Group 5 - Revision history

Cjcrowley at 14:00, 12 January 2012

2012-01-12T14:00:30Z

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	''Group members: Joshua Job, Ricky Patel, Nick Pritchard, Caleb Royer''		''Group members: Joshua Job, Ricky Patel, Nick Pritchard, Caleb Royer''

			[[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 ]]

	A classic example of a chaotic system is that of a dripping faucet. We have investigated in experiment and modeling the period doubling cascade of a dripping faucet.		A classic example of a chaotic system is that of a dripping faucet. We have investigated in experiment and modeling the period doubling cascade of a dripping faucet.

Cjcrowley at 16:16, 11 January 2012

2012-01-11T16:16:26Z

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			'''The chaotic faucet'''

	''Group members: Joshua Job, Ricky Patel, Nick Pritchard, Caleb Royer''		''Group members: Joshua Job, Ricky Patel, Nick Pritchard, Caleb Royer''

			A classic example of a chaotic system is that of a dripping faucet. We have investigated in experiment and modeling the period doubling cascade of a dripping faucet.

	[[File:Faucet.jpg \| thumb \| 400px \| alt = Chaotic dripping faucet \| A classic example of chaotic behavior]]		[[File:Faucet.jpg \| thumb \| 400px \| alt = Chaotic dripping faucet \| A classic example of chaotic behavior]]

Cjcrowley: Protected "Group 5" ([edit=sysop] (indefinite) [move=sysop] (indefinite))

2012-01-09T16:51:52Z

Protected "Group 5" ([edit=sysop] (indefinite) [move=sysop] (indefinite))

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(No difference)

Group5 at 15:56, 16 December 2011

2011-12-16T15:56:23Z

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	An example of the harmonic oscillator motion is shown in the video below:		An example of the harmonic oscillator motion is shown in the video below:
	<videoflash>~~http://youtu.be/~~0ELLw5PZq8E</videoflash>		<videoflash>0ELLw5PZq8E</videoflash>

	== Previous Research ==		== Previous Research ==
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	# When the drop fell through the laser, the top and bottom of the drop were double counted as crossings due to a phenomenon know as bouncing, often seen in mechanical contacts and switches. In order to remove this error, the data was processed to ignore the next immediate period measurement. The satellite drop is illustrated in the video below, taken using a high speed camera:		# When the drop fell through the laser, the top and bottom of the drop were double counted as crossings due to a phenomenon know as bouncing, often seen in mechanical contacts and switches. In order to remove this error, the data was processed to ignore the next immediate period measurement. The satellite drop is illustrated in the video below, taken using a high speed camera:

	<videoflash>~~http://youtu.be/~~qxbIOYBGDZY</videoflash>		<videoflash>qxbIOYBGDZY</videoflash>

	These changes were implemented during the data processing phase via MATLAB, and helped produce accurate that could be used to analyze the period doubling, chaotic windows, and chaotic regions.		These changes were implemented during the data processing phase via MATLAB, and helped produce accurate that could be used to analyze the period doubling, chaotic windows, and chaotic regions.

Group5 at 15:55, 16 December 2011

2011-12-16T15:55:03Z

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	where z is the position of the center of mass, v0 is the velocity of the flow, dm/dt=πa2v0 where a is the radius of the aperture, y is the drag force, and g is acceleration due to gravity. In their paper, Kiyono & Fuchikami<ref name="K&F" /> adopt a set of geometrized units such that g=1. For water at twenty degrees centigrade, these units are l0=0.27 cm, t0=0.017 s, m0=0.020 g, P0=270dyn/cm2. For a faucet 5 mm in diameter this yields a radius a=0.916 and the viscosity is 0.002. From their numerical simulations, they arrived at y=0.05, . When the center of mass passes through zcrit=5.5 with a positive velocity (i.e. traveling downward), they reset m, z, and the velocity according to the respective equations. Together, these equations and values compose their full model for a faucet 5 mm in diameter operating at twenty degrees centrigrade.		where z is the position of the center of mass, v0 is the velocity of the flow, dm/dt=πa2v0 where a is the radius of the aperture, y is the drag force, and g is acceleration due to gravity. In their paper, Kiyono & Fuchikami<ref name="K&F" /> adopt a set of geometrized units such that g=1. For water at twenty degrees centigrade, these units are l0=0.27 cm, t0=0.017 s, m0=0.020 g, P0=270dyn/cm2. For a faucet 5 mm in diameter this yields a radius a=0.916 and the viscosity is 0.002. From their numerical simulations, they arrived at y=0.05, . When the center of mass passes through zcrit=5.5 with a positive velocity (i.e. traveling downward), they reset m, z, and the velocity according to the respective equations. Together, these equations and values compose their full model for a faucet 5 mm in diameter operating at twenty degrees centrigrade.
	This model provided good agreement with Kiyoni and Fuchikami's more exact hydrodynamical simulations, and exhibited period doubling and the transition to chaos in agreement with experiment<ref name="K&F" />. The mass of the droplet at each pinch-off event uniquely determines the mass of the next droplet, making this model a one-to-one mapping (the simplest that can exhibit chaos). For a numerical prediction, they find that the period of one full cycle obeys the function Tn=1.276 v0-1 + 3.9 (actually, they find two equations that parameterize Tn and v0 based on the frequency of single-period oscillation in their bifurcation diagrams, this equation is arrived at by eliminating the parameterization). Their model also shows hysteresis, and multiple regions where there is a transition to chaos.		This model provided good agreement with Kiyoni and Fuchikami's more exact hydrodynamical simulations, and exhibited period doubling and the transition to chaos in agreement with experiment<ref name="K&F" />. The mass of the droplet at each pinch-off event uniquely determines the mass of the next droplet, making this model a one-to-one mapping (the simplest that can exhibit chaos). For a numerical prediction, they find that the period of one full cycle obeys the function Tn=1.276 v0-1 + 3.9 (actually, they find two equations that parameterize Tn and v0 based on the frequency of single-period oscillation in their bifurcation diagrams, this equation is arrived at by eliminating the parameterization). Their model also shows hysteresis, and multiple regions where there is a transition to chaos.

			An example of the harmonic oscillator motion is shown in the video below:
			<videoflash>http://youtu.be/0ELLw5PZq8E</videoflash>

	== Previous Research ==		== Previous Research ==
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	# The system was highly sensitive to nozzle diameter, with smaller nozzle sizes causing lateral movement of the droplets due to the imperfections in the tubing, a larger nozzle diameter is recommended		# The system was highly sensitive to nozzle diameter, with smaller nozzle sizes causing lateral movement of the droplets due to the imperfections in the tubing, a larger nozzle diameter is recommended
	# Satellite drops are drops that come off the main drop itself. This leads to incorrect period measurements as the satellite drops themselves are counted at main drops, which initially led to erroneous two, four and chaotic period data. This was corrected by establishing a laser crossing threshold to remove the satellite drop and keep the main drop data via MATLAB		# Satellite drops are drops that come off the main drop itself. This leads to incorrect period measurements as the satellite drops themselves are counted at main drops, which initially led to erroneous two, four and chaotic period data. This was corrected by establishing a laser crossing threshold to remove the satellite drop and keep the main drop data via MATLAB
	# When the drop fell through the laser, the top and bottom of the drop were double counted as crossings due to a phenomenon know as bouncing, often seen in mechanical contacts and switches. In order to remove this error, the data was processed to ignore the next immediate period measurement		# When the drop fell through the laser, the top and bottom of the drop were double counted as crossings due to a phenomenon know as bouncing, often seen in mechanical contacts and switches. In order to remove this error, the data was processed to ignore the next immediate period measurement. The satellite drop is illustrated in the video below, taken using a high speed camera:

			<videoflash>http://youtu.be/qxbIOYBGDZY</videoflash>

	These changes were implemented during the data processing phase via MATLAB, and helped produce accurate that could be used to analyze the period doubling, chaotic windows, and chaotic regions.		These changes were implemented during the data processing phase via MATLAB, and helped produce accurate that could be used to analyze the period doubling, chaotic windows, and chaotic regions.

Group5: /* Background */

2011-12-15T22:00:34Z

Background

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	where d(m)/dt=Q=flow rate, g is the acceleration due to gravity, k is the spring constant (coming from surface tension), b is the drag force (coming from viscosity), and v=dy/dt.		where d(m)/dt=Q=flow rate, g is the acceleration due to gravity, k is the spring constant (coming from surface tension), b is the drag force (coming from viscosity), and v=dy/dt.
	This model has been the most popular for the chaotic dripping faucet since its introduction <ref name="K&F">{{Cite journal \| last1=Kiyono \| first1=Ken \| last2=Fuchikami \| first2=Nobuko \| year=1999 \| title=Dripping Faucet Dynamics by an Improved Mass-Spring Model \| url=http://jpsj.ipap.jp.www.library.gatech.edu:2048/link?JPSJ/68/3259/ \| journal=Journal of the Physical Society of Japan \| volume=68 \| issue=10 \| pages=3259-3270}}</ref><ref name="Coullet">{{Cite ~~jounral~~ \| last1=Coullet \| first1=P. \|last2=Mahadevan\|first2=L.\|last3=Riera\|first3=C.S.\|year=2005\| title=Hydrodynamical models for the chaotic dripping faucet \| url=http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=283922 \| journal=Journal of Fluid Mechanics \| volume=526 \| pages=1-17}}</ref>. However, it is not very precise, and gives very little information about what are the appropriate values of k and b (as well as the critical y value, or the constant of proportionality between the momentum prior to pinch-off and the mass after pinch-off). This renders it a good qualitative model (exhibiting period doubling and transitions into chaos for some regions in the parameter space), and it is conceptually simple making it pedagogically useful, but it is not particularly accurate in an experimental context.		This model has been the most popular for the chaotic dripping faucet since its introduction <ref name="K&F">{{Cite journal \| last1=Kiyono \| first1=Ken \| last2=Fuchikami \| first2=Nobuko \| year=1999 \| title=Dripping Faucet Dynamics by an Improved Mass-Spring Model \| url=http://jpsj.ipap.jp.www.library.gatech.edu:2048/link?JPSJ/68/3259/ \| journal=Journal of the Physical Society of Japan \| volume=68 \| issue=10 \| pages=3259-3270}}</ref><ref name="Coullet">{{Cite journal \| last1=Coullet \| first1=P. \|last2=Mahadevan\|first2=L.\|last3=Riera\|first3=C.S.\|year=2005\| title=Hydrodynamical models for the chaotic dripping faucet \| url=http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=283922 \| journal=Journal of Fluid Mechanics \| volume=526 \| pages=1-17}}</ref>. However, it is not very precise, and gives very little information about what are the appropriate values of k and b (as well as the critical y value, or the constant of proportionality between the momentum prior to pinch-off and the mass after pinch-off). This renders it a good qualitative model (exhibiting period doubling and transitions into chaos for some regions in the parameter space), and it is conceptually simple making it pedagogically useful, but it is not particularly accurate in an experimental context.
	To rectify this, some<ref name="K&F" /><ref name="Coullet" /> have employed far more detailed hydrodynamical models of the flow, using simplified versions of the Navier-Stokes equations. These models are far more accurate, but due to their added complexity, they are of limited use for gaining an intuitive understanding of the dynamics. They are also prohibitively expensive computationally on readily available hardware. Kiyono and Fuchikami<ref name="K&F" /> used numerical simulations based on one such hydrodynamical model to produce a modified version of the mass-spring system, this time including well-defined parameters gleaned from the simulations. The new equation of motion in their model is:		To rectify this, some<ref name="K&F" /><ref name="Coullet" /> have employed far more detailed hydrodynamical models of the flow, using simplified versions of the Navier-Stokes equations. These models are far more accurate, but due to their added complexity, they are of limited use for gaining an intuitive understanding of the dynamics. They are also prohibitively expensive computationally on readily available hardware. Kiyono and Fuchikami<ref name="K&F" /> used numerical simulations based on one such hydrodynamical model to produce a modified version of the mass-spring system, this time including well-defined parameters gleaned from the simulations. The new equation of motion in their model is:

Group5 at 14:08, 15 December 2011

2011-12-15T14:08:59Z

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	[[File:expSetup.PNG\|200px\|thumb\|right\|The final experimental setup]]		[[File:expSetup.PNG\|200px\|thumb\|right\|The final experimental setup]]
	[[File:expSetup2.PNG\|200px\|thumb\|right\|A close up view of the final experimental setup]]		[[File:expSetup2.PNG\|200px\|thumb\|right\|A close up view of the final experimental setup]]
			[[File:ADC.PNG\|200px\|thumb\|right\|Analog to Digital (ADC) converter]]
			[[File:labView.PNG\|200px\|thumb\|right\|A sample VI output. Note: the downward peaks are laser crossings measured by the laser-photodiode system]]

	=== Parts list ===		=== Parts list ===

Group5 at 14:01, 15 December 2011

2011-12-15T14:01:34Z

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	== Procedure ==		== Procedure ==

	The ~~experiment will be set up as follows:~~		The final experimental setup consists of a photodiode, laser, reservoir, large diameter flexible tubing, and a flow regulator to dispense the fluid at a set flow rate. The laser is used to measure the period in between drops, while the reservoir returns the fluid back into the system in order to perpetually refill the reservoir. The tubing is used to transport the water. The resulting data is measured via an analog to digital converter, and fed into LabVIEW using a VI created by Nick Gravish, the TA for PHYS628, Fall 2011.
			[[File:expSetup.PNG\|200px\|thumb\|right\|The final experimental setup]]
	~~1. Feeder tank fills the~~ reservoir ~~tank.~~		[[File:expSetup2.PNG\|200px\|thumb\|right\|A close up view of the final experimental setup]]

	~~2. Reservoir tank drips though stopcock with needle nose valve. The flow rate~~, ~~or rate of drops falling~~, is the ~~controlled parameter in this setup. The feeder tank fills the reservoir tank~~ at ~~the same~~ flow rate ~~in order to maintain constant hydrostatic pressure in the reservoir tank.~~

	~~3. A circuit is implemented to amplify signals and reduce noise. This circuit uses a Schmitt trigger to reduce noise and op-amp to amplify the signal~~. The ~~signal~~ is ~~the read by an Analog~~ to ~~Digital Converter (ADC) and brought into~~ the ~~computer software realm. The sampling rate of the ADC will be high enough to capture individual drops at all flow rates investigated. This will likely be~~ in ~~the region of 500 Hz.~~

	~~4. Laser is aligned perpendicular to the direction of falling~~ drops~~. A phototransistor is positioned to receive~~ the ~~light from~~ the ~~laser. As~~ the ~~beam is broken by falling drops, this event is captured by~~ the ~~phototransistor~~. The ~~signal from the phototransistor~~ is ~~amplified by~~ the ~~amplifying circuit (if necessary) and the time history data acquired and digitized by the computer interface and computer~~. The resulting ~~time history~~ data is a ~~series of pulses~~, ~~with each pulse being a drop passing through~~ the ~~laser beam~~, ~~and the period of falling drops can then be measured~~.

	5. A ~~high speed camera will also be used to visualize the falling drops, in addition to the laser-photodiode timing mechanism.~~

	~~6. Data analysis then performed on the time history data with computer. This analysis shall primarily consist~~ of ~~period measurement between drops for~~ the ~~various flow rates, as well as plots of bifurcations, Poincare maps, and any other relevant phenomena.~~

	=== Parts list ===		=== Parts list ===

	# ~~Feeder tank~~		# Photodiode
	~~# Reservoir tank~~
	~~# Stopcock with needle nose valve~~
	# Laser		# Laser
	# ~~Phototransistor or photodiode~~		# Reservoir
	# ~~Breadboard circuit to amplify signals from phototransistor or photodiode~~		# Large diameter flexible tubing
	# ~~Power supplies for laser, photodiode or phototransistor, and circuit~~		# Flow regulator
	# High speed camera		# High speed camera
	# Analog to Digital Converter (ADC) Interface to computer		# Analog to Digital Converter (ADC) Interface to computer
			# LabVIEW VI

	=== Steps ===		=== Steps ===
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	== Results ==		== Results ==

			=== Error Analysis ===

			Some issues of note that were encountered during data collection were:

			# The system was highly sensitive to nozzle diameter, with smaller nozzle sizes causing lateral movement of the droplets due to the imperfections in the tubing, a larger nozzle diameter is recommended
			# Satellite drops are drops that come off the main drop itself. This leads to incorrect period measurements as the satellite drops themselves are counted at main drops, which initially led to erroneous two, four and chaotic period data. This was corrected by establishing a laser crossing threshold to remove the satellite drop and keep the main drop data via MATLAB
			# When the drop fell through the laser, the top and bottom of the drop were double counted as crossings due to a phenomenon know as bouncing, often seen in mechanical contacts and switches. In order to remove this error, the data was processed to ignore the next immediate period measurement

			These changes were implemented during the data processing phase via MATLAB, and helped produce accurate that could be used to analyze the period doubling, chaotic windows, and chaotic regions.

	=== Period 1 ===		=== Period 1 ===

Group5: /* Simulation */

2011-12-15T00:26:20Z

Simulation

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	As you can see, the simulation is able to reproduce one, two, and three periodic behavior quite well. One important feature of this model must be noted: it is not, strictly, chaotic. The final plot, "Chaotic Map", is actually a plot of ~10,000 drops. Unlike the physical chaotic attractor, or the attractor of some other maps such as the logistic map, we see that this attractor is actually one of very high periodicity. This can be deduced visually from the small number of visible points compared to the number plotted. It was determined in MATLAB that this model actually exhibits a 66-period motion at the v0=0.115 value quoted by Kiyono and Fuchikami to be a region of chaos<ref name="K&F" />. However, one must also note that the model is extremely sensitive to slight changes to the flow rate in this region. A flow rate less than 0.1% different, 0.1149, exhibits a 12-period, and v0=0.1151 exhibits a 14-period. In any physical system, there will be slight variations in the flow rate which would cause the period of the motion to change faster than a full period could be completed. Thus, the system would likely appear to be truly chaotic, even if it were fully described by this, non-chaotic, mass-spring model.		As you can see, the simulation is able to reproduce one, two, and three periodic behavior quite well. One important feature of this model must be noted: it is not, strictly, chaotic. The final plot, "Chaotic Map", is actually a plot of ~10,000 drops. Unlike the physical chaotic attractor, or the attractor of some other maps such as the logistic map, we see that this attractor is actually one of very high periodicity. This can be deduced visually from the small number of visible points compared to the number plotted. It was determined in MATLAB that this model actually exhibits a 66-period motion at the v0=0.115 value quoted by Kiyono and Fuchikami to be a region of chaos<ref name="K&F" />. However, one must also note that the model is extremely sensitive to slight changes to the flow rate in this region. A flow rate less than 0.1% different, 0.1149, exhibits a 12-period, and v0=0.1151 exhibits a 14-period. In any physical system, there will be slight variations in the flow rate which would cause the period of the motion to change faster than a full period could be completed. Thus, the system would likely appear to be truly chaotic, even if it were fully described by this, non-chaotic, mass-spring model.

			{\|style="background:white; border:1px solid dimgray;color:black" border="0" height="100" align="center" valign="bottom" cellpadding=10px cellspacing=0px

			<!--Start the first row and add row formats-->
			\|-align="center"

			\|[[File:Bigbif004to011.jpg\|border\|724x427px\|none]]

			<!--Start of a second row and add formats-->
			\|-align="center" valign="top"

			<!--Start of individual cell details for the second row-->
			\|width="100"\|'''''Bifurcation diagram from v0=0.04 to v0=0.11'''''
			\|}

			Finally, we produced a bifurcation diagram from a v0=0.04 to v0=0.11. One can clearly see that the overall trend is for the period to decrease. This is punctuated by periods of bifurcations to "chaotic" regimes, which then transition into a periodic window of period-two, and then another series of bifurcations to chaos and a collapse to single-period. Each of these multi-period regimes is seen to be roughly symmetric. We can attribute this behavior to the nature of our model's map. We can see in the return map for the chaotic case that the attractor intersects itself, which implies that the map is not unimodal. This is what produces the collapse of chaos to single-period, as discussed in Kiyono et. al.<ref name="K&F" />.

	== Discussion ==		== Discussion ==

Group5: /* Background */

2011-12-15T00:01:41Z

Background

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	== Background ==		== Background ==

	The simplest model for the chaotic dripping faucet is that of a mass-spring system described in “The Chaotic Behavior of the Leaky Faucet” (Martien, Pope, Scott, Shaw 1985)<ref name="Shaw">{{Cite journal \| last1=Martien \| first1=P. \| last2=Pope \| first2=S.C. \| last3=Scott \| first3=P.L. \| last4=Shaw \| first4=R.S. \| year=1985 \| title=The Chaotic Behavior of the Leaky Faucet \| url=http://www.sciencedirect.com~~.www.library.gatech.edu:2048~~/science~~?_ob=MiamiImageURL&_cid=271541&_user=655052&_pii=0375960185900659&_check=y&_origin=&_coverDate=12-Aug-1985&view=c&wchp=dGLzVBA-zSkzk&md5=68ada3999ef43eee3adf8a223d11c614~~/~~1-s2.0-~~0375960185900659~~-main.pdf~~ \| journal=Physics Letters A \| volume=110A \| issue=7-8 \| pages=399-404}}</ref>. This system uses the simple differential equation of a mass-spring system whose mass increases linearly with time. When the position of the center of mass of the drop reaches a critical value, the mass of the drop is reduced proportionately to the velocity of the drop, which simulates the pinching off of the droplet, and the model continues. Physically, this model is an intuitive description of the system: the water droplet is a mass, and it is being driven by the flow of water into it (which increases its mass linearly with time) and the force of gravity, there is some restoring force due to surface tension, and some drag due to viscosity. The equation of motion for this model is:		The simplest model for the chaotic dripping faucet is that of a mass-spring system described in “The Chaotic Behavior of the Leaky Faucet” (Martien, Pope, Scott, Shaw 1985)<ref name="Shaw">{{Cite journal \| last1=Martien \| first1=P. \| last2=Pope \| first2=S.C. \| last3=Scott \| first3=P.L. \| last4=Shaw \| first4=R.S. \| year=1985 \| title=The Chaotic Behavior of the Leaky Faucet \| url=http://www.sciencedirect.com/science/article/pii/0375960185900659 \| journal=Physics Letters A \| volume=110A \| issue=7-8 \| pages=399-404}}</ref>. This system uses the simple differential equation of a mass-spring system whose mass increases linearly with time. When the position of the center of mass of the drop reaches a critical value, the mass of the drop is reduced proportionately to the velocity of the drop, which simulates the pinching off of the droplet, and the model continues. Physically, this model is an intuitive description of the system: the water droplet is a mass, and it is being driven by the flow of water into it (which increases its mass linearly with time) and the force of gravity, there is some restoring force due to surface tension, and some drag due to viscosity. The equation of motion for this model is:

	:<math>		:<math>