  The nonlinear interactions between flexural and torsional modes of a
microcantilever are experimentally studied. The coupling is demonstrated by
measuring the frequency response of one mode, which is sensitive to the motion
of another resonance mode. The flexural-flexural, torsional-torsional and
flexural-torsional modes are coupled due to nonlinearities, which affect the
dynamics at high vibration amplitudes and cause the resonance frequency of one
mode to depend on the amplitude of the other modes. We also investigate the
nonlinear dynamics of torsional modes, which cause a frequency stiffening of
the response. By simultaneously driving another torsional mode in the nonlinear
regime, the nonlinear response is tuned from stiffening to weakening. By
balancing the positive and negative cubic nonlinearities a linear response is
obtained for the strongly driven system. The nonlinear modal interactions play
an important role in the dynamics of multi-mode scanning probe microscopes
