# Longitudinal Aircraft Dynamics #6 – worksheet implementation of the dynamics equations (a)

In this section, the parameters cl, cd and cm are scaled back to the force of lift, drag and the pitching moment of the aircraft. After that, the numerical  modeling scheme is described together with the macros behind it. At the end, the formulas for the angles of attack of the wing and the horizontal stabilizer are introduced.  ## Longitudinal Aircraft Dynamics #6- worksheet implementation of the real dynamics

by George Lungu

– This section proceeds with finalizing the dynamics formulas governing out 2D plane.
Scaling back the cl, cd, cm coefficients from normalized to real values:

– The data obtained from Xflr5 on the airfoils is two-dimensional data. This means that that the
program simulates a section of a wing of infinite span. The data obtained is associated with just the
airfoil and has no association with the span of the wing. Aerodynamic characteristics recorded include
the lift coefficient cl , the drag coefficient cd and the pitching moment coefficient about the quarter-
chord point (from the leading edge) cm. These coefficients are obtained by calculating the forces and
moments per unit length of the airfoil wing and expressing them as follows:
cl= l / qc

– where l is the measured lift per unit length of the airfoil wing, q is the testing dynamic pressure or
1/2*(rv2), r is the ambient air density, v the speed of the air relative to the airfoil and c is the chord
length of the airfoil section. Similarly:
cd= d / qc

– where d is the measured drag per unit
length of the airfoil wing
cm = m / qc2

– where m is the moment per unit length
acting on the airfoil measured at the quarter
point of the chord from the leading edge.

A cutaway through the flying wing Atlantica – www.wingco.com
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### Transferring the plane to a global system of coordinates (reference):

– The coordinates we have so far are in a preliminary system of reference, let’s call it local or static. In
this system the airplane is fixed and it stands in a perfectly horizontal position (the nose and tail of the
fuselage are at the same level). This is a “showroom” type system of reference with the origin in CG.

– As we model the flight of the glider we need to transfer the glider to a global or dynamic system or
reference (coordinates).

– We have the initial conditions given as parameters (coordinates, speeds – both linear and angular). As
the time simulation advances the new coordinates and speeds will constantly be calculated from the old
ones using Newton’s laws of dynamics and the aerodynamic forces and pitching moment.

– We need to introduce four more variables: x_plane, y_plane, Alpha_Plane and Alpha_Speed. It is
obvious that the speed and the plane will generally have different angles.
y (global)
Alpha_Plane
Alpha_Speed
x_plane x (global)
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### Dynamic considerations – how does the model work?

– Essentially this model is the 2D model of a stick having a mass and a momentum of inertia.

– The stick is being launched from known conditions (coordinates and speeds – both linear and angular)
and knowing the forces (aerodynamic and gravity) and pitching momentum acting on the stick, we can
calculate both linear and angular accelerations within a small time interval (Newton’s second law).

– From accelerations (both linear and angular) we can calculate speeds by integration and then from
speeds (both linear and angular) we can calculate new coordinates by integration. We can do this
incrementally, within small time steps and recursively calculate new coordinates and speeds from the
old coordinates and speeds as a loop.

– For small enough time steps we can use very simple approximations of the integrals so that the
resulting formulas are simple. A flow chart of the calculation loop principle is shown below. This
technique has already been used in several models on this site.

Linear speed and coordinates (old) Angular speed and coordinate (old)
Gravity Aerodynamic extracted functions (lift force, drag force and pitching moment)
Moment
Mass Newton 2nd law (linear) Newton 2nd law (angular)
of inertia
Linear Acceleration Angular Acceleration
Integration Integration Integration Integration
Linear speed and coordinates (new) Angular speed and coordinate (new)
The loop will be implemented as a “copy-the-present-
and-paste-it-in-the-past” infinite do loop macro.
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### Dynamic implementation:

– We saw that our method of
calculation involves an infinite loop
in which the data from the past is
used to calculate the present results
(speeds and coordinates).

– After the results are calculated,
they are copied and pasted in the
location of the past data. It is a recursive feedback scheme in which the model constantly feeds itself
with its own results and since the time step is small the recursive scheme can use simplified formulas
for derivatives and integrals.

-In the snapshot above, the initial conditions (constants) will be contained in the yellow ranges
(D47:G47 and I47:J47), the active formulas (present) will be contained in the green range (B51:J51)
and the past data (constants) will be in the orange ranges (D52:G52 and I52:J52).

– Range K51:K52 handles the index which is an integer equal to the number of loop iterations.

### The macros:

Dim runpause As Boolean
– The role of the Boolean “runpause” variable in the “Run_Pause” macro is to
allow this macro to be started of stopped using the same button. Sub Run_Pause()

– Any time by clicking on the Run_Pause button, the variable is toggled between runpause = Not (runpause)
Do While runpause = True
True or False. Because of this, the conditional Do loop will be stopped if it’s
DoEvents
running or started if it’s stopped. Otherwise, the macro does nothing more than
[D52:K1052] = [D51:K1051].Value
copying data from present calculation area into the past buffer and shifting all
Loop
the history back one time step. Since the copy-paste operation is done over 1000
End Sub
rows, 1000 time steps of historical data will be available for charting later (we
can plot an airplane trail for example, or some other historical data)
Sub Reset()

– The “Reset” macro first clears all the historical data then pastes the initial [D52:K1052].ClearContents
conditions in the “past” row. [D52:K52] = [D47:K47].Value
End Sub
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### The formulas for the angles of y (global) attack of the main wing and horizontal stabilizer:

Alpha_Plane

– The way we defined it, the static
angle of attack is the angle between Alpha_Speed
the fuselage and the airfoil.

– The angle of attack (dynamic)
however is the angle between the
speed of the airplane and the airfoil:
x_plane x (global)
Alpha_wing  Static_alp ha_wing  Alpha_plan e Alpha_speed 
Alpha_stabilizer  Static_alp ha_stabilizer  Alpha_plan e Alpha_speed 
Angles of attack – worksheet implementation:
– Everything in this tutorial is implemented in a new worksheet which is a copy of
“Longitudinal_Stability_Model_3” and it’s named “Longitudinal_Stability_Model_4”

– Since the attack angles will be used in the extracted XFLR formulas, which will be used in the “present”
line, we need to take the “Alpha_Plane” value from the “past line” (cell J52) in order to avoid a circular
reference.

– First let’s calculate “Alpha_Speed” in cell O64: “=IF(E52>0,1,-1)*180*ACOS(-
D52/SQRT(D52^2+E52^2))/PI(). I used this formula since any inverse trigonometric function will only
cover a half circle. This formula will cover a full circle having output values between -180o and 180o. We
need full circle coverage in order to be able to simulate full aircraft rolls. Name cell N64 “Alpha_speed”.

– B1: “=Static_alpha_wing+J52-Alpha_speed”

– B2: “=Static_alpha_stabilizer+J52-Alpha_speed”
To be continued…
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