Eulerian Angles - Revision history

Jledet at 02:30, 28 November 2016

2016-11-28T02:30:32Z

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	~~Claimed by Jonathan Ledet Fall 2016~~

	==Introduction==		==Introduction==

Jledet at 02:14, 28 November 2016

2016-11-28T02:14:34Z

← Older revision		Revision as of 22:14, 27 November 2016
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			Claimed by Jonathan Ledet Fall 2016

	==Introduction==		==Introduction==

Hjamal3: /* Introduction */

2015-12-08T05:06:33Z

Introduction

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	'''Converting Into One Frame'''		'''Converting Into One Frame'''
	Please view the image on the right to see how we described a bodies orientation into one frame even though it started off with three separate orientations.		Please view the image on the bottom right to see how we described a bodies orientation into one frame even though it started off with three separate orientations.

	[[File:Changing orientation to one frame.PNG\|thumb\|Changing orientation to one frame]]		[[File:Changing orientation to one frame.PNG\|thumb\|Changing orientation to one frame]]

Hjamal3: /* Example of a Gyroscope */

2015-12-08T05:06:00Z

Example of a Gyroscope

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	[[File:Gyroscope operation.gif\|thumb\|Gyroscope operation]]		[[File:Gyroscope operation.gif\|thumb\|Gyroscope operation]]

	~~===Simple===~~		[http://ocw.mit.edu/courses/aeronautics-and-astronautics/16-07-dynamics-fall-2009/lecture-notes/MIT16_07F09_Lec30.pdf Proof from MIT course]

	==Connectedness==		==Connectedness==

Hjamal3 at 05:04, 8 December 2015

2015-12-08T05:04:54Z

← Older revision		Revision as of 01:04, 8 December 2015
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	'''Converting Into One Frame'''		'''Converting Into One Frame'''
			Please view the image on the right to see how we described a bodies orientation into one frame even though it started off with three separate orientations.

			[[File:Changing orientation to one frame.PNG\|thumb\|Changing orientation to one frame]]

	==Example of a Gyroscope==		==Example of a Gyroscope==

Hjamal3 at 04:44, 8 December 2015

2015-12-08T04:44:16Z

@@ Line 27: / Line 27: @@
 ===Simple===
 ==Connectedness==
 #How is this topic connected to something that you are interested in?
-This topic is connected to airplanes which are a fascinating invention. It is hard to imagine that humans have been able to find a way to get a massive object to fly into the sky. When one really delves into all the physics that getting a plane into the air requires it really is fasccinating.
+All objects in the real world move in complex ways. Luckily for us, we only experience 3 dimensions. Hence, we can use the Eulerian Angles to classify all sorts of rotational behaviors. Knowing how much angular velocity an object has in any direction can give us its position at any time.
 #How is it connected to your major?
-This is not directly connected to my major. Although the chemical engineering field is vast I do not think there is yet, a connection with the concept of drag force.
+This topic is vastly connected to any major related to engineering mechanics. Mechanical engineering students and aerospace students will find this to be one of the fundamental building blocks of their field of study, as it is an important tool for engineers working in 3 dimensions. Imagine trying to position a satellite in space but not being able to determine where it's facing at a given time! Even video game creators and animators require Eulerian angles for mouse control and orienting animated objects (links below).
 ===Further reading===
-Books, Articles or other print media on this topic
+Euler Angles Related Examples
-===External links===
+Euler Angles for Animations
+[https://www.youtube.com/watch?v=zc8b2Jo7mno Euler (Gimbal lock) Explained]
 ==References==
 Dr. Wayne Whiteman
 DEFINE THE PROPERTIES OF ANGULAR VELOCITY FOR 3D MOTION
-Nave, R. "Terminal Velocity." Hyper Physics. N.p., n.d. Web. 7 Dec. 2015. <http://hyperphysics.phy-astr.gsu.edu/hbase/airfri2.html#c3>.
+[https://www.coursera.org/learn/motion-and-kinetics/lecture/nkGSe/module-13-eulerian-angles-for-3d-rotational-motion Coursera Lesson on Eulerian Angles]
-MATHEMATICAL COMPONENT, EXAMPLES (EASY AND MIDDLING)
+McGill, David J., and Wilton W. King. <i>An Introduction to Dynamics</i>. Monterey, CA: Brooks/Cole Engineering Division, 1984. Print.

Hjamal3: /* Eulerian Angles */

2015-12-08T04:32:13Z

Eulerian Angles

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	'''Angular Velocity'''		'''Angular Velocity'''
	Using the addition theorem (can be further studied at [https://www.coursera.org/learn/motion-and-kinetics/lecture/ohKPT/module-3-define-the-properties-of-angular-velocity-for-3d-motion Coursera]) we find that the body ''A'' with respect to the inertial frame ''I'' is defined to be '''w''' = θ' '''k (black/blue axis)''' + φ' '''j (blue/green axis)''' + ψ' '''k (green/red axis)'''.		Using the addition theorem (can be further studied at [https://www.coursera.org/learn/motion-and-kinetics/lecture/ohKPT/module-3-define-the-properties-of-angular-velocity-for-3d-motion Coursera]) we find that the body ''A'' with respect to the inertial frame ''I'' is defined to be '''w''' = θ' '''k (black/blue axis)''' + φ' '''j (blue/green axis)''' + ψ' '''k (green/red axis)'''. However, everything here is defined in different frames and we must consolidate our equations into one coordinate system.

	'''~~WHY AIR RESISTANCE DEPENDS ON DRAG COEFFICIENT~~'''		'''Converting Into One Frame'''
	Air resistance is affected by what is called the drag coefficient which is essentially the shape of the object. If the object has a certain shape such as a pointy edge rather than blunt edge then the air resistance is greatly reduced. For example a spherical object has a drag coefficient of .5 and irregularly shaped objects can even reach 2.

	~~'''WHY AIR RESISTANCE DEPENDS ON CROSS-SECTIONAL AREA'''~~
	It can be seen from practice that the bigger the cross-sectional area of the object the larger the effect that air resistance has on the object. This due to the fact that air resistance is the result of the collision of an objects surface with the air molecules. This means that the bigger the surface-area the more collisions with air molecules the object will experience and the faster it'll reach terminal velocity. For example a person going sky diving will fall much slower with an open parachute (more surface area, air resistance has a bigger impact) than with a closed parachute (less surface area, air resistance has a smaller impact).

	~~'''WHY AIR RESISTANCE DEPENDS ON SPEED'''~~
	Air resistance is affected by speed because it increases as velocity increases. This can be seen because there is always gravitational force acting on the object downward, but as the speed increases the air resistance increase making the net downward force much much smaller until it becomes 0. This means that at some point the object reaches terminal speed because there is no longer a net force acting upon it.

	==Example of a Gyroscope==		==Example of a Gyroscope==

Hjamal3: /* Eulerian Angles */

2015-12-08T04:30:28Z

Eulerian Angles

← Older revision		Revision as of 00:30, 8 December 2015
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	'''Angular Velocity'''		'''Angular Velocity'''
	Using the addition theorem (can be further studied at [https://www.coursera.org/learn/motion-and-kinetics/lecture/ohKPT/module-3-define-the-properties-of-angular-velocity-for-3d-motion Coursera]) we find that the body ''A'' with respect to the inertial frame ''I'' is defined to be '''w''' = θ' '''k (black/blue axis)''' + φ' '''j (blue/green axis)''' + ψ '''k (green/red axis)'''.		Using the addition theorem (can be further studied at [https://www.coursera.org/learn/motion-and-kinetics/lecture/ohKPT/module-3-define-the-properties-of-angular-velocity-for-3d-motion Coursera]) we find that the body ''A'' with respect to the inertial frame ''I'' is defined to be '''w''' = θ' '''k (black/blue axis)''' + φ' '''j (blue/green axis)''' + ψ' '''k (green/red axis)'''.

	'''WHY AIR RESISTANCE DEPENDS ON DRAG COEFFICIENT'''		'''WHY AIR RESISTANCE DEPENDS ON DRAG COEFFICIENT'''

Hjamal3 at 04:29, 8 December 2015

2015-12-08T04:29:49Z

← Older revision		Revision as of 00:29, 8 December 2015
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	Note, we moved from the black '''i, j, k''' axes to the blue '''i, j, k''' axes, where the k axis remained the same for both black and blue frames. Then we moved from the blue '''i, j, k''' axes to the green '''i, j, k''' axes, where the j axis remained the same for both the blue and green frames. Finally, we moved from the green '''i, j, k''' axes to the red '''i, j, k''' axes, where the k axis remained the same for both the green and the red frames.		Note, we moved from the black '''i, j, k''' axes to the blue '''i, j, k''' axes, where the k axis remained the same for both black and blue frames. Then we moved from the blue '''i, j, k''' axes to the green '''i, j, k''' axes, where the j axis remained the same for both the blue and green frames. Finally, we moved from the green '''i, j, k''' axes to the red '''i, j, k''' axes, where the k axis remained the same for both the green and the red frames.

	~~[[File:File.jpg]]~~		'''Angular Velocity'''
			Using the addition theorem (can be further studied at [https://www.coursera.org/learn/motion-and-kinetics/lecture/ohKPT/module-3-define-the-properties-of-angular-velocity-for-3d-motion Coursera]) we find that the body ''A'' with respect to the inertial frame ''I'' is defined to be '''w''' = θ' '''k (black/blue axis)''' + φ' '''j (blue/green axis)''' + ψ '''k (green/red axis)'''.
	~~p = density of the air~~

	~~C_D = drag coefficient (typically between .3 and 1.0)~~

	~~A = cross-sectional area~~

	~~v = speed of object~~


	'''~~WHY AIR RESISTANCE DEPENDS ON THE DENSITY OF THE AIR~~'''
	~~The density affects~~ the ~~air resistance for expected reasons, the denser the air is the larger the air resistance becomes~~. ~~The denser the air is the more air molecules the object collides with~~ and ~~faster~~ the ~~object reaches terminal~~ velocity~~. This implies~~ that ~~in areas~~ with ~~high altitudes where air~~ is ~~"thinner" or less dense such as Colorado there is less air resistance~~.

	'''WHY AIR RESISTANCE DEPENDS ON DRAG COEFFICIENT'''		'''WHY AIR RESISTANCE DEPENDS ON DRAG COEFFICIENT'''
	Air resistance is affected by what is called the drag coefficient which is essentially the shape of the object. If the object has a certain shape such as a pointy edge rather than blunt edge then the air resistance is greatly reduced. For example a spherical object has a drag coefficient of .5 and irregularly shaped objects can even reach 2.		Air resistance is affected by what is called the drag coefficient which is essentially the shape of the object. If the object has a certain shape such as a pointy edge rather than blunt edge then the air resistance is greatly reduced. For example a spherical object has a drag coefficient of .5 and irregularly shaped objects can even reach 2.
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	Air resistance is affected by speed because it increases as velocity increases. This can be seen because there is always gravitational force acting on the object downward, but as the speed increases the air resistance increase making the net downward force much much smaller until it becomes 0. This means that at some point the object reaches terminal speed because there is no longer a net force acting upon it.		Air resistance is affected by speed because it increases as velocity increases. This can be seen because there is always gravitational force acting on the object downward, but as the speed increases the air resistance increase making the net downward force much much smaller until it becomes 0. This means that at some point the object reaches terminal speed because there is no longer a net force acting upon it.

	==~~Examples~~==		==Example of a Gyroscope==

	~~Be sure~~ to ~~show all steps~~ in ~~your solution and include diagrams whenever possible~~		We can use the procedure outlined above to describe the orientation of a gyroscope at any time given the components of its angular velocity. An example of a rotation gyroscope is shown in the image.

			[[File:Gyroscope operation.gif\|thumb\|Gyroscope operation]]

	===Simple===		===Simple===
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	==References==		==References==

	~~Picture of the Airplane~~		Dr. Wayne Whiteman
	~~"What Is Drag?" What Is Drag? Ed. Nancy Hall. NASA, n~~.~~d. Web. 07 Dec. 2015. <https://www.grc.nasa.gov/www/k-12/airplane/drag1.html>.~~		DEFINE THE PROPERTIES OF ANGULAR VELOCITY FOR 3D MOTION

	~~HOW DOES~~ THE ~~DRAG COEFFICIENT AFFECT AIR RESISTANCE~~
	Nave, R. "Terminal Velocity." Hyper Physics. N.p., n.d. Web. 7 Dec. 2015. <http://hyperphysics.phy-astr.gsu.edu/hbase/airfri2.html#c3>.		Nave, R. "Terminal Velocity." Hyper Physics. N.p., n.d. Web. 7 Dec. 2015. <http://hyperphysics.phy-astr.gsu.edu/hbase/airfri2.html#c3>.

	MATHEMATICAL COMPONENT, EXAMPLES (EASY AND MIDDLING)		MATHEMATICAL COMPONENT, EXAMPLES (EASY AND MIDDLING)
	Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.		Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

Hjamal3 at 04:17, 8 December 2015

2015-12-08T04:17:26Z

← Older revision		Revision as of 00:17, 8 December 2015
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	==Introduction==		==Introduction==
	When you have a body that rotates in plane motion (2-D), it is easy to find the orientation of the body if you have the angular velocity as a function of time. You can integrate this value to find the orientation of the body at any position.		When you have a body that rotates in plane motion (2-D), it is easy to find the orientation of the body if you have the angular velocity as a function of time. You can integrate this value to find the orientation of the body at any position. In the image shown, we can find out the position of an object at any place in the ''xy'' plane.

	[[File:AngularVelocityPosition2D.PNG\|thumb\|Simple angular velocity equation in 2D.]]		[[File:AngularVelocityPosition2D.PNG\|thumb\|Simple angular velocity equation in 2D.]]

	===~~A Mathematical Model~~===		Unfortunately, you cannot integrate an angular velocity vector with 3-D components to get the orientation of the body in three dimensional space. This requires the concept of '''Eulerian Angles''', which are used in determining a body's orientation in space using angular velocity in the x, y, and z axes.
	~~Air resistance is~~ a ~~force that essentially opposes motion~~ and ~~dissipates energy~~. ~~Much like other opposing forces~~, ~~air resistance is dependent on~~ both the ~~speed and~~ the ~~size of~~ the ~~surface area of~~ the ~~object~~. ~~Many things go into what affects~~ the ~~force of air resistance~~, and ~~it can be defined by~~ the ~~following equation:~~
			===Eulerian Angles===

			Essentially, to describe the orientation of a body in 3 dimensions we must describe its rotation about three specific axes. In our example shown in the image, we will first rotate our body's reference frame about the '''k''' axis (black to blue), then the '''j''' axes (blue to green), and finally the '''k''' axis again (green to red).

			[[File:RotationsEulerianAngles3D.PNG\|thumb\|Rotationsabouteulerianangles]]

			Note, we moved from the black '''i, j, k''' axes to the blue '''i, j, k''' axes, where the k axis remained the same for both black and blue frames. Then we moved from the blue '''i, j, k''' axes to the green '''i, j, k''' axes, where the j axis remained the same for both the blue and green frames. Finally, we moved from the green '''i, j, k''' axes to the red '''i, j, k''' axes, where the k axis remained the same for both the green and the red frames.

	[[File:File.jpg]]		[[File:File.jpg]]