Melnikov s Method Applied to a MultiDOF Ship Model


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1 Proceedings of the th International Ship Staility Workshop Melnikov s Method Applied to a MultiDOF Ship Model Wan Wu, Leigh S. McCue, Department of Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, Blacksurg, VA, USA ABSTRACT In this paper, a coupled rollswayheave model derived y Chen et al (999) is studied. In order to address the small damping constraint, the etended Melnikov s method for slowly varying system is used y assuming the damping term is large. Using the etended Melnikov s method, the critical wave amplitudes are calculated. A phase space transport method has een applied. The ratios of erosion safe asin areas have een calculated ased on the Melnikov s method and were compared with the results from numerical simulations. KEYWORDS MultiDOF Melnikov; Slowlyvarying; Ship; Staility. INTRODUCTION Si degree of freedom (DOF) vessel motion prolems ehiit numerous compleities, particularly when studied analytically. Most previous work on multidof vessel motions either reduced the prolems to lower (one or two) DOF prolems or used numerical simulations. In the work of ship motion analysis, compared to DOF prolems, relatively little work have een done using analytical methods for multidof ship motion prolems. In this paper, the etended Melnikov s method (Salam, 987) is applied to a rollswayheave coupled ship model derived y Chen et al. (999). By changing the coordinates and applying the singular perturation technique, Chen showed the model can e simplified to a slowly varying system with three variales, which contain roll displacement and roll velocity as the fast varying variales and a slowly varying variale. This kind of system can e manageale using the Melnikov's method discussed y Wiggins and Holmes (987, 988). But similar to the planar Melnikov s method, the constraint of this method is the small perturation assumption. In order to address this constraint, the etended Melnikov s method for slowly varying systems is applied. The etended Melnikov s method developed in the literature y Salam (987) has een recently applied to ship motions prolems such as capsize (Wu and McCue, 007, 008, Wu, 009) and surfriding (Wu et al. 00 and Wu 009). The purpose of this work is to show the possiility of applying the etended Melnikov s method to multidof ship models. MATHEMATICAL MODEL The equations of motion for the coupled rollswayheave model in the earthfied coordinate system can e epressed in Eq.() my mz '' c '' c Y Z () '' I44 K in which, m is the mass of the ship, yc, zcand are sway displacement, heave displacement and
2 Proceedings of the th International Ship Staility Workshop roll displacement, respectively. YZand, K are generalized forces. The prime denotes the derivative with respect to time t. Chen et al. (999) transformed the model to a wavefied coordinate, in which the ship is viewed as a particle riding on the surface of the wave. The sway motion is now parallel to the local wave surface and the heave motion is perpendicular to the local wave surface. The equations of motion now can e epressed as in Eq.().. f, y g z z a y g z (,, y, z, z, ) (,, y, z, z, ) f ( z, z ) g (,, y, z, z, ) c () where, ', z z / h 0 in which h is the draft of the ship. y is a transformed coordinate which contains sway velocity and other variales. z 0 is small compared to h. (.) is the derivative relative to, where. r is the natural frequency of roll. In Eq.(), the heave motion is considered to ehiit fast dynamics compared to roll and y. Chen et al. (999) used the singular perturation theory to this system to show that z and z can e solved from the steady state equation and can e sustituted into the slow dynamics. The dynamics of the whole system Eq.() can e represented y the reduced system Eq.(3). f, y ga y g (,, y,, ) (,, y,, ) r t (3) Chen et al. found that for the reduced system in Eq.(3), roll motions are the fast varying variales while y is the slowly varying variale. Systems like this are called slowly varying systems. When 0, this is simply the planar roll motion with zero forcing and zero damping. When is a small positive numer, the y motion (which includes sway and other motions) ecomes relevant. Because the sway motion is stale, the system will trend towards the invariant manifold of roll dynamics. THEORETICAL BACKGROUND Melnikov s Method for Slowly Varying Systems Melnikov s method is one of few analytical methods that can e used to predict the occurrence of chaotic motions in nonlinear dynamic systems. Melnikov s method has een applied to a numer of ship dynamics prolems, such as capsize in eam seas (Falzarano, 990) and surfriding in following seas (Spyrou, 006). Most of these are treated as single DOF prolems. Melnikov s method for multidof prolems has een introduced in several references including the works of Wiggins and Holmes (987, 988), who derived the Melnikov s function for slowly varying system in Eq.(3). When 0, the unpertured system in Eq.(3) has a planar Hamiltonian, which contains a homoclinic (or heteroclinic) orit. The Melnikov s function for this system is M ( t ) ( H g)( q ( t), t t ) dt H ( ( z )) g ( q ( t), t t ) dt z (4) g [0, g, g ]. H is the Hamiltonian for the a unpertured system. q0( t) (, ) is the coordinates of the homoclinic orit for the unpertured system. And is the dot product. Melnikov s Method for Slowly Varying Systems with Large Damping When the damping term is assumed to e large, it is grouped in the unpertured system.
3 Proceedings of the th International Ship Staility Workshop 3 Therefore, the unpertured system is no longer Hamiltonian due to the presence of in f. f,, y (5) The homoclinic orit, which is essential in the formation of Melnikov s function, disappears as well. Since the homoclinic orit does not arise naturally, it has to e created artificially. Eq.(3) is then written in the form of Eq.(6)., a f, y g (,, y,, ) y g (,, y,, ) The Melnikov s function for this system is t M ( t0) ga ( q0, t t0) ep a( s) dsdt 0 t t f g( s t0) ds ep a( s) dsdt y 0 0 (7) in which, q0( t) (, ) is the coordinates of the new homoclinic orit of Eq.(5). as () is the trace of the Jacoian matri of Eq.(5). If the unpertured system in Eq.(5) is Hamiltonian, as ( ) 0. Eq.(7) can e reduced to the same form as Eq.(4). Phase Space Transport (6) As mentioned earlier, the unpertured system of Eq.(3) has a planar homoclinic orit, which contains a stale manifold and a unstale manifold. Wiggins and Holmes (987) pointed out that when is small enough, the pertured system is close to the local unpertured manifolds in a small neighorhood. Outside of this region, the pertured manifold is close to the unpertured manifold in finite time. The theory of phase space transport for planar systems is applied here to predict the safe region erosion in finite time. For the unpertured system, the inside of the homoclinic orit is the safe region. When the homoclinic orit is pertured, the manifolds will intersect resulting in loes. And some initial conditions initially inside the safe region may e outside the safe region for the pertured system (pseudoseparatri) after some time. This phenomenon corresponds to a special loe called turnstile loe (Wiggins, 99). The area of this loe is given in Eq.(8) (Wiggins, 99). T ( L0 ) M ( t0, 0) dt0 ( ) 0 O (8) in which M ( t0, 0) is the positive part of the Melnikov s function, L0 represents the loe, t 0 is the parameter in the homoclinic orit q0( t 0) denoting different time in the Poincaré map. 0 is the phase difference with the eternal forcing. T is the period of the eternal forcing. Phase space transport refers to the initial conditions transporting outside the safe region after several periods of eternal forcing. The amount of the transported phase space can e used to show the rate of safe area erosion. Chen and Shaw (997) derived the estimate of erosion ratio as shown in Eq.(9). 3 ( L ) 3 T 0 e M ( t0, 0) dt0 ( ) As A O (9) s 0 where As is the area of the unpertured safe region, e is the ratio erosion area divided y the original safe region, and ecause is a small positive numer, O( ) term can e ignored. In this work, Eq.(9) is used to show the erosion of safe asin for the capsize prolem. APPLICATIONS The data from twice capsized fishing oat PattiB are used here for numerical investigation. Chen et al. (999) proposed this model shown in Eq.(0).
4 Proceedings of the th International Ship Staility Workshop 4 f k k k 3 3 ga 4 cos 4 cos 4 y 44 (0) f ( )cos sin 44q 4 g cos y sin 4 f is the restoring moment in the roll motion, which includes the effect of ias. is the nonlinear roll 44 44q damping. is the nondimensional wave frequency. Other coefficients come from hydrodynamic forces, wind forces and wave forces. Melnikov s Function The etended Melnikov s method is applied here y assuming the roll damping terms are large. For the slowly varying system, it is essential to have a homoclinic orit in order to calculate the Melnikov s function (Wiggins, 987). If the linear damping term is assumed to e large, the center in the unpertured system will ecome a sink, which makes it impossile to have a homoclinic orit. In this work, the following damping term is assumed for roll B( ) c () 3 44 where and c are coefficients. Although it is physically unrealistic to have quadratic damping term in roll, it is used here to show the possiility of using the etended Melnikov s method to multidof prolems. In order to form the homoclinic orit for the unpertured system, the quadratic damping term is assumed to e large. The unpertured system is now 3 k k k3 where y cos 4 y () is the sway variale otained from averaging. is the coordinate of the saddle point, which can e calculated y setting 0 and 0. Eq() contains a homoclinic orit starting from a saddle connecting to itself, as shown in Figure. The solid line in the figure is the homoclinic orit for Eq.(), while the dashed line is the homoclinic orit for the unpertured system in Eq.(3) without the quadratic damping term. These two homoclinic orits start from the same saddle point, and are close to each other. The Melnikov s function can e calculated using Eq.(7). Numerical integration can e carried out without difficulty. Fig. : Homoclinic orit for the unpertured system Numerical Results Chen et al. (999) have found the hydrodynamic and hydrostatic coefficients in Eq.(0) for PattiB at wave frequency w 0.6 rad / s. In this work, the simulation is carried out for the case when the center of gravity has slight ias yg The wind forces are assumed to e zero. The quadratic damping coefficient is set to 0.. Melnikov s functions for oth the standard and etended methods can e calculated using Eqs (4) and (7), respectively. When Mt ( 0) 0, this corresponds to the critical wave amplitude a eyond which the chaotic motion and capsize may occur. The critical wave amplitude a has een calculated for oth Melnikov s methods listed in Tale. As shown in the tale, the etended Melnikov s method predicted the critical wave amplitude slightly higher than the standard Melnikov s method for the case studied here.
5 Proceedings of the th International Ship Staility Workshop 5 Tale : Critical wave amplitude for two Melnikov s methods Method a (m) Standard Melnikov 0.79 Etended Melnikov 0.86 Numerical simulations are carried out to otain safe asins of the 3DOF system with the damping terms shown Eq.() and with original damping term. The safe asins are calculated y integrating a grid of 0000 points in roll plane with yz, and z initial conditions equal to. Every initial condition is integrated until a roll angle is greater than the angle of vanishing staility ( rad ), thus capsize occurs or through 0 cycles of eternal forcing, thus deemed safe. Capsize was checked every dt 0.0s. Figure (a) is the system with quadratic damping and Figure () is the original system. In oth cases, the wave amplitude a 0. The ratio of erosion area has een calculated using Eq.(9) for oth Melnikov s function defined y Eqs.(4) and (7). Numerical simulations are also carried out for the 3DOF system to compare the results. Chen and Shaw (997) pointed out that in order to implement phase space transport methods, the dynamics should e studied on the invariant manifold where loes can e defined. Therefore, similar to their work, the initial conditions for the numerical simulations have een chosen as 70 points on the invariant manifold of roll dynamics, which are otained y numerically calculated the safe points for the unpertured system (asically the homoclinic orit). Two points are picked on every direction of yzand, z. A grid of (70 ) points are used as the initial conditions. For the numerical data, the ratio of erosion area is calculated using the points capsized in 0 cycles of eternal forcing divided y the total numer of points. (a) Fig. 3: The ratio of erosion area for different methods. () Fig. : Safe asins for different models (The white areas are the safe asins, and the dark areas are capsize area.) (a). Safe asin for system with quadratic damping included. (). Safe asin for original system. Figure 3 shows the ratio of erosion areas for different methods. The results from oth Melnikov s methods are conservative compared to the numerical simulation results. And the results from the etended Melnikov s method are more accurate than those from the standard Melnikov s method, especially for larger wave amplitudes. Compared to the time consuming 3DOF numerical simulations, the method of phase space transport ased on the etended Melnikov s method provides a fast way to estimate ratio of erosion with
6 Proceedings of the th International Ship Staility Workshop 6 reasonale accuracy. CONCLUSIONS REMARKS In this paper, the etended Melnikov s method has een used to a rollswayheave coupled model which can e reduced to a slowly varying system. In order to otain the homoclinic orit, a quadratic damping term is treated as large. Although it is physically unrealistic to have a quadratic term in roll damping, it is used here just to demonstrate the feasiility of the method. Coupled with the method of phase space transport, this results in a fast and effective way to estimate the ratio of erosion with apparently conservative accuracy. This work is the first step of applying the etended Melnikov s method to a special form of multidof dynamical systems. It provides the possiility of applying the method to other multidof prolems in ship dynamics. ACKNOWLEDMENTS This work has een supported y Dr. Patrick Purtell under ONR Grant N and Dr. Eduardo Misawa under NSF Grant CMMI References Chen, S.L.; Shaw, S.W. and Troesch, A.W.: A Systematic Approach to Modelling Nonlinear MultiDOF Ship Motions in Regular Seas. In: Journal of Ship Research. 43 (999) Chen, S.L. and Shaw,S.W.: Phase Space Transport in a Class of MultiDegreeofFreedom Systems. In: Proceedings of 997 ASME Design Engineering Technical Conferences (DETC97) Falzarano, J.M.: Predicting Complicated Dynamics Leading to Vessel Capsizing. PhD dissertation, University of Michigan, Ann Aror, 990. Salam, F.M.: The Melnikov Technique for Highly Dissipative Systems.In: SIAM Jounal on Applied Mathematics.47 (987) Spyrou, K.J.: Asymmetric Surging of Ships in Following Seas and its Repercussions for Safety. In: Nonlinear Dynamics. 43 (006) Wiggins, S.; Holms, P.: Homoclinic Orits in Slowly Varying Oscillators. In: SIAM Journal of Mathematical Analysis. 8(3) (987) Wiggins, S.: Chaotic Transport in Dynamical Systems. (99) SpringerVerlag, New York. Wiggins, S.; Holms, P.: Errata: Homoclinic Orits in Slowly Varying Oscillators. In: SIAM Journal of Mathematical Analysis. 5(9) (988) Wu, W. and McCue, L.S.: Melnikov's Method for Ship Motions Without the Constraint of Small Linear Damping. In: Proceedings of IUTAM Symposium on FluidStructure Interaction in Ocean Engineering, (007), Hamurg, Germany. Wu, W. and McCue, L.S.: Application of the etended Melnikov's Method for Singledegreeoffreedom Vessel Roll Motion. In: Ocean Engineering. 35 (008) Wu, W., Spyrou, K.J. and McCue, L.S.: Improved Prediction of the Threshold of Surfriding of a Ship in Steep Following Seas. In: Ocean Engineering. (00) doi:0.06/j.oceaneng Wu, W.: Analytical and numerical methods applied to nonlinear vessel dynamics and code verification for chaotic systems. PhD dissertation, Virginia Tech, Blacksurg, 009.
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