Measuring viscosity using the hysteresis of the non-linear response of a self-excited cantilever

A self-oscillating microcantilever in a feedback loop comprised of a gain, a saturator, and an adjustable phase-shifter is used to measure the viscosity of Newtonian fluids. Shifting the signal of the loop with the adjustable phase-shifter causes sudden jumps in the oscillation frequency of the cantilever. The exact position of these jumps depends on whether the shift imposed by the phase-shifter is increasing or decreasing and, therefore, the self-excited cantilever exhibits a hysteretic non-linear response. This response was studied and the system modeled by a delay differential equation of motion where frequency-dependent added mass and damping terms accounted for the density and the viscosity of the medium. Experimental data were obtained for solutions with different concentrations of glycerol in water and used to validate the model. Two distinct sensing modalities were proposed for this system: the sweeping mode, where the width of the observed hysteresis depends on the viscosity of the medium, and the threshold mode, where a sudden jump of the oscillation frequency is triggered by an arbitrarily small change in the viscosity of the medium.