Group 5 2014 - Revision history

Pendulum2014 at 23:20, 12 December 2014

2014-12-12T23:20:15Z

← Older revision		Revision as of 19:20, 12 December 2014
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	=Methods=		=Methods=
	===Experimental Set Up===		===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 2). 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.		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 2). The spheres we used for pendulum is $25\pm 0.2\,\mathrm{mm}$ in diameter and <math>1.2\pm0.2 g</math> 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.		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.
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	\end{equation}		\end{equation}
	We thus expect approximately a $2^\circ$ drift overall.		We thus expect approximately a $2^\circ$ drift overall.

			[[File:Triangle_in_Phase_Orbit.png \|thumb \| 300px \| right \| Figure 3: Triangle configuration orbit with in phase pendula.]]

	We observed that starting the system from essentially the same configuration with a slight perturbation		We observed that starting the system from essentially the same configuration with a slight perturbation
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	The tentacle structures in the phase space come from synchronization shooting pendula into other modes while it drifts.		The tentacle structures in the phase space come from synchronization shooting pendula into other modes while it drifts.
			[[File: X_Phase_Space_Triangle_in_Phase.png\|500px\|thumb\|center\|Triangle configuration X phase space]]


	~~[[File:Triangle_in_Phase_Orbit.png \|thumb \| 300px \| right \| Figure 3: Triangle configuration orbit with in phase pendula.]]~~


	=Appendix=		=Appendix=
	===Figures===		===Figures===
	<gallery caption="Triangle Configuration">		<center>
			<gallery mode=packed widths="350px" heights="250px" caption="Triangle Configuration">
	File: Triangle_in_Phase_Orbit.png \| Triangle configuration orbit with in phase pendula		File: Triangle_in_Phase_Orbit.png \| Triangle configuration orbit with in phase pendula
	File: Triangle_Phase_Shift_X_Position.png \| Triangle configuration showing beats and synchronization		File: Triangle_Phase_Shift_X_Position.png \| Triangle configuration showing beats and synchronization
	File: Z_Position_Triangle_in_Phase.png \| Triangle configuration Z position		File: Z_Position_Triangle_in_Phase.png \| Triangle configuration Z position
	</gallery>		</gallery>
			</center>
	<gallery caption="Velocity">		<center>
			<gallery mode=packed caption="Velocity" widths="350px" heights="250px">
	File: Y_Velocity_Triangle_In_Phase.png \| Triangle configuration Y velocity		File: Y_Velocity_Triangle_In_Phase.png \| Triangle configuration Y velocity
	File: Z_Velocity_Trianglein_Phase.png \| Triangle configuration Z velocity		File: Z_Velocity_Trianglein_Phase.png \| Triangle configuration Z velocity
	</gallery>		</gallery>
			</center>
	<gallery caption="Phase space">		<center>
			<gallery mode=packed caption="Phase space" widths="350px" heights="250px">
	File: X_Phase_Space_Triangle_in_Phase.png \| Triangle configuration X phase space		File: X_Phase_Space_Triangle_in_Phase.png \| Triangle configuration X phase space
	File: Y_Phase_Space_Triangle_in_Phase.png \| Triangle configuration Y phase space		File: Y_Phase_Space_Triangle_in_Phase.png \| Triangle configuration Y phase space
	</gallery>		</gallery>
			</center>
	=References=		=References=
	{{reflist\|35em}}		{{reflist\|35em}}

Pendulum2014 at 22:59, 12 December 2014

2014-12-12T22:59:56Z

← Older revision		Revision as of 18:59, 12 December 2014
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	==Appendix==		=Appendix=
	===Figures===		===Figures===
	<gallery caption="Triangle Configuration">		<gallery caption="Triangle Configuration">

Pendulum2014: /* References */

2014-12-12T22:59:26Z

References

← Older revision		Revision as of 18:59, 12 December 2014
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	</gallery>		</gallery>

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

Pendulum2014: /* Methods */

2014-12-12T22:57:31Z

Methods

← Older revision		Revision as of 18:57, 12 December 2014
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	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.		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.

	[[File:Final1.png \| thumb \| 300px \| ~~right~~ \| Figure 1: Coupled pendulum on the base plane.]]		[[File:Final1.png \| thumb \| 300px \| left \| Figure 1: Coupled pendulum on the base plane.]]

	=Results=		=Results=

Pendulum2014: /* Figures */

2014-12-12T22:56:43Z

Figures

← Older revision		Revision as of 18:56, 12 December 2014
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	==Appendix==		==Appendix==
	===Figures===		===Figures===
	<gallery caption="~~The singular points of the integral curves~~">		<gallery caption="Triangle Configuration">
	File: Triangle_in_Phase_Orbit.png \| Triangle configuration orbit with in phase pendula		File: Triangle_in_Phase_Orbit.png \| Triangle configuration orbit with in phase pendula
	File: Triangle_Phase_Shift_X_Position.png \| Triangle configuration showing beats and synchronization		File: Triangle_Phase_Shift_X_Position.png \| Triangle configuration showing beats and synchronization
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	<gallery caption="Velocity">		<gallery caption="Velocity">
	File: Y_Velocity_Triangle_In_Phase.png \| Triangle configuration Y velocity		File: Y_Velocity_Triangle_In_Phase.png \| Triangle configuration Y velocity
	File: Z_Velocity_Trianglein_Phase.png \| Triangle configuration Z velocity		File: Z_Velocity_Trianglein_Phase.png \| Triangle configuration Z velocity
	</gallery>		</gallery>
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	File: Y_Phase_Space_Triangle_in_Phase.png \| Triangle configuration Y phase space		File: Y_Phase_Space_Triangle_in_Phase.png \| Triangle configuration Y phase space
	</gallery>		</gallery>

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

Pendulum2014 at 22:55, 12 December 2014

2014-12-12T22:55:25Z

← Older revision		Revision as of 18:55, 12 December 2014
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			==Appendix==
			===Figures===
			<gallery caption="The singular points of the integral curves">
			File: Triangle_in_Phase_Orbit.png \| Triangle configuration orbit with in phase pendula
			File: Triangle_Phase_Shift_X_Position.png \| Triangle configuration showing beats and synchronization
			File: Z_Position_Triangle_in_Phase.png \| Triangle configuration Z position
			</gallery>

			<gallery caption="Velocity">
			File: Y_Velocity_Triangle_In_Phase.png \| Triangle configuration Y velocity
			File: Z_Velocity_Trianglein_Phase.png \| Triangle configuration Z velocity
			</gallery>

			<gallery caption="Phase space">
			File: X_Phase_Space_Triangle_in_Phase.png \| Triangle configuration X phase space
			File: Y_Phase_Space_Triangle_in_Phase.png \| Triangle configuration Y phase space
			</gallery>
	==References==		==References==
	{{reflist\|35em}}		{{reflist\|35em}}

Pendulum2014 at 22:41, 12 December 2014

2014-12-12T22:41:05Z

← Older revision		Revision as of 18:41, 12 December 2014
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	[[File:Final1.png \| thumb \| 300px \| right \| Figure 1: Coupled pendulum on the base plane.]]		[[File:Final1.png \| thumb \| 300px \| right \| Figure 1: Coupled pendulum on the base plane.]]

			=Results=
			By recording the position of the pendula we
			will were able to generate phase plots in addition to a host of other
			qualitative and quantitative results of the relative positions of the pendula.

			Through analyzing the configuration space of the initial positions of the pendula,
			we discovered quasiperiodic synchronization, beat formation, and chaos in the system.

			The phase space of the respective pendula showed tori that decayed toward the origin over time.
			Thus showing the effect of friction on the system.

			We also found drifting of the system over time when plotting every 15th point of the trajectory in phase space.
			This is due to the small effect of rotation of the Earth. During the course of our 7 minute experiment
			we observe an angular drift of
			\begin{equation}
			\delta \phi = \frac{7\cdot2\pi}{24\cdot60}.
			\end{equation}
			We thus expect approximately a $2^\circ$ drift overall.

			We observed that starting the system from essentially the same configuration with a slight perturbation
			led to radically different transient behavior, but similar synchronization and beat formation over
			the course of the experiment.

			The tentacle structures in the phase space come from synchronization shooting pendula into other modes while it drifts.


			[[File:Triangle_in_Phase_Orbit.png \|thumb \| 300px \| right \| Figure 3: Triangle configuration orbit with in phase pendula.]]

Pendulum2014: /* Methods */

2014-12-12T22:35:38Z

Methods

← Older revision		Revision as of 18:35, 12 December 2014
Line 159:		Line 159:
	=Methods=		=Methods=
	===Experimental Set Up===		===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.		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 2). 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.		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.

			[[File:Final1.png \| thumb \| 300px \| right \| Figure 1: Coupled pendulum on the base plane.]]



	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.]]~~

PatrickChang at 22:32, 12 December 2014

2014-12-12T22:32:53Z

← Older revision		Revision as of 18:32, 12 December 2014
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	Members: Caligan, Lucas, Li, Norris		Members: Caligan, Lucas, Li, Norris

	[[Media:pendulum.pdf\|Final ~~Paper~~]]		[[Media:pendulum.pdf\|Final Report]]

	=Introduction=		=Introduction=

PatrickChang at 22:29, 12 December 2014

2014-12-12T22:29:55Z

← Older revision		Revision as of 18:29, 12 December 2014
Line 1:		Line 1:
	Members: Caligan, Lucas, Li, Norris		Members: Caligan, Lucas, Li, Norris

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

	=Introduction=		=Introduction=