Simulation Mechanism Development for Additive Manufacturing

Simulation Mechanism Development for Additive Manufacturing

Available online at www.sciencedirect.com ScienceDirect Materials Today: Proceedings 4 (2017) 7270–7278 www.materialstoday.com/proceedings ICAAMM-2...

616KB Sizes 73 Downloads 166 Views

Available online at www.sciencedirect.com

ScienceDirect Materials Today: Proceedings 4 (2017) 7270–7278

www.materialstoday.com/proceedings

ICAAMM-2016

Simulation Mechanism Development for Additive Manufacturing Anupam Pandita*, Ravi Sekharb, Ram K. Revanurc a,b c

Symbiosis Institute of Technology (SIT),Symbiosis International University (SIU),Lavale, Pune - 412 115, Maharashtra State, India Manager, Software Development, Renishaw India, Raisoni Industrial Estate,Mann,Mulshi,Pune-411057,Maharashtra State, India

Abstract Selective laser melting process is an emerging additive manufacturing technology with promising presence in the medical (orthopaedics) industry as well as in the aerospace, high technology engineering and electronics sectors. Published literature shows that past research has been focused mainly on additive manufacturing of high quality parts depending on correct selection of material parameters, laser parameters for each point exposure and correct orientations of the part and its supports. However, there is a pressing need to simulate the build chamber of a selective laser melting machine, so that a close enough approximation of the actual build process could be obtained. In the present work, a number of experiments have been conducted to study the chamber atmosphere variability inside the build chamber during vacuum creation and inert/breathing cycles. Effect of other devices such as wiper, elevator, pumps andvalves are also studied. The obtained experimental data has been utilised to derive regression equations correlating the chamber pressure and oxygen levels with chamber build times under different conditions of peripheral devices. Subsequently, these equations were modelled to obtain the simulation model. Confirmation experiments were carried out to validate the simulation results. This work enables the end user to view the real time simulation of vacuum creation and inert gas environment building process. Values of parameters such as build chamber pressure, oxygen levels and conditions of other devices can be seen in real time. Thus, the developed simulation saves precious time and resources of an additive manufacturing user and enables him to make an informed decision by providing him the exact parameter values suitable for his manufacturing needs. © 2017 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility ofthe Committee Members of International Conference on Advancements in Aeromechanical Materials for Manufacturing (ICAAMM-2016). Keywords:Additive manufacturing; Selective Laser Melting; curve fitting; PLC simulator

* Corresponding author. Tel.: +91-8308578903 E-mail address:[email protected]

2214-7853© 2017 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility ofthe Committee Members of International Conference on Advancements in Aeromechanical Materials for Manufacturing (ICAAMM-2016).

Anupam Pandit,Ravi Sekhar & Ram K. Revanur / Materials Today: Proceedings 4 (2017) 7270–7278

7271

1. Introduction The additive manufacturing virtual machine simulator marks the beginning of an era of computer simulation for the additive manufacturing machine hardware. The best way to understand a complex mechanical process is nothing but simulation. Simulation has always been the answer to any process which is too expensive or dangerous to be performed. Virtual machine is interactive software which gives a feeling to the user as if he is interacting with the actual machine. Just as in the aviation industry, where it’s too risky and expensive for newer pilots to train in actual airplanes, the solution is flight simulator, similarly metal additive manufacturing machines are too expensive for researchers and new developers to train on. Hence the virtual machine simulator is the solution to this. Computer simulation is basically divided into three stages. The first stage is designing a model of the system. Second stage is to use a computer to execute the model and finally the last stage is analysing the output. The principle of simulation is “learning by doing” i.e. to understand a particular system, a model must be made first and then operate the model. Simulations have helped in improving researcher understanding of basic working of a system and investigate effects of problems that may occur during their research work [1]. Nomenclature AM SLM GUI

Additive Manufacturing Selective Laser Melting Graphical User Interface

The simulation mechanism talked upon in this research article is based on the metal selective laser melting method of AM which is a type of powder bed fusion technique [2]. SLM has emerged as the most widespread form of AM which has proved extremely efficient in producing complex geometries using metal powders [3,4,5,6].SLM process is digitally driven, directed from sliced 3D CAD data, in layer thicknesses ranging from 20 µm to 100 µm that form 2D cross sections. The process then builds the part by distributing an even layer of metallic powder using a recoater, then fusing each layer in turn under a tightly controlled inert atmosphere. The high energy and localized heat input produces a highly efficient thermal cycling of melting and solidification leading in formation of a microstructure with unique properties [7, 8,9]. 1.1. Stages in SLM SLM is basically a three stage process as shown in figure 1a.To begin a SLM process, vacuum is created to reduce the quantity of oxygen inside the chamber. High level of oxygen leads to oxidation of metal powders, hence vacuum creation is a vital step in SLM to reduce the risk of reaction between oxygen and metal powder inside the chamber. The vacuumed chamber is then filled with an inert gas, preferably argon. Argon being one of the cheapest and widely available noble gas is the most preferred by SLM manufacturers. A perfect inert gas environment helps in providing a moisture free environment with low oxygen levels inside the chamber for the fabrication process to be carried out. The fabrication process cannot be carried out by just creating an inert gas environment; the pressure inside the chamber needs to be kept between a particular range and at the same timeoxygen level needs to be below a particular level. This sub stage is termed as chamber stabilization. The pressure range and oxygen level varies for different metal powders. The above three sub stages form the first stage termed as the pre fabrication process. After the chamber is stabilized, the process moves to the next stage termed as the fabrication process. The fabrication process consists of four sub stages namely powder dosing, wiper movement, Laser firing and elevator movement. Powder dosing is the spreading of metal powder onto the substrate with the help of a wiper. Powder is deposited into the chamber from a powder hopper which is spread onto the substrate by a moving wiper, also termed as a recoater.After the powder is spread laser fire onto the metal powder layer melting it at the required co-ordinates. The laser is fed with a .stl file which has sliced cad model of the required geometry enabling it to fire at the coordinates required to generate the geometry. As the melting of a layer is over, the elevator moves down by a distance equal to the layer thickness of the geometry. Another layer of powder is spread on top of the previous layer by the

7272

Anupam Pandit,Ravi Sekhar & Ram K. Revanur / Materials Today: Proceedings 4 (2017) 7270–7278

wiper.The laser fires again melting the second layer of powder. The melted powder solidifies and fuses with the previously melted layer. This fusion of melted layers leads to the production of a homogeneous part. The support structures are removed and the end product is finally heat treated to obtain the finished part [10].

Fig. 1. (a) Stages in SLM (b) Renishaw AM 250

1.2. Need for Machine Simulation Fabrication process stage has been of major concern among researchers leading to development of several commercial softwares to simulate this stage. Renishaw Quantum,Materialize 3-Matic, Stratasys Insight, 3D-Sim are some of the major softwares available for fabrication process simulation. At the same time, pre fabrication process has been neglected, even though it has a vital role in producing quality parts. An optimized pre fabrication process means  Less time consumption for chamber stabilization leading to reduction in overall cost of the entire process  Increased quality of end products due to reduced fluctuations in pressure and oxygen levels inside the chamber.  Use of optimum parameters lead to reduced chances of machine failure. Obtaining an optimized pre fabrication process is a humongous task. Developers need to control the machine by coding in such a way that the chamber stabilizes within minimum time possible and at the same time maintain the chamber pressure and oxygen levels within the acceptable range for the fabrication process to be carried out. Several scenarios such as device failures, machine response to maximum and minimum set points, power failure etc. need to be handled by the developers in their code. For all this to be achieved, developers need to run several tests on the actual machine to visualize the response of the machine with their code change. Running the actual machine several times to check for minor code changes lead to enormous amount of power and time consumption. Large players like Renishaw have hundreds of developers working on different stages of the process. Being an extremely expensive machine, limited number of machines are available for testing purpose leading to developers queuing up to perform the tests. This leads to enormous waste of time, huge power consumption and heavy cost incurred due to enormous inert gas consumption. Repeated testing on machine also affects the health of the machine leading to reduced performance. The solution to all these problems is the AM Machine Simulator.

Anupam Pandit,Ravi Sekhar & Ram K. Revanur / Materials Today: Proceedings 4 (2017) 7270–7278

7273

1.3 Process Flow Model The actual process consists of the following steps. Any change in the UI is detected by the control logic(developer’s code) that commands the abstract (PLC).The PLC in turn commands the machine and obtains the feedback from the machine.This feedback is then updated back to the UI through the control logic. Similar to the actual process flow, in the simulation process flow,instead of the control logic commanding the PLC, it commands a PLC simulator and gets the feedback from a machine simulator which is then updated in the UI through the control logic shown in figure 2a.

Fig. 2. (a) Process FlowModel(b) Function of different devices in an AM machine

2. Experimentation Details Experiments have been carried out on the Renishaw AM250 SLM machine shown in figure 1b. The machine specifications have been shown in table 1.There are basically four devices which are used to obtain the perfect inert gas environment and a stabilized chamber. They are vacuum pump, gas pump, vent valve and recirculation pump. The functions of the devices have been shown in figure 2(b). The pressure and oxygen levels inside the chamber vary during the operation of the devices. Experiments were conducted to understand the variation in pressure and oxygen behaviour. The major effect on the chamber pressure and oxygen levels is due to the vacuum pump and gas pump. The vent valve is basically used as a pressure relief valve and recirculation pump is used to circulate the gas from the bottom of the chamber to the top of the chamber, hence both these devices do not play a major role in chamber stabilization. Hence the effects of vacuum pump and gas pump were studied to develop the simulation model. The following steps were followed to conduct the experiments: AM machine was switched on.  The alarm status for the vacuum pump on the GUI was checked to be healthy.  Vacuum pump isolation valve was opened.  Vent valve was closed.  Recirculation pump was turned off.  Vacuum pump was switched ON.  The chamber pressure and oxygen concentration were observed on the GUI.  Vacuum pump was switched off when the chamber pressure reached to -500mbar.  Vacuum pump isolation valve was closed.  Keeping the vent valve closed and recirculation pump off, the gas pump was turned on.  The chamber pressure and oxygen concentration were again observed on the GUI.  Gas pump was switched off when the chamber pressure reached back to 0mbar from -500mbar.  Logs developed by the GUI showing the variation in chamber pressure and oxygen concentration with respect to time were documented and saved for analysis purpose.

7274

Anupam Pandit,Ravi Sekhar & Ram K. Revanur / Materials Today: Proceedings 4 (2017) 7270–7278 Table1. Specifications of Renishaw AM250 Machine Specifications Manufacturer Renishaw Model AM250 Technology SLM Printing Material Metal Build Size 250 X 250 X 300mm Build Volume 18.8lit MinimumLayerThickness 0.02mm Power Input 230V, 1Ph and 16A Power Output 200 V Printer Dimensions 1700 X 800 X 2025 mm

Table 2. Experimental Layout Device Status

No. of Readings Vacuum Pump

Gas Pump

Vent Valve

Recirculation Pump

115

On

Off

Closed

Off

56

Off

On

Closed

Off

2.1 Experimental Layout The table 2 shows the experimental layout followed for conducting the experiments. It can be seen that during the operation of the vacuum pump or the gas pump, rest of the devices were kept closed. This assured that the variation in pressure and oxygen concentration inside the chamber was caused due to the particular device in operation. The vacuum pump was switched on when the chamber pressure was 0 mbar and switched off at -500 mbar. Negative sign shows that the pressure is below atmospheric pressure. The vacuum pump maximum pressure limit is -550 mbar, hence the reading for vacuum pump was taken up to -500 mbar so that the simulation model could be developed for the near to maximum cut off limit of the vacuum pump. This would enable the user of the simulator to get an approximate estimation of the pressure and oxygen concentration variation caused due to the vacuum pump up to its maximum cut off limit. The gas pump was switched on at -500 mbar and closed at 0 mbar i.e. when the chamber equalled the atmospheric pressure. 3. Results and Discussions 3.1 Observation The experiments conducted as per the layout was documented and graphs plotted showing the variation of pressure and oxygen concentration with respect to time as shown in the figures 3(a) & (b), figures 4(a) &(b) figures 5(a) &(b) and figures 6(a) &(b) .The below graphs show the trend in which the pressure and oxygen vary per second when the vacuum pump and gas pump is switched on. The graphs also show the time taken by the vacuum pump to create vacuum upto -500 mbar and the time taken by the gas pump to pump in argon inside the chamber to bring the chamber pressure back to the atmospheric pressure. It is observed that the pressure and oxygen in the case of vacuum pump drops gradually forming a curve. But when the gas pump is switched on, the pressure gradually increases to 0 mbar but the oxygen concentration increases. This is because the argon gas pumped has a composition of 95% argon and 5% oxygen. Oxygen is used in small amounts as an addition to argon because it reduces the surface tension of the molten metal during the fabrication process.

Anupam Pandit,Ravi Sekhar & Ram K. Revanur / Materials Today: Proceedings 4 (2017) 7270–7278 200000

Time (ms)

0 0

20000

40000

60000

80000

100000

180000

120000

160000

Top Oxygen

Oxygen (ppm)

-100

Pressure (mbar)

7275

-200

-300

140000

Bottom Oxygen

120000 100000 80000 60000

-400 40000 20000

-500

0 0

-600

20000

40000

60000

Time (ms)

80000

100000

120000

Fig.3. (a) Pressure vs. time (Vacuum Pump) (b) Oxygen Concentration vs. time (Vacuum Pump)

Time(ms)

200000 180000

0 0

10000

20000

30000

40000

50000

60000 160000

Oxygen (ppm)

Pressure (mbar)

-100

-200

-300

140000 120000 100000

TopOxygen

80000

BottomOxygen

60000 40000

-400

20000 -500

0 0

10000

20000

30000

40000

50000

60000

Time (ms)

-600

Fig.4. (a) Pressure vs. time (Gas Pump) (b) Oxygen Concentration vs. time (Gas Pump)

3.2 Curve fitting of output variables By using the principles of curve fitting [12], equations have been deduced predicting the variations in pressure and oxygen. Microsoft excel trend line option was used to deduce the curve fitting equation. For each device in operation, best fit curve was visualized in excel. The equations deduced have been used to obtain the simulation model. The equations obtained have been shown below. The output has been compared with the actual output obtained. For every reading the percentage error has been calculated. The percentage error has been then summed up and divided by the number of readings to obtain the average percentage error for each experiment. The R2 values have been mentioned on each graph and the R2values indicate that the simulated models are very good approximation. The values of R2 are acceptable as they lie between 90% to 100%. 1.

Vacuum Pump

y  3 *10

11

y  4 *10

12

x

3

x

3

y   3 * 1 0  1 3 x 3  9 * 1 0  8 x 2  0 .0 1 0 7 x

 10

5

 10

5

(Pressure)

(Top Oxygen) (Bottom Oxygen)

x

2

 1 .6 2 2 x  1 8 5 2 0 3

x

2

 1 .9 6 x  1 4 6 5 4 8

y   2 *10  8 x 2  0.0096 x  492.32 (Pressure) 2. Gas Pump (Top Oxygen) y   8 *10  6 x 2  1.8606 x  94768 10 3 5 2 y  9 *10 x  5 * 1 0 x  2 .5 7 5 x  2 8 2 8 1 (Bottom Oxygen)

7276

Anupam Pandit,Ravi Sekhar & Ram K. Revanur / Materials Today: Proceedings 4 (2017) 7270–7278

200000

Time (ms)

0 0

20000

40000

60000

80000

100000

Top O2: y = ‐3E‐11x3 + 1E‐05x2 ‐ 1.622x + 185203 R² = 0.9999 Top O2: Avg % Error = 3.24                  Max % Error=10.35 

180000

120000

160000

Pressure Simulator Pressure

-200

Poly. (Pressure) -300

Avg % Error = 3.94 Max % Error = 32.12

Oxygen (ppm)

Pressure (mbar)

-100

Top Oxygen Bottom Oxygen Top Simulator

140000 120000

Bottom Simulator

100000

Poly. (Top Oxygen) 80000

Bottom O2: Avg %Error =    4.09                        Max %Error = 18.44 

60000

-400

Poly. (Bottom Oxygen)

40000

Bottom O2: y = ‐4E‐12x3 + 1E‐05x2 ‐ 1.96x + 146458 R² = 0.9972

20000

-500

Pressure: y = ‐3E‐13x3 + 9E‐08x2 ‐ 0.0107x R² = 0.9987

-600

0 0

20000

40000

60000

Time (ms)

80000

100000

120000

Fig.5. (a) Pressure vs. Simulator Pressure (Vacuum Pump) (b) Oxygen Concentration vs. Simulator O2 Conc. (Vacuum Pump)

Time(ms)

TopOxygen

200000

Top O2: y = ‐8E‐06x2 + 1.8606x + 94768 R² = 0.9999

180000

0 0

10000

20000

30000

40000

50000

60000

Pressure (mbar)

-100

-200

Pressure Simulator Pressure

-300

Poly. (Pressure)

Oxygen (ppm)

160000

Avg. %Error = 3.52 Max %Error = 30.63

-500

Simulator TOP

Top O2: Avg. % Error = 0.177                 Max % Error = 0.483 

140000

Simulator Bottom

120000

Poly. (TopOxygen)

100000

Poly. (BottomOxygen)

80000 60000

-400

BottomOxygen

Bottom O2: Avg % Error = 5.112 Max % Error = 25.748

40000

Bottom O2: y = ‐9E‐10x3 + 5E‐05x2 + 2.575x + 28281 R² = 0.992

20000

Pressure: y = ‐2E‐08x2 + 0.0096x ‐ 492.32 R² = 0.9998

0 0

10000

20000

30000

40000

50000

60000

Time (ms)

-600

Fig.6. (a) Pressure vs. Simulator Pressure (Gas Pump) (b) Oxygen Concentration vs. Simulator O2 Conc. (Gas Pump)

3.3 Confirmatory Experiments To validate the result, 2 more runs of experiments were conducted following the same experimental layout under the same conditions for the same parameters. Similar steps were followed to conduct the confirmatory experiments as explained earlier. A 3D graph has been plotted showing the % errors at each point of time between the the actual and simulated model for each of the runs in figures 7-9 and in table 3.The confirmatory experiments very well validate the fact that the curve fitting equations deduced produce the best approximations for the simulated models. Table 3. % Error comparison between actual and simulated model No.Of  Readings  115  56 

Device

Vacuum  Pump  Gas Pump

Pressure 

Top Oxygen

Bottom Oxygen 

1  1.005

2 4.32

1 3.16

2 3.20

1 4.11 

2  4.05 

8.748

8.63

0.556

0.303

4.81 

4.96 

Anupam Pandit,Ravi Sekhar & Ram K. Revanur / Materials Today: Proceedings 4 (2017) 7270–7278

7277

25

20 18

20

14 12 10

Run 2

% Error

% Error

16

15 Run 2

10

8 Run 1

6

Run 1 5

4 2

Time (ms)

109000

103000

97000

91000

85000

1000 7000 13000 19000 25000 31000 37000 43000 49000 55000 61000 67000 73000

103000

109000

91000

97000

79000

85000

1000 7000 13000 19000 25000 31000 37000 43000 49000 55000 61000 67000 73000

Time (ms)

79000

0

0

Fig.7. (a) % Error comparison for pressure (vacuum pump)(b) % Error comparison for Top Oxygen (vacuum pump)

20

140

18

120 100

14 12 10

Run 2

% Error

80

Run 1

6

Run 2

60

8

Run 1

40

4

20

2

55000

49000

Time (ms)

52000

46000

40000

43000

1000 4000 7000 10000 13000 16000 19000 22000 25000 28000 31000

103000

109000

91000

97000

79000

85000

1000 7000 13000 19000 25000 31000 37000 43000 49000 55000 61000 67000 73000

Time (ms)

34000

0

0

37000

% Error

16

Fig.8. (a) % Error comparison Bottom Oxygen (vacuum pump)(b) % Error comparison for Pressure (Gas pump)

2.5

50 45 40

1.5 Run 2

1

Run 1

% Error

% Error

2

35 30 25

Run 2

20

Run 1

15 10

0.5

5

52000

Fig.9. (a) % Error comparison for Top Oxygen (Gas pump)(b) % Error comparison for Bottom Oxygen (Gas pump)

55000

46000

49000

43000

37000

Time (ms)

40000

55000

49000

52000

43000

46000

40000

34000

37000

1000 4000 7000 10000 13000 16000 19000 22000 25000 28000 31000

Time (ms)

1000 4000 7000 10000 13000 16000 19000 22000 25000 28000 31000 34000

0

0

7278

Anupam Pandit,Ravi Sekhar & Ram K. Revanur / Materials Today: Proceedings 4 (2017) 7270–7278

4. Conclusions In this work experiments were performed to investigate the effects of vacuum pump and gas pump on chamber pressure, top oxygen and bottom oxygen levels in an SLM additive manufacturing machine. Curve fitting was employed to simulate output parameters based on input parameters. R2 values were found to be greater than 90% proving that the required equations are statistically acceptable. Confirmatory experiments also showed very low average % errors thus validating the developed equations. So these equations can be utilized as a basis to generate simulation code for a SLM machine which will save the time and resources of designers. A comparative study between the actual model and simulated model shows the possibility of future study on optimizing the simulation model using optimization techniques. Genetic Algorithm has been of major interest among researchers working on curve fitting optimization [12, 13].Also the confirmatory experiments showed that the machine behavior is not same for each run even if operated under the similar conditions. The machine behavior changes dramatically due to change in atmospheric pressure and room temperature of the location where the machine is operated. The solution to this could be by calibrating the simulator [14, 15]. Hence, it is deemed necessary to develop a calibration technique to make the simulator fit for use in different scenarios. In conclusion, using this simulated model, a machine simulator can be developed which would be of great help to AM machine manufacturers in manufacturing optimized machines. 5. Acknowledgements The first author acknowledges the sponsorship and support by Renishaw India towards this research work. References [1]W. Trochim,J. Davis, Computer Simulation for Program Evaluation, Evaluation Review Vol. 5. 10 No. 5, October 1986 pp. 609-634. [2] J. J. Beaman, C. R. Deckard, Selective laser sintering with assisted powder handling, US Patents No. 4938816. [3] C. Taltavull, B. Torres, A.J. Lopez, P. Rodrigo, E. Otero, J. Rams Selective laser surface melting of a magnesium–aluminum alloy Mater Lett, 85 (2010), pp. 98–101 [4] B.C. Zhang, N.E. Fenineche, H.L. Liao, C. Coddet Microstructure and magnetic properties of Fe–Ni alloy fabricated by selective laser melting Fe/Ni mixed powder, J Mater Sci Technol, 29 (8) (2013), pp. 757–760 [5] Y.D. Wang, H.B. Tan, Y.L. Fang, H.M. Wang Microstructure and mechanical properties of hybrid fabricated 1Cr12Ni2WMoVNb steel by laser melting deposition Chin J Aeronaut, 26 (2) (2013), pp. 481–486 [6] D.D. Gu, Y.C. Hagedorn, W. Meiners, G. Meng, R.J.S. Batista, K. Wissenbach Densification behavior, microstructure evolution, and wear performance of selective laser melting processed commercially pure titanium Acta Mater, 60 (9) (2012), pp. 3849–3860 [7] H. Attar , M. Bonisch, M. Calin, L.C. Zhang, S. Scudino, J. Eckert, Selective laser melting of in situ titanium–titanium boride composites: processing, microstructure and mechanical properties. Acta Mater 2014; 76:13–22. [8]D.D. Gu, YF Shen,G.B. Meng, Growth morphologies and mechanisms of TiC grains during selective laser melting of Ti– Al–C composite powder. Mater Lett 2009; 63(29):2536–8. [9]F. Verhaeghe,T. Craeghs,J. Heulens, L. Pandelaers, A pragmatic model for selective laser melting with evaporation. Acta Mater 2009; 57(20):6006–12. [10]C. Sutcliffe,P. Fox., Manufacture of metal articles, US Patent - 20150135897 A1,(May 2015). [11] M. Hook, L. Junchen,O. Noriaki, S. Snowden, ”Descriptive and predictive growth curves in energy system analysis” Natural Resources Research, 2011, Vol. 20, Issue 2: 103-116. [12] M. Gulsen, A.E. Smith,D.M. Tate , A genetic algorithm approach to curve fitting, International Journal of Production Research ,1995; 33(7):1911-1923. [13] M.A. Farahat,M. Talaat, Short-Term Load Forecasting Using Curve Fitting Prediction Optimized by Genetic Algorithms, International Journal of Energy Engineering 2012, 2(2): 23-28. [14] B. Park, J.D. Schneeberger, Microscopic Simulation Model Calibration and Validation, Transportation Research Record 1856, Paper No. 032531. [15] M. Hofmann, On the Complexity of Parameter Calibration in Simulation Models, JDMS, Volume 2, Issue 4, October 2005 Pages 217–226.