This section continues the worksheet implementation of the dynamics formulas for aerodynamic forces and momenta.

## Longitudinal Aircraft Dynamics #8- worksheet implementation of the real dynamics

by George Lungu

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

Worksheet implementation of the force calculation formulas:

– We will calculate these forces in a new area of the worksheet. The formulas are shown to the right of

this page. Next page contains the detailed spreadsheet formulas used. A snapshot of the force formula

area is shown below (column N contains labels):

Lift wing x c l _ wing q d Chord wing Span wing sin(alpha speed)

Lift wing y c l _ wing d q Chord wing Span wing cos(alpha speed)

Lift stabilizer x c l _ stabilizer d q Chord stabilizer Span stabilizer sin(alpha speed)

Lift stabilizer y c l _ stabilizer d q Chord stabilizer Span stabilizer cos(alpha speed)

Drag wing x c d _ wing d q Chord wing Span wing cos(alpha speed)

Drag wing y c d _ wing d q Chord wing Span wing sin(alpha speed)

Drag stabilizer x c d _ stabilizer d q Chord stabilizer Span stabilizer cos(alpha speed)

Drag stabilizer y c d _ stabilizer d q Chord stabilizer Span stabilizer sin(alpha speed)

Drag fuselage x c d _ fuselage d q Frontal area fuselage cos(alpha speed)

Drag fuselage y c d _ fuselage d q Frontal area fuselage sin(alpha speed)

Gravity x 0

Gravity y g m

### Where q is the dynamic pressure expressed as:

q Air_density (v v )

d plane_ x plane_ y

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These are the spreadsheet formulas used:

– Let’s start with the dynamic pressure: O66, “=B4*(D52^2+E52^2)/2”

– Lift_wing_x, O67: “=T65*O66*Chord_wing*Span_wing*SIN(RADIANS(O64))”

– Lift_wing_y, O68: “=T65*O66*Chord_wing*Span_wing*COS(RADIANS(O64))”

– Lift_stabilizer_x, O69: “=W65*O66*Chord_stabilizer*Span_stabilizer*SIN(RADIANS(O64))”

– Lift_stabilizer_y, O70: “=W65*O66*Chord_stabilizer*Span_stabilizer*COS(RADIANS(O64))”

– Drag_wing_x, O71: “=U65*O66*Chord_wing*Span_wing*COS(RADIANS(O64))”

– Drag_wing_y, O72: “=-U65*O66*Chord_wing*Span_wing*SIN(RADIANS(O64))”

– Drag_stabilizer_x, O73: “=X65*O66*Chord_stabilizer*Span_stabilizer*COS(RADIANS(O64))”

– Drag_stabilizer_y, O74: “=-X65*O66*Chord_stabilizer*Span_stabilizer*SIN(RADIANS(O64))”

– Drag_fuselage_x, O75: “=B40*O66*B39*COS(RADIANS(O64))”

– Drag_fuselage_y, O76: “=-B40*O66*B39*SIN(RADIANS(O64))”

– Gravity_y, O77: “=-Mass*9.81”

### The general moment-of-force calculation formula in a 2D Cartesian reference:

– In a Cartesian system of coordinate we need the

coordinates of the moment reference (O), the force origin (P)

Moment

F and the x-y components of the force vector involved.

arm for Fx

– If we have all of them, the following formula applies (the

yP signs in the formula are so that a momentum which produces

x a clockwise rotation has a positive sign just like in Xflr5):

Moment arm for Fy

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### An inventory of the momenta acting on the airplane:

– There are a total of four moment-producing forces involved (they are the lift of the wing, the drag

of the wing, the lift of the horizontal stabilizer and the drag of the horizontal stabilizer). They have a

total of two components each (x and y) and are applied about the quarter chord point of the wing

(half of them) and the quarter chord point of the stabilizer (the other half).

– There are also two additional pure momenta to add to the resultant airplane moment. They are

produced by the wing and the horizontal stabilizer and their normalized interpolated formulas have

been extracted from Xflr5 curves. Being pure momenta (torques) there are no moment arms needed in

any computation and they just require a scale-back to be used in the final moment formula.

– The resultant moment will be calculated about the center of mass or gravity (CG) of the plane. The

center of mass is considered placed on the median fuselage line. The drag of the fuselage and the

gravity produce no moment since they are aligned to the center of mass (gravity) at all times,

regardless of the airplane conditions (for the drag this is a simplifying assumption).

– We already have the formulas for the eight x-y force components (mentioned on the top of this page

and implemented in the first two pages of this section) and we will use their outputs to further

calculate the moment components they generate.

– All we need to be able to do that are the two force arms (one for the wing and one for the

stabilizer) let’s call them levers. The first lever will connect the CG with the quarter point of the wing

and the second lever will connect the CG with the quarter chord point of the horizontal stabilizer.

-In order to be able to use the Cartesian moment calculation formulas from the previous page, we

need the x and y components for the two levers, therefore we need to calculate a total of four levers

(two levers will be parallel to the x axis and to will be parallel with the y axis of the global reference).

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### Calculating the arms of the momenta (levers):

– Let’s estimate the components of the levers in the original “static” system of reference, then using a

rotation by the angle of the airplane find the new lever components in the final “global” system of

reference.

– We can remember that the static system of reference was centered in the center of gravity (CG) of the

plane and in this system of reference the plane was positioned horizontally.

-Let’s estimate x-y coordinates of the levers in the static system of reference (since the static angles of

attack are small, for calculating the y-components of the levers, we will still keep a very good precision if

we assume that the wing and stabilizer are parallel with the fuselage):

y (static)

Chord_stabilizer

Center_Gravity* Length_Fuselage/100 Static_Alpha_Stabilizer

Chord_wing

Height_horizontal

_stabilizer

Static_Alpha_Wing

Height_wing x (static)

0 (center of mass/gravity)

Leading_Edge_Wing* Length_Fuselage/100

Length_Fuselage

Chord_wing/4 (3/4)*Chord_stabilizer

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### The lever components in the static x-y reference:

– The origins of the lever vectors are in the origin (coordinates [0,0]) of the system of coordinates, since

the static system is centered in the center of gravity.

– The tip of the levers are at the quarter-chord point of the wing and horizontal stabilizer respectively.

Because of this fact, the coordinates of the lever tips (which we will soon calculate) are equal to the

lengths of the levers x-y components.

– Analyzing the diagram in the previous page we can see the lever vectors as dotted brown arrows. The

quarter-chord points (where the tips of the lever vectors are) are represented as green dots on the

wing and stabilizer respectively.

– We can write the following relationships valid in the static x-y system of reference:

(Leading_edge_wing Center_Gravity) Chord_wing

Lever_wing_static_x Length_fuselage

100 4

Lever_wing_static_y Height_wing

3

Lever_stabilizer_static_x Length_fuselage Chord_stabilizer

4

Lever_stabilizer_static_y Height_stabilizer

– Since the levers originate in the center of gravity, in order to find the lever lengths in the global

system of reference we will only need to rotate them by “-Alpha_Plane” (the minus sign comes from the

fact that the plane faces left and a positive pitch corresponds to a negative trigonometric angle).

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### Calculating the lever components in the global x-y reference:

– The following general rotation formulas were derived on this blog before. If we rotate a point

(position vector tip) around origin in a x-y system of coordinates we can write the following formulas:

– Using the rotation formulas to the right, we can further

x x

write the final formulas for the x-y lever components in the rotated cos() sin() initial

global reference system (the formulas take in consideration

sin() cos()

y y

rotated initial

the fact that the angle is measured clockwise in our model):

(Leading_edge_wing Center_Gravity) Chord_wing

Lever_wing_x Length_fuselage cos( _ plane) Height_win g sin( _ plane)

100 4

(Leading_edge_wing Center_Gravity) Chord_wing

Lever_wing_y Length_fuselage sin( _ plane) Height_win g cos( _ plane)

100 4

3

Lever_stabilizer_x Length_fuselage Chord_stabilizer cos( _ plane) Height_stabilizer sin( _ plane)

4

3

Lever_stabilizer_y Length_fuselage Chord_stabilizer sin( _ plane) Height_stabilizer cos( _ plane)

4

Pay attention to the signs and the previous page

formulas. Remember that “cosine” is an even function

and “sine” is an odd function! In trigonometry the angles

are measured counterclockwise (we do the opposite).

To be continued…

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