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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

    Volume 19 Number 3 2013 144152

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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

    Marlon Wesley Machado Cunico and Jonas de Carvalho

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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

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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

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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

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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:

    [email protected]

    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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