Dynamic Stability and Post-Critical Processes of Slender Auto-Parametric Systems
High-rise structures exposed to a strong vertical component of an earthquake excitation are endangered by auto-parametric resonance effect. While in a sub-critical state, the vertical and horizontal response components are independent. Exceeding a certain limit causes the vertical response to lose stability and induces dominant horizontal response. This effect is presented using two mathematical models: (1) the non-linear lumped mass model; and (2) the one dimensional model with continuously distributed parameters. Analytical and numerical treatment of both leads to three different types of the response: (1) semi-trivial sub-critical state with zero horizontal response component; (2) post-critical state (auto-parametric resonance) with a periodic or attractor type chaotic character; and (3) breaking through a certain limit, the horizontal response exponentially rises and leads to a collapse. Special attention is paid to transition from a semi-trivial to post-critical state in case of time limited excitation period as it concerns the seismic processes.