Violent motions and capsizing warning system for oceangoing vessels

A system and a method to warn a ship crew of potential violent motions in the immediate near future for the oceangoing vessel when operating in seas. Violent ship motions not only discomfort ship crews, damage cargos and ship structures, but also pose potential capsizing risk to ships. The system includes motion sensors; computer hardware and software; and warning devices. The sensors measure the ship roll, pitch, yaw, and rudder motions. The time histories of these motions are stored in the hardware and constantly analyzed using Finite Fourier Transform by Fast Fourier Transform (FFT) to detect the nonlinear inertial coupling effect which is newly discovered by the inventor and believed to be the root cause leading to violent motions and capsizing. Based on the inventor's theory of nonlinear instability and inertial coupling effect, the invented method detects nonlinear yaw instability potential and inertial coupling events, and provides warnings to master to reduce potential yaw nonlinear instability, to avoid inertial coupling roll response, rudder induced oscillations, and broaching, and to prevent capsize in seas.

FIELD

The invention is generally related to violent motions and capsize avoidance for oceangoing vessels, more specifically, to systems for providing a warning for oceangoing vessels for the so-called inertial coupling effects which is believed to be the root cause for violent ship motions and capsizing.

BACKGROUND

Ship capsizing happens occasionally even nowadays. In 2014, Ro/Ro ferry, MV Sewol capsized due to “unreasonable sudden turn” with 304 people died according to Wikipedia. In 2015, cargo ship, EL FARO capsized with 33 people on board missing due to hurricane (Wikipedia). Despite long history of shipping industry, evidences indicate that ship broaching and capsizing in seas have not been understood satisfactorily. Why in following or quartering seas, wave crest amidships is dangerous; why when pitch frequency is close to roll frequency, sudden yaw (or turn) could happen and heading control could lose; why in perfect following seas, indicating no wave exciting roll moment, ship could have a large roll and even capsize; and why the wave exciting moments by linearization theory could not explain ship capsize. All those questions have not been answered satisfactorily. The reason for this situation is that a fundamental mistake has been made in dealing with the ship dynamics in naval architecture industry.

For a ship, the governing equations for its rotational motions (roll, pitch, and yaw) are given by Math. 1 in the vector form. They were obtained based on Newton's second law of motion in a body-fixed reference frame, see the reference, SNAME: “Nomenclature for treating the motion of a submerged body through a fluid”, Technical and Research Bulletin No. 1-5 (1950);
d{right arrow over (H)}/dt=−{right arrow over (ω)}×{right arrow over (H)}+{right arrow over (M)}Math. 1
wherein {right arrow over (ω)}=(p,q,r)=({dot over (φ)},{dot over (θ)},{dot over (ψ)}): the angular velocities of the ship; φ,θ,ψ: the roll, pitch, and yaw angle about the principal axes of inertia X, Y, Z, respectively; {right arrow over (H)}=(Ixp,Iyq,Izr): the angular momentum of the ship; Ix,Iy,Iz: the moment of inertias about the principal axes of inertia X, Y, Z, respectively (These parameters are constants in this frame); {right arrow over (M)}=(Mx,My,Mz: the external moments acting on the ship about the principal axes of inertia. In both the naval architecture academy and industry, the current practice to deal with Math. 1 is to make a linearization approximation first and then solve the equations because the nonlinear term −{right arrow over (ω)}×{right arrow over (H)} is too difficult to deal with. The linearization approximation makes the nonlinear term −{right arrow over (ω)}×{right arrow over (H)} disappear, and the equations become
d{right arrow over (H)}/dt={right arrow over (M)}.Math. 2
However, the equations are still considered in the body-fixed reference frame which is a non-inertial frame. The reason for this is that the external moments (Mx,My,Mz) acting on ships and the moments of inertia are needed to be considered in the body-fixed reference frame.

The fundamental mistake is that the nonlinear term −{right arrow over (ω)}×{right arrow over (H)} cannot be neglected because they are the inertial moments tied to the non-inertial reference frame which is the body-fixed reference frame in this case. This mistake is similarly like we neglect the Coriolis force which equals −2{right arrow over (Ω)}×{right arrow over (V)}, where {right arrow over (Ω)} is the angular velocity vector of the earth and V is the velocity vector of a moving body on earth. Then we try to explain the swirling water draining phenomenon in a bathtub. In this case, we are considering the water moving in the body-fixed and non-inertial reference frame which is the earth. The Coriolis force is an inertial force generated by the rotating earth on the moving objects which are the water particles in this case. Without the Coriolis force, we cannot explain the motions of the swirling water. Similarly, in the aircraft dynamics, the aircraft is rotating, and we consider the rotational motions of the aircraft in the body-fixed and non-inertial reference frame which is the aircraft itself. The difference between the two cases is that in the former the object (water particle) has translational motions (V) while in the latter the object (aircraft itself) has rotational motions ({right arrow over (ω)}) but they both have the important inertial effects which cannot be neglected because both the objects are considered in the non-inertial reference frames. In the former the inertial effect is the Coriolis force −2{right arrow over (Ω)}×{right arrow over (V)} while in the latter the inertial effect is the inertial moment −{right arrow over (ω)}×{right arrow over (H)} which are not forces but moments since we are dealing with rational motions instead of translational one. Without the inertial moment, we cannot explain many phenomena which happened to ship such as broaching and capsizing in following and quartering seas.

In the inventor's book: “Nonlinear Instability and Inertial Coupling Effect—The Root Causes Leading to Aircraft Crashes, Land Vehicle rollovers, and Ship Capsizes” (ISBN 9781732632301 to be published in November 2018 by Faiteve Inc), the equations Math. 1 have been solved analytically without the linearization approximation. It was found that the inertial coupling terms −{right arrow over (ω)}×{right arrow over (H)} can significantly change the roll motion. A summary of the findings is given below. The governing equations of rotational motions of a ship in following or quartering seas can be written in the scalar form as
(Ix+a1){umlaut over (φ)}+b1{dot over (φ)}1+k1φ=(Iy−Iz){dot over (θ)}{dot over (ψ)}+M11cos(ω11t+α11),   Math. 3
(Iy+a2){umlaut over (θ)}+b2{dot over (θ)}1+k1θ=(Iz−Ix){dot over (φ)}{dot over (ψ)}+M21cos(ω21t+α21),   Math. 4
(Iz+a3){umlaut over (ψ)}+b3{dot over (ψ)}=(Ix−Iy){dot over (φ)}{dot over (θ)}+M31cos(ω31t+α31),  Math. 5
wherein a1,a2,a3are the added mass for roll, pitch, and yaw, respectively; b1,b2,b3are the damping coefficients for roll, pitch, and yaw, respectively; k1and k2are the restoring coefficients for roll and pitch, respectively; M11,M21,M31are the wave moment amplitude for roll, pitch, and yaw, respectively; ω11,ω21,ω31and α11,α21,α31are the frequency and phase of the wave moment for roll, pitch, and yaw, respectively. These equations represent a dynamic system governing the rotational dynamics of the ship in seas. According to the current practice in the industry under the linearization approximation, these equations become
(Ix+a1){umlaut over (φ)}+b1{dot over (φ)}+k1φ=M11cos(ω11t+α11),  Math. 6
(Iy+a2){umlaut over (θ)}+b2+{dot over (θ)}+k2θ=M21cos(ω21t+α21),  Math. 7
(Iz+a3){umlaut over (ψ)}+b3{dot over (ψ)}=M31cos(ω31t+α31).  Math. 8
Therefore, the current practice says that the ship's roll, pitch, and yaw motions are not coupled. In fact, however, these motions are coupled as described by Math. 3, Math. 4 and Math. 5. To demonstrate the difference, we have performed numerical experiments for the two systems, one is represented by Math. 6, Math. 7, and Math. 8 which are under the linearization assumption, and another by Math. 3, Math. 4, and Math. 5 which are nonlinear equations without the linearization assumption. The two systems have identical wave exciting moments including identical amplitudes, frequencies, and phases.FIG. 10shows the time histories of roll, pitch, and yaw for the linearized system whileFIG. 11shows the results of the fully nonlinear system. As can be seen, the linearized system predicted the roll of only 4 degrees while the nonlinear system predicted 28 degrees of roll under the identical wave moments. The difference is caused by the nonlinear terms −{right arrow over (ω)}λ{right arrow over (H)} in Math. 3, Math. 4, and Math. 5. The linearization theory assumes that these nonlinear terms are small so that they can be neglected. The fact is that this assumption is not always valid. The reason is explained below. The roll dynamic system of the ship is a harmonic oscillation system as shown in Math. 3. As we know for a harmonic system, a resonance phenomenon can be excited by a driving force (or moment), no matter how small it is, as long as its frequency matches the natural frequency of the system. Let us explore this in detail by describing the motions in Fourier series as:
φ=Σi=1NA1icos(ω1it+β1i+φ0,  Math. 9
θ=Σj=1NA2jcos(ω2jt+β2j+θ0,  Math. 10
ψ=Σl=1NA3lcos(ω3lt+β3l+ψ0,  Math. 11
wherein N is the total number of terms of the three Fourier series, respectively; i is the index number for the roll's Fourier series; A1iis the amplitude of the ith mode of the roll's Fourier series; ω1iand β1iare the frequency and the phase of the ith mode of the roll's Fourier series, respectively; Φ0is the average value of roll; j is the index number for the pitch's Fourier series; A2jis the amplitude of the jth mode of the pitch's Fourier series; ω2jand β2jare the frequency and the phase of the jth mode of the pitch's Fourier series, respectively; θ0is the average value of pitch; l is the index number for the yaw's Fourier series; A3lis the amplitude of the Ith mode of the yaw's Fourier series; ω3land β3lare the frequency and the phase of the Ith mode of the yaw's Fourier series, respectively; ψ0is the average value of yaw.
Then the roll governing equation Math. 3 can be written as
(Ix+a1){umlaut over (φ)}+b1{dot over (φ)}+k1φ=(Iz−Iy)Σj=1NΣl=1N½A2jω2jA3lω3l{cos [(ω2j+ω3l)t+β2j+β3l]−cos [(ω2j−ω3lt+β2j−β3l]}+M11cos(ω11t+α11).  Math. 12
Note that this is a harmonic oscillation system with one wave moment and 2×N×N inertial coupling moments. The roll response amplitudes due to the inertial coupling moments can be derived as:

Iz-Iy2⁢Ix′⁢A2⁢j⁢ω2⁢j⁢A3⁢⁢l⁢ω3⁢⁢l[(ω2⁢j+ω3⁢⁢l)2-ω102]2+[b1′⁡(ω2⁢j+ω3⁢l)]2,⁢j=1,2,…⁢⁢N;l=1,2,…⁢⁢N;Math.⁢13Iz-Iy2⁢Ix′⁢A2⁢j⁢ω2⁢j⁢A3⁢⁢l⁢ω3⁢⁢l[(ω2⁢j-ω3⁢⁢l)2-ω102]2+[b1′⁡(ω2⁢j-ω3⁢l)]2,⁢j=1,2,…⁢⁢N;l=1,2,…⁢⁢N;Math.⁢14
wherein I′x=Ix+a1, b′1=b1/(Ix+a1), and the roll natural frequency ω10=√{square root over (k1/I′x)}. As can be seen if any of the frequencies ω2j+ω3lor ω2j−ω3lmatches the roll natural frequency ω10, the roll resonance will happen. Similarly, yaw responses can be also obtained as

In the inventor's book, a nonlinear instability phenomenon was also discovered. The nonlinear instability is always attached with the rotational direction where the moment of inertia is the intermediate between the other two inertias. For oceangoing ships, if the loading condition makes the pitch moment of inertia to be larger than the yaw inertia, i.e. Ix<Iz<Iy, the ship may expose to a risk of the nonlinear yaw instability. This kind of loading condition is a realistic loading condition for some cargo ships and could cause violent rolling in seas. For example, Crudu L. etc., “Ship stability in following waves: theoretical and experimental investigations”, Fifth international conference on stability of ships and ocean vehicles, November 1994. The nonlinear instability states that if a ship has only yaw oscillation and the yaw amplitude Ayaw(or yaw angular velocity amplitude Ayawω31) exceeds the yaw threshold AYTH(or the yaw angular velocity threshold rYTH) described below, the yaw oscillation alone becomes unstable so that the roll and pitch motions could grow from almost zero, especially the roll could grow to very large.

AYTH=2ω31⁢Ix′⁢Iy′⁢Z11⁢Z21(Iy-Iz)⁢(Iz-Ix),Math.⁢17⁢arYTH=AYTH⁢ω31=2⁢Ix′⁢Iy′⁢Z11⁢Z21(Iy-Iz)⁢(Iz-Ix),Math.⁢17⁢b
wherein ω31=|ω11−ω21| is the yaw frequency; I′x=Ix+a1; I′y=Iy+a2; Z11=√{square root over ((ω112−ω102)2+(b′1ω11)2)}/ω11; Z21=√{square root over ((ω212−ω202+(b′2ω21)2))}/ω21. Note that Math. 17b is based on the same formula Math. 17a, but described in terms of the angular velocity threshold.

In general, oceangoing ships have the relationship among the moments of inertia as Ix<Iy<Izwith Iyvery close to Iz. That is why Iyand Izare always assumed to be the same if they are not known. This conclusion applies to ships having a flat box transverse cross section, i.e. ship structure height is less than beam, like oil tankers and bulk carriers. However, for some type of ships Ix<Iy<Izis not true, but Ix<Iz<Iymay be true because those ships have tall transverse cross section instead of a flat transverse cross section. The larger vertical distance will increase the pitch moment of inertia since Iy=Σimi(xi2+zi2) and Iz=Σimi(xi2+yi2). As can be seen, when ziof a mass is increased, the effect of increasing in Iyis nonlinear and in a power of 2. Ro/Ro ships (or ferry) may belong to this category. For Ro/Ro ships, both the structure mass and the loaded mass are stretched in vertical direction (z direction), meaning that ziis increased but yikeeps the same. For example, Ro/Ro Ferry MV Sewol had a beam of 22 m but the roof of the 5thfloor was about 28 m from the baseline. This kind of design may make the pitch moment of inertia of a Ro/Ro ship being larger than its yaw moment of inertia. As a result, the ship may expose to yaw instability as described in Math. 17a and Math. 17b.

The above nonlinear instability and inertial coupling effects have been discovered by the inventor just recently. Therefore, there is a need to have a system and a method to warn a ship crew of potential violent motions in the immediate near future for the oceangoing vessel when operating in seas.

SUMMARY

This invention is designed to provide the ship crew with situational awareness of potential violent motions in the immediate near future for the oceangoing vessel when operating in seas. It provides alarming signals to warn the crew if potential dangerous situations detected, such as potential yaw nonlinear instability, potential broaching, inertial roll response exceeding, and rudder induced oscillation.

In one embodiment, a method is presented for identifying yaw nonlinear instability by comparing the moments of inertias for roll, pitch, and yaw.

In another embodiment, a calibration procedure is presented for identifying the inertial coupling roll response coefficient threshold by using Finite Fourier Transform (FFT) analyses for the most recent roll, pitch, and yaw time histories.

In yet another embodiment, a method is presented for identifying potential broaching situation by comparing the first roll frequency and the first pitch frequency obtained in the FFT analyses.

In still yet another embodiment, a method is presented for detecting potential violent roll motions in the immediate near future when a ship operating in seas.

In further still yet another embodiment, a method is presented for detecting potential rudder induced oscillation.

The features, functions and advantages discussed above can be achieved independently in various embodiments or may be combined in yet other embodiments. Further details can be seen with reference to the following description and drawings.

DESCRIPTION

The following text and figures set forth a detailed description of specific examples of the invention to teach those skilled in the art how to make and utilize the best mode of the invention.

Referring toFIG. 1, a violent motion and capsizing warning system100is part of or associated with the ship management computer (not shown). The system100includes a motion sensor module101, a computing service module200, and alarming device comprising113,115and116. The computing service module includes a data recording module102, a vessel specification module103, a yaw nonlinear instability check module104, a visual alarming module comprising105and106, a calibration module107, a broaching check module108, a calibration check module109, a detection computing module110, a broaching check module111, a roll response coefficient exceedance check module112, a rudder induced oscillation check module114.

Referring toFIG. 2, the motion sensor module includes at least a roll sensor101-1, a pitch sensor101-2, a yaw sensor101-3, and a rudder sensor101-4which measure the roll, pitch, yaw, and rudder motions constantly and send the time history data to the data recording module102where the time history data are stored in a readable hardware as shown inFIG. 3. The vessel specification module103as shown inFIG. 4includes loading condition specifications103-1such as moments of inertia for roll (Ix), pitch (Iy) and yaw (Iz), the roll characteristics103-2which includes the roll natural frequency u and the minimum roll damping coefficient b′1, and a pre-determined value module103-3which is a maximum roll peak value, Rollmaxused to trigger alarm. The vessel loading specification information, Ix,Iy,Izneeds to be current and as accurate as possible. The roll natural frequency needs to be consistent with the current GM condition and as accurate as possible. The minimum roll damping coefficient b′1required for the system is not the roll damping obtained in the roll decay test which is the common test for roll damping. This minimum roll damping coefficient represents the minimum value the roll damping could reach during heavy seas. In general, the minimum roll damping happens when high wave crest is amidships in a following sea such that the bow and stern of a ship are both exposed in the air. If the minimum roll damping coefficient b′1is not available, a small default value such as 10% of the roll damping obtained by a roll decay test may be used. The maximum roll peak value Rollmaxrepresents a tolerance level that the ship master wants to set to trigger an alarm. This value depends on the current ship righting arm curves, ship type, and the sea state the ship is going to encounter. Therefore, this value is vessel and sea state dependent. In general, this value is much smaller than the roll angle at which the ship could capsize. Preferably, Rollmaxis between 10-90% of the maximum roll to capsize a ship. The above information in103may be determined before the vessel leaving a port. In the check module104as shown inFIG. 1, the computing service200, which may be performed by a computer installed on board or through web service, compares the moments of inertial for roll, pitch, and yaw. If the yaw moment of inertia is found to be in the middle between the roll and pitch moments of inertia, i.e. Ix<Iz<Iy, a visual alarm signal is generated to alert the crew for a potential yaw nonlinear instability since sharp yaw velocity in maneuver may become large enough to exceed the yaw angular velocity threshold defined in Math. 17b and the ship may lose its yaw control and experience a sudden large roll motion. As an example, if the loading condition of Ro/Ro ferry MV Sewol made the pitch moment of inertia to be the largest one among the three moments of inertia, the warning system invented will give warning to the crew for a potential yaw nonlinear instability so that the crew needs to be very careful to avoid broaching and to turn the ship very slowly to prevent capsizing.

Referring toFIG. 1andFIG. 5, the calibration module107communicates with the module103to get the roll natural frequency ω10and communicates with the module102to get the time histories of motions for the most recent time span of TRwhich is the time duration for the FFT analyses and may be determined as
TR=2πn/ω10, n=2, 3, . . . 10.  Math. 18

The preferred number for n is 2, although the number could be larger such as 10 or higher. The maximum roll peak |Ψ|maxin the most recent time span TRis identified in107-1. The FFT analyses for the roll, pitch, and yaw data in the time span TRare performed in107-2. Since the waveform frequency resolution for FFT analyses is proportional to 1/TR, Lower frequencies than this resolution are not able to be detected. To solve this resolution problem, the data in the time span could be mirrored several times to increase the data samples. Therefore, by mirroring the data we increase the time span several times as well. Then the FFT analyses is performed for the mirrored data. Fast Fourier transform is used for the finite Fourier analyses. Therefore, the number of data samples is required to be in the power of 2, for example, 26or higher. The higher the power goes the more accurate the results would be. The roll, pitch, and yaw motions are described as finite Fourier series as given in Math. 9, Math. 10, and Math. 11, respectively. The amplitudes and frequencies in Math. 9, Math. 10 and Math. 11 are obtained by the FFT analyses performed in107-2. The roll responses due to the 2×N×N inertial coupling moments in Math. 12 can be obtained using Math. 13 and Math. 14. The number N is preferred to be 8 although it may be larger up to 20 or smaller than 8. If the maximum roll peak in this time span TRis equal to or greater than the value Rollmaxset in103-3, i.e. |Φ|max>Rollmaxwhich is checked in107-3, the maximum roll peak to trigger the alarm has been exceeded. The maximum roll response coefficient λmaxdefined in Math. 19 in the same time span TRis calculated in107-5.

φmax=λmax⁢Iz-IyIx′=max⁢{2×N×N⁢⁢roll⁢⁢responses},Math.⁢19
wherein the 2×N×N roll responses are given in Math. 13 and Math. 14. This λmaxis set to be the coefficient threshold in107-6, i.e. λThreshold=λmax. Then the calibration procedure is considered done in107-7. The system continues to108for broaching check, to109for calibration check, and to110for detection computing. On the other hand, however, if the maximum roll peak in this time span TRis less than the value Rollmaxset in103-3, i.e. |Φ|max<Rollmaxwhich is checked in107-3, the calibration is considered not done in107-4. Then the system continues to108for broaching check and to109for calibration check, and goes back to107for the next time span as shown inFIG. 1

For example,FIG. 12shows the time histories of roll, pitch, yaw and rudder of an experiment run by Pauling J. R. etc. 1972, “Experimental studies of capsizing of intact ships in heavy seas”, Technical report AD-753653, University of California, Berkeley. The data in Win1 ofFIG. 12has been extracted for FFT analyses. The roll natural frequency for this ship was found to be ω10=0.79. The time span TRin Win1 equals 15.9 seconds in this case. The maximum roll peak in Win1 was found to be |Φ|max=47 degrees. Note that the 47 degrees roll was quite large for the ship and the ship almost capsized in Win1. Let us set the Rollmaxin103-3to be 25 degrees for demonstration purpose.FIG. 13shows the FFT frequencies and magnitudes of the roll, pitch, yaw, and rudder for the data in Win1 ofFIG. 12. The largest eight components for each motion are identified as shown inFIG. 13. The time histories of motions and the largest eight components with the amplitudes and frequencies obtained from FFT analyses are shown inFIG. 14. The phases of these components are adjusted to best fit the original data in Win1 ofFIG. 12when the eight components are combined as shown inFIG. 15. The maximum of the 128 roll responses may be obtained in107-5as below for this case.

λmax=A24⁢ω24⁢A31⁢ω312⁢b1′⁡(ω24-ω31)=7.12,Math.⁢20
at the frequency ω24-ω31=0.79=ω10. The roll damping coefficient was assumed to be 0.4% of the critical roll damping for this case. The calibration check in107-3says that the maximum roll peak to trigger the alarm has been exceeded, i.e. |Φ|max=47>Rollmax=25. Therefore, the coefficient threshold was found to be λThreshold7.12 in this case.

Referring toFIG. 1andFIG. 6, the broaching check is performed in the module108. If the frequency ω11of the largest amplitude roll component obtained in107-2is equal to the frequency ω21of the largest amplitude pitch component obtained in107-2, a potential broaching can happen according to Math. 16. The system will generate audible and visual alarms in108-2, saying “potential broaching”. If these two frequencies do not match each other, the system goes to108-3and the alarm in108-2will be turned off if it is on.

Referring toFIG. 1andFIG. 7, the detection computing module110communicates with the module103to get the roll natural frequency ω10and communicates with the module102to get the time histories of motions for the most recent time span of TRwhich is the time duration for the FFT analyses. TRis determined by Math. 18. The preferred number for n is 2, although the number could be larger such as 10 or higher. The FFT analyses for the roll, pitch, yaw, and rudder data in the time span TRare performed in110-1. The roll, pitch, and yaw motions can be described as finite Fourier series as given in Math. 9, Math. 10, and Math. 11, respectively. The amplitudes and frequencies in Math. 9, Math. 10, and Math. 11 are obtained by the FFT analyses performed in110-1, respectively. The rudder motion can be also described as finite Fourier series as given in Math. 21 with the amplitudes and frequencies obtained in110-1.
η=Σm=1Ncos(ω4mt+β4m)+η40,  Math. 21
wherein N is the total number of terms of the Fourier series; m is the index number for the Fourier series; A4mis the amplitude of the mth mode of the Fourier series; ω4mand β4mare the frequency and the phase of the mth mode of the Fourier series, respectively; η40is the average value of the rudder motion.

The maximum roll peak |Φ|maxin the time span TRis identified in110-2. The roll responses due to the 2×N×N inertial coupling moments in Math. 12 can be obtained using Math. 13 and Math. 14. The number N is preferred to be 8 although it may be larger up to 20 or smaller than 8. The maximum roll response coefficient Amaxdefined in Math. 19 in the time span TRis calculated in110-3. The associated yaw frequency ωyaw, obtained also in110-3is defined as the yaw frequency associated with the inertial coupling moment which generates the maximum roll response coefficient Amaxobtained in110-3. If the maximum roll peak is greater than the value Rollmaxset in103-3, i.e. |Φ|max>Rollmaxwhich is checked in110-4, the maximum roll peak to trigger the alarm has been exceeded. The system goes to110-6to further check whether the maximum roll response coefficient λmaxobtained in110-3exceeds the coefficient threshold λThresholdobtained in107-6. If λmax>λThresholdthe system goes to110-5indicating the roll response coefficient λmaxexceeding. If λmax<λThresholdit means that the coefficient threshold λThresholdobtained in107-6is too high and needs to be updated by the value obtained in110-3. This update is performed in110-7. Then the system goes to110-5indicating the roll response coefficient λmaxexceeding. On the other hand, if the maximum roll peak is less than the value Rollmaxset in103-3, i.e. |Φ|max<Rollmax, the system goes to110-8indicating the roll response coefficient λmaxnot exceeding.

For example, in the demonstration case inFIG. 12the coefficient threshold λThresholdwas found to be 7.12 as shown in Math. 20 based on the data in Win1 ofFIG. 12. Similarly, FFT analyses are performed for Win2 which is a successive time span to Win1 inFIG. 12. Based on the same procedure as above, the maximum roll peak in Win2 is found to be |Φ|max=28 degrees which is larger than the preset Rollmax=25 degrees in103-3. According to detection computing110, the system goes to110-6inFIG. 7. The maximum roll response coefficient λmaxis found to be 3.56 in110-3. In this case,110-6check inFIG. 7shows negative. Therefore, the coefficient threshold needs to be updated, i.e., λThreshold=156 which is the new coefficient threshold. From now on, the detection computing110will use the new coefficient threshold for future detection until its next update. The system checks broaching in111(explained inFIG. 8below) inFIG. 1and it shows negative for Win2. Since |Φ|max=28>Rollmax=25,112roll response coefficient check inFIG. 1is positive in this case. Then the rudder induced oscillation is checked in114and it shows also negative for this case of Win2. Therefore, only alarm115is triggered in this case.

Referring toFIG. 1andFIG. 8, after detection computing110finished, the system goes to broaching check at111. If the frequency ω11of the largest roll component amplitude obtained in110-1is equal to the frequency ω21of the largest pitch component amplitude obtained in110-1, a potential broaching may happen according to Math. 16. The system will generate audible and visual alarms in111-2, saying “potential broaching”. If these two frequencies do not match each other, the system goes to111-3and the alarm in111-2will be turned off if it is on. This completes the broaching check.

Referring toFIG. 1andFIG. 9, after the broaching check111, the system goes to112to check again the roll response coefficient exceedance.112is to confirm the previous conclusion obtained in detection computing110. If the roll response coefficient λmaxexceeding is true, the system goes to114to further check rudder induced oscillation as shown inFIG. 9. If the associated yaw frequency ωyawis equal to anyone of the frequencies of the largest four rudder components as shown in114-1inFIG. 9, the associated yaw frequency ωyawmay be driven by that rudder frequency, indicating that the inertial coupling excited roll oscillation is driven indirectly by rudder. Therefore, the roll oscillation can be considered as rudder induced oscillation. If this is true, a rudder induced oscillation is detected in114-2. The system goes to116inFIG. 1to generate audible and visual alarms indicating “inertial coupling roll response exceeded, and rudder induced oscillation at frequency ωyaw”. The system then goes back to detection computing110to start the next cycle of detecting. If the roll response coefficient λmaxexceeding in112is true, but no rudder induced oscillation is detected, the system goes to115to generate audible and visual alarms indicating “inertial coupling roll response exceeded”. Then the system goes back to detection computing110to start the next cycle of detecting. If the roll response coefficient λmaxis not exceeding the threshold in112, the system goes to113to turn off the alarms generated in115or116whichever is on. Then the system goes back to detection computing110to start the next cycle of detection. The system is running continuously without stop until the system is turned off manually.

For example, in the demonstration case inFIG. 12the new coefficient threshold was found to be λThreshold=3.56 after the detection computing for Win2 inFIG. 12. We have performed detection computing110for Win3 inFIG. 12.FIG. 16shows the FFT frequencies and magnitudes of the roll, pitch, yaw, and rudder for the data in Win3 ofFIG. 12. The largest eight components for each motion are identified as shown inFIG. 16. The time histories of motions in Win3 ofFIG. 12and the combined results of the Fourier approximations using the largest eight components for each motion are shown inFIG. 17. The frequencies and amplitudes of the largest eight finite Fourier components for Win3 inFIG. 12are summarized inFIG. 18. The maximum roll peak was found in110-2to be |Φ|max=45 degrees. The maximum of the 128 roll responses has been obtained in110-3for this case as below.

λmax=0.54⁢A24⁢ω24⁢A36⁢ω36b1′⁡(ω24-ω36)=9.18,Math.⁢22
at the frequency ω24-ω36=0.79=ω10. Since the Rollmax=25 degrees and λThreshold=3.56,110-4and110-6checks are both positive. The roll response coefficient λmaxis exceeding, i.e. λmax=9.18>λThreshold3.56. Since ω11=ω21=1.38 as shown inFIG. 18, the broaching alarm111-2is triggered. The yaw frequency ω36involved with the inertial coupling to generate the maximum roll response coefficient as shown in Math. 22 is the associated yaw frequency, i.e. ωyaw=ω36. SinceFIG. 18shows that the yaw frequency ωyaw=ω36coincides with the first rudder frequency ω411.18,114-1check is positive. The rudder induced oscillation is detected in this case and the alarm116is triggered. Note that in Win3 ofFIG. 12, the three alarms, i.e. broaching alarm111-2, roll response coefficient exceeding alarm116, and rudder induced oscillation alarm116are all triggered in this case. This is a dangerous situation for the ship to encounter. The time span of Win2 is from 201.375 to 217.25 and the time span of Win3 is from 330.25 to 346.125 seconds. As shown inFIG. 12, the ship capsized at 348.4 seconds, just 2 seconds after Win3. Note that the maximum roll response coefficient λmax=9.18 in Win3 is about 2.5 times larger than the threshold value detected at Win2. If the ship can follow the warning detected in Win2 by the system to keep the inertial coupling roll response level low and to avoid broaching and rudder induced oscillation in an earlier time, the ship could prevent capsizing.

It should be understood that the above descriptions may be implemented to many types of ships, for example, such as oil tankers, bulk carriers, containerships, fishing vessels, Ro/Ro ships, boats, military ships, vessels in lakes or some other appropriate type of vessels. It should also be understood that the detailed descriptions and specific examples, while indicating the preferred embodiment, are intended for purposes of illustration only and it should be understood that it may be embodied in a large variety of forms different from the one specifically shown and described without departing from the scope and spirit of the invention. It should be also understood that the invention is not limited to the specific features shown, but that the means and construction herein disclosed comprise a preferred form of putting the invention into effect, and the invention therefore claimed in any of its forms of modifications within the legitimate and valid scope of the appended claims.