Nonlinear Dyn (2016) 84:1373–1382DOI 10.1007/s11071-015-2576-1

ORIGINAL PAPER

Research on chaos and nonlinear rolling stability of arotary-molded boat

Qiguo Yao · Yuliang Liu

Received: 12 May 2015 / Accepted: 18 December 2015 / Published online: 4 January 2016© The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract On the basis of nonlinear features analysisof restoring torque, damping torque and especially hulldeformation, a nonlinear roll equation of rotary-moldedboats has been established. The equation includes elas-tic deformation term which makes the equation dif-ferent from existing ones for steel boats. This paperdissected its dynamic characteristic of rolling motionfrom different aspects by using time domain diagram,phase graph, Poincaremap, aswell Lyapunov exponen-tial spectrum.And then, an exact feedback linearizationcontroller was designed using closed-loop gain shap-ing algorithm to make the boat get rid of chaotic state.Through numerical simulation, it is found that the rollfrequency and velocity increase when elastic defor-mation occurs; i.e., the roll stability margin shrinks,and the chaos phenomenon is much more remarkable.This conclusion laid the foundation for controlling orrestraining the boat’s rolling and can be used as thebasis of how to amend the evaluation criteria on rotary-molded boat stability.

Keywords Rotary-molded boat ·Elastic deformation · Roll motion · Nonlinear stability

Q. Yao (B) · Y. LiuSchool of Naval Architecture and Ocean Engineering,Zhejiang Ocean University,Zhoushan 316022, Zhejiang Province, Chinae-mail: yaoqiguo@163.com

Y. Liue-mail: lyl_zjou@126.com

1 Introduction

Rotational molding technology originated in the 1940sin the UK. Then, the technology was widely used in theUSA and Japan and became one competitive methodin the field of plastic forming [1–3] (Fig. 1).

A rotary-molded boat is often made of polyethylenematerial, which makes the boat not only pollution free,erosion resistant and alkali resistant, but also with largebuoyancy, high speed and low repair cost. However, theexisting boat construction methods are still based onsteel-material ships. Though the rotary-molded boat ismade of high-strength plastic, its rigidity will decreasewhen the load is too much or it rolls severely, and itsdeformation should not be neglected; thus, its stabilitywill decrease too. Therefore, there is a hidden dangerin stability evaluation in that the theoretical foundationfor steel-material ships is not suitable to rotary-moldedboats.

To the best of our knowledge, no literature on rotary-molded boat stability has been published. In this paper,we choose a rotary-molded fishing entertainment boatas the research object.We consider the elastic deforma-tion of boat in the new roll motion equation and studythe establishment of nonlinear roll motion equation andstability simulation.

Rolling is the most common ship motion. Rollingis also a crucial reason of overturning accidents [4–7]. In the nineteenth century, Krylov and Froude putforward early classical movement theories on ship lon-gitudinal and transversemotion, respectively. There are

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1374 Q. Yao, Y. Liu

Fig. 1 Contour diagram of a rotary-molded boat

voluminous literatures on ship stability. The theoreticalbasis of early studies was linear; i.e., many nonlinearfactors were not considered. Therefore, the reasonableexplanation for some phenomena such as sudden over-turning cannot be given [8,9].With the gradual matura-tion of nonlinear dynamics theory, many scholars havemade useful exploration on ship rolling. For instance,Li [10,11] used the multiple scale method to estab-lish ship nonlinear rolling equation in the case of regu-lar waves. He found the approximate analytic solutionof the nonlinear equation and succeeded in verifyingthe solution. Without considering the deformation andcoupling, the nonlinear single-freedom-degree rollingequation can be expressed as

(Ix + δ Ix )ϕ + D1ϕ + D3ϕ3

+Δ(GMϕ + C3ϕ3 + C5ϕ

5 + · · · )= ML + MF (1)

where Ix is the coherent rotation moment of inertia,δ Ix is the additional rotation moment of inertia, ϕ is theroll angle which is positive in clockwise direction andnegative in counterclockwise direction if looked fromstern to bow, D1 and D3 are damping coefficients, Δ

is displacement, GM is initial stability height, C3 andC5 are recovery coefficients, ML is wave disturbancetorque, MF is wind disturbance torque. Assume that atwo-dimensional irregular long crest wave propagatesalong a fixed direction, and it is superposed by manyregular waves with different frequencies, wavelengths,amplitudes and random phases. Also assume that thecrest lines are infinitely long and parallel to each other.Wave characteristic parameters such as wave heightand wave period change randomly. Its mathematical

expression is as follows:

ζ(t) =∞∑

i=1

ζi cos(ωi t + εi ) (2)

where ζ is the height of wave and ζi is the amplitudeof the i th harmonic wave.

In Eq. (1), the ML can be expressed as

ML = Iα0ω20π

∞∑

i=1

hiλi

cos(ωi t + εi ) (3)

where I is the total inertia moment which includes thecoherent rotation moment of inertia Ix and the addi-tional rotationmoment of inertia δ Ix , α0 is the effectivecoefficient of wave inclination, ω0 is the boat naturalfrequency, hi is the significant wave height, λi is thewave wavelength, ωi is the wave frequency, and εi isthe random phase angle which ranges from 0 to 2π[12,13].

Assume that the unsteady wind is composed of themean part and the fluctuating part. The wind speed is

uF = u0 +∞∑

i=1

ui cos(Ωi t + θi ) (4)

where u0 is the average speed, ui is the fluctuatingamplitude, Ωi is the pulsing frequency, and θi is thefluctuating random phase. Wind force on the boat canbe written as

f = 1

2ρair · u2F · S (5)

where ρair is air density and S is the upright-projectionarea of a boat hull above the waterline. Then, the winddisturbance torque MF on the boat is written as

MF = f · (H + d − zR)

= ρair · u2F · S · (H + d − zR)/2 (6)

where H is the distance between wind pressure centerand thewater surface, d is the boat draft depth, and zR isthe distance between the water resistance action centerand the baseline, which is approximately equal to d/2.Therefore, when not considering the boat deformation,we can get a single-freedom-degree rolling equation ofa rotary-molded boat as follows

(Ix + δ Ix )ϕ + D1ϕ + D3ϕ3

+Δ(GMϕ + C3ϕ3 + C5ϕ

5 + · · · )= I · α0 · ω2

0 · π ·∞∑

i=1

hiλi

cos(ωi t + εi )

+1

2ρair · u2F · S · (H + d − zR) (7)

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Research on chaos and nonlinear rolling stability of a rotary-molded boat 1375

2 A new rolling model of the rotary-molded boat

Because the material of a rotary-molded boat is poly-ethylene which has a small specific gravity value ofabout 900–960kg/m3, it is necessary to consider defor-mation in establishing amore realistic rolling equation.

Assume the boat speed is zero and the heeling is onlycaused by wind and wave. We draw boat rolling statediagram without considering deformation as shown inFig. 2, where point G0 is the center of gravity, B0 is theupright buoyancy center, B1 is the tilt buoyancy center,segment B1S1 is the reply lever, W0L0 is the uprightwaterline, and W1L1 is the inclined waterline. Let V1and V2 be the water wedge volume induced by boathull’s tilt. Let V0 be the floating displacement [14–18].So the volume below the waterline after being tilted is

V = V0 + V1 − V2 (8)

Assume that the boat hull’s deformation is onlycaused by crosswind. The wind pressure evenly dis-tributes along the high free-board. Assume that defor-mation amount is x as shown in Fig. 3. When defor-mation happens, the gravity center, buoyancy centerand displacement volume all change correspondingly.The gravity center changes from G0 to G1. The buoy-ancy center moves from B1 to B2. B2S2 is the newreversion arm. The position of the new gravity cen-ter G1 can be calculated based on the gravity centermoving principal. For simplicity, assume that damp-ing torque and response arm are constant. AssumeB1S1 ≈ B2S2. Then, the volume under water V ′ =V0 + (V1 + δV1) − (V2 − δV2), where δV1 is the vol-ume of trapezoidal block 5678 and δV2 is the volumeof trapezoidal block 1234 as shown in Fig. 3.

Here, 56 = tan ϕ · a, 78 = tan ϕ · (a + x) and12 = tan ϕ · (B − a − x), 34 = tan ϕ · (B − a).

δV1 =(1

2(a + x)2 · tan ϕ − 1

2a2 · tan ϕ

)· L (9)

δV2 =(1

2(B − a)2 · tan ϕ − 1

2(B − a − x)2 · tan ϕ

)· L(10)

Therefore, the change of drainage volume is,

δV = V ′ − V = δV1 + δV2 = L · B · x · tan ϕ (11)

and the change of displacement is as below,

δΔ = δV · ρwater = L · B · x · tan ϕ · ρwater

= k1 · x · tan ϕ (12)

Fig. 2 Rolling diagram without elastic deformation

Fig. 3 Rolling diagram with elastic deformation

Next,we describe the change of displacement as a func-tion ofwind speed.Because the deformation is assumedto be elastic, let f = k2 ·x . Because f = 1

2ρair ·u2wind ·S,so,

x = 1

2ρair · u2wind · S/k2 = k3 · u2wind (13)

That is to say, the change of displacement is

δΔ = k1 · x · tan ϕ

= k1 · k3 · u2F · tan ϕ = K1 · u2F · tan ϕ (14)

where K1, k1, k2, k3, k4 are constant numbers to a spe-cific boat. Thus, when elastic deformation is generatedonly by wind, the boat rolling motion equation is writ-ten as

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1376 Q. Yao, Y. Liu

(Ix + δ Ix ) · ϕ + D1 · ϕ + D3 · ϕ3

+(Δ + K1 · u2F · tan ϕ)

·(GM · ϕ + C3 · ϕ3 + C5 · ϕ5 + · · ·)= I · α0 · ω2

0 · π ·∞∑

i=1

hiλi

cos(ωi t + εi )

+ 1

2ρair · u2wind · S · (H + d − zR) (15)

Further study show that, the hydrodynamic impactwhich thewave acts on the boat should not be neglected,this impact will cause the boat’s bending and torsiondeformation [19,20]. Here, assuming that the wavedirection is in line with the direction of the wind and isvertical to the central lateral plane of boat.

According to the water elastic mechanics theory, theimpact force that the wave acts on the boat is [21,22]:

F = Δ · GM ·∞∑

i=1

πhi

λicos(ωi t + εi ) (16)

The deformation which is caused by the hydrody-namic impact is signed as x ′, for simplicity, hypothesisthe deformation is satisfied with the Hooke’s law, then,

F = k′ · x ′ (17)

x ′ = F/k′

= Δ · GM ·∞∑

i=1

πhiλi

cos(ωi t + εi )/k′

= k4 ·∞∑

i=1

hiλi

cos(ωi t + εi ) (18)

So, the total deformation can be expressed as:

X = x + x ′ (19)

And then, the change of displacement is:

δΔ= k1 · X · tan ϕ

= k1 · (x + x ′) · tan ϕ

= k1 · x · tan ϕ + k1 · x ′ · tan ϕ

= k1 · k3 · u2F · tan ϕ+k1 · k4 ·∞∑

i=1

hiλi

cos(ωi t+εi )

=K1 · u2F · tan ϕ+K2 · tan ϕ ·∞∑

i=1

hiλi

cos(ωi t+εi )

(20)

Thus, when thewind andwaves coupling is considered,the completed boat rolling motion equation is writtenas

(Ix + δ Ix ) · ϕ + D1 · ϕ + D3 · ϕ3

+(

Δ + K1 · u2F · tan ϕ + K2 · tan ϕ

·∞∑

i=1

hiλi

cos(ωi t + εi )

)

·(GM · ϕ + C3 · ϕ3 + C5 · ϕ5 + · · ·)= I · α0 · ω2

0 · π ·∞∑

i=1

hiλi

cos(ωi t + εi )

+1

2ρair · u2F · S · (H + d − zR) (21)

where K1, K2 are constant numbers to a specific boat.

3 Numerical simulation

3.1 The deformation is caused only by wind

Simulation parameters are given as: Boat length L is11.4m, type wide B is 2.51m, deep D is 1.305m, ini-tial stability height GM is 0.707m, design draft d is0.60m, drainage volume V is 8.14 cubic meters, dis-placement Δ is 8.34 tons, distance Zg between gravitycenter and midline is 0.304m, wind area S is 9.26m2,height H between wind pressure center and water sur-face is 0.882m, no-load weight is 2.9 tons, and full loadweight is 8.34 tons. The stability vanish angle ϕγ is1.57 rad. The area Aϕ surrounded by static stabilitycurve and stability vanish angle is 0.4112m rad. Takethe air density ρair = 1.2 kg/m3 and the effective coef-ficient of slope α0 = 0.729. Consider full load condi-tion. The total rotation moment of inertia is 1.820t m2.The coherent roll frequency ω0 = 1.80 rad/s. Damp-ing coefficient D1 = 0.04002 and D3 = 0.027288.Restoring factor C3 = −0.3352 and C5 = 0.019608.Restoring moment is only calculated up to the 5-powerterm ϕ5. Integrated deformation coefficient K1 = 0.1,K2 = 0. Waves are regular ones. The significantsea wave height, wavelength and wave frequency areh = 3m, λ = 90m and ω = 0.85rad/s, respectively.Also assume that there is no navigational speed andonly beam wind with speed uF = 10 + 10 cos(4t) isconsidered. Take the aforementioned parameter valuesinto Eq. 21 and get boat rolling motion equation asfollows

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Research on chaos and nonlinear rolling stability of a rotary-molded boat 1377

0 0.5 1 1.5 2 2.5 3 3.5 40

1

2

3

4

5

6

7

8

t/s

φ/ra

d

K1=0

K1=0.001

K1=0.1

Fig. 4 Roll angle curves changing with time for different coef-ficient K1 under K2 = 0

1.82ϕ + 0.04002ϕ + 0.027288ϕ3

+(8.34 + 0.1∗(10 + 10 cos(4t))2 · tan ϕ)

·(0.707ϕ − 0.3352ϕ3 + 0.019608ϕ5)

= 0.45 cos(0.85t) + 6.567(10 + 10 cos(4t))2 (22)

Let x = ϕ, y = ϕ, then Eq. 22 becomes

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

x = yy = −0.022y − 0.015y3 − (4.5824 + 0.055

·(10 + 10 cos(4t))2 · tan x)·(0.707x − 0.3352x3 + 0.019608x5)+ 0.25 cos(0.85t) + 3.608(10 + 10 cos(4t))2

(23)

Next, we discuss how the rolling motion changeswhen deformation coefficient K1, wind frequency andwave frequency are modified. We set wind frequencyto 4 rad/s and wave frequency to 0.85 rad/s. The ini-tial condition (x0, y0) = (0, 0). We get waveformdiagrams as shown in Figs. 4 and 5. We find that theroll angle decreases with the increasing of K1. On thecontrary, the roll angular velocity increases with thedecreasing of K1.

3.2 The deformation is caused by the coupling ofwind and waves

When the wind and waves coupling is considered,hypothesis the other parameters are the same as the for-mer, let K1 = K2 = 0.1, then the roll motion Eq. (21)will become:

0 0.5 1 1.5 2 2.5-60

-40

-20

0

20

40

60

80

t/s

φ ′/r

ad/s

K1=0

K1=0.1

Fig. 5 Roll angle velocity curves changing with time for differ-ent coefficient K1 under K2 = 0

0 0.5 1 1.5 2 2.5 3 3.5 40

1

2

3

4

5

6

7

8

t/s

φ/ra

d

K1=0,K2=0

K1=0.001,K2=0.001K1=0.1,K2=0.1

Fig. 6 Roll angle curves changing with time for different coef-ficient K1, K2

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

x = yy = −0.022y − 0.015y3 − (4.5824

+ 0.055(10 + 10 cos(4t))2 · tan x+ 0.002 tan x · cos(0.85t)) · (0.707x− 0.3352x3 + 0.019608x5) + 0.25 cos(0.85t)+ 3.608(10 + 10 cos(4t))2

(24)

Under different K1, K2, the roll angle and roll angu-lar velocity curve are shown in Figs. 6 and 7. Throughobserved, it is found that the roll angular velocity var-ied intensely and strengthened along with the K1, K2

increase.

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1378 Q. Yao, Y. Liu

0 0.5 1 1.5 2 2.5-80

-60

-40

-20

0

20

40

60

80

100

120

t/s

φ ′/r

ad/s

K1=0,K2=0

K1=0.1,K2=0.1

Fig. 7 Roll angle velocity curves changing with time for differ-ent coefficient K1, K2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1500

-1000

-500

0

500

1000

t/s

Lyap

unov

Exp

onen

ts

Fig. 8 Lyapunov exponential spectrum of system (24) underK1 = 0.005, K2 = 0.005

3.3 Validation of the chaos characteristic

The simulation parameters are the same as the former,consider the deformation is caused by the wind andwaves coupling.

Further study finds that system (24) may appear tobe chaotic with the change of K1, K2. For example,when K1 = 0.005, K2 = 0.005, the Lyapunov expo-nential is (0.0115, −0.0146, 0) by using the Gram–Schmit orthogonal method. Figure 8 shows its Lya-punov exponential spectrum, and Fig. 9 shows its phasegraph. There exists positive Lyapunov exponential. Itindicates that the system is in chaotic state.

It is well known to all that Poincare map is a goodkind of tool to distinguish chaos. The map of periodic

-60 -40 -20 0 20 40 60 80 100 1200

1

2

3

4

5

6

7

8

φ/ra

d

φ′/rad/s

Fig. 9 Phase graph of ϕ −ϕ′ of system (24) under K1 = 0.005,K2 = 0.005

4.65 4.66 4.67 4.68 4.69 4.7 4.71-40

-30

-20

-10

0

10

20

30

40

x

y

Fig. 10 Poincare map of system (24) under K1 = 0.005, K2 =0.005

motion is a finite number of discrete points, and thestandard periodic motion is a closed curve; the map ofchaoticmotion is a number of collection of points alongwith a line segment or a curve, and this collection ofpoints has self-similar fractal structure [9].

The simulation parameters are the same as the for-mer, considered the deformation is caused by the windand waves coupling. Set K1 = 0.005, K2 = 0.005, thePoincare map of system (24) is shown as Fig. 10, andobviously, the system is chaotic.

The above simulation result shows that the rollingwill getmore intense and the stabilitywill getweakenedwhen boat deformation is considered. At the same time,the ride comfort will get worse too. In particular, theappearance of chaos may increase the danger of the

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Research on chaos and nonlinear rolling stability of a rotary-molded boat 1379

boat [23]. So it is necessary to control chaos when boatdeformation is considered.

3.4 Control of chaos

With regard to the chaos control of boat rolling, doc-ument [24] achieved satisfactory results by using non-linear simple control method which is the combinationof the accurate feedback linearization method and theclosed-loop gain shaping algorithm to get rid of the boatchaotic state. Here, we use this method. Add a controlitem u to the second equation of system (24). Then, weget a controlled system (25):⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

x = yy = −0.022y − 0.015y3 − (4.5824

+ 0.055(10 + 10 cos(4t))2 · tan x+ 0.002 tan x · cos(0.85t)) · (0.707x− 0.3352x3 + 0.019608x5) + 0.25 cos(0.85t)+ 3.608(10 + 10 cos(4t))2 + u

(25)

According to the nonlinear simple control methodbased on the algorithm of accurate feedback lineariza-tion, u can be expressed as

u = 0.022y + 0.015y3 + (4.5824+ 0.055(10 + 10 cos(4t))2 · tan x+ 0.002 tan x · cos(0.85t)) · (0.707x− 0.3352x3 + 0.019608x5) − 0.25 cos(0.85t)− 3.608(10 + 10 cos(4t))2 + v

(26)

Using the method in document [24], we obtain a PDcontroller v = −6x − 3y. So the whole control law is

u = 0.022y + 0.015y3 + (4.5824+ 0.055(10 + 10 cos(4t))2 · tan x+ 0.002 tan x · cos(0.85t)) · (0.707x− 0.3352x3 + 0.019608x5) − 0.25 cos(0.85t)− 3.608(10 + 10 cos(4t))2 − 6x − 3y

(27)

In order to compare the effects before and after thecontrol, the condition of wind speed and sea waveformis given as below⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

0 ≤ t < 5s :uF = 10 + 10 cos(4t), ζ(t) = 3

15 cos(2t);5 ≤ t < 10s :uF = 15 + 10 cos(4t) + 5 cos(8t),ζ(t) = 3

15 cos(2t) + 103.8 cos(4t);

10 ≤ t < 15s :uF = 17 + 10 cos(4t) + 5 cos(8t) + 2 cos(12t),ζ(t) = 3

15 cos(2t) + 103.8 cos(4t) + 15

1.71 cos(6t)

(28)

05

1015

20

-50

0

50

100-2

0

2

4

6

8

t/sφ′/rad/s

φ/ra

d

Fig. 11 Phase graph of the controlled system (25)

Set K1 = K2 = 0.1. Other parameters are the sameas before. Add the controlling item u when t = 15s,end the control when t = 20s. The phase graph ofsystem (25) is shown in Fig. 11. After being controlled,the system gets rid of chaos and is controlled at theequilibrium point (0, 0).

3.5 Analysis of robustness

A controlled system is robust if it meets that (1) it hasa low sensitivity; (2) the system is always stable withinthe scope of the parameters may change; (3) the perfor-mance of the system can meet the requirements whenthe parameters change [25–27].

In fact, a real controlled system often exists modelerror and measurement noise, and an effective controlstrategy must have strong robustness. The robustnessof the controlled system (25) is analyzed as follows.

Firstly, when the system exists model error, we setthe wind frequency 4, 8, 12 rad/s, respectively, in dif-ferent times as expression (28), add the control strategyu at 15 s, and end it at 20 s, and the roll angle curveschanging with time of the controlled system (25) areshown in Fig. 12. From Fig. 12, we find the controlstrategy u can control the system very well even thoughthere exists model error.

Secondly, when the system exists measurementnoise, we add white noise enoise to the controlled sys-tem (25).

Here,

enoise = ψ(T ) · sin(�T ) (29)

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1380 Q. Yao, Y. Liu

0 2 4 6 8 10 12 14 16 18 20-1

0

1

2

3

4

5

6

7

8

t/s

φ/ra

d

Fig. 12 Roll angle curves changing with time of the controlledsystem (25) when exists model error

0 5 10 15 20 25 30 35 40 45 50-1

0

1

2

3

4

5

6

7

8

t/s

φ/ra

d

Fig. 13 Roll angle curves changing with time of the controlledsystem (25) when exists measurement noise

where

ψ(T ) = 0.05 ± 0.002r(T ) (30)

�(T ) = 0.5 ± 0.02r(T ) (31)

and r(T ) is a normally distributed random function[25]. Add the control strategy u from 15 to 50s, andthe time response curve of roll angle of the controlledsystem (25) is shown in Fig. 13.

At last, when the system exists both model errorand measurement noise, the other simulation parame-ters are the same as the former and the model errorand the enoise are the same as the former too. Thetime response curve of roll angle of the controlled sys-tem (25) is shown in Fig. 14.

0 5 10 15 20 25 30 35 40 45 50-1

0

1

2

3

4

5

6

7

8

t/s

φ/ra

d

Fig. 14 Roll angle curves changing with time of the controlledsystem (25)when exists bothmodel error andmeasurement noise

The simulation shows that the designed controllerhas good control effect, and the controlled system (25)is robust when the system is subject to parameter uncer-tainties and noise measurement.

3.6 Comparison with previously published work

The literature that specially researches the stability ofrotary-molded boat or rubber boat is very less. Eventhough there is a few of references which consideredits elastic deformation to study the dynamic character-istics of rolling of the rotary-molded boat, the researchobject of them is the steel ship. Compared with thereferences [6,8,9], the main work of our study can bestated as the following several aspects: (1) The model-ing method of us considered the wind and waves cou-pling, which is closer to the actual situation; (2) thesimulation shows that the rolling angular velocity andangular accelerationwill increasewhen considering thedeformation of boat; at this time, the ship’s stabilitydeserves more attention; (3) when considering its elas-tic deformation, the stability of boat will decrease, butregarding how to accurately calculate and evaluate howmuch of the stability decreased, as well how to improvethe stability, neither the references nor we consider it;this is our future research direction.

4 Conclusions

In this paper, we take a rotary-molded boat as theresearch object.Wefirst consider the hull’s deformation

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Research on chaos and nonlinear rolling stability of a rotary-molded boat 1381

to establish a nonlinear rolling equation. From simula-tions, it is shown that the stability getsweakened and thesystemgets chaoticwhen elastic deformation is consid-ered. In detail, the investigation leads to the followingconclusions:

(1) The rotary-molded boat has complex dynamiccharacteristics when the irregular wind and waveact on it.

(2) Along with the increase in wind and wave har-monic, the influence of the dynamic characteristicsof boat is significant. Some simulation methodsand results such as the time domain curve, phasediagram, Lyapunov exponential spectrum, as wellPoincare map reflect this changes from differentaspects.

(3) When the elastic deformation of boat is caused bythe wind and waves coupling, the rolling is muchmore intense, its stabilitywill decline furthermore,and the chaos phenomenon is more obvious.

(4) The controlling method is feasible and effective.It can control the system to the balance point. Fur-thermore, the controller has good robustness.

In addition, the rotary-molded boat technology is akind of innovation of boat material, and along with thedevelopment of social, there will have more and morerotary-molded boats. The rolling movement modelwhich is established by us is much more in line withthe actual than the related citations. The research of thispaper laid a foundation of improving the designmethodof boat, and the calculation method can be popularizedand applied to other ships. Therefore, it is significant toamend the stability inspection standard to control boatchaos more effectively, which is our research focus inthe future too.

Acknowledgments This study is supported by the NaturalScience Foundation of Zhejiang Province of China (Grant No.LY13E090004).

Open Access This article is distributed under the terms ofthe Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permitsunrestricted use, distribution, and reproduction in any medium,provided you give appropriate credit to the original author(s) andthe source, provide a link to the Creative Commons license, andindicate if changes were made.

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