nldlab - User contributions [en]

File:Faraday2014.pdf

2015-05-20T16:21:10Z

PatrickChang:

Main Page

2015-05-20T16:20:56Z

PatrickChang:

[[File: Howey_Building.jpg | thumb | Howey building. School of Physics, Georgia Tech]]

'''This is the wiki page associated with Physics 4268/6268 Nonlinear Dynamics & Chaos.''' Physics 4268/6268 is an undergraduate and graduate level nonlinear dynamics course taught by [http://crablab.gatech.edu/ Professor Daniel I. Goldman] in the [http://www.physics.gatech.edu School of Physics] at the [http://www.gatech.edu Georgia Institute of Technology]. The TAs are [http://people.seas.harvard.edu/~gravish/index.html Nick Gravish] and Feifei Qian. Wiki is maintained by [http://people.seas.harvard.edu/~gravish/index.html Nick Gravish]. The course is comprised of both a classroom component and a laboratory component. In the laboratory component student led groups will perform a nonlinear dynamics experiment and report their findings. Students thus develop intuition for nonlinear dynamics both on paper and in a hands-on laboratory environment.

'''The goal of this wiki is for students to interactively develop and compile a library of experiments that illustrate fundamental principles of nonlinear dynamics for further educational use.'''

== Site information ==

On this site you will find all content related to the student led nonlinear dynamics experiments. Announcements for the '''2012''' course can be found below. In addition all materials from previous years have been posted so that students may see examples of previous project wiki pages, papers, and presentations.

Each group is given access a project page in which they will edit and add content to a wikipedia styled entry detailing the experiment being performed. The page should include a brief summary of the experiment and the relevant scientific questions followed by a more comprehensive explanation of the theoretical and experimental details. Pages should contain references to outside information via journal or book citations, and external web pages. Once the lab has been completed the page should be edited to include information on experimental analysis, results, and a conclusion. Sample pages from previous years serve as useful references.

== Final papers / presentations 2014 ==

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Project
! Files
|-
|[[Group_1_2014 | Chiral Object]]
|[[Media: Chiral_presentation.pdf | Presentation]]. [[Media: Separation-chiral-particles.pdf | Final report]]
|-
| [[Group_2_2014 | Faraday Instability]]
| [[Media: Faraday2014.pdf | Presentation]]. [[Media: Faraday.pdf| Final report]]
|-
| [[Group_3_2014 | Astrojax Pendulum]]
| [[Media: Astrojax Pendulum.pdf | Presentation]]. [[Media: NLD.pdf | Final report]]
|-
| [[Group_4_2014 | Jump on Granular Media]]
| [[Media: Jump2014.pdf | Presentation]]. [[Media: Jumping-aerated-granular.pdf | Final report]]
|-
| [[Group_5_2014 | Pendulum Synchronization]]
| [[Media: Pendulum.pdf | Final report]]
|}
== Final papers / presentations 2012 ==

'''Congratulations to all the students!''' [[Pictures_2012 | Pictures and Videos]] from the semester have been uploaded as well as the final presentations and papers which can be found below.

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Project
! Files
|-
|[[Group_1_2012 | Duffing Oscillator]]
|[[Media: Duffingfinal.pdf | Presentation]]. '''Papers:''' [[Media: DuffingChampion.pdf | Andrew]], [[Media: Lodhi_finalProject.pdf | Aemen]], [[Media: DuffingGranowski.pdf | Ross]]
|-
| [[Group_2_2012 | Firefly Synchronization]]
| [[Media: Firefly_Final_Presentation.pdf| Presentation]]. '''Papers:''' [[Media: Morris Huang.pdf‎ | Morris]], [[Media: Mark Kingsbury.pdf | Mark]], [[Media: McInroe Final Paper.pdf | Ben]], [[Media: Wagstaff Project.pdf | Will]]
|-
| [[Group_3_2012 | Chua's Circuit]]
| [[Media: Experimental Characterizing of Nonlinear dynamics of Chua’s circuit.pdf | Presentation]]. '''Papers:''' [[Media: Patrick Chang-final paper.pdf | Patrick]], [[Media: CoyleGroup2Chua.pdf | Edward]], [[Media: Parker Report.pdf | John]], [[Media: Majid Paper.pdf | Majid]]
|-
| [[Group_4_2012 | Cricket Synchronization]]
| [[Media: Cricket Presentation.pdf | Main presentation]], [[Media: Charlie_Presentation.pdf | Presentation on pseudo-synchronization leading the pack]]. '''Papers:''' [[Media: Blythe report.pdf | Justin]], [[Media: Charlie Report.pdf | Charlie]], [[Media: Yuxuan Report.pdf | Yuxuan]]
|}

== Announcements: Fall 2012 ==

* Student groups will be posted shortly. Projects suggestions will be made in class 9/11/2012
* The room Howey S204 has been reserved for student groups to meet and discuss the following hours and dates:
** Mondays from Oct 22 - Dec 3 from 10am to 12pm
** Tuesdays and Thursdays from 11am to 12pm
** Wednesdays from Oct 31 - Dec 5 from 10am to 12pm
** Fridays from 3pm to 5 pm

== Final papers / presentations 2011 ==

'''Congratulations to all the students, the final papers and presentations were universally excellent!''' [[Pictures_2011 | Pictures]] from the semester have been uploaded as well as the final presentations and papers which can be found below.

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Project
! Files
|-
|[[Group_1 | Faraday waves]]
|[[Media: ProjectFaraday2.pdf | Main presentation]], [[Media:Corn_Starch_Slides.pdf | Non-newtonian presentation]]. '''Papers:''' [[Media: PaulCardenasLizana.pdf‎ | Paul]], [[Media: Orphee_Juan_Faraday.pdf | Juan]]
|-
| [[Group_2 | Plinko dynamics]]
| [[Media: Plinko_Compressed.pdf| Presentation]]. '''Papers:''' [[Media: Mass_Andrew.pdf | Andrew]], [[Media: Hardin_Charles_-_Group_2_-_6268_Final_Paper.pdf | Andrew]], [[Media: Cordell_Group2_Phys6268.pdf | Chris]]
|-
| [[ Group_3 | Inelastic bouncing ball]]
| [[Media: NLD_Presentation_Official_Final_Final_For_Real.pdf | Presentation]]. '''Papers:''' [[Media: Arora_Nitin_Paper_nitin.pdf | Nitin]], [[Media: Gray_Phillip.pdf | Phillip]], [[Media: Yunis_Jacob_6268_final.pdf | Jacob]], [[Media: Rodesneyinelastic_ball.pdf | Chris]]
|-
| [[Group_4 | Synchronization]]
| [[Media: Group4_Metronomes_Talk.pdf | Presentation]]. '''Papers:''' [[Media: Levenfeld_Vlad.pdf | Vlad]], [[Media: Jover_finalpaper_metronomes_jover.pdf | Luis]], [[Media: Taylor_Bradford.pdf | Brad]], [[Media: Tithof_Metronomes.pdf | Jeff]]
|-
| [[Group_5 | Chaotic faucet]]
| [[Media: PHYS6268 Group5 Presentation.pdf | Presentation]]. '''Papers:''' [[Media: Patel_Ricky_PHYS6268.pdf | Ricky ]], [[Media: Royer_Caleb_PHYS6268_Royer_FinalPaper.pdf | Caleb ]], [[Media: Job_Josh_ChaoticFaucetfinal.pdf | Joshua ]], [[Media: Pritchard_Peter_Phys6268_Final_Paper.pdf | Nick ]]
|-
| [[Group_6 | Ferrofluid]]
| [[Media: Presentation.pdf | Presentation]]. '''Papers:''' [[Media: Schoenwald_Group6_Final.pdf | Kipp]], [[Media: Potter_Ferrofluids_-_Potter.pdf | Daniel]], [[Media: Hamid,_Amir_Ferrofluid_Group_6.pdf | Amir]]
|-
| [[Group_7 | Inverted pendulum]]
| [[Media: The_Inverted_Pendulum.pdf | Presentation]]. '''Papers:''' [[Media: Marcotte_Dynamic_Stabilization_of_the_Inverted_Pendulum.pdf | Chris]], [[Media: Aguilar_Jeff_PendulumPaper.pdf | Jeff]], [[Media: LeeGustavo_Inverted_Pendulum_Final_Report_.pdf | Gustavo]], [[Media: Suri_Balachandra_NLD_final.pdf | Balachandra]]
|}

== Announcements: Fall 2011 ==

* Below is a list of the assigned dates for groups to work in the lab. If any groups have conflicts with the dates listed please email [mailto:nick.gravish@gmail.com me] immediately to resolve this.

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Date range
! Groups
|-
| 10/24 - 10/28
| Inverted pendulum ([[Group 7]])
|-
| 10/31 - 11/4
| Inelastic bouncing ball ([[Group 3]]) and dripping faucet ([[Group 5]])
|-
| 11/7 - 11/11
| Synchronization ([[Group 4]]) and Ferrofluid ([[Group 6]])
|-
| 11/14 - 11/18
| Faraday waves ([[Group 1]]) and Plinko dynamics ([[Group 2]])
|-
| 11/28 - 12/2
| Presentations beginning Thursday
|-
| 12/5 - 12/9
| Presentations
|}

* Information about the final presentations and paper can now be found on the [[Final]] page.
* Lab dates have been posted 10/19/11
* Added capability to embed youtube videos. Use the following code to embed <pre><videoflash>YouTubeFileName</videoflash></pre>
* Math rendering has been added using the [http://www.mathjax.org/ MathJax] program. This renders latex code typed into the page source and allows for copy/paste, scaling, and other features. For a Latex tutorial check [http://en.wikibooks.org/wiki/LaTeX/Mathematics here].
* Groups are only able to edit the page associated with their group.
* When you receive your group login and password you should change your password for security.
* For questions about this site email [mailto:nick.gravish@gmail.com Nick Gravish]

== About this wiki ==

See the [[About]] page for more information. This Wiki is open to the public to view but not to edit. However, we gladly make content available to other schools for non-profit educational use. Some links to copyright-protected references and software are not available to anyone without authentication as a Georgia Tech student or staff.

== Wiki 101 ==

Here is a link to mediawiki [http://en.wikipedia.org/wiki/Wikipedia:Cheatsheet cheatsheet].

Main Page

2015-05-20T16:20:34Z

PatrickChang:

[[File: Howey_Building.jpg | thumb | Howey building. School of Physics, Georgia Tech]]

'''This is the wiki page associated with Physics 4268/6268 Nonlinear Dynamics & Chaos.''' Physics 4268/6268 is an undergraduate and graduate level nonlinear dynamics course taught by [http://crablab.gatech.edu/ Professor Daniel I. Goldman] in the [http://www.physics.gatech.edu School of Physics] at the [http://www.gatech.edu Georgia Institute of Technology]. The TAs are [http://people.seas.harvard.edu/~gravish/index.html Nick Gravish] and Feifei Qian. Wiki is maintained by [http://people.seas.harvard.edu/~gravish/index.html Nick Gravish]. The course is comprised of both a classroom component and a laboratory component. In the laboratory component student led groups will perform a nonlinear dynamics experiment and report their findings. Students thus develop intuition for nonlinear dynamics both on paper and in a hands-on laboratory environment.

'''The goal of this wiki is for students to interactively develop and compile a library of experiments that illustrate fundamental principles of nonlinear dynamics for further educational use.'''

== Site information ==

On this site you will find all content related to the student led nonlinear dynamics experiments. Announcements for the '''2012''' course can be found below. In addition all materials from previous years have been posted so that students may see examples of previous project wiki pages, papers, and presentations.

Each group is given access a project page in which they will edit and add content to a wikipedia styled entry detailing the experiment being performed. The page should include a brief summary of the experiment and the relevant scientific questions followed by a more comprehensive explanation of the theoretical and experimental details. Pages should contain references to outside information via journal or book citations, and external web pages. Once the lab has been completed the page should be edited to include information on experimental analysis, results, and a conclusion. Sample pages from previous years serve as useful references.

== Final papers / presentations 2014 ==

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Project
! Files
|-
|[[Group_1_2014 | Chiral Object]]
|[[Media: Chiral_presentation.pdf | Presentation]]. [[Media: Separation-chiral-particles.pdf | Final report]]
|-
| [[Group_2_2014 | Faraday Instability]]
| [[Media: Faraday.pdf | Presentation]]. [[Media: Faraday.pdf| Final report]]
|-
| [[Group_3_2014 | Astrojax Pendulum]]
| [[Media: Astrojax Pendulum.pdf | Presentation]]. [[Media: NLD.pdf | Final report]]
|-
| [[Group_4_2014 | Jump on Granular Media]]
| [[Media: Jump2014.pdf | Presentation]]. [[Media: Jumping-aerated-granular.pdf | Final report]]
|-
| [[Group_5_2014 | Pendulum Synchronization]]
| [[Media: Pendulum.pdf | Final report]]
|}
== Final papers / presentations 2012 ==

'''Congratulations to all the students!''' [[Pictures_2012 | Pictures and Videos]] from the semester have been uploaded as well as the final presentations and papers which can be found below.

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Project
! Files
|-
|[[Group_1_2012 | Duffing Oscillator]]
|[[Media: Duffingfinal.pdf | Presentation]]. '''Papers:''' [[Media: DuffingChampion.pdf | Andrew]], [[Media: Lodhi_finalProject.pdf | Aemen]], [[Media: DuffingGranowski.pdf | Ross]]
|-
| [[Group_2_2012 | Firefly Synchronization]]
| [[Media: Firefly_Final_Presentation.pdf| Presentation]]. '''Papers:''' [[Media: Morris Huang.pdf‎ | Morris]], [[Media: Mark Kingsbury.pdf | Mark]], [[Media: McInroe Final Paper.pdf | Ben]], [[Media: Wagstaff Project.pdf | Will]]
|-
| [[Group_3_2012 | Chua's Circuit]]
| [[Media: Experimental Characterizing of Nonlinear dynamics of Chua’s circuit.pdf | Presentation]]. '''Papers:''' [[Media: Patrick Chang-final paper.pdf | Patrick]], [[Media: CoyleGroup2Chua.pdf | Edward]], [[Media: Parker Report.pdf | John]], [[Media: Majid Paper.pdf | Majid]]
|-
| [[Group_4_2012 | Cricket Synchronization]]
| [[Media: Cricket Presentation.pdf | Main presentation]], [[Media: Charlie_Presentation.pdf | Presentation on pseudo-synchronization leading the pack]]. '''Papers:''' [[Media: Blythe report.pdf | Justin]], [[Media: Charlie Report.pdf | Charlie]], [[Media: Yuxuan Report.pdf | Yuxuan]]
|}

== Announcements: Fall 2012 ==

* Student groups will be posted shortly. Projects suggestions will be made in class 9/11/2012
* The room Howey S204 has been reserved for student groups to meet and discuss the following hours and dates:
** Mondays from Oct 22 - Dec 3 from 10am to 12pm
** Tuesdays and Thursdays from 11am to 12pm
** Wednesdays from Oct 31 - Dec 5 from 10am to 12pm
** Fridays from 3pm to 5 pm

== Final papers / presentations 2011 ==

'''Congratulations to all the students, the final papers and presentations were universally excellent!''' [[Pictures_2011 | Pictures]] from the semester have been uploaded as well as the final presentations and papers which can be found below.

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Project
! Files
|-
|[[Group_1 | Faraday waves]]
|[[Media: ProjectFaraday2.pdf | Main presentation]], [[Media:Corn_Starch_Slides.pdf | Non-newtonian presentation]]. '''Papers:''' [[Media: PaulCardenasLizana.pdf‎ | Paul]], [[Media: Orphee_Juan_Faraday.pdf | Juan]]
|-
| [[Group_2 | Plinko dynamics]]
| [[Media: Plinko_Compressed.pdf| Presentation]]. '''Papers:''' [[Media: Mass_Andrew.pdf | Andrew]], [[Media: Hardin_Charles_-_Group_2_-_6268_Final_Paper.pdf | Andrew]], [[Media: Cordell_Group2_Phys6268.pdf | Chris]]
|-
| [[ Group_3 | Inelastic bouncing ball]]
| [[Media: NLD_Presentation_Official_Final_Final_For_Real.pdf | Presentation]]. '''Papers:''' [[Media: Arora_Nitin_Paper_nitin.pdf | Nitin]], [[Media: Gray_Phillip.pdf | Phillip]], [[Media: Yunis_Jacob_6268_final.pdf | Jacob]], [[Media: Rodesneyinelastic_ball.pdf | Chris]]
|-
| [[Group_4 | Synchronization]]
| [[Media: Group4_Metronomes_Talk.pdf | Presentation]]. '''Papers:''' [[Media: Levenfeld_Vlad.pdf | Vlad]], [[Media: Jover_finalpaper_metronomes_jover.pdf | Luis]], [[Media: Taylor_Bradford.pdf | Brad]], [[Media: Tithof_Metronomes.pdf | Jeff]]
|-
| [[Group_5 | Chaotic faucet]]
| [[Media: PHYS6268 Group5 Presentation.pdf | Presentation]]. '''Papers:''' [[Media: Patel_Ricky_PHYS6268.pdf | Ricky ]], [[Media: Royer_Caleb_PHYS6268_Royer_FinalPaper.pdf | Caleb ]], [[Media: Job_Josh_ChaoticFaucetfinal.pdf | Joshua ]], [[Media: Pritchard_Peter_Phys6268_Final_Paper.pdf | Nick ]]
|-
| [[Group_6 | Ferrofluid]]
| [[Media: Presentation.pdf | Presentation]]. '''Papers:''' [[Media: Schoenwald_Group6_Final.pdf | Kipp]], [[Media: Potter_Ferrofluids_-_Potter.pdf | Daniel]], [[Media: Hamid,_Amir_Ferrofluid_Group_6.pdf | Amir]]
|-
| [[Group_7 | Inverted pendulum]]
| [[Media: The_Inverted_Pendulum.pdf | Presentation]]. '''Papers:''' [[Media: Marcotte_Dynamic_Stabilization_of_the_Inverted_Pendulum.pdf | Chris]], [[Media: Aguilar_Jeff_PendulumPaper.pdf | Jeff]], [[Media: LeeGustavo_Inverted_Pendulum_Final_Report_.pdf | Gustavo]], [[Media: Suri_Balachandra_NLD_final.pdf | Balachandra]]
|}

== Announcements: Fall 2011 ==

* Below is a list of the assigned dates for groups to work in the lab. If any groups have conflicts with the dates listed please email [mailto:nick.gravish@gmail.com me] immediately to resolve this.

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Date range
! Groups
|-
| 10/24 - 10/28
| Inverted pendulum ([[Group 7]])
|-
| 10/31 - 11/4
| Inelastic bouncing ball ([[Group 3]]) and dripping faucet ([[Group 5]])
|-
| 11/7 - 11/11
| Synchronization ([[Group 4]]) and Ferrofluid ([[Group 6]])
|-
| 11/14 - 11/18
| Faraday waves ([[Group 1]]) and Plinko dynamics ([[Group 2]])
|-
| 11/28 - 12/2
| Presentations beginning Thursday
|-
| 12/5 - 12/9
| Presentations
|}

* Information about the final presentations and paper can now be found on the [[Final]] page.
* Lab dates have been posted 10/19/11
* Added capability to embed youtube videos. Use the following code to embed <pre><videoflash>YouTubeFileName</videoflash></pre>
* Math rendering has been added using the [http://www.mathjax.org/ MathJax] program. This renders latex code typed into the page source and allows for copy/paste, scaling, and other features. For a Latex tutorial check [http://en.wikibooks.org/wiki/LaTeX/Mathematics here].
* Groups are only able to edit the page associated with their group.
* When you receive your group login and password you should change your password for security.
* For questions about this site email [mailto:nick.gravish@gmail.com Nick Gravish]

== About this wiki ==

See the [[About]] page for more information. This Wiki is open to the public to view but not to edit. However, we gladly make content available to other schools for non-profit educational use. Some links to copyright-protected references and software are not available to anyone without authentication as a Georgia Tech student or staff.

== Wiki 101 ==

Here is a link to mediawiki [http://en.wikipedia.org/wiki/Wikipedia:Cheatsheet cheatsheet].

File:Jump2014.pdf

2015-05-19T15:25:53Z

PatrickChang:

Main Page

2015-05-19T15:24:32Z

PatrickChang:

[[File: Howey_Building.jpg | thumb | Howey building. School of Physics, Georgia Tech]]

'''This is the wiki page associated with Physics 4268/6268 Nonlinear Dynamics & Chaos.''' Physics 4268/6268 is an undergraduate and graduate level nonlinear dynamics course taught by [http://crablab.gatech.edu/ Professor Daniel I. Goldman] in the [http://www.physics.gatech.edu School of Physics] at the [http://www.gatech.edu Georgia Institute of Technology]. The TAs are [http://people.seas.harvard.edu/~gravish/index.html Nick Gravish] and Feifei Qian. Wiki is maintained by [http://people.seas.harvard.edu/~gravish/index.html Nick Gravish]. The course is comprised of both a classroom component and a laboratory component. In the laboratory component student led groups will perform a nonlinear dynamics experiment and report their findings. Students thus develop intuition for nonlinear dynamics both on paper and in a hands-on laboratory environment.

'''The goal of this wiki is for students to interactively develop and compile a library of experiments that illustrate fundamental principles of nonlinear dynamics for further educational use.'''

== Site information ==

On this site you will find all content related to the student led nonlinear dynamics experiments. Announcements for the '''2012''' course can be found below. In addition all materials from previous years have been posted so that students may see examples of previous project wiki pages, papers, and presentations.

Each group is given access a project page in which they will edit and add content to a wikipedia styled entry detailing the experiment being performed. The page should include a brief summary of the experiment and the relevant scientific questions followed by a more comprehensive explanation of the theoretical and experimental details. Pages should contain references to outside information via journal or book citations, and external web pages. Once the lab has been completed the page should be edited to include information on experimental analysis, results, and a conclusion. Sample pages from previous years serve as useful references.

== Final papers / presentations 2014 ==

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Project
! Files
|-
|[[Group_1_2014 | Chiral Object]]
|[[Media: Chiral_presentation.pdf | Presentation]]. [[Media: Separation-chiral-particles.pdf | Final report]]
|-
| [[Group_2_2014 | Faraday Instability]]
| [[Media: Faraday.pdf| Final report]]
|-
| [[Group_3_2014 | Astrojax Pendulum]]
| [[Media: Astrojax Pendulum.pdf | Presentation]]. [[Media: NLD.pdf | Final report]]
|-
| [[Group_4_2014 | Jump on Granular Media]]
| [[Media: Jump2014.pdf | Presentation]]. [[Media: Jumping-aerated-granular.pdf | Final report]]
|-
| [[Group_5_2014 | Pendulum Synchronization]]
| [[Media: Pendulum.pdf | Final report]]
|}
== Final papers / presentations 2012 ==

'''Congratulations to all the students!''' [[Pictures_2012 | Pictures and Videos]] from the semester have been uploaded as well as the final presentations and papers which can be found below.

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Project
! Files
|-
|[[Group_1_2012 | Duffing Oscillator]]
|[[Media: Duffingfinal.pdf | Presentation]]. '''Papers:''' [[Media: DuffingChampion.pdf | Andrew]], [[Media: Lodhi_finalProject.pdf | Aemen]], [[Media: DuffingGranowski.pdf | Ross]]
|-
| [[Group_2_2012 | Firefly Synchronization]]
| [[Media: Firefly_Final_Presentation.pdf| Presentation]]. '''Papers:''' [[Media: Morris Huang.pdf‎ | Morris]], [[Media: Mark Kingsbury.pdf | Mark]], [[Media: McInroe Final Paper.pdf | Ben]], [[Media: Wagstaff Project.pdf | Will]]
|-
| [[Group_3_2012 | Chua's Circuit]]
| [[Media: Experimental Characterizing of Nonlinear dynamics of Chua’s circuit.pdf | Presentation]]. '''Papers:''' [[Media: Patrick Chang-final paper.pdf | Patrick]], [[Media: CoyleGroup2Chua.pdf | Edward]], [[Media: Parker Report.pdf | John]], [[Media: Majid Paper.pdf | Majid]]
|-
| [[Group_4_2012 | Cricket Synchronization]]
| [[Media: Cricket Presentation.pdf | Main presentation]], [[Media: Charlie_Presentation.pdf | Presentation on pseudo-synchronization leading the pack]]. '''Papers:''' [[Media: Blythe report.pdf | Justin]], [[Media: Charlie Report.pdf | Charlie]], [[Media: Yuxuan Report.pdf | Yuxuan]]
|}

== Announcements: Fall 2012 ==

* Student groups will be posted shortly. Projects suggestions will be made in class 9/11/2012
* The room Howey S204 has been reserved for student groups to meet and discuss the following hours and dates:
** Mondays from Oct 22 - Dec 3 from 10am to 12pm
** Tuesdays and Thursdays from 11am to 12pm
** Wednesdays from Oct 31 - Dec 5 from 10am to 12pm
** Fridays from 3pm to 5 pm

== Final papers / presentations 2011 ==

'''Congratulations to all the students, the final papers and presentations were universally excellent!''' [[Pictures_2011 | Pictures]] from the semester have been uploaded as well as the final presentations and papers which can be found below.

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Project
! Files
|-
|[[Group_1 | Faraday waves]]
|[[Media: ProjectFaraday2.pdf | Main presentation]], [[Media:Corn_Starch_Slides.pdf | Non-newtonian presentation]]. '''Papers:''' [[Media: PaulCardenasLizana.pdf‎ | Paul]], [[Media: Orphee_Juan_Faraday.pdf | Juan]]
|-
| [[Group_2 | Plinko dynamics]]
| [[Media: Plinko_Compressed.pdf| Presentation]]. '''Papers:''' [[Media: Mass_Andrew.pdf | Andrew]], [[Media: Hardin_Charles_-_Group_2_-_6268_Final_Paper.pdf | Andrew]], [[Media: Cordell_Group2_Phys6268.pdf | Chris]]
|-
| [[ Group_3 | Inelastic bouncing ball]]
| [[Media: NLD_Presentation_Official_Final_Final_For_Real.pdf | Presentation]]. '''Papers:''' [[Media: Arora_Nitin_Paper_nitin.pdf | Nitin]], [[Media: Gray_Phillip.pdf | Phillip]], [[Media: Yunis_Jacob_6268_final.pdf | Jacob]], [[Media: Rodesneyinelastic_ball.pdf | Chris]]
|-
| [[Group_4 | Synchronization]]
| [[Media: Group4_Metronomes_Talk.pdf | Presentation]]. '''Papers:''' [[Media: Levenfeld_Vlad.pdf | Vlad]], [[Media: Jover_finalpaper_metronomes_jover.pdf | Luis]], [[Media: Taylor_Bradford.pdf | Brad]], [[Media: Tithof_Metronomes.pdf | Jeff]]
|-
| [[Group_5 | Chaotic faucet]]
| [[Media: PHYS6268 Group5 Presentation.pdf | Presentation]]. '''Papers:''' [[Media: Patel_Ricky_PHYS6268.pdf | Ricky ]], [[Media: Royer_Caleb_PHYS6268_Royer_FinalPaper.pdf | Caleb ]], [[Media: Job_Josh_ChaoticFaucetfinal.pdf | Joshua ]], [[Media: Pritchard_Peter_Phys6268_Final_Paper.pdf | Nick ]]
|-
| [[Group_6 | Ferrofluid]]
| [[Media: Presentation.pdf | Presentation]]. '''Papers:''' [[Media: Schoenwald_Group6_Final.pdf | Kipp]], [[Media: Potter_Ferrofluids_-_Potter.pdf | Daniel]], [[Media: Hamid,_Amir_Ferrofluid_Group_6.pdf | Amir]]
|-
| [[Group_7 | Inverted pendulum]]
| [[Media: The_Inverted_Pendulum.pdf | Presentation]]. '''Papers:''' [[Media: Marcotte_Dynamic_Stabilization_of_the_Inverted_Pendulum.pdf | Chris]], [[Media: Aguilar_Jeff_PendulumPaper.pdf | Jeff]], [[Media: LeeGustavo_Inverted_Pendulum_Final_Report_.pdf | Gustavo]], [[Media: Suri_Balachandra_NLD_final.pdf | Balachandra]]
|}

== Announcements: Fall 2011 ==

* Below is a list of the assigned dates for groups to work in the lab. If any groups have conflicts with the dates listed please email [mailto:nick.gravish@gmail.com me] immediately to resolve this.

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Date range
! Groups
|-
| 10/24 - 10/28
| Inverted pendulum ([[Group 7]])
|-
| 10/31 - 11/4
| Inelastic bouncing ball ([[Group 3]]) and dripping faucet ([[Group 5]])
|-
| 11/7 - 11/11
| Synchronization ([[Group 4]]) and Ferrofluid ([[Group 6]])
|-
| 11/14 - 11/18
| Faraday waves ([[Group 1]]) and Plinko dynamics ([[Group 2]])
|-
| 11/28 - 12/2
| Presentations beginning Thursday
|-
| 12/5 - 12/9
| Presentations
|}

* Information about the final presentations and paper can now be found on the [[Final]] page.
* Lab dates have been posted 10/19/11
* Added capability to embed youtube videos. Use the following code to embed <pre><videoflash>YouTubeFileName</videoflash></pre>
* Math rendering has been added using the [http://www.mathjax.org/ MathJax] program. This renders latex code typed into the page source and allows for copy/paste, scaling, and other features. For a Latex tutorial check [http://en.wikibooks.org/wiki/LaTeX/Mathematics here].
* Groups are only able to edit the page associated with their group.
* When you receive your group login and password you should change your password for security.
* For questions about this site email [mailto:nick.gravish@gmail.com Nick Gravish]

== About this wiki ==

See the [[About]] page for more information. This Wiki is open to the public to view but not to edit. However, we gladly make content available to other schools for non-profit educational use. Some links to copyright-protected references and software are not available to anyone without authentication as a Georgia Tech student or staff.

== Wiki 101 ==

Here is a link to mediawiki [http://en.wikipedia.org/wiki/Wikipedia:Cheatsheet cheatsheet].

File:Astrojax Pendulum.pdf

2015-03-09T14:58:34Z

PatrickChang: Final presentation from 2014 group 3 Astrojax Pendulum

Final presentation from 2014 group 3 Astrojax Pendulum

Main Page

2015-03-09T14:58:09Z

PatrickChang:

[[File: Howey_Building.jpg | thumb | Howey building. School of Physics, Georgia Tech]]

'''This is the wiki page associated with Physics 4268/6268 Nonlinear Dynamics & Chaos.''' Physics 4268/6268 is an undergraduate and graduate level nonlinear dynamics course taught by [http://crablab.gatech.edu/ Professor Daniel I. Goldman] in the [http://www.physics.gatech.edu School of Physics] at the [http://www.gatech.edu Georgia Institute of Technology]. The TAs are [http://people.seas.harvard.edu/~gravish/index.html Nick Gravish] and Feifei Qian. Wiki is maintained by [http://people.seas.harvard.edu/~gravish/index.html Nick Gravish]. The course is comprised of both a classroom component and a laboratory component. In the laboratory component student led groups will perform a nonlinear dynamics experiment and report their findings. Students thus develop intuition for nonlinear dynamics both on paper and in a hands-on laboratory environment.

'''The goal of this wiki is for students to interactively develop and compile a library of experiments that illustrate fundamental principles of nonlinear dynamics for further educational use.'''

== Site information ==

On this site you will find all content related to the student led nonlinear dynamics experiments. Announcements for the '''2012''' course can be found below. In addition all materials from previous years have been posted so that students may see examples of previous project wiki pages, papers, and presentations.

Each group is given access a project page in which they will edit and add content to a wikipedia styled entry detailing the experiment being performed. The page should include a brief summary of the experiment and the relevant scientific questions followed by a more comprehensive explanation of the theoretical and experimental details. Pages should contain references to outside information via journal or book citations, and external web pages. Once the lab has been completed the page should be edited to include information on experimental analysis, results, and a conclusion. Sample pages from previous years serve as useful references.

== Final papers / presentations 2014 ==

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Project
! Files
|-
|[[Group_1_2014 | Chiral Object]]
|[[Media: Chiral_presentation.pdf | Presentation]]. [[Media: Separation-chiral-particles.pdf | Final report]]
|-
| [[Group_2_2014 | Faraday Instability]]
| [[Media: Faraday.pdf| Final report]]
|-
| [[Group_3_2014 | Astrojax Pendulum]]
| [[Media: Astrojax Pendulum.pdf | Presentation]]. [[Media: NLD.pdf | Final report]]
|-
| [[Group_4_2014 | Jump on Granular Media]]
| [[Media: Jumping-aerated-granular.pdf | Final report]]
|-
| [[Group_5_2014 | Pendulum Synchronization]]
| [[Media: Pendulum.pdf | Final report]]
|}
== Final papers / presentations 2012 ==

'''Congratulations to all the students!''' [[Pictures_2012 | Pictures and Videos]] from the semester have been uploaded as well as the final presentations and papers which can be found below.

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Project
! Files
|-
|[[Group_1_2012 | Duffing Oscillator]]
|[[Media: Duffingfinal.pdf | Presentation]]. '''Papers:''' [[Media: DuffingChampion.pdf | Andrew]], [[Media: Lodhi_finalProject.pdf | Aemen]], [[Media: DuffingGranowski.pdf | Ross]]
|-
| [[Group_2_2012 | Firefly Synchronization]]
| [[Media: Firefly_Final_Presentation.pdf| Presentation]]. '''Papers:''' [[Media: Morris Huang.pdf‎ | Morris]], [[Media: Mark Kingsbury.pdf | Mark]], [[Media: McInroe Final Paper.pdf | Ben]], [[Media: Wagstaff Project.pdf | Will]]
|-
| [[Group_3_2012 | Chua's Circuit]]
| [[Media: Experimental Characterizing of Nonlinear dynamics of Chua’s circuit.pdf | Presentation]]. '''Papers:''' [[Media: Patrick Chang-final paper.pdf | Patrick]], [[Media: CoyleGroup2Chua.pdf | Edward]], [[Media: Parker Report.pdf | John]], [[Media: Majid Paper.pdf | Majid]]
|-
| [[Group_4_2012 | Cricket Synchronization]]
| [[Media: Cricket Presentation.pdf | Main presentation]], [[Media: Charlie_Presentation.pdf | Presentation on pseudo-synchronization leading the pack]]. '''Papers:''' [[Media: Blythe report.pdf | Justin]], [[Media: Charlie Report.pdf | Charlie]], [[Media: Yuxuan Report.pdf | Yuxuan]]
|}

== Announcements: Fall 2012 ==

* Student groups will be posted shortly. Projects suggestions will be made in class 9/11/2012
* The room Howey S204 has been reserved for student groups to meet and discuss the following hours and dates:
** Mondays from Oct 22 - Dec 3 from 10am to 12pm
** Tuesdays and Thursdays from 11am to 12pm
** Wednesdays from Oct 31 - Dec 5 from 10am to 12pm
** Fridays from 3pm to 5 pm

== Final papers / presentations 2011 ==

'''Congratulations to all the students, the final papers and presentations were universally excellent!''' [[Pictures_2011 | Pictures]] from the semester have been uploaded as well as the final presentations and papers which can be found below.

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Project
! Files
|-
|[[Group_1 | Faraday waves]]
|[[Media: ProjectFaraday2.pdf | Main presentation]], [[Media:Corn_Starch_Slides.pdf | Non-newtonian presentation]]. '''Papers:''' [[Media: PaulCardenasLizana.pdf‎ | Paul]], [[Media: Orphee_Juan_Faraday.pdf | Juan]]
|-
| [[Group_2 | Plinko dynamics]]
| [[Media: Plinko_Compressed.pdf| Presentation]]. '''Papers:''' [[Media: Mass_Andrew.pdf | Andrew]], [[Media: Hardin_Charles_-_Group_2_-_6268_Final_Paper.pdf | Andrew]], [[Media: Cordell_Group2_Phys6268.pdf | Chris]]
|-
| [[ Group_3 | Inelastic bouncing ball]]
| [[Media: NLD_Presentation_Official_Final_Final_For_Real.pdf | Presentation]]. '''Papers:''' [[Media: Arora_Nitin_Paper_nitin.pdf | Nitin]], [[Media: Gray_Phillip.pdf | Phillip]], [[Media: Yunis_Jacob_6268_final.pdf | Jacob]], [[Media: Rodesneyinelastic_ball.pdf | Chris]]
|-
| [[Group_4 | Synchronization]]
| [[Media: Group4_Metronomes_Talk.pdf | Presentation]]. '''Papers:''' [[Media: Levenfeld_Vlad.pdf | Vlad]], [[Media: Jover_finalpaper_metronomes_jover.pdf | Luis]], [[Media: Taylor_Bradford.pdf | Brad]], [[Media: Tithof_Metronomes.pdf | Jeff]]
|-
| [[Group_5 | Chaotic faucet]]
| [[Media: PHYS6268 Group5 Presentation.pdf | Presentation]]. '''Papers:''' [[Media: Patel_Ricky_PHYS6268.pdf | Ricky ]], [[Media: Royer_Caleb_PHYS6268_Royer_FinalPaper.pdf | Caleb ]], [[Media: Job_Josh_ChaoticFaucetfinal.pdf | Joshua ]], [[Media: Pritchard_Peter_Phys6268_Final_Paper.pdf | Nick ]]
|-
| [[Group_6 | Ferrofluid]]
| [[Media: Presentation.pdf | Presentation]]. '''Papers:''' [[Media: Schoenwald_Group6_Final.pdf | Kipp]], [[Media: Potter_Ferrofluids_-_Potter.pdf | Daniel]], [[Media: Hamid,_Amir_Ferrofluid_Group_6.pdf | Amir]]
|-
| [[Group_7 | Inverted pendulum]]
| [[Media: The_Inverted_Pendulum.pdf | Presentation]]. '''Papers:''' [[Media: Marcotte_Dynamic_Stabilization_of_the_Inverted_Pendulum.pdf | Chris]], [[Media: Aguilar_Jeff_PendulumPaper.pdf | Jeff]], [[Media: LeeGustavo_Inverted_Pendulum_Final_Report_.pdf | Gustavo]], [[Media: Suri_Balachandra_NLD_final.pdf | Balachandra]]
|}

== Announcements: Fall 2011 ==

* Below is a list of the assigned dates for groups to work in the lab. If any groups have conflicts with the dates listed please email [mailto:nick.gravish@gmail.com me] immediately to resolve this.

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Date range
! Groups
|-
| 10/24 - 10/28
| Inverted pendulum ([[Group 7]])
|-
| 10/31 - 11/4
| Inelastic bouncing ball ([[Group 3]]) and dripping faucet ([[Group 5]])
|-
| 11/7 - 11/11
| Synchronization ([[Group 4]]) and Ferrofluid ([[Group 6]])
|-
| 11/14 - 11/18
| Faraday waves ([[Group 1]]) and Plinko dynamics ([[Group 2]])
|-
| 11/28 - 12/2
| Presentations beginning Thursday
|-
| 12/5 - 12/9
| Presentations
|}

* Information about the final presentations and paper can now be found on the [[Final]] page.
* Lab dates have been posted 10/19/11
* Added capability to embed youtube videos. Use the following code to embed <pre><videoflash>YouTubeFileName</videoflash></pre>
* Math rendering has been added using the [http://www.mathjax.org/ MathJax] program. This renders latex code typed into the page source and allows for copy/paste, scaling, and other features. For a Latex tutorial check [http://en.wikibooks.org/wiki/LaTeX/Mathematics here].
* Groups are only able to edit the page associated with their group.
* When you receive your group login and password you should change your password for security.
* For questions about this site email [mailto:nick.gravish@gmail.com Nick Gravish]

== About this wiki ==

See the [[About]] page for more information. This Wiki is open to the public to view but not to edit. However, we gladly make content available to other schools for non-profit educational use. Some links to copyright-protected references and software are not available to anyone without authentication as a Georgia Tech student or staff.

== Wiki 101 ==

Here is a link to mediawiki [http://en.wikipedia.org/wiki/Wikipedia:Cheatsheet cheatsheet].

Main Page

2015-03-09T14:55:52Z

PatrickChang:

[[File: Howey_Building.jpg | thumb | Howey building. School of Physics, Georgia Tech]]

'''This is the wiki page associated with Physics 4268/6268 Nonlinear Dynamics & Chaos.''' Physics 4268/6268 is an undergraduate and graduate level nonlinear dynamics course taught by [http://crablab.gatech.edu/ Professor Daniel I. Goldman] in the [http://www.physics.gatech.edu School of Physics] at the [http://www.gatech.edu Georgia Institute of Technology]. The TAs are [http://people.seas.harvard.edu/~gravish/index.html Nick Gravish] and Feifei Qian. Wiki is maintained by [http://people.seas.harvard.edu/~gravish/index.html Nick Gravish]. The course is comprised of both a classroom component and a laboratory component. In the laboratory component student led groups will perform a nonlinear dynamics experiment and report their findings. Students thus develop intuition for nonlinear dynamics both on paper and in a hands-on laboratory environment.

'''The goal of this wiki is for students to interactively develop and compile a library of experiments that illustrate fundamental principles of nonlinear dynamics for further educational use.'''

== Site information ==

On this site you will find all content related to the student led nonlinear dynamics experiments. Announcements for the '''2012''' course can be found below. In addition all materials from previous years have been posted so that students may see examples of previous project wiki pages, papers, and presentations.

Each group is given access a project page in which they will edit and add content to a wikipedia styled entry detailing the experiment being performed. The page should include a brief summary of the experiment and the relevant scientific questions followed by a more comprehensive explanation of the theoretical and experimental details. Pages should contain references to outside information via journal or book citations, and external web pages. Once the lab has been completed the page should be edited to include information on experimental analysis, results, and a conclusion. Sample pages from previous years serve as useful references.

== Final papers / presentations 2014 ==

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Project
! Files
|-
|[[Group_1_2014 | Chiral Object]]
|[[Media: Chiral_presentation.pdf | Presentation]]. [[Media: Separation-chiral-particles.pdf | Final report]]
|-
| [[Group_2_2014 | Faraday Instability]]
| [[Media: Faraday.pdf| Final report]]
|-
| [[Group_3_2014 | Astrojax Pendulum]]
| [[Media: Astrojax Pendulum.pptx | Presentation]]. [[Media: NLD.pdf | Final report]]
|-
| [[Group_4_2014 | Jump on Granular Media]]
| [[Media: Jumping-aerated-granular.pdf | Final report]]
|-
| [[Group_5_2014 | Pendulum Synchronization]]
| [[Media: Pendulum.pdf | Final report]]
|}
== Final papers / presentations 2012 ==

'''Congratulations to all the students!''' [[Pictures_2012 | Pictures and Videos]] from the semester have been uploaded as well as the final presentations and papers which can be found below.

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Project
! Files
|-
|[[Group_1_2012 | Duffing Oscillator]]
|[[Media: Duffingfinal.pdf | Presentation]]. '''Papers:''' [[Media: DuffingChampion.pdf | Andrew]], [[Media: Lodhi_finalProject.pdf | Aemen]], [[Media: DuffingGranowski.pdf | Ross]]
|-
| [[Group_2_2012 | Firefly Synchronization]]
| [[Media: Firefly_Final_Presentation.pdf| Presentation]]. '''Papers:''' [[Media: Morris Huang.pdf‎ | Morris]], [[Media: Mark Kingsbury.pdf | Mark]], [[Media: McInroe Final Paper.pdf | Ben]], [[Media: Wagstaff Project.pdf | Will]]
|-
| [[Group_3_2012 | Chua's Circuit]]
| [[Media: Experimental Characterizing of Nonlinear dynamics of Chua’s circuit.pdf | Presentation]]. '''Papers:''' [[Media: Patrick Chang-final paper.pdf | Patrick]], [[Media: CoyleGroup2Chua.pdf | Edward]], [[Media: Parker Report.pdf | John]], [[Media: Majid Paper.pdf | Majid]]
|-
| [[Group_4_2012 | Cricket Synchronization]]
| [[Media: Cricket Presentation.pdf | Main presentation]], [[Media: Charlie_Presentation.pdf | Presentation on pseudo-synchronization leading the pack]]. '''Papers:''' [[Media: Blythe report.pdf | Justin]], [[Media: Charlie Report.pdf | Charlie]], [[Media: Yuxuan Report.pdf | Yuxuan]]
|}

== Announcements: Fall 2012 ==

* Student groups will be posted shortly. Projects suggestions will be made in class 9/11/2012
* The room Howey S204 has been reserved for student groups to meet and discuss the following hours and dates:
** Mondays from Oct 22 - Dec 3 from 10am to 12pm
** Tuesdays and Thursdays from 11am to 12pm
** Wednesdays from Oct 31 - Dec 5 from 10am to 12pm
** Fridays from 3pm to 5 pm

== Final papers / presentations 2011 ==

'''Congratulations to all the students, the final papers and presentations were universally excellent!''' [[Pictures_2011 | Pictures]] from the semester have been uploaded as well as the final presentations and papers which can be found below.

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Project
! Files
|-
|[[Group_1 | Faraday waves]]
|[[Media: ProjectFaraday2.pdf | Main presentation]], [[Media:Corn_Starch_Slides.pdf | Non-newtonian presentation]]. '''Papers:''' [[Media: PaulCardenasLizana.pdf‎ | Paul]], [[Media: Orphee_Juan_Faraday.pdf | Juan]]
|-
| [[Group_2 | Plinko dynamics]]
| [[Media: Plinko_Compressed.pdf| Presentation]]. '''Papers:''' [[Media: Mass_Andrew.pdf | Andrew]], [[Media: Hardin_Charles_-_Group_2_-_6268_Final_Paper.pdf | Andrew]], [[Media: Cordell_Group2_Phys6268.pdf | Chris]]
|-
| [[ Group_3 | Inelastic bouncing ball]]
| [[Media: NLD_Presentation_Official_Final_Final_For_Real.pdf | Presentation]]. '''Papers:''' [[Media: Arora_Nitin_Paper_nitin.pdf | Nitin]], [[Media: Gray_Phillip.pdf | Phillip]], [[Media: Yunis_Jacob_6268_final.pdf | Jacob]], [[Media: Rodesneyinelastic_ball.pdf | Chris]]
|-
| [[Group_4 | Synchronization]]
| [[Media: Group4_Metronomes_Talk.pdf | Presentation]]. '''Papers:''' [[Media: Levenfeld_Vlad.pdf | Vlad]], [[Media: Jover_finalpaper_metronomes_jover.pdf | Luis]], [[Media: Taylor_Bradford.pdf | Brad]], [[Media: Tithof_Metronomes.pdf | Jeff]]
|-
| [[Group_5 | Chaotic faucet]]
| [[Media: PHYS6268 Group5 Presentation.pdf | Presentation]]. '''Papers:''' [[Media: Patel_Ricky_PHYS6268.pdf | Ricky ]], [[Media: Royer_Caleb_PHYS6268_Royer_FinalPaper.pdf | Caleb ]], [[Media: Job_Josh_ChaoticFaucetfinal.pdf | Joshua ]], [[Media: Pritchard_Peter_Phys6268_Final_Paper.pdf | Nick ]]
|-
| [[Group_6 | Ferrofluid]]
| [[Media: Presentation.pdf | Presentation]]. '''Papers:''' [[Media: Schoenwald_Group6_Final.pdf | Kipp]], [[Media: Potter_Ferrofluids_-_Potter.pdf | Daniel]], [[Media: Hamid,_Amir_Ferrofluid_Group_6.pdf | Amir]]
|-
| [[Group_7 | Inverted pendulum]]
| [[Media: The_Inverted_Pendulum.pdf | Presentation]]. '''Papers:''' [[Media: Marcotte_Dynamic_Stabilization_of_the_Inverted_Pendulum.pdf | Chris]], [[Media: Aguilar_Jeff_PendulumPaper.pdf | Jeff]], [[Media: LeeGustavo_Inverted_Pendulum_Final_Report_.pdf | Gustavo]], [[Media: Suri_Balachandra_NLD_final.pdf | Balachandra]]
|}

== Announcements: Fall 2011 ==

* Below is a list of the assigned dates for groups to work in the lab. If any groups have conflicts with the dates listed please email [mailto:nick.gravish@gmail.com me] immediately to resolve this.

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Date range
! Groups
|-
| 10/24 - 10/28
| Inverted pendulum ([[Group 7]])
|-
| 10/31 - 11/4
| Inelastic bouncing ball ([[Group 3]]) and dripping faucet ([[Group 5]])
|-
| 11/7 - 11/11
| Synchronization ([[Group 4]]) and Ferrofluid ([[Group 6]])
|-
| 11/14 - 11/18
| Faraday waves ([[Group 1]]) and Plinko dynamics ([[Group 2]])
|-
| 11/28 - 12/2
| Presentations beginning Thursday
|-
| 12/5 - 12/9
| Presentations
|}

* Information about the final presentations and paper can now be found on the [[Final]] page.
* Lab dates have been posted 10/19/11
* Added capability to embed youtube videos. Use the following code to embed <pre><videoflash>YouTubeFileName</videoflash></pre>
* Math rendering has been added using the [http://www.mathjax.org/ MathJax] program. This renders latex code typed into the page source and allows for copy/paste, scaling, and other features. For a Latex tutorial check [http://en.wikibooks.org/wiki/LaTeX/Mathematics here].
* Groups are only able to edit the page associated with their group.
* When you receive your group login and password you should change your password for security.
* For questions about this site email [mailto:nick.gravish@gmail.com Nick Gravish]

== About this wiki ==

See the [[About]] page for more information. This Wiki is open to the public to view but not to edit. However, we gladly make content available to other schools for non-profit educational use. Some links to copyright-protected references and software are not available to anyone without authentication as a Georgia Tech student or staff.

== Wiki 101 ==

Here is a link to mediawiki [http://en.wikipedia.org/wiki/Wikipedia:Cheatsheet cheatsheet].

Main Page

2015-03-06T19:40:43Z

PatrickChang:

[[File: Howey_Building.jpg | thumb | Howey building. School of Physics, Georgia Tech]]

'''This is the wiki page associated with Physics 4268/6268 Nonlinear Dynamics & Chaos.''' Physics 4268/6268 is an undergraduate and graduate level nonlinear dynamics course taught by [http://crablab.gatech.edu/ Professor Daniel I. Goldman] in the [http://www.physics.gatech.edu School of Physics] at the [http://www.gatech.edu Georgia Institute of Technology]. The TAs are [http://people.seas.harvard.edu/~gravish/index.html Nick Gravish] and Feifei Qian. Wiki is maintained by [http://people.seas.harvard.edu/~gravish/index.html Nick Gravish]. The course is comprised of both a classroom component and a laboratory component. In the laboratory component student led groups will perform a nonlinear dynamics experiment and report their findings. Students thus develop intuition for nonlinear dynamics both on paper and in a hands-on laboratory environment.

'''The goal of this wiki is for students to interactively develop and compile a library of experiments that illustrate fundamental principles of nonlinear dynamics for further educational use.'''

== Site information ==

On this site you will find all content related to the student led nonlinear dynamics experiments. Announcements for the '''2012''' course can be found below. In addition all materials from previous years have been posted so that students may see examples of previous project wiki pages, papers, and presentations.

Each group is given access a project page in which they will edit and add content to a wikipedia styled entry detailing the experiment being performed. The page should include a brief summary of the experiment and the relevant scientific questions followed by a more comprehensive explanation of the theoretical and experimental details. Pages should contain references to outside information via journal or book citations, and external web pages. Once the lab has been completed the page should be edited to include information on experimental analysis, results, and a conclusion. Sample pages from previous years serve as useful references.

== Final papers / presentations 2014 ==

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Project
! Files
|-
|[[Group_1_2014 | Chiral Object]]
|[[Media: Chiral_presentation.pdf | Presentation]]. [[Media: Separation-chiral-particles.pdf | Final report]]
|-
| [[Group_2_2014 | Faraday Instability]]
| [[Media: Faraday.pdf| Final report]]
|-
| [[Group_3_2014 | Astrojax Pendulum]]
| [[Media: NLD.pdf | Final report]]
|-
| [[Group_4_2014 | Jump on Granular Media]]
| [[Media: Jumping-aerated-granular.pdf | Final report]]
|-
| [[Group_5_2014 | Pendulum Synchronization]]
| [[Media: Pendulum.pdf | Final report]]
|}
== Final papers / presentations 2012 ==

'''Congratulations to all the students!''' [[Pictures_2012 | Pictures and Videos]] from the semester have been uploaded as well as the final presentations and papers which can be found below.

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Project
! Files
|-
|[[Group_1_2012 | Duffing Oscillator]]
|[[Media: Duffingfinal.pdf | Presentation]]. '''Papers:''' [[Media: DuffingChampion.pdf | Andrew]], [[Media: Lodhi_finalProject.pdf | Aemen]], [[Media: DuffingGranowski.pdf | Ross]]
|-
| [[Group_2_2012 | Firefly Synchronization]]
| [[Media: Firefly_Final_Presentation.pdf| Presentation]]. '''Papers:''' [[Media: Morris Huang.pdf‎ | Morris]], [[Media: Mark Kingsbury.pdf | Mark]], [[Media: McInroe Final Paper.pdf | Ben]], [[Media: Wagstaff Project.pdf | Will]]
|-
| [[Group_3_2012 | Chua's Circuit]]
| [[Media: Experimental Characterizing of Nonlinear dynamics of Chua’s circuit.pdf | Presentation]]. '''Papers:''' [[Media: Patrick Chang-final paper.pdf | Patrick]], [[Media: CoyleGroup2Chua.pdf | Edward]], [[Media: Parker Report.pdf | John]], [[Media: Majid Paper.pdf | Majid]]
|-
| [[Group_4_2012 | Cricket Synchronization]]
| [[Media: Cricket Presentation.pdf | Main presentation]], [[Media: Charlie_Presentation.pdf | Presentation on pseudo-synchronization leading the pack]]. '''Papers:''' [[Media: Blythe report.pdf | Justin]], [[Media: Charlie Report.pdf | Charlie]], [[Media: Yuxuan Report.pdf | Yuxuan]]
|}

== Announcements: Fall 2012 ==

* Student groups will be posted shortly. Projects suggestions will be made in class 9/11/2012
* The room Howey S204 has been reserved for student groups to meet and discuss the following hours and dates:
** Mondays from Oct 22 - Dec 3 from 10am to 12pm
** Tuesdays and Thursdays from 11am to 12pm
** Wednesdays from Oct 31 - Dec 5 from 10am to 12pm
** Fridays from 3pm to 5 pm

== Final papers / presentations 2011 ==

'''Congratulations to all the students, the final papers and presentations were universally excellent!''' [[Pictures_2011 | Pictures]] from the semester have been uploaded as well as the final presentations and papers which can be found below.

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Project
! Files
|-
|[[Group_1 | Faraday waves]]
|[[Media: ProjectFaraday2.pdf | Main presentation]], [[Media:Corn_Starch_Slides.pdf | Non-newtonian presentation]]. '''Papers:''' [[Media: PaulCardenasLizana.pdf‎ | Paul]], [[Media: Orphee_Juan_Faraday.pdf | Juan]]
|-
| [[Group_2 | Plinko dynamics]]
| [[Media: Plinko_Compressed.pdf| Presentation]]. '''Papers:''' [[Media: Mass_Andrew.pdf | Andrew]], [[Media: Hardin_Charles_-_Group_2_-_6268_Final_Paper.pdf | Andrew]], [[Media: Cordell_Group2_Phys6268.pdf | Chris]]
|-
| [[ Group_3 | Inelastic bouncing ball]]
| [[Media: NLD_Presentation_Official_Final_Final_For_Real.pdf | Presentation]]. '''Papers:''' [[Media: Arora_Nitin_Paper_nitin.pdf | Nitin]], [[Media: Gray_Phillip.pdf | Phillip]], [[Media: Yunis_Jacob_6268_final.pdf | Jacob]], [[Media: Rodesneyinelastic_ball.pdf | Chris]]
|-
| [[Group_4 | Synchronization]]
| [[Media: Group4_Metronomes_Talk.pdf | Presentation]]. '''Papers:''' [[Media: Levenfeld_Vlad.pdf | Vlad]], [[Media: Jover_finalpaper_metronomes_jover.pdf | Luis]], [[Media: Taylor_Bradford.pdf | Brad]], [[Media: Tithof_Metronomes.pdf | Jeff]]
|-
| [[Group_5 | Chaotic faucet]]
| [[Media: PHYS6268 Group5 Presentation.pdf | Presentation]]. '''Papers:''' [[Media: Patel_Ricky_PHYS6268.pdf | Ricky ]], [[Media: Royer_Caleb_PHYS6268_Royer_FinalPaper.pdf | Caleb ]], [[Media: Job_Josh_ChaoticFaucetfinal.pdf | Joshua ]], [[Media: Pritchard_Peter_Phys6268_Final_Paper.pdf | Nick ]]
|-
| [[Group_6 | Ferrofluid]]
| [[Media: Presentation.pdf | Presentation]]. '''Papers:''' [[Media: Schoenwald_Group6_Final.pdf | Kipp]], [[Media: Potter_Ferrofluids_-_Potter.pdf | Daniel]], [[Media: Hamid,_Amir_Ferrofluid_Group_6.pdf | Amir]]
|-
| [[Group_7 | Inverted pendulum]]
| [[Media: The_Inverted_Pendulum.pdf | Presentation]]. '''Papers:''' [[Media: Marcotte_Dynamic_Stabilization_of_the_Inverted_Pendulum.pdf | Chris]], [[Media: Aguilar_Jeff_PendulumPaper.pdf | Jeff]], [[Media: LeeGustavo_Inverted_Pendulum_Final_Report_.pdf | Gustavo]], [[Media: Suri_Balachandra_NLD_final.pdf | Balachandra]]
|}

== Announcements: Fall 2011 ==

* Below is a list of the assigned dates for groups to work in the lab. If any groups have conflicts with the dates listed please email [mailto:nick.gravish@gmail.com me] immediately to resolve this.

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Date range
! Groups
|-
| 10/24 - 10/28
| Inverted pendulum ([[Group 7]])
|-
| 10/31 - 11/4
| Inelastic bouncing ball ([[Group 3]]) and dripping faucet ([[Group 5]])
|-
| 11/7 - 11/11
| Synchronization ([[Group 4]]) and Ferrofluid ([[Group 6]])
|-
| 11/14 - 11/18
| Faraday waves ([[Group 1]]) and Plinko dynamics ([[Group 2]])
|-
| 11/28 - 12/2
| Presentations beginning Thursday
|-
| 12/5 - 12/9
| Presentations
|}

* Information about the final presentations and paper can now be found on the [[Final]] page.
* Lab dates have been posted 10/19/11
* Added capability to embed youtube videos. Use the following code to embed <pre><videoflash>YouTubeFileName</videoflash></pre>
* Math rendering has been added using the [http://www.mathjax.org/ MathJax] program. This renders latex code typed into the page source and allows for copy/paste, scaling, and other features. For a Latex tutorial check [http://en.wikibooks.org/wiki/LaTeX/Mathematics here].
* Groups are only able to edit the page associated with their group.
* When you receive your group login and password you should change your password for security.
* For questions about this site email [mailto:nick.gravish@gmail.com Nick Gravish]

== About this wiki ==

See the [[About]] page for more information. This Wiki is open to the public to view but not to edit. However, we gladly make content available to other schools for non-profit educational use. Some links to copyright-protected references and software are not available to anyone without authentication as a Georgia Tech student or staff.

== Wiki 101 ==

Here is a link to mediawiki [http://en.wikipedia.org/wiki/Wikipedia:Cheatsheet cheatsheet].

Main Page

2015-03-06T19:40:25Z

PatrickChang:

[[File: Howey_Building.jpg | thumb | Howey building. School of Physics, Georgia Tech]]

'''This is the wiki page associated with Physics 4268/6268 Nonlinear Dynamics & Chaos.''' Physics 4268/6268 is an undergraduate and graduate level nonlinear dynamics course taught by [http://crablab.gatech.edu/ Professor Daniel I. Goldman] in the [http://www.physics.gatech.edu School of Physics] at the [http://www.gatech.edu Georgia Institute of Technology]. The TAs are [http://people.seas.harvard.edu/~gravish/index.html Nick Gravish] and Feifei Qian. Wiki is maintained by [http://people.seas.harvard.edu/~gravish/index.html Nick Gravish]. The course is comprised of both a classroom component and a laboratory component. In the laboratory component student led groups will perform a nonlinear dynamics experiment and report their findings. Students thus develop intuition for nonlinear dynamics both on paper and in a hands-on laboratory environment.

'''The goal of this wiki is for students to interactively develop and compile a library of experiments that illustrate fundamental principles of nonlinear dynamics for further educational use.'''

== Site information ==

On this site you will find all content related to the student led nonlinear dynamics experiments. Announcements for the '''2012''' course can be found below. In addition all materials from previous years have been posted so that students may see examples of previous project wiki pages, papers, and presentations.

Each group is given access a project page in which they will edit and add content to a wikipedia styled entry detailing the experiment being performed. The page should include a brief summary of the experiment and the relevant scientific questions followed by a more comprehensive explanation of the theoretical and experimental details. Pages should contain references to outside information via journal or book citations, and external web pages. Once the lab has been completed the page should be edited to include information on experimental analysis, results, and a conclusion. Sample pages from previous years serve as useful references.

== Final papers / presentations 2014 ==

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Project
! Files
|-
|[[Group_1_2014 | Chiral Object]]
|[[Media: Chiral_presentation.pdf | Presentation]]. [[Media: Separation-chiral-particles.pdf | Final report]]
|-
| [[Group_2_2014 | Faraday Instability]]
| [[Media: Faraday.pdf| Final report]].
|-
| [[Group_3_2014 | Astrojax Pendulum]]
| [[Media: NLD.pdf | Final report]].
|-
| [[Group_4_2014 | Jump on Granular Media]]
| [[Media: Jumping-aerated-granular.pdf | Final report]]
|-
| [[Group_5_2014 | Pendulum Synchronization]]
| [[Media: Pendulum.pdf | Final report]]
|}
== Final papers / presentations 2012 ==

'''Congratulations to all the students!''' [[Pictures_2012 | Pictures and Videos]] from the semester have been uploaded as well as the final presentations and papers which can be found below.

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Project
! Files
|-
|[[Group_1_2012 | Duffing Oscillator]]
|[[Media: Duffingfinal.pdf | Presentation]]. '''Papers:''' [[Media: DuffingChampion.pdf | Andrew]], [[Media: Lodhi_finalProject.pdf | Aemen]], [[Media: DuffingGranowski.pdf | Ross]]
|-
| [[Group_2_2012 | Firefly Synchronization]]
| [[Media: Firefly_Final_Presentation.pdf| Presentation]]. '''Papers:''' [[Media: Morris Huang.pdf‎ | Morris]], [[Media: Mark Kingsbury.pdf | Mark]], [[Media: McInroe Final Paper.pdf | Ben]], [[Media: Wagstaff Project.pdf | Will]]
|-
| [[Group_3_2012 | Chua's Circuit]]
| [[Media: Experimental Characterizing of Nonlinear dynamics of Chua’s circuit.pdf | Presentation]]. '''Papers:''' [[Media: Patrick Chang-final paper.pdf | Patrick]], [[Media: CoyleGroup2Chua.pdf | Edward]], [[Media: Parker Report.pdf | John]], [[Media: Majid Paper.pdf | Majid]]
|-
| [[Group_4_2012 | Cricket Synchronization]]
| [[Media: Cricket Presentation.pdf | Main presentation]], [[Media: Charlie_Presentation.pdf | Presentation on pseudo-synchronization leading the pack]]. '''Papers:''' [[Media: Blythe report.pdf | Justin]], [[Media: Charlie Report.pdf | Charlie]], [[Media: Yuxuan Report.pdf | Yuxuan]]
|}

== Announcements: Fall 2012 ==

* Student groups will be posted shortly. Projects suggestions will be made in class 9/11/2012
* The room Howey S204 has been reserved for student groups to meet and discuss the following hours and dates:
** Mondays from Oct 22 - Dec 3 from 10am to 12pm
** Tuesdays and Thursdays from 11am to 12pm
** Wednesdays from Oct 31 - Dec 5 from 10am to 12pm
** Fridays from 3pm to 5 pm

== Final papers / presentations 2011 ==

'''Congratulations to all the students, the final papers and presentations were universally excellent!''' [[Pictures_2011 | Pictures]] from the semester have been uploaded as well as the final presentations and papers which can be found below.

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Project
! Files
|-
|[[Group_1 | Faraday waves]]
|[[Media: ProjectFaraday2.pdf | Main presentation]], [[Media:Corn_Starch_Slides.pdf | Non-newtonian presentation]]. '''Papers:''' [[Media: PaulCardenasLizana.pdf‎ | Paul]], [[Media: Orphee_Juan_Faraday.pdf | Juan]]
|-
| [[Group_2 | Plinko dynamics]]
| [[Media: Plinko_Compressed.pdf| Presentation]]. '''Papers:''' [[Media: Mass_Andrew.pdf | Andrew]], [[Media: Hardin_Charles_-_Group_2_-_6268_Final_Paper.pdf | Andrew]], [[Media: Cordell_Group2_Phys6268.pdf | Chris]]
|-
| [[ Group_3 | Inelastic bouncing ball]]
| [[Media: NLD_Presentation_Official_Final_Final_For_Real.pdf | Presentation]]. '''Papers:''' [[Media: Arora_Nitin_Paper_nitin.pdf | Nitin]], [[Media: Gray_Phillip.pdf | Phillip]], [[Media: Yunis_Jacob_6268_final.pdf | Jacob]], [[Media: Rodesneyinelastic_ball.pdf | Chris]]
|-
| [[Group_4 | Synchronization]]
| [[Media: Group4_Metronomes_Talk.pdf | Presentation]]. '''Papers:''' [[Media: Levenfeld_Vlad.pdf | Vlad]], [[Media: Jover_finalpaper_metronomes_jover.pdf | Luis]], [[Media: Taylor_Bradford.pdf | Brad]], [[Media: Tithof_Metronomes.pdf | Jeff]]
|-
| [[Group_5 | Chaotic faucet]]
| [[Media: PHYS6268 Group5 Presentation.pdf | Presentation]]. '''Papers:''' [[Media: Patel_Ricky_PHYS6268.pdf | Ricky ]], [[Media: Royer_Caleb_PHYS6268_Royer_FinalPaper.pdf | Caleb ]], [[Media: Job_Josh_ChaoticFaucetfinal.pdf | Joshua ]], [[Media: Pritchard_Peter_Phys6268_Final_Paper.pdf | Nick ]]
|-
| [[Group_6 | Ferrofluid]]
| [[Media: Presentation.pdf | Presentation]]. '''Papers:''' [[Media: Schoenwald_Group6_Final.pdf | Kipp]], [[Media: Potter_Ferrofluids_-_Potter.pdf | Daniel]], [[Media: Hamid,_Amir_Ferrofluid_Group_6.pdf | Amir]]
|-
| [[Group_7 | Inverted pendulum]]
| [[Media: The_Inverted_Pendulum.pdf | Presentation]]. '''Papers:''' [[Media: Marcotte_Dynamic_Stabilization_of_the_Inverted_Pendulum.pdf | Chris]], [[Media: Aguilar_Jeff_PendulumPaper.pdf | Jeff]], [[Media: LeeGustavo_Inverted_Pendulum_Final_Report_.pdf | Gustavo]], [[Media: Suri_Balachandra_NLD_final.pdf | Balachandra]]
|}

== Announcements: Fall 2011 ==

* Below is a list of the assigned dates for groups to work in the lab. If any groups have conflicts with the dates listed please email [mailto:nick.gravish@gmail.com me] immediately to resolve this.

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Date range
! Groups
|-
| 10/24 - 10/28
| Inverted pendulum ([[Group 7]])
|-
| 10/31 - 11/4
| Inelastic bouncing ball ([[Group 3]]) and dripping faucet ([[Group 5]])
|-
| 11/7 - 11/11
| Synchronization ([[Group 4]]) and Ferrofluid ([[Group 6]])
|-
| 11/14 - 11/18
| Faraday waves ([[Group 1]]) and Plinko dynamics ([[Group 2]])
|-
| 11/28 - 12/2
| Presentations beginning Thursday
|-
| 12/5 - 12/9
| Presentations
|}

* Information about the final presentations and paper can now be found on the [[Final]] page.
* Lab dates have been posted 10/19/11
* Added capability to embed youtube videos. Use the following code to embed <pre><videoflash>YouTubeFileName</videoflash></pre>
* Math rendering has been added using the [http://www.mathjax.org/ MathJax] program. This renders latex code typed into the page source and allows for copy/paste, scaling, and other features. For a Latex tutorial check [http://en.wikibooks.org/wiki/LaTeX/Mathematics here].
* Groups are only able to edit the page associated with their group.
* When you receive your group login and password you should change your password for security.
* For questions about this site email [mailto:nick.gravish@gmail.com Nick Gravish]

== About this wiki ==

See the [[About]] page for more information. This Wiki is open to the public to view but not to edit. However, we gladly make content available to other schools for non-profit educational use. Some links to copyright-protected references and software are not available to anyone without authentication as a Georgia Tech student or staff.

== Wiki 101 ==

Here is a link to mediawiki [http://en.wikipedia.org/wiki/Wikipedia:Cheatsheet cheatsheet].

Main Page

2015-03-06T19:39:49Z

PatrickChang:

[[File: Howey_Building.jpg | thumb | Howey building. School of Physics, Georgia Tech]]

'''This is the wiki page associated with Physics 4268/6268 Nonlinear Dynamics & Chaos.''' Physics 4268/6268 is an undergraduate and graduate level nonlinear dynamics course taught by [http://crablab.gatech.edu/ Professor Daniel I. Goldman] in the [http://www.physics.gatech.edu School of Physics] at the [http://www.gatech.edu Georgia Institute of Technology]. The TAs are [http://people.seas.harvard.edu/~gravish/index.html Nick Gravish] and Feifei Qian. Wiki is maintained by [http://people.seas.harvard.edu/~gravish/index.html Nick Gravish]. The course is comprised of both a classroom component and a laboratory component. In the laboratory component student led groups will perform a nonlinear dynamics experiment and report their findings. Students thus develop intuition for nonlinear dynamics both on paper and in a hands-on laboratory environment.

'''The goal of this wiki is for students to interactively develop and compile a library of experiments that illustrate fundamental principles of nonlinear dynamics for further educational use.'''

== Site information ==

On this site you will find all content related to the student led nonlinear dynamics experiments. Announcements for the '''2012''' course can be found below. In addition all materials from previous years have been posted so that students may see examples of previous project wiki pages, papers, and presentations.

Each group is given access a project page in which they will edit and add content to a wikipedia styled entry detailing the experiment being performed. The page should include a brief summary of the experiment and the relevant scientific questions followed by a more comprehensive explanation of the theoretical and experimental details. Pages should contain references to outside information via journal or book citations, and external web pages. Once the lab has been completed the page should be edited to include information on experimental analysis, results, and a conclusion. Sample pages from previous years serve as useful references.

== Final papers / presentations 2014 ==

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Project
! Files
|-
|[[Group_1_2014 | Chiral Object]]
|[[Media: Chiral_presentation.pdf | Presentation]]. [[Media: Separation-chiral-particles.pdf | Paper]]
|-
| [[Group_2_2014 | Faraday Instability]]
| [[Media: Faraday.pdf| Final report]].
|-
| [[Group_3_2014 | Astrojax Pendulum]]
| [[Media: NLD.pdf | Final report]].
|-
| [[Group_4_2014 | Jump on Granular Media]]
| [[Media: Jumping-aerated-granular.pdf | Final report]]
|-
| [[Group_5_2014 | Pendulum Synchronization]]
| [[Media: Pendulum.pdf | Final report]]
|}
== Final papers / presentations 2012 ==

'''Congratulations to all the students!''' [[Pictures_2012 | Pictures and Videos]] from the semester have been uploaded as well as the final presentations and papers which can be found below.

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Project
! Files
|-
|[[Group_1_2012 | Duffing Oscillator]]
|[[Media: Duffingfinal.pdf | Presentation]]. '''Papers:''' [[Media: DuffingChampion.pdf | Andrew]], [[Media: Lodhi_finalProject.pdf | Aemen]], [[Media: DuffingGranowski.pdf | Ross]]
|-
| [[Group_2_2012 | Firefly Synchronization]]
| [[Media: Firefly_Final_Presentation.pdf| Presentation]]. '''Papers:''' [[Media: Morris Huang.pdf‎ | Morris]], [[Media: Mark Kingsbury.pdf | Mark]], [[Media: McInroe Final Paper.pdf | Ben]], [[Media: Wagstaff Project.pdf | Will]]
|-
| [[Group_3_2012 | Chua's Circuit]]
| [[Media: Experimental Characterizing of Nonlinear dynamics of Chua’s circuit.pdf | Presentation]]. '''Papers:''' [[Media: Patrick Chang-final paper.pdf | Patrick]], [[Media: CoyleGroup2Chua.pdf | Edward]], [[Media: Parker Report.pdf | John]], [[Media: Majid Paper.pdf | Majid]]
|-
| [[Group_4_2012 | Cricket Synchronization]]
| [[Media: Cricket Presentation.pdf | Main presentation]], [[Media: Charlie_Presentation.pdf | Presentation on pseudo-synchronization leading the pack]]. '''Papers:''' [[Media: Blythe report.pdf | Justin]], [[Media: Charlie Report.pdf | Charlie]], [[Media: Yuxuan Report.pdf | Yuxuan]]
|}

== Announcements: Fall 2012 ==

* Student groups will be posted shortly. Projects suggestions will be made in class 9/11/2012
* The room Howey S204 has been reserved for student groups to meet and discuss the following hours and dates:
** Mondays from Oct 22 - Dec 3 from 10am to 12pm
** Tuesdays and Thursdays from 11am to 12pm
** Wednesdays from Oct 31 - Dec 5 from 10am to 12pm
** Fridays from 3pm to 5 pm

== Final papers / presentations 2011 ==

'''Congratulations to all the students, the final papers and presentations were universally excellent!''' [[Pictures_2011 | Pictures]] from the semester have been uploaded as well as the final presentations and papers which can be found below.

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Project
! Files
|-
|[[Group_1 | Faraday waves]]
|[[Media: ProjectFaraday2.pdf | Main presentation]], [[Media:Corn_Starch_Slides.pdf | Non-newtonian presentation]]. '''Papers:''' [[Media: PaulCardenasLizana.pdf‎ | Paul]], [[Media: Orphee_Juan_Faraday.pdf | Juan]]
|-
| [[Group_2 | Plinko dynamics]]
| [[Media: Plinko_Compressed.pdf| Presentation]]. '''Papers:''' [[Media: Mass_Andrew.pdf | Andrew]], [[Media: Hardin_Charles_-_Group_2_-_6268_Final_Paper.pdf | Andrew]], [[Media: Cordell_Group2_Phys6268.pdf | Chris]]
|-
| [[ Group_3 | Inelastic bouncing ball]]
| [[Media: NLD_Presentation_Official_Final_Final_For_Real.pdf | Presentation]]. '''Papers:''' [[Media: Arora_Nitin_Paper_nitin.pdf | Nitin]], [[Media: Gray_Phillip.pdf | Phillip]], [[Media: Yunis_Jacob_6268_final.pdf | Jacob]], [[Media: Rodesneyinelastic_ball.pdf | Chris]]
|-
| [[Group_4 | Synchronization]]
| [[Media: Group4_Metronomes_Talk.pdf | Presentation]]. '''Papers:''' [[Media: Levenfeld_Vlad.pdf | Vlad]], [[Media: Jover_finalpaper_metronomes_jover.pdf | Luis]], [[Media: Taylor_Bradford.pdf | Brad]], [[Media: Tithof_Metronomes.pdf | Jeff]]
|-
| [[Group_5 | Chaotic faucet]]
| [[Media: PHYS6268 Group5 Presentation.pdf | Presentation]]. '''Papers:''' [[Media: Patel_Ricky_PHYS6268.pdf | Ricky ]], [[Media: Royer_Caleb_PHYS6268_Royer_FinalPaper.pdf | Caleb ]], [[Media: Job_Josh_ChaoticFaucetfinal.pdf | Joshua ]], [[Media: Pritchard_Peter_Phys6268_Final_Paper.pdf | Nick ]]
|-
| [[Group_6 | Ferrofluid]]
| [[Media: Presentation.pdf | Presentation]]. '''Papers:''' [[Media: Schoenwald_Group6_Final.pdf | Kipp]], [[Media: Potter_Ferrofluids_-_Potter.pdf | Daniel]], [[Media: Hamid,_Amir_Ferrofluid_Group_6.pdf | Amir]]
|-
| [[Group_7 | Inverted pendulum]]
| [[Media: The_Inverted_Pendulum.pdf | Presentation]]. '''Papers:''' [[Media: Marcotte_Dynamic_Stabilization_of_the_Inverted_Pendulum.pdf | Chris]], [[Media: Aguilar_Jeff_PendulumPaper.pdf | Jeff]], [[Media: LeeGustavo_Inverted_Pendulum_Final_Report_.pdf | Gustavo]], [[Media: Suri_Balachandra_NLD_final.pdf | Balachandra]]
|}

== Announcements: Fall 2011 ==

* Below is a list of the assigned dates for groups to work in the lab. If any groups have conflicts with the dates listed please email [mailto:nick.gravish@gmail.com me] immediately to resolve this.

{|cellpadding="5" cellspacing="0" border="1" width="60%"
! Date range
! Groups
|-
| 10/24 - 10/28
| Inverted pendulum ([[Group 7]])
|-
| 10/31 - 11/4
| Inelastic bouncing ball ([[Group 3]]) and dripping faucet ([[Group 5]])
|-
| 11/7 - 11/11
| Synchronization ([[Group 4]]) and Ferrofluid ([[Group 6]])
|-
| 11/14 - 11/18
| Faraday waves ([[Group 1]]) and Plinko dynamics ([[Group 2]])
|-
| 11/28 - 12/2
| Presentations beginning Thursday
|-
| 12/5 - 12/9
| Presentations
|}

* Information about the final presentations and paper can now be found on the [[Final]] page.
* Lab dates have been posted 10/19/11
* Added capability to embed youtube videos. Use the following code to embed <pre><videoflash>YouTubeFileName</videoflash></pre>
* Math rendering has been added using the [http://www.mathjax.org/ MathJax] program. This renders latex code typed into the page source and allows for copy/paste, scaling, and other features. For a Latex tutorial check [http://en.wikibooks.org/wiki/LaTeX/Mathematics here].
* Groups are only able to edit the page associated with their group.
* When you receive your group login and password you should change your password for security.
* For questions about this site email [mailto:nick.gravish@gmail.com Nick Gravish]

== About this wiki ==

See the [[About]] page for more information. This Wiki is open to the public to view but not to edit. However, we gladly make content available to other schools for non-profit educational use. Some links to copyright-protected references and software are not available to anyone without authentication as a Georgia Tech student or staff.

== Wiki 101 ==

Here is a link to mediawiki [http://en.wikipedia.org/wiki/Wikipedia:Cheatsheet cheatsheet].

Group 5 2014

2014-12-12T22:32:53Z

PatrickChang:

Members: Caligan, Lucas, Li, Norris

[[Media:pendulum.pdf|Final Report]]

=Introduction=
In 1665, the Dutch scientist [http://rspa.royalsocietypublishing.org/content/458/2019/563 Christiaan Huygens] discovered that two pendulum
clocks mounted on the same wall synchronize with one another---the bobs swing
with the same frequency but exactly out of phase.<ref name="bennett2002huygens">
{{cite journal
|last1=Bennett |first1=M.
|last2=Schatz |first2=M. F
|last3=Rockwood|first3=H.
|last4=Wisenfeld|first4=K.
|year=2002
|title=Huygen's clocks
|journal=[[Proceedings: Mathematics, Physical and Engineering Sciences]]
}}</ref> The
origin of this effect is weak coupling of the clocks mediated through the
wall’s vibrations. Ever since, the seemingly old topic of synchronization has
developed into one of the most actively studied phenomena in the fields of applied
mathematics, nonlinear dynamics, statistical physics and material science.

Modelling a system of interest using a system of coupled pendulums, is a very
general approach that one can observe in a number of different applications. It
can be used to take on very practical problems like studying the Millennium Bridge
problem and designing dampening systems for buildings to compensate for wind or seismic disturbances.<ref name="strogatz2005theoretical">
{{cite journal
|last1=Strogatz |first1=S. H
|last2=Abrams |first2=D. M
|last3=McRobie |first3=A.
|last4=Eckhardt |first4=B.
|last5=Ott |first5=E.
|year=2005
|title=Theoretical mechanics: Crowd synchrony on the Millennium Bridge
|journal=[[Nature]]
}}</ref> However it is also employed in the study of far more complex and
interesting systems where it use has ranged from modelling the signaling patterns
of insects (either visual or auditory), studying the interactions between large
ensembles of neurons in the brain during complex activities<ref name="resconi2015geometric">
{{cite book
|last1=Resconi |first1=SG.
|last2=Kozma |first2=R.
|year=2015
|title=Computational Intelligence
|publisher=[[Springer]]
}}</ref>,
and designing sensing elements for gravitational waves.<ref name="gusev1993angular">
{{cite journal
|last1=Gusev |first1=A. V.
|last2=Vinogradov |first2=M. P.
|year=1993
|title=Angular velocity of rotation of the swing plane of a spherical pendulum with an anisotropic suspension
|volume=36 |number=10 | pages=1078-1082
|journal=[[Measurement Techniques]]
}}</ref>

Crowd synchrony on the Millennium Bridge, as a specific example of
synchronization, has been investigated.<ref name="strogatz2005theoretical">
{{cite journal
|last1=Strogatz |first1=S. H
|last2=Abrams |first2=D. M
|last3=McRobie |first3=A.
|last4=Eckhardt |first4=B.
|last5=Ott |first5=E.
|year=2005
|title=Theoretical mechanics: Crowd synchrony on the Millennium Bridge
|journal=[[Nature]]
}}</ref> In that
study, pedestrians, while moving forward along the bridge, fell spontaneously into step
with the bridge’s vibration. This was due to the fact that in addition to their forward progress, a small component of their step was directed laterally to compensate for the small lateral sway of the bridge. After a transient state where the net lateral motion was small, the people on the bridge began to synchronize their steps, at least the lateral component of their steps, with the sway of the bridge due to the tendency to be thrown off balance in stride if your lateral motion was out of phase with the ever increasing sway of the bridge. The net result was a significant lateral deviation which grew with time and could, if left unchecked, potentially cause catastrophic failure of the bridge.

Understanding synchronization of pendulums has been a rather hot topic with a considerable amount of research tackling the problem of $N=2$.<ref name="wiesenfeld2011huygens">
{{cite journal
|last1=Wiesenfeld |first1=K.
|last2=Borrero-Echeverry |first2=D.
|year=2011
|title=Huygens (and others) revisited
|volume=21 |number=4 |pages=047515
|journal=[[Chaos: An Interdisciplinary Journal of Nonlinear Science]]
}}</ref> Furthermore, the majority of research completed considers simple, driven pendulums.<ref name="pena2014further">
{{cite journal
|last1=Peña |first1=R. J.
|last2=Aihara |first2=K.
|last3=Fey |first3=R. H. B.
|last4=Nijmeijer |first4=H.
|year=2014
|title=Further understanding of Huygens’ coupled clocks: The effect of stiffness
|volume=270 |pages=11-19
|journal=[[Physica D: Nonlinear Phenomena]]
}}</ref> Previous results have shown that various modes may occur during synchronization of pendula lying on the same plane.<ref name="czolczynski2009clustering">
{{cite journal
|last1=Czolczynski |first1=K.
|last2=Perlikowski |first2=P.
|last3=Stefanski |first3=A.
|last4=Kapitaniak |first4=T.
|year=2009
|title=Clustering and synchronization of n Huygens’ clocks
|volume=388 |number=244 |pages=047515
|journal=[[Physica A: Statistical Mechanics and its Applications]]
}}</ref> The potential configurations of the system are (i) complete synchronization, (ii) synchronization of clusters for three or five pendulums, and (iii) total antiphase synchronization.

The goal of this work is to expand upon previous efforts
which has analyzed synchronization of Huygens' clock to three
dimensional pendulums.<ref name="wiesenfeld2011huygens">
{{cite journal
|last1=Wiesenfeld |first1=K.
|last2=Borrero-Echeverry |first2=D.
|year=2011
|title=Huygens (and others) revisited
|volume=21 |number=4 |pages=047515
|journal=[[Chaos: An Interdisciplinary Journal of Nonlinear Science]]
}}</ref><ref name="czolczynski2009clustering">
{{cite journal
|last1=Czolczynski |first1=K.
|last2=Perlikowski |first2=P.
|last3=Stefanski |first3=A.
|last4=Kapitaniak |first4=T.
|year=2009
|title=Clustering and synchronization of n Huygens’ clocks
|volume=388 |number=244 |pages=047515
|journal=[[Physica A: Statistical Mechanics and its Applications]]
}}</ref> Particularly, we intend to extend previous
results by allowing for spherical pendulums which do not necessarily lie in a straight line. Rather, the pendula will rest on a base which is held up by Meissner bodies.<ref name="kawohl2011meissner">
{{cite journal
|last1=Kawohl |first1=B.
|last2=Weber |first2=C.
|year=2011
|title=Meissner’s mysterious bodies
|volume=33 |number=3 |pages=94-101
|journal=[[The Mathematical Intelligencer]]
}}</ref> By conducting physical, analytical, and
numerical experiments we hope to quantify what modes may occur and to
determine how various parameters impact the synchronization of multiple pendula.

To this end, we believe that a more generous
model of such synchrony on one base-plane will be of great interest and
benefit the study of all similar phenomena in future.

=Outline=
The phenomena of synchronization in pendula lying along the same plane has been studied extensively in [http://www.perlikowski.kdm.p.lodz.pl/papers/physa2009.pdf general]. It was shown that several modes can occur in general. The possible configurations are complete synchronization of the system, synchronization of clusters of three or five pendula, or total antiphase synchronization. The paper Perlikowski claims that these phenomena can be observed in the laboratory. An extension of this would be to allow spherical pendula to synchronize along a plane instead of requiring the pendula to be along a straight line of the plane they lie on. This will allow higher dimensionality in the phase space and phenomena that may be observed. A further extension will be to allow the motion of the plane to be controlled by placing it on top of [http://en.wikipedia.org/wiki/Reuleaux_tetrahedron Meissner bodies]. This will add more complexity to the system and hopefully more interesting dynamics as well.

=Theory=
It is easy to write down the system of differential equations for n simple harmonic oscillators coupled together. This takes the form
\begin{equation}
\dot{\theta}_k(t) = \omega + \frac{k}{n} \sum_{i=1}^n \sin\left(\theta_i(t) - \theta_k(t)\right).
\end{equation}
[[File:Coupled_Oscillators_11.png | thumb | 300px | right | Figure 1: Phases of 11 Coupled Harmonic Oscillators.]]
Synchronization phenomena of 11 coupled harmonic oscillators may be seen in Figure 1.
The differential equations for a single spherical pendulum are
\begin{equation}
\begin{aligned}
m r^2 \ddot{\theta} - m r^2 \sin\theta \cos\theta \dot{\phi}^2 + m g r \sin\theta &= 0,\\
2 m r^2 \dot{\theta} \sin\theta \cos\theta \dot{\phi} + m r^2 \sin^2\theta \ddot{\phi} &= 0.
\end{aligned}
\end{equation}
when the length of the pendulum is at a constant radius from its origin.

=Methods=
===Experimental Set Up===
Before studying the synchronization of coupled pendulum on same base plane, we will first study non-linear dynamics of one pendulum. A pendulum was positioned at the center of a base plane, with several glass spheres located randomly underneath the base plane. We found that the weight of the base plane strongly affected the dynamics of one pendulum, that is, when the base plane is relative light, it will move in large amplitude to accommodate with motion of pendulum. To this end, the synchronization of coupled pendulum could not be achieved. Thus we increase the weight of base plane in the final design of coupled pendulum on same base plane (shown in Figure 4). The spheres we used for pendulum is $25\pm 0.2\,\mathrm{mm}$ in diameter and $21.2\pm0.2 g$ in weight, which are coated with IR reflective tape for 3D position recording. The spheres were hanged to the hard frame by soft strings; the weight of strings was neglectable. The dimension of the base plane was 18 inches by 25 inches, and we attached two wood stick on its two sides to increase the weight.

We studied the synchronization of coupled pendulum by placing three pendulums on the base plane in triangular positions. We did not try more than three pendulums in our study due to the limited space of the base plane. However, we found that even with three pendulums, the synchronization behavior was complicated, partially because that the dynamics of pendulum was non-linear regime at the beginning stage (i.e., we manually released the pendulums at very large angle away from the center line). Notably, the motion of the base plane is subtle due to the large mass, thereby leading to a slow synchronization of the three pendulums.

The experiment will be conducted with several spherical pendula sitting on a plane. The plane will start out being rested on top of several spheres so that the only contribution to the dynamics of the system is from the pendula themselves. Once this portion of the experiment has been conducted and the result of Perlikowski have been observed, we will move to placing Meissner bodies under the plane. This will allow for an additional nonlinear effects to the dynamics of the system. The team will attempt to produce some of the synchronization phenomena found by theoretical means and possibly other phenomena that were not discovered theoretically.

Before studying the synchronization of coupled pendulum on same base plane, we will first study non-linear dynamics of one pendulum (Figure 2). A electromagnetic pendulum will be positioned at the center of a base plane, with several Meissner's bodies located randomly underneath the base plane. Experimental parameters such as Meissner's bodies (e.g., the numbers and position of Meissner's bodies), the mass of base plane, the height and weight of pendulum and the magnetic strength of electromagnetic pendulum will all influence the non-linear dynamic behavior of pendulum, those parameters will be evaluated individually.

Based on the results for the single electromagnetic pendulum on the base plane, we extend to the synchronization of coupled pendulum on same base plane. We propose that the positions of pendulums will determined the synchronization of coupled pendulum. Thus a series of arrangements of pendulums will be explored. Figure 3 shows one example of three pendulums locates on the base plane at the triangular points. Furthermore, four, five and more pendulums will be designed to position in various arrangement on the base plane. We believe that by such study, a general rule for synchronization of coupled pendulum on same base plane can be given.

[[File:Single2.png | thumb | 300px | right | Figure 2: One electromagnetic pendulum on the base plane.]]

[[File:Single3.png | thumb | 300px | right | Figure 3: Coupled electromagnetic pendulum on the base plane.]]

==References==
{{reflist|35em}}

Group 2 2014

2014-12-12T22:32:25Z

PatrickChang:

Members: Bollenbacher, Chambers, Cunningham, Putzel

[[Media:Faraday.pdf|Final Report]]

=Introduction=
First observed by Michael Faraday in 1831 \cite{Fpaper}, Faraday waves are surface instabilities that arise in a fluid undergoing vertical oscillations above a critical amplitude and/or frequency. Faraday himself measured that the resulting surface wave frequency was equal to half the driving frequency. Above this critical amplitude/frequency the fluid surface can exhibit an incredible variety of patterns as well as spatial and temporal chaos and combinations of the two, attracting the attention of mathematicians and physicists alike. The pattern symmetries and critical values are highly dependent on boundary conditions, and the properties of the fluid itself (for example viscosity and surface tension).

Beyond the mere satisfaction of mathematical curiosity, Faraday waves have several practical applications and made contributions to other areas of physics. They play a role in the amplification of earthquakes through looser sediments \cite{Equake} and can allow one to deposit thin films of material in a desired pattern because small particles in the fluid are pulled towards amplitude; such a skill has interesting applications in creating more precise optical instruments \cite{nanofilm}. In the field of quantum mechanics, Faraday waves have been observed in Bose-Einstein Condensates \cite{BEC} and there is a curious analogy one can make between a probability wave distribution in quantum mechanics and a the chaotic motion of a 'walking' droplet guided by a precisely tuned Faraday wave \cite{Walkers}. When confined to a circular container, an oil droplet can be made to bounce on a water surface supporting a Faraday wave within a small range of driving frequency and amplitude \cite{bouncingdrop}. Within an even narrower parameter range, the Faraday wave can guide the oil droplet in a random walk across the surface. This results in a probability distribution for the droplet reminiscent of the wave function of an atom confined in a similar circular geometry. The particle/wave guide pair also exhibits many effects similar to those predicted by quantum physics such as tunneling and bears a striking resemblance to wave/particle duality on a macroscopic level. These similarities have left many speculating that there may be a hidden variable theory for quantum mechanics that mimics the pilot wave in the walking droplet experiment \cite{Bush2014}.

=Methodology=

===Experimental Apparatus===
[[File:Both.jpg | thumb | 300px | Side view (right) and top view (left) of the experimental apparatus]]

The Faraday waves were generated by mounting a container of fluid onto a shaker table capable of producing an arbitrary, vertical, forcing function. The container was transparent, excepting one of the long vertical walls to use as a backdrop for imaging. The internal dimensions of the base were 25x190mm, and the height was 100mm. This narrow profile approximates the 2D case of the Faraday wave phenomenon. The container was filled to a depth of 20mm with water mixed with a small volume of rheoscopic fluid. The ratio of this fluid to the water was determined by adding the rheoscopic fluid in small amounts until the flow pattern could be clearly seen without making the water too opaque to image.

In order to image a slice of fluid flow, a laser sheet was shone through the top of the fluid, with the plane parallel to the larger side of the container. Using a high-speed camera, we were able to capture high definition videos which we used to extract information about the internal fluid flows. The depth of the laser sheet from the front surface of the container was chosen to be large enough to not observe the surface effects at the container boundary while maximizing the amount of light that could reach the camera through the slightly opaque mixture; in our case, this proved to be about 2mm. A black cloth covered the entire apparatus to block ambient light and allow us to only image the 2D plane illuminated by the laser sheet.

===Data Collection Procedure===
We ran a series of experiments over a range of amplitudes, from about 0.3-0.6 g's in .05g intervals. This range of values was selected after several trial runs as the first onset occurs around 0.3g, and above 0.6g the onset process becomes too fast to observe the qualitative stages distinctly. The lower amplitude was also limited by the length of video we were able to capture at such high resolution as the bifurcation onset takes longer for lower amplitude forcing functions. The aim of taking the range was to be able to observe different features of the same process at different amplitudes. For instance, at lower amplitudes, when the onset of the waveform is slower, so perhaps this could give greater temporal resolution of the bifurcation process. We chose to stay at a single frequency, 15Hz, as this resulted in a clean formation of an integer number of waves within the length of the container.

For each trial we began by mixing the fluid to homogeneity, and allowing a moment for the flows of the stirring to damp out. Once the fluid appeared on camera to be still and uniform, we turned on the shaker table, and quickly turned up the amplitude to our target amplitude for the trial. After reaching the target amplitude, we would not adjust the forcing function until the fluid had undergone the Faraday instability and settled into a steady oscillation. Once the fluid had reached a steady state (judged qualitatively by the profile of the forcing function and the images from the camera), we turned off the forcing function and allowed the waveform to damp out. We recorded the whole process with a highs-peed camera for each trial, and then saved the video. This video was the data we collected, and all other metrics were derived from it during post-processing.

=Data Analysis=
===Video Processing===

We extracted from each video two important functions: the effective forcing function, and the total reflected luminosity of the fluid as a function of time. The forcing function allowed us to account for particular transitions in the luminosity function, and the luminosity function gives us information about the internal fluid flows. The place to start to extract both functions from the video was to develop a surface tracking software. Given the position of the bottom of the container, we would be able to track the displacement in the vertical direction, and thus extract with forcing function. With the shape and location of the top surface of the fluid, we would be able to isolate the region of the fluid and integrate the total luminosity over that region.

To track these lines in the video, we looked at sharp changes in brightness, as we knew both the top and bottom surfaces were considerably brighter than their background due to reflection of the laser illumination. This method proved effective, and we were able to track both the top surface and the bottom of the container. However, because the illumination in the video was not completely uniform, we ended up cutting off the left and right edges of the video during processing, as these were slightly dimmer and disrupted the surface tracking method. This should not effect the results, however, as we were still able to have multiple full wavelengths in the frame.

After tracking these two surfaces, we were able to pull out both the forcing function and the total reflected luminosity within the fluid. We factored out the surface reflections and the visible meniscus of the fluid by chopping off a small buffer zone on the top and bottom of the region. These two functions were the major results of the current work, and proved to be quite illuminating.

[[File:WaveAnalysis.jpg | thumb | 300px | Top: raw image; Middle: surface tracking; Bottom: region of integration for luminosity function]]

===Analysis of Results===

[[File:WaveOnset.jpg | thumb | 300px | Pattern formation at the onset of the waves. These dots occur in the middle of convection nodes, where particles collide.]]

Of the two functions derived from each video, the luminosity function is the key one; the forcing function merely contextualizes it. The total luminosity reflected by the fluid is an interesting metric because of the way the particles within the rheoscopic fluid behave. These particles are tiny flakes, like fish scales, which align with the fluid flow. In a still fluid, they are oriented randomly. In a fluid with 2D flow perpendicular to the line of sight, the majority of the particles align parallel to the flow such that light shone from above will not reflect toward the observer i.e. the particles have the flat face towards the observer. Further, in areas of the flow with high sheer or curl (as in the center of a convection node), these particles collide and "tumble," and thus reflect light toward the observer. With this information about the way the particles behave in the fluid, we can decipher both what we see occurring in the videos, and the structure of the luminosity function.

In each video, we see several qualitative stages in the onset of the Faraday waves, delineated by visible changes in structure. First, after the forcing function is applied, we quickly see the formation of "dots," as in the figure below, which we suspect are the centers of convection nodes. Next, the surface waves begin to form. Finally we see these distinct dots in the centers of the convection nodes "fall" and then fade out, as if the convection nodes are growing in size and slowing in rate. Soon after this point, the wave settles into its steady state oscillation.

These qualitative transition points are reflected distinctly in the luminosity function. In the first stage, before the convection nodes appear, the luminosity function begins to oscillate in time with the forcing function. After the convection nodes appear, the mean of this oscillation begins to fall, as the fluid flows in the 2D plane set in. Then the surface waves form, and the oscillation in the luminosity function damps out in favor of steady flows. As the internal flows move deeper into the fluid (away from just the surface), the convection "dots" fall and fade out. After this time, the luminosity function increases as the shears in the fluid flow increase. As the surface waves oscillate (troughs becoming peaks, and vice versa), the internal flows also oscillate, creating shear forces as they switch direction. These shear forces create tumbling particles which increase the total reflected luminosity from that of the previous stages (though still lower than the initial, still, fluid, where many particles still align with the in-plane flows). The luminosity function then settles into a steady oscillation in time with the surface oscillations. When we turn off the forcing around 18s, the fluid settles, and the reflected luminosity increases slightly, back up to the high baseline of the still fluid where the particles are randomly oriented.

As the amplitude of the forcing increases, the profile of the luminosity function changes, especially in the early stages of onset. At higher forcing amplitude, the amplitude of the oscillations in the luminosity function grow much larger than in the lower amplitude case. While this makes the graphs look quite different at first glance, the behavior of the mean and relative amplitude is the same.

[[File:WaveGraph378.jpg | thumb | 300px | Luminosity function (top) and forcing amplitude (bottom) at .378 g. Yellow: convection "dots" first appear; Red: Surface waves form; Green: "Dots" fall and fade out.]]

=Conclusion=

These results give us a qualitative narrative of what the internal fluid flows are like at the onset of the Faraday instability. However, to do a more detailed analysis,the flows would have to be measured directly using Particle Image Velocimetry (or PIV). PIV uses the changing local densities of particles from frame to frame in the video to estimate the flow field. PIV techniques could be applied to videos similar to those produced by the current work. To do this, however, would require more precise control of lighting conditions, and specifically, greater uniformity of fluid illumination by the laser sheet. Future works may pursue this line of observation, and the subsequent theoretical explanations of what is observed.

[[File:WaveGraph547.jpg | thumb | 300px | Luminosity function (top) and forcing amplitude (bottom) at 0.547 g. ]]

=References=
{{Citation
| last = Faraday
| first = M.
| title = On a peculiar class of acoustical figures; and on certain forms assumed by a group of particles upon vibrating elastic surfaces
| journal = Philosophical Transactions of the Royal Society (London
| volume = 121
| issue = 2
| year = 1831
| pages = 299–318
}}

File:Pendulum.pdf

2014-12-12T22:30:53Z

PatrickChang: Final paper for 2014 pendulum sync group

Final paper for 2014 pendulum sync group

Group 5 2014

2014-12-12T22:29:55Z

PatrickChang:

Members: Caligan, Lucas, Li, Norris

[[Media:pendulum.pdf|Final Paper]]

=Introduction=
In 1665, the Dutch scientist [http://rspa.royalsocietypublishing.org/content/458/2019/563 Christiaan Huygens] discovered that two pendulum
clocks mounted on the same wall synchronize with one another---the bobs swing
with the same frequency but exactly out of phase.<ref name="bennett2002huygens">
{{cite journal
|last1=Bennett |first1=M.
|last2=Schatz |first2=M. F
|last3=Rockwood|first3=H.
|last4=Wisenfeld|first4=K.
|year=2002
|title=Huygen's clocks
|journal=[[Proceedings: Mathematics, Physical and Engineering Sciences]]
}}</ref> The
origin of this effect is weak coupling of the clocks mediated through the
wall’s vibrations. Ever since, the seemingly old topic of synchronization has
developed into one of the most actively studied phenomena in the fields of applied
mathematics, nonlinear dynamics, statistical physics and material science.

Modelling a system of interest using a system of coupled pendulums, is a very
general approach that one can observe in a number of different applications. It
can be used to take on very practical problems like studying the Millennium Bridge
problem and designing dampening systems for buildings to compensate for wind or seismic disturbances.<ref name="strogatz2005theoretical">
{{cite journal
|last1=Strogatz |first1=S. H
|last2=Abrams |first2=D. M
|last3=McRobie |first3=A.
|last4=Eckhardt |first4=B.
|last5=Ott |first5=E.
|year=2005
|title=Theoretical mechanics: Crowd synchrony on the Millennium Bridge
|journal=[[Nature]]
}}</ref> However it is also employed in the study of far more complex and
interesting systems where it use has ranged from modelling the signaling patterns
of insects (either visual or auditory), studying the interactions between large
ensembles of neurons in the brain during complex activities<ref name="resconi2015geometric">
{{cite book
|last1=Resconi |first1=SG.
|last2=Kozma |first2=R.
|year=2015
|title=Computational Intelligence
|publisher=[[Springer]]
}}</ref>,
and designing sensing elements for gravitational waves.<ref name="gusev1993angular">
{{cite journal
|last1=Gusev |first1=A. V.
|last2=Vinogradov |first2=M. P.
|year=1993
|title=Angular velocity of rotation of the swing plane of a spherical pendulum with an anisotropic suspension
|volume=36 |number=10 | pages=1078-1082
|journal=[[Measurement Techniques]]
}}</ref>

Crowd synchrony on the Millennium Bridge, as a specific example of
synchronization, has been investigated.<ref name="strogatz2005theoretical">
{{cite journal
|last1=Strogatz |first1=S. H
|last2=Abrams |first2=D. M
|last3=McRobie |first3=A.
|last4=Eckhardt |first4=B.
|last5=Ott |first5=E.
|year=2005
|title=Theoretical mechanics: Crowd synchrony on the Millennium Bridge
|journal=[[Nature]]
}}</ref> In that
study, pedestrians, while moving forward along the bridge, fell spontaneously into step
with the bridge’s vibration. This was due to the fact that in addition to their forward progress, a small component of their step was directed laterally to compensate for the small lateral sway of the bridge. After a transient state where the net lateral motion was small, the people on the bridge began to synchronize their steps, at least the lateral component of their steps, with the sway of the bridge due to the tendency to be thrown off balance in stride if your lateral motion was out of phase with the ever increasing sway of the bridge. The net result was a significant lateral deviation which grew with time and could, if left unchecked, potentially cause catastrophic failure of the bridge.

Understanding synchronization of pendulums has been a rather hot topic with a considerable amount of research tackling the problem of $N=2$.<ref name="wiesenfeld2011huygens">
{{cite journal
|last1=Wiesenfeld |first1=K.
|last2=Borrero-Echeverry |first2=D.
|year=2011
|title=Huygens (and others) revisited
|volume=21 |number=4 |pages=047515
|journal=[[Chaos: An Interdisciplinary Journal of Nonlinear Science]]
}}</ref> Furthermore, the majority of research completed considers simple, driven pendulums.<ref name="pena2014further">
{{cite journal
|last1=Peña |first1=R. J.
|last2=Aihara |first2=K.
|last3=Fey |first3=R. H. B.
|last4=Nijmeijer |first4=H.
|year=2014
|title=Further understanding of Huygens’ coupled clocks: The effect of stiffness
|volume=270 |pages=11-19
|journal=[[Physica D: Nonlinear Phenomena]]
}}</ref> Previous results have shown that various modes may occur during synchronization of pendula lying on the same plane.<ref name="czolczynski2009clustering">
{{cite journal
|last1=Czolczynski |first1=K.
|last2=Perlikowski |first2=P.
|last3=Stefanski |first3=A.
|last4=Kapitaniak |first4=T.
|year=2009
|title=Clustering and synchronization of n Huygens’ clocks
|volume=388 |number=244 |pages=047515
|journal=[[Physica A: Statistical Mechanics and its Applications]]
}}</ref> The potential configurations of the system are (i) complete synchronization, (ii) synchronization of clusters for three or five pendulums, and (iii) total antiphase synchronization.

The goal of this work is to expand upon previous efforts
which has analyzed synchronization of Huygens' clock to three
dimensional pendulums.<ref name="wiesenfeld2011huygens">
{{cite journal
|last1=Wiesenfeld |first1=K.
|last2=Borrero-Echeverry |first2=D.
|year=2011
|title=Huygens (and others) revisited
|volume=21 |number=4 |pages=047515
|journal=[[Chaos: An Interdisciplinary Journal of Nonlinear Science]]
}}</ref><ref name="czolczynski2009clustering">
{{cite journal
|last1=Czolczynski |first1=K.
|last2=Perlikowski |first2=P.
|last3=Stefanski |first3=A.
|last4=Kapitaniak |first4=T.
|year=2009
|title=Clustering and synchronization of n Huygens’ clocks
|volume=388 |number=244 |pages=047515
|journal=[[Physica A: Statistical Mechanics and its Applications]]
}}</ref> Particularly, we intend to extend previous
results by allowing for spherical pendulums which do not necessarily lie in a straight line. Rather, the pendula will rest on a base which is held up by Meissner bodies.<ref name="kawohl2011meissner">
{{cite journal
|last1=Kawohl |first1=B.
|last2=Weber |first2=C.
|year=2011
|title=Meissner’s mysterious bodies
|volume=33 |number=3 |pages=94-101
|journal=[[The Mathematical Intelligencer]]
}}</ref> By conducting physical, analytical, and
numerical experiments we hope to quantify what modes may occur and to
determine how various parameters impact the synchronization of multiple pendula.

To this end, we believe that a more generous
model of such synchrony on one base-plane will be of great interest and
benefit the study of all similar phenomena in future.

=Outline=
The phenomena of synchronization in pendula lying along the same plane has been studied extensively in [http://www.perlikowski.kdm.p.lodz.pl/papers/physa2009.pdf general]. It was shown that several modes can occur in general. The possible configurations are complete synchronization of the system, synchronization of clusters of three or five pendula, or total antiphase synchronization. The paper Perlikowski claims that these phenomena can be observed in the laboratory. An extension of this would be to allow spherical pendula to synchronize along a plane instead of requiring the pendula to be along a straight line of the plane they lie on. This will allow higher dimensionality in the phase space and phenomena that may be observed. A further extension will be to allow the motion of the plane to be controlled by placing it on top of [http://en.wikipedia.org/wiki/Reuleaux_tetrahedron Meissner bodies]. This will add more complexity to the system and hopefully more interesting dynamics as well.

=Theory=
It is easy to write down the system of differential equations for n simple harmonic oscillators coupled together. This takes the form
\begin{equation}
\dot{\theta}_k(t) = \omega + \frac{k}{n} \sum_{i=1}^n \sin\left(\theta_i(t) - \theta_k(t)\right).
\end{equation}
[[File:Coupled_Oscillators_11.png | thumb | 300px | right | Figure 1: Phases of 11 Coupled Harmonic Oscillators.]]
Synchronization phenomena of 11 coupled harmonic oscillators may be seen in Figure 1.
The differential equations for a single spherical pendulum are
\begin{equation}
\begin{aligned}
m r^2 \ddot{\theta} - m r^2 \sin\theta \cos\theta \dot{\phi}^2 + m g r \sin\theta &= 0,\\
2 m r^2 \dot{\theta} \sin\theta \cos\theta \dot{\phi} + m r^2 \sin^2\theta \ddot{\phi} &= 0.
\end{aligned}
\end{equation}
when the length of the pendulum is at a constant radius from its origin.

=Methods=
===Experimental Set Up===
Before studying the synchronization of coupled pendulum on same base plane, we will first study non-linear dynamics of one pendulum. A pendulum was positioned at the center of a base plane, with several glass spheres located randomly underneath the base plane. We found that the weight of the base plane strongly affected the dynamics of one pendulum, that is, when the base plane is relative light, it will move in large amplitude to accommodate with motion of pendulum. To this end, the synchronization of coupled pendulum could not be achieved. Thus we increase the weight of base plane in the final design of coupled pendulum on same base plane (shown in Figure 4). The spheres we used for pendulum is $25\pm 0.2\,\mathrm{mm}$ in diameter and $21.2\pm0.2 g$ in weight, which are coated with IR reflective tape for 3D position recording. The spheres were hanged to the hard frame by soft strings; the weight of strings was neglectable. The dimension of the base plane was 18 inches by 25 inches, and we attached two wood stick on its two sides to increase the weight.

We studied the synchronization of coupled pendulum by placing three pendulums on the base plane in triangular positions. We did not try more than three pendulums in our study due to the limited space of the base plane. However, we found that even with three pendulums, the synchronization behavior was complicated, partially because that the dynamics of pendulum was non-linear regime at the beginning stage (i.e., we manually released the pendulums at very large angle away from the center line). Notably, the motion of the base plane is subtle due to the large mass, thereby leading to a slow synchronization of the three pendulums.

The experiment will be conducted with several spherical pendula sitting on a plane. The plane will start out being rested on top of several spheres so that the only contribution to the dynamics of the system is from the pendula themselves. Once this portion of the experiment has been conducted and the result of Perlikowski have been observed, we will move to placing Meissner bodies under the plane. This will allow for an additional nonlinear effects to the dynamics of the system. The team will attempt to produce some of the synchronization phenomena found by theoretical means and possibly other phenomena that were not discovered theoretically.

Before studying the synchronization of coupled pendulum on same base plane, we will first study non-linear dynamics of one pendulum (Figure 2). A electromagnetic pendulum will be positioned at the center of a base plane, with several Meissner's bodies located randomly underneath the base plane. Experimental parameters such as Meissner's bodies (e.g., the numbers and position of Meissner's bodies), the mass of base plane, the height and weight of pendulum and the magnetic strength of electromagnetic pendulum will all influence the non-linear dynamic behavior of pendulum, those parameters will be evaluated individually.

Based on the results for the single electromagnetic pendulum on the base plane, we extend to the synchronization of coupled pendulum on same base plane. We propose that the positions of pendulums will determined the synchronization of coupled pendulum. Thus a series of arrangements of pendulums will be explored. Figure 3 shows one example of three pendulums locates on the base plane at the triangular points. Furthermore, four, five and more pendulums will be designed to position in various arrangement on the base plane. We believe that by such study, a general rule for synchronization of coupled pendulum on same base plane can be given.

[[File:Single2.png | thumb | 300px | right | Figure 2: One electromagnetic pendulum on the base plane.]]

[[File:Single3.png | thumb | 300px | right | Figure 3: Coupled electromagnetic pendulum on the base plane.]]

==References==
{{reflist|35em}}

File:Faraday.pdf

2014-12-12T22:29:21Z

PatrickChang: final paper for 2014 Faraday Instability group

final paper for 2014 Faraday Instability group

Group 2 2014

2014-12-12T22:28:30Z

PatrickChang:

Members: Bollenbacher, Chambers, Cunningham, Putzel

[[Media:Faraday.pdf|Final Paper]]

=Introduction=
First observed by Michael Faraday in 1831 \cite{Fpaper}, Faraday waves are surface instabilities that arise in a fluid undergoing vertical oscillations above a critical amplitude and/or frequency. Faraday himself measured that the resulting surface wave frequency was equal to half the driving frequency. Above this critical amplitude/frequency the fluid surface can exhibit an incredible variety of patterns as well as spatial and temporal chaos and combinations of the two, attracting the attention of mathematicians and physicists alike. The pattern symmetries and critical values are highly dependent on boundary conditions, and the properties of the fluid itself (for example viscosity and surface tension).

Beyond the mere satisfaction of mathematical curiosity, Faraday waves have several practical applications and made contributions to other areas of physics. They play a role in the amplification of earthquakes through looser sediments \cite{Equake} and can allow one to deposit thin films of material in a desired pattern because small particles in the fluid are pulled towards amplitude; such a skill has interesting applications in creating more precise optical instruments \cite{nanofilm}. In the field of quantum mechanics, Faraday waves have been observed in Bose-Einstein Condensates \cite{BEC} and there is a curious analogy one can make between a probability wave distribution in quantum mechanics and a the chaotic motion of a 'walking' droplet guided by a precisely tuned Faraday wave \cite{Walkers}. When confined to a circular container, an oil droplet can be made to bounce on a water surface supporting a Faraday wave within a small range of driving frequency and amplitude \cite{bouncingdrop}. Within an even narrower parameter range, the Faraday wave can guide the oil droplet in a random walk across the surface. This results in a probability distribution for the droplet reminiscent of the wave function of an atom confined in a similar circular geometry. The particle/wave guide pair also exhibits many effects similar to those predicted by quantum physics such as tunneling and bears a striking resemblance to wave/particle duality on a macroscopic level. These similarities have left many speculating that there may be a hidden variable theory for quantum mechanics that mimics the pilot wave in the walking droplet experiment \cite{Bush2014}.

=Methodology=

===Experimental Apparatus===
[[File:Both.jpg | thumb | 300px | Side view (right) and top view (left) of the experimental apparatus]]

The Faraday waves were generated by mounting a container of fluid onto a shaker table capable of producing an arbitrary, vertical, forcing function. The container was transparent, excepting one of the long vertical walls to use as a backdrop for imaging. The internal dimensions of the base were 25x190mm, and the height was 100mm. This narrow profile approximates the 2D case of the Faraday wave phenomenon. The container was filled to a depth of 20mm with water mixed with a small volume of rheoscopic fluid. The ratio of this fluid to the water was determined by adding the rheoscopic fluid in small amounts until the flow pattern could be clearly seen without making the water too opaque to image.

In order to image a slice of fluid flow, a laser sheet was shone through the top of the fluid, with the plane parallel to the larger side of the container. Using a high-speed camera, we were able to capture high definition videos which we used to extract information about the internal fluid flows. The depth of the laser sheet from the front surface of the container was chosen to be large enough to not observe the surface effects at the container boundary while maximizing the amount of light that could reach the camera through the slightly opaque mixture; in our case, this proved to be about 2mm. A black cloth covered the entire apparatus to block ambient light and allow us to only image the 2D plane illuminated by the laser sheet.

===Data Collection Procedure===
We ran a series of experiments over a range of amplitudes, from about 0.3-0.6 g's in .05g intervals. This range of values was selected after several trial runs as the first onset occurs around 0.3g, and above 0.6g the onset process becomes too fast to observe the qualitative stages distinctly. The lower amplitude was also limited by the length of video we were able to capture at such high resolution as the bifurcation onset takes longer for lower amplitude forcing functions. The aim of taking the range was to be able to observe different features of the same process at different amplitudes. For instance, at lower amplitudes, when the onset of the waveform is slower, so perhaps this could give greater temporal resolution of the bifurcation process. We chose to stay at a single frequency, 15Hz, as this resulted in a clean formation of an integer number of waves within the length of the container.

For each trial we began by mixing the fluid to homogeneity, and allowing a moment for the flows of the stirring to damp out. Once the fluid appeared on camera to be still and uniform, we turned on the shaker table, and quickly turned up the amplitude to our target amplitude for the trial. After reaching the target amplitude, we would not adjust the forcing function until the fluid had undergone the Faraday instability and settled into a steady oscillation. Once the fluid had reached a steady state (judged qualitatively by the profile of the forcing function and the images from the camera), we turned off the forcing function and allowed the waveform to damp out. We recorded the whole process with a highs-peed camera for each trial, and then saved the video. This video was the data we collected, and all other metrics were derived from it during post-processing.

=Data Analysis=
===Video Processing===

We extracted from each video two important functions: the effective forcing function, and the total reflected luminosity of the fluid as a function of time. The forcing function allowed us to account for particular transitions in the luminosity function, and the luminosity function gives us information about the internal fluid flows. The place to start to extract both functions from the video was to develop a surface tracking software. Given the position of the bottom of the container, we would be able to track the displacement in the vertical direction, and thus extract with forcing function. With the shape and location of the top surface of the fluid, we would be able to isolate the region of the fluid and integrate the total luminosity over that region.

To track these lines in the video, we looked at sharp changes in brightness, as we knew both the top and bottom surfaces were considerably brighter than their background due to reflection of the laser illumination. This method proved effective, and we were able to track both the top surface and the bottom of the container. However, because the illumination in the video was not completely uniform, we ended up cutting off the left and right edges of the video during processing, as these were slightly dimmer and disrupted the surface tracking method. This should not effect the results, however, as we were still able to have multiple full wavelengths in the frame.

After tracking these two surfaces, we were able to pull out both the forcing function and the total reflected luminosity within the fluid. We factored out the surface reflections and the visible meniscus of the fluid by chopping off a small buffer zone on the top and bottom of the region. These two functions were the major results of the current work, and proved to be quite illuminating.

[[File:WaveAnalysis.jpg | thumb | 300px | Top: raw image; Middle: surface tracking; Bottom: region of integration for luminosity function]]

===Analysis of Results===

[[File:WaveOnset.jpg | thumb | 300px | Pattern formation at the onset of the waves. These dots occur in the middle of convection nodes, where particles collide.]]

Of the two functions derived from each video, the luminosity function is the key one; the forcing function merely contextualizes it. The total luminosity reflected by the fluid is an interesting metric because of the way the particles within the rheoscopic fluid behave. These particles are tiny flakes, like fish scales, which align with the fluid flow. In a still fluid, they are oriented randomly. In a fluid with 2D flow perpendicular to the line of sight, the majority of the particles align parallel to the flow such that light shone from above will not reflect toward the observer i.e. the particles have the flat face towards the observer. Further, in areas of the flow with high sheer or curl (as in the center of a convection node), these particles collide and "tumble," and thus reflect light toward the observer. With this information about the way the particles behave in the fluid, we can decipher both what we see occurring in the videos, and the structure of the luminosity function.

In each video, we see several qualitative stages in the onset of the Faraday waves, delineated by visible changes in structure. First, after the forcing function is applied, we quickly see the formation of "dots," as in the figure below, which we suspect are the centers of convection nodes. Next, the surface waves begin to form. Finally we see these distinct dots in the centers of the convection nodes "fall" and then fade out, as if the convection nodes are growing in size and slowing in rate. Soon after this point, the wave settles into its steady state oscillation.

These qualitative transition points are reflected distinctly in the luminosity function. In the first stage, before the convection nodes appear, the luminosity function begins to oscillate in time with the forcing function. After the convection nodes appear, the mean of this oscillation begins to fall, as the fluid flows in the 2D plane set in. Then the surface waves form, and the oscillation in the luminosity function damps out in favor of steady flows. As the internal flows move deeper into the fluid (away from just the surface), the convection "dots" fall and fade out. After this time, the luminosity function increases as the shears in the fluid flow increase. As the surface waves oscillate (troughs becoming peaks, and vice versa), the internal flows also oscillate, creating shear forces as they switch direction. These shear forces create tumbling particles which increase the total reflected luminosity from that of the previous stages (though still lower than the initial, still, fluid, where many particles still align with the in-plane flows). The luminosity function then settles into a steady oscillation in time with the surface oscillations. When we turn off the forcing around 18s, the fluid settles, and the reflected luminosity increases slightly, back up to the high baseline of the still fluid where the particles are randomly oriented.

As the amplitude of the forcing increases, the profile of the luminosity function changes, especially in the early stages of onset. At higher forcing amplitude, the amplitude of the oscillations in the luminosity function grow much larger than in the lower amplitude case. While this makes the graphs look quite different at first glance, the behavior of the mean and relative amplitude is the same.

[[File:WaveGraph378.jpg | thumb | 300px | Luminosity function (top) and forcing amplitude (bottom) at .378 g. Yellow: convection "dots" first appear; Red: Surface waves form; Green: "Dots" fall and fade out.]]

=Conclusion=

These results give us a qualitative narrative of what the internal fluid flows are like at the onset of the Faraday instability. However, to do a more detailed analysis,the flows would have to be measured directly using Particle Image Velocimetry (or PIV). PIV uses the changing local densities of particles from frame to frame in the video to estimate the flow field. PIV techniques could be applied to videos similar to those produced by the current work. To do this, however, would require more precise control of lighting conditions, and specifically, greater uniformity of fluid illumination by the laser sheet. Future works may pursue this line of observation, and the subsequent theoretical explanations of what is observed.

[[File:WaveGraph547.jpg | thumb | 300px | Luminosity function (top) and forcing amplitude (bottom) at 0.547 g. ]]

=References=
{{Citation
| last = Faraday
| first = M.
| title = On a peculiar class of acoustical figures; and on certain forms assumed by a group of particles upon vibrating elastic surfaces
| journal = Philosophical Transactions of the Royal Society (London
| volume = 121
| issue = 2
| year = 1831
| pages = 299–318
}}

File:Jumping-aerated-granular.pdf

2014-12-12T15:52:53Z

PatrickChang: Final report for 2014 Jump on Granular Media group

Final report for 2014 Jump on Granular Media group

Group 4 2014

2014-12-12T15:52:18Z

PatrickChang:

'''Jumping on an Aerated Granular Medium'''

''Group Members: Alex Lind, Cristian Salgueiro, Casey Trimble''

[[Media:jumping-aerated-granular.pdf|Final Report]]

== Introduction ==
[[File:BoxApparatus.JPG | thumb | 150px | The jumping robot apparatus used in the variable spring experiments, courtesy of the Georgia Tech CRAB Lab.]]

Granular media present special challenges and benefits to terrestrial animals. When it is well packed, a granular medium may behave as a solid. When loosely packed, the particles in the medium are allowed to flow when forces are exerted on them. For a jumping animal or robot, this property of a loosely packed substrate reduces the efficiency of a jump.

The dynamics of a jumper in a granular medium are complex, and such systems have been studied in both human and robotic cases. In the current study, we expand the parameter space of the previous studies of a simple jumping robot in both granular media and hard ground runs. We also test the extreme case of jumps on an aerated medium, where airflow is introduced from below and allowed to fluidize the medium.

== Theory and Previous Work==
[[File:Simulation.png | thumb | left | 310px | Theoretical predictions of jump height normalized to forcing amplitude versus the nondimensional parameter, mg/kA]]

===Hard Ground===

Preliminary research carried out in the CRAB Lab on the dynamics of jumping on hard ground predicted an exponential relationship between nondimensionalized jump height and the nondimensional paramter mg/kA where A is the amplitude of oscillation of the motor acting on the jumper. This relationship can be seen in the figure below

In this case, jump height is nondimensionalized by the amplitude of the motor oscillations. For large values of $\alpha$, the single jump (solid) performs better than the stutter jump (dashed). A transition point occurs at $\alpha \approx 0.1$. Below this value, the stutter jump performs better.

The performance of the single and stutter jumps are also very dependent on the frequency of the forcing function. Theoretical predictions and past experiments have determined the optimal frequency of the forcing function for both kinds of jumps. In experiment, we will vary the frequency to further understand the dependence of jump height on input frequency.

===Granular Media===

The motion of an intruder in granular media can be modeled by:

\begin{equation} \label{eq:motion}
m_f\ddot{x}_f = F_{spring} + F_{GM} - m_fg
\end{equation}
and the force from the granular media is most simply modeled with the equation:
\begin{equation}\label{eq:response}
F_{GM} = k(\text{foot depth}) + \beta(\text{foot speed})^2
\end{equation}

The first term on the right side of \eqref{eq:response} acts as a one-way spring as the GM compacts beneath the foot of the intruder, and the second term is comparable to a drag force proportional to the intruder's velocity squared. Additional terms may be added to improve the accuracy of this model; one of the most important terms omitted from the above equation is an additive mass model. The idea is that more of the GM sticks to the foot as it spends more time in the medium, thus adding mass and affecting several of the forces. This term is crucial when studying stutter jumps in GM; however for the scope of this study, such higher order corrections can be ignored. This experiment will look at varying experimental parameters in order to observe the change of the coefficients in the equation of motion.

== Apparatus ==

[[File:ExptSetup.PNG | thumb | left | 150px | Simple schematic of the jumping robot apparatus.]]

The experiment utilizes an existing jumping robot in the Georgia Tech CRAB Lab. The diagram {DOSOMETHINGHERE} shows a schematic of the apparatus. The robot jumps on GM in a box resting above an air duct. In a simple jump, a linear actuator imposes a downward force on the rod, forcing the robot foot into the GM. The spring contracts, and the actuator is released causing the robot to jump. Both the actuator and the air flow are controlled and systematically varied in LabView.

In addition to varying air flow, we vary the spring constant of the foot. Our trials cover a range of flow rates for four different spring constants. We control the spring constant with a variable length spring, shown below.

Data is collected with a force sensor on the jumper and a high speed camera. The force sensor records the force of the motor on the rod, and the camera tracks the motion of a tracking marker on the rod. With force, position, and time data, we show the change in jump dynamics as air flow and spring constant are varied. We compare our results to the theoretical predictions of the CRAB Lab.

== Methods ==
We first summarize the construction of the spring, then define the parameter space of our trials.

===Automated Variable Spring===

[[File:varSpring.jpg | thumb | right | 310px | The variable spring apparatus.]]

We expanded on the design of a variable stiffness spring produced by the CRAB Lab using SolidWorks and a MakerBot 3D printer. The green chassis shown in Figure has a threaded interior, along which the white screw in the figure is allowed to turn. The spring is attached to the motor side of the chassis, and terminates in the blue foot. The coils on the motor side of the screw cannot be compressed, and the coils on the foot side of the spring contribute to the compression in a jump.

The white screw can be adjusted automatically using a stepper motor, mounted in the black motor case attached to the chassis. The motor is driven by a command in LabView, and turns a shaft that rotates the screw. The top of the motor mount is connected to the jump rod of the robot.

===Experiment Design===

The parameters varied in the experiments are spring stiffness, jump type, forcing frequency and amplitude, and the substrate parameters. Jumps are made on hard ground, as well as on GM that is continuously fluidized. For fluidized GM trials, the rate of air flow is varied.

A series of jumps were performed on hard ground using a constant spring stiffness.

For GM trials, four values of spring stiffness were chosen. For each stiffness, a series of 128 jumps were performed while alternating between single and stutter jumps and varying both the air flow and frequency of jumps. In total, 2048 jumps were conducted on the granular medium, including mistrials.

In the table below is a summary of the parameters swept through for each experiment. Each number corresponds to the number values assigned to each parameter. "N/A" means that parameter was not varied.

{| border="1" class="wikitable"
|+ Parameter Space
! Parameter
! Hard Ground
! Granular Media
|-
! Jump Type
| style="text-align:center;"|2
| style="text-align:center;"|2
|-
! Frequency
| style="text-align:center;"|10
| style="text-align:center;"|16
|-
! Spring Constant
| style="text-align:center;"|4
| style="text-align:center;"|4
|-
! Amplitude
| style="text-align:center;"|2
| style="text-align:center;"|N/A
|-
! Media Fluidization
| style="text-align:center;"|N/A
| style="text-align:center;"|4
|-
|}

==Experiment Videos==

<videoflash>j_CyJBfTNUc</videoflash>

<videoflash>ocFDIMlPMds</videoflash>

==Results==

Using the high speed camera setup, position versus time data was obtained for both the hard ground and GM jumping trials.From this data, the maximum height for each jump was obtained, and a series of plots showing how maximum jump height and the dynamics of the jump varied with the different parameters.

===Hard Ground Results===

[[File:HardGroundNormOne.png | thumb | left | 250px | Results of hard ground jumps. Plotting max height/A vs. mg/kA]]
[[File:HardGroundEquilTwo.png | thumb | right | 250px | Results of hard ground jumps. Plotting max height/(mg/k) vs. mg/kA]]

To the left and right are the plots for the hard ground jumps. On the left is a plot of the heights normalized to the forcing amplitude versus our nondimensional parameter, $mg/kA$. The plot on the right is similar, except the heights are normalized to the equilibrium position, defined as $mg/k$.

The plot nondimensionalized to forcing amplitude does not match well with theory. In our experimental results, the single jump outperforms the stutter jump at low $\alpha$, but the stutter jump outperforms the stutter jump at high $\alpha$. This is unfortunately contrary to theoretical results. The plot also has a strange decrease in amplitude as $\alpha$ decreases, which is also contrary to theoretical predictions. Part of this discrepancy could be better explained with a finer parameter sweep, specifically testing at more than just two amplitudes.

The plot nondimensionalized to equilibrium position is a much better match to theory. The frequency parameter was swept over a range of values from $f = 2 ~s^{-1}$ to $f = 12 ~s^{-1}$, but in these results, we only present two of those frequencies for clarity. Compared the the theoretical predictions, the plot is qualitatively similar, but not exact. There is still a discrepancy between optimal performance regions for the single and stutter jump. This may be resolved by a much finer parameter sweep given more time. Specifically, it would be beneficial to probe low-$\alpha$ more finely, as the transition between optimal jump type is supposed to occur around $\alpha = 0.1$, which our trials hardly reached.

During a number of the hard ground trials, the forcing amplitude was sufficiently high such that the tracking ball left the field of view of the high speed camera. As a result, the position data for those jumps does not contain the true maximum height of the jump. It was therefore necessary to extrapolate the maximum height for those runs. This was done using kinematics based on the time the jumper was in the air and the velocity with which it left the ground.
Systematically varying the frequency of forcing was found to have a strong effect on the time and amplitude of the maximum height of stiffness and the airflow had smaller effects.

===Granular Media Results===

[[File:HeightsPlot.jpg | thumb | 350px | Results for the granular media jumps. Plotting maximum height vs. time]]

In single jumps, the maximum amplitudes were plotted as the frequency of forcing was systematically increased. Shown in Figure , it is evident that there is an optimum jump frequency, which has been found in previous research. The frequency that yielded the highest jump was $5.375Hz$.

In stutter jumps, there is a range of frequencies at which the second jump in the sequence is higher than the preliminary stutter. Interestingly, the optimum frequency for a single jump is within this range of frequencies, which was from $2.25Hz$ to $3.75Hz$.

Varying the spring stiffness and the airflow in the medium had little effect on the value for the optimal frequency range.

==Conclusions==
The results for the hard ground experiment did not match what was expected from the simulations. The single jump performed better than the stutter jump at lower $\alpha$ values. In order to fully understand the discrepancy between the experiment and simulation, the experiment should be repeated with a finer parameter sweep.

By exploring the parameter space of a jumper in a granular media, it was shown that the frequency of the jumper has more immediate implications than either the stiffness or the airflow through the medium. It was also shown that there exists a critical frequency for granular media stutter jumps, below which the amplitude of the second maximum is lower than the first maximum; the stutter jump does not perform as expected in this frequency regime. It would be interesting to study this critical frequency further and understand the implications of such a bifurcation.

In broader terms, robots in granular media present an interesting physical problem. For instance, the moon is covered in a layer of granular regolith that can be anywhere from 3 to 20m deep, and a robot operating in such conditions would necessarily have to deal with the effects of granular media. Speculatively, it seems likely that robots would be the ideal tools for further exploration and scientific work on the moon, and further research into the motion of robots in granular media would be useful for future space missions.

Group 3 2014

2014-12-12T15:51:32Z

PatrickChang:

''Group members: Andras Karsai, Steven Harrington, and Colin Campbell''

[[Media:NLD.pdf|Final Report]]

[[File:astrojax_balls.png|thumb| alt+text| A simple diagram of the Astrojax<ref>http://www.ap-club.net/learning/Astrojax/basic_orbits/start_vertical_orbit.aspx</ref>]]
<blockquote>We seek to investigate the properties of a generalized double pendulum. Previous investigations on double pendulums often limit it to a planar case, and fix the lengths of each stage of the pendulum. The Astrojax is notable in that the pendulum stage lengths vary dynamically. We measure its behaviors in response to well quantified forcing using a perpendicular camera array to track position, and from that generate three-dimensional trajectories for each pendulum bob. The trajectories' mean chaotic lifetime and dominant oscillation frequencies are analyzed, and we find that this chaotic pendulum's oscillation frequencies are linearly correlated with the forcing frequency in the low-frequency regime. We also conclude that a naive periodic forcing of this system without a feedback mechanism is not sufficient to create stable, long-lived orbits of the Astrojax pendulum.</blockquote>

=Introduction=
An oscillation <ref>http://en.wikipedia.org/wiki/Oscillation</ref> is defined, in physics, as a regular variation in magnitude, position, etc. around a central point. Oscillatory mechanical systems are a vast subset of dynamical systems. They can be used to describe the evolution of states for nearly all physical phenomena. Oscillations of various types occur in real world mechanical systems, nearly all of which are thermodynamically irreversible (via damping, friction, energy due to heat loss, etc.). Even the light you see and the sound you hear are the results of the oscillations of a medium or a field. Here, we specifically study a set of coupled, chaotic oscillations that form from the forcing of a double pendulum.

As any good physics enthusiast knows, student, instructor, or otherwise, pendula are one of the most fundamental and ubiquitous systems studied, and while they may be common, they are only ideal and easily solvable in the most simple of cases. While the equations of motion for a single pendulum are easily attainable, especially for the low amplitude/small angle limit, a "double" pendulum is an example of the effects of coupling and gives rise to chaotic behavior.

Traditionally, a double pendulum has rigid axes, thereby defining the distance the two masses that can be from each other and also from the pivot. Also, more oft than not, a double pendulum is constrained to move in two dimensions via a constant, fixed polar angle. As such, the system has only two degrees of freedom: <math>\scriptsize{\theta_1}</math> and <math>\scriptsize{\theta_2}</math>, these being the azimuthal angles the axes of their respective masses form with the vertical.

It is well documented that such a system demonstrates chaotic behavior. However, we wish to examine the kinds of behavior that arise when we increase the degrees of freedom from two to '''''five'''''. How does one go about doing this? The answer is found very simply in a child's toy: the Astrojax.

=The Astrojax Pendulum=
The Astrojax is an assemblage of three weighted spheres on a string. Some versions of the toy allow all three spheres to move along the length of the string, but the case studied here does not. Two of the spheres are located at the ends of the string, and the third sphere is allowed to move along the length of the string between them. Some small amounts of damping are of course present. For the Astrojax, the polar angles are not constrained, allowing for motion in a three dimensional plane. This gives the Astrojax five degrees of freedom:
[[File:jax_pend.png|thumb| left | alt=text| A better view of the Astrojax. Note the five degrees of freedom.]]
*<math>\small{\lambda}</math> - The distance between the first and middle sphere
*<math>\small{\theta_1}</math> - The first axis' azimuthal angle
*<math>\small{\theta_2}</math> - The second axis' azimuthal angle
*<math>\small{\phi_1}</math> - The first axis' polar angle
*<math>\small{\phi_2}</math> - The second axis' polar angle

The lengths of the two pendula are defined as <math>\scriptsize{L_1=\lambda}</math> and <math>\scriptsize{L_2 = L- \lambda}</math> where <math>\scriptsize{L_1}</math> and <math>\scriptsize{L_2}</math> are the lengths of the first and second pendula, respectively, and where <math>\scriptsize{L}</math> is the total length of the string. It is important to note that because the length of the string is held constant and the length of the second pendulum depends on length of the first ''and'' the total length of the string, the length of the second pendulum is not an additional degree of freedom; it can be obtained through other lengths and variables already inherent in the system.

Hopefully, the addition of these three degrees of freedom will give rise to even more interesting chaotic behavior; if a rigid, double pendulum already possess chaotic properties, surely an unconstrained one will exhibit even more chaos, especially under forcing.

=Previous Work=
Surprisingly, the dynamics of three dimensional double pendulum are not well studied, not to mention those of a three dimensional variable length pendulum. Most studies of such systems constrain the lengths of the two pendula to be constant, giving them only four degrees of freedom as opposed to five for the Astrojax. Prior work is scarce, but there are some examples by [http://www.cds.caltech.edu/~marsden/wiki/uploads/projects/geomech/Dutoit2005.pdf Philip du Toit] and [http://www.katemaschan.com/uploads/2/4/4/5/24451800/astrojax_final_report.pdf Daniel Dichter and Kate Maschan].

The first differs from our experimental approach in that Dr. Du Toit simulated the system as whole. He simulated all three bobs, without any forcing, starting from some initial conditions. While his mathematical approach is still useful, his experimental process offers little in the way of the investigation of chaos, as he merely examined the dynamics of the system starting from some initial conditions, and did not infinitesimally vary those conditions to investigate how much differently the system evolved over time, which is one of the definitions of chaos.

Daniel Dichter and Kate Maschan completed much the same work as our group, however, they considered the string on which the Astrojax bobs were attached/moved to be a spring with some spring constant <math>\scriptsize{k}</math>, and they also took into account all viscous (i.e., damping) forces. Their paper is a qualitative approach to the system in much the same way ours ended up being a qualitative approach.

=Methods=
The motion of the Astrojax is, by nature, extremely complex. The motion of a double pendulum constrained to move in only two dimensions is already chaotic, and the sheer complexity of allowing motion in three dimensions can easily be extrapolated. It also bears stating that Astrojax is not purely a double pendulum. The center mass is free to move along the string, allowing the length of the two coupled pendulums to vary, but it is, fortunately, constrained to stay on the string itself, between the two end masses.

In order to proprerly observe, quantify, and analyze the data we made use of the Optitrack motion capture system and its proprietary software MOTIVE. A DENSO robotic arm held one of the end masses of the Astrojax and forced it, resulting in oscillations.

We forced the Astrojax in the vertical, or <math>\small{z}</math> direction in a trianglar wave rhythm using various speeds and amplitudes of the robotic arm. The software WINCAPS III was used in order to interface and program the robotic arm. The Astrojax were covered in infrared reflective tape for purposes of motion capture, along with a point on the robot arm. Thus, we could capture the three-dimensional dynamics of the two free bobs along with the forcing trajectories. Once captured and trajectorized, Optitrack's interpretation of marker disappearances and swaps in vertical position resulted in data gaps, marker identity loss, and marker switching. A tracking program in MATLAB was implemented (credit to M. Kingsbury) which used a ternary search algorithm which searched for least position differences in order to resolve each marker into a continuous path. Therefore we could achieve full 3-D trajectories for the middle & end bobs, along with the forcing marker.

In addition, we investigated the forcing motions used by a human in creating stable orbits with the Astrojax. To do this, we attached an infrared reflective marker to the human forcer's hand via a glove, and implemented the same methodology as with the robotic forcing, simply replacing the forcing marker.

[[File:denso.png|thumb|alt=text| An image of the DENSO robotic arm used for the experiement.<ref>http://www.globaldenso.com/en/</ref>]]

=====Physical Interpretation of the System=====
Previous analysis of the Lagrangian<ref>http://www.cds.caltech.edu/~marsden/wiki/uploads/projects/geomech/Dutoit2005.pdf</ref> and the Newtonian<ref>http://www.katemaschan.com/uploads/2/4/4/5/24451800/astrojax_final_report.pdf</ref> equations of motion have been derived in previous works. We provide a summary of these equations below.
The generalized cooridnates are given as:
<math>
\scriptsize{q = [x_1,y_1,z_1,x_2,y_2,z_2]}
</math>

This allows us to write the Lagrangian of the system as:
<math>
\scriptsize{L = \frac{1}{2}\dot{q}^TM\dot{q}-V(q)}
</math>
where <math>\scriptsize{V(q)=g(m_1z_1 + m_2z_2)}</math> and

<math>
\scriptsize{M = \begin{bmatrix}
m_1 & 0 & 0 & 0 & 0 & 0 \\
0 & m_1 & 0 & 0 & 0 & 0\\
0 & 0 & m_1 & 0 & 0 & 0\\
0 & 0 & 0 & m_2 & 0 & 0\\
0 & 0 & 0 & 0 & m_2 & 0\\
0 & 0 & 0 & 0 & 0 & m_2\\
\end{bmatrix}}
</math>

The Newtonian form of the system is

*<math>a_{2x} = \frac{Fs_{1x}-Fs_{2x}+Fd_{2x}}{m_2} </math>
*<math>a_{2y} = \frac{Fs_{1y}-Fs_{2y}+Fd_{2y}}{m_2} </math>
*<math>a_{2z} = \frac{Fs_{1z}-Fs_{2z}+Fd_{2z}}{m_2}-g</math>
*<math>a_{3x} = \frac{Fs_{2x}+Fd_{3x}}{m_3}</math>
*<math>a_{3y} = \frac{Fs_{2y}+Fd_{3y}}{m_3}</math>
*<math>a_{3z} = \frac{Fs_{2z}+Fd_{3z}}{m_3}-g</math>

where $\scriptsize{a_{ij}}$ is the acceleration of the $\scriptsize{i^{th}}$ mass in $\scriptsize{j}$ direction, $\scriptsize{g}$ is the acceleration due to gravity $\scriptsize{Fs_{\alpha\beta}}$ is the force of the string on mass $\scriptsize{\alpha}$ in the $\scriptsize{\beta}$ direction, and $\scriptsize{Fd_{\alpha\beta}}$ is the drag force and follows the same conventions as above.

While these equations provide a good interpretation of the Astrojax pendulum with a taut string, we find they do not cover a great deal of cases that occur in the real system we tested. For example, the Newtonian equations do not cover cases where the string goes slack unless the tensile force component is made piecewise active. This piecewise behavior can make the set of ordinary differential equations analytically intractable for traditional analysis methods.

In addition, the real system where Astrojax bobs with a finite radius are physically touching is not accounted for, as the actual degenerate state of the two bobs touching does not have them located at the same position.
==Results==
'''All Figures are located in the gallery below'''

We recorded an average of three takes each for each amplitude of robotic forcing, and five takes each for each type of human forcing (Horizontal, Vertical, Butterfly). In general, the naive method of having the robot arm force the Astrojax with a periodic motion with no feedback did not result in any stable orbits occurring, only quasi-stable and/or chaotic orbits that rapidly change or decay. In the robotic forcing, we found that a sufficient degree of acceleration was required in order to separate the two bobs from one another and cause the system to evolve away from the trivial state (i.e. the two bobs remain in contact with one another at the string's nadir).

Once separated in the robotic forcing, we found the Astrojax did not follow any predictable behaviors; their motion was mostly chaotic under this simple periodic forcing. The duration of time where the bobs were in this chaotic behavior vs. the time they spent in the trivial state increased overall as the forcing progressed, as shown in the plots below.

We also plot the mean active time of each take vs. each forcing's respective amplitude and frequency. As seen in Fig. 4, there is high variance in the mean active time solely for a forcing amplitude of 150mm, while the remaining amplitudes cluster around having 30 seconds. Figure 5 shows the same data with respect with the forcing frequency. Note that a small difference in forcing frequency near 2.5 Hertz seemed to generate a wider spread of mean active times.

Using MATLAB's Fast Fourier Transform (FFT), we found the most dominant frequency in the recorded oscillations in all three Cartesian dimensions. The resultant frequencies were often symmetrical about the Cartesian $\scriptsize{x}$ & $\scriptsize{y}$ coordinates, while FFTs of the $\scriptsize{z}$ coordinate often failed or were too noisy for some unknown reason. The plot below shows the resultant frequencies of each bob as a function of each take's forcing frequency.

As seen in Fig. 6, as the forcing frequency increases, the resultant dominant oscillation frequency for each bob tends to increase in a roughly linear manner, with the end bob increasing slightly faster than the middle bob.

With M. Kingsbury's tracking program, we created full 3-D plots and movies of the tracked trajectories. We provide one of the more interesting plots below: a trajectory plot of human forcing creating a stable horizontal orbit (Fig. 7).

The interesting part of the forcing is the green trajectory created by the human forcing marker. As seen in the plot, a complicated, helical path in order to maintain a steady horizontal orbit. The dynamic path that the human forcing creates with changing frequencies and amplitudes implies that some human 'feedback' mechanism is necessary to create such stable orbits.

==Discussion==
From our results, we can qualitatively confirm certain aspects of the Astrojax system. As seen in Figure 8, the resultant frequencies of the middle & end bobs increase linearly as the forcing frequency increases. We suspect that at higher forcing frequencies this linear relation will no longer be true as the forcing frequency becomes too fast and the end bobs no longer have time to fall sufficient distances to split.

Figures 1 thru 3 demonstrate that the active time of the end bobs increased as the Astrojax split from each other. These results match our qualitative observations with the robotic forcing. Often what would happen is that the forcing would generate small, short-lived splits of the jax that would either rapidly decay or create small, unstable orbits. It was only after some time of forcing and a few of these small splits that the jax would separate enough to create the full, rich, chaotic orbits with active times in the dozens of seconds.

Another observation we made in our experiments was that under certain smooth motions of the robotic arm's end affector, the Astrojax would not deign to split under any attempted forcing frequency or amplitude. This occurred because the type of motion interpolation used was too smooth, and the robot arm was slowing down before reaching its assigned endpoint. Using a different kind of interpolation generated a much jerkier motion and successful splitting of the Astrojax. Thus, we strongly suspect that a sufficient degree of acceleration is required to separate the bobs, an acceleration which most likely must exceed $\scriptsize{g}$. This acceleration is generated by a sufficiently jerky motion, since acceleration is the antiderivative of jerk.

==Conclusion==
The Astrojax is a very complex system with a great deal of dynamics and oscillatory patterns, especially when real-world constraints and perturbations are accounted for. From our results, we found that it is not feasible to generate stable orbits of any kind using a simple, naive periodic forcing. The best we could do was to create chaotic orbits that had some mean lifetime before eventually decaying. When examined alongside the complicated motions of human forcing that were necessary to create stable orbits, we conclude that some kind of feedback mechanism is needed to adjust the forcing frequency and/or amplitude on the fly to maintain a stable orbit.

Future work on the Astrojax system could involve creating a 3-D ordinary differential equation solver to simulate the trajectories of the Astrojax under our attempted forcings. We could then compare experimental vs. simulated trajectories and find differences in resultant frequencies and mean active times that arise from real-world constraints. Since we still have the raw data of the forcing trajectories and the initial conditions of the system, it would be simple to input those parameters into a ODE solver like MATLAB's Simulink library and examine differences in resultant trajectories that arise from the nonlinearities and chaos inherent in this system.

==Figure Gallery==
{|style="margin: 0 auto;"
| [[File: ActiveTimevsSplit90.png|thumb| alt=text| FIG. 1: A plot of the Astrojax's active time vs. the number of splits for 90 mm amplitude. ]]
| [[File: ActiveTimevsSplit150.png|thumb| alt=text| FIG 2: A plot of the Astrojax's active time vs. the number of splits for 150 mm amplitude. ]]
| [[File: ActiveTimevsSplit180.png|thumb| alt=text| FIG 3: A plot of the Astrojax's active time vs. the number of splits for 180 mm amplitude. ]]
| [[File: ActiveTimevsForcingAmp.png|thumb| alt=text| FIG 4: A plot of the Astrojax's active time vs. forcing amplitude. ]]
| [[File: ActiveTimevsFreq.png|thumb| alt=text| FIG 5: A plot of the Astrojax's active time vs. forcing frequency. ]]
| [[File: FreqPlot.png|thumb| alt=text| FIG 6: A plot of the Astrojax's oscillation frequency vs. the forcing frequency. Corresponds to the table in the Results section. ]]
| [[File: Take1Horiz3D.png|thumb| alt=text| FIG 7: A plot of the Astrojax's position in three dimensional Cartesian coordinate space. ]]
|}
==References==
{{reflist}}

File:NLD.pdf

2014-12-12T15:51:09Z

PatrickChang: Final report from 2014 Astrojax Pendulum group

Final report from 2014 Astrojax Pendulum group

Group 3 2014

2014-12-12T15:50:20Z

PatrickChang:

''Group members: Andras Karsai, Steven Harrington, and Colin Campbell''

[[Media:NLD.pdf|Report]]

[[File:astrojax_balls.png|thumb| alt+text| A simple diagram of the Astrojax<ref>http://www.ap-club.net/learning/Astrojax/basic_orbits/start_vertical_orbit.aspx</ref>]]
<blockquote>We seek to investigate the properties of a generalized double pendulum. Previous investigations on double pendulums often limit it to a planar case, and fix the lengths of each stage of the pendulum. The Astrojax is notable in that the pendulum stage lengths vary dynamically. We measure its behaviors in response to well quantified forcing using a perpendicular camera array to track position, and from that generate three-dimensional trajectories for each pendulum bob. The trajectories' mean chaotic lifetime and dominant oscillation frequencies are analyzed, and we find that this chaotic pendulum's oscillation frequencies are linearly correlated with the forcing frequency in the low-frequency regime. We also conclude that a naive periodic forcing of this system without a feedback mechanism is not sufficient to create stable, long-lived orbits of the Astrojax pendulum.</blockquote>

=Introduction=
An oscillation <ref>http://en.wikipedia.org/wiki/Oscillation</ref> is defined, in physics, as a regular variation in magnitude, position, etc. around a central point. Oscillatory mechanical systems are a vast subset of dynamical systems. They can be used to describe the evolution of states for nearly all physical phenomena. Oscillations of various types occur in real world mechanical systems, nearly all of which are thermodynamically irreversible (via damping, friction, energy due to heat loss, etc.). Even the light you see and the sound you hear are the results of the oscillations of a medium or a field. Here, we specifically study a set of coupled, chaotic oscillations that form from the forcing of a double pendulum.

As any good physics enthusiast knows, student, instructor, or otherwise, pendula are one of the most fundamental and ubiquitous systems studied, and while they may be common, they are only ideal and easily solvable in the most simple of cases. While the equations of motion for a single pendulum are easily attainable, especially for the low amplitude/small angle limit, a "double" pendulum is an example of the effects of coupling and gives rise to chaotic behavior.

Traditionally, a double pendulum has rigid axes, thereby defining the distance the two masses that can be from each other and also from the pivot. Also, more oft than not, a double pendulum is constrained to move in two dimensions via a constant, fixed polar angle. As such, the system has only two degrees of freedom: <math>\scriptsize{\theta_1}</math> and <math>\scriptsize{\theta_2}</math>, these being the azimuthal angles the axes of their respective masses form with the vertical.

It is well documented that such a system demonstrates chaotic behavior. However, we wish to examine the kinds of behavior that arise when we increase the degrees of freedom from two to '''''five'''''. How does one go about doing this? The answer is found very simply in a child's toy: the Astrojax.

=The Astrojax Pendulum=
The Astrojax is an assemblage of three weighted spheres on a string. Some versions of the toy allow all three spheres to move along the length of the string, but the case studied here does not. Two of the spheres are located at the ends of the string, and the third sphere is allowed to move along the length of the string between them. Some small amounts of damping are of course present. For the Astrojax, the polar angles are not constrained, allowing for motion in a three dimensional plane. This gives the Astrojax five degrees of freedom:
[[File:jax_pend.png|thumb| left | alt=text| A better view of the Astrojax. Note the five degrees of freedom.]]
*<math>\small{\lambda}</math> - The distance between the first and middle sphere
*<math>\small{\theta_1}</math> - The first axis' azimuthal angle
*<math>\small{\theta_2}</math> - The second axis' azimuthal angle
*<math>\small{\phi_1}</math> - The first axis' polar angle
*<math>\small{\phi_2}</math> - The second axis' polar angle

The lengths of the two pendula are defined as <math>\scriptsize{L_1=\lambda}</math> and <math>\scriptsize{L_2 = L- \lambda}</math> where <math>\scriptsize{L_1}</math> and <math>\scriptsize{L_2}</math> are the lengths of the first and second pendula, respectively, and where <math>\scriptsize{L}</math> is the total length of the string. It is important to note that because the length of the string is held constant and the length of the second pendulum depends on length of the first ''and'' the total length of the string, the length of the second pendulum is not an additional degree of freedom; it can be obtained through other lengths and variables already inherent in the system.

Hopefully, the addition of these three degrees of freedom will give rise to even more interesting chaotic behavior; if a rigid, double pendulum already possess chaotic properties, surely an unconstrained one will exhibit even more chaos, especially under forcing.

=Previous Work=
Surprisingly, the dynamics of three dimensional double pendulum are not well studied, not to mention those of a three dimensional variable length pendulum. Most studies of such systems constrain the lengths of the two pendula to be constant, giving them only four degrees of freedom as opposed to five for the Astrojax. Prior work is scarce, but there are some examples by [http://www.cds.caltech.edu/~marsden/wiki/uploads/projects/geomech/Dutoit2005.pdf Philip du Toit] and [http://www.katemaschan.com/uploads/2/4/4/5/24451800/astrojax_final_report.pdf Daniel Dichter and Kate Maschan].

The first differs from our experimental approach in that Dr. Du Toit simulated the system as whole. He simulated all three bobs, without any forcing, starting from some initial conditions. While his mathematical approach is still useful, his experimental process offers little in the way of the investigation of chaos, as he merely examined the dynamics of the system starting from some initial conditions, and did not infinitesimally vary those conditions to investigate how much differently the system evolved over time, which is one of the definitions of chaos.

Daniel Dichter and Kate Maschan completed much the same work as our group, however, they considered the string on which the Astrojax bobs were attached/moved to be a spring with some spring constant <math>\scriptsize{k}</math>, and they also took into account all viscous (i.e., damping) forces. Their paper is a qualitative approach to the system in much the same way ours ended up being a qualitative approach.

=Methods=
The motion of the Astrojax is, by nature, extremely complex. The motion of a double pendulum constrained to move in only two dimensions is already chaotic, and the sheer complexity of allowing motion in three dimensions can easily be extrapolated. It also bears stating that Astrojax is not purely a double pendulum. The center mass is free to move along the string, allowing the length of the two coupled pendulums to vary, but it is, fortunately, constrained to stay on the string itself, between the two end masses.

In order to proprerly observe, quantify, and analyze the data we made use of the Optitrack motion capture system and its proprietary software MOTIVE. A DENSO robotic arm held one of the end masses of the Astrojax and forced it, resulting in oscillations.

We forced the Astrojax in the vertical, or <math>\small{z}</math> direction in a trianglar wave rhythm using various speeds and amplitudes of the robotic arm. The software WINCAPS III was used in order to interface and program the robotic arm. The Astrojax were covered in infrared reflective tape for purposes of motion capture, along with a point on the robot arm. Thus, we could capture the three-dimensional dynamics of the two free bobs along with the forcing trajectories. Once captured and trajectorized, Optitrack's interpretation of marker disappearances and swaps in vertical position resulted in data gaps, marker identity loss, and marker switching. A tracking program in MATLAB was implemented (credit to M. Kingsbury) which used a ternary search algorithm which searched for least position differences in order to resolve each marker into a continuous path. Therefore we could achieve full 3-D trajectories for the middle & end bobs, along with the forcing marker.

In addition, we investigated the forcing motions used by a human in creating stable orbits with the Astrojax. To do this, we attached an infrared reflective marker to the human forcer's hand via a glove, and implemented the same methodology as with the robotic forcing, simply replacing the forcing marker.

[[File:denso.png|thumb|alt=text| An image of the DENSO robotic arm used for the experiement.<ref>http://www.globaldenso.com/en/</ref>]]

=====Physical Interpretation of the System=====
Previous analysis of the Lagrangian<ref>http://www.cds.caltech.edu/~marsden/wiki/uploads/projects/geomech/Dutoit2005.pdf</ref> and the Newtonian<ref>http://www.katemaschan.com/uploads/2/4/4/5/24451800/astrojax_final_report.pdf</ref> equations of motion have been derived in previous works. We provide a summary of these equations below.
The generalized cooridnates are given as:
<math>
\scriptsize{q = [x_1,y_1,z_1,x_2,y_2,z_2]}
</math>

This allows us to write the Lagrangian of the system as:
<math>
\scriptsize{L = \frac{1}{2}\dot{q}^TM\dot{q}-V(q)}
</math>
where <math>\scriptsize{V(q)=g(m_1z_1 + m_2z_2)}</math> and

<math>
\scriptsize{M = \begin{bmatrix}
m_1 & 0 & 0 & 0 & 0 & 0 \\
0 & m_1 & 0 & 0 & 0 & 0\\
0 & 0 & m_1 & 0 & 0 & 0\\
0 & 0 & 0 & m_2 & 0 & 0\\
0 & 0 & 0 & 0 & m_2 & 0\\
0 & 0 & 0 & 0 & 0 & m_2\\
\end{bmatrix}}
</math>

The Newtonian form of the system is

*<math>a_{2x} = \frac{Fs_{1x}-Fs_{2x}+Fd_{2x}}{m_2} </math>
*<math>a_{2y} = \frac{Fs_{1y}-Fs_{2y}+Fd_{2y}}{m_2} </math>
*<math>a_{2z} = \frac{Fs_{1z}-Fs_{2z}+Fd_{2z}}{m_2}-g</math>
*<math>a_{3x} = \frac{Fs_{2x}+Fd_{3x}}{m_3}</math>
*<math>a_{3y} = \frac{Fs_{2y}+Fd_{3y}}{m_3}</math>
*<math>a_{3z} = \frac{Fs_{2z}+Fd_{3z}}{m_3}-g</math>

where $\scriptsize{a_{ij}}$ is the acceleration of the $\scriptsize{i^{th}}$ mass in $\scriptsize{j}$ direction, $\scriptsize{g}$ is the acceleration due to gravity $\scriptsize{Fs_{\alpha\beta}}$ is the force of the string on mass $\scriptsize{\alpha}$ in the $\scriptsize{\beta}$ direction, and $\scriptsize{Fd_{\alpha\beta}}$ is the drag force and follows the same conventions as above.

While these equations provide a good interpretation of the Astrojax pendulum with a taut string, we find they do not cover a great deal of cases that occur in the real system we tested. For example, the Newtonian equations do not cover cases where the string goes slack unless the tensile force component is made piecewise active. This piecewise behavior can make the set of ordinary differential equations analytically intractable for traditional analysis methods.

In addition, the real system where Astrojax bobs with a finite radius are physically touching is not accounted for, as the actual degenerate state of the two bobs touching does not have them located at the same position.
==Results==
'''All Figures are located in the gallery below'''

We recorded an average of three takes each for each amplitude of robotic forcing, and five takes each for each type of human forcing (Horizontal, Vertical, Butterfly). In general, the naive method of having the robot arm force the Astrojax with a periodic motion with no feedback did not result in any stable orbits occurring, only quasi-stable and/or chaotic orbits that rapidly change or decay. In the robotic forcing, we found that a sufficient degree of acceleration was required in order to separate the two bobs from one another and cause the system to evolve away from the trivial state (i.e. the two bobs remain in contact with one another at the string's nadir).

Once separated in the robotic forcing, we found the Astrojax did not follow any predictable behaviors; their motion was mostly chaotic under this simple periodic forcing. The duration of time where the bobs were in this chaotic behavior vs. the time they spent in the trivial state increased overall as the forcing progressed, as shown in the plots below.

We also plot the mean active time of each take vs. each forcing's respective amplitude and frequency. As seen in Fig. 4, there is high variance in the mean active time solely for a forcing amplitude of 150mm, while the remaining amplitudes cluster around having 30 seconds. Figure 5 shows the same data with respect with the forcing frequency. Note that a small difference in forcing frequency near 2.5 Hertz seemed to generate a wider spread of mean active times.

Using MATLAB's Fast Fourier Transform (FFT), we found the most dominant frequency in the recorded oscillations in all three Cartesian dimensions. The resultant frequencies were often symmetrical about the Cartesian $\scriptsize{x}$ & $\scriptsize{y}$ coordinates, while FFTs of the $\scriptsize{z}$ coordinate often failed or were too noisy for some unknown reason. The plot below shows the resultant frequencies of each bob as a function of each take's forcing frequency.

As seen in Fig. 6, as the forcing frequency increases, the resultant dominant oscillation frequency for each bob tends to increase in a roughly linear manner, with the end bob increasing slightly faster than the middle bob.

With M. Kingsbury's tracking program, we created full 3-D plots and movies of the tracked trajectories. We provide one of the more interesting plots below: a trajectory plot of human forcing creating a stable horizontal orbit (Fig. 7).

The interesting part of the forcing is the green trajectory created by the human forcing marker. As seen in the plot, a complicated, helical path in order to maintain a steady horizontal orbit. The dynamic path that the human forcing creates with changing frequencies and amplitudes implies that some human 'feedback' mechanism is necessary to create such stable orbits.

==Discussion==
From our results, we can qualitatively confirm certain aspects of the Astrojax system. As seen in Figure 8, the resultant frequencies of the middle & end bobs increase linearly as the forcing frequency increases. We suspect that at higher forcing frequencies this linear relation will no longer be true as the forcing frequency becomes too fast and the end bobs no longer have time to fall sufficient distances to split.

Figures 1 thru 3 demonstrate that the active time of the end bobs increased as the Astrojax split from each other. These results match our qualitative observations with the robotic forcing. Often what would happen is that the forcing would generate small, short-lived splits of the jax that would either rapidly decay or create small, unstable orbits. It was only after some time of forcing and a few of these small splits that the jax would separate enough to create the full, rich, chaotic orbits with active times in the dozens of seconds.

Another observation we made in our experiments was that under certain smooth motions of the robotic arm's end affector, the Astrojax would not deign to split under any attempted forcing frequency or amplitude. This occurred because the type of motion interpolation used was too smooth, and the robot arm was slowing down before reaching its assigned endpoint. Using a different kind of interpolation generated a much jerkier motion and successful splitting of the Astrojax. Thus, we strongly suspect that a sufficient degree of acceleration is required to separate the bobs, an acceleration which most likely must exceed $\scriptsize{g}$. This acceleration is generated by a sufficiently jerky motion, since acceleration is the antiderivative of jerk.

==Conclusion==
The Astrojax is a very complex system with a great deal of dynamics and oscillatory patterns, especially when real-world constraints and perturbations are accounted for. From our results, we found that it is not feasible to generate stable orbits of any kind using a simple, naive periodic forcing. The best we could do was to create chaotic orbits that had some mean lifetime before eventually decaying. When examined alongside the complicated motions of human forcing that were necessary to create stable orbits, we conclude that some kind of feedback mechanism is needed to adjust the forcing frequency and/or amplitude on the fly to maintain a stable orbit.

Future work on the Astrojax system could involve creating a 3-D ordinary differential equation solver to simulate the trajectories of the Astrojax under our attempted forcings. We could then compare experimental vs. simulated trajectories and find differences in resultant frequencies and mean active times that arise from real-world constraints. Since we still have the raw data of the forcing trajectories and the initial conditions of the system, it would be simple to input those parameters into a ODE solver like MATLAB's Simulink library and examine differences in resultant trajectories that arise from the nonlinearities and chaos inherent in this system.

==Figure Gallery==
{|style="margin: 0 auto;"
| [[File: ActiveTimevsSplit90.png|thumb| alt=text| FIG. 1: A plot of the Astrojax's active time vs. the number of splits for 90 mm amplitude. ]]
| [[File: ActiveTimevsSplit150.png|thumb| alt=text| FIG 2: A plot of the Astrojax's active time vs. the number of splits for 150 mm amplitude. ]]
| [[File: ActiveTimevsSplit180.png|thumb| alt=text| FIG 3: A plot of the Astrojax's active time vs. the number of splits for 180 mm amplitude. ]]
| [[File: ActiveTimevsForcingAmp.png|thumb| alt=text| FIG 4: A plot of the Astrojax's active time vs. forcing amplitude. ]]
| [[File: ActiveTimevsFreq.png|thumb| alt=text| FIG 5: A plot of the Astrojax's active time vs. forcing frequency. ]]
| [[File: FreqPlot.png|thumb| alt=text| FIG 6: A plot of the Astrojax's oscillation frequency vs. the forcing frequency. Corresponds to the table in the Results section. ]]
| [[File: Take1Horiz3D.png|thumb| alt=text| FIG 7: A plot of the Astrojax's position in three dimensional Cartesian coordinate space. ]]
|}
==References==
{{reflist}}

Group 3 2014

2014-12-12T15:49:42Z

PatrickChang:

''Group members: Andras Karsai, Steven Harrington, and Colin Campbell''
[[Media:NLD.pdf|Report]]

[[File:astrojax_balls.png|thumb| alt+text| A simple diagram of the Astrojax<ref>http://www.ap-club.net/learning/Astrojax/basic_orbits/start_vertical_orbit.aspx</ref>]]
<blockquote>We seek to investigate the properties of a generalized double pendulum. Previous investigations on double pendulums often limit it to a planar case, and fix the lengths of each stage of the pendulum. The Astrojax is notable in that the pendulum stage lengths vary dynamically. We measure its behaviors in response to well quantified forcing using a perpendicular camera array to track position, and from that generate three-dimensional trajectories for each pendulum bob. The trajectories' mean chaotic lifetime and dominant oscillation frequencies are analyzed, and we find that this chaotic pendulum's oscillation frequencies are linearly correlated with the forcing frequency in the low-frequency regime. We also conclude that a naive periodic forcing of this system without a feedback mechanism is not sufficient to create stable, long-lived orbits of the Astrojax pendulum.</blockquote>

=Introduction=
An oscillation <ref>http://en.wikipedia.org/wiki/Oscillation</ref> is defined, in physics, as a regular variation in magnitude, position, etc. around a central point. Oscillatory mechanical systems are a vast subset of dynamical systems. They can be used to describe the evolution of states for nearly all physical phenomena. Oscillations of various types occur in real world mechanical systems, nearly all of which are thermodynamically irreversible (via damping, friction, energy due to heat loss, etc.). Even the light you see and the sound you hear are the results of the oscillations of a medium or a field. Here, we specifically study a set of coupled, chaotic oscillations that form from the forcing of a double pendulum.

As any good physics enthusiast knows, student, instructor, or otherwise, pendula are one of the most fundamental and ubiquitous systems studied, and while they may be common, they are only ideal and easily solvable in the most simple of cases. While the equations of motion for a single pendulum are easily attainable, especially for the low amplitude/small angle limit, a "double" pendulum is an example of the effects of coupling and gives rise to chaotic behavior.

Traditionally, a double pendulum has rigid axes, thereby defining the distance the two masses that can be from each other and also from the pivot. Also, more oft than not, a double pendulum is constrained to move in two dimensions via a constant, fixed polar angle. As such, the system has only two degrees of freedom: <math>\scriptsize{\theta_1}</math> and <math>\scriptsize{\theta_2}</math>, these being the azimuthal angles the axes of their respective masses form with the vertical.

It is well documented that such a system demonstrates chaotic behavior. However, we wish to examine the kinds of behavior that arise when we increase the degrees of freedom from two to '''''five'''''. How does one go about doing this? The answer is found very simply in a child's toy: the Astrojax.

=The Astrojax Pendulum=
The Astrojax is an assemblage of three weighted spheres on a string. Some versions of the toy allow all three spheres to move along the length of the string, but the case studied here does not. Two of the spheres are located at the ends of the string, and the third sphere is allowed to move along the length of the string between them. Some small amounts of damping are of course present. For the Astrojax, the polar angles are not constrained, allowing for motion in a three dimensional plane. This gives the Astrojax five degrees of freedom:
[[File:jax_pend.png|thumb| left | alt=text| A better view of the Astrojax. Note the five degrees of freedom.]]
*<math>\small{\lambda}</math> - The distance between the first and middle sphere
*<math>\small{\theta_1}</math> - The first axis' azimuthal angle
*<math>\small{\theta_2}</math> - The second axis' azimuthal angle
*<math>\small{\phi_1}</math> - The first axis' polar angle
*<math>\small{\phi_2}</math> - The second axis' polar angle

The lengths of the two pendula are defined as <math>\scriptsize{L_1=\lambda}</math> and <math>\scriptsize{L_2 = L- \lambda}</math> where <math>\scriptsize{L_1}</math> and <math>\scriptsize{L_2}</math> are the lengths of the first and second pendula, respectively, and where <math>\scriptsize{L}</math> is the total length of the string. It is important to note that because the length of the string is held constant and the length of the second pendulum depends on length of the first ''and'' the total length of the string, the length of the second pendulum is not an additional degree of freedom; it can be obtained through other lengths and variables already inherent in the system.

Hopefully, the addition of these three degrees of freedom will give rise to even more interesting chaotic behavior; if a rigid, double pendulum already possess chaotic properties, surely an unconstrained one will exhibit even more chaos, especially under forcing.

=Previous Work=
Surprisingly, the dynamics of three dimensional double pendulum are not well studied, not to mention those of a three dimensional variable length pendulum. Most studies of such systems constrain the lengths of the two pendula to be constant, giving them only four degrees of freedom as opposed to five for the Astrojax. Prior work is scarce, but there are some examples by [http://www.cds.caltech.edu/~marsden/wiki/uploads/projects/geomech/Dutoit2005.pdf Philip du Toit] and [http://www.katemaschan.com/uploads/2/4/4/5/24451800/astrojax_final_report.pdf Daniel Dichter and Kate Maschan].

The first differs from our experimental approach in that Dr. Du Toit simulated the system as whole. He simulated all three bobs, without any forcing, starting from some initial conditions. While his mathematical approach is still useful, his experimental process offers little in the way of the investigation of chaos, as he merely examined the dynamics of the system starting from some initial conditions, and did not infinitesimally vary those conditions to investigate how much differently the system evolved over time, which is one of the definitions of chaos.

Daniel Dichter and Kate Maschan completed much the same work as our group, however, they considered the string on which the Astrojax bobs were attached/moved to be a spring with some spring constant <math>\scriptsize{k}</math>, and they also took into account all viscous (i.e., damping) forces. Their paper is a qualitative approach to the system in much the same way ours ended up being a qualitative approach.

=Methods=
The motion of the Astrojax is, by nature, extremely complex. The motion of a double pendulum constrained to move in only two dimensions is already chaotic, and the sheer complexity of allowing motion in three dimensions can easily be extrapolated. It also bears stating that Astrojax is not purely a double pendulum. The center mass is free to move along the string, allowing the length of the two coupled pendulums to vary, but it is, fortunately, constrained to stay on the string itself, between the two end masses.

In order to proprerly observe, quantify, and analyze the data we made use of the Optitrack motion capture system and its proprietary software MOTIVE. A DENSO robotic arm held one of the end masses of the Astrojax and forced it, resulting in oscillations.

We forced the Astrojax in the vertical, or <math>\small{z}</math> direction in a trianglar wave rhythm using various speeds and amplitudes of the robotic arm. The software WINCAPS III was used in order to interface and program the robotic arm. The Astrojax were covered in infrared reflective tape for purposes of motion capture, along with a point on the robot arm. Thus, we could capture the three-dimensional dynamics of the two free bobs along with the forcing trajectories. Once captured and trajectorized, Optitrack's interpretation of marker disappearances and swaps in vertical position resulted in data gaps, marker identity loss, and marker switching. A tracking program in MATLAB was implemented (credit to M. Kingsbury) which used a ternary search algorithm which searched for least position differences in order to resolve each marker into a continuous path. Therefore we could achieve full 3-D trajectories for the middle & end bobs, along with the forcing marker.

In addition, we investigated the forcing motions used by a human in creating stable orbits with the Astrojax. To do this, we attached an infrared reflective marker to the human forcer's hand via a glove, and implemented the same methodology as with the robotic forcing, simply replacing the forcing marker.

[[File:denso.png|thumb|alt=text| An image of the DENSO robotic arm used for the experiement.<ref>http://www.globaldenso.com/en/</ref>]]

=====Physical Interpretation of the System=====
Previous analysis of the Lagrangian<ref>http://www.cds.caltech.edu/~marsden/wiki/uploads/projects/geomech/Dutoit2005.pdf</ref> and the Newtonian<ref>http://www.katemaschan.com/uploads/2/4/4/5/24451800/astrojax_final_report.pdf</ref> equations of motion have been derived in previous works. We provide a summary of these equations below.
The generalized cooridnates are given as:
<math>
\scriptsize{q = [x_1,y_1,z_1,x_2,y_2,z_2]}
</math>

This allows us to write the Lagrangian of the system as:
<math>
\scriptsize{L = \frac{1}{2}\dot{q}^TM\dot{q}-V(q)}
</math>
where <math>\scriptsize{V(q)=g(m_1z_1 + m_2z_2)}</math> and

<math>
\scriptsize{M = \begin{bmatrix}
m_1 & 0 & 0 & 0 & 0 & 0 \\
0 & m_1 & 0 & 0 & 0 & 0\\
0 & 0 & m_1 & 0 & 0 & 0\\
0 & 0 & 0 & m_2 & 0 & 0\\
0 & 0 & 0 & 0 & m_2 & 0\\
0 & 0 & 0 & 0 & 0 & m_2\\
\end{bmatrix}}
</math>

The Newtonian form of the system is

*<math>a_{2x} = \frac{Fs_{1x}-Fs_{2x}+Fd_{2x}}{m_2} </math>
*<math>a_{2y} = \frac{Fs_{1y}-Fs_{2y}+Fd_{2y}}{m_2} </math>
*<math>a_{2z} = \frac{Fs_{1z}-Fs_{2z}+Fd_{2z}}{m_2}-g</math>
*<math>a_{3x} = \frac{Fs_{2x}+Fd_{3x}}{m_3}</math>
*<math>a_{3y} = \frac{Fs_{2y}+Fd_{3y}}{m_3}</math>
*<math>a_{3z} = \frac{Fs_{2z}+Fd_{3z}}{m_3}-g</math>

where $\scriptsize{a_{ij}}$ is the acceleration of the $\scriptsize{i^{th}}$ mass in $\scriptsize{j}$ direction, $\scriptsize{g}$ is the acceleration due to gravity $\scriptsize{Fs_{\alpha\beta}}$ is the force of the string on mass $\scriptsize{\alpha}$ in the $\scriptsize{\beta}$ direction, and $\scriptsize{Fd_{\alpha\beta}}$ is the drag force and follows the same conventions as above.

While these equations provide a good interpretation of the Astrojax pendulum with a taut string, we find they do not cover a great deal of cases that occur in the real system we tested. For example, the Newtonian equations do not cover cases where the string goes slack unless the tensile force component is made piecewise active. This piecewise behavior can make the set of ordinary differential equations analytically intractable for traditional analysis methods.

In addition, the real system where Astrojax bobs with a finite radius are physically touching is not accounted for, as the actual degenerate state of the two bobs touching does not have them located at the same position.
==Results==
'''All Figures are located in the gallery below'''

We recorded an average of three takes each for each amplitude of robotic forcing, and five takes each for each type of human forcing (Horizontal, Vertical, Butterfly). In general, the naive method of having the robot arm force the Astrojax with a periodic motion with no feedback did not result in any stable orbits occurring, only quasi-stable and/or chaotic orbits that rapidly change or decay. In the robotic forcing, we found that a sufficient degree of acceleration was required in order to separate the two bobs from one another and cause the system to evolve away from the trivial state (i.e. the two bobs remain in contact with one another at the string's nadir).

Once separated in the robotic forcing, we found the Astrojax did not follow any predictable behaviors; their motion was mostly chaotic under this simple periodic forcing. The duration of time where the bobs were in this chaotic behavior vs. the time they spent in the trivial state increased overall as the forcing progressed, as shown in the plots below.

We also plot the mean active time of each take vs. each forcing's respective amplitude and frequency. As seen in Fig. 4, there is high variance in the mean active time solely for a forcing amplitude of 150mm, while the remaining amplitudes cluster around having 30 seconds. Figure 5 shows the same data with respect with the forcing frequency. Note that a small difference in forcing frequency near 2.5 Hertz seemed to generate a wider spread of mean active times.

Using MATLAB's Fast Fourier Transform (FFT), we found the most dominant frequency in the recorded oscillations in all three Cartesian dimensions. The resultant frequencies were often symmetrical about the Cartesian $\scriptsize{x}$ & $\scriptsize{y}$ coordinates, while FFTs of the $\scriptsize{z}$ coordinate often failed or were too noisy for some unknown reason. The plot below shows the resultant frequencies of each bob as a function of each take's forcing frequency.

As seen in Fig. 6, as the forcing frequency increases, the resultant dominant oscillation frequency for each bob tends to increase in a roughly linear manner, with the end bob increasing slightly faster than the middle bob.

With M. Kingsbury's tracking program, we created full 3-D plots and movies of the tracked trajectories. We provide one of the more interesting plots below: a trajectory plot of human forcing creating a stable horizontal orbit (Fig. 7).

The interesting part of the forcing is the green trajectory created by the human forcing marker. As seen in the plot, a complicated, helical path in order to maintain a steady horizontal orbit. The dynamic path that the human forcing creates with changing frequencies and amplitudes implies that some human 'feedback' mechanism is necessary to create such stable orbits.

==Discussion==
From our results, we can qualitatively confirm certain aspects of the Astrojax system. As seen in Figure 8, the resultant frequencies of the middle & end bobs increase linearly as the forcing frequency increases. We suspect that at higher forcing frequencies this linear relation will no longer be true as the forcing frequency becomes too fast and the end bobs no longer have time to fall sufficient distances to split.

Figures 1 thru 3 demonstrate that the active time of the end bobs increased as the Astrojax split from each other. These results match our qualitative observations with the robotic forcing. Often what would happen is that the forcing would generate small, short-lived splits of the jax that would either rapidly decay or create small, unstable orbits. It was only after some time of forcing and a few of these small splits that the jax would separate enough to create the full, rich, chaotic orbits with active times in the dozens of seconds.

Another observation we made in our experiments was that under certain smooth motions of the robotic arm's end affector, the Astrojax would not deign to split under any attempted forcing frequency or amplitude. This occurred because the type of motion interpolation used was too smooth, and the robot arm was slowing down before reaching its assigned endpoint. Using a different kind of interpolation generated a much jerkier motion and successful splitting of the Astrojax. Thus, we strongly suspect that a sufficient degree of acceleration is required to separate the bobs, an acceleration which most likely must exceed $\scriptsize{g}$. This acceleration is generated by a sufficiently jerky motion, since acceleration is the antiderivative of jerk.

==Conclusion==
The Astrojax is a very complex system with a great deal of dynamics and oscillatory patterns, especially when real-world constraints and perturbations are accounted for. From our results, we found that it is not feasible to generate stable orbits of any kind using a simple, naive periodic forcing. The best we could do was to create chaotic orbits that had some mean lifetime before eventually decaying. When examined alongside the complicated motions of human forcing that were necessary to create stable orbits, we conclude that some kind of feedback mechanism is needed to adjust the forcing frequency and/or amplitude on the fly to maintain a stable orbit.

Future work on the Astrojax system could involve creating a 3-D ordinary differential equation solver to simulate the trajectories of the Astrojax under our attempted forcings. We could then compare experimental vs. simulated trajectories and find differences in resultant frequencies and mean active times that arise from real-world constraints. Since we still have the raw data of the forcing trajectories and the initial conditions of the system, it would be simple to input those parameters into a ODE solver like MATLAB's Simulink library and examine differences in resultant trajectories that arise from the nonlinearities and chaos inherent in this system.

==Figure Gallery==
{|style="margin: 0 auto;"
| [[File: ActiveTimevsSplit90.png|thumb| alt=text| FIG. 1: A plot of the Astrojax's active time vs. the number of splits for 90 mm amplitude. ]]
| [[File: ActiveTimevsSplit150.png|thumb| alt=text| FIG 2: A plot of the Astrojax's active time vs. the number of splits for 150 mm amplitude. ]]
| [[File: ActiveTimevsSplit180.png|thumb| alt=text| FIG 3: A plot of the Astrojax's active time vs. the number of splits for 180 mm amplitude. ]]
| [[File: ActiveTimevsForcingAmp.png|thumb| alt=text| FIG 4: A plot of the Astrojax's active time vs. forcing amplitude. ]]
| [[File: ActiveTimevsFreq.png|thumb| alt=text| FIG 5: A plot of the Astrojax's active time vs. forcing frequency. ]]
| [[File: FreqPlot.png|thumb| alt=text| FIG 6: A plot of the Astrojax's oscillation frequency vs. the forcing frequency. Corresponds to the table in the Results section. ]]
| [[File: Take1Horiz3D.png|thumb| alt=text| FIG 7: A plot of the Astrojax's position in three dimensional Cartesian coordinate space. ]]
|}
==References==
{{reflist}}

Assignments

2014-10-30T15:36:29Z

PatrickChang:

==Assignments for 2014==

===Assignment 1===
Problems are from Strogatz, 2nd edition: 2.2.7, 2.3.3, 2.4.2, 2.5.6G, 2.6.2G, 2.8.8, 3.1.1, 3.4.11, 3.4.14, 3.5.6G, 3.6.6. Problems with a G at the end are problems added for the graduate students. Undergraduates may hand them in for extra credit.

HW is due in Prof. G's office on Thursday, Sept 11 by 5PM . Please put your name and if you are a graduate student/undergrad on the HW. Please staple all sheets together.

===Assignment 2 (Midterm)===
Midterm take-home exam, due 4PM, Oct 31 in Prof. Goldman's office.

The exam consists of a few parts: a preproposal, some problems from the book, a numerical exercise, a proposal like we discussed in class, and a wiki site. You must complete all parts to receive full credit.

1. (10%). Present your prepoposal talk on Oct. 9 in class.

2. (40%) Problems: 4.3.6, 4.3.10G, 4.5.1, 5.1.1, 5.1.9G, 5.1.10b, 5.2.13, 6.1.2, 6.3.6, 6.5.15, 6.6.8G, 6.7.2, where G denotes problems that must be done by graduate students. Unlike homework sets, you may not discuss these problems with your classmates.

3. (10%) For problem 6.3.6 write a program in Matlab, Octave, python, etc to sketch the phase plane and plot trajectories x(t) and y(t). Compare what you generate to solutions generated in pplane8.m (seehttp://math.rice.edu/~dfield/index.html). You must work on your own for this problem.

4. (30%) Collaboratively write a 1-3 page project proposal which will summarize the goals of your experiment, discuss experimental design, and cite any relevant literature. We would like these to be written using the LaTeX typesetting language. This can be downloaded and installed freely from http://miktex.org/ and free editors are available on the web as well (http://www.latexeditor.org/ for example).

5. (10%) Create a website for your project. Website content will be edited on the class wiki which is at http://nldlab.gatech.edu. Each group will be responsible for creating a wiki page for their experiment which will contain background information on the phenomena studied. Wiki pages should contain outside references such as journal articles.

==Assignments for 2012==

===Assignment 2===
Problems are from Strogatz (1994): 7.3.3, 7.6.15, 8.1.12G, 8.2.1, 8.2.3, 8.4.12, 8.7.10G, 9.3.9, 9.4.2, 10.1.12, 10.2.4, 11.3.8, 11.4.6G, 12.1.5, 12.3.2G, 12.4.3. Problems 8.7.3 and 10.4.10 can be submitted for extra credit for undergraduates and graduates students. Problems with a G at the end are problems added for the graduate students. Undergraduates may hand them in for extra credit.

HW is due Tuesday, November 20th by 5pm. Please put your name and if you are a graduate student/undergrad on the HW. Please staple all sheets together.

===Midterm===
The exam consists of four parts: some problems from the book, a numerical exercise, a proposal like we discussed in class, and a wiki site. You must complete all parts to receive full credit.<o:p></o:p>

1. (50%) Problems: 4.3.6, 4.3.10G, 4.5.1, 5.1.1, 5.1.9G, 5.1.10b, 5.2.13, 6.1.2, 6.3.6, 6.5.15, 6.6.8G, 6.7.2, where G denotes problems that must be done by graduate students. Unlike homework sets, you may not discuss these problems with your classmates.

2. (10%) For problem 6.3.6 write a program in Matlab, Octave, python, etc to sketch the phase plane and plot trajectories x(t) and y(t). Compare what you generate to solutions generated in pplane8.m (see http://math.rice.edu/~dfield/index.html). You must work on your own for this problem.

3. (30%) Collaboratively write a 1-3 page project proposal which will summarize the goals of your experiment, discuss experimental design, and cite any relevant literature. We would like these to be written using the LaTeX typesetting language. This can be downloaded and installed freely from http://miktex.org/ and free editors are available on the web as well (http://www.latexeditor.org/ for example).

4. (10%) Create a website for your project. Website content will be edited on the class wiki which is at http://nldlab.gatech.edu. Each group will be responsible for creating a wiki page for their experiment which will contain background information on the phenomena studied. Wiki pages should contain outside references such as journal articles.

Midterm is due at 5pm on Thursday, October 18, in the box outside Dr. Goldman's office. Please put your name and if you are a graduate student/undergrad on the exam and staple all sheets together.

==Assignments for 2011==
=== Assignment 4 ===
HW4: 9.1.4, 9.4.2, 10.2.4, 10.3.7G,11.4.6G,11.3.8,12.1.5, 12.3.2

Problems with a G at the end are problems added for the graduate students. Undergraduates may hand them in for extra credit.

HW is due <del>in class on Tuesday, Nov 29</del> Dec 1st at the beginning of class. Please put your name and if you are a graduate student/undergrad on the HW. Please staple all sheets together.

=== Assignment 3 ===
HW3: 7.3.3, 7.6.15, 8.1.1, 8.1.12G, 8.2.1, 8.2.3, 8.4.4G, 8.4.12

Problems with a G at the end are problems added for the graduate students. Undergraduates may hand them in for extra credit.

HW is due in class on Thursday, Nov 10. Please put your name and if you are a graduate student/undergrad on the HW. Please staple all sheets together.

=== Midterm ===
The exam consists of three parts: some problems from the book, a numerical exercise, and a proposal like we discussed in class. You must complete all parts to receive full credit.

1. Problems: 3.5.4, 4.3.6, 5.1.9G, <del>5.2.13</del>, 6.3.6, 6.5.15, 6.6.8G, 6.7.2, where G denotes problems that must be done by graduate students. Unlike homework sets, you may not discuss these problems with your classmates.

2. For problem 6.3.6 write a program in Matlab, Octave, python, etc to sketch the phase plane and plot trajectories x(t) and y(t). Compare what you generate to solutions generated in pplane8.m (see http://math.rice.edu/~dfield/index.html). You must work on your own for this problem.

3. Collaboratively write a 1-3 page project proposal which will summarize the goals of your experiment and cite any relevant literature. We would like these to be written using the LaTeX typesetting language. This can be downloaded and installed freely from http://miktex.org/ and free editors are available on the web as well http://www.latexeditor.org/ for example.

4. Create a website for your project. Website content will be edited on the class wiki which will soon be located at http://nldlab.gatech.edu/w/. Each group will be responsible for creating a wiki page for their experiment which will contain background information on the phenomena studied. Wiki pages should contain outside references such as journal articles. A sample page will be posted.

=== Assignment 2 ===

HW2: 3.5.6G, 3.6.6, 4.1.4, 4.3.3, 4.3.10G, 4.5.1, 5.1.1, 5.1.10b, 5.2.12, 5.2.13G.

Problems with a G at the end are problems added for the graduate students. Undergraduates may hand them in for extra credit.

HW is due in class on Tuesday, Sept 27. Please put your name and if you are a graduate student/undergrad on the HW. Please staple all sheets together.

=== Assignment 1 ===

Problems are from Strogatz: 2.2.7, 2.3.3, 2.4.2, 2.5.3G, 2.6.2G, 2.8.8, 3.1.1, 3.4.11, 3.4.14, 3.5.6G, 3.6.6. Problems with a G at the end are problems
added for the graduate students. Undergraduates may hand them in for extra credit.

HW is due in class on Tuesday, Sept 13. Please put your name and if you are a graduate student/undergrad on the HW. Please staple all sheets together.

Group 5 2014

2014-10-29T15:35:25Z

PatrickChang: Created page with "Members: Caligan, Lucas, Li, Norris"

Members: Caligan, Lucas, Li, Norris

Group 4 2014

2014-10-29T15:35:09Z

PatrickChang: Created page with "Members: Lind, Salgueiro, Trimble"

Members: Lind, Salgueiro, Trimble

Group 3 2014

2014-10-29T15:34:49Z

PatrickChang: Created page with "Members: Campbell, Harrington, Karsai"

Members: Campbell, Harrington, Karsai

Group 2 2014

2014-10-29T15:34:30Z

PatrickChang: Created page with "Members: Bollenbacher, Chambers, Cunningham, Putzel"

Members: Bollenbacher, Chambers, Cunningham, Putzel

MediaWiki:Sidebar

2014-10-29T15:33:53Z

PatrickChang:

Group 1 2014

2014-10-29T15:31:16Z

PatrickChang: Created page with "Members: Jansson, McMahon, Reitz, Savoie"

Members: Jansson, McMahon, Reitz, Savoie

Syllabus

2014-10-29T15:29:21Z

PatrickChang: /* Instructor & TA */

__NOTOC__

'''Class: Physics 4267/6268, Nonlinear Dynamics & Chaos, Fall 2012'''

==Instructor & TA==
'''Instructor:'''

Prof. Daniel I. Goldman, School of Physics, Georgia Institute of Technology 
'''Office:''' Howey C202 (office hours: by email) 
'''Phone:''' (404) 894-0993 
'''E-mail:''' daniel.goldman@physics.gatech.edu 

'''TAs:'''

Patrick Chang 
'''Office:''' MoSE G128 (office hours TBD) 
'''E-mail:''' pchang37@gatech.edu 

Feifei Qian 
'''Office:''' Howey W01 (office hours TBD) 
'''E-mail:''' qianfeifei_china@gatech.edu 

Nick Gravish 
'''Office:''' Howey W01 (office hours TBD) 
'''E-mail:''' nick.gravish@gmail.com 

Alexandros Fragkopoulos 
'''Office:''' Boggs B-55 (office hours TBD) 
'''E-mail:''' afragkopoulos@gmail.com 

==Course Description==

The course offers an introductory treatment of nonlinear dynamics and chaos, including first order ODE and their bifurcations, phase plane analysis, limit cycles, Lorenz equations, chaos, iterated maps, period doubling, fractals and strange attractors. Teams of students will also conduct one week of self-guided experiments in Prof. Goldman's laboratory and prepare final report/presentation of the results.

==Time and Place==

Tuesday, Thursday, 9:30-11AM, Howey S204

==Homework and grading==

Homework sets will be given every other week. Homework must be submitted at the start of class or it will be considered late.

'''Final Grades'''

Grades will be calculated using 40% homework scores, 20% from mid-term exam, and 40% from the final project

==Book==

"Nonlinear Dyanamics & Chaos", Steven H. Strogatz (Westview Press, 2001)

Syllabus

2014-10-29T15:29:07Z

PatrickChang: /* Instructor & TA */