Closed-form transient response induced by structural singularity (application to catenary–pantograph systems)

This study investigates transient wave responses in a catenary–pantograph system, where localized structural irregularities can generate disturbances that increase wear and maintenance costs. Although many analytical models exist for catenary dynamics, time-domain closed-form solutions for transients caused by localized singularities (e.g., connectors or splicers) are still scarce, and the effect of locally increased mass is often neglected despite its practical importance. To address this gap, we propose an analytically tractable model for a pantograph passing a singularity. The catenary is modeled as an infinite string with a localized mass–spring element, which is excited by a harmonic force traveling at constant speed. This moving harmonic excitation represents periodic contact force fluctuations arising from the interaction between the pantograph and the periodic catenary structure. We derive a time-domain closed-form solution and interpret it in terms of Doppler-shifted waves and impedance-governed reflection and transmission at the singularity. The solution clarifies the emergence of transition radiation and shows that its magnitude depends on the singularity location relative to the periodic structure, providing guidance for where to place connectors or splicers. The proposed framework provides practitioners with valuable insights to support the planning of full-scale tests and the interpretation of experimental outcomes.