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

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## Abstract

On the basis of nonlinear features analysis of restoring torque, damping torque and especially hull deformation, a nonlinear roll equation of rotary-molded boats has been established. The equation includes elastic deformation term which makes the equation different from existing ones for steel boats. This paper dissected its dynamic characteristic of rolling motion from different aspects by using time domain diagram, phase graph, Poincare map, as well Lyapunov exponential spectrum. And then, an exact feedback linearization controller was designed using closed-loop gain shaping algorithm to make the boat get rid of chaotic state. Through numerical simulation, it is found that the roll frequency and velocity increase when elastic deformation occurs; i.e., the roll stability margin shrinks, and the chaos phenomenon is much more remarkable. This conclusion laid the foundation for controlling or restraining the boat’s rolling and can be used as the basis of how to amend the evaluation criteria on rotary-molded boat stability.

## Keywords

Rotary-molded boat Elastic deformation Roll motion Nonlinear stability## 1 Introduction

Rotational molding technology originated in the 1940s in the UK. Then, the technology was widely used in the USA and Japan and became one competitive method in the field of plastic forming [1, 2, 3] (Fig. 1).

A rotary-molded boat is often made of polyethylene material, which makes the boat not only pollution free, erosion resistant and alkali resistant, but also with large buoyancy, high speed and low repair cost. However, the existing boat construction methods are still based on steel-material ships. Though the rotary-molded boat is made of high-strength plastic, its rigidity will decrease when the load is too much or it rolls severely, and its deformation should not be neglected; thus, its stability will decrease too. Therefore, there is a hidden danger in stability evaluation in that the theoretical foundation for steel-material ships is not suitable to rotary-molded boats.

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 boat as the research object. We consider the elastic deformation of boat in the new roll motion equation and study the establishment of nonlinear roll motion equation and stability simulation.

*i*th harmonic wave.

*I*is the total inertia moment which includes the coherent rotation moment of inertia \(I_x\) and the additional rotation moment of inertia \(\delta I_x\), \(\alpha _0\) is the effective coefficient of wave inclination, \(\omega _0\) is the boat natural frequency, \(h_i\) is the significant wave height, \(\lambda _i\) is the wave wavelength, \(\omega _i\) is the wave frequency, and \(\varepsilon _i\) is the random phase angle which ranges from 0 to \(2\pi \) [12, 13].

*S*is the upright-projection area of a boat hull above the waterline. Then, the wind disturbance torque \(M_F\) on the boat is written as

*H*is the distance between wind pressure center and the water surface,

*d*is the boat draft depth, and \(z_R\) is the distance between the water resistance action center and 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 of a rotary-molded boat as follows

## 2 A new rolling model of the rotary-molded boat

Because the material of a rotary-molded boat is polyethylene which has a small specific gravity value of about 900–960 \(\hbox {kg}/\hbox {m}^{3}\), it is necessary to consider deformation in establishing a more realistic rolling equation.

*x*as shown in Fig. 3. When deformation happens, the gravity center, buoyancy center and displacement volume all change correspondingly. The gravity center changes from \(G_{0}\) to \(G_{1}\). The buoyancy center moves from \(B_{1}\) to \(B_{2}\). \(\overline{B_2S_2}\) is the new reversion arm. The position of the new gravity center \(G_{1}\) can be calculated based on the gravity center moving principal. For simplicity, assume that damping torque and response arm are constant. Assume \(\overline{B_1S_1}\approx \overline{B_2S_2}\). Then, the volume under water \(V'=V_0+(V_1+\delta V_1)-(V_2-\delta V_2)\), where \(\delta V_1\) is the volume of trapezoidal block Open image in new window and \(\delta V_2\) is the volume of trapezoidal block Open image in new window as shown in Fig. 3.

## 3 Numerical simulation

### 3.1 The deformation is caused only by wind

*L*is 11.4 m, type wide

*B*is 2.51 m, deep

*D*is 1.305 m, initial stability height \(\overline{{ GM }}\) is 0.707 m, design draft

*d*is 0.60 m, drainage volume

*V*is 8.14 cubic meters, displacement \(\varDelta \) is 8.34 tons, distance \(Z_{g}\) between gravity center and midline is 0.304 m, wind area

*S*is 9.26 \(\hbox {m}^2\), height

*H*between wind pressure center and water surface is 0.882 m, no-load weight is 2.9 tons, and full load weight is 8.34 tons. The stability vanish angle \(\varphi \gamma \) is 1.57 rad. The area \(A\varphi \) surrounded by static stability curve and stability vanish angle is \(0.4112\,\hbox {m}\,\hbox {rad}\). Take the air density \(\rho _\mathrm{air}=1.2\,\hbox {kg/m}^{3}\) and the effective coefficient of slope \(\alpha _0=0.729\). Consider full load condition. The total rotation moment of inertia is \(1.820 t\, \hbox {m}^{2}\). The coherent roll frequency \(\omega _0=1.80\,\hbox {rad/s}\). Damping coefficient \(D_1=0.04002\) and \(D_3=0.027288\). Restoring factor \(C_3=-0.3352\) and \(C_5=0.019608\). Restoring moment is only calculated up to the 5-power term \(\varphi ^{5}\). Integrated deformation coefficient \(K_1=0.1\), \(K_2=0\). Waves are regular ones. The significant sea wave height, wavelength and wave frequency are \(h=3\,\hbox {m}\), \(\lambda =90\,\hbox {m}\) and \(\omega =0.85\hbox {rad}/\hbox {s}\), respectively. Also assume that there is no navigational speed and only beam wind with speed \(u_F =10+10\cos (4t)\) is considered. Take the aforementioned parameter values into Eq. 21 and get boat rolling motion equation as follows

### 3.2 The deformation is caused by the coupling of wind and waves

### 3.3 Validation of the chaos characteristic

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

The above simulation result shows that the rolling will get more intense and the stability will get weakened when boat deformation is considered. At the same time, the ride comfort will get worse too. In particular, the appearance of chaos may increase the danger of the boat [23]. So it is necessary to control chaos when boat deformation is considered.

### 3.4 Control of chaos

*u*to the second equation of system (24). Then, we get a controlled system (25):

*u*can be expressed as

*u*when \(t=15\hbox {s}\), end the control when \(t=20\hbox {s}\). The phase graph of system (25) is shown in Fig. 11. After being controlled, the system gets rid of chaos and is controlled at the equilibrium point (0, 0).

### 3.5 Analysis of robustness

A controlled system is robust if it meets that (1) it has a low sensitivity; (2) the system is always stable within the scope of the parameters may change; (3) the performance of the system can meet the requirements when the parameters change [25, 26, 27].

In fact, a real controlled system often exists model error and measurement noise, and an effective control strategy must have strong robustness. The robustness of the controlled system (25) is analyzed as follows.

*u*at 15 s, and end it at 20 s, and the roll angle curves changing with time of the controlled system (25) are shown in Fig. 12. From Fig. 12, we find the control strategy

*u*can control the system very well even though there exists model error.

Secondly, when the system exists measurement noise, we add white noise *enoise* to the controlled system (25).

*r*(

*T*) is a normally distributed random function [25]. Add the control strategy

*u*from 15 to 50 s, and the time response curve of roll angle of the controlled system (25) is shown in Fig. 13.

*enoise*are the same as the former too. The time response curve of roll angle of the controlled system (25) is shown in Fig. 14.

The simulation shows that the designed controller has good control effect, and the controlled system (25) is robust when the system is subject to parameter uncertainties and noise measurement.

### 3.6 Comparison with previously published work

The literature that specially researches the stability of rotary-molded boat or rubber boat is very less. Even though there is a few of references which considered its elastic deformation to study the dynamic characteristics of rolling of the rotary-molded boat, the research object of them is the steel ship. Compared with the references [6, 8, 9], the main work of our study can be stated as the following several aspects: (1) The modeling method of us considered the wind and waves coupling, which is closer to the actual situation; (2) the simulation shows that the rolling angular velocity and angular acceleration will increase when considering the deformation of boat; at this time, the ship’s stability deserves more attention; (3) when considering its elastic deformation, the stability of boat will decrease, but regarding how to accurately calculate and evaluate how much of the stability decreased, as well how to improve the stability, neither the references nor we consider it; this is our future research direction.

## 4 Conclusions

- (1)
The rotary-molded boat has complex dynamic characteristics when the irregular wind and wave act on it.

- (2)
Along with the increase in wind and wave harmonic, the influence of the dynamic characteristics of boat is significant. Some simulation methods and results such as the time domain curve, phase diagram, Lyapunov exponential spectrum, as well Poincare map reflect this changes from different aspects.

- (3)
When the elastic deformation of boat is caused by the wind and waves coupling, the rolling is much more intense, its stability will decline further more, and the chaos phenomenon is more obvious.

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

## Notes

### Acknowledgments

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

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