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Optimizing Geometric Parameters for Maximum Lift in Airplanes - Prof. Jean-Jacques Chattot, Study Guides, Projects, Research of Aerospace Engineering

University of California - DavisAerospace Engineering
Prof. Jean-Jacques Chattot

A project focused on optimizing the geometric parameters of an airplane to maximize lift capabilities. The parameters adjusted include the main wing chord, tail chord, tail span, location of the leading edge of the tail, and the total mass of the airplane. The project aims to ensure stability, limit the maximum coefficient of lift on the main wing, and maintain a minimum take-off velocity. Results include tables and figures illustrating the maximum take-off velocity for different total masses and settings and maximum payload for different configurations.

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EAE 127 Project 7
Airplane Equilibrium Analysis
Fall 2002 Due Thursday, December 5th (in class)
In this project, we will study the influence of some dominant parameters on the equilibrium and lift
capabilities of the Aerobrick’03 airplane, that will represent UCD at the SAE Aero Design West
competition next June. The rules can be found at http://www.sae.org/students/aerowest.htm .
The parameters that need to be changed are :
mass = mass of the airplane including payload (kg)
cxm = chord of the main rectangular wing (m)
bt = tail span (m)
cxt = chord of the tail horizontal surface (m)
xlt = location of the tail leading edge (m) < 1.4m
All the other data remain at the given value in the aerobrick.data file, in particular the span
bm=1.829m (72”), which is imposed this year.
The center of gravity is placed to satisfy a 4% positive static margin for the airplane and is
limited to the range
mxm
cg
5.03.0
(1)
due to fuselage constraints. The reference length is lref=1.4m, the length of the fuselage including
the tail boom.
The mass of the wing, hence the mass of the empty airplane varies with the chord cxm according
to
xmempty
cm 6667.60.1
(2)
The payload is
emptypayload
mmassm
(3)
The maximum lift coefficient of the wing is clm=1.8. Finally, the maximum take-off velocity (m/s) is
the result of the acceleration phase and depends primarily on the total mass of the airplane. The
velocity obtained at 55m, allowing rotation and take-off within 60m (200ft) is given by
32
0006.0041.017.10.25 massmassmassv
offtake
(4)
To leave the runway at take-off, the climb angle is required to be
deg3
(5)
We will not consider winglets at this stage.
Within these constraints, find a configuration (mass, cxm, bt, cxt, xlt) that can take-off. Find the
payload. Explore the design space and try to find the best possible configuration to maximize the
payload. Attach to your report (Appendix) the data file aerobrick.data corresponding to your best
configuration.
This project is fun! After a while you will get addicted to play with the parameters and gaining
more payload!
pf3
pf4
pf5
pf8
pf9
pfa

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EAE 127 Project 7

Airplane Equilibrium Analysis

Fall 2002 Due Thursday, December 5th (in class)

In this project, we will study the influence of some dominant parameters on the equilibrium and lift capabilities of the Aerobrick’03 airplane, that will represent UCD at the SAE Aero Design West competition next June. The rules can be found at http://www.sae.org/students/aerowest.htm. The parameters that need to be changed are : mass = mass of the airplane including payload (kg) cxm = chord of the main rectangular wing (m) bt = tail span (m) cxt = chord of the tail horizontal surface (m) xlt = location of the tail leading edge (m) < 1.4m All the other data remain at the given value in the aerobrick.data file, in particular the span bm=1.829m ( 72” ), which is imposed this year. The center of gravity is placed to satisfy a 4% positive static margin for the airplane and is limited to the range

  1. 3 mxcg  0. 5 m (1) due to fuselage constraints. The reference length is lref=1.4m , the length of the fuselage including the tail boom. The mass of the wing, hence the mass of the empty airplane varies with the chord cxm according to mempty  1. 0  6. 6667  c xm (2) The payload is m (^) payloadmassm empty (3) The maximum lift coefficient of the wing is clm=1.8. Finally, the maximum take-off velocity (m/s) is the result of the acceleration phase and depends primarily on the total mass of the airplane. The velocity obtained at 55m , allowing rotation and take-off within 60m ( 200ft ) is given by

v 25. 0 1. 17 mass 0. 041 mass^2 0. 0006 mass^3

take  off       

To leave the runway at take-off, the climb angle is required to be   3 deg (5) We will not consider winglets at this stage. Within these constraints, find a configuration ( mass, cxm, bt, cxt, xlt ) that can take-off. Find the payload. Explore the design space and try to find the best possible configuration to maximize the payload. Attach to your report (Appendix) the data file aerobrick.data corresponding to your best configuration. This project is fun! After a while you will get addicted to play with the parameters and gaining more payload!

Project 7 : Airplane Equilibrium Analysis

Fall 2002

University of California, Davis

EAE 127

Professor J. J. Chattot

December 5, 2002

Methods

In this project the flow was assumed to be incompressible, viscous, 3D, and

rotational. The viscous drag on both the tail and the main wing were assumed to be

approximately that of a flat plate.

The performance of the plane was determined for equilibrium conditions. These

force and moment equilibriums were:

 F^ z  Lm  Lt  W ^0 (7.1)

 Fx^  Thrust  Dtotal ^0 (7.2)

  m lm ^ xm t lt^ xt W xxg

c

L x

c

M L x (7.3)

d 

dcmo

Geometric constraints on the layout of the plane were also in place, for example

the span of the main wing was limited to 1.828 meters and the length of the fuselage was

limited to 1.4 meters. The to ensure stability, static margin was also a fixed +4%, the

center of gravity was located ahead of the aerodynamic center by 0.056 meters:

x cg  xac  0. 056 m (7.5)

This center of gravity was also limited to the range of:

0. 3 m x cg  0. 5 m (7.6)

The maximum coefficient of lift on the main wing was also limited to below 1.8 ensure

that the Selig airfoil would operate outside of stall.

The empty airplane mass as a function of solely the main wing chord was:

mempty  1. 0  6. 6667 c xm (7.7)

Thus the payload was:

m payload  mtotal  m empty (7.8)

Under a constant thrust of the engine down the fixed 55m runway, the maximum

attainable velocity before the plane must take-off was given by:

vtake  off   mtotal  mtotal  mtotal (7.9)

The velocity of the plane in equilibrium was thus required to be less than the maximum

allowable take-off velocity.

The climb angle was also limited to ensure take-off:

AR

d

dclm 

 

For the experiment, the code was first run to determine the aerodynamic center of

the configuration, from this with Equation 7.5, the center of gravity was determined and

put into the data file. The code was run again, and different tail setting angles were tested

until a coefficient of lift on the main wing was found to be just below 1.8 (~1.799). Once

this criteria was achieved, the climb angle was checked to ensure it satisfied Eq. 7.10, and

the velocity was check to ensure that it satisfied Eq. 7.9. If all the conditions were

satisfied, the total mass was increased, and the process repeated. Once the configuration

became invalid for a mass, the highest valid mass was recorded as well as the

configuration.

Table 7.2 Settings and maximum payload for different configurations.

Total Mass (kg) cxm (m) bt (m) cxt (m) xlt (m) ttd () Empty Weight (kg) Max Payload Weight kg Lb 14 0.3 1 0.1 1.3 3.00001 10.99999 24. 15.5 0.4 1 0.1 1.3 3.66668 11.83332 26. 14.5 0.32 0.75 0.2 1.2 -0.681 3.133344 11.36666 25. 15.5 0.4 0.75 0.2 1.2 -5.225 3.66668 11.83332 26. 13.5 .5 0.75 0.2 1.2 -10.595 4.33335 9.16665 20. 15 0.4 0.5 0.2 1.2 -15.58 3.66668 11.33332 24. 15.5 0.4 0.75 0.15 1.25 -7.515 3.66668 11.83332 26. 16 0.39 0.8 0.15 1.25 -5.749 3.600013 12.39999 27. 16 0.38 0.9 0.15 1.25 -3.293 3.533346 12.46665 27. 16 0.37 1 0.15 1.25 -1.43 3.466679 12.53332 27. 16 0.41 1.1 0.15 1.25 -1.85 3.733347 12.26665 27. 16.5 0.41 1.1 0.25 1.25 1.588 3.733347 12.76665 28. 16.5 0.4 1.1 0.25 1.25 1.846 3.66668 12.83332 28. 17.5 0.4 1.828 0.17 1.3 3.8258 3.66668 13.83332 30. 17.5 0.4 1.828 0.18 1.3 4.061 3.66668 13.83332 30. 17.5 0.39 1.828 0.17 1.3 3.9997 3.600013 13.89999 30. Payload vs Main Wing Chord 0 5 10 15 20 25 30 35 0 0.1 0.2 0.3 0.4 0.5 0. Main Wing Chord (m) Payload (lb) Figure 7.1 Plot of total payload weight versus main chord length.

Discussion

Although from this selection of configurations it was clear that many trends could

be seen in the final payload. Initially, from the first two configurations with identical

tails, the total mass increased by from 14kg to 15.5kg, while the payload increased by

almost 2 pounds, when the main chord length was increased to 0.4m from 0.3m. Even

though the larger main wing increased the weight of the empty plane by almost 1 pound,

its performance gain was clear. To see if this trend would hold again, this time the aspect

ratio of the tail was reduced, and chords of 0.3m, 0.4m and 0.5m were examined. Again

the largest payload was for a chord at 0.4m, with the performance of the 0.5m being by

far the worst. So from this point on, the remaining configurations were done with main

wing chords at or very near 0.4 meters.

With the main wing more or less determined, the tail became the next area of

focus. From the previous configuration with a 0.4 m main wing chord, the tail span was

reduced from 0.75m to 0.5m with the remaining values constant. Although the reduction

in performance was not significant it still reduced the total payload by 1 pound. Then the

tail span was made again 0.75m, and this time the tail chord was reduced. Surprisingly

the performance increased, with payload increasing by 1 pound, for a total mass of

15.5kg. The tail span was then increased to 0.8m from 0.75 and again the payload

increased. To see if this trend continued, the tail span was increased to 0.9m, 1.0m, and

1.1m. Although a payload gain was not immediately apparent, its benefit in reducing

equilibrium velocity was seen. Since for these configurations used positive setting angles,

the performance benefits were probably due positive lift being created by the tail, thereby

reducing the velocity needed by the main wing to generate lift. With a tail span of 1.1m,

the main wing chord was increased to 0.41m and the chord of the tail was increased to

0.25m. This change was enough for the three parameters of clm, climb angle, and velocity

to be satisfied for an increased total mass of 16.5kg. The main wing chord was reduced to

0.4m, and the same total mass was again lifted, but payload was increased with the

reducing in mass in the wing.

At that point it seemed that—at least aerodynamically—the larger the tail span,

the lower the equilibrium velocity required. So under the constraints and definitions of

empty weight given, the tail span was pushed to the same length of that of the main wing,

Conclusion

So from the results, it was clear that aerodynamically, the best performing

configuration utilized a tail that generated lift, reducing overall required velocity critical

during take-off. This configuration utilized an enormous tail with a span that of the main

wing. Just the tail itself would add approximately 0.5kg to the total mass of the plane

from a tail with half its span. When the added structural reinforcements are taken into

consideration, the actual performance gains, although present, not be significantly greater

than the best performing plane with a more conventional tail. Without the large tail, a

payload of 28.3 pounds was still lifted by the best conventional configuration.

Specifically for the configurations tested, the optimal chord size was 0.39m. But

as for overall trends, it was clear from the results that the optimal main wing chord length

was approximately 0.4 meters or 16 inches. Chords much larger increased the induced

drag as well as the weight of the airplane, and chords much small reduced the lift

generated by the main wing, so both were undesirable. Another finding was that a larger

tail can also be very beneficial when used as an additional lifting surface. Also, the

performance also appeared to increase, the further back the tail, as well as the

aerodynamic center and center of gravity.

Clearly for future experimentation, the code can be simply improved by

introducing a loop to calculate the center of gravity given a desired static margin, as

opposed to having to run the code and modifying the input file before useful results could

be found. Secondly, the equation of the mass of the empty weight can clearly be

improved to more accurately represent the mass of varying tail size as well. Even though

the flat plate approximation of viscous drag is a good one, it would obviously be better to

include a more accurate calculation of viscous drag once good configurations are

selected. As for obtaining more data, it would obviously be best to run more and more

configurations, fixing all but one parameter until whole families of performance curves

are created.

Appendix

Best Configuration:

mass=17.

xcg=0.

cxm=0.

bt=1.

cxt=0.

xlt=1.

Input File: aerobrick.data

101 =itx maximum number of iterations############################## 0.2 =omega relaxation factor for Newton's Method################### 1.225 =rho density of air (kg/m3)################################## 0.00001456 =anu kinematic viscosity (m2/s)############################## 17.5 =mass of airplane & cargo (kg) 0.481226 =xcg location of center of gravity from the nose tip (m) 1.828 =bm main wing span (m)######################################### 0.39 =cxm root chord of main wing (m) 0.241 =xlm location of main wing leading edge (m)####################

  1. =tmd setting angle of main wing from fuselage axis (deg)####### 0.086 =dm relative camber of main wing############################### 0.97 =em efficiency coefficient of main wing######################## 1.828 =bt tail span (m) 0.17 =cxt root chord of tail (m) 1.3 =xlt location of tail leading edge (m)
  2. =ttd setting angle of tail (deg)############################### 0.0 =dt relative camber of tail#################################### 0.96 =et efficiency of tail######################################### 1.77 =awit downwash coefficient near tail########################### 1.4 =lf length of fuselage (m)##################################### 0.095 =hf equivalent height of fuselage (m)##########################
  3. =Cdbrake airbrake drag coefficient############################# -5. =taud thrust angle (deg)####################################### '14x5N apc' =propeller reference########################################### 0.00000000 48.2972675 =vr(i) & tr(i) from theory (m/s) & (N)######### 2.17220473 44.9136963 ############################################### 4.30927229 42.0275803 ############################################### 6.42771482 39.3718071 ############################################### 8.54280853 36.9088402 ############################################### 10.6713619 34.6041908 ############################################### 12.8300104 32.4290695 ############################################### 15.0363255 30.3591923 ############################################### 17.3091049 28.3671875 ############################################### 19.6763783 26.4166374 ############################################### 22.1696682 24.4768085 ############################################### ###############lines marked with the # sign are not to be changed##############