This tutorial finalizes the implementation of the forces and momenta acting on the plane. It also initiates some hand testing and validation of the overall dynamics of the plane.

## Longitudinal Aircraft Dynamics #9- finalizing the dynamics – preliminary validation

by George Lungu

– This section continues with the dynamics formulas governing our 2D plane.

Worksheet lever calculation formulas:

– Copy the last worksheet and

(Leading_edge_wing Center_Gra vity) Chord_wing

name the new one “Longitudinal_ Lever_wing_x Length_fus elage cos( _ plane) Height_wing sin( _ plane)

100 4

Stability_Model_5”

(Leading_edge_wing Center_Gra vity) Chord_wing

Lever_wing_y Length_fus elage sin( _ plane) Height_wing cos( _ plane)

– There are three buttons, the 100 4

3

“Reset”, the “Run_Pause” and the Lever_stabilizer_x Length_fus elage Chord_stabilizer cos( _ plane) Height_stabilizer sin( _ plane)

4

“CG_visibility” buttons. Make sure

3

Lever_stabilizer_y Length_fus elage Chord_stabilizer sin( _ plane) Height_stabilizer cos( _ plane)

to reassign the right macros to 4

them, macros belonging to the current worksheet. By default, the custom buttons (created by the user

from a macro assigned shape) keep the macros from the original worksheet assigned to them.

Implementation:

– O80: “=(Length_fuselage*(Leading_edge_wing-Center_gravity)/100+Chord_wing/4)*COS(RADIANS(J52))+Height_wing*SIN(RADIANS(J52))”

– O81: “=-(Length_fuselage*(Leading_edge_wing-Center_gravity)/100+Chord_wing/4)*SIN(RADIANS(J52))+Height_wing*COS(RADIANS(J52))”

– O82: “=(Length_fuselage-3*Chord_stabilizer/4)*COS(RADIANS(J52) +Height_horizontal_stabilizer*SIN(RADIANS(J52))”

– O83: “=-(Length_fuselage-3*Chord_stabilizer/4)*SIN(RADIANS(J52))+Height_horizontal_stabilizer*COS(RADIANS(J52))”

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### The force generated moments:

-Using the Cartesian formula to the right (from the

M y y F x x F

previous section) we can identify eight moment O P O x P O y

components generated by the lift and the drag forces.

– All the forces and movements being in the x-y plane the moments will have z-components exclusively.

We can therefore later add the moment effects algebraically.

– Let’s group the moment components by the causing force:

Moment Lever Lift Lever Lift

wing lift wing x wing y wing y wing_ x

Moment Lever Drag Lever Drag

wing drag wing x wing y wing y wing_ x

Moment Lever Lift Lever Lift

stabilizer lift stabilizer x stabilizer y stabilizer y stabilizer_ x

Moment Lever Drag Lever Drag

stabilizer drag stabilizer x stabilizer y stabilizer y stabilizer_ x

In aerodynamics, momenta that have the tendency to raise the nose of the airplane are considered

positive and momenta which have the tendency to lower the nose of the airplane are negative. Pay

attention to signs, I messed it up at first and after seeing weird model behavior I had to come back and

fix it!!!

### Implementation:

– R67: “=-O80*O68+O81*O67“

– R68: “=-O80*O72+O81*O71“

– R69: “=-O82*O70+O83*O69“

– R70: “=-O82*O74+O83*O73“

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### The force generated moments

Scaling back the pure momenta of the wing and stabilizer (from Xflr5):

– So far we calculated the the momenta generated by the forces of lift and drag applied at the quarter

chord points of the wing and the horizontal stabilizer.

– We still need to scale back the cm (moment coefficient) components of the wing and horizontal

stabilizer and add them to the force generated momenta to find the resultant airplane pitching moment.

M Moment

Several sections ago we extracted the

m Span Span

c

real un-normalized pitching moment : m 2 2 2

qc v Air_densit ySpeed

2 2

c Chord

2 2

Moment c q Chord2Span

m d

Where q is the dynamic 1 1

2 2 2

q Air _density Speed _airplane Air_density(v v )

d plane_ x plane_ y

pressure expressed as: 2 2

– We need to apply the above moment formula for both the wing and the horizontal stabilizer.

### Implementation:

– R73: “=V65*O66*Chord_wing^2*Span_wing“

– R74: “=Y65*O66*Chord_stabilizer^2*Span_stabilizer“

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### The balance of forces and momenta around CG:

– Let’s calculate the x-y force resultant components and the

resultant moment:

– R77: “=O67+O69+O71+O73+O75”

– R78: “=O68+O70+O72+O74+O76+O77”

– R79: “=SUM(R67:R70)+SUM(R73:R74)”

### Timing your work – be slow, use your mind, experiment and play:

– Some decades ago, long before the cell phone era, it was fashionable for the young guys to build

electronics. It started with the ham radio craze when everything was based on tubes (I did not catch

that period). I caught however the end of the discrete transistor, the end of the TTL and the beginning

of the CMOS era.

– Most of people used to follow articles from some geek magazine. And of course whatever they built

this way did not work except by accident. This was because they were blindly following a cookbook

recipe and most of them did not have a very minimum background in electronics (Ohm’s law) or an

oscilloscope. Most of them refused to think, trusting the written word. To put it briefly, they were

walking blindfolded listening to a tune and hoping to reach a specific destination.

– When you build anything (mechanics, electronics, optics, software, etc) make sure to go one small step

at a time and after each step do a thorough testing. Play with your model in all imaginable ways before

proceeding with the next step.

– Some people (especially industry bosses) will try to rush you, there is no time, they say. They all do

this. And I tell you the truth, there is no bigger waste of time and resources than running through your

project, increasing the size of the steps, or even worse, finishing the project and leaving the testing and

troubleshooting to the end. The omissions, errors or misunderstandings that slowly creep into the design

this way are very hard (if even possible) to find at the end of the project.

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### A sanity check:

– During building of any of my models I spend most of the time at any stage playing with the model

and thinking. Not reading or coding, but playing with parameters, changing angles of perspective and

in general having fun with the model. Each new step follows after the “fun” from the previous step

starts to “die out”.

– The tutorials are written in a linear fashion and they miss that spirit. You need to be aware of this

and balance the content here with a lot of your own experimentation. Don’t trust anything you read

unless it passes your own experimental tests.

– Let’s verify the model we created so far:

– Cut range M66:R83 then select cell D19 and paste. We

do this as a provisional step to be able to see the forces and

moments while adjusting the input parameters.

-Let’s verify the dimensional parameters first. Make sure

the chart has a fixed x-scale. Modify one parameter at the

time. Try changing the angles of attack of both the wing

and the stabilizer, then adjust their chords, the x position

of the wing and the height of both the wing and the stab.

-Play with the fuselage length and with the center of

gravity. While adjusting CG the plane must shift along the

x axis since the origin of the reference is located in the CG.

-After testing the geometrical parameters of the plane

let’s adjust them while watching the resultant force and

moment.

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– First, be aware that the wing incidence of a normal plane is around +5 degrees and the incidence of

the horizontal stabilizer is slightly negative (-1 or -2 degrees) . Set that, then set the initial speed to 5

m/s and initial launch angle to 0 degrees and hit “Reset”. You must now see forces and momenta for

plane moving horizontally at a rate of 5 m/s. Later you will experiment with the speed values.

– Now you can adjust the angle of attack and see an proportional variation of the lift. If you increase

the main wing angle of attack past 7-10o you might start to see a drop in lift (stall).

– Play with the position of the center of gravity now. Notice that as you bring the CG forward the

momentum becomes negative and increases in absolute value. Exactly the opposite must happen as you

move the CG backwards. Bring it around the leading edge and start adjusting the angle of attack of the

horizontal stabilizer.

– You can see that you can now greatly change the overall moment on the plane between negative and

positive values. The negative values are obtained by setting higher positive angles of attack on the

horizontal stabilizer. Similarly the larger the negative angle of attack of the stabilizer is the larger the

positive resultant moment on the plane will be. Also move the CG a bit forward and try to cancel the

higher absolute value negative moment by giving the stabilizer an increased absolute value negative

angle of attack. Move it even more now, until you

cannot balance it anymore with the stabilizer. Increase

the chord or/and the span of the stabilizer and see

how you can now balance much more moment change.

– Play a lot with span and chord values of both the

wing and the horizontal stabilizer and see how all

the forces and moments are affected. Adjust the

length of the plane and check for the efficacy of the

horizontal stabilizer. Be creative.

To be continued…

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Atlantica – www.wingco.com 6