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Numerical simulation of deposition process for a new 3DP printhead design
Journal of Materials Processing Technology 161 (2005) 509–515
Numerical simulation of deposition process for a new 3DP printhead design Emanuel Sachsa , Enrico Vezzettib,∗ b
a Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Politecnico di Torino Dipartimento di Sistemi di Produzione ed Economia dell’Azienda, Croso Duca Degli Abruzzi 24, Torino 10129, Italy
Received 25 October 2002; received in revised form 13 January 2004; accepted 22 July 2004
Abstract The three-dimensional printing technology (3DPTM ) is a rapid prototyping process in which powdered material is deposited in layers and selectively joined with binder. The most common deposition method used in this rapid prototyping approach is the drop on demand. However, continuous jet deposition results in an order of magnitude increase in printing speed and that it is gaining popularity. A key component of a continuous deposition printhead is the catcher, which collects droplets that are not meant to hit the powder bed. Current catching systems face problems such as trapped air, and crystallisation that result in unwanted droplets hitting the powder bed. This work looks at redesigning the catcher, and addresses new control algorithms required for proper binder deposition. A mathematical model for binder flight trajectory is developed and validated by experiment. © 2004 Elsevier B.V. All rights reserved. Keywords: Rapid prototyping; Process simulation; Three-dimensional printing
1. Introduction Three-dimensional printing is an innovative rapid prototyping process that reduces the lead-time of prototype parts or machine tooling from weeks to days. Currently, prototype and tooling fabrication involve many labour-intensive, slow and costly steps, which 3D printing accomplishes in one step. The time and cost savings afforded by 3D printing have revolutionised prototyping and manufacturing [1]. The 3D printing process fabricates a part from a CAD model specified in three dimensions. A slicing algorithm divides the part into cross sectional layers, approximately 175 m thick, and creates detailed information about each cross-section. The 3D printing machine begins the layer building process by depositing a layer of spread powder. It then selectively prints a binder fluid onto the powder bed in the shape of the cross-section of the part, much like an inkjet printer prints a picture on paper. The powder bed is then ∗
Corresponding author.
0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2004.07.090
lowered, and a new layer of powder is spread. This process repeats until the entire part has been printed. After firing, the loose (unprinted) powder is removed. In addition to shortening lead-time, 3D printing can create complex geometries and under cuts which conventional manufacturing techniques cannot (Fig. 1). The droplet deposition process can employ two different technologies: drop on demand or continuous jet printing [2,3]. 1.1. Drop on demand In this process an electrical pulse is applied to the printhead for each drop printed. Typically, the drop is either created by the formation of an evaporative bubble on a small heating element or by the motion of a piezo element. This printing methodology needs a quiescent period between consecutive drops in order to allow the meniscus to reach capillary equilibrium. This limits the droplet generation rates to the range of 3–10 kHz. In order to increase the
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Fig. 1. 3D printing process.
production rate using this technique, multiple nozzles have been used [4]. 1.2. Continuous jet printing Continuous jet printing, on the other hand, continuously discharges a stream of fluid, which is broken into droplets by some form of vibration. Pressurised fluid binder is supplied to the body of the resonator. The system is driven at its resonant frequency by a sinusoidal wave input to two piezoelectric elements bonded to opposite sides of the printhead. The printhead is designed so that the fluid stream breaks into regular droplets inside the charging cell. Printing is accomplished by selectively charging certain droplets as they pass through the charging cell (Fig. 2). To print a droplet, a zero voltage (print signal) is sent to the charging cell. The print signal is synchronised with respect to the fast axis econder pitch. The position at which the droplet
Fig. 2. Components of the continuous printing system.
is printed is determined by a command file that contains the part geometry information. The droplets then pass through a high voltage deflection cell. The system is designed to print in either uni- or bidirectional mode. Bi-directional is faster, but it is also more sensitive to dynamic shift errors. Parts that require higher quality surface finish should be printed in uni directional mode. These two modes are similar to a “draft” and “final” mode in a conventional printer. During the printing phase the machine works in an analog mode. This way the voltage on the charging cell can be continuously varied to allow partial deflection of the droplets. Proportional deflection allows finer control of the droplet deposition location along the slow axis (the axis perpendicular to the main direction of printhead motion). The deflection process steers the droplets to the correct position on the powder bed. Drops that are not required to hit the powder bed are deflected into a catcher [5,6]. The most efficient aspect of this technology is increased droplet generation rates that can range from 60 kHz to 1 MHz.
2. The catching system In drop on demand printing, a droplet is selectively released when a specific nozzle, and consequently the printhead, are located in the right position on the working plane. However, in the continuous jet technology the fluid stream is continuous. The catcher is critical as it helps to achieve a selective printing strategy in the presence of a continuous stream. Two catching systems are in vogue. The first one (Fig. 3) consists of a stainless steel plate that curves away at the bottom. This part works out both as a deflection plate and a droplet catcher. Deflected droplets strike the plate at a shallow angle, follow the plate’s curve, and collect underneath. When the printhead is stationary, binder drips from the plate into a collection trough. When the printhead is moving, drops are flung off the plate as the printhead accelerates at the end of each traverse. The drops collect in troughs at either end of printhead as they travel and travel down the tubes to collection bottles. Binder that is collected in these bottles can then be recycled.
Fig. 3. First catching system that combines both the deflection and the catching components.
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Fig. 5. Binary deflection and proportional one. Fig. 4. Second catching system that separates the deflection and catching components.
The second catching system (Fig. 4) separates the deflection plate and the droplet catcher. Deflected droplets curve past the deflection plate and strike the catcher directly. They collect there until gravity or inertial forces cause them to fall into the collection trough. While this design functions basically in the same way, it has the advantage that the deflection plate never becomes wet with binder. This minimizes the chance of build-up occurring in the deflection cell. The drawback of this design is that it is less compact and has more parts [7]. Both catching systems discussed above have many disadvantages. Each nozzle has its own catcher, and there is only one pump that sucks off the binder from all these catchers. Since all the nozzles are not active at the same time, the pump is typically sucking binder from some catchers and air from the rest. This could affect the working of the pump and cause binder accumulation in some catchers. Another issue is the low cross-section of the individual catchers. Sometimes, with certain binders that have a high level of crystallisation, a partial or total obstruction of an individual catcher could occur and cause unwanted droplets to fall on the powder bed. To overcome these issues, it is proposed that a single catcher be used that is shared by all the nozzles (motivated by the ink jet experience [8]). 2.1. Design of proposed catching system The continuous jet printhead is characterised by two main degrees of freedom. The printhead follows a raster strategy, and covers the entire powder bed by travelling along the first direction, normally considered the X-axis, and then stepping along an orthogonal one (Y) for another run along the main working axis X. Normally the X-axis is called fast axis, because the machine has to travel along this direction quickly, while the step direction Y is called the slow axis because it represents only the secondary movement. Fig. 6a shows the normal configuration of a printhead. Notice that there is gap between the two nozzles, which is dictated by their physical size constraints. Proportional de-
Fig. 6. Standard deflection and modified one for the common catcher system.
flection makes it possible to fine tune droplet deposition along the y-axis between the two nozzles. Each nozzle can typically cover a distance of 1 mm along the y-axis on each side. Fig. 5 shows how proportional deflection gives the printing process a higher level of accuracy (Fig. 5). The configuration shown in Fig. 6a has the nozzles offset along the Y-axis. Clearly it is not possible to operate the printhead in this configuration using just one catcher. One configuration that comes to mind is shown in Fig. 6b. Here, the deflection plates are rotated 90◦ , maintaining the nozzles offsetted along the Y direction. This would make it possible to use a common catcher for all the nozzles. However, the power of having proportional deflection in lost in this configuration, for it is now aligned with the X-axis. In order to retain the advantage of proportional deflection and to have a common catcher for all the nozzles, it is proposed that the printhead be rotated 45◦ (see Fig. 7).
Fig. 7. The 45◦ deflection print head configuration.
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This new configuration imparts an additional velocity to the droplets along the X-axis, which is equal to the component of the deflection velocity. This has an impact on the timing algorithm that controls drop release. To better understand that, droplet flight is examined next section.
3. Droplet flight When a printhead is operating in the configuration shown in Fig. 6a, the only velocity the droplet has along the X-axis is that of the printhead. At some time (t = 0), a droplet is released from the printhead with an initial velocity, u0 , in the Z direction, proportional to the flowrate of binder through the system, and a velocity component, v0 , in the X direction equal to the fast axis speed (typically around 1.5 m/s). After a certain time, referred to as the time of flight (TOF), the droplet will land on the powder bed. During this time, the printhead will have moved a distance, P, and the droplet will have travelled an identical distance, X (Fig. 8). The distance, P, represents the offset of the printhead when the print command needs to be sent if the droplet is to be printed at the position, Pf . This offset is referred to as the “dynamic” shift as the motion of the printhead over the powder bed causes it. However with the proposed configuration (Fig. 7), dynamic shift is a little more involved. 3.1. Control law for new configuration Assume that the droplet leaves the deflection plates with a deflection velocity of v as shown in Fig. 9a and b. Depending on the direction in which the droplet is deflected, the droplet will have a different velocity along the X-axis. Hence, in Fig. 9a, the dynamic shift is higher than in Fig. 9b. Conse-
Fig. 9. (a) 45◦ configuration stream velocity in the positive direction, (b) 45◦ configuration stream velocity in the negative direction.
quently, the drop needs to be released in advance in Fig. 9a. This fact needs to be incorporated into the new printhead droplet release control algorithm. Assuming that the printhead velocity is v0 : If the deflection is along the positive X-axis, the control law for the droplet placement becomes: x = (v0 + vx ) · TOF where vx represents the X component of the deflection velocity. If the deflection is along the negative X-axis, the control law for the droplet placement becomes: x = (v0 − vx ) · TOF where vx represents the X component of the deflection velocity [9]. 3.2. Effects of drag forces
Fig. 8. Flight path of a droplet during printing.
But in order to have a more realistic vision of the entire printing process, and in order to understand which is the relative importance of the drag force on the dynamic shift the friction effect has been introduced in the first mathematical model. Starting from the consideration that the printing process is developed in air, it is important to consider that the drag force is a reactive one and that the magnitude of this phenomena depends on the entity of the force that cause the movement. So in order to find a mathematical formalisation of the drag force, the first step is to consider that the drag force has the same module of the active force, but an opposite direction.
E. Sachs, E. Vezzetti / Journal of Materials Processing Technology 161 (2005) 509–515
Following the formalisation process the droplet shape should be defined. Its geometry could be assumed as a cylinder if the entire stream is modelled, while it looks like a sphere, that maintains always the same dimension and shape during its path, if the attention is focused only on one drop. Working with this last model it is possible to assume that the drag force acts on the sphere in the same way on all the directions. So in relation with the drag force and the electric field the next step is to compute how is the magnitude of the force along the deflection direction and to evaluate the velocity of the droplet in that direction. So the mathematical equations necessary to analyse the droplet trajectory starts from the consideration that the drag force is a reactive force that uses this kind of formalization: 24 6 Cd ≈ + √ + 0.4 Re 1 + Re Re =
uD γair
Fdrag = Cd 21 γair u2
1 2 4 πD
for Re < 2 × 105
where Re is the Reynolds number; u the droplet velocity; γ air the air density (1.58 × 10−5 kg/m3 ); D the droplet diameter (80 m) V d where Fe is the electric force; q the droplet charge; V the deflection voltage (1500 V); d the deflection plates distance. Starting from these two main forces it is possible to compute the droplet trajectory. Considering that Fe is applied along a 45◦ axis, the value of Fe along x and y is: Fe = q
√1 Fe 2
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So dividing the path in different intervals along the z direction, and using some basic rules of the fluid mechanics it is possible to evaluate all the necessary parameters that characterize the droplet placement [10].
4. Experimentation First of all the experimentation has been developed with the standard machine configuration in order to obtain a first validation of the mathematical model developed. Working with a deflection value of 1 mm, with the basic machine configuration, the first test has been run firstly neglecting the presence of the drag force. Looking at the numerical results obtained and comparing the values with those sampled from the machine, it is possible to see that the simulation give a good representation of the real droplet trajectory for a deflection of 1 mm. Then, in order to have a more consistent representation of the real phenomena the drag force has been introduced in a second numerical simulation. The results obtained (Fig. 10, Table 1) in this second test show that the relative weight of the drag force on the dynamic shift and so on the droplet trajectory looks is very low. So this underlines the validity of the mathematical model developed because [11] it has been experimentally demonstrated that the standard continuous stream machine could efficiently controls the droplet release working only with the TOF logic and adding a constant value only in order to compensate the drag force effect. Starting from the good results obtained working with the standard configuration the model has been then applied to the 45◦ printhead set-up. In order to understand which is the relative importance of the drag force on the droplet placement and also in this case two different experimentations have been run. The first series has been developed without the drag force effect printing both in the positive and in the negative side.
Implementing with a numerical method the equations before the drop trajectory formalisation became: 1 Fe − Fdrag ki+1 = ki + uk,i t + 2 m where k = X or Y axis; u the velocity along k direction; m the droplet mass Fe − Fdrag uk,i+1 = uk,i + t m Using this kind of formulation the solution could be found employing an iterative method. But in order to simplify the solution research the droplet trajectory is considered in a discrete space instead of a continuous one. Working with little intervals, along the X and Y axis, thanks to the little drop mass that consequently reduce the drag force, it is possible to assume that the acceleration is constant inside every interval.
Fig. 10. Dynamic shift with drag force. Table 1 Dynamic shift obtained with the drag force model Printhead configuration 90◦
Dynamic shift 1.94 mm
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Fig. 13. X–Y trajectory in the 45◦ configuration. Fig. 11. Dynamic shift with negative deflection and with the drag force effect.
Considering that the drag force is expressed by a non linear equation and so if the effect of the drag force is significant the relation between the variables should be at least quadratic, it is possible to say that the drag force effect is negligible also in this configuration. So starting from these considerations the dynamic shift evaluation and the deflection value along the slow axis y could be obtained with the following formalization: x = vs + √1 vd TOF 2
y= Fig. 12. Dynamic shift with positive deflection and with the drag force effect.
The second series has been run with the drag force effect and also in this case printing in the positive and negative side with a deflection value of 1 mm. Looking at both the results with and without the drag friction effect it is possible to see that the drag force relative weight on the droplet placement depends on the deflection magnitude (Figs. 11 and 12, Table 2). As what has been said in the model presentation paragraph this phenomena is due to the deflection plane inclination that causes the drag force subdivision in two equal components along the slow and the fast axis. In order to get more information about the sensitivity of the mathematical model, different deflection values have been tested. So proceeding with the last experience the movement of the dynamic shift has been analysed in relation with the magnitude of the deflection (Fig. 13). Looking at the results of this last experimentation (Fig. 13), developed considering the friction effect, it is possible to see that the dependence between the X and Y magnitude is driven by a linear equation. Table 2 Dynamic shift values obtained with the drag force model Printhead configuration
Dynamic shift
45◦ + 45◦ −
2.65 mm 1.23 mm
√1 vd TOF 2
x = y + vs TOF in which vd is the deflection velocity; TOF the droplet time of flight; vs the printhead velocity along X With this experiment it is also possible to understand that neglecting the drag force in this machine configuration the worst case causes an error littler than 10 m in the droplet placement. Under this value the error is negligible because it represents the maximum resolution that the machine can reach in relation with the linear encoder that control the movement of the printhead along the fast axis. So also in this case the “control law” that can be used to decide the release point for all the deflection magnitudes in both the sides is always the TOF methodology.
5. Conclusions So analysing the results obtained and considering that the dependence between the X and Y magnitude is driven by a linear equation it is possible to understand that also in 45◦ configuration, as in the standard one, the droplet control system could be driven by the only TOF methodology. This fact is mainly due to the possibility to neglect the friction effect collecting its influence inside the TOF formulation with a constant value. More than this working with this printhead configuration, it is also possible to reduce the range of deflection along the slow axis of a square two factor.
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This effect is also interesting considering the necessity to reduce the distance between every nozzle. This distance represents in fact a problem for the 3D printing process because it is a very strong constraint for the possibility to maintain a certain contiguity in the line building process.
References [1] J. Heinzl, C.H. Hertz, Ink-Jet printing, Adv Electron. Phys. 65 (1985). [2] Johnson, L. Jerome, Principles of Non Impact Printing, Platino Press, 1998. [3] Paranjpe, C. Suresh, Ink Jet Printer with Secondary Cyclically Varying Deflection Field¨, U.S. Patent No. 4,544,930 (Date of Patent 10/1/85).
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[4] A. Curodeau, “Three dimensional printing of ceramic models with accurate surface macro-textures for investment casting of orthopaedic implants”, Master Thesis, MIT 1995. [5] Sachs, Brancazio, High rate, high quality 3D printing through machine design, on-line measurement, and control, Int. J. Mach. Tools Manufac. 38. [6] A. Milner, Time of flight controller and stream position compensation system for the three Dimensional printing process, MIT MS Thesis, 1991. [7] D. Brancazio, Development of a robust electrostatically deflecting printhead for three dimensional printing, MIT MS Thesis, 1991. [8] P. Suresh, Ink jet printer with secondary cyclically varying deflection field, U.S. Patent No 4,544,930 (Date of the Patent 10/1/85). [9] B. Jim, Binder stability and powder/binder interaction in three dimensional printing, PhD Thesis, Mech. Eng. MIT 1/95. [10] M. Potter, Fluid Mechanisms, Ronald Press Co., 1982. [11] W. Paul, Three dimensional printing: a new process to fabricate prototypes directly from CAD Models MS Thesis, 1990, MIT 5/90.
Numerical simulation of deposition process for a new 3DP printhead design Emanuel Sachsa , Enrico Vezzettib,∗ b
a Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Politecnico di Torino Dipartimento di Sistemi di Produzione ed Economia dell’Azienda, Croso Duca Degli Abruzzi 24, Torino 10129, Italy
Received 25 October 2002; received in revised form 13 January 2004; accepted 22 July 2004
Abstract The three-dimensional printing technology (3DPTM ) is a rapid prototyping process in which powdered material is deposited in layers and selectively joined with binder. The most common deposition method used in this rapid prototyping approach is the drop on demand. However, continuous jet deposition results in an order of magnitude increase in printing speed and that it is gaining popularity. A key component of a continuous deposition printhead is the catcher, which collects droplets that are not meant to hit the powder bed. Current catching systems face problems such as trapped air, and crystallisation that result in unwanted droplets hitting the powder bed. This work looks at redesigning the catcher, and addresses new control algorithms required for proper binder deposition. A mathematical model for binder flight trajectory is developed and validated by experiment. © 2004 Elsevier B.V. All rights reserved. Keywords: Rapid prototyping; Process simulation; Three-dimensional printing
1. Introduction Three-dimensional printing is an innovative rapid prototyping process that reduces the lead-time of prototype parts or machine tooling from weeks to days. Currently, prototype and tooling fabrication involve many labour-intensive, slow and costly steps, which 3D printing accomplishes in one step. The time and cost savings afforded by 3D printing have revolutionised prototyping and manufacturing [1]. The 3D printing process fabricates a part from a CAD model specified in three dimensions. A slicing algorithm divides the part into cross sectional layers, approximately 175 m thick, and creates detailed information about each cross-section. The 3D printing machine begins the layer building process by depositing a layer of spread powder. It then selectively prints a binder fluid onto the powder bed in the shape of the cross-section of the part, much like an inkjet printer prints a picture on paper. The powder bed is then ∗
Corresponding author.
0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2004.07.090
lowered, and a new layer of powder is spread. This process repeats until the entire part has been printed. After firing, the loose (unprinted) powder is removed. In addition to shortening lead-time, 3D printing can create complex geometries and under cuts which conventional manufacturing techniques cannot (Fig. 1). The droplet deposition process can employ two different technologies: drop on demand or continuous jet printing [2,3]. 1.1. Drop on demand In this process an electrical pulse is applied to the printhead for each drop printed. Typically, the drop is either created by the formation of an evaporative bubble on a small heating element or by the motion of a piezo element. This printing methodology needs a quiescent period between consecutive drops in order to allow the meniscus to reach capillary equilibrium. This limits the droplet generation rates to the range of 3–10 kHz. In order to increase the
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Fig. 1. 3D printing process.
production rate using this technique, multiple nozzles have been used [4]. 1.2. Continuous jet printing Continuous jet printing, on the other hand, continuously discharges a stream of fluid, which is broken into droplets by some form of vibration. Pressurised fluid binder is supplied to the body of the resonator. The system is driven at its resonant frequency by a sinusoidal wave input to two piezoelectric elements bonded to opposite sides of the printhead. The printhead is designed so that the fluid stream breaks into regular droplets inside the charging cell. Printing is accomplished by selectively charging certain droplets as they pass through the charging cell (Fig. 2). To print a droplet, a zero voltage (print signal) is sent to the charging cell. The print signal is synchronised with respect to the fast axis econder pitch. The position at which the droplet
Fig. 2. Components of the continuous printing system.
is printed is determined by a command file that contains the part geometry information. The droplets then pass through a high voltage deflection cell. The system is designed to print in either uni- or bidirectional mode. Bi-directional is faster, but it is also more sensitive to dynamic shift errors. Parts that require higher quality surface finish should be printed in uni directional mode. These two modes are similar to a “draft” and “final” mode in a conventional printer. During the printing phase the machine works in an analog mode. This way the voltage on the charging cell can be continuously varied to allow partial deflection of the droplets. Proportional deflection allows finer control of the droplet deposition location along the slow axis (the axis perpendicular to the main direction of printhead motion). The deflection process steers the droplets to the correct position on the powder bed. Drops that are not required to hit the powder bed are deflected into a catcher [5,6]. The most efficient aspect of this technology is increased droplet generation rates that can range from 60 kHz to 1 MHz.
2. The catching system In drop on demand printing, a droplet is selectively released when a specific nozzle, and consequently the printhead, are located in the right position on the working plane. However, in the continuous jet technology the fluid stream is continuous. The catcher is critical as it helps to achieve a selective printing strategy in the presence of a continuous stream. Two catching systems are in vogue. The first one (Fig. 3) consists of a stainless steel plate that curves away at the bottom. This part works out both as a deflection plate and a droplet catcher. Deflected droplets strike the plate at a shallow angle, follow the plate’s curve, and collect underneath. When the printhead is stationary, binder drips from the plate into a collection trough. When the printhead is moving, drops are flung off the plate as the printhead accelerates at the end of each traverse. The drops collect in troughs at either end of printhead as they travel and travel down the tubes to collection bottles. Binder that is collected in these bottles can then be recycled.
Fig. 3. First catching system that combines both the deflection and the catching components.
E. Sachs, E. Vezzetti / Journal of Materials Processing Technology 161 (2005) 509–515
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Fig. 5. Binary deflection and proportional one. Fig. 4. Second catching system that separates the deflection and catching components.
The second catching system (Fig. 4) separates the deflection plate and the droplet catcher. Deflected droplets curve past the deflection plate and strike the catcher directly. They collect there until gravity or inertial forces cause them to fall into the collection trough. While this design functions basically in the same way, it has the advantage that the deflection plate never becomes wet with binder. This minimizes the chance of build-up occurring in the deflection cell. The drawback of this design is that it is less compact and has more parts [7]. Both catching systems discussed above have many disadvantages. Each nozzle has its own catcher, and there is only one pump that sucks off the binder from all these catchers. Since all the nozzles are not active at the same time, the pump is typically sucking binder from some catchers and air from the rest. This could affect the working of the pump and cause binder accumulation in some catchers. Another issue is the low cross-section of the individual catchers. Sometimes, with certain binders that have a high level of crystallisation, a partial or total obstruction of an individual catcher could occur and cause unwanted droplets to fall on the powder bed. To overcome these issues, it is proposed that a single catcher be used that is shared by all the nozzles (motivated by the ink jet experience [8]). 2.1. Design of proposed catching system The continuous jet printhead is characterised by two main degrees of freedom. The printhead follows a raster strategy, and covers the entire powder bed by travelling along the first direction, normally considered the X-axis, and then stepping along an orthogonal one (Y) for another run along the main working axis X. Normally the X-axis is called fast axis, because the machine has to travel along this direction quickly, while the step direction Y is called the slow axis because it represents only the secondary movement. Fig. 6a shows the normal configuration of a printhead. Notice that there is gap between the two nozzles, which is dictated by their physical size constraints. Proportional de-
Fig. 6. Standard deflection and modified one for the common catcher system.
flection makes it possible to fine tune droplet deposition along the y-axis between the two nozzles. Each nozzle can typically cover a distance of 1 mm along the y-axis on each side. Fig. 5 shows how proportional deflection gives the printing process a higher level of accuracy (Fig. 5). The configuration shown in Fig. 6a has the nozzles offset along the Y-axis. Clearly it is not possible to operate the printhead in this configuration using just one catcher. One configuration that comes to mind is shown in Fig. 6b. Here, the deflection plates are rotated 90◦ , maintaining the nozzles offsetted along the Y direction. This would make it possible to use a common catcher for all the nozzles. However, the power of having proportional deflection in lost in this configuration, for it is now aligned with the X-axis. In order to retain the advantage of proportional deflection and to have a common catcher for all the nozzles, it is proposed that the printhead be rotated 45◦ (see Fig. 7).
Fig. 7. The 45◦ deflection print head configuration.
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This new configuration imparts an additional velocity to the droplets along the X-axis, which is equal to the component of the deflection velocity. This has an impact on the timing algorithm that controls drop release. To better understand that, droplet flight is examined next section.
3. Droplet flight When a printhead is operating in the configuration shown in Fig. 6a, the only velocity the droplet has along the X-axis is that of the printhead. At some time (t = 0), a droplet is released from the printhead with an initial velocity, u0 , in the Z direction, proportional to the flowrate of binder through the system, and a velocity component, v0 , in the X direction equal to the fast axis speed (typically around 1.5 m/s). After a certain time, referred to as the time of flight (TOF), the droplet will land on the powder bed. During this time, the printhead will have moved a distance, P, and the droplet will have travelled an identical distance, X (Fig. 8). The distance, P, represents the offset of the printhead when the print command needs to be sent if the droplet is to be printed at the position, Pf . This offset is referred to as the “dynamic” shift as the motion of the printhead over the powder bed causes it. However with the proposed configuration (Fig. 7), dynamic shift is a little more involved. 3.1. Control law for new configuration Assume that the droplet leaves the deflection plates with a deflection velocity of v as shown in Fig. 9a and b. Depending on the direction in which the droplet is deflected, the droplet will have a different velocity along the X-axis. Hence, in Fig. 9a, the dynamic shift is higher than in Fig. 9b. Conse-
Fig. 9. (a) 45◦ configuration stream velocity in the positive direction, (b) 45◦ configuration stream velocity in the negative direction.
quently, the drop needs to be released in advance in Fig. 9a. This fact needs to be incorporated into the new printhead droplet release control algorithm. Assuming that the printhead velocity is v0 : If the deflection is along the positive X-axis, the control law for the droplet placement becomes: x = (v0 + vx ) · TOF where vx represents the X component of the deflection velocity. If the deflection is along the negative X-axis, the control law for the droplet placement becomes: x = (v0 − vx ) · TOF where vx represents the X component of the deflection velocity [9]. 3.2. Effects of drag forces
Fig. 8. Flight path of a droplet during printing.
But in order to have a more realistic vision of the entire printing process, and in order to understand which is the relative importance of the drag force on the dynamic shift the friction effect has been introduced in the first mathematical model. Starting from the consideration that the printing process is developed in air, it is important to consider that the drag force is a reactive one and that the magnitude of this phenomena depends on the entity of the force that cause the movement. So in order to find a mathematical formalisation of the drag force, the first step is to consider that the drag force has the same module of the active force, but an opposite direction.
E. Sachs, E. Vezzetti / Journal of Materials Processing Technology 161 (2005) 509–515
Following the formalisation process the droplet shape should be defined. Its geometry could be assumed as a cylinder if the entire stream is modelled, while it looks like a sphere, that maintains always the same dimension and shape during its path, if the attention is focused only on one drop. Working with this last model it is possible to assume that the drag force acts on the sphere in the same way on all the directions. So in relation with the drag force and the electric field the next step is to compute how is the magnitude of the force along the deflection direction and to evaluate the velocity of the droplet in that direction. So the mathematical equations necessary to analyse the droplet trajectory starts from the consideration that the drag force is a reactive force that uses this kind of formalization: 24 6 Cd ≈ + √ + 0.4 Re 1 + Re Re =
uD γair
Fdrag = Cd 21 γair u2
1 2 4 πD
for Re < 2 × 105
where Re is the Reynolds number; u the droplet velocity; γ air the air density (1.58 × 10−5 kg/m3 ); D the droplet diameter (80 m) V d where Fe is the electric force; q the droplet charge; V the deflection voltage (1500 V); d the deflection plates distance. Starting from these two main forces it is possible to compute the droplet trajectory. Considering that Fe is applied along a 45◦ axis, the value of Fe along x and y is: Fe = q
√1 Fe 2
513
So dividing the path in different intervals along the z direction, and using some basic rules of the fluid mechanics it is possible to evaluate all the necessary parameters that characterize the droplet placement [10].
4. Experimentation First of all the experimentation has been developed with the standard machine configuration in order to obtain a first validation of the mathematical model developed. Working with a deflection value of 1 mm, with the basic machine configuration, the first test has been run firstly neglecting the presence of the drag force. Looking at the numerical results obtained and comparing the values with those sampled from the machine, it is possible to see that the simulation give a good representation of the real droplet trajectory for a deflection of 1 mm. Then, in order to have a more consistent representation of the real phenomena the drag force has been introduced in a second numerical simulation. The results obtained (Fig. 10, Table 1) in this second test show that the relative weight of the drag force on the dynamic shift and so on the droplet trajectory looks is very low. So this underlines the validity of the mathematical model developed because [11] it has been experimentally demonstrated that the standard continuous stream machine could efficiently controls the droplet release working only with the TOF logic and adding a constant value only in order to compensate the drag force effect. Starting from the good results obtained working with the standard configuration the model has been then applied to the 45◦ printhead set-up. In order to understand which is the relative importance of the drag force on the droplet placement and also in this case two different experimentations have been run. The first series has been developed without the drag force effect printing both in the positive and in the negative side.
Implementing with a numerical method the equations before the drop trajectory formalisation became: 1 Fe − Fdrag ki+1 = ki + uk,i t + 2 m where k = X or Y axis; u the velocity along k direction; m the droplet mass Fe − Fdrag uk,i+1 = uk,i + t m Using this kind of formulation the solution could be found employing an iterative method. But in order to simplify the solution research the droplet trajectory is considered in a discrete space instead of a continuous one. Working with little intervals, along the X and Y axis, thanks to the little drop mass that consequently reduce the drag force, it is possible to assume that the acceleration is constant inside every interval.
Fig. 10. Dynamic shift with drag force. Table 1 Dynamic shift obtained with the drag force model Printhead configuration 90◦
Dynamic shift 1.94 mm
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Fig. 13. X–Y trajectory in the 45◦ configuration. Fig. 11. Dynamic shift with negative deflection and with the drag force effect.
Considering that the drag force is expressed by a non linear equation and so if the effect of the drag force is significant the relation between the variables should be at least quadratic, it is possible to say that the drag force effect is negligible also in this configuration. So starting from these considerations the dynamic shift evaluation and the deflection value along the slow axis y could be obtained with the following formalization: x = vs + √1 vd TOF 2
y= Fig. 12. Dynamic shift with positive deflection and with the drag force effect.
The second series has been run with the drag force effect and also in this case printing in the positive and negative side with a deflection value of 1 mm. Looking at both the results with and without the drag friction effect it is possible to see that the drag force relative weight on the droplet placement depends on the deflection magnitude (Figs. 11 and 12, Table 2). As what has been said in the model presentation paragraph this phenomena is due to the deflection plane inclination that causes the drag force subdivision in two equal components along the slow and the fast axis. In order to get more information about the sensitivity of the mathematical model, different deflection values have been tested. So proceeding with the last experience the movement of the dynamic shift has been analysed in relation with the magnitude of the deflection (Fig. 13). Looking at the results of this last experimentation (Fig. 13), developed considering the friction effect, it is possible to see that the dependence between the X and Y magnitude is driven by a linear equation. Table 2 Dynamic shift values obtained with the drag force model Printhead configuration
Dynamic shift
45◦ + 45◦ −
2.65 mm 1.23 mm
√1 vd TOF 2
x = y + vs TOF in which vd is the deflection velocity; TOF the droplet time of flight; vs the printhead velocity along X With this experiment it is also possible to understand that neglecting the drag force in this machine configuration the worst case causes an error littler than 10 m in the droplet placement. Under this value the error is negligible because it represents the maximum resolution that the machine can reach in relation with the linear encoder that control the movement of the printhead along the fast axis. So also in this case the “control law” that can be used to decide the release point for all the deflection magnitudes in both the sides is always the TOF methodology.
5. Conclusions So analysing the results obtained and considering that the dependence between the X and Y magnitude is driven by a linear equation it is possible to understand that also in 45◦ configuration, as in the standard one, the droplet control system could be driven by the only TOF methodology. This fact is mainly due to the possibility to neglect the friction effect collecting its influence inside the TOF formulation with a constant value. More than this working with this printhead configuration, it is also possible to reduce the range of deflection along the slow axis of a square two factor.
E. Sachs, E. Vezzetti / Journal of Materials Processing Technology 161 (2005) 509–515
This effect is also interesting considering the necessity to reduce the distance between every nozzle. This distance represents in fact a problem for the 3D printing process because it is a very strong constraint for the possibility to maintain a certain contiguity in the line building process.
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[4] A. Curodeau, “Three dimensional printing of ceramic models with accurate surface macro-textures for investment casting of orthopaedic implants”, Master Thesis, MIT 1995. [5] Sachs, Brancazio, High rate, high quality 3D printing through machine design, on-line measurement, and control, Int. J. Mach. Tools Manufac. 38. [6] A. Milner, Time of flight controller and stream position compensation system for the three Dimensional printing process, MIT MS Thesis, 1991. [7] D. Brancazio, Development of a robust electrostatically deflecting printhead for three dimensional printing, MIT MS Thesis, 1991. [8] P. Suresh, Ink jet printer with secondary cyclically varying deflection field, U.S. Patent No 4,544,930 (Date of the Patent 10/1/85). [9] B. Jim, Binder stability and powder/binder interaction in three dimensional printing, PhD Thesis, Mech. Eng. MIT 1/95. [10] M. Potter, Fluid Mechanisms, Ronald Press Co., 1982. [11] W. Paul, Three dimensional printing: a new process to fabricate prototypes directly from CAD Models MS Thesis, 1990, MIT 5/90.