Nutation　gear　transmission　combines　nutation　drive　mode　with　double　arc　tooth　profile,　which　has　the　advantage　of　large　deceleration　ratio　and　solves　various　problems　in　the　traditional　involute　gear　profile　transmission.　In　his　pa-per,　the　numerical　solution　and　MATLAB　software　are　used　to　solve　the　free　vibration　equation　of　the　nutation　system.　And　the　undamped　free　vibration　natural　characteristics　of　the　nutation　transmission　system　of　bevel　gears　are　obtained.　Then　it　reveals　the　influence　of　different　physical　parameters　on　the　natural　frequency　of　the　system.　The　study　and　analysis　mode　shape　of　double　circular　spiral　bevel　gears　under　the　natural　frequency　are　carried　out　by　using　finite　element　software　ANSYS.　The　natural　frequencies　of　the　system　and　each　double　circular　arc　bevel　gear　are　compared　and　analyzed.　After　it　the　natural　frequencies　of　the　nutation　drive　system　with　150%,　200%　and　250%　average　meshing　stiffness　and　150%,　200%　and　250%　nutation　gear　support　stiffness　are　calculated　and　compared.　At　last　chapter,　we　get　the　orders　vibration　pattern　and　know　the　7th　order　is　the　most　influential　one　by　finite　element　analysis　in　the　ANASY.　The　natural　frequencies　of　the　system　and　each　double　arc　bevel　gear　are　compared　and　analyzed.　©　2020,　Springer　Nature　Singapore　Pte　Ltd.