Table 3 Auxiliary functions used in Proposition 1

Function definition

\(F\left ({\alpha,t} \right) = \frac {1}{{{R_{{\text {net}}}}}}\left ({t + \frac {{{z_{p}}\sin \alpha }}{{{m_{p}}\sin \alpha + \cos \alpha }}} \right){\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }} - \cos \alpha } \right)^{- 1}}\)

\(G\left ({\alpha,v} \right) = \frac {1}{{{R_{{\text {net}}}}}}\left ({v + \frac {{{m_{p}}{z_{p}}\sin \alpha }}{{{m_{p}}\sin \alpha + \cos \alpha }} - {z_{p}}} \right){\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }} - \sin \alpha } \right)^{- 1}}\)

\(\Gamma _{1}^{a}\left (\alpha \right) = \Theta \left ({\alpha \left | {\begin {array}{*{20}{c}} {F\left ({\alpha, - \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha, - \frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }} - \cos \alpha } \right)H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }} - \sin \alpha } \right)\)

\(\Gamma _{1}^{b}\left (\alpha \right) = \Theta \left ({\alpha \left | {\begin {array}{*{20}{c}} {F\left ({\alpha, - \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha, - \frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }} - \cos \alpha } \right)\bar H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }} - \sin \alpha } \right)\)

\(\Gamma _{1}^{c}\left (\alpha \right) = \Theta \left ({\alpha \left | {\begin {array}{*{20}{c}} {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha, - \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha, - \frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)\bar H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }} - \cos \alpha } \right)H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }} - \sin \alpha } \right)\)

\(\Gamma _{1}^{d}\left (\alpha \right) = \Theta \left ({\alpha \left | {\begin {array}{*{20}{c}} {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha, - \frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha, - \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)\bar H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }} - \cos \alpha } \right)\bar H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }} - \sin \alpha } \right)\)

\(\Gamma _{2}^{a}\left (\alpha \right) = \Theta \left ({\alpha \left | {\begin {array}{*{20}{c}} {F\left ({\alpha, - \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha, - \frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }} - \cos \alpha } \right)H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }} - \sin \alpha } \right)\)

\(\Gamma _{2}^{b}\left (\alpha \right) = \Theta \left ({\alpha \left | {\begin {array}{*{20}{c}} {F\left ({\alpha, - \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha, - \frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }} - \cos \alpha } \right)\bar H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }} - \sin \alpha } \right)\)

\(\Gamma _{2}^{c}\left (\alpha \right) = \Theta \left ({\alpha \left | {\begin {array}{*{20}{c}} {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha, - \frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha, - \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)\bar H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }} - \cos \alpha } \right)H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }} - \sin \alpha } \right)\)

\(\Gamma _{2}^{d}\left (\alpha \right) = \Theta \left ({\alpha \left | {\begin {array}{*{20}{c}} {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha, - \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha, - \frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)\bar H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }} - \cos \alpha } \right)\bar H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }} - \sin \alpha } \right)\)

\(\Gamma _{3}^{a}\left (\alpha \right) = \Theta \left ({\alpha \left | {\begin {array}{*{20}{c}} {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha, - \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha, - \frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }} - \cos \alpha } \right)H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }} - \sin \alpha } \right)\)

\(\Gamma _{3}^{b}\left (\alpha \right) = \Theta \left ({\alpha \left | {\begin {array}{*{20}{c}} {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha, - \frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha, - \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }} - \cos \alpha } \right)\bar H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }} - \sin \alpha } \right)\)

\(\Gamma _{3}^{c}\left (\alpha \right) = \Theta \left ({\alpha \left | {\begin {array}{*{20}{c}} {F\left ({\alpha, - \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha, - \frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)\bar H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }} - \cos \alpha } \right)H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }} - \sin \alpha } \right)\)

\(\Gamma _{3}^{d}\left (\alpha \right) = \Theta \left ({\alpha \left | {\begin {array}{*{20}{c}} {F\left ({\alpha, - \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha, - \frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)\bar H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }} - \cos \alpha } \right)\bar H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }} - \sin \alpha } \right)\)

\(\Gamma _{4}^{a}\left (\alpha \right) = \Theta \left ({\alpha \left | {\begin {array}{*{20}{c}} {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha, - \frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha, - \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }} - \cos \alpha } \right)H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }} - \sin \alpha } \right)\)

\(\Gamma _{4}^{b}\left (\alpha \right) = \Theta \left ({\alpha \left | {\begin {array}{*{20}{c}} {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha, - \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha, - \frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }} - \cos \alpha } \right)\bar H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }} - \sin \alpha } \right)\)

\(\Gamma _{4}^{c}\left (\alpha \right) = \Theta \left ({\alpha \left | {\begin {array}{*{20}{c}} {F\left ({\alpha, - \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha, - \frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)\bar H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }} - \cos \alpha } \right)H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }} - \sin \alpha } \right)\)

\(\Gamma _{4}^{d}\left (\alpha \right) = \Theta \left ({\alpha \left | {\begin {array}{*{20}{c}} {F\left ({\alpha, - \frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha,\frac {L}{2}\cos \alpha } \right)} \\ {F\left ({\alpha,\frac {L}{2}\sin \alpha } \right)}&{G\left ({\alpha, - \frac {L}{2}\cos \alpha } \right)} \end {array}} \right.} \right)\bar H\left ({\frac {1}{{{m_{p}}\sin \alpha + \cos \alpha }} - \cos \alpha } \right)\bar H\left ({\frac {{{m_{p}}}}{{{m_{p}}\sin \alpha + \cos \alpha }} - \sin \alpha } \right)\)