Suppression　of　noise　and　vibration　suppression　is　important　in　various　fields,　such　as　the　living　environment,　industrial　development,　and　national　defense　and　security.　The　bandgap　properties　of　phononic　crystal　metamaterials　provide　an　approach　for　controlling　and　eliminating　harmful　vibrations　in　equipment　and　noise　in　the　environment.　In　this　study,　we　used　two　types　of　two-dimensional　honeycomb　gyroscopic　metamaterials:　free　and　constrained.　The　dynamic　equations　of　the　two　systems　were　established　using　angular　momentum　and　Lagrange　theorems.　The　dispersion　relations　of　the　two　systems　were　obtained　based　on　the　Bloch　theorem,　and　the　influence　of　the　gyroscope　angular　momentum　or　gyroscope　speed　on　the　dispersion　relations　was　analyzed.　Numerical　simulations　were　conducted　to　analyze　the　wave　propagation　characteristics　and　polarization　under　different　excitation　conditions　in　a　limited　space　for　both　types　of　metamaterial　structures.　The　constrained-type　and　free-type　metamaterials　were　compared,　and　the　regularities　of　the　dispersion　relations　and　wave　propagation　characteristics　by　the　gyroscope　effect　were　summarized.　This　study　provided　a　comprehensive　and　in-depth　understanding　of　the　bandgap　and　wave　propagation　properties　of　gyroscopic　metamaterials　and　provided　ideas　for　the　design　of　bandgap　modulation　in　metamaterials.　　©　2024　American　Physical　Society.