User:Tohline/ThreeDimensionalConfigurations/ChallengesPt5 - Revision history

Tohline: /* Associated Velocities (2nd Try) */

2021-05-22T04:13:29Z

Associated Velocities (2nd Try)

← Older revision		Revision as of 04:13, 22 May 2021
Line 1,082:		Line 1,082:
	<td align="left">		<td align="left">
	<math>~		<math>~
	2\tan\varphi \biggl( \frac{x_\mathrm{surf}}{y_\mathrm{surf}} \biggr)^2 \biggl[ 1 + \tan^2 \varphi \biggr]		2 \dot\varphi \tan\varphi \biggl( \frac{x_\mathrm{surf}}{y_\mathrm{surf}} \biggr)^2 \biggl[ 1 + \tan^2 \varphi \biggr]
	=		=
	\frac{2\sin\varphi }{\cos^3 \varphi }\biggl( \frac{x_\mathrm{surf}}{y_\mathrm{surf}} \biggr)^2 \, .		\frac{2 \dot\varphi \sin\varphi }{\cos^3 \varphi }\biggl( \frac{x_\mathrm{surf}}{y_\mathrm{surf}} \biggr)^2 \, .
	</math>		</math>
	</td>		</td>
Line 1,202:		Line 1,202:
	<tr>		<tr>
	<td align="right">		<td align="right">
	~~ ~~		<math>~\Rightarrow ~~~ \dot{y}</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 1,209:		Line 1,209:
	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{ 1 }{2 ~~y_\varpi~~ (1 + F^2)}		\frac{ 1 }{2 (1 + F^2)}
	\biggl\{ \biggl[ \frac{1 }{y(1+F^2) - y_\varpi} \biggr]		\biggl\{ \biggl[ \frac{1 }{y(1+F^2) - y_\varpi} \biggr]
	\biggl[ y^2_\mathrm{surf} (1 + 2F^2) - y_\varpi^2 \biggr]		\biggl[ y^2_\mathrm{surf} (1 + 2F^2) - y_\varpi^2 \biggr]

Tohline: /* Associated Velocities (1st Try) */

2021-05-22T03:54:52Z

Associated Velocities (1st Try)

← Older revision		Revision as of 03:54, 22 May 2021
Line 906:		Line 906:
	</tr>		</tr>
	</table>		</table>
	And,		And, <font color="red">[NOTE: The last term in this next expression was corrected on 5/21/2021. Needs to be incorporated into Excel spreadsheet.]</font>
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Line 920:		Line 920:
	\frac{\dot{y}}{\tan\varphi} + \frac{(y_\varpi - y)}{\tan^2\varphi} \cdot \frac{d\tan\varphi}{dt}		\frac{\dot{y}}{\tan\varphi} + \frac{(y_\varpi - y)}{\tan^2\varphi} \cdot \frac{d\tan\varphi}{dt}
	=		=
	\frac{\dot{y}}{\tan\varphi} + \frac{\dot\varphi (y_\varpi - y)}{\sin\varphi \cos\varphi} \, .		\frac{\dot{y}}{\tan\varphi}
			+ \frac{\dot\varphi (y_\varpi - y)}{\sin^2\varphi}
			+ \biggl[ \cancelto{\mathrm{corrected}}{\frac{\dot\varphi (y_\varpi - y)}{\sin\varphi \cos\varphi}}\biggr] \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Associated Velocities (2nd Try) */

2021-05-22T03:10:38Z

Associated Velocities (2nd Try)

← Older revision		Revision as of 03:10, 22 May 2021
Line 1,174:		Line 1,174:
	<math>~		<math>~
	+ \frac{ 1 }{2(1 + F^2)^2}		+ \frac{ 1 }{2(1 + F^2)^2}
	\biggl\{ \biggl[ \frac{y_\varpi (1 + F^2)}{y(1+F^2) - y_\varpi} \biggr]\frac{y^2_\mathrm{surf}}{y^2_\varpi~~} \biggr\~~}		\biggl\{ \biggl[ \frac{y_\varpi (1 + F^2)}{y(1+F^2) - y_\varpi} \biggr]\frac{y^2_\mathrm{surf}}{y^2_\varpi}
	\biggl[1 + 2F^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} \biggr] \frac{d(F^2)}{dt}		\biggl[1 + 2F^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} \biggr] \biggr\} \frac{d(F^2)}{dt}
	</math>		</math>
	</td>		</td>
Line 1,182:		Line 1,182:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{y^2_\mathrm{surf}}{y^2_\varpi}\biggl[1 + \frac{ y_\mathrm{surf}^2 }{y_\varpi^2} \biggl(1 + F^2 \biggr) \biggl( F^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} \biggr) \biggr]^{-1 / 2} </math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 1,189:		Line 1,189:
	<td align="left">		<td align="left">
	<math>~		<math>~
	\~~mp~~~\biggl[ \frac{y_\varpi}{y(1+F^2) - y_\varpi} \biggr]\frac{y^2_\mathrm{surf}}{y^2_\varpi}		\frac{ 1 }{2(1 + F^2)^2}
			\biggl\{ \biggl[ \frac{y_\varpi (1 + F^2)}{y(1+F^2) - y_\varpi} \biggr]\frac{y^2_\mathrm{surf}}{y^2_\varpi}
			\biggl[1 + 2F^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} \biggr]
			- \frac{2y(1+F^2)}{y_\varpi}
			\biggr\} \frac{d(F^2)}{dt}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{ 1 }{2 y_\varpi (1 + F^2)}
			\biggl\{ \biggl[ \frac{1 }{y(1+F^2) - y_\varpi} \biggr]
			\biggl[ y^2_\mathrm{surf} (1 + 2F^2) - y_\varpi^2 \biggr]
			- 2y
			\biggr\} \frac{d(F^2)}{dt}
	</math>		</math>
	</td>		</td>

Tohline: /* Associated Velocities (2nd Try) */

2021-05-22T02:56:41Z

Associated Velocities (2nd Try)

← Older revision		Revision as of 02:56, 22 May 2021
Line 1,158:		Line 1,158:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~-~\frac{ 1 }{(1 + F^2)^2}		<math>~-~\frac{ 1 }{2(1 + F^2)^2}
	\biggl\{ \frac{~~y}{y_\varpi}~~ (1+F^2) \biggr\} \frac{d(F^2)}{dt}		\biggl\{ \frac{2y(1+F^2)}{y_\varpi} \biggr\} \frac{d(F^2)}{dt}
	</math>		</math>
	</td>		</td>
Line 1,173:		Line 1,173:
	<td align="left">		<td align="left">
	<math>~		<math>~
	+ \frac{ 1 }{2(1 + F^2)}		+ \frac{ 1 }{2(1 + F^2)^2}
	\biggl\{ \biggl[ \frac{y_\varpi}{y(1+F^2) - y_\varpi} \biggr]\frac{y^2_\mathrm{surf}}{y^2_\varpi} \biggr\}		\biggl\{ \biggl[ \frac{y_\varpi (1 + F^2)}{y(1+F^2) - y_\varpi} \biggr]\frac{y^2_\mathrm{surf}}{y^2_\varpi} \biggr\}
	\biggl[ ~~\biggl( F~~^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} ~~\biggr) + \biggl(1 + F^2 \biggr)~~ \biggr] \frac{d(F^2)}{dt}		\biggl[1 + 2F^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} \biggr] \frac{d(F^2)}{dt}
	</math>		</math>
	</td>		</td>

Tohline: /* Associated Velocities (2nd Try) */

2021-05-22T02:47:46Z

Associated Velocities (2nd Try)

← Older revision		Revision as of 02:47, 22 May 2021
Line 1,144:		Line 1,144:
	<math>~		<math>~
	\mp \frac{ 1 }{2(1 + F^2)}		\mp \frac{ 1 }{2(1 + F^2)}
	~~\frac{d}{dt}~~\biggl\{ \frac{ y_\mathrm{surf}^2 }{y_\varpi^2} \biggl[1 + \frac{ y_\mathrm{surf}^2 }{y_\varpi^2} \biggl(1 + F^2 \biggr) \biggl( F^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} \biggr) \biggr]^{-1 / 2} \biggr\}		\biggl\{ \frac{ y_\mathrm{surf}^2 }{y_\varpi^2} \biggl[1 + \frac{ y_\mathrm{surf}^2 }{y_\varpi^2} \biggl(1 + F^2 \biggr) \biggl( F^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} \biggr) \biggr]^{-1 / 2} \biggr\}
	\frac{d}{dt}\biggl\{\biggl(1 + F^2 \biggr) \biggl( F^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} \biggr) \biggr\}		\frac{d}{dt}\biggl\{\biggl(1 + F^2 \biggr) \biggl( F^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} \biggr) \biggr\}
	</math>		</math>
Line 1,159:		Line 1,159:
	<td align="left">		<td align="left">
	<math>~-~\frac{ 1 }{(1 + F^2)^2}		<math>~-~\frac{ 1 }{(1 + F^2)^2}
	\biggl\{ ~~1 \mp \biggl[1 +~~ \frac{ ~~y_\mathrm{surf}^2~~ }{y_\varpi^2} ~~\biggl~~(1 + F^2 ~~\biggr~~) ~~\biggl( F^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} \biggr) \biggr]^{1 / 2}~~ \biggr\} \frac{d(F^2)}{dt}		\biggl\{ \frac{y}{y_\varpi} (1+F^2) \biggr\} \frac{d(F^2)}{dt}
	</math>		</math>
	</td>		</td>
Line 1,173:		Line 1,173:
	<td align="left">		<td align="left">
	<math>~		<math>~
	~~\mp~~ \frac{ 1 }{2(1 + F^2)}		+ \frac{ 1 }{2(1 + F^2)}
	\frac{d}{dt}\biggl\{ \~~frac~~{ y_\mathrm{surf}^2 }{y_\varpi^2} \biggl[1 + \frac{ y_\mathrm{surf}^2 }{y_\varpi^2} \biggl(1 + F^2 \biggr) \biggl( F^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} \biggr) \biggr]^{-1 / 2} ~~\biggr\}~~		\biggl\{ \biggl[ \frac{y_\varpi}{y(1+F^2) - y_\varpi} \biggr]\frac{y^2_\mathrm{surf}}{y^2_\varpi} \biggr\}
	\~~frac{d}{dt}~~\biggl~~\{\biggl( F^2 -~~ \frac{y_\varpi^2}{~~y_\mathrm{surf}^2} \biggr) + \biggl~~(1 + F^2 \~~biggr)~~ \biggr\} \frac{~~d(F~~^2)}{dt}		\biggl[ \biggl( F^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} \biggr) + \biggl(1 + F^2 \biggr) \biggr] \frac{d(F^2)}{dt}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\frac{y^2_\mathrm{surf}}{y^2_\varpi}\biggl[1 + \frac{ y_\mathrm{surf}^2 }{y_\varpi^2} \biggl(1 + F^2 \biggr) \biggl( F^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} \biggr) \biggr]^{-1 / 2} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\mp~\biggl[ \frac{y_\varpi}{y(1+F^2) - y_\varpi} \biggr]\frac{y^2_\mathrm{surf}}{y^2_\varpi}
	</math>		</math>
	</td>		</td>

Tohline: /* Associated Velocities (2nd Try) */

2021-05-22T02:16:21Z

Associated Velocities (2nd Try)

← Older revision		Revision as of 02:16, 22 May 2021
Line 1,080:		Line 1,080:
	<td align="left">		<td align="left">
	<math>~		<math>~
	2\tan\varphi \biggl( \frac{x_\mathrm{surf}}{y_\mathrm{surf}} \biggr)^2 \biggl[ 1 + \tan^2~~(\dot~~\varphi t) \biggr]		2\tan\varphi \biggl( \frac{x_\mathrm{surf}}{y_\mathrm{surf}} \biggr)^2 \biggl[ 1 + \tan^2 \varphi \biggr]
	=		=
	\frac{2\sin\varphi }{\cos^3 ~~(\dot~~\varphi t)}\biggl( \frac{x_\mathrm{surf}}{y_\mathrm{surf}} \biggr)^2 \, .		\frac{2\sin\varphi }{\cos^3 \varphi }\biggl( \frac{x_\mathrm{surf}}{y_\mathrm{surf}} \biggr)^2 \, .
	</math>		</math>
	</td>		</td>
Line 1,090:		Line 1,090:

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>\frac{\dot{y}}{y_\varpi} \equiv \frac{1}{y_\varpi} \cdot \frac{dy}{dt}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~-~\frac{ 1 }{(1 + F^2)^2}
			\biggl\{ 1 \mp \biggl[1 + \frac{ y_\mathrm{surf}^2 }{y_\varpi^2} \biggl(1 + F^2 \biggr) \biggl( F^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} \biggr) \biggr]^{1 / 2} \biggr\} \frac{d(F^2)}{dt}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			\mp \frac{ 1 }{(1 + F^2)}
			\frac{d}{dt}\biggl\{\biggl[1 + \frac{ y_\mathrm{surf}^2 }{y_\varpi^2} \biggl(1 + F^2 \biggr) \biggl( F^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} \biggr) \biggr]^{1 / 2} \biggr\}
			</math>
			</td>
			</tr>

	<tr>		<tr>
Line 1,099:		Line 1,128:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{y_\varpi}{(1 + F^2)}		<math>~-~\frac{ 1 }{(1 + F^2)^2}
	\biggl\{ 1 \mp \biggl[1 + \frac{ y_\mathrm{surf}^2 }{y_\varpi^2} \biggl(1 + F^2 \biggr) \biggl( F^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} \biggr) \biggr]^{1 / 2} \biggr\} \~~, .~~		\biggl\{ 1 \mp \biggl[1 + \frac{ y_\mathrm{surf}^2 }{y_\varpi^2} \biggl(1 + F^2 \biggr) \biggl( F^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} \biggr) \biggr]^{1 / 2} \biggr\} \frac{d(F^2)}{dt}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			\mp \frac{ 1 }{2(1 + F^2)}
			\frac{d}{dt}\biggl\{ \frac{ y_\mathrm{surf}^2 }{y_\varpi^2} \biggl[1 + \frac{ y_\mathrm{surf}^2 }{y_\varpi^2} \biggl(1 + F^2 \biggr) \biggl( F^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} \biggr) \biggr]^{-1 / 2} \biggr\}
			\frac{d}{dt}\biggl\{\biggl(1 + F^2 \biggr) \biggl( F^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} \biggr) \biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~-~\frac{ 1 }{(1 + F^2)^2}
			\biggl\{ 1 \mp \biggl[1 + \frac{ y_\mathrm{surf}^2 }{y_\varpi^2} \biggl(1 + F^2 \biggr) \biggl( F^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} \biggr) \biggr]^{1 / 2} \biggr\} \frac{d(F^2)}{dt}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			\mp \frac{ 1 }{2(1 + F^2)}
			\frac{d}{dt}\biggl\{ \frac{ y_\mathrm{surf}^2 }{y_\varpi^2} \biggl[1 + \frac{ y_\mathrm{surf}^2 }{y_\varpi^2} \biggl(1 + F^2 \biggr) \biggl( F^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} \biggr) \biggr]^{-1 / 2} \biggr\}
			\frac{d}{dt}\biggl\{\biggl( F^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} \biggr) + \biggl(1 + F^2 \biggr) \biggr\} \frac{d(F^2)}{dt}
	</math>		</math>
	</td>		</td>

Tohline: /* Associated Velocities (2nd Try) */

2021-05-22T00:41:03Z

Associated Velocities (2nd Try)

← Older revision		Revision as of 00:41, 22 May 2021
Line 1,036:		Line 1,036:
	</td>		</td>
	</tr>		</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{y_\varpi}{(1 + F^2)}
			\biggl\{ 1 \mp \biggl[1 + \frac{ y_\mathrm{surf}^2 }{y_\varpi^2} \biggl(1 + F^2 \biggr) \biggl( F^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} \biggr) \biggr]^{1 / 2} \biggr\} \, .
			</math>
			</td>
			</tr>
			</table>

			Let's examine the time-derivative of y under the assumption that <math>~\varphi = \dot\varphi t</math>. First, note that,

			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{d(F^2)}{dt}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			2\tan\varphi
			\biggl( \frac{x_\mathrm{surf}}{y_\mathrm{surf}} \biggr)^2 \frac{d}{dt}\biggl[ \frac{ \sin(\dot\varphi t ) }{\cos(\dot\varphi t )} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			2\tan\varphi \biggl( \frac{x_\mathrm{surf}}{y_\mathrm{surf}} \biggr)^2 \biggl[ 1 + \tan^2(\dot\varphi t) \biggr]
			=
			\frac{2\sin\varphi }{\cos^3 (\dot\varphi t)}\biggl( \frac{x_\mathrm{surf}}{y_\mathrm{surf}} \biggr)^2 \, .
			</math>
			</td>
			</tr>
			</table>
			Hence,

			<table border="0" cellpadding="5" align="center">

	<tr>		<tr>

Tohline: /* Simpler Example */

2021-05-21T23:25:34Z

Simpler Example

← Older revision		Revision as of 23:25, 21 May 2021
Line 983:		Line 983:
	<td align="left">		<td align="left">
	<math>~		<math>~
	y_\mathrm{surf}^2 \biggl[ \frac{y_\varpi^2}{y_\mathrm{surf}^2} + F^2 \biggr] \, ,		y_\mathrm{surf}^2 \biggl[ \frac{y_\varpi^2}{y_\mathrm{surf}^2} - F^2 \biggr] \, ,
	</math>		</math>
	</td>		</td>
Line 1,001:		Line 1,001:
	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{1}{y_\varpi^2} \biggl[		\frac{v_\mathrm{surf}^2}{y_\varpi^2} \biggl[ (1 + F^2)\biggl( \frac{y_\varpi^2}{y_\mathrm{surf}^2}- F^2 \biggr) \biggr]
	(1 + F^2)\biggl( \frac{y_\varpi^2}{y_\mathrm{surf}^2}+ F^2 \biggr)		\, .
	\biggr]
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	~~Given that~~,
			The pair of roots (desired values of "y") are therefore given by the expression,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~~~\frac{4AC}{B^2}~~</math>		<math>~y</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 1,019:		Line 1,019:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>~\frac{B}{2A} \biggl\{\pm \biggl[1 - \frac{4AC}{B^2} \biggr]^{1 / 2} -1 \biggr\}</math>
	\frac{1}{~~y^2_~~\~~varpi y^2_~~\~~mathrm~~{~~surf} }~~\biggl[ ~~y^2_~~\~~mathrm~~{~~surf~~} ~~+ x_\mathrm~~{~~surf}~~^2 ~~\cdot \tan^2\varphi~~ \biggr]~~\biggl[ y_\varpi~~^2 - ~~x^2_\mathrm{surf} \cdot \tan^2\varphi~~ \biggr] \~~, ,~~
	</math>
	</td>		</td>
	</tr>		</tr>
	~~</table>~~
	~~the pair of roots are given by the expression,~~
	~~<table border="0" cellpadding="5" align="center">~~

	<tr>		<tr>
	<td align="right">		<td align="right">
	~~<math>~y</math>~~
	</td>		</td>
	<td align="center">		<td align="center">
Line 1,036:		Line 1,031:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{B}{2A} \biggl\{\pm \biggl[1 - \frac{~~4AC~~}{B^2} \biggr]^{1 / 2} -1 \biggr\}</math>		<math>~-~\frac{y_\varpi}{(1 + F^2)}
			\biggl\{\pm \biggl[1 + \frac{ y_\mathrm{surf}^2 }{y_\varpi^2} \biggl(1 + F^2 \biggr) \biggl( F^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} \biggr) \biggr]^{1 / 2} -1 \biggr\}
			</math>
	</td>		</td>
	</tr>		</tr>
Line 1,048:		Line 1,045:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{B}{2A}		<math>~\frac{y_\varpi}{(1 + F^2)}
	\biggl\{\pm \biggl[1 + \frac{~~1}{y^2_\varpi y^2_~~\mathrm{surf} }~~\biggl( y^2_\mathrm~~{~~surf} + x_~~\~~mathrm{surf}~~^2 \~~cdot \tan~~^2~~\varphi~~ \biggr)\biggl( x^2_\~~mathrm~~{~~surf}~~ \~~cdot \tan~~^2~~\varphi -~~ y_\~~varpi~~^2 \biggr) \biggr]^{1 / 2} -1 \biggr\} \, .		\biggl\{ 1 \mp \biggl[1 + \frac{ y_\mathrm{surf}^2 }{y_\varpi^2} \biggl(1 + F^2 \biggr) \biggl( F^2 - \frac{y_\varpi^2}{y_\mathrm{surf}^2} \biggr) \biggr]^{1 / 2} \biggr\} \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Associated Velocities (2nd Try) */

2021-05-21T21:51:55Z

Associated Velocities (2nd Try)

← Older revision		Revision as of 21:51, 21 May 2021
Line 983:		Line 983:
	<td align="left">		<td align="left">
	<math>~		<math>~
	y_\mathrm{surf}^2 \biggl[ \frac{y_\varpi^2}{y_\mathrm{surf}^2} + F^2 \biggr] \, .		y_\mathrm{surf}^2 \biggl[ \frac{y_\varpi^2}{y_\mathrm{surf}^2} + F^2 \biggr] \, ,
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
			in which case,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 994:		Line 994:
	<tr>		<tr>
	<td align="right">		<td align="right">
	~~ ~~		<math>~\frac{4AC}{B^2}</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 1,001:		Line 1,001:
	<td align="left">		<td align="left">
	<math>~		<math>~
	~~y^2~~ \~~biggl[ \underbrace~~{1 ~~+ \biggl( \frac{x_\mathrm{surf}~~}{y_\~~mathrm{surf}}\biggr)~~^2 ~~\tan^2\varphi~~}~~_{A} \biggr] + y~~ \biggl[~~\underbrace{ - 2 y_\varpi }_{B}\biggr]~~		\frac{1}{y_\varpi^2} \biggl[
	+ \biggl[ \~~underbrace~~{y_\varpi^2 ~~- x^2_~~\mathrm{surf} ~~\cdot \tan~~^2\~~varphi}_{C}~~ \biggr]		(1 + F^2)\biggl( \frac{y_\varpi^2}{y_\mathrm{surf}^2}+ F^2 \biggr)
			\biggr]
	</math>		</math>
	</td>		</td>

Tohline: /* Simpler Example */

2021-05-21T21:42:22Z

Simpler Example

← Older revision		Revision as of 21:42, 21 May 2021
Line 784:		Line 784:
	</table>		</table>

	====Associated Velocities====		====Associated Velocities (1<sup>st</sup> Try)====

	If we set <math>~\varphi = \dot\varphi t</math>, with <math>~\dot\varphi</math> constant, we appreciate that both of the coefficients, <math>~A</math> and <math>~C</math> will be functions of time. The time-derivatives of the fluid-element positions will therefore depend on the time-derivatives of these two coefficients. We find,		If we set <math>~\varphi = \dot\varphi t</math>, with <math>~\dot\varphi</math> constant, we appreciate that both of the coefficients, <math>~A</math> and <math>~C</math> will be functions of time. The time-derivatives of the fluid-element positions will therefore depend on the time-derivatives of these two coefficients. We find,
Line 921:		Line 921:
	=		=
	\frac{\dot{y}}{\tan\varphi} + \frac{\dot\varphi (y_\varpi - y)}{\sin\varphi \cos\varphi} \, .		\frac{\dot{y}}{\tan\varphi} + \frac{\dot\varphi (y_\varpi - y)}{\sin\varphi \cos\varphi} \, .
			</math>
			</td>
			</tr>
			</table>

			====Associated Velocities (2<sup>nd</sup> Try)====

			Let's simplify notation. Specifically, let's define,

			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~F^2</math>
			</td>
			<td align="center">
			<math>~\equiv </math>
			</td>
			<td align="left">
			<math>~
			\biggl( \frac{x_\mathrm{surf}}{y_\mathrm{surf}} \biggr)^2 \tan^2\varphi \, .
			</math>
			</td>
			</tr>
			</table>
			Then we can write,

			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~A</math>
			</td>
			<td align="center">
			<math>~\equiv </math>
			</td>
			<td align="left">
			<math>~
			1 + F^2 \, ,
			</math>
			</td>
			<td align="center">      and,       </td>
			<td align="right">
			<math>~B</math>
			</td>
			<td align="center">
			<math>~\equiv </math>
			</td>
			<td align="left">
			<math>~
			-2y_\varpi \, ,
			</math>
			</td>
			<td align="center">      and,       </td>
			<td align="right">
			<math>~C</math>
			</td>
			<td align="center">
			<math>~\equiv </math>
			</td>
			<td align="left">
			<math>~
			y_\mathrm{surf}^2 \biggl[ \frac{y_\varpi^2}{y_\mathrm{surf}^2} + F^2 \biggr] \, .
			</math>
			</td>
			</tr>
			</table>


			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			y^2 \biggl[ \underbrace{1 + \biggl( \frac{x_\mathrm{surf}}{y_\mathrm{surf}}\biggr)^2 \tan^2\varphi}_{A} \biggr] + y \biggl[\underbrace{ - 2 y_\varpi }_{B}\biggr]
			+ \biggl[ \underbrace{y_\varpi^2 - x^2_\mathrm{surf} \cdot \tan^2\varphi}_{C} \biggr]
			</math>
			</td>
			</tr>
			</table>
			Given that,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{4AC}{B^2}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{y^2_\varpi y^2_\mathrm{surf} }\biggl[ y^2_\mathrm{surf} + x_\mathrm{surf}^2 \cdot \tan^2\varphi \biggr]\biggl[ y_\varpi^2 - x^2_\mathrm{surf} \cdot \tan^2\varphi \biggr] \, ,
			</math>
			</td>
			</tr>
			</table>
			the pair of roots are given by the expression,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~y</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{B}{2A} \biggl\{\pm \biggl[1 - \frac{4AC}{B^2} \biggr]^{1 / 2} -1 \biggr\}</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{B}{2A}
			\biggl\{\pm \biggl[1 + \frac{1}{y^2_\varpi y^2_\mathrm{surf} }\biggl( y^2_\mathrm{surf} + x_\mathrm{surf}^2 \cdot \tan^2\varphi \biggr)\biggl( x^2_\mathrm{surf} \cdot \tan^2\varphi - y_\varpi^2 \biggr) \biggr]^{1 / 2} -1 \biggr\} \, .
	</math>		</math>
	</td>		</td>