We present here a numerical calculation for sensitivity comparison of single-and bimodal-excitation of micro-cantilever (owing rectangular shape) on the measurement of static acoustic force. We model the micro-cantilever as a point mass in our simulations similar to forced harmonic oscillator with damping. In bimodal operation, the micro-cantilever is excited at its first two flexural modes. Results of amplitude and phase shift obtained through bimodal-frequency excitation are compared with the ones obtained using single-frequency excitation scheme. Oscillation observables are calculated in both excitation schemes with respect to magnitudes of excitation forces at fundamental and second eigenmode. Influence of driving force for micro-cantilever actuation on amplitude and phase shift is explored so that optimum excitation parameters can be provided for highest observable sensitivity. Moreover, we relate our findings with virial and energy dissipation to understand the physics behind observed dynamics. Our results clearly indicates that phase sensitivity at the fundamental eigenmode to static acoustic force could be enhanced using multi-frequency excitation scheme. In addition, results of virial and dissipated power at second eigenmode point out the existence of interaction of flexural modes in the presence of static acoustic force in bimodal-frequency excitation scheme. Therefore, our approach can be used to increase sensitivity of dynamically excited micro-cantilever for measuring ultra-low (pN) acoustic forces.
Virial and energy dissipation, related to oscillation observable responses, possess complementary information regarding acoustic force measurements. In this paper, we introduce a mathematical framework describing the analytic relationship between oscillation observables and energy quantities at the second eigenmode in the measurement of dynamic acoustic forces. We utilize a bimodal-frequency excitation scheme for actuation of the micro-cantilever array to obtain high-sensitivity frequency bands. Herein, we analyze the virials of acoustic force interaction and the energy dissipation levels on the domain of acoustic force frequency. For our case, we obtain the high-frequency bands of around 200-270 kHz and 440-570 kHz for the force strengths in the range of 4.0-36.0 pN. In addition, results of virials and dissipated power with respect to acoustic force strengths are introduced for low- and high-sensitivity frequency regions. Therefore, the energy quantities can be robustly utilized to determine high-sensitivity frequency windows in the measurement of dynamic acoustic forces.
In this paper, we present a computational investigation to study amplitude sensitivities to acoustic forces in a wide frequency range. We utilize bimodal, trimodal, and tetramodal excitation schemes for the actuation of the Atomic Force Microscopy (AFM) micro-cantilever in the presence of dynamic acoustic forces. In multimodal operations, the micro-cantilever is driven by applying external excitation forces at eigenmode angular frequencies. The Equation of Motion (EOM) is constructed in the consideration of the driven and damped harmonic oscillator as a model and solved numerically to obtain the deflections of the micro-cantilever at flexural eigenmodes. In this current work, time-domain responses of the micro-cantilever to acoustic forces are introduced for different excitation schemes so that free oscillations are compared with the oscillations of the driven micro-cantilever undergoing acoustic forces. Then, we evaluate the results of amplitude responses at the first four eigenmodes with respect to acoustic force frequencies for diverse force strengths. For our case, we obtain the amplitude response at the fourth eigenmode of around 0.303 nm for the force strength of 2900 pN at the frequency of 1740 kHz. This result proves that acoustic forces at megahertz frequencies are measured by using resonant AFM micro-cantilever under tetramodal operation. Therefore, multimodal excitation schemes can be applied to enhance the amplitude sensitivities to dynamic acoustic forces.
We develop a theoretical framework describing numerical approach to explore dynamic acoustic force sensitivity using micro-cantilever array in monomodal and bimodal operations. The excitation force at the second eigenmode frequency is supplied to the micro-cantilevers in monomodal operation. Since we focus on measurement sensitivity of acoustic forces at higher frequencies, deflections of micro-cantilevers at higher mode, second flexural mode, are obtained. In bimodal operations, external driving forces at the first and second eigenmode frequencies are applied simultaneously for the actuation of micro-cantilevers in an array. Depending on acoustic force strength, the application of driving force at higher eigenmode frequency in bimodal excitation scheme increases the phase sensitivity in measurement of acoustic forces within a particular frequency scope. For both excitation schemes, oscillation observables such as amplitude and phase shift are determined with respect to acoustic force frequencies for diverse acoustic force strengths. Simulation results suggest that wider high-sensitivity frequency band could be acquired, utilizing resonantly excited micro-cantilever array. For our case, we obtain the high-sensitivity frequency band of around 200–270 kHz and 440–570 kHz for the acoustic force strengths in the range of 126–1138.5 pN.
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