Hilbert　spectrum　and　intra-wave　modulation　were　introduced　here.　For　a　SDOF　system　and　a　MDOF　system,　Hilbert　amplitude　spectra　and　energy　spectra　of　the　absolute　acceleration　responses　under　earthquake　for　both　their　tri-linear　stiffness　retrograde　systems　and　their　linear　systems　were　compared.　The　symbols　of　the　non-linear　mechanical　behavior　were　analyzed.　It　was　shown　that　band　broadening　and　dominant　frequency　drift　are　two　important　symbols　of　the　nonlinear　mechanical　behavior　of　the　dynamic　response　of　a　system;　the　acceleration　energy　of　a　linear　system　is　concentrated　in　a　narrow　frequency　band　near　its　natural　frequency　and　intra-wave　modulation　causes　band　broadening　in　Hilbert　energy　spectrum　of　a　tri-linear　stiffness　degradation　system;　due　to　influence　of　yield　frequency　and　cracking　frequency,　the　dominant　frequency　of　the　energy　spectrum　drifts　to　lower　frequency;　in　short　period　structures,　band　broadening　and　dominant　frequency　drift　are　obvious,　but　in　long　period　structures,　they　are　not　significant.