Background: Co-resonant coupling of a micro- and a nanocantilever can be introduced to significantly enhance the sensitivity of dynamic-mode cantilever sensors while maintaining the ease of detection. Experimentally, a low-stiffness nanocantilever is coupled to an easy to read out microcantilever and the eigenfrequencies of both beams are brought close to one another. This results in a strong interplay between both beams and, hence, any interaction applied at the nanocantilever alters the oscillatory state of the coupled system as a whole and can be detected at the microcantilever. The amplitude response curve of the microcantilever exhibits two resonance peaks and their response to an interaction applied to the sensor depends on the properties of the individual beams and the degree of frequency matching. Consequently, while an individual cantilever is characterized by its eigenfrequency, spring constant, effective mass and quality factor, the resonance peaks of the co-resonantly coupled system can be described by effective properties which are a mixture of both subsystem’s characteristics. These effective properties give insight into the amount of sensitivity of the nanocantilever that can be accessed and, consequently, into the sensitivity gain associated with the co-resonance. In order to design sensors based on the co-resonant principle and predict their behaviour it is crucial to derive a description for these effective sensor properties.
Results: By modeling the co-resonantly coupled system as a coupled harmonic oscillator and using electromechanical analogies, analytical expressions for the effective sensor properties have been derived and discussed. To illustrate the findings, numerical values for an exemplary system based on experimental sensor realizations have been employed. The results give insight into the complex interplay between the individual subsystem’s properties and the frequency matching, leading to a rather large parameter space for the co-resonant system’s effective properties. While the effective spring constant and effective mass mainly define the sensitivity of the coupled cantilever sensor, the effective quality factor primarily influences the detectability. Hence, a balance has to be found in optimizing both parameters in sensor design which becomes possible with the derived analytic expressions. Besides the description of effective sensor properties, it was studied how the thermal noise and, consequently, minimal detectable frequency shift for the co-resonantly coupled sensor represented by a coupled harmonic oscillator could be derived. Due to the complex nature of the coupled system’s transfer function and the required analysis, it is beyond the scope of this publication to present a full solution. Instead, a simplified approach to estimate the minimal detectable frequency shift for the co-resonant system based on the effective sensor properties is given.
Conclusion: By establishing a theoretical description for the effective sensor properties of a co-resonantly coupled system, the design of such systems is facilitated as sensor parameters can easily be predicted and adapted for a desired use case. It allows to study the potential sensitivity (gain) and detectability capabilities before sensor fabrication in a fast and easy way, even for large parameter spaces. So far, such an analysis of a co-resonantly coupled sensor was only possible with numerical methods and even then only with very limited capability to include and understand the complex interplay between all contributions. The outlined calculation steps regarding the noise considerations in a coupled harmonic oscillator system can provide the basis for a thorough study of that question. Furthermore, in a broader scope, the investigations presented within this work contribute towards extending and completing the already established theoretical basics of this novel co-resonant sensor concept and open up new ways of studying the coupled system’s behaviour.
Nonlinear Dynamics of Wave Packets in Tunnel-Coupled Harmonic-Oscillator Traps