Frd  ¼ KrdArdkvwk2hrd;

where Frd is the rudder force, and Krd and Ard are the rudder coefficient and the area of the rudder。 The symbol vw is used here to represent water speed。 For simplicity, the angle of rudder, hrd, is assumed to be the angle between the rudder and the surge direction, Xs。 The angle of rudder is limited to 35° to avoid stalling (Rawson and Tupper 2001)。 In reality, the rudder force  is not  a  linear  function of the rudder   angle。

However, since the rudder angle is limited to within [-35, 35], a linear function is a good approximation。 In (Rawson and Tupper 2001), the details of computing rudder force can be found。 As the rudder force enables the ship to yaw, the resistance of water produces the anti-force for yaw。 Assume the resistance is related to the yaw speed。 The net force for yaw is computed by:

Fy  ¼ Frd  — byIyxy;

where by is the resistance coefficient for yaw, Iy is the inertia moment for yaw, and xy is the angular velocity of yaw。 Based on Newton’s second law for rotation, the angular acceleration for yaw is obtained   by:

ay  ¼ Fy=Iy:

We assume that the ship is a beam structure as shown in part (a) of Fig。 4, and Iy is the same as the inertia moment of pitch, Ip:

Iy ~ ML2=12:

Then the velocity and magnitude of yaw are computed  by:

xy  ¼ ayDt; ð14Þ

hy   ¼ xyDt: ð15Þ

In reality, the ship will deviate a little from its course if the angle of rudder is changed because of the moment。 But it is difficult to simulate the process。 In our system, we set by to a small value。 Therefore, the ship will move straightfor- ward once the rudder angle changes to   0°。论文网

4。3Sway

Sway motion can be generated by winds, currents, or even propellers。 (Some large ships are equipped with propellers which can make sway motion inside harbors。) In our work,

only winds and currents are counted for sway。 To estimate the force of winds acting on the ship, the component of wind velocity in Zs direction is computed。 Then  the velocity component is scaled with a coefficient, Kw, to produce the wind force for  sway:

Fw  ¼ KwhVw; Zsi;

where Fw is the force of wind, h, i is the inner-product operator, and Vw is the wind velocity。 The coefficient Kw is related to the profile of the ship。 A larger ship profile will result in large wind coefficient。 By analogy, the force of current is estimated  by:

Fc  ¼ KchVc; Zsi;

where Fc is the force of current acting on the ship hull in Zs direction, Vc is the velocity of current。 and Kc is the coefficient of currents。 Let bsw be the resistance coefficient of sway and vsw the velocity of sway, then the net force for sway is:

Fsw  ¼ Fw þ Fc — bswMkvswk2:

The last term on the right hand side represents the drag force for sway。 It is similar to the drag force for surge。 Then the acceleration and velocity of sway are computed by:摘要：本文提出了一种有效的船舶运动计算模型。这个模型是用来模拟船舶的实时运动。与传统方法相比，这种方法具有根据不同的船舶形状，发动机和海上的能力，而不损失效率。基于我们的模型，我们建立了一个船舶运动仿真系统的两个娱乐、教育应用。我们的系统有助于用户了解一艘船遭遇的海浪、海流和风的运动等情况。用户可以通过调整发动机功率、舵等船舶设施运用图形化的用户界面来创建自己的船舶模型。也可以通过改变波的频率，改变环境波振幅，波方向，洋流，和风。因此，许多船舶和环境的组合，使学习变得更加有趣。在我们的系统中，一艘船被视为一个刚性体漂浮在海面上。它的运动建立在六个自由度：纵荡、横荡、艏摇、横摇、枞摇、垂荡。这些运动分为两类。前三个动作是由海浪引起的，最后三个是引起的由螺旋桨、舵、洋流和风引起的。基于牛顿定律和其他基本的物理运动模型，推导出的算法来计算的运动幅度。我们的方法可以进行实时并具有较高的保真度的模拟。根据船舶理论，船体外表面力的净效应取决于船体形状。因此，船舶的行为受其形状的影响。为了增强我们的物理模型，我们将船舶分为三种基本类型。它们是扁平的船，薄的船和细长的船。每种类型的船舶都与一些预设定的参数来决定它们的特性。用户可以通过不同的参数来调整船舶的行为，即使他们只有一点点船舶理论知识。