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Optimization of positioning system of FDMmachine design using analytical approach
Marlon Wesley Machado Cunico and Jonas de Carvalho
Department of Mechanical Engineering, University of Sao Paulo, Sao Carlos, Brazil
AbstractPurpose The purpose of this paper is to analyse the conception of the positioning system of fused deposition modeling (FDM) machines, optimisingdesign parameter and components accuracy to decrease mechanical errors of equipment, which, consequently, results in the increase of parts accuracy.This paper also reports studies related to analytical estimation of machine errors, describing a theoretical model which was used for the multivariablestudy. Additionally, an alternative conception is proposed, according with the result of this study.Design/methodology/approach For elaboration of the numerical model of equipment, the authors have focused on conception of first generationof FDM, specifying as design parameters, timing belt stiffness, linear bearing clearance, and accuracy grade of ball screw housing, support and pulley.In order to identify the main effect of each design parameter for the final error of machine, the authors have applied a multivariable method in additionto identifying the error budget of model. Also indicated are the two factors that promote more errors, undergoing a proposal of conception whichconsists in replacing one component of machine.Findings With reference to the evaluation of the numerical model, equivalency was found between the resultant error of model and the current FDMaccuracy. The result of multivariable study identified the main causes of errors in machine, implying on an optimized solution which decreases the initialerror in 69 mm. Similarly, the evaluation of the proposed conception resulted in the reduction of general error in almost 20 mm, even though the worstcase was studied for this comparison.Originality/value Although the number of applications for additive manufacturing has been growing in recent years, implying an increase ofdemand for high precision parts, there are still several challenges to be overcome, such as the improvement of equipment. For that reason, themotivation of this work concerns the contribution for development of new equipment, as well the improvement of current technologies. Furthermore,the authors focus was the reduction of mechanical errors through an analytical approach.
Keywords Advanced manufacturing technologies, Optimum design, Additive manufacturing, Optimization, Error budget
Paper type Research paper
1. Introduction
Along thelast years,the development of additive manufacturingtechnologies has been substantially increasing, even though
there are still significant challenges to be overcome. For
example, one of these challenges is concerned with mechanical
design of equipment, as such positioning and deposition
systems. Furthermore, the main purpose of this development is
to increase the precision of equipments, in addition to reducing
manufacturing costs (Gibson, 2010; Cunico, 2011).
Likewise, the main goal of this work is to analyse a
simplified design structure which is found in a FDM machine,
identifying the essential design parameters that influence the
final error of equipment. In addition, for pointing the main
machine elements that are suitable to be improved, this study
also intends to identify and reduce mechanical errors even in
preliminary phases of new projects.On the other hand, the object of our study is based on the
structure found in first generations of FDM machines, as such
3D Modelerw and FDM series (Wang, 2010). Alternately, the
schematic drawing of this layout is shown in Figure 1, wherein
can be seen the main conception of this equipment.
With regards to workability of this system, it can behighlighted that the motion of z direction is caused by the
simultaneous movement of two lead screws, which are placed
in the table through four screw nuts and guided by eight-
linear bearing. In addition, the motion of screws is provided
by a flexible transmission which is coupled in an electrical
motor. Similarly, the motion of x- and y-axis is directly done
by the displacement of a flexible transmission which is fixed
on extruder and axis housing (Swanson et al., 2001).
Having this conception in mind, we applied some methods
of precision engineering in order to elaborate a numerical
model which would have been able to point out final errors of
machine (Slocum, 1992). Additionally, it was defined housing
and supports accuracy grade, linear bearing clearance, timing
belt stiffness, pulley belt accuracy grade and ball screw
accuracy grade as the main design parameters that possibly
affect the final error of machine.
After evaluating the model through a comparison with a
current machine found in the market, we investigated the
The current issue and full text archive of this journal is available at
www.emeraldinsight.com/1355-2546.htm
Rapid Prototyping Journal
19/3 (2013) 144152
q Emerald Group Publishing Limited [ISSN 1355-2546]
[DOI 10.1108/13552541311312139]
The authors would like to thank the CAPES for financial support, as wellas the Department of Post Graduation in Mechanical Engineering of theUniversity of Sao Paulo (campus Sao Carlos), for providing access toinfrastructure and Laboratories.
Received: 22 August 2011Revised: 10 November 2011Accepted: 14 November 2011
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contribution of the main design parametersover the positioning
system error through multivariable methodologies. As result,
it was possible to propose an equivalent conception that
provides which replaces only one machine element, comparing
with the highest accuracy arrangement found in this study.
2. Material and methods
For the definition of values of parameters of our study, wehave preliminary analysed first generations of FDM machines,
selecting standard machine elements for composing the layout
of studied machine. In addition, it was also defined the main
design components and tolerances in order to formulate the
numerical model.
With respect to the optimization analysis proposed in this
work, it was used the software Matlab and Minitab as
computational tools which processed the numerical model
data and compiled its results.
2.1 Machine description
As this study intend to evaluate a numerical model through
reverse engineering of a current machine, we tried specifying
all machine components based on standard suppliers, as such
THK, Thomson or HIWIN for ball screw and linear bearing
besides Gates for pulley and timing belt. At the same way, the
international tolerance of housing and supports were defined
in accordance with accuracy grade for high precision parts.
For drive motor, it was specified DC motors and angular
encoder for the control system. Additional, it was also
considered that the preload of belts provides no backlash to
the system. In this case, the maximum resolution of
displacement is bounded by the encoder resolution, being
defined as 18,000pulsesper revolution. Therefore, defining the
diameter of pulleybeltsas 20 mm,the maximum resolution ofx-
y-axes would be equal to 6.9 mm. Furthermore, the maximum
accuracy recommended by fundamentals of precision
engineering is equal to 0.069 mm, which is the result of
10 per cent of resolution (Slocum, 1992).
In order to specify x-y-axes, it was considered that the
position system is composed by two linear bearing per shaft,
in addition to being feed by a flexible transmission which
concerns timing belt and pulley.
With reference to fixture of shafts and housing of linearbearing, we defined a single design parameter in accordance
with accuracy grade. This parameter was also applied to
z-axis, as such the fixture of lead screw on structure and the
screw nuts on table.
For the motion of z-axis, it was initially specified that the
linear displacement occurs through four ball screws whose
flanged nuts is directly coupled in the building table. It is
important to highlight that this conception applies no linear
guide and consequently, either the rigidity or straightness of
system might be compromised. In accordance with that
specification, it was initially selected a laminated ball screw
with no preload and whose tolerance quality is C7 for motion
of z-axis.
For this study, we initially considered a layer thickness
equivalent to the finest FDM machine (FORTUS 900mc from
Stratasys), which provides the minimal thickness layer equal to
0.127 mm(Stratasys, 2011a). Due to that, it was selected a step
motor as the drive motor that feeds lead screw. The operation
mode of this motor was defined to be half step, providing a
resolution equal to 800 steps per revolution. As result, if the
pulley diameter and the screw lead were, respectively,
considered equal to 20 and 4 mm, the minimal displacement
of z-axis would be 0.01 mm. And consequently, the main
maximum accuracy ofz-axis would be equal to 0.1 mm.
As the structural frame of machine was considered rigid
body, it was ignored errors caused by either deflection of
frames or housing bearing.
For definition of system basic dimensions, it was defined a
building area equivalent to FORTUS 360mc (355 254 254 mm) (Stratasys, 2011b), being shown in Figure 2. In that
figure, it is presented the schematic drawing of a simplified
FDM machine which contains the basic dimension that was
considered for this study. Additionally, it was shown the main
machine elements and the position of main axes which were
used in formulation of the numerical model.
Regarding the basic dimension of system, the distance
between shafts of x-axis is equal to 50 mm, while the supports
of both x- and y-axis 300mm. It is important to emphasise
that support ofx-axis and housing ofy-axis are features of the
same component, while the support of y- and z-axes are
features of machine frame.
Continuing the description, we defined the distance
between ball screw as 400 and 300 mm in both x- and
y-direction while the housing that places the screw nuts as
feature of building table.
On the other hand, it was determined the position of tool
axis is placed on the tip of extruder nozzle, while the x- and
y-axes on the centre of respective shafts. In contrast with that,
the z-axis can be found in the centre of building table, in
accordance with the zero point of machine.
As the main goal of this work is concerned with the
precision of positioning system, the influence of extrusion
head on machine accuracy was ignored. At the same way, for
estimating the contribution of errors from basic deformation
of shafts, it was attributed the mass 2 kg to extrusion head,
Figure 1 FDM machine scheme
X
Z
Y
Source: Swanson et al. (2001)
Optimization of positioning system of FDM machine design
Marlon Wesley Machado Cunico and Jonas de Carvalho
Rapid Prototyping Journal
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being placed this load on x-axis centroid, in order to divide
this load equally between the shafts.
It is also important to highlight that this study focused in
geometrical errors, considering being part of design process.
Otherwise, other sources of errors should be also studied as
kinematics, dynamics and thermal errors.
2.2 Numerical modeling
For elaborating the numeric model of machine errors, wedefined four main axes whose homogeneous transformation
matrix (HTM) includes the incremental coordinates of axis
translation and total errors of each axis, as exposed in
equation (1). Alternatively, it is also represented the position
of this axes in Figure 2:
RTzerror T Txerror
yTyerror zTzerror 1
With reference to the depiction of axes HTM, equation (2)
presents the general f ormulation of HTM and the
collaboration of both error matrix and translation matrix for
its composition:
aTberror aTb
aEb
1 0 0 X
0 1 0 Y
0 0 1 Z
0 0 0 1
2666664
3777775
1 21z 1y dx
1z 1 21x dy
21y 1x 1 dz
0 0 0 1
2666664
3777775
1 21z 1y X dx
1z 1 21x Y dy
21y 1x 1 Z dz
0 0 0 1
2666664
3777775
2
where:
aTberror is the HTM from axis a to axis b considering
errors.aTb is the translation matrix from axis a to axis b.
Figure 2 FDM machine scheme (schematic basic dimensions) and axes (X, Y, Z, T)
Source:Adapted from Swanson (2001)
Optimization of positioning system of FDM machine design
Marlon Wesley Machado Cunico and Jonas de Carvalho
Rapid Prototyping Journal
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aEb is the error matrix considering geometrical
interface from axis a to axis b.
X, Y and Z are the position elements, respectively, in x-,
y- and z-directions.
1x, 1y, 1z are the rotational errors along x-, y- and
z-directions.
dx, dy and dz are the linear error, respectively, in x-, y- and
z-directions.
Therefore, considering the main dimensions which were
shown in Figure 2, the HTM of each axis is defined as
equations (3)-(5):
TTxerror
1 21z1 1y1 2002X dx1
1z1 1 21x1 25 dy1
21y1 1x1 1 50 dz1
0 0 0 1
2666664
3777775
3
xTyerror
1 21z2 1y2 dx2
1z2 1 21x2 1252Y dy2
21y2
1x2 1
225 dz2
0 0 0 1
2666664
3777775
4
yTzerror
1 21z3 1y3 2200 dx3
1z3 1 21x3 2150 dy3
21y3 1x3 1 225 Z dz3
0 0 0 1
2666664
3777775
5
Additionally, for identifying the final error of system, it is
multiplied the current HTM (which considers errors) and the
inverse of expected HTM (which considers no errors), as
presented in equation (6):
ER RTzerror
21RTz !
dxdy
dz
1
2666664
3777775
1 0 0 X
0 1 0 Y
0 0 1 Z
0 0 0 1
2666664
3777775
21
1 21z 1y X dx
1z 1 21x Y dy
21y 1x 1 Z dz
0 0 0 1
2666664
3777775
6
2.3 Methods and investigation
In order to evaluate the numerical model of machine error, itwas defined initial values for design parameters, which was
selected in accordance with general accuracy grade for high
precision parts (IT3). In addition, standard deviation for pulley
beltand ball screw wasIT7, which indicates tolerancesrangefor
current manufacturing (Shigley and Mischke, 1996). With
reference with the final error of machine numeric model in the
centralregionof table (zero coordinates), it wascomparingwith
the general accuracy announced by a current technology.
For the estimation of errors of this initial study, the numerical
model of geometric errors was determined by the sum of errors
of transitional and rotational along the three axes. Where:dx, dy
and dz are transitional errors along x-z axes and 1x, 1y, 1z are
rotational errors along x, y an z-axes (pitch, yaw and roll,
respectively). Therefore, Table I lists the tolerance of both
general design parameters and main machine elements which
were used in our object of study.
In function of the number of variables that effect the final
error of machine, it was also applied a multivariable method
with purpose of identifying a systematic approach for selectingelements of positioning and tooling machines. For this, it was
applied the design of experiment methodology (DOE) in
order to analyse the collaboration of each design parameter
for the final machine error (Montgomery and Runger, 2010;
Cox and Reid, 2000).
For identifying the main contribution of each design
parameter and machine element for the final error of
machine, we defined the housing and support accuracy grade
(HS), pulleys accuracy grade (P), linear bearing accuracy (B),
timing belt stiffness (TB) and ball screw accuracy grade (BS) as
the main variables which vary the machine error in general.
Therefore, we defined two levels for each one of these
parameters, as it is shown in Table II.
On the other hand, we considered a full design (25) to
analyse the main effects of parameters against the final errors
of tool (dx, dy, dz) which were determined as the response
parameters.
Additionally, it was elaborated a contour diagram, showing
the relationship between the parameter that promote most
effect and the final error.
3. Results and discussions
With purpose of evaluate the numerical model of machine
errors, it was individually identified the geometric error of each
axis, being totalized by systematic and random errors
(10 per cent systematic error). On the other hand, it was also
indicated the main components which are related to axis error.
In accordance with this, the list of total, systematic andrandom errors are shown in Table III, which also presents the
main sources of geometrical error for each direction.
As result, the final error that was found for this arrange of
values can be seen in Table IV. In that table, it is possible to
identify the maximum error of 0.125mm, which is a value
that indicates the similarity with the maximum error
announced by F DM 3 60 mc (^0.127 mm) (Stratasys,
2011b). In that case, only 1 mm differ both cases, providing
a deviation of 1.5 per cent.
Then, with regards to evaluation of the numerical model,
the comparison between the maximum errors found proves
the equivalency between numerical model of geometrical
errors and current machines. Additionally, it is also important
to highlight that the model considers only geometrical errors,
ignoring kinematics, dynamics and thermal errors in function
of low efforts involved.
With respect to the analysis of main effect of general design
parameters, it was identified the final error resulted by
numerical model of machine errors. For doing this analysis, it
was varied the values of parameter in two levels according to
full design (25).
In Table V, it is shown the matrix design of study, wherein is
presented the levels and values of design parameters besides the
final error in each of three directions. Additionally, it is possible
to see that the range of error found in z-axis is higher than the
other axis errors, varying from 20.041 to20.349mm.
Optimization of positioning system of FDM machine design
Marlon Wesley Machado Cunico and Jonas de Carvalho
Rapid Prototyping Journal
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Another result that is important to be highlightedis the minimal
error for axes, being, respectively, 20.04895, 20.05615 and
20.04101 mmfor dX, dYanddZ. In thatcase,it wasconsidered
as design parameters, IT1 for general accuracy grade, IT5 for
accuracy grade of pulley and ball screw. Otherwise, it was
increased the stiffness of timing belt in order to reduce the
variation intrinsic to kinematic and dynamic loads, even though
the study focused on geometric errors. Additionally, stiffness of
belt was also determined as 3,000 N/mm, which infers the error
of 0.016 mm for a static load of 50 N.
With respect to the analysis of main effects of design
parameter, it is possible to see in Figures 3-5, the collaboration
of each factor for final error in x-, y- and z-directions,
respectively.
Table II List of levels of general parameters and tolerances for machine design
Level
Control factor 11 21 Description
Housing and shaft
support accuracy (HS)
0.013 mm (IT3) 0.007mm (IT1) .300 # 400 (Shigley and Mischke, 1996)
0.004 mm (IT3) 0.0015mm (IT1) .18 # 50 (Shigley and Mischke, 1996)
Pulley accuracy (P) 0.021 mm (IT7) 0.0009mm (IT5) .30 # 51 (Gates, 2011; Shigley and Mischke, 1996)
Timing belt
tolerance (TB)
0.020 mm
(2,200N/mm)
0.016 mm
(3,000 N/mm)
For load equal to 50 N (Gates, 2006, 2011)
Bearing tolerances (B) 1 1 Angular misalignment (Thomson, 2009b; HIWIN, 2006; THK, 2007)
0.004 mm 0mm Radial clearance (Thomson, 2009b; HIWIN, 2006; THK, 2007)
Ball screw tolerances (BS) 0.052 mm (C7) 0.023 mm (C5) Lead accuracy backlash (axial play) (THK, 2011; Thomson, 2009a; HIWIN, 2008)
Table III Description of combination of systematic, random and total errors for each axis, being detailed the main sources of geometrical errors andtheir contribution for each direction
Error dir Systematic Random Total error Error description
x-axis
dX (mm) 0.064 0.0064 0.0704 Timing belt pulley
dY (mm) 0.026 0.0026 0.0286 Extruder housing shaft supports
dZ (mm) 0.026 0.0026 0.0286 Extruder housing shaft supports1X (rad) 0.00052 0.000052 0.000572 Extruder housing shaft supports
1Y (rad) 4.33 1000 4.33 10201 4.77 1000 Extruder housing shaft supports
1Z (rad) 4.33 1000 4.33 10201 4.77 1000 Extruder housing shaft supports
y-axis
dX (mm) 0.026 0.0026 0.0286 Guide housing shaft support
dY (mm) 0.064 0.0064 0.0704 Timing belt pulley
d (mm) 0.026 0.0026 0.0286 Guide housing shaft support
1X (rad) 4.33 1000 4.33 10201 4.77 1000 Guide housing shaft support
1Y (rad) 8.67 1000 8.67 10201 9.53 1000 Guide housing shaft support
1Z (rad) 4.33 1000 4.33 10201 4.77 1000 Guide housing shaft suport
z-axis
dX (mm) 0.026 0.0026 0.0286 Table support shaft support
dY (mm) 0.026 0.0026 0.0286 Table support shaft support
dZ (mm) 0.066 0.0066 0.0726 Ball screw1X (rad) 0.000264 0.0000264 0.0002904 Table support shaft support ball screw
1Y (rad) 0.000264 0.0000264 0.0002904 Table support shaft support ball screw
1Z (rad) 0.197021697 0.01970217 0.216723867 Table support shaft support ball screw
Table I List of design parameters and machine elements tolerances
Tolerances Tolerance Description
Accuracy grade IT3 .300 # 400 0.013 mm Supports and housing
.18 # 50 0.004 mm supports and housing
IT7 .30 # 51 0.021 mm Pulley
Timing belt tolerance 0.020 mm Based on stiffness of belt
Bearing tolerances 1 Angular misalignment0.004 mm Radial clearance
Ball screw tolerances 0 mm Radial clearance
0.052 mm Lead accuracy (C7) backlash (axial play)
Optimization of positioning system of FDM machine design
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In Figure 3, it is possible to see the high effect of pulley accuracy
grade in comparison with other parameters. In addition, the
accuracy grade of ball screw was also found to be almost
irrelevant to x-y direction, in function of its low main effects.
On the other hand, Figure 4 points to pulley accuracy as
irrelevant for the error in z direction, evidencing the individual
collaboration of either pulley or ball screw accuracy grade for
the errors in x-, y- and z-directions, respectively.
Another point to be highlighted is the main effect of timing
belt stiffness, as well the accuracy grade of linear bearing,
housing and supports. Observing the collaboration of each of
these design parameters for the errors in three directions, it is
possible to identify almost the same effect. In spite of that, the
error caused by timing belt was shown to be the less relevant
among these parameters, when comparing their main effect.
Resulting from these observations, besides improving linear
bearing specification, the increase of housing and supports
accuracy grade would be the most important changes in order
to reduce the general errors of machine.
Another option for the improvement of this positioningsystem would be the replacement of linear bearing and shaft
by linear guides, which are composed by profiled rail and
auto-lubrificated block. Although this option tends to be more
expensive than linear bearing and shaft, it also provides higher
stiffness in addition to tighter angular tolerances.
Furthermore, the application of this conception also helps
to reduce error caused by housing accuracy, due to the
replacement of housing by the block of linear guide.
In fact, the importance of housing and support accuracy for
final error of system is highlighted by Figure 5, wherein is
presented the contour diagram of final error in x- and z-
directions as a function of these two design parameter.
Analysing these diagrams, it is possible to identify that even if
all the other parameter have been in high level (low accuracy),
Table V Description of combination of systematic, random and total errors for each axis, being detailed the main sources of geometrical errors andtheir contribution for each direction
Design parameters Responses
Trial H & S (mm) P (mm) TB (mm) B (mm) BS (mm) dX (mm) dy (mm) dz (m)
1 21 (0.006) 21 (0.009) 21 (0.016) 21 (0.000) 21 (0.023) 20.04895 20.05615 20.04101
2 1 (0.012) 21 (0.009) 21 (0.016) 21 (0.000) 21 (0.023) 20.28802 20.22641 20.34905
3 21 (0.006) 1 (0.021) 21 (0.016) 21 (0.000) 21 (0.023) 20.8062 20.08706 20.05151
4 1 (0.012) 1 (0.021) 21 (0.016) 21 (0.000) 21 (0.023) 20.10362 20.11421 20.0812
5 21 (0.006) 21 (0.009) 1 (0.022) 21 (0.000) 21 (0.023) 20.06262 20.06906 20.05151
6 1 (0.012) 21 (0.009) 1 (0.022) 21 (0.000) 21 (0.023) 20.08562 20.09621 20.08119
7 21 (0.006) 1 (0.021) 1 (0.022) 21 (0.000) 21 (0.023) 20.08662 20.09306 20.05152
8 1 (0.012) 1 (0.021) 1 (0.022) 21 (0.000) 21 (0.023) 20.10962 20.12021 20.08129 21 (0.006) 21 (0.009) 21 (0.016) 1 (0.008) 21 (0.023) 20.06582 20.06878 20.06485
10 1 (0.012) 21 (0.009) 21 (0.016) 1 (0.008) 21 (0.023) 20.08882 20.09593 20.09452
11 21 (0.006) 1 (0.021) 21 (0.016) 1 (0.008) 21 (0.023) 20.08982 20.09278 20.06485
12 1 (0.012) 1 (0.021) 21 (0.016) 1 (0.008) 21 (0.023) 20.11282 20.11993 20.09454
13 21 (0.006) 21 (0.009) 1 (0.022) 1 (0.008) 21 (0.023) 20.07182 20.07478 20.06485
14 1 (0.012) 21 (0.009) 1 (0.022) 1 (0.008) 21 (0.023) 20.09382 20.10093 20.09453
15 21 (0.006) 1 (0.021) 1 (0.022) 1 (0.008) 21 (0.023) 20.09482 20.09778 20.06485
16 1 (0.012) 1 (0.021) 1 (0.022) 1 (0.008) 21 (0.023) 20.11782 20.12493 20.09454
17 21 (0.006) 21 (0.009) 21 (0.016) 21 (0.000) 1 (0.053) 20.05662 20.06305 20.08151
18 1 (0.012) 21 (0.009) 21 (0.016) 21 (0.000) 1 (0.053) 20.07962 20.09019 20.11119
19 21 (0.006) 1 (0.021) 21 (0.016) 21 (0.000) 1 (0.053) 20.08062 20.08705 20.08151
20 1 (0.012) 1 (0.021) 21 (0.016) 21 (0.000) 1 (0.053) 20.10362 20.11419 20.1112
21 21 (0.006) 21 (0.009) 1 (0.022) 21 (0.000) 1 (0.053) 20.06262 20.06905 20.08151
22 1 (0.012) 21 (0.009) 1 (0.022) 21 (0.000) 1 (0.053) 20.10362 20.11419 20.111223 21 (0.006) 1 (0.021) 1 (0.022) 21 (0.000) 1 (0.053) 20.08662 20.09305 20.08152
24 1 (0.012) 1 (0.021) 1 (0.022) 21 (0.000) 1 (0.053) 20.10362 20.11419 20.1112
25 21 (0.006) 21 (0.009) 21 (0.016) 1 (0.008) 1 (0.053) 20.06582 20.06877 20.09485
26 1 (0.012) 21 (0.009) 21 (0.016) 1 (0.008) 1 (0.053) 20.08882 20.09591 20.12452
27 21 (0.006) 1 (0.021) 21 (0.016) 1 (0.008) 1 (0.053) 20.08982 20.09277 20.09485
28 1 (0.012) 1 (0.021) 21 (0.016) 1 (0.008) 1 (0.053) 20.11282 20.11991 20.12454
29 21 (0.006) 21 (0.009) 1 (0.022) 1 (0.008) 1 (0.053) 20.06582 20.06877 20.09485
30 1 (0.012) 21 (0.009) 1 (0.022) 1 (0.008) 1 (0.053) 20.09482 20.10191 20.12453
31 21 (0.006) 1 (0.021) 1 (0.022) 1 (0.008) 1 (0.053) 20.07182 20.07477 20.09485
32 1 (0.012) 1 (0.021) 1 (0.022) 1 (0.008) 1 (0.053) 20.11882 20.12591 20.12454
Table IV Final error found through numerical model for the initialvalues of design parameters
Direction Model error FORTUS 360mc accuracy
dX (mm) 20.119 0.127
dY (mm) 20.126 0.127
dZ (mm) 20.124 0.127
Average (mm) 20.123 0.127
Optimization of positioning system of FDM machine design
Marlon Wesley Machado Cunico and Jonas de Carvalho
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the arrangement of linear bearing, housing and support accuracy
would be able to provide a low range of error, as such 0.075-
0.085mm for both directions.
Above all, it is also interesting to emphasise that errors
caused by motion elements can be electronically minimized
by a controlling system and sensors, as such linear encoder or
interferometer. In that case, despite the geometrical error
provided by either ball screw or pulley and timing belt, the
final displacement of axis will depend on measurements of
sensors, allowing the computational compensation of error
Figure 3 Analysis of main effect of design parameter for final error in dX- and dY- directions
0.08
0.08
0.09
0.09
0.10
0.10
0.11
0.11
0.12
0.12
0.000 0.008
B (mm)
H & S (mm) P (mm) TB (mm)
Main Effects Plot forX and Y (mm)Data Means
BS (mm)
0.023 0.053
0.006
Mean
0.012 0.0210.009 0.016 0.022
Figure 4 Analysis of main effect of design parameter for dZ (mm)
0.08
0.08
0.10
0.10
0.12
0.12
0.000 0.008
B (mm)
H & S (mm) P (mm) TB (mm)
Main Effects Plot forz (mm)Data Means
BS (mm)
0.023 0.053
0.006
Mean
0.012 0.0210.009 0.016 0.022
Optimization of positioning system of FDM machine design
Marlon Wesley Machado Cunico and Jonas de Carvalho
Rapid Prototyping Journal
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through a control system. On the other hand, this situation
does not avoid completely the error of the final part, as the
geometrical errors are independent, as such straightness.
As housing and support accuracy and linear bearing were
indicated to affect more intensely the mechanical errors of
machine, it was analysed an option to replace just one machine
component in order to reduce errors which are caused by either
housing accuracy or linear bearing. As result of these analyses, it
was possible to propose linear guides with precision accuracy,
no preload andblockbasic dimensionof 15 mmas an equivalent
solution for linear bearing and machined shaft.
As result, the final error found in this proposal can be seen
in Table VI, wherein is also shown the comparison with the
current conception.
Finally,it is possibleto seethatthe simple replacement of thiselement were able to reduce the error in two axes (y and z),
in spite of increase of error in x direction. Nonetheless, it is also
interesting to be seen that the proximity between the values of
x errors, whereas the difference happens in order of microns. At
the same way, the general reduction of machine error can be
seen through the difference between the averages of errors.
4. Conclusion
Besides this work has evaluated an analytical model of errors
for first generation of FDM machines, it has shown the main
influence of the five principal parameters which are used in
positioning machines.
On the other hand, the analysis of main effect has also
allowed seeing the contribution of each parameter for eachaxis direction. Wherefrom is possible to be emphasised the
system elements that lack for additional improvements,
as such the increase of linear guide accuracy.
It was observed that the highest effect of error in x and y
direction is provided by pulley accuracy in spite of the timing
belt error. In fact, this highlights the importance of selecting a
suitable transmission element to minimize errors caused
by motion system. Similarly, the ball screw was indicated as
one of the main sources of errors in z direction even though
the housing and supports accuracy and the linear bearing
accuracy were generally found to be the main sources.
Figure 5 Contour diagrams of housing and support accuracy (HS) and linear bearing (B) vs final error in dX- and dZ-directions
Table VI Comparison between final error of actual conceptions (linearbearing shaft) and proposed conception (linear guide)
Direction Linear bearing Linear guide
dX (mm) 20.119 20.1177
dY (mm) 20.126 20.1135
dZ (mm) 20.124 20.0752
Average 20.123 20.1021
Optimization of positioning system of FDM machine design
Marlon Wesley Machado Cunico and Jonas de Carvalho
Rapid Prototyping Journal
Volume 19 Number 3 2013 144152
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Focusing on these two parameters, it was proposed an
alternative conception which concerns the replacement of just
one machine element. As consequence, the substitution of
linear bearing by linear guide provides the reduction of the
general error of system that was noted in almost 20 mm.
In spite of this, it is also important to be highlighted that
although the cost of equipment limit the development of this
sort of machine, this work opened a possibility for consideringthe cost of components, being at the same way performed an
optimization study which compares the cost and error
functions.
In conclusion, this work emphasises the importance of each
of the main design parameters on the final error of first
generation of FDM machines, allowing seeing ways for increase
of accuracy in addition to highlighting where is necessary to be
concentrated efforts.
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Corresponding author
Marlon Wesley Machado Cunico can be contacted at:
Optimization of positioning system of FDM machine design
Marlon Wesley Machado Cunico and Jonas de Carvalho
Rapid Prototyping Journal
Volume 19 Number 3 2013 144152
152
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