Authors' Reply

Abstract

conditions, which in some cases do not vanish, but instead lead to a limit cycle behavior. This is especially true at the parameter points on the transition curves separating stability from instability in which periodic motions are predominant. CONCLUDING REMARKSThe stability analysis of the pantograph}catenary system based on Hill's method of in"nite determinant clearly reveals additional unstable areas at lower values of r not mentioned in references [1, 2]. The new parameter plane depicting transition curves separating stability from instability is validated using the free dynamic response results. We also pointed out the misapplication of the Floquet theory to the damped Mathieu equation resulting in less conservative solutions and missing instability regions. Furthermore, the analytical solution for the steady state forced response derived from the straightforward perturbation method is shown to be limited to only small values of . Finally, we think these reported discrepancies do not actually alter the main conclusions of the earlier study [1, 2], but it does provide a more complete characterization of the stability behavior and points out an obvious misapplication of the Floquet theory.

Comments on “The Stability Analysis of Pantograph-Catenary System Dynamics”

“…This resulted in narrower unstable regions in the earlier study compared to the current one. We believe the error was made during the application of the Floquet theory in assessing the stability of the system based on the pair of eigenvalues of the monodromy matrix (also known as the characteristic multipliers [4]), which were computed from the simulated free dynamic response under two speci"ed initial conditions as suggested by Nayfeh and Mook [3]. The problem here is that this approach is meant only for the case of the undamped Mathieu equation.…”

“…Recall that equation (15) is applied earlier to demonstrate the calculation of the characteristic multipliers. Since it is well studied, one can simply apply the available published solutions including its parameter plane (generally known as the Strutt diagram [3,4]) to further approximate the dynamic characteristics of the pantographcatenary system. However, there is a limitation.…”

“…However, the presented stability analysis, based on the Floquet theory, appears to be slightly de"cient, 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 ( , r) by applying Hill's method of in"nite determinants [3,4]. Selected cases of stable and unstable free-responses based on our parameter plane result are also veri"ed by employing the 4/5th order Runge}Kutta integration algorithm.…”