Highly sensitive measurements of perturbations in stiffness of a resonator by virtual coupling with a virtual resonator

We propose a highly sensitive method to measure perturbations in stiffness of a resonator such as a cantilever, which is implemented in, for example, an AFM or stiffness meter. Such perturbations have been measured by detecting shifts in the eigenfrequency of the resonator. Recently, by coupling the (primary) resonator with another (secondary) resonator, measurements of much higher sensitivity can be performed owing to shifts in eigenmode. The method requires conditions whereby the secondary resonator is identical to the primary resonator in a state with no perturbation and very weak coupling. We realize these conditions by calculating in real time the dynamics of the secondary resonator in a computer and implementing weak coupling. In practical systems used in viscous environments, shifts in eigenmodes are identified by shifts in the amplitude ratio in a state in which damping is eliminated by velocity feedback. Experiments using small magnetic forces have demonstrated the high sensitivity of the measurements of perturbations in stiffness of the resonator.