Institute of Science Bangalore, India 17 June 2011 David R. B. Kraemer, Ph.D. University of Wisconsin Platteville USA B.S.: My background Mechanical

More information

Seakeeping characteristics of intact and damaged ship in the Adriatic Sea

Seakeeping characteristics of intact and damaged ship in the Adriatic Sea Towards Green Marine Technology and Transport Guedes Soares, Dejhalla & Pavleti (Eds) 2015 Taylor & Francis Group, London, ISBN 978-1-138-02887-6 Seakeeping characteristics of intact and damaged ship in

More information

Experimental Investigation of Inertial Force Control for Substructure Shake Table Tests

Experimental Investigation of Inertial Force Control for Substructure Shake Table Tests Experimental Investigation of Inertial Force Control for Substructure Shake Table Tests M. Stehman & N. Nakata The Johns Hopkins University, USA SUMMARY: This study investigates the use of inertial masses

More information

ARTICLE IN PRESS. Mechanical Systems and Signal Processing

ARTICLE IN PRESS. Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing ] (]]]]) ]]] ]]] Contents lists available at ScienceDirect Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp Excitation

More information

Nonlinear Time Domain Simulation Technology for Seakeeping and Wave-Load Analysis for Modern Ship Design

Nonlinear Time Domain Simulation Technology for Seakeeping and Wave-Load Analysis for Modern Ship Design ABS TECHNICAL PAPERS 23 Nonlinear Time Domain Simulation Technology for Seakeeping and Wave-Load Analysis for Modern Ship Design Y.S. Shin, Associate Member, American Bureau of Shipping, V.L. Belenky,

More information

3 Mathematical modeling of the torsional dynamics of a drill string

3 Mathematical modeling of the torsional dynamics of a drill string 3 Mathematical modeling of the torsional dynamics of a drill string 3.1 Introduction Many works about torsional vibrations on drilling systems [1, 12, 18, 24, 41] have been published using different numerical

More information

A Two-dimensional Mapping with a Strange Attractor

A Two-dimensional Mapping with a Strange Attractor Commun. math. Phys. 50, 69 77 (1976) Communications in Mathematical Physics by Springer-Verlag 1976 A Two-dimensional Mapping with a Strange Attractor M. Henon Observatoire de Nice, F-06300 Nice, France

More information

Linear Feedback Control Using Quasi Velocities

Linear Feedback Control Using Quasi Velocities Linear Feedback Control Using Quasi Velocities Andrew J Sinclair Auburn University, Auburn, Alabama 36849 John E Hurtado and John L Junkins Texas A&M University, College Station, Texas 77843 A novel approach

More information

Student name: This is a closed book examination. You are allowed 1 sheet of 8.5 x 11 paper with notes.

Student name: This is a closed book examination. You are allowed 1 sheet of 8.5 x 11 paper with notes. 13.012 Marine Hydrodynamics for Ocean Engineers Fall 2004 Quiz #2 Student name: This is a closed book examination. You are allowed 1 sheet of 8.5 x 11 paper with notes. For the problems in Section A, fill

More information

Ship Nonlinear Rolling and Roll Angle Reconstruction Based on FIR

Ship Nonlinear Rolling and Roll Angle Reconstruction Based on FIR Open Access Library Journal Ship Nonlinear Rolling and Roll Angle Reconstruction Based on FIR Jianhui Lu 1,2*, Chunlei Zhang 2, Shaonan Chen 2, Yunxia Wu 2 1 Shandong Province Key Laboratory of Ocean Engineering,

More information

Is the Hénon map chaotic

Is the Hénon map chaotic Is the Hénon map chaotic Zbigniew Galias Department of Electrical Engineering AGH University of Science and Technology, Poland, galias@agh.edu.pl International Workshop on Complex Networks and Applications

More information

16 Period doubling route to chaos

16 Period doubling route to chaos 16 Period doubling route to chaos We now study the routes or scenarios towards chaos. We ask: How does the transition from periodic to strange attractor occur? The question is analogous to the study of

More information

Experimental and numerical investigation of 2D sloshing with slamming

Experimental and numerical investigation of 2D sloshing with slamming Experimental numerical investigation of 2D sloshing with slamming A. Colagrossi C. Lugni M. Greco O. M. Faltinsen a.colagrossi@insean.it c.lugni@insean.it m.greco@insean.it oddfal@marin.ntnu.no INSEAN,

More information

Simple Estimation of Wave Added Resistance from Experiments in Transient and Irregular Water Waves

Simple Estimation of Wave Added Resistance from Experiments in Transient and Irregular Water Waves Simple Estimation of Wave Added Resistance from Experiments in Transient and Irregular Water Waves by Tsugukiyo Hirayama*, Member Xuefeng Wang*, Member Summary Experiments in transient water waves are

More information

ACCURATE FREE VIBRATION ANALYSIS OF POINT SUPPORTED MINDLIN PLATES BY THE SUPERPOSITION METHOD

ACCURATE FREE VIBRATION ANALYSIS OF POINT SUPPORTED MINDLIN PLATES BY THE SUPERPOSITION METHOD Journal of Sound and Vibration (1999) 219(2), 265 277 Article No. jsvi.1998.1874, available online at http://www.idealibrary.com.on ACCURATE FREE VIBRATION ANALYSIS OF POINT SUPPORTED MINDLIN PLATES BY

More information

Effect of Tethers Tension Force in the Behavior of a Tension Leg Platform Subjected to Hydrodynamic Force Amr R. El-Gamal, Ashraf Essa, Ayman Ismail

Effect of Tethers Tension Force in the Behavior of a Tension Leg Platform Subjected to Hydrodynamic Force Amr R. El-Gamal, Ashraf Essa, Ayman Ismail Vol:7, No:1, 13 Effect of Tethers Tension Force in the Behavior of a Tension Leg Platform Subjected to Hydrodynamic Force Amr R. El-Gamal, Ashraf Essa, Ayman Ismail International Science Index, Bioengineering

More information

Existence of super-harmonics in quarter-vehicle system responses with nonlinear inertia hydraulic track mount given sinusoidal force excitation

Existence of super-harmonics in quarter-vehicle system responses with nonlinear inertia hydraulic track mount given sinusoidal force excitation Journal of Sound and Vibration 313 (8) 367 374 Rapid Communication JOURNAL OF SOUND AND VIBRATION Existence of super-harmonics in quarter-vehicle system responses with nonlinear inertia hydraulic track

More information

DYNAMIC CHARACTERISTICS OF OFFSHORE TENSION LEG PLATFORMS UNDER HYDRODYNAMIC FORCES

DYNAMIC CHARACTERISTICS OF OFFSHORE TENSION LEG PLATFORMS UNDER HYDRODYNAMIC FORCES International Journal of Civil Engineering (IJCE) ISSN(P): 2278-9987; ISSN(E): 2278-9995 Vol. 3, Issue 1, Jan 214, 7-16 IASET DYNAMIC CHARACTERISTICS OF OFFSHORE TENSION LEG PLATFORMS UNDER HYDRODYNAMIC

More information

On the design of reactionless 3-DOF planar parallel mechanisms

On the design of reactionless 3-DOF planar parallel mechanisms Mechanism and Machine Theory 41 (2006) 70 82 Mechanism and Machine Theory www.elsevier.com/locate/mechmt On the design of reactionless 3-DOF planar parallel mechanisms Abbas Fattah *, Sunil K. Agrawal

More information

Simplified formulas of heave added mass coefficients at high frequency for various two-dimensional bodies in a finite water depth

Simplified formulas of heave added mass coefficients at high frequency for various two-dimensional bodies in a finite water depth csnak, 2015 Int. J. Nav. Archit. Ocean Eng. (2015) 7:115~127 http://dx.doi.org/10.1515/ijnaoe-2015-0009 pissn: 2092-6782, eissn: 2092-6790 Simplified formulas of heave added mass coefficients at high frequency

More information

Nonlinear Feedback Stabilization of High-Speed Planing Vessels by A Controllable Transom Flap

Nonlinear Feedback Stabilization of High-Speed Planing Vessels by A Controllable Transom Flap 2 American Control Conference June 8-1, 2. Portland, OR, USA FrB11.2 Nonlinear Feedback Stabilization of High-Speed Planing Vessels by A Controllable Transom Flap Handa Xi and Jing Sun Department of Naval

More information

Numerical Study of the Roll Decay of Intact and Damaged Ships by Q. Gao and D. Vassalos

Numerical Study of the Roll Decay of Intact and Damaged Ships by Q. Gao and D. Vassalos Session 7 Stability of Damaged Ships Numerical Simulation of Progressive Flooding and Capsize Numerical Study of the Roll Decay of Intact and Damaged Ships by Q. Gao and D. Vassalos Qiuxin Gao and Dracos

More information

Nonlinear Autonomous Systems of Differential

Nonlinear Autonomous Systems of Differential Chapter 4 Nonlinear Autonomous Systems of Differential Equations 4.0 The Phase Plane: Linear Systems 4.0.1 Introduction Consider a system of the form x = A(x), (4.0.1) where A is independent of t. Such

More information

Cause Investigation of Flooding & Sinking Accident of Ro-Ro Ferry Ship using Marine Accident Integrated Analysis System

Cause Investigation of Flooding & Sinking Accident of Ro-Ro Ferry Ship using Marine Accident Integrated Analysis System Cause Investigation of Flooding & Sinking Accident of Ro-Ro Ferry Ship using Marine Accident Integrated Analysis System Sang-Gab Lee, Jae-Seok Lee, Ji-Hoon Park and Tae-Young Jung Korea Maritime & Ocean

More information

Fuel-efficient navigation in complex flows

Fuel-efficient navigation in complex flows 2008 American Control Conference Westin Seattle Hotel, Seattle, Washington, USA June 11-13, 2008 WeB16.5 Fuel-efficient navigation in complex flows Carmine Senatore and Shane D. Ross Abstract In realistic

More information

WHEELSET BEARING VIBRATION ANALYSIS BASED ON NONLINEAR DYNAMICAL METHOD

WHEELSET BEARING VIBRATION ANALYSIS BASED ON NONLINEAR DYNAMICAL METHOD 15 th November 212. Vol. 45 No.1 25-212 JATIT & LLS. All rights reserved. WHEELSET BEARING VIBRATION ANALYSIS BASED ON NONLINEAR DYNAMICAL METHOD 1,2 ZHAO ZHIHONG, 2 LIU YONGQIANG 1 School of Computing

More information

AN ABSTRACT OF THE DISSERTATION OF. Tongchate Nakhata for the degree of Doctor of Philosophy in Civil Engineering

AN ABSTRACT OF THE DISSERTATION OF. Tongchate Nakhata for the degree of Doctor of Philosophy in Civil Engineering AN ABSTRACT OF THE DISSERTATION OF Tongchate Nakhata for the degree of Doctor of Philosophy in Civil Engineering presented on May 22, 2002. Title: Stability Analysis of Nonlinear Coupled Barge Motions

More information

On the Application of the Generalized Pareto Distribution for Statistical Extrapolation in the Assessment of Dynamic Stability in Irregular Waves

On the Application of the Generalized Pareto Distribution for Statistical Extrapolation in the Assessment of Dynamic Stability in Irregular Waves On the Application of the Generalized Pareto Distribution for Statistical Extrapolation in the Assessment of Dynamic Stability in Irregular Waves Bradley Campbell 1, Vadim Belenky 1, Vladas Pipiras 2 1.

More information

Path Following for Marine Surface Vessels with Rudder and Roll Constraints: an MPC Approach

Path Following for Marine Surface Vessels with Rudder and Roll Constraints: an MPC Approach 2009 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 10-12, 2009 ThC11.6 Path Following for Marine Surface Vessels with Rudder and Roll Constraints: an MPC Approach Zhen Li,

More information

Model Reference Adaptive Control of Underwater Robotic Vehicle in Plane Motion

Model Reference Adaptive Control of Underwater Robotic Vehicle in Plane Motion Proceedings of the 11th WSEAS International Conference on SSTEMS Agios ikolaos Crete Island Greece July 23-25 27 38 Model Reference Adaptive Control of Underwater Robotic Vehicle in Plane Motion j.garus@amw.gdynia.pl

More information

Lagrangian Coherent Structures (LCS)

Lagrangian Coherent Structures (LCS) Lagrangian Coherent Structures (LCS) CDS 140b - Spring 2012 May 15, 2012 ofarrell@cds.caltech.edu A time-dependent dynamical system ẋ (t; t 0, x 0 )=v(x(t;,t 0, x 0 ),t) x(t 0 ; t 0, x 0 )=x 0 t 2 I R

More information

The dynamics of the floodwater and the damaged ship in waves

The dynamics of the floodwater and the damaged ship in waves The dynamics of the floodwater and the damaged ship in waves Zhiliang Gao ( 高志亮 ) 1*, Dracos Vassalos 2 1 Research Center of Coastal and Estuarine Engineering, Tianjin Research Institute for Water Transport

More information

Chaos Control for the Lorenz System

Chaos Control for the Lorenz System Advanced Studies in Theoretical Physics Vol. 12, 2018, no. 4, 181-188 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2018.8413 Chaos Control for the Lorenz System Pedro Pablo Cárdenas Alzate

More information

6.2 Brief review of fundamental concepts about chaotic systems

6.2 Brief review of fundamental concepts about chaotic systems 6.2 Brief review of fundamental concepts about chaotic systems Lorenz (1963) introduced a 3-variable model that is a prototypical example of chaos theory. These equations were derived as a simplification

More information

Follow links Class Use and other Permissions. For more information, send to:

Follow links Class Use and other Permissions. For more information, send  to: COPYRIGHT NOTICE: Stephen L. Campbell & Richard Haberman: Introduction to Differential Equations with Dynamical Systems is published by Princeton University Press and copyrighted, 2008, by Princeton University

More information

u (surge) X o p (roll) Body-fixed r o v (sway) w (heave) Z o Earth-fixed X Y Z r (yaw) (pitch)

u (surge) X o p (roll) Body-fixed r o v (sway) w (heave) Z o Earth-fixed X Y Z r (yaw) (pitch) Nonlinear Modelling of Marine Vehicles in Degrees of Freedom Thor I. Fossen and Ola-Erik Fjellstad The Norwegian Institute of Technology Department of Engineering Cybernetics N-0 Trondheim, NORWAY (E-mail:tif@itk.unit.no)

More information

TTK4190 Guidance and Control Exam Suggested Solution Spring 2011

TTK4190 Guidance and Control Exam Suggested Solution Spring 2011 TTK4190 Guidance and Control Exam Suggested Solution Spring 011 Problem 1 A) The weight and buoyancy of the vehicle can be found as follows: W = mg = 15 9.81 = 16.3 N (1) B = 106 4 ( ) 0.6 3 3 π 9.81 =

More information

A NUMERICAL IDENTIFICATION OF EXCITATION FORCE AND NONLINEAR RESTORING CHARACTERISTICS OF SHIP ROLL MOTION

A NUMERICAL IDENTIFICATION OF EXCITATION FORCE AND NONLINEAR RESTORING CHARACTERISTICS OF SHIP ROLL MOTION ournal of Marine Science and Technology Vol. 5 No. 4 pp. 475-481 (017 475 DOI: 10.6119/MST-017-0418-1 A NUMERICAL IDENTIFICATION OF EXCITATION FORCE AND NONNEAR RESTORING CHARACTERISTICS OF SHIP ROLL MOTION

More information

Design and modelling of an airship station holding controller for low cost satellite operations

Design and modelling of an airship station holding controller for low cost satellite operations AIAA Guidance, Navigation, and Control Conference and Exhibit 15-18 August 25, San Francisco, California AIAA 25-62 Design and modelling of an airship station holding controller for low cost satellite

More information