Filmrechner
formeln
\[ \begin{align} b &= \text{Bobby (50) } \\ n &= \text{Wicklungen (910 )} \\ d_f &= \text{Dicke Filme(0.175)} \\ d_a &= \text{gesamter Durchmesser} \\ l &= \text{Länge Filme} \end{align}\]
\[ \begin{align} l & = \sum_{k=1}^n (b+2d_f k)\pi \\ & = \pi \sum_{k=1}^n b+2d_f k\\ & = \pi ( \sum_{k=1}^n b + \sum_{k=1}^n 2d_f k) \\ & = \pi ( nb +\sum_{k=1}^n 2d_f k ) \\ & = \pi (nb + 2d_f \sum_{k=1}^n k ) \\ & = \pi (nb + 2d_f \frac{n(n+1)}{2} ) \\ & = \pi (nb + \frac{2d_f n(n+1)}{2} ) \\ & = \pi (nb + d_f(n^2+n)) \\ & = \pi n(b + d_f(n+1)) \\ \end{align}\]
30mm Durchmesser
\[ \begin{align} n&=\frac{(300-b)}{2 d_f} \\ &=\frac{(300-50)}{2\cdot0.175}\\ &=714.285714286 \end{align}\]
\[ \begin{align} d_a&=3.141\cdot(50 n+0.175\cdot(n^2+n)) \\ &=157.05n+.549675(n^2+n)\\ &=157.05n+.549675n^2+.54967*n\\\\ &=157.500325n+.549675n^2\\ l&=392.317 \end{align}\]
Pro Schicht:
\[ \begin{align} l_n&=(b+2nd_f)\pi \\ l &= \int_1^n (b+2 n d_f) \pi \\ &= \pi d_f n^2+\pi b n \\ \end{align}\]
mit b= 50 and d=0.175
\[ \begin{align} &= 0.549n^2 + 157n \\ \end{align}\]
600m film = n windungen
\[ \begin{align} l &= 600 00mm\\ & = 0.549*n^2 +157n \\ n&=911 \end{align}\]