Liquefier Dynamics in FusedDepositionLayered manufacturing (LM) is an evolution of rapid prototyping (RP) technolwhereby a part is built in layers. Fused deposition modeling (FDM) is a particulartechnique in which each section is fabricated through vector style deposition of buiblocks, called roads, which are then stacked layer by layer to fabricate the final obThe latest improvements in this technology brought about the possibility of fabricatinonly a model but even the finished product. This paper presents the analysis oliquefier dynamics towards establishing control strategies for flow control duringextrusion phase, which is necessary to achieve the mentioned objective.@DOI: 10.1115/1.1688377#
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IntroductionFused deposition modeling has become a widely used r
prototyping technology. In the FDM utilized in this study~aStratasys 1650 Modeler! a plastic filament is supplied on a reand fed into a heated liquefier where it is melted. This melt is thextruded by a nozzle while the incoming filament, still in sophase, acting as a ‘‘plunger.’’ The nozzle is mounted to a mechcal stage~Fig. 1~a!!, which can be moved in thexy plane@1#. Asthe nozzle is moved over the table in a prescribed geometrdeposits a thin bead of extruded plastic, referred to as ‘‘roawhich solidify quickly upon contact with substrate and/or roadeposited earlier. Solid layers are generated by following aras-tering motion where the roads are deposited side by side withinenveloping domain boundary~Fig. 1~b!!. Once a layer is com-pleted, the platform is lowered in thez direction in order to startthe next layer. This process continues until the fabrication ofobject is completed.
Successful bonding of the roads in the deposition processcessitates control of the thermal environment. Therefore, thetire system is contained within a chamber, which is maintainea temperature just below the melting point of the material bedeposited.
Several materials are commercially available for this procincluding polymers such as ABS and investment casting wWhen needed, support structures are fabricated for overhangeometries using different materials, which are later removedbreaking them away from the object. A water-soluble support mterial, which can simply be washed away, is also available.
In FDM the part is built according to a pre-specified tool-pa~Fig. 2~b!!, usually determined during the design phase~i.e.Quickslice™! @2,3#. Each tool-path is conceptually divided intfive regions~see Fig. 2~a!! @4#:
I. Pre-movement: a prescribed volumetric flow rate is startebefore the deposition begins. A glob of certain size is generatecompensate for the intrinsic deposition delay, due to the intelength of the liquefier, which is the distance between the poinapplication of the pressure and the point of the material deliv
II. Start-up: as soon as the motion starts, an absolute flow-rhigher than the steady state flow rate, is established and mtained throughout the acceleration phase;
III. Steady-state: once the acceleration phase is completeconstant flow rate is specified;
IV. Slow-down: the main flow rate is stopped and a certa
Contributed by the Manufacturing Engineering Division for publication in tJOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript receivedDec. 2003. Associate Editor: C. Yao.
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amount of material~empirically determinated and dependentthe deceleration constant and the steady state flow rate! is broughtback revolving the motion of the rollers;
V. Exit-move: the flow control is turned off and the nozzlemoved to a pre-specified distance from the last point of the topath in order to avoid any further interferences.
Figure 2~a! schematically depicts some of the irregularities thare typical during road deposition. An ‘‘under-deposition’’ zonecan be observed during the acceleration phase and an ‘‘over-deposition’’ during deceleration.
In order to reduce these transient effects a better correlabetween deposition speed and flow rate is needed. The objeof this paper is to perform an analysis of the process taking pin the liquefier. A study based on mathematical modeling is psented first. Then a transfer function approach is developedcan be integrated as part of an overall control strategy foreffective flow control and improved part integrity@5#. These twomodels are then compared with experimental observations thaperformed by inducing step changes in deposition rates.
The dynamic of the liquefier~Fig. 1! is one of the most com-plicated phenomena to analyze in an FDM process. Insideliquefier the physical system exhibits a complex behavior duepecially to the unsteady movement of the viscous melting fluOther possible causes compounding this complex behavior cadue to some and/or all of the following reasons:
• Compressibility of the fluid;• Slippery contact conditions between the liquefier walls a
the melting flow;• Possible slippage between the feed-rollers and the filame• Uneven distribution of heat flux, which is provided by ele
trical heaters that consist of metal coils wrapped around theminum liquefier. Hence, it heats the thermoplastic material othe power is applied. A temperature regulator adjusts the powethe heater, based on temperature measurement at a single~0.59 away from the exit tip! @4#;
• Change of physical state of the melt. The thermoplastic fiment heats up and gradually melts as it travels down the lique@4#;
• Sticktion effect of the melt while the filament is pushed inthe liquefier.
Furthermore, the non-linear dependences of the material perties on the temperature and the shear rate~the melt can indeedbe modeled as a generalized Newtonian flow! bring additionaldifficulties to the modeling process.
e
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Liquefier Dynamics: Mathematical ModelThe extrusion is central to the fused deposition process du
which the thermoplastic filament is introduced, via mechanipressure, into the liquefier, where it melts and is then extrudSince the rollers are the only drive mechanism in the matedelivery system, the filament is under tensile stress upstreamthe roller and under compression@6# at the downstream side acing as a plunger. The compressive stress thus becomes the drforce behind the extrusion process~Fig. 3!.
The force required to extrude the melt must be sufficientovercome the pressure drop across the system, which strictlypends on the viscous properties of the same melt as well as ogeometry of the liquefier and nozzle~Fig. 4!. Because of the facthat melts adhere to the liquefier/die walls, the material is sjected to shear deformation during the flow.
g52dvdr
(1)
Fig. 1 Schematic of a liquefier in an FDM process
Fig. 2 Flow-control regions of a tool-path „a… and sample tool-path „b…
Fig. 3 Driving force
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Thermoplastic materials currently used in FD exhibit shear thning behavior, i.e., the viscosity decreases with increasing srate @7#. Although more extensive and complex models~i.e. BirdCarreau law, Cross law, etc.! are available, it can be assumed thadopting a power law for generalized Newtonian fluids for moeling of polymers such as ABS, is sufficiently accurate for moeling FD melts, especially in light of other high-order uncertaties in the system. Accordingly, the power law states:
t5S g
f D 1/m
(2)
wherem andf are material constants,t is the shear stress andgis the shear rate.
The power law parametersm andf indicate the flow exponenand the fluidity respectively. The general flow characteristic omaterial and its deviation from the Newtonian behavior isflected by the flow exponentm.
Applying the Power Law to the momentum flux balance onfluid element@8# respectively in the zone I, II, and III~Fig. 4!, itis possible to determine the total pressure drop:
DP1Iv52L1S vf D 1/mS m13
~D1/2!m11D 1/m
(3)
DP2Iv5S 2•m
3•tan~b/2! D S 1
D23/m
21
D13/mD S S D1
2 D 2
~m13!•2m13D 1/m
(4)
DP3Iv52L2S vf D 1/mS ~m13!~D1/2!2
~D2/2!m13 D 1/m
(5)
whereD1 andL1 are respectively the length and the diameterthe liquefier~section I in Fig. 4!, D2 andL2 are the length and thediameter of the tip of the nozzle~section III in Fig. 4!, b is theconvergence angle of liquefier-tip diameter transition. The velity v is the velocity of the filament at the entrance of the liquefiSince at this stage the material is still in solid form, this valueconstant and uniform over the cross section~plug flow!.
In Eqs.~2!–~5! the fluid is assumed to be isothermal at a teperatureT equal to the temperature of deposition. However, wha sudden change in the flow rate~i.e., step function! is applied atthe entrance of the liquefier, the steady state condition of the flis disturbed. With the introduction of new material in the systethe average velocity of the fluid increases and the averageperature drops. In order to stabilize new steady state conditithe heat flow rate must be increased. Since the temperature cois based on a single input from a thermocouple reading at theof the liquefier ~12 cm far from where the change has takplace!, the response of the system is delayed. Furthermore,temperature is regulated via heat input from electrical coil heatThe system continuously adjusts the power supplied to the caccording to the temperature difference between the desired v
Fig. 4 Regional decomposition of Liquefier ÕExtruder
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and the value detected by the thermocouple. The response timthe temperature controlling loops is thus of the order of secondue to the thermal inertia of the extrusion assembly.
The necessary heat flow rate, supplied through the wall ofliquefier,S52pD1/2L1 , can be approximately determined as:
q5~T2Ti !•r•v•A•cp
2•p•S D1
2 D •L1
(6)
whereTi is the temperature at the entrance of the liquefier,r is thedensity of the melt,A is the cross section of the inner liquefier ancp is the specific heat capacity of the melt. In order to simplify tcalculation, the melt is considered at the uniform temperaturT5Tf , that is the temperature at the end of the liquefier, andcphas been considered constant in the range of temperatureconsideration.
During the time of thermal adjustment, the flow is noisothermal, so temperature dependence of viscosity must be tinto account along with the shear-rate dependence@9,10#. Theviscosity expression can be factorized as follows:
h5H~T!•h0~ g ! (7)
whereH(T) accounts for temperature dependence andh0(g) isthe expression for viscosity at some reference temperatureTa~value of the temperature at which the parametersm andf of thepower law have been determined!. In this study the Arrheniusrelation has been adopted to model the behavior of ABS:
H~T!5expFaS 1
T2T02
1
Ta2T0D G (8)
wherea is the energy of activation,Ta is a reference temperaturfor which H(T)51 andT050 for absolute temperaturesT andTa .
Substituting Eqs.~8! and~2! into ~7!, various pressure drops ithe system can be expressed as:
DP152L1S vf D 1/mS m13
~D1/2!m11D 1/m
•e@a~1/T21/Ta!# (9)
DP25S 2•m
3•tan~b/2! D S 1
D23/m
21
D13/mD S S D1
2 D 2
~m13!•2m13D 1/m
•e@a~1/T21/Ta!# (10)
DP352L2S vf D 1/mS ~m13!~D1/2!2
~D2/2!m13 D 1/m
•e@a~1/T21/Ta!# (11)
DP5DP11DP21DP3 (12)
From Eqs.~9! to ~12! it can be observed that when a suddchange in the flow rate occurs, the average temperatureT dropsand the velocityv increases, thus resulting in considerablecrease for the pressure drop in the system.
Once total pressure drop is known, the compression forceplied to the filament in order to extrude the material can be cculated as:
F5DP•A (13)
whereA is the cross section of the filament~which is equal to thecross section of the liquefier!. Since the force is imposed by twdrive-rollers~see Fig. 3! driven by a pair of micro stepper motorsthe torqueG and the power required~to each motor! for the ex-trusion become:
G5F
2•Rr (14)
P5v r•G (15)
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whereRr is the radius andv r is the angular velocity of a roller.The expressions presented above are next arranged and deveinto a MatLab program.
In the initial approach the following assumptions were mad
1. Perfect adhesion between rollers and filament, thusv5v r•Rr ;
2. The angular velocity of the rollers abruptly changes~i.e.,step function! in order to provide the specified flow rate~see theExperimental Set Up section!;
3. The heat flow rate is increased following a ramp function~inorder to model the closed loop control!;
4. There is no upper limit to power and torque that drivmotors can provide.
Mathematical Model—Analytical ResultsIn Fig. 5 the results obtained with the first approach~no limi-
tation on power and torque supply!, have been grouped. The average velocity~Fig. 5~a!! of the melt is given according to thecommand provided by the SML file, thus it suddenly changeshalf the length of the road.
The heat flow rate is also given and it is assumed to chalinearly ~Fig. 5~b!!
dq5S 5402T0
pD1L1•r~vend2v0!•A•cpD • 1
80(16)
where 540 represents the exit temperature, in Kelvin, fromliquefier;
T05343 K is the temperature at the entrance;D151.8331023 m is the diameter of the liquefier;L150.15 m is the length of the liquefier;r5900 Kg/m3 is the density of the material;vend is the final velocity of the melt;v0 is the initial velocity of the melt;A5p/4D1
2 is the cross section of the liquefier;cp51500 J/Kg•K is the specific heat for the material.The average temperature, that is considered independent o
location, is then calculated using Eq.~6!. From Fig. 5~c! it ispossible to note that a sudden change in the inflow resultssudden reduction in temperature. This causes, according to~7! and~8!, an abrupt increase of the average viscosity of the mthus significant additional pressure~see Eqs.~9!–~12!! and force~Eq. ~13!! are needed to maintain the extrusion. Consequentlyallow continuity in the process, the motors have to provide excsive levels of torque and power. In Figs. 5~e! and 5~f! it is possibleto see that the required power increases from 0.0008 W to 15W at t52.3 seconds~when the flow rate is increased!. After thissudden change, the necessary power~and torque! decreases because the average temperature of the melt increases~the flow rateis kept constant, but the heat flow augments!.
With the assumptions presented earlier, the results presentFigs. 5~g! and 5~h! show that in order to provide a sudden increaof the angular velocity of the rollers, the stepper motors havesupply excessive levels of torque~because of the increase in thpressure drop, due to the increase in the viscosity! and power. Themotors that drive the rollers are MicroMo® Motors, typ2233T012S made by MicroMo® Electronics Inc. The maximuoutput power that they can supply is 3.66 W, the stall torque1.63331022 N/m @11# and the most efficient area of operationfor 1.63•1023,G,4.9•1023 N/m. This indicated that the modehas to be modified in order to take into consideration a limitatin power and torque supply.
In the modified approach the following assumptions have badopted:
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Fig. 5 Results for the initial model for a step amplitude Ä2.9851 mm3Õsec
the
l. In
1. Slip conditions between rollers and filament, accordingthe generalized Navier law@12#;
2. The angular velocity of the rollers changes in the middlethe deposition process~see the Experimental Set Up section! asmuch as possible, but without requiring an excessive power sply ~i.e. grater than 3.66 W!;
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3. The heat flow rate is increased following a ramp function~inorder to model the close loop control!;
4. There are prespecified upper bounds for the torque andpower supplied by the stepper motors.
Figure 6 presents the results obtained with the modified mode
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Fig. 6 Results from the modified model for step amplitude Ä2.9851 mm3Õsec
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Fig. 6~a! the theoretical response of the velocity of the rolleaccording to the command provided by the SML file is showHowever, because of the motor specifications, the speed ofrollers increases until a point when the maximum torque andpower limits are reached. Beyond this point the rollers maintaconstant velocity on an intervalDt ~Fig. 6~a!!, during which the
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desired temperature of the melt is reestablished due to the incrin the heat flow rate~Fig. 6~c!!. The necessary heat flow is calculated by specifying the exit temperature which must be kepttween 540 and 552 K~Fig. 6~d!!.
The slip condition is taken into account in calculating the mvelocity. The algorithm applied for this case is as follows:~1! the
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speed of the rollers is increased of a constant value, thus an ition is performed imposing that the melt is flowing at the sarate~no slip conditions!; ~2! the force applied by the motor is thecalculated using Eq.~13! and a check on the power and torquemade~Eqs. 14 and 15!; ~3! if the limit on the power and torque isreached, the program proceeds considering that the appliedis also used to overcome the friction between the rollers andfilament:
Fs5m•v r where m5m0•v re21 (17)
wherem051, e50.48 are material properties;~4! the forceF applied to extrude the melt is thus reduced
the amountFs ; ~5! the new value ofF is consequently used tocalculate the real average velocity of the melt~see Fig. 6~b!!.
These results also indicate that the power and torque neceto maintain the extrusion process are kept within an acceptrange ~Figs. 6~g! and 6~h!!. Thus, it can be concluded that thmodified model is more realistic and applicable to physical stems. Furthermore, with these more realistic assumptionsmodel became capable to better reproduce the experimentaservations as will be discussed in a later section.
Liquefier Dynamics: Transfer Function ApproachWhile the mathematical model gives an inside of the comp
dynamic taking place in the liquefier and is able to give an expnation to some observed phenomena, such as time-delaysteady-state error, it is of difficult implementation for further dvelopments. Since the final objective of this research is indeeimprove the quality of the end product, it is the focus of thproject to study the possible development of a close-loop consystem that, actively incorporated in the FDM software, can pdict and consequently avoid defects, such as under-deposstart-stop errors etc. For this reason a more direct approachalso considered, as explained in this section.
In the simplest case the liquefier can be modeled as a dynasystem subjected to the digital SML command, which indicathe desired flow rate, as the input signal and the real flow ratthe output from the system. In the present work the followiassumptions have been made:
• the communication between the PC and the controllerinstantaneous;
• the Asymtek digital controller is infinitely fast;• the ABS filament at the entrance of the liquefier is perfec
round, with a diameter of 1.78 mm and the material densityconstant;
• the filament is perfectly rigid and does not buckle betwethe rollers and the liquefier entrance.
In order to develop a model for the transfer function of tliquefier, the response to a step input function has been stuand analyzed. The main objective is that once the time constime delay and gain of the liquefier system are known, the flrate can be more accurately regulated according to the decetion and acceleration of the deposition speed. With this capabit will be possible to significantly reduce some of the irregularitdepicted in Fig. 2, particularly in zones II and IV. In this sectiothe experimental and analytical studies conducted to determithe liquefier transfer function are presented. In performingdynamic modeling for the liquefier, the following variables habeen taken in consideration:
• the time delayt between the signal~change in the materiaflow rate at the entrance of the liquefier! and the observation othe results~change of the flow rare at the end of the liquefier!;
• the slip conditions between the rollers and the filamentaccounted for as a delay term;
• the material properties of the ABS~i.e., viscosity, meltingflow, thermal conductivity, thermal capacity!;
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• the heat flux that needs to be adapted to the change inflow rate.
For a better understanding of the dynamics and developmena control strategy, an analog electronic circuit has been develoas shown in Fig. 7.
The specified flow rate can be represented as the applied curi ~or voltageu, due to the proportional relation between the twentities! at the nodeA; the slip condition between the rollers andthe filament can be modeled as a resistorR; while the materialproperties and the heat flux can be represented as an impedanL~they can be seen as an opposition to sudden changes in therate!.
The response of the liquefier can thus be mathematicallypressed using the following equations@13#:
H Ldi~ t2t!
dt1~R!• i ~ t2t!5u~ t2t!
y~ t !5 i exp~ t !i ~ t2t!5 i exp~ t !
(18)
where i exp(t) is the experienced response of the system, i.e. tcurrent registered at timet by the amperometer. It can be noticethat, due to the time delayt between the data acquisition ofi~current in the nodeC! and the imposed voltageu ~thus the currenti! between the nodesA andB, the analyzed response is:
y~ t !5 i ~ t2t! (19)
Substituting~19! into the first equation of the system~18!:
L• y~ t !1R•y~ t !5u~ t2t! (20)
Applying the Laplace transform~indicated with/! to Eq. ~20!:
/$L• y~ t !1R•y~ t !%5/$u~ t2t!% (21)
L•s•Y~s!1R•Y~s!5/$u~ t2t!% (22)
whereY(s) is the Laplace transform ofy(t).In order to develop a transfer function of the liquefier, a ste
function has been applied as follows
u~ t2t!5H m1 t<a1t
m2 t.a1t(23)
Defining H(t2k) the Heaviside step function:
H~ t2k!5H 0 t<k
1 t.k(24)
one can write@14#:
u~ t !5m1•H~ t20!1~m22m1!•H~ t2a! (25)
and similarly
u~ t2t!5m1•H~ t2t!1~m22m1!•H~ t2a2t! (26)
Fig. 7 The analog electric circuit for dynamic analysis ofliquefier
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Remembering the Laplace transform of an Heaviside step funcH(t)
/$H~ t2k!%5e2ks
s(27)
and applying the property of linearity for the Laplace transformEqs.~25! and ~26!, it is possible to derive:
/$u~ t !%5U~s!5m1
1
s~m22m1!
e2as
s(28)
Similarly
/$u~ t2t!%5m1
e2ts
s1~m22m1!
e2~a1t!s
s
5e2tsFm1
1
s1~m22m1!
e2as
s G (29)
Substituting Eq.~29! into ~22!:
~L•s1R!Y~s!5e2tsFm1
1
s1~m22m1!
e2as
s G (30)
Comparing Eqs.~28! and ~30!:
~L•s1R!•Y~s!5e2ts•U~s! (31)
The transfer function G(s) of the system can thus be expressas:
G~s!5Y~s!
U~s!5
1
R
S L
RD s11
•e2ts (32)
For simplicity, the following constant can be defined:
m51
R(33)
T5L
R(34)
Equation~31! can thus be rewritten in the more conventional fo@15#:
G~s!5m
Ts11•e2ts (35)
whereT is called equivalent time constant,t is called equivalent delay andm is the gain.As it is shown in Eq.~35!, the transfer function of the liquefie
is modeled as afirst order differential equation. This limits thevalidity of the studied model to low frequency responses. Tinvestigation of the second order effects is not indeed part ofgoals of the present paper.
Liquefier Dynamics: Experimental StudiesIn order to determine the dynamic response of the liquefier,
relation between the applied value and the actual response inflow rate during the deposition of a road has been studied. Mspecifically, a SML file has been written so to deposit a singlecm-long-road, at a constant speed of 2 cm/sec. In the same filcommand to change the material flow rate is given at the hlength point during building of the road. With accurate measurof the sectional areas along the extruded filament it was possto calculate the physical flow rate at the exit of the nozzle, thenabling comparison with the theoretical predictions.
In order to minimize measurement errors due to the impercontact between the extruded material and rough, porous, fplatforms typically used in FD processes, the roads have b
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deposited on a Plexiglas base. The measurements of the secof the roads have been performed using Raman spectromequipment by determining the road width and thickness resptively at various locations. More specifically, the CC/CD cameattached to the microscope~connected to the Raman equipmen!and a computer equipped with Renishaw WIRE software hbeen used, while the green and red lasers of the RAMAN hbeen not utilized. One of the Renishaw features, the WiRE VidViewer, allows not only to take pictures of the object set in tmicroscope stage, but also defines a Cartesian coordinate rence system to read the coordinates of a particular point, to arately move the stage inx andy directions using coordinate inputsetc. A two-speeds joystick is also connected to the microscoporder to manually adjust the position of the stage while observunder the microscope.
To generate the sectional data, the roads are first cut veryto the ends with a razor blade and are placed on the microscstage. Using a 53 magnification lens, the origin of the referencsystem is defined at one end of the fiber. Following this, the stcan be accurately moved within 1/10 of amm by enteringthe desiredx-value ~distance from one end of the road! in theprogram.
At each desired location they coordinates of two oppositepoints, i.e. pointsA and B in Fig. 8, are determined by simplymoving the cursor to follow the edges of the road. The widththe fiber is thus calculated as a difference between the twyvalues. These steps are repeated for different point locations~i.e.,pointsA8, B8, A9, B9 etc.! as well as for the side of the road~Fig.8~b!!, in order to also determine its height. For the purposeclarity and comparison with the schematic drawing of Fig. 8,Fig. 9 the actual micrograph of a deposited road shows the inment in width and high due to the suddenly imposed change offlow rate.
For accurate observation of the system response to thefunction, the sampling rate has been considerably increasedx5 l /2 which is the location corresponding to the instant of suddchange in the flow rate.
Mathematical Model: Comparison of ResultsIn order to verify the validity of the mathematical model, thre
simulations have been performed for different step amplitudethe input flow rate command, as described in the section dedicto the experimental results. In Figs. 10, 11 and 12, the theorechange in the flow rate~according to the SML file!, the experi-mental curve~determined as described in the previous section! aswell as the results obtained with the mathematical model hbeen assembled.
Following observations can be made from these graphs:
• there is a good agreement between the applied flow rate~i.e.theoretical curve! and the physical response of the system~i.e.experimental curve! for small magnitudes;
• there is a steady state error of approximately 12% betwthe input command and the resulting physical flow rate;
Fig. 8 „a… Top view of the road; „b… Side view of the road
MAY 2004, Vol. 126 Õ 243
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• there is a good agreement between the experimental cand the curve generated by the application of the mathemamodel.
The capability of the mathematical model to reproduce the expmental results is representative of the fact that:• slip conditions between the rollers and the filament can be csidered the cause for the existence of asteady stateerror when theflow rate is abruptly changed. It should be added at this pointthe FDM machine Stratasys 1650, used for the experimestudy, requires to be calibrated at the beginning according to
Fig. 9 „a… Top view of the road; „b… Side view of the road
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rveical
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operative conditions~i.e. flow rate!. This action, that results in theadjustment of the SML command so to annihilate the frictiforce between the rollers and the filament, avoids the steady serror when the machine operates at constant conditions, butnot eliminate it when the conditions changes during the consttion of the part;
• the limitation of power and torque of the motors, and tstrong temperature dependence of the viscosity of the materiabe considered the cause for the existence of a certaintime delayand of anequivalent time constantin the execution of the SMLcommand.
Transfer Function Approach: Comparison of ResultsThe results of the experiments described in the sections ab
have been also assembled in Figs. 13, 14 and 15. In these pthe curves for the step input for the flow rate as well as the curof the actually measured flow rates can be observed. Thedifference between the three experiments depicted in theseures is the amplitude of the input step functions, while all tother operating parameters are kept the same.
Based on the experimental data, conveniently summarizeTable 1, the transfer function for the liquefier can been obtain~see Eq.~36!! as
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Substituting Eqs.~36! and ~28! into ~32!, the functionY(s) waseasily calculated. Consequently, through the inverse of Laptransform,y(t) ~i.e., the observed flow rate! was determined andplotted as well in each of the graphs shown in Figs. 13, 14 and
Following observations can be made from these graphs:
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15.
• with the use of the studied transfer function, the responsethe system can be predicted with good accuracy; however
• the presence of higher order dynamics will likely improve tdynamic modeling while increasing the complexity of the analysubstantially.
The good prediction of the real flow, obtained when applyithe transfer function approach, suggests that the integration offunction as part of an overall control strategy will provide aeffective flow control and consequently will improve the part itegrity @5#. Moreover, a further development of a close-loop feeback system will lead to the following advantages:
• increase of the accuracy;• reduction of the sensitivity to disturbance;• reduction of the system induced noise and distortion.
ConclusionsCustomer-driven product customization and continued dem
for cost and time savings have generated a renewed intereagile manufacturing based on improvements on RP technoloand in particularly of fused deposition modeling~FDM!.
A shift from ‘‘prototyping’’ to ‘‘manufacturing’’ necessitateshowever the following improvements:
• part integrity and built-in characteristics to meet performanrequirements;
• improved surface quality.• These objectives can be achieved only through accurate
cess modeling, especially in the category of flow control durdeposition. For this reason an extensive study of the dynamicthe FDM liquefier was performed and explained in this paper.
• In order to have an insight of the complex phenomena toccur in the liquefier, a mathematical model based on physassumptions was developed at first. After comparison of thesults with the experimental data~see Figs. 10, 11 and 12! it wasconcluded that:
• slip conditions between filament and rollers can be consered the cause of a steady state error when sudden changeapplied to the flow rate;
• the limitation of power and torque of the motors as wellthe strong dependency of the viscosity of the material on the tperature can be considered the cause of the time delay inresponse of the system.
The mathematical model, which gives a physically explanatto the process taking place in the liquefier, demonstrated netheless to be of difficult implementation for further developmenFor the purpose of designing a close loop control system treducing the steady state error, will improve the accuracy ofdeposited flow, hence the final part integrity, a study based oninvestigation of the transfer function of the liquefier was devoped as well. In particularly, since the objective was primarily tinvestigation of first order effects, the system was subjectedstep function input and its response was accurately studied~seeFigs. 13, 14 and 15!.
The comparison of the experimental data with the resultstained using the transfer function of Eq.~36! shows a good agreement for the purpose of integrating a control function describthe dynamic response of a liquefier to changing input comman
Table 1 Parameters of the liquefier transfer function for a stepinput
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AcknowledgmentsThis project was sponsored, in part, by the Office of Na
Research~ONR! in MURI project, #N0014-96-11175. Many constructive discussions with other MURI team members, notawith Drs Danforth, Safari and Jafari of Rutgers University, aalso acknowledged.
References@1# Kai, C. C., and Fai, L. K., 1997,Rapid Prototyping: Principles and Applica-
tions in Manufacturing, Wiley ed., New York, ISBN 0-471-19004-7.@2# Bouhal, A., Jafari, M. A., Han, W., and Fang, T., 1999, ‘‘Tracking Control a
Trajectory Planning in Layered Manufacturing Applications,’’ IEEE TranInd. Electron.,46~2!.
@3# Langrana, N. A., Qiu, D., Bossett, E., Danforth, S. C., Safari, M., and SaA., 2000, ‘‘Virtual Simulation and Video Microscopy for Fused DepositioMethods,’’ Mater. Des.,22, pp. 75–82.
@4# Yardmci, A., 1999, ‘‘Process Analysis and Development for Fused Deption,’’ PhD Thesis, University of Illinois at Chicago, Chicago.
@5# Bellini, A., and Guceri, S. I., 2003, ‘‘Mechanical Characterization of ParFabricated Using Fused Deposition Modeling,’’ Rapid Prototyping Journ9~4!, pp. 252–264.
@6# Venkataraman, N., Rangarajan, S., Matthewson, M. J., Harper, B., SafarDanforth, S. C., Wu, G., Langrana, N., Guceri, S., and Yardimci, A., 20
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al-blere
ds.
ari,n
si-
sal,
, A.,0,
‘‘Feedstock Material Property—Process Relationship in Fused DepositioCeramics~FDC!,’’ Rapid Prototyping Journal,6~4!, pp. 244–252.
@7# Bird, B. R., Armstrong, R. C., and Hassager, O., 1987,Dynamics of PolymericLiquid: Second Edition-Volume 1: Fluid Mechanics, Wiley-Interscience, NewYork, ISBN 0-471-80245-X~v.1!.
@8# Michaeli, W., 1992,Extrusion Dies for Plastics and Rubber: Design and Egineering Computation—Second Edition, Hanser ed., New York, ISBN 3-446-16190-2.
@9# Bertoldi, M., 1998, ‘‘From Rapid Prototyping to Rapid Manufacturing: Analsis of the Fused Deposition Process,’’ Tesi di laurea del dipartimento di Ingneria Aerospaziale del Politecnico di Milano, Milan.
@10# Venkataraman, N., 2000, ‘‘The Process-Property-Performance RelationshiFeedstock Material Used for Fused Deposition of Ceramic~FDC!,’’ Depart-ment of Ceramic and Materials Engineering, Rutgers University, New Brswick.
@11# DC Micromotors—Brushless DC Sevomotors—Gearheads—Tachogerators—Encoders—Controllers, MicroMo catalog 1999–2000.
@12# Fluent Incorporation,Polyflow 3.8 Documentation—Tutorial Guide.@13# Melsa, J. L., and Donald, S. G., 1969,Linear Control Systems, McGraw-Hill
Book Company, New York, ISBN 07-041481-5.@14# Greenber, M. D., 1998,Advanced Engineering Mathematics—Second Editio,
Prentice Hall, Upper Saddle River, NJ, ISBN 0-13-321431-1.@15# Dean, F. K., and Chow, J. H., 2000,Feedback Control Problems: Using MAT
LAB and the Control System Toolbox, Brooks/Cole Thomson Learning, PacifiGrove, ISBN 0-534-37175-2.
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