This series of three Mathcad instructional documents permits students to explore the relationship between the time and frequency representations of vibrational motion in simple diatomic molecules. Molecular vibrations are described by a harmonic oscillator model in FTMolVib.mcd, an anharmonic "Morse" oscillator model in FTAnharm.mcd, and a damped harmonic oscillator model in FTLifeTime.mcd. In each case, a classical equation of motion is defined and is solved using the Runge-Kutta method, yielding the bond displacement as a function of time. A Fourier transform is subsequently performed to obtain the frequency spectra of these vibrations. Exercises included throughout and at the end of each document allow students to practice the numerical methods and further investigate how bond anharmonicity and vibrational damping affect the frequency spectrum. Upon completing these documents, students will recognize the link between bond anharmonicity and overtone spectra, and will be familiar with the concepts of anharmonic red shifts and lifetime broadening.