Research Article  Open Access
W. Zhao, M. Li, L. Xiao, "Nonlinear Dynamic Behaviors of a Marine RotorBearing System Coupled with Air Bag and FloatingRaft", Shock and Vibration, vol. 2015, Article ID 620968, 18 pages, 2015. https://doi.org/10.1155/2015/620968
Nonlinear Dynamic Behaviors of a Marine RotorBearing System Coupled with Air Bag and FloatingRaft
Abstract
To understand the nonlinear dynamic mechanism of a rotorbearing system coupled with air bag and floatingraft, the dynamic characteristics of the system are investigated. This work has two key objectives. First, the vibration mechanism of rotorbearing system coupled with air bag and floatingraft is investigated by developing a numerical model. Then, the nonlinear dynamics of the system and the effect of several parameters are studied, which includes the steadystate response and its spectrum, the orbit and its Poincaré map, the bifurcation diagram, and largest Lyapunov exponent (LLE). The results show that at low speed the dynamic behavior appears in a single periodic motion, and, with the increase of the speed, the motion becomes quasiperiodic and chaotic. These performances indicate that the air bag and floatingraft introduce some dynamic effects of marine rotorbearing system.
1. Introduction
Mechanical vibrations from power machinery are one of the main sources of noises in large vessels and how to reduce them is a matter of great concern. Nowadays, one of the most commonly adopted solutions in power equipment such as marine rotating machinery to suppress the excessive vibrations and reduce its disturbing noise is the use of air bag and floatingraft device.
In recent years, the nonlinear dynamic responses of rotorbearing system have been analyzed and described in literature [1]. Niu et al. [2] researched the behaviors and the design principles of active floatingraft isolation system and investigated their power transmission [3]. Niu and Song [4] analyzed the influence of the nonlinearity of floatingraft vibration isolation system, and Wang et al. [5] developed the dynamic modeling of elastic floatingraft system on theory of multibody dynamics and structural dynamics. Jiang et al. [6] studied the dynamic effects of elasticity on the raft body and its linear shock response based on the experimental method. Huang [7] studied the dynamic characteristics of marine vibration isolation systems and verified them by carrying out the experimental tests. Stephen [8] discussed the dynamics between a single and a twostage raft vibration isolation system based on the theory of matrix methodology. Using the finite element method (FEM), Wang et al. [9] analyzed the dynamic characteristics of the floatingraft vibration isolation system of pumping unit. All these studies mainly considered only the linear behavior of the floatingraft and rarely referred to the nonlinear one and do not consider the multimixed airfloating system combined with the rotorbearing. In addition, the structures of the models cannot be timely changed when the vessel is sailing under the complicated and changeable environment. Therefore, a deeper study of rotorbearing system coupled with floatingraft isolation device is auspicable [10].
The abovementioned literature mainly emphasized the linear dynamic behaviors such as the dynamic responses or the design of floatingraft isolation device [11–15]. However, the nonlinear dynamics of the rotorbearing system coupled with air bag and floatingraft isolation equipment is little mentioned. In previous studies, Zhao and Li [16] developed a mathematic model for a rotor system with vibration isolation equipment based on the theory of short bearing, and Song et al. [17] studied the dynamic reduction modeling and time delay for a vibration isolation system with a flexible base. Therefore, to investigate the effects of nonlinearities on nonperiodic motions of rotorbearing system, this paper paid more attention to the dynamics of the rotor system in long journal bearings coupled with air bag and floatingraft device. The nonlinear dynamic equations of the system are solved using computational methods. Steady response, dynamic trajectories, spectrum map, Poincare maps, bifurcation diagrams, and largest Lyapunov exponent are applied to analyze the dynamic behaviors of the system. The nonlinear dynamic mechanical model of the system is built and analyzed in depth to improve its performance of vibration isolation; these studies will lay down the requirements for the engineering design and application of air bag and floatingraft isolation equipment in large vessel.
2. Mathematical Model
Figure 1 shows the rotorbearing system coupled with multimixed air bag and floatingraft isolation equipment. Floatingraft and rotor are considered as a rigid body, while the air bag is regarded as a spring with vertical and horizontal deformations.
2.1. Motion Equations
According to Figure 1, based on Newton’s law, the motion equations of the rotorbearing system coupled with the air bag and floatingraft device can be expressed aswhere , are the mass of disc, the mass of floatingraft, is eccentricity, is the stiffness of air bag, is the damping, is the rotor speed, and and are the nonlinear oil forces in and directions, respectively. The mathematical model of the 4DOFs system is shown in Figure 1, and the coordinate system and notation are shown in Figure 2.
2.2. Oil Film Forces
Figure 2 illustrates the schematic diagram of journal bearing, in which and are the nonlinear oil forces in radial and tangential directions, respectively. This paper considers the hydrodynamic long journal bearing theory; thus, the Reynolds equation for the oil film pressure iswhere .
Under the semiSommerfeld condition , , and , , the oil film forces in radial and tangent directions are obtainedwhereThe oil film forces in (3) can be expressed aswhere and are the nonlinear oil film forces components of journal bearing in and direction, respectively.
2.3. Nondimensional Motion Equation
To simplify, the nondimensional notations are introduced and the bearing clearance, weight of disc, and time are considered. Thus, the nondimensional expressions of parameters are shown in Table 1. According to , . Equation (3) can be rewritten into a more convenient nondimensional form:Thereby, the nondimensional formulation of motion can be written asEquation (7) can be discussed on the state space and the four secondorder equations can be easily converted into eight firstorder oneswhere , , , , and are the nondimensional eccentricity, rotor speed, mass ratio, stiffness, and damping, respectively; is the Sommerfeld number; and are the nondimensional nonlinear oil film forces. Consequently, the dynamic characteristics of the system depend on the nondimensional parameters , , , , , and .

3. Nonlinear Dynamic Analysis
Equation (8) is strongly nonlinear and no analytical method is applicable. Thus, to solve (8) the present work adopts, the RungeKutta integration routine with variable step. The parameters of the system are , , , , and , and the following initial conditions are set as , , , , , , , and .
3.1. Numerical Results
Equation (8) describes a nonlinear nonautonomous system with multidimensional rotorbearing system, the nonlinear oil film, and nonlinear functions in terms of and . This section introduces the numerical solution using the RungeKutta method.
Figure 3 shows the bifurcation diagram of rotorbearing system coupled with the air bag and the floatingraft isolation equipment. The diagram refers to the displacement response when motion parameter varies from 0.600 to 1.370. The same figure also plots the largest Lyapunov exponent of the bifurcation diagram. The results indicate that the steadystate responses at low frequencies are synchronous with the rotor speed. When lies within the range 0.600–0.741, the diagram shows single periodic motions similar to dynamic behavior in the linear system. However, the bifurcation of quasiperiodic motion occurs at around 0.780–0.800 and 1.110–1.130; in these cases, the dynamic characteristics are mainly expressed by the quasiperiodic oscillation. With the rapid increments of rotor speed, the chaotic motion will rise; for rotation speed greater than 1.378, the chaotic motion clearly emerges, and at the same time the jump phenomenon of the amplitudes occurs and the rotor directly impacts the bearing bush. This bifurcation point and the type of bifurcation are the same as the above result obtained by numerical integration.
Figure 4 shows the bifurcation diagram of the same system for the displacement response with respect to the motion parameter , in which varies from 0.600 to 1.370. The same figure plots LLE of the bifurcation diagram. With the increase of the rotor speed, the dynamic behaviors change from singleone motion to quasimotion to chaotic motion. The largest Lyapunov exponents converge to positive values, confirming the rising chaos.
Figure 5 illustrates the dynamic response and its frequency spectrum and rotor orbit and its Poincaré map at about , which shows one point in the Poincaré map that the motion is in the state of a periodone motion at this rotor speed. Its largest Lyapunov exponent is −0.013731. Thus, we are sure that it is periodone motion at from the simulation results of steadystate response, frequency spectrum, rotor orbit, Poincaré maps, and the largest Lyapunov exponent.
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Figure 6 shows the oscillations at , in which the nonlinear dynamic behaviors are the quasiperiodic motion and the displacement response (Figure 6(a)) highlights the beat vibration. The frequency spectrum shows a harmonic components; the steadystate response of this system is confined to an annular region and the rotor makes complex movements as shown in (c). Points on the Poincaré maps are an approximate closed curve, the dynamic characteristic of the quasiperiodic motion at this rotor speed. Its largest Lyapunov exponent is negative and equals −0.00361. Therefore, simulation results identify the quasiperiodic motion of the system at .
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Figure 7 indicates the results of the simulations at . The analysis of the Poincaré map highlights that at this rotor speed the motion is chaotic. In fact, from the orbit graph, it can be seen that the steadystate response of this system is confined to an annular region and can make irregular movements, while largest Lyapunov exponent LLE = 0.007765.
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Figure 8 shows the dynamic behaviors of the system at . The orbit graph shows how this dynamic response confined to an annular region and the rotor makes complex movements. The points on the Poincaré map present a discrete and unclose curve, while its largest Lyapunov exponent is 0.00361, confirming that at the motion is chaotic.
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Figure 9 analyzes the oscillations at , in which the quasiperiodic motion is prominent. There exists some continuous spectrum in the frequency spectrum structure and rotor orbit trajectory is messy. Furthermore, the Poincaré diagram shows many irregular points, and its largest Lyapunov exponent diagram reflects the positive value showing that at the motion is chaotic.
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Figure 10 illustrates the steady response and its frequency spectrum and rotor obit and Poincaré map when rotor speed is . The frequency spectrum shows a series of continuous spectra, the orbit of rotor is more regular than the one at lower speed, and Poincaré diagram shows irregular points. In addition, its largest Lyapunov exponent is 0.009671. Therefore, one can conclude that when the system has a nonlinear dynamic behavior and shows chaotic motion.
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3.2. Parametric Analysis
As a structural parameter, the effect of the floatingraft mass is very important to be considered in the design.
Figure 11 illustrates the bifurcation diagram of floatingraft forced by an unbalanced excitation for the displacement responses and with respect to the motion parameter ; rotor speed varies from 0.200 to 1.300 in certain parameters for . With the increase of the rotor speed, the analysis of bifurcation diagram shows that dynamic behaviors in and directions can be considered as singleone motion, quasimotion, and chaotic motion. With the increase of rotor speed, the amplitude of airfloating increases gradually, and there is a sudden increment of both and at .
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Figure 12 illustrates the dynamic response and its frequency spectrum, phaseportrait plots and Poincaré map at . The points on the Poincaré maps are a close curve, the steadystate response of this system is confined to an annular region and the phaseportrait plot highlights the irregular movements of the rotor. Thus, the motion at this rotor speed is chaotic.
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Figure 13 indicates the dynamic response and its frequency spectrum, phaseportrait plots, and Poincaré map at . The steadystate response of this system is confined to an annular region and makes a complex movement from the phase diagram; the points on Poincaré maps are very messy. Thus, the motion states a chaotic motion at this rotor speed.
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Figure 14 analyzes the system at . The steadystate response of this system is confined to an annular region and movements are complex from the orbit graph, while points on Poincaré maps are very sophisticated. Thus, at this rotor speed the state is chaotic.
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Figure 15 illustrates the bifurcation diagram of floatingraft forced by an unbalanced excitation for the displacement responses and with respect to the motion parameter . Rotor speed varies within the range 0.20–1.30 in certain parameters for . The analysis of the bifurcation diagram shows that with the increase of the rotor speed the dynamic behaviors in and directions switch from singleone motion to quasimotion to chaotic motion.
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Figure 16 clearly shows that at the nonlinear dynamic behaviors are the chaotic motion. Phaseportrait plots are messy. Besides, many irregular points on the Poincaré maps are quite intricate, and therefore the motion at this rotating speed ratio is chaotic.
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Figure 17 illustrates the dynamic response and its frequency spectrum, phaseportrait plots, and Poincaré map at . The steadystate response of this system is confined to an annular region and makes a complex movement from the phaseportrait diagram. Besides, the distribution of the points on the Poincaré diagram demonstrates the chaotic motion. Thus, the motion is in the state of a chaos motion at this rotor speed.
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Figure 18 illustrates the bifurcation diagram of floatingraft forced with an unbalanced excitation for the displacement responses and with respect to the motion parameter , in which in certain parameters for . Increasing the rotor speed, the dynamic behaviors in and directions can be considered as singleone motion, quasimotion, and chaos motion. With the increase of rotor speed, the amplitudes of air bag and floatingraft increase gradually. Moreover, the diagram shows sudden increment of and at . The result indicates the relatively small reduction in displacements and of floatingraft with the increasing rotor speed, the specific situation in different case as shown in the flowing section.
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Figure 19 shows the dynamic behavior of the system at . The Poincaré diagram shows a nearly close curve distribution of the points. The steadystate response of this system is confined to an annular region and the phaseportrait plots show the system around circular orbit. Thus, the simulation results shown in Figure 19 confirm that at the motion can be considered quasiperiodic.
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Diagrams of Figure 20 summarize the results of the numerical simulation at . Phaseportrait plots are messy and frequency spectrum structure clearly shows that there exists a zero spectrum. The corresponding distribution of points in the Poincaré maps confirms the chaotic motion at , and therefore we may conclude that chaotic motion occurring at these rotating speed ratio .
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Figure 21 plots the bifurcation diagram of the floatingraft with an unbalanced excitation. (a) and (b) represent the displacement responses and with respect to the motion parameter , in which varied from 0.20 to 1.30 in certain parameters for . The bifurcation diagram shows how the dynamic behaviors in and directions change from singleone motion to quasimotion to chaotic motion, increasing the rotor speed. With the increase of rotor speed, the amplitude of airfloating increases gradually. At , one observes a sudden increment of both and values. The result indicates the relatively small reduction in displacements and of floatingraft with increasing rotor speed, the specific situation in different case as shown in the flowing.
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Figure 22 describes the dynamic response and its frequency spectrum, phaseportrait plots, and Poincaré map at . The distribution of points of the Poincaré maps follows a closed curved; similarly, the steadystate response of the system is confined to an annular region and the orbit diagram shows that it moves regularly. Thus, at this rotor speed, the system can be considered in a quasiperiodic motion.
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Figure 23 illustrates the chaotic motion of the nonlinear dynamics at . Phaseportrait plots are messy, frequency spectrum structure clearly shows that there exists zero spectrum, and corresponding points on Poincaré maps are messy with irregular distribution. Therefore, the analysis of the simulation results confirms the chaotic motion at .
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4. Conclusions
The nonlinear dynamical behavior of a rotorbearing system coupled with air bag and a floatingraft support is carried out. In this paper, the mathematical model of the 4DOFs system is developed according to the long bearing theory, and its nonlinear dynamic characteristics are studied by applying the RungeKutta method to solve motion equations. The result of this research greatly extends and enriches the understanding of the behavior of the system. Based on the results of the numerical simulations, the following main conclusions can be summarized.(1)The responses at low speeds show a synchronous single periodic motion, and the unbalanced force plays a dominant role in the vibration of the rotorbearing system. With the increase of rotor speed, the influence of nonlinear oil force has great effect on the system vibration, and quasiperiodic motion can be generated. Meanwhile, there exists a quasiperiodic bifurcation phenomenon. As a result, the system at high rotor speeds gradually moves into the complex and irregular state of chaotic motion.(2)The dynamic properties of floatingraft at different key system parameters along with ratio speed generate important nonperiodic motion. The result of numerical modeling of the system suggests that some designing parameters change the dynamic stability. Moreover, periodic motions, quasiperiodic motion, and chaotic motion can appear and enhance the understanding of the nonlinear dynamics of floatingraft and may help in the theoretical understanding of general nonlinear systems to help design ideal isolation equipment.
These dynamic properties can be used to provide the theoretical guidance of parameter design of a rotorbearing system coupled with air bag and floatingraft.
Appendix
Bearing clearance  
Eccentricity position of rotor centerline in journal bearing  
Dimensionless damping  
Damping damping  
,  Dimensionless oil film forces of journal bearing 
,  Oil film forces of journal bearing 
The acceleration of gravity  
Thickness of oil film  
Dimensionless stiffness  
Dimensional stiffness  
The mass of disc  
The mass of floatingraft  
The ration of mass  
Oil film pressure  
Time  
Vector variable  
,  Coordinates of the centers of disk 
,  Coordinates in width and circumference 
Directions of dimensionless eccentricity  
Bearing eccentricity  
Angular position of rotor centerline in journal bearing  
Sommerfeld of bearing  
The ratio of length diameter  
Dimensionless time  
Rotating speed of rotor  
Dimensionless rotor speed. 
1, 2:  Disc (or rotorbearing) and floatingraft. 
., ..:  First and secondorder derivatives with time 
′, ′′:  Firstorder and secondorder derivatives with dimensionless time 
Nondimensional time . 
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant no. 11372245) and the Natural Science Foundation of Shaanxi Province, China (Grant no. 2014JM1015). This support is gratefully acknowledged.
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Copyright
Copyright © 2015 W. Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.