Flexural Resonance Modes Research Articles

Ultrasensitive detection of lateral atomic-scale interactions on graphite (0001) via bimodal dynamic force measurements

Enhanced sensitivity to lateral forces on the nominally flat and inert (0001) surface of graphite is demonstrated via room-temperature dynamic force microscopy using simultaneous excitation and FM detection of the lowest flexural and torsional cantilever resonance modes. The site-independent long-range tip-sample interaction causes no significant lateral force variations except near atomic steps but unprecedented sensitivity to short-range forces is achieved on flat terraces in the attractive range. Two-dimensional bimodal force vs distance maps confirm the stronger distance dependence of the torsional frequency shift compared to the flexural resonance shift. This agrees with model calculations based on theoretical expressions. The lateral force gradient is extracted from the measured torsional shift, and a lateral force variation in at most $\ifmmode\pm\else\textpm\fi{}20\text{ }\text{pN}$ is obtained by integrating this gradient parallel to the surface. A further integration reveals a potential-energy variation in the attractive force range of only 3 meV.

Temperature-Insensitive Composite Micromechanical Resonators

Utilizing silicon and silicon dioxide's opposing temperature coefficients of Young's modulus, composite resonators with zero linear temperature coefficient of frequency are fabricated and characterized. The resulting resonators have a quadratic temperature coefficient of frequency of approximately -20 ppb/degC2 and a tunable turnover temperature in the -55degC to 125degC range. Reduction of the temperature dependence of frequency is shown in flexural-mode resonators (700 kHz-1.3 MHz) and extensional-mode ring resonators (20 MHz). The linear temperature coefficient of Young's modulus of silicon dioxide is extracted from measurements to be +179 ppm/degC. The composite resonators are fabricated and packaged in a CMOS-compatible wafer-scale hermetic encapsulation process. The long-term stability of the resonators is monitored for longer than six months. Although most devices exhibit less than 2 ppm frequency drift, there is evidence of dielectric charging in the silicon dioxide.

Reducing Anchor Loss in MEMS Resonators Using Mesa Isolation

In microelectromechanical systems resonators, dissipation of energy through anchor points into the substrate adds to resonator energy loss, contributing to low values of Q. A design for improving Q based on the reflection of anchor-generated surface acoustic waves is presented here. In this design, the resonator is surrounded by a trench, or a mesa, that partially reflects the wave energy back to the resonator. Depending on the distance from the resonator to the mesa, the reflected wave interferes either constructively or destructively with the resonator, increasing or decreasing Q. The proposed design is experimentally tested using a dome-shaped flexural mode resonator for a range of distances of the mesa from the resonator. Improvements in Q of up to 400% are observed. The resonator/mesa system is modeled using a commercially available finite-element code. Experiments and simulations compare well, suggesting that a finite-element-method analysis can be used in the preliminary design of mesas for the optimization of Q. The concept of using mesas to improve Q is simulated for both flexural and in-plane modes of vibration.

Size-dependent effective Young’s modulus of silicon nitride cantilevers

The effective Young’s modulus of silicon nitride cantilevers is determined for thicknesses in the range of 20–684 nm by measuring resonance frequencies from thermal noise spectra. A significant deviation from the bulk value is observed for cantilevers thinner than 150 nm. To explain the observations we have compared the thickness dependence of the effective Young’s modulus for the first and second flexural resonance mode and measured the static curvature profiles of the cantilevers. We conclude that surface stress cannot explain the observed behavior. A surface elasticity model fits the experimental data consistently.

Magnetoelastic/piezoelectric laminated structures for tunable remote contactless magnetic sensing and energy harvesting

In this letter, we report a method for a tunable magnetic field sensor based on the magnetoelastic coupling properties of a magnetoelastic/piezoelectric laminated composite structure. The magnetically and elastically tunable, flexural resonant mode in the bimorph FeNi36% (invar)/polyvinylidene fluoride clamped bilayer has been investigated by Doppler laser spectroscopy. Here we demonstrate that this bimorph structure can be used for low-frequency contactless detection of magnetic field fluctuation and magnetic field monitoring.

Multimode thermoelastic dissipation

In this paper, we investigate thermoelastic dissipation (TED) in systems whose thermal response is characterized by multiple time constants. Zener [Phys. Rev. 52, 230 (1937)] analyzed TED in a cantilever with the assumption that heat transfer is one dimensional. He showed that a single thermal mode was dominant and arrived at a formula for quantifying the quality factor of a resonating cantilever. In this paper, we present a formulation of thermoelastic damping based on entropy generation that accounts for heat transfer in three dimensions and still enables analytical closed form solutions for energy loss estimation in a variety of resonating structures. We apply this solution technique for estimation of quality factor in bulk mode, torsional, and flexural resonators. We show that the thermoelastic damping limited quality factor in bulk mode resonators with resonator frequency much larger than the eigenfrequencies of the dominant thermal modes is inversely proportional to the frequency of the resonator unlike in flexural mode resonators where the quality factor is directly proportional to the resonant frequency. Purely torsional resonators are not limited by TED as the deformation is isochoric. We show that it is possible to express the quality factor obtained by full three-dimensional analyses as a weighted sum of Zener formula based modal quality factors. We analytically estimate the quality factor of a cantilever and a fixed-fixed beam and corroborate it with data to show that the assumption of a single dominant thermal mode, which is valid in one-dimensional analysis, is violated. The analytical formulation described in this paper permits estimation of energy lost due to heat transfer in orthogonal directions. It is found that the entropy generated due to heat transfer along the beam becomes significant in beams with aspect ratio (length/width) below 20.

Thermoelastic Dissipation in MEMS/NEMS Flexural Mode Resonators

Understanding the energy dissipation mechanisms in single-crystal silicon MEMS/NEMS resonators are particularly important to maximizing an important figure of merit relevant for miniature sensor and signal processing applications: the Quality factor (Q) of resonance. This paper discusses thermoelastic dissipation (TED) as the dominant internal-friction mechanism in flexural mode MEMS/NEMS resonators. Criteria for optimizing the geometrical design of flexural mode MEMS/NEMS resonators are theoretically established with a view towards minimizing the TED for single-crystal silicon MEMS/NEMS flexural mode resonators.

Silicon carbide electromechanical resonators

Silicon carbide (SiC) flexural-mode resonators have been fabricated and actuated. Single-clamped beam (cantilever) and circular (ring) structures have been fabricated with top electrodes made of aluminium (Al). Ring structures have been shown to achieve higher resonant frequencies, in the megahertz range, compared to cantilevers. The bimorph Al/SiC structures have been actuated electrothermally by applying one input voltage to the Al electrodes. Electrothermal mixing of two input frequencies has been performed by applying two voltages showing that the devices can be used to convert an input frequency to a higher or lower one. Moreover, lead zirconium titanate (PZT) electrodes have been fabricated on top of SiC cantilevers. The ability of driving the PZT/SiC cantilevers piezoelectrically has been demonstrated by applying an input voltage to the PZT electrodes. The devices’ resonance has been detected electrically by measuring the electrode impedance. An enhancement of the electrical output has been obtained by decreasing the feedthrough capacitance. The results presented can be used for the implementation of SiC micro electromechanical mixer-filters with electrothermal actuation and piezoelectric sensing with robust and reliable characteristics.

Dimensional considerations in achieving large quality factors for resonant silicon cantilevers in air

This work aims to provide guidelines for designing rectangular silicon cantilever beams to achieve maximum quality factors for the fundamental flexural resonance at atmospheric pressure. The methodology of this work is based on experimental data acquisition of resonance characteristics of silicon cantilevers, combined with modification of analytical damping models to match the captured data. For this purpose, rectangular silicon cantilever beams with thicknesses of 5, 7, 8, 11, and 17 μm and lengths and widths ranging from 70 to 1050 μm and 80 to 230 μm, respectively, have been fabricated and tested. Combining the three dominant damping mechanisms, i.e., the air damping, support loss, and thermoelastic damping, the variation in the measured Q-factors with the cantilever geometrical dimensions is predicted. Also to better describe the experimental data, modified models for air damping have been developed. These modified models can predict the optimum length and thickness of a resonant cantilever to achieve the maximum quality factor at the fundamental flexural resonance mode in air.

A new magnetoelectric resonance mode in bilayer structure composite of PZT layer and Terfenol-D/epoxy layer

Magnetoelectric (ME) composites with a bilayer structure were prepared by easily combining Pb(Zr,Ti)O3 layer and Terfenol-D/epoxy layer. A new second flexural ME resonance mode was investigated and confirmed by vibration modal analysis. A giant ME coefficient at the first flexural resonance mode (2.79 V/cmOe at 35 kHz) is much larger than that at the transverse resonance mode (1.31 V/cmOe at 122 kHz). A flat ME response of about 360 mV/cmOe occurs in the range from 50 to 80 kHz, which shows a large bandwidth. This bilayer structure is expected to achieve a giant ME effect at relatively low frequency.

Magnetoelectric resonance behavior of simple bilayered Pb(Zr,Ti)O3–(Tb,Dy)Fe2∕epoxy composites

Magnetoelectric resonance behavior of a simple, bilayered structure of Pb(Zr,Ti)O3 ceramic chip and (Tb,Dy)Fe2∕epoxy mixture layer is investigated. Our results show that there occur three kinds of resonance, which are attributed to the first-order flexural, second-order flexural, and radial resonance modes, respectively. The measured phase spectra show a significant phase change around each resonance frequency. The thinner PZT chip is benefit for higher flexural resonance magnetoelectric effect. The magnetoelectric effects measured under open and short electric boundary conditions show different behaviors. This simple bilayered structure exhibits a large charge output of about 300pC∕Oe at the resonance frequency.

A Comparison of Dynamic Piezoactuation of Fiber-based Actuators and Conventional PZT Patches

This work aims to extend investigations of the dynamic flexural actuation of a cantilever beam to fiber-based actuators (e.g., active fiber composite (AFC) or macro-fiber composite (MFC)), which consist of a single level of unidirectionally aligned piezoceramic fibers surrounded by a polymer matrix and electrically loaded by interdigitated electrodes. The main advantage of such actuators is their poling in the same direction as the beam length, i.e., they are able to act with the piezoelectric strain constant d33 that is about two times higher than the piezoelectric strain constant d31 used in PZT patches. Modeling using the finite element method is based on the multiplication in both plane directions of a unit cell, which is taken as a fiber unit (with its matrix) within two pairs of half-electrodes of the same polarity. Modeling of the whole system is possible practically only with MFC actuators, whose fibers have a rectangular section, thus reducing the number of required finite elements. In addition to a non-uniform longitudinal free extension of the electrically loaded piezofiber-based actuator, a lateral free shearing has been calculated, which results in a non-planar deformation of the actuator. When two such MFC actuators are bonded on both sides of an aluminum cantilever beam and loaded in the antiphase mode, they conduct into all flexural resonance modes of the actuated beam, as expected. However, a slight interference by the torsion eigenmodes of the actuated beam has also been detected, which was not present in the actuation type using (leadzirconate titanate) PZT patches. The experimental validation of these FEM results first confirms the static values for free strain components of the MFC actuator depending on the loading voltage, then dynamic values regarding the eigenfrequencies, as well as the acceleration amplitudes of the MFC actuated vibrating beam.

Effects of coupled vibrations in cylindrical shell transducers

Coupled vibrations in cylindrical transducers can have unwanted effects on the resulting radiation or reception of sound. An investigation of vibrations of piezoelectric cylindrical shell transducers using analytical and experimental methods shows the presence of an additional flexural resonant mode that was not previously reported [see J. F. Haskins and J. L. Walsh, ‘‘Vibrations of ferroelectric cylindrical shells with transverse isotropy,’’ J. Acoust. Soc. Am. 29, 6 (1957)]. The additional flexural mode can be strongly coupled to radial and axial modes in cylindrical shells with finite thickness. The effect can be substantial, resulting in unwanted resonances that may be in the frequency band of interest for both projectors and receivers. An equivalent circuit is derived, which describes the cylindrical transducer operation taking into account the effect of coupled vibrations. The results of calculations in terms of resonant frequencies and frequency responses for both transmit and receive modes are in good agreement with experiments. The overall performance of the cylindrical shell transducer can be greatly enhanced by using cylinders of appropriate height to diameter aspect ratio. [Work supported in part by ONR 321SS.]

Coarse-Grained Molecular Dynamics for Computer Modeling of Nanomechanical Systems

Unique challenges for computer modeling and simulation arise in the course of the development and design of nanoscale mechanical systems. Materials often exhibit unconventional behavior at the nanoscale that can affect device operation and failure. This uncertainty poses a problem because of the limited experimental characterization at these ultra-small length scales. In this Article we give an overview of how we have used concurrent multiscale modeling techniques to address some of these issues. Of particular interest are the dynamic and temperature-dependent processes found in nanomechanical systems. We focus on the behavior of sub-micron mechanical components of Micro-Electro-Mechanical Systems (MEMS) and Nano-Electro-Mechanical Systems (NEMS), especially flexural-mode resonators. The concurrent multiscale methodology we have developed for NEMS employs an atomistic description of millions of atoms in relatively small but key regions of the system, coupled to, and run concurrently with, a generalized finite element model of the periphery. We describe two such techniques. The more precise model, Coarse-Grained Molecular Dynamics (CGMD), describes the dynamics on a mesh of elements, but the equations of motion are built up from the underlying atomistic physics to ensure a smooth coupling between regions governed by different length scales. In many cases the degrees of smoothness ofmore » the coupling provided by CGMD is not necessary. The hybrid Coupling of Length Scales (CLS) methodology, combining molecular dynamics with conventional finite element modeling, provides a suitable technique for these cases at a greatly reduced computation expense. We review these models and some of the results we have obtained regarding size effects in the elasticity and dissipation of nanomechanical systems.« less

Imaging of flexural and torsional resonance modes of atomic force microscopy cantilevers using optical interferometry

Commercial rectangular atomic force microscope cantilever beams made of silicon were set into vibration, using a piezoelectric ultrasonic transducer coupled to the chip of a cantilever. The transducer was excited with continuous rf in the frequency range of 100 kHz to 3 MHz. The vibrations were monitored using an optical Michelson heterodyne-interferometer allowing the surface of the cantilever under examination to be scanned with a lateral resolution of several μm. A number of free torsional and flexural vibration modes of the beams were imaged quantitatively. Comparison of the experimental resonance frequencies and the amplitude and phase distribution of the modes to theoretical models showed that asymmetries in the beam strongly influence the vibrational behavior of the beam. The consequences for quantitative local stiffness measurements are discussed.

Elastic properties of magnetostrictive thin films using bending and torsion resonances of a bimorph

The modification of the elastic properties of giant magnetostriction alloy films due to an applied magnetic field (the ΔE effect), has been studied. Two different types of films were deposited on Si substrates: (i) single layers of TbDyFeCo alloys typically 1000 nm thick and (ii) nanocomposite multilayer films of FeCo/TbFeCo each having a typical thickness of 6 nm. Both types of films were rendered magnetically anisotropic with a well defined in-plane easy axis. Rectangular samples were cut out of these bimorphs and firmly glued at one end to a heavy base to form a simple cantilever structure. The variations of film elastic moduli were deduced from the shifts of the cantilever resonance frequencies as a function of bias field for two basic configurations: (i) field applied along the easy axis and (ii) field applied along the hard axis. In contrast with previous work, both flexural and torsion resonance modes were excited and studied. As a result the field induced variations of both planar traction modulus and the shear modulus were obtained and new interesting features were discovered. In particular strongly negative values of the shear modulus were observed (at least in the nanocomposite films) in the vicinity of the divergence in the transverse magnetic susceptibility at saturation field along the hard axis. A simple but complete theoretical analysis shows that the uniaxial anisotropy model together with the assumption of isotropic magnetoelastic coupling gives a good semiquantitative understanding of all the experimental results.

Resonant response of complex shell structures

Resonances of shell structures play a prominent role in determining their acoustic characteristics. Among the most important of such characteristics are radiated noise, scattering, and self-noise levels. The nature of the structural resonances, for example the number of modes within a particular frequency band or the overall spatial dependence of the response, determines the importance of these phenomena in various contexts and what steps may be taken to modify their acoustic impact. In the past, knowledge of the resonant response of highly idealized systems has been used to guide the development of quiet platforms. The results of a scale model investigation of the effects of increased complexity on the nature of the resonances of submerged shell structures are reported. Effects presented here are flexural Bloch mode resonances, localized resonances resulting from structural irregularity, and the effects of non-axisymmetry on these phenomena. The implications of these results on acoustic design and the development of systems is discussed.

On the free and forced vibration of single and coupled rectangular plates

A mathematical model is developed for the coupling of two finite plates at an arbitrary angle for the prediction of the dynamic response and power flow at the coupling edge and at any cross section. The coupling at the joint edge considers bending, out-of-plane shear, and in-plane longitudinal (perpendicular to the joint edge) vibration. No constraint is imposed on the in-plane displacement perpendicular to the coupling edge. The exact solution for flexural mode shapes and resonance frequencies of rectangular plates with one edge free and the other edges simply supported is considered. This exact solution satisfies both the displacement and force boundary conditions, consequently it is used in the coupling of plate panels. An approximate relation is derived for prediction of the flexural resonance frequencies of higher-order modes with a reasonable accuracy. An approximate solution is presented for the in-plane response of the same plate panels when excited by in-plane forces perpendicular to the free edge. A simple expression is given for an approximate estimation of the cut-off frequencies for the different in-plane modes. The frequency response of the individual plate panels is presented as receptance functions for both flexural and in-plane vibration which are used in the plate coupling. The vibrational power injected into a plate by out-of-plane concentrated force and moment is computationally investigated. The importance of power input by a small moment excitation in some resonant bands even at low frequencies is highlighted. The spatial distribution of the components of flexural power transmitted by shear and moment across a section of the plate are investigated and the reason for the circulatory patterns of active power flow in plates is illustrated. The power flow characteristics in two plates coupled at an arbitrary angle is examined by a computational example. The effect of the coupling angle on the components of input power and power flow across the coupling edge is investigated. It is shown that the coupling of the two plates is mainly due to moment at frequencies up to the cut-off frequency of the first in-plane mode. Above this frequency, the coupling is due to out-of-plane shear and in-plane vibration with a diminishing participation of the moment in transmitting vibrational power through the coupling edge.

Finite-element analysis of both flexural and Helmholtz resonance modes in a flextensional transducer

During the last decade, several types of compact piezoceramic-driven barrel-stave flextensional transducers were researched and developed at the Defence Research Establishment Atlantic (DREA) for low-frequency high-power sonar applications. These transducers were designed to operate at the fundamental flexural resonance where they radiate sound efficiently and omnidirectionally. However, an important limitation common to almost all underwater flextensional transducers is that they require some form of depth compensation for operation at water depths exceeding a few hundred meters. One technique that may eliminate the need for depth compensation (and the associated design complications) for the barrel-stave flextensional transducer is to free flood the cavity inside the cylindrical ring-stack piezoceramic driver with the surrounding acoustic fluid medium and exploit the resulting Helmholtz resonances. To examine the feasibility of such a design approach, the flexural and Helmholtz resonances of a theoretical barrel-stave flextensional transducer with a free-flooded driver have been studied using finite-element techniques, and the results are presented in this paper. In addition, it is shown that the Helmholtz resonance in an experimental transducer does exist and may be practical.

Resonance frequency analysis by galerkin's method for clamped‐free, stepped flexural mode quartz crystal resonators and its application to a quartz crystal tuning fork

AbstractThis paper describes a stepped flexural mode quartz crystal resonator. The objective of this paper is to determine the resonant frequency, the motional series resistance R1, and the Q value of the stepped flexural mode tuning fork‐type quartz crystal resonator. The equation of motion is presented for a flexural mode resonator with nonuniform cross section with rotary inertia, and shear force is taken into consideration. Then, since the flexural mode quartz crystal resonator can be thought of as having two clamped‐free beams, the vibrational displacement computed from the boundary condition of clamped‐free ends and the condition of continuity of different cross sections are used to derive the frequency equation by Galerkin's method.In addition, the transcendental equation can be derived from the boundary conditions and the continuity condition. Further, from these relationships, the frequency constant (f·l1) and the resonant frequency are determined as functions of the width ratio W21 (width w2/‐width w1) and the length ratio L21 (length l2/length l1). As a result, it is shown that the frequency can be lowered without increasing the length of the resonator. These relationships are compared with the experimental results. It is shown that both agree very well. Hence, the present analysis results are effective for arbitrary frequency design. Finally, the typical values of the electrical equivalent circuit parameters are given. It is shown that, as L21 is increased, the motional series resistance R1 increases and the Q value decreases, although the characteristics are sufficient as a resonator. At the same time, the structure is resistant to vibration.