The approximate determination of the hunting movement of a railway vehicle by aid of the method of Krylov and Bogoljubov

Summary The author deals with the problem of the lateral oscillations of a railway vehicle moving on a straight track. Because of the possibility that the flange of each of the wheels temporarily contacts the rail belonging to it, the problem lies in the field of the self-sustained vibrations of non-linear mechanical systems with more than one degree of freedom. In the case under consideration, in which the vehicle and the track had been strongly simplified, the limit-cycle for the motion as well as its stability were determined approximately by the method of Krylov and Bogoljubov. This method is described in §§ 2 and 3. § 3 is more specially devoted to the so-called method of the equivalent linear equations. The latter equations are considered valid not only in the case in which the motion is purely periodic (in which a limit-cycle has been reached), but also in the neighbouring case, in which the motion slightly damps out or increases. This enables us to determine not only the limit-cycle but also its stability. In § 4 the method is applied to the problem of the hunting movement of the railway vehicle. The number of degrees of freedom is equal to 2, and we chose for the two generalised coordinates the lateral displacements of the leading wheel set and the trailing wheel set respectively. The linear creep law is assumed to be valid for the relation between the so-called slip coefficients and the tangential forces which the rails exert on the tyres and conversely. The limit-cycle obtained by applying Krylov and Bogoljubov's method corresponds to the limit-cycle previously found by means of Ritz' method. As to the stability it appears that the limit-cycle is always stable.Paper delivered at the Xth International Congress of Applied Mechanics at Stresa Aug. 31st to Sept. 7th, 1960.