  Mechanical resonators that possess coupled modes with harmonic frequency
relations have recently sparked interest due to their suitability for
controllable energy transfer and non-Hermitian dynamics. Here, we show coupling
between high Q-factor ($>\!\!10^4$) resonances with a nearly 1:1 frequency
relation in spatially-symmetric microresonators. We develop and demonstrate a
method to analyze their dynamical behavior based on the simultaneous and
resonant detection of both spectral peaks, and validate this with experimental
results. The frequency difference between the peaks modulates their ringdown,
and creates a beat pattern in the linear decay. This method applies both to the
externally driven and the Brownian motion (thermal) regime, and allows
characterization of both linear and nonlinear parameters. The mechanism behind
this method renders it broadly applicable to both optical and electrical
readout, as well as to different mechanical systems. This will aid studies
using near-degenerate mechanical modes, for e.g., optomechanical energy
transfer, synchronization and gyroscopic sensors.
