A two-ion pair in a linear Paul trap is extensively used in the research of the simplest quantum-logic system;however,there are few quantitative and comprehensive studies on the motional mode coupling of two-ion systems yet.This study proposes a method to investigate the motional mode coupling of sympathetically cooled two-ion crystals by quantifying three-dimensional (3D) secular spectra of trapped ions using molecular dynamics simulations.The 3D resonance peaks of the 40Ca+-27Al+ pair obtained by using this method were in good agreement with the 3D in-and out-of-phase modes predicted by the mode coupling theory for two ions in equilibrium and the frequency matching errors were lower than 2％.The obtained and predicted amplitudes of these modes were also qualitatively similar.It was observed that the strength of the sympathetic interaction of the 40Ca+-27Al+ pair was primarily determined by its axial in-phase coupling.In addition,the frequencies and amplitudes of the ion pair's resonance modes (in all dimensions) were sensitive to the relative masses of the ion pair,and a decrease in the mass mismatch enhanced the sympathetic cooling rates.The sympathetic interactions of the 40Ca+-27Al+ pair were slightly weaker than those of the 24Mg+-27Al+ pair,but significantly stronger than those of 9Be+-27Al+.However,the Doppler cooling limit temperature of 40Ca+ is comparable to that of 9Be+ but lower than approximately half of that of 24Mg+.Furthermore,laser cooling systems for 40Ca+ are more reliable than those for 24Mg+and 9Be+.Therefore,40Ca+ is probably the best laser-cooled ion for sympathetic cooling and quantum-logic operations of 27Al+ and has particularly more notable comprehensive advantages in the development of high reliability,compact,and transportable 27Al+ optical clocks.This methodology may be extended to multi-ion systems,and it will greatly aid efforts to control the dynamic behaviors of sympathetic cooling as well as the development of low-heating-rate quantum logic clocks.