In　this　paper,　a　Fourier　expansion-based　differential　quadrature　(FDQ)　method　is　developed　to　analyze　numerically　the　transverse　nonlinear　vibrations　of　an　axially　accelerating　viscoelastic　beam.　The　partial　differential　nonlinear　governing　equation　is　discretized　in　space　region　and　in　time　domain　using　FDQ　and　Runge-Kutta-Fehlberg　methods,　respectively.　The　accuracy　of　the　proposed　method　is　represented　by　two　numerical　examples.　The　nonlinear　dynamical　behaviors,　such　as　the　bifurcations　and　chaotic　motions　of　the　axially　accelerating　viscoelastic　beam,　are　investigated　using　the　bifurcation　diagrams,　Lyapunov　exponents,　Poincare　maps,　and　three-dimensional　phase　portraits.　The　bifurcation　diagrams　for　the　in-plane　responses　to　the　mean　axial　velocity,　the　amplitude　of　velocity　fluctuation,　and　the　frequency　of　velocity　fluctuation　are,　respectively,　presented　when　other　parameters　are　fixed.　The　Lyapunov　exponents　are　calculated　to　further　identify　the　existence　of　the　periodic　and　chaotic　motions　in　the　transverse　nonlinear　vibrations　of　the　axially　accelerating　viscoelastic　beam.　The　conclusion　is　drawn　from　numerical　simulation　results　that　the　FDQ　method　is　a　simple　and　efficient　method　for　the　analysis　of　the　nonlinear　dynamics　of　the　axially　accelerating　viscoelastic　beam.