The Ishlinskii oscillator model in the Keldysh shimmy problem

A.Yu. Ishlinskii's idea concerning the correspondence between a ‘vibrating system at the threshold of stability and a hypothetical one-degree-of-freedom linear system’ is used. For the case of linear shimmy, procedures are proposed for constructing an oscillator model, checking its correspondence to a prescribed system, and using it to find the vibration frequencies and amplitudes. Within the framework of M.V. Keldysh's most simple calculation scheme, the shimmy of a towed, pliant pneumatic tyre is examined. The obtained laws of turning of the towing gear are compared with the results of numerical integration of the initial equations of motion. It is established that the oscillator model fits the Keldysh system adequately, at least if the length of the towing gear is no smaller than the radius of the pneumatic wheel.