Table 1 Parameter design based on different loaded positions (L_i= length of goods in Part 1, \({m_{i}}\) = cargo mass in Part 1)

Parameters for Part 1

\(\left\{\begin{aligned} {L_{1}} &= \left( {{l_{v1}} + \frac{{{l_{r11}}}}{2}} \right) - \left( {{L_{c}} - \frac{{{L_{lc}}}}{2}} \right), \hfill \\ {m_{1}} & = {L_{1}}\frac{{{m_{c}}}}{{{L_{lc}}}}, \hfill \\ \end{aligned} \right.\quad {\text{if}}\left( {{l_{v1}} + \frac{{{l_{r11}}}}{2}} \right) - \left( {{L_{c}} - \frac{{{L_{lc}}}}{2}} \right) \geq {0}.\)

A portion of goods loaded on Part 1

\(\left\{ \begin{aligned} L_{1} & = 0, \\ m_{1} & = 0, \hfill \\ \end{aligned} \right.\quad {\text {if}} \; {\left( {l_{v1} + \frac{{l_{r11} }}{2}} \right) - \left( {L_{c} - \frac{{L_{lc} }}{2}} \right) < 0}. \)

No extra load on Part 1

Parameters for Parts 2 and 3

\(\left\{ \begin{aligned} L_{2}&= L_{lc} - L_{1}, \hfill \\ L_{3} &= 0, \hfill \\ m_{2} &= m_{c} - m_{1}, \hfill \\ m_{3} &= 0, \hfill \\ \end{aligned} \right. \quad {\text {if}} \; {L_{c} - \frac{{L_{lc} }}{2} < \frac{{l_{2} - l_{1} }}{2} + l_{1} \; and \; L_{c} + \frac{{L_{lc} }}{2} \le \frac{{l_{2} - l_{1} }}{2} + l_{1} }. \)

The load is not on Part 3

\(\left\{ \begin{aligned} L_{2}& = \left( {\frac{{l_{2} - l_{1} }}{2} + l_{1} } \right) - \left( {L_{c} - \frac{{L_{lc} }}{2}} \right) - L_{1}, \\ L_{3} &= \left( {L_{c} + \frac{{L_{lc} }}{2}} \right) - \left( {\frac{{l_{2} - l_{1} }}{2} + l_{1} } \right), \\ m_{2} &= \frac{{m_{c} }}{{L_{lc} }}\left[ {\left( {\frac{{l_{2} - l_{1} }}{2} + l_{1} } \right) - \left( {L_{c} - \frac{{L_{lc} }}{2}} \right)} \right] - m_{1}, \\ m_{3} &= \frac{{m_{c} }}{{L_{lc} }}\left[ {\left( {L_{c} + \frac{{L_{lc} }}{2}} \right) - \left( {\frac{{l_{2} - l_{1} }}{2} + l_{1} } \right)} \right], \\ \end{aligned} \right. \begin{array}{l} {\text {if}} \quad L_{c} - \frac{{L_{lc} }}{2} < \frac{{l_{2} - l_{1} }}{2} + l_{1} \\ {\text{and}} \\ {L_{c}} + \frac{{L_{lc} }}{2} > \frac{{l_{2} - l_{1} }}{2} + l_{1}. \\ \end{array}\)

The load on Parts 3 and 2

\(\left\{ \begin{aligned} L_{2}& = 0, \\ L_{3}& = L_{lc}, \\ m_{2}& = 0, \\ m_{3} &= m_{c}, \\ \end{aligned} \right. \quad {\text{if}} \; {L_{c} - \frac{{L_{lc} }}{2} \ge \frac{{l_{2} - l_{1} }}{2} + l_{1} }. \)

The load is completely on Part 3