Duct fabricated with additive manufacturing and having one or more curves

Abstract

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.

Claims

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 (i) a solid cross-section and (ii) a longitudinal axis {right arrow over (r)}(s) running through the solid cross-section and 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 duct axis {right arrow over (d)}(t) is characterized by a curvature (t) and a center of curvature {right arrow over (p)}(t) described by the vector function:
{right arrow over (p)}(t)=<(t), (t), (t)>; wherein the curvature (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.sub.1, t.sub.2] and s is a real number in the domain s: [s.sub.1, s.sub.2].

2. The duct 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 duct 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 duct of claim 1 wherein the evolute of {right arrow over (d)}(t) is a non-planar space curve over the interval [t.sub.1, t.sub.2].

5. The duct of claim 1 wherein the helix is a circular helix.

6. The duct of claim 1 wherein the range of j(t) is between 0.1 millimeters and 0.5 millimeters.

7. The duct of claim 1 wherein the run of material is chopped-fiber reinforced thermoplastic.

8. 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 stack of planar spirals, comprising a first planar spiral around the duct axis {right arrow over (d)}(t); wherein the first planar spiral is a rectangular-with-rounded-corners planar spiral; wherein the duct axis {right arrow over (d)}(t) is characterized by a curvature (t) and a center of curvature {right arrow over (p)}(t) described by the vector function:
{right arrow over (p)}(t)=<(t), (t), (t)> wherein the curvature (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.sub.1, t.sub.2], s is a real number in the domain s: [s.sub.1, s.sub.2], and k is an integer.

9. The duct of claim 8 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).

10. The duct of claim 8 wherein the run of material has an isolating axis i(t) at {right arrow over (r)}(s,k) that is constant.

11. The duct of claim 8 wherein the first planar spiral is a conjoined planar spiral.

12. The duct of claim 8 wherein the first stack of planar spirals further comprises a second planar spiral around the duct axis {right arrow over (d)}(t), wherein the first planar spiral and the second planar spiral are conjoined with each other.

13. The duct of claim 8 wherein a second stack of planar spirals is further formed, wherein the first stack of planar spirals and the second stack of planar spirals are conjoined with each other.

14. The duct of claim 8 wherein the range of j(t) is between 0.1 millimeters and 0.5 millimeters.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 depicts an illustration of the salient components of additive manufacturing system 100 in accordance with the illustrative embodiment of the present invention.

(2) FIG. 2 depicts an illustration of an orthographic view of duct 151, which is an illustrative embodiment of the present invention.

(3) FIG. 3 depicts an illustration of a cross-sectional view of duct 151.

(4) FIG. 4 depicts an illustration of the relationship of the conjoining axis j(t) and the isolating axis i(t) to the duct axis {right arrow over (d)}(t) and the longitudinal axis {right arrow over (r)}(s) of the run of material.

(5) FIG. 5 depicts a perspective view of a portion of a run of material that is enlarged for magnification purposes.

DETAILED DESCRIPTION

(6) FIG. 1 depicts an illustration of the salient components of additive manufacturing system 100 in accordance with the illustrative embodiment of the present invention. Additive manufacturing system 100 comprises: CAD/CAM system 101, build chamber 102, turn-table 110, deposition platform 111, robotic arm 121 (which itself comprises deposition head 122 and deposition nozzle 123), thermoplastic filament spool 130, and thermoplastic filament 131. The purpose of manufacturing system 100 is to manufacture duct 151.

(7) CAM controller 101 comprises the hardware and software necessary to direct build chamber 102, control robotic arm 121, deposition head 122, deposition nozzle 123, and turntable 110 to manufacture duct 151. It will be clear to those skilled in the art, after reading this disclosure, how to make and use CAM controller 101.

(8) Build chamber 102 is a thermally-insulated, temperature-controlled environment in which duct 151 is manufactured. It will be clear to those skilled in art how to make and use build chamber 102.

(9) Turn-table 110 comprises a stepper motor under the control of CAM controller 101 that is capable of rotating platform 111 (and, consequently duct 151) around the Z-axis. In particular, turn-table 110 is capable of: i. rotating platform 111 clockwise around the Z-axis from any angle to any angle, and ii. rotating platform 111 counter-clockwise around the Z-axis from any angle to any angle, and iii. rotating platform 111 at any rate, and iv. maintaining (statically) the position of platform 111 at any angle.
It will be clear to those skilled in the art how to make and use turn-table 110.

(10) Platform 111 comprises hardware on which duct 151 is manufactured. It will be clear to those skilled in the art how to make and use platform 111.

(11) Robotic arm 121 is a seven-axis arm capable of placing deposition nozzle 123 at any location in the build volume of duct 151 and from any approach angle. Furthermore, robotic arm can move deposition nozzle 123 in: i. the +X direction, ii. the X direction, iii. the +Y direction, iv. the Y direction, v. the +Z direction, vi. the Z direction, and vii. any combination of i, ii, iii, iv, v, and vi
while rotating the approach angle of deposition nozzle 123 around any point or temporal series of points. It will be clear to those skilled in the art how to make and use robotic arm 121.

(12) Deposition head 122 is hardware that heats and deposits filament 131 (which may partially or wholly contain one or more fiber strands) via deposition nozzle 123.

(13) Thermoplastic filament 131 comprises a continuous tow of carbon fiber that is impregnated with a thermoplastic, but it will be clear to those skilled in the art, after reading this disclosure, how to make and use alternative embodiments of the present invention in which thermoplastic filament 131 has a different fiber composition as described in U.S. patent application Ser. No. 14/184,010, which is incorporated by reference.

(14) Thermoplastic filament 131 is deposited as a run of material, which is not shown in FIG. 1 as distinct from duct 151. The physical and geometric properties of the runs of material are described below and in the accompanying figures.

(15) FIG. 2 depicts an illustration of an orthographic elevation view of duct 151 in accordance with the illustrative embodiment of the present invention.

(16) Duct 151 is a curvilinear duct that is capable of directing the flow of a fluid between opening 231 and 231. The curvature of duct 151 is defined by a three-dimensional space curve called the duct axis {right arrow over (d)}(t).

(17) In order to facilitate an understanding of the present invention, the duct axis {right arrow over (d)}(t) of the illustrative embodiment is confined to a plane. It will be clear to those skilled in the art, after reading this disclosure, how to make and use alternative embodiments of the present invention in which duct axis {right arrow over (d)}(t) is not confined to a plane (i.e., duct axis {right arrow over (d)}(t) is a non-linear and a non-planar space curve).

(18) In accordance with the illustrative embodiment, duct axis {right arrow over (d)}(t) is described by a vector function whose general form is:
{right arrow over (d)}(t)=<a(t), b(t), c(t)>(Eq. 1a)
where a(t), b(t), and c(t) are functions in a particular coordinate system (e.g., Cartesian, cylindrical, polar, etc.) and t is a real number in the domain t: [t.sub.1, t.sub.2]. It will be clear to those skilled in the art how to describe any space curve, and, therefore, any duct axis as a vector function. Furthermore, it will be clear to those skilled in the art how to represent the space curve of any duct axis using mathematical techniques other than vector functions.

(19) The particular vector function (in Cartesian coordinates) for duct axis {right arrow over (d)}(t) of duct 151 is:

(20) $\begin{array}{cc}a\left(t\right)=\frac{800}{3}\cos \left(\frac{t}{500}\right)& \left(\mathrm{Eq}.1b\right)\\ b\left(t\right)=0& \left(\mathrm{Eq}.1c\right)\\ c\left(t\right)=s& \left(\mathrm{Eq}.1d\right)\end{array}$
where t is a real number in the domain t: [0, 1000]. It will be clear to those skilled in the art, after reading this disclosure, how to determine the vector function for any duct axis in any coordinate system.

(21) Duct axis {right arrow over (d)}(t) comprises curves, and the general equation for the curvature (t) of duct axis {right arrow over (d)}(t) (expressed independently of any particular coordinate system) is:

(22) $\begin{array}{cc}\left(t\right)=\frac{.Math.{d\left(t\right)}^{}{d\left(t\right)}^{}.Math.}{{.Math.{d\left(t\right)}^{}.Math.}^3}.& \left(\mathrm{Eq}.2a\right)\end{array}$
When the vector function of duct axis {right arrow over (d)}(t) is expressed in Cartesian coordinates, the equation for the curvature (t) of is:

(23) $\begin{array}{cc}\left(t\right)=\frac{\sqrt{{\left(c^{}{b}^{}-c^{}{b}^{}\right)}^2+{\left(a^{}{c}^{}-a^{}{c}^{}\right)}^2+{\left(b^{}{a}^{}-a^{}{b}^{}\right)}^2}}{{\left(a^2+b^2+c^2\right)}^{2/3}}.& \left(\mathrm{Eq}.2b\right)\end{array}$
It will be clear to those skilled in the art how to compute the curvature (t) of any duct axis for any value of the parameter t.

(24) The radius of curvature (t) of duct axis {right arrow over (d)}(t) equals:

(25) $\begin{array}{cc}\left(t\right)=\frac{1}{\left(t\right)}& \left(\mathrm{Eq}.3\right)\end{array}$
which equals the distance from the duct axis {right arrow over (d)}(t) to the center of curvature {right arrow over (p)}(t), which is a space curve that is described by the vector function:
{right arrow over (p)}(t)=<(t), (t), (t)>(Eq. 4).
where (t), (t), and (t) are functions in the same coordinate system as that used for describing duct axis {right arrow over (d)}(t). It will be clear to those skilled in the art how to determine the center of curvature {right arrow over (p)}(t) for any duct axis {right arrow over (d)}(t). For example, it is well known to those skilled in the art that the center of curvature {right arrow over (p)}(t) is the point that lies at the radius of curvature (t) from duct axis {right arrow over (d)}(t) (i.e., (t)={right arrow over (d)}(t){right arrow over (p)}(t)) in the direction of the unit principal normal vector of duct axis {right arrow over (d)}(t) (into the curve). As space curve {right arrow over (p)}(t) is known as the evolute of {right arrow over (d)}(t). When duct axis {right arrow over (d)}(t) is a planar curve, the evolute of {right arrow over (d)}(t) is also a planar curve. In contrast, when duct axis {right arrow over (d)}(t) is a non-planar space curve, the evolute of {right arrow over (d)}(t) is also a non-planar space curve.

(26) In accordance with the illustrative embodiment, duct 151 comprises a continuous run of material whose longitudinal axis {right arrow over (r)}(s) is a space curve described by the vector function:
{right arrow over (r)}(s)=<f(s), g(s), h(s)>(Eq. 5)
where f(s), g(s), and h(s) are functions in the same coordinate system as that used for describing duct axis {right arrow over (d)}(t). It will be clear to those skilled in the art, after reading this disclosure, how to determine the vector function for whose longitudinal axis {right arrow over (r)}(s) for any duct, whether it comprises a helix, one or more planar spirals, or one or more conical spirals.

(27) For curved ducts and the curved portions of curvilinear ducts, the value of the conjoining axis j(t) is a function of t. In particular, the value of the conjoining axis j(t) at {right arrow over (r)}(s) is proportional to the distance between {right arrow over (r)}(s) and the center of curvature {right arrow over (p)}(t):
j(t){right arrow over (r)}(s){right arrow over (p)}(t)(Eq. 6).
This enables the run of material to remain conjoined and form a curved duct. In accordance with the illustrative embodiment, the range of j(t) is between 0.1 millimeters and 0.5 millimeters, but it will be clear to those skilled in the art, after reading this disclosure, how to select the values of j(t)in accordance with Equation 6for any duct.

(28) The value of the isolating axis i(t) depends on the deposition process and the desired mechanical characteristics of duct 151. For example, the value of the isolating axis i(t) can be a constant:
i(t)=I(Eq. 7a).

(29) Alternatively, the value of the isolating axis i(t) can also vary as a function of the distance from {right arrow over (r)}(s) to the center of curvature {right arrow over (p)}(t):
i(t)j(t){right arrow over (r)}(s){right arrow over (p)}(t)(Eq. 7b).
In any case, it will be clear to those skilled in the art, after reading this disclosure, how to select the particular values of i(t)in accordance with equations 7a or 7bfor any duct.

(30) It will be clear to those skilled in the art, after reading this disclosure, how to modify any duct taught in the co-pending patent application entitled Duct Fabricated With Additive Manufacturing (U.S. patent application Ser. No. 15/459,747) to have one or more curved segments. For example, any duct can be manufactured by depositing a continuous run of material in the form of a conjoined helixwith any profile including, but not limited to circular and rectangular-with-rounded corners. FIG. 3 depicts a cross-sectional view of duct 151 featuring a rectangular-with-rounded-corners profile. Additionally, any duct can be manufactured by depositing a conjoined stack of conjoined planar spiralswith each planar spiral having any profile including, but not limited to circular and rectangular-with-rounded corners. And still furthermore, any duct can be manufactured by depositing a conjoined stack of conical spiralswith each conical spiral having any profile including, but not limited to circular and rectangular-with-rounded corners.

(31) In reference to FIG. 4, the dimensions of the cross section of a run of material at location {right arrow over (r)}(s) are defined by two measurements: i. the conjoining axis j(t), and ii. the isolating axis i(t)
that are measured in relation to a plane (t) that goes through location {right arrow over (r)}(s) and is perpendicular to duct axis {right arrow over (d)}(t). In particular, the conjoining axis j(t) of the run of material at {right arrow over (r)}(s) is perpendicular to the plane (t) at {right arrow over (r)}(s). The isolating axis i(t) of the run of material at {right arrow over (r)}(s) is on the line in the plane (t) that connects {right arrow over (r)}(s) and {right arrow over (d)}(t). The relationship of the conjoining axis j(t) and the isolating axis i(t) to the duct axis {right arrow over (d)}(t) and the longitudinal axis {right arrow over (r)}(s) of the run of material is illustrated in FIG. 4. When the conjoining axis j(t) is a constant for all values of the parameter t in an embodiment of the present invention, it may be represented by J. Similarly, when the isolating axis i(t) is a constant for all values of the parameter t in an embodiment of the present invention, it may be represented by I.

(32) In order to facilitate a technical and accurate understanding of the illustrative embodiments of the present invention, FIG. 5 depicts a perspective view of a portion of a run of material 501 that is enlarged for magnification purposes. As shown in FIG. 5, the run of material comprises a longitudinal axis. A location along a longitudinal axis is designated location {right arrow over (r)}(s).

(33) The cross section of a run of material at location {right arrow over (r)}(s) is approximately an ellipse. It will be clear to those skilled in the art, however, after reading this disclosure, how to make and use alternative embodiments of the present invention in which the cross-section has another shape (e.g., a circle, a triangle, a rectangle, a square, a hexagon, an octagon, a dodecagon, an irregular shape, etc.).

(34) It is to be understood that the above-described embodiments are merely illustrative of the present invention and that many variations of the above-described embodiments can be devised by those skilled in the art without departing from the scope of the invention. It is therefore intended that such variations be included within the scope of the following claims and their equivalents.