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lehrkraefte:blc:math:vektoranalysis:jahrmarkt [2018/09/28 09:52] Ivo Blöchliger [Insider] |
lehrkraefte:blc:math:vektoranalysis:jahrmarkt [2018/09/28 14:17] (current) Ivo Blöchliger [Berg und Tal Bahn] |
#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> |
===== VIP VIP ===== | ===== VIP VIP ===== |
\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} $$ | \end{pmatrix} $$ |
| |
$$\left[ 2\,\left({{7\,\pi\,\cos \left({{\pi}\over{36}}\right)\,\sin \left({{9\,\pi\,t}\over{20}}\right)\,\cos \left({{11\,\pi\,t}\over{ 12}}\right)}\over{15}}-{{7\,\pi\,\cos \left({{\pi}\over{36}}\right) \,\cos left({{9\,\pi\,t}\over{20}}\right)\,\sin \left({{11\,\pi\,t }\over{12}}\right)}\over{15}}\right)-2.025\,\pi\,\cos \left({{\pi \over{36}}\right)\,\sin \left({{9\,\pi\,t}\over{20}}\right) , 2\, \left(-{{7\,\pi\,\sin left({{9\,\pi\,t}\over{20}}\right)\,\sin \left({{11\,\pi\,t}\over{12}}\right)}\over{15}}-{{7\,\pi\,\cos left({{9\,\pi\,t}\over{20}}\right)\,\cos \left({{11\,\pi\,t}\over{ 12}}\right)}\over{15}}\right)+2.025\,\pi\,\cos left({{9\,\pi\,t }\over{20}}\right) , 2\,\left({{7\,\pi\,\sin \left({{\pi}\over{36}} \right)\,\cos left({{9\,\pi\,t}\over{20}}\right)\,\sin \left({{11\, \pi\,t}\over{12}}\right)}\over{15}}-{{7\,\pi\,\sin \left({{\pi \over{36}}\right)\,\sin \left({{9\,\pi\,t}\over{20}}\right)\,\cos left({{11\,\pi\,t}\over{12}}\right)}\over{15}}\right)+2.025\,\pi\, \sin \left({{\pi}\over{36}}\right)\,\sin left({{9\,\pi\,t}\over{20 }}\right) \right] $$ | |
| |
</hidden> | </hidden> |