The critical excitation and response of simple dynamic systems

A method for defining the critical excitations and responses of dynamic systems is examined. The critical excitations are those functions which maximize some response norm with respect to the constraints placed on the admissible excitations. A class of critical responses for linear, elastoplastic and hysteretic single degree of freedoms systems is studied, showing the frequency and amplitude relations for these solutions. For linear systems it is shown that the critical excitations producing either a maximum displacement response or maximum energy input are harmonic and derivable from the harmonically excited response functions for the same linear system. The critical excitations for elastoplastic systems, however, are not harmonic and at low frequencies the response is significantly larger than the harmonically excited response. The critical response solutions require higher multiple frequency components to exist. Both periodic and inelastic offset types of critical response are examined for a hysteretic, elastoplastic system and the response characteristics for these solutions are discussed.