The following sections cover the formulas and calculations for the Extrusion process:

Material Cost depends on the following:

Raw material cost depends on the following:

Rough mass depends on the following:

Utilization depends on the following:

Utilization depends on the following:

Scrap material credit is 0 if the setup option Enable Scrap Material Credit is set to false. Otherwise, it is the product of the following:

Scrap mass depends on the following:

Scrap part credit is 0 if the setup option Enable Scrap Part Credit is set to false. Otherwise, it is the product of the following:

Cycle time is the product of the following:

Process time depends on the following:

Billet process time is the sum of the following:

Note that the extrusion process is assumed to be balanced, so that various activities are performed in parallel with the activities listed above, including stretching, cutting, racking, transporting, and applying release agent. Since they are performed in parallel, they do not add to the cycle time.

Contact time depends on the following:

Die change time per billet is the time required to change the die, amortized over the number of billets that can be processed before a change of die is necessary. It depends on the following:

This formula evaluates strain crack propagation. Based on this evaluation, the number of billets per die depends on the following:

Tool final crack length depends on the following:

Tool max stress depends on the following:

Tool die diameter depends on the following:

The maximum acceptable ram speed for each machine is calculated in order to guide machine selection (see Machine Selection for Extrusion). The maximum acceptable ram speed for the selected machine is also used to calculate cycle time (see Cycle Time).

For a given machine, acceptable ram speeds are constrained by a number of factors:

For a given machine, aPriori does the following to calculate maximum acceptable ram speed:

1. Find the maximum number of cavities that can fit in the die. See Maximum Number of Die Cavities.

2. Adjust the machine’s Max Ram Speed downward in order to account for part complexity and quench type. See Reduced Max Ram Speed.

3. Consider a die with each possible number of cavities, n, that can fit in the die. For example, if Maximum Number of Die Cavities is 3, consider a die with 3 cavities, a die with 2 cavities, and a die with 1 cavity. For each one, do the following:

The cost model finds, for a die with a given number of cavities, the highest ram speed that results an in-range exit speed by calculating the ram speed that results in the material’s Maximum Extrusion Exit Speed:

Max material ram speed depends on the following:

Extrusion ratio depends on the following:

The cost model finds the exit speed that results from Reduced Max Ram Speed as follows:

Exit speed, in this case, depends on the following:

As part of calculating Maximum Acceptable Ram Speed, aPriori adjusts the machine’s Max Ram Speed downward, in order to account for part complexity and quench type. The cost model uses the following formula:

Reduced max ram speed is the product of the following:

By default, the cost model assumes that the number of die cavities is the number of die cavities that results from the calculation of the Maximum Acceptable Ram Speed for the selected machine. This value is bounded above by Maximum Number of Die Cavities. Users can override the default with the setup option Number of Cavities in Die, but the override does not affect machine selection or the determination of the maximum acceptable ram speed.

This section discusses the calculation of the maximum number of cavities than can fit in the die. This quantity is used to help determine the Maximum Acceptable Ram Speed and the Number of Die Cavities.

For parts extruded at the same time, the centroid of the extrusions must be equidistant from the die centroid, as in the illustration below.

Because of this, the cost model determines the maximum number of cavities that can fit in a die as follows:

The number of circles that can fit can be determined from the angle, X, formed by the following two line segments:

Each SEC lies on a certain fraction of the circle (shown in black in the figure) whose center is the center of the usable die area and whose radius, R (shown in blue in the figure), is the difference between the radius of the usable die area and the radius, r, of an SEC. The fraction, f, of the black circle that is occupied by each SEC is (2 * X) / (2 * Pi). So the maximum number of SECs that can lie on the black circle (without overlapping) is given by the following:

The angle X can be determined from the SEC radius, r, and the black circle radius, R. X is the angle opposite the side of length r in the right triangle formed by the following line segments:

Note that the last two line segments form a right angle, since a line tangent to a circle at a given point is perpendicular to the radius that ends at that point. Therefore, X is given by the following:

The final value for maximum number of cavities is bounded above by the value of the cost model variable maxStandardNestedPartsOnDie (6 in starting point VPEs). That is, if the calculated value Max Number of Die Cavities (above) exceeds the value of maxStandardNestedPartsOnDie, the value of maxStandardNestedPartsOnDie is used instead.

Part complexity factor is used to determine the Maximum Acceptable Ram Speed. This factor is used to adjust the machine maximum ram speed in order to ensure that no deformation occurs during extrusion.

The complexity factor is bounded below by 0.1. More complex parts have lower complexity factors (and thus effect a greater reduction in the machine’s maximum ram speed).

The complexity score is bounded above by 10. More complex parts have higher complexity scores.

A part’s class complexity score is higher if there are CrossSectionHollow GCDs present, especially fully hollow or asymmetric ones. In the absence of CrossSectionHollow GCDs, thinner parts have a higher class complexity score than thicker ones. See Class Complexity Score .

A part’s tongue complexity score is higher for a part with a higher Critical Tongue Ratio. See Tongue Complexity Score.

Below are some sample cross sections arranged from most to least complex:

A part’s class complexity score depends on the characteristics of CrossSectionHollow GCDs present in the part or on the maximum thickness of the part cross section.

A CrossSectionHollow is a region of the part’s cross section for which one of the following holds:

The cross sections shown below have fully hollow regions:

The cross sections shown below have semi-hollow regions:

Note that the first two semi-hollow regions are symmetric, and the last one is asymmetric. The geometric property Symmetric indicates whether a CrossSectionHollow is symmetric.

The class complexity score is determined by cases:

If T is the part’s Critical Tongue Ratio, tongue complexity score is determined as follows:

The tongue ratio of a semi-hollow CrossSectionHollow GCD (see Class Complexity Score) depends on the geometric properties Area and Gap Length. (Fully hollow CrossSectionHollow GCDs have a Gap Length of 0.) Tongue ratio is the ratio of the Area to the square of the Gap Length:

The critical tongue ratio of a part is the tongue ratio of the CrossSectionHollow with the largest tongue ratio.

The tongue depth of a semi-hollow CrossSectionHollow GCD (see Class Complexity Score) depends on the geometric properties Perimeter and Gap Length. (Fully hollow CrossSectionHollow GCDs have a Gap Length of 0.) Tongue depth is the difference between the Perimeter and twice the Gap Length:

The critical tongue depth of a part is the tongue depth of the CrossSectionHollow with the largest tongue depth.

Critical tongue depth is 0 for a part with no semi-hollow CrossSectionHollow GCDs.

Quench speed reduction factor (see Reduced Max Ram Speed) depends on the current Quench Type:

This section discusses the calculation of four quantities:

One or more of these quantities are used in a variety of calculations, including the following:

By default, the quantities are determined as described in steps 1-6, below. If the setup option Billet Length overrides the default, they are calculated as described in step 7, below.

1. Determine the volume of the largest standard billet for the current machine. This depends on the following:

2. Calculate the maximum number of parts that can be produced by such a billet. This depends on the volumes of the following:

This step determines the maximum number of parts than can fit on a billet, given the number of cavities and the machine constraints on billet size. This provides an upper bound on the number of parts per billet assumed by the cost model.

3. Determine the number of billets required to produce one production batch, given that the number of parts produced from each billet is the quantity found in step 2. This depends on the following:

This is the minimum possible number of billets required for a production batch, given the machine constraints on billet size. The cost model assumes that this number of billets is used for each batch, provided this assumption results in a billet length that is greater than or equal to the machine’s Min Billet Length (see steps 5-7).

4. Determine the number of parts per billet, assuming that the parts in a production batch are evenly distributed across all the billets needed for the batch. This depends on the following quantities:

This might result in fewer parts per billet than found in step 2, and so result in a smaller billet size. The cost model assumes that the number of parts produced by each billet is the value found in this step, provided that this assumption results in a billet length that is greater than or equal to the machine’s Min Billet Length (see steps 5-6).

5. Determine the volume and length of the smallest billet that can yield the number of parts found in step 4. This depends on the following:

If Billet Length is greater than or equal to the Machine’s Min Billet Length, stop here.

6. If Billet Length is less than the Machine’s Min Billet Length, the billet length is assumed to be Min Billet Length. The Billet Volume is assumed to be given by the following:

Number of parts per billet depends on the following:

The number of billets needed for a batch depends on the following:

7. If the user overrides the calculated billet length with the setup option Billet Length, calculations like those in step 7 are performed:

Number of parts per billet is depends on the following:

The number of billets needed for a batch depends on the following:

The formulas below show the cost for one die. An evaluation of strain crack propagation is used to determine tool life—see the formula for Number of Billets per Die in Cycle Time. The final cost for hard tooling takes into account the number of dies required for a production batch.

Total tool cost is the sum of the following:

If the part has no CrossSectionHollow GCDs, tool hollow adjusted die cost is specified by the formula Tool Base Cost. Otherwise, it is specified by this formula, and depends on the following:

Tool center feed cost is 0 if the part has 0 or 1 CrossSectionHollow GCDs. Otherwise, it is given by this formula, and is the product of the following:

Tool center feed cost is 0 if the part’s Critical Tongue Ratio is less than the tool shop variable Shut Off Die Tongue Ratio Limit. Otherwise, it is given by this formula, and is the product of the following:

Tool base cost is the sum of the following:

Tool hollow die cost depends on the following:

Tool solid die cost depends on the following:

Tool die mass is the product of the following:

Tool die volume is the product of the following:

Tool die thickness is the value given by this formula, rounded up to a standard thickness, and bounded above by Tool Die Max Thickness (see below). It depends on the following:

This value is determined from a logarithmic function of billet diameter (specified by the machine property Billet Diameter), derived from a representative data sample.

Tool diameter is the value given by this formula, rounded up to a standard thickness. Standard thicknesses are governed by the values in the lookup table tblStandardSizes for extrusionToolDiameter and the unit system specified by the cost model variable defaultUnitSystem (US in starting point VPEs). Tool diameter is the product of the following:

Tool bolster cost depends on the following:

Tool bolster mass is the product of the following:

Tool bolster volume is the product of the following:

Tool bolster thickness is the value given by this formula, rounded up to a standard thickness, and bounded below by 0. It depends on the following:

Feeder thickness is determined based on the assumption that the feeder redirects flow at an angle of 45 degrees. It depends on the following:

The backer fills the non-bolster portion of the die stack, if necessary. The sum of the die and backer lengths is determined from a logarithmic function of billet diameter (specified by the machine property Billet Diameter), derived from a representative data sample. The backer is necessary only if this sum exceeds the Tool Die Thickness (see formula). If the backer is not necessary, tool backer cost is 0. Otherwise, tool backer cost depends on the following:

Tool backer mass is the product of the following:

Tool backer volume is the product of the following:

Tool bolster thickness is the value given by this formula, rounded up to a standard thickness, and bounded below by 0. It depends on the following:

Tool feeder cost is 0 if no feeder is required, that is, if the Outside Diameter of the part’s CrossSection is less than or equal to the machine’s Billet Diameter. If a feeder is required, tool feeder cost given by this formula, and depends on the following:

Tool feeder mass is the product of the following:

Tool feeder volume is the product of the following: