lehrkraefte:blc:math:vektoranalysis:jahrmarkt

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lehrkraefte:blc:math:vektoranalysis:jahrmarkt [2018/09/28 09:47]
Ivo Blöchliger [Insider] 		lehrkraefte:blc:math:vektoranalysis:jahrmarkt [2018/09/28 14:17] (current)
Ivo Blöchliger [Berg und Tal Bahn]
Line 128: Line 128:
 #end // while #end // while
 </code> </code>
 +Maxima Code (für die Ableitungen)
 +<code>
 +
 +/* K0 */
 +o0:[0,0,0];
 +x0:[1,0,0];
 +y0:[0,1,0];
 +z0:[0,0,1];
 +
 +alpha1:15/180*%pi;
 +h1:2;
 +
 +/* K1 schiefe Ebene */
 +o1:o0+h1*z0;
 +x1:-sin(alpha1)*z0+cos(alpha1)*x0;
 +y1:y0;
 +z1:cos(alpha1)*z0+sin(alpha1)*x0;
 +
 +/* Parameter K2 */
 +omega2:12/60*2*%pi;
 +alpha2:omega2*t;
 +
 +/* K2, drehende Ebene */
 +o2:o1;
 +x2:cos(alpha2)*x1+sin(alpha2)*y1;
 +y2:-sin(alpha2)*x1+cos(alpha2)*y1;
 +z2:z1;
 +
 +/* Parameter K3 */
 +r3:5;
 +alpha3:5/360*2*%pi;
 +f3:-alpha3*cos(3*alpha2);
 +
 +o3:o2;
 +x3:cos(f3)*x2+sin(f3)*z2;
 +y3:y2;
 +z3:-sin(f3)*x2+cos(f3)*z2;
 +
 +/* Passagier */
 +
 +define(p(t), o3+r3*x3);
 +define(v(t), diff(p(t),t));
 +define(a(t), diff(v(t),t)-9.81*z0);
 +
 +/* Betraege der Geschwindigkeit und Beschleunigung */ 
 +define(aa(t), sqrt(a(t).a(t)));
 +define(vv(t), sqrt(v(t).v(t)));
 +
 +plot2d([vv(t),aa(t), a(t).[0,0,1]],  [t, 0,10],
 +[title, "Geschwindigkeit und Beschleunigung"],
 +[legend, "v(t)", "a(t)", "az(t)"]);
 +</code>
 +
 +Das Bild dazu:
 +{{ :lehrkraefte:blc:math:vektoranalysis:bergundtal-v-a.svg |}}
 +$$\left[ 5\,\left(\cos \left({{\pi}\over{12}}\right)\,\cos \left({{2
 + \,\pi\,t}\over{5}}\right)\,\cos \left({{\pi\,\cos \left({{6\,\pi\,t
 + }\over{5}}\right)}\over{36}}\right)-\sin \left({{\pi}\over{12}}
 + \right)\,\sin \left({{\pi\,\cos \left({{6\,\pi\,t}\over{5}}\right)
 + }\over{36}}\right)\right) , 5\,\sin \left({{2\,\pi\,t}\over{5}}
 + \right)\,\cos \left({{\pi\,\cos \left({{6\,\pi\,t}\over{5}}\right)
 + }\over{36}}\right) , 5\,\left(-\cos \left({{\pi}\over{12}}\right)\,
 + \sin \left({{\pi\,\cos \left({{6\,\pi\,t}\over{5}}\right)}\over{36}}
 + \right)-\sin \left({{\pi}\over{12}}\right)\,\cos \left({{2\,\pi\,t
 + }\over{5}}\right)\,\cos \left({{\pi\,\cos \left({{6\,\pi\,t}\over{5
 + }}\right)}\over{36}}\right)\right)+2 \right] $$
 +Beschleunigung:
 +$$\left[ 5\,\left({{\pi^4\,\sin \left({{\pi}\over{12}}\right)\,\sin ^
 + 2\left({{6\,\pi\,t}\over{5}}\right)\,\sin \left({{\pi\,\cos \left({{
 + 6\,\pi\,t}\over{5}}\right)}\over{36}}\right)}\over{900}}-{{2\,\pi^3
 + \,\cos \left({{\pi}\over{12}}\right)\,\sin \left({{2\,\pi\,t}\over{5
 + }}\right)\,\sin \left({{6\,\pi\,t}\over{5}}\right)\,\sin \left({{\pi
 + \,\cos \left({{6\,\pi\,t}\over{5}}\right)}\over{36}}\right)}\over{75
 + }}+{{\pi^3\,\cos \left({{\pi}\over{12}}\right)\,\cos \left({{2\,\pi
 + \,t}\over{5}}\right)\,\cos \left({{6\,\pi\,t}\over{5}}\right)\,\sin 
 + \left({{\pi\,\cos \left({{6\,\pi\,t}\over{5}}\right)}\over{36}}
 + \right)}\over{25}}-{{\pi^4\,\cos \left({{\pi}\over{12}}\right)\,
 + \cos \left({{2\,\pi\,t}\over{5}}\right)\,\sin ^2\left({{6\,\pi\,t
 + }\over{5}}\right)\,\cos \left({{\pi\,\cos \left({{6\,\pi\,t}\over{5
 + }}\right)}\over{36}}\right)}\over{900}}+{{\pi^3\,\sin \left({{\pi
 + }\over{12}}\right)\,\cos \left({{6\,\pi\,t}\over{5}}\right)\,\cos 
 + \left({{\pi\,\cos \left({{6\,\pi\,t}\over{5}}\right)}\over{36}}
 + \right)}\over{25}}-{{4\,\pi^2\,\cos \left({{\pi}\over{12}}\right)\,
 + \cos \left({{2\,\pi\,t}\over{5}}\right)\,\cos \left({{\pi\,\cos 
 + \left({{6\,\pi\,t}\over{5}}\right)}\over{36}}\right)}\over{25}}
 + \right) , {{2\,\pi^3\,\cos \left({{2\,\pi\,t}\over{5}}\right)\,\sin 
 + \left({{6\,\pi\,t}\over{5}}\right)\,\sin \left({{\pi\,\cos \left({{6
 + \,\pi\,t}\over{5}}\right)}\over{36}}\right)}\over{15}}+{{\pi^3\,
 + \sin \left({{2\,\pi\,t}\over{5}}\right)\,\cos \left({{6\,\pi\,t
 + }\over{5}}\right)\,\sin \left({{\pi\,\cos \left({{6\,\pi\,t}\over{5
 + }}\right)}\over{36}}\right)}\over{5}}-{{\pi^4\,\sin \left({{2\,\pi\,
 + t}\over{5}}\right)\,\sin ^2\left({{6\,\pi\,t}\over{5}}\right)\,\cos 
 + \left({{\pi\,\cos \left({{6\,\pi\,t}\over{5}}\right)}\over{36}}
 + \right)}\over{180}}-{{4\,\pi^2\,\sin \left({{2\,\pi\,t}\over{5}}
 + \right)\,\cos \left({{\pi\,\cos \left({{6\,\pi\,t}\over{5}}\right)
 + }\over{36}}\right)}\over{5}} , 5\,\left({{\pi^4\,\cos \left({{\pi
 + }\over{12}}\right)\,\sin ^2\left({{6\,\pi\,t}\over{5}}\right)\,\sin 
 + \left({{\pi\,\cos \left({{6\,\pi\,t}\over{5}}\right)}\over{36}}
 + \right)}\over{900}}+{{2\,\pi^3\,\sin \left({{\pi}\over{12}}\right)\,
 + \sin \left({{2\,\pi\,t}\over{5}}\right)\,\sin \left({{6\,\pi\,t
 + }\over{5}}\right)\,\sin \left({{\pi\,\cos \left({{6\,\pi\,t}\over{5
 + }}\right)}\over{36}}\right)}\over{75}}-{{\pi^3\,\sin \left({{\pi
 + }\over{12}}\right)\,\cos \left({{2\,\pi\,t}\over{5}}\right)\,\cos 
 + \left({{6\,\pi\,t}\over{5}}\right)\,\sin \left({{\pi\,\cos \left({{6
 + \,\pi\,t}\over{5}}\right)}\over{36}}\right)}\over{25}}+{{\pi^4\,
 + \sin \left({{\pi}\over{12}}\right)\,\cos \left({{2\,\pi\,t}\over{5}}
 + \right)\,\sin ^2\left({{6\,\pi\,t}\over{5}}\right)\,\cos \left({{\pi
 + \,\cos \left({{6\,\pi\,t}\over{5}}\right)}\over{36}}\right)}\over{
 + 900}}+{{\pi^3\,\cos \left({{\pi}\over{12}}\right)\,\cos \left({{6\,
 + \pi\,t}\over{5}}\right)\,\cos \left({{\pi\,\cos \left({{6\,\pi\,t
 + }\over{5}}\right)}\over{36}}\right)}\over{25}}+{{4\,\pi^2\,\sin 
 + \left({{\pi}\over{12}}\right)\,\cos \left({{2\,\pi\,t}\over{5}}
 + \right)\,\cos \left({{\pi\,\cos \left({{6\,\pi\,t}\over{5}}\right)
 + }\over{36}}\right)}\over{25}}\right)-9.81 \right] $$
 +
 </hidden> </hidden>
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Line 314: Line 429:
  \pi}\over{36}}\right)\,\cos \left({{9\,\pi\,t}\over{20}}\right)+1.4  \pi}\over{36}}\right)\,\cos \left({{9\,\pi\,t}\over{20}}\right)+1.4
   \end{pmatrix} $$   \end{pmatrix} $$
-   +
-  $$\vec v(t) = \begin{pmatrix} 2\,\left({{7\,\pi\,\cos \left({{\pi}\over{36}}\right)\,\sin  +
- \left({{9\,\pi\,t}\over{20}}\right)\,\cos \left({{11\,\pi\,t}\over{ +
- 12}}\right)}\over{15}}-{{7\,\pi\,\cos \left({{\pi}\over{36}}\right) +
- \,\cos \left({{9\,\pi\,t}\over{20}}\right)\,\sin \left({{11\,\pi\,t +
- }\over{12}}\right)}\over{15}}\right)-2.025\,\pi\,\cos \left({{\pi +
- }\over{36}}\right)\,\sin \left({{9\,\pi\,t}\over{20}}\right) \\ 2\, +
- \left(-{{7\,\pi\,\sin \left({{9\,\pi\,t}\over{20}}\right)\,\sin  +
- \left({{11\,\pi\,t}\over{12}}\right)}\over{15}}-{{7\,\pi\,\cos  +
- \left({{9\,\pi\,t}\over{20}}\right)\,\cos \left({{11\,\pi\,t}\over{ +
- 12}}\right)}\over{15}}\right)+2.025\,\pi\,\cos \left({{9\,\pi\,t +
- }\over{20}}\right) \\ 2\,\left({{7\,\pi\,\sin \left({{\pi}\over{36}} +
- \right)\,\cos \left({{9\,\pi\,t}\over{20}}\right)\,\sin \left({{11\, +
- \pi\,t}\over{12}}\right)}\over{15}}-{{7\,\pi\,\sin \left({{\pi +
- }\over{36}}\right)\,\sin \left({{9\,\pi\,t}\over{20}}\right)\,\cos  +
- \left({{11\,\pi\,t}\over{12}}\right)}\over{15}}\right)+2.025\,\pi\, +
- \sin \left({{\pi}\over{36}}\right)\,\sin \left({{9\,\pi\,t}\over{20 +
- }}\right) \end{pmatrix} $$ +
-und +
-$$\vec a(t) = \begin{pmatrix} 2\,\left(-{{49\,\pi^2\,\cos \left({{\pi}\over{36}}\right)\, +
- \sin \left({{9\,\pi\,t}\over{20}}\right)\,\sin \left({{11\,\pi\,t +
- }\over{12}}\right)}\over{225}}-{{49\,\pi^2\,\cos \left({{\pi}\over{ +
- 36}}\right)\,\cos \left({{9\,\pi\,t}\over{20}}\right)\,\cos \left({{ +
- 11\,\pi\,t}\over{12}}\right)}\over{225}}\right)-0.91125\,\pi^2\, +
- \cos \left({{\pi}\over{36}}\right)\,\cos \left({{9\,\pi\,t}\over{20 +
- }}\right) \\ 2\,\left({{49\,\pi^2\,\cos \left({{9\,\pi\,t}\over{20}} +
- \right)\,\sin \left({{11\,\pi\,t}\over{12}}\right)}\over{225}}-{{49 +
- \,\pi^2\,\sin \left({{9\,\pi\,t}\over{20}}\right)\,\cos \left({{11\, +
- \pi\,t}\over{12}}\right)}\over{225}}\right)-0.91125\,\pi^2\,\sin  +
- \left({{9\,\pi\,t}\over{20}}\right) \\ 2\,\left({{49\,\pi^2\,\sin  +
- \left({{\pi}\over{36}}\right)\,\sin \left({{9\,\pi\,t}\over{20}} +
- \right)\,\sin \left({{11\,\pi\,t}\over{12}}\right)}\over{225}}+{{49 +
- \,\pi^2\,\sin \left({{\pi}\over{36}}\right)\,\cos \left({{9\,\pi\,t +
- }\over{20}}\right)\,\cos \left({{11\,\pi\,t}\over{12}}\right)}\over{ +
- 225}}\right)+0.91125\,\pi^2\,\sin \left({{\pi}\over{36}}\right)\, +
- \cos \left({{9\,\pi\,t}\over{20}}\right)-9.81 \end{pmatrix} $$+
 </hidden> </hidden>
 ===== Weitere Bahnen ===== ===== Weitere Bahnen =====
  • lehrkraefte/blc/math/vektoranalysis/jahrmarkt.1538120875.txt.gz
  • Last modified: 2018/09/28 09:47
  • by Ivo Blöchliger