No abstract

Here, we study the acoustic radiation generated by the vibration of miniaturized doubly clamped and cantilever beam resonators in viscous fluids. Acoustic radiation results in an increase in dissipation and consequently a decrease in the resonator’s quality factor. We find that dissipation due to acoustic radiation is negligible when the acoustic wavelength in the fluid is much larger than the bending wavelength. In this regime, dissipation is primarily due to the viscous losses in the fluid and may be predicted with the two-dimensional cylinder approximation in the absence of axial flow and substrate effects. In contrast, when the bending wavelength approaches the length of the acoustic wavelength, acoustic radiation becomes prominent. In this regime, dissipation due to acoustic radiation can no longer be neglected, and the cylinder approximation inaccurately characterizes the total energy loss in the system. Experiments are performed with microcantilevers of varying lengths in Ar and N2 to observe trends in the acoustic wavelength of the fluid and bending wavelength. Additional experimental results from doubly clamped nanoelectromechanical system beams are also presented. Experimental results illustrate an increase in dissipation, which is further analyzed with the use of three-dimensional finite element models. With the numerical simulations, we calculate the radiation efficiency of the measured devices and analyze the pressure fields generated by the vibrating resonators. This analysis allows one to estimate the effects of acoustic radiation for any resonator.

This paper presents a method to simultaneously predict the elastic modulus, axial load, and boundary conditions of a nanoelectromechanical system (NEMS) beam from a minimum of two measured natural frequencies. The proposed method addresses the challenges of the inverse problem at the nano scale, which include high natural frequencies, small geometric beam dimensions, and measurements limited to natural frequencies. The method utilizes a finite element model of an Euler–Bernoulli beam under axial loading to predict the response of the beam with axial loading and flexible boundary conditions. By expressing the finite element model in terms of dimensionless beam parameters, the proposed method may be applied to nano scale beams while maintaining numerical stability of the finite element equation of motion. With the stabilized finite element model, the NEMS beam properties are predicted by iterating through values of dimensionless beam parameters until the normalized error between predicted and measured natural frequencies is minimized. A key feature of the proposed method is the simultaneous prediction of the elastic modulus during the iterative search, resulting in a reduction of the search space and significant computational savings. Additionally, the proposed method readily accommodates an arbitrary number of measured natural frequencies without the reformulation of procedures and analyses. Numerical examples are presented to illustrate the proposed method’s ability to predict the elastic modulus, axial load, and boundary conditions. The proposed method is applied to experimental measurements of a NEMS beam, where the normalized error between predicted and measured natural frequencies is reduced below 10−3.

The oscillatory dynamics of nanoelectromechanical systems (NEMS) is at the heart of many emerging applications in nanotechnology. For common NEMS, such as beams and strings, the oscillatory dynamics is formulated using a dissipationless wave equation derived from elasticity. Under a harmonic ansatz, the wave equation gives an undamped free vibration equation; solving this equation with the proper boundary conditions provides the undamped eigenfunctions with the familiar standing wave patterns. Any harmonically driven solution is expressible in terms of these undamped eigenfunctions. Here, we show that this formalism becomes inconvenient as dissipation increases. To this end, we experimentally map out the position- and frequency-dependent oscillatory motion of a NEMS string resonator driven linearly by a non-symmetric force at one end at different dissipation limits. At low dissipation (high Q factor), we observe sharp resonances with standing wave patterns that closely match the eigenfunctions of an undamped string. With a slight increase in dissipation, the standing wave patterns become lost, and waves begin to propagate along the nanostructure. At large dissipation (low Q factor), these propagating waves become strongly attenuated and display little, if any, resemblance to the undamped string eigenfunctions. A more efficient and intuitive description of the oscillatory dynamics of a NEMS resonator can be obtained by superposition of waves propagating along the nanostructure.

A method for correcting multiple mechanical and acoustical properties in a finite element model using the Neumann series is presented and demonstrated with numerical examples. In previous work, the authors developed a method for estimating a model parameter in a complex structure using a Neumann series as an approximation to system response. The method computed the sensitivity of the system response due to changes in the parameter and subsequently determined the change needed in the parameter to bring model response into agreement with vibration measurements. This previous work demonstrated the accuracy and computational efficiency of the method when correcting one model parameter. In the present work, the authors extend the analysis to compute sensitivities for multiple parameters of a system, allowing for the correction of more than one parameter. The limits and accuracies of the method are explored for a canonical acoustic system in which a complex structure interacts with an acoustic medium. Two uncertain model parameters of the system are corrected by bringing model response into agreement with acoustic measurements. Results of these examples will be reviewed and presented to illustrate the accuracy and robustness of the method.