Graphics Reference
In-Depth Information
Figure 15.10.
Helical and polar filament winding.
Figure 15.11.
Laying tape on a rotating mandrel.
reader should consult appropriate engineering topics such as [RosG64] and [Schw97]
and related journals.
First, consider the mathematics involved in filament winding. Surfaces of revolu-
tion or similar surfaces whose diameter is small compared to the length are good
target objects here. Figure 5.10 shows the two common types of filament winding,
helical and polar. It is clear why filament paths need to stay close to geodesics. In fact,
on a frictionless convex surface it would be impossible to lay a filament in any way
other than along a geodesic since those curves correspond to a state of static equi-
librium. In the presence of friction the filament becomes “sticky” and its path can
deviate from geodesics. The amount of possible deviation would depend on the
amount of “stickiness.” In practice, the filaments need to be kept tight to minimize
slippage, something that is not a problem with tape. Additional problems arise with
concave regions in the surface that can cause “bridging.” Concave regions are also
less of a problem with tape.
Next, we consider tape laying and its mathematics. Figure 15.11 shows the laying
of tape along a rotating mandrel. The basic problem with laying a tape along an object
is that it may not unroll smoothly, but, depending on the curvature of the surface,
may form folds or crinkles in the process. To understand what is going on, one needs
to analyze the problem using differential geometry. A piece of tape is a developable
surface because it is isometric to a subset of the plane
R
2
. It follows from Theorem
9.15.5 in [AgoM05] that the only surface on which one can lay a tape in a problem-
free manner is a surface whose Gauss curvature is zero at every point. For example,
a cylinder is such a surface. On the other hand, since the Gauss curvature of a sphere
is nonzero, it is not possible to lay a tape smoothly onto a sphere.
