Rudder Design

Rudder Design

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6 Rudder Design

6.1 Assumptions and Equations Used in the Design and Analysis of the Rudders.

The rudder should have as small an area as possible to minimise minimise drag. However, if the rudder is too small the sailor will loose control of the boat at low speeds. Determining an acceptable rudder size is therefore an empirical exercise. A standard rudder design shall be analysed to provide acceptable characteristics. An equivalent rudder can then be designed that will also give an acceptable level of control. In the analysis a nalysis of the rudder desig de signs, ns, the following assumptions and equations are used:All rudders have an elliptical spanwise chord distribution and do not twist or deflect.  b = minor axis of the foil a = half the major axis of the foil

S = area of the foil S=pab/4 ARg = geometric aspect ratio ARg = a 2 / S

The effective aspect ratio of the rudder ARe is assumed to be twice the geometric aspect ratio due to the free surface boundary. Although the free surface is not a solid boundary it shall be assumed to mirror the rudder  and hence ARe = 2 ´ ARg = 2 ´ a 2 / S

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Rudder Design

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A transverse foil at the tip of the rudder (used to increase longitudinal stability - see Chapter 3 ) will increase the effective aspect ratio of the rudder. From figure 6.1, The foil increases the effective aspect ratio by approximately 1.3. \ ARe = 2.6 a 2 / S if the elliptical rudder has a transverse foil at its tip. The transverse foil will increase increase the t he total tota l drag of the rudder but this drag shall be ignored when rudder designs are compared. The rudders have elliptical plan forms so the following relationships are true. CL = k a / [ 1 + ( 2 / ARe )] where a = angle of attack  k = slope of the lift of the section with angle of attack graph = dL / da CL = coefficient of lift ARe = effective aspects ratio CDI = CL2 / p ARe CD = DDP + DDI where CD = total coefficient at drag CDI = coefficient of induced drag CDP = coefficient of profile drag evaluated from fr om graphs reprinted from "The Theory of Wing Sections" Sections" shown in figure 6.2 and 6.3 The maximum maximum section thickness is 12% of the t he cord length. "From the data for the NACA NACA four and fiv e digit  wing wing sections it appears appears that the maximum maximum lift coef ficients are the greatest for a thickness ration of 12 per  Sections. As the maximum lift is a function of area , and maximum lift lift coefficient coeff icient for  cent." Theory of Wing Sections. the minimum area, the sections with the maximum lift coefficients must be used.

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Rudder Design

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 b = 0.2m

S = p ´ 0.65 ´ 0.2 / 4 = 0.1021m 2 ARe = 2 ´ 0.65 2 / 0.1021 = 8.28 k = 1.3 / 12 CL = 1.3 a / [12 ´ ( 1 2 / 8.28 )] = 0.08726 a CDI = CL2 / ( p ´ 8.28 ) = 2.927´ 10-4 x a 2

Table 6.1

Standard rudder design lift and drag characteristics. a

CDP

CDI

CD

CL

CL/CD

0

0.0058

0

0.0058

0

0

2

0.006

1.171 x 10-3

0.00717

0.1745

24.3

4

0.0066

4.863 x 10-3

0.0113

0.3490

30.9

6

0.008

0.01054

0.0185

0.5235

28.3

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Rudder Design

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Design 1

Transverse foil fitted section = NACA 0012 a = 0.65 m  b = 0.20 m

S = p ´ 0.65 ´ 0.2 / 4 = 0.1021 ARe = 2.6 ´ 0.65 2 / 0.1021 = 10.75 k = 1.3 / 12 CL = 1.3 ´ a / [12 ´ (1 + 2 /10.75 )] = 0.09134 a CDI = CL2 / ( p ´ 10.75) = 2.470´ 10-4 x a

Table 6.2

Design Design 1 rudder lift and drag characteristics. charact eristics. a

CDP

CDI

CD

CL

CL/CD

0

0.0058

0

0.0058

0

0

2

0.006

9.881´ 10-4

6.988´ 10-3

0.1827

26.1

4

0.0066

3.953´ 10-3

0.01055

0.3654

34.6

2

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Rudder Design

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Design 2

Transverse foil fitted section = NACA 0012 a = 0.686 m  b = 0.189 m

S = p ´ 0.686 ´ 0.189 / 4 = 0.1021 ARe = 2.6 ´ 0.686 2 / 0.1021 = 12 k = 1.3 / 12 CL = 1.3 a / [ 12 ´ ( 1 + 2 / 12 )] = 0.09286 a CDI = CL2 / ( p ´ 12 ) = 2.287´ 10-4 x a 2

Table 6.3

Design Design 2 rudder lift and drag characteristics. charact eristics. a

CDP

CDI

CD

CL

CL/CD

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Rudder Design

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8

0.0095

0.01464

0.02414

0.7429

30.8

10

0.011

0.02287

0.03387

0.9286

27.4

Design 3

Transverse foil fitted section = NACA 64 - 012 a = 0.686 m  b = 0.189 m

S = p ´ 0.686 ´ 0.189 / 4 = 0.1021 ARe = 2.6 ´ 0.686 2 / 0.1021= 12 k = 0.9 / 8 CL = 0.9 a / [ 8 ´ ( 1 + 2 / 12 )] = 0.09643 a CDI = CL2 / p ´ 12 = 2.467´ 10 -4  2

Table 6.4

Design Design 3 rudder lift and drag characteristics. charact eristics.

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Rudder Design

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4

0.008

3.946´ 10-3

0.01195

0.3857

32.3

6

0.0095

8.879´ 10-3

0.01838

0.5786

31.5

8

0.012

0.01579

0.02779

0.7714

27.8

10

0.0145

0.02466

0.03916

0.9643

24.6

6.3 Rudder Design Discussion and Conclusions. It is clear from the graph in figure 6.2 comparing the designs that the effective aspect ratio has a large effect on the performance of the rudder. Of the designs presented, Design 2 is the best. Design 3 may be better when the angle of attack is below 3° (maybe 60% of the time) but NACA 64-012 section has a lower value of C L maximum; maximum; so for the same maximum lift the rudder would have to be larger. For this t his reason the NACA 0012 section would appear to be optimum. A limited limited factor fact or in the desig de sign n of the foils is is the structural struc tural consideration. considerat ion. Reducing b by 8% reduces the t he thickness of the foil by 8%. This means the second moment of area has decreased by (0.92 3) 22%. The centre of pressure has moved down the foil due to the increase in aspect ratio which increases the bending moment on the foil. The aspect ratio is therefore limited by structural considerations. It shall be assumed that an effective aspect ratio of 12 is achievable without an excessively heavy structure. The profile of the rudder should be modified to a crescent form as seen in "Letters to Nature". This type of   plan form is shown to have 4.3% less induced drag for 1.5% less lift than the standard rudder shape (with a straight trailing edge) at a = 4%. The theoretical theoret ical calculated values va lues of lift and drag are based ba sed on a plan form fo rm with a straight ¼ cord line (wing 1) and therefore must be modified to take account of plan form

Diagram of foil profiles

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Rudder Design

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Table 6.5

Correction factors for profile shape on rudder lift and drag coefficients. Wing

CDI

CL

f I

f L

1

0.00531

0.34157

1

1

2

0.00510

0.34236

0.96

1.002

3

0.00475

0.33714

0.89

0.987

The above data helps to support the analysis of the effect of angle of sweep in Chapter 5. Wing 3 has a larger  effective angle of sweep than Wing 1 and has less induced drag for less lift. Taking into account the profile of rudders, the lift and drag can be calculated as follows. L = ½ ´ r ´ V2 ´ S ´ f L ´ CL where r = density of water = 1025 kg / m 3 V = velocity taken as 5 knots = 2.87 m/s S = area of foil = 0.1021m 2 FL = correction factor 

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Rudder Design

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stronger winds where longitudinal stability is a problem.

Standard Rudder

(no transverse foil) ARe = 8.28 S = 0.1021 m2 section = NACA 0012

Table 6.6

Actual values of lift and drag for a Standard rudder at 5 knots. a

CDP

f I CDI

f L CL

D Newtons

L Newtons

0

0.0058

0

0

2.5

0

2

0.006

1.124´ 10-3

0.1748

3.1

75

4

0.0066

4.496´ 10-3

0.3497

4.8

151

6

0.008

0.01012

0.5245

7.8

226

8

0.0095

0.01798

0.6995

11.8

302

10

0.011

0.02810

0.8743

16.9

377

Design 2

(with transverse foil at zero angle of attack)

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Rudder Design

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D = ½ R V2 S ( CDP + f I CDI ) + 0.675

Table 6.7

Actual values of lift and dra g for a design 2 rudder with foils at 5 knots. a

CDP

f I CDI

f L CL

D Newtons

L Newtons

0

0.0058

0

0

3.2

0

2

0.006

8.143´ 10-4

0.1833

3.6

79

4

0.0066

3.257´ 10-3

0.3666

4.9

158

6

0.008

7.328´ 10-3

0.5499

7.3

237

8

0.0095

0.01303

0.7332

11.4

316

10

0.011

0.02035

0.9165

14.2

395

Design 2

(with no foil) Section = NACA 0012 S = 0.1021 m2 AR = 0.686 2/ 0.1021 = 9.22 CL = 1.3 a / [ 12 ´ ( 1 + 2 / 9.22 )] = 0.08902 a CDI = CL2 / ( p ´ 9.22 ) = 2.736´ 10 -4 a 2

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Rudder Design

10

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0.011

0.02435

0.8786

15.2

379

A graph comparing the calculated c alculated dat a above is shown in in figure 6.5. The tip of the foil will will have to be slightly slightly modified to accommodate t he transverse tra nsverse foil. The section may also have to be increased in thickness by using a NACA NACA 0015 section at the very tip. This will will make make the t he tip stronger and the foil less likely to break off.