Eigenfrequencies Of Vibrations Research Articles

On location of piezoelectric element in a smart-structure: numerical investigation and experiment

In this paper, based on some example problems it was demonstrated that in examining the possibilities of smart structure applications, the matter of considerable researchers’ concern is the problem of location of piezoelectric elements in the structure to allow effective realization of its smart functions in the framework of the specified strategy of structure control and target purposes (vibration damping, defectoscopy, etc.) The numerical and experimental investigations have shown that for structures with the elements made of piezoelectric materials, it is more convenient to use as a parameter, specifying the best location of the piezoelectric element for damping the vibrations at the prescribed frequency, the coefficient of electromechanical coupling, which is evaluated by the values of eigenfrequencies of the structure in the short-circuit and open-circuit regimes. The values of eigenfrequencies of vibrations are evaluated by solving the problem of natural vibrations of electromechanical systems by the finite element method using the applied ANSYS package. The investigation were conducted for a thin-walled aluminum shell in the form of half-cylinder.

Development of a perfect match system in the improvement of eigenfrequencies of free vibration

It is well known that the standard finite element method (FEM) with overly-stiff effect gives the upper bound solutions of natural frequencies in the free vibration analysis using triangular and tetrahedral elements. In this study, for the first time, this paper aims to improve the prediction of eigenfrequencies through the perfect match between the stiffness and mass matrices. With redistribution of mass in the system, we can tune the balance between stiffness and mass of a discrete model. This can be done by simply shifting the integration points away from the Gaussian locations, while ensuring the mass conservation. A number of numerical examples including 2D and 3D problems have demonstrated that the accuracy of eigenfrequencies is strongly determined by the location of integration points in the mass matrix. With appropriate selection of integration points in the mass matrix, even the exact solution of eigenfrequencies can be obtained in both FEM and smoothed finite element method (SFEM) models.

Optical bottle versus acoustic bottle and antibottle resonators.

The theory of slow acoustic modes propagating along the optical fiber and being controlled by the nanoscale variation of the effective fiber radius (analogous to the theory of slow optical whispering gallery modes) is developed. Surprisingly, it is shown that, in addition to acoustic bottle resonators (which are similar to optical bottle resonators), there exist antibottle resonators, the neck-shaped deformations of the fiber that can fully confine acoustic modes. It is also shown that an eigenfrequency of the mechanical vibrations of a silica parabolic bottle resonator can match the separation between the eigenfrequencies of a series of its optical modes, thereby enabling the resonant mechanical excitation of these series. The developed theory paves the groundwork for slow-mode optomechanics in an optical fiber.

Eliminate localized eigenmodes in level set based topology optimization for the maximization of the first eigenfrequency of vibration

Maximizing the fundamental eigenfrequency of vibration is an important topic in structural topology optimization. Previous studies of such a topology optimization problem should always be cautious of the “artificial localized mode” as it makes the optimization fail. In the present work, a level set based topology optimization is proposed to address such an issue. The finite element analysis is conducted on the actual structure by using a body-fitted mesh and without artificial weak material, thus localized mode that conventionally arises due to low-density region is prevented. In the present study, attention is turned to localized mode occurred gradually during the optimization. Such kind of localized mode results from the emergence of isolated area or cracked structure member produced by topological changes. A mode recognition technique based on the volume ratio of vibration-free region to the entire structure is proposed to identify such localized mode. Numerical examples of 2D structures are investigated.

Motion behaviour of magneto-sensitive elastomers controlled by an external magnetic field for sensor applications

The development of sensor systems with a complex adaptive regulation of the operating sensitivity and behaviour is an actual scientific and technical challenge. Smart materials like magneto-sensitive elastomers (MSE) are seen as one potential solution for this problem, since their mechanical properties may be controlled by external magnetic fields. The present paper deals with the investigation of elastic and damping properties of MSE containing magnetically soft particles under the influence of a uniform magnetic field. Based on the measurement of the first eigenfrequency of free bending vibrations of a fixed beam, the effective Young's modulus is evaluated theoretically and also numerically using Finite Element Method. It is shown that this parameter, as well as the first eigenfrequency of the beam, increases monotonically with the magnitude of the applied magnetic field. The results are aimed to develop an acceleration sensor with adaptive magnetically controllable sensitivity range for the detection of external mechanical stimuli of the environment.

Symmetric scrolled packings of multilayered carbon nanoribbons

Scrolled packings of single-layer and multilayer graphene can be used for the creation of supercapacitors, nanopumps, nanofilters, and other nanodevices. The full atomistic simulation of graphene scrolls is restricted to consideration of relatively small systems in small time intervals. To overcome this difficulty, a two-dimensional chain model making possible an efficient calculation of static and dynamic characteristics of nanoribbon scrolls with allowance for the longitudinal and bending stiffness of nanoribbons is proposed. The model is extended to the case of scrolls of multilayer graphene. Possible equilibrium states of symmetric scrolls of multilayer carbon nanotribbons rolled up so that all nanoribbons in the scroll are equivalent are found. Dependences of the number of coils, the inner and outer radii, lowest vibrational eigenfrequencies of rolled packages on the length L of nanoribbons are obtained. It is shown that the lowest vibrational eigenfrequency of a symmetric scroll decreases with a nanoribbon length proportionally to L –1. It is energetically unfavorable for too short nanoribbons to roll up, and their ground state is a stack of plane nanoribbons. With an increasing number k of layers, the nanoribbon length L necessary for creation of symmetric scrolls increases. For a sufficiently small number of layers k and a sufficiently large nanoribbon length L, the scrolled packing has the lowest energy as compared to that of stack of plane nanoribbons and folded structures. The results can be used for development of nanomaterials and nanodevices on the basis of graphene scrolled packings.

Vibrations of an elastic cylindrical shell near the lowest cut-off frequency

A new asymptotic approximation of the dynamic equations in the two-dimensional classical theory of thin-elastic shells is established for a circular cylindrical shell. It governs long wave vibrations in the vicinity of the lowest cut-off frequency. At a fixed circumferential wavenumber, the latter corresponds to the eigenfrequency of in-plane vibrations of a thin almost inextensible ring. It is stressed that the well-known semi-membrane theory of cylindrical shells is not suitable for tackling a near-cut-off behaviour. The dispersion relation within the framework of the developed formulation coincides with the asymptotic expansion of the dispersion relation originating from full two-dimensional shell equations. Asymptotic analysis also enables refining the geometric hypotheses underlying various ad hoc set-ups, including the assumption on vanishing of shear and circumferential mid-surface deformations used in the semi-membrane theory. The obtained results may be of interest for dynamic modelling of elongated cylindrical thin-walled structures, such as carbon nanotubes.

The theory of longitudinal vibrations of a conical elastic body in an elastic medium

Aim. To elaborate the theory of longitudinal vibrations of a solid elastic body with one fi xed end in the elastic medium. The example of such a body may be found in a sugar beet root in soil, the latter being elastic medium. Methods. The principle of stationary action of Ostrogradsky-Hamilton and the Ritz method were applied in the work. Results. The Ritz method was applied to obtain the Ritz frequency equation for the oscillating process under investigation. The analytic expressions were defi ned to determine the fi rst and second eigenfrequencies of vibration and the amplitude of constrained vibrations of any of its cross-sections. The values of the fi rst and second eigenfrequencies of the elastic body under investigation with specifi c geometric and physical pa- rameters were found. The dependency diagrams for the fi rst and second eigenfrequencies on the coeffi cient of elastic contraction of soil as the elastic medium, and the dependency diagrams for the amplitude of constrained oscillations of the mentioned body on the coeffi cient c of elastic deformation of soil and the distance of the cross-section of the body from the conditional point of fi xation were drawn. The dependency diagrams for the amplitude of constrained oscillations of the elastic body on the change in the amplitude and the frequency of perturbing force were obtained. Conclusions. The impossibility of resonance occurrence was substantiated as the frequency of the perturbing force cannot equal the frequency of eigenvibrations of the elastic body due to technological and technical reasons. It was proven that the breaking of the elastic body is impossible with lon- gitudinal deformations due to the shortness of the amplitude of longitudinal vibrations of the mentioned body.

Dynamic Analysis of Partially Filled Non-circular Cylindrical Shells with Liquid Sloshing

The paper investigates the dynamic behavior of thin-walled reservoirs containing an ideal liquid taking into account the effects of hydroelastic interaction and sloshing. A mathematical statement of the problem is based on the principle of virtual displacements, which accounts for the pre-stressed non-deformed state of the shell caused by various force factors. The behavior of compressible liquid is described by linearized Euler equations, which are transformed by the Bubnov–Galerkin method. The dynamics of partially filled circular and elliptical cylindrical reservoirs are investigated numerically using a finite element procedure. It has been shown that allowing for sloshing considerably reduces eigenfrequencies of vibrations of the examined systems but has inessential effect on the displacements of the structure under non-stationary loads. Based on the modal analysis we present a classification of eigenmodes of free surface oscillations of a liquid in vertical tanks. It has been found that due to consideration of the sloshing effect, the frequency spectrum of the system can split into two parts in the case when the vibration frequencies of liquid differ from the vibration frequencies of an empty shell. Moreover, under harmonic excitation consideration of liquid sloshing leads to a more complicated amplitude-frequency curve characterized by displacement jumps.

Principles of work of electronic durometers

Electronic durometers found a wideuse for control of hardness of details, both in the processof production and in the process of exploitation. The large variety of these devices is taken just tofour principles of measuring of hardness – on a rebound, on the difference of frequencies betweenthe eigenfrequency of vibrations of the diamond-pointed pyramid and frequency of the preliminaryloaded diamond-pointed pyramid, determinations of hardness on the size of electric resistance of thespecial diamond-pointed pyramid and realizing method of visualization of imprint of indenter.Most widely many firms and companies are present dynamic durometers (ASTM A956-06), workingson the method of rebound (hardness on Leeb, HL) and ultrasonic durometers (ASTM A1038) which use theresonance-impedance method of measuring of hardness (Ultrasonic Contact Impedance– UCI method).Unique method of Esa Test®, proper Din50 158 was developed and patented a firm Ernst(Switzerland).V present tense it is used only in the devices of this company. Principle ofdetermination of value of hardness on the method of Rockwell (En10109-3) was applied the samefirm due to the device patented by them, a stride in development of mobile durometers was becomeby the issue of durometer, realizing the method of visualization of imprint of indenter at measuringsof hardness of material on Vickers (TIV). The basic brands of modern durometers are resulted andprinciples and methods are described measurings of hardness, which they use.

Improving the mechanical performances of a multilayered plate with the orientations of its layers of fibers

We consider a symmetric composite multilayered plate whose fiber orientation varies from a layer to another. The plate model used is that of Mindlin. We are interested in determining the optimal fiber orientations that optimize, in the same time, two criteria: minimizing the compliance and maximizing the smallest eigenfrequency of vibration or minimizing both of the compliance and the smallest eigenfrequency or maximizing both of the smallest eigenfrequency and the smallest buckling load. In order to optimize one of the above criteria, a metaheuristic algorithm of Simulated Annealing type is used. While, in the case of optimizing two objectives, the Pareto front method is used. Numerical results are presented for a rectangular eight-layered plate.

Boson peaks in doped colloid glasses

We experimentally investigated the correlation between local structures and phonon modes in quasi-2D colloidal glasses. The glass samples consist of thermo-sensitive poly-N-isopropylacrylamide microgel (PNIPAM) particles, whose diameter can be tuned by small changes of sample temperature. A binary mixture of these particles is confined between two coverslips and forms a monolayer of quasi-2D glass. By changing the number ratio between large and small particles, the structure or the overall degree of disorder of the samples can be systematically tuned. We employ a video microscopy to record the motion of the colloidal particles in the sample for 11 min at a rate of 60 fps. The trajectories of individual particles are obtained by particle tracking software. Dynamical matrix is constructed using covariance matrix analysis, from which the eigenfrequency and eigenvector of vibrations are extracted. In this study, we focus on the evolution of the low-frequency quasi-localized phonon modes in glasses, as the system becomes more and more disordered from the increased dopants. To compare the results from different samples, we choose those with packing fraction of 86%, and rescale the eigenfrequencies by the median frequency of each sample. For the four doping levels investigated (2%, 9%, 29%, 61%), the density of states at low frequencies increases with the doping level, suggesting that the fraction of low-frequency modes increases with disorder, which is corroborated by the higher boson peaks at higher dopant fractions. We have measured the participation ratio of the obtained phonon modes, and find that the boson peak corresponds to quasi-localized vibration modes, or soft modes. We also examine the correlation between the soft modes and local structural parameter. Specifically, we have calculated the local orientational order parameter in our samples, and computed the correlation coefficients between the relative amplitude and the local orientational order parameter for each mode. The soft modes are found to have a significantly negative correlation with the local orientational order parameter, which implies that the soft modes are concentrated in regions with poor local order. We therefore conclude that the local disorder is probably the structural origin of soft modes in glasses.

Role of Polymer Graft Architecture on the Acoustic Eigenmode Formation in Densely Polymer-Tethered Colloidal Particles.

The concurrent evaluation of the vibration eigenfrequencies in densely polymer-tethered particle systems ("particle brushes") by Brillouin light scattering and elastodynamic theory reveals a distinctive change of acoustic eigenmode formation associated with polymer graft modification of colloidal particles. The eigenfrequencies of particle brushes reveal a characteristic red-shift compared to uniform core-shell particles that can only be rationalized by assuming imperfect boundary conditions and anisotropic elastic properties of the graft layer. The distinct characteristics of vibration modes in particle brush materials provide direct evidence for the implications of chain confinement on the nanomechanical properties of tethered chains. The results highlight a rich and hitherto unexplored parameter-space for controlling properties and interactions in particle-brush based systems that could spur the development of hybrid materials with novel functionalities.

Temperature Error Compensation in Two-Component Microelectromechanical Gyroscope

This paper presents the design and simulation of a microelectromechanical gyroscope that simultaneously determines two components of angular velocity. In this device, the silicon sensor is started by an electrostatic actuator to perform a linear harmonic motion at a controlled speed. The movable masses of the sensor move in two directions, orthogonal to the primary vibrations of the sensor under the action of Coriolis forces. This paper considers how temperature influences eigenfrequencies and informative vibrational magnitudes of the micromechanical angular velocity sensor. The parameters that have the greatest impact on the sensor output behavior are determined by a finite-element analysis method. Techniques to stabilize vibration eigenfrequencies of the sensing element are suggested.

Power flow response based dynamic topology optimization of bi-material plate Structures

Work on dynamic topology optimization of engineering structures for vibration suppression has mainly addressed the maximization of eigenfrequencies and gaps between consecutive eigenfrequencies of free vibration, minimization of the dynamic compliance subject to forced vibration, and minimization of the structural frequency response. A dynamic topology optimization method of bi-material plate structures is presented based on power flow analysis. Topology optimization problems formulated directly with the design objective of minimizing the power flow response are dealt with. In comparison to the displacement or velocity response, the power flow response takes not only the amplitude of force and velocity into account, but also the phase relationship of the two vector quantities. The complex expression of power flow response is derived based on time-harmonic external mechanical loading and Rayleigh damping. The mathematical formulation of topology optimization is established based on power flow response and bi-material solid isotropic material with penalization(SIMP) model. Computational optimization procedure is developed by using adjoint design sensitivity analysis and the method of moving asymptotes(MMA). Several numerical examples are presented for bi-material plate structures with different loading frequencies, which verify the feasibility and effectiveness of this method. Additionally, optimum results between topological design of minimum power flow response and minimum dynamic compliance are compared, showing that the present method has strong adaptability for structural dynamic topology optimization problems. The proposed research provides a more accurate and effective approach for dynamic topology optimization of vibrating structures.

Flexural model for a notched beam: Direct and inverse problems

Matching conditions simulating a transverse notch in a beam are proposed. These conditions are used to determine the location and size of the notch. The notch is identified from beam deflections at several points in the static case and from the first eigenfrequencies of the beam in the dynamic case. Dependences of the first eigenfrequencies on the characteristic parameters of the problem are obtained. The effect of the relative error of frequency measurements on the relative error of calculation of the notch parameters is studied. It is shown that the use of the first eigenfrequencies of flexural vibrations of the beam with respect to different axis provides a more accurate identification than the use of eigenfrequencies of flexural vibrations relative to one axis. Since local defects such as hollows, local corrosion, and an open crack can be simulated by a notch, the results can also be used to identify these defects.

Determination of the position and volume of a cavity in an elastic rod from its two vibration eigenfrequencies

The direct and inverse problems of the longitudinal vibrations of a cylindrical rod with a flaw in the form of a cavity with a small characteristic size are considered. An approach to the determination of the location and volume of a small cavity of an arbitrary shape, which is based on the representation of the cavity by a negative mass, is proposed. The results of numerical experiments and their analysis are presented.

Wear dynamics of fragments in the lower part of a fuel assembly with unattached rods

The dynamic characteristics and wear of unattached fuel rods in the lower supporting lattice are investigated. The vibrational eigenfrequencies and dynamic rigidity of the fuel rods in different configurations are determined. The vibrational wear at contact between the fuel rods and the spacing elements is studied.

Development of finite-element models of human skeleton bones and their application to the natural vibration problem

The capabilities of the previously developed algorithm for finite-element modeling of human body parts are demonstrated on tomographic measurement data. The obtained tomographic images of human shin bones, skull bones and pelvic bones are used to solve the problem of searching for eigenmodes and eigenfrequencies of vibrations of human skeleton bones by the finite-element method.

An approach to the problem of damage identification in an elastic rod based on the Timoshenko beam model

The development of simple and reliable methods for diagnosing the engineering state of critical struc� tures and their elements is one of the key problems for developing technologies for lowering risks and decreasing the consequences of natural and technogenic catastrophes included in the RF List of Critical Technologies. Some modern approaches to the solution of this problem are given in [1, 2], where the solu� tion to the problem of diagnostics of damage is based on using correlations between the parameters of the frequency vibration spectrum and the degree of the structural damage revealed using the resonance meth� ods of free (MFV) and induced (MIV) vibrations along with acoustic testing methods. In the majority of works, simple structural elements (rods, beams, and pipes) are analyzed. The damage identification algorithms discussed in [2] and in later works are based on the correlation of their parameters with variations in the eigenfrequencies of vibrations, modal characteristics, the curva� ture of mode shapes, and dynamic calculations of the matrices of flexibility and damping in faulty struc� tures. The key differences between these algorithms lie in the formulation of objective functions and opti� mization schemes. For example, the authors of [2, 11] use the Modal Assurance Criterion (MAC), which depends on variations in the shapes of vibrations in faulty structures. The original studies of damage in rods based on variations in the eigenfrequencies of vibrations due to the depth of a closing crack were carried out in the works of V.V. Matveev and A.P. Bovsunovsky [3–5].