This　paper　applies　the　asymptotic　perturbation　approach　(APA)　to　obtain　a　simple　analytical　expression　for　the　free　vibration　analysis　of　non-uniform　and　non-homogenous　beams　with　different　boundary　conditions.　A　linear　governing　equation　of　non-uniform　and　non-homogeneous　beams　is　obtained　based　on　the　Euler-Bernoulli　beam　theory.　The　perturbative　theory　is　employed　to　derive　an　asymptotic　solution　of　the　natural　frequency　of　the　beam.　Finally,　numerical　solutions　based　on　the　analytical　method　are　illustrated,　where　the　effect　of　a　variable　width　ratio　on　the　natural　frequency　is　analyzed.　To　verify　the　accuracy　of　the　present　method,　two　examples,　piezoelectric　laminated　trapezoidal　beam　and　axially　functionally　graded　tapered　beam,　are　presented.　The　results　are　compared　with　those　results　obtained　from　the　finite　element　method　(FEM)　simulation　and　the　published　literature,　respectively,　and　a　good　agreement　is　observed　for　lower-order　beam　frequencies.　(C)　2018　Elsevier　Inc.　All　rights　reserved.