 PARAGLIDER DESIGN HANDBOOK
CHAPTER 8. LINES

8.1 Geometrical design
8.2 Brakes and brake distribution
8.3 Elasticity

EN EDICIÓ

8.1. Geometrical design

The design of lines is simple, but it is necessary to apply mathematical knowledge of analytic geometry in space.

My method is based on the following concept:

"A line with its starting point in "1" (x1,y1,z1) and which branches ending in
"N" points located on the canopy, follows a 1-2 alignment, where "2"
(x2,y2,z2) is the center of gravity of the "N" points located in the canopy"

- This is only an empirical intuition, and probably prove mathematically
possible. But to me, it works correctly.

- This serves for any of the lines of the ramifications, starting by risers,
lower lines, intermediate lines,...

- It also presupposes that the shape of the canopy is a solid, meaning that
the final shape of the canopy is very similar to the project in 3D CAD.

- The center of gravity of the "N" points located in the canopy, can be
calculated using different approaches:

Case 0) I use parameter "0" in the program. to calculate the center of gravity
of the N points assuming that each anchor point in the canopy supports the

Case 1) I use parameter "1" in the program. to calculate the center of gravity
of the N points, using ponderated mean. Ponderation (weighted average) is
calculated by assigning different loads in anchors A, B, C, D, and in
proportion to the chord length of each profile... This is more real.

Case 2) Is it possible to do more realistic approach taking into account some
special weights in profiles (more weight in profiles adjacent to profiles
without anchors) and the inclinations of the lines. That I'm still studying.
But approaches 0) and 1) work,

CASE 0 in more detail:
---------------------

The first approach is to assume that the lift is uniform throughout the wing.
This is false, but it can work. This is the approach I call "0". This is not a
numerical value, only one label. This approach is determined by calculating
the geometric center of the set of points of anchorage.

See accompanying notes, page 1:

If 1,2,3....N is set of "N" points, calcule the geometric center or "center of
gravity" as:

G1=(G1x,G1y,G1z)

where:

G1x=(1/N)*(p1x+p2x+...+pNx)
G1y=...
G1z=...

I'm talking about the "center of gravity" or "center of mass", but in any case
I am referring to Earth's gravity, just mean the concept.

CASE 1 in more detail:
---------------------

The second approach is to asume that lift is not uniform, and in stabilized
flight, in each anchor point is:

A=35%
B=35%
C=20%
D=10%

This values specific to each model wing and also vary when using the brakes or
accelerate. This is specified in line 3 of section 18 of the data file:
http://www.laboratoridenvol.com/leparagliding/archives2/leparagliding.txt

Also assume that the lift, in each profile, is proportional to its chord.

Thus, it is possible to estimate the load on each anchor point.

To calculate the center of gravity G, necessary to make a weighted average
with the weight or "lift". If weight in the anchor point number "i" is "wi"
then:

G1x=(p1x*w1+p2x*w2+...+pNx*wN)/(w1+w2+...+wN)
G1y=...
G1z=...

Returning to the drawing on page 1: The point G is the center of mass of N
points. It is a virtual point. Used to calculate the "line of action" 1-G
between the starting point 1 of the line and its end point 2, in each segment
calculated.

Page 2 (above) shows a very intuitive form, the validity of the method
(examples a,b,c,d). Page 2 (below) tries to demonstrate the method validity,
but that unresolved (but not required).

Apologies if my explanations include an excess of mathematics.

I hope that the general concept is understood. It is possible to do all these
calculations analytically (only using mathematical formulas) or by geometric
form, using programs like AutoCAD in 3D.

(...)

"How do you determine the length of the dotted line from the lift centre to
the point where the lines from the nodes will join?"

Yes. Easy. The distance between point "2" and point "G" is defined as a data
input in the program (or method), chosen by the designer. If the point "2"
located near the point "1", then the ramifications are long. If the point "2"
located near the "G", then the ramifications are shorter.

Following the same reasoning. The point "2" may be considered as the starting
point "1" in the next level ramifications. At this level, the new point "G" is
calculated using only a subset of the previous anchors. The new distance
between the new point "2" and the new "G", also defined as data entry.

And so on.

8.2 Brakes and brake distribution

The ramifications of the brake lines are calculated geometrically from the  brake handle to the trailing edge.This calculation can be made exactly the same way as for the rest of the lines. Thus, by pulling the handle, the entire trailing edge is folded down uniformly.

However, it can sometimes be interesting to define a distribution of braking, which not act uniformly along the trailing edge. To do this, we can define various distributions of braking, so that braking begins acting first in one or several areas of the trailing edge along the wingspan.

The distribution of brake, which makes is elongate in some cm (vertical axis) the theoretical length that corresponds to a top branch at a point along the span (horizontal axis). Sometimes it may be desirable to extend the branches in the center wing, and in the wing tips. This is a very important, when adjusting a prototype. Sometimes it is advisable to test various distributions of brake.

8.3 Elasticity

(...)