When a lightning current flows between the lightning entry and exit points of a structure, the lightning current density varies in different parts of the structure depending on the shape of the structure and material variance. The structure can be discretized into parallel wires, called filament wires, running parallel to the current direction. Furthermore, using the filament wire method, we can calculate the current distribution among the wires. For a structure that has a low resistance material such as aluminum, current distribution can be calculated by considering self-inductance of the wire and mutual-inductance between wires but resistance is not considered. However, in modern aircraft, composite materials are used for parts of the structure because of their strength and weight. These composite materials have high resistance compared to metal, and resistance cannot be ignored. Thus, to solve a system of ordinary differential equations for a filament model, inhomogeneous structure, aperture, and resistance of each wire must be considered to obtain the correct current distribution of each part of the structure. However, the numerical solution of the filament wire model does not reveal the region of convergence and the accuracy of the given mathematical model. It also has high time complexity. This paper presents the analytic solution and stability condition for the mutually coupled resistive filament wire model using eigenvalues of given filament wire matrix model. The stability condition is rigorously calculated and the solution is also consistent with the numerical model.
Citation: Joonwoo Park, Raechoong Kang. Analytic solution for the lightning current induced mutually coupled resistive filament wire model[J]. AIMS Mathematics, 2023, 8(8): 19637-19655. doi: 10.3934/math.20231001
When a lightning current flows between the lightning entry and exit points of a structure, the lightning current density varies in different parts of the structure depending on the shape of the structure and material variance. The structure can be discretized into parallel wires, called filament wires, running parallel to the current direction. Furthermore, using the filament wire method, we can calculate the current distribution among the wires. For a structure that has a low resistance material such as aluminum, current distribution can be calculated by considering self-inductance of the wire and mutual-inductance between wires but resistance is not considered. However, in modern aircraft, composite materials are used for parts of the structure because of their strength and weight. These composite materials have high resistance compared to metal, and resistance cannot be ignored. Thus, to solve a system of ordinary differential equations for a filament model, inhomogeneous structure, aperture, and resistance of each wire must be considered to obtain the correct current distribution of each part of the structure. However, the numerical solution of the filament wire model does not reveal the region of convergence and the accuracy of the given mathematical model. It also has high time complexity. This paper presents the analytic solution and stability condition for the mutually coupled resistive filament wire model using eigenvalues of given filament wire matrix model. The stability condition is rigorously calculated and the solution is also consistent with the numerical model.
| [1] | SAE-ARP-5412, Aircraft Lightning Environment and Related Test Waveforms, 2013. Available from: https://www.sae.org/standards/content/arp5412/#: : text = Levels. |
| [2] | F. A. Fisher, J. A. Plumer, R. A. Perala, Lightning protection of aircraft, 2 Eds., 2004. |
| [3] | C. J. Hardwick, S. J. Haigh, B. J. C. Burrows, A filamentary method for calculating induced voltages within resistive structures in either the frequency or time domain, NOAA, International Aerospace and Ground Conference on Lightning and Static Electricity (SEE N89-10429 01-47), 1988, 401–407. |
| [4] | C. J. Hardwick, S. J. Haigh, The electromagnetic environment in CFC structures, NASA, Kennedy Space Center, The 1991 International Aerospace and Ground Conference on Lightning and Static Electricity, 1991. |
| [5] | W. C. Gibson, The method of moments in electromagnetics, Chapman & Hall/CRC, 2015. |
| [6] | D. K. Cheng, Field and wave electromagnetics, 2 Eds., New York: Addison-Wesley, 1992. |
| [7] | H. W. Ott, Electromagnetic compatibility engineering, New York: Wiley, 2009. |
| [8] | E. B. Rosa, F. W. Grover, Formulas and tables for the calculation of mutual and self-inductance (Revised), 3 Eds., Scientific Papers of the Bureau of Standards, 1916. |
| [9] | J. A. Brandão Faria, Electromagnetic foundations of electrical engineering, New York: Wiley, 2008. |
| [10] |
A. Storjohann, On the complexity of inverting integer and polynomial matrices, Comput. Complex., 24 (2015), 777–821. https://doi.org/10.1007/s00037-015-0106-7 doi: 10.1007/s00037-015-0106-7
|
| [11] |
P. Giorgi, C. Jeannerod, G. Villard, On the complexity of polynomial matrix computations, Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, 2003. https://doi.org/10.1145/860854.860889 doi: 10.1145/860854.860889
|
| [12] |
D. A. Harville, Matrix algebra from a statistician's perspective, Taylor Francis, 40 (1997), 164. https://doi.org/10.1080/00401706.1998.10485214 doi: 10.1080/00401706.1998.10485214
|
| [13] |
J. Sherman, W. J. Morrison, Adjustment of an inverse matrix corresponding to changes in the elements of a given column or a given row of the original matrix (abstract), Ann. Math. Statist., 20 (1949), 620–624. https://doi.org/10.1214/aoms/1177729959 doi: 10.1214/aoms/1177729959
|
| [14] | R. L. Burden, J. D. Faires, Numerical analysis, 3 Eds., PWS Publishers, 1985. |
| [15] | R. A. Horn, C. R. Johnson, Matrix analysis, 2 Eds., Cambridge University Press, 2013. |
| [16] | P. J. Olver, C. Shakiban, Applied linear algebra, Pearson Education Inc., 2006. |
| [17] | F. Zhang, Matrix theory, 2 Eds., New York: Springer, 2011. |
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Joonwoo Park, Raechoong Kang. Analytic solution for the lightning current induced mutually coupled resistive filament wire model[J]. AIMS Mathematics, 2023, 8(8): 19637-19655. doi: 10.3934/math.20231001
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