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Investigation into thermal characteristics of linear hotwire cutting system for variable lamination manufacturing (VLM) process by using expandable polystyrene foam
International Journal of Machine Tools & Manufacture 42 (2002) 427–439
Investigation into thermal characteristics of linear hotwire cutting system for variable lamination manufacturing (VLM) process by using expandable polystyrene foam D.G. Ahn, S.H. Lee, D.Y. Yang
*
Department of Mechanical Engineering, KAIST, Science Town, Daejeon 305-701, South Korea Received 31 August 2001; accepted 8 October 2001
Abstract The dimensional accuracy and efficiency of VLM-S, which is a new rapid prototyping process using hotwire cutter and expandable polystyrene (EPS) foam sheet, depends significantly on the thermal fields of EPS foam sheet when the hotwire cuts the sheet. The objective of this study is to investigate thermal effects of the hotwire cutting on the sheets and to find relationships between process parameters in order to obtain optimal conditions for hotwire cutting and improve dimensional accuracy of the process. Several experiments were performed to find the relationships between maximum cutting speed and heat input, and between cutting offset and heat input. Numerical analyses were carried out to investigate the influence of the cutting parameters on temperature distribution around the hotwire and to estimate the amount of the sheet melted away. Moreover, the size of the thermal front as the hotwire is about to lose its stiffness was predicted to propose the optimal cutting conditions. Based on the results, the optimal cutting conditions of the hotwire cutting system for cutting of an EPS foam sheet were found. In addition, the outcomes of the present study were reflected on the fabrication of a spanner shape and a clover punch shape. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Expandable polystyrene (EPS) foam sheet; Hotwire cutting system; Maximum cutting speed; Cutting offset; Transient heat transfer analysis; Optimum cutting condition
1. Introduction Rapid Prototyping (RP) techniques have unique characteristics according to their working principles: low building speed, the stair-stepped surface of a part due to layer-by-layer stacking, and additional post-processing to improve surface finish [1–5]. Moreover, it requires high costs to introduce and to maintain an RP apparatus. A new RP process, Variable Lamination Manufacturing using hotwire cutting system and expandable polystyrene foam (VLM-S), has been developed to overcome the unfavorable characteristics [6–8]. VLM-S process can be classified into the progressive type and the transfer type. All these types together have the shape generation unit including hotwire cutting of expandable polystyrene foam, as shown in Fig. 1. During hotwire cutting
* Corresponding author. Tel.: +82-42-869-3214; fax: +82-42-8695414. E-mail address: [email protected] (D.Y. Yang).
of EPS foam, a thermal field in the EPS foam is formed continuously in the cutting direction, as shown in Fig. 2. The thermal field depends on the operational conditions such as heat input, cutting speed, and material characteristics. The temperature distribution in the thermal field inferences dimensional accuracy of the cut parts and efficiency of hotwire cutting. Although hotwire cutting of EPS foam is a common process to generate an arbitrary shape using EPS foam, so far only a few research works in the physical phenomena of cutting and the effect of cutting parameters have been carried out to improve dimensional accuracy and efficiency except for a few papers [9,10]. Hotwire cutting of EPS foam sheets is similar to the laser cutting process, in which a heat source is moving and a workpiece is melted by heat conduction from the heat source to the workpiece. In a thermal analysis of laser cutting, the analysis of transient heat transfer using the moving coordinate, which is used to implement the moving heat source, is a common method to predict the temperature distribution of the workpiece [11–13].
0890-6955/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 0 1 ) 0 0 1 4 4 - 4
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Nomenclature VCMAX maximum cutting speed heat input QL coefficient of equation for relationship between maximum cutting speed and heat input Ai VCMAX.RD maximum cutting speed in rolling direction VCMAX.TD maximum cutting speed in transverse direction C.O. cutting offset coefficient of equation for relationship between cutting offset and heat input Bi k thermal conductivity of expandable polystyrene foam r density of expandable polystyrene foam c specific heat of expandable polystyrene foam cutting speed of hotwire Vtr h convection coefficient To initial temperature [C(T)] capacitance matrix [K(T)] conductivity matrix {V} velocity matrix for moving hotwire {Q} nodal heat flow vector q(r) heat flux through z direction M.D. measured dimensions of test part
Fig. 1.
Concept of hotwire cutting in VLM-S.
In this paper, the thermal characteristics during hotwire cutting of EPS foam sheet were investigated by experiments and numerical analyses in the case of normal cutting, that is, the hotwire being perpendicular to cutting surface, to obtain the optimal cutting conditions for the case and the dimensional accuracy of the parts of VLM-S. Empirical approaches were employed to find the relationships between maximum cutting speed and heat input, and between cutting offset and heat input. Analytical approaches using the finite element method
were carried out to study the influence of cutting parameters during hotwire cutting to estimate the melted width. The size of the thermal front when the hotwire loses its stiffness was also predicted to propose the optimal cutting conditions. In the numerical analysis, SYSWELD+ was used to predict the temperature distribution of an EPS foam sheet [14]. Based on these results, the optimal operating conditions of the hotwire cutting system were found during cutting of an EPS foam. In addition, the outcomes of this
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Fig. 2.
429
Schematic of thermal field during hotwire cutting.
study were reflected on the measurement of dimensional accuracy of a spanner shape and a clover punch shape.
2. Characteristics of hotwire cutting and EPS foam Considering hotwire cutting of EPS foam sheets, two interesting characteristics could be found: 1. The temperature distribution of the hotwire has no gradient in the wire direction except for the end region. 2. The melting zone of the EPS foam is very small, because it has a low thermal conductivity. The faster the speed of the hotwire cutter is, the smaller the melting zone becomes. The sheet type of EPS foam shows anisotropic characteristics in that it shows the difference in grain size in the rolling and transverse directions as shown in Fig. 3. It is because in manufacturing the sheet is passed through the multi-roller assembly to control the thickness of the extruded sheet, and then it is wound together and stored in a roll.
3. Experiments 3.1. Experimental set-up The experimental apparatus is composed of a hotwire cutter, XY table, and control software, as shown in Fig. 4. The cutting sequence of the apparatus is; 1. Point data and velocity of the point are input into the control software and then the electrical power is supplied to the hotwire of the cutter. 2. The control software generates the path of the XY table and the path data are transferred to the XY table. 3. The moving head of the XY table with a specimen set for test experiment moves in accordance with the predetermined path and cutting speed. Then a relative
Fig. 3. Microstructure of expandable polystyrene foam sheet. (a) 100 times enlarged photo of microstructure; (b) 400 times enlarged photo of microstructure.
motion between the hotwire cutter and the XY table cuts the specimen into an intended shape. The specifications of hotwire for the cutter are shown in Table 1.
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Fig. 4.
Experimental set-up.
VCMAX⫽A1⫻QL⫹A2
Table 1 Specifications of the hotwire Material
Diameter (mm) Resistance (⍀)
Ni 77%, Cr 20%, Mn 2%, Fe 1%
0.36
0.51
3.2. Experiment (I): maximum cutting speed (VCMAX) 3.2.1. Description of the experiment Because the major parameter, which determines the building time of VLM-S process, is the speed of hotwire cutting, the cutting speed has to be maximized to increase the efficiency of the process. But the maximum cutting speed of the hotwire cutter is limited by heat generation of the hotwire. In this experiment, the relationship between the maximum cutting speed and the heat input was derived from the regression method using the empirical data. The experiments were carried out in the rolling and transverse directions in order to consider the effects of material anisotropy of an EPS foam sheet. The details of cutting specimens are shown in Fig. 5. After the specimens are installed on the XY table and also the heat input is controlled by power supply, the specimen is cut according to each cutting speed. The relationship between the maximum cutting speed and heat input (QL) in each direction is expressed by Eq. (1):
(1)
3.2.2. Results and discussion of the experiment The results of the experiment in the rolling direction are shown in Fig. 6 and Eq. (2): VCMAX.RD⫽130.4⫻QL⫹0.4
(2)
Eq. (2) represents the regression curve of Fig. 6 and the microstructure of cutting section at each test point has a similar shape as shown in Fig. 6. The domain A in Fig. 6 is an available cutting region and the lower bound of Eq. (2) is the heat input of 0.4 Watt. The results of the experiment in the transverse direction show that the maximum cutting speed of the hotwire in the transverse direction is only attainable up to 64% of that in the rolling direction to due anisotropic material characteristic. VCMAX.TD⬵0.64⫻VCMAX.RD
(3)
This is because the linear density in the transverse direction is higher than that in the rolling direction, which is comparable to the ratio of anisotropy in two directions. As for polystyrene, higher density means more polymer chains and it needs more energy to cut the specimen in the transverse direction.
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Fig. 5.
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Details of cutting specimens for experiment I. (a) Rolling direction; (b) Transverse direction.
Fig. 6. Relationship between the maximum allowable cutting speed and the heat input of the hotwire.
In this experiment, the cutting offset that is the difference between the target size and the real size was measured by controlling the heat input of the hotwire and the cutting speed together. Also the results of the experiment I are reflected on the determination of cutting speed. The relationship between cutting offset and heat input at each cutting speed is derived from the regression method using the empirical data. The details of specimens and the cutting path are shown in Fig. 7. In this experiment, the target width, which is generated from CAD data, was defined to represent the target size and the real width (Wc), which is measured by the cut part, was defined to represent the real path. The cutting offset (C.O.) was defined to derive the empirical formulation, as shown in Eq. (4):
3.3. Experiment (II): cutting offset (C.O.)
W−WC ⫺1.0 C.O.⫽ D
3.3.1. Description of the experiment The dimensional accuracy of VLM-S parts depends upon the melted width, which is the maximum size of the melted zone in the normal direction (X direction) of hotwire cutting direction, as shown in Fig. 2. The melted width can be controlled by the heat input of the hotwire and the cutting speed.
Fig. 7.
Details of cutting specimens for experiment II.
(4)
The cutting offset, which is a non-dimensional parameter, is not influenced by the size of the specimen or the diameter of the hotwire, and C.O. can reflect the dimensional variation after hotwire cutting at each cutting speed.
Fig. 8. Relationship between the cutting offset and the heat input of the hotwire.
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Table 2 Coefficients of equation for the relationship between the cutting offset and the heat input Cutting speed (mm/s)
B1
B2
40 50 60
1.18 1.84 1.94
⫺0.54 ⫺1.47 ⫺1.76
Fig. 10.
Thermal conductivity of EPS foam.
4. Numerical analysis
Fig. 9.
Relationship between the cutting offset and the cutting speed.
The phenomenon of heat transfer during the hotwire cutting was simulated by SYSWELD+. The results of the analysis were compared with the results of the experiments to verify the numerical model. Using the modified finite element model, the final analysis was consequently performed. So, the size of the thermal front, which is the difference between the center of heat source and the front of the melted line in the hotwire cutting direction when the hotwire loses its stiffness, was estimated by using the results of the final analysis in order to propose the optimal cutting condition.
The linear function was selected to make a relationship between C.O. and heat input, as shown in Eq. (5):
4.1. Physical description of the numerical model
C.O.⫽B1⫻QL⫹B2
The following assumptions were made in the formulation of the finite element model:
(5)
3.3.2. Results and discussion of the experiment The results of the experiment are shown in Fig. 8 and Table 2. Table 2 is the coefficients of Eq. (5) at each cutting speed. As a result of the experiment, the relationship between C.O. and the cutting speed at each heat input could be derived from the regression method using the empirical data as shown in Figs. 8 and 9 and Eq. (6). Table 3 is the coefficient of Eq. (6) at each heat input. C.O.⫽C1⫻QL⫹C2
1. The hotwire is a line heat source according to the first characteristic of hotwire cutting in chapter 2. 2. The analyzed size of EPS foam was selected to have a size of 100 mm×60 mm×2 mm in the global analysis according to the second characteristic of hotwire cutting in chapter 2. 3. The workpiece is initially at 293K. Both the work-
(6)
Table 3 Coefficients of equation for the relationship between the cutting offset and the cutting speed Heat input (watt)
C1
C2
0.91 1.10
⫺0.024 ⫺0.025
1.42 1.80 Fig. 11. Specific heat of EPS foam.
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冉 冊 冉 冊 冉 冊 冎
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再
∂ ∂T ∂ ∂T ∂ ∂T ∂T k ⫹ k ⫹ k ⫹q⫽rc ⫺V ∂x ∂x ∂yM ∂yM ∂z ∂z ∂tM tr
Table 4 Thermal properties of EPS foam sheets Density (g/cm3)
Melting Temp (°C)
Convection coefficient (W/m2 °C)
6.26×10⫺2
240
1.0
(10)
∂T ⫻ ∂yM
The initial condition is piece and the mesh are fixed and the hotwire moves in the positive Y direction with the constant speed (Vtr). 4. The thermal conductivity and specific heat are considered to be temperature dependent as shown in Figs. 10 and 11 [15,16]. The convection coefficient and density are considered to be temperature independent. Table 4 summarizes the thermal properties of EPS foam [17–19]. 5. The temperature of hotwire is 973 K, which is measured by the infrared temperature-measuring device. The effects of radiation are not significant in the temperature range and ignored in computation.
4.2. Mathematical description of the model The transient temperature distribution T (x, y, z, t) satisfies the following differential equation for three-dimensional heat conduction in a D domain: →
→
∇·(k∇T)⫹q˙⫽rcT˙
(7)
The relationship between the moving coordinate and the fixed coordinate is represented as Eq. (8) and both coordinates are shown in Fig. 12. yM⫽y⫺Vtr⫻t, xM⫽x, zM⫽z
(8)
T(x,y,z,t)⫽T(x,y⫺Vtrt,z)
(9)
T(x,y,z,0)⫽To for (x,y,z)苸D The essential boundary condition is T(x,y,z)⫽To
(12)
on the boundary S1 for (x, y, z) 苸 S1 and t ⬎ 0. S1 represents those surfaces that are subjected to constant temperature condition. The natural boundary condition can be defined by kn
∂T ⫺h(T⫺To)⫽0 ∂n
(13)
on the boundary S2 for (x, y, z) 苸 S2 and t ⬎ 0. S2 represents those surfaces that are subjected to convection and imposed heat fluxes. 4.3. Finite element analysis The finite element analysis was carried out to find a solution, which satisfies the governing equation and boundary conditions in Section 4.2. SYSWELD+ was used to perform the finite element analysis. The SYSWELD + provides a convenient means of numerically modeling a heat source moving problem. The weak form of Eq. (10) can be derived and rearranged with integration by parts. The governing iso-parametric finite element equation of the transient heat transfer problem can be written in matrix form as follows; [C(T)]{T˙}⫹[K(T)]{T}⫹{V}⫽{Q(t)}
If the moving coordinate is applied to Eq. (7), the transient temperature distribution T (x, yM, z) satisfies the following differential equation for three-dimensional heat conduction in a D domain:
(11)
(14)
The generalized trapezoid algorithm is used to perform integration with respect to time. T(t⫹dt)⫽T(t)⫹dt[(1⫺v)T˙(t)⫹vT˙(t⫹dt)], v⫽0.5
(15)
The hexahedral element and skin element are used for finite element modeling. The hotwire was simulated by the volumetric heat flux that is a circular beam mode as shown in Fig. 12. In the case of a circular beam mode, the heat flux q(r) through the z direction of the workpiece can be expressed by the following equation. 4QL q(r)⫽ 2 pd Fig. 12.
Schematic of moving coordinate and fixed coordinate.
(16)
The amount of heat input, which is used for analysis, is 5.71 W/mm3.
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Fig. 13.
Geometry of global analysis.
4.3.1. Global analysis 4.3.1.1. Description of global analysis. The transient heat transfer analysis was carried out to estimate the temperature distribution and the effects of cutting speed and element size. The analysis geometry in the global analysis is shown in Fig. 13. The finite element mesh and the boundary condition are shown in Fig. 14. The cutting speed in the finite element analysis was set to be 10 mm/s, 30 mm/s, 40 mm/s and 50 mm/s. 4.3.1.2. Results and discussion of the global analysis. Fig. 15 shows the results of the global analysis in the case of cutting speed of 10 mm/s. The following outcomes are achieved from the results of global analysis: 1. The maximum heat affected distance, which is the maximum distance between the moving trajectory line
Fig. 15.
The results of global analysis.
of a heat source and the nearest isothermal line of room temperature from the moving trajectory line of a heat source in the normal (X direction in Fig. 16), was nearly 2 mm at the cutting speed of 10 mm/s as shown in Fig. 15. We could also define the heat affected region that is the inside region of the nearest isothermal line of room temperature (293 K) from the moving trajectory line of heat source, as shown in Fig. 16. 2. In case the cutting speed was increased, the heat affected region was rapidly reduced due to the reduction of an amount of heat transfer per unit time from hotwire to EPS foam. 3. The solution of finite element analysis oscillated in the cutting direction, because the aspect ratio of mesh was too large and the integration point at each time was not set on the nodal point. Based on the above results, it has been found that local analysis is necessary to obtain accurate temperature distribution and finer elements should be used, which has the side length of 0.5 mm and uniform arrangement. 4.3.2. Local analysis
Fig. 14. Finite element modeling and boundary condition of global analysis. (a) Finite element model; (b) Boundary condition.
4.3.2.1. Description of local analysis. The local analysis was carried out to estimate an accurate temperature distribution in the heat affected region and the size
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Fig. 16.
435
Schematic of thermal field to explain heat-affected region and maximum heat affected distance.
of a thermal front when the hotwire loses its stiffness in the cutting direction. The region of local analysis was selected to be ±2.5 mm from the center of the heat source in the X direction as shown in Fig. 16, because a practical cutting speed is higher than 30 mm/s and then half of the distance between bounding contour line of room temperature and the symmetric center line is smaller than ±1.5 mm. The geometry of local analysis is shown in Fig. 17. The mesh and boundary conditions are shown in Fig. 18. The starting point of hotwire cutting in the analysis was set on 5 mm from the end in the Y direction to minimize the transition effect of starting. Also, the time interval of numerical integration is adjusted to set the heat source on a node of element at each integration time so as to eliminate an oscillation of the solution. The cutting speed of the local analysis was set to be 30 mm/s, 40 mm/s, 50 mm/s and 60 mm/s. 4.3.2.2. Results and discussion of the local analysis. In local analysis, the results were compared with the empirical results to verify the finite element model and the final analysis has been subsequently carried out using the modified model. The results of the analysis are shown in Fig. 19. In consequence of the analysis, a stable temperature distribution appeared as the hotwire moved 3 mm from the starting point and then the effects of ends were eliminated. The analytical and empirical results show a good agreement with respect to the melted width, as shown in Fig. 20. Based on the outcomes of local analysis, the size of the thermal front at each cutting speed was calculated as shown in Fig. 21(b) and the size
Fig. 17.
Geometry of local analysis.
Fig. 18. Finite element model and boundary condition of local analysis. (a) Finite element model of local analysis; (b) Boundary condition of local analysis.
of the thermal front when the hotwire begins to bend in the opposite direction of the cutting direction was predicted by the regression method using the analyzed data. Fig. 21(a) shows the definition of the thermal front. In the experiment, the hotwire was about to lose its stiffness at the cutting speed of 55 mm/s. As the results of local analysis, the size of the thermal front at 55 mm/s was 0.41 mm and the size was 2.3 times larger than the radius of the hotwire. In spite of the sufficient size of the thermal front, the reason why the hotwire loses its stiffness at 55 mm/s was that the minimum distance “A” between the center of the hotwire and the melted line was 0.28 mm and then the cutting speed of the hotwire was too
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Fig. 19.
Results of local analysis. (a) Vtr =30 mm/s; (b) Vtr =40 mm/s; (c) Vtr =50 mm/s; (d) Vtr =60 mm/s.
fast to melt the EPS foam before the hotwire passes through the melted region. Hence, the cutting speed of the hotwire was limited to 50 mm/s and the lower bound of heat input was restricted to 0.95 Watt. 5. Evaluation of dimensional accuracy for VLM-S parts 5.1. Measuring the dimensional accuracy
Fig. 20. Comparison of analytical results with empirical results with respect to melted width.
Using the results of experiment II, the dimensional accuracy of test parts was evaluated. When the dimensional accuracy of test parts was computed, the expected dimension (E.D.) was introduced to predict the in-plane error of test parts. The E.D. could be estimated using the modified formula of Eq. (5) at each cutting speed,
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Fig. 21. Definition of the thermal front and relationship between size of thermal front and cutting speed. (a) Definition of the thermal front; (b) Relationship between size of thermal front and cutting speed.
as in Eq. (17) and the in-plane error could be predicted using Eq. (18). E.D.⫽W⫺(C.O.⫹1.0)⫻D Error(%)⫽
E.D.−M.D. ⫻100 E.D.
(17) (18)
A spanner shape and a clover punch shape, which are the practical examples to predict the dimensional accuracy, were manufactured by VLM-S and shown in Fig. 22. Fig. 23 shows the measured dimensions of each part. The test conditions are shown in Table 5. 5.2. Results and discussion of the evaluation of dimensional accuracy Table 6 shows the measured and the expected dimensions. Table 7 shows the dimensional accuracy of the parts. All results showed that the dimensional accuracy was less than 2%. Considering commercial RP processes with planar dimensional accuracy of 2% [20], it is known that the amount of the melted width has to be considered to obtain improved accuracy.
437
Fig. 22. Fabricated shape of test parts using VLM-S. (a) Spanner shape; (b) Clover punch shape.
6. Conclusion The thermal field of EPS foam during hotwire cutting influences the operational conditions of the cutting and the dimensional accuracy of VLM-S parts. The speed of cutting is thus determined by the operational conditions that affect the building time of VLM-S. In this paper, the thermal characteristics of hotwire cutting of EPS foam and the effects of the process parameters have been studied in order to obtain optimal cutting condition and dimensional accuracy of the VLM-S parts. Experiments were carried out by the 2-axis set-up and numerical analysis was performed by using the FEM code SYSWELD+. The empirical equations showing the relationships between the maximum cutting speed and the heat input at each material direction, and between the cutting offset and the heat input were acquired by the empirical data. It was also found that the maximum cutting speed in each material direction has a different value due to material anisotropy of EPS foam sheets. The transient heat transfer phenomenon during the hotwire cutting was simulated by an FE code using the twostep approach. In the global analysis, the effects of cut-
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Table 7 Dimensional accuracy of test parts Spanner Shape
Clover punch shape
Measurement Measurement Measurement Measurement I (mm) II (mm) I (mm) II (mm) Considering offset 1.14 (%) No considering offset (%)
1.55
1.05 Average=1.10 4.39 Average=2.97
Fig. 23. Geometry and the measured dimensions of test parts. (a) Open-ended spanner shape; (b) Clover punch shape.
Table 5 Cutting conditions of test parts
Heat input (watt) Cutting speed (mm/s)
Spanner shape
Clover punch shape
1.19 50
1.19 50
Clover punch shape
Measurement Measurement Measurement Measurement I (mm) II (mm) I (mm) II (mm) CAD 150 Expected 149.39 Dimension (E.D.) Measured 147.68 Dimension (M.D.)
2.60
0.61 Average=0.40 1.77 Average=1.92
ture distribution of EPS foam sheet during hotwire cutting was computed to estimate the melted region and to predict the size of thermal front when the hotwire loses its stiffness. The optimal operational condition such as the upper bound of the cutting speed and the lower bound of the heat input was proposed by considering the results of the local analysis and the experiment. Based on the results, a spanner shape and a clover punch shape were fabricated by VLM-S. The empirical equation for cutting offset, Eq. (5) for the case of 50 mm/s of cutting speed, was used to estimate the expected dimensions to predict exactly the dimensional accuracy of VLM-S parts. In future works, an additional experiment and simulation in case of three-dimensional cutting should be further performed in order to reflect the melted width and optimal cutting condition on the input data of VLMS. In addition, an additional module should be required to compensate for the melted width in VLM-Slicer that is the software to generate the input of VLM-S.
References
Table 6 Expected and measured dimensions of the test parts Spanner Shape
0.2
18.2 17.59
32.98 32.37
52.43 51.82
17.4
32.3
51.5
ting parameters and the mesh size on temperature distribution of EPS foam sheet were studied, and also important information such as the size of analysis zone, the mesh size and time interval for numerical integration was acquired. In the local analysis, the accurate tempera-
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Investigation into thermal characteristics of linear hotwire cutting system for variable lamination manufacturing (VLM) process by using expandable polystyrene foam D.G. Ahn, S.H. Lee, D.Y. Yang
*
Department of Mechanical Engineering, KAIST, Science Town, Daejeon 305-701, South Korea Received 31 August 2001; accepted 8 October 2001
Abstract The dimensional accuracy and efficiency of VLM-S, which is a new rapid prototyping process using hotwire cutter and expandable polystyrene (EPS) foam sheet, depends significantly on the thermal fields of EPS foam sheet when the hotwire cuts the sheet. The objective of this study is to investigate thermal effects of the hotwire cutting on the sheets and to find relationships between process parameters in order to obtain optimal conditions for hotwire cutting and improve dimensional accuracy of the process. Several experiments were performed to find the relationships between maximum cutting speed and heat input, and between cutting offset and heat input. Numerical analyses were carried out to investigate the influence of the cutting parameters on temperature distribution around the hotwire and to estimate the amount of the sheet melted away. Moreover, the size of the thermal front as the hotwire is about to lose its stiffness was predicted to propose the optimal cutting conditions. Based on the results, the optimal cutting conditions of the hotwire cutting system for cutting of an EPS foam sheet were found. In addition, the outcomes of the present study were reflected on the fabrication of a spanner shape and a clover punch shape. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Expandable polystyrene (EPS) foam sheet; Hotwire cutting system; Maximum cutting speed; Cutting offset; Transient heat transfer analysis; Optimum cutting condition
1. Introduction Rapid Prototyping (RP) techniques have unique characteristics according to their working principles: low building speed, the stair-stepped surface of a part due to layer-by-layer stacking, and additional post-processing to improve surface finish [1–5]. Moreover, it requires high costs to introduce and to maintain an RP apparatus. A new RP process, Variable Lamination Manufacturing using hotwire cutting system and expandable polystyrene foam (VLM-S), has been developed to overcome the unfavorable characteristics [6–8]. VLM-S process can be classified into the progressive type and the transfer type. All these types together have the shape generation unit including hotwire cutting of expandable polystyrene foam, as shown in Fig. 1. During hotwire cutting
* Corresponding author. Tel.: +82-42-869-3214; fax: +82-42-8695414. E-mail address: [email protected] (D.Y. Yang).
of EPS foam, a thermal field in the EPS foam is formed continuously in the cutting direction, as shown in Fig. 2. The thermal field depends on the operational conditions such as heat input, cutting speed, and material characteristics. The temperature distribution in the thermal field inferences dimensional accuracy of the cut parts and efficiency of hotwire cutting. Although hotwire cutting of EPS foam is a common process to generate an arbitrary shape using EPS foam, so far only a few research works in the physical phenomena of cutting and the effect of cutting parameters have been carried out to improve dimensional accuracy and efficiency except for a few papers [9,10]. Hotwire cutting of EPS foam sheets is similar to the laser cutting process, in which a heat source is moving and a workpiece is melted by heat conduction from the heat source to the workpiece. In a thermal analysis of laser cutting, the analysis of transient heat transfer using the moving coordinate, which is used to implement the moving heat source, is a common method to predict the temperature distribution of the workpiece [11–13].
0890-6955/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 0 1 ) 0 0 1 4 4 - 4
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Nomenclature VCMAX maximum cutting speed heat input QL coefficient of equation for relationship between maximum cutting speed and heat input Ai VCMAX.RD maximum cutting speed in rolling direction VCMAX.TD maximum cutting speed in transverse direction C.O. cutting offset coefficient of equation for relationship between cutting offset and heat input Bi k thermal conductivity of expandable polystyrene foam r density of expandable polystyrene foam c specific heat of expandable polystyrene foam cutting speed of hotwire Vtr h convection coefficient To initial temperature [C(T)] capacitance matrix [K(T)] conductivity matrix {V} velocity matrix for moving hotwire {Q} nodal heat flow vector q(r) heat flux through z direction M.D. measured dimensions of test part
Fig. 1.
Concept of hotwire cutting in VLM-S.
In this paper, the thermal characteristics during hotwire cutting of EPS foam sheet were investigated by experiments and numerical analyses in the case of normal cutting, that is, the hotwire being perpendicular to cutting surface, to obtain the optimal cutting conditions for the case and the dimensional accuracy of the parts of VLM-S. Empirical approaches were employed to find the relationships between maximum cutting speed and heat input, and between cutting offset and heat input. Analytical approaches using the finite element method
were carried out to study the influence of cutting parameters during hotwire cutting to estimate the melted width. The size of the thermal front when the hotwire loses its stiffness was also predicted to propose the optimal cutting conditions. In the numerical analysis, SYSWELD+ was used to predict the temperature distribution of an EPS foam sheet [14]. Based on these results, the optimal operating conditions of the hotwire cutting system were found during cutting of an EPS foam. In addition, the outcomes of this
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Fig. 2.
429
Schematic of thermal field during hotwire cutting.
study were reflected on the measurement of dimensional accuracy of a spanner shape and a clover punch shape.
2. Characteristics of hotwire cutting and EPS foam Considering hotwire cutting of EPS foam sheets, two interesting characteristics could be found: 1. The temperature distribution of the hotwire has no gradient in the wire direction except for the end region. 2. The melting zone of the EPS foam is very small, because it has a low thermal conductivity. The faster the speed of the hotwire cutter is, the smaller the melting zone becomes. The sheet type of EPS foam shows anisotropic characteristics in that it shows the difference in grain size in the rolling and transverse directions as shown in Fig. 3. It is because in manufacturing the sheet is passed through the multi-roller assembly to control the thickness of the extruded sheet, and then it is wound together and stored in a roll.
3. Experiments 3.1. Experimental set-up The experimental apparatus is composed of a hotwire cutter, XY table, and control software, as shown in Fig. 4. The cutting sequence of the apparatus is; 1. Point data and velocity of the point are input into the control software and then the electrical power is supplied to the hotwire of the cutter. 2. The control software generates the path of the XY table and the path data are transferred to the XY table. 3. The moving head of the XY table with a specimen set for test experiment moves in accordance with the predetermined path and cutting speed. Then a relative
Fig. 3. Microstructure of expandable polystyrene foam sheet. (a) 100 times enlarged photo of microstructure; (b) 400 times enlarged photo of microstructure.
motion between the hotwire cutter and the XY table cuts the specimen into an intended shape. The specifications of hotwire for the cutter are shown in Table 1.
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Fig. 4.
Experimental set-up.
VCMAX⫽A1⫻QL⫹A2
Table 1 Specifications of the hotwire Material
Diameter (mm) Resistance (⍀)
Ni 77%, Cr 20%, Mn 2%, Fe 1%
0.36
0.51
3.2. Experiment (I): maximum cutting speed (VCMAX) 3.2.1. Description of the experiment Because the major parameter, which determines the building time of VLM-S process, is the speed of hotwire cutting, the cutting speed has to be maximized to increase the efficiency of the process. But the maximum cutting speed of the hotwire cutter is limited by heat generation of the hotwire. In this experiment, the relationship between the maximum cutting speed and the heat input was derived from the regression method using the empirical data. The experiments were carried out in the rolling and transverse directions in order to consider the effects of material anisotropy of an EPS foam sheet. The details of cutting specimens are shown in Fig. 5. After the specimens are installed on the XY table and also the heat input is controlled by power supply, the specimen is cut according to each cutting speed. The relationship between the maximum cutting speed and heat input (QL) in each direction is expressed by Eq. (1):
(1)
3.2.2. Results and discussion of the experiment The results of the experiment in the rolling direction are shown in Fig. 6 and Eq. (2): VCMAX.RD⫽130.4⫻QL⫹0.4
(2)
Eq. (2) represents the regression curve of Fig. 6 and the microstructure of cutting section at each test point has a similar shape as shown in Fig. 6. The domain A in Fig. 6 is an available cutting region and the lower bound of Eq. (2) is the heat input of 0.4 Watt. The results of the experiment in the transverse direction show that the maximum cutting speed of the hotwire in the transverse direction is only attainable up to 64% of that in the rolling direction to due anisotropic material characteristic. VCMAX.TD⬵0.64⫻VCMAX.RD
(3)
This is because the linear density in the transverse direction is higher than that in the rolling direction, which is comparable to the ratio of anisotropy in two directions. As for polystyrene, higher density means more polymer chains and it needs more energy to cut the specimen in the transverse direction.
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Fig. 5.
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Details of cutting specimens for experiment I. (a) Rolling direction; (b) Transverse direction.
Fig. 6. Relationship between the maximum allowable cutting speed and the heat input of the hotwire.
In this experiment, the cutting offset that is the difference between the target size and the real size was measured by controlling the heat input of the hotwire and the cutting speed together. Also the results of the experiment I are reflected on the determination of cutting speed. The relationship between cutting offset and heat input at each cutting speed is derived from the regression method using the empirical data. The details of specimens and the cutting path are shown in Fig. 7. In this experiment, the target width, which is generated from CAD data, was defined to represent the target size and the real width (Wc), which is measured by the cut part, was defined to represent the real path. The cutting offset (C.O.) was defined to derive the empirical formulation, as shown in Eq. (4):
3.3. Experiment (II): cutting offset (C.O.)
W−WC ⫺1.0 C.O.⫽ D
3.3.1. Description of the experiment The dimensional accuracy of VLM-S parts depends upon the melted width, which is the maximum size of the melted zone in the normal direction (X direction) of hotwire cutting direction, as shown in Fig. 2. The melted width can be controlled by the heat input of the hotwire and the cutting speed.
Fig. 7.
Details of cutting specimens for experiment II.
(4)
The cutting offset, which is a non-dimensional parameter, is not influenced by the size of the specimen or the diameter of the hotwire, and C.O. can reflect the dimensional variation after hotwire cutting at each cutting speed.
Fig. 8. Relationship between the cutting offset and the heat input of the hotwire.
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Table 2 Coefficients of equation for the relationship between the cutting offset and the heat input Cutting speed (mm/s)
B1
B2
40 50 60
1.18 1.84 1.94
⫺0.54 ⫺1.47 ⫺1.76
Fig. 10.
Thermal conductivity of EPS foam.
4. Numerical analysis
Fig. 9.
Relationship between the cutting offset and the cutting speed.
The phenomenon of heat transfer during the hotwire cutting was simulated by SYSWELD+. The results of the analysis were compared with the results of the experiments to verify the numerical model. Using the modified finite element model, the final analysis was consequently performed. So, the size of the thermal front, which is the difference between the center of heat source and the front of the melted line in the hotwire cutting direction when the hotwire loses its stiffness, was estimated by using the results of the final analysis in order to propose the optimal cutting condition.
The linear function was selected to make a relationship between C.O. and heat input, as shown in Eq. (5):
4.1. Physical description of the numerical model
C.O.⫽B1⫻QL⫹B2
The following assumptions were made in the formulation of the finite element model:
(5)
3.3.2. Results and discussion of the experiment The results of the experiment are shown in Fig. 8 and Table 2. Table 2 is the coefficients of Eq. (5) at each cutting speed. As a result of the experiment, the relationship between C.O. and the cutting speed at each heat input could be derived from the regression method using the empirical data as shown in Figs. 8 and 9 and Eq. (6). Table 3 is the coefficient of Eq. (6) at each heat input. C.O.⫽C1⫻QL⫹C2
1. The hotwire is a line heat source according to the first characteristic of hotwire cutting in chapter 2. 2. The analyzed size of EPS foam was selected to have a size of 100 mm×60 mm×2 mm in the global analysis according to the second characteristic of hotwire cutting in chapter 2. 3. The workpiece is initially at 293K. Both the work-
(6)
Table 3 Coefficients of equation for the relationship between the cutting offset and the cutting speed Heat input (watt)
C1
C2
0.91 1.10
⫺0.024 ⫺0.025
1.42 1.80 Fig. 11. Specific heat of EPS foam.
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冉 冊 冉 冊 冉 冊 冎
433
再
∂ ∂T ∂ ∂T ∂ ∂T ∂T k ⫹ k ⫹ k ⫹q⫽rc ⫺V ∂x ∂x ∂yM ∂yM ∂z ∂z ∂tM tr
Table 4 Thermal properties of EPS foam sheets Density (g/cm3)
Melting Temp (°C)
Convection coefficient (W/m2 °C)
6.26×10⫺2
240
1.0
(10)
∂T ⫻ ∂yM
The initial condition is piece and the mesh are fixed and the hotwire moves in the positive Y direction with the constant speed (Vtr). 4. The thermal conductivity and specific heat are considered to be temperature dependent as shown in Figs. 10 and 11 [15,16]. The convection coefficient and density are considered to be temperature independent. Table 4 summarizes the thermal properties of EPS foam [17–19]. 5. The temperature of hotwire is 973 K, which is measured by the infrared temperature-measuring device. The effects of radiation are not significant in the temperature range and ignored in computation.
4.2. Mathematical description of the model The transient temperature distribution T (x, y, z, t) satisfies the following differential equation for three-dimensional heat conduction in a D domain: →
→
∇·(k∇T)⫹q˙⫽rcT˙
(7)
The relationship between the moving coordinate and the fixed coordinate is represented as Eq. (8) and both coordinates are shown in Fig. 12. yM⫽y⫺Vtr⫻t, xM⫽x, zM⫽z
(8)
T(x,y,z,t)⫽T(x,y⫺Vtrt,z)
(9)
T(x,y,z,0)⫽To for (x,y,z)苸D The essential boundary condition is T(x,y,z)⫽To
(12)
on the boundary S1 for (x, y, z) 苸 S1 and t ⬎ 0. S1 represents those surfaces that are subjected to constant temperature condition. The natural boundary condition can be defined by kn
∂T ⫺h(T⫺To)⫽0 ∂n
(13)
on the boundary S2 for (x, y, z) 苸 S2 and t ⬎ 0. S2 represents those surfaces that are subjected to convection and imposed heat fluxes. 4.3. Finite element analysis The finite element analysis was carried out to find a solution, which satisfies the governing equation and boundary conditions in Section 4.2. SYSWELD+ was used to perform the finite element analysis. The SYSWELD + provides a convenient means of numerically modeling a heat source moving problem. The weak form of Eq. (10) can be derived and rearranged with integration by parts. The governing iso-parametric finite element equation of the transient heat transfer problem can be written in matrix form as follows; [C(T)]{T˙}⫹[K(T)]{T}⫹{V}⫽{Q(t)}
If the moving coordinate is applied to Eq. (7), the transient temperature distribution T (x, yM, z) satisfies the following differential equation for three-dimensional heat conduction in a D domain:
(11)
(14)
The generalized trapezoid algorithm is used to perform integration with respect to time. T(t⫹dt)⫽T(t)⫹dt[(1⫺v)T˙(t)⫹vT˙(t⫹dt)], v⫽0.5
(15)
The hexahedral element and skin element are used for finite element modeling. The hotwire was simulated by the volumetric heat flux that is a circular beam mode as shown in Fig. 12. In the case of a circular beam mode, the heat flux q(r) through the z direction of the workpiece can be expressed by the following equation. 4QL q(r)⫽ 2 pd Fig. 12.
Schematic of moving coordinate and fixed coordinate.
(16)
The amount of heat input, which is used for analysis, is 5.71 W/mm3.
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Fig. 13.
Geometry of global analysis.
4.3.1. Global analysis 4.3.1.1. Description of global analysis. The transient heat transfer analysis was carried out to estimate the temperature distribution and the effects of cutting speed and element size. The analysis geometry in the global analysis is shown in Fig. 13. The finite element mesh and the boundary condition are shown in Fig. 14. The cutting speed in the finite element analysis was set to be 10 mm/s, 30 mm/s, 40 mm/s and 50 mm/s. 4.3.1.2. Results and discussion of the global analysis. Fig. 15 shows the results of the global analysis in the case of cutting speed of 10 mm/s. The following outcomes are achieved from the results of global analysis: 1. The maximum heat affected distance, which is the maximum distance between the moving trajectory line
Fig. 15.
The results of global analysis.
of a heat source and the nearest isothermal line of room temperature from the moving trajectory line of a heat source in the normal (X direction in Fig. 16), was nearly 2 mm at the cutting speed of 10 mm/s as shown in Fig. 15. We could also define the heat affected region that is the inside region of the nearest isothermal line of room temperature (293 K) from the moving trajectory line of heat source, as shown in Fig. 16. 2. In case the cutting speed was increased, the heat affected region was rapidly reduced due to the reduction of an amount of heat transfer per unit time from hotwire to EPS foam. 3. The solution of finite element analysis oscillated in the cutting direction, because the aspect ratio of mesh was too large and the integration point at each time was not set on the nodal point. Based on the above results, it has been found that local analysis is necessary to obtain accurate temperature distribution and finer elements should be used, which has the side length of 0.5 mm and uniform arrangement. 4.3.2. Local analysis
Fig. 14. Finite element modeling and boundary condition of global analysis. (a) Finite element model; (b) Boundary condition.
4.3.2.1. Description of local analysis. The local analysis was carried out to estimate an accurate temperature distribution in the heat affected region and the size
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Fig. 16.
435
Schematic of thermal field to explain heat-affected region and maximum heat affected distance.
of a thermal front when the hotwire loses its stiffness in the cutting direction. The region of local analysis was selected to be ±2.5 mm from the center of the heat source in the X direction as shown in Fig. 16, because a practical cutting speed is higher than 30 mm/s and then half of the distance between bounding contour line of room temperature and the symmetric center line is smaller than ±1.5 mm. The geometry of local analysis is shown in Fig. 17. The mesh and boundary conditions are shown in Fig. 18. The starting point of hotwire cutting in the analysis was set on 5 mm from the end in the Y direction to minimize the transition effect of starting. Also, the time interval of numerical integration is adjusted to set the heat source on a node of element at each integration time so as to eliminate an oscillation of the solution. The cutting speed of the local analysis was set to be 30 mm/s, 40 mm/s, 50 mm/s and 60 mm/s. 4.3.2.2. Results and discussion of the local analysis. In local analysis, the results were compared with the empirical results to verify the finite element model and the final analysis has been subsequently carried out using the modified model. The results of the analysis are shown in Fig. 19. In consequence of the analysis, a stable temperature distribution appeared as the hotwire moved 3 mm from the starting point and then the effects of ends were eliminated. The analytical and empirical results show a good agreement with respect to the melted width, as shown in Fig. 20. Based on the outcomes of local analysis, the size of the thermal front at each cutting speed was calculated as shown in Fig. 21(b) and the size
Fig. 17.
Geometry of local analysis.
Fig. 18. Finite element model and boundary condition of local analysis. (a) Finite element model of local analysis; (b) Boundary condition of local analysis.
of the thermal front when the hotwire begins to bend in the opposite direction of the cutting direction was predicted by the regression method using the analyzed data. Fig. 21(a) shows the definition of the thermal front. In the experiment, the hotwire was about to lose its stiffness at the cutting speed of 55 mm/s. As the results of local analysis, the size of the thermal front at 55 mm/s was 0.41 mm and the size was 2.3 times larger than the radius of the hotwire. In spite of the sufficient size of the thermal front, the reason why the hotwire loses its stiffness at 55 mm/s was that the minimum distance “A” between the center of the hotwire and the melted line was 0.28 mm and then the cutting speed of the hotwire was too
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Fig. 19.
Results of local analysis. (a) Vtr =30 mm/s; (b) Vtr =40 mm/s; (c) Vtr =50 mm/s; (d) Vtr =60 mm/s.
fast to melt the EPS foam before the hotwire passes through the melted region. Hence, the cutting speed of the hotwire was limited to 50 mm/s and the lower bound of heat input was restricted to 0.95 Watt. 5. Evaluation of dimensional accuracy for VLM-S parts 5.1. Measuring the dimensional accuracy
Fig. 20. Comparison of analytical results with empirical results with respect to melted width.
Using the results of experiment II, the dimensional accuracy of test parts was evaluated. When the dimensional accuracy of test parts was computed, the expected dimension (E.D.) was introduced to predict the in-plane error of test parts. The E.D. could be estimated using the modified formula of Eq. (5) at each cutting speed,
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Fig. 21. Definition of the thermal front and relationship between size of thermal front and cutting speed. (a) Definition of the thermal front; (b) Relationship between size of thermal front and cutting speed.
as in Eq. (17) and the in-plane error could be predicted using Eq. (18). E.D.⫽W⫺(C.O.⫹1.0)⫻D Error(%)⫽
E.D.−M.D. ⫻100 E.D.
(17) (18)
A spanner shape and a clover punch shape, which are the practical examples to predict the dimensional accuracy, were manufactured by VLM-S and shown in Fig. 22. Fig. 23 shows the measured dimensions of each part. The test conditions are shown in Table 5. 5.2. Results and discussion of the evaluation of dimensional accuracy Table 6 shows the measured and the expected dimensions. Table 7 shows the dimensional accuracy of the parts. All results showed that the dimensional accuracy was less than 2%. Considering commercial RP processes with planar dimensional accuracy of 2% [20], it is known that the amount of the melted width has to be considered to obtain improved accuracy.
437
Fig. 22. Fabricated shape of test parts using VLM-S. (a) Spanner shape; (b) Clover punch shape.
6. Conclusion The thermal field of EPS foam during hotwire cutting influences the operational conditions of the cutting and the dimensional accuracy of VLM-S parts. The speed of cutting is thus determined by the operational conditions that affect the building time of VLM-S. In this paper, the thermal characteristics of hotwire cutting of EPS foam and the effects of the process parameters have been studied in order to obtain optimal cutting condition and dimensional accuracy of the VLM-S parts. Experiments were carried out by the 2-axis set-up and numerical analysis was performed by using the FEM code SYSWELD+. The empirical equations showing the relationships between the maximum cutting speed and the heat input at each material direction, and between the cutting offset and the heat input were acquired by the empirical data. It was also found that the maximum cutting speed in each material direction has a different value due to material anisotropy of EPS foam sheets. The transient heat transfer phenomenon during the hotwire cutting was simulated by an FE code using the twostep approach. In the global analysis, the effects of cut-
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Table 7 Dimensional accuracy of test parts Spanner Shape
Clover punch shape
Measurement Measurement Measurement Measurement I (mm) II (mm) I (mm) II (mm) Considering offset 1.14 (%) No considering offset (%)
1.55
1.05 Average=1.10 4.39 Average=2.97
Fig. 23. Geometry and the measured dimensions of test parts. (a) Open-ended spanner shape; (b) Clover punch shape.
Table 5 Cutting conditions of test parts
Heat input (watt) Cutting speed (mm/s)
Spanner shape
Clover punch shape
1.19 50
1.19 50
Clover punch shape
Measurement Measurement Measurement Measurement I (mm) II (mm) I (mm) II (mm) CAD 150 Expected 149.39 Dimension (E.D.) Measured 147.68 Dimension (M.D.)
2.60
0.61 Average=0.40 1.77 Average=1.92
ture distribution of EPS foam sheet during hotwire cutting was computed to estimate the melted region and to predict the size of thermal front when the hotwire loses its stiffness. The optimal operational condition such as the upper bound of the cutting speed and the lower bound of the heat input was proposed by considering the results of the local analysis and the experiment. Based on the results, a spanner shape and a clover punch shape were fabricated by VLM-S. The empirical equation for cutting offset, Eq. (5) for the case of 50 mm/s of cutting speed, was used to estimate the expected dimensions to predict exactly the dimensional accuracy of VLM-S parts. In future works, an additional experiment and simulation in case of three-dimensional cutting should be further performed in order to reflect the melted width and optimal cutting condition on the input data of VLMS. In addition, an additional module should be required to compensate for the melted width in VLM-Slicer that is the software to generate the input of VLM-S.
References
Table 6 Expected and measured dimensions of the test parts Spanner Shape
0.2
18.2 17.59
32.98 32.37
52.43 51.82
17.4
32.3
51.5
ting parameters and the mesh size on temperature distribution of EPS foam sheet were studied, and also important information such as the size of analysis zone, the mesh size and time interval for numerical integration was acquired. In the local analysis, the accurate tempera-
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