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PARAGLIDER DESIGN HANDBOOK
CHAPTER 4. LONGITUDINAL EQUILIBRIUM
 

4.1 Introduction and definitions
4.2 Equations of static equilibrium
    4.2.1 Sum of moments from a  point is equal to zero
    4.2.2 Sum of vertical forces is zero
    4.2.3 Sum of horizontal forces is zero
4.3. Solving the system
4.4. Spreadsheet
4.5. Simplified equilibrium aproximation
4.6. Empirical methods

1. Introduction and definitions

The precise calculation of the balance of the wing in flight is one of the priorities of the designer of paragliders. This chapter details the calculation of the balance from a two-dimensional analysis. It also provides a simplified method and even empirical type.

The longitudinal study of 2D equilibrium in a paragliding, developed from the attached drawing
. The method is based in references [2] and [3], using the same notations.

Longitudinal equilibrium
Fig. 1: Longitudinal equilibrium (click to enlarge).

Forces, points, angles, and lengths involved. Vector represent bold.

Forces:

P = lift
T = drag
M = momentum of aerodynamic forces
Ts= drag of lines
Tp= drag of pilot
Wp = weight of pilot
Wa = weight of wing

Points:

A = leading edge
B = trailing edge
Cp = center of aplication of the aerodynamic forces
P = pilot position
C = calage point
F = foyer aerodynamic
Ga = center of gravity of wing

Angles:

γ = angle of glide
θ = angle between horizon and airfoil chord
α = angle of atack (AoA)

Lengths:

L = length of lines
dist(A,B)= l
dist(A,Cp) = δ
dist(A,C) = σ
dist(Cp,C) = ε

Understanding each of these elements is important to understand the balance of paragliding.

The paraglider is represented at a central profile representative of all the profiles of the wing. The geometry and aerodynamic properties of this profile represent the whole wing with sufficient accuracy, which is simplifying the model.

Forces P, T, amd momentum M, was the resultant of erodynamic forces acting on the profile, and applied in the center of pressure Cp. The values of aerodynamic coefficients Cz, Cx, Cm corresponding to different angles of attack α can be obtained through numerical models (program XFOIL).

Forces Ts and Tp depend on the lines scheme (diameters, lengths, pads) and the position of the pilot and his harness.

Wp and Wa apply in the centers of mass of the pilot and the wing to form a single center of mass, which should be balanced in flight under the cebter of pressure Cp of wing.

The calage point C, is an arbitrarily point defined by drawing a line perpendicular to the chord profile from the pilot position P. The AC distance is very important because it defines the relative lengths of lines and the behavior of the paraglider. Our main goal is to find the most appropriate calage.

The foyer point F. The foyer of not intervening directly in the calculations of balance, but their knowledge is important to know the stable or unstable profile chosen. Its definition is not simple, and can be performed as follows: The point of the chord from the moment produced by the coeficient of lift (Cz) balances (equal and opposite sign) the aerodynamic moment Cm. That is, the distance between the foyer F and the center of pressure Cp, multiplied by the coefficient of lift Cz is equal to Cm. For practical purposes it is enough to know that the foyer is located at 25% of the chord from LE in most profiles.

The angle of glide γ is the angle between the horizontal line and flight path.
The angle θ is angle between the horizon and the airfoil chord.
The angle α is the angle of atack (AoA) formed between the chord and the trajectory of the wing.


2. Equations of static equilibrium

There are three classical equations of equilibrium in two dimensions:

Σ M = The sum of moments from a  point is equal to zero.

Σ V = The sum of vertical forces is zero.

Σ H = The sum of horizontal forces is zero.

2.1 Sum of moments from a  point is equal to zero.

Σ M = 0

By convention we will take moments with respect to the leading edge point A

MA(P) + MA(T) + MA(Ts) + MA(Tp) + MA(Wp) + MA(Wa) + M = 0

(1) + (2) + (3) + (4) + (5) + (6) + (7) = 0

Moments positive counter-clockwise. We analyze in detail each term:

(1) = q0 * Cz * δ * cos (α)

q0 = (1/2) * ρ * S * V2    was the dynamic pressure, where  ρ is the density of air, S the area of the wing, and V the velocity.

P = k * Cz where Cz is the coeficient of lift and k was unitary vector ortogonal to trajectory

(2) = q0 * Cxa * δ * sin (α)

T = i * Cxa where Cxa is the coeficient of drag of the wing and i was unitary vector along trajectory

Cxa = Cxoa + Cxi 

where Cxoa is the drag of shape and friction of wing

and Cxi is the indiced drag  Cxi=Cz2(α)/(π * λ * e)

λ = b2 / S  is the aspect ratio, b span and S surface

e ≈ 0.9  Oswald factor


(3) = (L/3) * Ts * cos (α) + σ * Ts * sin (α) = q0 * Cxs * ((L/3) * cos (α) + σ * sin (α) )

where Ts = q0 * Cxs is the drag of lines

(4) = q0 * Cxp * (L * cos (α) + σ * sin (α) )

where Tp = q0 * Cxp is the drag of the pilot-harness

(5) = - mp * g * ( L * sin (θ) + σ * cos (θ) )

Wp = mp * g  is wheigt of the pilot, g=9.81 m/s2

(6) = - (l/3) * ma * g * cos (θ)

Wa = ma * g  and the center of gravity of the wing in the third the chord (l/3)

(7) = - q0 * Cm  is the aerodynamic moment

Making substitutions in each of the terms, the equation of equilibrium of moments becomes:

q0 * Cz * δ * cos (α)  +

q0 * Cxa * δ * sin (α) +

q0 * Cxs * ((L/3) * cos (α) + σ * sin (α) ) +

q0 * Cxp * (L * cos (α) + σ * sin (α) ) +

- mp * g * ( L * sin (θ) + σ * cos (θ) ) +

- (l/3) * ma * g * cos (θ) +

- q0 * Cm = 0

2.2 Sum of vertical forces is zero.

Positive in the direction of gravity.

Σ V = 0

mp * g + ma * g +

- q0 * Cz * cos (γ) - q0 * Cxa * sin (γ) +

- q0 * Cxs * sin (γ) - q0 * Cxp * sin (γ)  = 0

then,

g * (mp + ma ) - q0 * ( Cz * cos (γ) + ( Cxa + Cxs + Cxp) * sin (γ) ) =  0

Cxa + Cxs + Cxp = CxT is the total drag coeficient

 g * ( mp + ma ) - q0 * ( Cz * cos (γ) + ( CxT) * sin (γ) ) =  0

2.3 Sum of horizontal forces is zero.

Positive left to right

Σ H = 0

q0 * Cxa * cos (γ) +  q0 * Cxs * cos (γ) + q0 * Cxp * cos (γ) - q0 * Cz * sin (γ) = 0

then,

CxT * cos (γ) = Cz * sin (γ)

or,

tan (γ) = CxT / Cz

Glide ratio ( finesse ) = 1 / tan (γ)  by definition

then GR = Cz / CxT

Graphical notes:


Note 1
Fig. 2: Note 1

Note 2
Fig. 3: Note 2


3. Solving the system

Equilibrium equations:

[1]
q0 * Cz * δ * cos (α)  +
q0 * Cxa * δ * sin (α) +
q0 * Cxs * ((L/3) * cos (α) + σ * sin (α) ) +
q0 * Cxp * (L * cos (α) + σ * sin (α) ) +
- mp * g * ( L * sin (θ) + σ * cos (θ) ) +
- (l/3) * ma * g * cos (θ) +
- q0 * Cm = 0

[2]    g * ( mp + ma ) - q0 * ( Cz * cos (γ) + ( CxT) * sin (γ) ) =  0

[3]    tan (γ) = CxT / Cz

Auxiliar equations:

[4]    q0 = (1/2) * ρ * S * V2

[5]    Cxa = Cxoa + Cxi

[6]    Cxi=Cz2(α)/(π * λ * e)

[7]    λ = b2 / S

[8]    CxT = Cxa + Cxs + Cxp

[9]   Cxa = Cxoa + Cxi

[10]    γ = α + θ

[11]    GR = Cz / CxT

Analytical solution (to be developed)
(...)

Numerical method:

The solution of the system can be obtained by analytical or by numerical method. We use the numerical iterative way, based on the balance equation of moments to get their sum equal to zero.  The objective is to obtain a combination of parameters to meet the equations and provide the desired angle of glide.

Of the variables included in the equations above consider the following:

Inputs

ma, mp, g, e, ρ, S, b, L, l, α

numerical estimates

Cxs, Cxp

aerodynamic values obtained numerically

Cz, Cx, Cm, Cp

Results

V, γ, δ, θ, GR, Cxi, Cxa, CxT, q0


4. Spreadsheet


SOLVING THE EQUILIBRIUM EQUATIONS
Wing name: LAB_exa















Airfoil Laboratori






Surface 25.70 m2





Span 10.25 m





Chord 3.00 m





Considered chord 3.00 m





Aspect ratio 4.09






Lines length 6.00 m














Pilot masse 80.00 Kg





Wing masse 5.00 Kg














G 9.81 m/s2





RHO 1.11
Air density




Oswald 0.90
Oswald coeficient













CX pilot 0.015 Pilot drag





CX lines 0.020 Lines drag














Alfa 5.00 Deg 0.087 Rad Flight angle


Cz(alfa) 0.900 Lift





Correcció 3D Cz 0.624






Cz(alfa) correction 3D 0.562






Cx0a(alfa) 0.009






Cm(alfa) -0.042






Cm0 -0.038






Alfa0 -1.700






Cxi(alfa) 0.070

cxo+ci Cxo Ci Cp Cs
CxT 0.114 Total drag 100.00% 69.24% 7.65% 61.59% 13.18% 17.58%









GAMMA(alfa) 0.200 Rad 11.45 Deg 4.94 GR (Cz3D)





8.57 GR max

Theta 0.113 Rad 6.45 Deg



Q0(alfa) 1455.1






V 10.09 m/s 36.33 km/h Speed











Bl -0.046 Cm/Cz





Xcp 0.296 Cp (%1 chord)





CP 0.889 Cp position (m)





Calage 0.433 Calage (m)





Sigma 0.144 Calage chord fraction














Euilibrium of moments to LE
















M1= -725.02
Moment RFA




M2= -59.08
Moment lines




M3= -131.29
Moment pilot drag




M4= 866.82
Moment pilot weight




M5= 48.74
Moment wing weight













SUM= 0.17
Zero in equilibrium













FH= 0
Zero in equilibrium













FV= 0
Zero in equilibrium






Note: The spreadsheet is still in development (there are some details to be corrected). When functional the file was posted  in .ods .xls format for download.


5. Simplified equilibrium aproximation method

A more simplified approach, although based on the same principle of equilibrium is described below:

1. We draw the central wing profile in the planned equilibred flight.

2. We assume that the wing is flying with a glide ratio of GR (as expected for our wing), which corresponds to an angle γ

GR 1 2 3 4 5 6 7 8 9 10 11 12
Angle γ (deg) 45 26.57 18.43 14.04 11.31 9.46 8.13 7.13 6.34 5.71 5.19 4.76

3. We assume that the airfoil incidence corresponding  to max glide ratio or slightly higher (10%), is α. Depending on the profile of the whole wing, usually those angles ranging between 5 deg and 10 deg. It is preferable to a plan to estimate the glide at the highest speed, so we start by estimating the angle at 5 deg.

4. Then draw the representative profile of the wing in flight. The angle of the profile chord and the horizon is θ = γ - α

5. We locate the position of center of pressure Cp for the chosen profile and angle of incidence α.

6. We locate the position of the pilot P in the same vertical of the center of pressure Cp, at a distance equal to the lines length.

7. From the pilote point P draw a perpendicular to the line of chord wing to determine the calage C of the wing.

Simplified method
Fig. 4: The simplified equilibrium method

The method is simple and can be used for a first approximation. But has the disadvantage of having to estimate a priori the glide angle (deductible by the general characteristics of the wing) and the angle of attak at maximum glide ratio (more difficult to estimate). Some corrections on the length of the riser or lines in the prototype are necessary to determine the calage just ideal.

6. Empirical methods

Many designers use empirical methods based on their experiences with previous models. These methods are simple and effective.

A very effective method is to study the profile and the relative length of the lines of a existing paraglider with similar characteristics to the project. We only need to study the middle section and deduct graphically the calage employee, with and without speed system.


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