Inverted pendulum-2010 - Revision history

JeffAguilar at 21:47, 23 September 2012

2012-09-23T21:47:14Z

← Older revision		Revision as of 17:47, 23 September 2012
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	to the level necessary for the shaker. Acceleration data was also sent to an oscilloscope, from which the peak acceleration could be monitored independently of the controller.		to the level necessary for the shaker. Acceleration data was also sent to an oscilloscope, from which the peak acceleration could be monitored independently of the controller.

	[[File:ipsetup.jpg \| center \| ~~200px~~]]		[[File:ipsetup.jpg \| center \| 500px]]

	A high speed camera was mounted directly in front of the pendulum, and took photos at up to 200 frames per second (fps). The higher speeds caused erratic frame rates, so the camera was generally operated at 100 fps. The images were sent to a second computer with a Labview tracking program. The program applied a threshold of luminance to identify the spot, and tracked the centroid of the spot within a torus section selected by the user. The program was able to track the relative x and y position of the dot, from which the angle of the pendulum could be extracted. While the pendulum was in the stable inverted position, the vertical displacement of the pendulum equaled the vertical displacement of the shaker table, and so could be used to determine the amplitude of the forcing displacement. To determine the region of stability, we recapitulated our procedure from the numerical trials. We varied the driving frequency from 20Hz to 50Hz, and the eccentricity from 0 to 0.999. At each node, we attempted to increase the acceleration until the inverted pendulum became stable in the upright position. For those nodes that we could stabilize, we then decreased the acceleration slowly until stability was spontaneously lost. We then returned the pendulum to stability and recorded the acceleration and the displacement of the forcing function. This process had two main sources of noise: it was difficult to fine tune the acceleration to within 1g; at the boundary of stability, it was sometimes subjective whether the pendulum was truly stable.		A high speed camera was mounted directly in front of the pendulum, and took photos at up to 200 frames per second (fps). The higher speeds caused erratic frame rates, so the camera was generally operated at 100 fps. The images were sent to a second computer with a Labview tracking program. The program applied a threshold of luminance to identify the spot, and tracked the centroid of the spot within a torus section selected by the user. The program was able to track the relative x and y position of the dot, from which the angle of the pendulum could be extracted. While the pendulum was in the stable inverted position, the vertical displacement of the pendulum equaled the vertical displacement of the shaker table, and so could be used to determine the amplitude of the forcing displacement. To determine the region of stability, we recapitulated our procedure from the numerical trials. We varied the driving frequency from 20Hz to 50Hz, and the eccentricity from 0 to 0.999. At each node, we attempted to increase the acceleration until the inverted pendulum became stable in the upright position. For those nodes that we could stabilize, we then decreased the acceleration slowly until stability was spontaneously lost. We then returned the pendulum to stability and recorded the acceleration and the displacement of the forcing function. This process had two main sources of noise: it was difficult to fine tune the acceleration to within 1g; at the boundary of stability, it was sometimes subjective whether the pendulum was truly stable.
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	== Results ==		== Results ==

	~~Figure 5~~ is a scatter plot of stable points at the stability boundary for each value of m as a function of forcing acceleration (in units of gravitational acceleration - g) and radial frequency (rad/s). Best-fit lines for each value of m is fitted base on the data points. For each value of m, the forcing acceleration required for stability is a linear function of the frequency. Results also show that the lower the value of m, the smaller the region of stability and vice versa.		The figure below is a scatter plot of stable points at the stability boundary for each value of m as a function of forcing acceleration (in units of gravitational acceleration - g) and radial frequency (rad/s). Best-fit lines for each value of m is fitted base on the data points. For each value of m, the forcing acceleration required for stability is a linear function of the frequency. Results also show that the lower the value of m, the smaller the region of stability and vice versa.

	[[File:~~ipfigure1~~.jpg \| ~~right~~ \| ~~200px~~]]		[[File:ipfigure5.jpg \| center \| 400 ]]

	~~Figure 6~~ shows the scatter plot for the same experimental data points for each value of m, but as a function of vertical variation (cm) and radial frequency (rad/s). In this plot, for each value of m, displacement has inverse relationship with frequency. Results also show that the lower the value of m, the smaller the region of stability and vice versa.		The figure below shows the scatter plot for the same experimental data points for each value of m, but as a function of vertical variation (cm) and radial frequency (rad/s). In this plot, for each value of m, displacement has inverse relationship with frequency. Results also show that the lower the value of m, the smaller the region of stability and vice versa.

	Figure 7 shows the scatter plot for the same experimental data points for each value of m, but as a function of vertical variation (cm) and period frequency (Hz). Period frequency is equal to radial frequency/2π. In this plot, for each value of m, displacement has an inverse relationship with frequency. However, unlike earlier results, there are no significant differences between the regions of stability for the different values of m.		[[File:ipfigure6.jpg \| center \| 400 ]]

	~~We iteratively adjusted~~ the value of ~~Q in the equation~~ of ~~motion until a good fit is obtained between the simulated displacement~~ and ~~observed displacement~~ (~~Figure 8~~). ~~The~~ value of ~~Q is found to be 11~~ for ~~this system~~.		The figure below shows the scatter plot for the same experimental data points for each value of m, but as a function of vertical variation (cm) and period frequency (Hz). Period frequency is equal to radial frequency/2π. In this plot, for each value of m, displacement has an inverse relationship with frequency. However, unlike earlier results, there are no significant differences between the regions of stability for the different values of m.

	~~Figure 9~~ shows the stability diagram for the inverted pendulum as a function of relative amplitude and relative frequency, based on simulation results. For each value of m, displacement has inverse relationship with frequency. Similar to experimental results, there are no significant differences between the regions of stability for the different values of m. Symbols in the diagram represent the actual data points. Results clearly show that there is a constant offset between simulation and experimental results.		[[File:ipfigure7.jpg \| center \| 400 ]]

			We iteratively adjusted the value of Q in the equation of motion until a good fit is obtained between the simulated displacement and observed displacement (see below). The value of Q is found to be 11 for this system.

			[[File:ipfigure8.jpg \| center \| 400 ]]

			The figure below shows the stability diagram for the inverted pendulum as a function of relative amplitude and relative frequency, based on simulation results. For each value of m, displacement has inverse relationship with frequency. Similar to experimental results, there are no significant differences between the regions of stability for the different values of m. Symbols in the diagram represent the actual data points. Results clearly show that there is a constant offset between simulation and experimental results.

			[[File:ipfigure9.jpg \| center \| 400 ]]
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JeffAguilar at 21:40, 23 September 2012

2012-09-23T21:40:47Z

← Older revision		Revision as of 17:40, 23 September 2012
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	== Introduction ==		== Introduction ==
			[[File:ipfigure1.jpg \| right \| 200px]]
	The simple pendulum has only one stable state of equilibrium, which is the vertically down orientation (a). But if the axis undergoes vertical perturbation, three different equilibrium states emerge: (a) stable down, (b) stable up, and (c) Continuous rotation in either direction. The dynamics of an inverted pendulum while subjected to periodic forcing were investigated numerically and experimentally. The following questions were to be answered.		The simple pendulum has only one stable state of equilibrium, which is the vertically down orientation (a). But if the axis undergoes vertical perturbation, three different equilibrium states emerge: (a) stable down, (b) stable up, and (c) Continuous rotation in either direction. The dynamics of an inverted pendulum while subjected to periodic forcing were investigated numerically and experimentally. The following questions were to be answered.

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	== Modeling ==		== Modeling ==
			[[File:ipfigure2.jpg \| left \| 200px]]
	The following equation of motion was employed and integrated with a fourth order Runge Kutta approximation in matlab to compute numerical solutions.		The following equation of motion was employed and integrated with a fourth order Runge Kutta approximation in matlab to compute numerical solutions.

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	In order to verify the accuracy of our Matlab simulations, we set up an inverted pendulum arm on a shaker table with controllable acceleration. The shaker table was attached to a magnetic actuator that converted an electrical signal into mechanical force, similar to a speaker. An ADXL321 accelerometer was mounted to the underside of the shaker, in order to monitor the acceleration for feedback and recording. A signal of 50mV per g of acceleration was sent to a computer with a Labview testbench that controlled the waveform. The software controlled the amplitude of each harmonic component separately, permitting a wide range of waveforms, including Jacobi Elliptics of varying eccentricity. This signal was then broadcast to an VTS-500 high powered amplifier, which boosted the signal		In order to verify the accuracy of our Matlab simulations, we set up an inverted pendulum arm on a shaker table with controllable acceleration. The shaker table was attached to a magnetic actuator that converted an electrical signal into mechanical force, similar to a speaker. An ADXL321 accelerometer was mounted to the underside of the shaker, in order to monitor the acceleration for feedback and recording. A signal of 50mV per g of acceleration was sent to a computer with a Labview testbench that controlled the waveform. The software controlled the amplitude of each harmonic component separately, permitting a wide range of waveforms, including Jacobi Elliptics of varying eccentricity. This signal was then broadcast to an VTS-500 high powered amplifier, which boosted the signal
	to the level necessary for the shaker. Acceleration data was also sent to an oscilloscope, from which the peak acceleration could be monitored independently of the controller.		to the level necessary for the shaker. Acceleration data was also sent to an oscilloscope, from which the peak acceleration could be monitored independently of the controller.

			[[File:ipsetup.jpg \| center \| 200px]]

	A high speed camera was mounted directly in front of the pendulum, and took photos at up to 200 frames per second (fps). The higher speeds caused erratic frame rates, so the camera was generally operated at 100 fps. The images were sent to a second computer with a Labview tracking program. The program applied a threshold of luminance to identify the spot, and tracked the centroid of the spot within a torus section selected by the user. The program was able to track the relative x and y position of the dot, from which the angle of the pendulum could be extracted. While the pendulum was in the stable inverted position, the vertical displacement of the pendulum equaled the vertical displacement of the shaker table, and so could be used to determine the amplitude of the forcing displacement. To determine the region of stability, we recapitulated our procedure from the numerical trials. We varied the driving frequency from 20Hz to 50Hz, and the eccentricity from 0 to 0.999. At each node, we attempted to increase the acceleration until the inverted pendulum became stable in the upright position. For those nodes that we could stabilize, we then decreased the acceleration slowly until stability was spontaneously lost. We then returned the pendulum to stability and recorded the acceleration and the displacement of the forcing function. This process had two main sources of noise: it was difficult to fine tune the acceleration to within 1g; at the boundary of stability, it was sometimes subjective whether the pendulum was truly stable.		A high speed camera was mounted directly in front of the pendulum, and took photos at up to 200 frames per second (fps). The higher speeds caused erratic frame rates, so the camera was generally operated at 100 fps. The images were sent to a second computer with a Labview tracking program. The program applied a threshold of luminance to identify the spot, and tracked the centroid of the spot within a torus section selected by the user. The program was able to track the relative x and y position of the dot, from which the angle of the pendulum could be extracted. While the pendulum was in the stable inverted position, the vertical displacement of the pendulum equaled the vertical displacement of the shaker table, and so could be used to determine the amplitude of the forcing displacement. To determine the region of stability, we recapitulated our procedure from the numerical trials. We varied the driving frequency from 20Hz to 50Hz, and the eccentricity from 0 to 0.999. At each node, we attempted to increase the acceleration until the inverted pendulum became stable in the upright position. For those nodes that we could stabilize, we then decreased the acceleration slowly until stability was spontaneously lost. We then returned the pendulum to stability and recorded the acceleration and the displacement of the forcing function. This process had two main sources of noise: it was difficult to fine tune the acceleration to within 1g; at the boundary of stability, it was sometimes subjective whether the pendulum was truly stable.
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	Figure 5 is a scatter plot of stable points at the stability boundary for each value of m as a function of forcing acceleration (in units of gravitational acceleration - g) and radial frequency (rad/s). Best-fit lines for each value of m is fitted base on the data points. For each value of m, the forcing acceleration required for stability is a linear function of the frequency. Results also show that the lower the value of m, the smaller the region of stability and vice versa.		Figure 5 is a scatter plot of stable points at the stability boundary for each value of m as a function of forcing acceleration (in units of gravitational acceleration - g) and radial frequency (rad/s). Best-fit lines for each value of m is fitted base on the data points. For each value of m, the forcing acceleration required for stability is a linear function of the frequency. Results also show that the lower the value of m, the smaller the region of stability and vice versa.

			[[File:ipfigure1.jpg \| right \| 200px]]

	Figure 6 shows the scatter plot for the same experimental data points for each value of m, but as a function of vertical variation (cm) and radial frequency (rad/s). In this plot, for each value of m, displacement has inverse relationship with frequency. Results also show that the lower the value of m, the smaller the region of stability and vice versa.		Figure 6 shows the scatter plot for the same experimental data points for each value of m, but as a function of vertical variation (cm) and radial frequency (rad/s). In this plot, for each value of m, displacement has inverse relationship with frequency. Results also show that the lower the value of m, the smaller the region of stability and vice versa.

JeffAguilar at 21:29, 23 September 2012

2012-09-23T21:29:03Z

← Older revision		Revision as of 17:29, 23 September 2012
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	A high speed camera was mounted directly in front of the pendulum, and took photos at up to 200 frames per second (fps). The higher speeds caused erratic frame rates, so the camera was generally operated at 100 fps. The images were sent to a second computer with a Labview tracking program. The program applied a threshold of luminance to identify the spot, and tracked the centroid of the spot within a torus section selected by the user. The program was able to track the relative x and y position of the dot, from which the angle of the pendulum could be extracted. While the pendulum was in the stable inverted position, the vertical displacement of the pendulum equaled the vertical displacement of the shaker table, and so could be used to determine the amplitude of the forcing displacement. To determine the region of stability, we recapitulated our procedure from the numerical trials. We varied the driving frequency from 20Hz to 50Hz, and the eccentricity from 0 to 0.999. At each node, we attempted to increase the acceleration until the inverted pendulum became stable in the upright position. For those nodes that we could stabilize, we then decreased the acceleration slowly until stability was spontaneously lost. We then returned the pendulum to stability and recorded the acceleration and the displacement of the forcing function. This process had two main sources of noise: it was difficult to fine tune the acceleration to within 1g; at the boundary of stability, it was sometimes subjective whether the pendulum was truly stable.		A high speed camera was mounted directly in front of the pendulum, and took photos at up to 200 frames per second (fps). The higher speeds caused erratic frame rates, so the camera was generally operated at 100 fps. The images were sent to a second computer with a Labview tracking program. The program applied a threshold of luminance to identify the spot, and tracked the centroid of the spot within a torus section selected by the user. The program was able to track the relative x and y position of the dot, from which the angle of the pendulum could be extracted. While the pendulum was in the stable inverted position, the vertical displacement of the pendulum equaled the vertical displacement of the shaker table, and so could be used to determine the amplitude of the forcing displacement. To determine the region of stability, we recapitulated our procedure from the numerical trials. We varied the driving frequency from 20Hz to 50Hz, and the eccentricity from 0 to 0.999. At each node, we attempted to increase the acceleration until the inverted pendulum became stable in the upright position. For those nodes that we could stabilize, we then decreased the acceleration slowly until stability was spontaneously lost. We then returned the pendulum to stability and recorded the acceleration and the displacement of the forcing function. This process had two main sources of noise: it was difficult to fine tune the acceleration to within 1g; at the boundary of stability, it was sometimes subjective whether the pendulum was truly stable.

			== Results ==

			Figure 5 is a scatter plot of stable points at the stability boundary for each value of m as a function of forcing acceleration (in units of gravitational acceleration - g) and radial frequency (rad/s). Best-fit lines for each value of m is fitted base on the data points. For each value of m, the forcing acceleration required for stability is a linear function of the frequency. Results also show that the lower the value of m, the smaller the region of stability and vice versa.


			Figure 6 shows the scatter plot for the same experimental data points for each value of m, but as a function of vertical variation (cm) and radial frequency (rad/s). In this plot, for each value of m, displacement has inverse relationship with frequency. Results also show that the lower the value of m, the smaller the region of stability and vice versa.

			Figure 7 shows the scatter plot for the same experimental data points for each value of m, but as a function of vertical variation (cm) and period frequency (Hz). Period frequency is equal to radial frequency/2π. In this plot, for each value of m, displacement has an inverse relationship with frequency. However, unlike earlier results, there are no significant differences between the regions of stability for the different values of m.

			We iteratively adjusted the value of Q in the equation of motion until a good fit is obtained between the simulated displacement and observed displacement (Figure 8). The value of Q is found to be 11 for this system.

			Figure 9 shows the stability diagram for the inverted pendulum as a function of relative amplitude and relative frequency, based on simulation results. For each value of m, displacement has inverse relationship with frequency. Similar to experimental results, there are no significant differences between the regions of stability for the different values of m. Symbols in the diagram represent the actual data points. Results clearly show that there is a constant offset between simulation and experimental results.
	== Group Member Bios ==		== Group Member Bios ==

JeffAguilar at 21:23, 23 September 2012

2012-09-23T21:23:35Z

← Older revision		Revision as of 17:23, 23 September 2012
Line 28:		Line 28:

	where $I$ is moment of inertia, $b$ is damping coefficient, $\Omega$ is forcing frequency relative to natural frequency, $\omega_{0}$, and $cn$ is the Jacobi elliptic function used for non-sinusoidal forcing. In order to ﬁnd the boundary of stability numerically, we swept systematically through values of $\Omega$. For each value of Ω a binary search was used to narrow in on a value of $\epsilon$ lying on the edge of stability. In order to judge stability an initial condition $(\theta, \dot{\theta}) = (1e − 6, −1e − 6)$ was integrated for 1000 periods of the forcing function and checked for divergence.		where $I$ is moment of inertia, $b$ is damping coefficient, $\Omega$ is forcing frequency relative to natural frequency, $\omega_{0}$, and $cn$ is the Jacobi elliptic function used for non-sinusoidal forcing. In order to ﬁnd the boundary of stability numerically, we swept systematically through values of $\Omega$. For each value of Ω a binary search was used to narrow in on a value of $\epsilon$ lying on the edge of stability. In order to judge stability an initial condition $(\theta, \dot{\theta}) = (1e − 6, −1e − 6)$ was integrated for 1000 periods of the forcing function and checked for divergence.

			== Methods ==

			In order to verify the accuracy of our Matlab simulations, we set up an inverted pendulum arm on a shaker table with controllable acceleration. The shaker table was attached to a magnetic actuator that converted an electrical signal into mechanical force, similar to a speaker. An ADXL321 accelerometer was mounted to the underside of the shaker, in order to monitor the acceleration for feedback and recording. A signal of 50mV per g of acceleration was sent to a computer with a Labview testbench that controlled the waveform. The software controlled the amplitude of each harmonic component separately, permitting a wide range of waveforms, including Jacobi Elliptics of varying eccentricity. This signal was then broadcast to an VTS-500 high powered amplifier, which boosted the signal
			to the level necessary for the shaker. Acceleration data was also sent to an oscilloscope, from which the peak acceleration could be monitored independently of the controller.

			A high speed camera was mounted directly in front of the pendulum, and took photos at up to 200 frames per second (fps). The higher speeds caused erratic frame rates, so the camera was generally operated at 100 fps. The images were sent to a second computer with a Labview tracking program. The program applied a threshold of luminance to identify the spot, and tracked the centroid of the spot within a torus section selected by the user. The program was able to track the relative x and y position of the dot, from which the angle of the pendulum could be extracted. While the pendulum was in the stable inverted position, the vertical displacement of the pendulum equaled the vertical displacement of the shaker table, and so could be used to determine the amplitude of the forcing displacement. To determine the region of stability, we recapitulated our procedure from the numerical trials. We varied the driving frequency from 20Hz to 50Hz, and the eccentricity from 0 to 0.999. At each node, we attempted to increase the acceleration until the inverted pendulum became stable in the upright position. For those nodes that we could stabilize, we then decreased the acceleration slowly until stability was spontaneously lost. We then returned the pendulum to stability and recorded the acceleration and the displacement of the forcing function. This process had two main sources of noise: it was difficult to fine tune the acceleration to within 1g; at the boundary of stability, it was sometimes subjective whether the pendulum was truly stable.


	== Group Member Bios ==		== Group Member Bios ==

JeffAguilar: /* Modeling */

2012-09-23T21:11:34Z

Modeling

← Older revision		Revision as of 17:11, 23 September 2012
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	<math>\frac{\delta \theta}{\delta t}=v</math>, <math>Q = \frac{\omega_{0}I}{b}</math>		<math>\frac{\delta \theta}{\delta t}=v</math>, <math>Q = \frac{\omega_{0}I}{b}</math>

	where $I$ is moment of inertia, $b$ is damping coefficient, $\Omega$ is forcing frequency relative to natural frequency, $\omega_{0}$, and $cn$ is the Jacobi elliptic function used for non-sinusoidal forcing. In order to ﬁnd the boundary of stability numerically, we swept systematically through values of $\Omega$. For each value of Ω a binary search was used to narrow in on a value of $\epsilon$ lying on the edge of stability. In order to judge stability an initial condition $(/theta, \dot{\theta}) = (1e − 6, −1e − 6)$ was integrated for 1000 periods of the forcing function and checked for divergence.		where $I$ is moment of inertia, $b$ is damping coefficient, $\Omega$ is forcing frequency relative to natural frequency, $\omega_{0}$, and $cn$ is the Jacobi elliptic function used for non-sinusoidal forcing. In order to ﬁnd the boundary of stability numerically, we swept systematically through values of $\Omega$. For each value of Ω a binary search was used to narrow in on a value of $\epsilon$ lying on the edge of stability. In order to judge stability an initial condition $(\theta, \dot{\theta}) = (1e − 6, −1e − 6)$ was integrated for 1000 periods of the forcing function and checked for divergence.

	== Group Member Bios ==		== Group Member Bios ==

JeffAguilar at 21:10, 23 September 2012

2012-09-23T21:10:51Z

← Older revision		Revision as of 17:10, 23 September 2012
Line 27:		Line 27:
	<math>\frac{\delta \theta}{\delta t}=v</math>, <math>Q = \frac{\omega_{0}I}{b}</math>		<math>\frac{\delta \theta}{\delta t}=v</math>, <math>Q = \frac{\omega_{0}I}{b}</math>

	where $I$ is moment of inertia, $b$ is damping coefficient, $\Omega$ is forcing frequency relative to natural frequency, $\omega_{0}$		where $I$ is moment of inertia, $b$ is damping coefficient, $\Omega$ is forcing frequency relative to natural frequency, $\omega_{0}$, and $cn$ is the Jacobi elliptic function used for non-sinusoidal forcing. In order to ﬁnd the boundary of stability numerically, we swept systematically through values of $\Omega$. For each value of Ω a binary search was used to narrow in on a value of $\epsilon$ lying on the edge of stability. In order to judge stability an initial condition $(/theta, \dot{\theta}) = (1e − 6, −1e − 6)$ was integrated for 1000 periods of the forcing function and checked for divergence.

	== Group Member Bios ==		== Group Member Bios ==

JeffAguilar: /* Modeling */

2012-09-23T18:53:23Z

Modeling

← Older revision		Revision as of 14:53, 23 September 2012
Line 24:		Line 24:

	<math>\frac{\delta v}{\delta t}=-\frac{v}{Q \Omega}-\left[\frac{1}{\Omega^{2}}-\epsiloncn(t,m)\right]\sin{\theta}</math>		<math>\frac{\delta v}{\delta t}=-\frac{v}{Q \Omega}-\left[\frac{1}{\Omega^{2}}-\epsiloncn(t,m)\right]\sin{\theta}</math>
	<math>\frac{\delta v}{\delta t}=~~-\frac{~~v}{Q ~~\Omega}-\left[~~\frac~~{1}~~{\~~Omega^~~{2}}~~-\epsiloncn(t,m)\right]\sin~~{~~\theta~~}</math>
			<math>\frac{\delta \theta}{\delta t}=v</math>, <math>Q = \frac{\omega_{0}I}{b}</math>

			where $I$ is moment of inertia, $b$ is damping coefficient, $\Omega$ is forcing frequency relative to natural frequency, $\omega_{0}$

	== Group Member Bios ==		== Group Member Bios ==

JeffAguilar: /* Modeling */

2012-09-23T18:36:18Z

Modeling

← Older revision		Revision as of 14:36, 23 September 2012
Line 23:		Line 23:
	The following equation of motion was employed and integrated with a fourth order Runge Kutta approximation in matlab to compute numerical solutions.		The following equation of motion was employed and integrated with a fourth order Runge Kutta approximation in matlab to compute numerical solutions.

	<math>\frac{\delta v}{\delta t}=-\frac{v}{Q \Omega}-\left[\frac{1}{\Omega^{2}}-\epsilon cn(t,m)\right]*\sin{\theta}</math>		<math>\frac{\delta v}{\delta t}=-\frac{v}{Q \Omega}-\left[\frac{1}{\Omega^{2}}-\epsiloncn(t,m)\right]\sin{\theta}</math>
			<math>\frac{\delta v}{\delta t}=-\frac{v}{Q \Omega}-\left[\frac{1}{\Omega^{2}}-\epsiloncn(t,m)\right]\sin{\theta}</math>

	== Group Member Bios ==		== Group Member Bios ==

JeffAguilar at 18:35, 23 September 2012

2012-09-23T18:35:16Z

← Older revision		Revision as of 14:35, 23 September 2012
Line 23:		Line 23:
	The following equation of motion was employed and integrated with a fourth order Runge Kutta approximation in matlab to compute numerical solutions.		The following equation of motion was employed and integrated with a fourth order Runge Kutta approximation in matlab to compute numerical solutions.

	<math>\frac[\delta][\~~delta~~]</math>		<math>\frac{\delta v}{\delta t}=-\frac{v}{Q \Omega}-\left[\frac{1}{\Omega^{2}}-\epsilon cn(t,m)\right]*\sin{\theta}</math>
	== Group Member Bios ==		== Group Member Bios ==

JeffAguilar at 18:26, 23 September 2012

2012-09-23T18:26:44Z

← Older revision		Revision as of 14:26, 23 September 2012
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	[[Media: 2010_InvertedPendulum.pdf \| Main presentation]]. '''Papers:''' [[Media: Cheng_Invertedpend_.pdf‎ \| Yivei]], [[Media: hayward_pendulum.pdf \| Robert]], [[Media: Shapero_InvertedPendulum_Final.pdf \| Samuel]]		[[Media: 2010_InvertedPendulum.pdf \| Main presentation]]. '''Papers:''' [[Media: Cheng_Invertedpend_.pdf‎ \| Yivei]], [[Media: hayward_pendulum.pdf \| Robert]], [[Media: Shapero_InvertedPendulum_Final.pdf \| Samuel]]

	[[File:InvertedPendTeam.jpg \| right \| ~~300px~~]]		[[File:InvertedPendTeam.jpg \| right \| 200px]]
	This page is dedicated to the Inverted Pendulum Project. The simulations and laboratory experiments conducted in the inverted perdulum project are part of the requirements of the class: Phys 4267A/6268 Nonlinear Dynamics & Chaos, Spring 2010.		This page is dedicated to the Inverted Pendulum Project. The simulations and laboratory experiments conducted in the inverted perdulum project are part of the requirements of the class: Phys 4267A/6268 Nonlinear Dynamics & Chaos, Spring 2010.

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	The following equation of motion was employed and integrated with a fourth order Runge Kutta approximation in matlab to compute numerical solutions.		The following equation of motion was employed and integrated with a fourth order Runge Kutta approximation in matlab to compute numerical solutions.

	<math>\delta</math>		<math>\frac[\delta][\delta]</math>
	== Group Member Bios ==		== Group Member Bios ==