The research program includes development of high-resolution mechanical spectroscopy. Novel high-resolution computational methods are designed to estimate the resonant frequency, f₀, and logarithmic decrement, δ. Hilbert transform-based methods and interpolated discrete Fourier Transform methods are devised to estimate δ and f₀ from free decaying oscillations (exponentially damped harmonic oscillations) embedded in an experimental noise and biased by zero-point drift. New methods yield the smallest dispersion and scatter of experimental points δ and f₀, and higher density of experimental points in low-frequency mechanical spectrometer.