centre of gravity estimate
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CENTER OF GRAVITY LOCATION
Chapter 8
CENTER OF GRAVITY LOCATION
8.1 Introduction
An airplane in flight can be maneuvered using the aerodynamic control
surfaces; the elevator, rudder, or ailerons. As the control surfaces change the
amount offorce that each surface generates, the aircraft rotates about a point
called the center of gravity. The center of gravity is the average location ofthe weight of the aircraft. The weight is actually distributed throughout the
airplane, and for some problems it is important to know the distribution. But
for total aircraft maneuvering, we need to be concerned with only the total
weight and the location of the center of gravity.
8.2 Methodology used for calculating center of Gravity
An airplane is a combination of many parts; the wings, engines, fuselage,
and tail, plus the payload and the fuel. Each part has a weight associated withit which the engineer can estimate, or calculate, using Newton's weight
equation:
w = m * g
where w is the weight, m is the mass, and g is the gravitational constant To
determine the center of gravity cg, we choose a reference location,
or reference line. The cg is determined relative to this reference location. The
total weight of the aircraft is simply the sum of all the individual weights of thecomponents. Since the center of gravity is an average location of the weight,
we can say that the weight of the entire aircraft W times the location cg of the
center of gravity is equal to the sum of the weight w of each component times
the distance d of that component from the reference location:
W * cg = [w * d](fuselage) + [w * d](wing) + [w * d](engines) + ...
The center of gravity is the mass-weighted average of the component
locations.
http://www.grc.nasa.gov/WWW/k-12/airplane/elv.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/rud.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/alr.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/forces.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/rotations.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/cg.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/weight1.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/geom.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/turbine.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/fuselage.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/elv.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/wteq.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/wteq.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/cg.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/weight2.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/weight2.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/cg.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/wteq.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/wteq.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/elv.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/fuselage.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/turbine.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/geom.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/weight1.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/cg.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/rotations.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/forces.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/alr.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/rud.htmlhttp://www.grc.nasa.gov/WWW/k-12/airplane/elv.html -
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CENTER OF GRAVITY LOCATION
We can generalize the technique discussed above. If we had a total of
"n" discrete components, the center of gravity cg of the aircraft times the
weight W of the aircraft would be the sum of the individual i component
weight times the distance d from the reference line (w * d) with the index
Igoing from 1 to n.
W * cg = SUM(i=1 to i=n) [w * d]i
This equation says that the center of gravity times the sum of "n" parts'
weight is equal to the sum of "n" parts' weight times their distance. The
discrete equation works for "n" discrete parts.
In general the procedure adopted for calculating center of gravity of the
entire aircraft is as follows:-
Estimate the weights of individual masses
Determine the distances(arm) of the above masses from the particular
reference point in the aircraft
Calculate moments by multiplying weights with distances of the
masses
Add the moments of all the masses together
Divide the total moment by the total weight of the aircraft to give an
overall arm
8.3 The arm that results from this calculation must be within the arm limits
for the center of gravity that are dictated by stability conditions. If it is
not, weight in the aircraft must be removed, added (rarely), or
redistributed until the center of gravity falls within the prescribed limits.
Formulae Used:-
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CENTER OF GRAVITY LOCATION
Xcg =
Zcg =
8.4 Fuselage
Fuselage cross section is of rectangular shape. Its variation is as per the
3D view shown in the following figure.
8.4.1 Fuselage nose
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CENTER OF GRAVITY LOCATION
Mass of the section = 69.5 gm
Location of C.G:-
Xi from nose of fuselage = 187.438 mm
Zi from bottom of fuselage = 37.253 mm
Y from the reference point = 0 mm
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8.4.2Fuselage mid section
Mass of the section= 92.7 gm
Location of C.G:-
Xi from nose of fuselage = 475 .99 mm
Zi from bottom of fuselage = 50 mm
Y from the reference point = 0 mm
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8.4.3 Fuselage rear section
Mass of the section=112.27 gm
Location of C.G:-
Xi from nose of fuselage = 854.775 mm
Zi from bottom of fuselage = 63.396 mm
Y from the reference point = 0 mm
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8.4.4 Fuselage tail
Fuselage tail
Mass of the section=14.3 gm
Location of C.G:-
Xi from nose of fuselage = 1256 mm
Zi from bottom of fuselage = 82.5 mm
Y from the reference point = 0 mm
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CENTER OF GRAVITY LOCATION
For complete fuselage
Hence, Total mass of the fuselage, Mfuselage = 289 gm
Location of C.G:-
Xi from nose of fuselage = 592.638 mm
Zi from bottom of fuselage = 53.759 mm
Y from the reference point = 0 mm
8.5 Wing
Wing was designed with 1 spars and 20 ribs. Centre of gravity of aerofoil
is calculated using CATIA software. The co-ordinates of centre of gravity
were obtained in terms of chord length. For ribs, the c.g. is calculated by
multiplying with chord length. Hence total mass of the main wing,
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CENTER OF GRAVITY LOCATION
Mw = g.
Centre of gravity of entire wing was calculated by taking into account
c.g. of individual components.
8.5.1 wing ribs
Mass of the wing ribs = 124.7 gm
Location of C.G.:-
X from nose of fuselage = 437.921 mm
Z from base of fuselage = 76.491 mm
Y from reference point = 0 mm
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8.5.2 wing spar
Mass of the wing spar = 59.49 gm
Location of C.G.:-
X from nose of fuselage = 402.36 mm
Z from base of fuselage = 77.264 mm
Y from reference point = 0 mm
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8.5.3 wing skin
Mass of the wing skin = gm
Location of C.G.:-
X from nose of fuselage = 458.975 mm
Z from base of fuselage = 75.88 mm
Y from reference point = 0 mm
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For complete wing
Mass of the wing = 209 gm
Location of C.G.:-
X from nose of fuselage = 435.925 mm
Z from base of fuselage = 76.488 mm
Y from reference point = 28.179 mm
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8.6 Horizontal tail
8.6.1 horizontal tail ribs
Mass of the Horizontal Tail ribs= 70 gm
Location of C.G.:-
X from nose of fuselage = 1253.602 mm
Z from base of fuselage =82.441mm
Y from reference point = 0 mm
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CENTER OF GRAVITY LOCATION
8.6.2 horizontal tail spar
Mass of the horizontal tail spar. = 11 gm
Location of C.G.:-
X from nose of fuselage = 1236.401 mm
Z from base of fuselage = 82.5 mm
Y from reference point = 0 mm
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CENTER OF GRAVITY LOCATION
8.6.2 horizontal tail skin
Mass of the horizontal tail skin. = 12 gm
Location of C.G.:-
X from nose of fuselage = 1264.852 mm
Z from base of fuselage = 82.429 mm
Y from reference point = 0 mm
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For complete horizontal tail
Mass of the horizontal tail = 93 gm
Location of C.G.:-
X from nose of fuselage = 1257.897 mm
Z from base of fuselage = 82.442 mm
Y from reference point = 7.645 mm
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8.7 Vertical Tail
8.7.1 vertical tail ribs
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Mass of the Vertical Tail ribs. = 5.57 gm
Location of C.G.:-
X from nose of fuselage = 1243.829 mm
Z from base of fuselage = 251.01 mm
Y from reference point = 0.00251 mm
8.7.2 vertical tail spar
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Mass of the Vertical Tail spar = 5 gm
Location of C.G.:-
X from nose of fuselage = 1225.869 mm
Z from base of fuselage = 283 mm
Y from reference point = 0.018 mm
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For complete vertical tail
Mass of the Vertical Tail = 14.6 gm
Location of C.G.:-
X from nose of fuselage = 1250.113 mm
Z from base of fuselage = 263.863 mm
Y from reference point = 0.006 mm
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8.8 propeller
Mass of the propeller = gm
Location of C.G.:-
X from nose of fuselage = -43.971 mm
Z from base of fuselage = 20.36 mm
Y from the reference point = 1.415 mm
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8.9 Landing gear
X from nose of fuselage = 652.693 mm
Z from base of fuselage = -89.804 mm
Y from the reference point = .487 mm
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8.10 Complete Aircraft
Mass of the aircraft =
X from nose of fuselage = 568.506 mm
Z from base of fuselage = 16.067 mm
Y from the reference point = 73.804 mm
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Table-8.1: Tabulated Data For All Components
Component Mass (g)
X(mm) from
nose of
fuselage
Z(mm) from
Base of
fuselage
Y(mm) from
the reference
point
Fuselage 289 593.628 53.759 0
Wing 209 435.925 76.488 28.179
Propeller 22 -43.971 1.415 20.36
Landing Gear 50 652.693 89.804 -0.487
Horizontal Tail 93 1257.897 82.442 7.645
Vertical Tail 14.6 1250.113 263.8630.006
Payload 200 568.506 16.067 73.804
Motor 25 568.506 16.067 73.804
Battery 428 568.506 16.067 73.804
8.9 Motor, battery and payload is placed such that its c.g. coincides with the
c.g. of the aircraft. Whereas servos are arranged symmetrically about the c.g.
of aircraft so that they dont affect position of c.g.
Xc.g = 568.506 mm from nose of fuselage.
Zc.g = 73.804mm from bottom of fuselage.
Yc.g = 16.067mm from the reference point.
8.10 Conclusion:
The location of c.g. is obtained with the help of CATIA software (for
aerofoil) and considering other components separately.
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References:
1. Daniel P. Raymer Aircaft design: A conceptual approachAmerican Institute of Aeronautics and Astronautics, Inc. 370 LEnfant
Promenade, S.W., Washington, D.C., 20024
2. Dr. S. Kamle Aircraft Design lecture notes Department ofAerospace, IIT Kanpur.
3. Dr. E.G. Tulapurkara Introduction to airplane design(Aerodynamic- lecture notes) IIT Madras August 2008
4. Lloyd Jenkinson and Jim Marchman III Aircraft Design Projects forEngineering Students Butterworth-Heinemann Limited, 2003.
5.
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