Sizing Matrix & Carpet Plots - METUae452/lecture5_sizing_matrix_carpet_so.pdf · For sizing and...

Sizing Matrix & Carpet Plots

Serkan Özgen, Prof. Dr.

Middle East Technical University

Dept. Aerospace Eng., Ankara, Turkey

Outline

• Introduction

• Requirements

• Baseline Design

• Sizing and Trade

o Sizing matrix plot

o Carpet plot

• Optimum Design

• Conclusions

2

Outline

• Introduction

• Requirements

• Baseline Design



o Carpet plot

• Optimum Design

• Conclusions

3

Introduction

• Aircraft design: An intellectual process that combines engineering knowledge, creativity, and art.

• Aircraft design is not done by one person aimed at a single goal, but is an activity with multiple objectives.

4

Introduction

A successful design is:

• Technically sound,

• Feasible,

• Affordable,

• Safe,

• Reliable,

• Aesthetically pleasing.

5

Introduction

Contradicting requirements:

• A very safe airplane will not be feasible or affordable.

• A hi-tech airplane will be very expensive, etc.

Trade and optimization.

A good design is an efficient compromise of performance, safety, reliability, cost and aesthetics.

6

Introduction

• The basics of optimization of the performance and weight of a light sport airplane is outlined.

• The simple methodology is relevant for the conceptual design phase, where sizing and performance calculations constitute the major task.

• A well-optimized airplane is less likely to encounter unsurmountable weight and cost increases and performance deficiencies.

7

Introduction

The four main constituents of the presented method are:

• Weight estimation,

• Aerodynamics,

• Installed thrust,

• Performance.

8

Outline

• Introduction

• Requirements

• Baseline Design



o Carpet plot

• Optimum Design

• Conclusions

9

Requirements

• Each airplane is designed around a set of requirements that are imposed by the customer and/or by safety and certification requirements.

• The key to the success of a design is a set of realistic and consistentrequirements.

• The requirements involve purpose and operation of the aircraft, performance characteristics like speed, range, rate of climb, etc., and also mission characteristics like payload, low observability, etc.

10

Requirements - Customer

11

Definition Light Sport Aircraft

General

Number of engines

Occupants (80 kg each)

1

2

Performances

Optimized for

Speed range (km/h)

Altitude (m)

Range (km)

Rate of climb (m/s)

Take-off run (m)

Stall speed (km/h)

Cruise

≈ 280

2400

1000

> 5

< 300

< 100

Useful weight

Luggage/crew member (kg) ≈ 10

Miscellaneous

Comfortable

Green

Safe

Cheap

Requirements

EASA Certification Specifications that may be applicable are:

• CS-23: Certification Specifications for Normal, Utility, Aerobatic and Commuter Category Aeroplanes;

• CS-VLA: Certification Specifications for Very Light Aeroplanes;

• CS-LSA: Certification Specifications and Means of Compliance for Light Sport Aeroplanes;

• Also FAR-23: Airworthiness Standards: Normal, Utility, Acrobatic, and Commuter Category Airplanes is applicable.

12

Requirements

CS-23 and FAR-23 are applicable to:

• Normal, Utility, Aerobatic and Commuter Category airplanes.

• Normal, Utility and Aerobatic Category:

o Certified maximum take-off weight of 5670 kg (12500 lb) or less,

o Seating configuration, excluding the pilot seats of nine or fewer.

• Commuter Category:

o Propeller-driven twin engine aeroplanes that have a seating configuration, excluding the pilot seats of nineteen or fewer,

o Certified maximum take-off weight of 8618 kg (19000 lb) or less.

• Certification Specifications for Normal, Utility and Aerobatic category is further divided into two subcategories, namely airplanes heavier or lighter than 2722 kg (6000lb).

13

Requirements – CS/FAR

14

Definition FAA

FAR-23

EASA

CS-23

EASA

CS-VLA

EASA

CS-LSA

Applicability

Maximum take-off weight

Number of engines

Type of engine

Number of crew

Max. number of seats

2722 kg

one

reciprocating

2

9

2722 kg

one

reciprocating

2

9

750 kg

one

spark or compression

ignition

2

0

600 kg

one

non-turbine or electric

2

0

Performances

Stall speed, VSO1

Rotation speed, VR

Climb speed (@15m/50ft height), VCL

Climb rate or gradient @ VCL

Approach speed, VA

Climb gradient @ VA

61 kt

≥ VS12

≥ 1.2VS1

≥ 8.3 %

≥ 1.3VSO

≥ 3.3 %

113 km/h

≥ VS1

≥ 1.2VS1

≥ 8.3 %

≥ 1.3VSO

≥ 3.3 %

83 km/h

≥ 1.3VS1

≥ 2 m/s

≥1.3VS1

1:30

83 km/h

1 VSO: stall speed in the landing configuration.2 VS1: stall speed obtained in a specified configuration. In this case, take-off configuration.

Outline

• Introduction

• Requirements

• Baseline Design



o Carpet plot

• Optimum Design

• Conclusions

15

Baseline Design

Baseline Design is accomplished following the steps:

• Competitor study,

• First weight estimation,

• Airfoil and wing planform selection,

• Power-to-weight ratio (P/W) and wing loading (W/S) selection,

• Refined sizing and a better weight estimation,

• Geometry sizing and configuration,

• Configuration sizing,

• Aerodynamics,

• Empty weight estimation using statistical component weight estimation method,

• Installed and uninstalled thrust,

• Performance and flight mechanics. 16

Baseline Design

The configuration of the Baseline Design is as follows:

• Low-wing monoplane with conventional tail configuration,

• Tricycle fixed landing gear,

• Single naturally aspirated reciprocating tractor engine with a three blade constant speed propeller,

• Number of crew: 2 with side by side seating.

17

Baseline Design

18

Characteristics Value

Length (m) 7.54

Wing area (m2) 11.2

Wing span (m) 10.0

Aspect ratio 9.0

Maximum take-off gross weight (kg) 944.7

Wing loading, W/S (kg/m2) 84.3

Wing airfoil NLF(1)-0115

Engine Lycoming IO-360-L

Propeller 3 blade/76” diam.

Engine power (hp/kW) 160/119.3

Power-to-weight ratio (kW/kg) 0.126

g-limits +6, -3

Baseline Design

19

Performance characteristic Baseline Design Requirement Satisfied?

Range, R (km) 1050 > 1000

Rate of climb, R/C (m/s) 7.8 > 5

Take-off run, sg,to (m) 268 < 300

Stall speed, VSO (km/h) 94.3 < 100

Baseline Design

• The baseline design falls into the CS/FAR-23 Classification becauseof its weight.

• Also, the limit load factor selected puts it into the AerobaticCategory.

• The Baseline Design satisfies all the «Customer» Requirements by a comfortable margin.

• But, is this the best airplane satisfying the requirements?

• Can a lighter, i.e. greener and less costly airplane to acquire and operate be designed that can still meet the requirements?

20

Outline

• Introduction

• Requirements

• Baseline Design



o Carpet plot

• Optimum Design

• Conclusions

21

Sizing and Trade

• Sizing and trade studies constitute an important part of conceptual design.

• After this step is completed, the designer knows that the calculated design parameters represent a good combination of performance and weight.

• Since the weight of the airplane has a direct influence on cost, in a way, the cost is also optimized.

• For sizing and trade studies, two methods are outlined in the present study, namely the sizing matrix plot and carpet plot analyses. These methods are basically identical, they essentially represent the same data in different formats. 22

Sizing and Trade - Sizing Matrix Plot-1

• The parameters to be varied or optimized are selected, i.e. wing loading (W/S), thrust-to-weight (T/W) or power-to-weight (P/W) ratio, taper ratio (), aspect ratio (AR), etc.

• As the number of parameters increase, the number of combinations increase by at least 3n, n: the number of parameters chosen.

• Usually, wing loading (W/S) and thrust-to-weight (T/W) or power-to-weight (P/W) ratios are chosen for the sizing and trade studies since these have the greatest effect on performance and weight.

• Actually, in this study, S (trapezoidal wing planform area) and P (power) are chosen as the trade parameters because the weight estimation method utilized requires these parameters as inputs. 23


• The Baseline Design is perturbed by ±20% for W/S and ±20% for T/W or P/W.

• In this study, the trapezoidal wing reference area is perturbed by ±20% and power is varied such that more powerful and less powerful engines belong to the same engine family as the Baseline Design.

• While choosing candidate engines, choosing engines that have similar sizes to the Baseline Design simplifies the analysis significantly because the nacelle and the fuselage size will be unaffected from the engine choice.

• The trapezoidal wing planform areas and the engines that are studied are as follows:

Wing planform areas: 9.0 m2, 11.2 m2, 13.4 m2.

Engines: Lycoming O-235-F (125 hp), Lycoming IO-360-L (160 hp), Lycoming IO-360-F (180 hp).

24


• While perturbing the wing planform area, the horizontal and vertical tail sizes (SHT and SVT) are varied proportionately in order to keep the tail volume ratios (VHT and VVT) the same as the Baseline Design. This also relieves the designer from varying the fuselage length.

• Fuselage size is kept constant for all perturbed designs since the cockpit size and tail moment will remain constant, independent of the wing size and the powerplant.

• Fuel volume is initially kept constant for simplifying the analysis. If the available fuel is enough to satisfy the mission requirements (especially the range), the designer may keep the original fuel volume. If the range requirement is not satisfied, one needs to increase the fuel volume, which will have an effect on the total weight.

25


• The preliminary sizing matrix is constructed.

• Configuration 5 is the baseline design.

26

S=13.4 m2 S=11.2 m2 S=9.0 m2

P=180 hp Config.1 Config. 2 Config. 3

P=160 hp Config. 4 Config. 5 Config. 6

P=125 hp Config. 7 Config. 8 Config. 9


• The performance requirements that will be used for optimization are selected.

• For a sound analysis, at least three requirements shall be selected.

• The chosen requirements shall be balanced between the ones where wing loading (W/S) has a dominant effect like the stall speed (VSO) and the landing distance (sg,l), and those where power-to-weight ratio (P/W) has a dominant effect like the rate of climb (R/C), maximum speed (Vmax), and take-off distance (sg,to).

• In this study, the performance requirements chosen are the range (R), rate of climb (R/C), take-off run (sg,to), and the stall speed (VSO), which are the “customer” requirements. 27


• The empty (Wempty) and take-off gross weights (Wo) of each configuration is calculated.

• The empty weights are calculated using the Statistical Weights Method explained by Raymer. This method consists of 14 empirical relations for estimating the weights of:

o the wing, horizontal tail, vertical tail, fuselage, main landing gear, nose landing gear, installed engine, fuel system, flight controls, hydraulics, electrical system, avionics, air conditioning/anti-icing system and furnishings as a function of design gross weight, ultimate load factor, sizes and the geometric shapes of the major components, cruise speed of the airplane, dry engine weight, etc.

• It is highly recommended to use this method for sizing and trade, although it is a bit more cumbersome than other less comprehensive weight estimation methods since it results in a more detailed and accurate weight breakdown.

28


• The Aerodynamics module is updated for each configuration.

• The Aerodynamics module is a spreadsheet involving calculations of the maximum lift coefficient (CLmax), parasite drag coefficient (CDO), induced drag factor (K), and the drag polar as a function of speed and altitude.

• In these calculations, the effect of high lift devices and the landing gear are taken into account.

29


• The uninstalled thrust or power data obtained from the engine manufacturer or by using a scaling approach is analysed in order to obtain the installed thrust values.

• Here, the losses due to altitude, power extraction, blockage, compressibility, scrubbing drag, cooling and miscellaneous drag are calculated and the installed thrust is calculated as a function of speed and altitude.

• Performance calculations are performed for the requirements selected.

30


• Stall speed (VSO):

Here, the lift equation is employed with the wing planform size (S) selected for the given configuration, and the maximum lift coefficient (CLmax) calculated for that configuration. Here, the airplane is in landing configuration, where the flaps are fully extended.

31

SCV2

1LW maxL

2


32

• Rate of climb (R/C=dh/dt):

For the calculation of the rate of climb, the specific excess power approach is used, which can also be used in order to calculate other performance characteristics like the maximum speed, acceleration, service ceiling and the maximum sustained load factor that can be achieved at a given altitude and speed.

For rate of climb calculations, the load factor n=1 and take-off conditions are used. For take-off, flaps are partially extended and their effects on the parasite drag coefficient and induced drag are accounted for.

dt

dV

g

V

dt

dh

S

W

q

Kn

S/W

Cq

W

TVP 2DO

s


• Take-off run (sg,to):

Take-off distance is calculated using the expression given below:

In the above equations, μ=0.03 is the coefficient of rolling friction between the runway and tyres. The parasite drag coefficient, CDOincludes the additional drag of the partially extended flaps and the induced drag factor, K is corrected for the ground effect. During take-off, the airplane is fairly horizontal so CL≈0.1 is assumed.

33

T

2TOAT

Ato,g

K

VKKln

gK2

1s

,W/TKT

2LDOLA KCCC

)S/W(2K


• Range (R):

Range is calculated from the Bréguet Range Equation:

In this equation, ηp is the propeller efficiency, calculated as a function of the advance ratio, J=V∞/nD, n: propeller revolutions per second and D: propeller diameter.

The propeller efficiency is corrected for blockage, compressibility and scrubbing drag effects.

34

f

i

power

p

W

Wln

D

L

CR


The specific fuel consumption is denoted by Cpower and is dependent on the engine chosen.

For the candidate engines in this study, the specific fuel consumptions are obtained for the cruise condition from manufacturer’s data and have slightly different values for the three different engines:

Lycoming O-235-F, Cpower=0.4638 lb/h/hp=0.773*10-6 N/W/s,

Lycoming IO-360-L, Cpower=0.4500 lb/h/hp=0.750*10-6 N/W/s,

Lycoming IO-360-F, Cpower=0.4446 lb/h/hp=0.741x10-6 N/W/s.

35


L/D is dependent on the weight and the speed of the airplane.

In order to maximize range, a propeller-driven airplane must fly at L/D)max.

For the Baseline Design and the remaining configurations this occurs at a speed around 72 m/s (259 km/h) at the cruise altitude of 2400m.

However, it is seen in the calculations that, cruise flight at a speed of 280 km/h does not significantly alter the range performance of the airplane.

36

Sizing and Trade - Sizing Matrix Plot-15S=13.4 m2 S=11.2 m2 S=9.0 m2

P=180 hp

1

Wo=994.2 kg

VSO=88.5 km/h

R/C=7.9 m/s

sg,to=228 m

R=1006 km

2

Wo=965.4 kg

VSO=95.5 km/h

R/C=8.3 m/s

sg,to=259 m

R=1034 km

3

Wo=937.2 kg

VSO=102 km/h

R/C=9.3 m/s

sg,to=269 m

R=1080 km

P=160 hp

4

Wo=973.2 kg

VSO=87.6 km/h

R/C=7.4 m/s

sg,to=235 m

R=1028 km

5

Wo=944.7 kg

VSO=94.3 km/h

R/C=7.8 m/s

sg,to=268 m

R=1050 km

6

Wo=916.7 kg

VSO=101 km/h

R/C=8.2 m/s

sg,to=292 m

R=1088 km

P=125 hp

7

Wo=950.8 kg

VSO=86.6 km/h

R/C=4.9 m/s

sg,to=319 m

R=1041 km

8

Wo=922.6 kg

VSO=93.4 km/h

R/C=5.2 m/s

sg,to=365 m

R=1061 km

9

Wo=894.9 kg

VSO=99.8 km/h

R/C=5.6 m/s

sg,to=398 m

R=1100 km

37


• The range requirement is satisfied by all 9 configurations. This means that there is no need to vary the fuel weight between configurations but the range data cannot be used for sizing trade.

• There is 100 kg difference between the heaviest (configuration 1) and the lightest configurations (configuration 9).

• Configurations 3 and 6 do not satisfy the stall speed requirement.

• The take-off distance requirement is violated by configurations 7, 8 and 9.

• Configuration 7, which has the large wing but the small engine violates the rate of climb requirement also.

• Configurations 1, 2, 4 and 5 satisfy all the requirements. 38


• The next step is to crossplot the data in the Table.

• First, for each power value, the take-off gross weights are plotted.

• Then the data is transferred to a wing area (S)-engine power (P) graph.

39


40

Take-off weight, Wo Stall speed, VSO Rate of climb, R/C Take-off distance,sg,to

P=180 hp P=180 hp S=13.4 m2 S=13.4 m2

P=160 hp P=160 hp S=11.2 m2 S=11.2 m2

P=125 hp P=125 hp S=9.0 m2 S=9.0 m2


41


• Then, stall speeds, climb rate and the take-off runs are crossplottedin the most convenient manner.

• The combinations that exactly meet the requirements are transferred to the sizing matrix plot and joined by smooth curves.

42


43


• The Optimum Design is the one satisfying all the requirements, having the lowest take-off gross weight.

• The Optimum Design will therefore will be at the intersection of two constraint curves.

44

Outline

• Introduction

• Requirements

• Baseline Design



o Carpet plot

• Optimum Design

• Conclusions

45

Sizing and Trade – Carpet Plot

• The take-off gross weight plots are superimposed.

• When the data points corresponding to the same wing areas are connected with straight lines, the resulting shape looks vaguely like a carpet, hence the name!

• In the same figure, the data points that exactly meet the stall speed, take-off run and rate of climb requirements are plotted and joined by smooth curves, constituting the constraint lines.

• The optimum design is the lowest point in the carpet plot, i.e. the lightest airplane that meets all the requirements, at the intersection of two constraint lines.

46


47

Outline

• Introduction

• Requirements

• Baseline Design



o Carpet plot

• Optimum Design

• Conclusions

48

Sizing and Trade – Optimum Design

• Optimum Design corresponds to S=9.26 m2 and P=154.5 hp and Wo=915 kg, about 29.7 kg lighter than the baseline design.

• However, the engine found that delivers 155 hp is heavier and to be on the safe side (exceeding the requirements with a comfortable margin) the Optimum Design is chosen as S=9.7 m2, P=160 hp and Wo=926.4 kg.

49

Optimum Design

50

Characteristics Value

Length (m) 7.54

Wing area (m2) 9.7

Wing span (m) 9.34

Aspect ratio 9.0

Maximum take-off gross weight (kg) 926.4

Wing loading, W/S (kg/m2) 95.5

Wing airfoil NLF(1)-0115

Engine Lycoming IO-360-L

Propeller 3 blade/76” diam.

Engine power (hp/kW) 160/119.3

Power-to-weight ratio (kW/kg) 0.129

g-limits +6, -3

Optimum DesignGroup Weight Statementand Performance

51

Group Weight (kg)

Structures

Wing

Horizontal tail

Vertical tail

Fuselage

Main landing gear

Nose landing gear

101.1

7.4

4.7

111.3

57.5

18.7

Propulsion

Engine

Fuel system/tanks

176.4

17.4

Equipment

Flight controls

Hydraulics

Avionics

Electrical

Air conditioning & anti-ice

Furnishings

19.4

2.0

30.0

61.0

18.7

24.3

Total empty weight 649.9

Useful load

Crew

Fuel

Payload

160

96.5

20

Total gross weight 926.4

Performance characteristic Optimum Design Requirement Satisfied?

Range, R (km) 1069 > 1000

Rate of climb, R/C (m/s) 6.6 > 5

Take-off run, sg (m) 278.5 < 300

Stall speed, VSO (km/h) 98.0 < 100

Performance characteristic Optimum Design Requirement Satisfied?

Stall speed, VSO (km/h) 98.0 < 113 (CS/FAR)

Climb gradient @ VCL, γ 13 % > 8.3% (CS/FAR)

Climb gradient @ VA, γ 9.2% > 3.3% (CS/FAR)

52

Sizing and Trade – CompetitorsCharacteristic Optimum Design Grob G115E Grob G120A Slingsby T-67 Aermacchi Sf.260D Cirrus SR.20 Cirrus SR.22 Diamond DA.20

Performance

Maximum speed (km/h) 360 250 319 281 441 287 339 277

Range (km) 1069 1130 1537 753 2050 1454 1943 1013

Rate of climb (m/s) 6.6 5.3 6.5 7 9.1 4.2 6.5 5.1

Take-off run (m) 278.5 461 707 274 451 295 390

Stall speed (km/h) 98 96 102 100 109 104 111 78

Geometric

Length (m) 7.54 7.54 8.60 7.55 7.00 7.92 7.92 7.16

Wing area (m2) 9.7 12.2 13.3 12.6 10.1 13.7 13.7 11.6

Wing span (m) 9.34 10.00 10.19 10.69 8.22 11.68 11.68 10.87

Aspect ratio 9.0 8.2 7.8 9.1 6.7 10.0 10.0 10.2

Weights

Maximum take-off weight (kg) 926 990 1490 1157 1300 1386 1633 800

Wing loading (kg/m2) 95.5 81.1 112.1 91.8 128.7 101.1 119.1 68.9

Empty weight (kg) 650 685 960 794 675 965 1009 529

Empty weight fraction 0.70 0.69 0.64 0.69 0.52 0.70 0.62 0.66

Power

Engine power (kW) 119 139 190 194 195 149 230 93

Power to weight ratio 0.129 0.140 0.127 0.168 0.150 0.108 0.141 0.116

Outline

• Introduction

• Requirements

• Baseline Design



o Carpet plot

• Optimum Design

• Conclusions

53

Sizing and Trade – Conclusions

• These methods are applicable using widely used general purpose computational and graphical tools, without the need for costly, dedicated software.

• These aspects render the methods suitable for student projects, academic purposes and also for design of real airplanes in the conceptual design phase.

• The methods can be developed further to include a higher number of parameters but then their simplicity and visual aspects will quickly fade away and programming the methods as a software will become necessary.

• The methods are applicable to conventional configurations and should be used with care when the designed airplane configuration is unconventional like UAVs, tailless aircraft or aircraft powered by unconventional powerplants like electrical.

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„For an aircraft to fly well, it must be beautiful“

Marcel Dassault (1892-1996)

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Sizing Matrix & Carpet Plots - METUae452/lecture5_sizing_matrix_carpet_so.pdf · For sizing and...

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