A multi-modal nonlinear dynamic model to investigate time-domain responses of a micro-cantilever in fluids

Abstract

In this current work, a new nonlinear dynamic model based on the forced Van der Pol oscillator is introduced to demonstrate the time-domain sensitivities of the micro-cantilever to the varying properties of the surrounding fluids. Effects of diverse multi-frequency excitations on the hydrodynamically forced displacements are investigated for the Glycerol-water solutions with different concentrations. Driving forces at the eigenmode frequencies are applied simultaneously to actuate the micro-cantilever in multi-modal operations. The hydrodynamic force induces notable variations in the observables of high-frequency steady-state vibrations. To illustrate, the frequency of the displacements decreases with increasing dynamic viscosity and density of the fluids (among 55% and 85% Glycerol-water solutions) in bimodal- and trimodal-frequency excitations. Essentially, the observable responses are often used to distinguish the surrounding fluids in which the micro-cantilever operates. In addition, steady-state observables are achieved at only particular eigenmodes in single- and multi-frequency operations. It is highlighted that the periodic oscillations are obtained for the first and second eigenmodes with the highest value of forced Van der Pol parameter (μ = 1030). Clearly, higher eigenmodes require different values of the nonlinearity parameter to acquire periodic vibrations in multi-modal operations. In general, achieving steady-state observables is substantially critical in quantifying sensitivity to varying fluid properties. For instance, the vibration frequency of around 7.33 MHz and amplitude of around 0.03 pm are obtained at the first eigenmode for 75% Glycerol-water solution in tetra-modal operations. Note that femtometer amplitudes of deflections can be measured using quantum-enhanced AFM techniques. The frequency responses obtained in this work are compared with the measured ones in the literature and the results show satisfactory agreements. Therefore, a novel multi-modal nonlinear dynamic model enables to quantify observable sensitivity to micro-rheological properties at higher eigenmodes of the micro-cantilever.