In　practice,　it　is　difficult　to　avoid　the　axis　angle　deviation　when　some　regular　surfaces　are　in　micro-sliding,　such　as　gears　and　machined　surfaces.　In　order　to　better　investigate　the　micro-motion　contact　characteristics,　a　crossed　paraboloidal　contact　model　under　frictional　condition　is　proposed　to　simulate　both　tangential　displacement-controlled　fretting　and　the　evolution　of　the　energy　dissipation　in　a　load　cycle.　By　deriving　the　theoretical　of　the　normal　and　tangential　contact　course　of　the　model,　the　load–displacement　curves　during　initial　loading,　unloading,　and　reloading　stage　are　presented.　On　this　basis,　the　hysteresis　curve　is　then　obtained　by　integrating　the　closed　area　surrounded　by　load–displacement　during　unloading　and　reloading,　which　also　means　that　the　empirical　formulation　for　microslip　in　a　load　cycle　is　constructed.　This　study　also　reveals　the　plastic　yield　phenomenon　under　pure　normal　loading　and　plastic　shakedown　behavior　caused　by　cyclic　reciprocating　displacement　loads.　In　addition,　the　research　on　the　junction　growth,　the　evolution　of　tangential　load,　and　hysteresis　curve　with　different　COFs　under　multiple-cycle　load　is　also　carried　out.　The　implications　of　involved　parameters,　such　as　friction　coefficient,　axis　intersection　angle,　normal　load,　and　so　on,　are　discussed　with　respect　to　hysteresis　curve　shape　and　energy　dissipation.　The　difference　in　hysteresis　and　energy　dissipation　curves　between　the　paraboloidal　contact　model　and　other　classic　contact　models　is　then　presented.　It　is　discovered　by　comparison　with　other　models　that　the　paraboloidal　contact　model　presents　a　relatively　high　energy　dissipation　in　a　load　cycle.　©　2024　American　Society　of　Mechanical　Engineers　(ASME).　All　rights　reserved.