We present results of a systematic study of the c axis lattice dynamics in single layer Bi 2 Sr 2 CuO 6 (Bi2201), bilayer Bi 2 Sr 2 CaCu 2 O 8 (Bi2212) and trilayer Bi 2 Sr 2 Ca 2 Cu 3 O 10 (Bi2223) cuprate superconductors. Our study is based on both experimental data obtained by spectral ellipsometry on single crystals and theoretical calculations. The calculations are carried out within the framework of a classical shell model, which includes long-range Coulomb interactions and short-range interactions of the Buckingham form in a system of polarizable ions. Using the same set of the shell model parameters for Bi2201, Bi2212 and Bi2223, we calculate the frequencies of the Brillouin-zone center phonon modes of A 2u symmetry and suggest the phonon mode eigenvector patterns. We achieve good agreement between the calculated A 2u eigenfrequencies and the experimental values of the c axis TO phonon frequencies which allows us to make a reliable phonon mode assignment for all three Bi-based cuprate superconductors. We also present the results of our shell model calculations for the Γ-point A 1g symmetry modes in Bi2201, Bi2212 and Bi2223 and suggest an assignment that is based on the published experimental Raman spectra. The superconductivity-induced phonon anomalies recently observed in the c axis infrared and resonant Raman scattering spectra in trilayer Bi2223 are consistently explained with the suggested assignment.