Piecewise-smooth　models　are　widely　explored　to　investigate　occasional　contact　or　impact　systems,　such　as　gear　systems　in　mechanical　engineering　and　folding　wings　in　aircraft　structures.　However,　the　complicated　nonlinear　responses　of　non-smooth　features　may　result　in　unexpected　negative　consequences.　In　this　paper,　a　general　vibro-impact　single-degree-of-freedom　system　with　symmetric　and　asymmetric　constraints　is　identified　and　studied.　It　is　first　classified　into　several　categories　based　on　the　relative　placement　of　equilibrium　points　and　switching　manifolds.　The　geometric　structures　of　the　system＇s　phase　space　are　analyzed　to　derive　the　analytical　formulations　of　its　backbone　curves.　The　numerical　findings　show　that　the　theoretical　expression　of　the　backbone　curves　can　precisely　estimate　the　amplitude　of　the　limit　cycles　with　high　energy.　Analytical　and　numerical　approaches　are　used　to　study　grazing　bifurcations,　phase　angle　jumps,　and　energy　changes　in　switching　manifolds.　Finally,　the　dynamical　responses　of　these　systems　under　sinusoidal　period　external　stimulation　with　varying　frequencies　and　amplitudes　are　explored.　The　results　reveal　that　the　occurrence　of　subharmonic　resonance　at　low　frequencies　is　determined　by　the　magnitude　of　external　stimulation,　particularly　when　a　grazing　bifurcation　threshold　is　exceeded.　Notably,　this　criterion　is　not　present　under　free-play　systems.　With　increasing　external　excitation　frequencies,　the　response　amplitude　first　rises,　then　falls,　and　the　system＇s　energy　rapidly　diminishes　due　to　superharmonic　resonance,　potentially　resulting　in　transient　chaos　in　free-play　systems.　©　The　Author(s),　under　exclusive　licence　to　Springer　Nature　B.V.　2024.