Fundamental Flexural Mode Research Articles

Strong 4-mode coupling of nanomechanical string resonators

We investigate mechanical mode coupling between the four fundamental flexural modes of two doubly clamped, high-Q silicon-nitride nanomechanical string resonators. Strong mechanical coupling between the strings is induced by the strain mediated via a shared clamping point, engineered to increase the exchange of oscillatory energy. One of the resonators is controlled dielectrically, which results in strong coupling between its out-of-plane and in-plane flexural modes. We show both inter-string out-of-plane-in-plane and 3-mode resonances of the four coupled fundamental vibrational modes of a resonator pair, giving rise to a simple and a multimode avoided crossing, respectively.

Mode analysis of trilaminar bender bar transducers using an approximation method

Based on the vibration theory of a thin plate, an analytical treatment of the trilaminar bender bar with piezoelectric elements and inert substrate of various lengths is presented for mode analysis. Resonance frequency and effective electromechanical coupling coefficient are calculated by this method. The impacts of the geometries of the bender bar on the performance of its fundamental and third-order flexural mode are investigated in detail under rigid boundary conditions. It is shown that resonance frequency is extremely sensitive to the thickness of inert substrate. Moreover, the effective electromechanical coupling coefficient has peaks as the length of piezoelectric elements varies. The peaks are achieved when the length of piezoelectric elements equals the length between two nodes having zero strains in the x-direction. The trilaminar bender bar will be effectively excited when the strains on the piezoelectric element are in the same phase, which is important to disclose the vibration mechanisms of this kind of transducer. Also, analytical results are compared with the ones of numerical simulation. The results suggest that effective electromechanical coupling coefficient shares similar patterns with electrical conductance, which can be used to characterize transducer performance to a certain extent. It also demonstrates that the analytical treatment provides an efficient alternative way for optimizing the bender bar transducer design.

Experimental slowing of flexural waves in dielectric elastomer films by voltage

Shape and physical properties of dielectric elastomers are changeable by voltage. Theoretical works show that these changes can be harnessed to tune the propagation of superposed elastic waves. We experimentally demonstrate this concept by manipulating waves in a dielectric elastomer film, focusing on the flexural mode at low frequencies. To this end, we design an experimental apparatus to pre-stretch, actuate, excite waves at low frequencies in a VHB™ 4910 film, and measure the velocity of the fundamental flexural mode. Our results show that the excited wave velocity is slowed down by the applied voltage, and provide experimental proof of concept for the application of deformable dielectrics as tunable waveguides.

Wind-induced responses and dynamics characteristics of an asymmetrical base-isolated building observed during typhoons

Responses of an asymmetrical base-isolated building during the passage of Typhoon Roke 2011, Typhoon Guchol 2012 and a bomb cyclone strong-wind event on April 3, 2012 were investigated in this paper. Wind-induced accelerations of the building were recorded by dense array permanent monitoring system and analyzed in detailed by spectra analysis, time-frequency wavelet analysis and system identification. Results of analysis revealed that wind-induced responses of the buildings were dominated by the first mode which is the fundamental flexural mode coupled with torsion of the upper stories. Building responses demonstrated non-linear characteristics of natural frequencies, damping ratio and mode-shapes. Natural frequencies of fundamental modes decreased with the increase of acceleration amplitude, with the maximum reduction of 8–10% of natural frequencies observed during the peak wind speed. Damping estimates of the fundamental modes increased with the increase of accelerations, while characteristics of mode-shapes were also found to be dependent on the wind speed. Further analysis revealed that non-linear characteristics of building responses are related to isolator deformations caused by wind-induced static force.

Simultaneous dual-gas QEPAS detection based on a fundamental and overtone combined vibration of quartz tuning fork

A dual-gas quartz-enhanced photoacoustic spectroscopy (QEPAS) sensor system based on a frequency division multiplexing technique of a quartz tuning fork (QTF) was developed and experimentally demonstrated. Two beams from two independently modulated lasers are focused at two different positions between the QTF prongs to excite both the QTF fundamental and 1st overtone flexural modes simultaneously. The 2f-wavelength modulation technique is employed by applying two sinusoidal dithers, whose frequencies are equal to a half of the QTF fundamental and 1st overtone frequencies, respectively, to the currents of two excitation lasers. The resonance frequency difference between two flexural modes ensures that the correlated photoacoustic signals generated by different target gases do not interfere with each other. The proposed QEPAS methodology realizes a continuous real-time dual-gas monitoring with a simple setup and small sensor size compared with previous multi-gas QEPAS sensors.

Finite-time Stückelberg interferometry with nanomechanical modes

Stückelberg interferometry describes the interference of two strongly coupled modes during a double passage through an avoided energy level crossing. In this work, we investigate finite-time effects in Stückelberg interferometry and discuss the exact analytical solution of the double passage Stückelberg problem by expanding the finite-time solution of the Landau–Zener problem. Approximating the return probability amplitudes of the double passage in distinct limits reveals uncharted parameter regimes of Stückelberg interferometry where finite-time effects affect the coherent exchange of energy. We find the long-time limit of the exact solution to formally coincide with the well-established adiabatic impulse model which is, to the best of our knowledge, the only regime of Stückelberg interferometry reported so far. Experimentally, we study all predicted regimes using a purely classical, strongly coupled nanomechanical two-mode system of high quality factor. The classical two-mode system consists of the in-plane and out-of-plane fundamental flexural mode of a high stress silicon nitride string resonator, coupled via electric gradient fields. We exploit our experimental and theoretical findings by studying the onset of Stückelberg interference in dependence of the characteristic system control parameters and obtain characteristic excitation oscillations between the two modes even without the explicit need of traversing the avoided crossing. The presented findings are not limited to classical mechanical two-mode systems but can be applied to every strongly coupled (quantum) two-level system, for example a spin-1/2 system or superconducting qubit.

Tuning Nonlinear Mechanical Mode Coupling in GaAs Nanowires Using Cross-Section Morphology Control.

We investigate the nonlinear mechanical properties of GaAs nanowires with anisotropic cross-section. Fundamental and second order flexural modes are studied using laser interferometry with good agreement found between experiment and theory describing the nonlinear response under mechanical excitation. In particular, we demonstrate that the sign of the nonlinear coupling between orthogonal modes is dependent on the cross-section aspect ratio. The findings are of interest for applications such as amplitude to frequency conversion and vectorial force sensing.

Sympathetic laser cooling of graphene with Casimir-Polder forces

We propose a scheme to actively cool the fundamental flexural (out-of-plane) mode of a graphene sheet via vacuum forces. Our setup consists in a cold atom cloud placed close to a graphene sheet at distances of a few micrometers. The atoms couple to the graphene membrane via Casimir-Polder forces. By deriving a self-consistent set of equations governing the dynamics of the atomic gas and the flexural modes of the graphene, we show to be possible to cool graphene from room temperatures by actively (laser) cooling an atomic gas. By choosing the right set of experimental parameter we are able to cool a graphene sheet down to ~ 60 microkelvin.

Design of a quasi-2D photonic crystal optomechanical cavity with tunable, large x2-coupling.

We present the optical and mechanical design of a mechanically compliant quasi-two-dimensional photonic crystal cavity formed from thin-film silicon in which a pair of linear nanoscale slots are used to create two coupled high-Q optical resonances. The optical cavity supermodes, whose frequencies are designed to lie in the 1500 nm wavelength band, are shown to interact strongly with mechanical resonances of the structure whose frequencies range from a few MHz to a few GHz. Depending upon the symmetry of the mechanical modes and the symmetry of the slot sizes, we show that the optomechanical coupling between the optical supermodes can be either linear or quadratic in the mechanical displacement amplitude. Tuning of the nanoscale slot size is also shown to adjust the magnitude and sign of the cavity supermode splitting 2J, enabling near-resonant motional scattering between the two optical supermodes and greatly enhancing the x2-coupling strength. Specifically, for the fundamental flexural mode of the central nanobeam of the structure at 10 MHz the per-phonon linear cross-mode coupling rate is calculated to be g˜+-/2π=1MHz, corresponding to a per-phonon x2-coupling rate of g˜'/2π=1kHz for a mode splitting 2J/2π = 1 GHz which is greater than the radiation-limited supermode linewidths.

Experimental Evaluation of Three Designs of Electrodynamic Flexural Transducers.

Three designs for electrodynamic flexural transducers (EDFT) for air-coupled ultrasonics are presented and compared. An all-metal housing was used for robustness, which makes the designs more suitable for industrial applications. The housing is designed such that there is a thin metal plate at the front, with a fundamental flexural vibration mode at ∼50 kHz. By using a flexural resonance mode, good coupling to the load medium was achieved without the use of matching layers. The front radiating plate is actuated electrodynamically by a spiral coil inside the transducer, which produces an induced magnetic field when an AC current is applied to it. The transducers operate without the use of piezoelectric materials, which can simplify manufacturing and prolong the lifetime of the transducers, as well as open up possibilities for high-temperature applications. The results show that different designs perform best for the generation and reception of ultrasound. All three designs produced large acoustic pressure outputs, with a recorded sound pressure level (SPL) above 120 dB at a 40 cm distance from the highest output transducer. The sensitivity of the transducers was low, however, with single shot signal-to-noise ratio dB in transmit–receive mode, with transmitter and receiver 40 cm apart.

Analysis of overtone flexural modes operation in quartz-enhanced photoacoustic spectroscopy.

A detailed investigation of a set of custom quartz tuning forks (QTFs), operating in the fundamental and first overtone flexural modes is reported. Support losses are the dominant energy dissipation processes when the QTFs vibrate at the first overtone mode. These losses can be decreased by increasing the ratio between the prong length and its thickness. The QTFs were implemented in a quartz enhanced photoacoustic spectroscopy (QEPAS) based sensor operating in the near-IR spectral range and water vapor was selected as the gas target. QTF flexural modes having the highest quality factor exhibit the largest QEPAS signal, demonstrating that, by optimizing the QTF prongs sizes, overtone modes can provide a higher QEPAS sensor performance with respect to using the fundamental mode.

Quartz-enhanced photoacoustic spectroscopy exploiting tuning fork overtone modes

We report on a quartz-enhanced photoacoustic sensor (QEPAS) based on a custom-made quartz tuning fork (QTF) to operate in both the fundamental and the first overtone vibrational mode resonances. The QTF fundamental mode resonance falls at ∼3 kHz and the first overtone at ∼18 kHz. Electrical tests showed that the first overtone provides a higher quality factor and increased piezoelectric current peak values, with respect to the fundamental flexural mode. To evaluate the QTF acousto-electric energy conversion efficiency, we operated the QEPAS in the near-IR and selected water vapor as the target gas. The first overtone resonance provides a QEPAS signal-to-noise ratio ∼5 times greater with respect to that measured for the fundamental mode. These results open the way to employing QTF overtone vibrational modes for QEPAS based trace gas sensing.

Kelvin Probe Force Microscopy by Dissipative Electrostatic Force Modulation

We report a new experimental technique for Kelvin probe force microscopy (KPFM) using the dissipation signal of frequency modulation atomic force microscopy for bias voltage feedback. It features a simple implementation and faster scanning as it requires no low frequency modulation. The dissipation is caused by the oscillating electrostatic force that is coherent with the tip oscillation, which is induced by a sinusoidally oscillating voltage applied between the tip and sample. We analyzed the effect of the phase of the oscillating force on the frequency shift and dissipation and found that the relative phase of 90$^\circ$ that causes only the dissipation is the most appropriate for KPFM measurements. The present technique requires a significantly smaller ac voltage amplitude by virtue of enhanced force detection due to the resonance enhancement and the use of fundamental flexural mode oscillation for electrostatic force detection. This feature will be of great importance in the electrical characterizations of technically relevant materials whose electrical properties are influenced by the externally applied electric field as is the case in semiconductor electronic devices.

Experimental and finite element dynamic analysis of incrementally loaded reinforced concrete structures

This work investigates influence of damage in reinforced concrete (RC) structures on their dynamic properties through modal testing and non-linear finite element (FE) analysis. Five RC beams were designed with the fundamental flexural mode frequencies in the range of 6.5–18.0Hz for the uncracked state. Mechanical properties of concrete, such as static and dynamic elastic moduli were determined from standard tests and ultra-sonic pulse velocity readings. The beams were incrementally loaded until the span/250 deflection limit was reached and their natural frequencies were measured from the free decay vibrations. The progressive damage reduced fundamental frequencies of tested beams by up to 25%. The non-linear FE analysis was carried out for RC beams and one two-span slab and the calculated reduced frequencies of the 1st and 2nd vibration modes were in excellent agreement with measurements. This led to the conclusion that, given that the non-linear analysis can capture degradation of dynamic stiffness due to cracking, the future dynamic performance and damage identification on the RC structure can be reliably determined from the same FE model. The results reveal potential of the combined modal testing and FE analysis to improve inspection and assessment of the in-service RC structures.

Customized atomic force microscopy probe by focused-ion-beam-assisted tip transfer.

We present a technique for transferring separately fabricated tips onto tipless atomic force microscopy (AFM) cantilevers, performed using focused ion beam-assisted nanomanipulation. This method addresses the need in scanning probe microscopy for certain tip geometries that cannot be achieved by conventional lithography. For example, in probing complex layered materials or tall biological cells using AFM, a tall tip with a high-aspect-ratio is required to avoid artifacts caused by collisions of the tip's sides with the material being probed. We show experimentally that tall (18 μm) cantilever tips fabricated by this approach reduce squeeze-film damping, which fits predictions from hydrodynamic theory, and results in an increased quality factor (Q) of the fundamental flexural mode. We demonstrate that a customized tip's well-defined geometry, tall tip height, and aspect ratio enable improved measurement of elastic moduli by allowing access to low-laying portions of tall cells (T lymphocytes). This technique can be generally used to attach tips to any micromechanical device when conventional lithography of tips cannot be accomplished.

Acoustic formulation of elastic guided wave propagation and scattering in curved tubular structures

Recently, the use of guided wave technology in conjunction with tomographic techniques has provided the possibility of obtaining point-by-point maps of corrosion or erosion depth over the entire volume of a pipeline section between two ring arrays of ultrasonic transducers. However, current research has focused on straight pipes and little work has been done on pipe bends and other curved tubular structures which are also the most susceptible to developing damage. Tomography of curved tubes is challenging because of the complexity and computational cost of the 3-D elastic model required to accurately describe guided wave propagation. Based on the definition of travel-time-preserving orthogonal parametric representations of curved tubes, this paper demonstrates that guided wave propagation and scattering can be approximated by an equivalent 2-D acoustic model which is inhomogeneous and elliptically anisotropic. Numerical methods to solve the full wave equation and predict ray paths and travel times are introduced and applied to the case of a bend. Particular emphasis is given to the shortest-path ray tracing method, which is applied to the 2-D model to compute ray paths and predict travel times of the fundamental flexural mode, A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> , propagating across a curved pipe. Good agreement is found between predictions and experiments performed on a 220-mmdiameter (8-in-diameter) (D) pipe with 1.5D bend radius. The 2-D model also reveals the existence of an acoustic lensing effect which leads to a focusing phenomenon also confirmed by the experiments. The computational efficiency of the 2-D model makes it ideally suited for tomographic algorithms.

Acoustic formulation of elastic guided wave propagation and scattering in curved tubular structures.

Recently, the use of guided wave technology in conjunction with tomographic techniques has provided the possibility of obtaining point-by-point maps of corrosion or erosion depth over the entire volume of a pipeline section between two ring arrays of ultrasonic transducers. However, current research has focused on straight pipes and little work has been done on pipe bends and other curved tubular structures which are also the most susceptible to developing damage. Tomography of curved tubes is challenging because of the complexity and computational cost of the 3-D elastic model required to accurately describe guided wave propagation. Based on the definition of travel-time-preserving orthogonal parametric representations of curved tubes, this paper demonstrates that guided wave propagation and scattering can be approximated by an equivalent 2-D acoustic model which is inhomogeneous and elliptically anisotropic. Numerical methods to solve the full wave equation and predict ray paths and travel times are introduced and applied to the case of a bend. Particular emphasis is given to the shortest-path ray tracing method, which is applied to the 2-D model to compute ray paths and predict travel times of the fundamental flexural mode, A0, propagating across a curved pipe. Good agreement is found between predictions and experiments performed on a 220-mm-diameter (8-in-diameter) (D) pipe with 1.5D bend radius. The 2-D model also reveals the existence of an acoustic lensing effect which leads to a focusing phenomenon also confirmed by the experiments. The computational efficiency of the 2-D model makes it ideally suited for tomographic algorithms.

Signatures of two-level defects in the temperature-dependent damping of nanomechanical silicon nitride resonators

The damping rates of high quality factor nanomechanical resonators are well beyond intrinsic limits. Here, we explore the underlying microscopic loss mechanisms by investigating the temperature-dependent damping of the fundamental and third harmonic transverse flexural mode of a doubly clamped silicon nitride string. It exhibits characteristic maxima reminiscent of two-level defects typical for amorphous materials. Coupling to those defects relaxes the momentum selection rules, allowing energy transfer from discrete long wavelength resonator modes to the high frequency phonon environment.

A piezoelectric energy harvester with increased bandwidth based on beam flexural vibrations in perpendicular directions

We propose a new structure for piezoelectric energy harvesters. It consists of an elastic beam with two pairs of piezoelectric films operating with the fundamental flexural modes in perpendicular directions. A one-dimensional model is developed and used to analyze the proposed structure. The output power density is calculated and examined. Results show that, with simultaneous flexural motions in two perpendicular directions, the output power has two peaks close to each other resulting from the two fundamental flexural resonances. The distance between the two peaks can be adjusted through design to make the two peaks merge into one wide peak. Hence, the frequency bandwidth through which energy can be harvested is roughly doubled when compared with conventional beam bimorph energy harvesters operating with flexural motion in one direction and one resonance only.

Coherent control of a classical nanomechanical two-level system

The Bloch sphere is a generic picture describing a coupled two-level system and the coherent dynamics of its superposition states under control of electromagnetic fields. It is commonly employed to visualise a broad variety of phenomena ranging from spin ensembles and atoms to quantum dots and superconducting circuits. The underlying Bloch equations describe the state evolution of the two-level system and allow characterising both energy and phase relaxation processes in a simple yet powerful manner. Here we demonstrate the realisation of a nanomechanical two-level system which is driven by radio frequency signals. It allows to extend the above Bloch sphere formalism to nanoelectromechanical systems. Our realisation is based on the two orthogonal fundamental flexural modes of a high quality factor nanostring resonator which are strongly coupled by a dielectric gradient field. Full Bloch sphere control is demonstrated via Rabi, Ramsey and Hahn echo experiments. This allows manipulating the classical superposition state of the coupled modes in amplitude and phase and enables deep insight into the decoherence mechanisms of nanomechanical systems. We have determined the energy relaxation time T1 and phase relaxation times T2 and T2*, and find them all to be equal. This not only indicates that energy relaxation is the dominating source of decoherence, but also demonstrates that reversible dephasing processes are negligible in such collective mechanical modes. We thus conclude that not only T1 but also T2 can be increased by engineering larger mechanical quality factors. After a series of ground-breaking experiments on ground state cooling and non-classical signatures of nanomechanical resonators in recent years, this is of particular interest in the context of quantum information processing.