COMMENTS ON 'THE STABILITY ANALYSIS OF PANTOGRAPH-CATENARY SYSTEM DYNAMICS'

摘要

The authors of references [1,2] successfully formulated a single-degree-of-freedom linear dynamic model with time-varying stiffness coefficient expressed as to analyze the dynamics of the pantograph-catenary system typically used in high-speed electric trains. Here, τ is a non-dimensional time (equal to the nominal natural frequency times real time), y is the vertical motion of the pantograph component, f is the uplift forcing term, ζ is the damping ratio, α is the stiffness variation coefficient, and r>0 is a non-dimensional frequency that is proportional to the train speed. Even though the proposed dynamic model given above is relatively simple, the formulation does seem to contain all the critical elements of the problem. However, the presented stability analysis, based on the Floquet theory, appears to be slightly deficient, and does not provide the complete picture of the stable and unstable regions of interest. In this communication, we present the true nature of the actual stability boundaries including the missing transition curves separating stability from instability in the parameter plane (α,r) by applying Hill's method of infinite determinants [3,4]. Selected cases of stable and unstable free-responses based on our parameter plane result are also verified by employing the 4/5th order Runge-Kutta integration algorithm. Furthermore, equation (1) is transformed into the well-studied standard Mathieu form to reveal additional insight into the dynamic characteristics of the pantograph-catenary system.