PRACTICAL DESIGN OF CONTROL SURFACES
Om Prakash Sha Department of Ocean Engineering and Naval Architecture Indian Institute of Technology – Kharagpur, 721 302
1.
INTRODUCTION
Rudder and other control surfaces such as bow thrusters are crucial in achieving the controllability objectives. Different control devices can help in achieving the desired controllability characteristics of a vessel, but the rudder is the most simple and popular control device and hence this section will look into the design process of a rudder. During the concept and preliminary design stage, a naval architect has little information on which to base decision. Nevertheless he has to decide at this early stage, the hull form in terms of the shape of the underwater body, distribution of buoyancy, the shapes of sections and underwater profile. He then has to take decisions regarding the location and sizes of propeller, rudder and thrusters. All these decisions, which are interrelated, will affect controllability of the vessel. It is, therefore, important for the naval architect to evaluate at the preliminary design stage the type of rudder, its hydrodynamic efficiency, its structural supports, and clearances between propeller and rudder. The following are the four major constraints that limit the design of a rudder and any other control surface. (a)
In profile, the rudder should fit within the dimensions dictated by the shape of the hull. Its maximum span should fit within the vertical distance measured from the bottom of the deepest projection below the baseline of the ship permitted by draught or docking restrictions upward to the bottom of the hull immediately over the rudder or to the minimum prescribed depth below the water surface, whichever is lower. If the rudder is abaft the propeller, its chord should fit within the horizontal distance from the extremity of the ship to a line corresponding to a prescribed clearance from the propeller. (Control surfaces that extend significantly beyond the block dimensions of a ship, such as fin stabilizers, or the bow planes on some submarines, are almost always designed to the retractable).
(b)
The rudders, in maintaining a straight course, should minimise speed loss at every level of ship power plant output.
(c)
The rudder, the rudder stock, the rudder support, and the steering engine, considered together, should be of minimum size, weight, complexity, and initial cost, consistent with required effectiveness and acceptable standards of reliability and low upkeep costs.
1.
INTRODUCTION
Rudder and other control surfaces such as bow thrusters are crucial in achieving the controllability objectives. Different control devices can help in achieving the desired controllability characteristics of a vessel, but the rudder is the most simple and popular control device and hence this section will look into the design process of a rudder. During the concept and preliminary design stage, a naval architect has little information on which to base decision. Nevertheless he has to decide at this early stage, the hull form in terms of the shape of the underwater body, distribution of buoyancy, the shapes of sections and underwater profile. He then has to take decisions regarding the location and sizes of propeller, rudder and thrusters. All these decisions, which are interrelated, will affect controllability of the vessel. It is, therefore, important for the naval architect to evaluate at the preliminary design stage the type of rudder, its hydrodynamic efficiency, its structural supports, and clearances between propeller and rudder. The following are the four major constraints that limit the design of a rudder and any other control surface. (a)
In profile, the rudder should fit within the dimensions dictated by the shape of the hull. Its maximum span should fit within the vertical distance measured from the bottom of the deepest projection below the baseline of the ship permitted by draught or docking restrictions upward to the bottom of the hull immediately over the rudder or to the minimum prescribed depth below the water surface, whichever is lower. If the rudder is abaft the propeller, its chord should fit within the horizontal distance from the extremity of the ship to a line corresponding to a prescribed clearance from the propeller. (Control surfaces that extend significantly beyond the block dimensions of a ship, such as fin stabilizers, or the bow planes on some submarines, are almost always designed to the retractable).
(b)
The rudders, in maintaining a straight course, should minimise speed loss at every level of ship power plant output.
(c)
The rudder, the rudder stock, the rudder support, and the steering engine, considered together, should be of minimum size, weight, complexity, and initial cost, consistent with required effectiveness and acceptable standards of reliability and low upkeep costs.
(d)
Undesirable effects of the rudder on the ship such as rudder-induced vibration should be kept to a tolerable level.
Violation of any of the four listed constraints constitutes a misjudgement in rudder design. Because of the influence of the rudder on ship power [constraint (b)], adherence to a minimum total ship cost [constraint (c)] requires consideration of the entire ship design process.
2.
HYDRODYNAMIC CONSIDERATIONS FOR RUDDER DESIGN
The considerations for rudder design from hydrodynamic point of view are summarised as follows: (a)
Type of rudder and location
The type of rudder, its location and relative placement have significant influence on rudder effectiveness and ship controllability. Ideally, rudders should be located near the stern and should be located in the propeller stream for good controllability. Theoretically and from experience it can be shown that for dynamically stable forward moving ship at all speeds except dead slow, lateral control forces should be exerted at the stern and not at the bow. The formula for a ship’s dimensionless turning rate as derived from linear equation of motion for dynamically stable ships is Turning rate = where
L = R
⎡ Y ν ′ N δ ′ − N ν ′ Y δ ′ ⎤ = δ R ⎢ ⎥ R ⎣ Y ν ′ N r ′ − N ν ′ (Y r ′ − Δ′) ⎦ L
ship length
= turning radius
δ R =
rudder angle.
With conventional rudder location at the stern, the dimensionless turning rate is proportional to the sum of the magnitudes of the two numerator terms. But if the rudder is located at the bow, the sign of the factor N δ ′ is reversed, and the turning rate is then proportional to the difference in magnitudes of the two terms.
The physical effect of locating the rudder at stern and bow for ahead motion is illustrated in Fig 1.
Fig. 1 Effect of location of steering force [1]
When combined with forward ship motion these actions generate drift angle in the same direction, and drift angle brings into play the large hydrodynamic side force and consequent yaw moment that actually causes the turning. If, instead, the lateral control force acts at the bow, the contributions to drift angle due to yaw rotation and lateral translation are in
opposite directions and tend to cancel each other. Because both contributions are large, their difference is small, and turning rate is much smaller than in the rudder-aft case.
Locating rudders at the stern in the propeller race takes advantage of the added velocity of the race both at normal ahead speeds and at zero ship speed. This advantage is significant and may not require any increase in propulsion power over what would be required if the rudder were not in the race. The reason for this fortunate circumstance is that a properly shaped rudder in the race can recover some of the rotating energy of the race, which would otherwise be lost. There are, however, some negative aspects associated with locating a rudder in the propeller race. One is the possibility of rudder-induced ship vibration. For this reason, clearances of one propeller radius or more are common between the propellers and rudders of high-powered ships.
Submarines have horizontal bow planes and stern planes to control their motion in the vertical plane. Bow planes are moderately effective in this case because they either extend beyond the hull lines or are located on a superstructure above the main hull and hence do not interact too unfavourably with the hull. Bow planes extending beyond the hull lines are usually made retractable. The primary function of bow planes is to improve control at low speed at periscope depth under a rough sea. In the case of submarines that are very unsymmetrical about the xy -plane, bow planes are also useful to control depth at very low speeds deeply submerged; in this case the stern planes can cause ambiguous effects for reasons associated with the existence of the hydrostatic moment, M θ .θ .
Fig. 2 shows some of the major rudder type available to the designer. These are
•
All movable rudder
•
Horn rudder
•
Balanced rudder with fixed structure
•
All moveable rudder with tail flap
Each of these types has been used as single or multiple rudders or single and multiple screw ships.
Fig. 2 Various rudder arrangements [1]
All movable rudders are desirable for their ability to produce large turning forces for their size. With the possible exception of large fast ships, the all-moveable rudder is preferred for ships that possess control fixed stability without a rudder. For ships that are unstable without a rudder, the rudder area needed to achieve control-fixed stability may be larger than that necessary to provide the specified course-changing ability. In such cases, the horn rudder or balanced-with-fixed structure rudder is an attractive alternative to the all-moveable rudder. This is because the total (fixed plus moveable) rudder area of either of these rudders can be adjusted independently to provide the necessary controls-fixed stability. On the other hand, the moveable area can be adjusted independently to provide the required manoeuvring characteristics. The minimum total area generally satisfies the constraint (b) but not necessarily the constraint (c) of Section 1. The minimum moveable area should satisfy the constraint (c) of Section 1.
The main drawback of the all-moveable rudder is from structural considerations. Unless structural support is provided to the bottom of the rudder, the rudder stock of an allmoveable rudder has to withstand large bending moment as well as torque moment. The
bottom-supported type of rudder was common on slow and medium speed single-screw merchant vessels. But its use is avoided on high-speed ships as the cantilevered support is a potential source of vibration and its contribution to the support of the rudder may be structurally complicated.
The rudder stock size tends to become excessive on large fast ships. On these ships, a reduction in required rudder stock size can be achieved by extending the lower support bearing down into the rudder as far as practicable, or by the use of horn rudder or balanced rudder with fixed structure.
The bending moment on the stock for these rudders is
considerable reduced because bearing support is provided close to the span-wise location of the centre of pressure of the rudder. The horn rudder is also favoured for operation in ice.
Table 1 gives a rough first guide in selecting the balance ratio based on the block coefficient (CB) of the ship. The balance ratio is defined as
Area of rudder forward of the rudderstock total rudder area
Table 1 Balance ratio CB
Balance ratio
0.6
0.250 – 0.255
0.7
0.256 – 0.260
0.8
0.265 – 0.270
The preferred location of the rudder should aft of the propeller at the stern. Unless necessary, combinations such as single rudder with twin screws or single screw with twin rudder should be avoid as per as possible. At zero or low speed the propeller slip-stream increases the effectiveness of the rudder. The stern rudder is also more effective than a rudder placed at bow for manoeuvring ahead where as the bow rudder will be more effective in astern manoeuvring. The reason for this is the direction of drift angle which makes substantial contribution to the turning of the ship when the rudder is located aft.
(b)
Area, Size and Height of rudder
A suitable rudder area for a given hull form is to be selected so as to satisfy the desired level of dynamic stability and manoeuvring performance in calm water. Ships having higher block coefficient are less stable and therefore require larger rudder area for meeting stability requirements. It may also be noted that larger rudder areas have better performance under adverse conditions of wind and wave. The rudder area should be calculated and verified during the initial design stage. The proposed DnV formula for calculating the minimum rudder area is given as:
A R
L × T ⎡
⎛ B ⎞ ⎢ 1 + 25 ⎜ ⎟ 100 ⎢⎣ ⎝ L ⎠
≥
L × T ⎛ B ⎞
≥
⎜ ⎟ 15 ⎝ L ⎠
3
4
for
2
L B
⎤ ⎥ ⎥⎦ = 5 to 8
where A R
=
rudder area
T
=
draught
L
=
length between perpendicular
B
=
breadth of the ship
The above formula is to be used for aspect ratio ( AR ) of rudder around 1.6. If the aspect ratio of rudder is less than 1.6, the rudder area is increased by a factor given by
⎛ 1.6 ⎞ ⎜ ⎟ ⎝ AR ⎠
1
3
to
⎛ 1.6 ⎞ ⎜ ⎟ ⎝ AR ⎠
1
2
The DnV formula applies only to rudder arrangement in which then rudder is located the directly behind the propeller. For any other arrangement the DnV suggests an increase the rudder area of at least 30 per cent. The value of rudder should be compared with existing rudder areas for similar ship type and size. A table giving rudder area coefficients for different vessels is given.
Table 2 : Rudder area coefficients Sl. No.
Vessel Type
Rudder area as a percentage of L × T
1
Single screw vessels
1.6 – 1.9
2
Twin screw vessels
1.5 – 2.1
3
Twin screw vessels with two rudders (total area)
2.1
4
Tankers
1.3 – 1.9
5
Fast passenger ferries
1.8 – 2.0
6
Coastal vessels
2.3 – 3.3
7
Vessels with increased manoeuvrability
2.0 – 4.0
8
Fishing vessels
2.5 – 5.5
9
Sea-going vessels
3.0 – 6.0
10
Sailing vessels
2.0 – 3.0
A large number of potential manoeuvring troubles can be avoided by providing a margin of extra rudder area at the preliminary design stage. For some vessels the benefit of larger rudder area will diminish after the rudder area becomes greater than 0.2 × L T . The effectiveness of larger rudder area is directly dependent on the inherent dynamic course stability of the vessel. A vessel with positive inherent dynamic course stability will benefit least with increase of rudder area whereas vessel with instability will benefit most from increased rudder area.
The rudder height is usually constraint by the stern shape and draught of the vessel. However it is desirable to increase the height as much as possible so as to obtain a more efficient high aspect ratio for a given rudder area. The bottom of the rudder is kept just above the keel for protection. A higher value of the bottom clearance is preferred for vessels having frequent operations with trim by stern. Recommended propeller, hull and rudder clearances as given by LRS are shown in Fig 3.
Fig. 3 Propeller clearances [3]
(c)
Section Shape
For a given rudder location and rudder area, the choice of the chord wise section shape is governed by the following considerations:
•
Highest possible maximum lift
•
Maximum slope of the lift curve with respect to the angle of attack
•
Maximum resistance to cavitation
•
Minimum drag and shaft power
•
Favourable torque characteristics
•
Ease of fabrication.
A relatively higher thickness to chord ratio section shapes like NACA0018 and NACA0021 are preferred. This is because these sections have a relatively constant centre of pressure. Thicker sections offer reasonable drag characteristics and are also preferred from structural considerations.
The trailing edge of rudder has a noticeable thickness rather than taper to a knife-edge. This allows increased ruggedness of construction and is also beneficial for astern operations.
NACA section having any desired maximum thickness t , can be obtained multiplying the basic ordinates by the proper factor as follows:
± yt =
t
(0.2969 0.20
2 3 4 x − 0.126 x − 0.3516 x + 0.2843 x − 0.1015 x
)
where x is the chord length expressed in fraction of chord length along x -axis from 0 to 1. Fig. 4 shows the basic rudder foil chord wise cross-section with a table of ordinates for a rudder having a thickness of 20% of the chord.
Fig. 4 Basic ordinates of NACA family airfoils
(d)
Rudder deflection rate
The classification societies and regulatory agencies prescribe a minimum rate of 2 1 3 deg/sec and this value is independent of ship parameters. Whereas the design rudder deflection angle decides the desired steady turning characteristics, the transient manoeuvres (those manoeuvres in which the period of time the rudder is in motion is relatively long compared to the total manoeuvre time) determine the rudder deflection rate. The quickness of response in yaw and overshoot improve at increased rudder deflection rates. However, beyond a certain rate further improvements in transient manoeuvres are insignificant. The effect of an increase over the prescribed minimum of 2 1 3 deg/sec is the greatest on fast and response vessels. Large full-form ships benefit more from having large rudder areas than from an increase in rate of swing.
(e)
Maximum rudder deflection angle
The maximum rudder deflection angle could be
•
The maximum angle to which the steering gear can turn the rudder, i.e. the design maximum rudder angle
•
The maximum angle specified to be used for a particular manoeuvre, i.e. the manoeuvre maximum rudder angle
•
The maximum rudder deflection angle which when exceeded yields no significant improvements in the characteristics of the manoeuvre, i.e. maximum useful rudder deflection angle.
The maximum useful rudder deflection angle will decide the design maximum rudder angle and the manoeuvre maximum rudder angle.
Rudders experience a loss of lift at stall angles. Therefore, the maximum useful rudder angle will likely be just lower than the stall angle. However, the maximum useful rudder deflection may exist at angles of attack less than that of the stall angle.
Fig. 5 Orientation of ship and rudder in a steady turn to starboard
The possibility of the rudder achieving an angle of attack exceeding the stall angle is most likely during transient manoeuvres such as overshoot manoeuvre rather than during a steady turn. For example when a rudder is laid over in the opposite direction to check overshoot manoeuvre the angle of attack on the rudder may be larger than the deflection angle if the rudder deflection rate is very fast. On the other hand, during a steady turn the angle of attack on the rudder is far less than the deflection angle. Thus the useful maximum rudder deflection angle is likely to be far greater in steady turn than that of overshoot manoeuvre. The magnitude of the maximum rudder deflection angle will in almost all cases be determined by steady turn considerations. The angle of attack at the rudder during steady turn is (see Fig. 5)
α = δ R − β R where
δ R
=
rudder deflection angle
β R
=
actual drift angle at the rudder.
The geometric drift angle at the rudder is given by
β = β R + ε where ε = angle due to straightening influence of hull and propeller on the flow to the rudder.
The geometric drift angle is a function of the radius of turning circle. For a rudder located at a distance L 2 aft of the origin, β R + ε is related to the drift angle at the origin of the ship, β , by tan ( β R + ε ) =
tan β +
1 2 R cos β
where L is the length of the ship and R is the turning circle radius
Measurements of β made during the turning experiments of single-screw merchant ships models reported indicate
β ≈ 22.5 L
(where β is in degrees)
R
The straightening effect of the hull and propellers on the rudder is approximately a linear function of the geometric drift angle in the rudder, i.e.
ε ≈ j ( β R + ε ) ε =
j
1 − j
for 0 ≤ j ≤ 1
β R
Combining the preceding equations the following expression of angle of attack on rudder during steady turn is obtained.
⎡ ⎢ α = δ R − ⎢ tan − 1 ⎢ ⎢⎣
⎧ ⎫⎤ ⎪ ⎪⎥ L L tan 22 . 5 + ⎨ ⎬⎥ R L ⎛ ⎞ ⎪ 2 R cos ⎜ 22.5 ⎟ ⎪⎥ R ⎝ ⎠ ⎩ ⎭⎥⎦
(
)
(1 − j )
In most cases the steering gear capabilities tend to impose an upper limit on the rudder deflection angle, which is independent of turning considerations. Certain types of steering gears may not be suitable for mechanical reasons for deflection angles larger than 35 degrees. Most naval ships and Great Lakes ships are built with design maximum rudder angles up to 45 degrees.
3.
RUDDER DESIGN
The process of rudder design is usually conducted in two parts:
•
Selection of the geometric parameters and turning rate necessary to develop the desired ship characteristics, and
•
Calculations of torque loadings on the arrangement including the steering gear that must control the rudder movements
Determination of the hydrodynamic forces and torque on the rudder as the hull turns requires an accurate assessment of:
•
Hull wake
•
Hull drift angle
•
Change in the rudder angle of attack as the hull turns
In addition we need to know
•
The frictional losses in the rudder stock bearings
•
Steering gear drive mechanism
The computation of rudder-stock size requires knowledge of: (a)
The maximum design value of the resultant force on the rudder
(b)
The location of the span-wise centre of pressure
corresponding to the maximum
resultant force (c)
The location of the rudder bearings
The computation of the rudder-stock location and steering-gear torque for all the rudder requires knowledge of: (d)
The rudder normal force F and the location chord-wise centre of pressure (CP ) c as a function of rudder angle of attack at the maximum ship speed.
(e)
Bearing radii and coefficients of friction.
On ships that have no restrictions on the use of rudder in going astern, items (a), (b) and (d) have to be known for both maximum ahead and maximum astern speeds. On recent naval ships, there has been recognition of the following:
•
Typical combatant ships have large astern powers and hence are capable of correspondingly high astern speeds.
•
Adequate design for that astern speed would require large, heavy steering gear.
•
It seems reasonable to allow use of full astern power for crash stops, but there is no need to go astern at high speed after stopping.
•
Accordingly, instruction plates are provided limited the sustained astern shaft rotational speed to that which permits steering gear operation within the ahead limits.
•
The acceptance trials include demonstration of the workability of the Instruction Plate limit.
Thus, design practice for naval combatant ships bases the calculation of rudder-stock size, location, and steering gear torque on the ahead conditions.
Empirical formulas and experimental data are used for estimating rudder forces and moments. Details for computation of rudder forces and torque for spade and horn rudders have been given Harrington [2]. However, in order to use free stream data to compute the maximum design value of normal force, assumptions have been made concerning:
3.1
•
The maximum angle of attack the rudder is likely to encounter, α max .
•
The maximum flow velocity averaged over the rudder, . (V R ) max
•
Rudder effective aspect ratio, a .
Rudder Torque Calculations for a Spade Rudder – Ahead Condition
Number of rudders
=
Length on waterline (L)
=
m
Draught (mean) (T)
=
m
Max. Design Speed Ahead (V)
=
knots
⎛
⎞
V ∗ 0.5144 ⎟ Froude Number ⎜ F n = ⎜ 9.81 ∗ L ⎠⎟ ⎝
=
Thrust deduction fraction (t)
=
Wake fraction ( w )
=
Total appendaged resistance at V (R T)
=
Newtons
Density of water ( ρ )
=
kg/m
Propeller Diameter (D)
=
m
Maximum Astern Speed (V astern)
=
knots
Design speed ahead V = 0.5144 ∗ V knots
=
m/s
Speed of Advance (V A ) = V (1 − w)
=
m/s
=
Newtons
Propeller Thrust ( T ) =
RT
1 − t
3
Dynamic Pressure ( p ) =
1 2
ρ v A2
4 T
+
π D
2
=
2
N/m
Rudder angle of deflection (δ )
=
degrees
Rudder angle of attack α = M ∗ δ
=
degrees, where M =
o
o
o
o
o
o
o
o
o
o
Variation of δ to be considered
=
7 , 14 , 21 , 28 , 35
Variation of α to be considered
=
5 , 10 , 15 , 20 , 25
⎡ 2 (35 − δ )⎤ +⎢ 7 ⎣ 7 35 ⎥⎦ 5
Fig. 6 Stern arrangement and support details of spade rudder
The stern contour and propeller position must be available. Rudder shape, rudder stock centre line location and distribution of rudder area forward and aft of stock centreline must be determined as has been discussed before and the rudder diagram prepared similar to Fig. 6. Once the rudder geometry is known the following quantities must be noted in meters: X 1, X 2 , X 3 , X 4 , X 5 , X 6 , X 7 , X 8 and X 9 . The diameter of lower and upper stock bearings d 1 and d 2 in meters and type of bearing must also be noted from Fig. 6. Taper Ratio
:
λ =
X 5
( X 1 + X 2 )
Mean chord
:
c = 0.5 ( X 1 + X 2 + X 5 )
Sweep angle
:
⎡ 0.25 ( X 5 − X 1 − X 2 ) + X 2 − X 4 ⎤ Ω = tan −1 ⎢ ⎥ X 3 ⎣ ⎦
Rudder deflection angle in degrees
:
δ
Rudder angle of attack in degrees
:
α = M .δ
Rudder deflection angle in degrees
:
δ = 61.25 − 3751.5625 − 122.5α
Effective aspect ratio
:
a=
Data for uncorrected taper ratio
:
X 3 ⎡
⎢2 −
C ⎣
•
Lift coefficient (see Figs 7 and 8)
:
C L 1
•
Drag coefficient (see Figs 9 and 10)
:
C D 1
•
Centre of pressure (see Figs 11 and 12) :
where, M =
⎡ 2 ⎛ (35 − δ ) ⎞⎤ +⎢ ⎜ ⎟⎥ 7 ⎣ 7 ⎝ 35 ⎠⎦ 5
α ⎤
75 ⎥⎦
CPC 1
Lift coefficient C L 1 , drag coefficient C D 1 and centre of pressure CPC 1 can now be determined for various α values and the effective aspect ratio a for sweep angle Ω = 0 and 11 degrees from the graphs given in Figs. 7, 8, 9, 10, 11 and 12. Lift coefficient correction
:
1.63 λ − 0.73 ⎡ α ⎤ Δ C L = ⎢⎣ 57.3 ⎥⎦ a
Corrected life coefficient
:
C L 2 = C L 1 + Δ C L
2
C L2 2 − C L2 2
Drag coefficient correction
:
Δ C D =
Corrected drag coefficient
:
C D 2 = C D 1 + Δ C D
Uncorrected normal hydrodynamic coefficient
:
C N 1 = C L1 cos α + C D1 sin α
Corrected normal hydrodynamic coefficient
:
C N 2 = C L 2 cos α + C D 2 sin α
2.38 a
(
)
C M C 4 = 0.25 − CPC C N 1 − 2
1 2
Δ C L
0.25 − C M C 4
=
Corrected centre of pressure
:
CPC 2
Normal hydrodynamic force
:
F = p . c . X 3 . C N 2
Hydrodynamic torque
:
Q H = F ⎢c . CPC 2 −
Rudder stock bearing friction
:
QF
QF = μ 1 .
d 1
2
2
C N 2
⎡ ⎣
X 2 + X 4 ⎤
⎥⎦
2
⎡ 0.42 X 3 + X 9 ⎤ ⎡ 0.42 X 3 + X 8 + X 9 ⎤ d 2 + μ F . . ⎥ ⎢ ⎥ 2 2 X 8 X 8 ⎣ ⎦ ⎣ ⎦
. F ⎢
Rudder torque (displacing)
:
Q D = QF + Q H
Rudder torque (restoring)
:
Q R = QF − Q H
The coefficient of friction μ is given as = 0.01 for roller bearing = 0.1 to 0.2 for phenolic bearing = 0.05 to 0.1 for bronze bearing Thus, F , Q H , QF , Q D and Q R can be calculated and tabulated for various angles of attack a
The maximum bending moment (Q BM ) max and hydrodynamic torque (Q H ) max acting on the rudder can be computed as follows :
(Q H ) max = F d − CPc )
:
(Q BM ) max = ( L2 + D 2 )
1
2
( CP
s
+ b)
where F , L , D , CPc and CPs are determined at α = α max and maximum speed and b is the distance from the root chord of the rudder to the centre of the lower bearings supporting he rudder.
If the chordwise centre pressure on a rudder remained in a fixed location as the angle of attack on the rudder increased, it would be desirable to locate the rudder stock just forward of the centre of pressure. This would insure a low maximum torque value and in the event
that the rudder were inadvertently freed, the rudder would tend to trail at δ R = 0 deg as long as α = 0 deg. Unfortunately, on most rudders the centre of pressure moves aft as the angle of attack increases. Therefore, in order to reduce the maximum torque value, most ship rudders are not designed as trailing rudders. The practice is to determine the location of the stock on the basis that the hydrodynamic torque should be zero at an angle of attack of about 10 to 15 deg. A typical torque versus angle of attack curve takes the form as shown in Fig. 13. Therefore, if the zero point were taken at a larger angle of attack, the maximum torque at α = ± α max could be significantly reduced. The 10 to 15-deg zero point torque is used to minimise the power required for routine steering and course keeping, which on most ships seldom requires more than 10 to 15 degrees of rudder angle.
It can be seen from Fig. 13 that such a rudder is unstable at α = 0 deg. If the rudder was free at this point it would flip over to either ± 15 deg port or starboard. This instability may produce rattling, shock and excessive wear in gear mechanisms. Some designers therefore recommend that the rudder stock should be located at ( α = 0 deg) position of centre of pressure. However, this recommendation will lead to a requirement for a larger capacity steering gear.
Fig. 7 Lift coefficient, sweep angle 0 deg [2]
Fig. 8 Lift coefficient, sweep angle +11 deg [2]
Fig. 9 Drag coefficient, sweep angle 0 deg [2]
Fig. 10 Drag coefficient, sweep angle +11 deg [2]
Fig. 11 Chordwise centre of pressure, sweep angle 0 deg [2]
Fig. 12 Chordwise centre of pressure, sweep angle +11 deg [2]
Fig. 13 Typical torque versus angle of attack relationship
3.2
Rudder Torque Calculations for a Horn Rudder – Ahead Condition
Number of rudders
=
Length on waterline (L)
=
m
Draught (mean) (T)
=
m
Max. Design Speed Ahead (V)
=
knots
⎛
⎞
V ∗ 0.5144 ⎟ Froude Number ⎜ F n = ⎜ 9.81 ∗ L ⎠⎟ ⎝
=
Thrust deduction fraction (t)
=
Wake fraction ( w )
=
Total appendaged resistance at V (R T)
=
Newtons
Density of water ( ρ )
=
kg/m
Propeller Diameter (D)
=
m
Maximum Astern Speed (V astern)
=
knots
Design speed ahead V = 0.5144 ∗ V knots
=
m/s
Speed of Advance (V A ) = V (1 − w)
=
m/s
=
Newtons
=
N/m
Rudder angle of deflection (δ )
=
degrees
Rudder angle of attack α = M ∗ δ
=
degrees, where M =
Propeller Thrust (T) =
Dynamic Pressure (p) =
RT
1 − t 1 2
ρ v A2
+
4 T π D
2
3
2
o
o
o
o
o
o
o
o
o
o
Variation of δ to be considered
=
7 , 14 , 21 , 28 , 35
Variation of α to be considered
=
5 , 10 , 15 , 20 , 25
⎡ 2 (35 − δ )⎤ +⎢ 7 ⎣ 7 35 ⎥⎦ 5
The stern contour and propeller position must be available. Rudder shape, rudder stock centre line location and distribution of rudder area (fixed and moveable) forward and aft of stock centreline must be determined as has been discussed before and the rudder diagram prepared similar to Fig. 14. Once the rudder geometry is known the following quantities must be noted in meters: X 1 , X 2 , X 3 , X 4 , X 5 , X 6 , X 7 , X 8 , X 9 and X 10 .
Fig. 14 Stern arrangement and support details of spade rudder
The diameter of lower and upper stock bearings d 1, d 2 and d 3 in meters and type of bearing and their corresponding bearing friction must also be obtained Upper stock outer diameter d 1
=
m
Lower stock outer diameter d 2
=
m
Rudder pintle outer diameter d 3
=
m
Upper stock bearing type
=
Upper stock bearing fraction coeff. μ 1
=
Lower stock bearing type
=
Lower stock bearing fraction coeff. μ 2 Rudder pintle bearing type
=
Rudder pintle friction coeff. μ 3
=
Lower Rudder Section X 5
Taper Ratio
:
λ =
Mean chord
:
c = 0.5 ( X 1 + X 2 + X 5 ) :
Sweep angle
:
⎡ 0.25 ( X 5 − X 1 − X 2 ) + X 2 − X 4 ⎤ Ω = ⎢ ⎥ X 3 ⎣ ⎦
Aspect ratio
:
a1 =
( X 1 + X 2 )
X 3 ⎛ X 2 α ⎞ ⎜⎜ 2 − ⎟ . X 1 + X 2 75 ⎠⎟ c ⎝
Data for uncorrected taper ratio :
•
Lift coefficient (see Figs 7 and 8)
:
C L 1
•
Drag coefficient (see Figs 9 and 10) :
C D 1
•
Centre of pressure (see Figs 11 and 12):
CPC 1
Lift coefficient C L 1 , drag coefficient C D 1 and centre of pressure CPC 1 can now be determined for various α values and the effective aspect ratio a for sweep angle Ω = 0 and 11 degrees from the graphs given in Figs. 7, 8, 9, 10, 11 and 12. Lift coefficient correction
:
1.63 λ − 0.73 ⎡ α ⎤ Δ C L = ⎢⎣ 57.3 ⎥⎦ a1
Corrected life coefficient
:
C L 2 = C L 1 + Δ C L
2
C L 2 − C L 2 2
2
Drag coefficient correction
:
Δ C D =
Corrected drag coefficient
:
C D 2 = C D 1 + Δ C D
Uncorrected normal hydrodynamic coefficient
:
C N 1 = C L1 cos α + C D1 sin α
Corrected normal hydrodynamic coefficient
:
C N 2 = C L 2 cos α + C D 2 sin α
2.38 a
(
)
C M C 4 = 0.25 − CPC C N 1 − 2
Corrected centre of pressure
:
CPC 2
=
0.25 − C M C 4 C N 2
2
1 2
Δ C L
Normal hydrodynamic force
:
F l = p . c . X 3 . C N 2
Hydrodynamic torque
:
Q H l = F ⎢c . CPC 2 −
Rudder stock bearing friction
:
QF l
QF l = μ 3 .
d 3
2
⎡
. F l ⎢1 +
⎡ ⎣
X 2 + X 4 ⎤
2
⎥ ⎦
0.42 X 3 + X 12 ⎤
⎣
X 10 + X 11
F l ⎡ 0.42 X 3 − X 12 ⎤ ⎥ + ⎢ ⎥ [μ 2 . d 2 ( X 8 + X 9 ) + μ 1. d 1 X 9 ] X X + X 2 ⎦ ⎦ 8 ⎣ 10 11
Upper Rudder Section
Mean chord
:
cu =
Normal force coefficient (see Fig. 15)
:
C Nu
X 1 + X 6
2
Hinge moment coefficient (see Fig. 15) :
C HM
Normal hydrodynamic force
:
F u = p . c u . X 7 . C Nu
Hydrodynamic torque
:
Q H u = p cu2 X 7 C HM
Bearing friction
:
Q F u
QF u = μ 3 .
d 3
2
⎡
. F u ⎢1 −
0.42 X 7 + X 12 ⎤
⎣
X 10 + X 11
F u 0.42 X 7 + X 12 + ⋅ [μ 2 . d 2 ( X 8 + X 9 ) + μ 1 . d 1 X 9 ] ⎥ + X X X 2 8 10 11 ⎦
Total Rudder Section
Hydrodynamic torque
:
Q H = Q H l + QH u
Bearing friction
:
QF = QF l + QF u
Single ram correction
:
r =
Rudder torque (displacing)
:
Q D = QF + Q H + r QF + Q H
Rudder torque (restoring)
:
Q R = QF − Q H + r QF − Q H
μ 1 d 1 cos α 2 R
Fig. 15 Hinge moment and normal force coefficients of rudder area abaft horn [2]
Fig. 16 shows a graph of the torque elements ( Q H , QF , Q D and Q R ) during a simple manoeuvre. As can be seen, the frictional torque is significant. The Q D curve represents the sum of frictional and hydrodynamic components. Movement of the rudder from centreline would entail torques following the curve until the ordered angle is reached at Point a. The rudder is then held in position by the hydraulic ram pressure, and small movements tend to dissipate the effects of friction in making the transition to Point b on the Q H curve. A drift angle is assumed by the vessel, causing movement to Point c . If the rudder is then ordered to the centreline, the process works in reverse from the − Q H curve moving to the Q R curve.
Fig. 16 Rudder torque elements during a simple manoeuvre
3.3
Astern Torque Calculations – Joessel Method
Based on experiments conducted in the Loire river (having a maximum current of 1.3 m/s) with rectangular plate of span 30 cm and chord 40 cm, Joessel derived empirical relationships for the variation of torque and variation of centre of pressure with the angle of attack. These relationships, when corrected for larger density of sea water, are as follows: Q = 418.122 + A v 2 w sin α
and
x w
= ( 0.195 + 0.305 sin α )
where, Q
=
rudder torque about leading edge of the plate in N ⋅ m
A
=
area of the plate in m 2
v
=
2 velocity of water in m sec
w
=
width of the plate in m
α
=
angle of attack in deg
x
=
distance of centre of pressure from leading edge in m
By combining the above two equations, the resultant force on the plate is determined to be: F =
Q 418.122 A v 2 sin α x 0.195 + 0.305 sin α
The horn type rudder shown in Fig. 17 can be transformed into two rectangles as shown by the dashed lines. By applying the foregoing equations along with an inclusion of a Joessel coefficient, expressions for ahead and astern torque become as follows:
⎡⎛ w1 (0.195 − 0.305 sin α ) − b ⎞ ⎤ ⎟⎟ w1 h1 + w22 h2 ⎥ ⎣⎝ 0.195 − 0.305 sin α ⎠ ⎦
(Q H ) ahead = K ahead 418.122 v sin α ⎢⎜⎜ 2
(Q H ) astern = K astern
418.122 v 2 sin α 0.195 − 0.305 sin α
[(a − w (0.195 − 0.305 sin α )) w h 1
1 1
+ w22 h2 (0.805 − 0305 sin α )]
where K ahead and K astern are the Joessel coefficients or the experience factors
Compute the hydrodynamic torque in astern condition using the above Joessel’s formula for different rudder angles of attack α . Compute the normal hydrodynamic force on the lower and upper rudder sections in astern condition, i.e., ( F l ) astern and ( F u ) astern from the above equation.
Fig. 17 Model of a horn-type rudder used with Joessel method [2]
3.4
Steering Gear Torque and Power
Total steering gear torque
QT = Q H + QF + Q A
:
where, Q H
=
hydrodynamic torque
QF
=
bearing frictional torque
Q A
=
error allowance =
F max × 0.02 c
The error allowance for both lower and upper rudder section to be calculated and summed up. Calculate (QT ) ahead and (QT ) astern for all angles of attack.
Select the maximum torque (QT ) max = Maximum of { (QT ) ahead and (QT ) astern } Rudder deflection rate ( s ) in radians /sec is defined as s
=
=
deflection angle from hardover port to hardover starboard time required to move from hardover port to hardover starboard
2 δ max t
⋅
π 180
where, δ max
=
maximum rudder deflection on either port or starboard side
t
=
time require in seconds
The regulatory class requirements for minimum deflection rate ( s ) is 2 1 3 radians /sec. Power required for steering gear :
P=
(QT ) max s 1000η g
kW
where η g is the steering gear efficiency ≈ 0.75 to 0.85 (see Fig. 18).
Fig. 18 Efficiency of a Rapson-slide steering engine
3.5
Rudder Stock Diameter (Lloyds Rule Part 3 Ch 13)
Basic stock diameter, δ S at and below lowest bearing (ahead or astern) is given by: δ S to be greater of the following:
(a)
(b)
δ SF
δ SA
= 83.3 K R ⎡(V F + 3)2 ⎢⎣
V F
=
maximum service speed, in knots, in loaded condition
V A
=
actual astern speed, in knots, or 0.5V F , whichever is greater
A R
=
rudder area in m 2
K R
=
rudder coefficient
K R
=
0.248 for ahead condition and rudder in propeller ship stream
K R
=
0.185 for astern condition
N
=
coefficient dependent on rudder support.
N
=
A1
X PA , X PF
=
horizontal distance in m
(
2 X PF
+ N
(
2 X PA
+ N
2 A R
2 A R
2
2
)
⎤ ⎥⎦
)
⎤ ⎥⎦
0.333
= 83.3 K R ⎡(V F + 3)2 ⎢⎣
0.5
0.5
mm
(ahead)
0.333
mm
(astern)
where
(0.67 Y 1 + 0.17 Y 2 ) − A2 (Y 1 + 0.5 Y 3 )
(Refer to Fig. 19)
Astern condition X PA =
horizontal distance from centerline of rudder pintle to centre of pressure
Ahead condition X PF =
horizontal distance from centerline of rudder pintle to centre of pressure 0.12 A R as calculated but not less than , Y R
X PA , X PF
where Y R is the depth of rudder at stock centerline in metres.
Fig. 19 Rudder area distribution for stock diameter computation
3.9
Estimation of Turning Circle Diameter [4]
The procedure described below is given in Ref [4]. The results obtained from this procedure have been compared with field trials of fast vessels of both displacement and planing type. When a vessel takes a turn, the steady turning speed U C is less than the steady forward speed U A at that engine power. For estimating the reduced speed U C , the following procedure is followed: 1.
Estimate the resistance to forward motion of the craft including all appendages with rudder held at ship centerline position over the desired speed range.
2.
Estimate the drag of the rudder ( s ) at various rudder angles over the desired speed range as per the procedure described earlier.
3.
Estimate the rudder drag at zero angle using the ITTC friction line for frictional drag. Then, calculate the increment in rudder drag at various angles of deflection (or attack) over the drag at zero angle.
4.
Augment the appendage craft resistance to account for added drag due to yaw and heel of the craft in a turn. Since the effective angle of attack α eff of a rudder during a craft turn is less than the geometric angle (α ) due to yaw of the craft, the following relationship is assumed: α eff = M . α
where α eff and α are in degrees and
M =
⎡⎛ 2 ⎞ ⎛ (35 − α ) ⎞⎤ + ⎢⎜ ⎟ ⎜ ⎟⎥ 7 ⎣⎝ 7 ⎠ ⎝ 35 ⎠⎦ 5
Then craft angle of yaw, φ is given by φ = α − α eff
The drag of the craft with appendages in yaw can be given by φ ⎤ ⎡ D φ = D ⎢1 + 0.075 ⎥ 5⎦ ⎣
where D is the drag of the craft with appendages at no yaw condition. 5
For each rudder angle considered, add increment of rudder drag to resistance of yawed craft in turn to obtain the total resistance of craft in turn for that forward speed U A and rudder angle α .
6.
Draw the new speed power curve EHP versus U C assuming the same engine power for the corresponding speeds U A and U C .
Once U C has been estimated, the turning circle radius RC can be estimated from the following relationship:
L R C
⎡( =⎢ ⎢⎣ (
U A2
−
2 U C 2 K C U C
)⎤
) ⎥⎥⎦
1 2
where L RC UA UC KC
= = = = =
Craft length in feet, Steady turning radius in feet, Vessel forward speed in ft/sec, Vessel steady turning speed in ft./sec, an empirical constant which is 30 K C = F n2∇ × α
where F n2∇
=
Displacement Froude number, which is F n ∇ =
U A 1
⎡ ⎤2 3 ⎢g ∇ ⎥ ⎢⎣ ⎥⎦ 1
