Gelöste Aufgaben/Kita - Versionsgeschichte

Mechaniker am 15. September 2022 um 13:55 Uhr

2022-09-15T13:55:05Z

← Nächstältere Version		Version vom 15. September 2022, 13:55 Uhr
Zeile 364:		Zeile 364:
	/* ****************************************************/		/* ****************************************************/

	/* point "O" */		/* point "O" - geometry */
	BC: [subst(x= 0 , subst(sol[1][1],w(x)) )= 0 ,		BC: [subst(x= 0 , subst(sol[1][1],w(x)) )= 0 ,
	subst(x= 0 , subst(sol[2][1],v(x)) )= 0 ,		subst(x= 0 , subst(sol[2][1],v(x)) )= 0 ,
	subst(x= 0 , subst(sol[1][1],φ(x)) )= 0 ,		/* point "O" - bending moments */
	subst(x= 0 , subst(sol[2][1],ψ(x)) )= 0 ,		subst(x= 0 , subst(sol[1][1],M[y](x)) )= 0 ,
			subst(x= 0 , subst(sol[2][1],M[z](x)) )= 0 ,
	/* point "H" - geometry */		/* point "H" - geometry */
	subst(x= h[1], subst(sol[1][1],w(x)) )=W[H],		subst(x= h[1], subst(sol[1][1],w(x)) )=W[H],
Zeile 382:		Zeile 383:
	subst(x= h[1], subst(sol[2][1],M[z](x)) )=subst(x= 0 , subst(sol[2][2],M[z](x)) ),		subst(x= h[1], subst(sol[2][1],M[z](x)) )=subst(x= 0 , subst(sol[2][2],M[z](x)) ),
	/* point "H" - forces */		/* point "H" - forces */
	subst(x= h[1], subst(sol[1][1],Q[z](x)) )=subst(x= 0 , subst(sol[1][2],Q[z](x)) ) - sum(S[i]*subst(geo,e[i]),i,1,3)[1][3],		subst(x= h[1], subst(sol[1][1],Q[z](x)) )=subst(x= 0 , subst(sol[1][2],Q[z](x)) ) + sum(S[i]*subst(geo,e[i]),i,1,3)[1][3],
	subst(x= h[1], subst(sol[2][1],Q[y](x)) )=subst(x= 0 , subst(sol[2][2],Q[y](x)) ) - sum(S[i]*subst(geo,e[i]),i,1,3)[1][2],		subst(x= h[1], subst(sol[2][1],Q[y](x)) )=subst(x= 0 , subst(sol[2][2],Q[y](x)) ) + sum(S[i]*subst(geo,e[i]),i,1,3)[1][2],
	/* point "T" (Top) - geometry */		/* point "T" (Top) - geometry */
	subst(x= h[2], subst(sol[1][2],w(x)) )=W[T],		subst(x= h[2], subst(sol[1][2],w(x)) )=W[T],

Mechaniker am 15. September 2022 um 13:51 Uhr

2022-09-15T13:51:20Z

← Nächstältere Version		Version vom 15. September 2022, 13:51 Uhr
Zeile 308:		Zeile 308:
	\begin{pmatrix}0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\		\begin{pmatrix}0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
	0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\		0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
	0 & 0 ~~& 1~~ & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\		0 & -EI & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
	0 & 0 & 0 & 0 & 0 & 0 ~~& 1~~ & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\		0 & 0 & 0 & 0 & 0 & EI & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
	\frac{{{h}_{1}^{3}}}{6} & \frac{{{h}_{1}^{2}}}{2} & {h_1} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\		\frac{{{h}_{1}^{3}}}{6} & \frac{{{h}_{1}^{2}}}{2} & {h_1} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
	0 & 0 & 0 & 0 & \frac{{{h}_{1}^{3}}}{6} & \frac{{{h}_{1}^{2}}}{2} & {h_1} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\		0 & 0 & 0 & 0 & \frac{{{h}_{1}^{3}}}{6} & \frac{{{h}_{1}^{2}}}{2} & {h_1} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\
Zeile 329:		Zeile 329:
	0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {h_2} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\		0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {h_2} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
	0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\		0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
	0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{pmatrix} \cdot \underline{c}=\begin{pmatrix}0\\		0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{pmatrix}
			\cdot \underline{c}
			=
			\begin{pmatrix}0\\
	0\\		0\\
	0\\		0\\
Zeile 413:		Zeile 416:
	::<math>		::<math>
	\begin{array}{l}		\begin{array}{l}
	{c_1}=-\~~frac{5 {{2}^{~~\~~frac~~{~~15}{2}}} R {q_T} {{\alpha }^{2}}-85968 R {q_T} \alpha -98415 \sqrt{2} R {q_T}}{19200 EI {{\alpha }^{2}}+405 {{2}^{\frac{15}{2}}} EI \alpha +65610 EI~~}\\		{c_1}=0\\
	{~~c_2~~}=\frac{~~1280 {{R}^{2}} {q_T} {{\alpha }^{2}}-5373~~ {{2}^{\frac{9}{2}}} {{R}^{2}} {q_T} \alpha -~~196830~~ {{R}^{2}} {q_T}}{~~9600 EI {{\alpha }^{2}}+405 {{2}^{\frac{13}{2}}}~~ EI \alpha ~~+32805 EI~~}\\		{c_2}=0\\
	~~{c_3}=0~~\\		{c_3}=\frac{{{2}^{\frac{9}{2}}} {{R}^{3}} {q_T} \alpha -2187 {{R}^{3}} {q_T}}{180 EI \alpha }\\
	{c_4}=0\\		{c_4}=0\\
	{c_5}=\~~frac{184 {{3}^{~~\~~frac~~{~~5}{2}}} R {q_T} \alpha }{3200 EI {{\alpha }^{2}}+135 {{2}^{\frac{13}{2}}} EI \alpha +10935 EI~~}\\		{c_5}=0\\
	{~~c_6~~}=-\frac~~{23~~ {~~{2}^{\frac{9}{2}}}~~ {{3}^{\frac{5}{2}}} {{R}^{2}} {q_T} ~~\alpha~~ }{~~3200~~ EI {{\alpha }~~^{2}}+135 {{2}^{\frac{13}{2}}} EI \alpha +10935 EI}\\~~		{c_6}=0\\
	~~{c_7}=0~~\\		{c_7}=-\frac{{{3}^{\frac{7}{2}}} {{R}^{3}} {q_T}}{20 EI \alpha }\\
	{c_8}=0\\		{c_8}=0\\
	{c_9}=		{c_9}=\frac{5 R {q_T}}{3 \sqrt{2} EI}\\
	\frac{~~25 {{2}^{\frac{15}{2}}}~~ R {q_T} ~~{{\alpha~~ }^{~~2}}+17280 R {q_T} \alpha +10935~~ \sqrt{2} ~~R {q_T}}{3840 EI {{\alpha }^{2}}+81 {{2}^{\frac{15}{2}}} EI \alpha +13122~~ EI}\\		{c_{10}}=-
	{c_{10}}=-\frac{8 {{R}^{2}} {q_T}}{9 EI}\\		\frac{8 {{R}^{2}} {q_T}}{9 EI}\\
	{c_{11}}=-		{c_{11}}=-\frac{{{2}^{\frac{13}{2}}} {{R}^{3}} {q_T} \alpha +2187 {{R}^{3}} {q_T}}{180 EI \alpha }\\
	\frac{35 {{2}^{\frac{17}{2}}} {{R}^{3}} {q_T} {{\alpha ~~}^{2}}~~+~~584928~~ {{R}^{3}} {q_T} ~~\alpha +317115 {{2~~}^{~~\frac{3}{2}}} {{R}^{3}} {q_T}}{28800 EI {{\alpha }^{2}}+1215 {{2}^{\frac{13}{2}}}~~ EI \alpha ~~+98415 EI~~}\\		{c_{12}}=-\frac{243 {{R}^{4}} {q_T}}{5 \sqrt{2} EI \alpha }\\
	{c_{12}}=-\frac{~~621 {{2}^{\frac{15}{2}}}~~ {{R}^{4}} {q_T} ~~\alpha +178848 {{R~~}~~^{4}} {q_T}}{3200 EI {~~{\~~alpha }^{2}}+135 {{2}^{\frac{13}~~{2}}} EI \alpha ~~+10935 EI~~}\\
	{c_{13}}=0\\		{c_{13}}=0\\
	{c_{14}}=0\~~operatorname{,~~		{c_{14}}=0\\
	}{c_{15}}=-\frac{~~736~~ {{3}^{\frac{5}{2}}} {{R}^{3}} {q_T} ~~\alpha~~ }{~~3200~~ EI {{\alpha ~~}^{2}}+135 {{2}^{\frac{13}{2}}} EI \alpha +10935 EI~~}\\		{c_{15}}=-\frac{{{3}^{\frac{7}{2}}} {{R}^{3}} {q_T}}{20 EI \alpha }\\
	{c_{16}}=-\frac{~~23 {{2}^{\frac{15}{2}}}~~ {{3}^{\frac{3}{2}}} {{R}^{4}} {q_T} ~~\alpha }{3200 EI {{\alpha~~ }~~^{2}}+135 {{2}^~~{\~~frac{13}~~{2}}} EI \alpha ~~+10935 EI~~}\\		{c_{16}}=-\frac{{{3}^{\frac{7}{2}}} {{R}^{4}} {q_T}}{5 \sqrt{2} EI \alpha }\\
	{W_H}=-		{W_H}=-\frac{243 {{R}^{4}} {q_T}}{5 \sqrt{2} EI \alpha }\\
	\frac{~~621 {{2}^{\frac{15}{2}}}~~ {{R}^{4}} {q_T} ~~\alpha +178848 {{R~~}~~^{4}} {q_T}}{3200 EI {{\alpha }^{2}}+135 {{2}^~~{\~~frac{13}~~{2}}} EI \alpha ~~+10935 EI~~}\\		{V_H}=-\frac{{{3}^{\frac{7}{2}}} {{R}^{4}} {q_T}}{5 \sqrt{2} EI \alpha }\\
	{V_H}=-\frac{~~23 {{2}^{\frac{15}{2}}}~~ {{3}^{\frac{3}{2}}} {{R}^{4}} {q_T} ~~\alpha~~ }{~~3200 EI {{~~\~~alpha }^{2}}+135 {{2}^{\frac{13}~~{2}}} EI \alpha ~~+10935 EI~~}\\		{{\Phi }_H}=-\frac{{{2}^{\frac{13}{2}}} {{R}^{3}} {q_T} \alpha +2187 {{R}^{3}} {q_T}}{180 EI \alpha }\\
	{{\Phi }_H}=-		{{\Psi }_H}=-\frac{{{3}^{\frac{7}{2}}} {{R}^{3}} {q_T}}{20 EI \alpha }\\
	\frac{35 {{2}^{\frac{17}{2}}} {{R}^{3}} {q_T} {{\alpha ~~}^{2}}~~+~~584928~~ {{R}^{3}} {q_T~~} \alpha +317115 {{2}^{\frac{3}{2}~~}} {~~{R}^{3}} {q_T}}{28800 EI {{\alpha }^{2}}+1215 {{2}^{\frac{13}{2}}}~~ EI \alpha ~~+98415 EI~~}\\
	{{\Psi }_H}=-\frac{~~736~~ {{3}^{\frac{5}{2}}} {{R}^{3}} {q_T} ~~\alpha }{3200 EI {{\alpha }^{2}}+135 {{2}^{\frac{13~~}{~~2}}}~~ EI \alpha ~~+10935 EI~~}\\
	{W_T}=-		{W_T}=-
	\frac{~~62080~~ {{R}^{4}} {q_T} {{\alpha ~~}^{2}}~~+~~20871 {{2}^{~~\~~frac{15}~~{2~~}}} {{R}^{4}~~} ~~{q_T} \alpha +5845851~~ {{R}^{4}} {q_T}}{~~57600 EI {{\alpha }^{2}}+1215 {{2}^{\frac{15}{2}}}~~ EI \alpha ~~+196830 EI~~}\\		\frac{70 {{R}^{4}} {q_T} \alpha +2187 \sqrt{2} {{R}^{4}} {q_T}}{60 EI \alpha }\\
	{V_T}=-\frac{~~161 {{2}^{\frac{11}{2}}}~~ {{3}^{\frac{3}{2}}} {{R}^{4}} {q_T} ~~\alpha~~ }{~~3200 EI {{\alpha }^{2}}+135~~ {{2}^{\frac{13}{2}}} EI \alpha ~~+10935 EI~~}\\		{V_T}=-\frac{{{3}^{\frac{9}{2}}} {{R}^{4}} {q_T}}{5 {{2}^{\frac{3}{2}}} EI \alpha }\\
	{{\Phi }_T}=-		{{\Phi }_T}=-\frac{119 \sqrt{2} {{R}^{3}} {q_T} \alpha +2187 {{R}^{3}} {q_T}}{180 EI \alpha }\\
	~~\frac{185 {{2}^{~~\frac{~~15}{2}}} {{R}^{3}} {q_T} {{~~\~~alpha }^~~{2}~~}+843264~~ {{R}^{3}} {q_T} \alpha +~~885735 \sqrt{2}~~ {{R}^{3}} {q_T}}{~~38400 EI {{\alpha }^{2}}+405 {{2}^{\frac{17}{2}}}~~ EI \alpha ~~+131220 EI~~}\\		{{\Psi }_T}=-\frac{{{3}^{\frac{7}{2}}} {{R}^{3}} {q_T}}{20 EI \alpha }
	{{\Psi }_T}=-\frac{~~736~~ {{3}^{\frac{5}{2}}} {{R}^{3}} {q_T} ~~\alpha }{3200 EI {{\alpha }^{2~~}~~}+135 {~~{~~2}^{\frac{13}{2}}}~~ EI \alpha ~~+10935 EI~~}
	\end{array}		\end{array}
	</math>		</math>

Mechaniker am 15. September 2022 um 13:49 Uhr

2022-09-15T13:49:48Z

← Nächstältere Version		Version vom 15. September 2022, 13:49 Uhr
Zeile 244:		Zeile 244:
	==== ... aus Rand "<i>O</i>"====		==== ... aus Rand "<i>O</i>"====
	Das gelenkige Lager in ''O'' erlaubt keine Verschiebung und es nimmt keine Momente auf.		Das gelenkige Lager in ''O'' erlaubt keine Verschiebung und es nimmt keine Momente auf.
	Daraus ergeben sich die '''~~Geometrische~~ Bedingungen:'''		Daraus ergeben sich die '''Geometrischen Bedingungen:'''
	<ol start="1">		<ol start="1">
	<li><math>w_{ I}(0) = 0</math></li>		<li><math>w_{ I}(0) = 0</math></li>

Mechaniker am 15. September 2022 um 13:49 Uhr

2022-09-15T13:49:02Z

← Nächstältere Version		Version vom 15. September 2022, 13:49 Uhr
Zeile 230:		Zeile 230:
	<table style="background-color:white; margin-right:14px; border: 1px solid black; border-spacing: 30px;">		<table style="background-color:white; margin-right:14px; border: 1px solid black; border-spacing: 30px;">
	<tr><th>Richtung</th><th>z</th><th>y</th></tr>		<tr><th>Richtung</th><th>z</th><th>y</th></tr>
	<tr><td>Knoten T<br/>Auslenkung<br/>Kippwinkel</td><td><br/><math>W_T</math>,<br/><math>\Phi_T</math></td><td><br/><math>V_T</math>,<br><math>\Psi_T</math></td></tr>		<tr><td>Knoten T<br/>Auslenkung<br/>Kippwinkel</td><td><br/><math>W_T</math>,<br/><math>\Phi_T</math></td><td><br/><math>V_T</math>,<br/><math>\Psi_T</math></td></tr>
	<tr><td>Knoten H<br/>Auslenkung<br/>Kippwinkel</td><td><br/><math>W_H</math>,<br/><math>\Phi_H</math></td><td><br/><math>V_H</math>,<br><math>\Psi_H</math></td></tr>		<tr><td>Knoten H<br/>Auslenkung<br/>Kippwinkel</td><td><br/><math>W_H</math>,<br/><math>\Phi_H</math></td><td><br/><math>V_H</math>,<br/><math>\Psi_H</math></td></tr>
	<tr><td>Knoten 0<br/>Auslenkung<br/>Kippwinkel</td><td>%</td><td>%</td></tr>		<tr><td>Knoten 0<br/>Auslenkung<br/>Kippwinkel</td><td><br/><math> 0 </math>,<br/><math> c_3 </math></td><td><br/><math> 0 </math>,<br/><math> c_7 </math></td></tr>
	</table>		</table>
	In ''O'' hätten wir auch noch die Variablen <math>\Phi_O</math> und <math>\Psi_O</math> einführen könnten. Die sind allerdings identisch mit <math>c_3</math> und <math>c_7</math>.		In ''O'' hätten wir auch noch die Variablen <math>\Phi_O</math> und <math>\Psi_O</math> einführen könnten. Die sind allerdings identisch mit <math>c_3</math> und <math>c_7</math>.
	~~Die~~ Unbekannten fassen wir in
			Alle Unbekannten fassen wir nun in
	::<math>\underline{c} = \left(\begin{array}{l}c_{01}\\c_{02}\\\ldots\\c_{15}\\c_{16}\\W_H\\V_H\\\Phi_H\\\Psi_H\\W_T\\V_T\\\Phi_T\\\Psi_T\end{array}\right)</math>		::<math>\underline{c} = \left(\begin{array}{l}c_{01}\\c_{02}\\\ldots\\c_{15}\\c_{16}\\W_H\\V_H\\\Phi_H\\\Psi_H\\W_T\\V_T\\\Phi_T\\\Psi_T\end{array}\right)</math>
	zusammen.		zusammen.

Mechaniker am 15. September 2022 um 13:44 Uhr

2022-09-15T13:44:19Z

← Nächstältere Version		Version vom 15. September 2022, 13:44 Uhr
Zeile 234:		Zeile 234:
	<tr><td>Knoten 0<br/>Auslenkung<br/>Kippwinkel</td><td>%</td><td>%</td></tr>		<tr><td>Knoten 0<br/>Auslenkung<br/>Kippwinkel</td><td>%</td><td>%</td></tr>
	</table>		</table>
			In ''O'' hätten wir auch noch die Variablen <math>\Phi_O</math> und <math>\Psi_O</math> einführen könnten. Die sind allerdings identisch mit <math>c_3</math> und <math>c_7</math>.
	Die Unbekannten fassen wir in		Die Unbekannten fassen wir in
	::<math>\underline{c} = \left(\begin{array}{l}c_{01}\\c_{02}\\\ldots\\c_{15}\\c_{16}\\W_H\\V_H\\\Phi_H\\\Psi_H\\W_T\\V_T\\\Phi_T\\\Psi_T\end{array}\right)</math>		::<math>\underline{c} = \left(\begin{array}{l}c_{01}\\c_{02}\\\ldots\\c_{15}\\c_{16}\\W_H\\V_H\\\Phi_H\\\Psi_H\\W_T\\V_T\\\Phi_T\\\Psi_T\end{array}\right)</math>
Zeile 241:		Zeile 242:

	==== ... aus Rand "<i>O</i>"====		==== ... aus Rand "<i>O</i>"====
	'''Geometrische Bedingungen:'''		Das gelenkige Lager in ''O'' erlaubt keine Verschiebung und es nimmt keine Momente auf.
			Daraus ergeben sich die '''Geometrische Bedingungen:'''
	<ol start="1">		<ol start="1">
	<li><math>w_{ I}(0) = 0</math></li>		<li><math>w_{ I}(0) = 0</math></li>
	<li><math>v_{ I}(0) = 0</math></li>		<li><math>v_{ I}(0) = 0</math></li>
	~~<li><math>\varphi_{ I}(0) = 0</math></li>~~
	~~<li><math>\psi_{ I}(0) = 0</math></li>~~
	</ol>		</ol>
			und die '''Kraft- und Momenten-Randbedingungen:'''
	'''Kraft- und Momenten-Randbedingungen:'''		<ol start="3">
	* keine		<li><math>M_{z, I}(0) = 0</math></li>
			<li><math>M_{y, I}(0) = 0</math></li>
			</ol>

	==== ... aus Übergang "<i>H</i>"====		==== ... aus Übergang "<i>H</i>"====

Mechaniker am 15. September 2022 um 10:19 Uhr

2022-09-15T10:19:44Z

← Nächstältere Version		Version vom 15. September 2022, 10:19 Uhr
Zeile 326:		Zeile 326:
	0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {h_2} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\		0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {h_2} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
	0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\		0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
	0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{pmatrix} \cdot} \underline{c}=\begin{pmatrix}0\\		0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{pmatrix} \cdot \underline{c}=\begin{pmatrix}0\\
	0\\		0\\
	0\\		0\\

Mechaniker am 15. September 2022 um 10:18 Uhr

2022-09-15T10:18:01Z

← Nächstältere Version		Version vom 15. September 2022, 10:18 Uhr
Zeile 317:		Zeile 317:
	-{h_1} & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\		-{h_1} & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
	0 & 0 & 0 & 0 & {h_1} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\		0 & 0 & 0 & 0 & {h_1} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
	-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\frac{\left( {EA_3}+{EA_2}+4 {EA_1}\right) {{R}^{2}}}{\sqrt{{{R}^{2}}+{{h}_{1}^{2}}} \left( 4 EI {{R}^{2}}+4 {{h}_{1}^{2}}\, EI\right) } & -\frac{\left( \sqrt{3} {EA_3}-\sqrt{3} {EA_2}\right) {{R}^{2}}}{\sqrt{{{R}^{2}}+{{h}_{1}^{2}}} \left( 4 EI {{R}^{2}}+4 {{h}_{1}^{2}}\, EI\right) } & 0 & 0 & 0 & 0 & 0 & 0\\		-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\left( {EA_3}+{EA_2}+4 {EA_1}\right) {{R}^{2}}}{\sqrt{{{R}^{2}}+{{h}_{1}^{2}}} \left( 4 EI {{R}^{2}}+4 {{h}_{1}^{2}}\, EI\right) } & \frac{\left( \sqrt{3} {EA_3}-\sqrt{3} {EA_2}\right) {{R}^{2}}}{\sqrt{{{R}^{2}}+{{h}_{1}^{2}}} \left( 4 EI {{R}^{2}}+4 {{h}_{1}^{2}}\, EI\right) } & 0 & 0 & 0 & 0 & 0 & 0\\
	0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -\frac{\left( \sqrt{3} {EA_3}-\sqrt{3} {EA_2}\right) {{R}^{2}}}{\sqrt{{{R}^{2}}+{{h}_{1}^{2}}} \left( 4 EI {{R}^{2}}+4 {{h}_{1}^{2}}\, EI\right) } & -\frac{\left( 3 {EA_3}+3 {EA_2}\right) {{R}^{2}}}{\sqrt{{{R}^{2}}+{{h}_{1}^{2}}} \left( 4 EI {{R}^{2}}+4 {{h}_{1}^{2}}\, EI\right) } & 0 & 0 & 0 & 0 & 0 & 0\\		0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{\left( \sqrt{3} {EA_3}-\sqrt{3} {EA_2}\right) {{R}^{2}}}{\sqrt{{{R}^{2}}+{{h}_{1}^{2}}} \left( 4 EI {{R}^{2}}+4 {{h}_{1}^{2}}\, EI\right) } & \frac{\left( 3 {EA_3}+3 {EA_2}\right) {{R}^{2}}}{\sqrt{{{R}^{2}}+{{h}_{1}^{2}}} \left( 4 EI {{R}^{2}}+4 {{h}_{1}^{2}}\, EI\right) } & 0 & 0 & 0 & 0 & 0 & 0\\
	0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{{{h}_{2}^{3}}}{6} & \frac{{{h}_{2}^{2}}}{2} & {h_2} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\		0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{{{h}_{2}^{3}}}{6} & \frac{{{h}_{2}^{2}}}{2} & {h_2} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\
	0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{{{h}_{2}^{3}}}{6} & \frac{{{h}_{2}^{2}}}{2} & {h_2} & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\		0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{{{h}_{2}^{3}}}{6} & \frac{{{h}_{2}^{2}}}{2} & {h_2} & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\
Zeile 326:		Zeile 326:
	0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {h_2} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\		0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {h_2} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
	0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\		0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
	0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{pmatrix} \cdot \underline{c} =		0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{pmatrix} \cdot} \underline{c}=\begin{pmatrix}0\\
	\begin{pmatrix}0\\
	0\\		0\\
	0\\		0\\
Zeile 411:		Zeile 410:
	::<math>		::<math>
	\begin{array}{l}		\begin{array}{l}
	{c_1}=-\frac{5 {{2}^{\frac{15}{2}}} R {q_T} {{\alpha }^{2}}+85968 R {q_T} \alpha -98415 \sqrt{2} R {q_T}}{19200 EI {{\alpha }^{2}}-405 {{2}^{\frac{15}{2}}} EI \alpha +65610 EI}\\		{c_1}=-\frac{5 {{2}^{\frac{15}{2}}} R {q_T} {{\alpha }^{2}}-85968 R {q_T} \alpha -98415 \sqrt{2} R {q_T}}{19200 EI {{\alpha }^{2}}+405 {{2}^{\frac{15}{2}}} EI \alpha +65610 EI}\\
	{c_2}=\frac{1280 {{R}^{2}} {q_T} {{\alpha }^{2}}+5373 {{2}^{\frac{9}{2}}} {{R}^{2}} {q_T} \alpha -196830 {{R}^{2}} {q_T}}{9600 EI {{\alpha }^{2}}-405 {{2}^{\frac{13}{2}}} EI \alpha +32805 EI}\\		{c_2}=\frac{1280 {{R}^{2}} {q_T} {{\alpha }^{2}}-5373 {{2}^{\frac{9}{2}}} {{R}^{2}} {q_T} \alpha -196830 {{R}^{2}} {q_T}}{9600 EI {{\alpha }^{2}}+405 {{2}^{\frac{13}{2}}} EI \alpha +32805 EI}\\
	{c_3}=0\\		{c_3}=0\\
	{c_4}=0\\		{c_4}=0\\
	{c_5}=-\frac{184 {{3}^{\frac{5}{2}}} R {q_T} \alpha }{3200 EI {{\alpha }^{2}}-135 {{2}^{\frac{13}{2}}} EI \alpha +10935 EI}\\		{c_5}=\frac{184 {{3}^{\frac{5}{2}}} R {q_T} \alpha }{3200 EI {{\alpha }^{2}}+135 {{2}^{\frac{13}{2}}} EI \alpha +10935 EI}\\
	{c_6}=\frac{23 {{2}^{\frac{9}{2}}} {{3}^{\frac{5}{2}}} {{R}^{2}} {q_T} \alpha }{3200 EI {{\alpha }^{2}}-135 {{2}^{\frac{13}{2}}} EI \alpha +10935 EI}\\		{c_6}=-\frac{23 {{2}^{\frac{9}{2}}} {{3}^{\frac{5}{2}}} {{R}^{2}} {q_T} \alpha }{3200 EI {{\alpha }^{2}}+135 {{2}^{\frac{13}{2}}} EI \alpha +10935 EI}\\
	{c_7}=0\\		{c_7}=0\\
	{c_8}=0\\		{c_8}=0\\
	{c_9}=\frac{25 {{2}^{\frac{15}{2}}} R {q_T} {{\alpha }^{2}}-17280 R {q_T} \alpha +10935 \sqrt{2} R {q_T}}{3840 EI {{\alpha }^{2}}-81 {{2}^{\frac{15}{2}}} EI \alpha +13122 EI}\\		{c_9}=
			\frac{25 {{2}^{\frac{15}{2}}} R {q_T} {{\alpha }^{2}}+17280 R {q_T} \alpha +10935 \sqrt{2} R {q_T}}{3840 EI {{\alpha }^{2}}+81 {{2}^{\frac{15}{2}}} EI \alpha +13122 EI}\\
	{c_{10}}=-\frac{8 {{R}^{2}} {q_T}}{9 EI}\\		{c_{10}}=-\frac{8 {{R}^{2}} {q_T}}{9 EI}\\
	{c_{11}}=-\frac{35 {{2}^{\frac{17}{2}}} {{R}^{3}} {q_T} {{\alpha }^{2}}-584928 {{R}^{3}} {q_T} \alpha +317115 {{2}^{\frac{3}{2}}} {{R}^{3}} {q_T}}{28800 EI {{\alpha }^{2}}-1215 {{2}^{\frac{13}{2}}} EI \alpha +98415 EI}\\		{c_{11}}=-
	{c_{12}}=\frac{621 {{2}^{\frac{15}{2}}} {{R}^{4}} {q_T} \alpha -178848 {{R}^{4}} {q_T}}{3200 EI {{\alpha }^{2}}-135 {{2}^{\frac{13}{2}}} EI \alpha +10935 EI}\\		\frac{35 {{2}^{\frac{17}{2}}} {{R}^{3}} {q_T} {{\alpha }^{2}}+584928 {{R}^{3}} {q_T} \alpha +317115 {{2}^{\frac{3}{2}}} {{R}^{3}} {q_T}}{28800 EI {{\alpha }^{2}}+1215 {{2}^{\frac{13}{2}}} EI \alpha +98415 EI}\\
			{c_{12}}=-\frac{621 {{2}^{\frac{15}{2}}} {{R}^{4}} {q_T} \alpha +178848 {{R}^{4}} {q_T}}{3200 EI {{\alpha }^{2}}+135 {{2}^{\frac{13}{2}}} EI \alpha +10935 EI}\\
	{c_{13}}=0\\		{c_{13}}=0\\
	{c_{14}}=0\\		{c_{14}}=0\operatorname{,
	{c_{15}}=\frac{736 {{3}^{\frac{5}{2}}} {{R}^{3}} {q_T} \alpha }{3200 EI {{\alpha }^{2}}-135 {{2}^{\frac{13}{2}}} EI \alpha +10935 EI}\\		}{c_{15}}=-\frac{736 {{3}^{\frac{5}{2}}} {{R}^{3}} {q_T} \alpha }{3200 EI {{\alpha }^{2}}+135 {{2}^{\frac{13}{2}}} EI \alpha +10935 EI}\\
	{c_{16}}=\frac{23 {{2}^{\frac{15}{2}}} {{3}^{\frac{3}{2}}} {{R}^{4}} {q_T} \alpha }{3200 EI {{\alpha }^{2}}-135 {{2}^{\frac{13}{2}}} EI \alpha +10935 EI}\\		{c_{16}}=-\frac{23 {{2}^{\frac{15}{2}}} {{3}^{\frac{3}{2}}} {{R}^{4}} {q_T} \alpha }{3200 EI {{\alpha }^{2}}+135 {{2}^{\frac{13}{2}}} EI \alpha +10935 EI}\\
	{W_H}=		{W_H}=-
	\frac{621 {{2}^{\frac{15}{2}}} {{R}^{4}} {q_T} \alpha -178848 {{R}^{4}} {q_T}}{3200 EI {{\alpha }^{2}}-135 {{2}^{\frac{13}{2}}} EI \alpha +10935 EI}\\		\frac{621 {{2}^{\frac{15}{2}}} {{R}^{4}} {q_T} \alpha +178848 {{R}^{4}} {q_T}}{3200 EI {{\alpha }^{2}}+135 {{2}^{\frac{13}{2}}} EI \alpha +10935 EI}\\
	{V_H}=\frac{23 {{2}^{\frac{15}{2}}} {{3}^{\frac{3}{2}}} {{R}^{4}} {q_T} \alpha }{3200 EI {{\alpha }^{2}}-135 {{2}^{\frac{13}{2}}} EI \alpha +10935 EI}\\		{V_H}=-\frac{23 {{2}^{\frac{15}{2}}} {{3}^{\frac{3}{2}}} {{R}^{4}} {q_T} \alpha }{3200 EI {{\alpha }^{2}}+135 {{2}^{\frac{13}{2}}} EI \alpha +10935 EI}\\
	{{\Phi }_H}=-		{{\Phi }_H}=-
	\frac{35 {{2}^{\frac{17}{2}}} {{R}^{3}} {q_T} {{\alpha }^{2}}-584928 {{R}^{3}} {q_T} \alpha +317115 {{2}^{\frac{3}{2}}} {{R}^{3}} {q_T}}{28800 EI {{\alpha }^{2}}-1215 {{2}^{\frac{13}{2}}} EI \alpha +98415 EI}\\		\frac{35 {{2}^{\frac{17}{2}}} {{R}^{3}} {q_T} {{\alpha }^{2}}+584928 {{R}^{3}} {q_T} \alpha +317115 {{2}^{\frac{3}{2}}} {{R}^{3}} {q_T}}{28800 EI {{\alpha }^{2}}+1215 {{2}^{\frac{13}{2}}} EI \alpha +98415 EI}\\
	{{\Psi }_H}=\frac{736 {{3}^{\frac{5}{2}}} {{R}^{3}} {q_T} \alpha }{3200 EI {{\alpha }^{2}}-135 {{2}^{\frac{13}{2}}} EI \alpha +10935 EI}\\		{{\Psi }_H}=-\frac{736 {{3}^{\frac{5}{2}}} {{R}^{3}} {q_T} \alpha }{3200 EI {{\alpha }^{2}}+135 {{2}^{\frac{13}{2}}} EI \alpha +10935 EI}\\
	{W_T}=-		{W_T}=-
	\frac{62080 {{R}^{4}} {q_T} {{\alpha }^{2}}-20871 {{2}^{\frac{15}{2}}} {{R}^{4}} {q_T} \alpha +5845851 {{R}^{4}} {q_T}}{57600 EI {{\alpha }^{2}}-1215 {{2}^{\frac{15}{2}}} EI \alpha +196830 EI}\\		\frac{62080 {{R}^{4}} {q_T} {{\alpha }^{2}}+20871 {{2}^{\frac{15}{2}}} {{R}^{4}} {q_T} \alpha +5845851 {{R}^{4}} {q_T}}{57600 EI {{\alpha }^{2}}+1215 {{2}^{\frac{15}{2}}} EI \alpha +196830 EI}\\
	{V_T}=\frac{161 {{2}^{\frac{11}{2}}} {{3}^{\frac{3}{2}}} {{R}^{4}} {q_T} \alpha }{3200 EI {{\alpha }^{2}}-135 {{2}^{\frac{13}{2}}} EI \alpha +10935 EI}\\		{V_T}=-\frac{161 {{2}^{\frac{11}{2}}} {{3}^{\frac{3}{2}}} {{R}^{4}} {q_T} \alpha }{3200 EI {{\alpha }^{2}}+135 {{2}^{\frac{13}{2}}} EI \alpha +10935 EI}\\
	{{\Phi }_T}=-		{{\Phi }_T}=-
	\frac{185 {{2}^{\frac{15}{2}}} {{R}^{3}} {q_T} {{\alpha }^{2}}-843264 {{R}^{3}} {q_T} \alpha +885735 \sqrt{2} {{R}^{3}} {q_T}}{38400 EI {{\alpha }^{2}}-405 {{2}^{\frac{17}{2}}} EI \alpha +131220 EI}\\		\frac{185 {{2}^{\frac{15}{2}}} {{R}^{3}} {q_T} {{\alpha }^{2}}+843264 {{R}^{3}} {q_T} \alpha +885735 \sqrt{2} {{R}^{3}} {q_T}}{38400 EI {{\alpha }^{2}}+405 {{2}^{\frac{17}{2}}} EI \alpha +131220 EI}\\
	{{\Psi }_T}=\frac{736 {{3}^{\frac{5}{2}}} {{R}^{3}} {q_T} \alpha }{3200 EI {{\alpha }^{2}}-135 {{2}^{\frac{13}{2}}} EI \alpha +10935 EI}		{{\Psi }_T}=-\frac{736 {{3}^{\frac{5}{2}}} {{R}^{3}} {q_T} \alpha }{3200 EI {{\alpha }^{2}}+135 {{2}^{\frac{13}{2}}} EI \alpha +10935 EI}
	\end{array}		\end{array}
	</math>		</math>

Mechaniker am 15. September 2022 um 09:15 Uhr

2022-09-15T09:15:20Z

← Nächstältere Version		Version vom 15. September 2022, 09:15 Uhr
Zeile 276:		Zeile 276:
	Für die Kraft-Übergangsbedingungen in <i>y</i>- und <i>z</i>-Richtung müssen wir nun allerdings die Seilkräfte mit einbeziehen.		Für die Kraft-Übergangsbedingungen in <i>y</i>- und <i>z</i>-Richtung müssen wir nun allerdings die Seilkräfte mit einbeziehen.
	<ol start="15">		<ol start="15">
	<li><math>Q_{y, I}(h_1) = Q_{y,II}( 0 ) - \sum_{k=1}^3 S_i \cdot e_{i,y}</math></li>		<li><math>Q_{y, I}(h_1) = Q_{y,II}( 0 ) + \sum_{k=1}^3 S_i \cdot e_{i,y}</math></li>
	<li><math>Q_{z, I}(h_1) = Q_{z,II}( 0 ) - \sum_{k=1}^3 S_i \cdot e_{i,z}</math></li>		<li><math>Q_{z, I}(h_1) = Q_{z,II}( 0 ) + \sum_{k=1}^3 S_i \cdot e_{i,z}</math></li>
	</ol>		</ol>
	Dabei nehmen wir vom Einheitsvektor nur die Komponente der jeweiligen Raumrichtung, also z.B. <math>e_{i,y}</math> aus <math>\underline{e}_{i}=\left(\begin{array}{c}e_{i,x}\\e_{i,y}\\e_{i,z}\end{array}\right)</math>.		Dabei nehmen wir vom Einheitsvektor nur die Komponente der jeweiligen Raumrichtung, also z.B. <math>e_{i,y}</math> aus <math>\underline{e}_{i}=\left(\begin{array}{c}e_{i,x}\\e_{i,y}\\e_{i,z}\end{array}\right)</math>.

Mechaniker am 15. September 2022 um 09:12 Uhr

2022-09-15T09:12:51Z

← Nächstältere Version		Version vom 15. September 2022, 09:12 Uhr
Zeile 268:		Zeile 268:
	'''Kraft- und Momenten-Randbedingungen:'''		'''Kraft- und Momenten-Randbedingungen:'''
	Hier wird es spannend: durch die drei Stäbe werden Kräfte eingeleitet,		Hier wird es spannend: durch die drei Stäbe werden Kräfte eingeleitet,
	[[Datei:Kita-13.png\|180px\|right\|mini\|Schnittbild für Knoten ''H'']]		[[Datei:Kita-13.png\|180px\|right\|mini\|Schnittbild für Knoten ''H''<br/>hier ohne Schnittmomente]]
	Einfach sind die Momenten-Gleichgewichtsbedingungen am Knoten ''H'' - hier wird kein externes Moment eingeleitet, die beiden Schnittmomente müssen gleich sein:		Einfach sind die Momenten-Gleichgewichtsbedingungen am Knoten ''H'' - hier wird kein externes Moment eingeleitet, die beiden Schnittmomente müssen gleich sein:
	<ol start="13">		<ol start="13">

Mechaniker am 15. September 2022 um 08:58 Uhr

2022-09-15T08:58:55Z

Änderungen zeigen