  We demonstrate strong coupling between the flexural vibration modes of a
clamped-clamped micromechanical resonator vibrating at low amplitudes. This
coupling enables the direct measurement of the frequency response via
amplitude- and phase modulation schemes using the fundamental mode as a
mechanical detector. In the linear regime, a frequency shift of
$\mathrm{0.8\,Hz}$ is observed for a mode with a line width of
$\mathrm{5.8\,Hz}$ in vacuum. The measured response is well-described by the
analytical model based on the Euler-Bernoulli beam including tension.
Calculations predict an upper limit for the room-temperature Q-factor of
$\mathrm{4.5\times10^5}$ for our top-down fabricated micromechanical beam
resonators.
