Hopf-Friedrichs bifurcation and the hunting of a railway axle
Author:
R. R. Huilgol
Journal:
Quart. Appl. Math. 36 (1978), 85-94
MSC:
Primary 70.34; Secondary 34CXX
DOI:
https://doi.org/10.1090/qam/478858
MathSciNet review:
478858
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Abstract: After deriving the equations of motion which govern the lateral and yaw motions of a railway axle, these are cast in the form of a system of first-order nonlinear differential equations. To this system the Hopf-Friedrichs bifurcation theory is applied to determine when a periodic orbit will bifurcate from the equilibrium position. Sufficient conditions to guarantee the stability of the orbit are investigated.
F. W. Carter, Railway electric traction, Arnold (London), 1922
- Eberhard Hopf, Abzweigung einer periodischen Lösung von einer stationären eines Differentialsystems, Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Nat. Kl. 95 (1943), no. 1, 3–22 (German). MR 39141
- K. O. Friedrichs, Lectures on advanced ordinary differential equations, Gordon and Breach Science Publishers, New York-London-Paris, 1965. Notes by P. Berg, W. Hirsch, P. Treuenfels. MR 0224884
- A. B. Poore, On the theory and application of the Hopf-Friedrichs bifurcation theory, Arch. Rational Mech. Anal. 60 (1975/76), no. 4, 371–393. MR 404766, DOI https://doi.org/10.1007/BF00248886
- J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications, Springer-Verlag, New York, 1976. With contributions by P. Chernoff, G. Childs, S. Chow, J. R. Dorroh, J. Guckenheimer, L. Howard, N. Kopell, O. Lanford, J. Mallet-Paret, G. Oster, O. Ruiz, S. Schecter, D. Schmidt and S. Smale; Applied Mathematical Sciences, Vol. 19. MR 0494309
- Lamberto Cesari, Functional analysis and Galerkin’s method, Michigan Math. J. 11 (1964), 385–414. MR 173839
J. K. Hale, Applications of alternative problems, Lecture Notes, Center for Dynamical Systems, Brown University, Providence, 1971
R. P. Brann, Some aspects of the hunting of a railway axle, J. Sound. Vib. 4, 18–32 (1966)
F. W. Carter, On the action of a locomotive driving wheel, Proc. Roy. Soc. Lond. A 112 151–157 (1926)
J. J. Kalker, Survey of mechanics of contact between solid bodies, Zeit. angew. Math. Mech., 57, T3–T17 (1977)
F. W. Carter, Railway electric traction, Arnold (London), 1922
E. Hopf, Abzweigung einer periodischer Lösung von einer stationären Lösung eines Differentialsystems, Ber. Verh. Sächs. Akad. Wiss. Leipzig. Math. Nat. Kl. 94, 1–22 (1942) (see also the translation in [5])
K. O. Friedrichs, Advanced ordinary differential equations, Gordon and Breach (New York), 1965
A. B. Poore, On the theory and application of the Hopf-Friedrichs bifurcation theory, Arch. Ratl. Mech. Anal. 60, 371–393 (1976)
J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications, Springer-Verlag (New York), 1976
L. Cesari, Functional analysis and Galerkin’s method, Mich. Math. J. 11, 385–418 (1964)
J. K. Hale, Applications of alternative problems, Lecture Notes, Center for Dynamical Systems, Brown University, Providence, 1971
R. P. Brann, Some aspects of the hunting of a railway axle, J. Sound. Vib. 4, 18–32 (1966)
F. W. Carter, On the action of a locomotive driving wheel, Proc. Roy. Soc. Lond. A 112 151–157 (1926)
J. J. Kalker, Survey of mechanics of contact between solid bodies, Zeit. angew. Math. Mech., 57, T3–T17 (1977)
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Article copyright:
© Copyright 1978
American Mathematical Society
