Abstract
This paper studies the critical sectional area of surge tank (SAST) based on bifurcation and chaos behaviors of hydraulic-mechanical coupling hydropower station (HMCHS). Firstly, the model of HMCHS is established. Then, the criterion for the critical stable state (CSS) based on bifurcation behavior is derived. The critical SAST based on bifurcation behavior is determined and analyzed. Finally, the occurrence of chaos motion is identified. By analyzing the effect of SAST on chaos behavior, the critical SAST based on chaos behavior is determined. The results indicate that the critical SAST based on bifurcation behavior corresponds to CSS of HMCHS. Among nonlinearity of head loss of headrace tunnel, throttling orifice head loss of surge tank and nonlinearity of turbine characteristics, the influence degree of the nonlinearity of head loss of headrace tunnel is the greatest. Under the coupling effect of those three factors, the critical SAST based on bifurcation behavior becomes greater under load decrease and becomes less under load increase. For the chaos motion caused by low- and high-frequency oscillations together, there is the concept of critical SAST based on chaos behavior. With the decrease in proportional gain of governor, the critical SAST based on chaos behavior becomes less.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
References
Moran, E.F., Lopez, M.C., Moore, N., Müller, N., Hyndman, D.W.: Sustainable hydropower in the 21st century. PNAS 115(47), 11891–11898 (2018)
Kapsali, M., Kaldellis, J.K.: Combining hydro and variable wind power generation by means of pumped-storage under economically viable terms. Appl. Energy 87(11), 3475–3485 (2010)
Kishor, N., Fraile-Ardanuy, J.: Modeling and dynamic behaviour of hydropower plants. Stevenage: The Institution of Engineering and Technology (2017).
Shahgholian, G.: An overview of hydroelectric power plant: operation, modeling, and control. J. Renew. Energy Environ. 7(3), 14–28 (2020)
Guo, W.C., Zhu, D.Y.: A review of the transient process and control for a hydropower station with a super long headrace tunnel. Energies 11(11), 2994 (2018)
Singh, N.K., Bachawad, M.R.: A review on mathematical modeling of solar, wind and hydro pumped energy storage system. Int. J. Adv. Res. Electr. Electron. Instrum. Eng. 4(12), 9506–9512 (2015)
Vilanova, M.R.N., Flores, A.T., Balestieri, J.A.P.: Pumped hydro storage plants: a review. J. Braz. Soc. Mech. Sci. 42(8), 1–14 (2020)
Liu, Y., Guo, W.C.: Multi-frequency dynamic performance of hydropower plant under coupling effect of power grid and turbine regulating system with surge tank. Renew. Energy 171, 557–581 (2021)
Wu, R.Q., Chen, J.Z.: Hydraulic Transients of Hydropower Station. China Water Power Press, Beijing (1997)
Wu, F.L., Guo, W.C., Fu, S., Qu, F.L.: Hydraulic coupling vibration characteristics and control of hydropower station with upstream and downstream surge tanks. Sustain. Energy Tech. 50, 101842 (2022)
Xu, X.Y., Guo, W.C.: Stability of speed regulating system of hydropower station with surge tank considering nonlinear turbine characteristics. Renew. Energy 162, 960–972 (2020)
Guo, W.C., Yang, J.D.: Modeling and dynamic response control for primary frequency regulation of hydro-turbine governing system with surge tank. Renew. Energy 121, 173–187 (2018)
Guo, W.C., Yang, J.D., Wang, M.J., Lai, X.: Nonlinear modeling and stability analysis of hydro-turbine governing system with sloping ceiling tailrace tunnel under load disturbance. Energy Convers. Manag. 106, 127–138 (2015)
Mosonyi, E., Seth, H.B.: The surge tank—a device for controlling water hammer. Water Power Dam Construct. 27(2), 69–74 (1975)
Wang, B.B., Guo, W.C., Yang, J.D.: Analytical solutions for determining extreme water levels in surge tank of hydropower station under combined operating conditions. Commun. Nonlinear Sci. 47, 394–406 (2017)
Guo, W.C., Zhu, D.Y.: Setting condition of downstream surge tank of hydropower station with sloping ceiling tailrace tunnel. Chaos Soliton Fract. 134, 109698 (2020)
Allievi, L.: Theory of Water Hammer. Riccardo Garoni, Rome (1925)
Zhu, D.Y., Guo, W.C.: Setting condition of surge tank based on stability of hydro-turbine governing system considering nonlinear penstock head loss. Int. J. Electr. Power 113, 372–382 (2019)
Guo, W.C., Yang, J.D., Teng, Y.: Surge wave characteristics for hydropower station with upstream series double surge tanks in load rejection transient. Renew Energy 108, 488–501 (2017)
Guo, W.C., Wang, B.B., Yang, J.D., Xue, Y.L.: Optimal control of water level oscillations in surge tank of hydropower station with long headrace tunnel under combined operating conditions. Appl. Math. Modell. 47, 260–275 (2017)
Qu, F.L., Guo, W.C.: Robust H∞ control for hydro-turbine governing system of hydropower plant with super long headrace tunnel. Int. J. Electr. Power 124, 106336 (2021)
Guo, W.C., Liu, Y., Qu, F.L., Xu, X.Y.: A review of critical stable sectional areas for the surge tanks of hydropower stations. Energies 13(23), 6466 (2020)
Li, J.B., Feng, B.Y.: Stability, Bifurcations and Chaos. Yunnan Science and Technology Press, Kunming (1995)
Ji, K., Ma, Y.X., Wang, S.Q.: Study on the stable sectional area of surge tank under large fluctuation process. J. Hydraul. Eng. 5, 45–51 (1990)
Shou, M.H.: Study on the hydraulic turbine regulation with surge tank. Water Resour. Hydropower Eng. 7, 28–34 (1991)
Yang, J.D., Lai, X., Chen, J.Z.: The effect turbine characteristic on stable sectional area of surge tank. J. Hydraul. Eng. 2, 7–11 (1998)
Zhu, D.Y., Guo, W.C.: Critical sectional area of surge tank considering nonlinearity of head loss of diversion tunnel and steady output of turbine. Chaos Soliton Fract. 127, 165–172 (2019)
Dong, X.L.: Research on the stable cross-section area of surge tank of hydropower station. J. Hydraul. Eng. 4, 37–48 (1980)
Guo, W.C., Yang, J.D., Chen, Y.M., Shan, X.J.: Study on critical stable sectional area of surge chamber considering penstock fluid inertia and governor characteristics. J. Hydroelectr. Eng. 33(3), 171–178 (2014)
Guo, W.C., Yang, J.D.: Stability performance for primary frequency regulation of hydro-turbine governing system with surge tank. Appl. Math. Modell. 54, 446–466 (2018)
Peng, Z.Y., Guo, W.C.: Saturation characteristics for stability of hydro-turbine governing system with surge tank. Renew. Energy 131, 318–332 (2019)
Guo, W.C., Peng, Z.Y.: Hydropower system operation stability considering the coupling effect of water potential energy in surge tank and power grid. Renew. Energy 134, 846–861 (2019)
Guo, W.C., Zhu, D.Y.: Critical stable sectional area of downstream surge tank of hydropower plant with sloping ceiling tailrace tunnel. Energy Sci. Eng. 9(8), 1090–1102 (2021)
Ling, D.J., Shen, Z.Y.: Bifurcation analysis of hydro-turbine governing system with saturation nonlinearity. J. Hydroelectr. Eng. 26(6), 126–131 (2007)
Ling, D.J., Tao YShen ZY.: Hopf bifurcation analysis of hydraulic turbine governing systems with elastic water hammer effect. J. Vib. Eng. 20(4), 374–380 (2007)
Chen, D.Y., Ding, C., Do, Y., Ma, X.Y., Zhao, H., Wang, Y.C.: Nonlinear dynamic analysis for a Francis hydro-turbine governing system and its control. J. Frankl. I 351(9), 4596–4618 (2014)
Li, J.Y., Chen, Q.J.: Nonlinear dynamical analysis of hydraulic turbine governing systems with nonelastic water hammer effect. J. Appl. Math. 2014, 1–11 (2014)
Guo, W.C., Yang, J.D.: Hopf bifurcation control of hydro-turbine governing system with sloping ceiling tailrace tunnel using nonlinear state feedback. Chaos Soliton Fract. 104, 426–434 (2017)
Jiang, Y., Wang, S.X., Li, F.: Chaos optimization strategy for hydraulic turbine governing system. Chin. J. Sci. Instrum. 27, 845–847 (2006)
Jiang, C.W., Ma, Y.C., Wang, C.M.: PID controller parameters optimization of hydro-turbine governing systems using deterministic-chaotic-mutation evolutionary programming (DCMEP). Energy Convers. Manag. 47, 1222–1230 (2006)
Zhang, F.S.: Study and control for chaos phenomena of hydraulic turbine’s speed governor. J. Water Resour. Archit. Eng. 4, 69–73 (2012)
Chen, D.Y., Yang, P.C., Ma, X.Y., Sun, Z.T.: Chaos of hydro-turbine governing system and its control. Proc. Chin. Soc. Electr. Eng. 31(14), 113–120 (2013)
Xu, X.Y., Guo, W.C.: Chaotic behavior of turbine regulating system for hydropower station under effect of nonlinear turbine characteristics. Sustain. Energy Tech. 44, 101088 (2021)
Chaudhry, M.H.: Applied Hydraulic Transients. Springer, New York (2014)
Guo, W.C., Qu, F.L.: Stability control of dynamic system of hydropower station with two turbine units sharing a super long headrace tunnel. J. Frankl. I 358(16), 8506–8533 (2021)
Liu, Q.Z., Peng, S.Z.: Surge Tank of Hydropower Station. China Waterpower Press, Beijing (1995)
Bao, H.Y., Yang, J.D., Li, J.P., Fu, L.: Discussion on the setting condition of surge chamber based on operation stability of hydropower station. J. Hydroelectr. Eng. 30(2), 44–48 (2011)
Qu, F.L., Guo, W.C.: Coupling transient behavior of primary frequency regulation of hydropower plant. Sustain. Energy Tech. 52, 102227 (2022)
Wei, S.P.: Hydraulic Turbine Regulation. Huazhong University of Science and Technology Press, Wuhan (2009)
Guo, W.C., Yang, J.D., Chen, J.P., Wang, M.J.: Nonlinear modeling and dynamic control of hydro-turbine governing system with upstream surge tank and sloping ceiling tailrace tunnel. Nonlinear Dyn. 84(3), 1383–1397 (2016)
IEEE Working Group: Hydraulic turbine and turbine control model for system dynamic studies. IEEE Trans. Power Syst. 7(1), 167–179 (1992)
Chow, S.N., Hale, J.K.: Methods of Bifurcation Theory. Springer, Berlin (2012)
Iooss, G., Joseph, D.D.: Elementary Stability and Bifurcation Theory. Springer, Berlin (2012)
Hairer, E., Lubich, C., Roche, M.: The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods. Springer, Berlin (2006)
Hanselman, D.C., Littlefield, B.: Mastering Matlab 7. Prentice Hall, New Jersey (2005)
Lorenz, E.N.: The Essence of Chaos. University of Washington Press, Seattle (1993)
Peitgen, H.O., Hartmut, J., Saupe, D.: Chaos and Fractals: New Frontiers of Science. Springer-Verlag, New York (2004)
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Project No. 51909097).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix A
List for abbreviations
SAST | Sectional area of surge tank |
HMCHS | Hydraulic-mechanical coupling hydropower station |
CSS | Critical stable state |
WLF | Water-level fluctuation |
PID | Proportional–integral–differential |
DRP | Dynamic response process |
LE | Lyapunov exponent |
Appendix B
The expressions for the coefficients \(a_{i} \left( {i = 1,2, \ldots ,6} \right)\) in Eq. (11) are presented as follows.
$$ a_{1} = - \left( {\frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }} + \frac{{{\partial} \dot{x}}}{{\partial} x} + \frac{{{\partial} \dot{y}}}{{\partial} y} + \frac{{{\partial} \dot{u}}}{{\partial} u} + \frac{{{\partial} \dot{h}}}{{\partial} h}} \right), $$
$$ \begin{aligned} a_{2} & = \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{\partial} x} - \frac{{{\partial} \dot{q}_{H} }}{{\partial} z}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{\partial} y} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{u}}}{{\partial} u} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{h}}}{{\partial} h} + \frac{{{\partial} \dot{x}}}{{\partial} x}\frac{{{\partial} \dot{y}}}{{\partial} y} - \frac{{{\partial} \dot{z}}}{{\partial} h}\frac{{{\partial} \dot{h}}}{{\partial} z} \\ & \quad + \frac{{{\partial} \dot{x}}}{{\partial} x}\frac{{{\partial} \dot{u}}}{{\partial} u} + \frac{{{\partial} \dot{x}}}{{\partial} x}\frac{{{\partial} \dot{h}}}{{\partial} h} - \frac{{{\partial} \dot{x}}}{{\partial} h}\frac{{{\partial} \dot{h}}}{{\partial} x} + \frac{{{\partial} \dot{y}}}{{\partial} y}\frac{{{\partial} \dot{u}}}{{\partial} u} - \frac{{{\partial} \dot{y}}}{{\partial} u}\frac{{{\partial} \dot{u}}}{{\partial} y}{ + }\frac{{{\partial} \dot{y}}}{{\partial} y}\frac{{{\partial} \dot{h}}}{{\partial} h} + \frac{{{\partial} \dot{u}}}{{\partial} u}\frac{{{\partial} \dot{h}}}{{\partial} h} - \frac{{{\partial} \dot{u}}}{{\partial} h}\frac{{{\partial} \dot{h}}}{{\partial} u}, \\ \end{aligned} $$
$$ \begin{aligned} a_{3} & = \frac{{{\partial} \dot{q}_{H} }}{{\partial} z}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{\partial} x} + \frac{{{\partial} \dot{q}_{H} }}{{\partial} z}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{\partial} y} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{\partial} x}\frac{{{\partial} \dot{y}}}{{\partial} y} + \frac{{{\partial} \dot{q}_{H} }}{{\partial} z}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{u}}}{{\partial} u} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{\partial} h}\frac{{{\partial} \dot{h}}}{{\partial} z} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{\partial} x}\frac{{{\partial} \dot{u}}}{{\partial} u} + \frac{{{\partial} \dot{q}_{H} }}{{\partial} z}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{h}}}{{\partial} h} \\ & \quad - \frac{{{\partial} \dot{q}_{H} }}{{\partial} h}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{h}}}{{\partial} z} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{\partial} x}\frac{{{\partial} \dot{h}}}{{\partial} h} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{\partial} h}\frac{{{\partial} \dot{h}}}{{\partial} x} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{\partial} y}\frac{{{\partial} \dot{u}}}{{\partial} u} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{\partial} u}\frac{{{\partial} \dot{u}}}{{\partial} y} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{\partial} y}\frac{{{\partial} \dot{h}}}{{\partial} h} - \frac{{{\partial} \dot{z}}}{{\partial} x}\frac{{{\partial} \dot{x}}}{{\partial} h}\frac{{{\partial} \dot{h}}}{{\partial} z} \\ & \quad - \frac{{{\partial} \dot{z}}}{{\partial} y}\frac{{{\partial} \dot{y}}}{{\partial} u}\frac{{{\partial} \dot{u}}}{{\partial} z} + \frac{{{\partial} \dot{z}}}{{\partial} h}\frac{{{\partial} \dot{x}}}{{\partial} x}\frac{{{\partial} \dot{h}}}{{\partial} z} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{u}}}{{\partial} u}\frac{{{\partial} \dot{h}}}{{\partial} h} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{u}}}{{\partial} h}\frac{{{\partial} \dot{h}}}{{\partial} u} + \frac{{{\partial} \dot{z}}}{{\partial} h}\frac{{{\partial} \dot{y}}}{{\partial} y}\frac{{{\partial} \dot{h}}}{{\partial} z} - \frac{{{\partial} \dot{x}}}{{\partial} x}\frac{{{\partial} \dot{y}}}{{\partial} y}\frac{{{\partial} \dot{u}}}{{\partial} u} + \frac{{{\partial} \dot{x}}}{{\partial} x}\frac{{{\partial} \dot{y}}}{{\partial} u}\frac{{{\partial} \dot{u}}}{{\partial} y} - \frac{{{\partial} \dot{x}}}{{\partial} y}\frac{{{\partial} \dot{y}}}{{\partial} u}\frac{{{\partial} \dot{u}}}{{\partial} x} \\ & \quad - \frac{{{\partial} \dot{z}}}{{\partial} h}\frac{{{\partial} \dot{u}}}{{\partial} z}\frac{{{\partial} \dot{h}}}{{\partial} u}{ + }\frac{{{\partial} \dot{z}}}{{\partial} h}\frac{{{\partial} \dot{u}}}{{\partial} u}\frac{{{\partial} \dot{h}}}{{\partial} z} - \frac{{{\partial} \dot{x}}}{{\partial} x}\frac{{{\partial} \dot{y}}}{{\partial} y}\frac{{{\partial} \dot{h}}}{{\partial} h} + \frac{{{\partial} \dot{x}}}{{\partial} h}\frac{{{\partial} \dot{y}}}{{\partial} y}\frac{{{\partial} \dot{h}}}{{\partial} x} - \frac{{{\partial} \dot{x}}}{{\partial} x}\frac{{{\partial} \dot{u}}}{{\partial} u}\frac{{{\partial} \dot{h}}}{{\partial} h} + \frac{{{\partial} \dot{x}}}{{\partial} x}\frac{{{\partial} \dot{u}}}{{\partial} h}\frac{{{\partial} \dot{h}}}{{\partial} u} - \frac{{{\partial} \dot{x}}}{{\partial} h}\frac{{{\partial} \dot{u}}}{{\partial} x}\frac{{{\partial} \dot{h}}}{{\partial} u} + \frac{{{\partial} \dot{x}}}{{\partial} h}\frac{{{\partial} \dot{u}}}{{\partial} u}\frac{{{\partial} \dot{h}}}{{\partial} x} \\ & \quad - \frac{{{\partial} \dot{y}}}{{\partial} y}\frac{{{\partial} \dot{u}}}{{\partial} u}\frac{{{\partial} \dot{h}}}{{\partial} h} + \frac{{{\partial} \dot{y}}}{{\partial} y}\frac{{{\partial} \dot{u}}}{{\partial} h}\frac{{{\partial} \dot{h}}}{{\partial} u} + \frac{{{\partial} \dot{y}}}{{\partial} u}\frac{{{\partial} \dot{u}}}{{\partial} y}\frac{{{\partial} \dot{h}}}{{\partial} h} - \frac{{{\partial} \dot{y}}}{{\partial} u}\frac{{{\partial} \dot{u}}}{{\partial} h}\frac{{{\partial} \dot{h}}}{{\partial} y}, \\ \end{aligned} $$
$$\begin{aligned} a_{4} & = \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} u}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} \\ & \quad + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} x}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} y}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} \\ & \quad - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} \\ & \quad {\text{ + }}\frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} \\ & \quad - \frac{{{\partial} \dot{z}}}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}} + \frac{{{\partial} \dot{z}}}{{{\partial} y}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} \\ & \quad + \frac{{{\partial} \dot{z}}}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} \\ & \quad - \frac{{{\partial} \dot{z}}}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} + \frac{{{\partial} \dot{z}}}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{z}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{z}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} \\ & \quad - \frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} + \frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} \\ & \quad + \frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} + \frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} - \frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} + \frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} x}}, \\ \end{aligned}$$
$$ \begin{aligned} a_{5} & = \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} y}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}} \\ & \quad - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} x}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} y}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} \\ & \quad - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} x}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} \\ & \quad - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} \\ & \quad - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} x}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} x}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} \\ & \quad - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} \\ & \quad - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} \\ & \quad + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} y}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} y}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} \\ & \quad + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial}
\dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} \\ & \quad - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} \\ & \quad + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} + \frac{{{\partial} \dot{z}}}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{z}}}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{z}}}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{z}}}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} \\ & \quad - \frac{{{\partial} \dot{z}}}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} + \frac{{{\partial} \dot{z}}}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{z}}}{{{\partial} y}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{z}}}{{{\partial} y}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{z}}}{{{\partial} y}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} - \frac{{{\partial} \dot{z}}}{{{\partial} y}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} \\ & \quad - \frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} + \frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} - \frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} + \frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}, \\ \end{aligned} $$
$$\begin{aligned} a_{6} & = - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} \\ & \quad - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} y}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} y}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} y}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} y}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} \\ & \quad + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} \\ & \quad - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{z}}}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} \\ & \quad - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} \\ & \quad + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} x}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} \\ & \quad - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} x}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}+\frac{{{\partial} \dot{q}_{H} }}{{{\partial} x}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} x}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} x}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} \\ & \quad + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} x}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} y}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} y}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} y}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} \\ & \quad - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} y}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial}
\dot{h}}}{{{\partial} u}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} \\ & \quad - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{z}}}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}. \\ \end{aligned} $$
Appendix C
The expressions for the coefficients \(X_{j} \left( {j = 0,1, \ldots ,9} \right)\) in Eq. (14) are presented as follows.
$$ X_{0} = - \left( {\frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }} + \frac{{{\partial} \dot{x}}}{{\partial} x} + \frac{{{\partial} \dot{y}}}{{\partial} y} + \frac{{{\partial} \dot{u}}}{{\partial} u} + \frac{{{\partial} \dot{h}}}{{\partial} h}} \right), $$
$$ X_{1} = \frac{{Q_{0} }}{{H_{0} }}\frac{{{\partial} {\dot{q}}_{H} }}{{\partial} {z}} - \frac{{Q_{0} }}{{H_{0} }}\left( {e_{qh0} + \frac{{e_{qy0} y_{E} }}{{2\sqrt {h_{E} + 1} }}} \right)\frac{{{\partial} {\dot{h}}}}{{\partial} {z}}, $$
$$ \begin{aligned} X_{2} & = \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} y}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{u}}}{{{\partial} u}} \\
&\quad+ \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}} + \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} u}} \\
&\quad+ \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} + \frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\\
&\quad - \frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}} + \frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} \\ & \quad - \frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} \\ \end{aligned}, $$
$$ \begin{aligned} X_{3} & = - \frac{{Q_{0} }}{{H_{0} }}\left( {\frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{x}}}{{{\partial} x}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{y}}}{{{\partial} y}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{u}}}{{{\partial} u}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}} \right) \\ & \quad + \frac{{e_{{qh0}} Q_{0} h_{E} - Q_{0} e_{{qx0}} }}{{H_{0} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} \\ & \quad - \frac{{Q_{0} e_{{qy0}} \sqrt {h_{E} + 1} }}{{H_{0} }}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}} \\ & \quad + \left( {\frac{{Q_{0} e_{{qh0}} }}{{H_{0} }} + \frac{{e_{{qy0}} Q_{0} y_{E} }}{{2H_{0} \sqrt {h_{E} + 1} }}} \right) \\ & \quad \left( {\frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} + \frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}} \right) \\ \end{aligned} $$
$$ \begin{aligned} X_{4} & = - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} u}} \\ & \quad - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} \\ & \quad - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}} \\ & \quad - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} \\ & \quad + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}} \\ & \quad + \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}} - \frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}} \\ & \quad - \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} \\ & \quad - \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} \\ & \quad - \frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} + \frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} \\ & \quad - \frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} \\ & \quad + \frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} \\ \end{aligned} $$
$$ \begin{aligned} X_{5} & = - \frac{{Q_{0} }}{{H_{0} }}\left( {\begin{array}{*{20}c} { - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} u}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}} \\ { - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}}} \\ { + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}} \\ \end{array} } \right) \\ & \quad - \frac{{e_{{qh0}} Q_{0} h_{E} - Q_{0} e_{{qx0}} }}{{H_{0} }}\left( {\frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}} + \frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} + \frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}} \right) \\ & \quad + \frac{{Q_{0} e_{{qy0}} \sqrt {h_{E} + 1} }}{{H_{0} }}\left( {\frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}} + \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}} + \frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}} \right) \\ & \quad + \left( {\frac{{Q_{0} e_{{qh0}} }}{{H_{0} }} + \frac{{e_{{qy0}} Q_{0} y_{E} }}{{2H_{0} \sqrt {h_{E} + 1} }}} \right)\left( {\begin{array}{*{20}c} { - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}} \\ { + \frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} + \frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}} \\ \end{array} } \right), \\ \end{aligned} $$
$$ \begin{aligned} X_{6} & = - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}} \\ & \quad + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} \\ & \quad + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} \\ & \quad + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} \\ & \quad + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} \\ & \quad - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} \\ & \quad + \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} \\ & \quad - \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} \\ & \quad + \frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} \\ & \quad + \frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} \\ & \quad - \frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} + \frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} \\ \end{aligned}, $$
$$ \begin{aligned} X_{7} & = - \frac{{Q_{0} }}{{H_{0} }}\left( {\begin{array}{*{20}c} {\frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} y}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}} \\ { + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} h}}} \\ { - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}} \\ { + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} h}}} \\ { + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}} \\ { - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}} \\ \end{array} } \right) \\ & \quad - \frac{{e_{{qh0}} Q_{0} h_{E} - Q_{0} e_{{qx0}} }}{{H_{0} }}\left( {\begin{array}{*{20}c} {\frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} h}}} \\ { - \frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} + \frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}} \\ \end{array} } \right) \\ & \quad {\text{ + }}\frac{{Q_{0} e_{{qy0}} \sqrt {h_{E} + 1} }}{{H_{0} }}\left( {\begin{array}{*{20}c} { - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}} \\ { + \frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} - \frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}} \\ \end{array} } \right) \\ & \quad {\text{ + }}\left( {\frac{{Q_{0} e_{{qh0}} }}{{H_{0} }} + \frac{{e_{{qy0}} Q_{0} y_{E} }}{{2H_{0} \sqrt {h_{E} + 1} }}} \right)\left( {\begin{array}{*{20}c} {\frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}}} \\ { + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}}} \\ { + \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} - \frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} x}}} \\ { + \frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}} \\ \end{array} } \right), \\ \end{aligned} $$
$$ \begin{aligned} X_{8} & = - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} \\ & \quad + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} \\ & \quad - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} x}}, \\ \end{aligned} $$
$$ \begin{aligned} X_{9} & = - \frac{{Q_{0} }}{{H_{0} }}\left( {\begin{array}{*{20}c} { - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} h}}} \\ { + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} z}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} x}}} \\ { + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}{\text{ + }}\frac{{{\partial} \dot{q}_{H} }}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} y}}} \\ { + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} x}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} y}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} y}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} y}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} y}}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}} \\ { - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} x}}} \\ { + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} h}}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}} \\ \end{array} } \right) \\ & \quad - \frac{{e_{{qh0}} Q_{0} h_{E} - Q_{0} e_{{qx0}} }}{{H_{0} }}\left( {\begin{array}{*{20}c} { - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}} \\ { + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}} \\ \end{array} } \right) \\ & \quad {\text{ + }}\frac{{Q_{0} e_{{qy0}} \sqrt {h_{E} + 1} }}{{H_{0} }}\left( {\frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} h}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} h}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} h}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}} \right) \\ & \quad {\text{ + }}\left( {\frac{{Q_{0} e_{{qh0}} }}{{H_{0} }} + \frac{{e_{{qy0}} Q_{0} y_{E} }}{{2H_{0} \sqrt {h_{E} + 1} }}} \right)\left( {\begin{array}{*{20}c} {\frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} u}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} y}}\frac{{{\partial} \dot{u}}}{{{\partial} u}}\frac{{{\partial} \dot{h}}}{{{\partial} z}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} y}} + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} x}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} y}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}} \\ { + \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} z}}\frac{{{\partial} \dot{h}}}{{{\partial} x}} - \frac{{{\partial} \dot{q}_{H} }}{{{\partial} q_{H} }}\frac{{{\partial} \dot{x}}}{{{\partial} y}}\frac{{{\partial} \dot{y}}}{{{\partial} u}}\frac{{{\partial} \dot{u}}}{{{\partial} x}}\frac{{{\partial} \dot{h}}}{{{\partial} z}}} \\ \end{array} } \right). \\ \end{aligned} $$
Rights and permissions
About this article
Cite this article
Guo, W., Xu, X. Critical sectional area of surge tank based on bifurcation and chaos behaviors of hydraulic-mechanical coupling hydropower station. Nonlinear Dyn 110, 1297–1322 (2022). https://doi.org/10.1007/s11071-022-07672-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-022-07672-4