This paper presents a design methodology and experimental assessment of variable-geometry dies that enable the extrusion of plastic parts with a nonconstant cross section. These shape-changing dies can produce complex plastic components at higher manufacturing speeds and with lower tooling costs than injection molding. Planar, rigid-body, shape-changing mechanism synthesis techniques are used to create the links that comprise the variable-geometry die exit orifice. Mechanical design guidelines for production-worthy dies are proposed. Several dies were designed and constructed to provide significant changes in the cross-sectional shape and area of extruded parts. Experiments were conducted in a production environment. An analysis of the repeatability of the cross-sectional profiles along the length of the part is presented.

## Introduction

The commercial importance of plastic components has been increasing at a rapid rate during the past 40 years [1]. Extrusion is a preferred process for manufacturing plastic parts because of its low cost and high production rate [2]. A polymer extrusion machine is illustrated in Fig. 1(a). The extrusion process begins with pellets being introduced into a barrel and softened by friction with an auger to form a pressurized melt [3,4]. Augmented by auxiliary heaters, the polymer melt flows through the barrel into the bolster and die. The bolster directs the soft polymer toward the die land, which has an exit orifice that appropriately matches the intended product [5]. The bolster is commonly contoured to balance the melt flow, striving to achieve a uniform exit velocity across the entire profile while reducing back pressure [6,7]. The function of the die is to shape the molten polymer into the desired cross section as it is pushed through the die land exit orifice [8,9]. A puller draws the product leaving the die through a cooling trough, where the polymer solidifies.

Fig. 1
Fig. 1
Close modal

The conventional extrusion process uses a stationary die land to manufacture continuous plastic products with a constant cross section. A wide range of products are formed through conventional extrusion including pipes, drinking straws, gutter systems, household siding, weatherstripping, decorative molding, and window framing components. The stationary die illustrated in Fig. 1(b) consists of a flat plate with a constant cross-sectional exit orifice and bolster that smoothly directs the molten plastic flow to the die. Stationary dies cannot produce complex parts that require changes to the sectional profile. Giaier et al. [10] described design aspects of dies that have the ability to alter their exit geometry. Dies that exhibit gross changes to the exit orifice would allow the production of more complex extruded parts, thereby leveraging the associated cost and process time savings.

Rigid-body, shape-changing mechanisms have been identified as a fitting approach to variable-geometry dies. A typical shape-change problem seeks a device that approximates a set of specified shapes with the edge geometries of some of its components [11]. Rigid-body mechanisms are typically comprised of traditional mechanical components like rigid links, hinges, cams, springs, etc. Rigid-body mechanisms are able to create large changes in motion with readily predictable responses to a large variety of loading schemes. The rigid-body, shape-changing mechanism theory generates chains of links connected by revolute and prismatic joints that approximate a number of desired profiles [1214]. Applications of rigid-body, shape-change mechanisms include a morphing aircraft wing, deformable light reflector, automotive seat, etc. Established theory places a priority on revolute joints [1214]. Concepts for variable-geometry die design have shown that revolute joints are difficult to implement, whereas prismatic joints (straight or curved) have fewer mechanical design challenges [10]. To accommodate die design, shape-changing mechanism theory is extended to produce designs that place a priority on prismatic joints and to eliminate or reduce revolute joints [15].

This paper provides techniques for the design of variable-geometry, polymer extrusion dies including desirable design features and the methodology for synthesizing links that match desired exit profiles. To confirm the design application, an experimental assessment of the repeatability of extruded parts produced from moveable dies is performed. The remainder of this paper is organized as follows: Section 2 provides the design criteria of variable-geometry dies and joint designs. Section 3 reviews the fundamental rigid-body, shape-change theory. Section 4 describes the implementation of the shape-change methodology motivated by die designs. Section 5 presents several die designs, and experimental results are provided in Sec. 6.

## Variable-Geometry Dies

### Desirable Design Attributes.

The following list of desirable design attributes was compiled from die design literature [5,8,9] and in conjunction with numerous extrusion die design engineers having years of production experience.

#### Leakage.

Variable-geometry dies will inevitably contain moving parts connected by joints to exhibit relative movement. These moving joints should be created with the objective of minimizing or controlling leakage of polymer melt through interfaces. Accordingly, joint designs should strive for surface-on-surface contact, manufactured to tight fitting tolerances. Additionally, creating movement with a surface sliding across another surface restricts the leak path [16]. More versatile motion may be achieved by joints that have an edge on surface contact. To prevent damage to the part, material that is leaked must be channeled away from the extrudate [17].

#### Die Land.

Die land provides a uniform cross section passage just prior to the die exit, which relaxes the viscoelastic stresses in the melt before leaving the die [5]. Any cross-flow grooves or notches along the die land will disrupt the velocity profile and will likely reduce the dimensional stability and surface quality of the extruded part. Thus, moveable die features should not interrupt a smooth die land.

#### Die Exit Plane.

All components that form the die exit should lie on a common plane. A uniform die exit plane will help to maintain constant velocity throughout the flow field. Maintaining a uniform velocity throughout the profile section ensures higher quality surface finish [5]. Tests have confirmed that if the die exit occurs on multiple planes, extrudate curling can occur from the increased shear stress on the longer portion of the die land.

#### Die Exit Flow Area.

The volumetric flow rate produced by an extruder screw of diameter D with rotational speed ω is
$Q=π2D2Hω cos θ sin θ2$
(1)
where details of the metering section of the extrusion screw include the height of the thread (or flight) H and lead angle θ [18]. The average velocity of the extrudate is
$v=QAd$
(2)
which is nominally selected as the extrusion line speed. Noted from Eqs. (1) and (2) is that dramatic changes in die exit area Ad may require adjustments to the extruder line speed. At a constant screw speed ω, a reduction in die exit area will increase the velocity v of the extrudate. Accordingly, a reduction of die exit could be compensated by slowing the screw speed, increasing line speed, or opening bypass flow ports that keep a constant effective die area. Process development is required to resolve these issues and is left for future work. Initial tests on variable-geometry dies show no negative effects by limiting changes in Ad to 20%. Dies with greater area changes have been constructed to include a bypass channel to maintain a near-constant, effective die exit area. The material expelled through the bypass channel may be reground into pellets and placed back into the hopper within reason.

#### Actuation.

The forces required to actuate the moving components of the die should not significantly vary throughout the actuation sequence. That is, the linkages formed by the moving components involved in the shape change must not approach kinematic singularities. Single degree-of-freedom (DOF) systems can be controlled via a single actuator. Controlling the actuation of a system becomes more complex as the degrees-of-freedom increase, but expands the envelope of the die exit shapes. For actuation simplicity, single-DOF systems are preferred but multiple-DOF systems are viable [19].

### Joint Designs.

For general shape-change, the joints connecting the die land components must allow relative movement, including pure rotation (revolute joints) and pure translation (prismatic joints, potentially curved). The following joint designs have been developed and utilized in experimental dies [10].

The crescent joint is the preferred approach to permit rotation between adjoining die components because of its high resistance to leakage. This joint, shown in Fig. 2(a), uses a curved tongue and groove arrangement that permits rotation of adjacent components and is viable when a variable sharp corner is not required in the extruded part. The crescent joint can only be used when a rounded corner is permissible on the profile edge of the extrudate. While permitting rotation, the crescent joint on the die land surface is modeled as a curved-prismatic joint. The parts that comprise the joint contain geometric features that do not vary along one dimension, which facilitates manufacture. Design of a bolster must be such that the polymer melt does not flow directly into the groove or gap shown in Fig. 2(a). To restrict leakage, tight clearances are specified between the tongue and groove, which maintain surface-on-surface contact throughout motion. The length of the arc is based on the desired actuation limits.

Fig. 2
Fig. 2
Close modal

A corner joint shown in Fig. 2(b) exhibits pure rotation and has the advantage that it forms moveable sharp corners on the profile edge. Placing a curved tongue and groove outward of the die flow generates a center of rotation that lies directly on the die land surface. Like the crescent joint, the parts that comprise the corner joint contain geometric features that do not vary along one dimension, which reduces manufacturing difficulties. The drawback to the corner joint is that it relies on edge-to-surface contact to form a sealing interface. Thus, flow between the components should be expected and managed. Additionally, the bolsters must have features to eliminate the leakage path through the gap and groove as shown in Fig. 2(b).

A sliding joint permits linear translation between adjoining die components. A straight tongue and groove are incorporated as shown in Fig. 2(c). A sliding joint may also be formed along a constant radius arc. The joint exhibits surface-on-surface contact to impede leakage. Manufacturing of the straight grooves is a straightforward process. A drawback is that the interface runs along the die land, which may adversely affect surface finish of the extruded part.

## Rigid-Body Shape-Changing Synthesis for Extrusion Dies

Rigid-body, shape-change synthesis theory is used to identify appropriate geometry of the die exit components to approximate the desired orifice profile. The synthesis methodology generates a single chain of rigid bodies connected by revolute and prismatic joints that approximates a given set of curves, such as those shown in Fig. 3(a). Appropriately varying the angles of revolute joints and the lengths of prismatic joints, this rigid-body chain can morph to closely match the desired shapes as shown in Fig. 3(b). Observe that the corresponding red (darker-colored) links within the chain do not change shape as they are repositioned. The corresponding cyan (lighter-colored) links retract and lengthen along a constant radius, reducing to a length of nearly zero in some cases. The synthesis methodology from Refs. [1214] is adapted for extrusion dies and described in the subsections that follow: Specifically, adaptations include identifying corners (Sec. 3.2), establishing additional chain structures (Sec. 3.3), introducing homologous points in the segmentation process (Sec. 3.4), and introducing a connection vector and fusing segments (Sec. 3.5.1).

Fig. 3
Fig. 3
Close modal

### Acquire Design Profiles and Create Target Profiles.

The synthesis initiates with a set of p curves called design profiles, which are the outlines of the desired shapes to be obtained by the mechanism. Three design profiles are shown in Fig. 3(a). For extrusion work, the design profiles are specified as the desired shapes of the variable die exit.

Design profiles are converted to piecewise-linear target profiles. The jth target profile is formed by placing Nj points along the jth design profile at roughly equal lengths. Having the points placed equally along each profile facilitates comparisons among segments of the p target profiles necessary to form a suitable mechanical chain of rigid bodies that when repositioned will approximate all design profiles. The jth target profile is represented by a piecewise-linear curve connecting the ordered set of points with coordinates represented by a position vector $zji, i=1,…,Nj$. The jth target profile contains mj = Nj − 1 pieces, and its arc length can be quantified with mj.

### Identify Sharp Corners in the Design Profiles.

In die design, desired orifice profiles may have sharp corners or high-curvature regions. The presence of any sharp corners on the design profiles requires special attention in the subsequent synthesis processes and must be identified. For general piecewise-linear curves, the curvature at certain a point i is defined as the reciprocal of the circumradius defined by points i − 1, i, and i + 1. With target profiles being piecewise representations of design profiles, points on target profiles will not generally lie at the apex of the sharp corner on the design profile. As the number of pieces contained in a target profile changes, the circumradius near the sharp corner is unstable.

The angle between the two pieces adjacent to the sharp corner converges when using a sufficient number of pieces on the target profile. Therefore, the angle between pieces i − 1 and i + 1 can serve as a substitute measure to quantify the curvature of the target profile at piece i and is termed as the bend angle. The bend angle is a signed value: a positive value indicates the curve bends counterclockwise. The bend angle at a sharp corner is much higher than a smooth curve where it approaches 0 deg as the number of pieces is increased. Thus, it can be used to detect sharp corners on target profiles. Based on the bend angle information of the target profiles, the designer could specify a minimum bend angle that defines sharp corners. Once sharp corners are identified, an apices matrix Aj is constructed for the jth profile, with each column recording the target profile index position and bend angle of the sharp corners (or high-curvature regions) on that profile.

### Specify a Chain Structure.

As shown in Fig. 4, design profiles are categorized as open profiles (Fig. 4(a)), closed profiles, and profiles with fixed-endpoints. For a closed profile, the two endpoints coincide with each other (Fig. 4(b)). A set of fixed-end profiles have common fixed-endpoints (Figs. 4(c)4(e)).

Fig. 4
Fig. 4
Close modal

Extrusion profiles have relative geometries, as such the chain is commonly modeled as fixed-end profiles. As shown in Fig. 4(c), a set of fixed-end RR-type profiles have two stationary endpoints, hence it can be approximated by a chain connected to the ground with two fixed pivots (revolute joints). Since straight and curved prismatic (crescent) joints are preferred in variable-geometry dies [10], implementing fixed-end profiles with end links being prismatic joints is a necessity. Die design with crescent joints requires two additional types of fixed-end profiles: PP-type with both ends being prismatic joints with one fixed end (Fig. 4(d)), and RP-type (or PR-type) with one end being a prismatic joint connected to the ground and the other being a fixed revolute joint (Fig. 4(e)). An R-type end of the fixed-end profiles addresses position constraints for the end link of the rigid-body chain, while a P-type end has constraints on orientation in addition to position for the end link.

### Forming Segments.

A segmentation process identifies q segments of a target profile that when repositioned will approximate the other target profiles. There are two types of segments, $M$-segments and $C$-segments. An $M$-segment represents a single rigid body that is shaped to approximate a portion of the same number of pieces for all profiles. The red (darker-colored) segments in Fig. 3(b) are $M$-segments. A $C$-segment represents a pair of constant curvature links that contain a prismatic joint and approximates a variable number of pieces on each target profile. The cyan (lighter-colored) segments in Fig. 3(b) are $C$-segments.

#### Designate Segment Types.

$M$-segments are favored as they have better matching ability because they can take on any shape, while $C$-segments are restricted to a constant radius. The $C$-segments enable the chain to match a set of profiles of different arc lengths. For variable-geometry polymer extrusion dies, the crescent joint has emerged as the preferred mechanical design for relative movement and is represented by a $C$-segment. A corner joint is represented by two $M$-segments connected with a revolute. Understanding the intended joints in the extrusion die, the designer specifies the number q and types of the segments that will comprise the rigid-body chain with a segment-type vectorV. For example, the chain shown in Fig. 3(b) has q = 6 segments and is associated with $V=[C M M C M C ]$. For P-type fixed-end chain structures, the segment at the appropriate end of the chain specified in V must be $C$.

#### Generate an Initial Segment Matrix.

A p × q segment matrix (SM) is created to specify the number of pieces in each segment for each profile according to V. That is, for each target profile the total number of pieces mj, j = 1,…, p is divided into q segments. Each $M$-segment must have the same number of pieces for each profile. In forming segments, considering sharp corners as homologous geometric features among the set of target profiles is useful. Using a $C$-segment to match a region containing a sharp corner generates poor results because the average curvature is greatly affected by the sharp corner. Accordingly, a sharp corner region is approximated with two $M$-segments connected by a revolute joint or a single $M$-segment if the corresponding corner on all target profiles has roughly equal angles. If the profiles have a different number of apices, pseudo apices are placed on the profiles with fewer corners. The location of pseudo apices can be placed by inference from the other profiles or specified by the designer.

With the addition of pseudo apices, a set of profiles are considered to have an equal number of sharp corners on each profile. Note that the sharp corner regions are matched with either a single $M$-segment if the bend angles remain roughly constant, or two $M$-segments connected by a revolute joint at the corner if the bend angles have significant variation among profiles. If a corner is contained within a single $M$-segment, the corner point is considered homologous and divides the profile into subprofiles. Segmentation points (points that divide a target profile into portions to be matched by segments) next to a sharp corner must be placed at the same number of pieces from the sharp corner for all profiles. The designer specifies the segment and connection types for each subprofile, and initial SMs are generated accordingly.

As detailed in Ref. [20], the coordinates of each $M$-segment are formed as the mean shape of the corresponding portions of the target profiles, and the coordinates of each $C$-segment are formed as the average curvature defined by the points on the corresponding portions of the target profiles. The coordinates of the ith point on a segment in the jth profile are represented by a position vector $z¯ji$. Once the segments are generated, they are shifted to align with the corresponding portions of the target profiles. The point-to-point distance between each point on the proposed segment and the corresponding point on the target profile is calculated. For the ith point on a segment in the jth profile, the point-to point error is $‖z¯ji−zji‖$. A p × q error matrix is constructed to organize the maximum matching error of each segment.

#### 3.4.3 Optimize SMs.

Once an initial SM is formed, an iterative process creates the segments, evaluates how well they match the target profiles, and adjusts the SM to minimize the error. Information to guide the iterations is contained within the error matrix by identifying those segments that are acceptable and those that poorly match the target profiles. The SM is modified accordingly as the geometry (shape and length) of the bodies is iteratively optimized until a chain that best matches the set of target profiles is obtained. During the iterations, the points on the target profiles where the sharp corners occur are considered to be inviolable for $C$-segments, meaning that $C$-segments should not contain these points. At each step during the optimization, the SM is checked against the apices matrices to ensure that the inviolable points are not included in any $C$-segments. Otherwise, the SM is reset to its previous value.

### Connect the Segments.

Since segments are generated individually, their endpoints will not coincide. A process to locate segments such that they form a chain is described as follows.

#### Fused Connections.

Variable extrusion die orifice geometry applications exhibit an increased use of $C$-segments, corresponding with crescent joint preference. Accurate profile matching can be obtained by rigidly connecting a $C$-segment to another $C$-segment or an $M$-segment, without using revolute joints as was the previous strategy [1214]. A connection vector W is introduced to specify the manner in which the segments are connected, being either fused ($F$) or revolute joints ($R$). After the segments are generated and aligned with the corresponding portions of the target profiles, the angle between two adjacent segments that are to be fused according to W is measured for each profile. The two segments are fused at the average angle over all profiles at the connection point. Note that a group of fused segments should include at least one $C$-segment, otherwise it can be simplified to a single $M$-segment. After fused connections are established, all segments are shifted to align with the target profiles, and the matching error is measured.

#### Revolute Joint Connections.

After the appropriate segments are fused, revolute joints are added to assemble the segments into a continuous chain. The matlab nonlinear programming function fmincon is used to determine the final configuration of the chain with revolute joints added. An objective function to minimize the average matching error of all the points on the chain is
$f=∑j=1p∑i=1Nj‖z¯ji−zji‖p∑j=1pNj$
(3)

The output of fmincon defines the final configuration of the chain for all profiles j = 1,…, p. Specifically, the output includes:

1. (1)

The coordinates of the starting point of the chain, which is the coordinates of the first point on the first segment, $z̃j1$.

2. (2)

The orientation angle of each segment, except those following a fused connection.

3. (3)

The average piece length of each $C$-segment as adjusted for each profile. This is not necessary for open profiles as $C$-segment adjustments are not required to conform to end constraints.

Since the number of revolute joints is limited in die design, segments with constrained geometry might not be able to satisfy boundary constraints required by closed and fixed-end profiles. Thus, the arc lengths of $C$-segments need to be adjusted to assemble the chain. The mapping between points on the chain and points on the target profile must always be maintained; thus, the number of pieces in a $C$-segment is unchanged. The average piece length of the $C$-segment, $s̃je$, is subject to adjustment to change its arc length at each profile j, while the radius and number of pieces remain constant.

The starting search point for fmincon is the configuration (parameters 1–3 listed in the previous paragraph) of segments after the optimized segments are fused and aligned with the profiles at the error-minimizing location. Many local minima exist in the solution field; therefore, a pool of initial guesses that are slightly varied from the original values is investigated.

The constraints used for fmincon are as follows: The difference in the arc length of each $C$-segment between the unconnected chain and the assembled chain is limited to within a certain range (such as 10%) to preserve the current arc length and shape of the $C$-segments. This is specified by an inequality constraint in fmincon. Boundary conditions including the endpoint locations and end link orientations for fixed-end and closed profiles are also applied as equality constraints in fmincon. Closed profiles require the two endpoints of the chain to be coincident, which yields the following constraint:
$z̃j1=z̃jNj$
(4)
for each profile j = 1,…, p. For closed profiles that have a fused connection between its end links, the algorithm would first calculate the chain configuration as if the endpoints of the chain are connected but the angle between them is not fixed. Then, the configuration of the chain would be solved for again, with the connection between the endpoints now fused at the average angle computed from the previous result. Fixed-end profiles require both endpoints to be stationary and in a prescribed location
$z̃j1=z̃11, z̃jNj=z̃1N1 ∀ j=2,…,p$
(5)
If the fixed-end profiles are RP- or PP-type, then the link that approximates a P-end has additional orientation constraints. For example, the orientational constraints for fixed-end PR-type profiles are
$z̃j2−z̃j1‖z̃j2−z̃j1‖=z̃12−z̃11‖z̃12−z̃11‖ ∀ j=2,…,p$
(6)

In using fmincon, the maximum number of iterations is set to a high value to ensure a local minimum that satisfies these constraints are obtained. Multiple starting points are used to confirm that the same optimal solution is produced.

Once the geometry of the segments is optimized, all the segments must be connected to form a continuous chain as presented in Sec. 3.5. Finally, constraining rigid links are added to form a functional mechanism in the step called mechanization [19]. These added links guide the chain to obtain the prescribed shapes and provide potential actuation locations for the mechanism.

## Variable-Geometry Die Designs

The joints presented in Sec. 2.2 were combined to create moveable die designs for experimentation. Four die designs are presented in this section, three of which have been constructed and tested in a production environment.

### The Crescent Die.

A die is desired to change between shapes that resemble a parallelogram and rectangle, shown in the design profiles of Fig. 5(a) (the ground link outline is omitted). Termed the “crescent die,” crescent joints are used for all the four joints producing radii at the vertices of the shapes. As previously identified, the crescent joint is deemed as the best option to permit revolute motion because of manufacturing and die leakage considerations. Observe that significant shape changes occur during motion, but small changes to the exit area (<10%) reduce complications with extrusion line speed identified in Eqs. (1) and (2). In the rectangular configuration, Ad = 637 mm2, whereas the parallelogram configuration has Ad = 594 mm2.

Fig. 5
Fig. 5
Close modal

#### Synthesizing the Rigid-Body Chain for Crescent Die Profiles.

The outline of the movable parts of the crescent die exit can be modeled as fixed-end PP-type profiles. Three design profiles are obtained as the extreme and middle configurations for the die. The three profiles have the same arc length. Target profiles were created with 5000, 5001, and 5002 pieces (traced counterclockwise, numbered from the parallelogram to the rectangular shape). Approximation of design profiles by the target profiles, especially near apices, accounts for the small differences in the number of pieces. Using the method presented in Sec. 3.2, with a minimum sharp corner angle of 20 deg, the apices matrices of the crescent die profiles are constructed as
$A1=[5741750191330883662483789.4°89.8°35.7°35.3°89.4°89.8°]A2=[4721647191330893560473689.9°89.9°35.8°35.8°89.9°89.9°]A3=[3691545191430893458463489.8°89.8°35.7°35.7°89.8°89.8°]$
(7)
The apices matrices show that each profile contains six apices, and each corresponding apex has a nearly constant bend angle across the different profiles. Therefore, revolute joints are not necessary at the six sharp corners. Each profile can be divided into seven subprofiles, bounded by the six apices that are considered inviolable points. The number of pieces in each subprofile is presented in the matrix below
$[573117616311755741175164471117526611764711176266368117636911753691176369]$
Observe that the curved sections that correspond to the crescent joints (the first, third, fifth, and seventh subprofile) have a significantly different number of pieces for each profile. Therefore, a $C$-segment needs to be placed at each of these subprofiles. As for the straight sections (the second, fourth, and sixth subprofiles), their numbers of pieces on the three profiles are very close but not constant. Therefore, in a preliminary design, these sections would also contain $C$-segments. As such, the segment types for each section should be $[C M], [M C M], [M C M], [M C M], [M C M],[M C M]$, and $[M C]$, respectively. Since two neighboring $M$-segments at a sharp corner that has constant bend angle would be fused, they could be merged into one $M$-segment. The segment-type vector for the rigid-body chain is hence determined to have 13 segments, specified as $V=[C M C M C M C M C M CM C]$. After iterations as described in Sec. 3, the SM is obtained as
$SM=[286472380661872327103553742088912895518447237966119023271135527120889228915781472380661293232710355169208892289260]$
(8)
Note that the second, fourth, and sixth $C$-segments have very small length variation, hence the chain can be simplified to contain seven segments defined by $V=[C M C M C M C]$. The SM is modified to be
$SM=[286(472+380+661)87(232+710+355)374(208+891+289+1)(55−1)184(472+379+661+1)(190−1)(232+711+355−1)(271+1)(208+892+289)15781(472+380+661)293(232+710+355)169(208+892+289)260]=[286151387129737413895418415131891297272138915781151329312971691389260]$
(9)
Since die motion occurs solely through the use of crescent (curved-prismatic) joints, no revolute joints are included in the chain. Appropriately, each of the seven segments are joined by fused connections, specified by $W=[F F F F F F]$. Since no revolute joint is used, the lengths of $C$-segments are adjusted to satisfy the P-P chain-end constraints. The final chain configured to match each target profile is shown in Fig. 5(b), where the $C$-segments are shown in cyan (the lighter color) and the $M$-segments are shown in red (the darker color).

#### Mechanical Design of the Crescent Die.

Figure 6(a) illustrates the mechanical design of the crescent die with exit shapes that conform to the synthesis presented in Sec. 4.1.1. Four 25-mm long, moveable links were surrounded by an outer ring and placed between a bolster and cover plate (not shown in Fig. 6(a)). The link labeled 1 was pinned to the bolster, preventing any motion relative to the extruder. Links 2 and 4 are able to pivot, whereas link 3 serves as a floating coupler. An actuation handle was fastened to link 2. A 25 deg rotation of the actuation handle moves the die exit through the shapes shown in Fig. 5(b). The die exit shape resembling a rectangle is produced with the actuation handle in an upward position (U), and the parallelogram shape is associated with a downward (D) handle position, as identified in the extreme profiles of Fig. 6. The exit orifice has a 25-mm die land. The opening in the bolster was designed by observing the exit orifice created by the moving links between their limit positions. To restrict flow through each crescent joint, the groove was machined 0.10-mm wider than the tongue to permit a 0.05-mm clearance along each sliding surface. An axial clearance of 0.04 mm was used between the moving links and the bolster. The length of the tongue was selected to allow the bolster to conceal the gaps created in the groove.

Fig. 6
Fig. 6
Close modal

### The Crescent-Corner Die.

Since a crescent joint is unable to form a sharp angle in the extruded profile, the corner joint was evaluated as the best option to create this feature. After a rigid-body chain was synthesized in the same fashion presented in Sec. 4.1.1, a die containing a corner joint and three crescent joints was designed as shown in Fig. 6(b). As with the crescent die, the moving links were surrounded by an outer ring and placed between a bolster and cover plate. The link labeled 1 was fixed to the bolster plate with a pair of dowel pins. A corner joint is used to connect links 1 and 4. Crescent joints were used to connect links 1 and 2, links 2 and 3, and links 3 and 4. The actuation handle was fastened to link 2. Again, a die exit shape resembling a rectangle is produced with the actuation handle in an upward position (U), and the parallelogram shape is associated with a downward (D) handle position. An axial clearance of 0.04 mm was used between the moving links and the backer and cover plates. To assess the effect of a different crescent joint clearance, the groove was machined 0.25-mm wider than the tongue to permit a 0.13-mm clearance along each sliding surface.

### The Crescent-Corner-Prismatic Die.

A third die was created to assess the performance of a prismatic joint. After a shape-changing chain was generated, the die containing a corner joint, a gear-actuated prismatic joint, and three crescent joints was designed as shown in Fig. 6(c). As with the other dies, the link labeled 1 was fixed to the bolster. Crescent joints connect links 1 and 2, links 2 and 3, and links 3 and 4. A corner joint connects links 1 and 4. A prismatic joint joins links 4 and 5. The actuation handle was again fastened to link 2, creating different shapes when rotated upward (U) or downward (D). Being a 2DOF mechanism, a second actuation method is required. Gear teeth were formed on links 4 and 5. The rotation of the gear on link 4 is facilitated by rotating a hex-head shaft that causes the prismatic pair to extend (E) or shorten (S).

### A Die That Achieves Drastic Shape Change.

All the three dies presented in Secs. 4.14.3 exhibited modest area change for experimental assessment. A fourth die was designed and constructed to demonstrate more drastic area change with a single-DOF actuation.

#### Shape-Change Synthesis.

As shown in Fig. 7(a), three fixed-end RR-type profiles are desired to form the die exit orifice. Note that the set of design profiles contains common features that are intended to be matched by $M$-segments. Target profiles were generated with N1 = 5000, N2 = 5546, and N3 = 6145 pieces. With a threshold of 50 deg, the apices matrices are
$A1=[20023794−90.1°−67.9°], A2=[20093798−96.2°−68.9°], A3=[20043888−90.0°−64.7°]$
(10)
Fig. 7
Fig. 7
Close modal

Two inviolable points are set with the apices matrices, forming three subprofiles for each profile. Since the variations of the bend angle of the two apices are insignificant (6.2 deg and 4.2 deg), no revolute joint is needed. Also note that the subprofiles before the first sharp corner have roughly equal lengths (2001, 2008, and 2003 pieces); therefore, a single $M$-segment is used to approximate the first subprofile and the region around the first sharp corner. The segment-type vector is determined to be $V=[M C M C M C]$, where the first $C$-segment is located within the second subprofile, and the other two $C$-segments are located in the third subprofile. The connection vector is $W=[F F F F F]$.

After optimization, the final SM is
$SM=[201551778471663682015817783476637352015991778106631580]$
(11)
The assembled chain is shown in Fig. 7(b). Observe that the length of several $C$-segments (Eq. (11)) reduced to the point that they are not detectable in the chain. The average arc length of the target profiles is 785.28, the matching error of each segment is
$EM=[1.581.581.981.621.901.881.511.503.544.193.533.091.171.757.747.547.106.56]$
(12)
the overall mean error is $Ẽ=2.19$. Two reasons explain the higher matching error of the third to sixth segments on profiles 2 and 3. First, fusing segments at a common angle shifts segments away from their original optimal positions to align with the corresponding profile portions. Second, the endpoints of the final chain need to match those of the target profiles. Since no revolute joint is used, the lengths of $C$-segments are adjusted to satisfy chain-end constraints. The final chain configured to match each target profile is shown in Fig. 7(b), where the $C$-segments are shown in cyan (the lighter color) and the $M$-segments are shown in red (the darker color).

#### Mechanical Design.

According to the rigid-body chain, a die containing two corner joints, two crescent joints, and a prismatic joint was designed as shown in Fig. 8(a). As with the other dies, the link labeled 1 was fixed to the bolster. Corner joints were used to connect links 1 and 2 and links 1 and 5. Crescent joints were used to connect links 3 and 4 and links 4 and 5. A prismatic joint was created between links 2 and 3. The actuation handle was pinned to a fixed bolt on the bolster and a moving bolt on link 4. The smallest exit area is Ad = 208 mm2, whereas the largest extreme exit area is Ad = 374 mm2, an 80% increase.

Fig. 8
Fig. 8
Close modal

## Experimental Assessment

For all the dies, a fixed bolster is contoured to streamline the flow toward the shape of the die exit. The shape-changing dies will inherently have sharp changes in flow at the bolster/land interface that creates a stagnation region. The polymer melt in such regions may exhibit thermal degradation if it remains at elevated temperatures for extended periods of time. Therefore, only thermally stable polymers, such as polyolefin (polyethylene and polypropylene), silicone, and soft polyvinyl chloride (PVC), are appropriate for these variable-geometry dies. Other commonly extruded and less expensive polymer materials, such as rigid PVC, may potentially degrade in the dead spots [7] though no testing has been done. Experimentation described in Secs. 5.1 and 5.2 used soft PVC.

### Testing Protocol.

Figure 9 shows the crescent die from Fig. 6(a) installed on an extruder. Soft PVC was run with a 61 rpm screw speed and a 142 °C melt temperature, resulting in die pressure that fluctuated between 2.48 and 2.62 MPa. A line speed of 1.8 m/min was selected by consulting Eqs. (1) and (2) and inspecting that the polymer completely filled the die exit. For the crescent (C) and crescent-corner (C-C) dies, the following test protocol was formulated to assess the repeatability of the extruded part under different motion schemes:

Fig. 9
Fig. 9
Close modal
1. (T-1)

The die was repeatedly moved between extreme positions with a 1.0 s actuation, followed by a static period of 5.0 s between actuations.

2. (T-2)

The die was repeatedly moved between extreme positions with a 5.0 s actuation, followed by a static period of 5.0 s between actuations.

3. (T-3)

The die was repeatedly moved between extreme positions with a 1.0 s actuation, followed by a static period of 20.0 s between actuations to observe changes in part shape development when isolated from die actuation.

The test protocol was also conducted on the crescent-corner-prismatic (C-C-P) die. While actuating the handle, the prismatic joint was alternated between its extreme configurations. The intent of the fourth die was to demonstrate the potential of drastic shape change and large area differences. Visual inspection of the extruded part showed no anomalies, and additional reliability testing was deemed unnecessary.

### Testing Results.

After completing the test protocol outlined in Sec. 5.1, a laser profile measurement system was used to scan profiles along the length of the extrudate. The measurement system produces a planar point cloud representation of the profile. To facilitate point-to-point comparisons among the profiles, the same number of defining points is distributed at equal intervals around the perimeter of each profile to represent the scanned points. In this work, 3600 points are used to represent each profile.

For each test case outlined earlier with the C die and C-C dies, 15 scans were obtained with the actuator handle both in the up (U) and down (D) configurations. For the C-C-P die, 12 scans were obtained for each test case with the actuator at the U and D configurations along with the prismatic extended (E) and shortened (S). Four representative scans from the D configuration of T-1 for the C-C die are superimposed in Fig. 10 to illustrate the repeatability of the extrusion die.

Fig. 10
Fig. 10
Close modal

For each set of scan data, the first scan was arbitrarily selected as a comparison baseline for the other scans in the set. To assess the deviation, a point-by-point distance di was calculated between corresponding points along each profile and the baseline profile. The mean of the absolute values of the deviation $d¯$ was calculated from all the measurements. The standard deviation of the distances σd was calculated and used to determine a 95% confidence interval d95. The results for the C, C-C, and C-C-P dies are given in Table 1. For a part with nominal dimensions of 25 × 25 mm, a tolerance of ±250 μm is typical for a fixed-geometry die, polymer extrusion process. The repeatability, as seen by the d95 values, compares favorably with conventional extrusion processes. Also, the d95 values for the U (rectangular) and D (parallelogram) profiles are similar.

Table 1

Comparison between extrusion profiles created with the crescent die, crescent-corner die, and crescent-corner-prismatic die (in microns)

C dieC-C dieC-C-P die
TestHandle$d¯$d95Handle$d¯$d95PrismaticHandle$d¯$d95
T-1D114±272D118±137SD227±525
T-1U106±257U90±75SU164±372
ED128±332
EU127±292
T-2D134±161D139±323SD253±597
T-2U111±276U136±290SU188±497
ED97±235
EU104±267
T-3D135±242D71±157SD174±401
T-3U148±272U88±194SU293±744
ED205±487
EU206±476
C dieC-C dieC-C-P die
TestHandle$d¯$d95Handle$d¯$d95PrismaticHandle$d¯$d95
T-1D114±272D118±137SD227±525
T-1U106±257U90±75SU164±372
ED128±332
EU127±292
T-2D134±161D139±323SD253±597
T-2U111±276U136±290SU188±497
ED97±235
EU104±267
T-3D135±242D71±157SD174±401
T-3U148±272U88±194SU293±744
ED205±487
EU206±476

The C die was actuated to create the extreme profiles shown in Fig. 6(a). No effect of leakage through the crescent joint interfaces was observed on the extruded part. After completing the series of tests over a 5 h period, the die was disassembled showing that plastic melt had seeped into the joint clearance, but less than 50% of the land. The C-C die was also actuated to create the extreme profiles shown in Fig. 6(b). A considerable amount of leakage was observed through the corner joint interface after 3 h of operation. The plastic that leaked within the joint remained pliable and did not affect actuation force or limits. A small amount of leakage was detected through the crescent joints, which were manufactured with the larger clearances. Leakage did not affect the quality of the extruded product. The C-C-P die has the four extreme profiles as shown in Fig. 6(c). The motion schemes identified earlier were conducted using the angular actuation, while the prismatic joint was in the extended and shortened positions. After 2 h of operation, a significant amount of leakage was observed through the corner joint interfaces. Leakage did not affect the quality of the extruded product. For more extreme shape-change cases, exploration to permit and manage leakage by creating channels to divert, capture, and reuse material is being explored.

## Discussion

Variable-geometry extrusion dies permit the cross sections of an extruded part to change along its length. Plastic components that are more sophisticated than current fixed-die extrusion technology can be manufactured faster and with lower tooling costs than injection molding. Repeatability measurements show that the variability of the moveable dies is similar to fixed profile dies. A few suggestions for implementing the outlined die design method are listed later.

The bolster channels the plastic melt from the circular extruder exit toward the moveable die, as shown in Fig. 1(b). In any configuration of the die components, the shape of the fixed bolster exit should not be smaller than the die exit. The approach used by the authors is to form the bolster exit as the envelope of all the die exit profiles throughout the permissible motion. Creating the envelope is progressively more difficult as the die DOF increases.

Both mechanical joint styles that permit rotation between die exit components (crescent and corner) exhibit a groove and gap as illustrated in Fig. 2. The shape of the die components must prevent the gap and groove from appearing within the bolster exit, which would allow flow of plastic melt into the die joints and bypass the die exit.

Crescent joints are preferred for rotational movement between die exit components as they provide a high resistance to leakage through the joint interface. Corner joints allow for a sharp vertex in the die exit, but resistance to die leakage is significantly lower. Accordingly, it is advantageous to allow for a concave radius at corners of the extrusion profile.

Since the extruded product profile geometry is relative, any edge can be selected as fixed. The designer has the opportunity to wisely select the fixed edge to facilitate the mechanical design. When a corner joint is required, the mechanical design is simpler when the corner joint is placed adjacent to a fixed component. With a gap being close to the die exit, it is difficult to cover the gap with the bolster exit. This is particularly true when the corner joint is not adjacent to the fixed edge. A series of shields are required which complicates the die.

The designer should closely inspect the target profiles and judiciously identify a segment vector V and connection vector W. In the many cases experienced by the authors, intuition can be used to identify regions of the profiles that have similar shapes and can be appropriately designated as an $M$-segment. Additionally, connections between segments are obvious and should be specified. A carefully selected V and W leads to more rapid convergence for a shape-changing chain design.

Finally, a direct comparison between the die exit profile and the profile of the part is not usually made. Such differences are expected as the part swells as it exits the die and cools [9]. The amount and location of swell are dependent on the material and processing variables such as temperature and puller tension.

## Conclusions

This paper presented the development of variable-geometry dies that enable the extrusion of plastic parts with a varying cross section. The procedure for synthesizing planar rigid-body shape-changing mechanisms was presented. New categories of design profiles are introduced to shape-change design theory. Fused connections are enabled for designers to assign at the beginning of the synthesis process. A method for matching profiles containing sharp corners was addressed. Desirable design attributes were identified, and potential joint designs that could be used to join the interconnected parts that form the die exit were described. An array of joints allows for a wide variety of variable-geometry dies to be created. The different joint concept designs were used to create a variety of shape-changing dies that were implemented in a production environment. Experimental observations showed few adverse leakage effects of the moving die exit. Repeatability results indicate that these moveable dies were comparable to tolerances for a conventional extrusion die.

## Acknowledgment

Extrusion time and expert advice was provided, courtesy of Creative Extruded Products, Inc. Support with the data analysis was provided by Austin Fischer. Bingjue Li acknowledges the University of Dayton Office for Graduate Academic Affairs for supporting her scholarship.

## Funding Data

• Division of Civil, Mechanical and Manufacturing Innovation (Grant No. 1234374).

• University of Dayton (Graduate Student Su).

## Nomenclature

die exit area

•
• Aj =

the apices matrix of the jth target profile

•
• d95 =

the 95% confidence interval of the deviation for the experimental profiles

•
• $d¯$ =

the mean of the absolute values experimental profile deviations

•
• D =

diameter of the extruder screw

•
• DOF =

degree-of-freedom

•
• H =

•
• mj =

the number of pieces on the jth target profile

•
• Nj =

the number of points on the jth target profile

•
• p =

the number of profiles

•
• q =

the number of segments

•
• Q =

the volumetric flow rate of an extruder screw

•
• SM =

segment matrix

•
• v =

average speed of the extrudate

•
• V =

segment-type vector

•
• W =

connection-type vector

•
• $zji$ =

the coordinate of the ith point on the jth target profile

•
• $z¯ji$ =

the coordinate of the ith point on the chain at the jth target profile

•
• $z̃ji$ =

the optimized coordinate of the ith point on the chain at the jth target profile

•
• θ =

•
• ω =

rotational speed of the extruder screw

## References

1.
Biron
,
M.
,
2007
,
Thermoplastics and Thermoplastic Composites: Technical Information for Plastics Users
, 1st ed.,
Elsevier
,
Exeter, UK
.
2.
Muccio
,
E.
,
1994
,
Plastics Processing Technology
,
American Society for Metals International
,
Materials Park, OH
.
3.
Rauwendaal
,
C.
,
2001
,
Polymer Extrusion
,
Hanser
,
Munich, Germany
.
4.
Levy
,
S.
, and
Carley
,
J.
,
2005
,
Plastics Extrusion Technology Handbook
, 2nd ed.,
Industrial Press
,
South Norwalk, CT
.
5.
Kostic
,
M. M.
, and
Reifschneider
,
L. G.
,
2006
, “
Design of Extrusion Dies
,”
Encyclopedia of Chemical Processing
,
S.
Lee
, ed.,
Taylor & Francis
,
New York
.
6.
Sun
,
Y.
,
2006
, “
Optimization of Die Geometry for Polymer Extrusion
,” Ph.D. thesis, Michigan Technological University, Houghton, MI.
7.
Lebaal
,
N.
,
Schmidt
,
F.
,
Puissant
,
S.
, and
Schlafli
,
D.
,
2009
, “
Design of Optimal Extrusion Die for a Range of Different Materials
,”
Polym. Eng. Sci.
,
49
(
3
), pp.
432
440
.
8.
Michaeli
,
W.
,
1984
,
Extrusion Dies, Design and Engineering Computations
,
Hanser
,
Munich, Germany
.
9.
Covas
,
J. A.
,
Carneiro
,
O. S.
, and
Brito
,
A. M.
,
1991
, “
Designing Extrusion Dies for Thermoplastics
,”
J. Elastomers Plast.
,
23
(3), pp.
218
238
.
10.
Giaier
,
K. S.
,
Myszka
,
D. H.
,
Kramer
,
W. S.
, and
Murray
,
A. P.
,
2014
, “
Variable Geometry Dies for Polymer Extrusion
,”
ASME
Paper No. IMECE2014-38409.
11.
Lawson
,
P.
, and
Yen
,
J. L.
,
1988
, “
A Piecewise Deformable Subreflector for Compensation of Cassegrain Main Reflector Errors
,”
IEEE Trans. Antennas Propag.
,
36
(
10
), pp.
1343
1350
.
12.
Murray
,
A. P.
,
Schmiedeler
,
J. P.
, and
Korte
,
B. M.
,
2008
, “
Kinematic Synthesis of Planar, Shape-Changing Rigid-Body Mechanisms
,”
ASME J. Mech. Des.
,
130
(
3
), p.
032302
.
13.
Persinger
,
J. A.
,
Schmiedeler
,
J. P.
, and
Murray
,
A. P.
,
2009
, “
Synthesis of Planar Rigid-Body Mechanisms Approximating Shape Changes Defined by Closed Curves
,”
ASME J. Mech. Des.
,
131
(
7
), p.
071006
.
14.
Zhao
,
K.
,
Schimiedeler
,
J. P.
, and
Murray
,
A. P.
,
2011
, “
Kinematic Synthesis of Planar, Shape-Changing Rigid-Body Mechanisms With Prismatic Joints
,”
ASME
Paper No. DETC2011-48503.
15.
Li
,
B.
,
Murray
,
A. P.
, and
Myszka
,
D. H.
,
2015
, “
Designing Variable-Geometry Extrusion Dies That Utilize Planar Shape-Changing Rigid-Body Mechanisms
,”
ASME
Paper No. DETC2015-46670.
16.
Panchal
,
R. R.
, and
Kazmer
,
D.
,
2007
, “
Characterization of Polymer Flows in Very Thin Gaps
,”
ASME
Paper No. MSEC2007-31108.
17.
Gander
,
J. D.
, and
Giacomin
,
A. J.
,
1997
, “
Review of Die Lip Buildup in Plastics Extrusion
,”
Polym. Eng. Sci.
,
37
(
7
), pp.
1113
1126
.
18.
Morton-Jones
,
D. H.
,
1989
,
Polymer Processing
,
Chapman and Hall
,
London
.
19.
Funke
,
L.
,
Schmiedeler
,
J. P.
, and
Zhao
,
K.
,
2015
, “
Design of Planar Multi-Degree-of-Freedom Morphing Mechanisms
,”
ASME J. Mech. Rob.
7
(
1
), p.
011007
.
20.
Shamsudin
,
S. A.
,
Murray
,
A. P.
,
Myszka
,
D. H.
, and
Schmiedeler
,
J. P.
,
2013
, “
Kinematic Synthesis of Planar, Shape-Changing, Rigid Body Mechanisms for Design Profiles With Significant Differences in Arc Length
,”
Mech. Mach. Theory
,
70
, pp.
425
440
.