Higher Order Vibration Modes Research Articles

Wideband MEMS resonator using multifrequency excitation

We demonstrate the excitation of combination resonances of additive and subtractive types and their exploitations to realize a large bandwidth micro-machined resonator of large amplitude even at higher harmonic modes of vibrations. The investigation is conducted on a Microelectromechanical systems (MEMS) clamped-clamped microbeam fabricated using polyimide as a structural layer coated with nickel from top and chromium and gold layers from bottom. The microbeam is excited by a two-source harmonic excitation, where the first frequency source is swept around the targeted resonance (first or third mode of vibration) while the second source frequency is kept fixed. We report for the first time a large bandwidth and large amplitude response near the higher order modes of vibration. Also, we show that by properly tuning the frequency and amplitude of the excitation force, the frequency bandwidth of the resonator is controlled.

An integrated spectral collocation approach for the static and free vibration analyses of axially functionally graded nonuniform beams

A spectral collocation approach based on integrated polynomials is presented to investigate the statics and free vibrations of Euler–Bernoulli beams with axially variable cross section, modulus of elasticity, and mass density. The basic concept of the approach is the expansion of the highest derivatives appearing in the governing equations instead of the solution function itself by the truncated basis function. Then lower order derivatives and the function itself are obtained by integration. The constants appearing from the integrating process are determined by given classical or elastic restrained boundary conditions. Also, by incorporating the decomposition technique into the present approach, higher order vibration modes can be achieved even for stepped beams. Numerical examples including the statics and free vibrations of the beams with variance in geometry or material have been successfully solved, and the results are compared with those analytical or numerical solutions in the existing literature. The convergence and comparison studies show that convergent speed is rather rapid and the present approach can yield high accurate results with low computational efforts. Furthermore, the accuracy is not particularly affected by the adopted polynomials.

Higher order modes excitation of electrostatically actuated clamped–clamped microbeams: experimental and analytical investigation

In this study, we demonstrate analytically and experimentally the excitations of the higher order modes of vibrations in electrostatically actuated clamped–clamped microbeam resonators. The concept is based on using partial electrodes with shapes that induce strong excitation of the mode of interest. The devices are fabricated using polyimide as a structural layer coated with nickel from the top and chrome and gold layers from the bottom. Experimentally, frequency sweeps with different electro-dynamical loading conditions are shown to demonstrate the excitation of the higher order modes of vibration. Using a half electrode, the second mode is excited with high amplitude of vibration compared with almost zero response using the full electrode. Also, using a two-third electrode configuration is shown to amplify the third mode resonance amplitude compared with the full electrode under the same electrical loading conditions. An analytical model is developed based on the Euler–Bernollui beam model and the Galerkin method to simulate the device response. Good agreement between the simulation results and the experimental data is reported.

Multi-mode traffic-induced vibrations in composite ladder-deck bridges under heavy moving vehicles

Composite (steel-concrete) ladder-decks represent one of the most common solutions in road bridges nowadays. In these structures the Serviceability Limit State (SLS) of vibrations is traditionally ignored or roughly addressed by means of simple static deflection-based approaches, inherently assuming that the vibrations are controlled by the fundamental longitudinal mode. This work demonstrates that a wide range of high-order vibrational modes, involving the transverse flexure of the slab between longitudinal girders, govern the accelerations recorded in the deck and inside the vehicles. In addition, a new methodology for analysing the Vehicle–Bridge Interaction is proposed, including the approaching platforms, the transition slabs, and the bridge joints. The results suggest that the riding comfort for vehicle users is specially affected by direct effects on the wheels, like the road roughness and possible construction misalignments at the bridge joints, as well as low-frequency vibrations coming from the deck in short or slender bridges. The filtering effects resulting from the average of the response in time and in space when calculating the root mean square acceleration are also explored, and new design parameters are provided. In addition, several structural features (such as the depth and spacing of the longitudinal and transverse steel beams, the thickness of the concrete slab, and the stiffness of the cantilever cross beams at the diaphragm sections) have been studied, and a set of new design criteria has been established. It has been demonstrated that the transverse flexibility of the deck (specially influenced by the support conditions and the slab thickness) is critically important for the users’ (pedestrians and vehicle passengers) comfort, as it controls the aforementioned high-order vibrational modes which govern the dynamic response.

Optical signatures of electrically charged particles: Fundamental problems and solutions

The explicit solution to Maxwell׳s equations that satisfies the continuity equation is obtained for electrically charged spherical particles. The traditional separation-of-variables method (SVM) cannot be used to solve the vector wave equation for a non-uniformly charged spherical particle. In addition, a perturbation approach to the electromagnetic scattering problem fails if a spherical particle is occupied by electric charges that are not spatially homogeneous. By incorporating a correction to the conventional surface-current density, we have refined the conductivity model and found that the Rayleigh approximation (for mode n=1) is not a valid approach for modelling the optical effects by electrically charged particles much smaller than the wavelength of an incident radiation. Theoretical analyses indicate that peak enhancements of optical signatures are usually relevant in the long-wavelength limit due to the necessity to include higher-order modes of vibration (n>1).

Enhanced detection sensitivity of higher-order vibrational modes of gold nanodisks on top of a GaN nanorod array through localized surface plasmons

We report a method that enables the excitation of localized surface plasmons (LSPs) in a gold nanodisk array by placing each nanodisk on top of a GaN nanorod. When the rod length was much longer than the plasmon penetration depth inside the nanorod, the plasmonic field was found to be localized, and coupling between neighboring gold nanodisks was eliminated. The interaction between LSPs and acoustic vibrations in gold nanodisks was then investigated. Owing to the strong localization of the plasmonic field, weak, higher-order vibrational modes of gold nanodisk could be optically observed. Furthermore, such an LSP-based acoustic sensor could be operated at any angle of incident light. Our study not only provides an approach to excite LSPs in high-density metallic arrays, but also opens one of the possible solutions for the development of highly sensitive sub-terahertz hypersonic sensors with high angle tolerance of incident light.

Single walled boron nitride nanotube-based biosensor: an atomistic finite element modelling approach.

The unprecedented dynamic characteristics of nanoelectromechanical systems make them suitable for nanoscale mass sensing applications. Owing to superior biocompatibility, boron nitride nanotubes (BNNTs) are being increasingly used for such applications. In this study, the feasibility of single walled BNNT (SWBNNT)-based bio-sensor has been explored. Molecular structural mechanics-based finite element (FE) modelling approach has been used to analyse the dynamic behaviour of SWBNNT-based biosensors. The application of an SWBNNT-based mass sensing for zeptogram level of mass has been reported. Also, the effect of size of the nanotube in terms of length as well as different chiral atomic structures of SWBNNT has been analysed for their sensitivity analysis. The vibrational behaviour of SWBNNT has been analysed for higher-order modes of vibrations to identify the intermediate landing position of biological object of zeptogram scale. The present molecular structural mechanics-based FE modelling approach is found to be very effectual to incorporate different chiralities of the atomic structures. Also, different boundary conditions can be effectively simulated using the present approach to analyse the dynamic behaviour of the SWBNNT-based mass sensor. The presented study has explored the potential of SWBNNT, as a nanobiosensor having the capability of zeptogram level mass sensing.

Piezoelectric transduction of flexural modes in pre-stressed microbeam resonators

This paper reports on the optimization of the design of piezoelectric transducer elements integrated on doubly-clamped microbeam resonators utilized as (bio)chemical sensors. We report and emphasize the often forgotten influence of membrane stresses on defining the dimensions and optimal position of the piezoelectric transducer elements. The study takes into account stress induced structural changes and provides models for the equivalent motional parameters of resonators with particular shapes of the transducers matching the flexural modes of vibration. The above is analyzed theoretically using numerical models and is confirmed by impedance measurements and optical measurements of fabricated doubly-clamped beam resonators. We propose various transducer designs and highlight the advantages of using higher order vibration modes by implementing specially designed mode matching transducer elements. It is concluded that the paper describes and highlights the importance of accounting for the membrane stresses to optimize the resonator performance and the low power in electronic feedback of resonating sensing systems.

High-order face-shear modes of relaxor-PbTiO3 crystals for piezoelectric motor applications

The face-shear vibration modes of [011] poled Zt ± 45° cut relaxor-PT crystals and their applications for linear piezoelectric motors were investigated. Unlike piezoelectric ceramics, the rotated crystal was found to exhibit asymmetric face-shear deformations, and its two high-order face-shear modes degraded into two non-isomorphic modes. As an application example, a standing wave ultrasonic linear motor (10 × 10 × 2 mm3) operating in high-order face-shear vibration modes was developed. The motor exhibits a large driving force (1.5 N) under a low driving voltage (22 Vpp). These findings could provide guidance for design of crystal resonance devices.

A point-by-point impact vibration method for damage detection in beam-type structures

This paper develops the point-by-point impact vibration method, a damage diagnosis method for beam structures. The theoretical development and experimental verification were carried out on aluminum beams with single and double cracks in the vertical direction and on pre-stressed concrete beams with local breaks. The processing signals were the time histories of the acceleration responses of the beams and the impact force. The point-by-point impact vibration method is combined with the Dynamic Reciprocal Theorem, the Frequency Domain Decomposition method, and the Damage Index idea, all of which cover the modal curvature change (MCC) and the change of strain energy, as well as other measures. One of the advantages of this method is the ability to use fewer sensors to obtain higher order eigenmodes, which can be applied in the damage index. Experimental results indicate that higher order vibration modes exhibit great potential for damage identification.

Applications of in Situ Raman Spectroscopy for Identifying Nickel Hydroxide Materials and Surface Layers during Chemical Aging

The applications of in situ vibrational spectroscopy for identifying bulk and surface Ni(OH)2 are discussed. Raman spectra from α- and β-Ni(OH)2 samples immersed in water are generally similar to those collected from comparable dry samples. However, the Raman scattering intensities vary, and dry β-Ni(OH)2 additionally exhibits a surface O-H stretching mode at 3690 cm(-1). Using in situ Raman spectroscopy, the spontaneous transformation of α-Ni(OH)2 to β-Ni(OH)2 in room-temperature water was monitored. Such transformations are conventionally performed in high-temperature alkaline media. An intralayer OH-diffusion model is proposed. Internal stresses at the α/β-phase boundary caused shifted peaks, higher order vibrational modes, and a new water peak at 3520 cm(-1). We conclude that Raman spectroscopy may be applied to observe Ni(OH)2 materials in situ during chemical and electrochemical treatments. Such measurements provide information on the proportions of α- and β-Ni(OH)2 and their fine structural details with high sensitivity.

Boron nitride nanotube-based biosensor for acetone detection: molecular structural mechanics-based simulation

In this study, an ultra-sensitive biosensor based on a single-walled boron nitride nanotube (SWBNNT) structure is proposed for acetone detection. The molecular structural mechanics-based simulation approach has been used to model the atomic structure of SWBNNTs. The cantilevered and bridged configurations of SWBNNT-based biosensor have been considered for analysis. The resonant frequency shift due to attached mass has been analysed for the mass-based detection of acetone molecules. The present simulation approach is validated by comparing obtained simulated results with the continuum mechanics-based analytical results. Along with detection of the attached molecule, identification of its intermediate landing position along the length of the nanotube is equally important for the better performance of the biosensor systems. The frequency shift-based analysis has been reported for the mass-based detection of acetone molecules as well as its intermediate landing position along the length of the nanotube. The resonant frequency shift variations of the higher order modes of vibration for both the considered configurations of SWBNNTs have been assistive for the identification of intermediate landing position of the acetone molecule. The proposed molecular structural mechanics-based simulation approach is found to be very effectual in terms of simulation of the real atomic structures of the nanotube. The proposed biosensor can achieve extremely high sensitivity at molecular level and it can be potentially used for real-time sensing capability for the acetone concentration for future health monitoring.

Design methodology of piezoelectric energy-harvesting skin using topology optimization

This paper describes a design methodology for piezoelectric energy harvesters that thinly encapsulate the mechanical devices and exploit resonances from higher-order vibrational modes. The direction of polarization determines the sign of the piezoelectric tensor to avoid cancellations of electric fields from opposite polarizations in the same circuit. The resultant modified equations of state are solved by finite element method (FEM). Combining this method with the solid isotropic material with penalization (SIMP) method for piezoelectric material, we have developed an optimization methodology that optimizes the piezoelectric material layout and polarization direction. Updating the density function of the SIMP method is performed based on sensitivity analysis, the sequential linear programming on the early stage of the optimization, and the phase field method on the latter stage of the optimization to obtain clear optimal shapes without intermediate density. Numerical examples are provided that illustrate the validity and utility of the proposed method.

Development of robust control law for active buffeting load alleviation of smart fin structures

Aerodynamic buffeting load can lead to premature fatigue damage of aircraft vertical fin structures. This article presents a robust control law development strategy for active buffeting load alleviation of a smart fin structure. The impact of aerodynamic loads on the modeling uncertainties of the smart fin was investigated through extensive wind tunnel tests. Test results revealed that the airflow introduced higher damping ratio and caused frequency shift to the vibration modes. These aerodynamic effects may adversely affect the performance and robustness of active control laws. Based on the observations, the structured singular value synthesis technique was used to develop a robust control law for the smart fin using a truncated baseline dynamic model. A parametric uncertainty block was introduced to account for the changes in the modal parameters of the baseline dynamic model due to the aerodynamic effects. An additive uncertainty block was included to account for the unmodeled higher-order vibration modes as well as the modeling errors in the low frequency range. The robust performance of the control law was demonstrated through simulations as well as extensive closed-loop control experiments in the wind tunnel using various free airstreams and vortical airflows. This provided a verified control law design strategy for active buffeting alleviation applications.

An improved ground motion intensity measure for super high-rise buildings

Ground motion intensity measure (IM) is an important part in performance-based seismic design. A reasonable and efficient IM can make the prediction of the structural seismic responses more accurate. Therefore, a more reasonable IM for super high-rise buildings is proposed in this paper. This IM takes into account the significant characteristic that higher-order vibration modes play important roles in the seismic response of super high-rise buildings, as well as the advantages of some existing IMs. The key parameter of the proposed IM is calibrated using a series of time-history analyses. The collapse simulations of two super high-rise buildings are used to discuss the suitability of the proposed IM and some other existing IMs. The results indicate that the proposed IM yields a smaller coefficient of variation for the critical collapse status than other existing IMs and performs well in reflecting the contribution of higher-order vibration modes to the structural response. Hence, the proposed IM is more applicable to seismic design for super high-rise buildings than other IMs.

AN EFFICIENT FINITE ELEMENT MODEL FOR ANALYSIS OF SINGLE WALLED BORON NITRIDE NANOTUBE-BASED RESONANT NANOMECHANICAL SENSORS

In this paper, the dynamics analysis of single walled boron nitride nanotubes (SWBNNT) as a resonant nanomechanical sensor by using the finite element method has been reported. Molecular structural mechanics-based finite element model (FEM) has been developed by using three-dimensional elastic beams and point masses, such that the proximity of the model to the actual atomic structure of nanotube is significantly retained. Different types of armchair layups of SWBNNTs are considered with cantilevered and bridged end constraints. By implementing the finite element simulation approach, the resonant frequency shift-based mass sensitivity analysis is performed for both types of end constraints for considered armchair form of the SWBNNTs with different aspect ratios. For both types of end constraint, continuum mechanics-based analytical formulations, considering effective wall thickness of nanotubes are used to validate the present FEM-based simulation approach. The intermediate landing position of the added mass is analyzed, considering variations in resonant frequency shifts of the different fundamental modes of vibrations for both types of end constraints. The FEM-based simulation results for both types of end constraints found in good agreement with the continuum mechanics-based analytical results for the aspect ratio of range of 9–15. The mass sensitivity limit of 10-1 zg is achieved for SWBNNT-based resonant nanomechanical sensors. The resonant frequency shift for higher-order fundamental vibrational modes become stable as the attached mass moves away from the fixed ends for particular magnitude of attached mass. The present finite element-based approach is found to be effectual in terms of dealing different atomic structures, boundary conditions and consideration of added mass to analyze the dynamic behavior of the SWBNNT-based resonant nanomechanical sensors.

A Domain Decomposition Method for Vibration Analysis of Conical Shells With Uniform and Stepped Thickness

An efficient domain decomposition method is proposed to study the free and forced vibrations of stepped conical shells (SCSs) with arbitrary number of step variations. Conical shells with uniform thickness are treated as special cases of the SCSs. Multilevel partition hierarchy, viz., SCS, shell segment and shell domain, is adopted to accommodate the computing requirement of high-order vibration modes and responses. The interface continuity constraints on common boundaries and geometrical boundaries are incorporated into the system potential functional by means of a modified variational principle and least-squares weighted residual method. Double mixed series, i.e., the Fourier series and Chebyshev orthogonal polynomials, are adopted as admissible displacement functions for each shell domain. To test the convergence, efficiency and accuracy of the present method, free and forced vibrations of uniform thickness conical shells and SCSs are examined under various combinations of classical and nonclassical boundary conditions. The numerical results obtained from the proposed method show good agreement with previously published results and those from the finite element program ANSYS. The computational advantage of the approach can be exploited to gather useful and rapid information about the effects of geometry and boundary conditions on the vibrations of the uniform and stepped conical shells.

Analysis of vibration for regions above rectangular delamination defects in solids

We propose a semi-analytical approach to predict the natural flexural vibration frequencies of the material overlying a near-surface delamination defect in a solid. The formulation accommodates arbitrary length to depth ratio of the defect and higher-order modes of vibration. The material above the defect is modeled as a semi-clamped rectangular plate, where the dynamic edge effect factors, as deduced by Bolotin's asymptotic method, are estimated. The formulation results are evaluated through comparison to 3-D finite element (FE) simulation and experimental results obtained from impact resonance tests on concrete samples with controlled delamination defects. Good agreement with both experimental and 3-D FE results confirms the accuracy of the formulation in all cases.

Free and forced vibration analysis of uniform and stepped circular cylindrical shells using a domain decomposition method

This paper presents a domain decomposition technique for solving vibration problems of uniform and stepped cylindrical shells with arbitrary boundary conditions. Multilevel partition hierarchy, viz., stepped shell, shell segment and shell domain, is adopted to accommodate the computing requirement of high-order vibration modes and responses. The interface continuity constraints are enforced by using a modified variational principle and least-squares weighted residual method. The displacement components of each shell domain are expanded in the form of a double mixed series: Fourier series for the circumferential variable and polynomials/series (i.e., the Chebyshev orthogonal polynomials, Legendre orthogonal polynomials, and simple power series) for the axial variable. Free, steady-state and transient vibrations of uniform and stepped cylindrical shells are examined under different combinations of free, shear-diaphragm, simply-supported, clamped and elastic-supported boundaries. Comparisons with previously published results and finite element analyses show that the technique is computationally accurate and efficient. Effects of structural damping, stepped thickness and boundary conditions on the forced vibration responses of stepped cylindrical shells are also presented.

A domain decomposition approach for vibration analysis of isotropic and composite cylindrical shells with arbitrary boundaries

An efficient domain decomposition method is proposed for solving the free, harmonic and transient vibrations of isotropic and composite cylindrical shells subjected to various combinations of classical and non-classical boundary conditions. Multi-segment partitioning strategy is adopted to accommodate the computing requirements of high-order vibration modes and responses. The continuity constraints on the segment interfaces are incorporated into the system potential functional by means of a modified variational principle and least-squares weighted residual method. An arbitrarily laminated version of Reissner–Naghdi’s shell theory is employed to formulate the theoretical model. Double mixed series, i.e., the Fourier series and orthogonal polynomials, are used as admissible displacement functions for each shell segment. The utility and robustness of the method for the application of various basis functions are evaluated with the following four sets of orthogonal polynomial series, i.e., the Chebyshev orthogonal polynomials of first and second kind, Legendre orthogonal polynomials of first kind and Hermite orthogonal polynomials. To test the convergence, efficiency and accuracy of the present method, free and forced vibrations (including the harmonic and transient vibrations) of isotropic and composite laminated cylindrical shells are examined under different combinations of free, shear-diaphragm, simply-supported, clamped and elastic supported boundaries. The theoretical results are compared with those previously published in literature, and the ones obtained by using the finite element program ANSYS. Very good agreement is observed.