This　study　proposes　a　semi-analytical　method　for　free　vibration　analysis　of　typical　polyhedral　structural　elements　based　on　3D　elasticity　theory.　The　polyhedral　structures　incorporate　generalized　elastic　boundaries　modeled　using　artificial　springs　to　represent　boundary　stiffness.　The　material　properties　of　these　structures　vary　according　to　gradients　in　multiple　directions.　The　Lagrangian　energy　function　for　the　polyhedral　element　is　derived　within　its　original　irregular　physical　domain.　The　proposed　method　employs　a　trilinear　transformation　to　map　the　irregular　hexahedral　domain　into　a　regular　cubic　domain,　where　the　Jacobi　orthogonal　polynomials-based　Ritz　process　is　used　to　determine　vibration　characteristics.　Initially　developed　for　irregular　hexahedron,　the　presented　method　can　also　be　extended　to　pentahedral　and　tetrahedral　structures,　such　as　triangular　plates/prisms　and　quadrilateral/triangular　pyramids,　which　can　be　viewed　as　special　cases　of　hexahedron　with　some　coincident　vertices.　This　extends　the　applicability　of　the　proposed　method　beyond　the　regular　physical　domains　handled　by　the　classical　Ritz　method.　Mover,the　presented　method　leverages　three-dimensional　elasticity　theory　without　displacement　field　assumptions,　enabling　it　to　handle　polyhedron　with　various　geometric　characteristics　including　solids,　plates　and　prisms.　The　validity　of　method　is　demonstrated　by　comparing　its　results　with　finite　element　analysis　and　open　literature.　Numerical　simulations　on　MDFGMs　parallelogram　plate　reveal　the　effects　of　material　gradients,　geometric　configurations,　and　boundary　restraints　on　vibration　characteristics.　The　results　have,　for　the　first　time,　demonstrated　that　variations　in　material　gradients　within　structural　elements　can　alter　the　modal　sequence.　©　2025　Elsevier　Ltd