Curvilinear duct fabricated with additive manufacturing
Curvilinear ducts manufactured by depositing one or more runs of material in a conjoined helix, a conjoined plurality of conjoined planar spirals, and a plurality of conjoined conical spirals.
1. A duct with a duct axis {right arrow over (d)}(t) described by the vector function:
{right arrow over (d)} ( t )=< a ( t ), b ( t ), c ( t )>,
the duct comprising:
a run of material having a longitudinal axis {right arrow over (r)}(s) described by the vector function:
{right arrow over (r)} ( s )=< f ( s ), g ( s ), h ( s )>
that forms a helix around the duct axis {right arrow over (d)}(t);
wherein the helix is a rectangular-with rounded-corners helix;
wherein the duct axis {right arrow over (d)}(t) is characterized by a curvature κ(t) and a center of curvature {right arrow over (p)}(t) described by the vector function:
{right arrow over (p)} ( t )=<α( t ),β( t ),γ( t )>;
wherein the curvature κ(t)>0;
wherein the run of material is characterized by a conjoining axis j(t) at {right arrow over (r)}(s) such that j(t)∝∥{right arrow over (r)}(s)−{right arrow over (p)}(t)∥; and
wherein t is a real number in the domain t: [t 1 , t 2 ] and s is a real number in the domain s: [s 1 , s 2 ].
2. The article of claim 1 wherein the run of material has an isolating axis i(t) at {right arrow over (r)}(s) such that i(t)∝∥{right arrow over (r)}(s)−{right arrow over (p)}(t)∥.
3. The article of claim 1 wherein the run of material has an isolating axis i(t) at {right arrow over (r)}(s) that is constant.
4. The article of claim 1 wherein the evolute of {right arrow over (d)}(t) is a non-planar space curve over the interval [t 1 , t 2 ].
5. The article of claim 1 wherein the run of material is chopped-fiber reinforced thermoplastic.
6. A duct with a duct axis {right arrow over (d)}(t) described by the vector function:
{right arrow over (d)} ( t )=< a ( t ), b ( t ), c ( t )>,
the duct comprising:
a run of material having a longitudinal axis {right arrow over (r)}(s,k) described by the vector function:
{right arrow over (r)} ( s,k )=< f ( s,k ), g ( s,k ), h ( s,k )>
that forms a first conical spiral around the duct axis {right arrow over (d)}(t);
wherein the duct axis {right arrow over (d)}(t) is characterized by a curvature κ(t) and a center of curvature {right arrow over (p)}(t) described by the vector function:
{right arrow over (p)} ( t )=<α( t ),β( t ),γ( t )>;
wherein the curvature κ(t)>0;
wherein the run of material is characterized by a conjoining axis j(t) at {right arrow over (r)}(s,k) such that j(t)∝∥{right arrow over (r)}(s,k)−{right arrow over (p)}(t)∥; and
wherein t is a real number in the domain t: [t 1 , t 2 ], s is a real number in the domain s: [s 1 , s 2 ], and k is an integer.
7. The article of claim 6 wherein the run of material has an isolating axis i(t) at {right arrow over (r)}(s,k) such that i(t)∝∥{right arrow over (r)}(s,k)−{right arrow over (p)}(t)∥.
8. The article of claim 6 wherein the run of material has an isolating axis i(t) at {right arrow over (r)}(s,k) that is constant.
9. The article of claim 6 wherein the evolute of {right arrow over (d)}(t) is a non-planar space curve over the interval [t 1 , t 2 ].
10. The article of claim 6 further comprising a second conical spiral around the duct axis {right arrow over (d)}(t), wherein the first conical spiral and the second conical spiral are conjoined.
11. The article of claim 1 wherein the conical spiral is a rectangular-with rounded-corners conical spiral.