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Center of Gravity (CG) and Stability
What happens as the center of gravity is moved from the nose of the airplane
to the tail of the airplane?
Experiment Setup
To find out, we conduct a little experiment using
MultiElement Airfoils as a virtual lab. The experiment
starts by placing the leading edge of a NACA 0012 airfoil of chord length
c, at the origin, i.e. at (x,y)=(0.0, 0.0). The airfoil is rotated to an
angle of 0.0 degrees with respect to the x-axis. We will refer to this airfoil
as the main wing. Next we place the leading edge of another NACA 0012 of
chord length c/4 at a location (x,y)=( 1.5 c, 0.0 ). The airfoil is rotated
to an angle of -5.0 degrees with respect to the x-axis. We refer to this
second airfoil as the tail or horizontal tail. In fact, it is an all-movable
tail, i.e., the whole horizontal can rotate as opposed to just elevator rotation.
The following figure (Fig. 1) shows the setup of the experiment:
Fig. 1- Wing and Tail Configuration, AOA = 4.0 deg.
(Center of Pressure Contours)
Procedure
The experiment is conducted by placing the center of moment , (i.e. center
of gravity for our purposes) at various locations along the x-axis. The points
that we pick are Xcg = 0.0, 0.20, 0.40, 0.60 and 0.80. We then
compute the moment coefficient about these points for various angles of attack.
Next we ( the program, actually) graph the moment coefficient (Cm) versus
angle of attack (AOA) using the center of gravity point (Xcg)
as a parameter. This graph is show below in Fig 2.:
Fig 2. - Moment Coefficient Versus Angle of Attack (Linear)
A companion figure is the lift coefficient (Cl) versus angle of attack curve.
This curve is linear under the assumption of strictly inviscid flows. We
shall refer to this curve as Fig 3.
Fig 3 - Lift Coefficient Versus Angle of Attack (Linear)
Pause for Assumptions
Before moving on to results, lets look at some assumptions that we make both
explicitly and implicitly (due to the virtual experimental apparatus)
1. The flow is two-dimensional, i.e. both the main wing and tail are of infinite
span. Consequently, there is no tail downwash due to the main wing.
2. The tail is all movable.
3. We consider straight wings, (sweep back and twist could change the results
and conclusions).
4. A negative moment forces the nose of the airplane downwards (towards the
ground). A positive moment forces the nose of the airplane upwards (towards
outer space).
Results
Lets us refer to Fig. 2. We see that for Xcg = 0.0 and
Xcg = 0.2 the slope of the Cm versus AOA line is negative, i.e.
Cm decreases with an increase of angle of attack. Along this line, there
is a point where Cm is zero. For Xcg = 0.4, the Cm appears vary
slightly with angle of attack (for discussion purposes, let us assume that
it is constant with respect to angle of attack). Therefore this point, where
the moment coefficient is independent of angle of attack, is called the Neutral
Point of the airplane. It acts as the Aerodynamic Center for the wing-tail
combination. For Xcg = 0.6 and Xcg = 0.8 the slope
is positive, i.e., Cm increases with an increase of angle of attack.
The airplane is said to be TRIM at the angle of attack where Cm about
Xcg is equal to zero. At Cm = 0 there are no external forces acting
to change the pitch (angle of attack) of the airplane. At Xcg
= 0.0 the angle of attack for trim is 4.0 degrees and the corresponding lift
coefficient (from Fig. 3) is 0.28. At Xcg = 0.20, the angle of
attack for trim is 6.2 degrees and the corresponding lift coefficient is
0.54 degrees. At Xcg = 0.4, there is no angle of attack corresponding
to Cm = 0. This is the Neutral Point. At Xcg = 0.60, the airplane
is trim at -3.7 degrees with a corresponding lift coefficient of -0.63. Finally
at Xcg =0.80, the angle of attack for trim is -1.0 with a
corresponding lift coefficient of -0.31.
Since in the latter two cases (Xcg equal to 0.6 and 0.8), the
lift coefficient is zero, we must conclude that the airplane will not fly
with a negative lift coefficient and Trim cannot be achieved.
What happens, say, if a gust perturbs the trim angle of attack. For example
for the case where Xcg = 0, we see that if the nose of the airplane
is force upwards by a sudden gust of air, then the moment about the Center
of Gravity will be negative, i.e., the aerodynamics will act to force the
nose downwards and correct the problem. This is also true for Xcg
= 0.2. Likewise, if a gust forces the nose downwards, the moment about the
CG will become positive and force the angle of attack back to the trim location.
In general (according to our assumptions), the airplane will behave in a
similar manner if the CG is located at any point about which the Cm slope
is negative.
The above is a Stable situation.
In the cases of Xcg = 0.6 and Xcg = 0.8, we can conclude
that if a gust of air forces the nose upwards, the moment will not correct
the problem but will act to worsen the situation. The moment will act to
increase the angle of attack even more. The same is true for a downwards
gust.
This an unstable situation.
Conclusion
We can conclude (for our configuration), that if Xcg is designed
ahead of the neutral point then the slope of the Cm versus AOA curve is negative
and the airplane would be stable.
Food For Thought
1. What if the tail angle of attack is positive, how can we attain stability
(if at all). Can we attain both trim and stability?
2. What if different airfoils are used?
3. What if canards are used?
About MultiElementAirfoils
MultiElement Airfoils is an interactive Windows Software Tool for simulating
multiple airfoils within close proximity of each other. It is the easiest
way to estimate the lift and drag acting on slotted flaps, slats, biplanes,
airfoils in ground effect, multielement spoilers, hydrofoils, mast-sail
interactions and other airfoil configurations.
More Information.
How to Buy
MultiSurface Aerodynamics can be purchased
here.
The software can also obtained as part of Dr. Hanley Aerodynamics Suite-
Online. More information is available
here.
Compare Software
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Glider |
WingTail |
WAPlus |
VisualFoil |
MultiElement |
MultiSurface |
| Airfoil Analysis |
No |
Yes |
Yes |
Yes |
Yes |
Yes |
| Airfoil/Plain Flap |
Yes |
Yes |
Yes |
Yes |
Yes |
Yes |
| Airfoil/Slotted Flap |
No |
No |
No |
No |
Yes |
3-D/Camber |
| Multiple Airfoils |
No |
No |
No |
No |
Yes (20) |
3-D/Camber |
| Airfoil Stall Prediction |
Yes |
Yes |
Yes |
Yes |
Yes |
Yes |
| 3-D Wing (Tapered) |
Yes |
Yes |
Yes |
No |
No |
Yes |
| 3-D Wing (General) |
No |
No |
Yes |
No |
No |
Yes |
| Biplanes |
Yes |
Yes |
No |
No |
2-D |
Yes |
| Multiple 3-D Wings |
2 |
2 |
No |
No |
2D |
Yes (30 ) |
| Wing & Tail |
Yes |
Yes |
No |
No |
Yes (2D) |
Yes |
| Wing & Canard |
Yes |
Yes |
No |
No |
Yes (2D) |
Yes |
| Ground Effect |
Yes |
Yes |
No |
No |
Yes |
Yes |
| Atmospheric Table |
Yes |
Yes |
Yes |
Yes |
Yes |
Yes |
| Water Properties |
Yes |
Yes |
Yes |
Yes |
Yes |
Yes |
| NACA Airfoil Library |
Yes |
Yes |
Yes |
Yes |
Yes |
Yes |
| Custom Airfoils |
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Plus! |
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About Dr. Hanley
Dr. Patrick E. Hanley,
is the owner and founder of Hanley Innovations, a small business specializing
in the development of aerodynamics and fluid dynamics simulation software
for education and industry. Dr. Hanley earned his B.S. degree (summa cum
laude) in aerospace engineering from Polytechnic Institute of New York
and his S.M. and Ph.D. degrees from the department of Aeronautics and
Astronautics of Massachusetts Institute of Technology (MIT). He also completed
a minor in the area of management of innovation and technology at MIT's Sloan
School of Management.
After graduating from MIT, Dr. Hanley joined the Mechanical Engineering
faculty at the University of Connecticut where he formulated and taught
courses in aerodynamics, compressible fluids, introductory fluid mechanics
and heat transfer. As a faculty member, he won the highly competitive National
Science Foundation Research Initiation Award, the NASA-ASEE Summer Faculty
Fellowship and three consecutive research awards from NASA Lewis Research
center to study compressible viscous flows in turbomachinery using pseudospectral
methods. This research led to the successful education of four (4) Ph.D students
and four (4) Masters degree students. In addition Dr. Hanley can be credited
with a number of publications including the pioneering work in multi-domain
pseudospectral methods for compressible viscous flows entitled "A Strategy
for the Efficient Simulation of Viscous Compressible Flows using a Multi-domain
Pseudospectral Method" which can be found in Journal of Computational Physics,
Vol 108, No. 1, pp. 153-158, September 1993.
As owner and chief software author of Hanley Innovations, Dr. Hanley
has written a number of software packages including
AirfoilBrowser,
Airfoil Organizer,
Science Graphs,
VisualFoil,
ModelFoil,
Aerodynamics in Plain English,
Center of Gravity Calculator,
WingAnalysis,
SmockSoft,
PerpeturalPaper amongst other titles.
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