17
Manufacturability Constraint Formulation for Design Under Hybrid Additive-Subtractive Manufacturing (Supplemental Materials) Technical Report UIUC-ESDL-2018-01 Albert E. Patterson * , James T. Allison Engineering System Design Lab University of Illinois at Urbana-Champaign May 31, 2018 Abstract This supplementary document provides additional details relevant to the paper ‘Manufacturability Constraint Formulation for Design Under Hybrid Additive and Subtractive Manufacturing’ [1]. Detailed formulation and results of the case studies described in Section 5 of [1] are presented here. The full results and Matlab code can be found at [2]. * Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana- Champaign, [email protected] Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana- Champaign, [email protected] c 2018 ESDL 1

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Page 1: Manufacturability Constraint Formulation for Design Under

Manufacturability Constraint Formulation for DesignUnder Hybrid Additive-Subtractive Manufacturing

(Supplemental Materials)

Technical Report UIUC-ESDL-2018-01

Albert E. Patterson∗, James T. Allison†

Engineering System Design Lab

University of Illinois at Urbana-Champaign

May 31, 2018

Abstract

This supplementary document provides additional details relevant to the paper‘Manufacturability Constraint Formulation for Design Under Hybrid Additive andSubtractive Manufacturing’ [1]. Detailed formulation and results of the casestudies described in Section 5 of [1] are presented here. The full results andMatlab code can be found at [2].

∗Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, [email protected]†Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-

Champaign, [email protected]©2018 ESDL

1

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Patterson, Allison 2018 IDETC Supplement 2

Contents

1 Introduction 4

2 Case Study 1: Design of CNC Tool Shuttle Frame 42.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Manufacturing and Optimization Problems . . . . . . . . . . . . . . . . 52.3 Manufacturing Considerations . . . . . . . . . . . . . . . . . . . . . . . 52.4 Manufacturing Constraints . . . . . . . . . . . . . . . . . . . . . . . . 52.5 Manufacturability Constraints . . . . . . . . . . . . . . . . . . . . . . . 62.6 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.7 Solution and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Case Study 2: Design of Pulley with Brake 93.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Manufacturing and Optimization Problem . . . . . . . . . . . . . . . . 93.3 Manufacturing Considerations . . . . . . . . . . . . . . . . . . . . . . . 103.4 Manufacturing Constraints . . . . . . . . . . . . . . . . . . . . . . . . 103.5 Manufacturability Constraints . . . . . . . . . . . . . . . . . . . . . . . 103.6 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.7 Solution and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

References 17

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Patterson, Allison 2018 IDETC Supplement 3

List of Figures1 Case study 1 (a) configuration, (b) design variables, and (c) loading and

free-body diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Design problem formulation for case study 1 . . . . . . . . . . . . . . . 63 TO stress-mass curves for the (a) original and (b) SO-TO designs . . . 74 Case study 1 results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Case study 2 (a) configuration, (b) dimensions, and (c) free-body diagram 96 Compliance-mass curve for case study 2 . . . . . . . . . . . . . . . . . 117 Results for case study 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 11

List of Tables1 Shape optimization results . . . . . . . . . . . . . . . . . . . . . . . . 72 Shape optimization results . . . . . . . . . . . . . . . . . . . . . . . . 83 Topology optimization results . . . . . . . . . . . . . . . . . . . . . . . 11

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Patterson, Allison 2018 IDETC Supplement 4

1 Introduction

This supplemental document presents the detailed formulation of the case studies pre-sented in Section 5 the paper "Manufacturability Constraint Formulation for DesignUnder Hybrid Additive-Subtractive Manufacturing". The problem formulation and re-sults sections of each case study are reproduced exactly from the main manuscript inorder to ensure continuity. The two appendices (S1 and S2) contain detailed informa-tion referred to in the main body of this supplemental document. Red text refers tofigures or tables in the main manuscript which were not reproduced in this supplementaldocument.

2 Case Study 1: Design of CNC Tool Shuttle Frame

The first study examined here is the design of the frame on a CNC machine tool shuttle(Fig. 1a). Such carts are often used in manufacturing systems to shuttle tools aroundto various CNC machines during processing; this way, expensive or specialized toolscould be shared among several machines and mid-process tool replacements are easierto automate.

Figure 1: Case study 1 (a) configuration, (b) design variables, and (c) loading andfree-body diagrams

2.1 Problem Definition

The specific tool shuttle in question is designed to carry three tools at once, up to 3.5kg of mass each, along a linear rail via grooved track rollers; these rollers allow thecart to be tight and secure, while also allowing curves in the track. Figure 1b showsthe dimensions of a sample design that has not been optimized; Figure 1c shows itsmechanical configuration and free-body diagram. The cart must be able to carry thethree tools, as well as support the tool holder, a total weight of about 100 N distributedevenly along the cart; applying a reasonable 1.50 factor of safety, the design forceshould be 150 N. The main frame is to be made from ABS plastic and manufacturedvia a hybrid AM-SM process. The specified concerns of the user are the mass andstability of the carriage, and potential cracking of the plastic frame. To these ends, thedesigner concluded that the design should minimize mass as much as possible, while also

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Patterson, Allison 2018 IDETC Supplement 5

maintaining sufficiently low bending stress to avoid degrading or fracturing the plasticduring use. To ensure stability, a minimum web thickness x8 was set at one third of theupper deck thickness, x7/3. This is a constraint related to the use environment, notthe manufacturing conditions. In addition, it is a simplified constraint formulation, andcould be replaced with a higher-fidelity physics-based constraint. The mass reductionobjective was specified to be ten times as important as the stress objective, so long as thestress remains under yielding. An alternative approach would be to quantify the tradeoffbetween mass and stress objectives using a multi-objective optimization approach. Thecart has several other design constraints that arise from the configuration of the CNCmachines it will service:

1. The upper deck must retain its basic shape to interface with the tool rack, butthe thickness x9 can be modified.

2. The lower deck must retain its basic shape, but thickness x2 can be modified.

3. The overall height and length of the frame must be retained

4. The upper and lower surfaces of each deck must be parallel

5. The overall part must be symmetric to ensure balance

2.2 Manufacturing and Optimization Problems

The use of the hybrid AM-SM process to manufacture this frame allowed a two-stepsequential optimization problem, using not only shape optimization over the designvariables, but topology optimization as well. The hybrid process allowed for differentregions of the part be optimized differently, as the part could be manufactured usingboth AM and SM processes in different part regions. This concept was particularlyapplicable to this design, as large areas of the frame needed to be flat and smooth,while others could be more free-form in shape.

2.3 Manufacturing Considerations

In this type of problem, AM is enables fabrication of regions with complex topology,while the shape-optimized regions could be manufactured using subtractive processes.Due to cost and production time, it is usually best to avoid additive processes for simplegeometries, such as the decks in this part; subtractive processing will be needed to bringthe part to the surface finish requirements, so using it for manufacturing increases theefficiency of the process. The marriage of the two in this hybrid process also allowsthe hybridization of the shape and topology optimization problems to fit the intendedmanufacturing processes.

2.4 Manufacturing Constraints

With the list of manufacturing considerations completed, these were converted intomanufacturing constraints, shown in the central column of Fig. 9 in the main paperand is shown in detail in Appendix S1. There were eight AM- and three SM-specificconsiderations, as well as one that is common to them both. As a reminder, theseconstraints are those imposed on the use of the manufacturing processes by the natureof the manufacturing considerations. They are not design constraints, but will serve as

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Patterson, Allison 2018 IDETC Supplement 6

the basis for the formulation of the manufacturability optimization constraints (thirdcolumn).

2.5 Manufacturability Constraints

From the mechanics and limitations of the manufacturing processes, the design-relatedmanufacturability constraints can be derived, as shown in the right column of Fig. 9 inthe main paper and in Appendix S1. The listed constraints are the final set, with thedominated and inactive constraints eliminated. The problem is also subject to a set ofgeneral part manufacturability constraints, in addition to those on the specific designvariables, as summarized in Fig. 10 in the main paper.

2.6 Solution Method

As stated in the problem definition, the goal of the problem is to simultaneously minimizebending stress σ(x) in the frame and the mass m(x) of the frame. It is subject to fiveperformance-related constraints and twelve sets of manufacturing-related constraints,as discussed in the previous sections. The problem is a sequential shape-topologyoptimization problem; the formulation is shown in Fig. 2. As previously described, thisformulation is necessary since the entire frame must be optimized, but only a portioncan consist of free-form design. The constraints g(x) and h(x) for each problem arethose given in the problem statement and the manufacturability constraints. It shouldbe acknowledged that solutions to this sequential method will likely be different froma simultaneous approach. The sequential method supports easier solution and mapsintuitively to the hybrid process, but may be sub-optimal. A simultaneous approach mayrequire a fundamentally different design representation and other formulation elements,and may be very difficult to solve. Comparison of sequential and simultaneous methodsfor hybrid manufacturing is outside the scope of this article.

Figure 2: Design problem formulation for case study 1

2.7 Solution and Results

After formalizing the objective function in terms of the design variables, x = [x1, x2, x3, ..., x12],the composite objective function is:

f(x) = σ(x) + 10m(x) (1)

where each objective component was normalized to make it dimensionless. Objectivefunction weights may be varied to generate sets of non-dominated solutions to themulti-objective problem.

The full equations are too long to reproduce here; the full derivation and equationscan be found in Appendix S2, as well as in the attached Matlab codes. A shape

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Patterson, Allison 2018 IDETC Supplement 7

optimization was performed in Matlab using the fmincon routine with the interior-point algorithm. The initial and final values are shown in Table 1; fixed input values[x1, x7, x11, x12] are not shown here. The initial mass and stress values were calcu-lated using finite element analysis. Note that all the variables converged on the lowerconstraint bounds, which is logical for a problem that reduces mass and stress. As afeasibility check, the problem was also solved as an unconstrained problem and produceda non-trivial solution; the constraints were violated when they were removed, showingthat all the manufacturability constraints are active.

Table 1: Shape optimization results

Variable x0 x† Variable x0 x†

x2 10.0 4.0 x6 8.0 4.0

x3 8.0 4.0 x8 30.0 40.0

x4 8.0 4.0 x9 10.0 4.0

x5 6.0 4.0 x10 60.0 72.0

f(x) 5.9503 3.5519

exitflag 1 feval 529

Figure 3: TO stress-mass curves for the (a) original and (b) SO-TO designs

The shape optimization results x† were then used as the initial points for the topologyoptimization problem. The TO problem was solved using Pareto (Sciartsoft Inc.).Only the web and chamfers of the frame were considered during topology optimization,as previously described; surfaces subjected to SM were retained before being analyzedby Pareto. The stress-mass-fraction Pareto curve was generated for the TO problemFig. 3a, which was used to select the final volume fraction used to generate the optimaldesign. The final selected volume fraction is marked on the figure. Note that, for afull-density material, volume fraction is directly convertible to part mass. The lowestmass with a stress under the yield stress of the material and which produced a feasibleTO solution was considered to be the best solution. The best calculated volume fractionwas 0.49 for the SO-TO problem and 0.45 for the TO-only problem.

The TO problem was also repeated using the initial point (eliminating the shape

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Patterson, Allison 2018 IDETC Supplement 8

Table 2: Shape optimization results

Case Mass (kg) Max Stress (MPa) f(x)

Initial 0.5890 0.0603 5.9503

SO only 0.3244 0.6158 3.8599

TO only 0.3593 0.3544 3.9473

SO-TO 0.1752 1.1101 2.8621

Figure 4: Case study 1 results

optimization step) to see the effect on the TO problem Fig. 3b. Unfortunately, thedesign corresponding to the lowest calculated mass volume fraction could not producefeasible stl files for either case; since the design must be manufacturable, the bestfeasible case was taken as the best solution (0.54 and 0.61). A comparison of the mass-stress values for the original design, the shape optimization results, the hybrid problem,and the TO problem are shown in Table 2. Clearly, the hybrid problem, with shapeoptimization and then TO, produced the best overall feasible design, even though allare manufacturable. Figure 4 shows the geometries for these and the final manufacturedframe.

For the FEA and analysis, the yield stress for 3-D printed ABS is assumed to be 29MPa. The yield stress value was not used as a constraint, but as an input into the finiteelement problem for the topology optimization.

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Patterson, Allison 2018 IDETC Supplement 9

3 Case Study 2: Design of Pulley with Brake

The second case study examined a V-belt drive pulley for an electric generator illustratedin Fig. 5a. The configuration was a standard two-belt drive pulley, with the additionof a radial encoder to track its rotation and a pneumatic emergency brake. The pulleyis mounted on a frame, also containing the brake and encoder, and connected to thegenerator via a drive shaft.

Figure 5: Case study 2 (a) configuration, (b) dimensions, and (c) free-body diagram

3.1 Problem Definition

The generator pulley being designed in this case study is subjected to a torque loadduring operation, a load that could peak as high as 50 N-m during ramp-up and ramp-down. If the pulley is subjected to a shock load or the emergency brake is engaged, it isdesigned to break, while being retained by the pulley support bracket; a factor of safetyis not needed. The pulley must be made from ABS plastic and will be fabricated usinga hybrid AM-SM manufacturing process. The customer needs the mass to be reducedas much as possible, while also reducing the compliance of the pulley; the reduction ofmass is the most vital requirement and is a factor of five more heavily weighted thanreducing compliance, as long as the compliance remains under 1.0 N-m.

3.2 Manufacturing and Optimization Problem

The approach in this problem is identical with that of Case Study 1, except in this designproblem, the region on the edge (interfacing with the brake and the encoder), the driveshaft, and belt groves for the pulley are fixed and cannot be optimized. Therefore,only the web is subject to improvement. This can be done with a single TO problem,retaining the fixed surfaces.

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Patterson, Allison 2018 IDETC Supplement 10

3.3 Manufacturing Considerations

The frame was manufactured using a combination of fused deposition modeling andturning/facing/boring on a lathe, where the basic form of the frame was manufacturedusing the AM process and the interfaces and decks were cut to the proper size usingthe SM process. The manufacturing considerations are the same as those given in Fig.9 in the main paper, with the exception of those for the milling process. The additionalmanufacturing consideration from the lathe are shown in Fig. 9 in the main paper. Notethat Figs. 9 and 15 in the main paper should be used jointly in examining this problem,which will be subjected to the same nine AM-based manufacturing considerations.

3.4 Manufacturing Constraints

The manufacturing constraints, which are distinct from those shown in Fig. 9 in themain paper, are listed in the central column of Fig. 15 in the main paper. This listshould be used in conjunction with the list of constraints shown in Fig. 9 in the mainpaper.

3.5 Manufacturability Constraints

This problem is subject to the first three manufacturability constraints shown in Fig. 9in the main paper, as well that those shown in Fig. 10 in the main paper. There arethree additional manufacturability constraints, due to the mechanics of the lathe-basedprocesses. These are shown in the third column of Fig. 15 in the main paper.

3.6 Solution Method

As described previously, a shape optimization is not necessary for this problem dueto the fixed nature of the areas which will be subjected to SM. The objective of theoptimization is to reduce mass m(x) while also minimizing compliance c(x); the massobjective is more important when the compliance does not exceed the threshold givenin the problem statement. Otherwise, the setup and formulation of the study is thesame as described in Case Study 1.

3.7 Solution and Results

The objective function for this problem, based in the stated requirements for the prob-lem, is:

f(x) = c(x) + 5m(x) (2)

where c(x) is a compliance metric and m(x) is the mass of the pulley. The terms werenormalized to make them dimensionless. The topology optimization problem was solvedusing Pareto, similarl to Case Study 1. Two points were taken from the compliance-mass curve, one at the lowest mass which produced a feasible stl file, and one thatbalanced the mass and compliance. These are shown in Fig. 6. Table 3 and Fig. 7 showthe results of this study, including both the calculated designs and the manufacturedfinal parts.

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Patterson, Allison 2018 IDETC Supplement 11

Table 3: Topology optimization results

Case Mass(kg) Compliance(Nm) f(x)

Original design 0.262 0.298 1.608

Balanced 0.218 0.338 1.428

Min Mass 0.180 0.473 1.373

Figure 6: Compliance-mass curve for case study 2

Figure 7: Results for case study 2

Page 12: Manufacturability Constraint Formulation for Design Under

Patterson, Allison 2018 IDETC Supplement 12

App

endixS1

:Con

straintMapping

Table

ManufacturabilityCon

straintGenerationforAM:F

used

Dep

osition

Mod

eling

Manufacturin

gCon

siderations

Manufacturin

gCon

straints

ManufacturabilityCon

straints

DesignEff

ects

Generated

Con

straints

1Prin

torientation

Capture

materialprop

erty

effects

andcost

considera-

tions

ofprintorientation

Printorientationmustminim

ize

materialloss,minim

izesupp

ort

materialu

se,m

axim

izecontract

area

with

thebu

ildplate,anden-

sure

feasible

subtractivecutting

path

formilled

areas

Sincethedesig

nof

thesystem

understud

yhasfixed

overhang

angles,this

constraint

does

not

affectthedesig

nvaria

bles

Non

eforgivendesig

nNon

eforgivendesig

n

2Linear

printspeed

Printspeed

isalim

iting

fac-

torinFD

M,asitd

etermines

build

rate

andthepartqu

al-

ity

Linear

printspeedmustbe

high

enou

ghto

ensure

efficientbu

ildbu

tlow

enou

ghto

preventgaps

Forthechosen

desig

n,theprint

speedis

determ

ined

bythese-

lected

materiala

ndso

does

not

directly

affectthe

desig

nvari-

ables

Indirect

effectv

iathemate-

rialselectio

nNon

eforgivendesig

n

3Layerthickness

Factor

inqu

ality

andbu

ildrate;d

etermines

mechanical

andfra

ctureprop

ertie

s

Layer

thickn

ess

must

below

enou

ghto

ensure

acceptablesur-

face

finish

butthickenou

ghto

ensure

efficientprocessin

g

The

layerthickness

shou

ldbe

se-

lected

toensure

best

surfa

cefin

-ish

andincorporatesm

alld

esign

features

Affe

ctsthedesig

nvaria

bles

forthe

deck

thickn

essesa

ndtheminim

umfeaturesiz

e.How

ever,

itis

dominated

bythepart

shell/roof

con-

straints

andthereforeisno

tactiv

ein

thepresentd

esign.

Non

eforgivendesig

n

4Ex

trusionno

zzle

(1)

The

extrusionno

zzle

deter-

mines

thedepo

sition

rate,

shap

eof

theextrusionbead,

andsharpn

essof

corners

Extrusionno

zzlemustbe

anap-

propria

tediam

eter

forthe

de-

sired

processin

gparameters

The

basic

wall

thickn

ess

ofthe

features

mustbe

atleast

twicetheno

zzle

diam

eter

plus

the

machine-rem

oved

distance

toensure

that

astable

wallis

achieved.

The

minim

alwallthickness

istw

ice

the

nozzle

diam

-eter,

plus

some

infillfor

stability.

Forstand-alon

gfeatures,this

intotalwill

beab

out5x

theno

zzle

di-

ameter.

Clearly,the

wall

thickn

ess

constraint

dom-

inates

the

layer

thickn

ess

constraints.

Minim

umwalland

feature

leng

thscales

5Ex

trusionno

zzle

(2)

The

extrusionno

zzle

deter-

mines

thedepo

sition

rate,

shap

eof

theextrusionbead,

andsharpn

essof

corners

Extrusion

nozzle

must

beof

theprop

ermaterialfor

theun

i-form

heatingandpo

lymerization

needed

forthematerialu

sedto

manufacture

thepart

Nozzlematerialinflu

encespart

materialc

hoice

Sincethematerialis

fixed

andisno

tasevere/abrasive

material,

nodirect

effects

onthisdesig

nNon

eforgivendesig

n

Page 13: Manufacturability Constraint Formulation for Design Under

Patterson, Allison 2018 IDETC Supplement 13

6Pa

rtSh

ell(1)

The

shellisthe

outerskinof

thepart.

Itconn

ects

with

thebase

androof

andcon-

tainstheinfill

Part

shellm

ustbe

thickenou

ghto

ensure

that

thefin

alpart

isstable

andstrong

Sameas

#4

Sameas

#4

Sameas

#4

7Pa

rtSh

ell(2)

The

shellisthe

outerskinof

thepart.

Itconn

ects

with

thebase

androof

andcon-

tainstheinfill

Part

shellm

ustbe

thickenou

ghto

tolerate

cuttingwith

themill

inthesubtractiveph

aseof

the

process

Sameas

#4

Sameas

#4

Sameas

#4

8Pa

rtInfill(1)

The

infillisthe“fi

lling

”in-

sideof

partcontainedby

the

shellsandthebase/roo

fPa

rtshallbe

asclose

tofull-

density

aspo

ssible,1

00%

infill

The

minim

umleng

thscale

offeatures

mustallow

atleasttwo

shelllayers

and

asm

allinfill

amou

nt,a

sdescrib

edin

#4

Thisc

onstraintw

ould

affect

theminim

umfeaturesiz

e,bu

tis

dominated

bycon-

straint#4

Non

eforgivendesig

n

9Pa

rtInfill(2)

The

infillisthe“fi

lling

”in-

sideof

partcontainedby

the

shellsandthebase/roo

f.

Toensure

stability

ofthefin

alpart,everyinfilledregion

must

have

shell,roof,a

ndbase

region

sas

well

Baseistheinterfa

cebetweenthe

printbed

and

the

roof

isthe

layer(s)

which

finalizethepart

Sameas

#9

Sameas

#9

10Pa

rtBase/Roo

f(1)

Base

isthe

interfa

cebe-

tweentheprintbedandthe

roof

issectionwhich

finalize

thepart

Part

roof

mustbe

thickenou

ghto

ensure

that

thefin

alpart

isstable

andstrong

Sameas

#6-7

Sameas

#6-7

Sameas

#6-7

11Pa

rtBase/Roo

f(2)

Base

isthe

interfa

cebe-

tweentheprintbedandthe

roof

isthelayer(s)

which

fi-nalizethepart

Part

base

shallb

ethickenou

ghto

ensure

asecure

interfa

cewith

the

build

plate

and

consistent

warmingof

partfro

mbedheater

Sameas

#6-7

Sameas

#6-7

Sameas

#6-7

12Su

pportstructure

Supp

ortstructureprovides

supp

ort

for

complex

and

overhang

ingfeatures

ofthe

parts

Sameas

#1

Sameas

#1

Sameas

#1

Sameas

#1

ManufacturabilityCon

straintGenerationforSM

:Milling

Manufacturin

gCon

siderations

Manufacturin

gCon

straints

ManufacturabilityCon

straints

DesignEff

ects

Generated

Con

straints

13Workholding

/Fixtures

The

work

holding

inthis

prob

lem

isthebu

ildplate,

onwhich

thepart

will

beprinted

Partareassubjectedto

SMshall

allow

access

tothecuttingtool

Sameas

#1

Sameas

#1

Sameas

#1

Page 14: Manufacturability Constraint Formulation for Design Under

Patterson, Allison 2018 IDETC Supplement 14

14To

olSp

eed/

Feed

(1)

Speedandfeed

ofthetoolis

determ

ined

bytype

oftool,

the

material,

and

process

cond

ition

s

Part

features

shallbe

ofsuffi-

cientsiz

eto

dissipateheat

from

cuttingtoolswith

outdamaging

materialstructure

andprop

ertie

sSa

meas

#4

Sameas

#4

Sameas

#4

15To

olSp

eed/

Feed

(2)

Speedandfeed

ofthetoolis

determ

ined

bytype

oftool,

the

material,

and

process

cond

ition

s

Part

features

shallbe

ofsuffi-

cientsiz

eto

dissipatevibration

from

thecuttingtoolswith

out

cracking

orbreaking

thefeature

Sameas

#4

Sameas

#4

Sameas

#4

16ChipFo

rmation(1)

The

ability

ofthetoolto

cut

cleanly

and

effectiv

ely

de-

pend

son

chip-fo

rmingabil-

ity

Part

base

mustbe

thickenou

ghto

tolerate

cuttingwith

themill

inthesubtractiveph

aseof

the

processwith

outcuttinginto

the

infillorlosingmechanicalsou

nd-

ness

Sameas

#6-7

Sameas

#6-7

Sameas

#6-7

17ChipFo

rmation(2)

The

ability

ofthetoolto

cut

cleanly

and

effectiv

ely

de-

pend

son

chip-fo

rmingabil-

ity

Part

roof

mustbe

thickenou

ghto

tolerate

cuttingwith

themill

inthesubtractiveph

aseof

the

process

Sameas

#10

-11

Sameas

#10

-11

Sameas

#10

-11

18To

olDepth-of-Cut

DOC

ofthetool

isdeter-

mined

bytype

oftool,the

material,

and

processcon-

ditio

ns

The

smallest

available

cutting

tool

is1-mm

indiam

eter,so

anymachinedfeatures

mustal-

low

thismuchdistance

Sameas

#6-7

Sameas

#6-7

Sameas

#6-7

Page 15: Manufacturability Constraint Formulation for Design Under

Patterson, Allison 2018 IDETC Supplement 15

Appendix S2: Case Study 1 SO Objective FunctionDerivation

Calculating beam stress: Stress in a bending beam under a uniformly distributed loadis given by

σ(s) = W

2Slv(l − v) (3)

whereW signifies the total load (N), l is the length of the beam from support to support(mm), v is the position of the stress measurement (mm), and S is the section modulus(mm3), which is a function of design variables for the design at hand. Simplifying,assuming a uniform cross-section and center stress measurement:

σ(x) = Md

I(4)

where I is the moment of inertia (mm4), d is the distance form the neutral axis to theedge of the beam cross section (mm), and M is the total bending moment imposed.A is the area of each section. Note that, since the chamfers are symmetric, each pairof chamfers can be approximated as a single rectangle for the purposes of finding themoment of inertia.

The neutral axis is calculated as:

yn =∑Aiyi∑Ai

(5)

where the values are shown in the following table (note that some of the values of Iinclude the fixed values specified in the problem statement and therefore do have thecorrect units of mm4).

Page 16: Manufacturability Constraint Formulation for Design Under

Patterson, Allison 2018 IDETC Supplement 16

Section

Ay

Ay

Id

=y

−y n

Ad

2

160x

91 2x

9+x

10+x

230x

2 9+

60x

9x10

+60x

2x9

5x3 9

x2

+x

10+

1 2x

9−y n

Calcu

late

2x

8x10

1 2x

10+x

21 2x

8x2 10

+x

2x9x

101 12x

8x3 10

1 2x

10+x

2−y n

Calcu

late

3x

1x2

1 2x

21 2x

1x2 2

25 3x

3 21 2x

2−y n

Calcu

late

4x

5x6

x10

−1 2x

6+x

2x

5x6x

10−

1 2x

5x2 6

+x

2x5x

61 12x

4x3 3

x10

−1 2x

6+x

2−y n

Calcu

late

5x

3x4

1 2x

3+x

21 2x

4x2 3

+x

2x3x

41 12x

5x3 6

1 2x

3+x

2−y n

Calcu

late

Seeattached

Matlabfiles

forfullderiv

ationdetails

from

thistable

Page 17: Manufacturability Constraint Formulation for Design Under

Patterson, Allison 2018 IDETC Supplement 17

References

[1] A. E. Patterson and J. T. Allison, “Manufacturability constraint formulation for de-sign under hybrid additive-subtractive manufacturing,” in ASME 2018 InternationalDesign Engineering Technical Conferences (to appear), Quebec City, Canada.

[2] A. E. Patterson. (2018) Matlab files for supplement. [Online]. Available:https://github.com/pttrsnv2/IDTEC2018_Supplement