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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
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
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
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
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
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
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
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.
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.
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.
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
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
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
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
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).
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
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