manuevering of ships

8/18/2019 Manuevering of Ships

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Topics Covered: Various Kinds of Stability

Deriving Linear Equations of MotionNon Dimensional Linear EquationsControl Forces and MomentsStability Analysis Analysis of Hydrodynamic Derivatives


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Various Kinds of Motion Stabilit


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Straight Line Stability This is the ships ability to resume a straight-line path without

of control surface forces. This means, if the ship takes a straigh

after the disturbance is removed, then it is said to possess straistability


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Directional Stability This is the ship’s ability to resume a straight line path having th

direction as it had before the disturbance. There can be two po

during the disturbance phase: either it can be oscillatory or nooscillatory.


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Positional Stability Positional stability: this is the ship’s ability to resume a straigh

having the same direction and position it had before the distur

Here by position we mean that it follows essentially the same spath it had before the disturbance.


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Coordinate Systems Inertial Coordinate System – Fixed relative to the Earth Ship Coordinate System – moving with the ship


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Coordinate Systems


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Orientation of Fixed Axis and Moving


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Newton’s Law in Inertial Reference


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Forces and Velocities Expressed in

The motion of a ship is more conveniently expressed when refethe (x, y) system of coordinates fixed with respect to the movinThe ship fixed reference frame is always a right hand frame, witaxis pointing in the longitudinal direction, the y–axis positive sand the z–axis positive down. We can transform coordinate sysusing


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Velocity and Acceleration Ter

where the dot above the symbol signifies the first derivative of the quantity withrespect to time, and u and v are the components of V along x and y, respectively.Then


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Equations of Motion


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General Equations of Motion


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It is also convenient to write the above equations of motion wrespect to a ship fixed reference frame at amidships instead ocenter of gravity. If this is done we pick up a few additional te


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Characteristics of the Generalized Equations of Moti

Equations are cyclic u-v-w, p-q-r If O and G coincide we have X G=Y G=ZG=Ixy =I yz=Ixz=0

These Equations are derived purely from Newton’s 2nd law of m

Hydrodynamics has not come into the picture.


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Hydrodynamic Forces acting on s

The forces and moments (right hand side) of the equations of build up of four types of forces that act on a ship during a man

1. Fluid forces acting on the hull due to the surroundidesignated by the subscript F.

2. Forces due to control surfaces such as rudders, dive

planes, thrusters; subscript R.3. Various environmental forces due to wind, current, subscript E.

4. Propulsion force, T.


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Fluid Forces acting on the Hull (XF,


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Taylor Series


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Taylor Series In order to obtain a numerical index of motion stability, the fu

expressions shown in Equation (6) must be reduced to useful

mathematical form. This can be done by means of the Taylor ea function of several variables. The Taylor expansion of a functsingle variable states that if the function of a variable, x,, and aderivatives are continuous at a particular value of x, say x 1 , theof the function at a value of x not far removed from x 1 can be exfollows


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Taylor Series (1st Order approximatio If the change in the variable, 8x, is made sufficiently small, the

order terms of 8x in Equation can be neglected which can then

to

Similarly a linearized function of two variables x, y is simply a slinear terms as shown:


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Taylor Series- Linear Terms

The functional dependence of above equations can be quite cohowever, for usual maneuvering studies a significant simplifica

We are interested in the ship response around a nominal equilpoint designated by the subscript 1. Expanding the forces in a Taround the nominal point and keeping the first order terms on


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Taylor Series- Linear Terms

All of the partial derivatives are evaluated at the nominal condSimilarly as for Y F, expressions hold for X F and N F ca be derived


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Further Simplifications


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Fluid Forces XF, YF, NF


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Nomenclature of Hydrodynamic Deriv


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Equations of Motion- RHS (General-Nonline

Note: The RHS is General and contains 2nd order terms. For the term X F mvr , mxGr 2 are hi

terms, which have to be neglected. Further u can be written as u= U+u’ where u’ is small pein U. The terms in Y F mur will become mUr and in NF mxGur will become mxGUr .


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Linear Equations of Motion in Horizonta

Using the above notation, and substituting into equations, theequations of motion in the horizontal plane in the absence ofenvironmental disturbances and with the control surfaces at ze


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Non- Dimensional Equations

Every term of the first two equations in the previous slide has tdimensions of a force whereas every term in the third equationdimensions of a moment.

Therefore, to nondimensionalize , which is convenient for sevethe force equations are divided through by ½ ρL 2V 2 and the mequations by ½ ρL 3V 2


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Non-Dimensional Terms


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Non-Dimensionalized Linear Equati

Sway & Yaw

Note: The term U has disappeared as the non-dimensional term U/ U~ 1 for sma


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Rudder Induced Forces and Mom


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Rudder Induced Forces and Mom It is important to note that all of the terms of equations must includ

of the ship’s rudder held at zero. On the other hand, if we want to cpath of a ship with controls working, the equations of motion must

terms on the right hand side expressing the control forces and momby rudder deflection (or any other control devices) as functions of t Assuming that the rudder force and moment on the ship are functiorudder angle δ only, we have


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Rudder induced Forces and Mom


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Control Surface Linear Equations of M

where δ is the rudder deflection angle (see Slide 36) measured athe right hand sign convention; positive rudder deflection corra turn to port for rudder located at the stern. Y δ and N δ are the hydrodynamic derivatives. Including the rudder forces and momlinearized sway, and yaw equations of motion become


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Rudder Operation (Rudder to st


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Rudder Operation (Rudder to st


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Reasons for Rudder behind the Prop

The propeller does nothing but increases the velocity of the waflows out of its slipstream. And the lift generated (rudder forceproportional to the velocity of water falling on it. So if a ruddeat the aft of the propeller, the increased velocity of the proutflow results in a greater lift force. It is only for this reasorudder is placed aft of the propeller. However, if a rudder is pla

forward of the propeller, it will have the same turning effect wito direction, but the magnitude won’t be the same, given the faflow on the rudder is not as much as it would have been, had iplaced behind the propeller slipstream.


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Surge Equation Stability Analys


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Surge Equation – Stability Analys The Surge Equation can be written as :

The above Equation is an ordinary differential equation of first order with constant coefficiesolution is given by


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Dynamical Stability in Sway & Ya

The Equations of Sway and Yaw are as follows

The same can be rewritten as


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Stability Analysis (Cont)Theimage partwith relationship IDrId1 wasnot found in thefile.


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Stability Analysis (Cont)


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If both σ1

and σ2

are negative real numbers or imaginary numbers

negative real parts, we will have That means

tend to 0 as t approaches ∞.

The ship will finally tend to a straight course with constant forwar


N f C/A


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Nature of C/A

bili l i ( )


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S bili A l i (C )


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S bili A l i (C )


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St bilit A l i (C t)


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In conclusion, to ensure both σ1 and σ2 are negative real numbers

numbers with negative real parts, we must have B/A > 0 and C/A>

Stability Analysis (Alternate Metho


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Stability Analysis (Alternate Metho


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Nature of X and X


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Nature of Xu and Xu’

Similarly Xu will also follow the same trend as Xu’ i.e Xu


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Nature of Yv and Yv’

From the Figs above, we can easily see that Y v’ will always be negative and will be of veryforces acting on fore half part and on the aft half part of the ship induced by the transvein direction . Typical values of -Y v’ will be (0.9~1.2)m.Similar trend is shown for Y v too.

N f N d N


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Nature of Nr and Nr’

From the Figs above, we can easily see that Nr’ will always benegative and will be of very large value since themoments acting on fore half part and on the aft half part ofthe ship induced by the transverse rotation r’ are indirection .Similar trend is shown for N

rtoo.

Nature of Nv and Nv’


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v v

N v & N v’ are the lateral moments induced by a unit sway acceleration and by a unit sway velocirespectively. Since the moments acting on the fore half part and on the aft half part of the shipby the sway acceleration or by the sway velocity are in the opposite direction (see Fig above), thmoment will be very small (For a ship with fore and aft symmetry, it should vanish), and the sitotal moment will be the same as the larger one of the fore and aft moments, therefore N v & N very small magnitude and uncertain sign.

Nature of Yr and Yr’


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r r

Y r and Y r’ are the lateral forces induced by a unit yaw acceleration and by a unit yawrespectively. Since the forces acting on the fore half part and on the aft half part of tinduced by the yaw acceleration or by the yaw rate are in the opposite direction (seeabove), the total force will be very small (For a ship with fore and aft symmetry, it sh vanish), and the sign of the total force will be the same as the larger one of the foreforces, therefore Y r and Y r’ have very small magnitude and uncertain sign.

Magnitude and Sign of Linear Hydrody


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g g y y

Coefficients


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Stability Analysis We had seen in the previous class, that the condition for straight l

is that C/A>0 and B/A >0, where A,B,C are the coefficients of the D

Equation for v (sway velocity) and r (Yaw velocity) given by:

Further A, B and C are as follows


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Nature of A & B coefficients


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‘C’ Coefficient


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This means, for horizontal directional stability the necessary

sufficient condition (assuming xG =0) is that

In Non-Dimensional derivative form, the same is written as:

0>+−r vr v

N Y Y U m N

( ) 0''''' >+−r vr v

N N Y m N


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Criterion for Stability The Criterion for Straight Line Stability can also be written as:

Physically the above condition states that the Centre of Pressu

Yaw should be ahead of the Centre of Pressure for Pure Sway foto possess straight line stability.

v

v

r

r

Y

N

mY

N

'

'

''

'>

−


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Some Thoughts on St Line Stabi Usually for Ships N v is negative (Nbow is dominant over Nstern )

always negative. Thus to make the ship stable, N v is to be madenegative. How????

By adding Skeg

By moving the CG of the ship Forward

If the ship has got very high directional stability (i.e C has a hig value), the effectiveness of the rudder will be that much less. W


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Question 1


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Question 2


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manuevering of ships

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