The Study for the Influence of Nonlinear Foundation on Responses of a Beam to a Moving Load Based on Volterra Integral Equations

Abstract

Purpose

A new expression of closed-form solution is given to visually demonstrate the influence of the nonlinear foundation on the responses of a beam subjected to a moving load.

Methods

Greatly different from previous works, the nonlinear partial differential governing equation of the beam motion is discretized into sets of nonlinear Volterra integral equations by using the Laplace transform. Then the Adomian decomposition method combined with a simple iterative formula are employed to analytically solve the Volterra integral equations.

Results

The closed-form solution for responses of the beam can be expressed in the form of infinite series with mode shapes. A good agreement between analytical and numerical results of the responses of the beam for different parameters is observed. It is clear to visually demonstrate the influence of the nonlinear foundation in the closed-form solution.

Conclusion

Nonlinearity of foundation makes the modal coefficients in the closed-form solution, expressed in the form of infinite series with mode shapes, become more complex. When the damping coefficient of the foundation is very small, the beam has more possibility of resonance for the case of nonlinear foundation.

Appendices

Appendix A

$$\begin{aligned}&E_{11}=\frac{3}{8}\frac{\alpha _{21} r_{1}\left[ i\left( 3 q_{31} q_{41}^{2}+4 q_{41} q_{32} q_{42}+4 q_{41} q_{33} q_{43}+q_{42}^{2} q_{33}\right) +q_{31}^{2} q_{43}\right] }{(\alpha _{21}+i\alpha _{1}-\beta )(\alpha _{21}-i\alpha _1+\beta )}-\frac{iq_{31}}{2},\nonumber \\&E_{12}=-\frac{3}{8}\frac{\alpha _{21} r_{1}\left( 4 q_{31} q_{32} q_{42}+4 q_{31} q_{33} q_{43}+3 q_{31}^{2} q_{41}+q_{32}^{2} q_{43}+i q_{41}^{2} q_{33}\right) }{(\alpha _{21}+i\alpha _{1}+\beta )(\alpha _{21}-i\alpha _1-\beta )}+\frac{q_{41}}{2},\nonumber \\&E_{13}=\frac{3}{8}\frac{\alpha _{21} r_{1}\left[ -2i\left( q_{31} q_{41} q_{43}-q_{31} q_{42}^{2}-q_{32} q_{42} q_{43}\right) +q_{41}^{3}\right] }{(\alpha _{21}+i\alpha _{1}-3\beta )(\alpha _{21}-i\alpha _1+3\beta )},\nonumber \\&E_{14}=\frac{3}{8}\frac{\alpha _{21} r_{1}\left[ 2(q_{31} q_{41} q_{33}- q_{32} q_{42} q_{33}- q_{41} q_{32}^{2})-i q_{31}^{3}\right] }{(\alpha _{21}+i\alpha _{1}+3\beta )(\alpha _{21}-i\alpha _1-3\beta )},\nonumber \\&E_{15}=\frac{3}{8}\frac{\alpha _{21} r_{1}(2iq_{31} q_{43}^{2}-q_{41}^{2} q_{43}+2 q_{41} q_{42}^{2})}{(\alpha _{21}+i\alpha _{1}-5\beta )(\alpha _{21}-i\alpha _1+5\beta )},\nonumber \\&E_{16}=\frac{3}{8}\frac{\alpha _{21} r_{1}(i q_{31}^{2} q_{33}-2i q_{31} q_{32}^{2}-2 q_{41} q_{33}^{2})}{(\alpha _{21}+i\alpha _{1}+5\beta )(\alpha _{21}-i\alpha _1-5\beta )},\nonumber \\&E_{17}=\frac{3}{8}\frac{\alpha _{21} r_{1}(2 q_{41} q_{43}+q_{42}^{2})q_{43}}{(\alpha _{21}+i\alpha _{1}-7\beta )(\alpha _{21}-i\alpha _1+7\beta )},\nonumber \\&E_{18}=-\frac{3i}{8}\frac{\alpha _{21} r_{1}\left( 2 q_{31} q_{33}+q_{32}^{2}\right) q_{33}}{(\alpha _{21}+i\alpha _{1}+7\beta )(\alpha _{21}-i\alpha _1-7\beta )}. \end{aligned}$$

(A.1)

$$\begin{aligned}&E_{20}=\frac{3}{4}\frac{\alpha _{22} r_{2}\left[ -q_{31}^{2} q_{42}-q_{31} q_{32} q_{43}+i\left( q_{41}^{2} q_{32}+q_{41} q_{42} q_{33}\right) \right] }{\alpha _1^2+\alpha _{22}^2},\nonumber \\&E_{21}=\frac{3i}{8}\frac{\alpha _{22}r_{2}\left( 4 q_{31} q_{41} q_{42}+2 q_{41} q_{32} q_{43}+3 q_{32} q_{42}^{2}+4 q_{42} q_{33} q_{43}\right) }{(\alpha _{22}+i\alpha _{1}-2\beta )(\alpha _{22}-i\alpha _1+2\beta )}-\frac{iq_{32}}{2},\nonumber \\&E_{22}=-\frac{3}{8}\frac{\alpha _{22}r_{2}\left( 4 q_{31} q_{41} q_{32}+2 q_{31} q_{42} q_{33}+3 q_{32}^{2} q_{42}+4 q_{32} q_{33} q_{43}\right) }{(\alpha _{22}+i\alpha _{1}+2\beta )(\alpha _{22}-i\alpha _1-2\beta )}+\frac{q_{42}}{2},\nonumber \\&E_{23}=\frac{3}{4}\frac{\alpha _{22}r_{2}\left[ q_{41}^{2} q_{42}+i\left( q_{31} q_{42} q_{43}+q_{32} q_{43}^{2}\right) \right] }{(\alpha _{22}+i\alpha _{1}-4\beta )(\alpha _{22}-i\alpha _1+4\beta )},\nonumber \\&E_{24}=-\frac{3}{4}\frac{\alpha _{22}r_{2}(iq_{31}^{2} q_{32}+q_{41} q_{32} q_{33}+2 q_{42} q_{33}^{2})}{(\alpha _{22}+i\alpha _{1}+4\beta )(\alpha _{22}-i\alpha _1-4\beta )},\nonumber \\&E_{25}=\frac{3}{8}\frac{\alpha _{22}r_{2}(2 q_{41} q_{43}+q_{42}^{2})q_{42}}{(\alpha _{22}+i\alpha _{1}-6\beta )(\alpha _{22}-i\alpha _1+6\beta )},\nonumber \\&E_{26}=-\frac{3i}{8}\frac{\alpha _{22}r_{2}\left( 2 q_{31} q_{33}+q_{32}^{2}\right) q_{32} }{(\alpha _{22}+i\alpha _{1}+6\beta )(\alpha _{22}-i\alpha _1-6\beta )},\nonumber \\&E_{31}=\frac{3}{8}\frac{\alpha _{23}r_{3}\left[ -2 q_{31}^{2} q_{43}+i\left( -q_{31} q_{41}^{2}+2 q_{41} q_{32} q_{42}+2 q_{42}^{2} q_{33}\right) \right] }{(\alpha _{23}+i\alpha _{1}-\beta )(\alpha _{23}-i\alpha _1+\beta )},\nonumber \\&E_{32}=\frac{3}{8}\frac{\alpha _{23}r_{3}\left( -2 q_{31} q_{32} q_{42}+q_{31}^{2} q_{41}-2 q_{32}^{2} q_{43}+2i q_{41}^{2} q_{33}\right) }{(\alpha _{23}+i\alpha _{1}+\beta )(\alpha _{23}-i\alpha _1-\beta )}. \end{aligned}$$

(A.2)

$$\begin{aligned}&E_{33}=\frac{1}{8}\frac{\alpha _{23}r_{3}\left[ -q_{41}^{3}+3i\left( 4 q_{31} q_{41} q_{43}+ q_{31} q_{42}^{2}+4 q_{32} q_{42} q_{43}+3 q_{33} q_{43}^{2}\right) \right] }{(\alpha _{23}+i\alpha _{1}-3\beta )(\alpha _{23}-i\alpha _1+3\beta )}-\frac{iq_{33}}{2},\nonumber \\&E_{34}=\frac{1}{8}\frac{\alpha _{23}r_{3}\left[ -3(4 q_{31} q_{41} q_{33}+4 q_{32} q_{42} q_{33}+q_{41} q_{32}^{2}+3 q_{33}^{2} q_{43})+i q_{31}^{3}\right] }{(\alpha _{23}+i\alpha _{1}+3\beta )(\alpha _{23}-i\alpha _1-3\beta )} +\frac{q_{43}}{2},\nonumber \\&E_{35}=\frac{3}{8}\frac{\alpha _{23}r_{3}(2 q_{41}^{2} q_{43}+q_{41} q_{42}^{2})}{(\alpha _{23}+i\alpha _{1}-5\beta )(\alpha _{23}-i\alpha _1+5\beta )},\nonumber \\&E_{36}=-\frac{3i}{8}\frac{\alpha _{23}r_{3}\left( 2 q_{31}^{2} q_{33}+q_{31} q_{32}^{2}\right) }{(\alpha _{23}+i\alpha _{1}+5\beta )(\alpha _{23}-i\alpha _1-5\beta )},\nonumber \\&E_{37}=\frac{3}{4}\frac{\alpha _{23}r_{3}q_{42}^{2} q_{43}}{(\alpha _{23}+i\alpha _{1}-7\beta )(\alpha _{23}-i\alpha _1+7\beta )},\nonumber \\&E_{38}=-\frac{3i}{4}\frac{\alpha _{23}r_{3}q_{32}^{2} q_{33}}{(\alpha _{23}+i\alpha _{1}+7\beta )(\alpha _{23}-i\alpha _1-7\beta )}. \end{aligned}$$

(A.3)

$$\begin{aligned}&Q_{1j}=\left\{ \begin{array}{l}{\frac{\alpha _{21} r_{1}R_{1j}}{[\alpha _{21}+i\alpha _{1}-j\beta ][\alpha _{21}-i\alpha _1+j\beta ]}},\ j=1,3,5,7, \\ {\frac{\alpha _{21} r_{1}R_{1j}}{[\alpha _{21}+i\alpha _{1}+(j-1)\beta ][\alpha _{21}-i\alpha _1-(j-1)\beta ]}},\ j=2,4,6,8. \\ \end{array}\right. \end{aligned}$$

(A.4)

$$\begin{aligned}&Q_{2j}=\left\{ \begin{array}{l}{\frac{\alpha _{22}r_2R_{20}}{\alpha _{22}^2+\alpha _1^2}},\ j=0, \\ {\frac{\alpha _{22} r_{2}R_{2j}}{[\alpha _{22}+i\alpha _{1}-(j+1)\beta ][\alpha _{22}-i\alpha _1+(j+1)\beta ]}},\ j=1,3,5, \\ {\frac{\alpha _{22} r_{2}R_{2j}}{[\alpha _{22}+i\alpha _{1}+j\beta ][\alpha _{22}-i\alpha _1-j\beta ]}},\ j=2,4,6. \\ \end{array}\right. \end{aligned}$$

(A.5)

$$\begin{aligned}&Q_{3j}=\left\{ \begin{array}{l}{\frac{\alpha _{23} r_{3}R_{3j}}{[\alpha _{23}+i\alpha _{1}-j\beta ][\alpha _{23}-i\alpha _1+j\beta ]}},\ j=1,3,5,7, \\ {\frac{\alpha _{23} r_{3}R_{3j}}{[\alpha _{23}+i\alpha _{1}+(j-1)\beta ][\alpha _{23}-i\alpha _1-(j-1)\beta ]}},\ j=2,4,6,8,\\ \end{array}\right. \end{aligned}$$

(A.6)

where

$$\begin{aligned}&R_{11}=\frac{3}{4}(3 E_{12} q_{41}^{2}-3 E_{13} q_{31}^{2}-2 E_{14} q_{41} q_{43}+2 E_{14} q_{42}^{2}+2 E_{15} q_{31} q_{33}-2 E_{15} q_{32}^{2},\\&\qquad +2 E_{16} q_{43}^{2}+4 E_{22} q_{41} q_{42}-4 E_{23} q_{31} q_{32}+2 E_{24} q_{42} q_{43}-2 E_{25} q_{32} q_{33}-E_{32} q_{41}^{2},\\&\qquad +E_{33} q_{31}^{2}+4 q_{41} q_{43} E_{34}+E_{34} q_{42}^{2}-4 q_{31} q_{33} E_{35}-E_{35} q_{32}^{2})+\frac{3i}{2}(3 q_{31} q_{41} E_{11},\\&\qquad +2 q_{32} q_{42} E_{11}+2 E_{11} q_{33} q_{43}-q_{31} q_{43} E_{12}-q_{41} q_{33} E_{13}+2 q_{31} q_{42} E_{20}+q_{32} q_{43} E_{20},\\&\qquad +2 q_{41} q_{32} E_{21}+q_{42} q_{33} E_{21}-q_{31} q_{41} E_{31}+q_{32} q_{42} E_{31}+2 q_{31} q_{43} E_{32}+2 q_{41} q_{33} E_{33}),\\&R_{12}=\frac{3}{4}(-3 E_{11} q_{31}^{2}+2 E_{13} q_{31} q_{33}-2 E_{13} q_{32}^{2}+3 E_{14} q_{41}^{2}-2 E_{15} q_{33}^{2}-2 E_{16} q_{41} q_{43},\\&\qquad +2 E_{16} q_{42}^{2}-4 E_{21} q_{31} q_{32}-2 E_{23} q_{32} q_{33}+4 E_{24} q_{41} q_{42}+2 E_{26} q_{42} q_{43}+E_{31} q_{31}^{2},\\&\qquad -4 q_{31} q_{33} E_{33}-E_{33} q_{32}^{2}-E_{34} q_{41}^{2}+4 q_{41} q_{43} E_{36}+E_{36} q_{42}^{2})+\frac{3i}{2}(-q_{41} q_{33} E_{11},\\&\qquad +3 q_{31} q_{41} E_{12}+2 q_{32} q_{42} E_{12}+2 E_{12} q_{33} q_{43}-q_{31} q_{43} E_{14}+2 q_{41} q_{32} E_{20}+q_{42} q_{33} E_{20},\\&\qquad +2 q_{31} q_{42} E_{22}+q_{32} q_{43} E_{22}+2 q_{41} q_{33} E_{31}-q_{31} q_{41} E_{32}+q_{32} q_{42} E_{32}+2 q_{31} q_{43} E_{34}),\\&R_{13}=\frac{3}{4}(3 E_{11} q_{41}^{2}-2 E_{12} q_{41} q_{43}+2 E_{12} q_{42}^{2}+2 E_{14} q_{43}^{2}-3 E_{15} q_{31}^{2}+4 E_{20} q_{41} q_{42},\\&\qquad +2 E_{22} q_{42} q_{43}-4 E_{25} q_{31} q_{32}-E_{31} q_{41}^{2}+4 q_{41} q_{43} E_{32}+E_{32} q_{42}^{2}+E_{35} q_{31}^{2}),\\&\qquad +\frac{3i}{2}(-q_{31} q_{43} E_{11}+3 q_{31} q_{41} E_{13}+2 q_{32} q_{42} E_{13}+2 E_{13} q_{33} q_{43}-q_{41} q_{33} E_{15},\\&\qquad +2 q_{31} q_{42} E_{21}+q_{32} E_{21} q_{43}+2 q_{41} E_{23} q_{32}+q_{42} E_{23} q_{33}+2 q_{31} q_{43} E_{31}-q_{31} q_{41} E_{33},\\&\qquad +q_{32} q_{42} E_{33}+2 q_{41} q_{33} E_{35}),\\&R_{14}=\frac{3}{4}(2 E_{11} q_{31} q_{33}-2 E_{11} q_{32}^{2}-3 E_{12} q_{31}^{2}-2 q_{33}^{2} E_{13}+3 E_{16} q_{41}^{2}-4 E_{20} q_{31} q_{32},\\&\qquad -2 E_{21} q_{32} q_{33}+4 E_{26} q_{41} q_{42}-4 q_{31} q_{33} E_{31}-E_{31} q_{32}^{2}+E_{32} q_{31}^{2}-E_{36} q_{41}^{2}),\\&\qquad +\frac{3i}{2}(-q_{41} q_{33} E_{12}+3 q_{31} q_{14} E_{14}+2 q_{32} q_{42} E_{14}+2 q_{33} q_{43} E_{14}-q_{31} q_{43} E_{16},\\&\qquad +2 q_{41} q_{32} E_{22}+q_{33} q_{42} E_{22}+2 q_{31} E_{24} q_{42}+q_{32} E_{24} q_{43}+2 q_{41} q_{33} E_{32},\\&\qquad -q_{31} q_{41} E_{34}+q_{32} q_{42} E_{34}+2 q_{31} q_{43} E_{36}),\\&R_{15}=\frac{3}{4}(-2 E_{11} q_{41} q_{43}+2 E_{11} q_{42}^{2}+2 E_{12}q_{43}^{2}+3 E_{13} q_{41}^{2}+2 E_{20} q_{42} q_{43},\\&\qquad +4E_{21}q_{41}q_{42}+4q_{41}q_{43}E_{31}+E_{31}q_{42}^{2}-E_{33}q_{41}^{2})+\frac{3i}{2}(-q_{{31}}q_{{43}}E_{{13}},\\&\qquad +3\,q_{{31}}q_{{41}}E_{{15}}+2\,q_{{32}}q_{{42}}E_{{15}}+2\,E_{{15}}q_{{33}}q_{{43}}+2\,q_{{31}}E_{{23}}q_{{42}}+q_{{32}}E_{{23}}q_{{43}},\\&\qquad +2\,q_{{41}}E_{{25}}q_{{32}}+q_{{42}}E_{{25}}q_{{33}}+2\,q_{{31}}q_{{43}}E_{{33}}-q_{{31}}q_{{41}}E_{{35}}+q_{{32}}q_{{42}}E_{{35}}),\\&R_{16}=\frac{3}{4}(-2\,E_{{11}}{q_{{33}}}^{2}+2\,E_{{12}}q_{{31}}q_{{33}}-2\,E_{{12}}{q_{{32}}}^{2}-3\,E_{{14}}{q_{{31}}}^{2}-2\,E_{{20}}q_{{32}}q_{{33}},\\&\qquad -4\,E_{{22}}q_{{31}}q_{{32}}-4\,q_{{31}}q_{{33}}E_{{32}}-E_{{32}}{q_{{32}}}^{2}+E_{{34}}{q_{{31}}}^{2})+\frac{3i}{2}(-q_{{41}}q_{{33}}E_{{14}},\\&\qquad +3\,q_{{31}}q_{{41}}E_{{16}}+2\,q_{{32}}q_{{42}}E_{{16}}+2\,q_{{33}}q_{{43}}E_{{16}}+2\,q_{{41}}E_{{24}}q_{{32}}+q_{{33}}E_{{24}}q_{{42}},\\&\qquad +2\,q_{{31}}E_{{26}}q_{{42}}+q_{{32}}E_{{26}}q_{{43}}+2\,q_{{41}}q_{{33}}E_{{34}}-q_{{31}}q_{{41}}E_{{36}}+q_{{32}}q_{{42}}E_{{36}}),\\&R_{17}=\frac{3}{4}(2\,E_{{11}}{q_{{43}}}^{2}-2\,E_{{13}}q_{{41}}q_{{43}}+2\,E_{{13}}{q_{{42}}}^{2}+3\,E_{{15}}{q_{{41}}}^{2}+2\,E_{{21}}q_{{42}}q_{{43}}\\&\qquad +4\,E_{{23}}q_{{41}}q_{{42}}+4\,E_{{33}}q_{{41}}q_{{43}}+E_{{33}}{q_{{42}}}^{2}-E_{{35}}{q_{{41}}}^{2})+\frac{3i}{2}(2\,q_{{41}}{E_{27}}\,q_{{32}}\\&\qquad +q_{{42}}{E_{27}}\,q_{{33}}-q_{{31}}q_{{43}}E_{{15}}+3\,q_{{31}}q_{{41}}E_{{17}}+2\,q_{{32}}q_{{42}}E_{{17}}+2\,E_{{17}}q_{{33}}q_{{43}}\\&\qquad +2\,q_{{31}}E_{{25}}q_{{42}} +E_{{25}}q_{{32}}q_{{43}}+2\,q_{{31}}q_{{43}}E_{{35}}-q_{{31}}q_{{41}}E_{{37}}+q_{{32}}q_{{42}}E_{{37}}),\\&R_{18}=\frac{3}{4}(-2\,E_{{12}}{q_{{33}}}^{2}+2\,E_{{14}}q_{{31}}q_{{33}}-2\,E_{{14}}{q_{{32}}}^{2}-3\,E_{{16}}{q_{{31}}}^{2}-2\,E_{{22}}q_{{32}}q_{{33}}\\&\qquad -4\,E_{{24}}q_{{31}}q_{{32}}-4\,E_{{34}}q_{{31}}q_{{33}}-E_{{34}}{q_{{32}}}^{2}+E_{{36}}{q_{{31}}}^{2})+\frac{3i}{2}(2\,q_{{31}}-q_{{41}}q_{{33}}E_{{16}}\\&\qquad +3\,q_{{31}}q_{{41}}E_{{18}}+2\,q_{{32}}q_{{42}}E_{{18}}+2\,q_{{33}}q_{{43}}E_{{18}}+2\,q_{{41}}E_{{26}}q_{{32}}+E_{{26}}q_{{42}}q_{{33}}\\&R_{20}=\frac{3}{4}(-4\,E_{{13}}q_{{31}}q_{{32}}+4\,E_{{14}}q_{{41}}q_{{42}}-2\,E_{{15}}q_{{32}}q_{{33}}+2\,E_{{16}}q_{{42}}q_{{43}}-2\,E_{{21}}{q_{{31}}}^{2}\\&\qquad +2\,E_{{22}}{q_{{41}}}^{2}-2\,E_{{23}}q_{{31}}q_{{33}}-3\,E_{{23}}{q_{{32}}}^{2}+2\,E_{{24}}q_{{41}}q_{{43}}+3\,E_{{24}}{q_{{42}}}^{2}-2\,E_{{25}}{q_{{33}}}^{2}\\&\qquad +2\,E_{{26}}{q_{{43}}}^{2}-2\,E_{{33}}q_{{31}}q_{{32}}+2\,E_{{34}}q_{{41}}q_{{42}}-4\,E_{{35}}q_{{32}}q_{{33}}+4\,E_{{36}}q_{{42}}q_{{43}})\\&\qquad +\frac{3i}{2}(2\,E_{{11}}q_{{41}}q_{{32}}+E_{{11}}q_{{42}}q_{{33}}+2\,E_{{12}}q_{{31}}q_{{42}}+E_{{12}}q_{{32}}q_{{43}}+2\,E_{{20}}q_{{31}}q_{{41}}\\&\qquad +3\,E_{{20}}q_{{32}}q_{{42}}+2\,E_{{20}}q_{{33}}q_{{43}}+E_{{21}}q_{{41}}q_{{33}}+E_{{22}}q_{{31}}q_{{43}}+E_{{31}}q_{{32}}q_{{41}}+2\,E_{{31}}q_{{33}}q_{{42}}\\&\qquad +E_{{32}}q_{{31}}q_{{42}}+2\,E_{{32}}q_{{32}}q_{{43}}),\\&R_{21}=\frac{3}{4}(4\,q_{{42}}q_{{41}}E_{{12}}+2\,q_{{42}}q_{{43}}E_{{14}}-4\,q_{{32}}q_{{31}}E_{{15}}+2\,E_{{20}}{q_{{41}}}^{2}+2\,E_{{22}}q_{{41}}q_{{43}}+3\,E_{{22}}{q_{{42}}}^{2}\\&\qquad -2\,E_{{23}}{q_{{31}}}^{2}+2\,E_{{24}}{q_{{43}}}^{2}-2\,q_{{31}}E_{{25}}q_{{33}}-3\,E_{{25}}{q_{{32}}}^{2}+2\,q_{{42}}q_{{41}}E_{{32}}+4\,q_{{42}}q_{{43}}E_{{34}}\\&\qquad -2\,q_{{32}}q_{{31}}E_{{35}})+\frac{3i}{2}(2\,q_{{42}}q_{{31}}E_{{11}}+q_{{32}}q_{{43}}E_{{11}}+2\,q_{{32}}q_{{41}}E_{{13}}+q_{{42}}q_{{33}}E_{{13}}+q_{{31}}q_{{43}}E_{{20}}\\&\qquad +2\,q_{{31}}q_{{41}}E_{{21}}+3\,q_{{32}}q_{{42}}E_{{21}}+2\,q_{{33}}E_{{21}}q_{{43}}+q_{{33}}E_{{23}}q_{{41}}+q_{{42}}q_{{31}}E_{{31}}+2\,q_{{32}}q_{{43}}E_{{31}}\\&\qquad +q_{{32}}q_{{41}}E_{{33}}+2\,q_{{42}}q_{{33}}E_{{33}}),\\&R_{22}=\frac{3}{4}(-4\,q_{{32}}q_{{31}}E_{{11}}-2\,q_{{32}}q_{{33}}E_{{13}}+4\,q_{{42}}q_{{41}}E_{{16}}-2\,E_{{20}}{q_{{31}}}^{2}-2\,q_{{31}}q_{{33}}E_{{21}}-3\,E_{{21}}{q_{{32}}}^{2}\\&\qquad -2\,E_{{23}}{q_{{33}}}^{2}+2\,E_{{24}}{q_{{41}}}^{2}+2\,E_{{26}}q_{{41}}q_{{43}}+3\,E_{{26}}{q_{{42}}}^{2}-2\,q_{{32}}q_{{31}}E_{{31}}-4\,q_{{32}}q_{{33}}E_{{33}}\\&\qquad +2\,q_{{42}}q_{{41}}E_{{36}})+\frac{3i}{2}(2\,q_{{32}}q_{{41}}E_{{12}}+q_{{42}}q_{{33}}E_{{12}}+2\,q_{{42}}q_{{31}}E_{{14}}+q_{{32}}q_{{43}}E_{{14}}+q_{{33}}q_{{41}}E_{{20}}\\&\qquad +2\,q_{{31}}q_{{41}}E_{{22}}+3\,q_{{32}}q_{{42}}E_{{22}}+2\,q_{{33}}q_{{43}}E_{{22}}+q_{{31}}E_{{24}}q_{{43}}+q_{{32}}q_{{41}}E_{{32}}+2\,q_{{42}}q_{{33}}E_{{32}}\\&\qquad +q_{{42}}q_{{31}}E_{{34}}+2\,q_{{32}}q_{{43}}E_{{34}}),\\&R_{23}=\frac{3}{4}(4\,q_{{42}}q_{{41}}E_{{11}}+2\,q_{{42}}q_{{43}}E_{{12}}+2\,E_{{20}}q_{{41}}q_{{43}}+3\,E_{{20}}{q_{{42}}}^{2}+2\,E_{{21}}{q_{{41}}}^{2}+2\,E_{{22}}{q_{{43}}}^{2}\\&\qquad -2\,E_{{25}}{q_{{31}}}^{2}+2\,q_{{42}}q_{{41}}E_{{31}}+4\,q_{{42}}q_{{43}}E_{{32}})+\frac{3i}{2}(2\,q_{{42}}q_{{31}}E_{{13}}+q_{{32}}q_{{43}}E_{{13}}+2\,q_{{32}}q_{{41}}E_{{15}}\\&\qquad +q_{{42}}q_{{33}}E_{{15}}+q_{{31}}q_{{43}}E_{{21}}+2\,q_{{31}}E_{{23}}q_{{41}}+3\,q_{{32}}E_{{23}}q_{{42}}+2\,q_{{33}}E_{{23}}q_{{43}}+q_{{33}}E_{{25}}q_{{41}}\\&\qquad +q_{{42}}q_{{31}}E_{{33}}+2\,q_{{32}}q_{{43}}E_{{33}}+q_{{32}}q_{{41}}E_{{35}}+2\,q_{{42}}q_{{33}}E_{{35}}),\\&R_{24}=\frac{3}{4}(-2\,q_{{32}}q_{{33}}E_{{11}}-4\,q_{{32}}q_{{31}}E_{{12}}-2\,q_{{31}}q_{{33}}E_{{20}}-3\,E_{{20}}{q_{{32}}}^{2}-2\,E_{{21}}{q_{{33}}}^{2}\\&\qquad -2\,E_{{22}}{q_{{31}}}^{2}+2\,E_{{26}}{q_{{41}}}^{2}-4\,q_{{32}}q_{{33}}E_{{31}}-2\,q_{{32}}q_{{31}}E_{{32}})+\frac{3i}{2}(2\,q_{{32}}q_{{41}}E_{{14}}\\&\qquad +q_{{42}}q_{{33}}E_{{14}}+2\,q_{{42}}q_{{31}}E_{{16}}+q_{{32}}q_{{43}}E_{{16}}+q_{{33}}q_{{41}}E_{{22}}+2\,q_{{31}}E_{{24}}q_{{41}}+3\,q_{{32}}E_{{24}}q_{{42}}\\&\qquad +2\,q_{{33}}E_{{24}}q_{{43}}+q_{{31}}E_{{26}}q_{{43}}+q_{{32}}q_{{41}}E_{{34}}+2\,q_{{42}}q_{{33}}E_{{34}}+q_{{42}}q_{{31}}E_{{36}}+2\,q_{{32}}q_{{43}}E_{{36}}),\\&R_{25}=\frac{3}{4}(2\,q_{{42}}q_{{43}}E_{{11}}+4\,q_{{42}}q_{{41}}E_{{13}}+2\,E_{{20}}{q_{{43}}}^{2}+2\,E_{{21}}q_{{41}}q_{{43}}+3\,E_{{21}}{q_{{42}}}^{2}\\&\qquad +2\,E_{{23}}{q_{{41}}}^{2}+4\,q_{{42}}q_{{43}}E_{{31}}+2\,q_{{42}}q_{{41}}E_{{33}})+\frac{3i}{2}(2\,q_{{42}}q_{{31}}E_{{15}}+q_{{32}}q_{{43}}E_{{15}}\\&\qquad +q_{{31}}E_{{23}}q_{{43}}+2\,q_{{31}}E_{{25}}q_{{41}}+3\,q_{{32}}E_{{25}}q_{{42}}+2\,E_{{25}}q_{{33}}q_{{43}}+q_{{42}}q_{{31}}E_{{35}}+2\,q_{{32}}q_{{43}}E_{{35}}),\\&R_{26}=\frac{3}{4}(-2\,q_{{32}}q_{{33}}E_{{12}}-4\,q_{{32}}q_{{31}}E_{{14}}-2\,{q_{{33}}}^{2}E_{{20}}-2\,q_{{31}}q_{{33}}E_{{22}}-3\,E_{{22}}{q_{{32}}}^{2}\\&\qquad -2\,E_{{24}}{q_{{31}}}^{2}-4\,q_{{32}}q_{{33}}E_{{32}}-2\,q_{{32}}q_{{31}}E_{{34}})+\frac{3i}{2}(2\,q_{{32}}q_{{41}}E_{{16}}+q_{{33}}q_{{42}}E_{{16}}\\&\qquad +q_{{33}}E_{{24}}q_{{41}}+2\,q_{{31}}E_{{26}}q_{{41}}+3\,q_{{32}}E_{{26}}q_{{42}}+2\,q_{{33}}E_{{26}}q_{{43}}+q_{{32}}q_{{41}}E_{{36}}+2\,q_{{33}}q_{{42}}E_{{36}}),\\&R_{31}=\frac{3}{4}(-E_{{12}}{q_{{41}}}^{2}+E_{{13}}{q_{{31}}}^{2}+4\,E_{{14}}q_{{41}}q_{{43}}+E_{{14}}{q_{{42}}}^{2}-4\,E_{{15}}q_{{31}}q_{{33}}-E_{{15}}{q_{{32}}}^{2}\\&\qquad +2\,E_{{22}}q_{{41}}q_{{42}}-2\,E_{{23}}q_{{31}}q_{{32}}+4\,E_{{24}}q_{{42}}q_{{43}}-4\,E_{{25}}q_{{32}}q_{{33}}+2\,E_{{32}}{q_{{41}}}^{2}-2\,E_{{33}}{q_{{31}}}^{2}\\&\qquad +2\,E_{{34}}{q_{{42}}}^{2}-2\,E_{{35}}{q_{{32}}}^{2}+3\,E_{{36}}{q_{{43}}}^{2})+\frac{3i}{2}(-E_{{11}}q_{{31}}q_{{41}}+E_{{11}}q_{{32}}q_{{42}}+2\,E_{{12}}q_{{31}}q_{{43}}\\&\qquad +2\,E_{{13}}q_{{41}}q_{{33}}+E_{{20}}q_{{31}}q_{{42}}+2\,E_{{20}}q_{{32}}q_{{43}}+E_{{21}}q_{{32}}q_{{41}}+2\,E_{{21}}q_{{42}}q_{{33}}+2\,E_{{31}}q_{{31}}q_{{41}}\\&\qquad +2\,E_{{31}}q_{{32}}q_{{42}}+3\,E_{{31}}q_{{33}}q_{{43}}),\\&R_{32}=\frac{3}{4}(E_{{11}}{q_{{31}}}^{2}-4\,E_{{13}}q_{{31}}q_{{33}}-E_{{13}}{q_{{32}}}^{2}-E_{{14}}{q_{{41}}}^{2}+4\,E_{{16}}q_{{41}}q_{{43}}+E_{{16}}{q_{{42}}}^{2}\\&\qquad -2\,E_{{21}}q_{{31}}q_{{32}}-4\,E_{{23}}q_{{32}}q_{{33}}+2\,E_{{24}}q_{{41}}q_{{42}}+4\,E_{{26}}q_{{42}}q_{{43}}-2\,E_{{31}}{q_{{31}}}^{2}-2\,E_{{33}}{q_{{32}}}^{2}\\&\qquad +2\,E_{{34}}{q_{{41}}}^{2}-3\,E_{{35}}{q_{{33}}}^{2}+2\,E_{{36}}{q_{{42}}}^{2})+\frac{3i}{2}(2\,q_{{41}}q_{{33}}E_{{11}}-q_{{31}}q_{{41}}E_{{12}}+q_{{32}}q_{{42}}E_{{12}}\\&\qquad +2\,q_{{31}}q_{{43}}E_{{14}}+q_{{41}}q_{{32}}E_{{20}}+2\,q_{{42}}q_{{33}}E_{{20}}+q_{{31}}q_{{42}}E_{{22}}+2\,q_{{32}}q_{{43}}E_{{22}}+2\,q_{{31}}q_{{41}}E_{{32}}\\&\qquad +2\,q_{{32}}q_{{42}}E_{{32}}+3\,E_{{32}}q_{{33}}q_{{43}}),\\&R_{33}=\frac{3}{4}(-E_{{11}}{q_{{41}}}^{2}+4\,E_{{12}}q_{{41}}q_{{43}}+E_{{12}}{q_{{42}}}^{2}+E_{{15}}{q_{{31}}}^{2}+2\,E_{{20}}q_{{41}}q_{{42}}+4\,E_{{22}}q_{{42}}q_{{43}}\\&\qquad -2\,E_{{25}}q_{{31}}q_{{32}}+2\,E_{{31}}{q_{{41}}}^{2}+2\,E_{{32}}{q_{{42}}}^{2}+3\,E_{{34}}{q_{{43}}}^{2}-2\,E_{{35}}{q_{{31}}}^{2})+\frac{3i}{2}(2\,q_{{31}}q_{{43}}E_{{11}}\\&\qquad -q_{{31}}q_{{41}}E_{{13}}+q_{{32}}q_{{42}}E_{{13}}+2\,q_{{41}}q_{{33}}E_{{15}}+q_{{31}}q_{{42}}E_{{21}}+2\,q_{{32}}E_{{21}}q_{{43}}+q_{{41}}E_{{23}}q_{{32}}\\&\qquad +2\,q_{{42}}E_{{23}}q_{{33}}+2\,q_{{31}}q_{{41}}E_{{33}}+2\,q_{{32}}q_{{42}}E_{{33}}+3\,E_{{33}}q_{{33}}q_{{43}}),\\&R_{34}=\frac{3}{4}(-4\,E_{{11}}q_{{31}}q_{{33}}-E_{{11}}{q_{{32}}}^{2}+E_{{12}}{q_{{31}}}^{2}-E_{{16}}{q_{{41}}}^{2}-2\,E_{{20}}q_{{31}}q_{{32}}-4\,E_{{21}}q_{{32}}q_{{33}}\\&\qquad +2\,E_{{26}}q_{{41}}q_{{42}}-2\,E_{{31}}{q_{{32}}}^{2}-2\,E_{{32}}{q_{{31}}}^{2}-3\,E_{{33}}{q_{{33}}}^{2}+2\,E_{{36}}{q_{{41}}}^{2})+\frac{3i}{2}(2\,q_{{41}}q_{{33}}E_{{12}}\\&\qquad -q_{{31}}q_{{41}}E_{{14}}+q_{{32}}q_{{42}}E_{{14}}+2\,q_{{31}}q_{{43}}E_{{16}}+q_{{41}}q_{{32}}E_{{22}}+2\,q_{{33}}q_{{42}}E_{{22}}+q_{{31}}E_{{24}}q_{{42}}\\&\qquad +2\,q_{{32}}E_{{24}}q_{{43}}+2\,q_{{31}}q_{{41}}E_{{34}}+2\,q_{{32}}q_{{42}}E_{{34}}+3\,E_{{34}}q_{{33}}q_{{43}}),\\&R_{35}=\frac{3}{4}(4\,E_{{11}}q_{{41}}q_{{43}}+E_{{11}}{q_{{42}}}^{2}-E_{{13}}{q_{{41}}}^{2}+4\,E_{{20}}q_{{42}}q_{{43}}+2\,E_{{21}}q_{{41}}q_{{42}}+2\,E_{{31}}{q_{{42}}}^{2}\\&\qquad +3\,E_{{32}}{q_{{43}}}^{2}+2\,E_{{33}}{q_{{41}}}^{2})+\frac{3i}{2}(2\,q_{{31}}q_{{43}}E_{{13}}-q_{{31}}q_{{41}}E_{{15}}+q_{{32}}q_{{42}}E_{{15}}+q_{{31}}E_{{23}}q_{{42}}\\&\qquad +2\,q_{{32}}E_{{23}}q_{{43}}+q_{{41}}E_{{25}}q_{{32}}+2\,q_{{42}}E_{{25}}q_{{33}}+2\,q_{{31}}q_{{41}}E_{{35}}+2\,q_{{32}}q_{{42}}E_{{35}}+3\,E_{{35}}q_{{33}}q_{{43}}),\\&R_{36}=\frac{3}{4}(-4\,E_{{12}}q_{{31}}q_{{33}}-E_{{12}}{q_{{32}}}^{2}+E_{{14}}{q_{{31}}}^{2}-4\,E_{{20}}q_{{32}}q_{{33}}-2\,E_{{22}}q_{{31}}q_{{32}}-3\,E_{{31}}{q_{{33}}}^{2}\\&\qquad -2\,E_{{32}}{q_{{32}}}^{2}-2\,E_{{34}}{q_{{31}}}^{2})+\frac{3i}{2}(2\,q_{{41}}q_{{33}}E_{{14}}-q_{{31}}q_{{41}}E_{{16}}+q_{{32}}q_{{42}}E_{{16}}+q_{{41}}E_{{24}}q_{{32}}\\&\qquad +2\,q_{{33}}E_{{24}}q_{{42}}+q_{{31}}E_{{26}}q_{{42}}+2\,q_{{32}}E_{{26}}q_{{43}}+2\,q_{{31}}q_{{41}}E_{{36}}+2\,q_{{32}}q_{{42}}E_{{36}}+3\,E_{{36}}q_{{33}}q_{{43}}),\\&R_{37}=\frac{3}{4}(4\,E_{{13}}q_{{41}}q_{{43}}+E_{{13}}{q_{{42}}}^{2}-E_{{15}}{q_{{41}}}^{2}+4\,E_{{21}}q_{{42}}q_{{43}}+2\,E_{{23}}q_{{41}}q_{{42}}+3\,E_{{31}}{q_{{43}}}^{2}\\&\qquad +2\,E_{{33}}{q_{{42}}}^{2}+2\,E_{{35}}{q_{{41}}}^{2})+\frac{3i}{2}(2\,q_{{31}}q_{{43}}E_{{15}}-q_{{31}}q_{{41}}E_{{17}}+q_{{32}}q_{{42}}E_{{17}}+q_{{31}}E_{{25}}q_{{42}}\\&\qquad +2\,E_{{25}}q_{{32}}q_{{43}}+2\,q_{{31}}q_{{41}}E_{{37}}+2\,q_{{32}}q_{{42}}E_{{37}}+3\,E_{{37}}q_{{33}}q_{{43}}),\\&R_{38}=\frac{3}{4}(-4\,E_{{14}}q_{{31}}q_{{33}}-E_{{14}}{q_{{32}}}^{2}+E_{{16}}{q_{{31}}}^{2}-4\,E_{{22}}q_{{32}}q_{{33}}-2\,E_{{24}}q_{{31}}q_{{32}}-3\,E_{{32}}{q_{{33}}}^{2}\\&\qquad -2\,E_{{34}}{q_{{32}}}^{2}-2\,E_{{36}}{q_{{31}}}^{2})+\frac{3i}{2}(2\,q_{{41}}q_{{33}}E_{{16}}-q_{{31}}q_{{41}}E_{{18}}+q_{{32}}q_{{42}}E_{{18}}+q_{{41}}E_{{26}}q_{{32}}\\&\qquad +2\,E_{{26}}q_{{42}}q_{{33}}+2\,q_{{31}}q_{{41}}E_{{38}}+2\,q_{{32}}q_{{42}}E_{{38}}+3\,E_{{38}}q_{{33}}q_{{43}}). \end{aligned}$$

Appendix B

$$\begin{aligned} M_{12}&=3\int _0^{t}{B_1(t-\tau )[3\,{u_{{10}}}^{2}u_{{12}}-\,{u_{{10}}}^{2}u_{{32}}+3\,u_{{10}}{u_{{11 }}}^{2}-2\,u_{{10}}u_{{11}}u_{{31}}}\nonumber \\&\quad -2\,u_{{10}}u_{{12}}u_{{30}}+4\,u_{{10}}u_{{20}}u_{{22}}+2\,u_{{10}}{u_{{21}}}^{2}+4\,u_{{10}}u_{{30}}u_{{32}} +2\,u_{{10}}{u_{{31}}}^{2}\nonumber \\&\quad -\,{u_{{11}}}^{2}u_{{30}}+4\,u_{{11}}u_{{20}}u_{{21}}+4\,u_{{11}}u_{{30}}u_{{31}}+2\,u_{{12}}{u_{{20}}}^ {2}+2\,u_{{12}}{u_{{30}}}^{2}\nonumber \\&\quad +\,{u_{{20}}}^{2}u_{{32}}+2\,u_{{20}}u_{{21}}u_{{31}}+2\,u_{{20}}u_{{22}}u_{{30}}+\,{u_{{21}}}^{2}u_{{30}}]\mathrm{d}\tau ,\nonumber \\ M_{22}&=3\int _0^{t}{B_2(t-\tau )[2\,{u_{{10}}}^{2}u_{{22}}+4\,u_{{10}}u_{{11}}u_{{21}}+4\,u_{{10}}u_{ {12}}u_{{20}}+2\,u_{{10}}u_{{20}}u_{{32}}}\nonumber \\&\quad +2\,u_{{10}}u_{{21}}u_{{31}}+2\,u_{{10}}u_{{22}}u_{{30}}+2\,{u_{{11}}}^{2}u_{{20}}+2\,u_{{11}}u_{{ 20}}u_{{31}}+2\,u_{{11}}u_{{21}}u_{{30}}\nonumber \\&\quad +2\,u_{{12}}u_{{20}}u_{{30}}+3\,{u_{{20}}}^{2}u_{{22}}+3\,u_{{20}}{u_{{21}}}^{2}+4\,u_{{20}}u_{{30} }u_{{32}}+2\,u_{{20}}{u_{{31}}}^{2}\nonumber \\&\quad +4\,u_{{21}}u_{{30}}u_{{31}}+2\,u_{{22}}{u_{{30}}}^{2}]\mathrm{d}\tau ,\nonumber \\ M_{32}&=3\int _0^{t}{B_3(t-\tau )[-\,{u_{{10}}}^{2}u_{{12}}+2\,{u_{{10}}}^{2}u_{{32}}-\,u_{{10}}{u_{{ 11}}}^{2}+4\,u_{{10}}u_{{11}}u_{{31}}}\nonumber \\&\quad +4\,u_{{10}}u_{{12}}u_{{30}}+2 \,u_{{10}}u_{{20}}u_{{22}}+\,u_{{10}}{u_{{21}}}^{2}+2\,{u_{{11}}}^{2} u_{{30}}+2\,u_{{11}}u_{{20}}u_{{21}}\nonumber \\&\quad +\,u_{{12}}{u_{{20}}}^{2}+2\,{u_{{20}}}^{2}u_{{32}}+4\,u_{{20}}u_{{21}}u_{{31}}+4\,u_{{20}}u_{{22}}u_ {{30}}+2\,{u_{{21}}}^{2}u_{{30}}\nonumber \\&\quad +3\,{u_{{30}}}^{2}u_{{32}}+3\,u_{{30}}{u_{{31}}}^{2}]\mathrm{d}\tau .\nonumber \\ M_{13}&=3\int _0^{t}{B_1(t-\tau )[3\,{u_{{10}}}^{2}u_{{13}}-\,{u_{{10}}}^{2}u_{{33}}+6\,u_{{10}}u_{{11 }}u_{{12}}-2\,u_{{10}}u_{{11}}u_{{32}}}\nonumber \\&\quad -2\,u_{{10}}u_{{12}}u_{{31}}-2\, u_{{10}}u_{{13}}u_{{30}}+4\,u_{{10}}u_{{20}}u_{{23}}+4\,u_{{10}}u_{{ 21}}u_{{22}}+4\,u_{{10}}u_{{30}}u_{{33}}\nonumber \\&\quad +4\,u_{{10}}u_{{31}}u_{{32}} +\,{u_{{11}}}^{3}-\,{u_{{11}}}^{2}u_{{31}}-2\,u_{{11}}u_{{12}}u_{{30 }}+4\,u_{{11}}u_{{20}}u_{{22}}+2\,u_{{11}}{u_{{21}}}^{2}\nonumber \\&\quad +4\,u_{{11}} u_{{30}}u_{{32}}+2\,u_{{11}}{u_{{31}}}^{2}+4\,u_{{12}}u_{{20}}u_{{21} }+4\,u_{{12}}u_{{30}}u_{{31}}+2\,u_{{13}}{u_{{20}}}^{2}\nonumber \\&\quad +2\,u_{{13}}{u _{{30}}}^{2}+{u_{{20}}}^{2}u_{{33}}+2\,u_{{20}}u_{{21}}u_{{32}}+2\, u_{{20}}u_{{22}}u_{{31}}+2\,u_{{20}}u_{{23}}u_{{30}}\nonumber \\&\quad +\,{u_{{21}}}^{2} u_{{31}}+2\,u_{{21}}u_{{22}}u_{{30}} ]\mathrm{d}\tau ,\nonumber \\ M_{23}&=3\int _0^{t}{B_2(t-\tau )[2\,{u_{{10}}}^{2}u_{{23}}+4\,u_{{10}}u_{{11}}u_{{22}}+4\,u_{{10}}u_{ {12}}u_{{21}}+4\,u_{{10}}u_{{13}}u_{{20}}}\nonumber \\&\quad +2\,u_{{10}}u_{{20}}u_{{33}}+2\,u_{{10}}u_{{21}}u_{{32}}+2\,u_{{10}}u_{{22}}u_{{31}}+2\,u_{{10}}u_ {{23}}u_{{30}}+2\,{u_{{11}}}^{2}u_{{21}}\nonumber \\&\quad +4\,u_{{11}}u_{{12}}u_{{20}}+2\,u_{{11}}u_{{20}}u_{{32}}+2\,u_{{11}}u_{{21}}u_{{31}}+2\,u_{{11}}u_{ {22}}u_{{30}}+2\,u_{{12}}u_{{20}}u_{{31}}\nonumber \\&\quad +2\,u_{{12}}u_{{21}}u_{{30}}+ 2\,u_{{13}}u_{{20}}u_{{30}}+3\,{u_{{20}}}^{2}u_{{23}}+6\,u_{{20}}u_{{ 21}}u_{{22}}+4\,u_{{20}}u_{{30}}u_{{33}}\nonumber \\&\quad +4\,u_{{20}}u_{{31}}u_{{32}}+\,{u_{{21}}}^{3}+4\,u_{{21}}u_{{30}}u_{{32}}+2\,u_{{21}}{u_{{31}}}^ {2}+4\,u_{{22}}u_{{30}}u_{{31}}\nonumber \\&\quad +2\,u_{{23}}{u_{{30}}}^{2} ]\mathrm{d}\tau ,\nonumber \\ M_{33}&=3\int _0^{t}{B_3(t-\tau )[-\,{u_{{10}}}^{2}u_{{13}}+2\,{u_{{10}}}^{2}u_{{33}}-2\,u_{{10}}u_{{11 }}u_{{12}}+4\,u_{{10}}u_{{11}}u_{{32}}}\nonumber \\&\quad +4\,u_{{10}}u_{{12}}u_{{31}}+4\,u_{{10}}u_{{13}}u_{{30}}+2\,u_{{10}}u_{{20}}u_{{23}}+2\,u_{{10}}u_ {{21}}u_{{22}}-\frac{1}{3}{u_{{11}}}^{3}\nonumber \\&\quad +2\,{u_{{11}}}^{2}u_{{31}}+4\,u_{{11}}u_ {{12}}u_{{30}}+2\,u_{{11}}u_{{20}}u_{{22}}+\,u_{{11}}{u_{{21}}}^{2}+2 \,u_{{12}}u_{{20}}u_{{21}}\nonumber \\&\quad +\,u_{{13}}{u_{{20}}}^{2}+2\,{u_{{20}}}^{2} u_{{33}}+4\,u_{{20}}u_{{21}}u_{{32}}+4\,u_{{20}}u_{{22}}u_{{31}}+4 \,u_{{20}}u_{{23}}u_{{30}}\nonumber \\&\quad +2\,{u_{{21}}}^{2}u_{{31}}+4\,u_{{21}}u_{{ 22}}u_{{30}}+3\,{u_{{30}}}^{2}u_{{33}}+6\,u_{{30}}u_{{31}}u_{{32}}+ \,{u_{{31}}}^{3} ]\mathrm{d}\tau . \end{aligned}$$

(B.1)