SlideShare a Scribd company logo
PRESENTATION ON RAPID
     PROTOTYPING
What is Rapid prototyping?
• The first commercial rapid prototyping process was
  brought on the market in 1987.
• Rapid prototyping is a technology which considerably speeds
  up the product development, by allowing corrections in early
  stages.
• It consists of various manufacturing processes by which
  a solid physical model of part is made directly from 3D
  CAD model data without any special tooling.
• It is also referred as computer automated manufacturing.
• Now a days, more than 30 different processes (not all
  commercialized) with high accuracy and a large choice
  of materials exist.
Why Rapid prototyping?
•   Visualization.
•   Complex shapes can be build.
•   Decreases lead time.
•   Facilitates concurrent engineering.
•   No tooling.
•   Testing and evaluating the design.
•   First to market(product).
•   Patentable details.
DIFFERENCE BETWEEN RAPID PROTOTYPING AND TRADITIONAL
                             FABRICATION
                     RP                                       Traditional
1. RP processes are “additive”. Parts are built 1. Subtractive processes, as the name
   up by adding, depositing, or solidifying in a    implies, create objects by removing
   horizontal layer-wise process.                   unwanted material from a large block
                                                    or sheet in the form of chips.
2. RP technologies are able to create one- 2. Part is made in assemblies.
   piece part geometries.
3. Undercuts, overhangs, difficult shapes can 3. Useful for making simple shapes only.
   me made.
4. No tooling required.                          4. Tooling is always required
                                                    (jigs and fixtures).
5. Part can be fabricated in hours.              5. Take more time.

6. Rough surface finish.                       6. Good surface finish.
7. Less human interaction.                     7. More human interaction.
8. Non dangerous in nature                     8. Dangerous in nature.
   (dust free, noise free).
ADVANTAGES OF RAPID PROTOTYPING
•   No planning of process sequences.
•   No specific material for material handling.
•   No transportation between machines.
•   No need of jigs and fixtures.
•   High accuracy.
•   Tool for visualisation and product verification.
•   CAD data files can be manufactured in hours.
Rapidprototyping
SEQUENCE OF RAPID PROTOTYPING
SOLID MODELING AND TESSELATION
• Solid model is generated with the help of softwares (eg: pro
  engineers, solidworks, etc).
• Surfaces of the CAD model are tessellated and STL file is
  exported.
• Tessellation is piecewise approximation of surfaces of CAD
  model by using series of triangles.
• STL file=An stl file is a standardized computer exchange file
  which contains a 3d model, it entirely composed of triangular
  faces, further the faces have assigned normals which
  indicates their orientation (inside/outside).
• The name “STL” is taken from its extension .stl .
Solid model
Tessellated model
Slicing
• After selecting part orientation, the tessellated model is
  sliced.
• The pre-processing software slices the stl model into a no of
  layers from 0.01mm to 0.7mm thick, depending upon the
  build technique.
• The program may also generate an auxiliary structure to
  support the model during the build. Supports are useful for
  delicate features such as overhangs, internal cavities.
slicing
Rapidprototyping
GENERATION OF LASER SCANING PATHS

  Deposition of Layers: various technologies for
  layer deposition are used.
• Liquid based: Stereolithography
• Powder based: Selective Laser Sintering
• Solid based: Fused Deposition
  Modeling, Laminated Object Manufacturing
  etc.
Post processing/cleaning
• The final step is post-processing. This involves removing
  the prototype from the machine and detaching any
  supports.

• Some photosensitive materials need to be fully cured
  before use.

• Prototypes may also require minor cleaning and surface
  treatment.

• painting the model will improve its appearance and
  durability.
Rapidprototyping
METHODS OF RAPID PROTOTYPING
•   Shape deposition manufacturing(SDM)
•   Stereolithography(SLA)
•   Solid base curing(SBC)
•   Fused deposition modeling(FDM)
•   Ballistic particle manufacturing (BPM)
•   3D Printing
•   Selective layer sintering(SLS)
•   Laminated object manufacturing(LOM)
Shape deposition manufacturing
Shape deposition manufacturing
• Shape Deposition Manufacturing (SDM) is developed at Carnegie
  Mellon University in the early 1990's.
• Deposition of material(eg: welding).
• After deposition each layer is precisely shaped with a cnc milling
  machine.
• Due to thermal deposition the internal residual stresses are build
  up, due to contraction of internal gradients of previous layers on
  the new layer, the internal stresses can lead to warpage, early
  failure of the material.
• Shot peening is required to control the build up of stresses(small
  round metal spheres are called “shot”)are projected at high velocity
  against the surface of layer it imparts the compressive load on the
  upper layer, this process is used to balance the stress.
SDM
• Special components can be embedded inside the structure.
• The filler material is removed to get the finished part.
STEREOLITHOGRAPHY(SLA)
STEREOLITHOGRAPHY

• Stereolithography started the rapid prototyping revolution.
  The technique builds three-dimensional models from liquid
  photosensitive polymers that solidify when exposed to
  ultraviolet light.
• Elevator lowers hardened cross section below liquid surface.
• Laser prints the next cross section directly on top of previous.
• The self-adhesive property of the material causes each
  succeeding layer to bond to the previous one and thus form a
  complete, three-dimensional object out of many layers.
• Objects which have overhangs or under cuts must be
  supported during the fabrication process by support
  structure.
SOLID BASE CURING
Rapidprototyping
SOLID BASE CURING

• Cross section shape is printed onto a glass mask
• Glass mask is positioned above photopolymer tank
• UV lamp shines through mask onto photopolymer light only
  can pass through clear part, polymer solidifies
  there, polymer in masked areas remains liquid
• All excess polymer is removed-part is again hit with UV light
• Melted wax is spread over workpiece filling all spaces
• Workpiece is precisely milled flat
• Glass is erased and re-masked, workpiece is placed slightly
  below surface in photopolymer repeats
• After fabricating part, wax is melted and removed
• Note : No support or post cure needed
FUSED DEPOSITION MODELING
          (FDM)
FDM
• FDM is the second most widely used rapid prototyping
  technology, after stereolithography.
• A plastic filament is unwound from a coil and supplies
  material to an extrusion nozzle.
• The nozzle is heated to melt the plastic and has a
  mechanism which allows the flow of the melted plastic to be
  turned on and off.
• The nozzle is mounted to an X-Y plotter type mechanism
  which traces out the part contours, There is a second
  extrusion nozzle for the support material (different from the
  model material).
• Materials used mostly are ABS and PC(polycarbonate).
Ballistic particle manufacturing (BPM)
Ballistic particle manufacturing (BPM)
  Employs a technology called Digital Microsynthesis

• Molten plastic is fed to a piezoelectric jetting mechanism, similar to
  those on inkjet printers.
• A multi-axis controlled NC system shoots tiny droplets of material onto
  the target, using the jetting mechanism.
• Small droplets freeze upon contact with the surface, forming the
  surface particle by particle.

  Process allows use of virtually any thermoplastic.
3D PRINTING
3D PRINTING
•   The process starts by depositing a layer of powder object material at
    the top of a fabrication chamber. To accomplish this, a measured
    quantity of powder is first dispensed from a similar supply chamber by
    moving a piston upward incrementally.
•   A roller or scraper then distributes and compresses the powder at the
    top of the fabrication chamber the jetting head subsequently deposit a
    liquid adhesive (binder) in a two dimensional pattern onto the layer of
    the powder. The binder bonds the powder particles together where it
    has been deposited, solidifying it to form a layer of the object.
•   Once a layer is completed, the fabrication piston moves down by one
    layer thickness, and the process is repeated until the entire object is
    formed within the powder bed. After completion, the object must be
    removed from the chamber, and excess powder is brushed off.
•   No external supports are required during fabrication since the powder
    bed supports overhangs.
SELECTIVE LAYER SINTERING(SLS)
SELECTIVE LAYER SINTERING(SLS)

• SLS or Selective Laser Sintering is another process that begins
  with raw material in a powdered state. Only instead of binding
  with some extra solvent (as was done in 3-D Printing) powder
  particles are sintered together, by heating with a laser until
  adjacent particles sinter together.
• So process begins with a cartridge feeding system spreading a
  thin layer of heat fusible (sinterable) powder into workspace
  container. This layer of powder is heated to just below its
  melting point.
• A carbon dioxide laser then traces the cross sectional shape.
  Wherever the laser hits, enough extra heat is generated to sinter
  the particles together. Upon completion of one layer, the
  process can repeat.
LAMINATED OBJECT MANUFACTURING
             (LOM)
LAMINATED OBJECT MANUFACTURING
• A solid physical model is made by stacking layers of sheet
  stock, each an outline of the cross-sectional shape of a CAD
  model that is sliced into layers.
• Starting     material      =      sheet   stock,   such      as
  paper, plastic, cellulose.
• After cutting, excess material in the layer remains in place to
  support the part during building.
REFRENCES

• Google.com
• http: / /paniit.iitd.ac.in/~ pmpandey
• RAPID PROTOTYPING - Principles and
  Applications (2nd Edition) World Scientific
  Publishing Co. Pte. Ltd.
• Manufacturing Processes and Engineering
  Prof. J.S. Colton © GIT 2011
Rapidprototyping

More Related Content

PPT
Rapid Prototyping
PPTX
Selective laser sintering
PDF
Additive Manufacturing
PPTX
Laminated object manufacturing
PPTX
Rapid prototyping
PPTX
selective laser sintering;a rapid prototyping technology
PPTX
Rapid prototyping( additive manufacturing)
Rapid Prototyping
Selective laser sintering
Additive Manufacturing
Laminated object manufacturing
Rapid prototyping
selective laser sintering;a rapid prototyping technology
Rapid prototyping( additive manufacturing)

What's hot (20)

PPTX
Design for Additive Manufacturing Essentials
PPTX
rapid prototyping
PPTX
Module 1 Additive Manufacturing
PDF
Rod, wire and tube drawing
PPTX
Additive manufacturing file formats or 3D file formats
PDF
Product Development & Design for Additive Manufacturing (DfAM)
PPTX
Lecture: An introduction to additive manufacturing
PPTX
Super plastic forming and explosive forming
PDF
Selective Laser Melting versus Electron Beam Melting
DOCX
Solid ground curing
PPTX
Rapid prototyping
PPTX
Stereolithography
PPTX
FUSED DEPOSITION MODELING
PPTX
Surface engineering
PPTX
REVERSE ENGINEERING
PPTX
Laminated object manufacturing
PPTX
HydroForming
PPTX
SURFACE MODELLING .pptx
PPTX
3D PRINTING - REVERSE ENGINEERING
Design for Additive Manufacturing Essentials
rapid prototyping
Module 1 Additive Manufacturing
Rod, wire and tube drawing
Additive manufacturing file formats or 3D file formats
Product Development & Design for Additive Manufacturing (DfAM)
Lecture: An introduction to additive manufacturing
Super plastic forming and explosive forming
Selective Laser Melting versus Electron Beam Melting
Solid ground curing
Rapid prototyping
Stereolithography
FUSED DEPOSITION MODELING
Surface engineering
REVERSE ENGINEERING
Laminated object manufacturing
HydroForming
SURFACE MODELLING .pptx
3D PRINTING - REVERSE ENGINEERING
Ad

Viewers also liked (15)

PPTX
Rapid prototyping technology
PPT
Rapid prototyping seminar
PPT
TYPES OF RAPID PROTOTYPING - ADDITIVE PROCESS
PPTX
Prototyping
PPTX
Rapid Prototyping
PPT
Presentation for Software Development Methologies
KEY
Rapid Prototyping With jQuery
PPTX
Seminar Presentation on 3D Printing
PDF
Rapid Prototyping,Rapid prototype,SLA prototype
PPTX
4.selective laser sintering (by Hari Prasad)
PDF
Rapid Prototyping
PPTX
Ion beam deposition, wtth great uniformity.
PPT
Rapid Prototyping and its Applications
PPT
Rapid prototyping technology
Rapid prototyping seminar
TYPES OF RAPID PROTOTYPING - ADDITIVE PROCESS
Prototyping
Rapid Prototyping
Presentation for Software Development Methologies
Rapid Prototyping With jQuery
Seminar Presentation on 3D Printing
Rapid Prototyping,Rapid prototype,SLA prototype
4.selective laser sintering (by Hari Prasad)
Rapid Prototyping
Ion beam deposition, wtth great uniformity.
Rapid Prototyping and its Applications
Ad

Similar to Rapidprototyping (20)

PPT
rapid_prototyping classification.ppt
PDF
rapidprototyping-130214024311-phpapp01-140307000659-phpapp01.pdf
PPTX
Rapid-prototyping ( mechanical )
PDF
Lecture (SLS) RP Rapid Prototyping Method
PPTX
MS Lecture 11 Mechanics of Mfg processes.pptx
PDF
4. rapid prototyping
PDF
RPT- ppt.pdf
PPTX
Rapid prototyping
PPTX
AM.pptx
PPTX
rapid prototyping ppt.pptx power point prsesntation
PPTX
prototype , type of 3D printer and MFG.pptx
PPTX
Rapid Prototyping Technology BY NRAH SINGH MEENA K10926
PPT
Rapid-prototyping
PPTX
Sohan rapid prototyping.ppt.x
PDF
Rapid prototyping/ 3D Printing
PPT
PPT
Upfront 3D printing technology for students
PPTX
180180708010 - RAPID PROTOTYPING
PPTX
Rapid Proto.pptx
PPT
rapid_prototyping in the enggineering field-1.ppt
rapid_prototyping classification.ppt
rapidprototyping-130214024311-phpapp01-140307000659-phpapp01.pdf
Rapid-prototyping ( mechanical )
Lecture (SLS) RP Rapid Prototyping Method
MS Lecture 11 Mechanics of Mfg processes.pptx
4. rapid prototyping
RPT- ppt.pdf
Rapid prototyping
AM.pptx
rapid prototyping ppt.pptx power point prsesntation
prototype , type of 3D printer and MFG.pptx
Rapid Prototyping Technology BY NRAH SINGH MEENA K10926
Rapid-prototyping
Sohan rapid prototyping.ppt.x
Rapid prototyping/ 3D Printing
Upfront 3D printing technology for students
180180708010 - RAPID PROTOTYPING
Rapid Proto.pptx
rapid_prototyping in the enggineering field-1.ppt

Recently uploaded (20)

PDF
GENETICS IN BIOLOGY IN SECONDARY LEVEL FORM 3
PPTX
Orientation - ARALprogram of Deped to the Parents.pptx
PPTX
Tissue processing ( HISTOPATHOLOGICAL TECHNIQUE
PDF
Chapter 2 Heredity, Prenatal Development, and Birth.pdf
PDF
01-Introduction-to-Information-Management.pdf
PDF
Black Hat USA 2025 - Micro ICS Summit - ICS/OT Threat Landscape
PPTX
Radiologic_Anatomy_of_the_Brachial_plexus [final].pptx
PDF
Trump Administration's workforce development strategy
PPTX
UV-Visible spectroscopy..pptx UV-Visible Spectroscopy – Electronic Transition...
PPTX
Cell Structure & Organelles in detailed.
PDF
Module 4: Burden of Disease Tutorial Slides S2 2025
PDF
LNK 2025 (2).pdf MWEHEHEHEHEHEHEHEHEHEHE
PDF
Microbial disease of the cardiovascular and lymphatic systems
PDF
Computing-Curriculum for Schools in Ghana
PDF
grade 11-chemistry_fetena_net_5883.pdf teacher guide for all student
PDF
ChatGPT for Dummies - Pam Baker Ccesa007.pdf
PDF
Complications of Minimal Access Surgery at WLH
PPTX
UNIT III MENTAL HEALTH NURSING ASSESSMENT
PDF
Updated Idioms and Phrasal Verbs in English subject
PPTX
Final Presentation General Medicine 03-08-2024.pptx
GENETICS IN BIOLOGY IN SECONDARY LEVEL FORM 3
Orientation - ARALprogram of Deped to the Parents.pptx
Tissue processing ( HISTOPATHOLOGICAL TECHNIQUE
Chapter 2 Heredity, Prenatal Development, and Birth.pdf
01-Introduction-to-Information-Management.pdf
Black Hat USA 2025 - Micro ICS Summit - ICS/OT Threat Landscape
Radiologic_Anatomy_of_the_Brachial_plexus [final].pptx
Trump Administration's workforce development strategy
UV-Visible spectroscopy..pptx UV-Visible Spectroscopy – Electronic Transition...
Cell Structure & Organelles in detailed.
Module 4: Burden of Disease Tutorial Slides S2 2025
LNK 2025 (2).pdf MWEHEHEHEHEHEHEHEHEHEHE
Microbial disease of the cardiovascular and lymphatic systems
Computing-Curriculum for Schools in Ghana
grade 11-chemistry_fetena_net_5883.pdf teacher guide for all student
ChatGPT for Dummies - Pam Baker Ccesa007.pdf
Complications of Minimal Access Surgery at WLH
UNIT III MENTAL HEALTH NURSING ASSESSMENT
Updated Idioms and Phrasal Verbs in English subject
Final Presentation General Medicine 03-08-2024.pptx

Rapidprototyping

  • 1. PRESENTATION ON RAPID PROTOTYPING
  • 2. What is Rapid prototyping? • The first commercial rapid prototyping process was brought on the market in 1987. • Rapid prototyping is a technology which considerably speeds up the product development, by allowing corrections in early stages. • It consists of various manufacturing processes by which a solid physical model of part is made directly from 3D CAD model data without any special tooling. • It is also referred as computer automated manufacturing. • Now a days, more than 30 different processes (not all commercialized) with high accuracy and a large choice of materials exist.
  • 3. Why Rapid prototyping? • Visualization. • Complex shapes can be build. • Decreases lead time. • Facilitates concurrent engineering. • No tooling. • Testing and evaluating the design. • First to market(product). • Patentable details.
  • 4. DIFFERENCE BETWEEN RAPID PROTOTYPING AND TRADITIONAL FABRICATION RP Traditional 1. RP processes are “additive”. Parts are built 1. Subtractive processes, as the name up by adding, depositing, or solidifying in a implies, create objects by removing horizontal layer-wise process. unwanted material from a large block or sheet in the form of chips. 2. RP technologies are able to create one- 2. Part is made in assemblies. piece part geometries. 3. Undercuts, overhangs, difficult shapes can 3. Useful for making simple shapes only. me made. 4. No tooling required. 4. Tooling is always required (jigs and fixtures). 5. Part can be fabricated in hours. 5. Take more time. 6. Rough surface finish. 6. Good surface finish. 7. Less human interaction. 7. More human interaction. 8. Non dangerous in nature 8. Dangerous in nature. (dust free, noise free).
  • 5. ADVANTAGES OF RAPID PROTOTYPING • No planning of process sequences. • No specific material for material handling. • No transportation between machines. • No need of jigs and fixtures. • High accuracy. • Tool for visualisation and product verification. • CAD data files can be manufactured in hours.
  • 7. SEQUENCE OF RAPID PROTOTYPING
  • 8. SOLID MODELING AND TESSELATION • Solid model is generated with the help of softwares (eg: pro engineers, solidworks, etc). • Surfaces of the CAD model are tessellated and STL file is exported. • Tessellation is piecewise approximation of surfaces of CAD model by using series of triangles. • STL file=An stl file is a standardized computer exchange file which contains a 3d model, it entirely composed of triangular faces, further the faces have assigned normals which indicates their orientation (inside/outside). • The name “STL” is taken from its extension .stl .
  • 11. Slicing • After selecting part orientation, the tessellated model is sliced. • The pre-processing software slices the stl model into a no of layers from 0.01mm to 0.7mm thick, depending upon the build technique. • The program may also generate an auxiliary structure to support the model during the build. Supports are useful for delicate features such as overhangs, internal cavities.
  • 14. GENERATION OF LASER SCANING PATHS Deposition of Layers: various technologies for layer deposition are used. • Liquid based: Stereolithography • Powder based: Selective Laser Sintering • Solid based: Fused Deposition Modeling, Laminated Object Manufacturing etc.
  • 15. Post processing/cleaning • The final step is post-processing. This involves removing the prototype from the machine and detaching any supports. • Some photosensitive materials need to be fully cured before use. • Prototypes may also require minor cleaning and surface treatment. • painting the model will improve its appearance and durability.
  • 17. METHODS OF RAPID PROTOTYPING • Shape deposition manufacturing(SDM) • Stereolithography(SLA) • Solid base curing(SBC) • Fused deposition modeling(FDM) • Ballistic particle manufacturing (BPM) • 3D Printing • Selective layer sintering(SLS) • Laminated object manufacturing(LOM)
  • 19. Shape deposition manufacturing • Shape Deposition Manufacturing (SDM) is developed at Carnegie Mellon University in the early 1990's. • Deposition of material(eg: welding). • After deposition each layer is precisely shaped with a cnc milling machine. • Due to thermal deposition the internal residual stresses are build up, due to contraction of internal gradients of previous layers on the new layer, the internal stresses can lead to warpage, early failure of the material. • Shot peening is required to control the build up of stresses(small round metal spheres are called “shot”)are projected at high velocity against the surface of layer it imparts the compressive load on the upper layer, this process is used to balance the stress.
  • 20. SDM • Special components can be embedded inside the structure. • The filler material is removed to get the finished part.
  • 22. STEREOLITHOGRAPHY • Stereolithography started the rapid prototyping revolution. The technique builds three-dimensional models from liquid photosensitive polymers that solidify when exposed to ultraviolet light. • Elevator lowers hardened cross section below liquid surface. • Laser prints the next cross section directly on top of previous. • The self-adhesive property of the material causes each succeeding layer to bond to the previous one and thus form a complete, three-dimensional object out of many layers. • Objects which have overhangs or under cuts must be supported during the fabrication process by support structure.
  • 25. SOLID BASE CURING • Cross section shape is printed onto a glass mask • Glass mask is positioned above photopolymer tank • UV lamp shines through mask onto photopolymer light only can pass through clear part, polymer solidifies there, polymer in masked areas remains liquid • All excess polymer is removed-part is again hit with UV light • Melted wax is spread over workpiece filling all spaces • Workpiece is precisely milled flat • Glass is erased and re-masked, workpiece is placed slightly below surface in photopolymer repeats • After fabricating part, wax is melted and removed • Note : No support or post cure needed
  • 27. FDM • FDM is the second most widely used rapid prototyping technology, after stereolithography. • A plastic filament is unwound from a coil and supplies material to an extrusion nozzle. • The nozzle is heated to melt the plastic and has a mechanism which allows the flow of the melted plastic to be turned on and off. • The nozzle is mounted to an X-Y plotter type mechanism which traces out the part contours, There is a second extrusion nozzle for the support material (different from the model material). • Materials used mostly are ABS and PC(polycarbonate).
  • 29. Ballistic particle manufacturing (BPM) Employs a technology called Digital Microsynthesis • Molten plastic is fed to a piezoelectric jetting mechanism, similar to those on inkjet printers. • A multi-axis controlled NC system shoots tiny droplets of material onto the target, using the jetting mechanism. • Small droplets freeze upon contact with the surface, forming the surface particle by particle. Process allows use of virtually any thermoplastic.
  • 31. 3D PRINTING • The process starts by depositing a layer of powder object material at the top of a fabrication chamber. To accomplish this, a measured quantity of powder is first dispensed from a similar supply chamber by moving a piston upward incrementally. • A roller or scraper then distributes and compresses the powder at the top of the fabrication chamber the jetting head subsequently deposit a liquid adhesive (binder) in a two dimensional pattern onto the layer of the powder. The binder bonds the powder particles together where it has been deposited, solidifying it to form a layer of the object. • Once a layer is completed, the fabrication piston moves down by one layer thickness, and the process is repeated until the entire object is formed within the powder bed. After completion, the object must be removed from the chamber, and excess powder is brushed off. • No external supports are required during fabrication since the powder bed supports overhangs.
  • 33. SELECTIVE LAYER SINTERING(SLS) • SLS or Selective Laser Sintering is another process that begins with raw material in a powdered state. Only instead of binding with some extra solvent (as was done in 3-D Printing) powder particles are sintered together, by heating with a laser until adjacent particles sinter together. • So process begins with a cartridge feeding system spreading a thin layer of heat fusible (sinterable) powder into workspace container. This layer of powder is heated to just below its melting point. • A carbon dioxide laser then traces the cross sectional shape. Wherever the laser hits, enough extra heat is generated to sinter the particles together. Upon completion of one layer, the process can repeat.
  • 35. LAMINATED OBJECT MANUFACTURING • A solid physical model is made by stacking layers of sheet stock, each an outline of the cross-sectional shape of a CAD model that is sliced into layers. • Starting material = sheet stock, such as paper, plastic, cellulose. • After cutting, excess material in the layer remains in place to support the part during building.
  • 36. REFRENCES • Google.com • http: / /paniit.iitd.ac.in/~ pmpandey • RAPID PROTOTYPING - Principles and Applications (2nd Edition) World Scientific Publishing Co. Pte. Ltd. • Manufacturing Processes and Engineering Prof. J.S. Colton © GIT 2011