M.Salai Mathi Selvi·L.Rajendran·Marwan Abukhaled
Abstract A steady-state roll motion of ships with nonlinear damping and restoring moments for all times is modeled by a second-order nonlinear differential equation.Analytical expressions for the roll angle,velocity,acceleration,and damping and restoring moments are derived using a modified approach of homotopy perturbation method(HPM).Also,the operational matrix of derivatives of ultraspherical wavelets is used to obtain a numerical solution of the governing equation.Illustrative examples are provided to examine the applicability and accuracy of the proposed methods when compared with a highly accurate numerical scheme.
Keywords Nonlinear damping.Steady-state roll motion.Ultraspherical wavelets.Homotopy perturbation method.Analytical solution
Ships,in general,may experience three types of displacement motions(heave,sway or drift,and surge)and three angular motions(yaw,pitch,and roll)as depicted in Figure 1.Ship roll reduction or stabilization consists of two main parts:The first is the roll reduction or stabilization method,and the second is the evaluation or modeling method of the roll performance.Started with Froude(1861),roll motion and related topics have been studied since the late-mid 1800s.For example,Tanaka(1961),Kato(1965),and Ikeda et al.(2004)studied roll damping based on theory,numerical calculation,and experimentation and suggested an empirical formula including the effect of a bilge keel.A simple method for predicting the roll damping of a ship at forward speed was proposed by Himeno(1981)while Ikeda suggested a modified model(Ikeda et al.2004).
The dynamic roll characteristic has been one of the most central topics in sea keeping study.However,due to the complexity induced by high nonlinearity,potential flow methods cannot solve the roll motion problems effectively.Up to date,many research groups have adopted various methodologies,including the advanced experimental process,empirical formulas,or computational fluid dynamics(CFD).One of the topics is related to the characteristics of roll motion for bare hull and with different appendages.Another branch of related research has focused on the analysis method for roll damping,which includes a higher polynomial roll damping model,finite differential model,or hyperbola model.Recently,CFD-based methods have been used for the analysis of local phenomena such as vortex shedding around bilge keels(Yeung et al.1997).
Figure 1 Schematic diagram of ship showing the six degrees of freedom
Several authors have studied roll damping experimentally for local flow visualization(see,for example,Aloisio and Felice 2006;Bassler et al.2007;Oliveira and Fernandes 2012).On the other hand,mathematical modeling of roll damping has been investigated by researchers,for example,Oliveira and Fernandes(2014)used the bilinear or hyperbola fitting method,and Agarwal(2015)used a fractional differential equation model.However,modeling damping motion in terms of a second-order ordinary differential equation(quadratic model)is widely accepted.Recently,Oliveira and Fernandes(2012)discussed the nonlinear roll damping of FPSO hull.Although these methods have advantages,especially for newer hull designs or nontypical appendages in terms of shape and size,the traditional methods for a typical design have not yet been fully validated.
Behavior analysis of a ship affected by external forces and analysis of ship stability in waves are based on more complex mathematical models as a result of strong nonlinear terms in the governing equations of these systems.Nonlinear analysis makes it possible to comprehend the theory of ship behavior for forecasting stability changes in various conditions of operation.In addition,ship turning motion is an important part of ship maneuverability and is directly related to the safety of a sailing(Lihua et al.2018).The flow in the vicinity of 2D ship sections carrying out forced roll motions is simulated using the solution of the Navier-Stokes equations(Lavrov et al.2017).A new coarse and fine-tuning fixed grid wavelet network is proposed for online predicting ship roll motion in regular waves(Huang et al.2018).Kianejad et al.(2019)improved prediction of a ship roll motion in parametric roll and dead ship using CFD.Another important related topic that has been under investigation is fin stabilizer control.Demirel and Alarcin(2016)used LMbased H2and H∞state-feedback controller for fin stabilizer of a fishing boat and also presented a backstepping control design procedure for nonlinear fin roll control of a trawler(Demirel et al.2017).
The second-order nonlinear differential equation modeling the ship roll motion does not have a closed-form solution.Despite the fact that reliable numerical and experimental methods have been implemented to find approximate solutions,the need for analytical solutions is still a necessity in order to understand the effect of parameters’variation on the governing system.Of the well-established methods that can be employed to obtain analytical expressions of roll angle,velocity,and damping and restoring moments,we mention the variational iteration method(Lu 2007;Abukhaled 2013),differential transformation method(Chen and Liu 1998),Green function based method(Khuri and Abukhaled 2017;Abukhaled 2017),finite element method(Comini et al.1970),series method(Salomi et al.2020),homotopy perturbation method(He 1999),and homotopy analysis method(Liao and Chwang 1998;Liao 1997;Liao 2004).Another approach that can be employed is that the differential equation may be analyzed based on the formulation of an equivalent integral equation(Jang et al.2009;Jang 2011;Jang 2013).
In this project,the governing equation will be solved analytically by using a modified form of the homotopy perturbation method and numerically by using a high-order ultraspherical wavelets-based method.Wavelet-based methods using operational matrix of derivatives are accurate and reliable and have been intensively used in many fields of science and engineering applications.For example,waveletbased methods are used to find approximate solutions for nonlinear differential equations(Salai Mathi Selvi et al.2017),for Michaelis-Menten enzymatic reaction equation Salai Mathi Selvi and Hariharan(2017),and for solving integrodifferential equations(Tavassoli Kajani and Hadi Vencheh 2004).Wavelet-based methods have also been employed to solve optimal control and fractional optimal control problems(Razzaghi and Yousef 2002;Sadek et al.2007;Abualrub et al.2018).Recently,Salai Mathi Selvi and Hariharan(2016)applied the Chebyshev wavelet algorithm for solving steadystate concentration of packed-bed reactor model,Zheng and Wei(2016),used Legendre wavelets to estimates the error of approximation when solving integral equations,and Abualrub and Abukhaled(2015)employed wavelets to regulate cellular processes for the growth or regeneration of a tissue within an assigned terminal time.
This paper is organized as follows:Section 2 describes the mathematical formulation of the problem.Section 3 discusses the properties of ultraspherical wavelets and a wavelet-based solution method.In Section 4,the homotopy perturbation method for obtaining an analytical solution is explained.Illustrative examples are presented in Section 5 with their discussions in Section 6.Finally,Section 7 presents our conclusions.
Under the assumption of independent oscillation,the nonlinear rolling motion of a ship is described by the differential equation(Cardo et al.1981)
and the righting moment is expressed in the normalized form
Introducing the angle and time scalesφnand Tn,the following dimensionless parameters can be defined
then,using these parameters and rearranging terms gives the following cubic damping moment of nonlinear roll motion(Cardo et al.1981)
In this section,we use the function presentation via shifted ultraspherical wavelets to construct a wavelet-based method to obtain a numerical solution for the governing Eq.(5).Appendices 1 and 2 provide simple prelude to this section(see Doha et al.2016 and the references therein for more indepth details).
where m=0,1,2,…M,n=0,1,2,…2k-1,m is the constructed polynomial order,αis a prescribed parameter,ξthe normalized time,and
A function f(t)defined on[0,1]may be expanded in terms of ultraspherical wavelets as
where
Using the truncated finite series,f(t)can be approximated in terms of ultraspherical wavelets as
where
The operational matrix of derivatives is given by the following theorem.
Theorem 1 For the ultraspherical wavelets vector(12),
where D is a 2k(M+1)×2k(M+1)operational matrix given by
in which F is an M+1 square matrix whose(r,s)th entry is given by
Proof See Doha et al.(2016).
The following theorem states that the ultraspherical wavelet expansion of a function f(t),with a bounded second derivative,converges uniformly tof(t).
Moreover,
Proof See Doha et al.(2016).
The wavelet-based method for finding a numerical solution for a homogenous or nonhomogenous nonlinear equation describing the roll motion of a ship begins by using ultraspherical wavelets expansion along with a spectral collocation method to reduce the governing initial value problem into a system of algebraic equations with unknown coefficients.
For finding an approximate numerical solution for Eq.(5),we begin by expanding x(t)and the function g(t)=cosωt in terms of ultraspherical wavelets as follows
Using the operational matrix of derivatives gives
Now substituting Eqs.(17)-(19)into Eq.(5)gives the residual written explicitly as
A brief introduction to the basic concept of the homotopy perturbation method is delineated in Appendix 3.
As per Eq.(4)(Appendix 3),the homotopy of Eq.(5)takes the form.
where p∈[0,1]is an embedding parameter.The approximate solution of Eq.(21),expressed in a power series,is
By letting p=1,an analytical solution,in the form of series expansion,is derived.
In this section,we apply the analytical HPM method and the numerical ultraspherical wavelet method to cubic damping moment of nonlinear roll motion.
Example 1 Consider the nonlinear BVP(Cardo et al.1984)
Using the ultraspherical wavelet method,we substitute Eq.(17)and Eq.(19)into Eq.(23)to obtain
Also using Eq.(17)and Eq.(19),the initial conditions given in Eq.(24)become
where the operational matrix of derivatives is
in which
Now,the approximate wavelet solution x(t)is
where the vectorψ(α)(t)is given by
and the constants vector C is given by
To find the HPM analytical solution for Eq.(23),we use the homotopy in Eq.(21)to find that the coefficient associated with p0is the nonlinear equation
Now x0is obtained by solving Eq.(32)subject to initial conditions
In a similar fashion,the nonlinear differential equation associated with p1can be obtained and then solved for x1.
Using only two terms of Eq.(22)with p=1,we obtain the following approximate analytical solution of Eq.(23)
Example 2 Consider the cubic damping moment of nonlinear roll motion represented by the IVP(Cardo et al.1984)
We will consider three sets of experimental values for the parameters.
Case 1 Assume that
Using the ultraspherical wavelet method,we substitute Eqs.(17)and(19)into Eqs.(42)and(43)to obtain
As detailed in the example above,the approximate numerical wavelet solution,x(t),is expressed in the form
Applying the proposed approach of the HPM to Eq.(35)gives the following analytical expression for the roll angle at any time t:
The velocity and acceleration are,respectively,given by
Figure 2 Comparison of UWM,HPM,and NM for roll angle decay curve in Example 1
Rearranging terms gives the cubic restoring moment formula
and the cubic damping moment formula
The numerical wavelet solution is given by Eq.(39)where Ψ(α)(t)is obtained by solving the equation
Figure 3 Comparison of UWM,HPM,and NM for roll angle decay curve in Example 2,Case 1
Figure 4 Comparison between the HPM and NM for the velocity curve in Example 2,Case 1
subject to the initial conditions given in Eq.(38).The HPM analytical solution is
Figure 5 Plot of analytical expression of the acceleration curve in Example 2 Case 1
Figure 6 Plot of restoring moment versus time,Eq.(43)
The numerical wavelet solution is given by Eq.(39)where Ψ(α)(t)is obtained by solving the equation
subject to the initial conditions given in Eq.(38).
The analytical solution obtained by the HPM is given by
Figure 7 Plot of damping moment versus time,Eq.(44)
Figure 8 Comparison of UWM,HPM,and NM for roll angle decay curve in Example 2,Case 2
Example 3 Consider the cubic damping moment of nonlinear roll motion represented by the IVP(Liao and Chwang 1998)
where the experimental values of the parameters areη=0.1,b=0.5,and l=0.2.
Using the ultraspherical wavelet method,we obtain the approximate solution
Figure 9 Comparison of UWM,HPM,and NM for roll angle decay curve in Example 2,Case 3
Figure 10 Comparison of UWM,HPM,and NM for roll angle decay curve in Example 3
whereΨ(α)(t)is obtained from the equation subject to the initial conditions take the from
In order to apply the homotopy perturbation method,we first construct the following homotopy equation
The solution and the coefficient of the linear term are expanded in the forms in Eqs.(3)and(4)
Figure 12 Roll angle decay curve with its envelope from Eq.(34)
By substituting Eqs.(55)and(56)into Eq.(54),the following linear differential equation system is obtained
Solving Eq.(57)for x0gives
Substituting Eq.(59)into Eq.(58)leads to
or
Figure 13 Roll angle decay curve with its envelope from Eq.(40)
Figure 14 Roll motion vs.frequency for Example 3
No secular term in x1requires that
or
The first-order approximate from Eq.(55)is
and hence from Eqs.(63)and(64),the following frequency is readily obtained
Also,from Eq.(59),we obtain the velocity expression
Figure 15 Velocity vs.frequency for Example 3
Figure 16 Roll motion vs.frequency for Example 3
The proposed methods are applied to investigate a steady-state roll motion of a ship with nonlinear damping and restoring moments.A comparison between the numerical results obtained by the ultraspherical wavelet method(UWM),the analytical results obtained by a new approach of the homotopy perturbation method(HPM),and numerical simulations obtained by the fourthorder Runge-Kutta method(numerical)is discussed.
Roll angle decay curves in Example 1 are obtained by UWM(Eq.(29))and the HPM(Eq.(34)).Figure 2 shows that these methods are in excellent agreement with the numerical solution obtained by RK4.Figure 3 also shows an excellent agreement between the UWM solution(Eq.(39)),the analytical solution(Eq.(40)),and RK4 for the roll angle decay of Example 2,Case 1.The curves in Figures 2 and 3 resemble the large amplitudes roll angles due to cubic damping moments and show that the motion continues indefinitely when only conservative forces act,and thus,the mechanical energy remains constant.Figure 4 shows that the velocity curves in Example 2 Case 1obtained by the modified HPM(Eq.(41))and RK4 are identical.In Figures 5,6,and 7,the curves of acceleration(Eq.(42)),restoring moment(Eq.(43)),and damping moment(Eq.(44))are plotted,respectively,against time using the analytical HPM.Roll angle decay curves for Example 2,Cases 2 and 3,are shown,respectively,in Figures 8 and 9,while Figure 10 represents the roll angle decay curves for Example 3 using the proposed methods.Figure 11 depicts the displacement and velocity with weak nonlinearity and small amplitude.The envelopes of nonlinear ship roll motion,velocity,and acceleration are depicted in Figures 12 and 13.The frequency curves versus roll motion,velocity,and acceleration are given in Figures 14,15,and 16.
In this paper,a mathematical model of the roll motion of ships with nonlinear damping and restoring moments and exciting moments is discussed.To predict the roll motions of ships in irregular or regular waves and to identify the damping and restoring moments in the model,an analytical solution using a modified form of the HPM and a numerical solution using ultraspherical wavelet-based method are obtained.The accuracy of the analytical and numerical solutions is confirmed by a direct comparison with the highly accurate and widely used fourth-order Runge-Kutta method.Analytical expressions for the roll angle,velocity,acceleration,and damping and restoring moments were also derived.
AcknowledgementsThe authors are thankful to Shri J.Ramachandran,Chancellor,Col.Dr.G.Thiruvasagam,Vice-Chancellor,Academy of Maritime Education and Training(AMET),Deemed to be University,Chennai,for their support.
Appendix 1:Ultraspherical polynomials
where
are the eigenfunctions of the following singular Sturm-Liouville equation
The following integral formula and the theorem that follows are needed to establish the convergence of the expansion of the ultraspherical wavelets
Theorem 3 The following inequality holds for ultraspherical polynomials
Appendix 2:Shifted ultraspherical polynomials
The shifted ultraspherical polynomials are defined on[0,1]by
All properties of ultraspherical polynomials remain valid for the shifted polynomials.
For more properties of ultraspherical polynomials,see Rainville(1960).
Appendix 3:Basic idea of HPM
Consider the nonlinear differential equation
with the boundary condition
where A,B,f(r),andΓare a general differential operator,a boundary operator,a known analytical function,and the boundary of the domainΩ,respectively.Expressing A(u)as the sum of linear(L)and nonlinear(N)parts,Eq.(74)becomes
The homotopy technique begins by defining
v(r,p):Ω×[0,1]→R,such that
where p∈[0,1]is an embedding parameter and u0is an initial approximation of Eq.(74)that satisfies boundary conditions(Eq.75).Evidently,Eq.(77)implies that
As p changes from 0 to 1,v(r,p)changes from u0to ur,a process known as a homotopy.The solution of Eq.(77)may be expressed in terms of a power series in the form:
An approximate solution to Eq.(77)is given by the following:
Journal of Marine Science and Application2021年1期