Bistable　energy　harvesters　have　been　extensively　studied.　However,　theoretical　research　on　the　dynamics　of　bistable　energy　harvesters　based　on　asymmetric　bistable　composite　laminated　plate　and　shell　structures　has　not　been　conducted.　In　this　paper,　a　theoretical　model　on　the　dynamics　of　an　energy　harvester　based　on　an　asymmetric　bistable　composite　laminated　shell　is　established.　The　dynamic　snap-through,　the　nonlinear　vibrations　and　the　voltage　output　with　two　potential　wells　of　the　bistable　energy　harvester　are　studied.　The　influence　of　the　amplitude　and　the　frequency　for　the　base　excitation　on　the　bistable　energy　harvester　is　studied.　When　the　frequency　for　the　base　excitation　with　a　suitable　amplitude　in　the　frequency　sweeping　is　located　in　a　specific　range　or　the　amplitude　for　the　base　excitation　with　a　suitable　frequency　in　the　amplitude　sweeping　is　located　in　a　specific　range,　the　large-amplitude　dynamic　snap-through,　nonlinear　vibrations　and　voltage　output　with　two　potential　wells　can　be　found　to　occur.　The　amplitude　and　the　frequency　for　the　base　excitation　interact　on　each　other　for　the　specific　amplitude　or　frequency　range　which　migrates　due　to　the　softening　nonlinearity.　The　vibration　in　the　process　of　the　dynamic　snap-through　behaves　as　the　chaotic　vibration.　The　nonlinear　vibrations　of　the　bistable　system　behave　as　the　periodic　vibration,　the　quasi-periodic　vibration　and　the　chaotic　vibration.　This　study　provides　a　theoretical　reference　for　the　design　of　energy　harvesters　based　on　asymmetric　bistable　composite　laminated　plate　and　shell　structures.