Rudder Design

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

The rudder should have as

The rudder should have as small an area as possible to minimismall an area as possible to minimise drag. However, if the rudder se drag. However, if the rudder is too small theis too small the sailor will loose control of the boat at low speeds. Determining an acceptable rudder size is therefore an 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 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.

equivalent rudder can then be designed that will also give an acceptable level of control. In the a

In the analysis of the rudder denalysis of the rudder desigsigns, the following ns, the following assumptions and equations are assumptions and equations are used:- used:-All rudders have an elliptical spanwise chord distribution and do not twist or deflect. All rudders have an elliptical spanwise chord distribution and do not twist or deflect. b = minor axis of the foil

b = minor axis of the foil

a = half the major axis of the foil a = half the major axis of the foil

S = area of the foil S = area of the foil

S = p a b / 4 S = p a b / 4 ARg = geometric aspect ratio

ARg = geometric aspect ratio

ARg = a ARg = a22/ S/ S

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

and hence

ARe = 2 ´ ARg = 2 ´ a ARe = 2 ´ ARg = 2 ´ a22/ S/ S

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A transverse foil at the tip of the rudder (used to increase longitudinal stability - see Chapter 3 ) will increase 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

the effective aspect ratio of the rudder. From figure 6.1, The foil increases the effective aspect ratio by approximately 1.3.

approximately 1.3.

\ ARe = 2.6 a \ ARe = 2.6 a22/ S/ S if the elliptical rudder has a transverse foil at its tip.

if the elliptical rudder has a transverse foil at its tip. The transverse foil will i

The transverse foil will increase tncrease the totahe total drag of the rudder l drag of the rudder but this drag shall be ignored when rudder designsbut this drag shall be ignored when rudder designs are compared.

are compared.

The rudders have elliptical plan forms so the following relationships are true. The rudders have elliptical plan forms so the following relationships are true.

C CLL= k a / [ 1 + = k a / [ 1 + ( 2 / ARe )]( 2 / ARe )] where where a = angle of attack a = angle of attack

k = slope of the lift of the section with angle of attack graph = dL / da k = slope of the lift of the section with angle of attack graph = dL / da C

CLL = coefficient of lift= coefficient of lift

ARe = effective aspects ratio ARe = effective aspects ratio

C CDIDI= C= CL2L2/ p ARe/ p ARe C CDD = D= DDPDP+ D+ DDIDI where where C

CDD = total coefficient at drag= total coefficient at drag C

CDIDI = coefficient of induced drag= coefficient of induced drag C

CDPDP = coefficient = coefficient of profile drag evaluated frof profile drag evaluated from graphs reprinted from "The Theory of Wing Som graphs reprinted from "The Theory of Wing Sections" shownections" shown in figure 6.2 and 6.3

in figure 6.2 and 6.3 The maxim

The maximum section thickness is 12% of tum section thickness is 12% of the cord lengthhe cord length. "From the data for the . "From the data for the NACNACA four and fivA four and fiv e digit e digit wi

wing sections it appeang sections it appears that the maximurs that the maximum lift coefm lift coef ficients are the ficients are the greatest for greatest for a thickness ration of 12 per a thickness ration of 12 per cent."

cent." Theory of Wing STheory of Wing Sections. As the maximum lift is a function of areaections. As the maximum lift is a function of area , and maximum li, and maximum lift coeffft coefficient for icient for the minimum area, the sections with the maximum lift coefficients must be used.

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b = 0.2m b = 0.2m S = p ´ 0.65 ´ 0.2 / 4 = 0.1021m S = p ´ 0.65 ´ 0.2 / 4 = 0.1021m22 ARe = 2 ´ 0.65 ARe = 2 ´ 0.6522/ 0.1021 = 8.28/ 0.1021 = 8.28 k = 1.3 / 12 k = 1.3 / 12 C CLL= 1.3 a / [12 ´ ( 1 2 / 8.28 )] = 0.08726 a= 1.3 a / [12 ´ ( 1 2 / 8.28 )] = 0.08726 a C CDIDI = C= CL2L2/ ( p/ ( p´ 8.28 ) = ´ 8.28 ) = 2.927´ 102.927´ 10-4-4 x ax a 22 Table 6.1 Table 6.1

Standard rudder design lift and drag characteristics. Standard rudder design lift and drag characteristics.

aa CCDPDP CCDIDI CCDD CCLL CCLL/C/CDD 0 0 00..00005588 00 00..00005588 00 00 2 2 00..000066 11..11771 1 x x 1100-3-3 00..0000771177 00..11774455 2244..33 4 4 00..00006666 44..88663 3 x x 1100-3-3 00..00111133 0..303449900 3300..99 6 6 00..000088 00..0011005544 0..000118855 00..55223355 2288..33

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Design 1 Design 1

Transverse foil fitted Transverse foil fitted section = NACA 0012 section = NACA 0012 a = 0.65 m a = 0.65 m b = 0.20 m b = 0.20 m S = p ´ 0.65 ´ 0.2 / 4 = 0.1021 S = p ´ 0.65 ´ 0.2 / 4 = 0.1021 ARe = 2.6 ´ 0.65 ARe = 2.6 ´ 0.6522/ 0.1021 = 10.75/ 0.1021 = 10.75 k = 1.3 / 12 k = 1.3 / 12 C CLL = 1.3 ´ a / [12 ´ (1 + 2 /10.75 )] = 0.09134 a= 1.3 ´ a / [12 ´ (1 + 2 /10.75 )] = 0.09134 a C CDIDI = C= CL2L2/ ( p ´ 10.75) = / ( p ´ 10.75) = 2.470´ 102.470´ 10-4-4x ax a 22 Table 6.2 Table 6.2 Desig

Design 1 rudder lift and n 1 rudder lift and drag charactdrag characteristics.eristics.

aa CCDPDP CCDIDI CCDD CCLL CCLL/C/CDD 0 0 00..00005588 00 00..00005588 00 00 2 2 00..000066 9.881´ 109.881´ 10-4-4 6.988´ 106.988´ 10-3-3 00..11882277 2266..11 4 4 00..00006666 3.953´ 103.953´ 10-3-3 00..0011005555 00..33665544 3344..66

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Design 2 Design 2

Transverse foil fitted Transverse foil fitted section = NACA 0012 section = NACA 0012 a = 0.686 m a = 0.686 m b = 0.189 m b = 0.189 m S = p ´ 0.686 ´ 0.189 / 4 = 0.1021 S = p ´ 0.686 ´ 0.189 / 4 = 0.1021 ARe = 2.6 ´ 0.686 ARe = 2.6 ´ 0.68622/ 0.1021 = 12/ 0.1021 = 12 k = 1.3 / 12 k = 1.3 / 12 C CLL= 1.3 a / [ 12 ´ ( 1 + 2 / 12 )] = 0.09286 a= 1.3 a / [ 12 ´ ( 1 + 2 / 12 )] = 0.09286 a C CDIDI = C= CL2L2/ ( p / ( p ´ 12 ) = 2.287´ 10´ 12 ) = 2.287´ 10-4-4x ax a 22 Table 6.3 Table 6.3 Desig

Design 2 rudder lift and n 2 rudder lift and drag charactdrag characteristics.eristics.

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8 8 00..00009955 00..0011446644 0..00022441144 00..77442299 3300..88 1 100 00..001111 00..0022228877 0..00033338877 00..99228866 2277..44 Design 3 Design 3

Transverse foil fitted Transverse foil fitted section = NACA 64 - 012 section = NACA 64 - 012 a = 0.686 m a = 0.686 m b = 0.189 m b = 0.189 m S = p ´ 0.686 ´ 0.189 / 4 = 0.1021 S = p ´ 0.686 ´ 0.189 / 4 = 0.1021 ARe = 2.6 ´ 0.686 ARe = 2.6 ´ 0.68622/ 0.1021= 12/ 0.1021= 12 k = 0.9 / 8 k = 0.9 / 8 C CLL = 0.9 a / [ 8 ´ ( 1 + 2 / 12 )] = 0.09643 a= 0.9 a / [ 8 ´ ( 1 + 2 / 12 )] = 0.09643 a C CDIDI = C= CL2L2/ p ´ 12 = 2.467´ 10/ p ´ 12 = 2.467´ 10-4-4 22 Table 6.4 Table 6.4 Desig

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Start Free Trial Cancel Anytime. 4 4 00..000088 3.946´ 103.946´ 10-3-3 00..0011119955 00..33885577 3322..33 6 6 00..00009955 _{8.879´ 10}_{8.879´ 10}-3-3 00..0011883388 00..55778866 3311..55 8 8 00..001122 00..0011557799 00..0022777799 00..77771144 2277..88 1 100 00..00114455 00..0022446666 0..00 033991166 00..99664433 2244..66

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 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 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 the angle of attack is below 3° (maybe 60% of the time) but NACA 64-012 section has a lower value of CLL maximum

maximum; so for the ; so for the same maximum same maximum lift the rudder lift the rudder would have to be larger. For twould have to be larger. For this reason the NACA 0012his reason the NACA 0012 section would appear to be optimum.

section would appear to be optimum. A lim

A limited factited factor in the deor in the desigsign of the n of the foils ifoils is the strucs the structural considerattural consideration. Reducing b by 8% reduces tion. Reducing b by 8% reduces thehe thickness of the foil by 8%. This means the second moment of area has decreased by (0.92

thickness of the foil by 8%. This means the second moment of area has decreased by (0.9233) 22%. The centre) 22%. The centre of pressure has moved down the foil due to the increase in aspect ratio which increases the bending moment 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 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.

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 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 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%. straight trailing edge) at a = 4%. The theoret

The theoretical calculated vaical calculated values of lift and drag are balues of lift and drag are based on a plan fosed on a plan form with a straight ¼ cord line (wing 1)rm with a straight ¼ cord line (wing 1) and therefore must be modified to take account of plan form

and therefore must be modified to take account of plan form

Diagram of foil profiles Diagram of foil profiles

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Table 6.5 Table 6.5

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

W Wiinngg CCDIDI CCLL f f II f f LL 1 1 00..0000553311 0..30344115577 11 11 2 2 00..0000551100 00..3344223366 00..9966 11..000022 3 3 00..0000447755 00..3333771144 00..8899 00..998877

The above data helps to support the analysis of the effect of angle of sweep in Chapter 5. Wing 3 has a larger 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.

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. Taking into account the profile of rudders, the lift and drag can be calculated as follows.

L = ½ ´ r ´ V L = ½ ´ r ´ V22´ S ´ f ´ S ´ f LL´ C´ CLL where where r = density of water = 1025 kg / m r = density of water = 1025 kg / m33 V = velocity taken as 5 knots = 2.87 m/s V = velocity taken as 5 knots = 2.87 m/s S = area of foil = 0.1021m

S = area of foil = 0.1021m22 F

FLL= correction factor = correction factor

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stronger winds where longitudinal stability is a problem. stronger winds where longitudinal stability is a problem.

Standard Rudder Standard Rudder (no transverse foil) (no transverse foil) ARe = 8.28 ARe = 8.28 S = 0.1021 m S = 0.1021 m22 section = NACA 0012 section = NACA 0012 Table 6.6 Table 6.6

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

aa CCDPDP f f IICCDIDI f f LL CCLL DD Newtons Newtons L L Newtons Newtons 0 0 00..00005588 00 00 22..55 00 2 2 00..000066 _{1.124´ 10}_{1.124´ 10}-3-3 0..101774488 33..11 7755 4 4 00..00006666 4.496´ 104.496´ 10-3-3 0..303449977 44..88 115511 6 6 00..000088 00..0011001122 00..55224455 77..88 222266 8 8 00..00009955 00..0011779988 00..66999955 1111..88 330022 1 100 00..001111 00..0022881100 00..88774433 1166..99 337777 Design 2 Design 2

(with transverse foil at zero angle of attack) (with transverse foil at zero angle of attack)

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D = ½ R V

D = ½ R V22S ( CS ( CDPDP + f + f IICCDIDI) + 0.675) + 0.675

Table 6.7 Table 6.7

Actual values of lift and dra

Actual values of lift and dra g for a design 2 rudder with foils at 5 knots.g for a design 2 rudder with foils at 5 knots. aa CCDPDP f f IICCDIDI f f LLCCLL DD Newtons Newtons L L Newtons Newtons 0 0 00..00005588 00 00 33..22 00 2 2 00..000066 _{8.143´ 10}_{8.143´ 10}-4-4 0..10 1883333 33..66 7799 4 4 00..00006666 _{3.257´ 10}_{3.257´ 10}-3-3 0..30 3666666 44..99 115588 6 6 00..000088 _{7.328´ 10}_{7.328´ 10}-3-3 0..50 5449999 77..33 223377 8 8 00..00009955 00..0011330033 00..77333322 1111..44 331166 1 100 00..001111 00..0022003355 00..99116655 1144..22 339955 Design 2 Design 2 (with no foil) (with no foil) Section = NACA 0012 Section = NACA 0012 S = 0.1021 m S = 0.1021 m22 AR = 0.686 AR = 0.68622/ 0.1021 = 9.22/ 0.1021 = 9.22 C CLL = 1.3 a / [ 12 ´ ( 1 + 2 / 9.22 )] = 0.08902 a= 1.3 a / [ 12 ´ ( 1 + 2 / 9.22 )] = 0.08902 a C CDIDI = C= CL2L2/ ( p ´ 9.22 ) = 2.736´ 10/ ( p ´ 9.22 ) = 2.736´ 10-4-4 aa 22 R

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1

100 00..001111 0..00022443355 00..88778866 1155..22 337799 A graph comparing

A graph comparing the cthe calculated datalculated data above a above is shown iis shown in figure 6.5.n figure 6.5. The tip of the

The tip of the foil wifoil will have to be ll have to be sligslightly modified to accommodate thtly modified to accommodate t he trahe transverse foil. The section may alsonsverse foil. The section may also have to

have to be increased be increased in thickness by using a NACin thickness by using a NACA 0015 section at A 0015 section at the very the very tip. This wtip. This will mill make take the tiphe tip stronger and the foil less likely to break off.

stronger and the foil less likely to break off.

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