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Why tubing and piping design has
been an art
The reason that tubing design has been an art is because
there is no mathematical formula to calculate what straights,
bends and rotations are required to get from one xyz position
in space to another xyz position in space. In
mathematical circles, this is
called an "inverse" problem and is described more
fully below.
Forward Problem
The "forward problem", although messy, can be
solved directly. Consider the picture at the right
showing a simple piece of tubing consisting of 4 segments:
Color
Segment Parameters
_______
Blue
straight length
= 6.080 inches
Green
bend 70.234 deg, radius = 5.000 inches
Magenta bend 103.574 deg, radius =
3.000 inches
Red
bend 19.646 deg, radius = 3.000 inches
If you know the above tubing segment parameters as well as
the relative rotations between the tubing segments (not
shown), and the xyz location where the tubing solution begins,
as well as the direction it is pointing initially (the
direction vector), then given these parameters of
straights, arc angles, bend radius and rotation angles between
the segments, it is possible to directly calculate the ending
xyz position of the tube as well as the xyz direction it is
pointing. This consists of solving pipe segment
parameters, direction vectors and requires solving 3d
trigonometric rotation transforms shown on the right.
So, if one has a tubing solution that is already
designed, you can calculate the ending xyz
positions and direction vectors directly knowing the starting
point, direction and tube segment parameters.
Inverse Problem
Unfortunately, the converse is not true. In other
words, if the problem is to find a tubing solution given some
desired xyz starting and ending points and directions, there
are no formulas to solve the problem -- and most tubing
problems are precisely this way. The reason is that
there are an infinite number of solutions of tubing bends,
twists and rotations that would work. In
mathematical circles, this is known as an "inverse
problem".
Worse yet, most practical applications require multiple
tubes which must not collide, must fit within packaging
constraints, and must be constructed from available bend
radius's. In addition, many automotive applications
require minimizing flow loss, achieving certain overall tubing
lengths, or other requirements that are often daunting.
Solution
Spectrum5's principals have worked extensively in
solving inverse problems related to photonics and
communications signal processing and have applied this
knowledge to achieve a practical solution to the inverse
tubing design problem.
Although there are no direct solutions to inverse
problems, many can be solved using various gradient descent or
genetic algorithms. The inverse tubing problem is
particularly difficult because it is highly nonlinear and
requires handling hundreds or thousands of variables for
typical problems. Even so, methods exist that can
solve these problems and the convergence properties can be
proven.
Software Capabilities
The Spectrum5 software is a suite of tools that can solve
very simple or extremely complex tubing problems. The
major capabilities of our tools are listed below:
- Flexible Coordinate Systems and Units
- Supports database of available tubing components:
Solutions can converge upon a set of available tubing bend
radius's, arc angles, lengths, etc.
- Powerful simultaneous solution of all tubes, position
and direction vector objectives, all inter-tube collision,
packaging constraints, flow, cost constraints,
etc.
- Tree structured tubing segments allow handling of tee's,
pipe splits and other complex structures
- Components and pipe splits may be in fixed positions, or
may be allowed to "float" in 3d space during the
solution
- Flexible space constraints: Boxes, planes, point
clouds or irregular surfaces are easily handled
- Flexible solution tolerances: Convergence to
acceptable errors for position and direction vector
objectives and minimum collision avoidance
- Export capability to Solidworks to provide integration
with components in 3d CAD environment and support of
popular file formats
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