Higher Order Flexural Modes Research Articles

Piezoelectric-AlN resonators at two-dimensional flexural modes for the density and viscosity decoupled determination of liquids

A micromachined resonator immersed in liquid provides valuable resonance parameters for determining the fluidic parameters. However, the liquid operating environment poses a challenge to maintaining a fine sensing performance, particularly through electrical characterization. This paper presents a piezoelectric micromachined cantilever with a stepped shape for liquid monitoring purposes. Multiple modes of the proposed cantilever are available with full electrical characterization for realizing self-actuated and self-sensing capabilities. The focus is on higher flexural resonances, which nonconventionally feature two-dimensional vibration modes. Modal analyses are conducted for the developed cantilever under flexural vibrations at different orders. Modeling explains not only the basic length-dominant mode but also higher modes that simultaneously depend on the length and width of the cantilever. This study determines that the analytical predictions for resonant frequency in liquid media exhibit good agreement with the experimental results. Furthermore, the experiments on cantilever resonators are performed in various test liquids, demonstrating that higher-order flexural modes allow for the decoupled measurements of density and viscosity. The measurement differences achieve 0.39% in density and 3.50% in viscosity, and the frequency instability is below 0.05‰. On the basis of these results, design guidelines for piezoelectric higher-mode resonators are proposed for liquid sensing.

Persistent Response in an Ultrastrongly Driven Mechanical Membrane Resonator.

We study experimentally and theoretically the phenomenon of "persistent response" in ultrastrongly driven membrane resonators. The term persistent response denotes the development of a vibrating state with nearly constant amplitude over an extreme wide frequency range. We reveal the underlying mechanism by directly imaging the vibrational state using advanced optical interferometry. We argue that this state is related to the nonlinear interaction between higher-order flexural modes and higher-order overtones of the driven mode. Finally, we propose a stability diagram for the different vibrational states that the membrane can adopt.

A Low-Noise High-Order Mode-Localized MEMS Accelerometer

This paper reports a precision mode-localized accelerometer operating in a higher-order flexural mode. The accelerometer consists of two symmetric resonators coupled by a central rigid coupler to generate an ultra-weak coupling factor. A reduced noise floor is observed when the resonators operate in the higher-order flexural mode compared to the basic lower-order mode. The mode-localized accelerometer working in the fifth-order mode demonstrates an input-referred bias instability of 130 ng and noise floor of 85 ng/ √Hz, which are the best results obtained for accelerometers employing the mode localization paradigm to date. These results indicate that the performance of the mode-localized sensors can be improved by operating at a higher working frequency if the coupling factor and quality factor do not drop significantly. [2020-0365].

Reducing dissipation in piezoelectric flexural microplate resonators in liquid environments

While piezoelectric microplates are emerging as a promising MEMS liquid-based sensing platform, often acoustic radiation losses limit device performance. This paper presents a new analysis of microplate acoustic radiation losses through the use of traditional analytical models for the fundamental mode and a new finite element model that is used to analyze trends affecting Q for both the fundamental and higher order flexural modes. Results from these models are compared with experimental measurements of frequency and Q measured using both electrical characterization and the laser Doppler vibrometer for multiple modes of the microplate in water as compared to air and vacuum. Finally, a microplate mode with a high initial Q of around 150 in air and that demonstrates an increase in Q going from air to water is presented as a strong candidate for use in droplet based sensing applications.

Evidence for structural damping in a high-stress silicon nitride nanobeam and its implications for quantum optomechanics

We resolve the thermal motion of a high-stress silicon nitride nanobeam at frequencies far below its fundamental flexural resonance (3.4 MHz) using cavity-enhanced optical interferometry. Over two decades, the displacement spectrum is well-modeled by that of a damped harmonic oscillator driven by a 1/f thermal force, suggesting that the loss angle of the beam material is frequency-independent. The inferred loss angle at 3.4 MHz, ϕ=4.5⋅10−6, agrees well with the quality factor (Q) of the fundamental beam mode (ϕ=Q−1). In conjunction with Q measurements made on higher order flexural modes, and accounting for the mode dependence of stress-induced loss dilution, we find that the intrinsic (undiluted) loss angle of the beam changes by less than a factor of 2 between 50 kHz and 50 MHz. We discuss the impact of such “structural damping” on experiments in quantum optomechanics, in which the thermal force acting on a mechanical oscillator coupled to an optical cavity is overwhelmed by radiation pressure shot noise. As an illustration, we show that structural damping reduces the bandwidth of ponderomotive squeezing.

Effect of Photoresist Coating on the Reusable Resonant Cantilever Sensors for Assessing Exposure to Airborne Nanoparticles

Abstract Silicon cantilever resonators were designed and fabricated in various shapes and dimensions to be used in an airborne nanoparticle (NP) sensing application. For a repeatedly usable sensor, the cantilever surface was covered by a sacrificial layer of photoresist prior to the airborne NP sampling. Using the electrostatic precipitation method, airborne NPs were sampled on the cantilever. Within a lift-off process of wet cleaning, the photoresist layer was removed together with the trapped NPs. The resonant frequency and the quality factor ( Q -factor) of the fundamental and higher order flexural modes of the silicon cantilevers were characterized for different thickness given by viscosity of the photoresist layer. It was found that the proposed recycling technique was most effective when a thin photoresist film was used. By using higher order resonant modes, Q -factors of more than 1000 in air were maintained even after the photoresist coating.

Note: Exciting higher-order flexural modes of freestanding microstructures with square wave driving signals

We examine using the harmonics of low fundamental frequency (f(1)) square waveforms to electrostatically excite higher-order flexural modes of a micromechanical structure in higher (>f(1)) frequency regimes. Square wave signals swept between a fundamental frequency range of approximately 0-333 kHz demonstrated an effective high magnitude spectral excitation of approximately 0-1 MHz. This enabled optical heterodyne detection of structural resonant frequencies above 333 kHz. Low frequency square wave signals were also used to selectively excite modes at higher frequencies.

The determination of the optimal length of crystal blanks in quartz crystal resonators

It is well understood that the strong coupling of thickness-shear and flexural vibrations in piezoelectric crystal plates only occurs at specific length at which the vibration mode conversion, like the flexural mode gradually converting to thickness-shear mode while the thickness-shear mode converting to higher-order flexural mode, happens. It is important to avoid the strong coupling of modes in a crystal resonator that uses thickness-shear vibrations to enhance the energy trapping. To achieve such a design goal, the length of a crystal blank should be carefully chosen such that the coupling is at its weakest, which usually is in the middle of two strong coupling points. Through a closer examination of the frequency spectra, or the frequency-length relationship in this study, we can see that the strong coupling points appear periodically. This implies that we can find exact locations with the plate theory that predicts the resonance frequency. Based on this observation, we first use the first-order Mindlin plate theory with the precise thickness-shear frequency, which is normalized to one, to find corresponding wavenumbers. Then the length as a variable is solved from the coupled frequency equation for exact coupling points in a crystal plate of AT-cut quartz. The optimal length of a crystal blank in the simplest resonator model is calculated for the coupled thickness-shear, flexural, and extensional vibrations. The solutions and the method will be important in the determination of optimal length of a crystal blank in the resonator design process.

Resonant harmonic response in tapping-mode atomic force microscopy

Higher harmonics in tapping-mode atomic force microscopy offers the potential for imaging and sensing material properties at the nanoscale. The signal level at a given harmonic of the fundamental mode can be enhanced if the cantilever is designed in such a way that the frequency of one of the higher harmonics of the fundamental mode (designated as the resonant harmonic) matches the resonant frequency of a higher-order flexural mode. Here we present an analytical approach that relates the amplitude and phase of the cantilever vibration at the frequency of the resonant harmonic to the elastic modulus of the sample. The resonant harmonic response is optimized for different samples with a proper design of the cantilever. It is found that resonant harmonics are sensitive to the stiffness of the material under investigation.

High-resolution imaging of elastic properties using harmonic cantilevers

We present a micromachined scanning probe cantilever, in which a specific higher-order flexural mode is designed to be resonant at an exact integer multiple of the fundamental resonance frequency. We have fabricated such cantilevers by reducing the stiffness of the third order flexural mode relative to the fundamental mode, and we have demonstrated that these cantilevers enable sensing of non-linear mechanical interactions between the atomically sharp tip at the free end of the cantilever and a surface with unknown mechanical properties in tapping-mode atomic force microscopy. Images of surfaces with large topographical variations show that for such samples harmonic imaging has better resolution than standard tapping-mode imaging.

Enhanced sensitivity to force gradients by using higher flexural modes of the atomic force microscope cantilever

Higher flexural modes of a Si cantilever are used in order to increase the operating frequencies in non-contact mode scanning force microscopy. By recording frequency responses at different tip–sample distances and by using different flexural modes, long-range tip–sample interactions are studied. The experimental results reveal that operating at higher resonant modes makes it possible to improve sensitivity to force gradients and to avoid overlap between hydrodynamic interactions and attractive interactions in the range of 100-nm tip–sample distances. Using the second-harmonic, a sevenfold-enhanced sensitivity is demonstrated. Finally, discussion on the appropriate technique related to force gradient measurement, i.e. slope detection or frequency modulation technique, is reported. During the last decade, atomic force microscopy (AFM) [1] has atttacted a great interest in a wide field of applications including biological living cells [2]. For this kind of application, dynamic mode is the most used operating mode and avoids damage to such samples. There are numerous advantages in developing an AFM operating at high frequencies. Mainly, it allows fast scanning and reduces the thermal noise, and therefore the ultimate sensitivity to force gradients. Imaging systems with true atomic resolution as well as high-speed data storage are the main driving forces in this field. In order to obtain very high operating frequencies, fabrication of nanometric-scale cantilevers of small mass has been already reported [3, 4]. Due to the small size of the cantilever (typically sub-micron size), it requires more complicated detection systems. However, another alternative is to use conventional micromachined AFM cantilevers vibrating at higher flexural modes. Several groups are investigating this possibility. Rabe et al. [5] published a detailed analysis of cantilever vibration mechanisms, detecting more than 10 modes using silicon cantilevers. Recently, tapping mode images were performed by ∗ Permanent address: Laboratoire de Physique et Metrologie des Oscillateurs (LPMO-CNRS UPR 3203) Centre National de la Recherche Scientifique, 32 Avenue de l’Observatoire, 25044 Besancon Cedex, France, Fax: +33-3/8166 6998, E-mail: hoummady@lpmo.univ-fcomte.fr Minne et al. [6] using higher flexural modes. The piezoresistive cantilever that was used was driven at 132 kHz, which corresponded to the second flexural mode of the clamped-free beam. A quantitative analysis of sensitivity enhancement using higher modes has not yet been achieved. In this paper, we will demonstrate the ability of higher modes to enhance sensitivity to force gradients. For our experiments, we used a home-made force microscope with an original detection technique based on a two degrees of freedom optical detection system: interferometry and optical beam deflection [7], and a silicon cantilever driven at higher modes from the first to the fourth flexural mode (up to 6 MHz). By recording the quasi-instantaneous frequency response at different tip– sample distances, the two main long-range interactions were distinguished, i.e. air damping and attractive forces. The results reveal that higher order flexural modes allow us to avoid overlap between these two interactions. For measurement of force gradients, two techniques have been investigated, the frequency modulation technique [8] and slope detection [9]. At a tip–sample distance of 10 nm, the measured sensitivity of the frequency modulation technique was found to be sevenfold-enhanced using the second flexural mode. Therefore, sensitivity in terms of a decrease in amplitude at the initial frequency (corresponding to the slope detection technique), which depends mainly on the quality factor, remains roughly the same [10]. The results reveal also that higher order flexural modes allow us to avoid damping due to frictional forces between the cantilever and sample.

The Effect of Liquid Storage Tanks on the Dynamic Response of Offshore Platforms

The sloshing of liquids in storage tanks on fixed offshore structures affects the structures' natural frequencies and damping. Analytic procedures by which one may account for these effects are presented. A procedures by which one may account for these effects are presented. A method for designing tankage that suppresses the dynamic response at the fundamental flexural natural frequency of the structure is proposed. Supporting data from full-scale field studies are presented. Introduction The ability to measure accurately and to predict the dynamic response properties of fixed offshore platforms is an important concern of the industry. In platforms is an important concern of the industry. In particular, it is important to be able to measure and particular, it is important to be able to measure and predict changes in the natural frequencies of a predict changes in the natural frequencies of a structure and to measure and predict the damping of that structure at its natural frequencies.Structural damage is known to cause small shifts in the natural frequencies of a structure in flexure and torsion. However, detection of the damage by monitoring the changes in frequency is possible only if all other sources of change are understood, predicted, and accounted for. One little-understood predicted, and accounted for. One little-understood source of change in natural frequency is the dynamic response (sloshing) of liquids in storage tanks.Small changes in tank depth have been shown to result in substantial changes in response behavior. This paper provides a method by which the effects of liquid storage tanks on the measured flexural natural frequencies of a structure may be predicted and accounted for.The modal damping associated with each natural frequency also is important because it controls the dynamic amplification at the flexural and torsional natural frequencies. This amplification must be considered when estimating the fatigue life of a structure.The damping on jacket structures is very low, typically 1 to 3%, and comes from various sources-structural, viscous hydrodynamic, wave radiation, and soils. An additional, not insignificant source generally overlooked is the damping contributed by the sloshing of liquids stored in on-board tanks.This paper quantitatively assesses the damping contributed by the motions of stored liquids and also provides the methods by which tanks may be provides the methods by which tanks may be designed to suppress adverse dynamic properties and to increase the damping available to the structure.Our analysis emphasizes the two first-order, end-on and broadside flexural modes of a structure with a rectangular platform. It is, therefore, applicable to most steel structures in the Gulf of Mexico and the North Sea. The first-order flexural modes are of principal concern because they are the most principal concern because they are the most important contributors to dynamic response and because they are most affected by deck-level storage tanks. This analysis may be extended to torsional and also to higher-order flexural modes. An example is presented in the Appendix. presented in the Appendix. Another topic of current concern is the response of a platform to earthquake excitation. The analytic model and methods presented may be extended to estimate the effects of liquid storage tanks on the dynamic response of the structure to earthquakes. JPT P. 1231