Two key issues in earthquake resistant design of structures are resonance and damping. Nonlinear resonance problems originally encountered around 1960 are re-formulated as critical excitation problems for elastic-plastic structures under a simple impulse sequence. The double impulse with two impulses of positive and negative directions is introduced as a simple representative model of near-fault pulse-type ground motions. The interval of such impulses is treated as a variable to derive a critical input. Furthermore, another critical excitation problem is considered for elastic-plastic structures under multiple impulses representing long-period, long-duration ground motions. These critical excitation problems are solved by taking full advantage of the energy balance approach between the kinetic energy and strain/dissipated energy.