2.2. Equivalent Model for A Piezoelectric Co-Resonant Vibration System

The direct excitation and measurement of a nano-resonator is challenging; therefore, we used and extended the co-resonance concept. The main idea was to couple a smaller and a larger resonator in such a way that a change of the vibration properties of the smaller resonator could be measured by the vibration properties of the larger resonator. For this purpose, two vibrational structures were selected (typically two bending beams) and independently tuned to the same resonance frequency. Then, the smaller beam was placed on the larger beam at a position where the selected eigenmode of the larger beam had the highest amplitude (anti-node). At this position, the impact of the smaller beam on the larger beam was maximized. Due to the physical combination of the two structures, a new system was created with new properties, with respect to eigenfrequencies and mode shapes.

For analytical purposes, we applied a simplified model based on a system with ideal lumped parameters; two degrees of freedom; and discrete stiffnesses, damper, and masses. The piezoelectric effect—which is used for actuation and sensing—can be included through the second electro-mechanical analogy, e.g., see [25]. By utilizing a parameter comparison of the equations for electrical and mechanical discrete components, the analogy was traditionally derived. In the herein-used second analogy, displacement is equivalent to electric charge and force is equivalent to current. An ideal transmission element such as a lever can transfer energy from one domain to the other. However, the lever parameter consequently has a unit. The model is depicted in Figure 3.

Simplified equivalent model for the co-resonant mass detector.

The model has three coordinates: the absolute position y of the lager mass M, the position x of the smaller mass m with respect to y, and the electric charge Q. The larger mass M is inertially connected though the stiffness kM and the damper dM. The stiffness k and the damper d connects M and m. The analyte mass is mn adding to m. The lever with the unitized factor α couples the mechanic and the electric domain. The capacity of the piezoelectric element is C_p, and the system is driven by the voltage U. Utilizing complex amplitudes x(t)=Re(x^ejφejΩt)=Re(x^_ejΩt) and the imaginary unit j and Ω as angular excitation frequency, for the equations of motion follows for the harmonic excitation:

Additionally, we defined the two decoupled eigenfrequencies with mn=0:

For further analysis, transfer-functions were also needed; here, we used the current amplitude i^_=jΩQ^_: