1 Introduction

The control of foiling during the America’s cup appeared to be a determinant point in the success of Oracle Team USA(OTUSA for short).In particular during upwind legs,when the boat had to avoid the waves created by the wake of the preceding boat,the automatic stabilization is a fundamental advantage that OTUSA exploited in a smart way and finally won the competition.Such situations are represented on the Figures 2 taken from TV transmissions during the America’s cup in San Francisco(September 2013).

如果孕妇没有深海鱼的足量摄取或额外补充DHA，在此前提下不建议选用单不饱和脂肪酸，而应该选用多不饱和脂肪酸以利于DHA的合成。因为人体需要的2种必需脂肪酸都是多不饱和脂肪酸，即亚油酸和α-亚麻酸。其中α-亚麻酸在代谢过程中可合成EPA（脑白金）和DHA（脑黄金）。就α-亚麻酸的含量而言，在烹调油中大豆油和菜籽油相对比较高，所以建议孕妇可多选用大豆油或菜籽油烹饪。

In this paper,we have tried to give a simple and precise ma the matical model and analysis of such boats.Even if a more industrial 3D analysis would be obviously more realistic,,and in order to be as simple as possible,we restrict our analysis to a bi-dimensional case.Hence,only two movements of the ship are taken into account:The heaving which is a normal displacement to the surface of the sea,and the pitching which is the rotation around a horizontal axis transverse to the main direction of the ship.Hence,the yawing angle and the rolling are eliminated from our model.Obviously they are meaningful,but according to our mind,not necessary for the understanding of our purpose.

为提高助教效果，教师应该对家长助教活动进行评价，撰写评价心得或者案例分析，把自己的意见与助教家长进行交流，争取与家长达成共识，让今后的家长助教活动达到更高的水平，也为以后更好地利用家长资源做好准备，提高家园合作的质量。

The inclination of the main foil should be manually driven but a hydraulic ram can be used for the control process(rules of the race)using the high pressure collected from a small hole in the foil.Because the system is a second order one(with inertia,damping and stiffness),only a phase control can lead to optimal results(see[1,2,10]).This driven angle is named the rake.It appears,in the numerical simulations,that the regulation law strongly depends on the ship velocity.Even if the experimental data that we introduce in our numerical model could be improved,they are sufficient in order to give an idea of how things work.

The aim of this paper is to obtain a faithful dynamical model of the movement of the boat which respects the following facts:

(1)the existence of the foiling velocity under which the boat can not stand up on its foil,

(2)the possibility that the velocity of the boat can be greater than the wind velocity,

(3)the possibility to discuss the stall flutter phenomenon of the foils.

The plane of our work is the following one:We first compute the aerodynamic propulsion force due to the wind and which is applied on the sails bearing in mind its importance since it is the unique energy source of the boat.We then establish a nonlinear model which take into account the apparent velocity of the water flow at the rear and main foils.We then analyze the steady state of the nonlinear model and a numerical study will point out the overs peed phenomena.We finally study the stability of the linear model and discuss the stall-flutter phenomenon.A part of the work was the characterization of the foiling velocity at which the boat takes off.All our the oritical work is illustrated with numerical results with computations performed with Matlab.The fundamental tools for the numerical analysis are developed in the book of Ciarlet[3].

2 Dynamical M odel of the Boat

2.1 Descrip tion of the boat and main notations

The orthonormal basis of R3 is denoted by(e x,e y,e z).The velocity of the boat is−ue x where u>0 and therefore the velocity of the flow of the water in the basis connected to the boat is ue x.As said before,the movement is assumed to be represented by two functions(see Figure 3):The heaving z and the pitching angleγin the plane(e z,e x).The equilibrium is written at an arbitrary point—say O.For sake of convenience,it is chosen to be the center of rotation of the main foil.

The forces applied to the ship and implying an evolution of these two previous functions,are those due the rear and the main foils.The local hydrodynamical coefficients(c z for the lift and c m for the pitching moment)depend respectively on the apparent local angle of attack of each foil.For the rear foil,it is denoted by(β + γ)a and(α+γ)a for the main one.

Characteristics of the boat,of the air and of the water:

a mass density of the air,

在吴钧陶《爱丽丝奇境历险记》汉译本中，他在翻译地名的时候采取了音译加注释的方法，既可以保留源语言的文化信息又可以在一定程度上解决文化冲突。

e mass density of the water,

⋄g=9.81 m/s2 is the gravity,

⋄−ue x velocity of the ship,

⋄M is the mass of the ship,

⋄G center of mass of the boat,

⋄J G is the inertia around the center of mass G in the pitching,

⋄J O is the inertia around the center of mass O in the pitching,

⋄M o is the moment of the external forces at point O in the pitching,

⋄ d s=O′S is the length of the stick supporting the steering rudder,O′being the anchor point of the rear foil,

⋄d f=OF is the length of the foil in the depth direction,

⋄S s,S f are respectively the cross sections of the foils at the extremities of the rudder and the main foil,

⋄a(respectively b)is the distance between the center of mass and O(respectively O′),

⋄h=a+b=OO′,

⋄d og is the distance from the rotation point of the foil to the center of mass of this foil.

Variables for the description of the movement of the boat:

⋄z is the heaving,

⋄γis the pitching angle,

For the angles,apparent velocities and forces:

⋄αis the angle of attack of the mail foil,

⋄βis the angle of attack of the rear foil,it is supposed to be fixed,

First of all,let us characterize the equilibrium position of the ship.The term(α0,β0)corresponding to the equilibrium of the ship over the water(γ=0)is the solution of

⋄c m f and c m s are the pitching hydrodynamic coefficients at points F and S.They are continuous functions in their variables,

⋄v is the absolute wind velocity:It is in the plane(e x,e y),

The Figure 5a is concerned with a floating boat(which has its bows in the water)whereas Figure 5b and Figure 5c illustrate the case of a flying boat.

人心有许多情感足以战胜死亡——战场上的仇忾压倒死亡，相爱者的忠诚战胜死亡，自由的荣耀蔑视死亡。一个流放的时代，曾经冲动而不自如，后来有罪而不自觉，现在纵观统筹、高谈阔论。我说过我真正感兴趣的是人们在危急时刻的表现。

is the angle between the velocity of the wind and the direction in which the boat is moving forward,

⋄V a is the modulus of the apparent velocity of the wind,

⋄V the modulus of v,absolute wind velocity,

⋄V as,V af are respectively the apparent flow velocity at the two foils:One on the rudder and the other one which,is the main one,supported by the dagger board,

⋄u is the modulus of the velocity of the ship,

踏着皎洁的月光，迎着微凉的晚风，拖着疲惫的身躯，我晚上八点多才到他家。孩子的父亲听说我要来，老早买了水果和瓜子在家等着，很热情地接待了我。小龙爸爸是个话语很少略显羞涩的人，一见面就开门见山谈孩子的缺点，有点恨铁不成钢的感觉。他坦然承认，孩子不听话时，孩子妈妈会采取粗暴的行为，用威胁性的语言厉声斥责，甚至动手打，而他和孩子的奶奶对孩子极其温柔、慈祥，无原则地宠爱，对孩子的要求大多会无条件满足，所以小龙在家里是小皇帝。

The following notations are used.

2.2 The ap parent velocity of the wind

Even if it is a side subject for our main purpose,it is worth to recall how the apparent wind velocity can induce overs peed for particular positions of the boat with respect to the direction of the wind.The formulae used in this section,are not original.Our goal is only to show with a simple numerical simulation,the influence of various parameters on the boat velocity and mainly the one of the sailing position and of the drag coefficient of the bows or the foils in the water.

2017年，黑龙江省用水总量控制在353.05亿m3，万元GDP用水量降幅8.8%，万元工业增加值用水量降幅22.7%，农田灌溉水有效利用系数提高到0.5998，全国重要江河湖泊水功能区水质达标率62.8%，化学需氧量、氨氮排放量分别比2015年削减5.26万t、0.49万t，减排比例为3.51%、5.44%，比2016年分别减少1.88%和2.48%，全面完成国家下达的控制目标[2]。

Let us consider the situation represented on Figure 4.Following the notations of this figure,the apparent wind velocity is given by

and its modules is equal to

The apparent angle between the normal to the sail plane and this apparent wind velocity isand it satisfies

and therefore

Using(2.3),we get

⋄c f andξare respectively the stiffness and the damping coefficient of the system used for the stabilisation of the main foil.

and thus

另外，我们也可以通过刘姥姥与贾雨村形成对比，我们知道他们都相当于打秋风的人，但是两个人对恩人的表现却十分不同，贾雨村尽管有时会在利益的诱惑下帮助恩人，但大多是助纣为虐，而最终在没有利益的时候，在要付出巨大的代价的时候，贾雨村选择了背叛恩人，还落井下石，让我们看到了人性丑恶的一面，而刘姥姥却不一样，她用自己的一切选择了帮助恩人，尽管恩人当时已经不能再给她任何好处了，而且刘姥姥还要付出巨大的代价，所以人性的善恶通过这两个人我们就可以清晰地看到，此外，这两个人的安排也十分巧妙，都是从头到尾的线索人物，让我们看到了曹雪芹写作技巧的高超。

The propulsion force due to the wind denoted by F x,is the projection on the direction e x of the aerodynamical force applied to the sail.For sake of simplicity it can be written(the square of the cos(µa)takes into account the normal component of the apparent wind velocity and n is the unit normal to the sail plane)as

In fact,a correction coefficient is included in the surface S a which takes into account the aerodynamical coefficient c z(µa)of the sail.With(2.3)we get

The drag force is the sum of two contributions:One due to the sail and another one due to the drag in the water of the bows(zero during the flight)and the foils which are always immersed.Furthermore,the last term depends on bothβ+γandα+γ.Let us assume that this drag force can be evaluated by

where S e is the crosss ection immerged into the water and c x e the corresponding drag coefficient.In fact,the important number is the product S e c x e.It is about 1.5 m2 for a ship floating and about 1 m2 for a flying one as far as the profil of the foils are correctly drawn.

2.3 The overs peed phenomenon

We deduce that the velocity u of the boat is obtained by solving the equation

微波的传输情况可参考图1.当有入射波b从端口B输入，并通过透镜L1聚焦后，一部分能量被样品S反射成为出射波b1.另一部分能量透过样品S传输到端口A成为出射波b2[11].

Due to the complexity of this equation,it is easier to perform numerical tests.We have drawn on Figure 5 the sign of the function F x−T x with respect to the two variables u on the abscissa which is the velocity of the ship andζon the ordinate which is the angle of the sail plane with the direction e x(velocity of the boat).The boundary between the two areas(black and white)are the solutions.

肝肾综合征治疗：（a）可用特利加压素（1 mg/4～6 h）联合白蛋白（20～40 g/d），治疗 3 d血肌酐下降＜25%，特利加压素可逐步增加至2 mg/4 h。若有效，疗程7～14 d；若无效，停用特利加压素。（b）去甲肾上腺素（0.5～3.0 mg/h）联合白蛋白（10～20 g/L）对1型或2型肝肾综合征有与特利加压素类似结果[44-47]。

For the floating boat,it appears,with the set of data used,that the absolute wind velocity can not be overtaken with our choice for the physical data.For the flying boat,the absolute wind velocity can be exceeded.This is due to the reduction of the drag force on the bows in the water.In fact the pictures on these figures show that a tacking for upwind sailing is much better with large angles concerning the velocity because it enables to make the boat flying above the water using the foils.And even if the distance covered is more important,the time necessary can be smaller.But the flight must be stabilized similarly to what is done with an aircraft.Because,even if the flight is stable,there can be perturbations due to the gravity waves for instance.This is more critical if the boat has to cross over the wake of a preceding boat.In fact the phenomena are very close for a simple reason:The ratio between the aerodynamical forces on the wing of an aircraft is similar to the one applied to the foil of a flying ship.In fact,the ratio between the mass density of the water and the one of the air is about 1000/1.2≃833 and the one between the square of the velocities is about

And the equivalence is deduced from the fact that the forces are proportional to the mass density times the square of the velocity.

3 Dynamical Equations

3.1 The ap parent flow velocities on the foils

The apparent velocity of the water on the foils implies the terms˙γand˙z.It is the difference between the wind velocity and the one of the boat.First of all,let us give the expressions of the velocities of the points S and F corresponding respectively to the rudder and the main foil where the hydrodynamic forces are given from hydrotunnel tests.We refer to Figure 3.

One has

In another way,we have

and therefore the velocity of the flow at point S is equal to

信息的统计解析设计主要针对的是统计解析监管逻辑方面，其使用的方式为创建数据分析函数、计算统计函数、搜索查询函数及信息传输函数等。此外，在此功能的设计过程中，系统管理工作者在搜索数据的时候，首要工作就是发出信息搜索请求，然后信息会直接传递到页面，再通过搜索条件，将最终所需要的数据返还回来。

We get analogous results at point F where the velocity of the flow is

The computations of the hydrodynamic apparent velocities are performed at points S and F in the axis(e x,e z).These apparent flow velocities are defined as the difference between the absolute flow velocity and the one of the point considered.Let us notice that there are two different notions for the apparent velocity.The one of the wind and the one of the hydrodynamic flow on the foils.From now on,it is the second one which is taken under consideration.It is given by the following formulae(see[6,9]):

and we then obtain

and

Furthermore,the hydrodynamic apparent angleof attack of both thee xtremity of the rudder and the one of the driven foil,which are denoted by(β + γ)a and(α + γ)a,are given by the following expressions((·,·)3 is the scalar product in R3):

3.2 The equations of the movement of the boat

The equations of the movement are the following ones(the right-hand-side is derived from the factor of andin the expression of the power of the hydrodynamical forces):

If the point O waschosen asthe center of hydrodynamic forces,then one would have M 0=0.

改革开放不仅促进了经济的开放化，也促进了思想的开放化，如何在开放的思想环境中做好企业思想政治工作至关重要。现在是网络化的时代，企业政工人员应当积极把握时代发展，创新工作理念和工作方式，以此来做好思想政治工作，提升企业员工的思想政治素质。

3.3 On the steady state

⋄c zf and c zs are the lift hydrodynamic coefficients for the main foil and the rear foil.They are continuous in their variables,

It can happen that there is no solution to system(3.7)and there may be several reasons for this:For instance,it could be because there is no solution at all or because there is no acceptable solution.Indeed,the anglesα0 and β0 must be small enough and the interval??seems to be reasonable.Of course,in order to solve(3.7),we have to know the different hydrodynamic coefficients involved in it.We assume in the following computations that these coefficients depend linearly on their variable.It means that the anglesα0 andβ0 are assumed smaller than the static stall values.We prove the following theorem.

Theorem 3.1(i)There exists u f>0,such that for uf,system(3.7)has no solution.

(ii)In the case of linear hydrodynamic coefficients,one can choose u f>0 such that system(3.7)has a unique solution for u≥u f and u f can be explicitely computed.

Proof(i)Let us define

目前，独立学院法学专业基本上选用与普通高校法学专业相同的本科系列教材。这些教材大多是普通高校法学本科专业的规划教材、统编教材，知识体系完整、系统，“内容过于理论化，缺乏案例分析，缺乏法律职业技能训练内容”［2］30，对独立学院法学专业的学生来说难度可想而知。

System(3.7)can be written asLet us denote by∆the first bisector of R2.System(3.7)has at leat one solution if and only if

The setis bounded since G is continuous.Let us set

由于医养结合养老服务运营流程及模式都会根据时代的进程发生不同程度的改变，故在设计指标体系时，应充分考虑能反映医养结合养老服务水平的本质指标，构建具有普适性的评价指标体系，以应对不断优化和改进的发展模式。

A necessary condition for having solution is

and thusThe nonnegative real is convenient in order to prove the point(i).

(ii)We assume that the four hydrodynamic coefficients are linear.Let us write

By simplifying equation(3.7),we can easily prove thatα0 is the solution of

where coefficients A,B et C do not depend on the velocity u.Indeed,their values are

and

Let us setA necessary condition for existence of solution of(3.9)is that u satisfies

Taking into account an analogous condition for the angleβ0,we introduce a critical speed named foiling velocity and defined by

The value u f is the minimum speed for which the boat can stand up on the foils.It is obvious that,ifthe function f is one to one in a neighborhood of 0.Hence,the best value of u f can be explicited:It depends of all the parameters that appear in A,B and C.The proof of(ii)is complete.

The graphs that follow on Figures 6–7 provide the function|f|and the steady anglesα0 and β0 for a=0.2.The foiling speed is nearly 7.2 m/s for a 1000 kg boat which seems reasonable.Of course,the foiling speed depends on the mass of the boat.

Figure 6 Graph of the function|f|.

Figure 7 The evolution of the steady equilibrium angles.

3.4 Linearization of the equations

The discussion on the dynamical stability will be organized from a linearization of these equations around the steady anglesα0,β0 solutions of(3.7).The variables are z and γ.The first step is to formulate the linearized model around an equilibrium position solution of(3.7).Let us recall thatβ is assumed to be fixed(equal toβ0),and the evolution of the pitching angle of the rear foil is only due to the global pitching of the boat-sayγ.We writeδ=α−α0.In a formal way one can write this linear model as follows:

In practice,the coefficients C11,C12,C21 and C22 could be computed using a symbolic computation software if for instance a three dimensional modelling would be concerned.In our two dimensional case it is still possible to perform a hand computation.

We introduce the several matrices of the system—say—M,C and K,

and the right-hand-side F 0,B,E are defined by

System(3.11)can now be written:Findsatisfying

with initial data X(0)=X 0∈R2 and.Forξ1 andξ2 small enough,we set

Obviously,we get on the one hand

and on the other hand

Let us define

and

the right hand side of the system(3.6).

The expressions in(3.11)are(recall that the steady state is often denoted by 0)F1=F(0),F2=G(0)and partial derivatives of the functions F and G with respect toat the equilibrium point(in apparition order).With(3.7),we can assume F(0)=G(0)=0 thus F1=F2=0 and F 0=0.The following lemma is useful for the computation of the derivatives.

Lemma 3.1 We get at the equilibrium point(α0,β)the following expressions:

The proof,which rests on a simple but long computation,is left to the reader(see[7]).

3.4.1 Computation of the stiffness matrix K

One obtains

with

and

We introduce the opposite of the dynamical stiffness:

and

The stiffness matrix is then

3.4.2 Computation of the matrix C

The coefficients are given explicitely in the following table.

Table 1 Expressions of the coefficients C ij aroundγ=0 versusαandβ.

4 Stability of the Boat

4.1 Static stability

We assume in this subsection that the apparent flow velocity is negligible(C=0),and that the angles of attack at the main and the rear foils are solution of(3.7)with a velocity larger than the foiling one(it is the one for which the foiling appears).The model is

Let us setWe obtain Y∈R4 and

We introduce the 4×4 matrix A:

The solutions of

are

where

The stability is governed by the sign of the real part of the eigenvalues of the matrix A and more precisely the stability is acted if(σ(A)is the spectrum of A),

A vector v 6=0 is an eigenvector of A if one can findλ∈C such that A v=λv.One can write withwhere v i∈C2(i=1,2),

or equivalently

with.We then deduce that the eigenvalues are also the solutions of

Let us write(recall that

The valueλ0=0 is an eigenvalue because of the heaving which is not restricted,and the other are the solutions of

If aR1+R2>0,one of them is nonnegative and therefore an instability(which is coupling between heaving and pitching)may appear.The stability is ensured if

In that case,the solutions arewhereµis the pulsation of the movement and we have

The frequency is

Let us notice that the existence of the instability depends only on u via the anglesβ0 and α0.There is no direct dependence.For a small enough,the sign of aR1+R2 is the same as the sign of R2 and is therefore equal to the sign of the trace of the matrix(K)which is nonnegative.We deduce that the eigenvalues of K areλ1=0 and

If a=0,it is easy to check that

Therefore

Since

and since the initial data must satisfy(3.7),one can see that the angle of attack of the main foilα0 must be adjusted in order that the boat resists capsizing.

The Figures 8–9 illustrate the numerical computation of the eigenvalues when the hydrodynamics coefficients linearly depend on their variables for sake of simplicity.The velocity u begins at 1 m/s and up to 10 m/s with 300 steps in time.At each iteration,the values of the anglesβ0 and α0 at the equilibrium are computed with the formulae(3.7).Once computed,the spectrum of the matrix A is obtained.We have drawn the real part(damping or increase)and the frequency of the movement f with respect to the velocity u.We can notice an instability in heaving due to the double null eigenvalue.The pitching is stable since 0 is the only real eigenvalue.Moreover,a>0 means that the center of mass of the boat is behind(but not far from)the point O.We have drawn the graphs for a=0.2 and a=0.8 with h=5.The frequencies seems to be near 1Hz or 1.5 Hz for large values of u.There is an instable region for small speeds for a=0.8 m.

Figure 8 The evolution of the spectrum versus the velocity of the boat with a=0.2 m and h=5 m.

Figure 9 The evolution of the spectrum versus the velocity of the boat with a=0.8 m and h=5 m.

The Figure 10 presents the same graphs for h=7.One can notice that the movement is stable for both values of a=0.2 and a=0.8.

The Figures 11–12 correspond to different hydrodynamics coefficients,smaller at point S than at point F,for h=5.

The Figures13–14 concern the previous phenomenon.On Figure13,we have drawn R2+aR1 with respect to a and u(recall that the stability is ensured if R2+aR1<0).Onecan notice that for large value of the parameter a (the distance between the center of mass and the dagger board),there isa critical value u c=u c(a)under which instability may occur.The Figure14 is the graph of u c with respect to a.These graphs are achieved with h=5.The Figures 15–16 illustrate the same phenomenon in the case of different hydrodynamics coefficients.

Figure 10 The evolution of the spectrum versus the velocity of the boat with a=0.8 m(left)and a=0.2 m(right)with h=7 m.

Figure 11 The evolution of the spectrum versus the velocity of the boat with a=0.2 m(left)and a=0.8 m(right),h=5 m and different hydrodynamics coefficients.

4.2 Dynamic stability

Let us notice that the dynamical behavior of a single foil is presented in[8]where the hydroelastic response and stability of both rigid and flexible 2D hydrofoils in viscous flow are discussed from experimental and numerical aspects.There are in general four kinds of dynamic instabilities which can occur in general.

(1)One is well known by the sailors.It concerns induced vibrations on the rudder due to vortices created by the main foil.But,this so called buff etingeffect(see[4,9]),can occur only for very particular cases.In our case it would be due to a vortex shedding from the main foil onto the one of the rudder.And it would appear if the frequency of the vortices is close to the one of the rudder and its foil.It is quickly detected and should be suppressed by an ad hoc conception of the ship.It is not necessarily destroying but can reduce considerably the efficiency as far as it takes energy from the kinetical energy of the boat.

Figure 12 The evolution of the spectrum versus the velocity of the boat with a=1 m(left)and a=0.8 m(right)with h=7 m and different hydrodynamics coefficients.

Figure 13 The criterion stability with respect to a and u.

Figure 14 The critical velocity with respect to a is sensitive to small value of a.

(2)The second one is the classical flutter which is violent and corresponds usually to the unlimited exchange of energy between two movements with the same eigen frequency(here for the heaving and the pitching).Clearly the secure flight of a ship would be seriously compromised by such an instability implying an exponential increase of the movement of the ship(see[9]).It would be difficult to control it using the main foil without additional lifting supplementary wing.Furthermore,because the phenomenon is very quick and complex,its control requires an automatic loop driven by an electronic computer.In fact as far as the boat is flying over the water,there no stiffness on the heaving excepted the one due to the hydrodynamical forces acting on the foils which are fully immersed in our model.Maybe it would be different if the bows were in contact with the water and the Archimède forces would operate.

Figure 15 The stability criterion with respect to a and u.

Figure 16 The critical velocity with respect to a is sensitive to small value of a.

(3)The third possibility of instability is due to a fluctuation of the wind velocity.It is a rather complex phenomenon implying the apparent flow velocity but mainly the perturbation of the flow due to vortex shedding.The rudder is the part mainly concerned by this phenomenon due to the turbulence generated from the main foil.It is under the skipper/helmsman control.It is discussed in Subsection 2.2.It could be compared from the mathematical point of view to the buff eting phenomenon.

(4)The fourth dynamical instability and the last one in our discussion,is due to the apparent water velocity on the foils.It could be compared to a stall flutter phenomenon as the one encountered in the breakdown of the famous Tacoma-Narrows bridge which collapsed on November 1940.This accident was correctly explained forty years later by Scanlan[7]and the final explaination rests on the apparent flow velocity.From the mechanical point of view,the phenomenon can be understood as a negative damping.See also[5]for a similar collapse of a model of a military aircraft.

We are mainly interested in this paper in the case of stall flutter phenomenon(see(4)).We then consider the model

We still setWe have

and

Following the previous section,we have to consider the spectrum of the following matrix A:

Let be an eigenvector of A.One has

and withwith v i∈C2(i=1,2),

which leads to

withTherefore

which is an equation like

The stability of(4.3)is ensured if the solutions are simple and with a negative real part.We haveλ=0 or

If one considers the pulsation,thenλ=iµandµis the solution of

and the stability condition becomes

The Figure 17 concerns the dynamic stability.On the upper graph of the Figure 17(left),we have drawn the maximum and minimum of the real part of the spectrum.On the centered graph,we have drawn the frequencies,and the down graph is the one of the equilibrium angles.On the Figure 17(right),we have drawn the maximum of the real part of the spectrum A with respect to the speed.The parameters are a=1m and h=5m.The Figures 18 concerns the case a=0.2m and h=5m.On Figures 22–24,the criterion of stability is drawn with respect to a and u respectively in the case of identical hydrodynamic coefficients for h=5m,h=7m and different hydrodynamic coefficients(c zs and c zf):The stability is ensured under the plane z=0.

The Figures 19–20 concern the case of different hydrodynamic coefficients for decreasing value of the parameter a and h=5.As shown on Figure 19,the stability is ensured but one can see that the eigenmodes are quite different.

The Figure 21 concerns the case of different hydrodynamic coefficients for a and h=7.Same phenomena occur.

Figure 17 The spectrum(left)and the stability criterion(right)for a=1 and h=5.

Figure 18 The spectrum(left)and the stability criterion(right)for a=0.2 and h=5.

5 Conclusion

Figure 19 The spectrum(left)and the stability criterion(right)for a=0.2 m

Figure 20 The spectrum for a=1 m and a=0.8 m:Stable in both cases.

Figure 21 The spectrum(left)for a=1 m and the stability criterion(right).

In this article,the influence of the apparent flow velocities has been studied.It concerns both the effect of the wind on the sails and the one of the water on the foils.For the apparent wind effect it appears that the velocity of the boat can be almost three times the one of the absolute wind,as far as the bows are out of the water.For the foils the apparent water flow plays a different role.It generates a damping effect as far as the boat is stable.The numerical examples show that the instabilities can only appear at very low velocity,but for which the foiling is impossible,or for high velocities which are at the limit of the performance of such ships.Hence the control from the foils can be introduced for any speed of the ship between the foiling velocity and the maximum velocity of the boat.Clearly,a control on the rudder would certainly be the best method but it was not permitted during the 2013 America’s cup.

Figure 22 The stability criterion with respect to a and u for h=5.

Figure 23 The stability criterion with respect to a and u for h=7.

Figure 24 The stability criterion in case of different values of the hydrodynamic coefficients.

References

[1]Bellman,R.,Dynamic Programming,Princeton University Press,Princeton,1957.

[2]Cea,J.,Optimisation,Théorie et Algorithmes,Dunod,Paris,1968.

[3]Ciarlet,P.-G.,Introduction à l’analyse numérique et à l’optimisation,Masson,Paris,1985(in France).

[4]Destuynder,Ph.,Introduction à l’aéroélasticité et à l’aéroacoustique,Hermès-Lavoisier,Paris-Londres,2007.

[5]Destuynder,Ph.,Analyse et contrôle des équations différentielles,Hermès-Lavoisier,Paris-Londres,2010.

[6]Destuynder,Ph.and Ribereau,M.T.,Nonlinear dynamics of test models in wind tunnels,Eur.J.Mech.A/Solids,15(1),1996,91–136.

[7]Dowell,E.H.,Curtiss,H.C.,Scanlan,R.H.and Sisto,F.,A Modern Course in Aeroelasticity,Mechanics:Dynamical Systems,Kluwer Academic Publishers Group,Dordrecht,1989.

[8]Ducoin,A.and Young,Y.-L.,Hydroelastic response and stability of a hydrofoil in viscous flow,Journal of Fluids and Structures,38,2013,40–57.

[9]Fung,Y.C.,An introduction to the theory of aeroelasticity,John Wiley and Sons,New York,1955.

[10]Lions,J.L.,Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués, Masson, Paris,1988.