1

Designing A Conventional Aircraft

Arash Sonei KTH 9 June 2014

Abstract This paper is explaining the important design phases of dimensioning an unmanned

conventional aircraft from scratch and will also design one according to a few chosen

requirements. The design phases discussed will be all from wing dimensioning to stability

and spin recovery, aircraft performance requirements and how to select a motor which

overcomes these. As well as the optimal rate of climb for improved efficiency is discussed. In

the end an aircraft which manages the set requirements and is stable in pitch managing spin

recovery with no problem will have been dimensioned.

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Contents Abstract ................................................................................................................................... 1

Introduction ............................................................................................................................ 4

Initial Plane Sketch and Mission Plan ..................................................................................... 5

Approximating the take-off weight ........................................................................................ 6

Fixed-Engine Sizing .............................................................................................................. 6

Rubber-Engine Sizing .......................................................................................................... 6

Empty-weight Fraction Estimation ..................................................................................... 7

Fuel-weight Fraction Estimation ......................................................................................... 7

Main Wing and Tail Geometry and Configuration ................................................................. 9

The Main Wings geometry .................................................................................................. 9

Wing tips ........................................................................................................................... 11

The tail Arrangement and Its Geometry ........................................................................... 11

Tail geometry .................................................................................................................... 12

Selecting the Appropriate Airfoils ........................................................................................ 14

The Center Of Gravity of a Stable Aircraft ............................................................................ 17

Landing Gear ......................................................................................................................... 18

Relating the drag to the lift .................................................................................................. 20

The Component Buildup Method ..................................................................................... 20

Skin friction coefficient .............................................................................................. 21

The Component Form Factor ........................................................................................ 22

The Interference Factor ............................................................................................... 23

The Wetted Surface Area ....................................................................................... 23

........................................................................................................ 25

Drag-due-to-lift Factor .................................................................................................. 26

Analyzing the Aircraft ........................................................................................................... 26

Engine Performance .......................................................................................................... 27

Engine Selection ................................................................................................................... 28

Propeller ............................................................................................................................ 29

Refined sizing ........................................................................................................................ 32

Spin Recovery through Rudder ............................................................................................. 33

Rate of climb ......................................................................................................................... 36

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Conclusion ............................................................................................................................ 39

Appendix A. Polar Plots ........................................................................................................ 40

Appendix B. Motor Specifications ........................................................................................ 46

References ............................................................................................................................ 48

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Introduction This paper explains how to design an unmanned conventional aircraft from scratch and

mainly follows the empirical formulas provided by Raymer [1]. In order to be consistent and

be able to use empirical formulas which exist for different aircraft types, the selected type

here will be homebuilt-composite. In order to have something to design a few requirements,

such as payload weight, cruise speed and altitude as well as range, will be set.

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Initial Plane Sketch and Mission Plan

In order to start the design analysis one must first have an idea from what to start from and

set some desired requirements for the aircraft which is going to be designed. Figure 1 shows

the initial sketch of a conventional aircraft with a tricycle landing gear configuration, which

will the basis of this paper.

The mission profile seen in figure 2 shows the desired flight plan for the aircraft. Here a

simple cruise without any loitering is chosen.

Figure 1 Initial Sketch

1. Take-off

2. Climb

3. Cruise

4. Landing

Figure 2 The aircraft’s mission plane: simple cruise

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The desired requirements for this design are:

Must be able to carry a payload of

Cruise at an altitude of at the cruise speed of

(0.1 Mach)

Range for

Approximating the take-off weight The first step before one is able to calculate any components for an aircraft is to determine

its initial weight at take-off. There two methods usually used for this when making a first

design Rubber-engine sizing and fixed-engine sizing according to Raymer [1, §6.1].

Fixed-Engine Sizing

Fixed-engine sizing is a method where an existing engine is selected for which the take-off

weight is approximated from. This method is widely used due to the fact that the engine

taken into account already exist, developing a whole new piston engine, which is the most

common engine for a propeller propulsion aircraft, is quite expensive. However using the

fixed-engine sizing method often results to the fact that either the mission range or the

performance will not meet the set requirement. This results to that the designer has to use

optimization methods to improve either the range or performance of the aircraft.

Rubber-Engine Sizing

This method suggests that the engine can be “rubberized” meaning that the take-off weight

should be calculated in order so that the aircraft meets all set requirements, and then a

brand new engine should be designed for this method. However an existing engine can still

be chosen for the aircraft using this method if the engine is not too heavy and meets the

minimum power required conditions which shall be discussed later in this paper. It is this

method which is used in this paper.

The design take-off weight noted can be broken into components existing of crew

weight, payload, fuel weight and the remaining empty weight which includes structure,

engine, landing gear and anything else not included in the three other weight components.

The take-off weight would then be a sum off all its components:

(1)

For an unmanned aircraft the crew weight is automatically equal to zero and the payload is

given in the requirements for the aircraft. Two components yet remain unknown and they

both are highly dependent on the total weight . Therefore they must we rewritten so that

can be obtained through iteration. In order to avoid messy calculation while doing so

both components can be expressed as fractions of :

7

(

) (

)

(2)

Solving for yields:

( ⁄ ) ( ⁄ )

(3)

Empty-weight Fraction Estimation

The aircrafts class has a strong impact on its empty-weight fraction, Raymer [1, §3.3] suggest

that this factor should be approximated using statistics from historical trends depending on

the aircrafts type. He suggests that the empty-weight fraction is an exponential function of

the total take-off weight

(4)

For a homebuilt-composite aircraft for a fixed sweep

else it should be set to 1.04. Here is set to 1 as no variable sweep is considered. As

equation (4) is an empirical formula based on the statistics of manned aircrafts, a reduction

of 5% is considered here as the aircraft under design is unmanned.

Fuel-weight Fraction Estimation

The fuel-weight needs to be calculated so that the aircraft is able to carry enough fuel to

carry out its entire mission. This is best done by calculating the mission-segment weight

fractions.

The mission-segment weight fractions are the percentage of fuel the aircraft has consumed

between the segments of its mission. For a simple cruise mission without any loitering the

segments are as follows:

1. Warm-up and takeoff

2. Climb

3. Cruise

4. Land

The statistical trend for segment 1, 2 and 4 are so similar throughout historical data that an

average will suffice for these for the initial sizing.

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Mission Segment: ( ⁄ )

Warm-up and takeoff 0.970

Climb 0.985

Landing 0.995

In table 1 descend is ignored and is assumed that the cruise segment ends with descending

and that the distance traveled during descending is a part of the cruise range.

Using the Breguet range equation (which is out of the scope of this paper) the cruise

segment mission weight fraction according to Raymer [1, §3.3] is approximated as:

⁄

(4)

At this stage ⁄ are guessed or approximated, the take-off weight will be

refined further below where more accurate number are inserted in the equation.

Typical values seen in propeller aircraft should be used for the specific fuel consumption ,

like

and the propeller efficiency is seen usually somewhere between 0.7-0.8

for this initial weight approximation it shall be set 0.8.

⁄ cannot simply be guessed like and , it must be estimated using the initial sketch of

the aircraft and the equation

⁄ √

⁄

(5)

Where for a nonretractable propeller aircraft and ⁄ is read from Figure 3.6

Raymer [1, §3.4], using the initial sketch yields ⁄ . A typical value for the aspect

ratio of an aircraft of type homebuilt-composite is , which is also used here as an

additional requirement. This yields ⁄ .

Now that all mission-segment weight fractions are known the fuel-fraction can be estimated

as:

( ∏

)

(6)

Table 1 Historical Mission-Segment Weight Fractions

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All that remains now at this stage is to calculate using iteration as shown in table 2.

, guessed ⁄

⁄ , calculated

100 0.7069 0.053 70.6942 85.3

85.3 0.7069 0.053 58.889 84.57

….. ….. ….. ….. …..

83.22 0.7069 0.053 59.87 83.22

As can be seen in table 2 the result is that kg where 5.3% of it is used for fuel and

59.9 kg for building it including the engine.

Main Wing and Tail Geometry and Configuration

The Main Wings geometry

With the take-off weight set the geometry of the wing and tail can be decided. Earlier in the

paper it was stated that the main wing should have an aspect ratio of 6 which is common for

this class of aircraft. The aspect ratio is defined as following

(7)

Here is the wing span and its reference area which is the entire area of the wing

including the part hidden in the fuselage.

Table 2 Iterating for the take-off weight

Figure 3 Indicating the reference area

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In figure 1 an unswept tapered rectangular wing is drawn. As the aircraft is designed to

cruise at Mach it would be highly unnecessary to sweep the wings at all. As the main

function of a wing sweep is to reduce the adverse effects of transonic and supersonic flow.

This is only considered for aircrafts cruising at much faster speeds. The optimal wing

geometry would be an elliptical wing as it reduces the induced drag. However these forms of

wings are difficult and expensive to manufacture. An alternative is to use a tapered

rectangular wing; with a taper ratio of it produces a lift distribution very similar to the

elliptical wing. Now if one where to take account to weight reduction a taper ratio of

would be optimal for most unswept wings which is also considered here.

The taper ratio is defined as the quotient between the tip chord and root chord of the

wing.

(8)

In figure 4 several other variables are defined such as the mean aerodynamic chord . The

mean aerodynamic chord also called MAC has a very special property which allows the

entire wing to have its aerodynamic center approximately at the same point on the MAC as

the airfoil alone. For subsonic flight this point is from the MAC’s leading edge. The Y-

bar is simply the MAC’s distance to the center line, with a symmetrical wing this is at half

the wing span ⁄ . Furthermore following relationships exist between :

Figure 4 Illustrating important parts of the wing

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( )

(9)

(10)

(11)

For a vertical tail a factor 2 must be applied to equation (11) in order to be used.

Once the reference area is know the wingspan can easily be calculated and once again this is

accomplished by the usage of historical data for this aircraft type. Raymer [1, §5.3] suggests

that the wing load

⁄

for a homebuilt aircraft. Using this value a reference area

of and a wing span of is obtained. Alternatively the wing loading can be

analyzed for different parts of the mission plan where it would be considerable to choose

the smallest value. However a smaller wing loading means a larger reference area which

results to a larger wing span which could eventually lead to construction issues if is

not large enough. If one where to find the optimal wing loading and wing span a trade study

would have to be performed where exact knowledge of the construction material and

additional loads just as potential gust loads and other structural loads would be required,

which exceeds the scope of this current paper.

Wing tips

Wing tips are usually shaped so that they can reduce the aircrafts drag and come in many

different shapes. However for low subsonic flight it is not really necessary to shape the wing

tips in any particular form, in other words a simple cut-off wing tip will do. A cut-off wing tip

is the most simplest form of wing tips and shows the exact form of the airfoil as if one had

cut open the wing, therefor the name cut-off. As a matter of fact, a cut-off wing tip

generates less drag than a rounded one.

Now that the geometry of the main wing has been decided the tail arrangement and its

geometry remain.

The tail Arrangement and Its Geometry

The tail, similar to the main wing also generates but this lift can be neglect compared to the

amount of lift produced by the main wing. The tail has instead other main functions. The

main function of the tail is to provide stability and control. These functions are discussed

later throughout this paper.

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Tail geometry

Unlike the main wing some variation is shown for different aircrafts type’s aspect and taper

ratio, table 3 provides data for common .

Horizontal tail Vertical tail

Fighter

Sailplane

Others

Equation (7) cannot yet be used to determine the tail spans as the areas are still lacking.

With no wing loading to directly determine the tail areas one has to use another method,

approximating them through the fuselage length.

The area for corresponding tail is defined as:

(12)

(13)

Here and are the so called volume coefficients for corresponding tail, Raymer [1,

§6.5] suggests that for a homebuilt aircraft and . The tail moment arms

and are the distance from the wing’s 0.25 MAC to the corresponding tail’s 0.25 MAC.

Table 3 Common tail aspect and taper ratio

Figure 5 The horizontal tails moment arm

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For an aircraft which has a propeller mounted in the front the tail arm is usually set to

of the total fuselage length. Here is set to and to

the reasons to this choice are explained later in this paper when discussing

stability.

The question yet remains how one should size the fuselage’s geometry. When designing the

fuselage a few things have to be taken into consideration such as cockpit, payload shape and

fuselage shape, so that the motor will fit in the front. When it comes to designing the

fuselage of an unmanned aircraft the approach gets simpler as one does not have to take the

cockpit and pilot’s comfort into consideration. The design requirements state that the

aircraft must be able to carry a payload of so the maximum diameter will be relatively

small, when taking to account that the aircraft needs space for its avionics and other

components a maximum diameter of is chosen.

Raymer [1, §6.5] suggests with data from ‘Jane’s All the World’s Aircraft’ that the fuselage

can be approximated as a function of the take-off weight such as:

(14)

For an aircraft of the type homebuilt-composite the constants correspond

to . Using this, a fuselage length of is achieved hence the tail reference

areas defined in equations (12) and (13) can be shown to become and

. Now the rest of the geometry of the stabilizers can be decided using equations

(7), (8), (9), (10) and (11) along with table 3 choosing a taper ratio of for both stabilizers

as an aspect ratio of for the horizontal tail and for the vertical tail. These values will

result to that both the tails have similar MAC values. This allows them to be analyzed

together later on. Table 4 summarizes the most important results obtained for the geometry

of the wing and the stabilizers.

Figure 6 The vertical tails moment arm

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Selecting the Appropriate Airfoils

The airfoil’s characteristics are heavily dependent on the Reynolds number for which it

will be operating at. For instance an airfoil which has great characteristics at

cannot be used at as it would show different characteristics. This could have

serious consequences, like for instance making the aircraft stall at speeds it should not. The

Reynolds number is dependent on the flight speed, altitude and length of the component

being analyzed. It is defined as:

(15)

Where for the main wing and stabilizers is the corresponding mean aerodynamic chord.

Figure 7 showing the airfoils anatomy [2]

Table 4 The most important results obtained for the wing and stabilizers

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For low subsonic speed aircrafts the best performance is usually shown with airfoils of a

thickness of around 12%. However in modern times the same performance can be achieved

with airfoils with maximum thicknesses of 15 and 18%.

So how can one tell which airfoil suits ones design? The easiest way to attain this is to study

the airfoil’s polar plots along with the vs plots for the corresponding Reynolds number

which it will operating at and make a table, rating its characteristics and compare it to similar

airfoils. When comparing airfoils one must take in account to what happens to the 2-

dimensional lift coefficient after the stall angle . The most desirable property here

would be a gradual drop in after stall.

When grading the stall property the airfoils will be graded with either A, B or C. Where A =

gentle drop in , C =absurd drop in and B of course is between A and C. Figure 8 shows

stall characteristic A (left) and outrageous stall characteristic C (right).

The main wing has a and when constructing a table comparing several airfoils

selected randomly with the same thickness ratio from [12], following resulted:

Figure 8 desirable and undesirable properties of after stall [2]

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Joukovsky 0018

NACA66-018 NACA0018 Score

Thickness ratio (high is best )

(highest is best )

AOA of

Stall characteristics

(lowest is best) ( ⁄ ) (highest is best)

of ( ⁄ ) (low is best)

Point summation

As the horizontal and vertical stabilizers have roughly the same mean aerodynamic chord

they can be analyzed together where a mean of the two mean aerodynamic chords is used.

By this a Reynolds number of is achieved. For this analysis 3 typical NACA

airfoils have been chosen. Using the same method just used to select airfoil for the main

wing one achieves following when comparing the airfoils at :

NACA0010 NACA0015 NACA0018 Score

Thickness ratio (high is best )

(highest is best )

AOA of

Stall characteristics

(lowest is best) ( ⁄ ) (highest is best)

of ( ⁄ ) (low is best)

Point summation

From table 5 and 6 it is clear that a reasonable choice is NACA66-018 for the main wing and

NACA0015 for the stabilizers.

Table 5 Comparing different airfoils for the main wing (These values were derived from Xfoil. For more

information regarding the polar plots see appendix A.)

Table 6 Comparing different airfoils for the stabilizers (These values were also derived from Xfoil. For

more information regarding the polar plots see appendix A.)

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The Center Of Gravity of a Stable Aircraft As with any other vehicle the position of the center of gravity is of great importance, for an

aircraft it will determine its pitch stability.

For a paper which analyzes the outer parts of the aircraft it can be hard to determine a

stable center of gravity as little is known about the placement of the inner components such

as avionics. A way to approach this is by looking at the stable case at set the center of gravity

accordingly as a requirement. Pitch stability is often describes in terms of the stability

margin which must be positive according to the stability criteria and usually ranges

from .

(16)

Here is the center of gravity’s position away from the main wings MAC as shown in

figure 9. is the position of the point on the aircraft which gives neutral pitch stability

meaning it’s neither stable nor unstable. This point is also the aerodynamic center of the

whole aircraft.

the neutral points position can be shown with several approximations to become:

(

)

(17)

The yet unkown terms in equation (17) can for unswept wing be estimated from:

(18)

Figure 9 Center of Gravity’s position

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(19)

( √

)

(20)

(21)

The higher the stability margin, the more stable is the aircraft considered to be. So if a

stability margin of is desired, must with . The center of

gravity for the aircraft can be seen in figure 10.

Landing Gear One of the functions of the landing gear is to prevent the propeller from striking the ground,

and unless the aircraft is launched and does not need to land in one piece, it is essential for

take-off and landing. There are several components making up the landing gear but to

simplify this paper considers the landing gear as 2 components, the strut and the wheels.

As seen in figure 1 a tricycle configuration, which is quite common for this type or small

aircrafts in general, is chosen.

One might wonder what the dimensions tires of the wheels should be, as they carry the

entire aircraft. Raymer [1, §11.2] suggests that this can be approximated using the empirical

equation composed through historical data of various aircrafts:

Figure 10 Displaying the position of the aircrafts center of gravity

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[ ]

(22)

Where is the weight on the wheel and are constants taken from table 7.

(23)

(24)

Where is wheel ’s distance to the lateral axis which goes through the center of gravity.

Using equations (23) and (24) the corresponding weight on each wheel is:

(25)

Figure 11 Showing the force balance of the wheel configuration

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(

⁄

)

(26)

Setting and yields that and , the

dimensions of corresponding wheel can now be calculated using table 7.

Aircraft type Diameter Width

General aviation

Because the aircraft under investigation is relatively small, it would be only reasonable to

have nonretractable landing gear as there would be no space for them in the fuselage. This

will cause the landing gear to contribute additional drag to the aircraft which will be seen

below. From table 7 a diameter of and a width of are obtained for the

frontal wheel and a diameter of with a width of for the two other wheels.

Relating the drag to the lift The common name for the existing relationship between the drag and lift coefficients is

called drag polar. This relationship states that the drag coefficient is actually a function of

the lift coefficient, and is usually approximated as:

(27)

Where is the so called zero-lift drag coefficient and the drag-due-to-lift factor both

which can be estimated in the design phase of a new aircraft.

There are several methods which can be used to approximate some giving better

accuracy than others. In this paper is approximated using the component buildup

method.

The Component Buildup Method

This method implies that one shall split the aircraft to its components i.e. fuselage, main

wing, etc. and analyze them separately before approximating the total for the entire

aircraft. The total of the aircraft is given as:

∑[ ]

(28)

Table 7 Parameters given by Raymer [1, §11.2] to be used in the empirical equation (22)

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This formula is used for subsonic flow which the aircraft brought forth in this paper is

designed for. Here there are 4 components which the aircraft is split into: fuselage, main

wing, horizontal stabilizer and the vertical stabilizer. In this equation is the total skin

friction coefficient of component c. is a form factor to include contribution due to viscous

effects on the pressure distribution. The interference factor depends on what kind of

component is being analyzed and is discussed later, is the total wetted area of the

current component.

is the contribution from miscellaneous irregularities or objects sticking out of the

plane like the landing gear for instance. is other contribution from leaks which

cannot be directly detected but is rather estimated as a percentage of the total . Raymer

[1, §12.5] suggests that for a propeller plane this value is somewhere between 5%-10% of

.

Skin friction coefficient

is highly dependent on the Reynolds number for component c. In this analysis the

Reynolds number should be the one which the aircraft is flying with during the cruise phase.

Reynolds number is given by equation (15), where for the fuselage is the overall length of

that component and for the main wing and stabilizers shall be set as the aerodynamic mean

chord of the current component.

is defined differently for laminar and turbulent flow. To determine if the flow is either

laminar or turbulent, a rule of thumb is required. For flat plates the standard rule of thumb is

that . This is the Reynolds number for which the flow becomes turbulent.

However this value can be either larger or smaller for boundary layers along a curved surface

depending on the sign on the pressure gradient. In this paper we neglect this and use

Surface type [ ]

Camouflage paint on aluminum Smooth paint

Produced sheet metal Polished sheet metal

Smooth molded composite

The surface of the component being analyzed influences the character of the flow. Assuming

that all the components have the same surface material, is set to be from

table 8. This is only suiting as the aircraft being analyzed is the type of homebuilt-composite.

Table 8 skin roughness for different materials suggested by Raymer [1, §12.5]

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This variable can now help to approximate a cut-off Reynolds number which at subsonic

speeds can be estimated as:

( )

(29)

If the skin friction coefficient equals

√

(30)

and when which means that the entire flow now is fully turbulent

[ ]

( )

(31)

If then in equation (31) and if , it is

which must be used as . is the Mach number during the cruise, which is the

velocity during cruise divided by the speed of sound during cruise:

(32)

The Component Form Factor

This factor is used as a correction factor to adjust the skin friction coefficient to take in

account to pressure drag. Once again using empirical formulas suggested by Raymer [1,

§12.5]:

For fuselage and similar components:

(33)

Where is the ratio between the length and maximum diameter for that component:

√( ⁄ )

(34)

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For wings, stabilizers, struts, pylons and similar components:

[

( ⁄ ) (

) (

)

] [ ( ) ]

(35)

Here

is the is the maximum normalized thickness of the component which is often given as

a percentage. ( ⁄ ) is the chord wise position of the maximum thickness and is the

sweep angle of that line which stated earlier for aircrafts at Mach should be set to zero.

The Interference Factor

When two or more components intersect with each other, their boundary layers interfere

with each other resulting into increased drag. How much they interfere with each other is

also dependent of the component. The fuselage has in most cases a negligible effect hence

one can set its interference factor . The tail arrangement also affects the magnitude of

this factor. Raymer [1, §12.5] suggests that 3% is enough for a clean V-tail and ranges up to

8% for an H-tail, while for a conventional tail it can bet set to 5%. Even the configuration of

the main wing affects . For a main wing which has a configuration of either a high-wing,

well-fitted low wing or a midwing the interference is negligible i.e. . While for an

undiluted low wing the interference factor ranges from 10 to 40%. In this paper a midwing is

considered.

The Wetted Surface Area

The wetted area is just as the name suggests the area of the component which comes in

contact with air and get “wet”. For the main wing and tail their corresponding wetted area

can be approximated from their surface area which is exposed to the air as shown in figure

12.

Figure 12 The exposed area of the main wing

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How much the wetted area is for these components are is fully dependent on their thickness

ratio

.

If

(36)

If

[ (

)]

(37)

For the fuselage the wetted area can be approximated using the top and side of the

fuselage.

Raymer suggests that such approximation can be yield with:

( )

(38)

Where is a factor depending on the fuselage cross section shape. For a long circular cross

section and for a rectangular . If the cross section is somewhere in between

Figure 13 Illustrating the top and side area of the fuselage

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circular and rectangular, Raymer [1, §12.5] suggests that should be set to . In this

design we consider a circular cross section therefor . When calculating and

one must not forget to subtract the area from intersecting components such as the main

wing, horizontal and vertical stabilizer.

As stated before is additional drag contribution from irregularities or objects

sticking out of the aircraft which in this case is only the landing gear. The magnitude of

can be calculated by multiplying the tire and the struts frontal area with a factor ,

depending on the shape, divided by the aircrafts main wings reference area . The values for

depending the strut and tires can be found in table 9.

∑

(39)

Strut or tire shape

Regular wheel and tire

Second wheel and tire in tandem

Streamlined wheel and tire

Wheel and tire with fairing

Streamlined strut

Round Strut or wire

Flat spring gear leg

In this case of the aircraft being analyzed, there are 3 round struts and 3 regular wheels with

tires. The easiest way to obtain the struts frontal area and other areas in general is with a

CAD program such as solid edge.

By designing the struts in Solid Edge a frontal area of is noted for each. The frontal

area of the wheels is simply the corresponding diameter multiplied with the corresponding

width.

Earlier it was noted that was usually set as a percentage of of the total

. Assuming a leakage of about and following each step explained a total of is

obtained for the zero-lift drag coefficient , as seen in table 10.

Table 9 Different values of the factor E applied in equation (39) depending on the strut or

tire shape suggested by Raymer [1, §12.5].

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Wing Fuselage Stabilizers Miscellaneous (Landing gear)

Leaks

of total

Total

Drag-due-to-lift Factor

At subsonic speeds the drag-due-to-lift factor is commonly expressed as:

(40)

Here is called the Oswald efficiency factor and for an unswept-wing aircraft can be

estimated using empirical formulas such as:

( )

(41)

With the aircrafts aspect ratio being set to equations (40) and (41) give

.

Analyzing the Aircraft

With the approximation that the thrust force is roughly parallel to the velocity vector ,

force balance demands that:

Figure 14 displaying force balance for the aircraft during steady level flight

Table 10 illustrating the different contributions to

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(42)

(43)

and in equation (42) and (43) are the lift and drag forces which are related to each other

through the drag polar relationship discussed before. With this relationship along with the

force balance the drag force can be rewritten as:

(44)

In order for the aircraft to be able to fly at a steady level the thrust must equal the drag. This

is usually called the thrust required In terms of power this can be related to the power

required . The relationship between thrust and power is:

(45)

During the steady flight the required power will equal the power available . This is a

necessary condition for an aircraft with a propeller propulsion system. For aircrafts with a

propeller propulsion system, the available power is delivered from the engine and is heavily

dependent on the propeller through the propeller efficiency . The power available is the

product of the propeller efficiency and the power delivered from the engine at the current

altitude.

(46)

Engine Performance

Most propeller propulsion system aircrafts receive their power usually from a piston engine.

The piston engine, which is also used in this paper, is a combustion engine and is greatly

dependent on the amount of oxygen that enters its cylinder. This means that the piston

engine will give out less power with increasing altitudes. The amount of power a piston

engine is able to deliver is a function of its maximum power output at sea level and the

density of the altitude which it is currently operating at. This relationship can be shown to

be:

(

⁄

)

(47)

Here in equation (47) it is ISA values for the altitude which must be used.

28

Engine Selection As stated earlier, a rubberized engine has been considered so far. Now if one wishes to

select an existing engine one must take 2 points into consideration:

1. The engine must weigh as little as possible and be as small as possible to make the

aircraft as light as possible.

2. The engine must deliver a power such that the power available exceeds

the minimum power conditions set for the steady flight at the selected altitude.

Which in this case is at .

So what are the minimum power conditions set for steady flight? From equations (43), (44),

(45) and (46) the following relationship is obtained:

(48)

Seen as above, equation (48) is a function of the speed. So one must first find the minimum

speed required for minimum power. This is achieved by taking the derivative of equation

(48) with respect to and setting it to zero.

[

(

)

]

⁄

(49)

Inserting this into equation (48) yields the minimum power required for steady flight:

[

(

⁄

)

⁄

]

⁄

(50)

For the design this paper is analyzing the minimum power and speed required for steady

flight are

and .

29

Propeller

There are several configurations which one may take in account to when designing the

aircraft propeller. In this paper configuration A (shown in figure 15) also known as the

tractor configuration is analyzed as seen in the sketch.

Now that one is able to calculate the power delivered from the engine at a selected altitude,

only one more variable remains in equation (46) in order to select an engine which

overcomes the minimum power requirements. This variable is namely the propeller

efficiency which in return is a function of the advance ratio that is defined as:

(51)

Here is the diameter of the propeller and is the revolutions per second (rps), which the

engine will be operating at. Nearly all aircrafts do not use their engine at 100% capacity as

this would wear out the engine quite rapidly. Instead aircrafts tend to fly with 50-80% of

their engine’s maximum capacity; this also affects the rps which it is operating at.

The engine’s rps can easily be read from its specifications delivered by its manufacturer.

While on the other hand the propeller diameter has to be estimated. This estimation will be

dependent on the propeller material, number of blades, engine power and flight speed

which is explained below.

One way to estimate the propeller diameter is to analyze the helical tip speed of the

propeller, which is the vector sum of the rotational speed and the aircraft’s speed.

Figure 15 Showing common propeller configurations [2]

30

√ ( ) ( )

(52)

For a metallic propeller which is considered here ( ) must not exceed

as a

rule of thumb. Raymer [1, §10.4] suggests that the propeller diameter can be estimated by

the engine’s delivered power in kilowatt and the number of blades. Table 11 shows

coefficient values for the number of blades. Here a propeller with 2 blades is analyzed as a

designer’s choice.

√

(53)

No. blades

From equations (52) and (53) two propeller diameters are calculated for which the lesser is

chosen.

The propeller efficiency is not always only a function of the advance ratio but can also be a

function of the power coefficient :

(54)

Gudmundsson [2, §14.4] provides a table for which the propeller efficiency can be

approximated by using through the usage of typical observed data as seen in table

12.

Table 11 for equation (53), inserting in kW gives in m

31

Now that it is clear how to calculate the power available for a piston engine, one can begin

selecting reasonable engines and compare its available power with the requirements set for

steady flight.

Through trial and error the engine chosen is LIMBACH L 275 E, a piston engine which only

weighs . It will be operating at of its maximum capacity producing at

sea level at 4000 rpm. The results obtained by using equations (50), (51), (52), (53), and (54)

along with table 12 are recorded in table 13 below.

For full specifications of the selected engine please refer to appendix B.

Table 12 Observed typical values of propeller efficiency for a constant speed propeller [2]

Table 13 Showing the results of the engine analysis

32

Refined sizing Now that the drag polar has been approximated and the engine selected, one can now

refine the sizing method discussed before with more accurate parameters in order to yield

an improved estimation of the take-off weight.

The selected engine will of course have its own specific fuel consumption which differs from

the approximated one. For the set condition which the engine shall be operating at, this will

be

according to the data provided by the manufacturer found in appendix B.

The estimation of the ratio between the lift and drag during cruise also needs to be

improved. As the weight varies from start to cruise due to fuel consumption it needs to be

adjusted accordingly. According to Raymer [1, §6.3.7], for a propeller propulsion aircraft this

can be shown to be:

⁄

[

⁄

⁄ ⁄

⁄

⁄ ⁄

]

(55)

In equation (55) is the dynamic pressure during the cruise segment:

(56)

During the cruise segment equals according to force balance, so that ⁄ can be

written as the inverse of the drag divided by This yields ⁄ .

Once again the method of iteration shall be used to determine the approved take-off weight

. Using the same methods as the first approximation with these new parameters, a new

take-off weight of is achieved. Where of this shall be used for fuel

storage and as empty weight.

As the weight changed one must once again check the minimum speed and power condition

for steady flight as they are dependent on . Through equations (49) and (50) one

can see that the needed requirements decrease with decreasing take-off weight. As the

analyzed aircraft exceeded the previous requirements one can for sure say that they will do

the same now because power available and the set cruise speed is independent of the

calculated take-off weight.

33

Spin Recovery through Rudder When an aircraft stalls it spins around its vertical axis due to that there is more lift produced

by the inner wing than the outer which is more fully stalled. A rudder on the vertical

stabilizer is able to reverse this effect. However only parts of the rudder which are out of the

wake (shown in figure 16 help) to reverse spin. The wake is the part where stalled air from

the horizontal stabilizer flows. A rule of thumb to determine the wake is to draw a line of

from leading edge and a line of from the trailing edge of the horizontal stabilizer,

(also shown in figure 16). As another rule of thumb at least one third of the rudder area

should be outside the wake.

The stall speed is given by:

√

(57)

When not taking any regards to any flaps and taking into consideration

that the weight changes from take-off to cruise due to fuel consumption a stall speed of

is derived. is the 3-dimensional lift coefficient, while on the other hand is the

2-dimensional lift coefficient. It is which is obtained from the airfoil data.

Figure 16 Rule of thumb for analyzing how much of the rudder is outside the wake [2]

34

The spin generated will be opposed by damping forces, mostly from portions of the aft

fuselage and vertical tail underneath the horizontal stabilizers denoted . As seen in figure

17 the part of the rudder outside the wake is denoted . It’s here the decision to make the

tail moment arm for the vertical tail to be smaller than for the horizontal becomes

noticeable. With this decision one can increase without changing the entire configuration

of the tail in case one might need more rudder space for spin recovery.

To analyze the rudders effect, empirical formulas and figure 16.32 brought forth by Raymer

[1, §16.10.3] are used. In order to do this first the tail damping ratio , unshielded rudder

volume coefficient and the airplanes relative density parameter µ needs to be

decided.

( ⁄ )

(58)

( ⁄ )

(59)

Figure 17 The rudder of the aircraft brought forth in this paper

35

⁄

(60)

As the load on the airplane is mainly given by the pressure distribution on it, and

should be set to the distance from the aircrafts center of gravity to the pressure’s point of

attack for each corresponding part. However as pretty much nothing is known about the

pressure distribution around the aircraft at initial design, one can approximate these points

with the center of gravity of each corresponding point instead as shown in figure18 .

Now all that remains is the calculate the tail-damping power factor ,

( ) ( )

(61)

and the spin recovery criterion ,

⁄

(62)

then to check with figure 16.32 Raymer [1, §16.10.3] if the rudder designed will provide spin

recovery.

are the mass moments of inertia of corresponding axis and are for a single-

engine propeller aircraft given as:

Figure 18 Displaying and

36

(63)

(64)

From equations (60) and (62) one yields that for the current aircraft that’s being analyzed

that and . From figure 16.32 Raymer [1, §16.10.3] one can read

that in order to achieve spin recovery with rudder alone.

With a of is

yielded, so this aircraft will have no problem with spin recovery with the designed rudder.

Rate of climb One might wonder how this aircraft will perform during the climb and what the ideal rate of

climb ⁄ and the climb angle might be (Note that here ⁄ is one variable and not the

ratio between two). To do so one must analyze the force balance during climbing.

(65)

(66)

Figure 19 Force balance during steady climb [3]

37

This will give us:

(67)

⁄ is the vertical component of the velocity which is reach by multiplying to .

⁄

(68)

Now we can calculate the climb angle by setting the relation for the power available and

making use of the drag polar:

(

)

( ⁄ )

(69)

⁄

(

)

( ⁄ )

(70)

38

From figure 20 one can that see the best rate of climb at all these selected altitudes occur at

and the corresponding climb angles can be seen in table 14.

Altitude [m] Maximum climb angle

As seen in table 14 the optimal climb angle decrease with increasing altitudes. So the aircraft

has to adjust its climb angle accordingly at each altitude to achieve best ⁄ in order to be

as efficient as possible.

Figure 20 Rate of Climb against speed at different altitudes

Table 14 Displaying the maximum climb angles at corresponding altitude

39

Conclusion Through these methods an unmanned conventional aircraft has been dimensioned weighing

with a wing span of and an overall fuselage length of . An engine

which exceeds the minimum power requirements by a good margin was chosen. By this

engine choice, one is still able to use the rudder-engine sizing method, so that both range

and performance meet the set requirements. The aircraft is both very stable in pitch and

manages spin recovery with no problem. If better spin recovery is desired, the tail

placements were chosen so that one can increase the rudder area without changing the

aircraft’s configuration. Additionally the best climb conditions have been taken forth so that

the aircraft can perform as efficient as possible. As seen in figure 21 below, the final design

of the aircraft, managed to look very similar to the initial sketch, which it was designed after.

Figure 21 CAD models of the designed aircraft

40

Appendix A. Polar Plots All the polar plots have been calculated with [12] which works with data obtained from Xfoil.

Figure 22 Displaying the polar plots for the Joukovsky 0018 airfoil analyzed at

41

Figure 23 Displaying the polar plots for the NACA66-018 airfoil analyzed at

42

Figure 24 Displaying the polar plots for the NACA0018 airfoil analyzed at

43

Figure 25 Displaying the polar plots for the NACA0010 airfoil analyzed at

44

Figure 26 displaying the polar plots for the NACA0015 airfoil analyzed at

45

Figure 27 Displaying the polar plots for the NACA0018 airfoil analyzed at

46

Appendix B. Motor Specifications

47

Figure 28 Displaying the selected motors specifications [14]

48

References

[1] Daniel P. Raymer. Aircraft Design: A Conceptual Approach. AIAA Education Series, 5th edition, 2012.

[2] Snorri Gudmundsson. General Aviation Aircraft Design. , 1st edition, 2014.

[3] Arne Karlsson. 'Steady climb performance with propeller propulsion'. Dept. of Aero-nautical and Vehicle Engineering, KTH, 2013. [4] Arne Karlsson. 'The aeroplane – some basics'. Dept. of Aero-nautical and Vehicle Engineering, KTH, 2012. [5] Arne Karlsson. 'Steady level flight of an aeroplane with propeller propulsion'. Dept. of Aero-nautical and Vehicle Engineering, KTH, 2013. [6] Arne Karlsson. 'Aeroplane weight, balance and pitch stability'. Dept. of Aero-nautical and Vehicle Engineering, KTH, 2013. [7] Arne Karlsson. 'How to estimate and in the simple parabolic drag polar

'. Dept. of Aero-nautical and Vehicle Engineering, KTH, 2013.

[8] Arne Karlsson. 'Cruise performance of aeroplanes with propeller propulsion'. Dept. of Aero-nautical and Vehicle Engineering, KTH, 2013. [9] http://aerospace.illinois.edu/m-selig/ads/coord_database.html. Visited May 7, 2014. [10] http://en.wikipedia.org/wiki/Tail_configuration. Visited May 7, 2014. [11] http://en.wikipedia.org/wiki/Flight_dynamics_(fixed-wing_aircraft). Visited May 7, 2014. [12] http://airfoiltools.com/compare/index. Visited May 23, 2014. [13] http://en.wikipedia.org/wiki/Reciprocating_engine. Visited May 9, 2014. [14] http://www.limflug.de/de/products/engines-15kw-40kw.php. Visited May 12, 2014.