Optimization of Positioning System of FDM Machine Design Using Analytical Approach

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 69mm. Similarly, the evaluation of the proposed conception resulted in the reduction of general error in almost 20mm, 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 the last years, the development of additive manufacturing

technologies 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 be

highlighted 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

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Rapid Prototyping Journal

19/3 (2013) 144–152

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

144

contribution of the main design parameters over the positioning

system error through multivariable methodologies. As result,it was possible to propose an equivalent conception thatprovides which replaces only one machine element, comparingwith 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 maindesign components and tolerances in order to formulate thenumerical model.

With respect to the optimization analysis proposed in thiswork, it was used the software Matlab and Minitab ascomputational tools which processed the numerical modeldata 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 specifyingall machine components based on standard suppliers, as suchTHK, Thomson or HIWIN for ball screw and linear bearingbesides Gates for pulley and timing belt. At the same way, the

international tolerance of housing and supports were definedin 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 alsoconsidered that the preload of belts provides no backlash tothe system. In this case, the maximum resolution ofdisplacement is bounded by the encoder resolution, beingdefined as 18,000 pulses per revolution. Therefore, defining the

diameter of pulley belts as 20 mm, the maximum resolution of x-y-axes would be equal to 6.9mm. 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 linear

bearing, we defined a single design parameter in accordancewith 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 whoseflanged nuts is directly coupled in the building table. It is

important to highlight that this conception applies no linearguide and consequently, either the rigidity or straightness of

system might be compromised. In accordance with thatspecification, 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 fromStratasys), which provides the minimal thickness layer equal to

0.127 mm (Stratasys, 2011a). Due to that, it was selected a stepmotor as the drive motor that feeds lead screw. The operation

mode of this motor was defined to be half step, providing aresolution 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 displacementof z-axis would be 0.01 mm. And consequently, the main

maximum accuracy of z-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 offrames or housing bearing.

For definition of system basic dimensions, it was defined abuilding 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 simplifiedFDM machine which contains the basic dimension that was

considered for this study. Additionally, it was shown the mainmachine 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 300 mm. It is important to emphasisethat support of x-axis and housing of y-axis are features of the

same component, while the support of y- and z-axes arefeatures of machine frame.

Continuing the description, we defined the distancebetween ball screw as 400 and 300 mm in both x- and

y-direction while the housing that places the screw nuts asfeature of building table.

On the other hand, it was determined the position of toolaxis 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, inaccordance with the zero point of machine.

As the main goal of this work is concerned with theprecision of positioning system, the influence of extrusion

head on machine accuracy was ignored. At the same way, forestimating 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



Volume 19 · Number 3 · 2013 · 144–152

145

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, we

defined 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:

RTzerror¼T Txerror

· yTyerror· zTzerror

ð1Þ

With reference to the depiction of axes’ HTM, equation (2)

presents the general formulation of HTM and the

collaboration of both error matrix and translation matrix for

its composition:

aTberror¼ aTb · aEb

¼

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

¼

1 21z 1y X þ dx

1z 1 21x Y þ dy

21y 1x 1 Z þ dz

0 0 0 1

2666664

3777775

ð2Þ

where:

aTberroris 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)




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146

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):

T Txerror¼

1 21z11y1

200 2 X þ dx1

1z11 21x1

25 þ dy1

21y11x1

1 50 þ dz1

0 0 0 1

2666664

3777775

ð3Þ

xTyerror¼

1 21z21y2

dx2

1z21 21x2

125 2 Y þ dy2

21y21x2

1 225 þ dz2

0 0 0 1

2666664

3777775

ð4Þ

yTzerror¼

1 21z31y3

2200 þ dx3

1z31 21x3

2150 þ dy3

21y31x3 1 225 þ Z þ dz3

0 0 0 1

2666664

3777775

ð5Þ

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 ¼ RTzerror·21RTz !

dx

dy

dz

1

2666664

3777775

¼

1 0 0 X

0 1 0 Y

0 0 1 Z

0 0 0 1

2666664

3777775

21

·

1 21z 1y X þ dx

1z 1 21x Y þ dy

21y 1x 1 Z þ dz

0 0 0 1

2666664

3777775

ð6Þ

2.3 Methods and investigation

In order to evaluate the numerical model of machine error, it

was 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

belt and ball screw was IT7, which indicates tolerances range for

current manufacturing (Shigley and Mischke, 1996). With

reference with the final error of machine numeric model in the

central region of table (zero coordinates), it was comparing with

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 whichwere 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 methodwith purpose of identifying a systematic approach for selecting

elements of positioning and tooling machines. For this, it was

applied the design of experiment methodology (DOE) inorder 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), pulley’s 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 errorsof 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 alsoindicated the main components which are related to axis error.

In accordance with this, the list of total, systematic and

random errors are shown in Table III, which also presents themain 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 toidentify the maximum error of 0.125 mm, which is a value

that indicates the similarity with the maximum error

announced by FDM 360mc (^0.127 mm) (Stratasys,2011b). In that case, only 1mm 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 tofull design (25).

In Table V, it is shown the matrix design of study, wherein is

presented the levels and values of design parameters besides thefinal 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 to 20.349 mm.




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147

Another result that is important to be highlighted is the minimal

error for axes, being, respectively, 20.04895, 20.05615 and

20.04101 mm for dX, dYand dZ. In that case, it was considered

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) 0.007 mm (IT1) .300 # 400 (Shigley and Mischke, 1996)

0.004 mm (IT3) 0.0015 mm (IT1) .18 # 50 (Shigley and Mischke, 1996)

Pulley accuracy (P) 0.021 mm (IT7) 0.0009 mm (IT5) .30 # 51 (Gates, 2011; Shigley and Mischke, 1996)

Timing belt

tolerance (TB)

0.020 mm

(2,200 N/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 þ 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-axisdX (mm) 0.064 0.0064 0.0704 Timing belt þ pulley

dY (mm) 0.026 0.0026 0.0286 Extruder housing þ shaft supports

dZ (mm) 0.026 0.0026 0.0286 Extruder housing þ shaft supports

1X (rad) 0.00052 0.000052 0.000572 Extruder housing þ shaft supports

1Y (rad) 4.33 £ 10þ00 4.33 £ 10201 4.77 £ 10þ00 Extruder housing þ shaft supports

1Z (rad) 4.33 £ 10þ00 4.33 £ 10201 4.77 £ 10þ00 Extruder housing þ shaft supports

y-axisdX (mm) 0.026 0.0026 0.0286 Guide housing þ shaft support

dY (mm) 0.064 0.0064 0.0704 Timing belt þ pulley

d (mm) 0.026 0.0026 0.0286 Guide housing þ shaft support

1X (rad) 4.33 £ 10þ00 4.33 £ 10201 4.77 £ 10þ00 Guide housing þ shaft support

1Y (rad) 8.67 £ 10þ00 8.67 £ 10201 9.53 £ 10þ00 Guide housing þ shaft support

1Z (rad) 4.33 £ 10þ00 4.33 £ 10201 4.77 £ 10þ00 Guide housing þ shaft suport

z-axisdX (mm) 0.026 0.0026 0.0286 Table support þ shaft support

dY (mm) 0.026 0.0026 0.0286 Table support þ shaft support

dZ (mm) 0.066 0.0066 0.0726 Ball screw

1X (rad) 0.000264 0.0000264 0.0002904 Table support þ shaft support þ ball screw

1Y (rad) 0.000264 0.0000264 0.0002904 Table support þ shaft support þ ball screw

1Z (rad) 0.197021697 0.01970217 0.216723867 Table support þ shaft support þ 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 misalignment

0.004 mm Radial clearance

Ball screw tolerances 0 mm Radial clearance

0.052 mm Lead accuracy (C7) þ backlash (axial play)




Volume 19 · Number 3 · 2013 · 144–152

148

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 positioning

system 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 þ1 (0.012) 21 (0.009) 21 (0.016) 21 (0.000) 21 (0.023) 20.28802 20.22641 20.34905

3 21 (0.006) þ1 (0.021) 21 (0.016) 21 (0.000) 21 (0.023) 20.8062 20.08706 20.05151

4 þ1 (0.012) þ1 (0.021) 21 (0.016) 21 (0.000) 21 (0.023) 20.10362 20.11421 20.0812

5 21 (0.006) 21 (0.009) þ1 (0.022) 21 (0.000) 21 (0.023) 20.06262 20.06906 20.05151

6 þ1 (0.012) 21 (0.009) þ1 (0.022) 21 (0.000) 21 (0.023) 20.08562 20.09621 20.08119

7 21 (0.006) þ1 (0.021) þ1 (0.022) 21 (0.000) 21 (0.023) 20.08662 20.09306 20.05152

8 þ1 (0.012) þ1 (0.021) þ1 (0.022) 21 (0.000) 21 (0.023) 20.10962 20.12021 20.0812

9 21 (0.006) 21 (0.009) 21 (0.016) þ1 (0.008) 21 (0.023) 20.06582 20.06878 20.06485

10 þ1 (0.012) 21 (0.009) 21 (0.016) þ1 (0.008) 21 (0.023) 20.08882 20.09593 20.09452

11 21 (0.006) þ1 (0.021) 21 (0.016) þ1 (0.008) 21 (0.023) 20.08982 20.09278 20.06485

12 þ1 (0.012) þ1 (0.021) 21 (0.016) þ1 (0.008) 21 (0.023) 20.11282 20.11993 20.09454

13 21 (0.006) 21 (0.009) þ1 (0.022) þ1 (0.008) 21 (0.023) 20.07182 20.07478 20.06485

14 þ1 (0.012) 21 (0.009) þ1 (0.022) þ1 (0.008) 21 (0.023) 20.09382 20.10093 20.09453

15 21 (0.006) þ1 (0.021) þ1 (0.022) þ1 (0.008) 21 (0.023) 20.09482 20.09778 20.06485

16 þ1 (0.012) þ1 (0.021) þ1 (0.022) þ1 (0.008) 21 (0.023) 20.11782 20.12493 20.09454

17 21 (0.006) 21 (0.009) 21 (0.016) 21 (0.000) þ1 (0.053) 20.05662 20.06305 20.08151

18 þ1 (0.012) 21 (0.009) 21 (0.016) 21 (0.000) þ1 (0.053) 20.07962 20.09019 20.11119

19 21 (0.006) þ1 (0.021) 21 (0.016) 21 (0.000) þ1 (0.053) 20.08062 20.08705 20.08151

20 þ1 (0.012) þ1 (0.021) 21 (0.016) 21 (0.000) þ1 (0.053) 20.10362 20.11419 20.1112

21 21 (0.006) 21 (0.009) þ1 (0.022) 21 (0.000) þ1 (0.053) 20.06262 20.06905 20.08151

22 þ1 (0.012) 21 (0.009) þ1 (0.022) 21 (0.000) þ1 (0.053) 20.10362 20.11419 20.1112

23 21 (0.006) þ1 (0.021) þ1 (0.022) 21 (0.000) þ1 (0.053) 20.08662 20.09305 20.08152

24 þ1 (0.012) þ1 (0.021) þ1 (0.022) 21 (0.000) þ1 (0.053) 20.10362 20.11419 20.1112

25 21 (0.006) 21 (0.009) 21 (0.016) þ1 (0.008) þ1 (0.053) 20.06582 20.06877 20.09485

26 þ1 (0.012) 21 (0.009) 21 (0.016) þ1 (0.008) þ1 (0.053) 20.08882 20.09591 20.12452

27 21 (0.006) þ1 (0.021) 21 (0.016) þ1 (0.008) þ1 (0.053) 20.08982 20.09277 20.09485

28 þ1 (0.012) þ1 (0.021) 21 (0.016) þ1 (0.008) þ1 (0.053) 20.11282 20.11991 20.12454

29 21 (0.006) 21 (0.009) þ1 (0.022) þ1 (0.008) þ1 (0.053) 20.06582 20.06877 20.09485

30 þ1 (0.012) 21 (0.009) þ1 (0.022) þ1 (0.008) þ1 (0.053) 20.09482 20.10191 20.12453

31 21 (0.006) þ1 (0.021) þ1 (0.022) þ1 (0.008) þ1 (0.053) 20.07182 20.07477 20.09485

32 þ1 (0.012) þ1 (0.021) þ1 (0.022) þ1 (0.008) þ1 (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




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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-

0.085 mm 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

–0.120.000 0.008

B (mm)

H & S (mm) P (mm) TB (mm)

Main Effects Plot for δX and δY (mm)Data Means

BS (mm)

0.023 0.053

0.006

Mea

n

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 for δz (mm)Data Means

BS (mm)

0.023 0.053

0.006

Mea

n

0.012 0.0210.009 0.016 0.022




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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 and block basic dimension of 15 mm as 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 possible to see that the simple replacement of this

element 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 each

axis 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 þ 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




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Focusing on these two parameters, it was proposed analternative conception which concerns the replacement of justone machine element. As consequence, the substitution oflinear bearing by linear guide provides the reduction of thegeneral error of system that was noted in almost 20mm.

In spite of this, it is also important to be highlighted thatalthough the cost of equipment limit the development of thissort of machine, this work opened a possibility for consideringthe cost of components, being at the same way performed anoptimization study which compares the cost and errorfunctions.

In conclusion, this work emphasises the importance of eachof the main design parameters on the final error of firstgeneration of FDM machines, allowing seeing ways for increaseof accuracy in addition to highlighting where is necessary to beconcentrated efforts.

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Corresponding author

Marlon Wesley Machado Cunico can be contacted at:

[email protected]




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Optimization of Positioning System of FDM Machine Design Using Analytical Approach