There　is　inevitably　an　angular　offset　during　contact　of　mechanical　structures　such　as　roller　bearings　or　gears.　Most　of　the　previous　contact　models　involve　the　contact　between　two　parallel　cylinders.　This　paper　introduces　the　angle　parameter　into　the　model　constitutive　relation　for　the　first　time.　A　new　cylindrical　contact　model　is　proposed　to　simulate　the　contact　at　different　angles.　In　this　paper,　a　cylinder　contact　model　with　different　axis　crossing　angles　is　proposed.　The　load-displacement　curves　of　the　contact　model　during　loading　and　unloading　are　provided　via　theoretical　derivation　of　the　normal　and　tangential　contact　processes　of　the　contact　model.　On　this　basis,　the　hysteresis　curve　model　of　the　contact　model　is　established.　Through　finite　element　modeling　and　finite　element　analysis,　the　cylinder　hysteresis　curve　models　under　different　axis　crossing　angles　are　verified.　Compared　to　other　common　models,　the　energy　dissipation　of　the　cylindrical　model　is　higher.　By　introducing　the　angle　parameters　into　the　model　constitutive,　it　provides　ideas　for　solving　the　macro　contact　problems　such　as　cylindrical　gear　contact　and　rolling　bearing.　At　the　same　time,　on　the　micro-contact　problem,　it　provides　a　theoretical　basis　for　the　tangential　contact　process　of　the　regular　joint　surface.