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DESIGNING AND DETAILING OF BUILDING SYSTEMS. MECHANICS IN CIVIL ENGINEERING

Asymptotic expansion of the integral with two parameters

Vestnik MGSU 7/2014
  • Kuzmina Lyudmila Ivanovna - National Research Institute “Higher School of Economics” Candidate of Physico-Mathematical Sciences, Professor Assistant, Department of Higher Mathematics of Moscow Institute of Electronics and Mathematics, National Research Institute “Higher School of Economics”, 20 ulitsa Myasnitskaya, Moscow, 101000, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Osipov Yuriy Viktorovich - Moscow State University of Civil Engineering (MGSU) Candidate of Physico-Mathematical Sciences, Professor, Department of Information Sciences and Applied Mathematics, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 34-40

In our work we study a classic pursuit problem in which two material points - a Pursuer and a Pursued - move in a plane at constant velocities. The velocity vector of the Pursued does not change its direction and the velocity vector of the Pursuer turns and always aims at the Pursued. If the Pursuer moves at a higher speed, it will overtake the Pursued for any initial angle between velocity vectors. For example, a crane system simultaneously producing three movements: rotation, extension/retraction and luffing, may seize the cargo moving in a straight line while the crane is standing motionless. In the coordinate system of the Pursuer, the path length of the Pursued is given by an integral, depending on two parameters: the ratio of the initial velocities of two points and the initial angle between them. The theorem on the asymptotic integral expansion is formulated and proved considering the speed of the Pursuer is much greater than the speed of the Pursued. The first two nonzero terms of the asymptotic expansion provide fast convergence to the exact value of the integral because of the absence of the first- and the third-order asymptotic elements. The third nonzero element of the fifth order allows to determine the difference of path lengths corresponding to the adjacent initial angles between the velocities of the points.

DOI: 10.22227/1997-0935.2014.7.34-40

References
  1. Nahin Paul J. Chases and Escapes: The Mathematics of Pursuit and Evasion. Princeton University Press, 2007, 270 pp.
  2. Mungan C.E. A Classic Chase Problem Solved from a Physics Perspective. European Journal of Physics. 2005, vol. 26, pp. 985—990.
  3. Simoson A.J. Pursuit Curves for the Man in the Moone. The College Mathematics Journal. Washington, 2007, vol. 38, no. 5, pp. 330—338.
  4. Rikhsiev B.B. Differentsial'nye igry s prostym dvizheniem [Differential Games with Simple Motion]. Tashkent, Fan Publ., 1989, 232 p.
  5. Bernhart A. Curves of Pursuit. Scripta Mathematica. 1954, vol. 20, pp. 125—141.
  6. Krasovskiy N.N. Igrovye zadachi o vstreche dvizheniy [Game Problems on the Meeting of Movements], Moscow, Nauka Publ., 1970, 420 p.
  7. Azamov A.A., Kuchkarov A.Sh., Samatov B.O. O svyazi mezhdu zadachami presledovaniya, upravlyaemosti i ustoychivosti v tselom v lineynykh sistemakh s raznotipnymi ogranicheniyami [The Relation between the Chase, Controllability and Overall Stability Problems in Linear Systems with Heterogeneous Constraints]. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics]. 2007, vol. 71, no. 2, pp. 259—263.
  8. Barton J.C., Eliezer C.J. On Pursuit Curves. The Journal of the Australian Mathematical Society. Ser. B41, 2000, pp. 358—371.
  9. Kuzmina L.I., Osipov Yu.V. Raschet dliny traektorii dlya zadachi presledovanya [Path Length Calculation in the Pursuit Problem]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2013, no. 12, pp. 20—26.
  10. Silagadze Z.K., Chashchina O.I. Zadacha presledovaniya zaytsa volkom kak uprazhnenie elementarnoy kinematiki [The Dog-and-rabbit Purcuit Problem as an Exercise in Introductory Kinematics]. Vestnik NGU, Seriya Fizika [Bulletin of Novosibirsk State University, Issue Physics]. 2010, vol. 5, no. 2, pp. 111—115.
  11. Silagadze Z.K., Tarantsev G.I. Comment on ‘Note on the Dog-and-rabbit Chase Problem in Introductory Kinematics’. European Journal of Physics. 2010, vol. 31, pp. 37—38.
  12. Kuzmina L.I., Osipov Yu.V. Calculation of the Pursuit Curve Length. Journal for Computational Civil and Structural Engineering. Moscow, ASV Publ., 2013, vol. 9, no. 3, pp. 31—39.
  13. Kuzmina L.I., Osipov Yu.V. Asimptotika dliny traektorii v zadache presledovaniya [Path Length Asymptotics in the Pursuit Problem]. Voprosy prikladnoy matematiki i vychislitel'noy mekhaniki [Problems of Applied Mathematics and Computational Mechanics]. Moscow, MGSU Publ., 2013, vol. 16, pp. 238—249.
  14. Maslov V.P. Asimptoticheskie metody i teoriya vozmushcheniy [Asymptotic Methods and Perturbation Theory]. Moscow, Nauka Publ., 1988, 310 p.
  15. Olver F. Introduction to Asymptotics and Special Functions. New York, Academic Press, 1974, 375 p.

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