Vibration Mode Shapes Research Articles

Square Membrane Resonators Supporting Degenerate Modes of Vibration for High-Throughput Mass Spectrometry of Single Bacterial Cells.

In nanomechanical mass spectrometry, sensing devices are commonly placed in the vacuum environment and a stream of analytes is directed toward the sensor surface for measurement. Beam structures, such as double-clamped nanobeams and nanocantilevers, are commonly used due to their low inertial mass and the simplicity of the analytical models for mass extraction. The drawback of such structures is their low capture areas, compromising the capture efficiency and throughput of this technique. Bi-axisymmetric resonators, such as ultrathin square or circular membranes, arise as an optimal geometry to maximize capture efficiency while minimizing the device inertial mass. However, these structures present degenerate mechanical modes, whose frequency perturbations upon analyte adsorption are not well described by commonly used models. Furthermore, prior knowledge of the vibration mode shapes of the sensor is crucial for the correct calculation of the analyte's mass, and the mode shape of degenerate modes may change significantly after every adsorption event. In this work, we present an accurate analytical theory to describe the effect of mass adsorption on the degenerate modes of square membrane resonators and propose two different methods based on the new theory to update the vibration mode shapes after every adsorption event. Finally, we illustrate the problem experimentally obtaining the mass and adsorption position of individual Escherichia coli K-12 bacterial cells on commercial square silicon nitride membranes fabricated with very small tolerances.

Vibration of a thin panel exposed to ramp-induced shock-boundary layer interaction at Mach 2

Flight vehicles that operate in the supersonic regime can be subject to adverse fluid–structure interactions due to their lightweight design. The presence of geometric obstructions such as control surfaces or fins can induce shocks that can interact with the boundary layer, leading to flow separation. This work investigates experimentally the interaction between a compliant panel in a Mach 2 flow under a compression ramp-induced shock-wave/boundary-layer interaction (SBLI). Thin brass panels of different thickness are investigated in a wind tunnel. Tests are performed both with and without a 20° compression ramp installed. This direct comparison allows characterization of the effect of the SBLI on the system dynamics. High-speed stereoscopic digital image correlation (DIC) and fast-response pressure sensitive paint (PSP) are used to obtain simultaneous measurements of full field deformation and surface pressure of the panels. The panel vibration is dominated by the first bending mode. Despite the forcing of the separation shock foot, the presence of the SBLI does not significantly modify the operational deflection shape, frequency, and amplitude of the dominant vibration mode, beyond what is observed for the no-SBLI case. On the other hand, analysis of the shock foot motion shows that the shock primarily oscillates at the first natural frequency of the panel. This leads to the conclusion that the shock foot oscillation is driven by the panel vibration in a one-way coupling mechanism. The SBLI does modify the higher modes, which is likely due to localized forcing by the separation shock foot. Full-field surface pressure predictions are made using first order piston theory. Results show that the fluid–structure interaction is dominated by the large region of attached flow upstream of the shock foot.

The effect of auxeticity on the vibration of conical sandwich shells with ring support under various boundary conditions

Auxetic materials, materials demonstrating negative Poisson's ratio, have revolutionized the use of materials in industries, as they demonstrate superb acoustic response, fracture resistance, and energy absorption. For the first time, this study embraces the free vibration of conical shells consisting of an auxetic core with and without ring support under various boundary conditions. First, the material characteristics of the auxetic core are calculated by means of a micromechanical approach. Afterwards, the kinematic motion equations of the conical shell are derived utilizing the first-order shear deformation theory. Finally, the governing equations are solved using the powerful generalized differential quadrature element method (GDQEM). The primary goal of this paper is to study the role of implementing an auxetic core as well as ring support in determining the vibrational behavior of the structure. The results of the study showed that the honeycomb interior angle and the presence of ring support can significantly affect the natural frequency of the structure. Lower frequencies can be reached as the interior angle increases. The importance of ring position is found to be highly dependent on the longitudinal mode shapes of vibration. The impact of ring position on natural frequencies is affected by the semi-vertex angle of the cone, and a shift in frequency peaks can be observed by increasing the semi-vertex angle.

Supersonic flutter of variable stiffness circular cylindrical shells

Curvilinear reinforcement fibres allow to tailor the elastic properties of composite laminates in space, leading to Variable Stiffness Composite Laminates (VSCL). In this work, the aeroelastic stability of VSCL cylindrical shells under a supersonic airflow along the axial direction is analysed in order to verify if curvilinear fibres can be used to increase the flutter speed. A linear mathematical model for thin circular cylindrical VSCL shells is developed for that purpose. The aerodynamic pressure caused by the supersonic airflow is modelled recurring to the linearised piston theory. The p-version of the finite element method is used to transform the differential equations of motion from partial to ordinary. Using the latter, eigenvalue problems are defined to obtain the natural frequencies and mode shapes of vibration, and the critical flutter conditions. Various types of curvilinear fibre paths are explored, to find paths that result in higher values of critical flutter free stream pressure. The values are compared with the ones of conventional, constant stiffness, composite laminated shells.

Frequency and modeshape evaluation of steam turbine blades using the metal magnetic memory method and vibration wave propagation

Identifying frequency and modeshape of vibration to which a given mechanical structure is subjected most of its operational time is a key issue in diagnostics of the technical condition of the machine. By evaluating these modal parameters, the sources of vibration, and thus the possible sources of damage can be identified. The current paper presents a new diagnostic technique based on the metal memory method (MMM), which can be used for identification of operational frequency and modeshape of steam turbine blades. The MMM method includes the measurement and analysis of blade's residual magnetic field (RMF) distribution, which reflects directions of main stresses resulting from working loads. It is demonstrated that based on the results of RMF measurement, vibration frequency and modeshape, to which the blade is subjected most of the time, can be evaluated. To calculate these modal parameters, stress waves propagating along the blade, and the resulting standing wave have been considered. Clear correlation between blade’s magnetic field distribution, stress distribution, and FEM results of frequency and modeshape calculation has been observed. It has been confirmed that the frequency and modeshape at which the blade vibrates during its normal operation is close to the blade's 10th natural frequency, which is about 1014 Hz.

Wave propagation behaviors of a low-symmetry reentrant chiral structure with mass inclusion in a single material

This work proposed a low symmetry characteristics metamaterial in a single material. The structural configuration of the metamaterial is composed of the reentrant structure, chiral structure and mass inclusion. The wave propagation characteristics are investigated in terms of the Bloch-Floquet wave propagation theorem. Firstly, the band diagram of the metamaterial is analyzed. The wave behaviors of the structure are changed due to the low symmetry characteristics. The irreducible Brillouin zone (IBZ) of the proposed structure is changed. The band diagram in the whole IBZ is analyzed to get the accurate distributions of the bandgaps. Some extremum values of the dispersion surface are deviated from the boundary of the IBZ. The formation mechanisms of bandgaps are illustrated based on the analysis of the vibration mode shapes. Then, the phase velocity and group velocity of the elastic wave are calculated, showing the strong anisotropic properties. Moreover, the effects of the geometrical parameters on the bandgaps are investigated. The locations and width of the bandgaps can be tuned for by the rational design. The existence of bandgaps of the metamaterial are validated by a finite element model. Finally, a metamaterial beam sample with seven cells is made and experiment test are conducted for the vibration attenuation performance and error analysis. The proposed structure can be used for vibration attenuation in the fields of underwater vehicles.

Principal component analysis of vibration response of a rod bundle subjected to jet cross-flow

Flow injecting through loss of coolant accident (LOCA) holes in the baffle plates of some pressurized water reactors (PWRs) can potentially induce vibrations of reactor fuel assemblies. For reactor fuel rods in close proximity to LOCA holes, the jet flow transverse to the rod axis may lead to vibration levels significantly higher than those for rods located away from the LOCA holes. The jet cross-flow-induced vibration could, in turn, induce fretting wear of the fuel rod cladding at the fuel assembly supports, also known as spacer grids.Fuel rods have axisymmetric flexibility, testing the real condition of the fuel assembly is important to know the most dominant vibration direction relative to the jet flow. This information could be used to suppress the dominant vibration by increasing the stiffness of spacer grids in the dominant direction. In this study, the effect of jet cross-flow on a 6x6 square rod bundle axisymmetric vibration is experimentally investigated. This research aims to answer the following important question: Does there exist a generalized eigenvector model that could be used to predict rod bundle vibration in jet flows, in addition to characterizing the stability effect of the diameter ratio? The dynamical behavior of the rod bundle is studied for three jet-to-rod diameter ratios; 1.6, 2.3, and 3.0. Flow-induced vibration tests are performed to determine the stability threshold of the bundle as well as to extract the bundle vibration mode shapes for the three nozzles tested. The experimentally measured mode shapes identify the dominant vibration direction which provides valuable insight into the fluid–structure interaction dynamics. The experiments show that increasing the diameter ratio lowers the critical velocity at which fluidelastic instability occurs. The maximum vibration amplitude in the bundle, on the other hand, decreases as the diameter ratio increases.Analysis employing PCA and SVD confirmed the existence of a set of robust modes that exist for the range of system parameters tested. Hence, modes shape obtained with different nozzles are the same for the same acceleration ratio of the jet flow. The existence of these structurally stable (in parameter space) fluid–structure modes has significant implications for the stability analysis of reactor fuel rods subjected to jet cross-flow. The results show that reduced order models of the high-dimensional fluid–structure can be developed and that the models can realistically capture the dynamics of the jet cross flow-induced vibrations of the fuel rod bundle.

Analytical solutions of free vibration for rectangular thin plate and right-angle triangle plate on the Winkler elastic foundation based on the symplectic superposition method

Abstract Based on the symplectic superposition method, the free vibration models of rectangular and right-angle triangle plates on the Winkler elastic foundation are established in the present paper, and the modes and frequencies are studied. In addition, the theoretical calculation model and finite element analysis model of rectangular thin plate and right-angle triangle plate on elastic foundation are established by using Mathematica software and ABAQUS software. It proves that the symplectic superposition method converges very fast and has a good consistency with the finite element simulation results. Analytical results show that foundation stiffness, aspect ratio, and boundary condition have great influences on vibration frequency and mode shape for structures. This paper solved the free vibration problem of rectangular plate and right-angle triangle plate on elastic foundation by using symplectic superposition method. Compared with the inverse or semi-inverse method, this method avoids the process of assuming the form about the solution, hence the result of this method is completely rational.

Modal testing with a pedestrian as a vibration exciter

A novel approach, for identification of modal properties using a dynamic excitation induced by a human walker, has previously been proposed to identify natural frequencies, damping ratios and modal masses. The developed approach is now expanded to also estimate vibration mode shapes, thus fully characterising the dynamic behaviour of a structure in a simple and affordable way. Additionally, the reliability of the method is parametrically analysed and validated against simulated and real world data of a footbridge. Results indicate that, given a reasonable initial estimate of the target modal properties values and with careful consideration of the number and location of measured responses, it is possible to estimate all modal properties of a structure in a reliable and simpler way than conventional modal testing methods: handheld impact hammer, electrodynamic shaker or ambient vibration tests.

Consideration of sensing method for aperture cracks focusing on vibration mode shape changes in Timoshenko beam

Consideration of sensing method for aperture cracks focusing on vibration mode shape changes in Timoshenko beam

A novel vibration analytical model for thin-walled box-like long cylindrical structures combining elastic plates and straight and curved beams

This paper proposes a novel analytical model for elastic vibrations of thin-walled box-like long cylindrical structures, commonly observed in railway vehicle carbodies, aircraft, and buses. Here, we focus on railway vehicle carbody, where three-dimensional elastic vibrations are a significant problem for improving ride quality. The carbody is modeled as a box-like structure consisting of elastic plates and beams. The condition to connect those structural members is satisfied by introducing artificial springs at their joints. From observations of some actual carbody's elastic vibration modes, it has already been found that the relative vertical deformation between the roof and floor interacts with lateral deformation of side structures. To express this three-dimensional deformation simply, the authors introduce curved elastic beams as pillars in the side structures in this study; therefore, the newly proposed model is called the curved box model. The free vibration of the curved box model has been investigated in this paper by applying the Ritz method to derive the eigenvalue problem. The modal characteristics (natural frequencies and mode shapes) of elastic vibrations are calculated. The vibration analysis procedure is validated by comparing those results with those obtained from a finite element model. As some numerical examples, the effects of cross-sectional shape changes on the natural frequencies of elastic vibration modes are presented.

Axisymmetric bending vibration analysis of bidirectional functionally graded rotating micro-disk under thermo-mechanical loading

The free vibration behavior of a bidirectional functionally graded rotating micro-disk that is subjected to uniform transverse pressure and high-temperature thermal loading has been studied. The micro-disk is functionally graded along the radial and thickness directions. The problem is mathematically formulated using two distinct but interrelated steps within the framework of Kirchhoff plate theory and modified couple stress theory. The first step determines the time-invariant deformed configuration of the micro-disk under centrifugal, pressure, and thermal loading using minimum potential energy principle. The second step determines the free vibration behavior of the micro-disk in the neighborhood of the deformed configuration using Hamilton’s principle. The solutions of the governing equations for both these steps are obtained using the Ritz method. The mathematical model is successfully validated with various reduced problems. The numerical results for the first four axisymmetric bending vibration modes are presented to investigate the effects of wide range of parameters such as rotational speed, applied pressure, thermal loading, size-dependent thickness, volume fraction indices, and radius ratio. The mode shapes of vibration are illustrated through surface and contour plots.

Study on the dynamic behaviour of Bafgh-Bandar Abbas lane railway bridge for structural health monitoring purpose

ABSTRACT This paper uses the vibration-based damage detection method for structural health monitoring of a Railway Bridge. Hence, the bridge had been exactly modelled in finite element analysis software. Due to the elastomeric bridge bearing, direct calculating of the natural frequencies and extracting the mode shapes of the bridge is not suitable and effective. Therefore, a time-dependent transient analysis of the train movement on the bridge is done and then the vibrations of all truss cells during the train crossing period were extracted. Afterwards, the relevant data is transformed, which represents the natural frequencies of the bridge vibration and their amplitude at each point. In the following, the mode shapes of vibration and the distribution of vibration energy are calculated. Applying any artificial damages in the model, occurring changes in the natural frequencies, mode shapes, and the vibration energies of the modes, are examined. Consequently, detecting and locating damages in the structure has been done with acceptable accuracy. Finally, for validation of the results, an accelerometer is installed on the bridge truss in the middle part of the bridge to extract the acceleration of bridge vibration at the train crossing time interval. In the end, the results are compared and presented.

Bio-inspired non self-similar hierarchical elastic metamaterials

Hierarchy provides unique opportunities for the design of advanced materials with superior properties that arise from architecture, rather than from constitutive material response. Contrary to the quasi-static regime, where the potential of hierarchy has been largely explored, its role in vibration mitigation and wave manipulation remains elusive.So far, the majority of the studies concerning hierarchical elastic metamaterials have proposed a self-similar repetition of a specific unit cell at multiple scale levels, leading to the activation of the same bandgap mechanism at different frequencies. On the contrary, here, we show that by designing non self-similar hierarchical geometries allows us to create periodic structures supporting multiple, highly attenuative and broadband bandgaps involving (independently or simultaneously) different scattering mechanisms, namely, Bragg scattering, local resonance and/or inertial amplification, at different frequencies. The type of band gap mechanism is identified and discussed by examining the vibrational mode shapes and the imaginary component of the wavenumber in the dispersion diagram of the unit cell. We also experimentally confirm this by performing measurements in the lowest frequency regime on a 3D printed structure.Hierarchical design strategies may find application in vibration mitigation for civil, aerospace and mechanical engineering.

Effect of fibre volume on the natural frequencies of laminated composite plate

Resonance is an important phenomenon related to natural frequencies that could lead to catastrophic failure, and therefore investigating this phenomenon is crucial. There have been many studies regarding the effect of stiffness on the natural frequencies of laminated composite plates; however, studies relating the fibre volume to the natural frequencies have still not been established. This paper aims to investigate the effect of fibre volume on the natural frequencies of laminated composite plates. A 300 mm × 300 mm plate model made of carbon/epoxy composite laminates under Clamp-Free-Clamp-Free (CFCF) boundary conditions was developed using Finite Element Software (ANSYS). Different lamination schemes (fibre orientations, symmetry and anti-symmetry lay-up) with various fibre volume fractions were the analysed parameters. The natural frequencies were calculated for each scenario using finite element simulations. The mode shapes of vibration of the plates were also examined. To establish the precise finite element model for free vibration analysis of laminated composite plates, convergence analysis and numerical verification were conducted. The present numerical simulation results were in good agreement with published numerical results. The findings demonstrated that increasing the fibre capacity results in an increase of all vibrational modes and their shapes. Aside from that, the fibre orientation at 90°had the highest natural frequency value (910.52 Hz, 915.66 Hz and 1332.2 Hz) and the lowest natural frequency value at 0° (244.84 Hz, 245.90 and 627.95 Hz) for the first three modes shape of vibrations. In general, this study has made important contributions on how a fibre volume affects the natural frequencies of laminated composite plates.

Bimorph sensor based in-line inspection method for corrosion defect detection in natural gas pipelines

Recently, bimorph sensors based on the in-line inspection (ILI) method have shown considerable promise for detecting surface defects (e.g., corrosions) in gas pipelines. However, the effect of physical parameters such as bimorph position and scanning speed on the performance of the bimorph sensor has not yet been investigated. As a result, the capability of the bimorph sensor for defect detection in pipelines is not clear. This study aims to optimize the physical parameters and investigate the influence of physical parameters on the performance of the bimorph sensor. For this purpose, first various corrosion defect sizes were considered and analyzed using piezoelectric theory. Next, an analytical relationship between the vibration response of the bimorph sensor and defect is derived using the Euler-Bernoulli beam theory. Finally, a real-time field test was conducted based on the installation of the proposed method on a developed smart PIG to demonstrate the detection efficiency of the bimorph sensor. The novel contributions of this study are as follows: (1) The proposed analytical method uses cantilever beam vibrational mode shapes together with the piezoelectric effect to detect and quantify the corrosion defects. (2) The current study proposed a bimorph sensor method capable of performing high scanning speed (0.5 m/s) in real-time pipeline inspection. (3) The performance of the proposed method on the various position of bimorph on the cantilever probe and scanning speed is studied analytically and experimentally. (4) Multiple corrosion defects and pipeline joints were detected and localized successfully in complex pipeline networks. • Proposed a bimorph sensor method to detect corrosion defects in gas pipelines. • Performance of the proposed method is investigated analytically and experimentally. • Bimorph sensor is designed and tested in a laboratory testbed. • Effectiveness of the proposed method on various physical parameters is presented. • Field test is conducted with the designed bimorph sensor array.

Inertia mass bio-sensors based on snap-through phenomena in electrostatic MEMS shallow arch resonators

This research examines on a new class of MEMS inertia mass sensors that are simple, sensitive, and selective in possibly detecting tiny objects. The sensor consists of two beams attached by an end-plate and is mounted on an electrostatically actuated shallow micro-arch. The presence of the end-plate overcomes the shortcoming of building inertia mass sensors using in-plane beam resonators. It gives more room to deposit detector material and, therefore, controls and mobilizes the quantity of the detector toward sensing a target. The design exploits the bistable behavior, resulting from the combination of the snap-through instability and the nonlinear force to move from one stable equilibrium to another. The transition can be controlled statically or dynamically depending on an operational modes. The eigenvalue problem assessment shows a considerable reduction in the first and third symmetrical resonant frequencies under DC voltage when a few picograms of the object substance are introduced. It is also corroborating that placing the added mass at the center of the end-plate and operating the sensor at vibration mode shape that dominated by the platform are more effective for mass detection through measuring the change in its frequency and bifurcation points. We found that superimposing the excitation signal to a small AC harmonic load, linear dynamic responses show a shift in the neighborhood of the first resonant frequency. On the other hand, increasing the actuation signal, dynamic responses show nonlinear trends offering possibilities to use the proposed design as a bifurcation-based type inertia sensor. This added mass leads to significant shifts at the locations of the bifurcation points compared to those in the absence of the object.

Multiple particle identification by sequential frequency-shift measurement of a micro-plate

• A resonant cantilevered micro-plate is proved to be a promising candidate that can sequentially detect multiple particles. • The identification of multiple particles is regarded as an optimization problem via genetic algorithm to replace the widely-used least squares method. • Better identification accuracy is illustrated by exquisitely updating the modal shape changes of a micro-plate after each particle adsorption. • An experimental strategy based on focused ion beam and scanning probe microscopy verifies the applicability of the theoretic framework in designing and evaluating micro/nano sensors. The detection of multiple particles has been mainly studied by examining the flexural vibration properties of a one-dimensional Euler-Bernoulli beam under the assumption that the vibration mode shapes are unchanged. In fact, with the torsional vibration of the detector, valuable pieces of information can be derived, while the detection process is facilitated. Besides, the mode shapes of the detector can be easily changed by measuring multiple adsorbates or heavy analytes, which can fundamentally affect the detecting accuracy. To the best of our knowledge, these two issues have not yet been systematically considered in the literature. Along these lines, in this work, a theoretical framework is introduced to detect both the masses and positions of multiple particles absorbed on a two-dimensional micro-plate, where the history of the mass deposition events is sequentially measured from the recorded frequency shifts. The detection of multiple particles was realized by using an optimization process via the genetic algorithm, which circumvents the initial value and the local optimum problems that the widely-used least-squares method could be confronted with. In addition, an improved detection accuracy was ensured by exquisitely updating the modal shape changes of a micro-plate after the adsorption of each particle. To validate the proposed method, the frequency shifts were obtained by running modal analysis, where 20 μ concentrated masses were added sequentially on the top surface of the micro-plate modelling. As was expected, all particles could be effectively identified with enhanced accuracy. Furthermore, experimental validation was performed on Si-based micro-plates cantilevered, which were fabricated by focused ion beam milling and sequentially loaded by carrying out Pt depositions from the measured frequency shifts of the micro-plate structure before and after each sequential Pt deposition, the loaded masses and their positions in both the length and width directions were extracted. The results obtained from the model are in good agreement with the estimation based on scanning electron microscopy. The proposed method is anticipated to be further applied to multiple particle detection applications in many fields including biology, medicine and chemistry.

Structural and modal analysis of gas turbine blade using ansys

Gas turbine technology has advanced significantly during the last twenty years. The advancement in materials technology, new coatings, and novel cooling techniques are driving the expansion. Material selection is critical since turbomachinery design is complicated, and material performance is directly connected to efficiency. The materials should be able to withstand high vibrations, high temperature, and pressure. In this work, the maximum stress and the deformation induced due to pressure were determined to study the vibrational response, mode shapes, and natural frequencies. Four different materials have been considered, namely Inconel 617, Titanium alloy, Rhenium alloy, and Nimonic 80A. A suitable material for a gas turbine blade was suggested after analyzing these materials. The 3D CAD model is modeled in CATIA V5 and analysis is carried out using FEM-based software ANSYS

Natural Vibration Characteristics Analysis of a High-Rise Reinforced Masonry Structure Based on Field Test Data

In structural response array observation, the vibration response of a structure during an earthquake or from the natural environment is recorded and stored using high-sensitivity strong motion seismographs, and the dynamic characteristics of the structure are analyzed and determined using random signal data processing technology. Due to the use of field test data for analysis, this may be the most accurate and effective way to obtain the actual characteristics of the structure, which can be further used to verify the accuracy of theoretical analysis, experimental results, and numerical simulations. Therefore, this technique plays an important role in earthquake prevention and disaster reduction, with the application of strong motion observation data. In this paper, field vibration tests were performed on the highest reinforced masonry structure in China. With the test environmental vibration data, the natural vibration frequency values and mode shapes of the structure were identified using the peak picking method. A numerical modal analysis was then performed to verify the accuracy of the field test results. In addition, the structural response records obtained during an earthquakes in Songyuan were also used to identify the natural vibration frequency of the structure and the changes in the natural vibration frequency before, during, and after the earthquake. The results showed that the structure was not damaged during the earthquake and remained in an elastic state.